A SUPPLEMENT TO ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

William F. Mccombs

TAELE OF CONTENTS

Article and Main Topic Page Article and Main Topic Page

A5. 2Ja Beam-Columns 1 c·11. 29a Tension Field Beam Holes A5,24a Beam-Column Formulas 1 Cll.24a Rivet Design A5. 25a Beam-Column Deflections 1 Cll. Jla Stringer Construction A5. 26a Beam-Column .Formulas 1 Cll. J2a Diagonal Tension - Stringers A5.27a Margins of Safety 1 Cll,JJa Stringer System Allowables A5,28a Truss Analyais 1 Cll,J4a Stringer Construction Example A5,29a Tangent Moduius 1 ·c11.J5a Diagonal Tension - Longerons A5.Jla Multispan·Beam~Columns 1 Cll,J6a Lorigeron System Analysis A5. 32 Approximati! Buckling Formula 1 · Cll, J7a Longeron Construction Example A6, 7a Plastic Torsion 1 Cll, J8a Diagonal Tension Summary A7,la Beam Deflections 1 C11,J9a Problems for Part 2 A7,12a Numerical Analysis 1 C11,40a Problems for Part 1 All, 2a Moment Distribution· 2 ·CS, la Monocoque .Shell Buckling Data All.5aa Stiffness/carry Over Factors 2 CS,2a Shell Axial Comp, Euckling All,lJa Moment Distribution Factors 2 CS,5a Effect of Interna1.Pressure Alf. 15a Truss Analysis 2 CS, 7a Shell Bending Buckling A11, 15b Truss Analysis

65. CS, Sa.. Effect of Internal Pressure

AlJ,lla Curi(ed Beams CS,9a External Hydrostatic Pressure AlS, 5a Column Analysis 7 CS, lla Shell .~orsion Buckling Al·B, Sa Colum.'l Analysis 7 cS, 12a Effect of Internal Pressure A18, Sb Torsional Buckling 7 CB, 15a Combined Loads Buckling A1B,27a Column/Eeain-Column Data 8 CB,19a. Con.ical Shell Buckling AlB, 28a Material Properties 8 CS, 20 Euckling of Spherical Caps A19,2J Rib Cruelling Loads 8 D1,2a Fitting Design Cl, lJa Margins ot' Safety 8 Dl, Ja Fitting Margins of Safety Cl,lJb Dealing With Tolerances 11 ·Di,4a Factors of Safety C1,6a Combined Stresses 11 D1,5a Eolts C2,la Column Cu):'Ve Construction 12 D1,6a Nuts C2, 2a Free-Ended Columns 12 Dl, Baa Bushings . C2,Ja Shear Effect on Buckling 14 Dl,12a Transversely Loaded Lugs C2,Jb Multispan Columns 14 'Dl,lJa Obliquely Loaded Lugs C2,6a Stepped Columns 16 Dl.2Ja Rive.ted Splices C2,6b Numerical Column Analysis 17 Dl,27 Thread Design and.Strength C2,6c Initially Bent Columns 21 DJ,5a Filler (Shim) Effects C2, 7a Column Design Data 2J DJ, 7a Curved Beam Data C2,8a Column Elastic Supports 23 D3,Ja Tension Clip Allowable Data C2,10a. Need for Successive Trials 2J Dl,21a Flush Rivet Joints C2,13a Column Elastic End Restraint 2J Dl,22a Elind Rivet Joints c2,16 Tangent and Effective Moduli 25 Dl,7a Eolt Strengths C2, 17 Buckling Load Data 26 C2,1B End Friction Effects 30 Design Check List

References ·for Chapter C2 JO Margins of Safety CJ,4a Plastic Bending Data 31 Tension Clips CJ,7a Plastic Bending Example 31 •Preload Torque Factors for Eolts CJ, lOa Plastic Bending Procedure · Jl Rockwell and Erinell Hardness Data CJ, lla Complex Plastic Bending 31 Minimum I for Stiffeners CJ,12a Shear Stresses in Plastic Range32 Stress-Strain Curves c3,15 Yield Stress Bending Modulus 33 .Sheet and Other Buckling Data CJ,16 Residual Stress After Bending 3J .Material Propert~es c3,1S Beam-Column Analyses 34 .Fastener Joint Design Notes c4, 20a Plastic Torsion 38 Fastener/ Joint Allowable Data C4,2Ja Apparent Margins of Safety 39 Nut/Collar Tension Allowables c5,7a Flat Plate Shear Buckling 39 Eolt Allowable Bending Moments c5,Sa Flat Plate Bending Buckling 40 Beam Formulas c7,lOa Unequal Angle Leg Thicknesses 41 Additional References c7,JO Crippling Method Three 41 Bulkhead, Frame and Arch Analyses C7, Jl Torsional Buckling ···' er. 1;4 •Column Elastic Support De sigh C9.13a Frame Stiffness Criteria 47 Operating Devices · C10, 15b Thick-Web Beam Analysi.s 47 Doubler and Splice Desie;n

© 199B William P. Mccombs

i

49 50 51 51 54 57 60 61 63 66 67 67 67 67 67 67 67 67 68 68 6S 72 7J 74 74 74 74 74 74 74 74 74 75 76 77 77 77 77 77

At Al At A2 A2 AJ AJ A4 A6

AtO AlO A21 A21 A21 A21

Bl ES E9

.ElO.·

PREFACE

The widely used and.recommended'college/industry textbook HAnalysis and Design of FlightVehtcle Structuresff by Dr, E.F. Bruhn .has had only one revision .i:;ince its inception in. 1965,, That was the 1973 ,ed1: tion in,. which Chapter A23 W!J..$., .. Iff!!Vi.sed and expanded~ Chapter Cl) was .completely rewriJ;ten b;V' ~hother author; .. and. a few minor ch~nge,s were made. i.n Chapte.r 1{1, .. Aside from these the book· .remains in its original form. ··•o:.·r.

'.l'l'ie purj)g13e .of tJ:i1,s supple111ent is to incre'as!! .t~~ ~'(5qpe and usefulness, .of the textbook in .numerous .$peCifid ... areas Of analys,is • These inc;Lude .. columns, beam-column~,. ben4,h1g . .· strenir.th, margins of.·safety, tension field analyses, fastener/ joint data; arches,:·bl1lkheads and numerous others •... The Pl'a9J1:~~l.·· usf!! .. pf 'tl'ie ~.)tPP~emen-t is discusse.d, ~P ~?l" ,Introd.uc tton. Only Ql'!e or. ;!;w9 ... aPPl:·i:;c;ia tiqrts of , the Supple.ment;'s, ;,<?,C>ntents ,can .·, be· worth much more than its cost. ,, ,

'·, ·.· .,. Th,e 's~~pl:~,Jri'etl~· fri!;i.i be,: expanded in some futu~~.:.~ea,f;:c so ,any ';, suggest. ioni;; f(>r. this., or for corrections or chaMes· in ... J ts current

. -.... ,-.,.' •;' ... . - '' ' : "-; ., ( ·' ;""·"" • .. ;.•\•,•'' ' ' ' ' ·. text will. be1;•/il.PPreciated." Thqse readers who w1,13,!1jtp; b.e;,,1,1:1forl!led of any ·future.:.rev.isions or who ,have. suggest1ons.·.can/,c·ontact the: author. at p:.o. BO:i 763576, Dal1as, TX 7537·6-J57·6(TeTJ214'J37-5506).

. ,, - _h·" ,• '' .

The author is'. one'. Of the. coauthors Of the te'.Jitbook, His career incluQ.es over f;orty ;years, of experience. Hi:,structural analysis and design of·numerous aircraft and missiie'<projects in the aerospace industry~· A1so included <are:1te9hn1c'al'' papers and the prep?,;i:·a t io!l .. an,d, t~;~~~i:tlg or p:l:jactical c.C>url\l.l:l~J"fi;l:'"~~ili€'tU,t~i design and analysis .for .e_1:1gineer!l. work1,ng in the aerospa:Ce .. al:ld other industr .. ie's, · · · · · ·· ·· · ·· ·· · ·' · · ·· ·· ·· ·· ... · ···· 1 •·•· ·•

',·1 •'· ,T';· ·;;'·"[Xi:['._.'..:>·~

'· ··" · William F. Mccombs . •' ~, .

0 1> ···;

FOREWORD

I am please.~;;tq::11.~re ·.th~.jo,P1portuni.ty to fe~<;!i0¥~,n.c;l. ';tfiis Supplement to my late, father's widely used c,ollegE!lind)lstrY textbook "Analysis ai:liiibes ign 10'.f\ Flight YE! lii~l~ ; $true ti.ires H.

I hope its practical applicat<iorts will be a benefit to all who have the textbook'. The e..aa1t'!o?l9.l da.ta contained in this · Supple111ent ca,n ''tie appf~~'4 tO: iioth' the' ,S,k~lit,~ ~#,4, t)ie work of structural desig?l anci ana].ysi,13. Thf;!! :~upPJ,t;m~P,t,,131'lould also be of interest· to, ·and eventually benefit, those.;.:W'ho may be. considering purchasing the textbook.

, .. -· .-··'

Patricia Bru'nh Beachler

ii

INTRODUCTION

The purpose of thls Supplement ls to lncrease the scope and usefulness of the widely used and recommended college/industry textbook "Analysis and Design of Flight Vehicle Structures" by E.F.Bruhn. As such it ls by no means a revision of that book. Rather, it ls, an expansion and clarification of numerous topics and data in the book, along with the introduction of additional topics and data.

For best coordination with the textbook the following has been done. Where an existing article such as, for example, Art. Cl.13 has been expanded or otherwise changed or corrected, the Supplement includes it as Art. Cl.13a, the letter indicating a change or addition. When a new topic is added to a chapter it is given, an article number which is subsequent to the last article number in that chapter and includes no letter. For example, the last article in Chapter C3 ls Art. c3.14. Two additional topics, "Yield Stress 'Bending Modulus" and "Residual Stresses Following Plastic Bending" have been added, so they have been given article numbers c3.15 and c3.16 ,respectively with no letters.

The same thing has been done with figure numbers and table numbers. Where a figure has been changed or added to, it retains the same figure number with. an added letter. For example, Figure 2.27 has added information ~o it has the designation F1g.2.27a in the Supplement. ,When a new figure, is added it ls given a number which ls subsequent to the last figure number in the chapter, so no letters are used with its number in the Supplement. This procedtire also applies to tables.

Although the Supplement provides an Index, for most usefulness the textbook must be marked in such a manner as to guide the reader directly to revised, corrected and new topics, figures, tables and references in the Supplement. , A highly recommended scheme for doing this is provided later.

Structural design and analyses are based on theory, empirical methods and data, various assumptions and individual judgement. The assembled structure is a result of various specific manUfacturing methods and procedures. Because of these things and also the possl­bili ty of inadvertent calculation errors, it is always necessary to prove the adequacy and safety of the completed structure by means of a sufficient test program before it ls put lnto use. Such tests must demonstrate the structure's adequacy as to ultimate and yield , strength, fatigue life, fracture and stiffness. The test results must be properly evaluated sin,ce the test article 1 a materials usually have properties in excess of the minimum requlred values.

Procuring agencies such as the military, the airlines and other government and private: organizations usually specify the desle;n and te.st .,requirements and other crl teria which· must be·, used ''dr met. ,

iii

Coordination of the Supplement and the Te:x:tbook.*

In order to easily guide the reader from the te:x:tbook to the Supplement, the following marking of the te:x:tbook is recommended,

1.. On. the te:x:tbook' s Table of Contents place an asterisk, in red ·ink, after. ~Contents" and add the following footnote at the bottom of the page1

* A red asterisk preceding any articie or figure number in.the te:x:tbook means that additional material is available in the same-article or figure in the Supplement,

2, t<or each· te:x:.tbook•. article listecf'iri''the Supple111ent • s Table ' · of Contents, place a large asterisk:; in red ink, just to the left of eac.h .·corresponding article nUiilber in the te:x:tbook (e,g,,,'*All,2)/ ··

J, The 12 articles i~the Supplement's Tabl~ of C:onterl.ts riot having a letter in the article number a:r.e 11ew .a;:i;ic.les, (e,g., CJ,18 Beam-column An9.1yses'),< Their nuriibe'fs and titles should be written in at the. e!ld of. their chapters, . with a red asterisk' Just to the left of their artic'le numbers,

' '• --<-·· ,' ,'","' _,.

4, The following figures ln ;the te:x:tb.o()k shoul_d have' .an as:l;erlsk, in red ·ink·, placed' just ·to th~. left qf'tlieir figure numbers, to indicfate that a revision:· or ·additional data is in the Supplelnentii · · · ·

A5,1 c2.17 c2.26 c5.14 c8,l5 C8,28 Cll,47 .· .. A18•, 8. C2,18 C2;, 27• C8,8a c8,20 c8,29 Cl.1,48

Cl,8 C2,19 CJ,27 cs,11 c8~·25 c10.15 Dl,,15 c2.2 C2:, 20 · CJ,28 c8, 13 c8,26 d11.4J c2.16 C2,25 cs.11 c8,14 c8.27 C11.44

5, The .. a9.ditional references shown on p. Jo (:for,Chi;i:pter C2) and .. onip,A21 (for.Chapters CJ, c4, c7·· and.' C:11) shoul(i be written in at the end Of the list .. Of referertces for. these chapters• ·

'- ' '

6, Put. a. black asterisk after "RINGS" bn the title of p,A9.1 . and , at the bottom of the ri~ht hand column add the footnote * See Appendix B of the Supplement f9r alternative analyses .. o~ bulkheads, frames,· arches and bents~ · · ·

* To increase the scope· and usefulness of other such textbooks, put appropriate asterisks in their texts with footnotes saying which article to see in the Supple111ent,

iv

A SUPPLEMENT TO "ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES" l

A5.2}a Introduot1on

Add the following at the end of Art. c5.23. For a discussion of all types of beam-columns, including non-uniform mem­bers, with numerous example problems see "Engineering Column Analysis" described in Art.Al8.27a. A5.24a Et'teots ot Combined Ax1&1 and Lateral Loads

Add the following at the end of Art. c5.24. Formulas and example ·problems for beams in tension are available in the book described in Art. Al8.27a,

A5·25& Equat1ons tor a Compreaaive Axiallf Loaded Strut w1th Unltormly D1at~1buted Load.

Add the following at the end of Art. A5.25. The deflection, y, at any beam station can be calculated. as follows.:¥

14 = 14Q +- Mp = l4Q + Py therefore

y = -(M 14Q)/P

5 (MQ - M)/P

The negative sign is introduced since a positive beam bending moment produces a negative (downward) deflection. Mo is the moment due to the lateral loadJ only and 14 is the final moment;·

To calculate the slope at any sta­ticn calculate the deflection at a point very slightly to the left of the stat ion and then very slightly to the right, the same amount, AL. The slope is th.en

slope = (YR -· ~L)/2AL A5.26a Form.ulaa tor Otner Single Span Load.in&a<ft«

Table A5.la presents numerous addi­tional cases of single span 1oad.1ngs.""'

The case of. a beam-column with a varying EI or a varying axial loading re­quires a numerical analysis, just as does a column of this nature. The numerical analysis procedure is presented in .03.18. A5.27a Combinations ot Load Systems, K&ratna ot

Safety and Accuracy ot Calculations For a beam-column the true margin of

safety must be calculated as discussed in Art. C4.23a. The allowable stresses or bending moments are calculated as dis­cussed in Art. C3.l8. A5 .• 28& Example Problem11

In Example Problem 2 member BD is considered to be pinned to ABC at joint B This is why it does not pick up any ofthEI 36 ,ODO in lb bending moment at joint B .•. AS.29& Str~asea Above :Proportional Limtt· St~eas

If a column curve is not available -----.. See footnote on -p,J7, L.H. column

for the beam material, the axial stress, P/A, is calculated, ·f c/Fo. 7 is calculated Fi~. C2.16 is entered with this value and Et/Eis obtained. Then Et = E(Et/E). E' in Art. A5.29 is actually Et. A5.3la Beam-Co~umna 1n aont1nuoua Structures

Multispan beam-columns require a mo­ment distribution analysis to determine the end m~ents acting on each span. Once these are known the applicable single span formulas in Table·A5,1 and A5.la can be used to determine the bending moments ·at any station within a span. This is discussed in Art. C3.l8. A5.J2 App~ox1ma~e F~rmuJ.a. tor Beam.-Oolumna

For preliminary sizing when there are no end moments the following formula can be used to determine ·the final bend­ing moment at any station

where 14Q is the moment due to the trans­verse loads only and Per is the critical load as a column (Chapter 2). This is most accurate for a uniform lateral load and.least accurate for a concentrated load. Per is calculated assuming the member has a stable cross-section even if this is not the case. For a·beam in tension the negative sign in the formula is replaced with a positive sign (ten­sion makes the bending moment smaller).

¥i _• 7~ _ TOra1mi ,,r Soi.id. Jlon .. Q1r~ular Sb.ape•

All of the previous formulas are based on the shear stresses being in the elastic range. With ductile materials failure (rupture) does not occur until the shear deformation has gone well in­to the plastic range (similar to the plastic bending case). The torsional moment at which rupture occurs can be predicted as discussed in Art. C4.20a.

For the special case of tubes ha­ving a circular cross-section the failing torque ean be calculated as discussed in Art, C4.20 and its associated Figures c4.17 to C4,30 and in the example prob­lems of Art, C4.21. A7.~a Intro4uat1on,

For practical purposes the deflec­tion and elopes of beams are calculated as discussed in Art. A7.12a, standard formulas being used for uniform beams, and for varying section beams using tables such as Table 03.3.and the "End Fixity" discussion. A7 .12& Detleat1on• and.-·~lar Changea _ ot:

_Beama "by Jletnod. or "El.aatla 11e1gtltaA

For beams of uniform section def­lections and slopes are most easily cal-

•• Por other d1str1"buted loadln~s replace them w1th several sto,~1c•1.ty equlvalent concentrated loade and ueo eu~rpoe1t1on tor these. Por any csse of one end f1%ed one s1mpl:r su~DOrted not shown, use tho both ends fl%od case and then multl"Pl:T the moment at the fl:s::l!ld end by (l • COP}; ~ ls zero at tho other end. Tho •%eept1on to this ls e~se 19 •hero (l - COP) must be used.

?

;o ...

·' ll

JZ,

'

'

/J .. ·

)6

···17. ,., -·

18 '

'

~·,· \ .

Or U•• ··'.

"•·•11.sz. 20

21

SINGLE-SPAN BEAM-CO.LUMN FO.RMULAS

Table .A5.1a Cont1?)uat1on of Table A5. l

!'i'i:. C1s1nx/j + C2ci;>sx/j + f(w)

LOAOllfO

·~ .:c~ P1i9d Erld.-B.Ur.-.. --.

Un1Cor"lll Load~·

a Coe b/J .&ca;- !in I;J

• <.L/2:· -1/<!., J

'p ·i ·-,.-t.12: ''\ ,~ ~:_: '1' 1;2·~·j'·-:(2Co• Ll2J

,Pl••d £nd, Bilaa_-c ,l'.'c•n_ ;c •. ".'; rated Loi.d. at Canter

c, t (•)

0 ..,, 2

•J

Sin "J 0 .. 0 -.".- \ 'O

-wJ 2 CCIII a/J ... J~

0

0 0

0

."w L J 2 .'.fa'n l./2J

...... "°"'NT

De term.in• ~ plattin9

[.,·

• 2[ 1 -C·o·."'I J ~ 'J . '""?Oi'"172J .

At M14.:pjln

If b~41'j/2,

-1\t.u •t. -·

If b>.,.j/2, l\y:x at L- 'ftj/l

........ iir.J[bnfr'.2J-W~JJ

iint;;'J - QJ .... d'*•,r,tJj

At :!t •.O

•'

0 "-·... .

'~ [JTa~lf;looW2J•l J . ~n ,J -. L

o· .. -'At x·· o' : ,- ~-'. ' ' ' .

···:Qr :u:¥ie CJt.,se __ ::v .... w1 th· r,..2L; W"''-2W and a~.~t,, there for same r.e.sul ts

·.,·-, '•!·.··''

Oi' use·· CSse III with t.=2L'B.nd same w there for same re~ults

Or cai.CU1ate ·M=6Eia/L"- ( 2{1-~) See Art •. ;11.15a

"'"' U;em,.,6. CG=></L)

'

,.-1\.a:..• •J [J_(l-!•c.L/J)~ . . L Ta~ i../~ At .it • L

'~ ··,

a P 'nln w21,

At it • 0

'<'.°2•M1 (c&s• II) And - ....... At JI. • L

MA-WL(b/1cosL/ -s1nL/1+si:na/ 1+s1nb/ f~L/ 1oosb/ 1+a/ 1 l ~'·''·''"<~OBLtJ~L1JsinL1J1·

M-= as above but s-t-tehina a and b

MMax = Wjs1na/j ~()s~ij-~ii;b/j(Hco~L/j )/~irt,L!] .. ·· (O~_c;u:c-s .g?)de:c- ... the ... J,oads .. W.) . , ..... ,

TM.s --1s a--supe-os1 tion,.,oJ\ Cases V ·

'.

MOMENT DISTRIBUTION WITH AXIAL LOAD 3 culated by using beam deflection and slope formulas widely available in the structural literature. These formulas are based on bending stresses only and are accurate unless the web is very th·in or otherwise quite flexible due to holes eta .. , in which cases there may be signi­ficant additional deflection due to shea For beam-columns the deflections and the slopes at any station can be calculated as discussed in Art. A5.25a.

For beams with a varying EI a num­erical analysis is necessary to deter­mine the deflections and slopes. This ca be done as discussed and illustrated in Art. c3.18 where Table 03.3 shows the de­flections due to the transverse loads on­ly and Tables c3.4 and 03.5 .show the ad­ditional deflections due to the axial loads, the final bending"moments being i 1'able C).o The deflections are obtained using the data in the tables and .the geo­

'metric series, p.J5, as was done· in Table ··cr;i • .s for moments. Tables for end fixity are discussed in Art. c3.18. With ·any elastic end restraint a moment distribu­tion analysis is required to determine the end moments on the beam etc. as dis­cussed in Ref. 3 (Art. Al8.27a). All.2& Det1n1t1ona and Der1V&t1ona of' Term.a

Add the followins at the end of the article. In the moment distribution pro­cedure, where adjacent spans "meet" (at support) there is assumed to be a "Joint" even thoughthe member may be continuous across the Joint• ··The sketoh in Fig·. All.92 shows the direction of (+) and (-) moments as they act upon the spans and also upon the Joint. As seen there (+) moments act clockwise on the span and counterclockwise on the Joint. This is different from conventional beam sisn conventlon where a (+) moment produces compression in the "upper" surface,

-M +M . ..___f> (f) (J 1 ~11Jo1nt 11

Fig. All.92 Siifl Convention*

All.5aa. Example Problems. The stiftness factor, K, as used in

Art. All.Sa and subsequently,· is a "rel­ative" factor rather than a true one. It has the value EI/L for the far end fixed and ,75 EI/L for the far end pinned. As such it applies only when there is no ex­ternal elastic restraint, k .. (in-lbs per radian)at the ends or at any Joint, and when there is no axial load in the spans. In such cases 11 correctian fa.otors 11 ·must be applied to it, as shown in Art.All.14 for example. In general, to avoid inad-

"Some books u~e an opposite sign conven_~ion.

vertent calculation errors it is best to use the true value of the stiffness fac­tor, SF, which is in in-lbs per radian and is

SF = 4SCEI/L

where SC is the stiffness coefficient and can be obtained from Fig. All.47 (as "a") or calculated per Art. All.l3a. Doing this eliminates the need for introducing the correction factors otherwise needed. The carry over factor, COF, can be ob­tained from Fig. All.Ji6 or calculated per Art. Al1.l3a.

In doing moment distribution .calcu­lations, unless the "far and" of a span is pinned (or free) it is assumed to be fixed for determining the values of the stiffness and carry over factors.

All.l'a F1x•d End Moment•, St1ttneaa and carry OV'•r 7aator• tor Beam­Column• ot Canatant Croaa-3ect1on

Sometimes it is necessary to use the values of the SC and the COF (see Art. All.Saa) for larger values of L/J than are given in Fig. All.46, All.47 and All.56 (for 20-~). Those values can be calculated as follows for compression members:"

SC (far end pinned) SC (far end fixed) COF :d./213 ·.

= 3/ 4,!."" <! = 3(31(...., - o( )

where o<_ = 6(-Jccsec; - l)/(L/J )a.

f.3 a 3( l - jcot~/( L/ J ))a

Fo.r members in tension the same formulas can be used, but the ·trigonometric functions are replaced with the hyperbo­lic functions, cosech and coth. Exten­sive table's of the SC and COF values (to 6 significant figures) from L/J c 0 to 217' for compression members and from 0 to 50 for tension members are in the book described in Art. Al8.27a.* '* A11.l5a Secondary B•nd1n;; Koment1 1n

True••• with R1s1d Joints

Art. All.15 gives a procedure for determining the secondary bending moments in such trusses but no illustrative exam­ple is provided. The following illus­trates the procedure except that in step one one finds the relative rotation of each member usini the method of virtual work, step 2 is omitted and in step 3 "its relative rotation" is used in place of "these transverse displacements". That is, due to the applied loads the truss Joints move, and therefore the members •In Fig.All.56 2/1 -ot ·can be calculated as follows~

6 [ L (' + cosL/i i'] , #(3-ri"' iL/TJ" 2 - j sinL/j ~ d' ,.,.

1 . .,

J • j ,; j31J. .:lJ1 IV / >l ~ /1-/J;;, ,poh,,M~)"' 7J-eJJ't. j>• .

4 ~J;:CPNDARY. BEND.ING MOMENTS IN. TRUSSES

rotate relative to each other. These ro­tation• generate fixed end moments at their .. ends. (just as a beam with unequal-· ly deflecting supports will undergo ·an angular rotation and develops fixed end momenta). The following illus­trates. the procedure using the truss shown in Fig.All.92.

-5774 5774

60'. ··. ·-21'17 , set,. c. cei;so E 5000~ c::::=:;:·i.~2~0·~· =···=·i···~··=·:;· ·~·:C:•:::i.f'· ·t~o~"=··=· ~,..., 861i660#

Fig. All.92 Applied and Internal Loads

l. For tlia aiit>fi'ed load.a the' resulting internal loads. in the members· are de­termined as shown in the figure.

2. M!jml:)er BO ie a~b1trarily.1'eJ,ected as the •ipase niem~'er,11 :f'r.o'm wl:i1cli t!ie ,ro"'. tations of all other members are cal­culated.

3, For each other member ( onE1 'a.t ,a ,.,tl,mE!) a clockwise l in-lb moment in the for of couple loa.<is ( l/L) •at.. :+,t.1>. '!\'!d,'! 1,!! applfeii. and·.reac.ted. V11tb, a, ,ciount'!r::. clo.okwise couple at the en4EI' of the, base member BO. The resulting loads in the tru~a. iUlembers are,: .• t.hen .... dE!te.r­minea. See Fig. All.93 anii note that only a fE!W (3 of 4) meml:)ers are loaii-. in this procedure,

4. ·Then tor.each. ioadE!d.' meml:)~r.,;t.he, quan­tity SuL/ J).ji; 1 a ,.oalc;L!la ted and . the re­,aults stimJllei!.Ao ,ot>,t.S:.in the, rel1>tiv:e ro,tat1on .• :·:ei o:t' .1'hE1.,member, to which tlie .c+ockV1ise. coup),E!, wa!! f,i,pplied.

- . '•'' ., .... ·'"" :·"""'',•··· .-,,.-., " .. ·-··. ., .·. 5 ~ Ste!JS .. ( 3f ancf (4) 8,~,e J;'epeB.ted for .

eB.ch 'of the 'renia1nin~ U1enibera ',to ,get their relative l:'otationa, e; Table · All. 4 presents the b":sic lij.ata anlil. Table All.5 si:iinriiari:Z:ee' ;t\).e ·calcu-lations. ·

Table All.·4 Bas1c Da.ta ' '

Member :· _J:L 111·L ;·~-.a. E .s~/AE· :· ~ 3:./ . ....!-;,, •;.: ,.: ,.·'o-'6-· AB. I· 577.4 20··: ,.563' :.10:_·' ; .0205 o;,'0264' 6;76 AC -2887 20 ,563, .10 -.0103 ,0264 9.56 sa· · -5774 20 .563 '10 - .0205 .0264 6.76 BD 5774 20 ,563 10 .0205 '.;0264 6,16 CD 5774 20 ,563 -10. '.0205 • 0264 6.76 . CE -8660 10 .360 10 -.0200 ..• 0100 3,53 "E. 0 l"'.""2 • O<;o 10 0 ·.ooi:;-~' """

05

F1g. ·All._~;, R~l.1.t.1ve _Rot~-_t;'l_on Loa:ds ..

F.or, each of the above membera••·the fixe<i encl,•moments are calculated as

~EM= ,-6EIS/L( 2/J - ot) .

I./ J

2,91 2.0, 2.96 2.96 2.96 2.83

0

where 2,13 -o( if< ol:r~ail\,ed. from . Fis;. AllS6 and accounts for the effect of the .·.a;><ial•.:loadf Si .1n .. the member, ·A Po.s1:ti:v:e. {.clpckw:l;se,,J ·r.elative ·'.rotation, 8 •' .prodUC.E!S,'.;gegati:V.8 ,•(,o ount'.erclockW1,se)

..FEM' .. s (per Example" 2 ·sketch on ·p•All.2) · l:<eno.e:· .. the.·m1nua':sign·'.:1r,:'.the,..·f·ormula•' Ifo'a memb.er. has •. :one end·,:pinned .a fixed end mcip1ent occurs only• at the other end. and is calculated as

FEM=·-6EIS(l--.. COFJ/L(-2/3 - o{)

where OOF is from the pinned end to . th•i."fixed end. Table Atl,5 summarizes the· calculations for' the' va·lues· of e.

6. The fixed ertd."in'om~nta at all truss joints are now known and the moment distribµtion procedure can be carried out as 1llui!trated in Fig. All,43 to obta,in: ,the ,final moments. at the ends of each member. Th<!:n sue A~t:. A5,3ia;..

Table All.5 Oalculation of Relative ~otat;ona, e

·Mam:. SLf . .AE Counle 'at AB Coirnle at AC Couole a:.t en· noun le 'at f!D Ooi.nole at CE :i: •. :.:.:·- uSLIAE ''uSL/AE uSLf AE u " ' uSL/AE u uSLlAE u u u

AB .0205 -;0289 -593 ', ". 0577 -1183 0 0 0 o.· 0 0, :Ac'. -.Ol03 .0577 -594 .0289, - 2,89 0 .Q 0 0 0 0 BC - .0:205 -.0289 593 .0289 ~ 593 .0289 -592 -.0289 592 -.02~9 593 BD. .0205 0 0 0 0 .0289 592 .0577 1183 .0577 1183 OD I '.0205 0 0 0 0 ' -.0577 -1183 -.0289 - 592 :...1155 -2367

" ·o~'" ·:.. .o2off · 0 0 0 ·o 0 0 O· 0 0 ,0 ·oE, ' ,, 0 0 0 0 0 Q .. - 0 '.0 0 .1000 0

E;f& _n.r; ~""'LILL . 'rA Ft : -.21 in5', e-- = -llH'• . Q- - : . ll><'I 9,,,,,: -~ 91 -Note: M»ltiply eacn e by 10~.: DE <ioean't rotate since it is held by the supports.

Seep.:.Bll. for the- critical (buckling) loading calculation and p.Bia-13 for gusset plate data.

SECONDARY BENDING MOMENTS IN TRUSSES 5 All.15b Th• Ettect1 ot Inoroaaed Internal

Load• on Secondary Bending Momenta Although the secondary bending mom­

ents in the truss of Art. All.15a were relatively small, they can become quite large as the internal axial loads in the truss members increase. As the applied loading on the truss approaches the crit­ical loading these moments will approach infinity. This is why the theoretical critical loading for a truss can never be attained; bending failure will precede i~ The following example, using the simplest rigid Joint truss illustrates this.

LAc= 17.3 11, EIAc= 5x1o!A=Z,0"1

LAB= 30.011 , EIAB = 5xl06..iA .. ,a1Ci

F1.~. All.94 Rigid Jo:tnt _Truss with Piruled Ende

A is a rigid Joint and ends B and C a.re pinned. Assuming A to be a pinned Joint only to determine the axial loads in AC and BC, 'they are as shown in the figure. With the axial loads known thia simple truss can be analyzed for stability (buckling) as a two-span oa,lumn with pinned ends, as •is illustrated and dis~ cussed in Art. C2.3b, Example 2. By suc­cessive trial calculations, when P 101700 lbs, PAB = 88072 and PAC= !)0850.

(L/J )AB :-,o/;/5 x 10°/88072 =}.9816

SCAB= -l.5428

SFAB = 4(,.,,1.5428)(5 x 106 )/30

=-1028500

(L/J )Ac= 17.3/V5 x lo'/50850 =l.7446

SCAC : • 89059

SFAc .:4( .89059)(5 x l0')/17.'J

=1029600

Hence, at Joint AZSF = 1029600-1028500::!< 0 so 101700 is the buckling (critical) load for the truss.* If A were assumed to be a pinned Joint then an applied load of only 63315 would cause AB to buckle as a pin­ended column, since for pinned ends PcR for AB is 54831 and .866 x 63315 ::. 54831.

To illustrate the effects of applied loadings near the critical loading on secondary bending momenta, assume that P = 100000 and find the resulting momenta. ~ .t'BI' at &n.J Joint £ O there 1• no re•1at.e.noe te rot.at10n

n an 1nt1n1teamal aomen't. will eauae Nt.at.10'1 and t&1lur•~

Let AC be the base member, apply, a l in­lb couple on AB and react it with an op­posite couple on AC as in Fig. All.95.

l/17.3 /10

l/ .3 l/. Fig. All.95 Virtual Work Loads

C B

The relative rotation of member AB is then calculated as follows.

Membe

AC AB

The only FEI.! is at A since end B is pinned (and AC has no relative rotation).

FEl.!AB:: -6( 5xl06 )(-5803xl0°6 )(1.,.. 2.323)/

= -!Ss3i. in lbs 30(1.44)

Doing the moment distribution fo;i:,the 'final moments at A

.. ,.. (q>'\.(i:~"'

lJ>. ~ SC ,8884/-1.394 SF 1027100 -932670 SF 94430 DF 10,8771·9.877

FEM 0 ·5331 lle.1 2ZID • H$~~ . F1nal M'om 3'7953" -

Fig. All,96 Mo~ent D1atr1but1on

The extremely lar~e final moments are, of coursa, unacceptable since bending fall­ure would occur (if P were 101700 the mo­ments would be infinite). Therefore, when the applied leading on a truss is near the cv1tical loading bending failure will eccur (and prevent the critical loading from being reached), due to the * deflection of the truss Joints under loa~

Repeating the above calculat1ens for suocesaively smaller valuea of the app­lied loading, P, results in the final mo­ments shown in Table All.6. Nete that as the applied loading decreases from near the critioal loading the final moments decrease very rapidly at first. When the loading deoreasea to the value which is the critical value assuming pin ·Joints

* The maxi?aum bending in AJ3 is ??.840 1n-lbs, occuring 18.3~ from A (using Case 2 in Table A5.1)

6 ' CURVED BEAMS

Table All.6 Variation "'o.r F·ins.l Moments with Applied Loading

' ,,

Appl.ied % of Final Loading Critico.l Momenta

Loading At A

101700 lOO.o <:><>

10000'0 98.4 -57985 99000 97.3 -24983 98000 96,4 -20522 95000 93.4 - 8876'.' 63315 62.3 0 4000,,0 39,3 - 60

(63315) the final moments 'are reis.tively, sms.lJ;;- Tllia is why an a,s.sumed,,P,i)j'":J0int analysis ,,which iSl'lores the:, .s:epcindary mom­ents', a-· .. ~ellimon ·Preioedu;r:-e., +l.'s ·unl1keiy te r~sult 'hi strength fa,ilufea. Fatigue life ni:r:ght -be a. ·cfotl'i:erri tor very' light structure·&'. If a:' truss member is subject to a lateral,: :j.qading, ths.t. as.uses an add­ition!"l .. type 1>.:f' secondary Il\Om!l?\t t,cf occur and additional beaoi-co1umn·af'feots;

'AltH<;ugh thi~ 1li.'uatrati ve exafui;iie'' use• only . the simple,st type qf rigid joint truss, the results would be &imilar fer a oonve!ltional type of .. ilrua,,.. The analysis would, of courae1· ;t>e mo.i;e t¥,,­dioua. Yore about truss'eii l"s 1h the book mentioned in Art.AlB.27s. (and Fig.All.43i

Al3.lla Ourved Beam.•

For c0mpao.t cross-se'ctions the bes.m is net aubject.te fl11nge instability ef­fects (Chs.pter 7} .•.. 'l'herefore, with duc­tile materials,>.~l;\EI 1J.l'\..1ms.te bE1nding · strength can be' calculated s.s discussed in Chapter 03, ignoring the curvs.ture.

·Whan the' ·cross-section he.a rals.tiva- · ly ,thin .,fla,rig!'&.,, ,s.s W/.,th,, ,s.n ,I ,., .. chS.!lJ:la.l, z etc. :there,i.s:another.' effect.or '.ourva-·. ,- ' •. ,.,, -:,· .,,, -·-·-:l '!".'.'"-''''' -· ... ,.,,._ ___ ... , . ._,,,!'•'";,.,,,., ..•• , -. ·'•

ture:.; ·•. I·t •c;;,;us.!>'li. ;,the flange ... j;0 ' l:>an\l,, a.a. sho~p ,,iq, ,:i;:ig. A,13 .•. 22a,, .. and, th9rafora l:>a,-:-

E.or symmetrical. c.ross-sect1ons .the cir­i:umfe,rentfs.l bendihg stress a:t any point on the sac.tlon c,an be ca:i:o·ulated as

fb = ..A! 11+ 1 ( ~ 0. ' : ARL Z R.,.Yi.j

wherEI .• • i.. = ~rii.~ ot'. c,rosa.:'aeot'ron , R:Ra,dius of ciUr\iature at the

oentr.oidal axis ' ' 14 "'Applied momept; Positive for '' t.eh,sfarl iri 't.he outer .:fibers

and vice-versa y: distance from centrciidal a.xis,

being -1- outw.ard from this a.xis and ·• if inward

z ,,._1 n.zdA - ~ 1 (E/.S:L AJ R ... y - A) R+ y

where w =- cross-aeot1ons.l . wi<ith at distance y

Ts.bl& Al3.:1vprea·ents 'formulas for z for sEl'\l,ers.l f;poas-seiqttqns,.': WhE1,re, a, flange width is, .. required, it i,s nqt the aot.ual l'(,t~th, J1,· l:>.lJ.t. ,r~~re'r: &I\'affE1<:t.1ye. w1~t.~. beff•'"due tq the, <i~flectiqn shown in' Fig. Al3.• 22S,,• This can oe .. os.louls.ted s.s

l:>err.,:: i:':Lb", .·

wh.ere d1 is ob}al.ned f~'!m Fig. Al3.22b. · .. beff is d:!,sr,.d ,t.pi; ii",t.'!1"1!\~!!.ing, A an\'.),,Z,,, , whan· .. :flangss: ar;.e,. present, .. , Being less:.· thari the -s.ctual b. it results iri' high~ er bending.a~tresjesl. f,.'h• '

I.]' ' . 1 1 • 1'

•• • •. .:-1., ·::: ~~ :,·,·,·,. ,··.·.' . .. ·v ·1 .. ·1 ,

F1g.Al3.22b

curved 'Beam Bending

·.coefficients

"" .:·.' . . ::. · .. !·l=d···1 f.I

l!:: ,: ,, ,:ii:.r ~ E .: ~.o 1- -... ~ .!: .. :~ •• -; ~ .1 ··t;: =1: . ~"!:':~u •f -I l .I .. ... . ·:.: ."''. . '·I·- ;c .: :

:: · 4o .• .1 l.2 l.t 2.0 l.• 2.t l.2

INR,;i: The ,tranayerse bend,ing stress in

the fls.nge, fpt• can be os.lculated s.s , come l<>e.s effect~v,e,, ,I'.e9ultlng in higl:\er bend1n's 'iitre~saarcirthe l:>'ea.m; Its.lac'' gene.rs.;te,s, l:>Eil\41-iig.:~tfasse,~, \jl t,l;l~ flang9a il\, a .dii;,~oticih.noryial to. the:,,plii.ne. of the

. well w,h1oh a.re a maximum s.t '!:!he flange-to- where C2 is obtained from Fig.Al3. 22b web int•l'rsection. ' ···· · · ' · . an.d fb ~.•ttf!e 9ti;~,s9 · c,a.lculat,ed previ-

'c;."'~~;;,:;::;;=;_,.._;:'~'.'.';,; -~~!:;~ /,' :"'-> oµsly us~ng 91~:·: Jl.g":in,, .th.is d.1souss1on s.ppl1e9 only,,;tCI '8,Ymmetr1qs.l oro.ss~

·_· -·c •• p.;'" 011-b.r // seot1a:ns. as .in Fig. Al3.22a. l'l11.~6 11

" · .. - ' \1-- --Ce..;tp. ;,.-1;.11•" l\

FJa119a \1 ,, .. · i:.::.::::.~- -:-;..c.~ 4' -~.. \1

_;;~"'-"'°"'S·"""'-tr~;a'''

rig,Al3.2Zs. Curved Beam Section Bend:ir!g

When weight is important and rela­tively thin flanges result in high ·atr1>sse&,t the stresses can be redu.oed . by using thin, closely; spaced,,,~acb,Lhed in pls.oe "bulkheads," between the· flanges, or be1.111ath a. T'\member'.s .flan!!is• . This reduoaa

CURVED BEAMS,

the flange deflection and therefore re­reduces the beam bending stresses and the flange's transverse bending stress. Un­fortunately, there is apparently no de­sign criteria for this, so one must rely on Judgement and tests as to, both, lim­it and ultimate load adequacy.·

Unsymmetrical crass-sections should be avoided in fittings and hook& with lar5e curvature .sim:e .there are no formu;. las or data for predicting bending stress in such flanged members (tests required).

Finally, much curvature with flanged members can cause significant compressive ("crushing") stresses on the web and the thinner the web the worse the effect, These are generated as discussed for Fig. 03~28 and at.1ftener• are needed as illus­there to prevent crushing or buckling. The machined bulkhead• also do.this in in the case discussed for machined i tema .•

AlB.5a Buo~l1n& Loada ot Column•

Equation 16.(b) applies only when the. axial stress is iri: the elastic range and when the crass-section of the column re­mains stable• (Le,, no local. buckling oc­curs before Par is attained). When these conditions do not exist the critical load will be less than predicted by Eq. 16(b), as discussed in Art, Al8.8a.

Al8.8& Im.pe~teot Column.a. Tangent llod.\Ll.ua 1'beoey

Add the following at the end of Art. Al8,8, Since Et (in Eq. 16b, 30 and 34) decreases as the compressive stress in­creases above the proportional limit (Et being the slope of the stress-strain curve), a successive trials solution for Per is indicated. That is, Et must be the value corresponding to Der• The suc­cessive trials can be avoided if a column curve is available as in Fi5. Al8 •. ll, en­tering with L/r and reading o0r on the ordinate. If a column curve is not av­ailable the procedure illustrated in Art. 02.10 can be used, ·

Equations l6b, 30 and 34 also do not apply, nor does the column curve, if any part of ·the cross-section is thin enough to have a local buckling stress which is smaller than 6"0r as predicted by the a­bove equations. In this case a special column curve which accounts for this lo­cal instability must be constructed and used as shown in Art. 07,26 and C2,16,

TORSIONAL BUCKLING 7

Table Al3,4 Some Formulas for Z y

~· z~-1+2!_ (~

1:11 -11:11

-~)

rmtr:y . .L ' ..

y

I :--7-tf­hlP.-lT· 2R [[ ·-·· •] Z•-1+-- b,+-(K+c,

Cb•b1lb h

r-1- • ..i {' . .•. (::::)-(·-··)) Z • -t+~(t log.(ll+c1)+ (b-t)

log.(R-c:J) -b·I~ (R-c:1J)

z- -1+~ [b, 1as.n1+C,1+tt·b1>

lal'.~R":c.•) ... lb·~llor,CR-c1 J -b to1.cR-c:111. .,.

Al.6.8b Torsional. Buckl.ing

The previous discussions are for the conventional form of general instability involving only a bent (buckled) shape, also referred to as bending buckling. There is another form of general instabi­lity which can result in smaller values of Par • This form of buckling involves a twisting of the column (even though there is no applied twist~g moment) and is called "torsional buckling". Depend­upon the cross_- sectional shape, the .buck­led shape may be either a pure twisting or a combination of twisting and bending about one or both axes. It occurs for open cross-sections having thin elements and is usually more critical than bend­ing buckling in the short to medium range column rS:nge,4" Torsional buckling is dis­cussed further .. in. Art. ,07 .. ~J and in de­tail in the book described in Art.Al8.27a

~Sa,._ Pij. C:.7,73

8 RIB CRUSHING LO.ADS,

AlS.27& Comprehena1Ve Traat.ment ot . Column Strength

The broad covera5e (180 pages) of column design referred to in Art. 18.27 as being in ah originally planned "Volume 2 11 was not included when it was decided to publish only the current single volume rather than two volumes because of space requirements. However, .this material is available irf the book "En!!;lneeririg' Column Analysis~.' by W.F. Mccombs, Datatec , P.O. Box .. 76.3576., .. Dallas, TX 75376-3576. Top­ics include columns, beam-columns, truss members, bents, arches, torsi:ona·i buck­ling, local buckling, crippling,buckling of shells and members on elas'ti'c or oth­erwise sagging supports. Both, uniform and varying section members are included.

Al8;28&.K6ch&ilical and Physical Proper~ t.iea or· some A1rcra.!'t IC&teriala

The last para51'apba of Art. ·Al8, 28. ref er to a planned Volume 2 which was finally included as Parts B, C'and D of th'e textbook-. Therefore, .it f( Chapters Bl .and.,B.2 .0f .the textbook to .vihicn'.the reference is made:l'"

AlSl.23 Cruab.1n; Load.a on R1ba Du.• t.o W1n~ Bend.in"

Chapter Al9 provides methods for de'­termining ·the stress in the flanges, shear webs, stringers and skin· panels of: the wing structure. Chapter·A.21 discus­

FACTORS OF SAFETY

,, .... .Stl-1"1.,-

, i1.~ .. ;u9.44~. -~ln&'iub··c;~9h1ng··_~&4·; ·q

ai:ted by the rib load. Q, :bi the amount ~' - . . ~--·· ' .-. ,_ ' " ·- .

Q = 2Psih( e/2) = P9 or, Q = PL/R

Since R " EI/M Q = PLM/EI

If L1 iiJ:id. ·1.2 are dif{!lx~~nt rib spacings

Q "' P( L1 + L2 )M/2EI : ,-:- - ; .. ! .. :'", •

Where Q is the arushl.!lg' loaii. on the rib web a.t .. the strin5er, M is the bendin5 mo­ment on the wing···a:t ·the rib sta'tf6n ·a.na J;;I +,ii, .• th!'-t ,C!f.. ,tfl.~ .• ~,il)g: at;..}he, .. Fi.l;L station. Tb.ere fore, ea.oh si:.r'illger wil:). )ieed a clip tq the_ rib (.or iis equl~a.ieni) 'which can !'.a.ea' \~e Jl:)ad -~.·into the 'rib 1'eb, . ""ld .if tt\,~"'iq!¢..' is iB;ri;e, eriougfi to cr:ush the i;:ib t~!" c'.p.p w;ill. also· n.,;ea to ·ce .;xte'nded. a.rid fastened spas to sery~ a,s _a stif-fen­r6r' the' r11:>; to pr\ivent crushing Of the rib web. "

ses the shear loads and stresses. in win!' cl.lJa Fa.otor• ot aarety ri.nd. "Kar&iiia · ribs. Nei th"r of these chapters discus- or aaret:r · sea the crushing loads pn, the rib webs. a.t .• a:'.!'Jb"..Stringer joint .which a.re· ca,used by' .. ' The!ii{ itiiins a.,r'e' dl.scu~sed ln articles the bending of the wing 'or of any box A4, 2 ~RP. 1,n ;c:i: .• :i,~,,'.:.tfl..z:o]lgl'f' Ci, 15, ilt,\ ~lJ:eiy beam having strp1g'jI'.S .. B}l~PoI'.te<J, p;i:, ribs. d_o, hot; prov.fa,~, ~:Pl\~+:r+c .. ,a,;~app~a,J;ea,)l~~ ... s.!.nce the ri be provide simple support for bers for factors of safety or methoO,s .for the stringers in compression, 1f' a. rib 'ca.1c'ulatin5. the margin. of safhy for iilem­fa'ils ·loca•lly 1ts'siinple· 'support ·tor the be~a···)li;:der.inany .. combl.ried i9'!-li:sY.steiila.···

· striri!Ser w.111 'vil.nish aria ti:B stri.ni;er will .Tlie · pl,lrpose .. of. tlie followihe( discussion failaa 'a: cpl#·!: When the rib'. web' ls ' 'ta t"o'''a.o. that •. ·_·' ' ·""·' '' '· ... , . ' thin wi'th n<i' focal reiriforcieinent and''tlie' . '' .. : ; ' .. cru_shiri5 ,l_pads a.re' large S\jCh fa.LLur'e' can Factors . of Safety. occiur~ The'Iiianiier' 'iri' which these crush-' . , .. ·: •.-·····.. . .. .. -•, .. :1:h¢ 10'a,lis .'a:I'.:i:se,,,1,s a,s fol,lo.ws_. · · tJiti~(l.te,,,Joa,ds, .also ca.lJ,ed design

' .. ,._,,. ··. ,. ''.· '. _ ' ' ' '· · . ·. _ ·· .. · ··" . _ . . loads, a.re obtained by my.ltip).ying the· . ' Ffg.·At9',44 shbws a· .front'' v'iew of a ·l1mi t loads (the actual or expected' '·. wlhi(ln it's bent form (greatly, el<a.e;gera.:o loads) by a,,fa.qtpr of. sa.fe1;.y •. For pilo­tedJ; ·the 'upper stri!lgers''being' in com- ted aezfui~~e vehicles. :tile ,fa.cit or of sa;\e-pi'essfon, the low~F ones .. be1n5'1li'ten~ ty' is usUaiiy, 1~5 •••.... F,o'i.:mi..s.siles, .which· si§p aHd the load ~ine, ff,om rib :to r_+b are n~.Kc pilpt~d; the' facrf.p/'". .of sa.fet,y i,~ shown by the br_oken line. Due to the _. .is usually 1.25 .ex~ept. for, any load con-

. l:iEi~dlng :fa.di us of curvature. Fl1 the angle di ti on' Wh!'r.~ the. sa.fety of' people is in~ E(·w~ll:be 9=-L/R, .a very small. a.n,gle.. . volved where it is 1.5 (e.~., for the eJ-s1nc.e. R is very large in a ·pra._ct1c:!fl win ection load .. s from a carr:ier airplane), struct~re;· The load, P, 1ri ·B:r\y str.ing,er Na.val airplanes are. designed for a very loc~tep:· at a rib will_ have_ cp,iJlpbneii:t•iJ· ·of Ffast·· s1'!1kXlfg · spe'ed';"' about"' 26 · f 1fot p'er se­loa.d parallel to the rib which a.re re- !cond, so no safety factor is applied to

r-:lo•• •Uclt d.&ta an al•o on p • .&.6-A.9.

l'\ARGINS OF SAFETY 9

these loads. However, the landing gear must continue to function after such a landing and the major attach fittings (landing gear to wing or fuselagj3, wing to fuselage and major fuselage section splices) must show a margin of safety of .25 for this landin~.condition. In gen­eral, factors of safety are specified by the procuring agency.

Margins of Safety

The margin of safety for a structur­al member subject. to a single load or stress is calculated as

Allowable Load or Stress 1 0 M.s.=~*-u~l~t~i~m~a~t~e'-"Lo""'a~d;-i~o~r'-'s~t~r~e~s~s"+- •

and must be zero or !JlOre. However, in some cases it must have a specified posi­tive value as mentioned for. Naval air­planes and .15.for fittings. This margin is also required if additional fitting or casting factors have been specified. For shear Joints a minimum margin of safety

tests rather than theory, are used to show· structural adequacy. A typical in­teraction equation ls of the form

'to b c: 11_ 0 R1 + R2 + R3 + ----- Rm - l.

Where Rl= f1./Fi. R2 :: f2/F2 , R3 = f3/F3 and Rm= fm/i'm, f being the ul"timate stress (or load) and F being the allowable stress (or load), R is called the "stress ratio". When the.left side of the equation ie l.O the M.S. is zero. When it is less than l.O the M.S. is pos­itive but undefined and when it is more than 1.0 the M.S. is negative but unde­fined.

Example:

f1 = 10000 and Fl= 30000, so R1 = • 333 f2 = 40000 and F2 = 60000, so R2 = .667

Interaction Equation: R1+ ~; l.O ;a .

,333 + .667 = ,778 ("'-l.O)

of .15 is .required and for tension jointa ·Hence the M.S. is positive but undefined. it is .501'" Also for these.joints there must be no yielding at limit load,. For shear Joint, .attachments if the bearing yield strength f's less than 2/3 of the ultimate bearing strength; the bearing allowable strength is taken as 1.5 times the bearing yield strength and the joint is said to be "yield critical". For .riv­eted shear and tension joint allowa,ble · data see Chapter Dl. .Also see Append~x A .•.

The prev.ious simple formula for the M.S. gives the decimal fraction by which the load may be increased and still have a M.S. of zero. For example, if M.S.:.20 the applied loads may be increased by a factor of l.20 •. However, the simple for­mula applies only when all of the .follow­condi tions exist

l. The member is subject only to a single type of loading, not to several types such a~ .shear plus compression etc.

2. The internal loads vary linearly with applied loads.

3. All applied loads are variable, i.e., none are fixed in magnitude such as a constant pressurization.

There are two oases for which the M.S. can be calculated directly using the in­.teraotion stress ratios, R, as follows.

a) When all exponents (a, b, o, n) have values of l.O and/or 2.0. In th1s case

2. -1.0

b) When all exponents have the same value, n, in which case

- ] 1 0 :11.S. - (Rf+ RI! + .. ,. R)k)'i?I - •

For all other oases the M.S. is found by us1ng. plofa·of the applicable interaction equation (discussed later) or by succes­sive trial calculations. The latter is done by finding, by successive trials, by what common factor, A, all of the loads must be multiplied to satisfy the interaction equation. The M.S. is .then A - l.O, Or, when

"' .b " J. (AR1) -t- (AR2) -t- (AR3) 4- (Afi4) = 1,0

M. S. = A - l.O

These conditions are discussed as folloWB. Example;

l. For example a bolt may be subject ta shear, bending and tension simultan~ eously. In such oases so.,. called ''1nteraC.t1on equations 11

1 derived from

For the previous example

A X· .333 + (A X".667f·""· l·;O

10 MARGINS OF·SAFETY, IRTERACTION EQUATIONS

After several trial,s the equation is' sa:t'-. Monocogue Cylinder (Various) Table C8,l isfied wheri A= l.171, so"

Curved Sheet 14.S .• "'.A - l,O = l.171- l,O.:: ~ Compression or Te,nsion and Shear

C9.5

Sip.ca all exponep..ts are l.O and/or the M.s. can .also be calculated by formu'.1~;. as .. ~

2.0 .Stiffened Cylinder Shear and.Compression

Torsion and Bendin15

c9.ll

-1,0 = ;171 ,667" · · -- • Shear and Bending

C< ~- • __ ' ' - '· ., :·

c9.13

c9.13

Whan llairig this. for!Dula. with other more.• . lerigt!,iy' 1nt.eraot'1.o~ equations' ,tlle ·. paren;;.· theses around the· t'erm'£F(1 must be noted· and used. l:R1 is the sum of all stre1s ratios having the exponent 1.0 an.d %'R2 is .the sum of those having the exponent 2.0.

. sb.eliS .in Diagonal Tension · Stringer Primary and ' Secondary Stresses

Ring (Frame). Primary and Secondary Stresses (for

· Stringer, System) wiieri plots of interaction equations

are available the M.S. can be determined .Longez:on p·rimaioy and. by a graphical construction, Thia is il'- . Stre.a·.l'les .

secondary

·::·1_ -; !'' \ ·"' ', ' _·· Rins ( F,rame) ,Prill!a:r'Y ,and sec:pnciary st:r~sseii (for

Oll •. 36a lustrated 1ri Art •... c4.24a iri· the textbook using Fig •. 04,36, .. However, graphical so­lutions are not .needed whefr the prior formulas apply, and in many other. oa~e a the· successive triB.l'a· procedure may be• · · easier with calculators or when the inte"- 2.

, ~ollgi;,J:'oll; sy ~j;em J'. . . ' :~ 1- -- ,' : - :. ... ' ,',';,. ;·. . TJ:iere are cases where the internal loads.1hamembe:r dop.q;t vary' linearly w.i;tn. the ap,pl1ed. lpa<J:s. (k\e eica.uiple ii the beam-column wl;ler,e the .. ;beridi?lS mom~ erit inore.ases faster: ;t&il,!i does.' ,the ap-

ractioh curves are not available.

Listing of Interaction Equations

Numerolls intera.ction equations ap-. · pear in vari.ous·parts of the book per the following listing.

Bend.ins, Shear and CompresSion

Bending•and Shear

Bending and Bending Bea,m~C:olumn Bending & Axial Load .·

·TuMne;: ·Bending and Compression

Bending and-Tension ''!

Bending and Torsion

Compression, Bending and Torsion

Bending .and Shear

Compression, Bending , Shear and· Torsioi:i · Tension and Torsion Flat $heat Bending and Compression

Bending and Shear

Shear and Tension or Compression ",_ "'~--;

Compression, Bending and Shear

c3.13

03;12

03,8 c11.22

04.22

c4,23a

04.24

C4o24a

c4.25

C4.26

c4,27

05,9

C5.10

C5.ll

05.12

·. plied,, .lo!iil,1riil; •. ,,Afuo.:ther,. eic~p.le is .the ·. aic:iai str.es.8 .. i'n .:at'r.irigers due: to ten ..

, • ,-_ . 0 ... , __ .,.,,'.,• .• /. ,, _ ,. :.• .•.' ,. •., i ·1• .. i _ •. ' "· ,. ,., ; :

aiO:!l. fie):d ,a,gtion. , .f1>r slloh. }:.ases ,the 14, s. as PICtiiViously .caJ;culat.ed would be an'" 1ili,ppS.r6iit 11'·M.s;·. not a trUe"M.s·.· Whet.he.r the. apparent M.S. is. positive o?; nee;S.f~,V.e:~ .the, true. ll .• s. w1;i.;i. always tie olose:rt_t() zero. than the a,ppar•mt 11. s.:, .The· true 14,s. must be calcU:la.ted. as' ~ollowa·. . ' ' ' "

By successive trials find the com­mon· faotor, A,' of' which the applied.· loads ( whi·ch prciduoe t&a· internal loads in the members) .must all b<!. multiplied :tci give a, <iii.1cuiated M. s, of z'er9. The

' true 14;s. is therf A ,_ i.o '. For. a peam­column tne·applied 'loads ·are the axial load. a.n.d the transverse ~oadil. lolulti­plyl.rig these by a:ny factor, F, will generate a bending moment in the beam­oolumn which.increases faster than the oommori facitbr, .F. A if!. t!ie value of F for which the"oalculated M.S. is zero, and then the true M.S. is A - 1.0. This procedure applie• for any inter .. aotiori equation that· may be applicable. Note that this requires a greater ef­fort than that in (l) previously. This is because iric.reasing the applied loads requ:l'.rl!lf · anotli'~r artaly ai a to determine the internal loads, since they do not

DEALING WITH TOLERANCES,

vary linearly with the applied loads.

This is important because mar­gins of safety are reported for struc­tural members and are user:tto see if the members can withstand an increase in the applied loads. If the applied loads increased by a factor of, say, l •. 20 and the report showed a M. S. of, sB.y, l.20 it would be judged to be ac­ceptable as an increase. But if this happened to be an apparent M.S. the true 14.S. would be smaller and the in­crease in applied loads would not be acceptable. Hence, the true values. of the M. S. should be in the report. .If nqt they should be "flagged" as being apparent ones so that a proper eval­lia tfon can be made when needed.

The internal loads in a member are due to the applied loads. Some­times the applied loads consist of "fixed" loads which do not vary, such as, for example, constant pres­surigation loads. When such are pres­ent the internal loads or stresses in the member due to them should not be multipli"ed by the common factor, A, in (l) previously or by the factor, F, in (2) s_ince they are constant. If such w.ere done the calculated !.!.$, would be conservative (too Slllall) if these loads were "additive" and vice-versa

COMBINED STRESS EQUATIONS 11

or -,25 for redundant members. These ·are arbitrary limits, some companies allowing more negative values suoh as -,19 and -,39 respectively. Such negative margins caused by tolerances are acceptable since the probability is qUite low that, simul­taneously, the material will have minimum properties, the tolerances will be as large as allowed, the loading condition will be achieved and the internal loads in the members will be as large as pre­dicted. If the M.S. had to be gero or more bas·ed on such minimum, rather than nominal,dimensions considerable weight would be added to the structure.

When dimensioning structural members care should be used to prev.ent the build-up. of large tolerances affecting the final dim­ension. Unacceptable tolerance effects are most likely to occur when dimension­ing small_ machined or cast protuberances or holes, where small internal corner radii are present and for thin machined .or chem-milled parts,

However; the above should in no way be construed as sanctioning negative mar­gins based. on nominal, dimensions or en­dorsing the salvage of parts having less strength than required per the drawing.

c~.6& -C~mb1ned ltHaa Equations

if they were "subtractive". This also For practical calculation purpcfses applies when the M.S. is calculated by ·the following summary and example problem the formulas in (la) and (lb), and are helpful. Fig.Cl.Ba(~ shows the posit­when.the M.S. is determined graphical- ive direction of known or given applied ly, which makes these calculations not stresses. Fi5, (b) shows the resultin~ applicable for for such cases_ (the stresses,'IE> anda'e , on any plane S (posi-successive trials procedure is then tive directions shown), which can be needed). calculated as follows. ·

When calculating margins of safety for a structural member nominal (mean) dimensions are used to determine the stress in the member and its allowable load. Ideally, the M.S. is gero, gener­ally. However, all drawings specify the manufacturing tolerance which accompan­ies each dimension such as, for example, "!: .03 11 and these are considered as follo""

In aerospace structures it is com­mon practice to consider only the two worst tolerc\Ilces affecting any dimension and to compute the reduced margin of safety based on the reduced dimensions. This will give a negative M.S. when the nominal M.S. is zero. Such negative margfn·s a:re acceptable i:E they do not exceed -.15 for single load path members

d,. =·i(d:o< -t-dy) + ·Ho'..,. -d~ ) cos2S + 't-'llsin2S

~ = t(o'.., -dif ) sin 2S - 'l"")b' cos2S

1. The principle stresses, cf_ 1 are cal-culated as follows. n.

d,, = t(d;i, + dy ) ± 1/ t(cf ;y, - d '.1 t' +'txif 2. The plane for the l~rgest principle

stress, Sp, is measured from the plane of the larger of d;,; _or dv and is oalc-ulated as follows.

Sp:>::: tarctan(2~;i/(d,.. -d!;J ) )

The plane of the smaller principle stress is 90"' away from this.plane.

3; The maximum··she'ar str·ess, Tmax• is calculated as follows.

'L max = -Yr-t,_(-c:r';c.,..._ ---cf-~-);;_.--+-T-xy~;i.

12 .. COMBINED STl\ESSES.

4. The plane on which """"' is l.oc13.ted is e,,.':=. tarctan( (o'~ -d'v l/-21".;i:u ), "~1"Y~: .,.-, 4·-· J, __

.;,: 1Dr-t;:)' .Pos1t1ve d1rect1ons or ~ ~ a.pplied stresses shown

r,., :· , 0

• Tenai~n ia(+l, comp. 1a'H

<. a"); . · I "'.·~ ..... ' ·t"'•·:. ' .!, .~i

\.1'e- . ·.-· · ~· · dy :and 'f",b. are ·known·-, .. ~ ~e- v~lues,: po.s*_t1ve as·.~.how~.

2'· ~nd.'1f, ai-Ei pos:CtH.ve·· as shown

""' ·,··1 ·--'1''1l'

' . -r. . .•.. . .(b):'"·f~ ld'.)i' "''' . ' '" ··. ·' ·'• ' "

Sh9a~ 8.nti-- N(!rmai Stresses'·.

Examl?'i'<i Problem ·

pooo

lQP?~, ·JI -: l ~. i?o .. oo. . .· . ~4500

"'"T:lbao .. . ·

For the stresse's. shown in the sketch above (note that <JY. i,s oompre,aai;v;p" hence negative) what ared and 't on a plane at .. e.,,:= 60Q? What ~re d11 .• Sp, 't'max and <it?

. F.REE-ENDED COLUMNS mininl!' the value of the largest~ and of the largest ~max•

02.l& Ke.tb.ode-ot_-.co~\llZID. J'a1lur• • Qolt.mi.Jl ·~u&,t.10Xi8 · · ·

, ... The :taat parae;i,;aph is extended to include the following. Predicting fail­ur:e ~u.e, ~p ,J,pcaJ,, iµ~t;':J:lilf.~Y't~qu~res that, the oollJlllil curve .be reqonstructed in th,~ short ';t;d ip.t!)rmeciiate ranges. ' •''.J'hi s

.reiionstrt1C'tiori'"'is discussed' an'.d .illus­t:c'a't.ed 1n: Art'r 01 .25 "tJhrougii c7.. 27 '·

. ~· . i·.'. - ,., ' - • J ' .;~ •

. Fig;o2.2 shows only a special case of the fi-ee-ended column, one eh'ci being fuily fiX:eci aricj. :1;,he ,J,qad remaining llaral­lel td th'ii' axis' O'f the member in' '!"ts' straight (unbuckled). form. In general, hq)'(~ver', . the lb'a\i::t P,: may b .. e. dfrec1;.ed either t<;i .oz. fr6¥·a 'giveri. point, "o", as shqivn in rig. 02,; 2a.. ... . . . '

··.' _iJ.:_;:;_~--:~~~.:!-. .. "{$} Load.dii-~.o.ted towa_rd_-11

01!.

m = a/ta.+ L) · · .. _ '!',

. . ~ ·: " - " - .. - ' - ' .. ' . ., -: . ' :· -' -, "

"' · .. 1·, ::~:;:~"'~;?~"';_,,: :::·;,l,;; <fa , :: ,f< looqo - 2099 l+ f(ioooo'+ 2oqo}co's120' ·· · l (il) Lo~d· 41recit.•d froin "o" · .

.. ··· , 'in:~ (L + Lo)/Lo '. . , O"'·• ,,, : .+ 4500a1nl20, ""' .. 4897 .: .. '• ' ._,.,,. ---.. ,,

~ ,,: J<~qodo + ~ooo):s-J:ni20°:::45oo~os · i2o"

= 7446

d11 :: t(lOOOO - 2000) :t 1{ -!'( 10000+2000 )~45002

::. 115 00 ·and '-35 00

6P:: tarctan(2 x 4500/(iqooo + 2ooof) 0

"'18.43

7,'max ::= 'ft;: ( 10000 + 2000 f' + 45 002 : 75 00

s-r .. :;;f>r,9~?-ll( lopoo +-'2008/-2(4500) = .-26 .51'

9'7;' is always 1+5° aviay from. ef;'· If any of: the above· calculated values had been negative they would be acting or lo­cated iri .a dire 0tion oppoai te to that shown in Fig.O ... lJla. ·

Usually one is in'tere.sted only in deter-

Fis. 02.2a<Fre~· ... £:nded ·001Umn, F,ixe·a Eiid . ·, -. . ". '• .

As'sl+owii'in Jl.ef'.4 (Art.A18;;17a), the bu&Ji:'ling ioad is defined by the trans-cendental equation (·· ,,

( L/ J )ctnL/ .l =' 1 - ( l/mf"

Fbi::.~/:+Y g~yep. V:a~¥t'f8.~ m JF~g.02.;iil). Por is :(ouncj.b;j[ aucqe.s~iJ\~.:tr~aJ,s •... ,Ti}";~,is, oµe a,saumes a value, foi:: .;p, calc:;ula1;.es

. J'. (~ .l/EJ/J>) ~d .ua,es.'. thls in .tiie equation. )\'hen'.P''°''Pcr ~he equatiori. wili ce aatis­fied, •. ,.Hii'wever; this. effort can be great­ly' :i"ela:uoed bf using F1e;; 02 .2b, entering the fig):II'e,, .... ;~h m· .. and obtaining (L/j lcr· Per 1 s then calc\l.lated as

• . 2 ·· a Per= \L/J) 0r(EI/r; )

Net's that .when "' or L.,= Oo • m = i:o and Per= trr/2f(EI/Ii'-); When. a"' o,m = o and Per:: 'Tl'il(EI/L'"); when L.,= o, m :o<> and Per.::;;. O. Tha't. is .• as a decreases Per, incr'eases, but ... a,a .. h,.<l.,e,oilease.s Per· decreases. · · · · · ~e;. (a) an alternate .fQ~ ia TanL/.J · ... a/J

and fOl' P'l.g 0 (b) TanL/J • (L + Lc,)/J .

•• Fo-r f'rti-t-•nded. 'be&11•cclU1".tl• ••• .Art • .UB.27& book

FREE-ENDED COLUMNS WITH ELASTIC END RESTRAINT 13 ----- - _.:__ --

I I

I ·--~ -·-·-i

- ~----:--~·--

'• ·;::i, ~ I .. ..,

"' ,.., ' M

:11 ., -·-·-• I:: .!!

,':3-- ::;-,- -r;r -;- i ~

,:--,--- ---

:;; ~ I I

I ' -·- --·It\

I ' 1. I

I I' v

-·

'f !..----" /~ -. I ,_ 0

. Elastically Restrained End

• • " • • • • • 0

Instead of beinB fully fixed, the end may be elastically restrained as shown by the "torsion spring", k, in Fig. C2.2c, which has a value in in-lbs/radian Thia results in a smaller buckling load than when fully. fixed ( k ""'<><>) ,

r--• L - .,,..--]-. '*n" -ur-------:.,-_-;;;?-:-.~---~ w ,,-- p

(&) Load d~rected t·oward. "o"

· m = a/(a + L)

I L --.:-?"-- - Lg p k f.(g ;.. --=::..:=:.. - ~ p- ""-:-.~" - { b) Load dl_rected from "tl 11

m = (L + Lo)/Lo Flg. C2.2c Free-Ended Oo1umn H&vlng a.n

El&stlcall.y Restrained End

For thls case, as shown ln Ref,.lf, the buckling load is given by the equation*

k PmL - 0 - l - m(l - (LfJ)ctnL/J)-

P er· is found by successive trial calcu-

• Der1Ted 07 us11l.6 the t1101111nt d11tr1but1on "'l:'Ocedure,

When m "' 1 the equatioo .beccmes. simply k - PjtanL/j ,, o

lations as follows. For a given k and m one assumes a value of P, calculates J and uses these .. values in the equation. When P = Per the equation is satisfied,

For the special case where a or w0

is infinity, so that m = l,O, P0 r can be de­termined directly by entering Fig, C2,44 with kL/EI, obtaining the value of C and calculating P0r as

_j! .. Per = 011 EI/I;

Note that when k=""" (fully fixed) C=.25 .as given 1n Fig,C2.2.

It must be remembered that in all column calculations E is Et, which de­creases as the compressive stress ex­ceeds the proportional limit. Aiso, when the column has an unstable (thin) cross­section E is an "effective" modulus~ To determine it one finds the buckling stress from the modified column curve (Art.7,26) and then calculates it as

Eerr = F;,..(L/ff ;~a For free-ended columns Fig.02.17 is aP­plicable only when m = l,O

If the column has lateral loads or an initially bent shape lt becomes a beam-column. Free-ended beam-columns are discussed in Ref' .. 4,also Table AS.1a,case 17.

Example ·Problem l

For a column as in Fig,C2.2a(a) assume that a-= 30", L = 30 11

, E.-=- 10.5 x·106 . (7075-T6 Extrusion, p.Bl.11), I= .191, A -= ,50 sq in and a stable cross-section What .is Per?

m = 30/(30+ 30) = .. 50. Per Fig.C2.2b form= ,5, (L/Jlcr= 2,03. Therefore,

Per= 2.o:f(10.5xiob)(.191)/30a= 9183 For = Cf 183/ ,50 = 18366 -

Since Far< Prop. Limit, Et= E as assumed. Example Problem 2

Repeat Example l assuming the fixed end .is replaced with an elastic restraint of k; 50000, as in Fi5.c2.2c. This must be .solved by succe .ss1 ve trials.

Trial l: Assume Pg~;= .err2EI/.lf:ri2-= 4399 Then J =~"'IJLO. x 10"( .191)/4399 = 21,35 and L/ J .: 1.405 . Then 50000- 4{99( ·iol(3oJ .~.

l - ,5 1 - • 05ctnl,lf05) - : -56849 (;6 0) . • Stn1 Art. c2.16

14 EFFECT OF SHEAR ON'' COLUMN BUCKLING. MULTISPAN COLUMNS

Afte.r .sevez'al more tria~'s it. is, found. that: when p is 26,22. j:.he .,equa.tiorl )_s sa•f;,'­isfied, so, this is the c.ritical load, as follows;

j = )/10.5 x 106( .191)/2622 = 27.656

L/ j •·= 30/27 .656 := 1,0848 ·, ' . ., .... _ ,., ;.

50000 m 2622( ,5){30) 1 - .5.(1 - l.0848ctnl.0848)

= -4.7 wh.ich is essentially z.ero.

So this particular elastic restraint has reduced. the critical load from 9148 for a fully fixed erid t.o''onl;Y' 2622 lbs.

middle where dy/dx, the slope, is zero. Consequently there is a shear in the mem­ber which, in effect, makes the column more flexible and thereby reduces the buckling load or stress.

As discussed in Ref.5, for uniform columns th<'I buckling load, considering shear, is ·

p' ',,_ "Vl + 4hPg"1AG -1.0 er , 2nAG

where Per :ls the. buckling load ignoring shear, n is the form factor for the cross-section, A is the cross-sectional area and G is the shear modulus of elas­ticity (G = E/2(1 + µ))where µis Pois­son's ratio• The,.values of n for several cross-sections are shown in Table c2.2.

Table C2 2 Cross-Section Form Fa t • . c ors : cross-·Sec+-.·on n

Rectangle . 1.200 Solid Circle 1.100 Thin Round Tube 2.000 I-Beam.or Rectan~ular Tube Area/Web Area

. . .. . .

, Equ~tions (l) and {2) of Art.c2.l consider bnil.Y the :bending· stiffn6'ss 'of colU!ima.· Wlien a beam bends ,because of applied trarisver~e loads arid the result­ing 'beridiri5 :·moments, ·the usual de flEi a"' tion formulas consider only the bending moment. Tliere j)~ •' hovi'ever.1·, an addition-. al deflection due to shear. For example, n can be calculated for other cross-a simply support'ed oealli 'of uniform EI .section• as having a load; Q, at' mid~'span will have ·a J:: maximum deflection given by n :::. (A/I ) Q dA/b . A

. y ~·. QJ1/48EI ~ nQL/4AG . ,-.,,, ·,::_~ _,_,_. ... :·. ,"j ~-::··,;· .:·.'.\'•.:- -', h·'··, ~-~ ' , ..

where A .is cross-sectional area, Q is the static moment ,Cl,~ ll,fJ!'!!:·'ll"•Yond :dA•,;about '.the neutral axis arici·b···1a 'the'wld.th"a"i:."the-· n'!,i,;tral ,,.xis ·/·:For,.;cplumns having i.lat­t1c'!d at.r,ut.s,, (.t?Jussed:,colllliins or ·"batten" plat.Els) ,;or 'h,av:ing perforated webs see·

The :t'l.rst term :!.~' due'. t'o 'bending and the second term is due to shear, where n is a form factor which depe4.ti!l)\lP<:!n t.hE> Sh~PE> of the cross_,sectiol1• This is usually.· ' negligible,, but as 'll increa.,eil due to '..a thiri ::or per:ffora tad "or. o·tnerw1•iie mar&' flexible 'web ,1t can becoine ·s15n1fica:rit.

. ,R,~~·•:"5·: r .It, is when ·the web becomes thin .qi;:,.oth.er,.i.s.1>.,:;has .. less shes.r sti:ffne ss .. ' that the column .. buckling load is sie;nifi­ca.ntly reduced.* - '

For a column there is rio .applied transverse load, but a shear load is 5eri-· c2.3b llu1t1epan Columns erated bY the· .axial load as the ... column .. ,, ... ··· , .. : ·:. . takes. :on .a• bent (buckled) .shape, as shoiin Unforti,mately; .thi~ .. and most other in Fig, C2. 2d textbooks do not' di'scuaii co1unins havine;

• .. , more than one span. Such columns are eas-

F1g.

·-~~':: =-= = ? ::====::=::~=;ar::,.f _..1_,

02.2d Genaration of Bh:ear .11! a· Column

AtJa:riy station the bending '14:::. ~Y and .. the shear iri ~i,.e

v = dJ.!/d.x = Pdy/d.x

moment i.s member.' is

where dy/dx .is the slope of the bent shape; Hence, for the uniform simply supported column the' shear varies from max.imum at the ends 'to zero a·t· the"'""''

ily checked for stability by applying a 1 in-lb·couple at anY "Jolrit" (support) not having full fixity and carrying out the moment· di,stribut1on procedure S:s dis­cussed and illustrated in Art. All.13 -All•l'f and 1 ts exampl·e problems. If the successive ~arr,y:over moments at all joints become smaller ( oonv'erge) the col­.umn is stable, The larger the axial load the .!'slower'·' the convergen·ce. , If they diverge at any joint the column is Un­stable. The critical load 1.s that for which they do neither, found only by auc-

a cessive .. trial analyses, varyin5 the axial ,load:., used., in. ,each. momen.t. di atribution, ""'

•· Por·'~.;arf1iig EI c~l~ri~ the .&.bi.~ tn Art.c2.6b can 'be amended to account tor ahear1 mee Bet.4.

MULTISFAN COLUMNS 15

There are some helpful techniques for doing this in Ref. :4, including. some · "quick checks" for detecting 1nstabil1 ty · sometimes without the above procedure. Ref, 4 also contains numerous example prob:J,ems for multispan columns including those on elastic or otherwise deflecting supports, those having free-ended members snd those having varying loads and EI values within the spans.

The alternati~e to doing the moment distribution analysis is to conservative-· ly assume that each span is simply. sup­ported and check. each span individually. Thia is accurate only when all spans are identical, the two ends are·simply sup­ported and the axial load is constant.

"Quick Check" In.stability Criteria

cal load is that for whichl:SF = O, found by successive trial values of the axial load.*

Example Problem l

Ia the three-span column shown in the sketch below stable or unstable?

r-2!,'"",-'*' .... __ ,,,. ___ ...._ __ "°. l40oclO EI= 2xl0"' El=-,~lo"

ea SF

:II.SF

DI' cor .... 0 B&l COM S.l COii Bal COM B•l COM .. l

0

. l) The following can. be done when each span is uniform and the axial load does not vary along the span (J is constant). If any span has a value of The analysis is carried out as shown, ap­L/J greater t.han shown in Table c2.3 plying a l in-lb couple to the Joint at the column is known to be unstable. B. Note that the SC, SF and COF are om-If less than these values it may or itted at the ends, A and D, since they may not be unstable, but .a moment dis- are simply supported and no iniftial mom-tribution analysis is required. ents (FEl4 1 s) are present there. · As the

Table C2,J Inatabilit Criteria"'~ moment distribution process shows, the ~-.=;;;;;:;:;.....;,;..:.;;;,.,.;:;:;50"u:::p:::o:::r:;;;::..::L....::::.=.;;;L;.;.:~-..., I s uc ce s si v e carry over moments {COii: ' s ) at h--..,~eµ:.e;n,,.;;.=-=e~a;s;,.;:~c~a~·~yc.__~-fl::l'.'IT~~2~q1ai1 joints are decreasing. Therefore the

restrained, one end free column is known to be ... stable ..... ,. ·

B, Both ends simply "fl' Actually,· one should· always calculate the supported / c. One end simply supported, 4.Ji9T compressive stress, P A, for each span one elastically restrained and use the corresponding value of Et. (or

c. Both ends elastically 21T Eeff) in calculating the value of J • Al-restrained · so, Ref. 4 contains extensive tables giv­

Elastic restraint is provided by an adjacent span in a multispan column or by a torsion spring as in Fig.C2.2o

2) If at any Joint the sum of the stiff­ness factors, ZSF-, is negative t.he column is unstable (see the footnote for Art. All.15b),

The Two-Span Column

The following applies·not only to a two-span column but to any number of spans meeting at a common Joint (support~ If the outer ends of the members are ei­ther simply supported or fully fixedo•f~~ (no .elastic supports) the column is known to be stable if at the common joint ESF is positive.* I.f it is negative the col­umn is known to be unstable. No moment distribution is,therefore,required so the analysis is quite simple. The cflti'­• And 1r ~r1ter1•, (1) aboTe 1s mat

·••The 11.sted. t/J ·Talues are for P • Per • errZEI/L&

the values of SC and COF to six signifi­cant figures (better than Fig.All.46-47).

Example Problem 2

Is the two-span column shown in the sketch below stable or unstable?

~40· 60"·-----'

EI::: l z l.O

I/J & 4.]8 J: }.79

,0781--897 .}12 x lil" -1.794 x lo"

-l.462 x. iQ6

Proceeding as discussed before for the two-span column, it is seen that ~SF at the common joint, B, la negative. There­fore, the column is known to be unstable.

By successive tr1ala·1t is found that when the axial load is 93040 lba::?lSF = O, so. tha,t is the buckling load.

If the axial load were 168750 to 219600

* Por a tree-ended co1umn when m :z 1 (Ft°g.c2.2aJ the 8,P, ie -PJtanL/J

t:.if.,•!"or any 3-span column, ABCO, havinq fixed 11.nd/or pinned ends -·It.nuid. ha• •la.st1c t>.nc!lng :N•t:ralnt (•·&'••a tol'rlon 1 rl l {p.15) tile stability ean l:la checked olS follows. uaa 1t1 k Ta.lu1 •• lt. .SP a.:id lni:ludi thh

1114 aa anatha:rpJo~t

If Df"tc CO~-.: DFca COFca < 1.0 thl!I column is stable and vice versa. 111 ti'"· ac1Mnt d.l1trlbllt1on J1:c"Do::1<1~- (k °t.-11:1&' th• IJl?'lng coa3t.-.nt-1

·7.f. span BC is llni!orm than COFaC' • COF.:• .......,_____~.a. -'I:

S.TEPPED COLUkNS

2SF would be positive, indicating stab1l- . Far ( 7 Per/A) should, bEI calculated, for 1·ty; but' thls would not''be appl:l:c,.bl!t{ be- ''·eaeh «itep to be sure that 'the values' used cause L/J for span BC would exceed the·· for· Et (Fig.C2.16') or Eerr \Art.c2.16) u1aximum (4.49+) allowed in Tab.le c2.3. corre.,pond to·Fcr• which they must:°' This is why the Table 02.3 criteria ·· should always be checked first, before Siniie•·it· is ·possible for more than proc•eeding with any oth~r analyses. one •value of'P' to'•sattfify"the: equations

(qui•te different values), it is best to In .. awnmary, the moment distnibution use' a 'lent\;tli~weighted average ''EI ,··'cal·cu­

procedure can be used to check any multi- late a2

corresponding value·;t'cir Pczi as span column for atabili ty. But l:)efore ~EI/IJ . and u~e this a.a the .initially as-d·oing ,this ip.stabiU ty 'carf''be detected .r1sumed ·value· for·•p. ""Fcir''tlie: special cases quickly peJ:'·t.(!",,.crit<>l"~s of ,Table c;2.3 · of::one-step a'.nd symmetrical twoC:stsp col-and if at a.D.,Y:: Joint .ESE 1·8' negative; umns .graphical plots •a.re available in Then, for any two-span c9lumn if ~F 'is ·Ffg·• 02.21 and C2.22 for a direct deter-posi tlve at the commol'l Joint the 'column mination of Per. is· known to be stable i.f criteria ' ( 1 ) is met. Unitorm Two-sp!Ui Column Formula

For the special cli.se of a uniform member on tlii::ae:· simple supports an ap­proximate formula. for. the buckling load: is

,. Per = efEI:( 2 - b/a)/aZ

whe.re. a .i·s .the longer span· and .b ·is the shorter span. When a ,: b (=c:L/2) the: for­mula. is exact •. When b becomes .. zero (1.·e. as ·a ,becomes L) the,..srror: i's only 2·•44%' and is. 1conservat1ve .(one: end' be .. come.s:

. essentially fixed as ·a bece11es. Lh · · '"' ·.:··!/\_... .. ,,. :-_!r_·a_·_. ... ·'tl· ~- ,,

PX ..... ' ... ' Th~ }C>~~i~ c~" .b~ verl'.~ied nume';ii:ally ):>y calaula,tions for. a ,,two.,span column as pria;v:iously, illustrated·; .'or. ;:theoretically as .1·ri the ·.book ",Theory,:::of Limit Deai.e;n" by. J, .A.,. Van .Den Broek {iJ. Wiley and Sons),

"1>,

c2.6a Steppe4 Oolumn•

One-Step Column*

;._,,_,_,

Two-Step ,column '

L

.. ... ,;.:E2I2 . ,. . . E}I; li:l.ll. ... - .' .-: .. ·_ ·-.. ·:· .,,. __ , ..

i'1g. c2:.:'i9a :.TWd~-6-teP:- .. ColUD:in

- -t~ntiV~i~r~ L1i/ifii ,;

"·i.·

wher.e ·~ =

Two.,,Step· Symmetrical ·Column'.'''

p

These are a special case f'Or \,;11{(ih. form,W,l\E1,.(l>l.\!il!':Lip,g ,eq\,\ations) are. avail.­able as di.scu,s.sed .in, Ref, •. 7 •.. Form.ulas a.re pre'sent~d "ror the cases of' slmply · sUPPO.J:'ted members having cine, two and tl:lree steps iri EI. . For:,:lllore .. than three stej)s, the' r.ormu.,la.s p.~00!"~· to.q l~ngthy for pra.ct·ical Useae;e •·· .,sO:.'.I\ simple tabular calculation form 1s ':Presented with. an ex­ample problem sho1'in:.

, rtf:' .··•. ~ , ,,. , . i.1 .·

!' A E1l1 ; . Eai~,:' : E1I2i;,p F1g.C2.20a. Two-Step 8ymm:atr1caal Column~~ tan( L1 VP/E1I1) x" tan( L2 VP/E2I2f 2)

...,. -VE2I2/E1l1

: .,sin~~ the ,equations ~re ti:~nscenq.eO:- ·Three-step, Column .tal. in .f'.OI'Jll:• a successive t.rials prooe- · · · ''·

~tie l~=~~"c1~~:r~o~~i~~!e~i~:s:~,;~r~t1- ~cJ; . 'I 1.4--J value for P (and. Etl ,ea.loti.lating any P I· 1

P associl\ted parameters and using these ... v .. - E;I; E4I4 i--'---lues in the equat'ions. When the assumed P is Per' tl'le equation will be satisfied·~- .. """"' ,;115.,,2.i!oi>·=••-•tep column

• 't'hese equations are eu11J- prograU,ed--tor r*pid. •olut1on_ b)' oomputer or au.1table cal_culators •

.. "',An ,,,./l:C.t: .... &_,;,,,, he-!/' n.·r r. uck u; j e't"• tJoJ1 ,, t)),tl :f,,,t- st-<h1/(tj. + C'ob'-4 ' { -4 air( Jkz ;>.' 0

v. 1, JJ ~i'·.,:n '

STE:PPED COLUMNS,

l -

= _ taJ!4 whore Sn= lJ P/Enin anoi ~n = /3 nLn

NUMERICAL COLUMN ANALYSIS

c2.27 or other similar type data, a num­erical analysis is required to determine tlie critical load or stress. Art.c2.6a .-- - presented a numerical solution for the special case of stepped columns on simple supports. The following pre­sents a procedure for any variation in shape and for simple and fixed supports.

17

The equations are easily programmed These prooedures are in ·tabular form., ra-for rapid evaluation using & suitable ther than as computer programs, since calculator (or computer), The derivation this gives a better understanding of what of the equations is available in Ref. 7, is being done. The .procedures are, of

course easily programmable for solution Columns Having More than· Three Steps by computer. As before, the critical

load is determined byeuccessive·trials, For these cases (and also for'·2 or 3 .. which means successive tables of calcu-

steps) the following tabular numerical lations. Example problems illustrate the procedure can be used, as illustrated in procedures.* Table C2.4. It·, too; is a successive trials procedure (successive tables). The basic procedure uses the method One assumes a value for P (and E) as sug- of discrete elastic wei~ts (also called gested later and carries out the tabular "Mohr's Method-" and the "Conj~ate Beam calculations as shown. When the assumed P Method") to replace the I.I/EI diagram and is Por the value in Col. 6 for the last ·calculate deflections and is due to New­segment, N, will be equal to the value in mark, Ref, 6. The formulas for the el­Col. 8 for the segment N-1 multiplied by ast .. ic wei~ts are discussed in Ref. 4, -1. Any number of segments can be used. The example below shows the last of eave- Simply Supported Columns ral .successive trials with different va-lues for P, the last being 79200 lbs. Referring to Fig.C2.20d and Table

As discussed previously; to start with a reasonable value for Pa length­wei~ted average EI should be used and p calculated as 'fi'EI/L2

• When Col. 6N is leas than -Col. BN-i a larger value for P should be assumed for the next trial and vice-versa for 6N > -BN-l•

20-· .... 1-.101 : -

p Ji. x 107

:

F15.C2.20o Data for Table C2.4 Example Problem

Tab.le C2.4 Stepped Column Analysis Q) © IS> ® (/) .. "' n

l 2

. ' • Sine•

L

Since~~ ~. Per= P"' 79200 lbs.

As discussed for the formulas in Art. c2.6a, the proper values of E must be used for each segmen~.*

C2.6b Numerlo&~ Col1.11n Anal.Ji••

Except for those oases discussed in Art.C2.l through c2.3, Fig.C2.2l through

'* To check for stability under a given load, P, use that , load. It 6N 0::. -8N-1 the column is stable, and vice-

- versa. A similar check can be made with the torm.ula~. ·

C2.5, the procedure is as follows.·

l.. Divide .the column into several "equal length segments, S, at least five or six segments, but ten or more will give a more accurate value of Par•

2. Assume (sketch) an initial buckled (deflected) shape. Any initial shape will do (even two stra1~t lines), but the more realistic it ls the soon­er the effort will be completed.

3. At the station in (2) above which has the largest deflection, y, ·1et its def­lection be taken as l", and let the deflections at the oth'er stations be proportional to this (per the sketoh).

4. Let P be one lb. Then at any station, n, the bending moment will be lln = Py0 = y0 , eo the values in (3) above are en­tered in Col. 2. Positive deflections are upward and positive bending mom­enta produce compression in the upper surface, hence the minus sign used in the headings for Col. 2 and 4.

5, At each station enter the value of EI 1n Col. 3, If there 1s a "step" in EI an adjustment is made to EI at the atation·nearest to the step, discussed later, and its EI is '°flagged" with an astel"isk to indi.cate this adjustment.

• '!he•e (and. other) table• &?'II •u117 pt'Ogl'UIM4 tor rap1.d •olut1on b7 coaputer or aul table calculator•,

Tapered portions can be sul:divided. into several stepped portions

18 VARY'ING SECTION .COLUl'INS NUMERICAL' COLUMN ALALYSIS

6. The "equivalent .. concentrated el.a.st~,c · loads 11

, ( 11 e'l!astic. vieight~ 11 ) are then'

oal!culated per the formulas below the table for ·each statfori · an.d. ar~ en,tered

. in Coli, 5. These f''ormulas ar'e dis­.cussed" .J:ri· Ref •. 4.

Th.e+tabula.r opera.ti'ons a.re then car­ried out as shown, and a range of Per va.lue,s i.s• :obt;dned in .CoL 10. If. 'the a.i<;la.l :load :is, below ,,this ra.ne;e the:.i'co~ lumn ,i·a ·,s,~:able, a.nd H above· H .the :' 'column, is•·unsta.ble; An a.veI'.ag;e va.·lue for Per is Qa.l:culated a.a shown; '44;~~0 lbs·" Note· that seve1'al checks a.re.· ' ma.de 'to detec.t ·any .. ·errors ma.de after 'the ,Col. 5 .da.ta. a.re oa.J:culated/ Co'-:'

:, 1 \l!l1ll 9 , ( a.nd 8) ·defines a new sha.pe. , which will be different from that as~ sumed in Col. 2.

!) • . Usi~i t.he deflections, :l!n ·Col•'9 o:(il'l.

~ J~P F"-+j, ~k--~---n;:,~ m,··

F1e;.c2.20a Adjustment fo~. ·a. .Step Geometry . : ... : -l·''' .

1. Let the station nearest to the step be "n 11 and tha.t on the other side of the ate:p .be 11 m11

.,.

2. caicuiate ·•· .,..;

, Eiav ';:

3. Ca],cula te. 2 ( En In, x Emimj

Enin + EJlllm '

R = 'Eiav - EnII1 4. Calcula.te. . . . , ..

, , (Enin)eff ::o Enin + ( l-2a/S)R

5. Us~: (EI1rI1ieff i~:c~1~ 3 fo1' sta.t16n n,,, :.Np\,e thS,t•;,:l.!'1.:tl).e,,step: is midway b.~tweeri. n ,aI\d m• CEttinlef1'''"'' Enin so no a.djustm9nt ... is cnece;ssary; If··a·::o 0

· 1£h~n (Enin l·eff .::: Eiav • Col.,: .. :.8.,,: let the large st. of thi(sei· b,e ,pne

inch .·and get the others' by d:l'vidirt~· tl«ii:r. ·deflections by, the'" ·1are;~st· de'-' E~ample . .,,, :•"

· f1e .. ctiort •, En.tar· these :l!n a.· s'ecOii.ci' m ' , . , . , ,., . 1 • table Is: Col.. 2 and complete' the' ta.ble. Ftjf'tli~' co'iumn'''iri' Fig•'C2.20d station

.3 'is neares,t .to the; step,, which ,_Lg :be- ,. 'tween,,.sta.tiQns,;;3,,a,nd,i,4, a,,;=;,·3,811• and ,, 9. Repeat (8) as::needed·unt:l:l',the ,range.

of Per values is quite small,. ~nd Per can then1 .. be taken· as· the average of, the va.lues in Col. 10• · "" · '

s ·.,. .'.9 ,411,;, •• Tllenef1ore ,: .EI at· .station 3 i•s

adjusted· a.a follows where n :.3 and m = 4 il'l ,tl;le abov,e ·r,ormuJ:as:. 'E = i:o;5 ·x· 107 •

lQ .. If at any,, sta.tion"•F'ilr (=·Pcr/A)iis sig- l. n. = 3 'and m =.:4;; ''''" ,, '';··;·, '' · •:: · nificantly a.bove the.•proportiona.l lim- · , · ,,,,;·;. .. f ,,, >•

7.: ;,,, . ''

, i,t,. str.e.ss ·(or a.bove an:( local .bu)cklin 2; E,I, .. 2

(\1:.5(10 .} x .5(10' J.'i· _. 7·5

xlOT : ~tre·ss ). :the. value· of Et (or Eef'r ·'used .. av=··. 1.5:( lO'xi + .5(107 - •

ih the term EI must correspond to F0 ,, , . , . "· . ·

.(or to the. local buckling stress)' 3. R = .75(~07 ) ~· l.S(lOr) = -.75(107 ) wh,ich; a.an require more· succesirive trial tabular ca1·culations'('Art,c2.16~ 4.

E3I3eff Tables: c2.5 th,,ough c2. 7 illustrate

the procedure. Three or fotir tables are usually,:. sufficient.· For the'' exampl'e shown Per-:: ,4-5000,lbs iper Tab'le C2.~7.

,,,..,, ; ,,,;: ( actua.Uy between 44-200 and' 45900 lbs)• If the appli"'d load were lessc .. than,: J9JOO the .column, ·would be known to be stable a.fter only .the first .table• ,Or, if more tha.n (1-7'60'0 it w~uld .be known to be unatabli!;''wfi\hotit further

·'effort.· Using ten segments instead of five resu.lts in a mC>re aacuriite val1le of Per·= ·44200·lbs;. ·.The value!' bf :Et and

· .Of".Eeff ;at any' stress· level· are obtained .as· discu·ssed in Art.02.16~

Adjustment for a Step in EI

'Referrin15 to Fig.C2.20e, this is done to keep the tabuJar. O:QJ".rations

"simple: ·

= 1.5( 10 1 )'t( l-2x3. 8/9, 4)(-,. 75) (101 )

"= l.36'x· lo"·

5. Ther.e::t:ore · 1.36 Table ... c2.5 fo.r

Coiu\in Fixed, at tlfa '£aft End f· I"'• , <:

The' f'o1iBwing ,procedure :i,a used for a column fUllY .. ,fixed .a.t .it left end. If the right end is fixed, .. rotate the, colUIIln so that it' i>econies the left' end. ··The pro­cedure. is the same as before exc~pt that a longer table i B' needed to account for the fixed"end moments at·the left end. The table is shown.as Table C2.8.

The data in the table ... are for the membe1• shown in Fig";c2·:2o<i''except that the left end .. 1Sf.ully fixed. Therefore,

,.-,.,

NUMERICAL COI.UMN ANALYSIS

Asaum9u Deflect8d Sha.pe-::::v-_ s = 9 .411 EI=.4 xl07

--·- --- -- = x 107 "':"' ......

', I I --===rr::::::;~r-JP~ 11 ,2 p" •4 ·k j-=9 ·Ji.--..j

1

.

1

EI ".5 xlo' , 20. 12 ' 15___..j

Fig. Simply Supported V&rying-Section Column

lo f 1 ·~ 1' f'• ts .. FR=L==:::!:====±====:::!j--F=~s==~,==:::::::\iRR

Elast1c Weigh.ta a.net Reactions

Table 02.5 Determination of Per· (First Trial)

<D ® @ .

Sta. M= -Py EI

n Data Data

P= 1# x 10-"

0 0 .40

I - .40 .92

2 - .80 1.43

3 -LOO 1.36.olf.lf

4 - .80 . . 50

5 0 .50

•see formulas below

... Adjusted for step.

© © -~ Eq. Cone.•.

El El. Loads

-®!@ ©inEqs.

x 101' x 10"

0 1.03

.435 4.91

.559 6.76

.736 9.59

1.600 16. 73

0 4.43

:& • 43.45

® CD © Mom. of Unit · Unit

Loads Slope o.r.

@xCD RL-:&@n :i:(i) •.

x IO"' X lOT x 10., . • . .

0 .. 0 15.17

4.91 15.17 10.26

13.52 25.43 3. 50

28. 77 28.93 - 6.09

66.92 22.84 -22. 82

22.15 .02

136.27

19

N = No. of Segments s 5 S o::: L~ngth o~ Segment ""9. 4.

·s• rr= 7.36

© @ True

Pea De!.

@x s•/12 -®!® x 10"'

0

101. 7 39 300

187 42 800

213 olS. 900

168 47.&IO

0 -.4JI•6,600

PcaAv = · 44, 150

R !.ill,_ 136. 27 - 27 25 R• N ---5--===.. CHECKS:

-©N~I + @N = Ra EQUIV. CONC. EL. LOADS OF COL.@ ©i =0

a) At Ends, Sta. 0 & N,

®o = 3.5 x ©o + 3.o ~ ©1 - .• 5 x © 2 @N = 3.5 x ©N + 3.0 x ©N-l - .5 x ©N-2 True slope =Unit slope x s/12

Slope at.left end = RL K S/t2. Slope at right end = RR" S/12.

b) At Other Stations, n,

®n = ©n-1 + IO " @n + ©n+l

Beam Sign Convention

(+) Loads act upward (•) Reactions act downward (+) Deflections are upward (+) Slop~ is up to the right (+) M puts top in compression (+) Axial load is compression

The· development of the calculation tables is"'shown · itl Ref, 4

20 NUMEll..ICAL COLUMN ANALYSIS

Table 02.6 Determination of Per {Second Trial) N • No. of Sepncnts • S S • Length of Serment • 9.. 4••

CD ® 0 © ® © (fJ © ® @)

Sta. M =-Py El MQ Eq. Corle.• t-tom. ol Unit Unit True

Pea "Er El. Loads Loads Slope De!. Def.

n Data Data -@!@ ©in Eqs. ®~CD RL- z.@n z(fJ. @xS'/12 -®!® P= IH X lQ•T ·- X'10 7

0 0 .40 ,.Q.,

1 - .478 .92 ' . 519

2 .-, 1· "' .879' 1. 43 .613

'

3 -1.000 ' 1. 36 • 736

4· - • 790' • 50 1. 580 :',,:

5· 0 • 50 0 *,.Use applicable fo:rµiula

** AdJusted:for step

Ra. z;J>. 137504 • ~ ·

. ·-- ---"-•"' '

"'!''' -;x 10,.,,, ·- X'10·~ ' ,.~ X 1()"-T ·x lO'l'

.. .l.25 0 •' '' ' 0

' 16. 14 ' 5.80 5.80 16. 41

10. 34 7.29 14. 58 26.75

3.05 9.-55 28.65 29.80

·- 6. 50 16. 54 66.16 23. 30

-23. 0.4 4.37 21.85 . 26

TS.bl& c2~7 ResU.its or Third Table ·or C&leulat1on8

x 10 7

0

121 39 500

197 44 600

219 45, 'jQQ

171 ,45, 200 :

0

41176,000

PcRAv ,. 44,~

CHECKS: Same as before

1~t?~:~~.r,~ h4~gl44~gg·l~u~g 145jg~1·· ·~ r·· when sketching the assumed initial .. b;,.ck~.· 1· be zero, to be. realistic. The unit va ... led shape its slope at the· left end. shoUld luea (due to the left end fixity) in

© Sta.

n

0

1

, 3

• 5

® @ © J.t•--Py _:r.JML El

, .. .·,·:.-.

I· Dali Unit Dau

:x 10- 7

0 1.0 '40

- .20 •• ·" - .60 .6 l. 43

-1.00 .4 1.3g

- .90 " . so

0 0 .50

Table 02.8 Determination of Per (First Trial)

® © -M/iI -Mt/EI·

'_,._.,, .- . .. '

·@l© -01,© x 10"' x 10·

0 -2.50

.218 - .87

.420 - .42

.735 - .29 ' '

1. BOO - . "' 0 0

0 ® ® @ @ @. '@ l13l . Eq. Cone"! Eq. Cone~ Mo?,) of Ma~·ar 'ML Total Unit Unit

Loads Loads Loads Loads Slope Del. ·.,·_,;

@tnEqs. @1n Eqs. G:lf!i-<DJ ®~~<DJ ©•ML ©·@ ,i:@ i:@ x IO" x 10"' :"C 10" x 10"' x 10 7 x 10"' x 10"' x 10 7

.43 -l 1..15 2.-17 -55.75- " -5. 64 - 5.20 0 . '" ,-, -~- ·-·-· 5.20

2.60 -11.62 10. 40 -46.48 -5.88 - 3.28 5.20 ,-, •. , .. ,,, ,-,--, "' '.·:.c ' 8,48

5.15 - s. 36 15. 45 -16.08 .:.2. 71 2.44 13.68 . ' 6.04

9.57 ... :.: 3.-72-; . 19.-14 - 7,44 -1. 88 7.69 19. 72 · ... ' ' - 1. 65

lB. 74

5.0:l

- 4.29 18. 74 • -1.·29 -2.17 16.57 18.07 -18.22

- l.04 0 0 . - • 52 4.51 - .15

CHECKS' i:G)'~, !:@ • l:@; •@~:, ·~· @N • ~l:@; o&I.52-18.80 •22.'13 i -18.22;. 4..51 • -22.'13;

•·' @ @

Tro• Pea J:?ef. . Q'3>·x s•/12 -@/@

:ic 10 7

0

38.3 52. 100

101 59. 300

145 69. 000

' 133 I 67 600 I

0 I

• J 248,QOO

·~<_:RA.Ui~-.~ 2:-002:: i13lN•O ·· . -.15 ot:S 0

llote that Mom.or© (and@) is ca1cu1ated d.1~~~rently tor this case (about the right end) 1

NUMERICAL COLUMN ANALYSIS, INITIALLY BENT COLUMNS 21

Table 02,9 Determination of Per . ( Secona Trial) (!) ® ® © © © © © ® @ .@ @ @) @ @ @

Sta. M• lofML El -M/EI -Mt/EI Eq. Cone~ Eq. cone-: Mom. of Mo{i) oC ML Total Uoit Unit T~

PcR Loa"" . ,,,. .. <Tl . ' . Loads ,,,.., Slope Der. Del;

)'[ 10-:r x !0 7 )'[ 10'" x 10 7 x 10" x 107 x 101' y 10 7 x 101 x 101' x 10 7 x 107

0 0 1.0 . "' 0 -2. 50 .60 -11.15 3.00 -55. 75- -6. 20 - 5. 60 0 0 5 60

1 - .26 . ' .92 .28 - .87 3.29 -11.62 13.16 -46. 78 -6. 46 - 3.17 5.60 41. 2 63 100

" " 2 - .70 • 6 1.·43 .49 - . 42 5.91 - 5.36 17.13 -16.08 -2. 98 2. 93 14.37 105. 7 66 200 5.84

3 -1.00 .. l. 36 . 73 - .29 9.63 - 3.72 19. 26 - 7.14 -2.07 7,56 20.21 14B.7 61 200 . l. 72

• - ,92 .2 .50 l. 84 - . 40 19.13 - 4.29 19.13 - 4.29 -2. 39 16. 74 18.49 136.J 67 600 . -16.46

5 0 0 . so 0 0 5.16 - 1.04 0 0 - . SB 4. SB .03 0

* See Table c .:i . 43. 72

** H

72.28 -130.04 -20. 68 . 23.04 • 264 100 ~ Ma)te same checks as before PcaAv. • s&.025•

Columns 3, 6, 8, and 10 do not change in successive tables, For any station, n, the entry in column 3 is l - n/N where N is the number of segments used. Hence, these decrease uniformly from LO for station O to zero for station N. Columns l through 10 are completed, than ML is calculated a·s shown and the rest of the,. tabla is completed. For the member in Fig. C2.20f using five.segments P0 r is found to be 66025 after two tables are completed. A third table would reduce., the spread in the Col. 16 values for Par•

Column Fixed at Each End

The procedure is the same as for the previous left end fixed case except that additional columns are needed to account for the fixed right end. The initial as­sumed buckled shape should be sketched in with zero slope at each end to be realis­tic. Table C2.l0 shows the procedure.

The values entered in ~1,J,4,7 1 8;1~ '11,1'.J& 14 do not change ln successive tables. For any station, n, the entry in Column 4 is n/N, where N is the number of segments used. Hence, these entries increase uniformly ··from zero at station O to l.O at station N. Columns 1 through 14 are completed; then ML and Ma are cal­culated and the rest of the table is com­pleted. Using five segments the buckling load, Par• is found to be 172250 lbs after two tables are completed. More tables would give more accuracy. The suggested. checks should be made to detect any errors made after Column ll.

These tables, with a minor change, are also used to determine the bending moments in beam-columns with a varying EI as disc.ussed in Art. 03.18. The value of Per for too member as a column can also be calculated as shown in Table :cJ,6*~ •see roatnote on p.14.

Other Uses of the Tablas

For a complete discussion of the .tabular method development, the "equiv­concentrated elastic loads" and addition­al examples see Ref. 4 (described in Art. Al8.27a) which also discusses.the follow­ing applications.

l, Axial loads between the ends

2. A column consisting of two pie'ces oonnected by a torsion spring (which could also be a splice).

3. Multispan oolumns having one or more spans of a varying EI

4. Free-ended columns having a varying EI

5. The numerical determination of carry­over factors and stiffness factors

6. Multispan columns on elastic supports which may be present as either dis­crete isolated supports or another beam or column.

. 02.60 Col\llna Having an In1 t1al Bent Shape

When a column has an initially bent shape as in Fig. 02.40 the axial load will generate a bending moment along the column in the amount Iii= Py, where y is the deflection of the bent shape at any station. The column may be uniform or it may have a varying EI.

p

F1g.C2.40 ColUIUl With In1t1a1 Bent Shape

Therefore, the column is actually a beam-column, as discussed in Art.c3.l8, s'~am-Cblumna .Having an· Init"ially Bent' Shape" except that there are no trans-

22 NUMERICAL COLUMN ANALYSIS

Aasumed Deflected EI =;~4,, xlD~;-:.-'

!f • No. ,cl Sep:qeuts · "' 5

Tab le· 02 .10 Det.erminat'J. oh of Per (Firllt Trial) ·s .. i.e.11~ f:!. ~p::i.tm ~ T. ...

®.© ® © 0©®®@®@@

a Data Ua.lt Un.&l 0.~ I

·';', 1·,·:' xlO"·"' ,(:10"· ':it-iO'.- x·to'' i-IO": X··l(l''' x'lO'!··'x 10 1

0 .. .-o loO 0 ~40 ·o -2.50 ·o .. 73. -n.15 ... :.s1-. o x':·io"···- 'X'tO"

O". 0

~ ,n 'Tota.I Unit. . Untc ~&ML '<;;:f:r.M.i:t ~, Slope:· Del •

..

!.:·to'"-'- J;-'JO" x·ia·7,' - ·:w:'10' -2.il· o.- ·-0

. ·,; 1 ·-:·,3· .a ·,._~:z .92 ",326 --.87 •• ·217 3075 -'11:62• 2,·45 3. ~- .;;U;62: - 2.45 -l.JJ· -1.26 --.&4 2. 73 20.09 149 Ot.""l

2 - ,7"' .6 ·.4 1,·43 · .489 •• 42 •. ;:zyg -5,95 ~ 5.35-'.3.45 .llo90 -J0.70 ~-6.90· -1.54. -1;7'f-, 2.64 .. 6.-30 - 46,JT 151·000 ; .,·. -, .. ' .93

.1 -1.0 ·"' .s 1.J .735 -.29-.·.t41 8.84 -3.76-6.2tu2&.~2 -n.2s ~ltl:B7· ···1.os---· .. 3;.z3-- 4.5J___ .1.·23. SJ:21,,~ .12J3·0<10

t - .5 .2 .8 .50 LOOO - ,40 ·1;600 10.-74 - -.4,:z9 -18.44 -42.96 -17.·16 -7li76 '-1.23 :..9.·"8 · .03 "3.63 . 26.12 li:2 OO'J '', _,., .. ,

, 0 0 1.0 .50 0 D ·.

Table c2: l.L Det.ermination or ·p ·'·'· '(Sec ... ciliii .Ti'iaif'. .. er, .. " .. 'i' .. @®.©®

,,:·.. . : ,. . -

Q Data Ui:Ut Ui:Ut Da.ta

'','.,\).!-': "'-'IXlo·' xlo.,·· :ii:'10-' x'1(11' :ii.-10' :1110' x'IO' ·x1or. xio• x10' x.10• ·xlO~ xto' xJO' xto• xl07' o -·" .o o. ·"° o -2.:iu o ;93 -·11.-1 _, . .,1 o o. o ,.J.94 -· .26 -J.:n o

. ... ' -·. .-92 .40 - .8 .·.211· 4;·12--11.sz-·z.45 4.·12 .. 11.62 2,45 -•.11 -1.2" • .ISS ,_,

.. :. !"; ._o .· 3. 92 2 .. • K .. .4.> .610 - .42 - ·' 9 1 • .t:4 .. :t.3:> • .>.-4 14.98 -iO. 10 """'"'!IO ... I.·89·, -1.71 5?.90i-,I !165 •

• 34 ', 3 -1.00 .4 .6 l.3it ,_,730_ ... 29 -"441 .B.91.-·J.·-76-·6·.z9,z&.73 -11.28 ·-18.87 :1•];33 .. J.23 '4.'23 T.SJ' · -5;;;,«J" 'tBQ·Qo;:.J

•"" " "' -3.89 .. ' 4 - .50 .2 .s .so i.ooo - .40 -1.soo 10.TJ - 4.29-IB.-44 -42.-9? .n.-16 · .. 1J:T6 -1.52 -9.48 - .21 3.64 .2&.eo -ns.cou . . -3;62. .. . ..

,0 0 l.D .so D • -2.000 2.1c - 1.os-11.se 13.Jo - s.25 -51.&0 - .31. -5.95 .. .:.3.68 _, ,.,,- ;· .02· ''""· 0 .... '"' ·'·'·· , . y,:.'

,HL.·.:'" .. -~S-4. Ma·~ x • ~. -3T.~2-4~.12 .. 1.02.1s_ ~-~.:~":!!:..!! ;!!--..· . -· M:ake •am_e cbec~.u &bar•~-···

v~rs~ fdads 'and' th~J'.r' ini. hai beriaing mil~. gin ia based on using Per• ments,, MQi .·Thus,, 'the column woJ.l~ci be .an­alY:zed as discussecj. .1ihere using. an equiv­alent transverse loading such as that in Fig.c3.33 or.tbareabouts.

Actll!llly, no co.lumn is perfectly strai/3ht although this is assumed in us­ing the numerous buckling load formulas and.data for Per• Because of this.many analysts· and designers· maintain a small positive margin or safety when the mar-

There fore.; .. an"•· alterna tiva· .·approach is· to assume an ihit1albent shape having a "bowl!. or L/800inches or,thereabouts, or or what the drawing specifies such as "straight within X inches".where ·X is us­ually on the order of L/800 or so. X is actually a tolerance for deviation from straightness of tX inches, so the mean wo11ld oe····ix/:1" anftbl.s would be used to'· show a .zero or positive margin of sar.ety.

COLUMNS WITH ELASTIC SUPPORTS 23

Then a check is made using X. to be sure that there .1s not a negative margin of safety ~reater than -.15 or -.25 (per .Art,C1,13b). Sometimes x., rather than X./2, is used to show a positive margin, which is a conservative procedure. In any case a beam-column analysis is made,

C2.7a Design Co1umn Currea tor Columns with Non­Unlform Croaa-Sectlona

For columns with other steps the procedures in Art.C2.6a are used; For columns who.se taper does not meet .. the requirements of Fig. C2.23 or C2.24 ei­ther a conservative assumption (adjust­ment) for the taper must be used or a numerical analysis as in Art. C2.6b is required.

C2.Ba Column Fixity Coefficients C·for Use with Col'.l?lns with Elastic Side Bestralnts and Known End. Bend~ng Restraints

Fig.C2.25a and C2,26a provide the envelopes (for q=<><>) for Fig.C2.25· and C2 .26.

40

1.

4

1~~~~~;11~~fl~Llti 20

1.2 ~

1.0 0.1 Fig,c2;25;,·

c

0.1. 0.2- 0.3.

Fig,C2,26a

0.2 0.3 . xlL

0.4

0.5 0.6 O.'l 0.8 0.9 1.0

xlL

C2,10a Solution Without' Using Column Curves

The inside scales for Fig. C2.l7 are not shown in the textbook. Fig.C2.l7a shows the inside scales for the ordinate and for the abscissa •.

It must be understood that, like a column curve, Fig.C2.l7 (and C2.l7a) can­not be used when determining the critical load requires using a parameter which is also a function of E. This occurs, for example, with Fig.C2.25, c2.26, c2.27 (and C2.27a) where the parameter C is a function of E. This also occurs for other such data. In these cases a suc­cessive trials procedure must be used (as in Art.C2.l2, Case 2 Inelastic Fail­ure, Portion 2). That is, the value of E used in determining Per per the figures must also be the.value of E corresponding to Fer= Per/A.when the stress is in the plastic range (E is Et.l or "hen local in­stability is present lE is Eeff). Et and Eeff are discussed in Art.C2.16.

C2.l3a Column Streagth With Jtnown Ind Be•tralnlng loaetata .

Fig.C2.27a is much more useful than Fig.C2.27 since it provides curves' for numerous··valuea of k B.:Od kl, where the symbol "k" is "u" in Fig.C2.27 and in Art C2.12. The curves can be obtained by us­.the 1roment distribution procedure,

Referring to .. the example shown for Fig.C2.34, for each of the ·restraining members, AC, AE and AF the value of k is calculated as k= 4SC(EI)/L where SC is obtained as C in Fig.All.47, conserva­tiveiy using the "far end pinned" curves. for tension or compression in the member. The total restraint at end A of the mem­ber AB will then be

The same tlline; would be done at end B ·for members BF, BG and ED to obtain kB• Then having the values of k Per for mem­ber AB is obtained by using Fig. C2. 27a. Note in Fig.All.~7 when L/j exceeds'fl' for compression members C becomes neg­tive for the pinned end case,

This procedure ls more accurate than that in Art.c2.13, but it is still an approximate one. For more exactness the procedure discussed in Art.All.15b should .be used, Many trusses are designed assu­mlmg pl:nned joints which is generally conservative as discussed in All,l5b.

-;:::;:l. 00

.: ~ ,!!. .• 90

~ N ..,; .80

E' 'E _, -,70

0

0

' ~

'

.. ·'

'!

-- ~ ---.. ~ '..._ "' " 1' I' ..._ I"- ' ..._

" ...__ I"

... - 1.0 ,Q.l· ......

' , .. ,

. '

'

'

,··

NON~DIMENSIONAL DATA FOR COLUMNS

'°' '"" .... ...... -., I'\. .. •• 0 1.00

'

' " " " .10 \ \ °'iO -· ·-K ,'&, ~\. 9 ·~t, \ ... I

'. " 90

' K '1~ I\'" ,\ \

' ' 1'8' .\ ~ \\ I ' '

'

80

-i::- ·H!' \ ,\t\ I

'2:5 r:::; ...... ix' f.,\ "' ' .. \; ' .. ··+:·' .. --1 70

"" ~ I\\ ~ •'i1''. ., ·',

' ,.._ ,_ '

60

'··.· ' I ,.-.... ,.___ r- "'-- ~: 2.1 O<

50

~ )--.... ~ ,_ 40

' ' ' ~ r--~ 2,. ,. •·· ' " r--. 0 Et 1 "',_ ' ..._, ,... -.. ,, I••'

E• 1 +~n·(~n-1 . ' '

~ K I°"' r--.. 0 , _ Oo T I I I I I I ' ' ,' -·~ l'. ~ ' '. -·,,. "" ,·,'. ,,', .-:- .

' ;3 30.

20

.I (Re!. 'NACA,,T. N.902)

~ ~ K"-< '-.....; k·'fl r--.. ~ 0 1. ... ~' (ff'..::~ '

~~ ,", :.;_;·.; ' ···.

' ' 'I-.;~ p:;: 39. I '

• 10

0 .10 .20 .30 .40 ,SO .60 .'i_O .80 .90 1.00 1.10 1.20 1.30 F/FO,T .

Dimensionless Tangent Modulus Curies

1.40 1. 50

Fig,C2,16a ' '·'···· - ····-...

,,,,. '' '' ' '

1

.3

JrlW.~-! -~i\ ;~! ~1 ~~~. · ~:!;;;: 1 ·~ ll\U}.!.ft iil- ~~: .·· ·: .. ·~" ... :;::

. ' " . .. . . . :~:. . . : ~ : . : .

0.9 -~~:~.; ·.ri · ·-~!-! . . · 5;i ~itl';,i~ ~ij F 2;0 n:·i~·- .1., .· .. , .. i,:k.; k.- -V,~ .·.1 -:-: · .: .. 1!

··'•

' ,0, 0.2 .0.4 0.6 0.8 .1.0 1.2 1.4 1.6 1.8 2.0: 2.2, 2.4· 2.6 2.8 3.0 3.2 3.4 3.6

Fig. C2.17 B ·~ff(r>/p) ,

Fig,C2, l7a Non-'bimensiona:l Column Curves

Bis also'/Fo,7/Fcr where Fer= 1l"'E/(Lij>)"-, E being Youngs Modulus (not Et)

TANGENT AND EFFECTIVE MODULI 25

E:irn.mple Problem

Preceding as discussed, recalculate Fer for member AB of F1g,C2,J4 assuming that the loads in the restraining members at ends A and B are due to an applied up ward load of J,000 lbs at joint D, react­ed at A and B, The following table shows the calculations for k at ends A and B,

·:abJ.e t02.12 Generation or 3t.1!!neaa Fa.otora, k

.U.em- Load EI . J L 1/J SC Jt o::. 4SCEI/L bor xlO~

AC 9000C l.121 11.16 ,o,o 2.69 .30 448,000} AE 4243T .eos 13.77 Ji.2 • .\ 3.oe 1.13 ese,ooo t=l;.984,ooc AF 30000 .697 15.24 ,a.a l.97 ,73 676,000 =ti.. BO 30000 1.121 19.33 30.0 l.55 .62 927 ,OOOJ . BF 4243T .eos 13.77 42.4 ),OB 1.13 asa,ooo z.:.2,"63,00< BG 30000 .697 15.24 30.0 l.97 .73 678,000 :kB

Using F1g,C2,27a, k1 = kB (the larger k) and k = kA• .

kiL/EI = 246JOO(J0)/1,121,ooo =·6,59 so k/k1 = ,80 and C ls obtained from Fig, C2, 27a as 2. 2,

For this member D/t = 1,25/,058 = 21,6 and per F1g,C4,9 its crushing (crippling) strength ls quite high, 67500 psi, There­fore, the column curve of F1g.C2,J for RT can be used without modification, For member AB

L = L/yc = Jcy'\12,2 = 20, 2 L/f = 20.2/,422 = 47,9

Then, per the column curve, for. member AB Fer= 5),000, This ls slightly less than obtained per the Art,C2,1J procedure.

If pinned ends were assumed for all joints then C = 1,0, L/j> = J0/,422 = 71,1 and Fer= 4),000 (too small); but words· about bending fa1lu~e in Art,A11,15b also applies here, If the truss has members subject to local instability see Art. c2,16, Also if the members' b/t were large, say 9 or more, and of open section, torsional buclrling might be critical (Art, C7,3l),

C2,16 The Use ot Et and. Eerr The Tangent Modulus, Et

As the axial stress in a column be­comes increasingly greater than the pro­portional limit stress, the tangent modu­lus which ls the slope of the stress- · strain curve becomes successively smaller. This reduction in Et ts shown in F1g,Bl,5 (the curves on the right being Et) and also in the follow•t•ng"'sketch,

As discussed in Art,Al8,8, Shanley

has shown and experimental data has veri­fied (F1g,Al8,11 and A18,12) that the use of Et ls justified for determining the buckling stress of columns, It ls also used for beam-columns having significant axial stresses, Hence, the use of Et in Art,C2,l and subsequent articles ls un­derstood,

However it must be understood that Et is applicable only for columns of stable cross-sections (reasonably thick flanges etc,), That is, there must be no flange having a local buckling stress smaller than the column's buckling stress

If there is then Et does .not apply and an 11effective" modu1us, Eerf• must be used instead, ~t is smaller than E~,at anyt.'/f' yiz/us. · The Effective Modulus, Eeff

.When any flange has a local buck­ling stress smaller than Fer for the col­umn,assuming a stab1e cross-section,an "effective" value for E, Eeff• must be used, Eeff is obtained as follows,

A column curve is constructed as discussed in Art,c7,25 and illustrated in Art,c7,26, Method 1 there is the'':most frequently used, fo1lowed by Method 2. Having this adjusted column curve'''which accounts for the reducing effect of lo­cal instability, Fer can b<i obtaii)ed for any value of LI~, '

However, when the buckling stress is determined for other than the above simple case it is necessary to use Eeff which can be gotten as follows. For any value of Fer the va1ue of ~If ls abtained from the column curve and Eeff is calcu­lated as

Eeff = FcrCL/'.f)2 hr~

Pc _ :PrQl)ort1ona1-Lim1.t-5tre.as.. __

It is for cases where Fer is determined using a "constant" which is a function of E that Eeff must be determined and used, Examples of such cases are F1g,C2,25, C2,26, C2.27, C2,4l··'etc·;'' In·''the•S-e'·cases a successive trials procedure is needed

26 COLUMNS WITH ELASTIC SUPPORTS, COLUMNS WITH INTERMEDIATE LOAD

(just as wquld be.if Et were applicable), That, is, one assumes a value .for'Eeff• de­termines Fer .and then determines the value of ~err correspo.nding to thi,s Fer per the , above formula, If it is not the same (es­sentially) as the assumed Eeff another V$"7. where lue ·ror Eefr .is assumed. and the proqedure

=

is repeated, This is repeated as, ne,cessa- The critical load combinl'tion, ry until the assumed value is essentially,, ·~alled y1 ·and p'2 must be' in the same the. value corre.sponding to Fer•, 'J;his pro- ratio as the applied loads P1and Pz and cea'il:;:,e f,$,,,simiiar 't~ .t)lat .~equir~d whel! are .found as follows·, . For tne, given us1i;ig 'Et ,e,s !!1.squs5~d in. Art. C2,10a, Ee ff loads calcula,te a =· Pf/Pz; Assume F'2 is is,nq~ ):\Se~.~ith,beam7columns, .(only Et is) P2 and per t,l1e formulas ,find'bY s71ccesi'fe s1,i;ice 1;~e ~e,'1;"1-~'1.l,u~n al:J,~wable, :St;pess , trials the 't'alue of F'1 which satisf.ies considers· any loca.l buckling effect when si · Method 2 therein l'.s' used, However, with, ' s !;j Method 1 EeffWOulcJ,,.be us!'d whon local in­stability is prese,nt,,, .see Art,.CJ,,18,

• '' • ' : '· ~' · ••. J ' ' ' . • • .•

This article p·rovides add1 tional buc)cling lo.ad data for various .t:i'pes of

· co1\.t~ris, It is g~nerally self~explana• tory; . · · . . . . ,

·• .• (:. ,,,, .···• './ ,, --ii:· · I ''.z1 r;, u SD:~. 1.1;1,ti::oMJ.1,~puS--

/J'" k/a lbs/in p<ir Ln<;:I\

p ..... 2E,iiti/1, 21 .r.' Cc0. t~le "l'?"' ., . -- . '

., .. ·

s lD .-15 , 20 1 ,JO 40.· -SO, 15,.100

'ii;1~:::' 1 ,, .!2~ ~·&t9: -i~"i'.: ._61~ .!.517' .... ~_itl .43.7, __ .i421 ,.406' ._l76 .J.~l 8L411_~,6£1 2qi:i' loo -500 _·_70(! lOOO 1~.oo 2~ JOOO 4000 SOOD _eooo _19~

L'/L

Fig,C2,41 Columr;\,On Elastic Foundation

.. l~.itlli'.h?tttm:nmf~~l· P•k/a lb•/ln P9t'-.1ncl\ k

1~41cr-.· ff'~r,tr/IL'l.~_', L' !re. t&Dla ti.low

.i. S L'/J.

L'/L

Fig,C2,42 · Co:J,Ul)ln On Elastic• Fi:iundat.ion. .· 'with' a Distributed Altial Load

P~ .. L.cc .. --"z_r_z _____ __,~ Fz Pi ,,J; ~ E:r, ·' nJ;: '\:0---- Lz -----;-- ~l..-~

Fig, C2;4J Column With Intermediate Load

(~

·' Note, that the intermediate load,, .P.2.h• "" in Fig,C2,4J must be located at the step, if .one. exists,i! The buckling equation is

·' dee •Formulas ·ror Stress and Strain•'· Boa.rk and Young, 5th Edlt1on. tor raptd aolat1on tor numerous ca.sea and end oon~1t1ona.

"' 0 . .. ... •' • • Q

• ... .. ~

"t

'l,,, ~·.:,

·t 0 ~

II

" .~o.,

1---+--+--+--t--;"'

.\ ' ' \

.. ''"

'' ··. ·''

\ 11--+---+-'rl--t--\ "' ". ' "" ... ·.~.

"' I! .. ...

. ' "' .ll ,.·

\"\, ' I'' ~' c

0 ~

~ ., ,,_,,,_,

~s • ' " "'"' O' II "' "' -1h II c c ~ ':' ~ • ,_, ,, • "' ...:11~ " 0 • " " "' ~

l--Jl---1---+---+---+-t-< ~ @

"' ,, '"!

COLUMNS WITH ELASTIC END BENDING RESTRAINT 2? the equation and then cal~ulate a'= P]./F'2 If a~ a assume a different value of P2, calling it Pz. and repeat, Repeat as necessary until a = a. Pt and F2 are then the critical load combination~* The margin of safety is then

M.S. = Pi/P1 - 1,0 (or P2/P2 - 1,0)

portion then the value of Et must be used, and the value .of Et corresponding to Fer must be the same , essentially, as the value used in calculating it,{successive trials). If local instability is present Eeff would be used instead of Et. The procedure is, of course, much simpler for the special case where there is no step {EI1 = EI2),~.- "'P""' l';:,.=-t>•

If Fer is in the plastic range for either · .(? i. ..r= t"'fC . Per= Crr'EI/L2 . .. e.'i: "'trTC Co T

4.D ~~~~~~~~~~~~~~~~~~~~~"T"~~~..-..,,t......::'-To~~~.-~~-.

c

,cEI k

pt4,~ p

L >1

0 ;!.O 50 100 . 150 200 ~ EI k1 is the larger of the two springs

Fig, C2,27a Constant Section Column Having Elastically Restrained Ends

Note that interpolation for C "between the lines" is ·non-linear. For an exact value here, or for k1L/EI > 200, Per can be found by successive trials, as follows

1.Use the figure to get an approximate value of C and calculate Per•

2, Get the corres.ponding.values of J, L/j COF and SF (= 4SCEI/L).

3, Enter the values in the following buckling equation {see . Ref, 4 for derivation of the equation).~

SF" {COF'f j((k1 + SF){k +SF))= 1,0

4,If the left side of the equation

is< 1,0 assume a slightly larger va­lue for Per and repeat steps (2)-(4)1 if > 1,0 assume a smaller value,

5.Repeat steps (2)-(4) until the equation is satisfied and Per is as assumed,

rhe value used for E must cor-respond to Et for Fer when F0 r is in the plastic range, or to Eerr.

To check a column's adequacy under a given load enter its values of SF and COF in the equation, If the left side is < 1.0 it is stable1 if> 1,0 it is un­stable, See A:IJ •. l.3a. for SC and COF .values. The: k/k1 curves··1n'the above'···r1gure we~e generated using the buckling equation • .... If k1 =ca (fully fixed) the buckling equation 1~

0 SF'+ k = O, so for stability L/j< L1'f a.nd SF-+ Ft>

.§. m - 1 +'IT" m

28 ·COLUMNS· I/ITH DISTRIBUTED AXIAL WADS

.Table c2 •. 13. Uniform Coltll1lll with Distributed Load anct End Load· ', .. ·,?,,,··~ '--''-"-~-~-'-'-""-'-'-"'--~-'--"'-'--~ '''·'' -,,., •'• ' ··'

FIG. 1· , , , ,

iJ t 1 • . I I I

'l ' • ., I

. I

t t t t '. + .iil , + "'

.. --+ -,.L P + tiL

c:,r;.st I CASE 2 CASf. 3 CASE A

HINC;EO.HIHGEO flXEO-flU flXEO.HIHCUO f1xus·~PIXEO

FIG. 2· CRITICAL (&ucK~ING) LO AO$ ..

' .

•-•--l~--'-l----.l----1

'----'-'-~-...... _,

'

-1.0

-· ~J _, -1.s -I.II

-0.5

• 0.25

I.SO

0.75

1.a' ;,'1,5,

' ·l

•• •YI-'

TABLE 1

CRITICAL(suCKLiN~ LOADS.

L •. CA.SE .1 - c.u! 2 · ' b.S! 1

2.70 2.11] 2.lOl t.7U

. 2.151 1.UI 1,tl1 2.:UI

1.Jll 1.515 1.Ml 1.Uf

1.711 1.os 1.•tl 1.1u

1.Cll· 1.ln ·:· 1.lll 1.•U

1.2'6 1.U7 1.110 1.246

I I 1 1

0.87• 0.TU 0.913 11.17'

a·:1~ o.1sa o.a11 ' · 0.7" Cl.61' ' 11.7U 0.7ll· 0.614

O.•U 0,212 C.?IS ,

-0.061 O.JI• 0.1511 -11.1161

' -D.6'1 o.ost '-4.17l -o.ns -0.277 ·- ; .a.629

'. -2.Sll

~J.125. !

'~-L'

'·"" ; 11'~ Jl.471

'l~'"Z.:. ~\ '

,,,;, ·- .,_ ··-·-· -.. l.176 J.IG 1.1t0 ,.

I

The above data :are used as follows;·· For a given p·,, the cr1 t­i cal value of. q is found. Or, for a given value of q the critical value of P i a found. . Po ~iti ve value a for the loads act as shown; 'ne!gative values acr/iri the opposite dire~tlon. If P >Pe then q 0 r will be negative. If q > q0 r (for P== 0) then Per will be

. negative. '. _·.,I ••• , •

Compute Pausing the equation.in T/;.blei (20th row) Compute 'qL/Pe. observing the: prope,r. sit?ri fo.r q . Obtain P/Pe from either Table· 1 ( interpoliiting) or Fig; 1 Compute Per::: P~(P/Pef

!- • ' ' -.. ",~ --1

For a given P q 0r (the maxi.mum allowable above, but using P/P~· 1~ ate~ ( 2), qL/P9 qcf = Pe/L x ( qL/Pe J· 'in ·step ( 4);

value' of q) i~ foUnd as in,step. (3l,,·":Il\l co@putil}3

.·, ··''' ,, .. .. . ': : . ' .: . .. :· ' ' Example 1 (For all cases L= 50.• EI=1 o') Exam~le 3 . . .. , ,. . . .

,p=_;;oooo,. Case· 3, I/hat is q 0 r? Pe=2.04 q=lO o;·qase 1. , .wru.t .1s .. P0 I?· · ('It ·EI/.L~) =80773, P/Pe=,619. Per Fig,.2 Pe=39478, qL/Pe=l,6Q;,. Per .F g,1 qL/Pe=l, 05, So q 0z.=1,05(80773)/50=1696 P/Pe=,Jlj., So Pcr=•::JQ;(39478): 13423

""· · Examnle.,2· · P=·•J0.000', ''Ci!fse 1, What is q0 r? . Pe""ff~EI/L' =39478, P/Pe'='-;76,. Per Table

qL/Pe;=J.17, So qcr=J,17(39478)/50=2503

Examile 4 ·.· ···p = "3300·0·,···case J, What .is q0 r?

Pe=80773, P/P~=i,65, Per Table l qL/Pe=-2.02, q0 r=-2,02(80773)/50=J263

,· -···; .. "::·- -.

, 'r·

Example

COLUMNS WITH UNIFORM ELASTIC SUFPORTS

.4 .6 .8 l.O

I IJ r-1" ~ ~

.a v --rr // ~ y

VJ ~v -. 7 I/ h V: ~ For the example ot an

I 8 Bp&ll coluzm, D • .625 I .6

,f ' I/ ' '

l.1 I ., r.1 i

11 I) I I

' ' ' I/ ' .3 .

.2

.1

0 0 l

B 2 3

To Determine Per:

l. Compute value or B. 2, Compute all possible values ot D .. 3. Interpolating betveen the above

plotted values ct D vith the actual ~ v&l.ues or D 1n (2) &lid the value or ::t B, tind the small.est value or Per/PE. j

4. Cooz:pute Per• (Por/Pll:lm.tn(l'J:)

NOTE: I~ doing step (3) above it is easiest to first simply tind vhich one ot the plotted values ot D give tb.e smallest Per/PE· 'lben consider only the 2 actual values of D that are closes~ to this for interpolating.

A quick conservative estimate can be obtained by only usi::ig B and tbe lOver "envelope" or the ~lottPd D values.

.. 9 • D (Values or n/m)

4

D • 1.0 is the top 11ne

lt • Suppcrt spring comltant, #lin .

L • Distance betveen au:Dt>Orts

D • n/m

m • 'l'btal number ot. 1pans or lei:gtn L

n • A positive integer havitl& values from l tnrougll m-1

Ir k and/or L varies see Rer.4·

I • I

'

Tb.is data is ;used to determine the mini­m.um support spri.r:g constant required, kmt111 to obtain !!-im­pie support through­out.

'km1n • Xkm1nLIPe)<Pa/L) 0.1..,.~~-"-~~~~~~~~

0 5 10 15 20

No. of el~stic suppo!"'ts, m-1.

F1g,C2,45 Uniform Column on Equally Spaced Elastic Supports Having Simply Bupported Ends

m:8, so D=l/8, 2/8, J/8, • • • .. . For an eight span column having L = 20", Per the above figure,P

0r/Pe = ,705

EI = 1rf, k = 1510 lcs/in what ls the cuckling load? Proceed1ng as outl1ned1 Therefore, Per=· 24674(',705) = i7J95

29

7/8.

Pe =11~ (10)6

/2rf·=-· Z4674" Tei o"'btatn simple support thrcfaghout km1n

B = 1510(20{'24674 = 1,22 = J.82("24674)/20 = 4713 (then Per= Pel•

30 END FRICTION EFFECTS .•.

C2.18 End Friction·. Et'tec'ts'

STATIC AND MOVING COLUMNS

Th~, two cranks. and the .. rod, BC, are a mechanism: which moves· ·as shown by the

For pin-ended 'columns' the presence broken lines, For each case, (a) and (b) of end fixity due to friction will affect the "input" torque, -T, on crank AB causes the axial load carrying ability in two the mechanism to move to the right and is different ways, One is to increase it reacted; .. by .the "output" ·,torque at D, This and the other is to decrease it, Such ,piJ,ts ·rod BG in compression, The broken

~~i~~n~n~l~~~~1!it~;eaa~1:~~ f~;t!~er;;· ~~~eio[~~hlit~~~~~~~~1~-E~~!t!:~~a~i~:~~ ical) bearing or a roller (or ball:l . bear- P; and .:t.he e,nd mome,nts, M;, due to fr ic­ing with a connecting· bolt or pin. Wi_t,l:t_ tion are .shown .. for each case, Note that plain or spherical bearings the end fri'c- 'ciise (b) is' 'more' severe than is case (a). tion can be quite •significant, whereas The magnitude of the moments are calcu-wi th ball or roller bearings it is' es sen- .lated as M = P(uD/2 ) tially negligible, ·

The effect depepds Upon the manner in which the column func'.tfons, If it is a stationary column with no end rotation relative to its supporting structure (such as a column in a tes;ting machine) the effect is to increase the buckling : load. If it is part of a .mechanism or '· structure where such·a relative rotatio'n at the ends occurs the effect, is to gen_,, erate a be.l'!ding moment ·whieh reduces the axial load·carr;Ying"'abl:l1ty.

Stationary Columns

For these the effect is to oppose'.' any rotation at the ends (which occurs when a colum.n bl!ckleisT, Therefore i the buckling load 'increases; fa{t th1'!f canriO.t' be used in design or analysis since it is not reliable,

Moving Columns (End·Rotat·1on)

These cases must consider the end friction sfo.ce -it' causes end moments which re'du6Ei''its 'strength. Fig,C2,46 1llustrat~s- ti,t~ two possibilities here,

-uD/2 is the radius of the "friction circle*' where

u = coefficient o·f friction between the pin and the bearing

D = Diameter of 'the connecting pin

For most cases ii ::it ; 20 out can be much higher if high-friction materials are used, :_'l'_he ·~ial load ,.,,.P., actually acts t,.ngent to .the radius of:'ithe friction circle' as shown'in Fig,C2.47 for the above two cases.,;

p _·,_ ~Load L1ne CC::Fr1et1on Cirele (a}

p 0

· __ ~·j·,ctoad'"Lln'e (b)

Rod~

Rod:> E'4-t

".18'. C-~~~7, __ Lol!-i1:,.l'..1ne .rfl.ng!nt· to Friction Circle

Hence, the rod BC is a beam-column, due to the friction .e.nd ·moments., and there­fore will fail b!O':Core its buckling load, Per can be attained, Th,e bending moment along the ro4· ·can be gotten per Case I (Cir ~I) irf'Tab1.,·A5,1, and the stress

'analysis mad" per Art.c3,18. If the friction e'ffec,:t;.~ al;'.e ignored there will be a bad "surprise" when an earlier than predicted failure occurs, particularly

Coririeot·1ng· Pin-18.nd Bsa%-1ngs' Rod: e f1'====::=::=:=::::::::::;=::=;:=====~=i~£ ~he.n :!'.,approaches Per•

/ f--=- =- -=--=---= =--:...c.:: ::. --=.-- - -=---- ·-:~~~f .. ' A~;Lcrank Column (Rod) T~~'.:

p qH~ C~-am-Column (Sod:) ·-?-~-. (a.) ~c

T

~.r--crank ,, -~=-=---­i;.t=======::::::::5:~===:;;;;=;;;~~::;il)~~ ,,,

Column

" T . . ,

" -~=:=·============,--====Be==am=:=.:=:='"=""=~=;.!~=.Ro=d=)=======M='=c?i):;B2P'.,_

P'1g.C2.46 Column as Part of a Mechanism

Re_ferences 1 (Add to those _on p. C2, 18)

4) E:iigr,;~ering co1un{~ Analysis, (see Art.A18,27a)

.S,L TMQ.r;)". of .Ele,stic Stab1li ty, Timoshen.:. ko\S.P,arid GE!re,J,M,, McGraw HillGo,.

6) Numerical Procedure for Computing Def­lections; Moments ,.nd Buckling ·Loads, Newmark,N.W.,Trans, ASGE, Vol,108 01943

7) •critical Loads of Columns of Varying cross Secti:oii:"Tn6mpson~ w.T., Journ. Of App, Mechanics,, Ju~e, 1950

PLASTIC BENDING 31

C3.4a The Cor.,z:~e S1mpllr1ed Procedure

The following presents. a correction and an addition to Fig.C3.8.

1. The value of K for the tube in the figure should be 1.27 to 1.7.

2. The first two sentences of the final paragraph are revised as follows. ·

If the calculated value of K is g:i'ea t­er than the ranges shown or greater than, say, 2.0 for combinations an er­

.ror has been made. For the concave diamond shown in Fi~, C3.7a, if the shape is given as y = tax (for the left side) then.K = (9n T 3)(4n T 2). For example, if n = 1· (a diamond) then K = 2.0; if n = 1.2 (a slight concavity) then K = 2, 03.1 if n = 2 (a large con­cavity) then K = 2.11 if n ="°(the maximum,but impractical,concav1ty) then K = 2,25, the maximum value. ·A four-pointed star would have similar values, The Fig,C3.7·a data should be added below the diamond in Fig.C3.7.

C.).?a E:zample Problem.a in Finding Bending· Strength

The following should be added after the end of Solution 1 of Example Problem No. 3. .

32,750 in-lbs· not the predicted bending strength of the member. It is the first approximation, This is beca~se the com­pression and tension loads (due to the compression and tension stress distri­butions) are not equal, They must be eq­ual since no applied axial loads . are present (see Art.CJ.3, second paragraph, second sentence). In this case the stress distributions produce a compres­sion load of 32,780 lbs on the portion above the N,A,, but a tension load of only 23,350 lbs on the portion below the N.A. Therefore more calculations are needed assuming the N,A, is farther up

In doing the plastic bending analyses the allowable compressi.ve unit atrain-.is E = Fcc/E, and this is at the mean line of the compression flange.

However, a more realistic and larger prediction or the bending strength for members like the. above cases is presented in Art,C10,15b. It is based upon experi­mental data and does not base failure on­ly on the flange allowable stress. c).10& th11J1DM't.r1c&l Seot.1.:on With Ho

· · .ui• ··or Sfmm•t.?T

If the procedure in Art C3.10 does not give a pos.itive margin of safety, the more exact procedure outlined in Art, C3.lla should be tried,

The steps used in carrying out the more exact procedure suggested in Art. C3.11 are as follows (for an unsymmetri­cal section with no axis of symmetry) as shown in Fig.C3.27a. Referring to Fig. C3.27a, xx is the axis about which the .applied bending moment is given,

r y

\ ~

~--~x

F1g.CJ.27a Unsymmetrical Section_

1. Assume a neutral axis. position, ol ,

2. Find the allowable plastic bending mo­ment about axis x•x•. This will re­quire several iterations, moving x'X' parallel to itself until the tension and compression loads on the cross­section are essentially equal, as in Art,CJ.7a. This gives the allowable bendin,s. moment about X' x.', Ma.11 x'·x·•. from the section's centroid which was as­

sumed to be the N,A. After additional trials, it is found that when the N ,A. is 3. assumed to be ,42" down from the top, the compression and tension loads are essen­tially equal, and the allowable bending moment is 31,080 in-lbs.

Find the moment of the stresses in (2) a'l5out axis. y'y'. This gives the corresponding allowable moment about y'y'' Me.lly'y' •

This applies to all allowable bend­ing moment determinations when the cross­sec tion is not syminetrical (or when the tension and compression stress-strain * curves are .. nq,t· assumed to be identical), This discussion also applies to the sol­ution of Example Problems 3 and 4,

Fig,CJ,?a ~+ OonoaTO D1 .. ond -v-

*Also see Art.C3.J

2.00 to

2,25.

4. Resolve the moments Mallx'x' and Mally'y' into moments about axes xx and yy to get Mallxx and Mallyy.

Mallxx = Mallx~cosoC.,,!,, M'!tly•y•siru{.

Mallyy = Mallx'x' s ino( T Mally• y •cos I(.

32 FLEXURAL SHEAR STRESS.IN THE .PLASTIC RANGE

·The le.ft hand rule, 'shown iri'·Fig. C2. 27b, is used for ·any·•se't 'of 'axes,

5. Plot the point'''Mallxx• ·Mallyy as shown in F1g.C2,27b,

'

f 0 and calculate·s fb = fm 'i- 'CK - i)f0 ,

This. is,.repeate~ untiJ tM value of fb equals that calcu:i,e.ted as. Mc/I, The pro­per values "of fm, f 0 , fb and E are then known for . the applied bending moment, M. . ' )' ' . ' ; . .

' , •·. '. . . ··~r ;""

6. :ae'peat steps { 1 )· ·- ('5) for several Then:,' at the point fm ·and E on the other 'assume·d values•'''of (one ·of· t;m ,cury11

1 on:e dl'.'11'!.i'· a st;r:,'\igh;t :line ··which

which 'can •be' o/.:' = 0) to 'e'stablish all 'i's -~!"ngep,t j;o ~!l~· curve ,and wh.ich, .inter-allowable curve (\.f 'Mallxx vs, Mi:illyy sects the ordinate , calling the value of as in"Fig,'C2,·27b• · :rtr ts only necess- t:m,1'-t. the q;rdi,ra,,t~ fil\•,,,:}h,i.s i.s rep.eated ary to obta.in· that.portion o.f· the ' •' using' fo an:d. tl)e fo .curve,,calling the va­curve (the solid line)'• wherein the ap- !u~· ·of f 0 at. tiie ordlnate f 0, This pro­plied moments are acting, as shown by' "c.edltre is lllustrated in Fig,CJ,280., the point 'mx. and my·'' in the figure. fm'

' ,. ' ' ' ' ' ~--;:_.---,---

!~ 7, Using this' plot, for any applied mo­ments, mxxand myy• the margin ''of safe­ty can be determined as

'

or,

' '

M,S, = ob/oa .;. 1,0

M.S, = ob'/oa' 1.0

' --

+Myy,. ·".

my

--·---Myy

M.S. :.. ob/oa - 1 : 08/oa: - l

, , ,

Fig. C3.;·27b Y..S;. for Unsj"mmet.rlcal 5Act1ona.

,In Ar.t• CJ, 2 it was·• stated that in the _plastic ranger.:tM equatton for· 'the ,shear•stress. is:·:·CVQ/It;:: ·rh the' e'lastic region C =' 1·~·0,· but···w1th••pJ:as'tic bend­ing 1 t is largevthan•l. O'·in the neutral axh.,,region and•,, less· tnan h O near the out"r f~bers·, The••procedure in Art. CJ. 12 for' calculating c is an approximate one, Tl)is is because flexural shear is 4,M/4X, a!ld .when M changes in the plastic regJe>n .. fm ,and f 0 :also change, which the ' Art.CJ,12 procedure does not conslder, Following is the more exact procedure,

', ···,· . For a given bending moment at a

given beam static;in one determines the corresponding values of fm and fo by suc­cess). ve trials, as 'J?reviously shown ( fb is calculated as Mc/I), Or, for a symmetri­caJ, section. one calculates fb = Mc/I and then assumes a strain,€ ' obtains fm and

fo

Oil-'":f:--7".,.._,~~~,_,,~~~~-

,, • .- •• , 1:-"r-15 • ~3-~~6~ -Det8~mJ.nin6:.-,t'~· !Ild.·_ r~· ·.~-.; !" "·· -- -;,· .. ,· _'· _-,~-~--CJ!;;.;.··. f.., .. ;c,._;,::•; SP. . The" corresponding value of ff, is then calculated using fm and .. f 0 in the :pre­vi'Ous 'form\i:la, One then caiculates.

i:ifm= r;;,,.r,n, L..r0 = r 0 :.. ~,;.and Lirb,,;,_,fb-fb

Fig. c3.28d ~lasti_c· and' Linear Streases :-· ~'

I Li' _., 'M . .if- .. cr:--1 C.rc f...

" . ... ' . - '' . ' ' ·-, ·• '

' ' ' .. ·: : .. - . ' ' ,. .- -~-. •• ' . ., ' .' ,·, ' •.. --, ... _ .. <·;t·f ( '-· . . . :·· - ' . '

·~~- - - ... - .

..!!... ' -' ' ' ,• ' ' ''' -! A-t 0 j.... ,· ' '" • Flg. c3.2Be Net'P:l,as'i.1.9,and··;7J.p.ear Stresee~:-,.t

One then calculat.es the axial load on the se.c.tion (on .Qne side e>f. the N.A") due to the Llfm and"Llf0 stre~s distributions, calling this load Pa and repea·ts this us­ing the Llfb and .<iff,.,,,s,tress distribution, calling this. load' f!), .. c;: i.s ·tneri calcu1a-·"' ted as ''

C = Pa/Pb

RESIDUAL STRESSES AFTER PLASTIC BENDING 33

Example Problem

Using the data from Example Problem 2 on p~C3.10 and its associated curves in Fig.C3.19, calculate the shear stress at the N,A. given in Fig,C3.28 and also at the intersection of the web and flange.

Per the example fm = 35,800, f 0 = 19700 and€= ,0222. Constructing the straight lines tangent to the fm and f 0 curves at€ = ,0222, these intersect the ordinate at ffil = 23,000 and f 0 = 13,000. fb = 39150 (p, c3, 10) and ·

f~ = 23000 + (1,17 - 1)(13000) = 25,210 . Using the above stresses as prev­

iously discussed,

Llfm = 35,800 - 23,000 = 12,800 t:>fo = 19,700 - 13,000 = 6,700 Afb = 39,150 - 25,210 = 13,94-0

These are used as in Fig.C3.28d to cal­culate the axial loads. At the N.A,,

Pa = 1,375(,125)(12,800 + 11,930)/2 + .875 (,125)(11,930 + 6,700)/2

= 3,143

Pb = 1.375(,125)(13,940 + 12,198)/2 .875(,125)(12,198 + 0)/2

+

= 2,913

Therefore, C = 3,143/2,913 = 1.08

This is slightly smaller than the 1,09 value obtained using the approximate proc~dure (p.C3.11), However, for other sections and other materials C is fre­quently much larger than.when the.approx­imate procedure is used,

At the web to flange intersection,

where fm and f 0 are the values listed in the table under "Yield". These are the values at the unit strain,t, correspon­ding to the yield stress on a f m and f 0

vs € plot.

For a material not included in Table C3.2 (or not in any other similar avail­able tables) the basic fm and f 0 vs e curves are needed. Fby can then be det­ermined by locating the values of Fty or Fey on the fm curve and dropping verti­cally from this point to obtain the cor­responding value. of f 0 on the fo curve, Fby is then calculated as previously shown using these fm and f 0 values (fm .here is the yield stress, Fty or Fcyl,

CJ.16 Residual S~reaaea Fol.lew1ng l'laat1o Banding

When a beam is loaded so that the bending stresses are ·in the plastic range and the applied loads are then removed, there will remain a residual stress and a residual strain in the beam. This will be largest at the extreme fibers. Those fibers which were loaded in compression will haTe a tensile residual stress, and vice-Tersa for those which were loaded in tension.. The residual stresses can be of' importance when doing fatigue or fracture analyses and can be determined as\·follows

It is necessary to have a 11 K-curv:e" where K is the shape factor ( 2Qc/I) for the cross-section being analyzed. For such a curve see Fig,C2,19 where the curve for K = 1,17 is shown. It is also necessary.to have the fm (stress-strain) and fo curves. The K-curve is generated as follows for any selected value of K.

1. Select a unit strain value, € •

Pa= 1.375(,125)(12,800 + 11,930) = 2,125 2. Obtain the fm and f 0 values at the se­lected€.

Pb= 1.375(.125)(13,940 + 12,198) = 2,246

Therefore, C = 2,125/2,246 = .946

C:5 ,15 rield. Streae ijend1ng Modulus

At limit load there must be no det­rimental yielding (Art.Cl.14) which gen­erally means limiting the applied stress to the yield stress, Fty or F y• deter­mined as shown in Fig,Bl.1. ~his allows the use of a yield bending modulus, Fby• which will be slightly larger than the yield stress, For the materials listed in Table C3.2 Fby can be calculated as

Fby = fm + f 0 (K - I)

3, For the selected K value calculate the bending modulus at this strain as ss

Fb = fm + f 0 (K - 1)

4. Repeat steps (1) - (3) for each of several values of€ .

5, Plot the points Fb• € , draw a curve through them and label it K = (selec­ted value). This is shown in Fig, c. 29 (the fm curve is for K = 1.0),

For this K-curve the residual· stress iincr strain in ·the outer -fibers can then be determined as follows.

· BEAM-COLUMN ANALYSIS K

fm (i<=l)

f

._,'

a E F1g.C3.29 Determ1na.t1on·:or~--Ra·a1dua.l ·streae

Symmetrical Cross-Section~. . .. · . ·.··. Referring t() .. pig,CJ,29, calculate

the'' bending stress fb = Mc/I 1 enter the figure with fb a,r:\l, go,!\cross j;o the K­curve establishing the po1n•t A,· Draw the straight line AB parallel to the linear , portion of,. the stress-strain.c.curve and

.esfa.blisl1, j;he. point ·B, Then go down from poii:i.t,,A to· esta])lish,.,fm1 .. on.''the s.tress.~ strai11. cui:-ve., , The residual stress is·

lish point A, Then, in the s·ame manner as for the symmetrical section, draw the line AB and determine the residual outer fiber stress ·and strain for portion 1 of the cross-section; Then repeat step (5) using €'2 for portiOn 2 Of the cross-section, · · ·

This article' presents '·tl'\e procedure for doing the ·stress analyses' for uniform and also varying section: members,

.Uniform Single-Span Members

For uniform single-span members num­erous formulas for the bending moments along the span are presented in Tables A5.1 and AS.la. For aiiy case not avail­able there (or'by' superposition of those cases), the bending moments can ·always be calculated by using the ta.bularprocedure presented' later·'•,for• beam-qolumtis 'of non­uniform·'sparis i ''· Allowable -stre's.ses ·and margins of safety are discussed.later,

• • • ' ~- • - ,, - - - ' ' - '• : ' <

f res ~·: f~~· ~' fb Multispan Members of ·Uniform Spans

The ·~\!sldua:J.. s,traill, Eres•· ,,1s, that· at poinj;, B, . T!lese,,are .,j;J:1e outer .fiber vs.- FOJ:' multispan members where. the lues, .. j;he .. ,.largest•·yal.ues; The same •pro- spans are uniform: but 'EI maY v.i.ry from c.edure w0uid. .establlsh·the smaller•value·s span to span, 'the''moment d1strib,ut;ion at.anyJn?le.r,,fiber•,' ''"' , .. ,,,,,, procedure with axial load is used,· as

· illustr.ated in Art,4ll, 14, to obta,in the Unsymmetrical .. Cross-Sections be.nding·:moments·:at' the ends of 'each 'ci6n-

, ·. ,/ .. ,; ·'····· .,,,,,,.,,, ·rf\ ·. · tinuous span, The bending moments with-Referriri"i j;'?,Fig,(\>);of Art,CJ,7, more inatiy span are 'then' obtal'ried by tlsing effo:r;j; \~ i:;~quire(l. ,since the neutral .ax- these oending••m~me'nts as app'lied loads

.. is,i~ not the,,,ce.I1troida]. •axis, and a sue,-· (eig, ;••Case'•IIr Table •.A5,1'0)together with ?~ ~~ iyE! , ~J;'ials J1rO£,e.ciUre , is . the re.fore. any applic'able• fload ·,case'' (e ;'/t ~ , Case. III,

· i;ie9.de,d, .. ,as .fQ.llow,s, ,, ·" v·etc,); 'using superpos:ttloij 'of the re­sults (for the same axial 'l'oad) . ,

1. Assume S:.n outer fiber strain, e,, for the pc;>rtion ,with the ,Il)ost depth •from Varyini>:-Section Single"SJ?an Members the 'centroid, y, G2 is then known for the o,the:r; ,port,~oll ,a.s

11,sho.wn in the Fig, When EI •varies a18ng 'tlie span ihe form.::

ulas for uniform members in Table A5,1 2, Proceed as discussed for Example Prob- and A5; 1a .cannot be used, I?l such cases

l~lll ). ,111 AJ;'.t •. ,c,3,,7 aJ:ld, a,lso .Jn Ar,t, CJ, a numerical procedure is required to de-7~ tp ,d,etermipe .M = m1 :t m2 (finding termine the moments within the span, the correct neutral axis), This re- This is done by using the same tables of quires successive trials, calculation as ·for· varying-section col­

J, R,epea1;. s:l?~·PS (1) and, (?),,for eacl1; of;. several 'other as.sumed.values of E1 so. that a curve of 'M vs E 1 can be drawn,

4-.· Enter this curve with th';' applied'. mom" ent an,d obtain ,the corresponding value of € l • I: 2 will then also be known · (step 1 ),

5,· EnJ;er .the appropriate .. plastic bending curve with E 1 • g.o vertic11-],ly, to inter~ sect the applicable K-curve to estab-

umns in f.rt,C2.6b.{Tables c2,5, C2,0, and C2·;/0)T eiceptuthat the fonding mom.en ts,

·· M,. along , the' 'span due. to the" applied. · loads axe' •used iri Col', ( 2•) Of th,e' first table of .calculations,'· A't' this 'pOiiit'the reader should review Art,2,6b as to liow the tables and data are used there·,

For the' beam-column the .difference is tha:.t fos'l:ea:dw''dI~Using· an assumed shape for the first table one enters in Col, (2) the bending moment at each station due to the applied lateral loads and any end mo-

BEAM-COLUMN ANALYSIS 35

men ts. These .bending moments are deter­mined assuming simply supported ends even though end fixity may .actually be present The procedure is -as follows and applies for each type of table,

1. Calculate the bending moment at each station assuming simply supported ends and enter these moments in Col, (2) •of the first table,, c,o.lling them Mg since they are for the applied lateraI loads

2, Complete the table (as discussed in Art,C2,6b1;but eliminate the last col­umn for Per•

3, Multiply the true deflection at each station by the axial load, P, and en­ter this moment in Col, (2) of the next table, . Table I, reversing the slgn (because of the sig?\ convention)

4, Carry out another table of calcul-ations, Table n; as "in (2) and (3),

5, Repeat again, Table III, ate,

After several tables are completed the final bending moment at each station, n, can be computed alfl!1,

Mn= MQ +Mr+ Mrr + ····MN/(1-MN+1IMN)

This is a geometric series where the su scripts refer to the. calculated table, Q being the first and N b&ill$ .• the last, The value of M is taken from Col. (2) of each table, and MN+1 is obtained by mul-· tiplying .the true deflection in .. the .last calculated table, N, by the axial load, usually three or four tables are adequate

I---- 20 ;001

JOOOO# I 17

Same beam as 1n F1g.C2.20d

;oo# 1000#

The bending moments at the six beam sta­tions, 0 - 5, are, from statics

Mo = 0

Mi = 4,700 M3 = 10,000 M5 = 0

These values are entered in Col, (2) of Table C3,3 and the table is ·completed as shown, Note that this table is of the ·same ·form as Table C2, 5 (Art. C2. 6 'b)" ex­ce~t that the above moments are in Col, (2) and Col, (10) is eliminated, The de­flections in Col,(9) are then multi­plied by the axial load, 30,000 lbs, and these moments are entered in Col,(2) of Table CJ,4, reversing the sign, and that table is completed as shown,

Table c).) s.aa .. Ce>lman llwMrloal Anal7a111 1 Plrat .Table

0 0 .... • - 1.21 . 0 0, ~~''' 0

4700 -17.17

1. .92 .. _ .s11 - 5.11 - 5,77 -17.17 -.126 -11.40

2 9400 1.43 - .657 - 7.e2 -tS.64 -2•.s1 -.210 - ).SB

) 10000 1.:)6 - .73S - 9.89 -29.67 -32_.15 -.2)7 6.31 . 4 9400 ,so -1.880 -19.54 . -78.16 -2S.B4 -.190

25.85 Unless the axial load is quite large the W...L-L.L.~•i!.1-....ll.-L..:..:wi:..._=w:u.. __ .L-..ll...--L--'"--_J de flee tions decrease fairly rapidly in see 'l'able c2.s tor oalclllatlon noa• and procedurea. (Art.C2.,b)

the successive tables (if P is greater than Per the deflections will increase and are meaningless-)• If the axial Table C::J.4 !llllam-Collllln lllmlrloal An&lr11111, S11oond Table

stress, P/A, on any segment is greater than the proportional limit stress .the tangent modulus, Et (not Eeff) must be used, The geometric series converges only if P< Per as a column,

The procedure is illustrated for a simply supported member in .the following example, For members with end fixity the procedure .is the same e.xcept that the table will be longer.~~

Example Problem

The member in Eig,CJ,32 is a beam­column s.ub,Ject ... to the applied loads., as .. • shOwn.

41• Se,. footnotl' t1n p.14.

• Can also be used for final deflections and slope·s,

0 0 ••• • Q 0 0

1 )780 - .41~ - •.55 -11.95

.92 - •.s5 -11.95 -.088

2 6)00 1.4) - •• 441 - .7.40

-.142 - 5.)4 .. 10.68 -19.)S - 2,06

) 7110 1.:)6 - .52) - 6.81 -20,4) -21 .• 41 -.158 4.75

4 5700 .so -1.140 -11.92 -4?..68 -16.66 -.123 16,67

0 6 -See Table CZ.5 ror c•lculatlon note:s and proced.u:res (J.rt,C2.6bl

Two more tables were completed and the re.suli.ts .f.or all four· tables are summar­ized in Table CJ, 5, showing only the va­lues of M and true deflection for each station.

J6 BEAM-'COLUMN:ANALYSIS

Table C-::t, i; Results of Tabular Calculations Sta TablS Q' Table I · 'Table II Table .III

·.: -... Mr. oen, Mr 0er1. Mn ·ner1.· Mr II Defl. u u 0 0 0 0 0 0 0 1 4700 -.126 J780 -,088 2640 -.059 1770 -,040 2 9400 -.210 6JOO -.142 4260 -,095 2850 -,06J :i ·10000 -,2J7 7110 -.158 4740 -.105 J150 -.070 4 9400 -.190 5700 -.12J J690 -,081 24JO -,053 5 0 0 0 0 ·O 0 0 0

The fl~l bending moments at each station are calculated using the geomet­ric series,: .. as .. shown·in,Table. c3, .. 6, · Col.~10) shows the calculation of Per• discussed later,

Table: CJ ."6 :Determ1nat1on ,-:·o·r< Final ,Moments :and ,:p

'',:: o.. a ... _ o· o-: :·:. o ;:o o .. 4700 )7BQ, 2~.40 1770 l200 16617 44JOO. , .

':· 940.0 6JOO 4260 2850' 1890. 28417 45JOO .

No,te that .the::,,small•:: .range 1.n Per values 1s similar to ·that •in Table' C2, 7, The f::ormula fo·r Per .is. discussed ~n· Ref,J , (Art·,A18. 27a'l ·and applies onl•y when all segments of the member have the· same value of E (i.e., in the elastic range), fo.r P/;A),, ·

Salvadng Tabular:• ·caleula t iohs · fot'' Changes, in the Applied Loads:•; · ·

i f-' ''. {;'. -~

The results of the tabular calcu­lations can' be used fo.r changes in ap­plied loads' as long as two ·'eonditfons are met,

1. E is the same for all segments (P/A 1s' ih the elastic· range ) • ·

2, All lateral loads change by the same fac·tor~,

•.

10000 7110 4740 3150' 2100 J1J09' 45000 9400 5700 J690 24Jo 1590 2567.5 45700 This··catl save much eff'ort by riot having

Li..L_;_· _:!·0L·.L'0LL-"o!...1'·-"0L.LJl0_L....;01!..·'.;.' .....;l.::-"""-.:0.,,.;;;"'!----'I to repeat the t'abular ' calcula tiiins, The _,"! ,,~e~. ,--~~X.t ... ,·}t,* See te_~,~'.; - -~ve~~~1 .Pcr ~, 45pz_~ procedure is a~ fo~~ows, ,

'> e" ·'\'c!•'' '· ; <' , ·:, "!' (0, ,. .: • ·,;•,.\ ·"_; ;.'/~- (i ,; .'.: ' ''

End p'ixity

If one or both ends are fixed the same· procedure 1s used '!<i th ·the longer calculation tables, like Table C2.~:: and ci;10.. '

'i' ,,;;-,'.. ' ' - .. ,. '''

crit1ca1 '1t:if~i ~~ii'··

The ~ri ticai axial loa,.d for a peam-, co1UJl)h' is. the sliJne load' as 'if' it were a:

1, Let the final momeni<s from the tagular calculatioris •be

2, Let the fe.c tot for . the 'change in (all ) lateral loads be L

'"°':'"\' ' :~i "-"' '• -~~ - . -'~ :::

J. Let the 'fap"tcl'r f,o~. t .•. h.; ch~nge, in the axial ·'.toad• ·ce'<'K ' '"'

c.qlumn (no bending moments.)'; The ·maximulll. 4,, The' new ·bending Iliomi!'lrts' along the sp.an allo'!<abl<! axial loa,d .f.Q.r,e.::beam,-column: is,: ae· 'each statfort, will' 'theri' be ies's th!.:n the critical load .sfnce earlier ' "'·': •:: : •;: fatl1,1re' will occur by bli'nd.i:i'ig .due to the MO: = LMQ: : Mir' = LK2 M£I: applied lateral loads or moments, The • : · · • · · • · critical load for •the above member is Mr = LK:Mr l>lrn s LK. Mnr 45000 lbs and the axial load is JOOOO, If it were larger th" peJ:ldi,ng mom.,n.~s .would increase', and ·e:s 'i 't ··approached. the 6riti-, cal load the moments wouldapproacli inf' ity, .OJ:, course 1 b.!ndlJig f'a,ilure 'would be~

'cur lie:rore thh''coi.ila:.\:occur.·,''.;::: ' ·" ( ' - ,, .,.,..,-' -· ... ' ·'• ' ' - ,., ...

caICul~ting P~r /~;om· B;.S.ln.:.co1W.n Data· ,

These are used ti{ <leterillin~: the new firia:F be'riai.rig momehts' (i:';~'. ,'by re',;. . reptaci.iig 'the: va:lires in Table CJ. 6). . -- ,i.1:~1l/ J ·": ., ' :.:'-' . • . •

Example Problem '.,,·,; ,., .. .The data in the beam-column ca1cu­

lation tal:J.les can also be used to easily •,:: 'ReR~e.'t1 'tne'preif§Ji~ ~fruhple pro):>J,em ap,i. qulcJ<::ly calculate Per as follows,, At ·if P ls ini:rea:sed'to be'J9,ooo lbs and. any s~.a,tion the lateral loads, are.' ip.()reased. by a fac-

Pcr,;, P(MN/MN+'1) tor of 1,2, For this case· L = 1.2 and K = J9000/30000 = 1,J,

where N is from the last calculation Table and MN+l is as discussed before, For .the above example Per is calculated in.the· le.st:: column"·of··Ta:bl!f OJ•;6, and' is .• for e·x­ample, at station 2

Per = J0,000(2850/1890) = 45,JOO lbs

• ~hlch can alao be uaed to.calculate the SP, COP and PEX'e aa 1llU3trated 1n the book Of Art.~1~0 27& .

Applying t.hese . .factors in the spec­ified manner to the bending moments in Table CJ, 6,. the new· ffoil.1: reridfrig mom,,nts become the much larger values shown in Table CJ, 7, Note that al though the e.pi­plied load~. ha,ve been .incree.s .. ed by an . , '"" .- ·... ., ' . '.

BEAM-COLUMN ANALYSIS 37

!Jue to e eased III v n.i

1 5640 5897 5354 1'666 •u3 25670 Same 2· 11280 9828 8639 7514 6478 43739 a5 1n 3 12000 11092 9613 8305 7197 48207 Table CJ.6 4 11280 ·5592 7483 6406 5449 39510 5 0 0 0 0 0 0

average factor of only 1, 25, the bending moments for those four stations affected by .t.he axia,l loads .have increased by a much larger average factor of 1,61,

'

Beam-Columns Having an Initial Bent Shape

Sometimes for various reasons a beam-column may not be "straight" even bef.ore any loads are applied, For uni­form members, for which various formul­as apply as iri Table A5,l arid A5,fa, the member should simply be considered to be straight and a, la,teral loading applied which produces the same bending· moment along the span (assuming simple support.­ad l.nds) as d.oes ·the .axia.l .load times the bent shape deflections at the stations, This is in addition to any actual lateral loading. Proper. signs must be .observed in doing this,· That is, downward (-) de­flections. result in a positive bending moment, Fig,C3,33 illustrates the pro­cedure, A single bend is shown in (a), a. reversed bending in (b) an.d a smooth con­tinuous bend .in (c), ~he. "equivalent" loads for a straight beam are shown be­neath each of. these, That in (c) is a good approximation, Thus, the loads con­sist of ~he equivalent loads and the ap­plied loads, and.the formulas of Tables A5, 1 and A5, la apply,*

For the case of a varying section member it is best to complete the first table (for the lateral applied loads), Then the bent shape deflections at each station are added (~ or -) to the true deflections calculated and their totals are multiplied by P to obtain the bend­ing moments for Col,(2) of the second table, This is done only once, for the second table, Additional tables are com­pleted in the routine manner,

Approximate Formula for Bending Moments

When there are no end moments the following formula can be used to estimate the bending moments along the span for a uniform, simply supported beam-column, At any station

Per is the critical load if the.member were a column, If Fer (=Per/A) is in the

+ Since there is no actual transverse loading Mq; z O, so Yr1na1=. -Mr1na1/P at anY station.

p

Bent Shape and Axial Load

~M>Pe Bendlng.lfoment Diagram

j Q • Pe(~~ t~')

Equivalent Load (a)

Bent Shape and Axial Load

~ Bending Moment Diagram

~ s~+P(e1tt21 1,Q -~+P(e1+e21 _[1 Lt Lz ! 2 L"f. 12

Equivalent Loads (b)

I • p ;A; T

Bent Shape and A::z:lill ~ad

t

<::::::::::::::-~----~IC~P~•~~~::::::::;=='." Beinding Moment Diagram

p

· ~tr,. 8Pe/L2 . t~'PProximatel. . l i f f f 1 f 1 1 1 t f f t f f t f 1 t t t '.f RL =4Pe/L Equivalent Loading Ra= 4Pe/L

(c)

Fig.C).)) Bent Shapes and.Equivalent Loads

plastic range Et is applicable (but not Ee ff), hence a column curve can be used, but not one having any crippling adjust­ment, MQ is the moment due to the later­al loads only, The formula is most accu­rate for a unii,ormly distributed load and least accurate for a concentrated load,

The formula can also be used when tte axial load is tension, adjusted as shown •.

Mfinal = MQ/(1 + P/Pcrl

A tensile axial load reduces the moments,

Other Beam-Column Considerations

There are numerous other consider­ations which affect the bending moments generated in a beam-column, These are discussed in detail with example problems in Ref,J (Art.A18.27a) and include the following topics, ·

* But conservative

JS BEAM-COLUMN MARGINS OF SAFETY·; .. TORSIONAL MODULUS.OF RuPTURE

1. Axial loads within a span

2, Ela,s.tic (rotational) end fixity

J, M"embers with internal (flexible) splices

4, Multispan beam-columns with ends spliced toge.tber;, affecting fixed end m_oments ,. carry· over an·d stiffness factors ·

'

5, Multispan beam-columns having varying EI spans require numerleal evalu­ation of .fixed end moments, carry over and stiffness factors '

6, Mul.tispan bealll'-columns on elastic ports and on. s.a,gging supports

7, Free-ended beam-columns

sup-

8, Beams in tens ion ha17ing a varying EI 1 the 'tabular procedure can be used but requii'es modifications

All'!:>wable Loads' or .. stresses and ~rei'iris of Safety

,i~ de.termined b:f inter.pglating lin­earlcy between (a) and. )b) above;

d) If the section has ·~ local bu~kling ·stress (,Art.c5,4, c5,5 .and c6.l+l less than ln ·(a) • Jc.) then Fe is the· outer fiber stress correspond­ing to ~he, lo,qal buckling s.~r~s~ at 'the.·· :r1a.nge··· s "me·an·l.··:1. n~·.·'.C(.'YJ.1f J'~e·- •' 10, '18 ,ca.,, l:hofe.·· .' ., .. ,, · ·'

· e) For •cases where rio' .. }:ierinanent. set is . •a1lowed at• some'' part'l'cular ':load.

level, such as at proof load for · .actuators, Fb' Ys"cthe. proportional 111111 t stress;" Meth,od 1 above is not applicable for .this case,

The margins of safef;ir as calcul~ted aqoye are app,,,rerit ori,es, 'The.true M: s; is' cal­culated: per Art .• c4 ..• 2J~(ap.d C1.1Ja,.

Ci ;1<·,:'

Art,C4;20 .. diScussecl. tors1on s.tresses for cinly the .. pase'O~ ~1I'c~iar c:r;-pss:.. .. · ... · seCti'cihs•'· For othet ;crossHsectl:i>ns pias­'tl.'cF t'oI'sfori arid a forsl.ohai'inddulus of rupture' also occur for.cl.uct11~ mater,ials. Foll?'wing are: methods 'for, d.,!!'term1ning,, :the

Following are two different. proce- ultimate streni!;th. ih torsi:cin, ,ror such: , dures for C?al,,ulating j;he margin of safe• members\·'' The"rdrmµ1fu.s';'' cia:ta•· and dis6'us·;:. ty for a beam"-cciluhiri, · For a given case ·s1oris '1n A:rt;A6:4 througti"'A6;7' and in'' the larger of the two ~hould be used, Tabler A6.} .";PP1Y j:in1y'whe'ir; th,e'sh~!i.r ·

'. ,,, . < '• ·' stress•'is''fo·the''elastic: range~: '' 1. The follow'ing interaction equation ... ,, ·; ·' ''·' ·•'·

( frqnj,,tests .of cfrclilar tubes} is 'Torsion' l:ri ~ll~,, pl;\.st).c r'fi:ige; is clfs-

:P/1>0 r + M/Mult = 1,0 ,.·1· "'"

ciisse·if•"fo'. Ret\4· an\l: 5 •!'·For'non•ch'ctilar ' ' -... ' ' -, . -· ;' ,., ' . ', -- ' .

cross'-sections the. u:J,~l:mate tors.ional str~ngth is projlortiona.:L, t? .. t~~ ·· vilJ,ume of ,

where P is the axial load, Per is the dry sand (or table salt) that can be bll_c\<Cling .load ,,as a column, .M is the heap~d upon. the cros?,:-s~ction, , If.. the final beam-column bending moment and cross'-'sifotlon J;lii.s holes 'a' special :tech-Mult .. is .. the. ·allowable bending mome.nt nlque· is· reqii.1re'd, as discussed later .• per Art, c4,t?, ,CJ,4. or c10,15b, Tl;ll!-· .. ·'· ·· ,,, · · ,, appll,r~;,t .. 11 .• s, .··.(Art, c1 .1Ja) is then · · i This. volume 'or' ~ancl.'' can b6 ca'lcili-

. . ' ' ' ' ..... ,,. ' • "'' ated• for only three types ~t c~ds~- ' .·. ji;s; ,,,,. i7(~!Pc~ + M/Multl.'"' LO i• sections,' a reqtii.ngle; a circle ,and a

;,, - :_,:' - j"

2 ;· An ,alt,erna te ,pro,ce.dure is . ·

···;,• F0 /(P/A + Mc/I) -1.0 .,,._.,_ .. _· ,,, ··.,,1; :"· >:?-':!; \.'• . · where P and M are as above, F0 is the

outer. fi be .. r.. allowable compre.ssive stress l ''-,d·etermiried as fOilows, and A i.~, ~h•t cross-sectiona,l area

a) If.the beam section shape factor, K, is' 1, 1 or less then Fe 1S the proportional limit stresS* or that in (d) 1f smaller

•• --.,, ... - c ... , ••

1'.LJ'f · .. ~ ,ts., 1', 5 or more then F~.''is '"Fey

c)' fi"K ti between 1.1; and 1.5 .. then Fe • Tbe ·p.rop. ·11a1t ·atriil• c&li be foqn4 u Whlin Pc/Po.7 to·r·th11 ·~ oun:e 1• ~ant to th• n ·-cum Ul-·Plc;CZ.17a·'- 0

triangle, as shown in Fig,Cl,f,31, ·,' , .. ,. .. ,, . ' . ('· " ,\·; ..

The equation for calculating the ultimate torsional strength, Tult• when no local buckling can ·occur, as with very .. ' thin, "' · · "" tubes etc, is

Tult = 2VFrs

SHEAR BUCKLING OF FLAT RECTANGULAR.PLATES 39

where Frs 1.s a r·eference stress which can be taken as .. Fsu. for the material or cal­culated per· Table c4, 9. V is the sand· heap volume in cu, in,

For other cross'-sections without holes a cardboard template of the actual cross-section is made and sand is slowly poured onto the template until any addi­tional sand runs off, The sand is then weighed and the volume is calculated as V = weight/weight per cu in of the sand, Having V, Tult 'can then be calculated per the· formula,

If the cross-section has holes the following is done to determine v. Snug fitting thin right cylinders made from thin paper are inserted in othe holes, long enough to project above the volume of sand that can be piled on, The sand is then poured on as above, and the pap­er cylinders 'slowly pulled downward un,... til their edges become contour lines at at the max, elevation, (a bit of practice is helpful here), That is, none of the side area of the paper cylinders will be exposed, The weight of the heaped sand is then determined, To this weight of of sand must be added the weight of sand required to fill the cylinders from the template up to their tips, The sand vol­ume can then be calculated as discussed previously and Tu1t is then calculated, It is generally unconservative to deter.-. mine V simply by deducting from the strength of the cro.ss-section without its.holes the strength of solid sections having the cross-se.ction of the holes, but this is sometimes done for rough pre­liminary estim~tes,

Instead of using Fsu for Frs in cal­culating Tult•. Frs can be determined as follows. For the alloys in Fig,C4,17 through C4,JO the torsional modulus of rupture, Fst for solid circular cross­sections can be obtained at D/t = 2, Then, s.lnceTuu= 2VF;r~· Fst = Tu1tR/J, J = 'lfR~ /2 and V = 'l1D" /24 one can solve for Frs as .a function of Fst• By alge­braic manipulation

Frs = .75Fst

Table c4,9 shows the values of Fst• Frs• and Fsu for several alloys from the Fig­ures shown, Fsu is from MIL-HDBK-5.

Table C4.q ~ompar1son of Frs and Fau

F1g~e All'of F-~t Frs:;:. -75F 8 t •au

c4.17 95 ksl Steel 71.4 53.5 55.0 C4.20 180 ka1 Steel 142.5 106.9 109.0 c4.30 7075-T6 Forging 58.7 44.o 45.0 C4.28 6061-T6 Tubting 34.8 26.l 27.0 C4,27 2024-T4 Tubing 49,3 37.0 37.0

The values of Fst were obtained from Art, c4,20: They could also be gotten by do­ing torsion failure tests of rods having a circular cross-section,

C4.2}& Illustrative ?reblems Involving Combined Bending And Comp.resa1on

I~ Art,C4,22 Equation .C4,14 does not give the true lllS.rg~n, but rather an apparent one (see 'Art.1,lJa), Therefore, ln Art,C4,2J the calculated M.S. of ,09 is not the true margin which will be clo­ser to zero. (but not· negative), To find the true M.S. one must find, by succes­sive trials I the Common factor by .which !J.11 . ·applied loads must be mul tipl1ed 89 that 1:'ne ca1culated M,S, is zero, Calling this factor F, the true margin is F - 1.0 This applies to any beam-co:J,umn, or to any other case where the internal loads or stresses do not vary linearly with the applied loads,

05.7& Bucltl1ng ot Flat Ract.angul&r Pl.a.tea Under Sb.ear I.Gade

Add the following.to Art.c5,7,

The use of the secant modulus · .. for shear buckling stresses in the plastic ran12:e. has no theoretical -basis; 1 t Sim­ply-happens to match experimental results,

It is helpful to redefine the shear buckling. coefficient as Ks = ks'11' 2/l2.(1-v") Then Eq, (5) becomes

1cr = KsE(t/bf or 1cr = E/(b/tVKsl"

The abscissa in Fig.C5,13 will then be either of the above div.ided by Fo,7,

, PANEL ASPECT RATIO alb

---- ---- c c s ~I~=- I~ c~cs~s cj' 2.20'

------ c c s

Fig, C5 •. lla Shear Buckling Coef,, ijK;,

ll-0 BENDING.BUCKLING OF FLAT RECTANGULAR PLATES ' '

The value of"\l'K;, is given in Fig,C5,11a. Fig,5,14b is for('- plate in. bending F~g. AU.: gives F~cr (='tcrl f,or se~eral aQli compression (not tension), with the

,COII)lllOn materials .and includes the effect .ccimpr9ssion (upper} edge' ,free and the of plasticity, . Note that "cut-ciff" lines ·other three edges simply supported, This are used, These are at ·shear stresses of can also be useful, for example, in esti­approximate.~:f • 5Fo,, 7 (or "":5Fcy),, This is mating the buckl,i~ stress,.,.ror the up-a fairly co,mmo.n c9nservat1ve practice. ' stan~ing free edg~ of a r.'.'.be.f<.ru in bending

wi.th the Upper edge in compression,. The ,F0r qther:m"ct.enals Fif,c5,1J can be 'buc)<lirig s~ress i~ ca.'.).cuJ11-ted,.as

usecl.:.:.entering ·witl'CK8E(t/1')9i<0 , 7 .,..tNit~ E/ Ii. -;i (bft\lf<,5)?/0 .. 7 ~ ,ob.tairiihg .. Fsc;.r/Fo 7 and c:.a,1- Fccr = l'..'efEc/12!1-v"jCt/b{ culating Fscr =e.FO,f:~,l'.sc,;rt:Fo,7)., SJ,n~~, n and Fa, 7, a,r,e ,lieelie.<i: t.o do this th<1y' are .. If Fer is in· the 'piitstic ril.h~e' it is· cor-obta~ned from 'Tabl,e Bl,l, or for other ''rected as discussed for the previous case IIiateri,al~ ... n cari .. be' calculated .peir Art, Note that the figure shows cases ranging

, Bl.1 .. or read from .Fig, B1.1J (Obtaining from pure bending to pure compression , Fa, 7 ·,17nd,,.F0,.85 fj::.om,,the wa~erial' s stress lb/h = 2 to b/h = 0), For tension and strain: ,curve per Fig, B1;:12; b8nding b/h > 2 and is not covered by the

, dat"c in th(l figure. . h. is the distance c5.,6a .Bu~k:lins·. ~i -:riat:· .. ·Re.ctanguiar Plat6s f;rqm the. free edge to tl:le li:g_e, of z8ro

Uiider Bendi'.ng Loli.dB $"tress . ,.,

tisefui data for two cases of plat~s subject to bending stresses are shown fn Fig,5.14a 1;tnd Fig.5.14b, That in (a} is for all edges simply support!!d·wit)'I KB being the value k1t'Ec/12(1-'ll'}. The buakling stress is calculated as FBor = KBEc(t/b}", where Ee is the compressive value c;>f Young'sM()dtilus, If FBcr is in the plastic rai\ge one' cialqulates

F:scr/Fo, 7 ·

e.nt;er~, ,J;:ig .• C5, 8 wi tl} this. !\nd obtains · ,

~g~~Fge.i~u1r~:l iir~ci:';~ F;~?~d. ~{ FBi~r is a/b values less than ,JO, KB can be calculated as

.3•

.KB ;g

,,

"' ·.J .4

Fig,C5.14a

" ' " " '

.5 q_, b .• ·' Flat Plate Compression Buckling Coefficient, ·KB

. '

••

CRIPPLING 41

What 1s the buckling marg1n of safety for sider th1s to be Fee• However, a recom.­the plate shown below? mended Method J is provided 1n Art.c7,30,

I' f 1

II 7075-T6 Plate Mater1al -· E=l0,5x1o', n=9.2,

t=·,.125 Fo 7 = 64,500 M J-4---1 M=iOOOin lbs, P=1000 l'.:>s

A= 3(.125) = ,375, I = ,125(3)/12 = ,281 fb = 1000(1.5)/,281 = 5338: fc = 1000/.375 = 2667 :f.u = 5338 .+ 2067 = 8005 (comp.) ft = 5338 - 2667 = 2671 (tens,) b/h = ( 8005 + 2671 )/ ·8005 = 1.3J4 a/b = 1.33Jr Per Fig,C5,14b k = 1,4

Fer= 1.4('11)' (10.5x10• )(,125/Jf /12(,89) = 23, S84 . (not .in plastic range)

Buckl1ng M,S, = 2J,5B4/8005 - l = 1.95-1/

Example Problem 2

What is the buckling margin of safety for the upstanding leg of the T-Eeam below?

Beam length = 6" 1 1 Material is 7075-T6 Ext,

__t_ n = 16, 6, Fo 7, = 72, ooo 36 Ee= 10.5 x 10"

.J!-:-•·S'--j T P = .2J00lbs, M"' 1000

A= ,JO in, I = ,0763 in , cu= 1,15 ln, CL • ,35 (to bottom of upper leg)

fbu= 1000(1.15)/;0763= 15072 (comp.) fbL= 1000(,)5)/,0763 =· 4587 (tens,) fc = 2100/,30 = 7000 (compJ fu = 22072 (comp,), fL = 2413 (comp.)

b/h = ( 22072 ~ 241) )/ 22072 = ,89 (Fis., a/b = 6/1. :5 = 4 C0.1'1-lo; Per Fig,C5,14b k = ,62

Fer= ,62('!1)2

(10,5x106 )(,1/1,5f/12(,89) = 26,738 (not in plastic range)

Buckling M. S, = 26738/22072 -1 = , 21 * Had Fer be.en in the plastic range for this case, Fig,C5,7 would be used,

07.1oa I1lustratlve Problems 1n Calculating . Crippling Stresses .

The previous Methods 1 and 2 show only a uniform thickness for the cross­section and do not discuss how to ac­count for different element thickness­es, One approximate way to do this where angle units are involved ts as follows, First, assume the two elements to have the thickness of the thinner leg .. and com­pute Fee• Then assume them to have the larger thickness and.compute Fee• Then calculate the average of these and con-

07.30 KETHOD 3. ror Crippling Strea1 Caloul.at.loa.1

The following Method 3 for predict­ing crippling stresses is recommended, since it is more representative of what is used in the aerospace indust'ry. It consists of calculating Fee for each el­ement of the cross-section and obtaining the weighted average of these as the crippling stress for the cross-section, The crippling load, Pee• is then calcula­ted as this average Fee times the actual area of the cross-section, This is done in a tabular manner as discussed later,

Crippling Stress of Elements*'

For any ductile alloy at.any temp­erature the crippling stress for each el­ement of the cross-section is calculated as follows,

a) For one edge free elements

66 ·"' - I .eo Fee = ,5 Fey E (b/t)

b) For no edge free elements

Fee = 1,425 F0y~0 E'~0 /(b/t)'80

The manner in which b is obtained is shown in Fig,C7,37, For clad material

.the bare thickness and the secondary modulus are used, ·Also see "cut-off" stress discussion.

~~t:2 -"If-ix rT Use bare When t2 > t

I tb.ickn•oa and tx=t/2 : secondary modulus f'or When t 2 < t

1 clad mater1al t.x=t2/2 1

_J_ j to got Fee J_ j

t ~E!e Pree tj '•••~ ... :I F1g. C7 ,37 Crippling Elemett.t Geometry

Table C7.e shows the calculation pro­cedure, The example in the table is for the zee section shown in Fig,C7, 38. For this case, 7075-T6 Ale. the ·applicable properties.and other data are. shown .in

Tabla 07 .z Crippling Streaa and !.cad calotU.a.t.lona

S:lemant b t b/t. 1 !~~~~ Foe bt btFco

l .60 .0'!6 13.0 l . 3...,.= .027b 939 2 1.25 ,046 27 .2 0 47457 ,0576 2729 ,4 ,70 .046 15.2 0 62000 .0322 1996

0 28 .046 6,1 1 62000 .Ol2Q 800 I:= .1303 6464

_Fco = 6"64/ .1303 : +2609 pal Pee= ~609(,123):_ 61.Q? l'ba .

... The or1p"P11ng strength at bv.1lt-up <at"t&ohed) aem'bera * f'or ult'imitte bendlni:t: (only} 'Strength see p~4_8,"U:-Sect1ons" 1• 110re than that ot the 11111blU'll.te •m'bera, •ee Bet',14

42 CRIPPLING ' if

the figure, ..

J_ p.. 7cr--j

·t: ® - Q)-'! d 7075-T6 Alclad 6 I t :,05

, Esecd.y -.-... 9.7 x 10 ® l.25

~ ... ~.~.a .... ~P.:.d •. i~.46

• " •. 09.

2

,· ·.· .. J· .. · ... Area '* .123 sq 'in ., , . . . F~; •. ~2600 ·,· I ~ ....

,•',,·,,,_ .. :, :;' ·'"· ' . , ______ •"

ho-J Fig; C7 ,38. cr1pp1fug :~~arii'P:i~, ~eometry

,-. '., ... ·_, ' .

There"are other considerat-ions•which may be present. Thes·e concern 'the abil­i ~;:,: ... p·f a,i;i,, e l ... me.11t; to ... , be . able· to provide siinpl~ support.,.for its. adjacent· element,

· ·T!le·:r:e, are three .cases here, as follow,

Bulb Supporting Element

Example Problem

In Fig,C7.39 let bw = 1,25, tw = • 046, tr =·1, 046, . What"iS the smallest• value of bf which will .Provide· simple · ·support· for element· 2?

After several tr~als . when . br .= · • 247 the above eqilatl.on is·'satis_fied,··a:s"

14~ •. BB ")~37:;::::: 1j5;B7.

He.nee, , 247 is ·the least . value· for bf' that will prov1de s inlple •1iupport for bw.

Reduction in Fee for Short Flanges

When a flaqge is too short.to pro­vide, simple support •'for its adjacent e:1-ement, a reduced value for Fee. for the two elements ( 1 and 2 ~n.Fig.,c7,39) 'can be determined as follows.

r :··· .. ,, f· ·•

il A'ssU:me that •b'r has the mini:mum value requir!"d aqd, ce.l?u,laJ;" an aveJ:.,.ge Fee for the' two'' element's"'·as follows. ' Feel'= Fct:2 i= (·bl.'tfF!);61 + bzt2Fc;p2l/

, I ; ( b1 ti + ; b2t2) This cas~''f;tdi;~~~s~d:'~n Art,C?.9, in Fig,C?,10 .and l:n Fig,C?,12, and in Ex-ample 10 on p;c7,9, If the bulb is not 2) large enough to do. this i •a .•reduced .crip­pling stress oan be estimated as illustra

calculate ari Fee tor element tng t t has on~, edg~; free·;

2 assum-

ted fc>J;· ;a short •fla11e, (lip), using Dbulb J) instead of bf• Abul\i instead of bl tl• Fc,y for Fee and .tD/Dmin )~instead ·of t'B'r/brminf

Using'the actual·viiitie of bf, inter" pol.,.te. "cubica:J,ly'\betwee11, the Fee vaities in ·er> and .(2) above to get the' rina1 va:J.t!e roi',.'.'Fci~ 1 '"and l'cc 2 • tha J; is , ' ' · ·· · · · ·· · Effecf'~r S~~~t.ou;~tanding···Flang~s (Lfks)

As'·;,n outst';,,.;ding flange is made'• p:r'o gressively shorter, it will eventually, be unable,· to provide simpl,e. support· for .its adjacent element, Thfa ·flange lerigth,' bf, can be' determined as follows. referring to Fig. C?. 39, The minimum f:Lang~· :J,"'!igth for providing simpl~ ~µpport i~,,-. t~a,~ ;\<ihich sa­t is fie~ the fci.(l.+()wing eql.1ation, • ,

,91 (bf/tw~• - br/~w = Sb,;,/tf

where i;w,., ti: :IJ.n\i'. t\.i are 'k:riawn and ,bf can be f01ci11d,by s:uccessi)'!l trials, Fig.c7,11 applies' only when tr = tw-.

Fig. c'7 .3·9 Miilimum Flane:e .Geometry

joggled. ria?iB·~-~ see tlg. c·.4~(~~-!3°'"~?: ,.,,.-. ,-;.•,'/!.;:oi;'.:!;'J> •.

* For

,;.,,, .... ·"'

l'cc 1=Fcc2= 0'?r/bfmin f(Fcc ( 1 )-Fee ( 2) ) + Fcc(2)

4) Use Fee in (3) for elements l and 2 in the table for crippling calcu­lations (i.e., in Table C7;2),

E:x:amp l_e Problem

.. Whai(,,is the; ,c;rippling stress and the cripp~+ng,,;lolJ.d,. f.or .the. member in , Fig, C7 038 if the length of element 4 is only .15~· .. ~ i .. ;')e,:,,.,

As illustrated previously, the min­imum Ji'ng'l(ti.,,pr, e.l,~meµ.j;, 41• i:<h~~h. will: pro­vide ,simpl,e; support, fpr ,element 3 is found· 'to bii bJ ;,, :, 205", ,Proceeding as outlined..and.using Fee data· from Table c7. 2. • ·

= (Boo +1996)/ (. 0129' + • 0322)

1) Fcc3 = Feel; = 62000

2) Fcc3 (for one.edge-free) = 30026 '.···'

CRIPPLING 43

3) Fcc3 = Fcc4 = (,15/,205f(62000 -30026) + 30026 = 42552

4) Using 42552 for elements 3 and 4 in Table c7.2 and .15 for b4, the re­duced values for Fee and Pee are

Fee = 42896 Pee = 5019

Outstandin5 Flan5es not Perpendicular to Adjacent Elements~

On occasion a supporting flange must be inclined from the perpendicular di­rection, as shown in Fig.c7,4o.

,A

~e_l~+ Fig. C7 .40 Flanges at Small Angles*"

In such cases the "effective 11 va.lues, beff and teff• must be used when checking to see if it has adequate supporting length for its adjacent element, as dis­cussed previously, These are calculated as follows,

beff = bsinQ teff = t/ s inll

teff is then used for tr in the cubic equation to determine the minimum flange length, If beff is less than this then beff is used in step (3) in place of bf·, The actual flange length, bf• and thick­ness, tr, are used to calculate Fee for the two elements. in step (1), It is rec­commended that 9 be kept within the range of 2r! to 160°, If Q is either quite small or large, the least I will be quite small for an angle section and this will, of course, result in a small buckling load for the member as a column,

Example Problem

Repeat the Table C?.2 example prob­lem assuming,tliat element 4 is at an angle of 25°rather than at 90°,

beff = , 28 sin25° = , 118 terr= .o46/sin25°= .109

using ,109 for tr in the cubic equation, after several trials, when bf is ,156 the equation ls (essentially) satisfied. Therefore, beff is not large enough to provide simple support for .element 3, Using .118 for· bf· in the previous 9tep (3) and the .value of , 156 for bmin gives

• ,i!er NACA WRL 28 (1944) such angles can be consid@red to be 9rf for calculating :c.~ , so the following procedure is ,c::onservative e1nd not necessary for calculating Fee ~

3 Fcc3 = Fcc4 = (,118/,156) (62000-~0026)

+ 30026 = 43864

This results in Fee = 4327.7 and Pee = · 5323 for the cross-section.

Large Curved Elements

The previous crippling calculation methods do not include data for cross­sections with large curved elements as in Fig, C7, !fl.

;;,;~

Fig,C?,41 Large Curved Element

·Flg,C?,42 provides such data for ane edge free and no edge free curved elem­ents, The procedure would be the same as in Table c7,2, but with the following restriction, No elements in the cross­section are allowed to. have a crippling stress greater than that of the curved element, That is, the curved element's

_.L_ ... L ___ _I __ i- -~ 1 i I r 1 I ~ I : i • , a!ie["---'

-i -!-+ !1 -i ir~ ~

····-· 1-·-i···· ··+·- +·- \ l· [

+-++i\'.' ~ i ' ' 1- ,\ ' I -·-r ·-·-r- --·~-· --1 ·· ···i, · -r-- · ··1 ..

.-t-. -L. _11 ____ ,_ --1--. -l-. \L--~ ....

l ! ! I I I •.

_L_L IL,_,_ _L_l ___ L,_ L i ! --~ : ! -·:\

~--i -~~ -1- -t-+ :-f ,_ ~ ' ' - ~

-f-+ +-P~-i-+ + . -l---t-+A-~: +14-L\!-

:o

. ~

44 CRIPPLING

cr.ippling stress is also a 11 cut-off 11, or

maximum', crippling stress for all other elements. The reason for this is that tests. show that when the large curved el ement.buckles failure of the section wil follow almost immediately,

Crippling "Cut-Off" (Maximum) Stresses :'. -··.1·'·,

In· using the.formulas for crippling stresses, when•·J>/t is very small .the va­lue for Fee Pecomes unrealistically large exceeding Ftu (=Fcul for the material. There fore, it is. ne_pesse.ry to use a real­istic maximum or ..'1ci+t-·orr 11 value for F00 ,

There are three main ways that this has been done in the aerospace industry over the years·.

1. Fee max = Fey

2. Fccmax = Fco (per Art.c4.4 and c4.J and c4.2)

Tables

). Fccm~x- = Ftu

Using F,cy (J.:tem 1J,.1s .. the. most cons9rya-. .tive, but. bas proba1lly-.Peen .. used·• by·. most Mrospa,9e .,co!Jlpe.nies· and.is·. used in the . examples of this Supplement,

· Th~s!l cut-off values ·.are· also. used for Fcmax' for columns· not·•subJect to ·

, crippllµg ... qr .. loc"l:: bu.ckli!)g: (stable··:"· cross-sections),. ·and also •for columns· subject to torsional buckling, discussed bel.ow. · ·

07 .,l Tor~1on&l lh.1.ckl.1ng·

TORSIONAL BUCKLING

of such cros's-se.ct1ons are .zees, channels ·angles, tees, ·.I-sections etc. The buckled shaRe is either a pure twisting or a combination .. of. twisting and Pending about one or .two axes, as follows.

1. For a cross-section having two axes of. symmetry or point symmetry it is a pure twisting only.

. .· .

2. F~r one axis Of sy:nllnetr'y It b a 99m­bine.tion Of .twisting and Of bending about •the axis Of• symmetry.

) , For no. axis of symmetry i .t· is a :como.. bine.tion of .twisting and of bending about each principal axis.

For case (1) one computes the torsional buckling load and the conventional buck­ling load about the minor principal axis. The smaller of. these is .the critical load. For case (2) it is the smaller of the .tors.i.onal buckling 101<d and the conven­tional buckling load about the axis of non,-symmetry .•. :For ·case:·()) ·the .torsional buc)tl1ng.,.1oad. is :always: cr1·t1cal, · As .. the colmnri ·lel'lgth :increases it will' approach the conventional buckling load as ce.lcu­la ted •about the. minor principal· ax:l!s • ~iii:.C?.43 and C7.41t show for the case·s o! an equal leg angle and a channel section how ,tors.i'one.l bucl<;ling can bec·ome cri ti-. cal. Note that as t decreases (and b/t inc·ree.sesJ the loss:"in ;;s:trength •due· 'to tor'sional .-bucklin.o: -can· •become very larp;e. For ·".thtck•·:'sect:i:Ons it w1J.l not be cri·t-. l:cal· (b/t •about·B or :1ess).

,-,·· '·'-J'

coriventional · ( benil.irt~) buckling is a The :;;eviol.\s ·d~~c1lssfori: is for "free" type of general instability which e.ssum!)s .struts.·.·· For rest:r'al.i:ied st·~~ts such as that the buckled shape is. due to beinding stringers attached to .skin. (as for. wing only,. as. in Fig.c2.2. The buckling structures) torsfonal .bucklfog involves .stress ;qr load for a uniform columri is as both twisting· and 'ilendfrig, ··and ·!JlaY ·b'.;·· de.fl;ned °in equations (3), (5) and :(6) of less than the· conventional buckling load. Ar.t/C2. 2· for stable cr'oss-sect'ion shapes, · ·.:r:. ' · ··. . ., .. '" '' •. . ... . .· .

. ' .. · - ... - . .

There is ano.ther. :type . of. general· in­s 1;abi1i 1;y called. "tqr'si.onil.i." buckling for which th,e buckle.d. s\iape inyolves .tw.isting although . .ther~ is i;i~/ e.Jlplied : t)'f1s.ting mo­ment. T .. orsional buckling is •likely to be critj.ca+ for members he.ving·open type

. cross-se.ctions ·With• relatively thii) Walls . II). EJ:.dd1t1q!), it is usually most. likely .to

be. criti·ce.l when the column is ;rele.tiye,ly short, .. that is, short and intermediate . length column range. It is riot a problem for :closed cross-sections such as tubes, because of their hi.gh resistance .to twisting d.eflections. It is not likely to be cr1 t-ice.l when• the .b/t (thinness) of no member of the cross-section is more than about B.o (see Fig.C'i'.'13.). Examples

l'ractical methods for calculating the torsional buckling stress o{ r'reei struts and restrained struts are present-. in Ref.·l"H~see A:i:-t.AlB,27.i). ·

Ref. 15 1 16and17 also discuss tor­sional buckling. · Ref. 15 shows a compari­son with test results for the channel shown .in Fig.C?.44, case (a.), Ref.16 e.lsq 'shows some te s.t results. Ref •. 1 7 has de.ta . for st.ringers a tt,e.ched to s.kins .>¥- .

Noter Chapter CB articles are located

after Chapter C11 art.icles.

A Reference:! are on' 'P.A2i. '.' ·

H

.,,

'

0 J.. I: 1· ' ' I'' '::Vht "I J·· 1I'I'1111 :J)h'' •_ 11:.;I'11 1 l!l IJ!Mr n 111 'I''' '11"213' 1t1• 111• 11 · '"~:J~ · ., r · rr· '' • n.)'~~· •

Note tha.t

= 72,000 = 7J,ooo = l0,5x10

·= 26.

1, When .t • ,O) torsional buckling ls crltlcal when L/p < 175

2, When t • ,06 torsional buckling is or1t·1cal when L/p < 86

<J• When t' > ,10 torsional buck.ling ls no longer critical ·

'"Ii 1 i • • • •· . i Ii 1111!111111In111IfhII1I•,111111111111 If 'the cross-seot1.on were a Tse (with base.• 2.0 - _t) the res_ults

'would be about the same. '·r If 1t were 'a cruclrorm (height •

width• 2.0 - ··t} the torsional ·buckling stress would be the local buckllh.g stress

F1p;ure C7,43 Buckling of an Equal Leg Angle Column

8 0

Ell H 0

r;; I:"

gJ ~ I:" H

~ 0 '-.:!

:.> Z'l t;,; {IJ 8

~

~ '-"

H rn ~

1-t .

~

- ----- --·

1-2.0-J 1-2.0_,

.. '•t~ ll~:11·:J,!~i~l:i!~f ;,. ·~,~~tl:';~J·.'·.:~ . ~f_if~ .. c.~I ·~~.i~~-l~.:b.il:.:-~:-.~_···.l.~-~;.-!.i~ ... ·.'_.~_!,~-.. ~.-~-.-~.---~ .. i Jl al r·1an9•,-·; rc·r: • ,2~;9oo

:i!. b) s~cti~O_Fccf-~---_21,_a.1.o

i~tJTTITlTTITITTT .

·:~;l\1':1 ~x,,..;~~,,.-!'t11ur•• ·.t' <IJllillm :-:l; 1 ·, 33, :specimens:_.;: ·- , __ -_, it_:_ ., .- ,, , .... , •

. see Ref. 15 _or. P.A21:. . i11Ji11ldlflttflllttilttfflllllflll,I ,j.f · f.t 111li'I

11!

5_ UUl!llJJHlllHflmm1

'

l!. Ir m 11 ~ ' q! Jc ' 1

t:ritF-f~rirrf'.r ~1J.J··:l:Jl.AE1-.1.J.LJ LX~.1.,·_ ..... 0 20 40 60

L'//' ..,.,, ~ ':4/r,o/QJJefl:J.;tc..IJ.,1=~ Ve/, 2 J J'/'I 3'

tt; les Md /leiJt:!!I ;T,;,i;h ;1/il"'J' fSbn<>,

. r- 2.0-i _r-¥8

n[83•··· ... <T 2,_c i · •. J:f• ,1.ci .• _c T ,,:.. ; ;.

(cl

Local ln•tab&lity Str•••e•1 ·•) :_Fl8~9e __ ,- Fer·-. 14;000 b) ,s-:.ctioh Fee.• l,l_,200

.2o 40

L'/p

.

60E.: :80t ~.:100 120 ,:°'' -.-

01- NACA T'N 73j . Flgure C?.44 Buckling of Channel Struts

-!=' °'.

1-3 0

~ 0

~ t-<

gi ~ t-< H

1:5 0 ~

0

r:: z 1ii t-<

Cll 1-3

~

GENERAL INSTABILITY.

c9.13a Calculation of General Instability

Art. C9.13 pre.sents an equation (C9.7) for the required EI of light cir­cular frames so that they will provide simple support for the stringers without collapsing (as illustrated in Fig.C9.?b), called "Shanley's Criteria"which is ap­parently greatly based upon work by N •. J. Hoff. For elliptical frames it is sug­gested that for preliminary sizing pur­poses or checks the following be done, For a bending moment about the x-axis, M,1p let D in equation C9 .• 7 be 2b, For be_rding .. about the !:I-axis, Mt, let D be 2b /a, referring to Fig.C9.9.

y '

x -- -x

y F1g,C9.9 Elliptical Frame

The frames must also have enough stiff­ness so that there will be no instabil­ity due to the shear in the skin panels, per Fig,Cll,39.

Shanley's criteria applies only for a bending moment acting on a shell of stringer construction. When there is tension field action due to shear loads this will generate additional compression loads in the stringers, (Art .. Cll, 31-, 32), To account .for the effect of this ad­ditional stringer loading it is recom­mended that the right side of equation C9.7 be multiplied by the factor (Pc+ PDT )/Pc where Pc is the most highly stressed stringer load due to M,and PDT is the .largest of the diagonal tension loads in this stringer and in the .string­ers on each side of it (3 stringers), The diagonal tension load in a stringer is calculated as

PDT = . 5 (kf shtcotoOa + • 5 (kf shtcoto()b

as discussed in Art.Cll.32, item 3. a and b are the panels on each side of the stringer being considered.

ClO.lSb Analysis of Th.1ok-Web Beams

This textbook and others do not dis­cuss the strength prediction for thick­we b, shear resistant beams. These are quite commonly· used in areas other than in the primary structure, such as for supporting numerous types of equipment,

THICK-WEB BEAMS 47

There are two types of beams, thin­web and thick-web•

Thin-Web Beams

Thin web beams are shown in Fig, ClO, l, ClO. 2 and C10, 3, For these. the web thickness is much smaller than that of the flanges, Thin-web beams may have either a non-buckling ("shear resistant") web, as in.Chapter C10 or a buckling web ("semi-tension field") as in Chapter Cll, Fig,Cll,3 and Cll,4, In general the deeper the beam relative to its length the more likely it is that the tension field type will be the lightest, Al­though a trade study is really needed to determine which is lightest, Wagner pro­posed that the following rough criteria for this be used, When the "index" K = VV/h is more. thane1even the shear re­sistant type is probably lightest, and .when the index is l•ss· than ~even the semi-tension field type is probably the lightest. In the index V is the shear in lbs and h is the beam depth between the flange centroids in inches,

The shear re·sistant type is usually the simplest from a manufacturing con­sideration, particularly if the web needs out ... outs Qr ho1es to a11ow wire btfnd1es, rods etc, to pass through it (Fig,Cl0,19, C10,22 and D3,27), For thin-web beams the crippling strength.of the compression flange is used in calculating the.;.margin of safety, .Art,Cl0,3, Cl0,5 and Cl0,15 on p,Cl0,13 (but too conservative for.thick­web·. types as discussed later). However, if the beam's deflection (stiffness) is of importance, the initial buckling stress ofthe flange's. outstanding legs.should be used.at limit load, This is because when such buckling occurs the beam's I will be become much smaller as to additional loads, so much stiffness will be lost, See Art, C6,1-C6,4 for initial buckling,

Thick-Web Beams

Thick-web beams have webs which are of about the same thickness as that of the flanges. Examples of thick-web beam. cross-sections are shown in Fig,C10,15a,

DClLTUO Vig. Cl0,15a Some Thick-Web.Beam Cross-Sections

·In· general·~:· thi.ck '"'w9b De'ams are of one piece and are shaped by forming sheet metal, by extruding or by machining.

48 THICK-WEB BEAMS ,

When subjected to bending the allow-, able compressive stress ,for strength 'Cal­culations is frequently taken as the crippling stress of the flange, but this usually gives a quite conservative esti­mate.•. as bending:·tests of such .members show. When deflect.ion or stiffness is important, . the .flange• s initial buckling· stress.:: (and:, limit load) should be"'tised ::a's discussed for the thii\-'web beam• · For-' the

·· case.s.· .of",cee, zee. and! U sec:tions the ulti­mate,, bend'ing' strength:': can:· be•'moreaccur-. ately de.termined by using the following

'empi·rical formulas•· These are ·based 'Upon bending, streng.th· tests'' made· by 'the' cur"­tiss,.Wright Co. and·reported in their

, Report :R-162. , In: these the:,,thickness was uni.form., ·the.: two flanges"' had the same wia:th"!l,nd·'"j;he zee' section 'Was prevented from• de'fle'c t·ing •late rally,"=" ·' ' ·

Cee, and,. ·Laterally Restrained· Zee Beams

These are shown in Fig,C10.15bbelow.

. Fie;.Cl.O.l5b Ch~el and Zee 'Beain28~'i;a; :'.(·''> -.. . ,,. •-"· ·' ·:'!

· For, al.uminum alloy~ having Fey = J9 0 000 · and Ee·= 1.0..7 x 10 , · · ,, ..

Fbu = 4s~ooo(li.>1iJ'~7 . /(b/t/'"' so-,.-.o

Mall = FbuI/c·,

,: Fpr other all;ys;,Fbu is multiplied by a factor. of ,( Fcy/E~·""° /, 0.60'T where the prime mark·s refer ... to the other alloy's proper­tie.s. , Un1mpported· compression· flanges mus.j; .. , also:::; , be•.che·cked far. lateral··· buckl.1ng, assuming a web. height of, h/6 to be integral with them when doing this.

- '.'

u-section Beams (Also fo:r: T-s'ections)

For the,s9me alloy as above

Fbu = 730,000/(~/t)1·' 7

where ~ is the smaller of c and d, The formula is for compression in the out~ standing, flange, If it is in tension then Fbu = Ftu. For other alloys the same correction factor as for cees and zees is use.d •. The formulas can be used also for ex:t'ruded ··and machined shapes if they have a uniform thickness" and' the' flanges of cees .and ze~E! are of the same widths·,

* When a compress1ve load 1s also present these are beam-Columns with M.S, per p,J8, but··.do not use Method :?d fo:t, U or T-Sect1ons,£J..5e Fi9, C'5.14D Cc ah t:a-1>? I-<:. .

·r·.

Cee and Zee Members Attached to a Skin in Tension

For these .. cases .the following for­mulas can be used to calculate, the allow­able bending moments for any:·materials and: at .any temperature.,:J:f .:h :!: .8" and the beam,.spacirtg"is 5" .or more;., .:The' skl:n' thi'ckness·i· t, does'::not appearsince :the' ·skin is assumed ·to, take· a:Ir "Of ·the tiin-. ston load.'' The ... beridi:ng morilertt "therefore is .·calcu·lated••,about the base· !of the '•beam. The skin is in. tension"•and the free ,flan,s;e of the' beam is in compression.

For h/t ::: 791

Mall= ~684t"h(b/tl2

r E·OOJOJ(h/t) +

For h/t ~ 791

-~OJ (FcyEc) J 1. 2~

Mall [684.t"h(b/tizS' (F0 yEclJ

x

x

f. 0005 (h/t + i. o~ where b/t = (b + o. 7h,J'/2t. Attachment to a skin provides lateral restraint for

"

'the ·beam; "'

Cee and Zee Members Attached .fo a Skin in Compression

lFor:· these cases the··fo:l'lowrrig for-' mula can:be,used for any temperature and ma·terial · . .if 8 :!. b/t ~ · 20 'anct .'the beam and Skin are' Of, the same material!. If they are differ'ent materials the "skin thick~ ness; t, •must be··bonverted into 'an e·ff­ective thickness;'ts; using the beam's material;· The."equaticirt l:s· valid for • 50::: t/ts~ 2.0.. ., · '

Mail = ~t:(J;'syE0 fn> /b/~)';~~ x

[si6(Ec/Fcyl25

+ ·• 462 ( b/t (23

(t/t~ )~ Unequal Flange Widths

When one flange is, shor,ter than the other, use th7shorter width for.b (and d) in tl1il "previous 'foriliil.1a:.

;. :'··:· .. ,·,I ·.· - . ,,,,, ·:'.",

Other Flanged Beams Symmetrical About the X-X Axis

With extruded or machined members the flanges might.be.thicker than the webs , and with .an>. r.~,beam" the re is a flane;e portion on ·.;oacli.'side of the web. Although not covered by"1;es1;s, the fol­lowin:iprocedure .is suggested for an

CIRCULAR HOLES IN FLAT TENSION FIELD BEAMS 49

approximate value for the allowable mom­ent, Mall.

1. Determine the crippling load for each compression element outboard of the web and sum these to get Pee•

2, Calculate the allowable bending momen carried by the flanges as

Mr1ange = Pcc(h - tflangel

J, Calculate the bending buckling strength of the web as

Fbcr = KbEfweb/(h - tr1angef

4, Calculate the allowable moment due to the .web ··as

Mweb = 1,45Fb0 riweb/(h/2)

" . = 1 .. 45 fbcrtwebh /6 ·

5, Finally,· calculate the beam• s allow­able bending moment as

Mall = Mflange + Mweb

Fig, c5,14a can be used for obtaining Kb in (3) above, For most thick-web beams a/b >1,0, so Kb= 21.7 (web stiffener:i,. Could result in smaller values of a/b),

Other Flanged Beams Unsymmetrical About the X-X Axis

l, If the compression flange area is equal to or smaller than the tension flange area proceed as above (for sym­metrical sections)

2, I.f the 'compression flange area is larger than the tension flange area·pro­ceed as in Art,C3,7, Example Problem 4 (but also see Art,CJ,4a), Then assume it to have the same area as the tension flange and proceed.as for the symmetri­cal section above (for different thick­nesses), Use the larger of the.allow­able.moments calculated, If the beam is also unsymmetrical about the Y-Y axis, supports will be needed to prevent lat­eral movement, just as for a zee.

Interaction Effects of Shear and Bending

If shear is present it will promote earlier failure because of web buckling and the ensuing tension field action,

w,s, = 1/1/f.rs/Fscrf"'+ '(M/Mauf- i.o

The previous formulas do not apply when the web has significant holes,

~ ?or I-bellll. trpo •embers (flanges on eaon aide or web) or · oonatant th1cknoas a aore reallstlc procedure la as rollowst 1) Using the web and 2 flanges (a Coe-section) calcuJ.ate Kall• 2) For the other flange calculate the crippling load, p00 ,

and then Mtlange • Pcc(h-t). )) The total moment ls then Ma.11 • M(l) ~ M(2)

since th.e tests did not include such, Holes will reduce the strength but no test data are available, In such cases unless the holes are.quite small and near the beam centerline, it is probably best to base the strength upon the crippling stress of the flange, as for thin-web beams.

Cll.29& Circular Hole• 1n Flat Tena1on Field. Beam•

According to "Stress Analysis I1an­ual", AD-759199, AFFDL-TR-69-42, a single unreinforced, unflanged centrally locat­ed circular hole in a web will reduce the strength of the tension field beam's com­ponents as follows,

1. The reduced allowable web strength is

F~ = Fs(tD(d-D) + AueDC*)/C5td2

where Fs is the allowable with no hole, C4and c5 are per Fig,Cl0,17 and .18, D is the hole diameter, d is the ·upright spacing and Aue is per Art,Cll.19, equation (58), The parameter limit­ations based upon test data are

• 02 :S t '!: • 1J2 ,04"' tu~ ,079 7,4,,; h ~ 19.4

7,0 ~ d:S 18.0 2. 375 :S D~ 5, 875

where h is the web height between flange centroids and tu is the thick-ness of the upright, ··"

Some aerospace company data indi­cate that a "small" hole without a doUbler results in a reduced web. strength of

' Fs = Fs(l - 1,J)D/d)

with the limitations that D ~.75, D/d~ ,15, hole center 1s located 2,5D away from any fastener line, hole is not in a panel next to an upright introducing a significant load into the beam and the material is either 2000 or 7000 series aluminum or Ti-6Al-4V sheet,

2, The reduced allowable upright (adjac­ent to the hole) forced crippling stress is

F~ = Fu/(l + D/d)

3, The tension loads in the web-to­upright fasteners due to prying action of .the buckled web (Art,Cll,24, eituation 66) should' be"multiplied by a factor of 1 + D/d,

The strength of the web (and other compo-

~a · .. RIVET DESIGN

provide running load strength per inch l.O

/ a

• of 2Fst x 10 between tangent lines oh ~1 . ' ·-·' each side Of the hole, In the figure 0.1·

/ tr = reinforcing ring thickness o.6

I C4 . t = web thickness 0.4 ./,. .. Fs = Allowable web stress in Ksi 0.2

/ Ao = Cross.-sectional area Of ·the rein-0 0 1. 0 2.0 l.O •. o forcing ring = 2Wtr sq in,

. \,It AH = Cross-sectional area of the web d7" removed· by hole = Dt sq in

Fig, Cl0, 17 Reduced Web Strength Factor,C4 (required) Using the figure, for a given

Fs and hole diameter, D, one ·can deter-LS •. I .'.I mine by successive trials the ·required ~ size of the reinforcing ring and the re-

L I/ quired fas·tene-.rs •.. _Or, for a gfven hole,

c

,, fe._stener pattern and reinforcing ring one 1.l

I can determine dir.ectly the value Of Fs, s Instead of being a sepa~ate piece, the

1. 2

I . • reinforcing rl:ng could be integ'ral with ,. the web ( chem.,.milled · or ·machined).

. •. . '

J. 1 7, Cll.24& Rivet .Design I.

0.2 ••• o.• o.a 0.10 0 With· a shear resl.stant (non-bucklin l>'· web· the 'skin· (or· web) ·does· not pry on the h ri:"ets •and put thein in tension. The only

Fig,Cl0,18 Reduced Web Strength Factor,C5 ·tension .in· the rive·t··••theri 1s···tliat ·due· to

g)

' '' -~ "''''" -the 11 cl8JJ1p-up· 11 fo:rce, Qt ~s. sh~~n 1_n _Fig, nents can be maintained·by install.ing a 01; 43c{ WMn·.•.the .. we bier .. skin/·buckles .. , ring doubler whi.ch· meets the requirements the resulting "folds" try 1;o•·~•go···across of Fig,C10,19 and the following limi- the restraining upright or frame, and tat ions .• this puts the ·rivets ·in ·tension; The

- .. , more severe· the :.tensi·ori• ·field ( 1, 8 i I the

I~' D/hc :S ,50 . D/d ;$ ,67 larger the.·•factor··k.), ·the large»r ~s the

.tension in the rivets;

r W/D :S ,50 . . .. Equations (65) and· (66) specify the

+1:_+ Hole center within h/8··from rivet joint tension strength'·requfred to ..,,..-. bealil horizontal'ceriterline prevent the folds i!' thebuckied'web or

i.'~" q '' .. , '"" . sklrtfrom·lifting.off'of~he upright or

. . .. frame, This criteria•·was·•suggested by P • ·" JJ.t_~-::r ,. I I '•~' I !,."I. 'j Iii

' ' Kuhn•in NACA TN 266i; '''The ·criteria does ., .L....;... • ,-1 -.,! ! I. 1 ' 1 ,~' ! L.! i I ;,,! ,. ·l ; not involve the web'stress cir'k; 'so rt ... ~4·' ·j .. i f--1 .. i.,L .. 1 .. :-r ... ; ' ' must· be ''adequate for a;··very highly de'• k; ""'I i· "!." 1· ! ' l 2024•Tl Clad

.12,

4""'-;-~: ; "j .. •! _- I · 1 2024.::Tl B1ic11-:!:::;z ~ i....::.... veloped tension field, a large value Of k 1 : , < • ) I I ' ' Hence, it will be 'quite conservative for ·" ··-~·:f~~-i : a· lesser tension field effect, a smaller

:fi.l::Ot . I ; I 1 Tl ~ 1' ;

k, and for •a.k of•;zerd .the required ri-:,,-'.! : •1 I i . . .l., ' ' ~-°' ! ! ' l '

I·' 1075·T6 Clad ' ,. I vet tension strength (for this effect) .,. : I '1: ' 1075-T6 aace ./; ' : 1 would be -- ze_ro .• In view 'of thiS •. some ~

'' I .. ' : l· ·I ! . '!

' analysts that the .M .. , I j ... ::i ·: . I I ;-i i assume above criteria ' : ' '1 I . ! : 1 • I ; for rivet strength applies down to a k 0 .. ,

_i.:.I ! I ' •.I· I ' , ! ; ' ; I ' ,50 and then decreases linearly L~· or so, ' 1 ~ I. ' ' ' ; I ; ,1

with k for lesser values of k, becoming .» .~ ,. ..., "" .w ·" 1.0

w zero for k = 0 (no we.b buckling) • No ;; further such data was publl:sheoPby ·the

f

Fig,C10,19 Circular Hole Dou bier Data NACA, although some ·tests (unpublished')

companies have run for smaller k values.

The rivet pattern is to be uniform and to *or Art.c11.24·

CURVED WEBS IN DIAGONAL TENSION 51

C11.)1a General D1ac~sa1on

On p,Cll,JO; item 5, at the end of the last sentence add the following• put­ting the riTets in tension

On p~ C11. JO, item 2, agd 1 (e) the string­er bending moment (item. J) will increase,

On p. cii. Jl, R. H, Col., after the end of the 18th line down, add1 For the case of stringers a .. more direct alternatiTe me­thod is in Ref,9, discussed later,

o:i p. C11, J1 reTise t)'te "Floating rings sentence. to be as follows 1 "Floating" ring!!, support only t)'te. stringers and are therefore not attached to the skin, so the;r 40 "not determine the spacing; d,

On p;t11.J1, t~xt line 26 down, put an * after A20 and add the footnote• *How-

. eTer, the effectiTe skin area (per Art, A20,J) should be reduced per Eq, (SJ) as giTen later in Art,C11,J2a,

on -p. cfi. J2,. R.H. Coi., 4th line up.,. re­place "before• with "assuming no"1 put an * after •center" and add the. footnote 1 * or better; at.:.a_ point;. 70%. of_ the way up from ,_the l_<!west stressed edge .(stringer.) to the highest a.tressed edge. (stringer). see· Example Problem .1. Then "t thE! end of the l.ast lip.e add "panel• s •. , C11.)2a stringer Systems ln Diagonal Tension . Remainder of Part 2, Curved Web Systems

Beginning with·the first line on p, C11.JJ, the remainder of Part 2 has been completely rewritten as follows,

compression stress increases more slowly, Thus one can write·

or fc/fs = B

fc = Bfs -··- - - - - (7J)

Substituting these formulas for Fccr and fc 9ack ·into the interaction formula, (71), one obtains

.2 Bfs/Afscr + (fs/Fscrl = 1,0

SolTing .this quadratic for fs

- (74)

where f s is the actual shear stress at which th!panel buckles due to the pre­sence of compression stresses. Calling this stress F'scr and calling the expres­sion in the brackets Re

F'scr = FscrRc - -(75)

where . i./ . . Re = -B/li. + v(B/A)

2 + 4 - - - (76)

Re is always less than 1,0 slon stresses are present. lated for each panel,

when compres­Rc is calcu-

Next, consider a panel subject to a shear stress and a tension stress, ft. For this case it has been experimentally shown,.Ref,6, that the interaction form­ula for buckling is

fs/Fscr - ft/2Fccr = 1.0 - - - - (77)

The shear stress, fa, at which t)'te panel will buckle when tension is present is obtained from Eq;(77) as

fscr/Fscr= 1. 0 + ft/2Fccr or

fs~r = (1,0 + ft/2FccrlFscr - - .- (78)

Calli~ the expression in parenthesis in Eq, (78) RT 01-!J-d.."alling fscr l~scr

where F'scr = FscrRT - - - - - - - (79)

RT = (1 + ft/2FccrlFscr (So)

Since Bir is· greater than 1, 0 the ... actual shear buckling stres~ will always be greater than Fscr when tension' is-. present and when the tension stress is large en­ough there wi~l be no buckling,

Just as .the compression stress re­duces the panel's shear buckling stress, the shear stress also reduces the com­pressiTe buckling stress; Therefore, a factor, Rh, is applied to the panel •.s ef­fectiTe area (items (1) and (2), p,A20.3) to obtain a reduced effectiTe area (like having a smaller. thickness). That is, the reduced co~pressiTe buckling stress is

fccr = ~Fccr - - - - (81) where

so, ~= Rc(B/A) - - - -

~skin = AeskinRc - - -

( 82)

(SJ)

and Aeskin is used .. in all section proper­ty calculations and whereTer effectiTe skin areas are used <Aeskin is per Art,A20,J, items (1) ana. \Z)),

2, Diagonal Tension Factor, k

The next step for. each .panel is. the. determination of the diagonal tension factor, k, This is a function of fs/F~cr where , as discussed aboTe,

' ' Fscr = FscrRc or Fscr = FscrRT

52 STRINGER SYSTEMS IN DIAGONAL TENSION

depending Upon whe_t~er comp,ression or' tension a-tresses are' p·reSent·.- ·

Fore. curvedpe.nel the formula. for

fp (kfshtcotcl)'' + · fstr = - 2Astr +.~5(fit(l-k Rcla +

· (kfshtcotol.}~ .5\fit(i-k)Rc b - ·- ·- -(87) k e.•fdetermined·from ll!llnY tests, Ref,J,

is the emp'irice.l relationship· ' · · ·

k = tan··· h Ii, 5 + .. '.3. O.?RJt, td __ l.l_ og_ .. lo/scsJ .. :: .. -.·.·:. _ ( 84) i~ ~~~~~p:i:;..:s;;~~~;1~t! ii~:i:~~e~~ : . L . present in ea.ch panel Eq,(81) becomes ' .. '" ,,, - ' .. ;· '.' :·· -

with the auxiliary rules that ' •.-o," ,, ... ;-:,r· ,... .. -, c:.:··· ..

a) if d,/h > 2 use oli1Y' 2

b) 1f h >_ d ( longeron system). replace ' d/h w1th h/d and 1f h/d>.a use only 2

k ce.n also be found directly from F1g, C11.19, ' '

'.·.I I ..... ,'

· " · '' · • kf sci>tOC. ' •' ·· • fstr = fp - Astr/ht + •;5{1-l<JR/' . .,.. ,.,.(88)

' .. - • " •. .• ' '• ,, --l ' ' ,- J,_

The two pa?lels t::an have . identical pe.ra­meters only when part of ·a:· cii'cul.l:r s!iell subject only to torsion'e.nd compression,

•not to shear or bending which cause Te.i:y­ing stresses, k and ficTalu'!s,' 'Howe'ier, for closely spaced •strrngers •using iiTe'r-

3. Stringer Loads, stresses and, Strains _age values and. Eq, (88) is reasonable, The stringer sti:e.1ri; needed to determfoe

As in tl;le case of the plane web sys-· rA later; l.s• :: ·· ..... ' · ,. · tem, the total stringer load will con-sist of the primary ax~e.:J, loe.cJ,~ • .l''P• .. Aue E. str = fatr/',.Estr - - ,;. (89') to the· applied 1Jending momemts. arid7or ax-ial loa,ds~ .plus the diagoniil 'ti!nston th- All terms in Eq, (87 l and (88) .a:r:~ .known duced loads, Po, T, except:'~'''• •'whfoh is 0<;i'sc;is·sed late'r;' ,. ·

PsTR =Pp - Po,T. - - - " (S5)

If Pp 1s tenshe it wo~ld be a positive number 1n the .above equation, Pp c,e.n be dete:i;mined .as:.in'Che.pter A2Q,2 - :A?O'· .. ·. Po,t · Hi cce.loulated'.as''foi:towg',ure!'efifing to.·: l;g;•Cll,JS,~; .··'"~ha:'.,·····_·.,'

'"' . X:'' ·'' .. ,., hb

"'"' •Y' ;.;

··.·• Per Art°'91.Jf,. it'em J, the diagon­al teniiion''generate1i'oonci;ng of the . ' ~tr1ngers• 'bowing theni iryward, ' P•H-' Ref;3 .the res1,1'1,,t is ~ pe~ ;i::e'naJ'.'g niome,nt at the cen,tei:'. or the. :str1n,gei: .'!'rid e.t. tl\e ends (at -tlie rings) "1n the.'amcl'tiri:t .. ' ' '

.,., ,,,,1·,'·: -··;"';: ,,; '<l'

Miit:r•:,,,. fshtd"ktan~·/24R - - ,_ •.:: -'(90)

where R is •the '·ra:dfus 'or cuI'fature. •' .1-,,,.,, . ··~· •' ;·-0 ) '<, ' - '·~·,.'I

This produces tension 'ori 'tlia irtriEOi: suI';.. face of the stringer at its middle and compressi•on •:on tlie' forie:f surfacie '8£ 'the'

•Fig;1.c11·,J5a'Stringer Diagonal Tension Load rings• This bending i's :aue to the diag-·, ,,,,. · • one.l tension and is inqreased when pri-

Po T is the ·diagone.i tension i'daci 'in .;, mary compression loads "are a'lsci present str1nger'-bourided·'cyip'iine1''"1L'•:·0 n: d'rie 's':fde (but not in defined a.mount., 1,e;, a beam-e.nd·:by. parie1• "b" •ori•tfie•:otliei'','' Llit :tile'·'' ' -c«:ilumn effect), " · , ":i,dth of panel a, ·l:>e ha e.n,d that of b be hi> ,(for eque.llY· spaced stringers ha =. hb) Let the shear flow_ in panel a be qe. and· that- in b be ql:), 'Tlien·

(86)

Ref,J shdws that a:' panel's effective ar'ea (at ea.ch stringer bordering it) for the diagonal tension load, Po, T., is · ·· ·

Aeo,T, = .5ht(1-klRc/2

. ""The•r,.·fore, the stringer stress is

. ··,.-:

where _fp, is negative when P 1s compre·s­sive and positive when P is tensile, so,

Per J:lef .3 .fo .T is ~~. a'.'f~r'a:e;~ s,.tr~ss There ts "alii'o 'e. peal1: cir max_imum stre·ss, fD,T.MJ.:x• calculated as

fo,T.MAX. f:[),T, (fo,T.MA~/ro.T.) -(91)

where fD,T;Mu,lf'o.'T,: ·is' 'obtained from F1g,CU,21. fD,T, _is the. le.st term in Eq,(88). fD,T,MAx iS·Used for a local strength check lsee Example Problem),

4, Stresses and Strains in Rings

.. ,, . ., ·The.re .. are,, .. two;,types, of,,.;,rings, Con­V";,!11;ional ring'!o-,intel;'sect the str1ngers­and, a,re notched .to let t,heni pass through, and are attached j;o the•i,sk1n.-· :_So-called "(loating" rings are located ,inside of

• the stringers e.nd are nbt .at:t'ached to the

STRINGER SYSTEMS IN DIAGONAL TENSION 53

skin, The resulting stre~ .. ~e~.'ln these two types of rings are therefore differ­ent. ConTentional Rin5s

Just as the tension field produces a. compressiTe stress, r0 , T., in the str~ng­ers it also produces a compressiTe stress frg.in the rings, For circular shells this is, per Ref, J, for conTention":.l riligs

.... ,-.

kfs tan o< (92 l frg = Arg/dt + .5(1"-k) - - - - "'

Just as for stringer~, Eq,(88),.this as­sumes that the panels on each side Of the ring haTe the same parameters (k, fs, d, t, 0( ), If not, Eq, (92) wo~ld be of a form similar to the last term 1n Eq,(87), HoweTer, as for stringers, aTerage Talues can be used in Eq, (92) when the Talues are slightly diffe.rent, Tlie axial strain in the rings will be

Ring stiffness (EI) Requirements

Since the ringssupport the string­ers against general instability, see Fig,C9.7b, they must huean adequate EI, This Talue is affected by the diagonal tension loads .. in the' stringers and is discussed in detail. ir Art.9.1Ja,

5, Strains in the Skin Panels

The'strain in the skin panels is given in Ref,J as

Esk =ls E [ 2k + ~in 2p sin2DC(1-k) (t+..u]-(97)

where µ = Poisson's ratio terms are known except ct, of help, giTing the value ed term

(,33) and all Fig. c11. J6 is

of the bracket ·

Erg = frg/Erg 6, Determination ofo<.

- ( 93)

The strain is used to determine c/.,

Rings can also haTe a m~ximum or peak stress, like stringers (Eq, 91), us­ing frg in place of fO,T, in 'that eqtla­tion, HoweTer, per Fig,C11.21, frg/frgMAX will be i; o for typical stringer construction where d/h> 1.0, so there is then no peaking of the stress. •e1oat1ng• Rings

For "floating" rings concentrated radial loads pushing inward. on the rings· are generated at the stringer locations, Fig,C11,JOe, This produces a hoop comp­ression in the circular ring and also a bending moment, since the radial loads are concentrated, not distributed, The axial compression in a floating ring is

kfs tano( frg = Arg/dt - - - - - - - (94)

Note that there is no "effectiTe" area term for this case, The axial strain in the ring, due to frg• is

(95)

The maximum bending moment generated in the ring is

Mrg = kfsthed tano<./12a - - - - - ((96)

It occurs at the ring-stringer Junction, producing compression in the outer flange of the ring, There is a secondary bend­ing.,.moment, half as large, midway .between· the stringers, There is no "peak" stress for floating rings, As for conventional rings, Eq, (94) and (96) assume that the panels on each side of the ring haTe the

. . same. parameters.

For the stringer system (d > h) Ref,J' shows thatoC is related to the stringer strain, the ring strain and web or skin strain by the formula

the

(98)

where Esk is a tension (+) strain and Erg and Estr are compression (-),

«. is determined by successive ap• proximation as follows, The strain eq­uations are (for conTentional rings)

_.Lr - kfstanol 1 €rg = E0 LArgfdt + ,5(1-k!J

1 [ kfs cotoC l €str = Ee fp - Astrfht + ,5(1-k)~

Esk =-fa ~i~2D\+ (1-k)(1+µ)s1n2~

(93)

(89)

(S7)

fp is negative when in compression, To determineol. for any panel use the values of k, .fs, Re, t a.nd d for. that particular panel in the aboTe strain.equations, Al­so, for Es tr use th.e stringer bordering the panel whi.ch has the larger compres­sive Talue of fp,

• • Assume a Talue ford,, say JO to 35 to start, and.compute the values of the three strains, Then enter these into Eq;(98) and computeo(, Compare with cl. assumed, Repeat as necessary until the calcula.ted· Talue of c< is· essentially the same as that assumed, and o<. is then known. Note that €rg and E-str are negative, being in com.,re.ss1on, For "floating" rings Eq,(94) is used in place of Eq,(9J). Once 0( is known all of the stresses

.STRINGER.SYSTEMS IN )'.>IAGONAL TENSION

·and effects involTing 1t a:re .known,

7, Loads on the Attachments

The cmly ~eine.ining i~ads to be cai­culated .are those on .. tlJti e.ttachments, Thesi('are o:f two j;ypes i those in the plarie of the)ii<ills arid· those.· no:rll!'il to the skins' which ar<i. "prying" 'fofoea''t'ry,;, ing to "pop off~ .. the riv<;!t heads. or to pull the skin tipward·around'the'riTets, The~e normal lc;>e,.ds a:re mo~t lilte1y to be cri tical'wh'en coimtersunk rtT!i~s 'are pre­sent and are of small diameter.

The rivet loads· due to the applied. loads are called "primary" loads and those due to tens+on field action are "secondary"· loadS~ ,., · · •• 1 ·;.- '.'' • _, \ _.;.· ,.

a) The' secondarY' riTet shear loadi, must be considered wheneTer a shear-buckled skin is spliced or e.nds along ·e,n ?pen­ing or "cut.,..out" • such ·as along a si(l"ing!!r or a .ri~, . .i\-t .. a .~Jl.liC.e or opening along a stri'nger 'the load per inc'h is · · · · · · •• · ., .. , •.... ,,, ···'

Lbs/in= fstG +~(c~sd.- ~TI~·. -(99)

Along a ring (a !el'tice,.1 direction) the loading is

·.Lbs/in = 'fst ff ~ ~(s'in.;. • 1]-' -.(100)

f 6 is the total paIJ,e,l !!he,e,r· l.oad and includes any redistributfon effects due 'to a .cut-out;" as' in.Fig;n3,21. .If the ·panei ''is ncin 'bUcki1iig then k = o;

b) The normal or "prying~ .. loads are not 'determinal)le by any formula from theory, but based on. test data Ref,} recoll)lllends. the ron>wing e,.s the loa.ding on .the ·rh'ets, ·

When the .. skin .is continuous across a·. ring· or stririger

'Lbs/in = , 22tFtu': .. _;;.·. - (101)

. When the 'skin stcips along a ring or stringer {due to a' cut.;out)

Lbs/in= .15tFtu -·'-. (102)

wher".. t is the skin t.h.;,ckne~ .. s e,.nd Ftu is •the skin ultimate tensi.le' strength,

These criteria are' no doubt cd?lserTa­t1ve and are discussed further in

· Art,C11,24a, Soine data: fol' the · ·stire'nl>'th?"ot''"riTift'.;sheet' coinl5fnat:fons

· . ·in the' normal direction are' 'in Fig; .. c11,37a·, c11,37b and Art,n1.26, Flush heads are most likely to be .. cr1t1cal;'

.,,,. .- '·::""' ...

Cll.))a Allowable_ ._.S_tre•••• (and ·· · · Iatera~t1on) .

1, Stringers

Three methods of stringer. analysis .. are .. i>:r,e'~ent~d •

. .-;:. _· .. (.;·; ;··: . ' ,\ '.'., ···_ . .. al.A '.'.Quick .. .J•!ethod", for preliminary

'i?lg and quick .checks, siz-

b) The N.A.C.A •. Method wli.ichlS quite tedious and therefore not. recommended

c) The 'Melc,one-Ensrud Method. which is the recommended one

The Quick Meth.od

,.. b'-6 analyses' 'are med.e, ohe. tor i~cal strength and pne for,bUCkling, ·The local strength analysis uses the. int.eraction equation

fp/Fcc + (fn,T.Ms.x/FsTl"5

= 1.0 - -(103)

where fp ~ M..Yh, ~cc is the "na tu;S.1 • crippling,st,i;ess, .Art, .. 9z •. 2, .. c7,4 or, bet-tel".• . Ar.·.·'~'.c7,3p, fn·,T MB.xh per. Eq,,.(9.1) and FsT is per Fig, Cll,.JS,. .. · ,

The bU:~kll'.ng analysis• uses tne inter-Elc,ti<?P. equa~:).on , ...

-(10'+)

' ~· ' • - ' ·' ') ' ' - J'

= buckling stress due to diagon-al tension load using an er~ fecti ve skip. area of , 5ht(1-k )Hc;c1n cii.1Culahng Pstr and :C: ~ L/Y2 when string­er .. is ... continuous·. at both ends or,r: 7 .. Llv'f.3, .. when it is con­tiriuou·s et only one end, .

However, in either case an effectiTe skin area is not used in calculating P if this results in .. a ,larger p than for the string­er al cine~· ··This is because buckled skin · has an undefined reduction effect on the stringer's buckling strength. (Fig,C?.36),

The N,A, C,A, Method

. This is discussed only briefly since Method. c •·, next, 1~. :recq,~,en4ed.. Ref, 3 suggests the following .,analysis,

a) Local Strength Che.ck ·:<"

Corl~ider 'tiie structure ;to be. su·bJect

STRINGER SYSTEMS IN DIAGONAL TENSION 55

only to its axial (or -bending l applied loads, not to any" shear., a,11d calcu.late fp and Fee for 1;he string.era., ·. Then consid,er the structure to be subject only to its shear arid· \;or's ion loads . (no ·axial:· or ben­ding loads) so that fq:r all str1nge:rs fp = o;· For any stringer the aTa,fage shear in the two panels bordering it ·1s fs• Then using the last term (fn,T,l in Eq, (88) find by success~Te ,trials what Talue Of fs results in

fn. T. M.a,x = FllT

This is a Tery considerable effort, since for'each assumed Talt!e off~ one must de­termine numerous paramel;ers k, IC:• s, o<. et Then le.t t,his Ta,lue of .fl! jJe 013.l:led Fso• The margin of safety is .then found from the 1nteract1on equation :."

·~p/Fcc + (fs/Fs0 f0 = ~.~ - -(105)

by using Fig,C4,J4 wit.h. Rb and Rt there being replaced with fp/F00 and fs/F60 re­spectively, In Eq,(105) fs is the actual (average) shear stress. iri the two panels bordering the stringer,

b) Buckling Check

. The same approach as used in (a) ap­plies, except that the allowable stress terms.are Fer instead of Fee.and F~ 0 in­st,ead .qr Fs0 and are d,etermined .as follows,

Fer is determined in the same manner as for Eq, (104), As in (a), by succes'­siTe trials find what value of fs results in

f -FnT· D.T. - · • •Cr

where Fn,T.cf is as defined. for Eq, (104) in (a), Aga n, this is a Tery consider­·able effort as mentioned in (a), Then, let thts value of fs be called Fs 0 and the margin of safety is determined from the interaction equation

I ( I ' •·S fc Fer+ fs Fs0

l =: 1,0

and can be found as suggested for Eq. (105).

The !1elcone-Ensrud Method

-(106)

This method is recommended since it is much more direct than method (b) and also considers the torsional stiffness of the stringer, As shown in Ref,9, the stringer margin of safety is calculated as

1 M.s. = [<fp/F0 rY"8 +((f(, + r'b)/F~)'-2•]•&o

-1.0 - - ~ - - - (107)

where . . . . f'p = compre·ssiTe stress in stringer due

to. the primary .shell loads (My/I + P/A), bending· ·and axial, as· present,

Fer = stringer critical load as a calculated as discussed for Eq, (104),

column

F~ = bucklinp: strength of the stringer itself, considering crippling, a,nd using a, fixity• of 2 wh,en t,he st:i;ing­er · is continuous acros.s rings a·t ea.ch end and 1,5 when continuous at orily one erid,

r~ = See Table c11;1

fu c = See Table C11,1

Table c11.1 inTOlTes ftve· parameters .. ,which are defined as follows

T = 1 + (d/R) [(Istr/Jstr)(t~h) ]'2

5'

where J str = torsiona.l stiffness factor

= Astrt"str/J for open sections

= 4A2tstr1P for closed section<

where A = enclosed are&

p = perimeter .25

~ = 1/(1 + f 0 g/fs)

where fog = applied primary compressiTe stress based on stringer area plus total skin area (fully effectiTe), When fc~ = 0, i\ = 1,() (no applied compres-,· siTe loads), That ls, fog= My/Igross + ·P/Agross•

F~ = (1f/4s'2.-)(Et/R)

where S = d2 /Rt

F'~ = 5,25E(t/hf

Table C11,1 Determination of f ~ and f ~

f~ = 0 when fs< AFscr (no buckling)

when fs 2:,\Fscr

f~ = 0

(Continued)

56 STRINGER SYSTEMS Iii' DIAGONAL. TENSION

.... ·fc=

f " -c -

. Table. Cll, 1 . (Con' ll.) -.·.'

·.. I .'-•''II. ,"

'- }l.Fs) ( llFsdt) (, 05Jh J"" Fs As tr d ·: ., -

when .ill'.s ~.fs ,, '' ,\F8 dt (. 053h)m

Astr.~ . when f s ;::: Fscr

~h:en .t~~Te,lu"' or .. fsY!n,i~s 1n.,.the. panels ab9Te arill. P';:lo'! the, s~rliiger use the av­erage'·· Or,,m9re, Cpnservat1velY, US!',.the larger·. If h or· t varie~ use tli1i. avera~

The follow1ng 1s app:J,1ce,ble to all .. three methods. If a· str1nger has a neg• at1ve margin of safety it doe.s n()t mean . failure 1f the othe:i..' strirtge\:-s c!iri carry e~_?ue;i;i. o~,:t.h'l,. s)\.,,11' 11. applied bending .. momerttto preTent this. (see Att.Al9.11), Howe.ver, each stringer must cari:y 'its own diagonal.tension load, PD,T.•

' . Al~? l· ~inc~ 4l,11e;9~:J,. t'\'.n~iop lo!!-d!I.

do not vary l1nearlY'wi th the applied loads, the margins of safety as calcu­lated are actually apparent ones rather than true ones (seeArt;.C1,1Ja, item 2), but this.ls not always considered. This also .. applies . to other parts loaded by the· tension f1eld 'actfori~

It''' is i"of" course, best to determine Fer from compression t"'sts .. o.f. a segm"'n~ , of the shell having four or f1Te'strf~­ers, but this. is expensive to do al~ though it eliminates 1lncerta1nties. ·,

2; :Rings :r ··

Rings' have allowable .stresses simi_; lar to those'fqr,stringers. For con­ventional·rings the margin of safety is obtained using the equation•·

/I '· ·' ·- :

M,S, = I -1.0 --(108) .fP Fee+ frgMQxlFrg

-, -- ·J. ;

where fp is the largest compressive mess in a flange due t 0 any primary load1ng on the ring (bending and cqmpre.ssion) if any, and Fee is the crippling stress for · that flange • · .

frgi.\a): is f rg (no peaking here for tne stringer system) , Eq,(92),

Frg is the allowable. forced crip­pling stress for the. ring, per Fig,Cll,)8,

Since conventional rings are· notched to let the stringers pass thro1l~h, it is important to use an adequate clip at the notch to prevent local ·buckling of

·the ring's notched web,

...... iioatl.ng r1ng~ .. are .)lo~, subject .to any .forcied' cirippl,1ng,, .,only,. .to .any axial stresses o.r bendillg s.fresse~ .cause<i by primary loads, r~ .any, arid .. to ... •the. a:z:1al sti:ess. ?f Eq,(Q~).).nd ~p, Mrg o(ilq. (96), The. ma:t"giri of S!!.f.ety is c11lculated f.or tn,e .flan~!\ tiritlc,~l tor thes!i. loads as'

1~0

-(109)

3. General Instab1l'ity

A g-;;l')erai' in~'tabil1 ty. ·check for the .s,hell .can bi? made .)lSing the emp1r1cal criteria, p:t"esiinted in Fig,C11.J9, , Th1s is obtal'.ned from. test ,data and. ?'.!"commen­dations of Ref,7 'an<i· .Re.f,;3. The allow­ables are based upon"pure tcirs1on tests. The Jl!al'.gin or ,s!!.fet:r is calcu1ate4, ·as '··. ,. . ..

The allowabl~· .slf'i.n stres,s can s11,'-' ply be read from Fig,CU.4:2 .. '!-s ,i;:9 , 'rh!! marg1n of safety is calculated as

M.s;~ ... F~{i's ,.:r;o ~ - .. -<111)

'i's ts' 'th:~'r1;r~ss .~kiri ~.'t'~ii'ss' (1g;i6~~~··:;·:' r1Tet 110:iesl'. Thei' n~t' shear str'ess (be~ tween r1vet holes) can be carr1ed"up t'o .l"su.1,, t.l1E!, ;ultimate .11hear sti:.,,ss for the sk1JCmater1'iil · · · ··

_.;.·

5.' ~rg1'n of s~r~t:r f~r A.t't;.chneni:s

For an;i' at'tachm'e nt ( r i Te t ) the mar­gin of safety, is. ca,lcµlated as

',,.· ,,; " : ,•).,\' !'J.~.i ; ,., : .... ·'.:·,;'.',, .. c' .

v s Joint .A. 11.owable Load,··· .··· .. ;.1 ,. 0 l"h • = , . Attachme.zit ,~pa~.ing \.Load/in) , .,

' ' ' ''' ' ' ' ' ' ' ' .•.• ..; ... , ~ - -(112.)

where the Load/in 1s as. fofio;;s' a.) At a splice or ed!!'e along astr1nger

(horizontal) Eq, (99) applies, •

b) At a splice or edge along a ring (vert1cal) Eq,(100) applies

c) For a change' in stress, ~r shear flow, across a ringer stringer (no edge or.~plice)Afs is used in­.stead or fa inEq,(99) .or (lOQ),

Eq, (}12) appl.ies for, both, in-plan,e and normal' loads, The joint· allowable··•· load is the allowable load per r1Tet,

The following E~lllllple Problem 11-:i,ustrates the analy~is, procedure,

S.TRINGER SYSTEMS IN DIAGONAL TENSION 57

c11.)4a ~ple Prob1ema

Determine the following margins of safety for the structure shown in Fig, C11,4Ja. All loads are ultimate,

a) Skin rupture for panels a, b and c

b) ·General instability in shear based on the shear iri ·panel b

c) InterriTet buckling of skin along stringer 2 in bay .AB

d) Stringer 2 between panels a and b

e) The fasteners along stringer 1, as­suming the skin is spliced there

f) The ring between panels a and c and stringers 1 and 2

g) The fasteners along the ring at A in panel b assuming the .skin is spliced

h) The fasteners in panel b along ring B

Internal Loads

The first step is to determine the internal loads in the structure due to the applied (external) loads, This can be done as discussed.in Chapter A,20, or it may be done by finite element analysis For tension field analysis ·pu%-;ioses, it is necessary to do this for two structur­al conditions, first assuming the skin

does not buckle •and then w.i th the skin in the .buc1<led condition, For the latt.er condition this is a successiTe trials procedure,• since buckled pan<fls haTe ef­fective widths which depend upon the pan­els stress, and the stresses depend upon the amount of effectiTe skin area,

It is assumed here that the internal loads have been determined for the she11 and its applied loads'• which. S.re a shear, a torsion and a bending moment, .The fol­lowing table summarizes. those internal ·loads andr the geometry needed for the re­quired analyses, ·All loads 'are ultimate,

~able C11.2 Internal Loads and Geometry .

qa.S: qo s qc °, .. ,

. ' i J Cond1t1on y Ya Yb Ye ot Skin lb/i~ lb/1n lb/in 1n 1n 1n 1n

No Bucklin, 111 121 111 0 14,40 11,)4 14,40

Buckled 116 125 116 2.71 17.11 14,05 17,11

Table C11 2 ( Corit1nued) . Condition J'strt 1'str2 ~BC , MAB'" 1•

or Skin 1n 1n ·in-lbs in-lbs 1n•

No Bu:ckl1ne 14.61 12.65 171,000· 216-, 000 )82

Buckled 17.J2 15.J6 171,000 216,000 276

~ 1, For panels y is measured from .the. shell's neutral axis to a point 70%" of the distance from the lower edge to the upper edge ·(rather than to the midpoint),

2. For stringers y is to the· stringer+ effectiTe skin centroid,

a. • .A 2 --- - - £ - --....jio IZ.

II .te.=15"

'f

8 7

h

A c

A// mat_,,.;;,; is 2D2'f-T3 .f'),eet £. :/0.]A/dh Fc,y ..: J9,0C6 Ec.=/tJ,s;&.JO'- ~ll=

F_n. = 6~a'10 121V.as l/J"e. vi· C1A.IY4r.S4/ Aead ll5201f.JtJ AO-'f at: .7$"" ,;-pe,;,J"

This is the same structure as was .used by. the N.A.C,A·, 1n deTSlop1ng ·their tension field criteri~, except that tstr was ,080 there,

Fig,C11,4Ja Shell with Stringer Construction for Example Problem

58 STRlNqEI( SYSTEMS IN DIAGONAL TENS ION

3. MAB and Mac are; bend1·ng• momenta at 'the Rcb = .69, kb= .46, Fsb = 23,500,0 m1P,dle, of;, bay .. AB imd'·BC:crespec't1·:reln· fsb = 5,000, Rcb·. "'· ,52, o<b .= 2~,8,

M,S,b '= Hi!jih 4, The"sa~~;· I is used for ba~ ~B :and BC although :CAB '.would be slightly smaller ,s.ince .. t.he,, larger bending •moment would re-sult in:,.:i:_ess ,e,ffective skin area. ''

'"'·"''' ·Skin Panef' Analys;s•,.

. '~he ~'i;;;;f;sis ~~\'111us~·;~ted for pan-el..·a· only., ... The results· of,· sim:ilar· analy­ses, ,for panel_S,,. b •and '!C are ·presented;. , '

alb ... = 151.7.85 .. 1:91, .. z ;,,,.,158 (1~.)i)'~ 15(,025) :;;· 1551 ks: J2((F1~;C9,4)i kc "' .56, (F:l,g, C9 ,'1) ·. ' ' 1" • • '

.-•}"·• - ·-·····

F~~r .. 32.ntc'1f.7x106H.625.>"f12<1-.33~) · . ;··. (7 • .8,S.) = J,205 (Fig,C9,4) .

Fccr = 56'.!1'.~('10, 7x10& ),( ,.o2$f/12( 1-,Ji) (7,85)2 = 5,609• (Fig;c9.4·J ; .. · .. · ·

' J ' ' • :•: ~- ' •

A= Fccrfi,s~; :5609/3205 "' 1:1.sc(Eq.72)

For the non-buckled skin (to. determine Re, Fscr and Re)

Rec= ,.67, kc= ,45, Fae= 23,000 0

fsc = 4,640, Rec= .56 1 olc = 24,o, M.S.c'= High

Interri"ie1f bucklii\g along 'Stringer 2 · 1n bay AB

Fir= 'rr2x10;7x10 /(;7,51;58~.b?S)~ = 39,473 (EiJ.,C7,22) '· ·

fc = ~7gkin9:7~~;Ts~~·~t;i~~~~2)!52~~o48 M,S, = 39473/22048;.. 1,0 = ,79

General Instabl.Uty

The folloir1ng .·ls. bas(/d :upon !?~el 2 (1 t is more' crl.t1ca1· tor panels at ~he. neu'tral axi:S. where; pahei shear stre'ss is larger)~ Per F1g.C11,39 anci its notes .r,r·: ,~g.,:;-: ·",, .,,..._,,;. .·. . ~.;;;.-· ~ .. ,; .. f'strxfx:.,x10= .409 x.765 x~~ . 44 8 (iififr !F* (15x.785T'x 15· = •

FsrnstlE"-> 3,000 'I

Fsinsi:'> 3,000 x 10;7 x, 10 > J2,ooo fc;,,, ~yfi'.'= 2i!$,ao8(14;4)/383 = 8o121c

fa·',. q/t .. ,111/.025-,;, 4.440· '"'i' ,. ,) · • r9 =•5,ooo·c.···, ··

'·' :.:•::.:,_,:':';"<"<;··· .. ,,. ...

B =, fc/fs .=)3121/4440 .= 1"83. (Eq;73)

B/A = 1,83/1,75 = 1,05 .. :;_; .. ,. J

Re =(-i.05 +-V1 .• 05" + !f)/2 = .• 60 .. (Eq.,76>

Jl6 = Rc(B/A) = ,63 (Eq,82)

F'scr = J,20,5(;60) = 1,923 (Eq,75)

For the skin in the buckled, condition

f s = q/t ,;; ,~'16/, 025 = 4,640

fs/Fscr = 4,640/1,923 = 2.41

300td/Rh = 300(.025)(15)/15(7,85)= .955

k = ,51 (Fig;Cll,19)

Fs = 23 ,000 (Fig,Cll,42)

M,S•rupture = 23000/4640 -1,0 =High

Eek= fs 12x,51/sin2o\ + ,652sin2o£f/e: · = ,oci'o442(1,02/sin2~+ ,652s1n2~)

- - -(Eq,97)

o\ = · 25.4° fsee "Determination of~!')

Proceedfng ~imi1arly' for panels b' and c

M,S; > 32'iOOO/S,6()() -·; -- ·· '!," • ·- ·c'.(·l

Stringer 2· (between panels a and b)

. Th1s will be ehecl;ced using the three methods dispussed'and using an ~verage of the Talues for'data in panels a and b

f!!tr = fp - fD,T.

·. :. J!l.. kf8 cotoC · " · .. , I - ;!Ac±.ss:tr~/~h<:t!.-+-, 5""'(""'1'""-.,.k7) "'""Re (Eq,88)

-216000(15,36) _. .485(4820) x 276 • 094/7. 85 .'x • 025 +

' cot(24,8°) ,5(1 - ',485)(,645

= - 12,021 "" 7,736 = -19,757

fo,T,Max .. -7,986(fn;T,Maxlfn,T,) (Eq,91)

= -7,936(t;23) = -9,761

FsT = NxC = 21,800L883),., 19.,249 (F1g,~l1

'.Pf,;tr only) = ,4o9 , Fee = 28,600 •39

>

.. '.\'.\'e three t;y:pes of anal:v,se.s follow,

STRINGER SYSTEMS IN DIAGONAL TENSION 59

Quick Analysis

Local Strength t.S

12,021/28,600 + (9761/19,249) = .776 (Eq,103)

M.S. = .23 (Fig,C4,J4)

Buckling Strength

Per Eq, (104.) and its discussion.

Fer = 28600 - 28600• (15/ ,409 f;4'rf~10. 7x1o'

= 25,995 (Eq,C7,J1)

FDTcr =' 28, 600 - 28, 6002

( 15/VZ x • 348) I 2 . ,,

4'fl x10,7x10 = 26,801 · (Eq,C7.31)

M.S. 1 = 12021/25995 + 7736/26801 -l.O

(Eq,104)

N.A.C.A, Method

Local Strength

For an applied torque loading only it is found af.ter several successive trials t~at wh~n fs is 10,600 psi the resulting valu.,,. of fD.T.MAJC is equal to FsT (Fig.Cll.38), Then, per Eq,(105) with Fso = fs = 10,600 ·

,,,-12021/28.600. + (4820/106ocp = • 727(Eq,105)

M,S, = ..:1.£. (Fig, C4,34)

Buckling Strength

As above, it is found that when fs is lJ,.400 psi the resulting values of fD.T and Fn.T.cr are equal, Then, per Eq, do6 with F'so = fs = 13,400

!.If 12021/25995 + (4820/12400) = ,678 (Eq,

M,S, = ..u2_ (Fig,C4,J4)

Melcone-Ensrud Method

fc = 12,021, Fe = 25,995, F'c = 27,300, fs = 4,820

106)

• Istr = ,0157,.Jstr = 094(,040)/J =,000050

V = 1-f- (15/15) n,0157/,00005)(.025/7,8.r = 2.0

fcg = My/Igross = 21600(12,55)/383 = ·7,078

. ·2> fl.= 1/(1 + 7078/4820) = ,798

s = l{ /15(,025) = 600

.2s- ' F's =!il'/4(600) (l0,7xlO x,025)/15 = 2,777

F~ = 5,25x10,7x10' (,025/7~85) = 559

Fscr = 2777 + 559 = J,336

~Fscr = .798(3336) = 2,662

r'c = 2.0(4820 - 2662)(7,85)(,025/,094 = 9,011 .. . ,,,

r'~ = ,798(559)(15)(.025)(,053%'7.85/15l/ . = 539 . ,094

1 - 1,0

( 9011 + 539{'2J]'80

27300 '/ J

Summary of M,S, .. Results for Stringer 2

Item Quick if.A. C.A •. Meli:iom M.S. Anal"sis Ensrud

Local. Strength ,23 ,JO ,37

Buckling ,32 ,39

Ring B (Between Stringers 1 and 2)

f _ £L_ _ k.t;..tan rg - Fee .Arg/dt + .5(1-k) (Eq.92)

This adds the fp ratio not shown in the Eq,(92) previously but which applies when any primary loads are present on the ring, Using average values of the two panels (a and c) on each side of the ring, (fp and Fee would be for the critical flange) .

,48(4640)tan24.1' frg= -,134/isx.025 + ,5(1-.48) = -7· 844

frgMax = l,O, frgMax = 7,844 (Fig,Cll .z1)

Frg = NxC = 14000x,883 = 12,362 (Fig,Cll ,JB)

M,S, = 12362/7844 -1,0 = ,58

Attachments

Assuming a splice in the skin along stringer 1, the load/in is larger for panel a

Load/in= 4640x,025.1+,51(1/cos25,4-1)

= 122 (Eq,99)

Load/attachment= 122(,75) = 92 lbs

P~1i/attachment = 356 :lbs (joint data)

M,S, = 356/92 .,. 1,0 = High

There is also the normal loading/in of

60 LONGERON .SYSTEMS IN DIAGONAL TENSION

Load/in = , 22(. 025) (64000) = ·352 lbs/in €str = -. 001123 - • 00032.3 cotC>l (Ea,102)

Load/at.tm't = 352(.75) = 264 lbs €sk = ,000438/sin2ol. + ,000342sin2ol

Pa11/attm't = 197 lbs (Art.Dl,26, Table A)

M,S. = 197/264 -l.O = -.25

Fo;r-. the. r,equired M,s, .. of •·~5 .the·.rivet spac fng would lia"i"e to be reduce.d to about ,48 in,

'Fora 'spitee along Ring ll in th~ region of panel b

Load/in = 5ooox, o'25[1+.46(l/sin24,s" -1 B = 205 lbs/in (Eq,100)

Load/rivet = 205(,75) = 154 lbs

M.S .• = 356/154 -1,0 = 1.31

The tensiorf M,S, here is the same as for the J,ong.itud.inal ~plic..e .above (-,25) and w.Ould. also require,,the above ,48 spacing

Determination ofcX in. the Panels

Panel a1 k = ,51, fs = 4640 Re = ,60, M = 21,p.,OOO

E _ 1 [: .. ,51(4640) tan<>< a · rg - - 10,7xl,<J6 ,134/15.x.025 + .5(1-.51

. . .. . (.Eq,93) = -. 000367 tane!'. ... . . .

. ·-, ·- '' 1-'. -

· ······ i c· 216. 000(17. 32) €str =; 10. 7:x:ltfC .~76,.r

. ·.· · .• 51(46!fo) cotoC..... ·····~· , 094/7, 85x, 02·5· .. · ·. +.5~ 1-. 51) ( •.. 60

' > . (Eq.89 = -• 001,267 - , OOOJS3 'tano< ·

tsk = ig;~xi<f·~l~~g/ + (1-,51)(1 .t .3'.f)

·· .. . . .· x s in2'>f] .(Eq, 97)

= , 000451/sin;zoi, + , g,OQ288 sin2o<

tan2.o< = €sk Estr :e Esk - trg - (7.85/15) /24

= Esk - Estr 'Gsk -€rg - ,011412

(Eq,98)

After several trials it is found that., when'Dc =· 25.4° is' used in the strain · equations · the resulting value of o{ from E~.(98) is also 25,4~ hence ol = 25 ,4° for panel a

Panel bi'' k"'·= ,46, Re = • 69,

and

fs = 5,000 M= 216,000

Erg = -. 000342 tan o<

Proceeding as for panel a, o{b = 24. 5~ Panel c1 k = ,45, fs = 4,640

Re.,; ,69, M = 171,000 Proceeding as for panel .":• .. oCc = .24_.o:

These values of IX were used in th,e previous calculations of 'stresses· due to diagon":l t.ension action.

Final Note

To illustrate the analyses, only the region in the upper portion of the shell has been considered, where the stringers are more critical. The shear is higher in the panels near the neut­ral axis, so the panels, . rings .and splice attachmerit·s 'will 'be more critical there.

LONGERON SYSTEM

Cll.35a Longeron System 1n D1agona1 ~ns1on

The longeron ~tructural system is somewhat simpler fromcca 1j'otal' .,;nalysis stand-point~ • Thi•s ts pec":~.~e::t,~~re ar.~ fewer .members carrying. the· a:r1a1, load.'! and not as ~ny, Sheli,r paj'i,'el~: wi,tll,,, rary:,. irig .shear• loads,· \':ri,tis: 'type,.: of stfuct•., ure may, or may ncit;' be the optimum ar• rangement.fro'": a,weigli:t and manufactur'." ing, 'c6st' considerat1:oil" 'for' a' p'articular airplane1 that. requires an optimization study,,'' Tlie methods· Qranalygis presen­ted here would, howeTer, have a place in calculations for such a study,

.. Some typical types of longeron strtictural systeinscross:.sections·for a fuselage are sho\m in Fig;cq.44t... and m.470.:u 0

(a) (b) (c)

Fig.C1l.44a Longeron System Structures

Fig, (a) shows the minimi,un arrange­ment as to the number of longerons, since at least three axial members are needed for equilibriuih when bending mom­ents in more than one plane are present, HoweTer, this do~s not provide a "fail­safe" design,, si.hoe __ t:ailure. ot: any orie member will not leaTe a structure cap.•· able of withstanding '.·some .. arbitrary re.,. quired -percentage .(:Usuuly · 60;_67%) of the design ultimate. loads, . Tlie. system

LONGERON SYSTEMS. IN DIAGONAL TENSION 61

shown .. 1n Fig., 44b is capable of_ ·doing _this and ls _the minimum ,type acceptable from the .. fall-safe_· standpoint. (four longerons) Four t~ e 1ght,. longerons are normally used dependlJl>upon other design and manufact­uring · fac_tor_s,

__ The longeron .. system, howeTer, re ... quires more_ clO<Jely spaced rings than does the stringer system, The spacing is set after trade-oft' studies are made and 1S usually four to eight in. depending up-· on the size of the fuselage; ·The rings support the skin, diTiding it into small­er panels, T!tey als() supp~rt the longer" ons in the same manner that •uprights• support the flanges in a plane web beam, but not in a radial direction, There are no "floatilii; • rings ·1n the longeron sys­tem.

After buckil.ng" the skin has tension folds which_, .. rather than being "flat".as in tlzte .. st;:inge?" system, l1e on a, hYPerbo­loi_d of -reTolution. That is, they .. [lat­ten diagonally between the closely spaped rings (see Ref, 3 and 7), In this system d<h, see Fig,Ctl.J4,

C11.J6a Anal.ysls or LongeroD S7stem 1n D1agona.l·Tension

Tlie ~ngiMerlng procedure for calc:u­la ting the stresses and allo:wables for the longeron system is s1m1lar to that for the stringer system, The reader will note the differences.

1. First, at any bay being checked ·one· determines the primary internal.loads distribution in j;he,longerons and.shear Penel~,; due to the primary applied loads; uSing"engineering bending theory or a' fi­nite element analysis, EffectiTe skin per Eq,(83) is used with the longerons in c·ompression.

2, Next, one determine.a the shear buc'lc­ling stresses in the skin panels, Since compression stresses are often present,. pure shear buckling does not occur, Thu~ as discussed for stringers,. ·some rational interaction is used to obtain a "reduced" shear buckling stress. This can be done, for example, by using some ."aTerage • com­pression in the panel, weighted toward· the high side for conservatism. Thereby the interaction.method of Art,C11;32 can be used where ·

- 1l. + Yi.JJ.f + ii; Re = A A

2 · , as 1n Eq, (76)

and A is determined for a curved panel of length "d" between rings anl.height "h" between longerons measured along the cir­cumference, as in Fig,C11,J4, Bis the ratio fc/f s for the part;1cular loading

condition being investigated, The comp­ression stress is calculated for no skin buckled, Then, as in Eq,(75), the re­duced shear buckling stress is

F~cr = RcFscr

The effective skin areas are re_d,uced by the shear stresses, per Eq, (SJ) ·

When tension strains are present with the shear then, as in Eq, ( 77) ·

RT ~ 1,0 + ft/2Fcr and

F~cr = RT Fscr , as in Eq, (79)

), Next, the loading ratio, f9/F~cr• can be calculated using F'scr as determined. in 2 above.

4, The diagonal tension factor, k, is ob­tained. from Fig,Cll,19,

5, The total axial stress in the longeron can be written as follows

fL = fp _._ fo,T,

= fp _ kfsht coto/) 9 · +" ZAL + _(.$ht(l-k)fic)a +

kfsht coto<)h !c-(l13 ) 2AL + (,5ht(1-k)Rc)b ·

where

and

fp = primary stress froni step,, (1), (+) if tendon and. (-) if comp,

AL = Longeron Area

k, fe• cot«, Re, RT, h_, e.nd t are as preTiouslY defined, One set• subscript (a), is for the panel aboTe the longeron , and the other, (b),_ for the panel below 1t,

6; The aTerage stress in the supporting ring (or frame) d.ue to diagonal tension is ginn by the following formula (and. note that this is different from that in the stringer system)

frg = -

where (as

kfs tano< Arge + .5(1-k) dt .

ih Eq, (58 ))

- - - - ( 114)

Arge = Arg/(1 +. (e/p )2 )

This formula 1S similar to that for the effectiTe area.of a single u~right in a plane web sy:stem, Art, C~;t",1 20) where

e = distaµce from ring c,g, to the skin meanline

!'=radius of gyration of the ring

62 · LONGERON SYSTEMS IN DIAGONAL TENSION

cross-section about an axis.parallel to' the skin,.

If ,the longeron is "straight" between the frames and has no radial loads. f:fom

. ·'· •any."·ring, ·tt can,be·anal·yzed''as a"col\iinn, 'E:<lue:t1on (114) assumes, that k, fs• , I:f ·Lt has curTature• ·between frames or'

d, t, etc, are the same for the panels radl:al,,•loads from any: ring''''it' is.a'oeam-on each side of the ring, If not, then column, In either case, although it may the average value , should be . used, oth- ·. be continuous across supporting frames, e.rwise,·frg must, be wrHten as the sum· of it is ttana•ll!y conse:M'at1vely' assumed to two effect~, as, was, do.ne ,f()r the. longeron be pin ended and of lehgth'L,. ·' .... , in Eq, (112)·. ~;:.' ,.~·:.;::;~.:\ .,,. .~·-'1·-"·::; -·,-''.' ~-:)y<•.:."'''' 1

If thi> ring (or fra111e) .als~ has in• ternal loads due' t'o the primary applied ' loads, then, a. term fpshouldbe inelude.d in Eq, (114, as dlscussed for Eq. (108); This is most like,ly to ,;bt3 significant for the larger "frames• rather than· for rings

. ' . '.''-'··,

,·. , ... ;t:!ie .~x,imU111,~tr1>,~'s, ring will be ·, ,,,

f . - f cfrgMax) rgMax - rg . f.rg'

frgM!lx ,, ... in the

- -- - (115)

. where frgMa:r/frg is from Fig,Cll,21,

7, The iii~gtinal, te~s iop, ~ngl~ ,.J. , .'.for.: each panel (needed.for the various eq­uations} can be calculated 1n two ways for the longeron system, The quickest method, for preliminary sizing particu­larly, .. 1s to determim o< directly from Fig,C11•'4), The more accurate, but more tedious .•. , 1s s1m1lal" to that shown for the stringersyst!!ll!i'as.fo11ows

-; • ::.'·; \, ·r_,_,__ i '·.>·'·'-.'. ·- , . .-_ ... , ... ,

Erg= -lE .·. kfs coto1 l "'.., - (~16) Ee. Argeld.t t •5<~:kt] .

EL _ 1 {;, . kf 5 co:to< ::1 . - E;°(.1' - Ai/ht+ ,5(1-k)~ - -(~l?)

'Es~ =~N~~ 2u. + (l-k)(1+µ),~i~2~ -'(118) -· -.... .

!\'

cl.. is determined by successiTe,, trials as discussed for the stringer system, For the initial trial a Value for 0( Of JO' - , J5° is reasonable. or 'use the value from Fig,Cll,45, ,, ,

8; Strength , checks can then be made as follows· ··· Longeri:ins·

The 1ongeron acts as either a column or a· beam-column,. stipport.ed ,by the major frames ( not by the,' rings )'"which have. sore spa~ ing, · L ( wh;ch i~ in,uch, :J,a,rger than d) ,

Column" Analysis'' ",. P;r.c;~g = Pp ~, Pn:r:' ' '' . . .·.

',.'fp(AL·+ Aesk) - ITk;sht cotot)<t/2

. +- (kf 8 ht c~tol )b /~ Per = Fcr<AL + Aes\•) - '"'.' -'(121)' y ---(120 l

where fp ,;,, . primary. axl.al compression (-'-) stress, Aesk is per Eq; (BJ)• Ft;r is per Eqi(C7~J1) •or. F1g;c7;JJ;''witlj '.Fels ':" Fee per<Art,c?~JO; with L distS:iic~,,:~t,w<!.~') .. ·major f:rames and P ca1cli.lii:ted :with Ae·sk• 'Theil, .,, "::; Ci:: ,··-/' . ,~ .- V·"' - '·

:.: ,, _ M,Si • 'Pcr!Ppong - ,7;·0', .,,.,,

Beam-Column Ap,alysi,s

If the longeron is a be.am-column as described' above, •local faf'lure' ~tis t be checked:for ,:tlie cri:t1ce.1 c'ompres'sfifo flange '•stres's' which iS'' · · · · , ,--,,'

•· 1:;:;~·,~r-~c:t. · ···~·''' - : ·::·

f = fc + fn,'T','''+ 'M6'/r · ":.;- '-(f22')

where fc;::is:th,e•primafY compressfon (-) stre .. s;and':M'·fa,.the bending moment ·due' to beain,-:column.:action· 'including Aesk• . M' .)(il!li cause:ntenS'iori.'{+): in one fJ.ange ·and ·compression•··(- h in: the other; Then·· . ·

·,"1'1 '' ',_

M,s, = 1/(•Rc + 'RD,T. + Rben~l '"'. Lo

- 1,0

Sk.in • : ...

L"-"-"-...,,_." / b_ .(L.o"S'eron)

F1g.C11.47a P'lll.17 Supported. Longeron (b)

LONGERON SYSTEMS IN DIAGONAL TENSION

M.S. = FrglfrgMa.:t· - 1.0

Where Frg is per Fig. Cll. JS

If the rings have stresses, fp• d11i> to any primary loads then

M.s. s 1/(fp/Fcc + frgMa.x/Frg) - 1.0

§.!tlM

Strength check for skin rupture

M.S. • F8 /fs -1.0

where Fs is per Fig.C11,42

Permanent Buckling

Attachments

The margins of safety for the at~ tachments are calculated in the.same man­ner as discussed tor the stringer system,

.C11,J?s. Example Problem

Determine the following margins of safety tor the ·structure in F1g,C11.4Ba and its ultimate applied loads,

a) Skin rupture for panels a and b

b) Skin permanent buckling

c) Interr1vet buckling .along longeron 1

d) General instability based pn panel b

el Longeron 1

Usually no· permanent buckling is as- f) Ring at A between longerons 1 and 2 lowed at limi.t load.

M.S. = FsP.B./fs - 1.0

where Fsp, B, is pe.r Fig •. C11,46

General Instability

M,S, "Fsrnst/fs - 1,0

where Fsinst is per Fig,C11,J9, Note that no errective skin is used and that f'Long is limited to ,)4 at most, A M.S. ot at least ,15 is recommended,

Freme

:l

.Jr11 fT .o _j~ . "[$'

g) Fasteners along longeron 1 assuming skin is spliced there

h) Fasteners along ring A, assuming skin is spliced there

t) Same as .(g) assuming no splice there, .,£

As for Example Problem 1, the primary loads are determined f!lr, both, the con­dition of no skin buckling and then for the. skin buckled, The following table presents those internal loads and geome­.try needed for the above analyses,

I I I 0 I' I I I t 1 t , 1 I I

i-1-11-11-1 ' I

SJtiil and H.in?5 Lon1ercJts

·1 V;J,600/b<

M:Z"3,oi:5'£> io·/lu

.il02't •7'3 Sii<d: 7a7S-773 E:<tJ-P{:'On ..e. = ,10.S"J< '"'" c ::: /0·'"''0 Cc= /0.71' 10" ~ .. ,10.~)l./4(,. r'rv .. 6~()()0 Ew·· ts:cao F~y=.3'!,,000 Fc.,.::70.,aa"

• .15r.7S"JllJ,,5'°0l.o":f~l?Jll .]S'.,.,75"' /~]:1'£/11.;r· AL=.Zz.88/,,i. ;A/l&=."78¥

/l.i,n:d~ 4r~ 1/a~ VAirer.s4/ /,eqd M.J'ZC'f70 AO.:-¥ at 175'"' .sp4cin9

I,. •. o7tio ,;,• .l~r,..= .o,72.

Fig.C11.4Ba Shell with Longeron Construction for Example Problem

64 ·LONGERON SYSTEMS IN DIAGONAL'TENSION

Table cil.J Internal, toads- ~nd,_Ge·~·metry .

COn,_d.itio·_n_ .<,la __ 5 , .. qb .. q_e,O: a. :. ~a,_ Yb. J;c · ~ 0·r· Skin:: lb/1n· lb/ln lb/tn 1n ··1n·· in in

No Buok11ng 212 26) 26J 0 14.)9 9.09 ·9.0

Buckled 212 264 264 J.JO 17 .• 70 l2.J9 12. ·'' • ' . ,).''' • ;; \', !, ·' -~ L· •'. , . , , "' . "·' ,, '·'"

'-'';;.;·:· ~,· ~-·_,-'!':;.ab::1::.• _:C::,11;.:·~J;;_;·· ';;.;·::,;< C::.o•::;t::l~nu::;•:;.d·~Lt;.,;!. ·;:.___:·:;.;_;· <';_:,·';·---4." .. ,.···~ Condition YLong:( 1torig'2.: <l<Ma, b 2

''"·'. Mc-'· Ia, 1:5 of' Skin in in 1n_;lba in-lbs in"

No Buekl.lng · 'i2."JJ o · 32.f;Ooo' 310/5(>0 4()2

.Buckled 15.6) . J.Jo J24,ooo .Jio.,50.0 261

Notes• foe.me e.s for Table C11.2) . ' ··.r ... ~~'-: . ' - ' . '

Skin Panel Analyses

Permanent Buckling (At Limit Lead)

fsLimit = fslJ1t:llo5;. 8,480/1,5 = 5,653

4Fs xlO /EcFcy = 4x5JJ4x108

/10.1x10' x er 39,000 = 5,11

Fsp, B/Fscr ,. 1. 3 . q•!g i c1i. 46).

F = 1.~(5,334) • 6,934 s_p, B.

M,S, a 6,934/5,653 - 1,0 = ..._gi

Interrivet Buckling, Along Longeron 1,,;,/b

Fir ='11 2 xlO, 7x106/! i75h 58i, 024 )~ •''39«473 (Eq,C7,22)

fc = fcsk + l"n,T = Jl7,-250·(16,29)/261 + 10,445 {see "Lcngeron") = 30,246

·.. . ''· . . '- .

M,S, • J9,47J/J0,246,- 1,0 ".:l!.

The analysis is illustrated only for General Ins:e.b1l~ :tr,, J:le.s~!J. µpo:n. Panel 1> panel e., but results of a .similar analy~ sis for panel b are presented later on.

Pe.riei a

Z = b-'(1,.>f'J' .. /Rt = 4,5 (l-',JJ'•i/15(,025) = e./b = 15,71/4,5 = 3,491 ks = 17•51 511 Fscr = 17,51T"x10,£xl0"(,025/4,5)2./

. , 12(1-,JJ ,) _a. 5,334psi (Fig,C9,5) kc = 19,5dFig,C9,1 )i

6 a

Fccr = 19,5rrx 10,7x10 (;025/4,5) I· ,.. i2U~;333J =·5•944ips1 .,,

F~;, t~~ no~..:fuic~l~~ skin ~'~ndit~~n (to obi;e.i?) Be.•• k,e.nd. Be),,,,,· ·. ,,,

fc = Me.Ya/fa m 317 ,250 (14,J9)/4o2 = 0

11,356 ~ ·:

fs = cte./t =212/,e25"' 8,480

~: ~~7¥~f=0ri)5gi§t~o = l.~J9 B/A = 1;20 ····

Re ;. (-'1, 2p. + "V:i.20" + 4 )//=: , 57 R'c = .5?(1.20) = .68

Proceeding per Fig, c11. J9 iiotes and. data1 ;{"' f; x 10" = • 3{{,~:z:; 6894?x10": -(dh ;,,n (4,5x,15,71y/iJt543~B:3

Fsrnstxl03

/Ee > 3

Hence, Fs:i:nst >3xl0,7x10• /103 > j'2;ooo psi

fs = 10,560 .,

·· M~s; = 32,qoo/io56o .,:l..o =''ai!i;h '

~- (Between :i'S'ne1s b and cl

Arg = ,07841 Ilg = ,OJ721 Prg = ,689 e = ,875(,0125 = ,888.

Ar~~ = ,0784/(1+(;8S~/;689'f)=,0295 (Eq.~

r .s57<1os60) te.n36( os0

8 801 ,rg = ,0295 4,jx,025 + ,5 1-,555l(iq,il4)

frgMe.:z:/frg = 1,30 (Fig,CU.21 for ke.v = ,555)

F1" ·' .. ,, "°F •"' °*O Ser "'"C SCI\_ J' .

, ,~,rgMe.x = 1,30(8801) = 11,441 . , .

For the buckled skin condition,

fs a C[e,/t = 212/,025 = 8,480

fs/F'scr = 8,480/3,040 = 2,79

JOOth/Rd = JOO(, 025) ( 2, 0 )/15 = 1 .• 0 k = .59 (Fig,Cl.1,19)

Skin Runture

,.f'it'"' B!;.80r .Fs =' 21,100 (F-l:g,Cll,42)

'M~S. • 21,100/8,480 - 1,0

' = 1.49 ·-

M,S, = 11,743/11,441 - 1,0 = ..J1l

Lcngeron 1

The longeron is suppoz:ted by the ma­jor frames, J6 in ape.rt, This is the onr ly supnort in,the re.dial direction, Per

· EC! , ( 113) and de. te. for pane ls e. and b,

rp - .rn,T.

-317250 .(15, 6J) "·'"''"=' 261 ' ''

'j;'

r.:t9(8480)(15 71) -l! .2~88) +,5(15,71)

1 (. 025)

LONGERON SYSTEMS IN DIAGONAL TENSION

( .025) cotJ6; 10J . • . (1-.56)(,7}) :i= - 1(·~p9)9, " 10445

• (fo,T.)

= -29 ,4.44 psi

. . . If ·Longeron 1 were supported in two directions (which .it isn't). l.ike the one (b) ln Fig, 11,47a, then Eq·, (123) would apply with fp = 371,250(15,63)/261 • 22, 2J2 (al'. the .. lef1; end); Then

1

Buckling Check

This uses Eq;(120) and (121), For a longeron the effective sk1n:· width is (Eq,1, p,A19.t2)

2w = 1,9tVE0 /f0

= 1,9x.025-V10,5x10 /2944* = so

Aesk = ,89(,025) = ,0223 1n2 .

and using Fig,c7,26,

,89 in

For the Longeron Fee = 62, 690 (Art, C7, 3q,

So, Fer= 62690 - 6269t" (61)2

/4'!fx10,5 x = 27412 10'

Since this is< F0 c/2, use

Fer = 'f1"~10,5x10'/612 = 27850

Per= 27850(,2288+,02/l.J) =6,993 lbs (Eq,1

PLong •-1S999 (, 2288~, 02'f J) - G_°59 ( 8480

(15.71)(,025)(1.3137)/2 +

,56(10560)(15.71)(.025)(1,371)/2].

=-7736 lbs (eomp•) So

M, S, • 6993/77 J 6 -1. 0 = -, 10

So, the longeron would need to be made larger (more .thickness).

Beam-Column Check

The Example Problem does not have any beam-column loadings, If it did, the M,S, would be much less than the -,10 for buckling (beam-column margins are always less than. for the. column). To find· the· M,S, for this particular case (where

. P >Perl one. must find the common factor, A, by which all of the shell applied lo.ads must be multiplied to give a M,S,

of zero (and A wi.11 be <1,0), The M,S, will then be'.:A-1, o (Art·,c!J.18),

However, to illustrate the' general procedure assuine that only the'' applied bending moments on the shell are reduced JO% to .be 184,275 and 259,875;· and that the data in Table Cli.J'st1ll ap'!'l1es (some of it will change slightly), Then, proceeding as before

fL: -245025(15.63)/261 - 104455

= -1467J - 10445 =,25118

2w = 1. 9 (. 025l-V10. 5x:l.o' /25118 = , 97 in

Aesk ".' • 97 (. 925) = • 024 J sq ii;

PLong = -14673(,2288+,024J) + 2965=6679

and Buckling M,S, = 7049/6679-1,0 = ,o6

Fo~ the beam-column check assume that the longeron has a slightly curved installed shape, as in Fig,CJ,JJc with e = ,10 in, Then, per the "approximate formula"

Per Fig,C10,15b formula lh ,~• ;nl

Mult "'!:5000(,75x1.51 /(,75/,08),~ •ID .

(70000/10,5x10') /,0604 = 10717

Then, per Method 1 of Art,C3,18,

M,S, = QQ22_+ 12224 - 1,0 =

~ 10717

-,53

Using Method 2, Art,c3,18

K = 1.19 (Fig,CJ,8)

F0 = 50000*+ 1 19- 1 1 (70000 - 5ooocrt) 1,50-1,1 = 54500

. 66 f = P/A +Mc/I=~+ 1427J(,75)

• 2531 • 0765 = 1663iO psi (comp)

So Ms = 54500 - 1,0 = -, 67 • • 166320

Per Art,CJ.18, the "better" M,S, of Method 1,-.53, would be used, Thus the longeron must be made larger (more thickness .. ) , Note the large negative margin caused by beam-column action, whereas the buckling margin was a .. small ,12..ositive one.

·.- ~1• 1• ti. Pr'OJl• 11a1t. •tn••I ••• ,f)age J8 tootDo.te.

66 riJ::A.aoN'AL'TJ;;!;srcrn suMMARY REMiRKs Shear Panel,b Cal~ulated i.lafa'. 8486 l}(,59') . ·· • · Al

E'sk = l0,7xlO'~in20\ + (1:-.59)(1+,33)s1n"".'.j The, follol!'ing data .for panel b were

· calculatil'i:t. as was· 111ustrate·d for panel a = , 000935/sin2o< + , 6oci431sin2o(

tan• ti, = E sk E'sk · [. ···· . ··

- ~rg + (4,~(15)2 t,;.n2,IA/6 ·"·

frg - , 011250 tan2 Q(.

Attachments ' . After seve.ral. successive tirials it is found tha·t \rhen d. • J5.lf is used in the

The allowable load/:i-ivet is the same three strain equations, the ·resulting va­as for the stringer system calculations, lue of o< per the last equation is 35, 8~

hence p( = 35, 8° a) For an assumed splice along Longeron 1 between panels a and b, the. largest 'shi!a f.low. ~s }63 U1s/).n

Load/in~ 264(~j-.56(1/cos36,1•'."1l)= 299 · Load/rivet • 299 (. 7 5 ) = 224 . ·· · "

}!:,S, ,. 356/224 -'1,0 = ,5<}

Iiu<ir to 'ten1,1,ion fl.~~d,, prying .. ~cti.On ... ·P· ·, ,-_ · ,.,,_ . ! • ) ., , • o ,.~ -·-,, - , · - •

Load/in• ,15(,()25)(64000) • 240 Load/rivet • 240(,75Y= 180

M,S, ioi 191/180 -1,0 = ...Q2

b) Along Ring, ·assuming panels b and c are spliced, there

Lo~d/in =·264(1+,.56(1/sin36"~0 -1 l) • )67 Load/rivet • j67(,75) • 275 ·. · 1

M,s, • ~97/275 - 1,0 • -.28 ,_-· ':,.:

cl Along Bing . for no J~if~e th:e'rl!l u no shear transfer, only a prying load

Load/in• ,22(,025)(64000) • 352 Load/rivet = 352(,75) • 264 . ·•

. · .... · M.S, •< 26.4/352 .. ~ 1, 0 • ::....il. Hence, for (b)' and. (c)' a smaller rivet spacing is need.ed., about ;60 in;

Diagonal.Tension .b,g1e,

Panel a:

Erg • 1 . r .· -.59(8480) tan~ l 10.7x~Q& h"0295/45:z:.025 + .5(1-.59}j

• -. 0()1001 tan oc

Panel b1

.Proceeding as ·for panel a, the fol.­lowing d.ata are obtained

k=,561 fs=105601 llc=.73 which result in

Erg = -, 001134 t1m Q(. . .. . • ,. '°L • -, 001809- ·, 000758 cote< !: sk • , 001105/s1n2ol. +. , 000578 s1n2o(

~. , ..

Proceed,,1ng as for panel a, o{b =· 36.1°,

Panel c 1

k=,551 fs=.105601 Rc•.7441. ~310,500 » ' ' '·" •":''.-'',,:<- ')'·' ,'' ' .

. ,-. 0

Proceeding as for panel a, o<c • 36, 0 •

Cll. :J8a .summary

The effect Of dtagc:inaltension is to generate additional c9mpressiv11 stresses in' the stringer•N iongerons and. rings or.· frames, which if not a.ccounted for·w111 cause early failure;" The effect

.of compression stresses along with the shear stresseif ·increases these ·effects, while tension stresses will reduce them,

' - ' ' ,\,-' ": -, ' .

The exact magnitude of th.e Tar1ous diagonal tension effect.a ·throughout a network of skin panels d.11fies simple 11v­aluat.1on, par.ticularly when both shear . and. a:i:1al stresses var:r. from panel to pa­nel. However,.·. some rational approach is nec11ssary to .complete the design, or spe­cimens for test programs, and the ap­proaches given in this ehapter represent one such procedure, Where margins are small, element tests. ·are •in· order, .

.. ·. i · 1-317250(15,61) ,59(8480) · The stringer system is usually found, ,. .. 10,5x10• [ 261 ,2288 /· ·for example, in fuselage and wing struct-

·~•-- ,,. .. · ...... " ures··where .. ·there .. are'·re·:lat,1ve17 .. f11w ·large ... Coto< · .·. ~ •cutouts .. • .. , to disrupt the stringer cont1nU-

15, 71x, 025 +:sc1-.;;9)C.56) tty, .... This. ts more 11kel:r· for transport, bombe.r .. and.;other ·cargo carrying aircraft,

= -. 00181 - , 000683 cot o< The· longeron•. system' is more effic1en't and • .ln4 ah~ ·1nareue4 •tin. panel and. rutener ~-u

BUCKLING OF CIRCULAR CYLINDERS 67

suitable where a large number of "quick access" panels and doors o·r other "cut­outs• are needed· to service various sys­tems rapidly, These would "chop up" a stringer system very severely, making it inefficient and also more expensive from both a weight and manufacturing stand­point,. Therefore, longeron systems are usually found 1n fighter and atta.ck type aircraft•: and in others having such fea­tures.· There are, of cou.r::;_e, other f_~c­.tors influencing the choic·e of structural arrangem0rft. ·

Some.further notes concerning this general subject are inc.luded in Chapter DJ, These include "beef,,-up" of panels and axial members bordering cut-outs and non-structural doors, which is also re­lated to "end bay" effects discussed in Art.c11,29,

Although tension field analysis has been discussed only for primary structure it can also significantly effect secon­dary structure such as' flaps, ailerons and other control surfaces. For. example, such surfaces usually have a. relatively light trailing edge, Significant buck­ling of their skin can.generate addi.,. tional compressive'loads in the trailing edge promoting buckling or local failure, A normal loading which bends the edge member inwards and generates additional loads in the supporting ribs is also present, Hence attention is needed here,

Cll.)9• Probl•SI tor Part 2

1, Recalculate the margins of safety for the stringer example problem assuming that the.torsion on the shell is in­creased from 150,000 to 300,000 in-lb For the stringer analyses use only the "Quick" and Melcone-Ensrud methods,

2, Recalculate the margins of safety for the longeron example problem assuming that the torsion on the shell is in­creased from J00,000 to 450,000 in-lb Use .only Fig, Cll,45 for determining o(,

C11.40a ~blems tar Part 1 .

The problems for Part 1 remain the same as in the textbook·. (p,Cll,48),

The.reader is encouraged to consult the references (p,Cll,49), particularly (3) ,. (4) and ('9) for a broader under­standing of the subject,

CS.la Introduct1ois

NASA CR-912, "Shell Analysis Manual presents a large amount of data and dis­cussions for the buckling of monocoque

shells, Some of the more helpful data are presented in the following articles and figures, The reader is encouraged to consult CR-912 for a broader understand­ing or· .the subject,

cB.2& Buckling or Monocoque Cylinders Und.er A.:1:1&1 Compression

a'0 r/'I\ is calculated per Fig, CS.Baa, Then Fer is found by entering Fig,CS,9 with (a' er/ti.) /Fo, 7, obtaining Fc/Fo, 7 and then calculating Fccr E Fo,7(Fc/Fo,7), The shell is then capable of withstanding an axial ·1oad of

Per = adRFccrt

ce.-sa Bu.ckling ot c1:rcU.lar C711nders · Under Ax1a1 Loacl aDd. ·Internal :Press1,ll'O

When there is an internal (burst) pressure, p, along:w;1;11 an axial compres­sive load, the internal pressure increas­es Cc by an amountACe per Fig,C8,Ua ·whieh increases dc:r:fri• Fccr is then tained as in Art,CH,a. above •. Pc~ becomes

Per = 2fTRFc0 rt +'If R1 p

C8.?a 4Y&11able Design CUrTe• tor Bending Based. on !xper1aental Besults

dcrhi is ealculated per Fig,C8,1Jaa, Fb is then found by entering Fig·, CS, 9 wi%fi (dcr/'tl)/Fo,7, obtaining Fc/Fo,7 and then ca1culating FbcI .= Fo .• 7 ( f.·.c/.Fo. 7 l. The shell can then w thstand a bending moment of

· · Mer '= 'ITJfFbcrt

ca.Sa Buckling Strength ot clrcul.a:' Cylinders ln Bending with Internal Pressure

When there is an internal (burst) pressure, p, along with bending stresses the internal pressure· increases Cb by an amount ACb per Fig,C8,14a, which increas­es a'crhi. Fbcr is the.n obtai.ried as in Art. cs. 7a abov.e.. The shell is theref_ore capable of withstanding a bending moment of 2

Mer • 'fl'H Fbcr t + '!l'pR-1 /2 ·

c8.9a ltxternal H7d.roatat1c Preasure·

When there is " hydrostatic (col.­lapse) pressure, p·; o'crh\ 1s calculated per Fig,C8,15a (the lateral plus axial pressure curve), Fccr is then found by. entering Fig,q8,!1 with (o'crl'tll/Fo,7, ob-· taining Fc/Fo,7 and then calculating Fccr • Fo, i(Fc/Fo, 7), The axial stress is one-half of the circumferential stresa Fccr is in the .circumferential directfcin so Per • Fccrt/R

I

68 .... · BUCKLING' OF CIRCULAR CYLINDERS

• · .

•

' 'J'•

Fig, C8, 15a

When th~;re is a: r.idfal Ciat~;al) collaps7: J:i.r'!s~11re., p,' onl:)i, .. o"crl"•! i.s .cal­culated per Fig, c8, 15a using the, upper· curve. Fc0r is the!l: .oqi;ained,. in the same manner as ln Art,C8,9a above, Fccr is in the circumferential direction, so . ,- ,, ,,_ -

.. ·:·•;•· .·-·t!;;: i' ,; ·:~_,-,,--·;. :·· •. ·.: ;'L' £,:,:,~. .: -··.: .:. t,·'·. C~~.~:~-~ ... _.-Bllck;_~ng _of.. q1rcular C711n'1ers -

' - •• ·-· 0 Under 'Pure -Tors10n: · ,.

·· .... · '•For' th,f~,,pa'.k!!.,'.10:,,/~ i~,6a1C:µ1ated, P~i:: F.1. g •..... cs .•.. 2 .. o ..... ~ .... ···.·.F·s·.··0.• :t' .. '.'1.s. fo. \irid by. enter• i~· ~~g'. C8, 19a ~i ~h \Tcr/??Yt;;,~· obtaining Fs/.Fo · 7 and then calculatfrig Fscr = Fo,7\fi'scr(Fo,7). The allowable torsion is then' · · ..

Tcr = Fscr2'!R t

cS.13.&;., Suokl.ing .or Circu,lar C71.1n~~a Under ' '. ~ .. .'~.ors ion W-1 th Interna;. Pressure

l .,t,., 0 •

· When there is an internal (burst) p:i;esor11re·~ 'p( a?-e>ng wi-t;h the torsion Cs is increas~d by an amount ACs pe:r ·" · ·" Fi~.~ CS~ 20])''··. Th.is increases 'l"crli} and Fscr .. is ob);a,ined,, in th". same JDaJ?-ner. as in Art.C8,11a,~b0ve,, as is Per•

,,.,-= :::1 ' I . .. . I' .. 7' • ~ ... _.: ·:; : .. - :~~-~ ;~· .:'.- t; ~-~i- -~

.; ;., :l.;-~·: .. u:. ,I I •

11 "!: ,, , I

-'',

For the case of transverse shear, V, one proceeds as follows, Determine Fscr :for·1torsion1 as.1n"Art,C8,11a 'or·C8 013a .. and multiply. it by,,t.-25 to determine Fscr for shear;:·: Since ·fa = ·vQ/It orie c'ari solve this.for v,«:ror Fscr in·place Of f s, For a cylinder this gives

,,_ .. '-.,., ,-,,,,, __ • --,\;, ,_f .,,.

Ver = Fscr'ifRt

CS.15·a· BUok1~- Or· ~1ro~la:i- ·-c,.i_t~de~s U~er .· · Combined'· Load 97.steu · · ,. · ·'

The . .t'o110..:1n:g .:interaction curve ap­plies for the general case, and can be used to determine the margln Of' Safety· by successive trials (Art,Cl,1Ja),

•" '.•!,',' .:~.. ' ,I ; ,-, ''. ;, , ,· _; ('. ~/

Re+ Hp+ Rsf +•:(a;+ R~) 3 = i.o

Tlul subsc.:ripts c·• p,. st, s, and b refer to .compressio;n, collaps_e pressu~e:, tor­sion, ·transverse. shear and .bendfng re­spectively, Including Rp to only the first power is probably conservative, bl,lt· collapse pressure is "seldom· present with the others.

.. Fo,r certain cases, where some of the loadings a:re .absen.t.; the margin of safety

BUCKLING OF CIRCULAR CYLINDERS 0.45 . .. . . . .. . .... -·· ..... .

. . . . . . . . . . . . . . . . . . l :: : : : :: : :: : ; : .. : : . : : .. : : . : . . . . . . •·· ......... ~-. . . . . . .. .

0.40 ::~: :::.; ;~:; :::: :::: :::: :::: ....

O.Jf

O.JI

. . . . .. ... ·-.. . . . . . .....

........................

. : : : :1: .. : : : : : : : : : : : ; . : . . . . . .

::: "" : <: "i'' : ::: .... ~~ :::: c:: :: !'': .,::

o'er/~ = CcEt/R : :: I ...

z = r! V1 -)-1-2 /Rt -- ..... . . . . -... .

t -- t--+--11--t

.. --· -·~-+--•

:I ·: .

. ' ' ... : .. l.-" a.JS' t--t--+-+--+-+--+-~ Valid for

'Z '> 25 for z· >so for

---­' . .... . .

Cc r.:~~·cr-··~·;·-··-t-f--t-'--~·~·

0.20 ~-='. ~:: :-'.:-~ .. ~·: .. ~ ·! ::· . ·· .. :

Simply Supported Edges_·+--+--< Clamped edges ..

· .. ·o:.~ .. 1 .. :: ·:.::,. !::.·• : j :: :::: ::- :J our.·:..,·~· f"'-+~'-"'.L--LL1--~-1f-'-'+-1.....:.-"'-'-'+'-'-. ·+·;.o·;..o··;;·;;.··;.o·f-·'-··+-'--1f--,.:....._: '.:~ .:.....:..:.

. ~::· . .. '1. .:.1 : : .. ... ··:. :. :.::.: ... ,: '::· i':: .:.:.: :.:.: .. :.::.:::.:.:.~. ~:~ .. .:::J~ ; ~:: : .. :: . : t ,, .,;.: .. . .... .

0.10 .. .. . . .·; i' : r:::: .:.. ... ··1· ~~-11--l-~-·-t--+--'-'""-'-+

' • I • • • .. ··~.;_ .... ~

. . . . . . . . . . .. ~:: : : : : I: I I : . i . ·' 0.0df.-"Cr·~fc:--+'-::-:!"-+-:-+--""--+--"-1--..l-'"-l--~~~·~··~·~··:..,·.i:-:··~ .. +·~·~"+'~·~··+·:..,··'-+':..,'~·~·:..,·~·~'!:l'.l

. .. . . . .. :. ::1 ....... :.::: ~· :.;.;: i.;.: ............. :: :::: ·.l:. .:·::-:.:· ....... :;:. :;: ·: .::: ::1: :::.: :::~ :1:: ::I :j:; .. .. : : : ! !: . "•,: :1: . :; ·1:; ;:i.: ;:: ::;; :;;; ;~ :fi- ;;

. . : .i. • . . :; I . u

2000 .... . 4000

R/t

Fig.CS.Saa. Buckling Stress Under Axial Load

10 . ~ --a

• - ·- . .. • l

1.0 B

• •

.<l.Cc l

~ ...... _,

0.10 B

• . ...

• / .

, l

0.01 l 4 6 a J 469 .z 468 l • • •

0,DI 0.10 1.o(p/E)(R/t)2 10

Fig,CS.tia Increase in Cc Due to Burst Pressure, p

69

70 BUCKLING.OF CIRCULAR CYLINDERS

····:-!;.·

... t .:..:.:.: ::.::

····--· ., ... ·:::-:

:·:: ::: \ :'..: ·· !::: ;:.; ;:::: .. Valid f.or . · o.J4 :::: :~:: :~, .. ::. ·~ ·· :.:: :.": ::.: ~: Z>10'0. for Simply, su·pported. Edges

Cs ~j i'.;'. :;,, ::. ;: ;:, ::;: ,;;: =::: z >too roi clamped Edges 1

.,, •••••

o.nj :. :i; :::. ~''4 j~; : : ::j.: ''.: ·::: ::: .... :;,: ::;: :;,: F': :::· '::: :-.: : : ·-··.. .. ·.;, ........ ·• .. ... .. .. . .. ,.. .... .. . .. . .. . .... . i ' 1:::: :,;: ::;: ::: \. :::: ..

o.so ... ;;; .... .... .... ~ .............

. . . .. ::·:: .. :· ::: ..

O.tl. : : "1 . :\ ::J'X i:·; -:: ::': :::: :::: .:~: :!:: "': ;;, .. :.;

"

''I'',,, .. ,-. •· .•... ;· '' '''' ••• :.:;il::~ :1•.: ;:.:: -1::· ::•: . '.'····:.·i ·: I·:'' ·::. ::·: :::: .... i··· "" -··· ....... . F±''~·'+· ·:.::··""·r·4·· ':"''""'···+c'::":·:..i· -····-31.: " •· ••· ••........••• :: .......... . •• ' ••• .. • •• • "·"-· 1• .:!· ::, :.::: ;::: .:r: 1:1• i:::'!!:: ::,: '.·. :.·.· 1;·,

"0.46 ":::: .. . ... ··I ~ .. ::. ;:;: ;.:. ;:~; :.: ! .. ::: ··:·: .. . i:: ;:: : : :1; ... : .. ; I . ""......_ : .: ': .: : ••. : I ' 11 ·; :•:.: •:I.: ... . ·- ~ .. :.~::,:_._; .. .'.:· ;.!•· ·,-,,-, '·•-!' , ......... , '""·T .......... , ... , •··f" ~. '' ·l•·1r:-1····J~·!J ' l~il• . ••

Ii::::::: '·" 'i' :::: :; ; ,:: ::. :: ::;, .,.: :;.: ::: ..... , . ;~· #;;' ·=~· o.4-~ .. ;:·! :.:·-;,. 1>:-:. : .. ·: ·:,1•. ·::·_:· '-'·--:: -: -: : : .-.::- .... ·,'-····~ti .... .-: U,:' : !H· II·

0 ,... - R/t - 4QDQ

Fig,C8,20a J;uckling,Due to Torsion

" • .. ,,

1: ''.' ' ~ ,I' I

/ "-'"

'

w·· 466 2 468 4 6 B

1.o(p/E) (R/t):z 10 1o2

Fig .. C8, 20b Incre.?<se in. C8 . Du.e to Bllrst Pre.ssure, p

BUCKLING OF CIRCULAR CYLINDERS

!-=J S: :..:i! :::: :::: !!::· ::: . .. _ .... --·· ···- ............. , p ::..:.J. :::·: =~:- :t:~ ::;~: ::-::

..... •• - -•·Ii'-·• ,, .,

_ _;_

~: ! : • - . . ; :.:...: 1.:..:..:.:. . --- . . . . . .. ... . !;l:X ==~=::::: :;:~ :::: ;::! O.lS ltF.,~, :.E'""~' ~:::~_;F-!~.~--f~-~:~i\:: :~;::~: ;::;J:;

....... ---

Q.)O

, ................. , ........ . . . .... .... -- --.. ... -···

IOOI

cfcr/Y/ = CbEt/R

Z = L2 J/1 - -'f /Rt

JOOD R/t

. . . . . . . . . ..... __ _

•ooo

Fig, C8, l:Ja Buckling Stress Under Bending Moment ';-;.

To

1.0

a 6

• l

a

1

.

M ...

@~~ t b-L~

,/ /

--·- --

- - ·--· No External A.rial Load

(P = 0) -

- .

- -

-·--· -- .

- -E xternal Axial Load Balances

0.10 a 6

• 1

0.01

0.01

-Longitudinal Pressure (P = pnR "-)

·-

... . 4 6 8 l ,,6 2 448

0.10 Lo (pfE)(R/t)z 10

.

Load

-• 6 0

101

Fig,C8,14a Increase in Cb Due to Burst Pressure, p

71

72 BUCKLING :OF CONICAL SHELLS·

can be,_ calculated directly· by formui~.

a) When Rs =O

M. S', = Rc+Rp+Ro+ VR~+Rp+~~ )2 ~4Rst -1. O

b) Wheri Rb : 0 . '

M, 3

;' = Rc+Rp~Rs+ Y<Rc+Rp+Rs 'f +4Rst -1. 0

c) When Rst . ., 0 = .Re =

. . I ' 3 ,. •J• M,s,, - 1 {Rs + Rb) ·: .. , " .

c8.19a A:dd.ltional. Deslgr\" Bu6.k11ng CUn'~:S tor -ror ~~?-W~~~~ ,Conl.ca;.-.~-9-~~~~ .. ,; ; j ' : ' : ! ': : .. - .; ; ·: ~ : ; : . ' . ' - . ' .

~r/'1 1:s ~alc;ul11-ted per Fig,C8,;;?5a, Fccr t.s 1<hen obtaineq. as for a cylinder in Art;qB, J~l F6~r is at the small end. of the:· co,ni~al ,sl:lel.l'• and ·

PCr: = FCC~2'11" Hit cos 0c· p

'.; .. ;(,,'.

pt o'er/"! = CcEt/Re Re = Rt/cos o<.

z = rf-1/1 :.:_,f21Rat \:·;

1._p. ·.T: ....

f P Flg,C8,25b

r:fcrq ;=. (Cc + 6Ce ) Et/Re

Tq obtain 1cc enter l'ig,c:B;.11ai w:ith Re/t

/'

Ihciea:~Ei'' 1.n '6c with Burst,Pre~sure, p

'Re.= Rl/coso{

'dcrl>/ · = cbEt/Re

z = r?-1/1 -)I.a. /Rat

Val1:d •.for ... •. , ·. . .... z > 26 for 31.mpiy Supported Edges z :> 80 for Clamped. Edges . o( < 60°

·. '. 'l'o1obJ:e.in ,cb enter. Fig •. CiB.13e. with Re/t

F1g,C8,26a Buckling Stress Under ' , . aehbng Moment

wneri ai'i int'ernar (bu~st) pressure, p 'is prf!>se.nt:· along with the bending moment, :ft 1iicrease.s Cb'.bf an amount ~Cb per Fig,

· :cs. 26b, Tl)is i11cz:~ase!I ifcr/'r;, and Fbcr Valid for ii .,. .. ,,·.:. , ,_,,,/>., •. is. tl:l.en .,qbtained.; ··as before, by entering z >25 for Simply Supp9rted Ed.g_es• Fig,CB.9 with 0(d"cr/>i)/Fo;•7 and obtaining Z :>Bo for Clamped. Ed.ge~r• : . · " FclFo, 7 , ... FbQr is then calculated. as: o<...::: 75° . . . .. · •i;' j : , . Fo, 7 (Fc/Fo, 7 ~ •. an4 ·

,i ': - - .: '-· - • ' ·- ' '"; .::. _,\: •. :. "' ·_ ""._ " ._'

To obtain ,Cc. ~p:~e.f .. F1g:;¢s.~'.~~)iff;h 'llelt · ".fc£-~ ·J%g~ 1i'Rf~ c~sci. + p11 Rt/2

Fig, CB, 25a1 OOd~).l.ili; .Str~s!i ,l;lnder . ''i: i. , ···. ··: F·or cdnfoal shells subject to tor-. .u1a.1 Load . '.; · · •·si,on·:'l'erl?/ is ciilciilated peJ:' Fig, cs. 27a · · · · ·•·· ..... ·. 'arid>F~ci-· ~s. ,then .. obtained. by entering

When S,,rnt~rnal (burst) pressure, p, FJ,g,Q!i·19 ~l,th.J"fcr/'17)/Fo,7 and. getting is present along with the axial compres- Fs/l'o 7 •. Fs'_Qr is then calculated. as sive stress; it incre_a1>es Cc by !in amount Fo, 7(Fs/Fo,';lI1 and. ACc per Fig,C8,25b, which 1.ncreasessOCrfri; ·• .,.-.; : • · •· · F00 ?" is thi:n obtained. as ab9ye, The . . , . 'i''l'cr = Fscr~11' Rf t critical axial 19ad.1s·t~e:n __ ~ .. :.i. ':.' :,,: w~~n·-1~"-::i~it,e~~a.·i··c~urst) pressure, p

Per·= F,ccr2'l'!'Rlt. cosO{ +· p11Rf is also. present with torsion stresses the pr'e~isure 111cre8.ses Fccr• However there

For conical, sheiis subject, to bend- is no· -data for calculating the increase, ing o"crl'l i.s caicuiated per Fig, CB, 26a, and Fbcr f'ir ehen• obtained a,_i( in Art, CB, Ja Fbcr is at the small, end. of the shell, The bending, stress is fb = Mc/I, so. ·

Mer = Fi.cr11' aft cos;,(.

Fdr· c'cinl:ce.:r'_iihells 's\10Ject ·to hydro­static (coliapse) pressure dcr/'1 is cal­culated., .per Fig, CB, 2ia·j · Fccr is then ob­tained by entering Fig,C8,9 with

. ( dt;r/'Y/l/Fo;7, obtaining Fc/Fo,7 and. then

BUCKLING OF CONICAL SFELLS AND SPHERICAL CAPS 73

To obtain 4Cb enter Fig, C8, 14a with .ffe/t

Fig,C8,26b Increase in Cb Due to Burst Pressure, p

'1'cr/'I • CsEt~/R12 Be?J'" He = Q. + ( 1 + R2/R1 T5 N2

(1 + R2/R1f",,-\1'2"] R1cos0(

Z • L/i/1 - _,d- /He t

Valid for z > 100 for Simply Supported Edges z >100 for Clamped Edges 0( < 60°

To obtain Cs enter Fig,C8,20a with Belt

Fig,CB,27a Buckling Due to Torsion

Fccr as Fo,7(F0 /Fo,7), Per is then

Per= F0 rt cosot/R2

For conical shells subject to a transverse shear, V1 .the buckling stress is obtaine.d by determining the buckli_ng . stress due to torsion and .then multipiy­ing this by 1,25, Then, s~nce fs =·VQ/It and with• fs being at );he neutral axis,

Ver = Fs0 r'11' llt/4

where Fscr = 1, 25 times Fscr for torsion,

I•" .t ft 6 I tpl

Fig,C8,28a Conical

p~

er

RJ + R.z

Z·cos ~

ITcr t cosCI(.

Rz ·• L Z l /Z Z=>· -•- (I -l'Z)

a.1

z z 4 __ 6. • uj z

-

• • • ... Shell; Collapse Pre.ssare

CS.20 Bu.ck11ng ot Spher1ca1 Caps For a spherical cap subject to a uni­

form collapse pressure; Fig;ca, 29, the buckling stress can be calculated as

Fccrh-/ = ,175Et/R Fccr is found by entering Fig,C8,9 with (F cr/7?)/Fo,7• obtaining Fcor/Fo,7 and then ca!i'.culating Fccr as Fo,7(Fccr/Fo,7), The critical pressure, Per• is then

Per = Fcor2t/R .·

Fig,C8,29 Spherical Cap,Collapse Pressure

NASA CR-912 is recommended to the reader ror additional da_ta and discussions not only for shell buckl1i:ig, but for much other helpful data about shell structures,

74 BOLTS" NUTS, .BUS.RINGS, TRANSVERSELY :·LOADED LUGS AND CLAMP-UP

Dl.Za Enonomy in Pitting Design

Also s~e Art,DJ,9, .items 8 and 12,

Dt.'.}a Pitting Design Loads. M1niieum Margins ot Saret7

Also see Art.Cl.lJa about design loa,ds and margins· of.safety, n1)+;,. iij:,:;~·ial o·r Rlgl'te.t P&~:~ora or Satet;r

Also see Art.clr1Ja'abo'i1t design loads S.i)il margins of', safety,

Di.Sa Atrcratt Bolts

Unfo'rtunately, Art,D,15 and subseq­uent articles do not discuss single shear bolted :Jo1n:t;s •. These are widely used in

. . -·~:a,e~r"OSi)B.'ce···::S:tI-\ib.ttif_6s'.-,-... s-U:ch a·s .flor. attach--. ..' - ·-.. ' '·•o ·," . ."i, :··u- ,. ~ · :: ·.' ,., : ;"·_ » ''.'· ':':: · ",-,,. -·' .';" ·~). ,, ,.,,,' ,., -.-~« , __ , '·· " · , ·• •' .,., _' · ' ,

.··:~l:jjg:51<:,:l,:i;i:s•.:to 'sp,;.r;::C:aps, skin splices etc, See :Ar:t;:;:Dt•.28 ,: Df,8ai\ and DJ,ja for. help-

~1~~~2~~l~p~:~~~r<f~bo'ut the.se caii~s, and

. ,-D~~ •. ~~~·-'.':.~1~,;£;;;.;;:;~· ....... 'to.:_·· ... . •"'••~ n14 ; .'·,i:i;;.~" ·_:- ·,,~'.:·i:Lt'

i< •.•• :1riii;.~~#9'i~;~il ~Q.t;t1J~'L27 ;1f.iai¥,r~£i~ A46

Df•:~·~>_aui:h1~•: .. __ :· __ -. _,;.~·- _.-:._:-: _:-'·_ :_._ .. :~ .... _ .... <'., :·\.::·;_·-.::

s\&:~~~~~;;~~~~~;~~~f,~1R~~i~~~l~~~~~d~f a:afe;t;i:.~il'.P. '"¢,; p;t'e19eii,t: .. tf:' 'the; .next size bushing· is irisfalled, :(f no bushing is present, the ).ug. or. fittJ,ng i;.nould· be, de-

•'sigri,ed'" to''' show· th.r'·requirei'd' margin u' a bushing is irn•talled la,te.r·•on,

D1.1~& L~ ·"a·r.~~~tii ~~11~1_,~ ~~~-r ' ·:Tranave:rae'.Lead.l.iie; · ·

' _, ;:) ' '-~ ' -' :1 ·"

For the·case of concentric lugs (Fig,p1,11,,,~.,wl):ere .e .=; l!/2) :see Art, ,

, .Dl.lJa; ,·F\?i:-.,:the allowable _load at 90° , ( a transvei:-~e load) bas.ed on test data see Art,'Iii,\1Ja,below, ·

Dl.1J• Lug Strength.·· ~l7Sia Under Oblique Loada

type lug, the curve gives an unconserva­tive allowable load,

1,0 --!':--- ....._ ,·

t-~ - - +-.........__

ii--.·· ~ ' ,(,,.d_.·-

~--f- ~·Po ' , . Pe .6

~

ol:'. 0

I I 20

I I "° 9• 60

. ··'

.~ ~ -

80 F1g.D1.15a Concentric Lug Oblique Load Strength, Pg

D1•27& 'l'h• Importance ot··COOd.. Claap•Up :·-·''·

It is import.;_'i\t .. that the stress an­alyst and 'structural designer have a good understanding of the importance of clamp­up action as to shear joint strength, particularly for single shear joints which are So Common in aerospace struct­ures, Prevl,.o,us articles a:bout r.1veted joints, Dl,18, to·Dl.22, and bolted joints Dl, 16, have nift discussed this important subject, As ,discussed later, lack of good clamp-up .is why"'blind or countersunk or shaved head rivets and and blind, countersunk, thin head bolts or bolts with soft collars always r"su.lt , ln, joints hav­ing less strength/stfrrne's's than 'do pro­truding head z:i vets. ·or, bpl ts with th.lck heads and nuts, It is also' why the al­lowable jqirit :+cads, w,ith, these must al­ways "'bi{ determined 'b;V' tests· rather than by calculations."" The:. following discus­sion discusses the rEiason's for. this.

A :f.aste11er. t:ra,nsfer:ri.ng a shear load in a typical s,i,~.~e sP,~,ar ,joint• is ·shown' in Fig,Dl.41.' If' the fastener does not have a head (i,e,, a pin), or otherwise do!ls not clamp. the:.two, sheets ,together, the Joint equilibrium and bearing

F.or ,tl:le case ::of concentric lug,s stresses will be .like that sketched in (Fig,Dl .• ll~a where e = W/2.) Fig,D1;15 can Fig,Dl.42;· The extremely high bearing b'! .·use'd to quickly determine the allow- stresses in the sheets at the faying sur-able load, Pg for any angle from 0° to 90~ faces will result in early yielding, a Only the calculated allowable axial load, permanent set arid a greatly reduced joint P0 , is needed (Art.Dl.11), Pg is then strength and yield .load, Thus the car-calculated as dinal rule iS to never use any headless

Pg = P0 (Pg/P0 ) type pin or ~.-~olt, .or rivet. not pro,vid­ing good clamp-tip,

The' -curve'· is based. on a great number of tests and with different materials, As Clamp-'Up show:n '-·_,the_ ,:rf3dtH~_t,i.on factqr varies from 1.0 for g ~ O"to,',;75 fore= 90~ If.the ,, Whe·n a buc}<ed pro:truding,.}')eact_,z:i.vet lug W:idth increases as in.,Flg,Dl;ll-b, .... o:r a conventional bolt with a tightened

. tp,e, curY~ gives a conse.rvative allowable nut is used, the joint equilibrium and ··load, ·If it narrows, as with an eye-bolt the resulting bearl'ng s:tress distri-

* See p.AlO - A21 tor bolted Joint allowable ilata. • For· much 11u~ii data· ·aee p. Aio ~ \A2i

SINGLE SHEAR JOINTS AND CLA~~UP. DUCTILE JOINTS 75

._,,, Flg,Dl.41 A Single Shear Joint

' (a) (b)

Fig,Dl,42 lloaring Stresses Without cl.aiiip-Up

butipn are qulte different, as ·13howri in· Fi.g,01';43, The presence of th~··tlg)1t head provides clamp-up and_therebyallows the load~ Q to be generated. an( balance th'e mpmerit due to the. bearing stresses.

~- .. · .

~- . Qt p

To (, -jel-

Flg,])1.43 Bearing Stresses With ~lSlllp-Up

Thus there exists

1, A prying loaa. on the' fastener heads

2, A tension in the fastener

3; A near uniform bearing str'ess on 'the fastener (and sheets)

With good clamp-up the b0aring stresses are near uniform, but even a "finger­tight" clamping of a bolt's nut is far better than hone (a "loose• nut):.' Those fastenefs,previously discussed, whfoh do not develope as mlicli clamp.:.up,,,have less joint strength and stiffness, arid their joint allowables must be obtained from tests rather than from calculations, Blind· fasteners are t_he wor.st in this re­spect, A steel collar is much better than an aluminum one but is heavier-.

Shims ·(Fillers) Between Sheets 'in Single Shear Joints

When a shim is .. present .• be.tween the fastened sheets, its effect is to in­crease the loads Q (Fl.g,Dl,43), Th.ls re­sults in more. bending and tens ion in the

fastener which can cause less strength and stiffness, .. wli.eri .-the shim is. fairly thin, say less than 1.0-12~ of the. fasten­er diameter, its effect is usually ig­.nored, Otherwise, see .Art, DJ, 5a, Art, DJ.5 a_nd Fig;o3,18 as to how to pro­ceed • .

,Dl..23& Riveted ,Sll,et. Splice Intorma.t1on * The follo)"ing l..nf'orm>i.t'l.on should be

added at the end of "Proper Rivet Size to Use 111

:

StructurS.1 joints should be. made critical in sheet bearing, not in shear, A.bearing critical.· Joint has ductility, whereas a shear critical joint is of a camparativel;)!'. "brittle"- ,nature, This is because once the fastener begins-to shear failure :f'olle>ws qti,itl! quickly, but when bear.ing ... 0r1tical,_therl!. is considei:-able 4efo:i;mat1on within ·the joint before fail­·ne occurs:"* This can be S:een from the comparison of load-deflection curv'es for t. joint as sho~. in l."ig,Dl,44, The joint deflection here is that within the joint, between the upper .. and ,le>l"er sheets in the direction of the shear load, A shear eritlcal joint Dlight be. desirable when a piece 11f structure, is .. to be ejected <>r "bl<1wn Off", SUCh 9.S the e Jection-'' Of a nose cone by an explosive charge,.

..... shear Critical Joint t·Br1ttle)

E<iaring Cri tlc'al Joint (Duct1le) ··

(Spot welded joints are the mo.st brittle)

Deflection·

Fi0 ,Dl.LJ.4 Brittle and Ductile Joints

Dl.27 Dee1gn and Strength at 1?b7e&d.a

The purpose here _is to provide help­ful information for the design of threads to transmit· a given load, Applications

are for bolts in tapped holes, bolt-nut combinations and hydraulic cylinder end caps, The general question is how many threads are required, The answer depends upon the shear strength, Fsu•, of the ma­terial 1n which the ,threads will strip ·and the cylindrical area which will be .sheared; The following .. assumptions are made for bolt-nut combinations and tapped hole combina~ions,

• See p.A10 • .1.21 tor r1Tated jo1n1; &l.lowa'bl.e data. Bee p.AlO ror fastened sheet ed.<!je dlstanc~•-

- X::s:sent1ally 111&nd&t1:n:7 for &nJ" m:ult1tutaner Jiattern 1n.sta.ll&t1on

76 STRENGTH OF THREADS

1. If<the bolt me,ter1al has a higher · ·s·trengtlr tha,ri the hut material:' 'the ·threads will be strlpped· from the. nut, ' i ,' . ·:>

2. If «the nut material has ·the· higher strength the ·threads will be" stripped from the bolt;

3. rr the ''J:lut 9.t;.d: b~1i: material~ ''ai'e ·or· equal shear strength the, bolt' threads will usually ~·stripped first. ; .. -,; ' :( /,,,.,_·1.y·/· :. ' ~ .,-_., .>.: ··:' ,_, .. ,.--·: - . . \,.,;

The strength of the combination· is' calcu­lated for the three .above assumptions as follows', · · ' · · · ·.··

\C','

1.' Load at wh'ich 'tlir.e'ads 'strip' f'rom··nu:t ·

n' Pr\u i: =. 'KnOj:)LF su 'wliere

K = A ·i=· 'etric'l!ericy 'of nut threads ·Db'•= bolt thread' ln•fjor"diameter' ,, ,, '

L ""'''='· le'ngth of· thread ·engagement' · Fsu· = Shear stre!lgth, of 'riu't ·material , .. /:"''"" •'• ·' . ::,,/:'.''';<·'5:>··· '':-_;,i,:.c.X ,, _,·::~,_,:_ ':_ ' ...... , __ .'fii··:: .. ,

2.· Load at 'wliich.ithreads"'s'trip' fro'm .,bolt ,. ;' ,:·. , • .;._ -, .-, •. _ .. _,1·',i >~; ,·· .,,_q ,•.,··: .. ~· '? '';···· ' ·n··,_

where K' = .4 ='efficiency of bolt threads on· = nut'' thread mirier iUame'ter' ' L '=> le'rigtli Of thread eng'ag'emerit Fsu = shear ,s'tre'ngth of ·bolt''lliaterl'al

3. same as (2) above

The formulas apply for typical bolt-nut C'?!J!.bip!>,tio.l'ls o,r b():J..ts in tapped holes:'. ' where:f.,i~,lijore than 2/3 Of the bolt cJ.i­ameter; 'For smaller values of L, K will increase slightly, as shown in F1g.n1;45, because of a more uniform thread loading, but such short thread engagement is usu-

. ally not good pract1ce, since it may. not develope the strength of the bolt. '

Hydraulic Cylinder End Caps

There are many,types of hydraulic cylinders used·in aerospace vehicles. In general, the larger their diameters and the thinner their walls the more critical 'the end cap thread design becomes.·

' For the end cap to cylinder.threads there is a tendancy (due to the' angle Of the threads} for, the part with the inter­nal threads to expand (due to hoop ten­s1on} and for·the part with the., external 'thre·ads to contract (due to hoop compres­s1on}. Th1s lessens the thread contact area. With quite flexible (thin) parts onl;y the. tip portion ()f the threads may be.ene;agecl:, ard f!!.llure,,wil:).. o~~u:r !".Joe,

·. tC>w 'Perc~.ntage ··of the ''calc];i1'1 t~d.~ :Loe.CJ,,' everi le~s' than 20% 1ri S<'lme case's'; ,,. In.,ad­di t1ori' to this the' cylinder explincfs . under the .. internal pressure, and this wtll"' les­sen' ;tihe thread engagement even :ll!cire 'i.f the' 'threads on the cylinder are ·internal

. one's:. This is most significant. rqt cyl­:.tnd:e'rs having thiri' walls and larg~ 'd1am-: ete_·rs.. .. ·'

·In general it 'ls suggested that for cylinder end cap combinations a K factor 'of;. :.3 .be used, and :.that the design be carefully consider•fd 'to minimize the re­lative expansion.Ej'.nd contraction of the ,me,ting parts,.,, '.l'J:lrE!ad.~ ,9p,,piston,,rods, '8:ha cB.pS~ roa· 'enas arid oo.rrels are ·'usu­ally machine cut. MacJ:liPe4 thr~ads .)1ave a lower fatigue life than do rolled threads, so this sho11ld be kept in mind when' de sighing parts ~hich might be 'fa­tigue critic~l in,the.threaded region.

DJ •.Sa Filler.~ { .Lnd .~ma}

As discussed' irl Art .• D3. 5; e.· filler (or shim} between .. Johe two shE!ets .. of a, sbigle' s~i'ar JC>i11t ~ .. d\ic~'!<j;he 'si:r.,rigth of th~ fasteners'' becaus'e ' 9f j;he increased

~~~===r,,,,,,,.,-,,....-i=-,,,,-.,,,-,-,,.,,,=====ml prying load.· . Extending tJ:le filler and. adding m:C>r" attachment~. tis shoj<n in'

~#.l";;F.8-"'f.,f;;f.8"4.'$'hi'4"2H±t§H5tt§'tifutl Fig.D~';1B'; re~tores the strength; but in 'many cases' tlii's is not feasible of mices-

HB'i±*-~'u.:~'5":f7E':b!Hc'tt\ff7"H'fuf':rtifif:ttl sarY:: ·········· . ' ' . .. ... .· . .

Another apprpacJ:l ts' to, det~.~mine the reduced jolrit strength "and use this , if

P.4\f+'\4~;;:,ts+AS~h<"f~+\CSl±i'±'+'i+i:i!t!e,;::in1 it is'"adeqiiate •.. The following is a sug-li+.,,\,,;;t;J gested method f.or doing this for single

shear bolted joints •.. It can be particu­larly .. helpful when shims' are' found 'to ·be necessary 1n manufacturing and salvage P~?ce_dures.

For a conven1;1onal single :shear joint having a protruding 'head bolt with

,'a stiff collar or nut,' the allowable load

SHIMMED SINGLE SHEAR JOINTS (FILLERS) 77 is calculated as· the lesser· of a shear calculate. ts 2 a l/0/l')l~zD failure of the bolt or a bearing failure of.tit<> she.et (for all other types of 4, If ts1 > t1 use _ts1 = tl and recal-bolts and nuts, or collars, the joint strength must. be ·determined by tests culate V0 = Fbru1Dt1 rather than by calculations, as discussed in Art.D1,27a-), · If ts2 > t2 use ts2 = t2 and recal-

Pl;~ing a shim between the sheets causes an increased tension and bending moment in the bolt, so there .may·then be 5, a, third. type or failure, bending and .ten~ sion in the bolt, The thicker tne 'shim,

culate. V0 = Fbru2Dt2

Use the smallest of the calciiiated loads in (1), (2) and (4) aboTe as allowable shear load, V0 ,

Vo the

the lower the joint allowable·srear load, v0 , bec_omes. The interaction or shear · and bending is not cri t1cal• so t,he joint allowable' sheaf' load, V0_, 'ts_ calculated as foll9ws, referring t? Fig,DJ,19a. Also see Fig.Ill ,'t-3. for more about-, fasten-er equ1~i.br1wn, ·

Dis Bql .. C1at1'11

F~g,DJ.19a Shimmed Single Shear Bolted Joint, Loads on Bolt (and.Sheets)

F~r all other fasteners.a Joint allowable load, V0 , is gotten by assuming it· varies inversely -with the- pr;y;ingload (moment) on the fastener heS:d. Hence, .·v~ is the allow­able load for no filler (Table JJ-44 and MIL HDBK 5) multiplied by t 5 /(t9 +-tf/2), where ts is the average of the two sheet· thicknesses. ,. ·

The a)loTe proced,ures are ~1i~·:,.eC1 'Co be reasonable, but _they have not been checked by aingle' she~ joint.; test's, which are need13d_ ;!'or 'substantiation •

.. --·:e.

For r1Teted jo.int.i a. sim_ilar proced­ure, could b1J considered;.wi.th'substant_i-

. a:tirig test~ or single: shear •joints. . . .- ' -",,, __

A suggei!'t~d test- llpeci.i.en for>~ingle shear joints is sketched in Fig,D,46,

The sheet bearing stresses generate a ~~~16~ moment 1n the bolt. which is, from !{=-:;::=1--::====i;:=5';s~---;;;:'=ii

Mo =V.<ts1/2 + ts2/2 + tr )/2 ~ where ts is the required thickness of the Fig.D.46 Shimmed Single Shear Test Specimen sheet to develo1>,e the applied shear load, V0 , and tr is the thickness of the filler 0~.7a Special Cao•• or Bea• Do•1sn (shim).

hence 2. 1 -1 Mo = V0<2..... D + 2F D +trl/2

•oru1 bru2 Letting A = 2D/(1/Fbru1 + l/Fbru2) Then :i!.

Vo /A + Votf ·- 2Mo = o or,

~ + VotfA - 2MoA = 0

1, SolTing for Vo (the allowable shear load) and letting M0 be JI!' B where JI!' B • ; 88MB""

Vo = (-Atr + -V<Atrl ... + 8Ml3AJ/ 2

See Art.A13,11a for additional curved beam discussions and data.

D). Ja Tens10n Cl1pa

Additional allowable data for sever­al types of such clips is in Fig.Al - A~.

D1.21a Strength ot R1Tttta nu•h ~

See p. AlO - A21 for allowable data

D1.22a Bl.ind. B1Teta

See p., A10 - A21 for allowable data

2. If Vo> VB use Vo = VB where VB.is tlJ.e _, D1.7a &,it Shear. Tension a. Bonding Stringtho·

bolt single shear allowable. load See Table A45

J. Calculate ts1 = V0 /Fbru1D and • MB • bolt 11hank t!i.lo.wable bet1d1~· aoiii~nt (Table A4$)

and .88 accounts for the tension in the bol~.

Al APPENDIX A

DRAWING. CHECKLIST

..

.. ..........

ADDITIO.NAL .DESIGN AND .,ANALYSIS DATA

.

Jl!AH,GINS OF SAFETY .AND DESIGN CHECKLIST 'MprQln 'QI ·Safely

A checklist ·01' minimum margiiis of safety aMd·:uemS' tO b'e consid'ined wh.en de!Jlgnl~g ,~nd analyzln_g st~1fCtur1l.Jgint~ ia pre~entad below.

Margins. of Safe"ty (M;S.) lot fasten•. is .'. n.d 'tol.nli • • M.S •. :: 0.15 on all.shear Joints.at ultlmat.e •. ;

M.S-; = o.oo on all joints at yleld. ')·"-' ; . M.S. = 0.50 on •II tension join ta at u1Uma111 .. · . M.S. • D.15 on lugs. M.S . ., 0.25 on critlcal Joints (wing pivot, landing. gear a_ttach lugs, -!~~.), •• ... ~1lgn_a1ad ._~y· 1~,.:_•lrc.r~fl ·d_ealg_!:' p~_!!os_ophy; ;_

.'Th't 1lr•s1 1na1ylt .sh0Ulci take.ctre. liOt ''.10 'be ~llmlted.b)'' the ltcltns:1is\ad'8nd ,~o,"~I!~• g.o~ ·~·g·l~klg,Judg~e~t-at au tlm••· · ·;r•: . , ';' .

'.,,,.,.-., ~ ·M_•k•Jo.lnt f:rltlc11_1. In -~~ . .t:be1rln11. rather th•n. •.h.aar-of· fastener. S~clty. 'torque v1lun on pr.loaded bolts and apecial _appllcatlons. AvOld ·Using rtv111 (solid ind blind) In primary tenSlon applications. Ho~ IOl•111nctl Ind flll IO"IUll the ln1t11llat1on~~-· ;· '.' ; '

" Corrosion and dl11lmlllr mttals,protactlon (Contact M &.P Group}. 6'."- ,. .. . ,. ... ~- ~ -·wear and:frteUon qu81ltle1 cf moving parts. ' ·

- -= ....... ':' Exca11lv•"be1rlng 1tre11e1.-ln rotall"!G:P•rts. ,.

t·"~""·':H-t-+t-t+H~t+r++H+~++i-++1-++H++i-++H++I-+~ Stress raisers' I!" highly -~l~••••~-•r.ea~ (lhl!•!i•_, g~~~-vea, U·1 ,,,, .. ·.t'lubrlcat1onflUlng1).·•e·· • · ·- ----·,, ·'--·-·-'·' · · Local dlacontlnullHll and eccenrrlcUi.1._ •. , .. • _ _ , .:· Possible 11rength reductions -due 10 adverse tolerancaa. Minimum lhlcknd• of atMat for counteraunk holea.

use WAS ERS UNOER BOLT "*=AO .l. ' ~)~~;\CHMENT POINT SP.ACING:

•-+-_,...- (•I~ 1 IN., "'P'" IS lSS. PER' INCH OF ANGLE

(b) ,.11N., "P"'lS LBS.PER ATTACHMEN'f'.-:POINT __ ,,,._. " ., - ., ''

(2) • FOR ANGLES OTHER THAN' 2Q24-T3 f-C---'--j+-i- CL.AO MULTIPLY"P" BY:

. Ftyy ~,

''"""' (3) WHEN TWO ANGLES ARE PLACED BACK

TO BACK MULTIPLY"P'" BY 2;5·· '­f-C.,,--'--j+,+,-,_(4fTHE u°LTIMATEAi..LOWABLE i..OAO, "P:'\IS,

DEFINED AS l,STIMES THE LOAD AT Q.005 INCH PERMANENT SET.

NOT APPLICABLE r0 MAGNESIUM ALLOYS,

=.-t J. ·-l ·!

!

.. l.,~~~J .. =.~~~~~~~~~~~~~~~~-~t·~J Q.8 1.2 1.6 2.0 ECCENTRICITY, C (JNS.)

see PIG.A) FOR BACK TO BACK ANGLES ULTIMATE ALLOW, LOAD FOR 2024-T3 CLAD SHEET METAL ANGLEf

;'J.-'

Effects ol deflac1~n1 •. 1 c~·· :;.-·· Compa!lbUlty ot a1r1lnt:<' . Stlttneaa 'requlramer:it•i· - _:;:_· ·-·':~:1 ' ,. ··"' _,

< • Local deflectton1,-.:,(~ .. thin !airings), .rala~I!~ to aerodynamic· ' ·.-tolarancas; __ . -~···{· - ·~.:···' ·:~ --···-·- .:··;.: ,

Panel flutter on large, thin gauge, flat external panel•. · Fatigue re'qulremanta. . ' ,._,

: • , Thermal stresses at- ctralns. ··~ RedUclion· 1n·.a1_1ow1bles lor:·1'!rnpe·;a:n1fii. :_ ·.-;--· .-~:-;. ,. ~,. ... , , Proof load of pans per speclflcatlon:~~qulrement;'C(•rr•attng hooks, hydraulic ac1uator1, 'ate.). · · ·· · - .

l'IG,A~ ALIDWABLE TENSION LOADS !'OB CLIPS c (l!X:rB!JDED ANGLES)

(US_E~ASH~~'~!"CJE.~ ~OLT;"l_U.D) ·

NOTES: 111 ATTACHMENT POINT SPACl_NG:

3500 . (~) s I IN~ ·ris·Las: PER lNCH Of ANGLE ..

; '.j }~l ;~~';_:~:;:is L~~:,~ER A,"f!ACH,MENT

,... f-'--'',-f,,.-!--jc'i-.: ... ;:',2)' ~ri°R':i~::L~~n:~~~;!f A:*~Q24•T,4 . ' ~· .l .. ! ·-1:: .. (3) WHENTWO~~ES ARE PLACED aACK:, .... I·- T08ACKMULTIPLY"P'"BY2.5

• "' - "7" ,(4):-~~~~~r:;r .. R~A3~~';.f~'·DEDT~,es. (Si THE ULTIMATE ALLOWABLE LOAD, "P", IS

~~":.Npe:R~~~-:~Msi;:1"HE LOAD AT a.005.

o l±fil~.L~~ll±!J a.o o,4 a.a 1.2 1.s 2.0

~CCENTIUCITY,C(INS,) " (;'>"

·1SEE · 'Pl:C;:i) FOR BACK TD ·BACK-ANGLES OR TEE secT1'0~s1 Ut.·TIM4TEA!..LOW.-t.OAO FOR'2024·T4 EXTRUDED ANGLE

~PEND!X A ADDITIONAL DESIGN AND ANALYSIS DATA A2

• = •

, l'IG.A~ ALLOWABLE LOADS FOR TENSION CLIPS

AUOWABLE TENSION LOADS 1. SINGLE ANGLES:

P'tgurii• At and J.2", GIVE THE ULTIMATE ALLOW ABLES LOADS FOR VARIOUS GAGES OF 2024-1'3 AL CLAD FORMED SHEET ANO 2024-T4 EXTRUDED ANGLES. ULTIMATE LOADS FOR ANGLES OF OTI-IER HEAT·TREATED ALUMINUM ALLOYS, mANlUM, STEEL. OR ANY MATERIAL ~ MAGNES!!IM ALLOYS, MAY BE OBTAINED BY DIRECT RATIOING WITH THE MIL·HDBK·S TRANSVERSE YIELD STRENGTH AS NOTED IN THE FIGURES.

2. BACK TO BACK ANGLES: THE AUOWABLES (ANO CONDITIONS) USED FOR SINGLE ANGLES ARE ALSO USED TO DERIVE ALLOW ABLES FOR DOUBLE ANGLES USING lHE MULTIPLYING FACTORS SHOWN BELOW:

.. < i.zs P·

2.5 p.

FORMED OR EXTRUDED

3. TEE SEc:TIONS:

...

WHEN EXTRUDED TEES 'ARE LOADED SYMMETRICALLY IN BOTH ounrrANOING FLANGES. THE ALLOW ABLES OBTAINED FROM na.il ARE TO-B~ MULTIPLIED BY THREE (3) AS SHOWN BELOW.

3.0 p.

l'IG.A4 ~OCKWELL AND BRINELL l!AllDNESS DATA'

110

APPROXIMATE HARDNESS· TENSILE STRENGTH RELATIONSHIP OF CARBON AND LOW ALLOY STEELS•

... 100

... ..

... 80

... ~10 • • ~ 350 I~-.,

z :l,. :l ""' Ill,,.

• ~·· .. z ~ 250 8'•o

a: .. _'

200 " '" 20

100 10

50 .. .. 100 . .. ... 160 ,.. TENSILE STRENGTH • KSI

.~FOR CARBON AND ALLOY STEELS IN ANNEALED. NORMALIZED, AND OUENCEO.ANO.TEMPEREO CONDITIONS

TABLE Al TOR<!UE RATIO FACTOl!S FOR TENSION PRELOAD ON BOLTS .

Fiu MIN DIAMETER DRY (AS LUBRICATED SPECIFICA TIOH

90LTINUT ~TERIALS (KSI) saES (IN) REC'D)" {MIL·T·5544) 208o 1 J..233 REf R FACTORS STRESS (Fiu:KSfl

SEE HOTE(21

1. STEEUSTEEL ,,. 3111 • 5116 • 165 . ... ., (ALLOY OR CRES) 318·1118 .170 .0731 .,

2. STEEUSTEEL ... 114·5116 .165 .ooo 5"67 (ALLOY OR CRES) 318-1118 .170 .0731 50,67

3. STEEUSTEEL 180 1/4. 5115 .165 AOO STA (ALLOY OR CRES) 318·11/8 .170 .on1 57A

4. Tl/STEEL 180 114·5.'16 .180 .ooo 50.67 3/8-1118 .100 .DllS 50.87

Tiie "•• receh1ed"' condlllo11 mey v1ry lr11m no lubrlc:11nt {"dry"'} to v1rlou1 d1gl'Ms ol unknown "llghr' 1u11rlc1nl such as machine oll. T1111n11 I• r.qulrld for uaknowa conditions when 1st1Dll1lllng prllNda lor crlt1c:11l appllc:11lloM.

NOTES: 1. The raUo 18ctoni and torqu11qu1!1on1 ere b•Md 011 19faetfon of a tension tYJN"nut

hl"lllG 8d1qu1i. stranglll to dlvelop th• f•1l1n1r with• prope_r ""rgln ol .. 11ty. T•l'lslll 1tr1111gths for common nu'l lypes •re pN11ntad on T~ A46. u111m111 t1MH1 slr.ngtha ol lhnlldld fa1l•nen 111d th• thrnd root,,.,, st• P"•Hntad on tJSL uz

2. TM nileNnn slrn1n shown 1r1 IM p,.leH tenall• •tr•"'• lor th• 1tenderd torq111 v1lu11of1pedllcallon 20&-13-233. Th ... W•ni usld lo d•.,•lop th• A lecloni ua1no th9 111uauon:

T•R(d_.)ftu.('-t); or T•R(dnom)llt

whwl: T • torq111 (In-lbs.) d- • nomln .. al'lank d .. l'Mllf' ltu • desired 1trn1 lev11 al thNld root AlllOI • t11re1d roo1 er11 ol bon A • ••111 ten•ll• pr9'oad on bolt

3. Tiiis fftlthod I• not 1ppllcallfl to HJ.Lok collers hsvlng controlled torquHtf lfttuN• and YlrlOUI lubrlcantL SH Hf.Lok ltMUtdl AIL

FIG.A5 AREA 01' 9¢' BENDS

0.090

0.080

OJJ70

N!i 0.060 /fd:;oif;f;hl'i/iflitiftit\;~rY?l~~~f,~11:'80. < w

~ 0.050 lf''f;Sl'-"-+''47.-:-f.-"""'-*',tj,9f.74¥j.!;<l!f'fj~ l!·! f. i! u ;·:~ [

o ....

U30

0.020

0.015

0.010

0.005

0 inl ~... n-~-· Jtn

0 0.01 0.02 0.03 0.04 0.05 0.06 O.ll7 ll.08 0.09 0.11l 0.11 ll,12 ll.13

SHEET THICKNESS • IN.

::::

20•f:ffif1:b+···· ::

~ , .. Rll"~i~".t11~·11il:~1~;lifil!l~l~[i~IJ~i~;~r~\~-'i:ffii•:~1~@~I '

! , .. H~l·W.~l~C°" ::.f:-::1:::iJ;;_:_,;_;;,_

:;.,

ao1'·1H~;1::;:1::::=1::·

11rT·r· 40'""":·:: ==~·

.. ti > "' ~ ~ ~ "' to ;;; H

&; " .. ., .. w .. "' E t; .. I

l; !;? H .. 0

0.02 .... .... 0.12 .... 0.10 I • STRAIN· IN.JIN.

.... · .. , ~~a-· fi:il

~1--" <

I~ 2::2:: ::"'O ~ ~ i;i:: .c:§~ :at ~~: .g;~;;:~

- .-. _. . ..,. --~-..... o1· .. ~o?<>.:! o N ... "' 0 ~ l!l-.s S 8 ~ ~ § 8 § o ~zji· m_-~ .. 1-1 : "' ..,i .. '.'<·.,, .. '1

t • , • • .• , l I I I 1 z,.~ - . !i •·

·:··t-i --:--- ..... 1---r-: ~ 5i - ,, •• ·- < ""'3•-rr··, s -, -1 ···1 m

..... ·----'""·-·/·1-.. -'f-.-'r~· ,t ~ts i: · z • ~:; Sl ·+. +n·--'--·---r----·-· ........... ,. ------·--/ .... ,.--t-f~..!..i,-!::;::a- l!!o,Q~i -'-'·l-..L---.,---i-----i·- S _

·~-=t~r=+~t± _ii: i.;~-1: ~ ! ~ ~; ~f..~(t-J=~:=~·~-=~~1~~t-~t11+~1 '-------'----'--'---. . , -~ ~ i"i ~ ~.'" · ! : · r , ;. ii '.

- n ... ,.o.,~ " 0 §Z ... llo

~ ~=!!f ... .; ~ ·-~~ ~ ~ 'il' ii' ·.·Q· t:" ~~ !~ •I' •P-i I: ~ ... !..1 .·~_I

i -~·f:r--~· .....

~·~ g ..... '

<>·: t~~:i~8 '-~ s ~ ~~~~i~~ tJ " n,.,. 11 ~ 0 i s:i,. .. fl tlJ ~ ,.i!i!i,.11:= z

<> ......... :a~g w ~ e ~~~~~;~ fg

8 '"! .... I: .. ~ p ~!1:::;;1! iii j;!!f~~ i !! ~;~:i!it~ · I m 1r

-~ !ii ~ I i

"' H

" 1X.

~ ~ ~ !'ii s

" to 1_ .. 0.5 ~ 5.oO" ! 8 I = 2.00-· 4.00 _ e! I S-2.00 111 i.s.0.250. ~

. .. IS0.250 ~ IS3.00 It

. "'

0.02 . ... STRA~~~l:.:w::cccLlo::.·L .. J§il!)§i§!H,,fE4§~1)liJi I

> "'

~ ttj

8 H >< >

g H 8 H 0

!~ ..., t1 ttj

"' H

"' z

1§ > ~ t' ..., "' H

"' ~

APPENDIX A ADDITIONAL DESIGN AND ANALYSIS DATA A4

FIG.AlO

•• .•

COHPI!ESSION.BUCKLING CONSTANT !'LAT Slll!BT

' ' PANEL ASPECT RATIO alb

' cEf ,., c

' ·EJ· ' ' cE}c • ' ·G· • • cGc. • •

·6· . . ' cGc I

. ' ·~·

• . cGc I .. ·~

I c U f.ULEll

c ••

1 12 EULEll 1 c ..

FIG.All Sl!EA.l! BUCKLING C111M!S l'O!! SUET llA'rERIAL

50 ;;;

" I • y ._w I

"' "' w

" ~ "' CJ s " (.)

" m 20

" .. w

"' "' 10

REFERENCE . ,M.ELCON ·COZZONE BUCKLING METHOD

A B c D E F G H I J K

.. . ~ ,_ . (--""-)' tl'KI

AMS4901 TI 18-8 TYPE 301 7075-T& ALC. 7075-T& ALC; 2024-181 ALC. 2024-181 ALC •. 6061-16 2024-TJ ALC~' 2024-13 ALC. 2024-142 ALC. AZ31B-H24

"". b •

I •

30 ·->.:_ t<KI

SHEAR BUCKLING CONS.TANT_.·'

SHORT SIDE OF. RECTANGULAR PANEL

112 HARD 0.040. 0.249 0.012- 0.039 t~0.063

t < 0.063 I -c 0.250 0.063. 0.249 0.010. 0.062 I< 0.063 0.016-0.250

SD l"IG.AU ·FOR STEEL & TI

FIG.A12 COHPI!ESSION BllCKLlHG CURVES l'Ol! SKEET MATERIAL WIT!! NO EDGE FREE ..

• '

A ' a- .. c

·•'

•

· 111 H-~

-· -t· •.. ,

' ·' ·~ ~ '·

10 20

..... •

•• •

_§_

ltk.;f COMPRESSION BUCKIJNQ CONST ANT

PANEL DIMENSION PERPENDICULAR TO DIRECTION OF LOAD

SHEET THICKNESS

TANGENT MODULUS : i

.-t-, CUAVES;t.REMiNIMUM .... . · i- GUARANTEED AND ARE FOR ft.

-· +· ~:.-'..~~-~-EAAT~RE USE::·. J.~ . ... +:

. ' . '

.:+ . ' ··~±$1 '" ~·rt.j:: "1': "b;.l

t. :µ::~ - · ·tr: +Tt+

+·: ~:!±~ ·'-+ : 1:t l: : :c ~1- 'ffH tH· !

30 .. .. .. __._ A" AM54110t11 I~ F 2024 ·T3 ALC 1 c0.250 B 7075 •Tl At.C a.IMO• D.2'9 C . 7075 • l'I .A.LC D.D12 • D.039

a. H

2024 • T.f.2 A.LC ....... 6061 ·_T& .......

p t8•eTTPE:I0111%HARD I AZ318. H24 0.0115 - 0.250 E 2024 ·Tit ALC l -c G.OS3

FIG.Al) BUCKLING CUl!VES l'Ol! TITANIUM AND STEEL

"'

h .~

.,. ,.,

,.,

~+f!irmiHHfH\~*lj.H!Hltlltl ~~:~~~:o~=~:E: ~lfffitititfttlit ~~: :~-i TEMPERATURE !HJl!l\il~!HJ,i!! COLUMN CiJRVES ARE FOR

10 20 30 40 50 60

bftt'f¢: sit: L'/p: bftnt;

. COMPRESSION INTER·RIVET COLUMN SHEAR PH1.5·7MoRH950SHEET A B C D TI·IAl·4VH.T.SHEETta0.181-0.2SQ E F 0 H TI-IAl-4VANN.SHEET I J K l.

A5 Al'PENDIX•A ADDITIONAL DESIGN AND.ANALYSIS.DATA

AMS 4901 Tl .. ; . F 18-8 TYPE 301 114 HARD 2024-T41ALCtc0.250 6061•Tlit'~0.251! . '_AZ31 B;H2.4. OJ!1 ~ ~ 0.250

707S. Ti ALC 0.040 - 0.249 ·: G 7075-TS ALC 0.012 ~'o.o:i; H 2024-T81 ALCi'co.0&3 ·1' 2024-Tl ALC f~D.250

PIG.A1$. qQLUllH CURVES :POR EXTRUSIONS AND FORGING

·- "· .

L'lp

A 1o7s. T6511 Eli:TR t .. 0.25CI F 8 7D'15-T6DIEFORGIS3.IJ'' G C 7079-T6,•T652.HANOFORG lc6,Q H

,O 7079-T601EFORGIS:6.0 ·1 E ,7075-T73DIEFORG ts::l.tl J

:m1 ... T6DIEFORG tS4.0 2014-T6,·T652HANO FORG tsli.O,As36 202 ... T4 EXTR I c 0.250 202'-T3511 EXTR lc0.250 ZK60.L-TS EXTR A c 2.00

PIG.A16 TITANI1111 AND STEEL ALLOY COIJJllH CURVES

~· w

~

'''·'

,.

10

.L"/p

TABLE OF WELD CUTOFF FOR STEEL TUBING H, T. AFTER WELDINQ

HEAT TREAT 'fiu

-·- ,_ 2&0,000 ,· '200,000

180,000 : ;' ,/ 150,000

,-., __ --125,000",:,, ;, NORMALIZED'

15,000 ·.' '.,.

, Fe .. ~

CUTOFF Foo

"'·""' 160,000 . 1,Y,000 . :120,000

~5i000

67,500

1·+ •+·' "·"' 1'''T" ··tl'±hT;~l~1 ·• \. ~'''f''"I i· -l +FH :tt+1-1J:t=! :f t.t ·ri·t!· iTb: t:rn ~:. ·,::11 N! .:itl~ 2 CUPVES.ARE!i'INIMUM .J..!i-"· LJ!.!. f'jj:•J .. q.l.J. ·1+, ~! i..;"""I ' GUARANTEED_AND_ARE . +1 ·1 -;·~+c r . !- !+•t -··i-j+lh,.......:..._' · FORRO.OMTEMPERATURE ·~ ·· ·!•·1··- 1···•·•·· 1 ·'-•--·····~·-•-·•·l·!·•· ,.--

1 USEONl.Y.'. _'_'·.. . !ii! Jill· ;ri1 ''I' Jiiii ~1·1· Jjil dlif!Li!!U~l:.!~.qj!JI ~!Ii!! . !J.IJ Ii!.[ ii i:J. ll!i ' ~~~~~~~~~~~~~~~~~~~~~

0 10 ,. 30 .. " .. • ' llt , .. ,: . ·- .

~ _rpr;- a'sA"' .d1m1?ie"'"/ -:Ji' spotw~1.::f Jam"t:~ !Y<41t/j>ly S Joy 1.10.

APPENDIX A ADDITIONAL DESIGN AND ANAL7SIS DATA

TA.BL!! A2. llAT!IUAL. PBOPEl!TIES.

GENERAL NOTE: Date shown are the most commonly used otner mated a ls, . temper$ _ otherwise, the sizes shown are and thl_ckness ranges are shown In MIL·HDBK-5. Unless speclfled

thickness dimensions.

l•l

lb)

l•I

(di

(t)

(I)

(g)

L·Longltudlnal, T·Tranaverse, LT·Long Transverse, ST-Short Transverse- grain -dlrec1lon. Values In the ."A""column are minimum_ guaranteed properties (either. "A_" or "'$" valuesJ. Values In the "B" column are based on prObablllty data ·(lowest 'expected- In 90--"'· of the material).~ Average modulus of 1lastlclty-_ln tension (Ed ·and compression (Eel~- - : : _· . ·· See. SOM Manual (or--MIL·HDBK•5) for secondary modulu_s In compression (Ee-secondary) loi' clad materlal.s. .' _ __ ... , ., . .. · :

lh)

(11 •

Percent .elongation Is_ expressed In terms. ol the specification "S" · values._ usually lot 2.0 Inch gage l_enath,-·~n 11!11 .sp~lmens end 4D lot round .. specimens, See note (h) or (k) when noted. Coefficient o_t_:lh!rmal expansion .o. In tann.s 01, lnllnl"F 1,10·.6) aiid _assumed Hn'elr tor·reng& 01 10• to '212·F. For other terriperature data. see MIL·HDBK-_S. · Consul\ Materlals·.-and. Processes __ Unff Jn regiln:I to 'reduced lransversB···propertles 10; Secllons of,· thickness over_ ~.:inc:~•s.:~r _a.r~_as.grealer than '40

·.squ.are lnches .•. :-.•-i\-:.<i.'·:,;';_·/ :·-· .. _.-: :.:.. "".' - .> These e_longa_llon_ .values .•.r• for gag11 length of 4 umes_ .. lh• ·d1arri&1er~;.,' -· :·,,._;~- ... -· -" · .. , Hand_ lofglngs s_han.:nol be used. wHhou(the specific approval ol the Engineering Materials and .. process Unit. _ ·;·::; ... .. "·_1 ·;·,,_. i'" ·-- · . , '. -,_,· ". . -,.,

(J)

lk)

For aluminum Casllrlgs the -data are appllcablit to spectal mold; pennanent mold. or.Investment ri'lold proces_~e~~ ... Thls.:data Is applicable to 11tength cless number t which muat be n1gotlated wllh the vendor. All of the casting -atrenglh ·data ar1 representative of speclmen1 taken from the castings. Elongation value la 1 minimum tor lhls thickness range. For 1 speclllc lhlcknes~ SH MIL·H!JBK·S.

i I i I 1111;;;:;;

; ~ :II I'!

s :II ::; ::; ::; ::; lll

~ s ! : .. ~ s ....

•• • •

(I)

(m)

Maenan1c11 properties It 1ne .200 ksl ana 260 ksl strength level •PPIY: io .fi340 m•lerl•I only. Relel'9nce MIL-HDBK·5 tor correlation· ot thk:knea•1s, diameters. Ind .. sh1pea· rel1Uve to lhe m1tert111 1nd heel treat .. vela •hown.

TEMpEB DESIGNATIONS F As·labrlcated ·wrought material. O Anne1led, recrystamzed wrought materlal. N Normalized. . T 3 Solutlon heal treated and ·cold Worked by

flattening. _ . T 4 Solutlo~ he1t_ lreated and naturally aged. T42 Solutlon hell treated by user. T 6 Solution tieat trealed snd · artltlclally aged. T62 Applicable ·-lo ·sheet and plate heat treated and"'

1ged-by the user.· ..... ,_.,., ___ .. , ., _. .. ,,. T7- · Solutlon_._heat. tre1ted and_ stablll:l:ed,' T73 Solution tieat treated;· speclai_.aG'lng~ T74.) SollJtlon _heat treated,'. speclal . .-_1glng and m.ch. T76 property · dlflerences. '-T 8 t Solutlon heat_ treated, cold_ workee! and

anltlclally--'•ged~';'· ,. ·

sTaess~-·-ij-et·1EF-·· .. y'Afi1Af10Ns T·St Stress relieved by stretching (plate, rOlled or

cold-finished ·rod 1nd bitr). T51 o Stress relieved by ·stretching (extruded rod,

bet, shapes and- tublrig)~ T 51 t Same as T·510 e1:cept minor straightening

alter stretching. · T-52 Stress relieved by compression. T·S4 Sires• rellaved by combined stretching and

compr1sslon.

A6

'·:

A? APPENDIX A ADDITIONAL' DESIGN AND ANALYSIS DATA

ll'nd OH'( ilil!C

APPENDIX A ADDITIONAL DESIGN AND ANALYSIS DATA AB

TABLE A2 MATERIAL PROPERTIES {Con'd)

S: It I I

! a e !! I 11

" ~ = .. ! i i • • ~ ... 1;i • a ! g•. ! 8 ~°"'"• ?i" <C ~"Jr;)$~ 5 ~ ::i : 5 • • 5 : 5 ~ q : 5 : : 5 ~~~ 0 0

a a ,., ~ :I "' "' ; a = = ; ; ; ; a ; ; ; ; ; ~-==

~ I I I I I 11 ~~· J J ~ ~ ~ ~ j J. ~ z ~

2 .,,;.,"' = = = ~ ~ " " :; :; • :; :; ~ ~ ~ : : i~.., .. : ; ~ ; ; ; ; .~ ' ' ; ; ~ i ~ ' ' ;;'~~ .. =ft; ~.;: ; ; ; ; ; ; ; ; ; ; ; : : : ; ; a~'~*' .. i ~ • : • • • "ill • ii'"'~"-9 I = • • • • • : • • • • • • • • ; :' .~ I ;; - : !i . • : . ~:-~ ,., .. " JI e ! a ! a ft ! • e e - • • • ~ ! r'.::.' . • • • • • • ~~It • • • • • • • • • • • • • • • • . ...., I 11 ... ~~ -~~'1~ . . .

Ill~ ~ ,.,, "i. !~- I 11 • • ~,~·1 : . !i ~ .. "" 11 II I.;

~ ~ ' ' I

:II =~lft. • ~ • ~ i ~~ §! ~ • • i~i·i ~· - ~ ~ •• ! 5

~~ ·~· . ' <

• ! a• ~-~ 2 e. ~ • >,;;;. • ...... !. • ,. 2 ;

~~; !.

-~ .,. :::5!S

*~I~~ i~j ~ s

i::11:1 ;::~~ .. , , 3< ~<C 3< < < • . " < SOH!D "°'., liDlllDWOol sHorsnw.u;i

A9 APPENDIX,A' AJ:)DITIONAL,DESIGN,AND ANALYSIS DATA

TABLE A2 , MATERJ'.AL PROPERTIES (Concluded)

'

a~ aNw w11 a;niow

,' ,,

; " ·~ ' • - ~:-: ·,: .. __ ; " ; - -.;·,;_' < • ' ' : •

~,,,~~:;,r':8~R' C01l!ITBaaumt READ DEPTH <IN. i' ' , .. i,.-....:.:,~: ' .

, .... ·

.1

c··:;·; ·ig··" . ' c• c• c• c·

-C.'.

... , .. -BLIND ·RIVET

0

.1

1/2 '---"'"'--....J'---.,,.,,,_._,...J...,..-,i----J...-.J,. .. Appl_lcabJe head sizes par specification.

APPENDIX A ADDITIONAL DESIGN AND ANALYSIS DATA AlO

FASTENED/JOINT DESIGN PROCEDURES AND DATA

PASTlllll!lllJOINT DESIGN NOTES GENERAL NOTES Tn• t•bl•• In tn• ls11t•n•r 1ectlon pre,.nl dsl• nec1sury lo obt1ln the str1natn1 ol th• morl com""'" l11t1n1rs ind l11t1n1r•lh11t combln1tlons. Addltlon1I d• .. •r• pr111ntlt<I In Mll.·HDBIC.5, or the p1ttlcul1r "cu11tom1r" d11lgn -manual wh•n 1ppllc1bl1.

B13lr; y111d.!J!!imat1' Re!prJqn The b111lc yl1ld-ultlm1t1 r1\1tlon tor l111t1n1rs 111 ultlm•I• = 1..5 x yl1ld. Th• ylsld lo1dlng Is crlllc1I If 1.5 x yl.td str.ngth Is c u1t1m111 Slreflgth. Wh1n lhls oecurs 1n 1dlu1t1d u1Um111 value must 1M1 c1lcul1t1d by liking 1.5 x yl1ld lo1d. TM Joint 1u1n;th t1bl1s h1v1 1lrudy bHn 1d]u11t1d 1nd us ld•nlllled by In a1t1tlsk.

Note that when e11h:ul1tlng th• shut b11rlng 111r.ng1h for protrudlna h11d l111tan1n1 (I.a., !hos• lltltd on P•ll• 26) lh• yleld-ultlmll• r•l•llon mu11t IM con11ld•rld. An 111nl1k (1 l• pieced on lhl b1srlng r•tlo l•ctors ot Tab1• A9 or· A.10 to \ndlcall yl•ld crlllc•l ca1•1.

S!ellc lglnl An!!ly,11 Qnll F1st11nn Sh1111r su1ngth: Obtain rrom Joint str•noth t1blH wh1n 1ppllcab11 or from 11h1111r s'uer!gth ·11bl1.1 16 and A7 (for solld rlv•tsl, Ind p•o•· 32 lor LK-Bolts 1nd thr11d1d l1H1n1rs. Sh•!"I B••rlng Sl~•nQth: Obt1fn lrom Joint str•ngttt l•bl•• •h•n 1ppllc•bl1 or lrom cslc.~l,stlons. •h•n using protruding hod ls11an.rs

Sh91tl Slr1nii1h:· Chack lh"t lor n•t "'' t1nslon •nd sh1ar .11aroul. Also v•rlfy adequ111 •dg1 .m1rgln.s, no knll• tdg1 conditions and u1• of equlvat1nt hole dlam111rs lar countersunk slM•ls.

· • Shffl lnterrlv•t buckling or Vl'l'lnkUng •nd 1H1cl1 ol t1mp111t1ir• on joint 1tr•n111h . Mi"rglna or, s.11ty '" sno•n on p1g1 ·A1

Edga 0!1tnne1 Yarlnt!gnJ . . All al the joint 11rangth !•bias ire appHcabla lo lolnls hevlng •n ldq• dl1tanc. l•I of two dlamaters (20).- Whtn olh•r itdqa dlstanc1s are tMllng anelynd, th• labia· v1h.t11 ere to bot reduc9d Hn11rly uslnq lh• tMl1rlnQ 1Uow1blas IF1,..1 1nd ftiryl Dras•nl•d In MIL-HDBK-5 01' 011ng. t1otor1 1n 'l'abl• .19 .nd. tor •ID . ot 1.so .nd. 2.0 ·

Itn:jlgn Agnllgallgn3 SOHd or bHnd rlvlll and bllnd bolts '" not •llowld In Drlmary tension 1pp11c1tlons. Tl'lly ere ellowed tn secondary t11111lon cases such 1s, skins with Hrodyn1mlc Hn tarc•s or lntern•I pr ... ur•, attachment of shear or comprHSlon webs, end strlngars to tram••.

For s1cond1ry tension en effective l1nsl\a Rpu!l-lhruR •llowaR!I• ol 20% of lh• ullllllll• 101n1 strength .i1owabla (11 the 1peclllc shfft thickness) 11 considered to bl consenrallve. This approech Is f9Comrnend•d tor 1ny l•stener type ll.IOl.ll1 tho111 hiving sheer haeds 1nd rlfducad head l••luru or If spaclllc d11l1 his bl•n developed. For NAS 1097 rtducea snaer !Med rtva11 us• HI% 01 tn. ~rtlculer Ultlm•I• lolnt tlrenglh 1Uowable.

FLUSH HEAD FASTENERS

Th• joint strenglh allowables for all flush head _fasteners ara based on t••t data. Calculated joint strengths are not permitted.

When selecting flush head fasteners and sheet thickness, Rknll..ad119" conditions are not allowed. Th• recommended design rallo (countersink depth/sheet tklckness) Is 0.8 for secondary structure and 0.7 lor primary structure with reversed loading. Saa Tab1"• -.AJ '"4 A4 lor the countersink depths of common fasteners. ii-1s:.A16 and. .US- · ptovlde equivalent hole diameters (De In count•rsunk sh11111t) for several lasteners and head styles. These diameters ~re used when dete~lnlng net "hola-out" In shHl m1tarial.

PROTRUDING HEAD FASTENERS

Th• Joint strengths for the following fastener types and conditions must be baMCI on test data:

Any fastener Installed In sheet material where tlD Is c 0.18. Aoy /olnt with protruding shear head or reduced head fast1ners. Any olnt with protruding head blind rivets or blind bolts (with al1h•r •h•ar or t•nslon type h111ads).

TIM! Joint str111ngth allnwables for protruding "tension" ha1d flstan• ... · may be calculated as described below:

Tna to•d per 1asten111r at which fastener shear or shnl bearing lype of f11lure occurs I• determined separately and th• lower of Iha two govarns th• design. 1 • Shear strengths for prot. hd. so/Id ri'fl!lts of any IJ!atarlal, .l.G.l.R1

plumlnym, are determined dlractly rro1t Table A6 For grotn1dlng sg!ld ah1mlnym rivets (tension tv~s) •n effactlvti rivet strengtk Is obtained_by mulllplylng lhf"'Tab;to Ao 1h111r 'i'alun by th• reduction factors or Table A7

2. Shear strengths lor protruding head Hl-J.o.!ss, Lk·B9l,a, and threaded fasteners are obtained l1•ted. 1n Tabl" ,.,

3. Calculate th• eff1ctlve sh1111t bearing strengths _for...!!.!...qrqtr11d!ng btnd :sgl!d rlyll!!! by multlplylng tha Tabllll AB "raluea : by th• ratio of allowable bearing stress/100,000 (K valu111a} . ·l)r Tabl" A9 or A10 for the applicable shnt m111t111rlal.

4, Calculi.ta the aHectJve sheet bearing strengths for ~ .L.k:. .B.ala ind solid !hr11gd1d tyoes by multiplying the Tabl• Ta1u"" by the ratio !actors (K valu••I 1n Tables A9 cir .A1·0 for th• applicable ah••I m111terlal.

TABLB A4 PAl!Tll!ID COUNTSllSUNK READ DEPTH fIN,)

Hl-LvKSILO..,KBvLT SCFIE.WSl8LJNOBul1:.. SHE.AR HEAD SHI FLUSH HEAD CFH) OR TENSION ITHJ

cA u TABLr. o Pc cAl 1AaLr. .... Hl11(<>•.,) • """"" HL13(~•w) " HO••v• GPL:JSC ,. LOCKBOLT LGPLBSC I LOCKBOLT LGPL2SC ,. NAS1456-62 I NAS143M2 ,. . NAS151S.18 I . LGPL9SC G . NAS7024-32 J' .

'" CSR924 F SWAGED PIN LGPL4SC J . '" HSR101 E . MS90353 K {1)Bll"!,DBOLT

HSR201 E . . MS21140 I' NAS1581 ,. SCREW AN509(STDJ H' SCREW NAS1620 •• . NAS517 H' NAS1670L H' f1lBLINDBOl T NAS1580 K' .

FASTENER DIAMETER IJn. E F G H .

I JI~ K ,,, .041°

"'' .041/.039 .04:1 /.034 ·'"' .070/.0b8 .074 /.073 .051•/,049 ·" 3116 .047/.045 .048 /.046 .045 .081/.079' l.u82 /.0511 ·'"' .oeo 1" .0611.059 .063'/.060 .059 .1081.106 .11Dl.108' .078°/.074 .111'1.105 ~" .0681.066 .070 /.067 -·~ .l:l5/.1l3 ,140'1,137 .095 /,09 ,140 1.1:11

~· .0781.07 1 /.0 .o ,1 .1 .IDB .11)5 .111 /,11u .1671.1'>:>

7116 .0971.094 .100· .190/.187 .· ,195"1.193 1'2 .1071.104 .111- .2161.213 .222 1.220

NOTES:

1)

2)

81\mf bolls {H', r 1nd KJ 1'9 lntand9d !or lht•r lppllc.ilon Ind HCOndll'y 1'1'nslon. lnt•~late hemcl. Prlm1ry shear ind ncondary lenston 1ppllc:allons. (Rallt.~Jj

Taoie Aj is loca~ea on p,A9

PIG.A18 llQUIVJ.LllllT BDLB.DL\l!l!Tl!BS (CSK JIOLES)

!:; 'W 0 d m ~ ;. :>; :l 0 ~ ~ :;; ;;; "' ~."' ~ e. ~ "

., " H c:: ~ b

"' ~ ;i w tu t; e ~ ::! " !il < ~ d 1S 0 "' w z w § ,;

N

~ " m " ...J m z ::; m " u

ii 0 ., " ;; 0 a

~ ..

:i: " a t; =j :;; 0 ~

" w ! c:: "' :i: ~ II! .... !:; ~ " .. 0 0

:: m ,; 0

I/) ;. ,; " ;;; a w ~ z ~ :: " 0. i! w " ~ ~ .. ~ b "' c:: ~

w z w d "' .. '1 .. 'I' :! w ~ ~ < .... j:: w 5 g I/) > < ,.: ~ ti i! u.. " => N ~

< u < !l " iii ;l II ~ .. ~ " H .. ·•n·::n ..

:! TMIHa 1~nYMrm0

All APPENDIX A ADDITIONAL DESIGN AND. ANALYSI~ .DATA

FIG.A19 l!QUIVALBHT HOLE DIAllETEBS (CSK HOLES)

··~.;::-:·-

·.<./·',·; ::·i'{'' , .. , '.,,·;: "' : ' " ' ' ~.··~~ .. ~:,

::LE;~5 ~lf~fuiI:~ ~;·~~~~~ FOR· ........ ·· .. • CALCULATED JOINT STllENGTH ' .

Jolnl ~:\~~··ri~~~~~:.:fo-1'.:;_h:~:'.f~:i,;o~j-~~- f~~1an·e~ types.are to, be:ca1d~1ated as descrlbad.BEFQRE ·.:AND QN,:TBB TABLES LISTED

;,j_:· :(·K~·6}~:J6:1·~,~~::·:.~~~-~$1d;N;:;:OR .UNIVERSAL HEAQ _s;~L~,~ "' ,;r::;·sa1)0:-»'R!VETs '.:_;: SEE TABLES. A6

.TBRU_ ·Al"O_., · .. '''."; -;-.<-·>'<,'' , ...

·;·.··_:.'. ,';

MS2047D AD & B MS20615M

"NAS1198 CSR903B -

MS21250 MS14181 NAS1003

UNIV. HD. UNIV. HO. UNIV. HD •. U~IV. HD.

ALUM. MONEL"~1 A286 Tl/Cb

NAS1303 (SERIES) NAS6203, 63.03, 6403, 6703 & 6803 (SERIES

· NAS464 '

TABLE A6 SOLID liIVl!T SllEAR STllENGTH

"'SHEAR ST~ENGnt OF SOLID RIVETS·

RIVET SIZE IN.

D~_l~~f!4?.·, ''" 1• -NOMINAL HOLE DIAMETER. IN.

DESIGNATION

RIVET MATERIAL: '·.··,_:,'

sosa-Hl21·F~~:.2a-uiti · e 2111.n·F,u'• 3tiksib AD

2017·131 F,u,,.:Mksl" D 20~_T·T3 F,u • 38 1ts111 D

1116 Jr.J2 I i/8 I &32 ! ·3111· 1 '114 I 5116 I 318

51 41 30 21':: r'•it1 ·;) i;:.:F:-_, "P

,;:097 6:095'. o:·failS' 0~15'9·: <:o~19'f·1- ·o.2sr· ·o.323 ' .-: , .. SINGLESHEARSTREHGTH<lbsa.d

w 0.386

·203 363:• 551 ':·802 1450 - 2295 327!1

217 "388" '59': ·~?:' 1¥!~. '24to 3510 247 . 442 S7!1 977 1755 2785 JHO

215 4M 755 1090 t970 3115 444s

29~. 531 814_ ,1175. ,2125 3360 ~800 296 531 . 114 1175 2125 3390 4800

311 558 a5' 1230 2230 3525 5030

:Z0.2~)':1'1 Fsu.~ 41 ~II DD 7050-T73F11u•41k.r:' E.KE 1asiJ.:m1 F1u • 43 q111 E.K~

MoMI Fsua '"' k•L 355 635. 973 1405 2540' '4015 5735

' Tl-Cb Fw • 53 QI ·384 687 1050· ··1520<' 2750 43'4(1" 5200 A.-286 F,;;.;·t1fk11 i51 t17o'· 1790 :·2590· - .q10'. ri7tl 111500

.. • b

' d

BaNd on nominal hole dlameler ~~ .. ~·. . , ,,,, v.iu.. - lat lhe • drw.n condition, on• probablLlty bUls'(B

. ~;~.~.v~lu-~:.~ e .'!'.~~, ~ .. ~ .~'iii~iµ;~b~t.d.-"' Apply 1hn1-ltreoglhreductioi'll~.;~BI.- ~' .wMf't 11.ing pnitnJdng 119-1 lllllmlnim lllloy ~ rtvsi.,

TAD A7 .. PROTRUDING:.l!i!AD .RIVET. Bl!:oili:TION PACTOBS.

..

'· .. ;,,."

PROTRUDING Hi:'.AO 'A.LiJM.·sbLiD ifrViri' sHi!AR'STA"eNGTH AEouCT10N l'~CT0RS (3), (4) (FOR ALUM. RIVETS ONLY) MS20470 HEAD STYLES '

DIA OF RIVE 116 ' 3132 "' 5132 "3116' 1/4 :. S/l&C 3/8, ,.. . . 0.01&, OS , 0.687 '"

.

!: ' 0.090 ~~ .. ':, ',.~'·'·':"'. -':~.:: :. i'.~·, 0.969 0.924 ~:=. g:~~ g:::~ ~.1_00

SS 0.995 0.972 DS.. o.992 o.951 o.870 o.789 . 11,roa

0.125

11.160

11.190

,~ .. ss. DS

·;;; SS DS .. ···'

"' ' ; ·~;

11.992 11.941 0.691

0'.981 0.939

.... Ul()O 1.CIO<)

NOTES: ., _ . ·., ... _ , " I. :U9:~ ~\:~'.u' is !hat, °:I. !hi! thin~ ~_ti,eet in s~l~,~ar ~ ol tti- midc:lla ~hoot in

2. Values nsted are' !or room temPerature uSe oni"V. " · 3. Values based on te1ts or aluminllm rivets. 4. Do not UH this t.able lor monel, Tl. or steel rivsts.

APPENDIX A ADDITIONAL DESIGN AND ANALYSIS DATA A12

TABLE A8 SHEET UNIT llEAJiING STRENGTH - LllS BASED ON Fbr • 100,000 PSI FOR PROTRUDING TENSION HEAD SOLID RIVETS

(~S 2D470 AND HEAD STYLES OF TA.atl!: A!))

Unll Bearln Strength for Rivet Diameter Indicated, lbs RJvETSIZE 1116 3132 1/8 5132 3116 114 5/16 ,,, HOLE DIAMETER 0.067 0.096 0.1285 0.159 0.191 0.257 0.323 0.386

ai .; "' w ·~

!..! I' ... w w :z: "'

0.012 so . 0.016 107 . 0.018 121 173 I 0.020 134 192 ' 0.025 168 240 321 0.032 214 307 411 509 0.036 241 346 462 572 ... 0.040 268 384 514 636 764 . 0.045 302 432 578 716 860 . . 0.050 335 480 642 795 955 1285 0.063 422 605 ' . 810 1002 1203 1619 2035 0.071 476 682 912 1129 1356 1825 2293 D.080 536 768 1028 1272 1528 2056 2584. 0.090 603 864 1156 1431 1719 2313 2907 0.100 670 960 1285 1590 1910 .2570 3230 0.125 938 1200 1606 1988 2388 3212 4038 0.160 1072 1536 2056 2644 3056 4112 5168. 0.190 1273 1824 2442 3021 3629 4883 6137 0.250 1670 2400 3210. 3975 4ns· 8425 8075

.· Where Dlt Is greater than s.s, tesls. are required 10 substanllate yield and ulllmata bearl~g su111ngths.

NOTES:

. .

. . .

. . 2741 3088 3474 3860 4826 6176 7334 9650

1. Desl9n values of sheet bearing strength are obtained by mult1~lylng the bearing strength (lbs) given above by the rallo ol design sheet bearing stress• 100,00Q. Determine values tor both' yield and ultimate. It 1.5 IC Pbry Is less than Ptiru the joint Is yield crillc;al arid Pbru :-1.S x Pbry. Bearing ratios are presented on TABI.83 A~ AND A10.

. 2. Values for rivets are based on hole diameters shown. See 'l':si. J.13 lor solid shank threaded fasteners/Jockbclts.

TARLE A9 BEARING STRENGTH RATIO FACTORS TO BE USED WITH TABLE.AS AND TABLE Al) DATA

~

~e~~~~~ 1111~~11~::;~~~~~~11111~1 .. ;: q ... &~:i! 800 I I I 1•• I ·r:; ::p:i ... ~~I I I I 151 3

< > ~

• • O! <>! O! ':: ~"!"!

w~ .· · ..

~~ 8"' I I I 19~ I 1:?~~ ... :; ~ I I I I I~ I ~c ....: O! O! CJ! O! <>!

p 5 Cl

;~ -M NOO I I I 159 I 11:3~!!3 ·-· -· I I I I 1: I

• !J !J ~~== ::: ~

~"~~~~~~~~~~:=~~gg:::s~~~~~ggg 9S · · · · · · · · · · ·· · ··· ··· · ·· · · · · ·· oo•r+~+-~.+~~-+-+-~+-~-+-~.+~~~t-.+--1 w1?;c:i. . ~~~~~~~~~~~~~q~:;~~~~~~~~~~~~~~~ > ~ --~r+--t--t~-t-'-~.++-~t-~t-~t--~~+--+--1 ' "'! • • • • • ••••

8~&~~~~~~~~~~~~~~~~~~~~~~~~~~:;~ :c~--- ........... ...... ...

5ci • • • . ' • . •..•. ~~~5:::::::s595:9s:s:!::~:sss~

"

I ~ ~

TABLE A10 BEARING STRENGTH FACTORS

• •• ~-

~~t:l:;~~:; l~~~ ~~~ I [~~~ I I I ~~~ I I I I I

0

• " ~~:ii~~!! 1-N• 5~3 1 1~!5 I I --· I I I I I w ~= :;: :::: ::!:::: ~ •• " • > " i> ~~ ~Vi~~::: 1 55~ 533 I I~~~ I I ~:~ I I I I I

•

~ "~

~i E iii J

::::;~ iii • ~S' ~ ~ ~a ~6' ~6 w

""""" ~ ~· ~ "' "~ -~ "" HU~ ~ .. ~!l ~!:!. ~!2. • --

TABLE All SHEAR STRENGTH OP THllEADED/SOLID SHAllK FASTENERS

SHEAR STRENGTH.OF THREADED/SOLID SHANK FASTENERS

F1u{MIN) 125 (KSI) 160 180 210 220 Fsu fMINl 75{KSI) 95 108 125 132

FAST. DIAM. IN. AREA IN2 ULTIMATE SINGLE SHEAR STRENGlH, LBS

3116 0.188 0.02761 2071 2623 2982 3452 3645

#10 0.190 0.02835 2126 2694 3062 3544 3743

1/4 0.250 0.04909 3682 4660 5300 6140 6480

5/16 0.312 0.0767 5750 7290 8280 9590 10120

318 0.375 0.11045 8280 10490 11930 13810 14580

7116 0.438 0.15033 11270 14280 16240 18790 19840

112 0.500 D.19635 14730 18650 21210 24540 25920

9116 0.562 0.24850 18640 2361$1 26840 31060 32800

518 0.625 D.30680 23010 29150 33130 38356 40500

314 0.750 0.44179 33130 42000 <Cnoo 55200 58300

7/8 0.875 0.60132 -45100 57100 84900 75200 76400

1 1.000 0.78540 58900 74600 84800 98200 103700

NOTES:

1. Shear values above are based on the basic shank diameters shown and ara also applicable to solid shank pins, I.e., lackbolta. The values may vary tor some fastener types when the procur111ment 1peclllcat1on lists different dlamaler1 and '· _,.:,· material shaar strengths (F1u).

AlJ APPENDIX A ADDITIONA,L DESIGN .AND ANALYSIS DATA

TABLE A12 ULTIMATE .. TENSILE. STRENGTH OF THREADED STEEL FASTENERS (MIL-S-8879 ROLLED THREADS) . .

FASTENER DIAMETER

FASTENER TENSILE STRENGTH~KSI 160 180 220

1D-:12· 1/4-28

511$-24 . 318-24 ·7116-20 . 112·20 9116-18

518-18 314-18 ,_, 7/8-1'4

1•12 1~1/8-12 .; ,,.,.,.

0.190 ,0.250

0.312 o.31s: 0.438· , o.500· 0.562

0.625 0.750 0.875 1.DDD 1.125

1iso· 1.37s''

.. 1.500

, MAXIMUM MINOR AREA IN

. '0.018602 0.004241

D.054005 , . 0.083879 :,

0.11323 ' 0.153511 0.19502

0.24700 0.38082 0,411327 0,64158 0.83129

1.0456 1.2844 1.51477"'

21178 .... 17!10

13 420 18 120 24 570 31200

311 520 57700

""' 102 600 ,133 000

187 300 205 500· 247 600

;3348 •s 1ec:i .

·11880 ;s 100 · .

i•'.2e13s9_"'' '27640

'3.~--,-~ .... ,. '&4''llOO 88 800

11s soo· ~·'~·. 188 200 231 200' 218 600.

12 oaO 1,8'450 2•110 ~3"~

'•"'' "·~,~ 54'3ci0

'"" 108500 1"(100 182900

TABLE A13 SHEET/PLATE UNIT BEARING STRENGTH --. LllS BASED ON Fbr ~ 100,000PSI (PROT, llEAD SOLID SHANK FASTENERS/PINS)

niAEAOEO TYPES: NAS 1303, B:Z03/ll303, M03, l1'03, '803, MS21250 HI LOKS;Ta!SIOH HEAD STYLES SUCH AS HL 12, HLT •n LOCKBOLTS; TENSIOH HEACI ST't'US MAS 1"65-72.AHD MAS 1~32

~=::·'l11n. ·5132 :111.•~ ··;,~ 5111 :Lii 1111 11.2 9111 w

, n;~tw;:.. r '.'~ IU541 D.1111 0.250 O~f2 0.lTS 0."38 0.500 05112 0.US

0.036•. · 563 ·ins

O.CM5 •. ·., 7fM &45 ·" •. 0.050 781 940 1.:t50 O.oa:J · 915 l,1llO 1,$75 "1,"59 0.071 1,110 1.330 1,n5 2,219 2,&&2 O.llllO 1,250 I.SOC 2,000 2,500 3,000 .3.500 0.090. 1,407 1.690 2,250 ·2.112 3,:IT5 3.!1311 4,500

:JI( 711

0.750 ·0.~75

!·'

: ., r--::":::::-"-1-;;;~::~o'r.;~~';:'r.::~:::=r~~~::.=·r.~:'=:;:::1-::~:~~=::r.:~::::i,;,:~=· ,. '''·""';-,"·"·:;-+-,-c1-,c-c1 0.160 ,.,.. 2.500 3,000 4.000 5,000 6.000 7,000 8.000 9.000 10.000 12,000

C1.20D 3,US 3,7$0 5,000 1,%511 7.500 1,750 10,000 U.250 U,500 1.!i,000 7,5DCI 211,000 0,250 3.!116 4.6118 6,250 7,112 9.315 10,!MQ 12,500 1',DSO tS,52!1 11,750 1.875 2..5.000

'' 0.312 4.867 S,866 7,llOO 9,T.M 11,700 13.570 15,600 17,530 lt.500 23,400 ,:JOO 31,20:! . 0375 s.aso 1.oso 9,375 11,100 1•,oe.:1 "·'25 11,150 21.075 n,'°"° 28.125 2,a10 37,500

7,llOO ,,,00 12,.500 15,600 11,750 21,900 25.000 28,100 31,250 37,SOll ,750 50,000 9,750 1,750 15,625 19,500 :0,440 27,375 31,250 35.125 D,Cll52 41,B1S ,190 U.!IOO

0.750. 1.700 t,100 ta,750123,Q 21,1u 32,850 37,500 •z.150 ol&,175 5&,250 lS,525 n,ooo · O.B1S 3.150 6,tsO 21,B15 27,300 3%.110 31 .. US '3,750 ti.17$ 511,l!IO 65,525 1.560 17,500

1.CIOIJ 5.600 a.BOO 25.000 31,?0ll 31.SOO '3.800 50,00ll 511.200 G2.500 75.000 !7,500 00,000

Wt..,. 01111 gQlttt' lhanS.5 IHll.,. qqiked 10 substan\IM• \Wfd M!d ul\llNl•b91'1!111 str•nqU'I. I. S'-1 IMIMl!lll MNftq\hl llW obtllned by mulllplytnf the-. v.luH (LBI by the .-IO ol <Hlllgn $11Mt

i....i1111 otnu .. 100,DOQ. °*l•rmlM ~•UH iorbolh yleld lndulllrnal•. N 1.5 1~1:1 lotu lhln PtmJ, the Joi~ II )'leldcrlllc.J>l •nd P"""'"°'.5 •hir:,-;: llorftlg ~1cq.,. Pf9M'll.-cl on 'rilLICil ·A9 AJID 110

z. v.iu.s tor 90llr:I .-ll'IQ- Wt.,,.,. - '*"' .,. bl-.d on 1111 -11lnll dlameltt'lo lhowrl.

TABLE A14 SOLID RIVETS.CSK IN 2024~T42 CLAD

',·,· ..

TABLE A15 SOLIQ SllEAR HEAD RIVE'rS CSK IN CLAD ..... •. ,.·" .. '·I"' .. ,. · ...

RIVET

SHEET MATL DIAM. (HOM) SHEET

(th. 0.025 0.032" 0.640·.' 0.050 0.063. 0.071 0.080 0.090 0.100

. 0425 0.160 0.190 0.250 SHEAR

; ' . ", ,,, '• ,, .... " .... , ' <: 1"'< i . ' ,' ·1,.

ULTIMATE STRENGTii,OF HAS1D97.·E RIVETS COUNTERSUNK IN CLAD ALUMINUM AL.LOY (1 oo•:sHEAR HEAD~

·. ··,\ "· ·"'

NAS1097·E (FSu = 41 ksl) (7050-173) . · 1 .. · .•. . . .........

Clad2024"-T3 : Clad 7075-T6 .,· .·' . >'•

1/8 5/32. 3/16 1141 1.8 5132 ·, "3/16 (0.1285) (0.159) (0.191) (0.257) {0.1285) (o·.1s9f' . (0.191)

., . UL~MA'fE: S!RENGTH, LBSC2)

= -· = 326 run ' 354 !illl - - -~· 437. 505 !WL ,627. 439 547. fil1l

466 679 m l2lW 456 674 823 ·485 717 1005 ··1275 •n 700 980 497 731 1025:. .1500'. .49i) 716 . 999 507 747 ~ .1045' . 1750 505 734•, ,102~

521 765. 1065: ',.1640. .520 . 754 1045 531 781 .. 1085 1870 53L n4 '107,0

- 814" :·:1135': 11i:is:: - i:i"14 . 1130 1175 203°' ., . 1175 - - - 2110 - - -

•·. ''2125:. 531 814 1175. 2125 531 814 1175

. ··. ... ' · .. .

1/4 . (0.257

--""" 1330 1570 1760 1790 1825 1905 2020 2115 2125 2125

Values are for ioOm 1e·mp.r~i.Ure use only. · . :Values lhat are fundarf!nedl are knife e_dge condlllon. Sea page .. 21 tor ~.0:mma11ded ;i;heet thlCknesses.

APPENDIX A ADDITIONAL DESIGN AND ANALYSIS DATA A14

TABLE A16 SOLID SHEAR HEAD RIVETS CSK IN CLAD

RIVET

.~ "·---0.025 0.032 0.040 0.050

···-0.071 o .... ' 0.090 u.1w 0.125

SHEAR

ET

.~ 0:032. 0 • ..., 0.050 0.063 0.071

0:090 ~100 0.125 U.160

I 0.1DO ::iHEAR

NOTES:

ULT;~ATE .. Si-REN~~ OF. MS1~21e~o R:~~ COUNTERSUNK JN ALUMINUM {120° SHEAR HEAD)

MS14218AO !F,.0 ::: 30 ksO (2117-TJl 0 2024-13

UL Tl MATE STRENGTH LBS 141

3132 118, : 5/32 ;L'.16 7Q2

""' -""' 12121 ' - - -'" '" - - -216 322 "' lllil ,-;;"' 217 390 "' '" "' ~· '51 '" - - 5'6 917 1015 - '" 1125 - - - - 1205 - -· - 1225 - - - - -217 "' "' "' 1225

ULTIMATE STRENGTH OF MS14218E RIVETS COUNTERSUNK IN ALUMINUM {120" SHEAR HEAD)

MS1421BE (Fsu,. 43 ksl) (70SO-T73) cl 2024•13

UlllMATESTRENGTH LBS

v.J2 ~16 7Q2 114 OQ2

- - - -lllil - - - -479 - .. ::-.. - -"' 732 - -'" 10'5 1135 = Ullil 903 1110 1365 "" 15:l0

'" - 1140 Hi65 1735 1835 ... 1175 1605 ,,,. 2200 - 1205 1"45 "" 2525 - 1230 1740 2140 2650 - - 1'55 2- 2S20 - - - - 2B"

"' "" 1755 2230 2840

1. V•luH- i.r• lor R.T. uH only. 2. V•lun lt••I ••• Wndu!\nt.d) ••• knll...tgi• """""'""· S.. p•gi• A 10.

r•commended ahMl l~lckMHH.

114

--, .. ,. 1--1155 1"0 1425 1---1555 1555

'"' ----...... ,,,. 2320 2725 3205 --3"' 3525

TABLE A17 SOLID TENSION HEAD RIVETS CSK IN ALUM.

RIVET ·SHEET

ULTtMATE STRENGTH OF MS1"219E RIVETS COUNTERSUNK IN ALUMINUM (120" TENSION HEAD)

MS14219E (F ... '""3 ksl) (7050-173) CLAD 202.t-T:J

ULTIMATE STRENGTH, LBS'l21

.~ 5132 3116 7132 114 ''"

'

0.050 0.063 0.071 0.090 0.090 0.100 0.125 0.160 0.190

SHEAR

RIVET SHEET

---....:: '·"' 0.050 0.0<3 0.071 0.080 0.090 0.100 0.125 0.160

SHEAR· .. NOTES:

lillJ """ -,., ..,., - - -798 "' = - -"' 1105 1290 - -'" 11&5 1520 = !lllil

1230 "" 1800 = - - 1755 2145 25&0 - - - 2230 2540 - - - - -'" 1230 1755 2230 2840

ULTIMATE STRENGTH OF MS14219E RIVETS COUNTERSUNK IN ALUMtNUM..{120" TENSION HEAD)

MS141219E (Ftu"' "3 ksl) {7050-T73) CLAD7075·T&

ULTIMATE STRENGTH, LBS (21

5132 ~" 7/32 114 "" """ - -- - - - -903 = = - -"' 1140 = Uilll -... 1180 1600 .. = = 1230 1650 = = - mo 1700 """ = "" 2230' "'' - - - - "" • 854• • 1230 1755 2230 ""

1. Values •r• lor R.T. usa only, 2. ValuH lh•t llt• (~ 11re knlfe-«!g• condition. S.. page Al D,

recommtondlld shHt thlckneu•s. ·

"" -----= 2965 3415 3525

""

"" -----

l2illl <llilll 3295 3525 3525

TABLE A18 SOLID T1-45Cb RIVETS CSK IN ALUMINUM AND IN T1-6AL-:-4V

ULTIMATE STRENGTH OF COUNTERSUNK BRFS. T RIVETS IN ALUMINUM AND TITANIUM 1120" SHEAR HEAD\

RIVET BRFS-T (F•u"' 53 k1J) {TJ..45Cbl

SHEET CLAD 7075-T& I Tl-8Al-4V ANNEALED

ULTIMATE STRENGTH. LBS 121

.~ 1ra v.J2 3116 1ra v.J2 3116

,..,. ~

""' - """' -'·"" 360 lill1 '" (ill) -0 • ..., 461 572 ..... ... 7911 """ 0.050 "' 713 "' 602 967 ,, .. o .... 910 '" 1 ... 650 '" 1270

0.071 928 ,,. 1220 690 ... 1310

0.080 ... "' 1300 "' 1005 1360 ..... 671 "' 1'30 1050 1420 0.100 "' "' 1370 - - 1"70 0,125 - 1050 1450 - mo 0.100 - - 1520 - - -

SHEAR '" 1050 1520 "' 1050 1520

Ul. TIMA TE STRENGTH OF COUNTERSUNK CSR902B RIVETS IN ALUMINUM 202.t-T81 CLAD 1100" FLUSH HEADI

!WET I CSRll02B (F., • 50 k1I) (TI ... SCb) "'! SHEET ' CLAD 2024·T81

ULTIMATE STRENGTH .l.BS!2l RIVET DIAMETER 3132 1ra "" ~"

SHEET THICKNESS (IN.)

0.020 0.025 0.032 ,.,. " o . ..., "" 1'Sz..:l - -0.050 361 458. .... " -0.063 - I' ... "" -0.071 ... 910" ...• " 0.080 ... 1172 •

0.090 ,.,, RIVET SHEAR 362 ... ... ,.,,

NOTES: t. V1lun "•for R.T. u11 only. z. vatuet ti.t ••~••for Mlle«I~ condition.

Slit •rU'ftKa 113T&!i" 11!11 Ul:!Olllll:llDm Slll!rl' TllICDllSSU 1 • DenCltff UIUITlllll l'•luee ti.Md on 1.5 l )'Mkl. ·

TABLE A19 SOLID MONEL RIVETS CSK IN SS SHEET,

t~ ~i E • .2 ;; •• iii£

A15 APPENDIX .A APDITIONAL DESJ;GNAND ANALYSIS DATA

TABLE A20 TENSION 3TJ!ENGTH OP SOLID ALUM, RIVETS

EFFECTIVE ULTIMATE TENSION.'STRENGTH. OF ALUMINUM RIVET~. I~ .Al;-UMINUM SH.EET..f\O~--~E~ONOARV LOADING)

,020

' '.025 • 032

.O<O ~so

~" .071 .080

.090 .. , .109 .. ,, ";125

RIVET

SHEET

o05Q. ~OGJ .011' .080

.09?. . ;100

1'0

180 240 300

375..i

"'

'"' 220

'260 . ., '350

430:

560:( 642

3116

320 '20 520

,c84Q

700

"' "'

1/C

570 700 860 870

1100

mo 1380 1650

BARE & CLAD 2024·T3, To1, T&, T8X, 707S-.T6

118

ULTIMATE STRENGTH- LB

·-· 5132

360 '-4'0

- ... 535 590 578 ... :715

1/C

MZ 925 .125

.16(!

1350 -. -.-,·- ' 1485,,

'' - ' ·• .

TABLE A21 BLIND PROT, HEAD RIVETS IN ALUM,

ULTIMATE STRENGTH OF BLIND PROTRUDING HEAD ALUMINUM - ·• ALLOY RIVETS /N·ALUMINUM-SHEET 111

RIVET TYPE '. ;~~ .. ~~:.~~~-~: .~-'; ;_NAHSA~~::~D, ; ~ "'!1.~~~~~D CODE A IF • 31) k•I) CODE A (f..J • 311 hll tF • 3• hi)

HEETNATERl.aL CLAD 2024-T3 AND HIGHER STRENGTH ALUMINUM ALLOYS RIYET DIAMETER 118 5132 3/!tl 114 1/8 5132 3118 •. 1/4 ; 111 5132 3111 HOt.E OIAMETER ll.1'30 .lU G.IH ll.2511 0.130 0.182 .1H .2511 0.14' D.171 D.2a7

' ' .

ULTIMATE stR~NGlH ~~ Bll~·o pfi6TRUDING HEAD MON~L: RIVETS IN ALUMINUM SHEET (1)

RIVET' NAS 1398 MS MW ANO CODE A F-· "55 ks! SHEET CLAD 707S. T6 & HIGHER ALUM ALLOYS

ULTIMATE STRENGTH LBS.

,---LR .. ,, ~/B' 5132 .. ~ .. 0.020 ; '! '" '"· 0.025 . ,,, 361 394 0.032 ; "' '" '" 0.040 '" 624 m 0.050. '" 720 "' 0.063 647 '" 1072 0.071 ' 710 '" 1168 0.080' .. 1009 1272 0.000 . ·1090' - 1387

·0.100 .. "" 0.125 1580

RIVET SHEAR 710 ,. .. 15110

NOTES: 1. Values .re !or room 11mp.-ralur• uu only.

'(i~ld str•n;th It I nexcess of 80% GI ullllrMll

TABLE A22 BLIND. PROT,. HEAD .Rrl'ETS IN ALUMINUM

ULTIMATE STRENGTH OF BLIND PROTRUDING HEAD ALUMINUM ALLOY {2017} RIVETS IN ALUMINU~ SHEET (1) .

' . " "(OVERS1Z_E DIAMETER) '

FllVETTYPE

SHEET MATERIAL RIVET DIAMETER HOLE DIAMETER

SHEET THICK 0.020 . ·,

0.025 0.1Xl2 0.040 0.050 0.063 0.071 0.080 0.090 0.10!!

RIVET SHEAR

··.···· •.:

SHEET MATERIAL . RIVET DIAMETER·

HOLE DIAMETER

. SHEET THICK

0.032 i\'1'·• o.040 >'· .,,,

.,·::· '.'" . ._·:.: .. '_'

' CLAD 2024·T3 -118 . 5132 .• 1-·- 3116

10.1441 10.11a1 _ ·:m.20"7\ . UL TIM ATE STRENGTH las.

373. 429 ... '" 6_19

"'

307.

"'' 577 659.. .. 767 830 ... "' 935

'. N~S:176BD (FiU,. 3B'k'sl) . . .... ... .. CLAD 7075-T6

"' 677 810 934

.1011 1097 1195 1260 1260

..

·118 "' 5132 ... , 3/16 10.1441 I0.1781 ; ,;.: m.zoTI

ULTIMATE STRENGTH LBS.

.... 579°

"' ' 676

'" 963· Gut" 0.071 627 1028

t-~...,o;·";;;'c~·'--1-~~~~-1-'-~'~'°o-~+-'-';"~'~-1' 0.090 , ... ;· ". ' 935 1176 0.100 1260

RIVET SHEAR 619 935 1260 .. NOTES:· - ' ·. . 1. Va1U•~ Ir• IOf room l•m!M'tllur• u5e only.

TABLE A2J BLIBll l'ROT, READ RIVETS IN ALUMINUM 't'.. -.·'·~";:' ~-·-~· ·~··

...

. ';f - •"( ,,!' ,,, •. :

FASTEHER DIAMETER. IN. ' 'HCM!HAL CIAMETER IN.' -.,.

SHEET THICKNESS, JN. O.Dte 0.0211 0.025 0.031 a.IMO 0.050 O.OG3 o.on , __ 0.090 . ,,,

111 . '0.1440"'

122 '

"" ''2H

'" -, •n . q'

"' ... "' -- :::

C::R32'3 PROTRUDIHQ -2024-T3CLAO

·. o~:O·.-l·o:::o I ULTllU.TESlRE GTH LBS.

_.- 240 · .. ~:· 317 .,, ··u2 ...

·~ "' q' "' '" •» "' "" 1150

'" 0.2530

'" 0.2730

~~-1D25 ·= "" 1.5115 ,, 1095· . llOO .....

o.ns D.U50 O.!to 0.250 0.312

· c7H.', n>•

"' "' '" -'" 105!•', 1H3• ,,,,..

·~ 1!1S-'­Ut5" 1575°

. "r~:~·~?

FASTENER SHE.aR STREtrGTH no .. -,-NOTES: •. "1 •. Vah.lnaN IOftooml..rtp9ra1ur1onl)I. """ ~.

•• • <HllOI" u~lmat• nlua ~Hd on 1.s a ri.111. · 3. PtellmlnaryallQw~ ChMTyT1rtron 1n11ng.

·-· "'' n•s· ·-· U25

APPENDIX A ADDITIONAL DESIGN Aml ANALYSIS DATA A16

TABLE A24 BLIND PROT, HEAD RIVJ!TS IN ALUMINUM

ULTIMATE STRENGTH OF BLIND PRO~UDING HEAD MONEL RIVETS IN ALUMINUM SHEET ...

RNETT'l'~I! .. '· . CR.1523 !Fou • es k•O SHEF.T MATERIAL 0.AD 71115 .. TS ,

RIVET DTANETER, FM. .. I ~' j,, .::,~~ •. •NOMIHAL HOLE DIAMETER IN,• 10.130' a.1n• ULTIMATE STIIENGTll LB9.

SHEET THICKNESS, IN. ' • .,,, ,~ . "' .

'·"' "' ,.. ··- m <M "' ··- "'

,., ~

""' "' '" m .,., "" "' "" .·. om1 "' ·~ mo

"'" "' •m• "" '·"' - "" ·~ 0.10) "' "" 151111,, 0.1M ~ '"' 1725 0.190 "" .... 0.1to 1120, o.:rsa

RIVET SHEAR STI!ENGlll '" ·- 1920

ULTIMATE: STRENGTH O.F BLIND PROTRUDING HEAD MONEl RJV~~.JN .ALUMINUM SHEET (OVERSIZE DIAMETER)

' AIVETTYPE HAS 17AM CF.., •' S5 k .. I SHEET M.tl TIORIAL CU.0707~11

'" O.l581

on

'"' .. u HD

·~ "" ""' mo ... "" ,.,, ,.,.

9--RIVET DIAMETER, IN, •• I ~' ...

•NO .. NAL HOLE DIAMETER IN,1 ·'"'"' '0.178' ut. T1MAT1' STRENGTH LBS.

SHEET THICKNESS, IN. 0.,. "' "'" 0.025 ,,,

"' 0.1131 ... '" '" 0.- "' - "' 0 .... '" "" ... 0 .... "' ... "" 0.071 "' .... "" ..... "' "" UH o,,,. ... UIZ "" .... ""' "" D.125 "" "" RIVET !HEAR ...

·~ ""

TABLE A25 BLIND Pl!OT, HEAD RIVETS IN ALUMINUM AND AISI 301

ULTIMATE STRENGTH OF BLIND PROTRUDING HEAD A-286 .. RIVETS IN ALUMINUM

RIVET TYPE CAU23 (F.., • 75 k .. I

SHEET MATERIAL CLA0707.S.T8 RIVET OIAMETER, IN. •• I ~·

... /NOMINAL HOLE DIA'""TER IM.' I0.1:10l "'.1H 0.19'

ULTIIAATE STRENGT Ll!IS.

St\EET THICKNESS, IN.

··= "' 0.025 ~ "' D.032 "' "' -.... ... ... "' ., .. .. o m ~

o.ou m ... '"' 0.071 ... '"" "" o.~ "' "'' "" ·- ... "" oM• 0.1tQ "' .... ·-D.125 "" "" 0.160 2215 .... 0.250

RIVET SHEAR STAEHGTH "' .... 2215

ULTIMATE STRENGTH OF BLIND PROTRUDING HEAD MONEL RIVETS IN STAINLESS STEEL SHEET

RIVET TYPE NAS 13H MS OR WW AND HAS 13H MS OR MW CODE A IF111 • !5 k1tl

SHEET MATERIAL AtSI X11•HALF HAllD RIVET DIAMETER, tN. •• ~' ...

I 'NOMINAL HOLE DIAMETER IN.• "'.1:10 '".182 10.1!M• ULTIMATE STRENGTH LBS.

Sl'iEUTHICKNESS, IN. o.= '" ... "' o.= ., ~ "' 0.032 - ~ '" , ... ... "' ·-o ... '" "' 1270 o.ou "' "' "" 0.071 "' '"' ·~ ..... '" ·~ "' 0.- ·- 1•71 0.100 ·= 0.125 ·-RIVET SHEAR "' ·- ....

NOTES: VII~• ar• for room l•m1>9r1ture us- only.

I

.

,,. "'.251'

"' 1185 ·-·-1t25 2175 uzs -=o

"'' mo 3920

Yl9ld 11r1nglh Is In ••c••• of BO% of umlm•ll

TABLE A26 BLIND CSK RIVJ!TS IN ALUMINUM

ULTIMATE STRENGTH OF BLIND.COUNTERSUNK HEAD RIVETS 1.N CLAD ALUMINUM ALLOYS (100 .•HEAD) 11)

RIVET TYPE HAS 1391 BAND HAS 13111 0 AND

·. ," N4s 1319 B, CODE A (5058) NAS 1311 0, CODE A (2111n tf_,,.30 kill ... lf'-·•38k90

SHEET MATERIAL CLAD 202(.T3 AND HIGHER STRENGH ALUMINUM ALLOYS RIVET DIAMETER ,,,

'"' 3116 ... ··io~1~,J1' : 5132 . CNOMINAL HOLE! ·'co.1301 10.1121 10.1u1 '10.1121

... ULTIMATESTRENGnt LBS. SHEET THICKNESS

0.032 .•

'"'"' UllL:l llJICl .... 171° = 171° = 0.063 .... 273' W1'l ,... 273' 0.071 "' 368. 330' 311' 308.

'·"" ... 474. "'. 423 474° O.OllO 571 ..... "' 5114° 0.100 "' 740° ... '" 0.125 .. , 755 0.160.

RIVET SHEAR 36B "' '" ... 755

•'•" RIVET TYPE NAS 13111J _,.. vR MW

MON EL · AN.D NAS 1311J MS OR MW, CODE A

" IF.u:o55k911 .. -~- .... ,,..., .. Ar11u I .· ... ··. ALLOYS

""" ··-···· 1m 10~11'21 ~ .. !NOMINAL HOL~ f0,1301 . f0.1Ml

I A11:. ::11R SHEET THICK ...

0.032 ..· ..... = 0.050 ,,.. l>ZI!l

'""" 4117° ,,.. ! = 0.011 557 '". 677' 0.080 "' '" ... ..... "' 873 1031 • 0.100 "' "' 1175 11.125 71Q ~:a -"'c·· 1 ••• 0.160 1505 0.190 "" RIVE ~H 710 MO ""

NOTES 1. V•h••• ••• for room l•mpmntur• 'u1• 11nly. 2. • Oenot1s ullimll1 values blhd on 1.5 x yL•ld. 3. V•lu11 lhll •• lllmlHllDUl •r• knlle IKlg:•.

TABLE A27 BLIND CSK l!IVJ!TS IN ALUMINUM

ULTIMATE STRENGTH OF BLIND COUNTERSUNK HEAD RIVETS IN CLAD ALUMINUM 2024-T3 ALLOY (100° HEAD) (1)'

NAS 1739 B :t RlVETTYPE AND NAS 1739 E (F.,~ 34 ksl) '

SHEET MATERIAL CLAD 2024-T3 AND HIGHER RIVET DIAMETER, lH. "' .. ''" ~ ..

/NOMINAL HOLE DIAMETER. IN.I I0.'144) 10.1781 10.2011 ULTIMATESTRENGTH Les.

SHEET THICKNESS, IN. 0.020 0.025 0.032 lllil ..... 266 """ o.oso ... "' , . ., '41 533 lilll 0.071 "' 608 ,,. D.030 554 "' 794

'"'" "' ... ~100 837 1015 0.125 1128

RIVET SHEAR STRENGTH '" 837 1128

RIVET TYPE .. ·. HAS 1761 D T .. "'38 ksl) SHEET MATERIAL CLAD 2G24· T3 ·

RIVET DIAMETER, IN. 1ra "'' ~ .. I /NOMINAL HOLE DIAMETER IN.I 10.1441 ro.1111 f0.2071

UL TJMA.TE STRENGTH LBS. SHEET THICKNESS, IN.

0.032 l1ll'1 ..... ,,.. W1'l 0.050 36' 447. "1lt:l 0.063 ... '" 675' 0.071 ''" 624 70 0.080 ... '" ... D.090 "' "' "' 0.100 871 1020 0.125 '" 1260

RIVET SHEAR STRENGTH '" '" 1260

NOTES: 1. V1lues 111 lor room t1mpera11rre uu only. 2. • Deno111 u1Um111 v1lu1s bls•d on 1.5 x yield. 3. V1lu111lult•1~1reknll•lldg1.

~" I0.11Ml

..,,_,, 330'

'". 511° 740. ...

"" ""

A17 APPENDIX.A, Al)DITIONAL.DESIGN AND ANALYSISDATA

TABLE ·A28 BLIND CSK RIVETS· IN ALUMINUM

. UL·TIMATE STRENGltt'oF"tiLIND coiJNTEn;S'UNK HE.Ao RIVETS' IN CLAD 7075-TB ALUMINUM ALLOY (100•-HEAO) 11J

-SHEET MATERIAL · -· · CLAD 707S-T6 -'118 . 5132 3116

f0.1<1141 ' 10.1781 10.2071

. ,~ SHEET THICK~ess.:·:~: ULTIMATE STRENGTH ·Las •

... 0.032'"' ---0.040 o.oso;-.·' 0.063 ''

'.' 0.07,1 , .. ... Q.080

: 0.090 ,.0.100 ,:·'

·,::;-.·-~:~:~---}' ;. ' .

-.RIVET SHEAR STRENGTH

•.

:'ii'

""' """' 382 ""' 485 592

"' '" . 569'" 7411. 619' '" '• "' > '35

'19 9J5

SHEET MA TER\Al.:' · • ' ·."! :''-, CLAD 707S-T6

0.011· _, o.oeo'

"· " 0.090:

0.160 --· 0.190'·,

. RIVET.'SHEAR STRENGTH

686_:·: 724 765 .805. , .. 895

',.

895

«ti•1 528·'·'' 732• 117.

1007 ..

"" - '., -11011.

'"'' '"" ,', '•

1353

NOTES: ·-.· .. "- ' -;,.-.:· - . _::_·::·: . 1. ·Velu1a 1ro tor room lempin1ture u1e· Only.

'

..

2. • 01nol11 ultlmall V•lues blind on 1.5 x yleld. 3. ValUHlh•l"'•-~··knltt_Hge.

'"""' m

'" 9 ..

"" 1237

"" -.1250

. ... 1170° 13.ls

·- 1405 1551 1757 11123

""

TABLE,A?9• BLIND.CS!( RIVETS:IN ALUMINUM

Ui.TiMAfE'Sl.ReNGt;H.·6~ 'lfl~i.Nci:d6'UNTER~U'~:~-..i~Ab- Ri~~s IN.CLAD 7075-T6.ALUMINUM (100" HEA.0)'111

.,_ •• ,._,_, • . ...... ·-·~· • ····«-·"'"'""'

...

RIVET TYPE \ ,;-

NAS 19:Z1 ~J~~ .~--36 kal) ~

SHEto MATEAlAL 'i' CLAD 7075-T&

SHEE·f'fi.t1CKNESS, It( 0.032' "' .....

ULTIMA E.STAENGTH LBS.

0.050: 0.063 0,071 0.0110' .0.090 •

' 0.100' 0.125 1

·'•'

li.160.";: '." !

(165 °) 232 313•··

"" ''416,

"' . '494'

1257,")

'" <27 '4911,' 571

'" 75~,

RIVET SHEAR STRENGTH 495 ~ 755

' -.. '~~{.,, .... '

SHEET THICKNESS, IN:·: !' ·: · ... o.oso.- ,.. """ ,;531 ~,.·

0.063''. 0.011 o.oeo o.090. 0.100 0.125 0.1&0 o.1110

'671. 756°-

;~ .. 11311•

1020

''

831. ,,.. >oo> 1111-'4. 1328. 1'453. 1565.

5'48. '• 6911°

1 ••2 1-428° ·a111· 15llO. 1159. 18118 10211 • 2145. ....

. RIVET SHEAR STRENGTH '"' 1565 ~ .. NOTES: ..... '" .. " .- -,,._' ;;':>.'.<·-·-:: .. ,:.:;:,::: .. 1. V•lun 1r9 for f(lom l1fnP9111tur• UM only. 2. • Denot•• ultlm•I• valOH b11ed on 1.5 x yl1ld. 3. VllUH !hal-lr.• fundpr!!nttfl ••knife ldge. ·

('4051 f4941

5" "' 748

.1178 '1090 1090

..,. ... 1108. 1310.

'"'. . 1'485.

13117. 1658. 1598' 1988.

2-4~·

'"'0

TABLE AJO BLIND CSK RIVETS IN ALUMINUM

·uL'r1MA1e·smEHatlt"oF BL'~&-eOuHTERsuNK HEAo r:uvrn ICHERRYMAX't IN 20i-4~TJ ALUMINUM.ALLOY 1100" HEAOl (1)

RIVET TYPE CR3212 (Fi;...• 50 ~I) . . .. ,

SHEET THICKNESS, IN. ii.=· 0.032 ;~

' ·,-,,, ~

0.040''• l2ll.:l: W1'l .

0.050 ·: 302•·- lill.'1 .,..!~!!~? 0.063". 380"". '47'4•

o:~ .. ::,·, "' ,~, -· "1 iili.:l '474. 591 ,- '. '725.

0.090" 511 . IS61 • 810° ' 0.10Q._ . ... , ... ,., ••J.

0.1~.-:·-~; '". 852. '"1085 0.160 "' .... 077• 1268.

.·· 0.190 1007 "" 0.250'" 1'466. ' 0.312 ... 1030 1'480

RIVET :TYPE'" .

1197. 1490. 1875 2100 2453. 2565. 2615

SHEET TH1CKffEss, .,.;;·. __ :.

00.25

•0~1~' . : · r·-312..'... . I '0~~3'

R

0.032 w.n · . 0.040· 275•

0.050

0:100·" -"<'·" 0.125 0.160 Q,1QO 0>50

'" «5 St• 545 .... "" "' "' ... I

543 .. _f.HCl 605 71'4 701 71M -'" 921

;.;.; '- ,,, •1035

·', ·1165 ·'''13~

1"25·

"" ... ' - :;:( ~:;~~ ... ,-. ... -~:: .. 1 • .. ·: 190:»

>H "' 814 12-45 "" lfQ1'1:;::J;' ·' l ' "-t. ..,..._ ... ""._l~Ullllftfr. 1.- • Dwm" urthlM• r.iu.1bllldO!l1.ll • yi.ld. :a.. v-....11111·- tumlllmtio- •"'19 "'9•· C. · P'.-mn.ry1~C~T..:tronlffllna.

~

•• '!!1-2 '200 1315 1890 2000 mo mo ,,. . 2925

""

TABLE AJ1 BLIND dsK.~iVETS IN ALUMINUM

"""""

RIVET TYPE CR4522 'F · ::'65 kall , · ·SHEET MATERIAL -- '"' _.:.· · "' Cl;An 7075-T6

. RIVET DIAMETER; !ff.' """· 'NOMINAL HOLE DIAMETER IN.

SHEET THICKNESS, IN, 0.050

n"·

0.063 0.07,1 o.osq. 0.090 0.100 0._125 0.160 0.190 0.250.: ·"I"' 0.31:i

RIVET SHEAR STRENGTH

RIVET TYPE

... ','

,,•

.... 5111"-,.,. ,,. 778

"' 852

"'

"' SHEETM.t.TERIAL. , ... ' .. ,.

SHEET THICKNESS, IN, 0.050 ' 0""3 0.071

' 0.080 0.090.,

~:!~ ,'_ 0.160

·~····· o:m·· 0,250

RIVET SHEAR STAENGTH

NOTES:

576. 734.

"' .., '" '95

995

'•

••

. ULTIMATESTRENGTH LBS.

I'.".'•'•

·-410• 612. ·•··. .. ,. 725.

1095 1032. 1170 1332" 1240 169( 1335 1810

''" 1910· .... 13'0 1920

.. CR-4622 'F- "'75 ksll

CLAD 707S.T6

... •o~~;'·:.·:· -- _, '3/16 · 10.2on'

ULTIMATESTRENGTH LBS. ,.

509° 705. 930. 861'

1181. 1161. 1420. 1-455. 1525 "" .15-4~. 2215

1545 2215

.1.. .ValUH ara for room l1mptr1lur1 Uh only. 2. 'DtnotH ulUm111 values blsed on 1.5 •yield,

•

2033. 2975 3105

"" 3'00 .... -- 114

'' IQ,273'

12110. 2258. 3605 ;,.• 11'· 3810 3920 3920

.;;,:

APPENDIX A ADDITIONAL DESIGN AND ANALYSIS DATA Al8

TABLE AJ2 BLIND CSK RIVETS IN AISI J01 SS

·ULTIMATE STRENGTH OF BLIND COUNTERSUNK MONEL RIVETS JN ~ISi 301 SHEET (100" !-1E~0).511

HAS 139" MS OR MW ANO RIVETTY"E HAS 1399 MS OR MW CODE A IF-·" 55 ksll

SHEET MATERIAL AISI 301 ·HALF HARD . RIVET DIAMETER, IN.. , "' '"' ~"

I !NOMINAL HOLE DIAMETER IN,l 10.130) f0.1621 10.1941 ULTIMATE STRENGTH UIS.

SHEET THICKNESS, IN, 0.032 0.040° •' ·-·· ., '·"° 383 .,,...,, .... 0.063 491 ... 0.071 ... '"

.... .,. .. 0.080 657 "' "' 0.090 "' "' 1032 0.100 1019 1182 0.125 '"" 1580

RIVET SHEAR STRENGTH 710 "" 1580

NOTES: 1. Valu•!I 1r1 tor room lemp11r.tur1 use only. 2. •Denotes ultlm1111 v11lu1s b«slMI on 1.5 ll: yleld, 3. V1h1es !hat ate Uulll.adl.D.mll at• knll1 edga.

TABLE A34 BLIND BOLTS CSK IN ALUMINUM

ULTIMATE STRENGTH OF BLIND BOLTSl100•HEAD} IN CLAD 7075 TS ALUMINUM l'h

FASTENER TYPE MS21140(F.,a95k!il)

SHEET MATERIAL CLAD 707S.T6 FASTENER DIAM, IN. 5132 ~" "' "" /NOMINAL SHANK 01AMI 10.1631 10.1981 10.2591 10.3111

ULTIMATESTRENGTH LBS. SHEET THICKNESS ·.

0.050 0.063 0.071 . ..... , 0.080 876. WJ..:l O.O!IO 1053. 1095. 0.100 1229. 1352. 11a111 •1"

0.125 1673. 1890. 2153' ~:llJL'.l 0.160 1972. 2640. 3135. -- 3428' 0.190 2880. 3983. ""48. D.200 .. of260' - 4785. 0.250,· ..... 1480' 0.312

FASTENER SHEAR 1980 2925 .,,, 7215

RIVET TYPE NAS 1671M. -

SHEET MATERIAL CLAD 7075-T& FASTENER DIAM, IN. 5132 ~"

,,. "" /NOMINAL SHANK DIAM! 10.1631 ro.19111 102591 (0.311)

ULTIMATE STRENGTH LBS. SHEET THICKNESS

0.063 IZ"1.'.l 0.071 902' '"" " 0.080 1067. 1182. W!IU 0.090 1203. 1of12. !lli!!.:J 0.100 1331' 1628. ...... , ·-··-0.125 1658. 2010. 2655. """"' 0.160 1678 ,. ...

"'" ofOBO.'-0.190 2620 3983. of800. D.250 .... ""' 0.312 0.375

FASTENER SHEAR 1678 "" "" 6000

NOTES 1. V•lu.s ar• for room l•mP9ratur1 w• only. 2.. • O.not1.1 ultlm.tt v1lun bald on 1.5 x yl1ld. 3. VllUll th1t 111 Wm1.ar.Jlnl:lU 1r1 knll1 Id~.

~ ,,.

10.3731

1'ill.:1 4853. 5285. 7290. 96Qf).

10380

~

~· C0.3731

"""" ....... 7530. .. ,.. "" ""

TABLE AJJ BLIND BOLTS CSK IN ALUMINUM

ULTIMATE STRENGTH OF BLIND BOLTS (100 •HEAD) IN ALUMINUM ALLOY (1)

FASTENER TYPE MS 90353(F.,•112 ksl)

SHEET MATERJAL CLAD OR BARE 7075-TG OR T651 FASTENER DIAM "" 3118 ,,. 5'16." /NOMINAL HOLEl f0.1631 ro.1911 m.2591 I0.3111

. ULTIMATE STRENGT LBS.• SHEET nilCKNESS

0.050 0.063 •'

0.071 Wi.!l

"'" '"" . = O.O!M> 1181. 1313-~ ~ 0.100 1364. '"'. . ..... , 0.125 1823. ""'' 2460. = 0.160 "" 2865' 3473. '''" 0.190 "'" '""'' "935. 0.250 '"' 7020. 11.3'2

'""' 0,375 FASTENER SHEAR 2340 "'" . '"' ....

.

NOTES

'· Valu.1 ••for room lernP"f"•tu11 •re use only. • O.not11 ulllm11t1 v1lues baud on 1.5 111 yllld. VllUll thll 111Undlrllned1r1 knlf.l ldg1,

~

3.

TABLE AJ5 BLIND BOLTS CSK Ill ALUMINUM

~ .. 10.3731

l!Z7.~ 5531 • 8018.

10613. 12200 12200

ULTIMATE STRENGTH OF BLIND BOLTS (100• HO,) IN ' CLAD 2024 AND 7075 ALUMINUM {1)

FASTENER TYPE """' FF•2110 FF-311

SHEET MAniRlAL CLAD· """ cuo "" "'-" "" :mc.'ru 7075-TI Zll24-T•2 7019--TI 20ZC.TU 7117$-Tt

FASTCHER DIAM, IH, 0~9~• I ... •a::,, I '" . .. ... 'NOMINAL SHANK' "'.Hf' •D.259• •0.311• 111311•

ULTIMATE STRENGTH, I.BS.

SHEET TH1CKNES!I ,.,, .... .... ,..., """' = a.on , ... nss• IJH'!. - "'" ·~·

..~ 1470° 11515' """" """" o.1u - .... , ... ,.,.. ·-· """' """" 0.1IO

"" ·- 2SSO' 3015' 1730' ,.,.. O.tto •. _ ·- "'' l015' 31115' -· ,_. .,,. ""'. ,., . .... "" -..2s· ..... D.312 - . .,, .... .

,ASTENER SHEAR "" MZO - ,.,, - -FAS!EffER TYPE

SSHFA-ZDO SSHFA·21SO - •50k•IJ· lf-·•!IOkt!I

SHEET MATERIAL cuo "'-" cuo CLAD Z02C.TU '1G7S.TI Zll24-TU 101$-TI

FASTENER DIAM, IN. ... ~· "' "' •NOMINAL SHAHKl m.1u1 !0.1981 10.2591 10.2!191

. ULTIMATESTAENGTI1 LBS . SHEET THICKNESS

O.DU 0.040

'"" ..... ·~· ""' .... '·""' "" "" ..... "" "" D.100 .... ·~ """ """' O.t:ZS '"' '"' "" "" O.tlO """ '"' 0.190 "" 0.25'l "" FASTENER SHEAR .... '"' "" ""

"""' 1. ValuH ar. lol' room l.....,.iuta.,.. only. 'L 'D.,,<11•1Ulllmllll¥Min-onl.5ayi.ld. 3. Y9'w1lhllltnlMWll:r:IL!llll.,.knll•lod1111.

A19 .APPENDIX A ADDITIONAL DESIGN AND ANALYSIS< DATA

,TABLE AJ6 BLIND . BOLTS CSK IN ALUMINUM

ULTIMATE ST_RENGi:H Of BLIND BOLTS (100~ !-tEAD) IN ' . ALLUMl~ll~.ALLOY (1) ·

.. FASTENER TYPE "-'·'' .,. ....... PLT-150(Fau•112kll) .. 1H-11 Nut 1nd 1cr1w lnc(ln•I X-750 or"A·2B6 ·s1ffv1'

SHEET MATERIAL FASTENER OIA~,IN, !NOMINAL SHANI(\

' t .. !... -~''·. -· -', /C :· -.·. i . SHEET THICKNESS

'0.083 . ' 0.071

0.080 0.090 0.100 0.125

·0.160 0.190 ' . 0.250 0.312·

--· FASTl;HEA S~EAR

"'11l' 1123.

1058. 12118. 13511 ~-·' 1853.

·:· 2318'

2340

· · Cl.A.07075-TB .·:

ULTIMATESTRENGTH LBS.' .. · ...

=· 1245_; . ..... ; ·1430'' 1613,' L2l21B.l ~-2085. 2625 ._ 2865. 3465. .....

5760.

3'50 5900

NOTES :.'.:1-.C \'.•l~_.,._1,lor roOili lemper8tur9 u11 only.

2. • Denot11 ulllmate v1JuH bned on 1.5 11 yt1ld. 3. Values lhllt lt91lmdlt1lllmtl ,,.. knit• 9d;.;

.

··-·-·;

sns•. 80113 ·-

10478' 12250

FASTENER TYPE GPL3SC·V Pin, 2SC-3C Collar (Fsu.,. 95 kll)

·;;

SHEET M.t.TEAIAL FASTEN.EA DIAM, IN • . fNOMlNAL·SHANK\

'SHEETTiilCKNESS ' ''•. 0.050 ' ;<~·;·

·'.g:~' ·:·:._ o.oao 0.090 0.100 .

' -·· ', >'0.125 ..... .. .. , .• 0.16Q., , .. .

"0.190 0.25(1 ..... 0.31~ 0.375

FASTENEA SHEAR ... FASTENER TYPE SHEET MATEA1AL

. ..

FASTENER DIAM, IN. /NOMINAL SHANKl

SHEET THICKN~ ~ 0.050,

O.D63 0.07:1 0,080 0.090. 0.100, .... 0.125 0.160 0.190 0.250

FASTENER SHEAR

NOTES

'•< •.

.,_-,-,,ULTIMATESTRENGTH LBS; • \\I" ,y " ~- .. , . .-, i ' . "··.-·- .. ,, , .. '

'··· ""' ...

"" ,. 1455

I::·; ~=:· '.,i '-··1980.

··--2318 ~ 2611,4 •<•'

2694 ',_.,'

WJ5)•'· 1795; 2085 ...... 2410 2735 3308'

. ... 3930.• 4463. 4660

4660

2«0

'"" 3245. '·" "'4270

5243 ~., .. 5903!.' 7230

' ·7290 . ,,

(2740) ' 3230 I - -3725

~4930

"" 7515'" 9113.

10490.

10'90

GPL3SC-V Pin, 2SC..'.lC (Fsu·: 95 ksl) CLAD 7075·T6 '•'

3116. ' '114 ' '"" 5/16' ~· 10.1901 10.250' f0,3121 '0.3751 ULTIMATESTAENGTH LBS.

·'.ii'.i = _·;i·Y. 1500 l10JllU 1740' 2125 w>Oi 21)20 2485 2865 ...... 2200 I 2885 ~:~--,: .. 3780 2355 ___ 3310 -43ia 2694 3945 5135 5880

4660 6245 8005 7010 8955

""' 10490

"" 4660. 7290 ""0

1. Valuu .,• lar rGOm l•m~ratur• UM an1y. 2. • O.nat•• UllllNll• Y•IUH bHad an 1.5 x yield. 3. Vlllue1 that •r• ~ ar• knll• ldga.

•. TABLE AJ8 I.OCKBOL_TS 9SK IN ALUMINUM

ULTIMATE STRENGTH OF 100~ F.LUSH SHEAR HEAD Tl LOCKBOLTS IN ALUMINUM ALLOY 11)

RIVET TYPE LGPL2SC·V Pin (Fsu .. 95 kSl), 3SLC·C Collar SHEET MATERIAL,. .,_, CLAD 2024· T3,

-;N~~7~1D~~~:.;~~:-:,·::·::'; .. >. " .3116 •', ,,_,' .114. 1.~: ~~3\~. '0~~51 : ,. ro.1901 f0.2501

i """ ' UL Tl MATE STRENGTH 'LSS. S11EET)''HICKNESS . ·:.,·;-~W5)''.' ,,.,,:.,.1.'.': .

0;050 0.063 1180 lllilll 0.071 1395 1630 ,.n.,,.: 0.080

,- ~~~~--,- ,950 . 2155:.:: =·· o:ci1l'if 2300 2595' ,-, 2800 0.100 2070. 2650 3035 3335 0.125 "'" 3~20. 4140 4640 o.160 2655 "" 5460. 6500 .

0.190 ' 2694 4355 6085 7800. 0,250 4660 6965 D180 0.312 7290 10270 0.375" ·' •' - 10490

. FASTENER SHEAR 2694 4660 7290 10490

RIVET TYPE LGPL2SC·V Pin (F!lu = 95 k!ll), 3SLC·C Callar SHEET MATERIAL .· ' CLAD·7075-T6

RIVET OIAM, IN, . ' ~,, '~; '..::;~~=~·;·:,·;';· :'.j . 10~3\621 'NOMINAL SHANKI _ .. · .. /0,190\ L

ULTIMATESTRENGTH LBS. SHEET THICKNESS

0.050 """' 0.063 1370 '"'"' 0.071 1575 1980 """ o.oeo 1805 2280 2715 0.090 2060 2615 3130 0.100 2315 2950 3550 0.125 "" 3700 4605 0.160 2694 4430 6070 0.190 4660 6750 0.250

'•. .

; ·- ·~ 7290 ' 0.312 ' : -~ .

FASTENER SHEAR 2694 4660 "" :·NOTES•,-,.,

1. Vir.lu~S:11r•:lar:rOom'·1emperahi'r• iHH• only. 2.: • O•~a_te.s ultf11111.1i Vii1U95 based· an is• ~Ii.Id_. i -ValuciS th8f iire l1iodc1Uotd! are knife ldga,' -

'" "•:"),;-\',':!:'·"'' "' ,-__ . """"°'"""""' "y•-'-

FASTENER})'~~.-

SHEET MATERIAL FASTENEA DIAMETER ~.- NOMINAL SHANK

0.080 . .;.,:.:;·.; 0.090 0.100 _;~·

0,125 ·-_0.160 '";~','_,,., ;. 0.100',;'·>:"·•'

FASTNENER SHEAR

·,807 ',fQ20""•" - 1150

,300

"" .1630

2007

.

2694

1970 ,··,:_;2220

:2470,, . :·:. ,3095 ; .

3975 -4660 4660

•• 10,3751

= 3620 4130 5375 7150 8660 10~5' . 10490'

""0

, F~.ST~ERJYl;'E': CSR 925.(F..," 9~ ksl) ""!-SHEET MATERIAL .. CLAD 7075--16

: F~~~~=ci~~~~~R "'' '3116 10~~:'01·' fO.i6-'ll . (0.1901 -

,,,, ULTIMATE STA ENG TH LBS. SHEET THICK

0.050 095 0.063 1227 1442 0.011'' 13if 1607 0.080 1532 1792 2415 0.000 1711 2001 ,.,. 0.100 -1890 2205 2'60 0.125, 2007 ""' 3641 0.1&0"' 4595 0.190 '660

FASTENER SHEAR 2007 2694 '660

NOTES: 1. Valu!s er• tar room t•mperatur• us• only,

APPENDIX 'A '.ADDITIONAL DESIGN AND ANALYSIS DATA A20

TA,BLE A4,0 CRERRYBUCK FASTENERS CSK IN ALUMINUM

·uL l'IMATE. STRENGTH O'F-'1ilo".FLUsH-sHEAR HEAo Ti. "CHERAYBUCK" FASTENERS IN ALUMINUM ALLOY (1) _, - ,,, .. __ , .~.-

. _,_ _,

FASTENER TYPE ·-" -- CSR 924 {f.,,.·,. 95 kslj =t-SHEET'MATERIAL . CLAD 2024-1':3

FASTENER DIAMETER ~32 ~" 11' I NOMINAL StiANKl' ' '" ···10;11;4{' 10.1901 10.2so{'

·UL TIM ATE STRENGTH LBS. SHEET THICKNESS -·. , - .

0.050 "' 0.063 10111 1118 0.071 1152 1319 o;oeo 1260. '1509 1837 0.090 1350. 1664 '·· 2168 0.100 1A40' 1167 •• 2""' 0.125 1665. 2028 .- 2969-'. 0.160 1992. 2394 •• 3450.'

"0:190 2694'' 3863. 0.250 ' 4660"'

fASTNENER SHEAR 2007 2694 4660

' -· --· "FASTENER'T'fl'~···. <•"·: ... '.-~R 924 .. (·F~-., 8.5 ksl) ,·_-"'.::;-

' SHEET MATERIAL· -- CLAD 7075-T& NOMINAL DIAMETER ~32 3116 1" 'NOMINAL SHANKl IQ,1641 ro.1901 C0.2501

ULTIMATESTRENGTH LBS. SHEET THICK .

0.050 '" D.063 1207 1303 0.11[1 - 1305 1588'

22a1 0.08-0 1557 1779 0.090 1775 2050 "" 0.100 1876 2263 2919 ' • 0.125 .... •• 1950 2542 3765 0.160 2007 2660

I 4387 .

0.190 "" "" 0:2so 4660 FASTENER SHEAR 2007 2694 4660

.. NOTES:

-1. -values •r• far raam _lamperatura usa anly. 2. ' 0.natas ulllm•I• valu•s b•...t an t 5 )( ylald ••

TABLE A41 HI-IDK FASTENERS IN ALUMINUM

ULTIMATE STRENGTH OF. 100• FLUSH SHEAR HEAD Tl Ht.LOK IN CLAD 7075-TB ALUMINUM (1)

FASTENER TYPE HL 11 PIM (F111 ~ 95 ksl), HL 70 COL~R; '.!!j-SHEET MATERIAL CLA0.7075•T6

FASTENER DIAMETER :'. 1~:41 ... r -~o~~!i,, 5116

/NOMINAL' SHANK\ 10.1901 (0.312} . . ULTIMATE STRENGTH LBS •

SHEET •THICKNESS, IN. . . .

~:~=·~· ""' ""' 9'1 1083 = 0.063: 1207 1393 llilll' . = 0.071 1385 1588 2012 ·.•' 2463 0.080 1557 1779 2281 . 2823 0.090 1775 2050 25'4 3193 0.100 1876 2263 2919 3631 0.125 1050 2542 3765 .... 0.160 2007 2660 3970 5890 0.190 2694 4165 6105 0.250 4530 6580 0.312 .... 7050 0.375-'' .. 7290

FASTENER SHEAR·, . 2007 2694 4660 7290 __ , "' ono"' 'o"u """" 0000•

- -HI LOK IN CLAD 7075 T6 ALUMINUM (1) .•.

FASTENER TYPE HL 18 PIN (F.., .gs ksl), HL 70 COLLAR '."!-SHEET MATERIAL CLAD 7075-T6

FASTENER DIAMETER 5ra2 MS "' ~16 •NOMINAL SHANK! 10.1641 fD.1901 10.2501 - C0.3121

ULTIMATE STRENGTH, LBS. SHEET THICKNESS, IN. . ,,

0.050 1078 0.063 1353 1559 0.071 1520 1776 0.080 '1718 1957 2593 D.090 .. 1890 2224 2937 0.100 1930 2473 3250, <050 0.125 ""' 2580 4063 5075 0.160 2694 '450 6509 0.190 '620 6880 -· -0.250 4660 7290

FASTENER SHEAR 2007 2694 4660 7290

NOTES: 1. Values 1r11 !or raam 1empar1tur1 use only. 2. V1lu11 thU era Cumln~llQ!tdl ar• knll1 edge.

.

..

··•

TABLE A42 HI-IDK FASTENERS CSK IN ALUMINUM

., uLTIMArE'sTRENciTH OF 100° FLUSH SHEAR HEAD ALLOY STEEL Hl-LOKS IN CLAD 7075-16 ALUMINUM _{1).. . ... ..... -

FASTENER _TYPE ·. HL 19 PIN {Fsu. _95 k•I), HL 70 COLLA~. - -, ' ~ SHEET MATERIAL CLAD 7075-T&

FASTENER DIAMETER· 502 3'16 114 ~" !NOMINAL SHANKl 10:1641 I0.1901 10.2501" , .10.3121

ULTIMATE STRENGTH LBS« SHEET THICKNESS

-.-,'._' 96S '0.050

0.083 1251 1408 0.071 '' 1400 1606 0.080 1595 1823 23 .. 0.090 ,.:·· 1815 2050 . 2675 ·'., 0.100 1903 2300 3000 3660 0.125

•.•• 2005 2570 3781 4885

0.160 26'• 4420' 6051

·- . 0.190 4625 ,. 6'32 0,250 4660 ', "'°.

FASTENER SHEAR · 2077 2694 ' 4660· i 7290

.. F_ASTENER 'f'l'.PE _HL 719 ~IN (Fw~ 108 kal),HL 79 .!?OLLAR.-· -~!4-SHEET MATERIAL . 7075-16· --~~

FASTENER DIAMETER ~· :v16··.

1",;, ''10.250 . 5116 • 10.3121 ,•

,,-,. 318 fNOMINAL SHANIO'- 10.1641 f0.1901 10.3151

.., ' .,, ·" .. , - ~ - ' ULTIMATE STRENGTH LBS. SHEET.THICKNESS ·.·· .. ···-

0.040 ""' 0.050 1044 = 0.083 ·- "" 1585 .11l1'1

0.071 1518 1620 2216

,, 0.080 "" 1998 "" 2918' .. 0.090 "" 2193 3015 :1532·~, .. 3n< 0.100 1825 "'' 3330 4059· 4518 0.125 "" "" 3980 522.9 ·' 6167 0.160 2195 2n~ . 4350 6347';:· .7928 ~190 2989 4634 •"6702'-· 9067 0.250 3002 5200 7412 .... 0.312 5300 .. . 8146 10810 0.375 8280 11760

FASTENER SHEAR 2261 3062 5300 .. · 8280 11930

NOTES t. ValU91 ar• far u•om lllfl'lp!lrll1ur11 u•• only. 1. V•luaa lh•l •r• luqsf![flnndl.,.. knll1 ldg1.

TABLE A4) THREADED PASTENEBS 'csK. IN.ALUMimJM

ULTIMATE STRENGTH OF 100~ FLUSH 5HEAH HEAD (SS, p·H13-BMO-H1000) FASTENERS IN Tl,·6Al-4V_,_ALLOY (1).

FASTENER TYPE PDF 11IF4',=125 ksl SHEET MATERIAL ANNEALED Tl·6A1-4V

FASTENER DIAMETEfl 502 ~" 316 1'2 !NOMINAL SHANKl 10.1641 10,2501 (0.3751: 10.500\

ULTIMATE STRENGTH,.LBS • SHEET THICKNESS, IN.

'·"' = 0.0.50 1963 O.o6'1 2528 """ 0.071 2640 4213

,,, .. .;;.., D.080 4813 C.090 '"' 7818 0.100 6140 8T75 ililllll 0.125 11264 14575 0.1t.O 13810 19250 0.190

,, 23200

D.200 24540 FASTENER STRENGTH 2640 6HO 13810. 24540

.

ULTIMATE STRENGTH.OF 100.° FLUSH HEAD ALLOY STEEL NAS'1620 (SERIES) FASTENERS IN CLAD 7075-16 ALUMINUM (1)

.

FASTENER TYPE NAS 1620 IF = 95 ksO SHEET MATERIAL CLAD 707S.T6

FASTENER DIAMETER ~" 1/4 ' 5116 ~· fNOMINAL SHANK\ C0.1901 10.2501 f0.31~ 0.375 ULTIMATE STRENGTH LBS.

SHEET. THICKNESS, IN, . 0.080 = 0.090 1748.

0.100 1920. ··~·--0.125 2340. 3195. 'lillfil. 0.160 2690 3975. . 4350 • l'1lllll • 0.1110 4590. 5160. 6180.

'""' 4650 6675" 8130. 0.312 7300 9945. 0.375 """' RIVET SHEAR STRENGTH 2690 4650 7300 10500

.

HO TES: 1. Values are tar roam l•mperature US• only. 2.. •Denotes ultlmata valllfls based an 1.5 x yleld. 3. V•lues Iha\ art lynder!!ood) fir• knlle adg•.

A21 . APPENDI:lC. A .ADDITIONAL, D,ES.IG.N.,.AND ANALYSIS' .DATA

TABLE A44 SLEEVI! ,B<lLTS ,CSK. IN ALUl!IHUK

ULTIMATE STRENGTH OF 100•_ REDUCED FLUSH HEADJALlOY STEEL ·, ::.: -_· .. ,p1NJALUMIHUMSLEEVE)IN~LUMINUM'l1l 21 ·,1 ·_i;,,,

'~) ,. ':

FASTENER TYPE_ MIL·8~31/4 (F ... , • 108 ksl SHEET MATERIAL Cl.AD 2024-r.I

fASTENER DIAMETER 3110 10.~21 10~~=!1\ 0 , • l0.~1-. 10~:21: /NOMINAL HOLEI''._' ' f0.23901 ,.- ULTIMATE STRENGTH LBS.

. SHi:Er.nt1c·KNES's :-2175 .... . 0:,100;-.'· ... : '

0.125 2720 3450 4205 O.HIO 3"0 .... '! 5380' .. ,, '7315 0.190 5240 "" 7525 .... 0.250 ···.··.·, .... '7945 '"' ,--~~~: 0.312 ·"t:

"" "'8185 ~ 11085 -0.375 ·- ":.: 5670 . 8385 11345 14845 0.500 .. , 8760. 11885 '15445 0.825 {, ,,

12385- :-16045

k·, 0,7$0 1"40 16645 0.175' '\') ., ''7'00 1.000 .- -

' "' ' · FASTENER SHEAR 3290 5670 •'-8760 12640 17100

. • FASTENER TYPE ·· " , .. "~ ---MIL·B-883114 F .1oe ksl SHEET MATERIAL--: ""' · -CLA.D 7075-T& , .. ,.

FASTENER DIAMETER - .-(NOMINA('H6LEI'.· .. - ... ci~~:O; .. ,.: .. 1/4.

-'io.30321 f0.~~:5i '•o.~I'~ 'io:iJ~; -;;,"-,,·,:·.-·. ,.:;~._,,, ULTIMATE STRENGT11 LBS •.

SHEET THICKNESS ' '0.100 2585

·0.125 ' 3205 .... 503if ~ ... .', 3290 "'" 6385; 7580 8710 0.1!f0,._, 5670 7535· •.. "" 103&o 0.250'"'· a16d' ..... ,,. .. 0.312 ·.·

,·· ihr ,., \\,,· i 12395 16195

·;,;; _;g~~·i;\~', '. ; ,, ... 18:625 17100

I 'FASTENER-SHEAR "'3290 5671) 8780 ' 12640 17100 ' NOTEs ;·--·-_ . . :-:-,,"

1. ;", V1lun 1 ... 1or roomtH1pe ... lure use only.· 2~ -~1 .FasC•n•t1 lnsttlled In ln_t1rter1ni::1 lev1l1 o(O.DD25.0.0011 In. 3." . . Sito .'!'l.IL_·B .. -8831/~ .!!-J".. ~,mlnll 11o'-.cl~ll~.

'

',,,,,

112 10.57351

"20 13050 18285 19070 111755 20440

'""' 21805 22250

"""

112 10.57351

11oqo 15480 19180 21213;5 22250

""'°

, TABLE. A4S STEEL B<lLT/SCBEW BENDING MOMENTS

ULTIMATE ALLOWABLE.SHANK BENDING MOMENTS.FOR STEEL BOLTS ,. ' :.· .. '-_·. AN~. sc~~i._ys:(R,9.9~ TEMPE~ATURE)

DIAMETER (INCH)

0.190 0.250 0.312 0.375 0.438·--,,

0.500·' 0.562 1'

o.e:2ii· o.750' i:: o.a1s-. 1.000---

125• ·- · · I ·· 160'"'°"'-1 ·-•,'· ·180 · ' I" · · 220 103 -\ 142;. ·- I 163 ·1 180 197 -. I -., 259 I 294 ;:·.I."·;-. -346

. ,- ' ' ' : '' ,,_''·'·· :·. ULTIMATE ALLOV(ABLES_SttANK BEND_l!llG M.OMEHT

··. 1 ,;~:ii'";:;!l_N·LBS)· ' - .

"' 302 5i0''

1020 . 1620',',\·;

0

2415 'n''

3440 4720·,., ....

12970 ' '' -· 19340

776 1340' ,_ .. 213o· -:-.· 3180.'

4525 8210

10700

17060 25420

198 •: .. .. , 801

1520 2415 3605

"" , .. , 121'50 '

111360

"'"

'233 .,,;~, .... •'•,1790

'2845 0

4245 .... .8295

""° 22790 33960

SECTION MODULUS {UC){IN3)

.000673

.001534

.002996 Jl0518 .ri;oa22 .01227

.01747

.02397

.04142

.06S87

.09817

NOTES: ···-- ............... ., .• "" 1: ·- CAES .1 .. 111 recornm1.td_«1 when 1ubj1c1ed to •pp ... cilbl• ti.ndlng • 2.- Th• u1• oJ bolls with en F.:, 2·1ao k1_1r1qul1111pprov11 by Struc1urat lnl•grlly, 3,' TH u .. of,boltl •nci •cr_1w1 hlVlng 0.190 Incl\ dl11Mt1~ Of '91• 11 not

11con11T11ndld li:irbendlng'ippllcallons. ·· · .. ·

! • :.:,.,:,; .;,~ . ' .. ~

c

·J ,., ...

I• il~ ,: '

~~ • ~· i1· .•. • ' ·-·· ~· il' ·~~ il

i •'•'

• .. ;:! • ~· r ;( ] .} J I -

R :~ I ,J.

'J: I 'T 'T l ·r .. i! ..

J. • • " J. ,• ,"-

BEF!!llENCESI, Add to .Ch&J>ters .indicated

Cha;t,o'r ~4 ,, . , , 4) Plastic ".Analys1s, Nada1,A., McGraw Hl-ll' co. 5) An :Ex.te_nsion_. _of _..the Sand_ }i~_a1f_.Ana._10gy. 1n

Pla~t1c:_·lj'.'or_s1on·: ·Having- ·one_. or.· More··Holes, _?,:,~.~rn_~~---~f_ App~~~-d ~eQ~nlc·s, .. Dec:. 1944

. , . ''·I:' ChaPter"c7·:.~.: .. _,_,_. ;.:::-.-· .. ->::·, -_· --- ·. _ ....... _. '":.r: __ . _ .- ,,_.. 1

14 )Engineering Column ·Analysis (see Art.;.o\18 .• 2?a)' ~5_) Airplane .Struct;ures, Vc1-.·2,_ 1943'1 N1les.~A.s.

and .. Newellt _J .s. i; John Wiley and Sons i6) "Primary Instability cf Exti'usions Under

Compressive LoadsH, Ramberg,W and LaVy, s., Journal of Aeronautical Sciences, Oct., 1945

17) Primary Instability or Open-Section String-. ers Attached, to Shell, Levy,s. and Kro11,W t Journal__;,~,f_,~~ronaut1cal Sc1enoest .•. Oc.t."' 1948'

Chapter c11·(p,C11,49) 10) Engineering ,Column Analysls. (se0 Art,A18.2?a)

Cha-pter C) 3) Eng1neer1!1~ ~~l~ ,:~l?:~;ysls _ '{ a·ee Art. A18 .-2?a)

APPENDIX B BULKHEADS, FRAMES, ARCHES AND BENTS Bl

Following is an alternative analysis to that presented in Chapter A9. It is based upon the minimum energy principal instead of upon deflections as ih Art.A9.2

A loaded bulkhead or frame,· in a fus­elage for example, is supported by the fuselage skin panels as in Fig.Bl, which shows the general (but uncommon) case of

The values of M0 , Vo and P0 must be such that U is a minimum. Therefore

Differentiating (2) as indicated the three simultaneous equations

n .. N 4811

yields

L 1MQ,q +11tJ+v0 111 +P07,,,1~-o n• I II • II IS

(3)

n•N 4B1t"Xn

L IMQ,q +lllo+Vo:rn+ P.,1,.1~•iJ II :11 1 II ll II

····--·.(4)

(5)

an unsymmetrical bulkhead. The applied loads consist of the loads, Q, and the re­acting shear flows, q (determined by VQ/I and T/2A per Art.A20.3-A20.10). To deter­mine the internal loads in the bulkhead it is "cut" at the top, and the unknown intel'o · nal loads Mo, 'vo and Po there are shown acting on the left side of the cut ('on segment 1) in Fig,Bl(b). Opposite loads act on the right side of the cut (on seg­ment 20), · Mo, Vo and Po are redundant loads and are ,positive for the directions shown. x and y are the moment arms for

·All terms in the equations are known ex­cept the redundants and these are found by solving the three eqyations, The final values of M at the center of each segment are then calculated using Eq,(l), the forces Vo and Po respectively., posi­

tive as shown. The bulkhead is then divi­ded into a large number, N, of segments of length As, at least twenty, The segments are usually of the same length, . . Q;

~t '• ~ +.)/

.J_. " ,, ) "

~· ,, ~

~. " " ,, I

~ n.•~±J ~ ~

'" '" Fig.Bl Unsymmetrical Frame Analysis Data

The bending moment in each s~gment, n, is

Mn= MQn··+ Mqn +Mo+ Voxn + PoYn ·-·····(1)

where Mn is at the center of the segment. The moment of the shear flows, q, about each segmen~s center is calculated as shown on p,A19,19. Positive moments pro­duce compression in the segments outer· flange, The bending energy in each seg­ment will be

. Un = ~Asn/2Enin and the total energy in the frame will be

• Bulkheads are relatively highly loaded and sturd7 rramea< For ver7 light tramee also see Fig.C9.1)a.

The analysis is most easily carried out by using a tabular form (Table Bl), and this is illustrated for the bulkhead shown in Fig,B2. The values of MQ,qn• ~S/Enin, x and y for the bulkhead are shown in Columns 3, 2, 4, and 5 respect­ively of Table Bl. The final bending mom­ents are shown in Col. 14. The values of V and P at the center of the segments would be calculated as noted in Col, 16 and 15 of Table Bl (by statics),

All shear flows are

,,, I-HI----+--- I

I ,, I

I

Fig.B2 Unsymmetrical Frame Analysis Data

.Symmetrical Frame and Symmetrical Loads

For this case,. due to symmetry, V0 = 0 so only Eq,())and (5) apply, and it is only

·necessary to use. o!\E!-half of' 'the frame. This procedure is illustrated for the frame

.of Fig.B2 assuming that its right side is 'the same as its left side, Table B2 shows

B2

<D ® Se1me11i O.SIEI'. .

l/Lh·ln

'. I .900 2 ,900 3 ,900

• .900 6 .9"0 6 9<W 7 .900 8 .900 9 .900 10 .900

" I 154

" I 598 13 I 864 .. 2 130

'" . i,2:308 -.: 2.308 16

,·. 2x:i'o "t'

17 .. 'f 1.864,.,

" d:i:' '20;

I

.APPENDIX· ll BULKHEADS, .FR(l.MES, ,ARCHES ANJ? JlENTS

Tabl:e Bl ·unsymmetrical Frame Alialysis '

@ © © © @ © ® @ ''• @' @ @ @'

o.s~• , ,ASiry Mq.q6Sy l!.~y':-· 1 Mnnoi1 ' . ~ff· Mq,"ti.S·. ••• .. , 'Mq,q0.81: . ' - -- ~ + M.,.+'Vux© li:l '1•' El .'..·El " El ; El. El 'El' El ~.@_• '··: p~;~@"

Lb· In In, In, ®•® rvz© ®•@ ©11@ @11:@1 ®•©11@ ·®·© ',' ,, . ·. 'l.·1°' ' ,·~y;:.; .'•-1 r, - '.'·,,--·;,·,. ,,

' 3.0: ' .7 ·o·' 2.70 ·" 0 ll.10 1.89,' ,-"' ' ... -21,649_ 30.89 9,477 13.69 -38,854_ 2,700 ••• 3,9 2,430 7.92 3.51 21,384 69.70

· 1.co.sa · 101.26 671,290 12.90 · .. ::1,6Hi 70,900 12.6 9.0 63,810 tl.26 8.10 797,625 t:l,23 1,853,262 , 199.BQ ~:r:!~, .. '1,828,3~, 19,:1.48 . 31,666,' 138,200, 14.9 14.7 124,381 13.41 '·.;

14.22 18.46 2,526,472 . 224.68 3,276,'120 3'18:23 .. 36;384 1'1'1,600 15.8 Z0.5. 159,a.tl :' 209,200, 15.8' 26,6' 188,280 . 14.22 23.94 2,9'14,82' '224:G8 - 3'18.25 5,008,248 636.81 , 3\,'163

443.6'1 7,456,43'1 986.06 3.~.406 13.41 29.79 3,356,623 "::~:=~·,' 250,300 1.f..9 33.1 226,270

432.00 8,975;232 1,327.1 12,708 11.25 3.f..56 2,921,GZS. 259,100 12.5 38.4 233,730 ' "\ '69.70 342.14 9,761;104 1,679.6 -24;118 8.8 43.2 226,7~0 7.92 38.88 l,986,33S 250,800

124 • .f.7 _9,970,041 1,912.7 -50,934 240,300 3.0 46.J 216,270 2.70 41.49 648,8U B.10 -3.0 46.0. 277,3(}6 -3.48 53.08 -831,918 10.38 -159.25 12,756,076 2,Ht.9· -49,406 240,30Q

-13.10 68.11 -3,251,005 !07.45 -563.45 17,047,952 2,964.7 -22,981 248,100 -8.2 43.0 396,464 246,400 -11.6 38.3 •59,290 -21.62 71.39 -6,327,764 260.82 -828.14 . 17.690,807 2,734.3 3,'157

382.48 -947.69 16,342,468 2,347.B 19,021 231,IOO -13 ... 33.2 492,243 -28.64 70.72 -6,696,056 -8'1'1.92 11,732,169 '1,633.0" 18,352 191;100 ~H.3. 26.6 441,069 -33.00 81.39, -6,307,1•3 471.96

47i.98 7,433,013 91!9.9 20,514., 157,100, :t;:: 20.6 382,688 -33.00 47.31 -5,184,980 -676.li9 109,500 14.8 233,231i -28-.li:i '31.52 ' -3;125,349 382.46 :..:42f42'· '3,451,818 466.li 6,680

-21.~2 16.116 -1,636,818 260.82 1-196.75 1,284,056 15".3 6,446. 75,700 -11.6 8.1 141,105 i4.'3 -34,:JO(I' 3,9 7,361 -13.10 6.23 ..:.S0,218 107.45 -51.10 28.669 4,600 -8.2

' .:a.o ,7 0 ·. -3.46 .Bl ~" 10.36 .. ·-2.42 0 ,8 -20,714 -

·- 2 0 369 - _, 2 0 '11"2 """"' -~ w=

Table B2 Symmetrical Frame with Symmetri~al Applied Loads '

CD ® '' m © .®. ©. ' ® ® ®

Selment ·.-.as.·.-·· ,. MQ;q .... , .''Mq,qAS ··ASy ' ···M~ .. ~·s·). , ASyl Mft.w. : - ,

~ ©+ M• + P• z(! El El .. :EI •• Given Given Given' @•® @z@ @z@ j\z@•

',

I I 0 ' •'" . 7 0 .7 0 • -24,907 2 1'"" 1 .. '"2;ioO:: ' 3.9 '·7~::-·, 3.9 11,000 15.2 -41,063 3 ·I 70,900 9.0 - ·' ;..1( 63~000 81.0 -2,191

' 1 138,200 '14.7 -·1138,200

' .·· I I 177,600 '20.5. 177,600 :'6 " I ; 209,200 ~.6,. 209,200

'7 I . 250,300 33.1: 250,300 8 I 259,700 .... ' 259,ZOO

' I 250,800 .. 3.2 ·260,800 IO I 240,300 •e.1 240,300

- -}; - 10 I:.,. l,li99,700

I:@ +M.J! @+P,t@ ~0··1··.··. t (!) +M,t@+P,E@•O ·.

14.r 2,032,~ . 216.1 30;794 .... 3,841,000 420.3 36,018 28,8 lli,565,000 -·101.6 31,8.70 33.1 8,285,000 l,095.6 ,:w.••6 38.4 9,972,000 1,47~.6 12,634 43.2 10,835,000 1,866.2 -24,51)1 46.l ll,D78,000 2,105.2. -52,140

• ...

2.36.2 lli2,0li7,000 8,001.8

1,599,700 + HIM0 + 236.2P0

: :} M, ~ -20,782 62,0li7,000 + 236.2 M0 .f. 8,001 P,

@ l[j)

Az:l•I' Shear Load .....

This is the Same•sror component-" the ax iii load parallel to bu't using the the neut.ral component 1xi10Cthe normal.to the vector a um of neutral axia all loads rrom at the U1t "cut" to segment the segment center • center, inchidi':lg.P0 •

'

P. = -5,892.8

@ @

A1i:lal Shear l.oad l..oad

'

Thi• i~ thr Suinea~ cump.,. for the llt!ll 1<xlal l11i<d pur1illt!I~ but using tlil: "' nc1:11r~I , Cllnlp<I•

11xi111flhe nenl Yec'tOr norm11l lo sulnof11ll "' lo11dsfrom ne11lr11I the "cul~ 11xi~11lthu to the 9Cl1m1:nt 1e1menl center. center, lneludlng r~an~ v~.

the calculations for the left side, The loads on the right side are the same as for the left, be.oaus·e. of .symmetrY, e.g., seg­ment 11 same as 10, 12 same as 9 etc, The' two simultaneous equations for Mo and Po are

SymmetricalFrame and UnsYmiiletricai Loads ',' ,. '!, I''.' "·' • . -,;

ThtS · ca.·se · 'C'Ould be a.ne.1YZed as was done for Fig,B2, using.Table Bl, but an eas~er proOedUre: is .. as. fpll9wS~:'.~hich uses only one'-half of the frame ( s~!fments i-10 ),

••N I•Mq+M+

It ,. 1 ' .q" 0·

These are obtained from Eq, (2) (with V0 =0) by differentiating it with respect t6 M0 .

and to Po. '

An unsymmetrical· loading can be re,­s.olved into a symmetrical loading and an antisymmetrical one, as shown in Fig.B3. An antisymmetrical loading is one which when rotated 180° and the load directions

•. :i;eviars.e.d will. give the original loadlng, The final internal loads are found by de­termining those due to the antisymmetrical and then adding these to those due to the

APPENDIX B BULKHEADS, FRAMES, ARCHES AND BENTS BJ

J5,000 I~•

10,000 Iba

+

t•I Cbl

Fig.BJ Resolving Unsymmetrical Loads

Fig.BJ Resolving Unsymmetrical Loads

symmetrical ones, The antlsymmetrlcal loads will add to the symmetrical ones on one·· half of th.e frame and subtract from those on the other half, to give the final internal loads,

To illustrate this procedure assume that the frame in Fig, BJ ls the same as that in the ~reviouS ·exafilple (in_Tab16 B2) The symmetrical loads will be the same. as in Table B2, Those due to the antlsymmet­rical loading are found as follows, Due to geometric symmetry'and loa~ing asymmetry Mo and Po are zero,·· so the· only redundant ls Vo. Therefore,'·the only equation ls Eq,(4) with Mo= 0 and Po·= 0, or

·n=N , n-¥i(MQn + Mqn + Voxn)ll.Snxn/Enin = O·{~

This is solved for V0 , and the final anti­symmetrical loads are obtaine·d for each segment by' using the terms inc the paren­theses of Eq_, (6), These. antisymmetrlcal

loads are determined as shown in Table BJ, The final internal bending moments are shown in Table B4 0 Col,8, It is seen that the asymmetrical loads add to the symmet­rical ones on the left side and subtract from them on the right side,

Two-Lobe Frame

Such a frame is shown in F1g,B4, For an unsymmetrical "-frame there Eire·· siX redundants P.s shown, hence there wo.uld be six simultaneous equations, derived in the same manner-as was done for the previous bulkhead, Aside from this the procedure is the same a·s. before, For the more com­mon symmetrical frame Vo and Va are zero, so there would be only four equations for· a· symnietrical loading involving· Moo Pao Ma and Pa. For an asymmetrical load­ing there would be only two simultaneous equations involving Vo and Va, since Mo,

Table BJ Symmetrical Frame with .Antisymmetrlcal Applied Loads

@ ® @ @ ® ® Gl ® ®

.... •• Mq,q • Mq,qAS:it AS:12 Mn.• Adal Shear ment - -- Load Load

EI EI EI Data D•ta Data ®•@•@ ®·©· @+Vo•©

I I 445 3.0 1,300 9,0 -15,131 This i.s the Same as for the

2 I 1,197 8.8 10,500 77.4 -44,493 component axial load but

3 I 37,509 12.5 468,900 156.3 -27,391 parallel to the using the 4 I 70,851 14.9 l,055,700 222.0 -6,510 neutTal a:.:is of component

5 I 85,916 15.8 1,357,500 249.6 3,882 the vector sum of normal to the 6 I 94,598 15.8 l,494,600 249.6 12,56°4 all loads from the neuti-al a:tis at 7 l 99,671 14.9 l,485,100 222.0 22,310 "cut" to the the segment

8 l 85,216 12.5 1,065,200 156.3 20,316 segment center, center. 9 I 52,122 8.8 458,700 77.4 6,432 including V 0.

10 l 6,539 3,0 19,600 9.0. -9,037

l: 7,417,1-00 1,428.6

t@ +V0 t@ •O 7,417,100 + 1,428.6 V0 = 0

· APPENDIX B ·BULKHEADS, FRAMES·, ARCHES AND BENTS

Table B4 Symmetrical Frame.with Unsymmetrical Applied Loads ....

0 ® ® 0 @ @ G> @ ® @

Sef·. M•r• Axial•r• Shear•.r• M.,1• Ax~a1 .. ,. Shear .. ,. Mu11•.r• Axial.,,..,.. Shear.,,..1.,, merit

ll~s in-lbs lb• lh• In-lbs ,., lbs· in-lbs ,., . . . .. .. .. ®+@ ® + ® © + G> I -24,907 Thia is the com.'; · Sa~1e a.!I for the -15,131 'fhis is the Same·asfor -40,038 This is thl! c111n: · Sumi! 1:u1 for the

' -41,063 ponent parallel·. alli&I load but -44,493 component · the aii:ial: -85,556 ponent parallel' axial 1011.d but I. 3 -2,197 to the neutral using the com po· -27,391 parallel i.o ioad but -29,588 to the neutral W:ling th.I! compo-

• 30,794 axis of the v'ect.d~ neni noi-1nal to ~.510 the neutral .: using:tne·' 24,284 axisofthe vector nent normal tu ,. 36,016 aumofalllriada the neufral axis 3,882 axis of the component 39,898 sum of all foad11: the neutr11l 11•ii1 6 31,670 from the "~ut" .tO at the segment 12,564 vector sum br normal to 44,234 from the "cut""iii at the 11egmenl 7 34,466 the segme.nl cente'r, 22,310 all loads the ne·~.tral . 56,776 the segment center. 8 12,634 center, iricluding 20,316 from the "a:ii:isstlhe 32,950 center, i~cluding 9 -24,551 P,. 6,432 "cut" lo ttie segment -18,119 Pu1tnd Yu• 10 -52,140 -9,037 segment center. ' -61,177 11 -52,140" . 9,037 center, -43,103

" -24,551 -6,432 including V 0. . -30,983 13 12,634 -20,31,6 -7,682

" 34,466 -22,310 12,156_ ,. 31,670 -12,564 19,106 16 36,016 -3,88~· 32,134 17 30,794 6,510 37,304 ,. -2,197 27,391 25,194 ,. 19 -4 l,Ofi3 44,493 3,430' 20 -24,907 . .. 15,J;ll . ' -9,7-?6 . .

,·'"";. •·;! C'•· . · . ; · ··· ,.--.- ··· - '" ·, ·'( ,.,.•" '' '.· 'T"h'·i··; ,, .. ,, .. , .. - :. ''.-'r'·,·:;-.,·i•":•·.. ; ·'-"!-- ·'.;r.-' ·-· -_.-· ,,·!:!

P<? '·· ~a.· ~fl:d ~a ~!"._'!!. zelr_q ,. .J;.r . 1:(.he .... .Qi'oss'-;.l:?Sam." Inner_ .Flange Lateral .Supports were pinned at .its en.ds Ma a.!jd Va would be zero,· s'd the above s.ix eq11ations ... _Wo.ui·d .be­come only 'four·'.tthe ·fOU.r e·qua:·t'iOri~r woUld become on1Y"' "three and the two equations .. would become only one,

Calculati.ons aha' DatS:

Ir\ al:r ·or the"preCed.1hg ce:lculatl~h tables·{and also· in Table•f..9,5, C()~·B) the most labo:r'lous effort fs' caicU:iatlng 'the ·. values or. tl\e applicable shlar flow ~om­ents. o !'!<!. o aqd Of the ~Oncept;iil:t~d applied load.s, .. ~·~Q_,.:'about e·ach segmen;t.-~·s_ C_eri·t_e.r .... · _. This 1.s' dOhe' sta:i-ting with .·segmept one• and . proceedih@.; Sounterclockwl:Se (Mc;' ·fie.ii be cal­culated per p;A19.19 or per p~A9,l2), Once the'se mom~nts_ ·are. Ca_lc.u1Et.te~--- :and··. 9fiter!3d .. in Co:t:3 the' remaining calchia tfor\.s are. routine so they can also be programmed for calcu­lation by ·computer .. or\ suitable calculators; In Col.3 and elsewhere positive moments produc~ comp.ression i.n the outer flange. Positive values of x and y and of M0 , V0 .and P0 are as shown in Fig. Bl. ' .,•' . v

0

Unlike the outer flanges, the lnner flange o~" freme.ls.not usually supported ag~l?l.~,t; '.b'\~tling.pY,~k;l?l PB,)J,~ls, There-, ro;:~:_.'.'1f3.te_r~1 _}:1'_~p-'Rc:>.;t€t;~:r .s,_¢ine _type -are nee.d~d. \f.he.P. t;he flange .compressive, .load.s

·~:r:~'. ~fa}]iri,ii'f"t· ;:t;f\e'.A\il'i>ort ~p~ci.9g. mu:st be such that .l'\teral. buq]l:Ung wtll not oc­cur~::_; .,.~Th'9: '$,Up_p_q_.:f~.ii-.ar.e· .. usu_~1i.Y .. tube.s . (i!t:r;'\f .P'§f 1ni;erc'ostiis e;s .. .in, .Flg·. B5 . , ... ,, .. , . '. . . ' '

Fig.~S

.. ·~ ; \;L;oads

Inner Flange

QJ.ter: Skin

Frames With Inner · Flanges Stabilized Ag:iinst (lut-Qf-Plane Buckling Uslni'!: Axial_ Members (Tubes) and Fittings or IntercostaJs.

: fl!lnge Must be Attached to Supporting Member in Stlif Manner.

Lateral Supports

Circular Fr.imes of ul'll.rorm .EI

Two-Lobe Frame (a)

Redundant Loads (b)

Two-Lobe Frame Fig, B4

When' thesei are !iupported by skin pan­el·s as in.:. previ:ous analyses'., data is av­atlable in Ref';.is of p.A21 for a more dl­.re·qt solution ':(usinp; ·the "Wlse Coefflc­_ie'!i'ts11, rrom J6u_r. Ae'ro Sci9nces' Sept .• 39) The data can also be used when any self­·equl11bra.ting loads (no supporting sklns) are prese,,t, by superposition of the ap­plied and. reacting .. l.Oed .cases, The data ls of graphical form, giving the internal mo­ments, axial and shear loads for applied normal,· tangeptial and moment loadings.

APPENDIX B BULKHEADS, FRAMES, ARCHES AND BENTS B5

Accuracy of Analyses

Since the previous analyses do not include the stiffnesses of the structure adjacent to the bulkheads (skin panels, stringers and adjacent frames) the results are approximate and usually conservative for the bulkhead, but unconservatlve for the other structure, A discussion of this with examples is available in "The Analy­sis of Structures", N,J,Hoff, John Wiley and Sons. A more accurate analysis re­quires an involved finite element analysis which model s all of the structure, How­ever, the previous analyses are commonly used, and the results can also be used to size the frame for a finite element model,

Arches

Arches are beams with significant curvature, as in F1g,B6. They can be an~ alyzed in the same manner as previously illustrated for frames, The internal loads depend upon the end fixities, so the analyses are as follows. Also see Art.

A1J,11a Both Ends Fixed

This case ls shown in Fig,B6, the re­dundants being taken at the right end, 0, The applied loads, Q, can be in any direc­tion. The arch is 11ke a half-frame. The internal loads are determined in Table B5. The segment EI values may be uniform or · varied, Table B5 is the same as Table B2 exceut·that x.and y are interchanged since the arch is shown in a horizontal posi t1on1 and there are no shear flows, q,

'for a symmetrical arch, loads,reactions and e.n even number of segments only t of the arch is needed for the analysis with segment 5 fixed at its left end, Po: ZQy/2 so only che first 2 equations are used with the Po -lE terms omitted, Loads in 6-10 are same as 5-1.

S,OoO lbs

:n.1---.,... i ~.----.--r----fl0,000. lba +' ~. -t'S

_l.~--'~~L• .

• p 0

Fig,B6 Arch with Both Ends Fixed

One End Fixed e.nd One End Pinned,

This is as shown in F1g,B7, there be­ing only the two redundants Vo e.nd Po at the right end, so only two simultaneous equations are invo1ved. These .ar~

i; @> +v,i; @> +P,i: @ =6

1:@ +V,E@ +P,1: @ =O:

Therefore, Table BS would be used except that Col, 7 and 8 and their sums are elim­* J1 th Ji inned ends Mo= 0, so only the first equation

applies with the Po and Mo terms omitted.

Table B5 Arch with Both Ends Fixed

CD @ @ © @ @ © ® ® @I @ @ @ @ @ @J

.... ••• ... . ' •• •• •• •• •• • • •• • • •• Mo~ 111ent El Mq ii - . El)' Mq ij z .. ii zy Mq-Y @+

Adal Shear .. El Ei y M., + V.,•@ ..... ... .. •• I @z@ ®"@ ©"@ ©"@ ®•©J ®•©•@ @·@ ®"@I + r.,,.@ lb-In ln-lb1 .. ..

I LO 0 .7 3.0 0 .7 3.0 0 ·" '·' 0 9.0 24,666 ·rhi11 ia the &ameo•

' 1.0 0 3.9 ••• 0 3.9 ••• 0 15.21 34.32 0 77 ..... -32,829 compo· rorlhc 3 1.0 25,000 9.0 12.5 25,000 9.0 12.6 225,000 81.0 112.60 312,500 158.25 -17,669 ncnl ui1111 .. 11d

• 1.0 .. 9,000 U.7 14.9 49,000 .... 7 14.9 720,300 216.09 219.03 '130,100 222.01 20,937 parollol to bulu~in¥

' 1.0 0 20.5 15.6 0 20.5 16.8 0 .. 20.26 323.90 0 20.8-t 9,552 the neu· lhec .. no·

' 1.0 -61,000 26.6 15.8 -61,000 26.6 U.8 -1,622,600 707.56 420.28 -963,800 249.64 2,078 ln1Ju.i• pun~nl

7 1.0 -135,000 :13,l 14.9 -1:15,000 33.I 14.9 --4,468,500 1,095.61 493.19 -2,011,500 222.01 -1,607 ohhe narm11llt1

' 1.0 -212,000 38.4 12.5 -212,000 38.4 12.5 -a,140,800 1,474.56 480.00 -2,660,000 156.25 3,310 vecl.<lr Lhencu· 9 1.0 -321,000 43.2 6.8 -:121,000 43.2 ••• -13,867,200 1,866.24 360.16 -2,874,800 77.44 --8,890 •umu!all lriol11·i• 10 1.0 --422,500 46.1 3.0 --422,500 46.I . 3.0 -19,477,:?:iO 2,125.21 138.:10 -1,267,500 9.0 ... IOtld• from 11Uh"

lhe HcutH Hgm.,nl tolheHg· cmnler. menl cenler, indudin1 r •• nc1 v •.

I: - 10.0 E 1,071,500 236.2 110.0 --46,631,050 8 002.22 2 603.8 ... ' 15,000 1,428.'J KWffc;.,r1 af1q~ Ht--

• When~Sor Eor Ii• i:on•t.antovu the frame llc•n be eon.tde~ to be unity fcweolumn @. For Lhl••umpl• all oftheH ar. i:onetanL

I:@ +M1 t@+V.I:@ +.P,tG') =O} M.=82,787

t@ +M,l:@+v.t @+P,t@=O v.::::.14,764

I:@ +M1 I:G)+v,:c@+P,:C @=o l'o=!,774.8

B6 APPENDIX B BULKHEADS, FRAMES, ARCHES AND BENTS

5,000 lbs

Fig,.B7 Arch, One End Fixed, One Pinned

1n!>ited, the sum of Col, 6 is not need.eel, Mo is ·zero and only the .. above two equat'ions are.·. U.Se·d. 'to .. det6i'inih9' Vo and Po. , . Proceeding as in the above manner the fin­al internali.loads are as shown in Table B6

(r,;:-1f,J7,? ,,.,.,,x. P,, "'~.j'-~'S) Table B6, one• End Fixed and One End Pinned

Seament ~

. ,, @+ V••@ 1+P0 1:{~)

-26,901 2 -61,435 3 -29,519

• 21,60,2 5 17,433

• 14,160 7 11,730

• ll,707 . 9 -10,519 10 -19,759

This iS the cO'mponent parallel to the neutral u:ieorthe vector 9um or all load.I from the '.'CUt" to tb~ ~gm~~t cent.er, including P0

an~V0 ,

Shear Load

'-'·1'

s.i:·~. u·-roir~th1 axi~I load but using the component normal to th1 neutral air: is at the Segment center.

One End'Fixed. and One' End on' Rollers

For•' t;ht~:·· c~s<;! .• ·Fig,, BB •· .. there 'ts oply the redundant Po at the right. end·'·· s.o the only minimum energy equati'Ori ··1s ~ -rer·e'ri1n·g to .. Tal:lle B7 ,

E@+P,£@ =O

5,000 lbs

' 5 10,000

~v,, 47.0 ------'--~.;

Fl.g,BS Arch, One End Fixed, One· Rollers

The final loads are de termir\ed usi'ng Table B7.

Both Ends Pinned

This' case is shown in Fig; B9, Siilce Vo -pas.Sas: throup;ti __ .poiri.t::<Ai ·Pde.an.: b~ ciil·cul­'l,ted 'directly as Po :'~A/L"'1eii:vfog V0 as the·· on1y·redundiirtt, It. is de·te'rml:ned··rrom th¢ mt~imum-' energy eqUB:t1on, .>rererrrng to Table ES,

i:@+v,i:@.:o

The final internal loads are' determined U~~tlg Table.~-,, B.8· shdwrj,

5,000 lbs

!-""-'--""--- .-2'] .. 7'-~--i . ''f' lb• .

47.0 --------~; 'o

Fig,B9 Arch with Both Ends Pinned

0

Table B7 Argh with One End' F·ixed and:' One on' Rollers .

CD. © qi 0 ' .. ©.· .,©, © ~ ,· © ®. .. ~ .

Seg;· •S Mo. • Mq.1.S.jlC. ti.S,sl M ... .. AJ:lal shear ment - -. ,_--. @+P;z0 i:Oad! Lolid

El El El

Do ta Data' Dota ®·@·0 ®·0· . .

I 1.0 D

" D '" <&,oya Thia is th1 co~po- Same as for the

' i.o' D 3.9 0 15;.i, 22, 725 nent parallel to the axial IOad but 3 1.0 25,000. 9.0 225,000 81.0 n,-443 neutral axis or the usingthecompo· 4 1.0 "9,000 14.7 720,000 216.l . 134,657 vector sum or all ne.nt nor ma_ I to

' 1.0 0 20.5 0 420.3 119,453 loada from the "'cut" the neutral axis

' 1.0 -61·,000 26.6 -1,622,600 707.6 93,998 to the aegrnent at the segmen~ 7 t.O -135,000 33.l . -4,-468,500 1,095.6 57,87~_ center, including center .

• 1.0 -212,000 38.4 -8, 1-40,800 1,474.6 11 ,757 • .. -i~i ,OOfi .

... .. ·' ··:<-.. :'°<'· .

9 1.0 43,2 ·-tl,867,200 1,866.2 -69,274 10 1.0 . -422,500 46,l -19,-471,250 2,125.2 -15~,875 .. ..· ..

t = -46,631 ,050 Po=- 5,827 lbs

APPENDIX B BULKHEADS, FRAMES, ARCHES AND BENTS B7

Table BB Arch with Both Ends Pinned

CD @ @ © @ @ <V @ ®

•• * Mq,p0

ASy O.Sy2 M,.. Seg· MQ,Po y Adal Shear ment - - -- @+V,•© Lo•d Load

El EI El

D•ta D•ta D•ta ®•®•© @·©· 1 1.0 6,940 J.o 20,820

2 1.0 ' 38,699 6. B - _340,551

J 1.0 ·.114,235 ~2.5 1>4?7,938

4 1.0 1'94, 751 14.9 2,"901,790

5 1.0 203,258 15.a 3,211,476

6 1.0. 202,.139 15.B : 3,203,276

7 1,0· 193, 187 14.9 : 2,878,486 8 1.0 ' 168, 736 12,5 : 2, 109,200

9 1:0 107,328 8',8 944,486

10 1,0 ~4,582 J,0 103, 746

i; .... ~7,141,769

One End Pinned and One End on Rollers

This case is statically determinate since Po = )::!MA/L, so the moments at all segments can be calculated directly, For the previous applied loads

Po = 466,000/47 = 9,915 lbs

9.·o -29;os1 77.lj.. -66,892 This i11 the com po- Same as for the

nent parallel to the axial load but 156,J -3,575

neutral a:r.is ohhe using the com po-222.0. 15,966 vector sum of all nent normal to 249.6 13, 764 ' loads from the ·cut• the neutral a::i:is

249,6 13, 155 to the segment at the segment

222,0 14,402 cent.er, including P 0 center.

155,J 18,74.9 andV0 •

77,4 1,737

9,0 -1,415

since the effects of curvature of the arch were not considered, There is no analyt1c11./ way to include these effects"!", · The fol­loWing successive a~proximat1on procedure can be used to see if deflections matter.

which is the same as for the previous case,

1) Use the final bending moments from the arch analysis as the moments in Table CJ, 3 and calculate the deflections. These mom­ents are at the segment centers, but they must be at the segment ends in Tabl"e C3. 3. Therefore, plot the moments versus dist­ance along the arch, draw a curve through them and then obtain the moments at the ends of the segments for Table CJ. Jc,

Bents"'

Examples Fig,BlO, The presented for

4>.a

A 0

of these are shown in same discussionr and tables arches also apply for these,

4'..,

A ---

,., Mo

'" Fig, BlO Bents""

The discussions and calculation tables for arches and bents enable one to quickly proceed with the determination of their internal loads, All one needs to do is to enter the applicable data in Col,l through 4 (or 5) and then carry out the calculations as indicated in the tables, The same also applies for bulkheads and frames. Stress analyses can then be per­formed for the structures.

Accuracy of Analyses

2) The calculated deflections are at the se~ment ends, so they must be found at the

·segment centers in the manner described .above, The de .. flec-tions are normal to the arch neutral axis, positive deflections being outward (or "up"). This gives a new shape for the arch and the applied load locations, Recalculate the arches final moments for this new geometry. If they are not significantly different from the original ones they are used for design purposes, ·

3) If the recalculated final moments are significantly different.successively re­peat (1) and (2) until they are not so~

It is for relatively long and slender arches that the deflections are most like­ly to be significant, The following rough guide can be used to see if a deflection analysis is needed (see 11 Advanced Struct­ural Analysis", Borg & Gennaro,D.Van Nost­rand Co,). Calculate the following factor, F, If F is less than 3 a deflection an­alysis is probably necessary,,"

For the usual ns.t_urdy 11 arches (and bents) the calculated" fiftal ""bendinp; mom­ents are suitable for engineering design purposes, even tfiough they are not "exactu, F = CPcr/P

~ Does not account forsidesway effects, see Art.All.ll and the • Sucoessivei7 1ncreae1ng detleotlonu indicate instability reference below. Fpr axial load effects artd buckling- of bents,•• Ignoring' curvature results in an e 0 f bo t "• see Enqineerinq Column Analysis by W.F.McC0111b11, published by rr r O a u ... ,.. catatec.2106 Siesta Dr., Dallas, Tx 75224.

BB APPENDIX B BULKHEADS" FRAMES;. ARCHES AND BENTS

Per is the buckling load assuming the arch to be a pin-ended columri>Wh6.Se'·:1~n.gth·,1s the arc length ·or the arch, For ·pinned ends C = !j, and for fixed ends C = B, For a uniform EI Per = 'fl"EI/L~ For a non-uni­form EI Per is per Art,C2,6a - C2·;6b, P· is the average compressive load in the arph

Carrying out (1) .above :for th.e pin­ended arch of Fig, B9 ";nd. Table BB·, where ~ is 6, l" and. t is 61"'; assuming• EI is 16 x 10• th\' cierl~ctio'n,s at'. the segment centers (using, Table CJ,J) ";re .found to be

Seg, 1 2 3 4 5 6 7 ' 8 9 . 10 De~ ·oa .22 ,15 ,05 -.06·•.12 -,16 -.16 -.10 -,03

These deflections are so· small .. that they would cause a very minor change in the arches sl:lape, _and 1 ts res.ul ting effec'ts_ on the final moments would be incomiequentfal if (2) were carried out (the arch in. Fig, B9 is. the ref.ore a "sturdy" one), Also, for this cilse F ·is much larger than'), which•' indicates no deflection· analysis is

1) For· pinned ends C = (1T"'/-<a) -1. O 2) For fixed. ends C = k" - l, 0 where k

is the value satisfying the equation ktano< cotko\ = 1. 0

J) For fixed ends or pinned ends having a center pin,C is.obtained r:r;om Table B9,

1..e ":eQ ci.rcU.1ar''Arcn. with 8. center P1n . "" 'c<. P1nned •• 15 '·"'.162 _,,, 108 JO .. 40,2 ' 27,6 45 1?.4' 12.0 60 10.2· ' 6,?S ?5 6~56 ~:~t no 4 6• .• 0

Fo"i 8.1,'1._uniform ·parabolic arch and loading as in.· .Fig,Bl) .. ? I I I \l I l-4.i._

Wcr = CEI/L3 where I I.. ~ C is per Table B10 Fig,Bl)

Table B10. P9.rabo11c A ch IFi" B1'\l Oat

h/L "~ Pt-a ' ••n 2 "'1-s ,

, 1 00,7 JJ,o 28,5 ~~:~ ,2 101. 59 45, 5

·' 115 ., 4-6, 5 6.5 ,4 111 96 4).9 4§.9 .5 97.4- - 91 )8,4 ' ,4 .6 _8),8., -_ 80 ., ,30.s J0.5

·' 59,1· - 59.~1 ' . 20.0 20.0 t.0 43,7 4).7 14.1 14, 1

needed: (as the deflection calculations ill- For a· t Qliil ~D'1J::I !al:9 h ·see- T bl a e B11. · so showed·)';.

',"•' Buckling of Arches and Bents*

Some ·.arches and -bents and· their ap­plied loadings'.are such·that no- bending is present, 'b111'' t)lere ·is· cons1derable'•axial stress-·;::,..:· For silch ca:ses".'-ther:.e:11owa_.ble" )·•"· load:ll'lg .. ,.: ls'·' the •buckling • load ·•for·, the 'me·m­bere'··1·' .. Wheno:there ;:ei_S· ·also ';\sonie bending ·"'1l're_­sent it is ··magnified· by·. the ci ''.beam-'colunhi ' effect_"~:_. The ·;·buckl1ng.~:no·rma:·lly-·IodcurS ···as·' a "sidesway.11

, as shown for two typical cases .. in F1g,Bl1, (a) .beirig·a circular. arch and• (b) •being a ·bent 0

···'

--,-,:·-

Q4 r;i;t ,.- ~~---~-~"~'-'--. "'~.---.: I I •

I

I I I

Fig, Bll Buckling of an Arch and a Be'nt

The buckling of arches and bents is discussed in the book of Art,AlB,27a,*<i For a·· few cases such as the following, the critical value of the loading can be de­termin!'.d: by formulas and related data,

I

R

Wcr = CEI/R3

where· C is per the text

items (1)-(3) . •?.!;-:,·

Fig,Bl2 Uniform Circular Arch and Uniform . , . Radial Distributed Loading ... ··

• For additional d11cu11111ons and other o&ae• 11•• -n.oq ot.... . naatlo ·s~b111t1• •. T1111011henko ·and ;c.:re·o MoG:raw-1111·· Boole ·co.·

•• 'fo dehct 1nltab111t1 (p,B7 toatnote) tlwH mast b9 110• aoM,.­-tr7 1n th• arch or 111 th• load., Ir not, add a ••ll latanl load., aboY.t !i:C or the total load, at the centliZ! ot the arch.-

'

Ta'!Jle:Bl·1·-'· Catenar:v Arcih w1th'.a Center P In

,_h/L Plxed'Ends .·. .Pinned ·En"e' ,1 .-59,""' . 2tl,lf. ,2 96,4 4),2

·' 112 41,9 ,4 920) :;5.4 ,5 B0,7 27,4 ..

1.0 2' ' 1;06 * Sn me load an'd formula. as in F1g, St)

For a yery shallow arch, Fii;.Bl4, buck­ling o~n o_c_c~ur _as 1:5: _ 11 snap-thr.011~h 11 move­ment as sh'owri by th'e 'broken'}'irie and is due "to 'shortening' ·C'aused by compress ton,

.. w·k~~)~ ?:··. Gtk I . 1l l ! * r 1 • 1 ie,Q I i 1.J_J I e • .. 1Jt1i L•

.,J.,.1. - - --.-- - -- - - =-.re , ----,_i2. ____ _c __ l.-----~I;--- ~

F1g;B14. Very Shallow Uniform Arch

When there are several loads express them in terms of only one, Q, as shown, The cri­tical value of ~ is defined..·. by .the equation

$ /e = L + 1/,4(1~m)'3/27m3 ·where £ •= d<lflection at center of. arch and

is written in.terms of Q (or of w if:• only w •is present)

m = 4I/Ae" {A = cross'-sectional area) '.

The. above equation.is solved to get the cri­tical va.lue of Q (or of w if oply w is present;),. '"'' ""' When m' ~" l, O th19 type of buckl1rig· ca.nnot o;cur, ~ut the ;:eneraJ: !:_ype (sid.=sway) c!-n•

I,A

T· .....-Buck}lng Equation*

L __LLJ__ :6LI1 [ . 1 1 1 taTiL7j - bI ij + 24LI1/ AitJ

wh.ere j = 'VE'fJP Fig, B15 speciai case of Bent Buckling

·•,_;solv9', :bY: ·,auCc~&·a·~.v~' ·:'tr1&1~·. t~"r,_ ]?_.- .Whep p • p~l" the equat1c:in -:111::'be:a:.at~•t1ed. AlsO. apP11es within IL·

.ery rew percett:~ tor-- &n.T ~.~: ¢Q, L • lo,.d.s having _.. _to~l. or 2P_, tna.>1· J::,e- 1.1 risy 'lf1. ttJ et ,.f att.r• ..

For pinned ends .the equation is (L/j )tanL/j • 61;. L.jzb and tha above _co111111ents apply ·

APPENDIX B FLEXIBLE SuPPORTS, SECONDARY STRUCTURE AND OPERATING DEVICES B9

Effect of Column End Support Stiffness

when the lateral'end 'supports fo; a column'are significantly flexible, this can reduce the allowable axial load to be below the buckling load, as follows, For the pin-ended case the allowable axial load due to support stiffness i~, in Fig,B16 (see the book of Art,Al8.27a,for a derivation).:

_P4j::l'""'~~~~~=L====E=I ::::::jl:-"-P , Pall• L ~ ktk~ :f1 ''k£ k1 + kz Fig, B16 Effect of End Support Stiffness

When kl = oo (simple support) Pall • k2L and vice-versa' if k2 = 00 • For, the c,olllmn itself the allowable load is 1T"'EI/rf. There­fore, mJ-n11Dum values 'of ki and k2 needed to provide adequate support are given, by ,

kik2/Ck1 + k2) = 'l'T"EIIL3

If less than this; Per cannot be reached bu,t. Pall may be adequate' for the abtual loail,P,

,For any multispa~ column,the ability of flexible ,supports to provid&' simple sup­port for a given axlal ,load, P, can be checked using the ,formula in Fig:B16 with successive calculations from end A to end E' as in Fig, Bl 7, This is discussed and illus; trated in the above mentioned book and also in NACA TN 871, Pinned Joints are assumed,

P of A i~ -1c l•o kE1P Fig.Bl? Multispan Column Flexible Supports

Structural Requirements for Secondary Structure ·and -.-Opera tins Devices

Troubles for these items are caused by a lack of strength or insufficient stiffness and usually ShOW Up during proof testing or during serV'.ice,* S,ome suggestions for avoid­ing such troubles are as follows.

1. All devices must maintain full strength , at limit loading, Therefore, the effects of limit deflection, whether due to limit load on the device or to limit load on ocher parts of the structure, must be checked, All operating devices must function properly at deflections due to limit loading,

2, For areas of engagement of locking de­vices where retention depends upon bearing pressure between two parts, keep the bearirg stresses below 10,000 psi and/or provide for ,25• misalignment on assembly,

J. As to location and direction of loads, assume them to be anywhere within a J0° cone centered on the computed direction, When the load direction ls based on a properly orter>­ted machined fitting face assume' a lO"mis-• And are then ~er7 erpenslTe, both

dollar and tlmo-•lae, to r1%.

alignment to exist, For analysts of back­Up structure, assume the most adverse po­

, sition of load, e,g., at the tip end of lugs (which, is universal practice in gear tooth design), Where strength depends on a dead-center mechanism, provide both strength and mechanical advantage from the operating standpoint to take care of a 10" misalignment either way, , '

4, Design limit switch brackets and insta­. llations to deflect a maximum of ,OJ" under a 100 lb load ,in the direction of switch actuation and to have no permanent set un­

, der a load of 100, lbs in any direction,

5, Design external power plug receptacle installations for a limit load of 1500 lbs

,along the plug axis and, simultaneously, 500 lbs, normal to the plug axis in any di­rection. and at the largest moment arm which the plug permits. External power plugs are subject to rough and carele,ss handling.

6. Gaps provided to prevent secondary ,structure from loading up must be checked for adequacy against ,gap cl,osure due to structural load and/or temperature

, deflections,

7, Provide adequate hinges and warp for doors to remain closed and sealed under

'proof loads and, where applicable, design ,1,. the hinges for superposition or proof loads and warp loads,

8, The structural analysis oi' control sur-:':., faces hinged to primary structure must in-,,,

, elude, the, ~ffects of ,the, deflections of the supporting structure. This also applies to power Or coriti-ol system s.hafting. See ex­amples in the book of, Art.A18, 27a.

,9, When control surface skin, panels are al­lowed to bµck~~· the secondary loads due to tension field 'action must be considered, Otherwise these loads may cause either a failure or exc'esslve deformation, particu­larly for thin trailing edges (see p,67),

It is recognized that procedures (2) through (5) may run into difficulties, in­consistencies and designs which, when in­stalled properly, will be considerably ov­er strength, The design penalty may seem to be large large in the layout stage, but actually it is only a small fraction of the cost, dollar ,and time-wlse, of making the changes at any later date, particularly in the field. , The weight penalty is more th,<>n Justii'ied in terms of increased reliability,

, Adding macer1al to a structure, never reduces its strength but can reduce its life,

With a redundant ductile structure (and joints, Flg,Dl,44) the redundant loads can be assumed, the member loads calculated and the structure will deform to this load dist­ribution prior to strength failure, This ioes not apply.to strµctu'[al life, however. nr to nc y;eJd'm9 a:t: !!mi Mad.

B10 ·· APPENDIX B AXIAL LOAD TRANSFER - SPLICES AND DOUBLERS

,:rhei,~- are two . maJ:n cases Of axial· ' 10ad transfer,;: by ·shear1· between. "m~mbeZ.s whic;h require detailed analyses._, -'These_ cases are the installations of doublers and spl'ices.

Spl1ces

Splices involving the transfer of load, by-, shear_,,~--•f.rom ·~ne·: membe·r ·to' Bno.ther shou;Ld• _be ·kept iis:·,short• as 'poss,1b:l.!' •.•• .. _For . example 0: when• .two: '"skins~ •-are· •'spliced to- · gether an:: ideal:. spl1ce•·would involve '·on1y · a single•-line of ·.fasteners, but ·two 'or · . three lines may sometimes be needed.

. '-~ .- ' -, . Doublers••

,, , .

. :, ::. nouble_rs are'relatively lorig niembl!:rs tnstalled·-·on:··a !tba:s·e 1'· mei:D.bei-~.auCh as' On···a wing .. .-skin :;for example, for various pur7, poses stich :·as the following,· ' ·

a) Strength

1-) Td strengtren ,.,; e;if~Hng ,-~ J;r## r-~*~ 2) To• siilvag•» e. damaged 'area .· , . • ·· : J) To strengthen an axially loaded mem­

ber having a hple ,or a out7 out or a non·-stru·ctu:ra:I :-ao~_r-. '.

~'" ·,c·; I'."

doubler p1aking up axi"':L lead, so a - cheek. on a: possible harmful effect on fafague:life at.j;he fasteners may be nefress~ry',

e) The·erfect_of endirig~riaX'fal.member, such 9,s ~ stringer,, on a skin or sheet, mate'rlal; · This may 'be desirable from a manufacturing or ·salvage standpoint, but .its eff.ecj;_ on fatigue .life: must be ·considered or inv~stigated.

The main effort in.analyzing doublers or splices is the determination of the loads in the attachments, Once these are. kn!'wn the 'lo~!ls and stresses, in the mem­bers' ·are knolin and can be evaluated as to strength or fatigue iife, or .si:.irrriess.

ii~tire Bi8 -she.is ii coDl;~;fs,on of the fastener (and member} lOiid.s fn a' doubler an\i. in ,._ long(undesi]'.'able) ,splice••: Note thi>: t the l<>,.ds !l:i:-e le,rgeos.t ir:1 . ,_the end fasteners'; so· the members must be designed an!l ,'tr11-ilo:r:"d ">j;o. r,ed.UC!!:' J;h~s: ".peaking" of 10,acH•; · ,Tl!'.: 1,'as);_l!Oneor 1;1.()"'d, :d_i'!'t!'ibution also depencis upon. the•,_ spring. constants of the : fastener . joints (determined: experimentally.), , . , .... ' '·'·"' .

'''-'·'·!·,,.

A 'd.isciisston o~ J;he analyses and de_, sJgn '!.:Cd,()'\IJ>_leri! .. and .spli.ce.s,is:beyond· the: scope o.f t!tis l;>o.()k. .,However, ·.a>:deta1led

. .···· .. _. , and practical discussiori of the subject is 1) To increase the life' 01' ari exist'f\'l'g ,avail ... J:>le in the_·/'ir .Force.: .. F.l1ght. Dynamics

design in a fatigue critical region. Laboratory'Report AFFDL7TR.;67-1Jl4; jiinuary _ 2) iTo .. properly salvage e.: 'daiila~e'a 'str11,c'- , 1968, ~Arialytioal. Desigri Methods for Air­

·ture: for. ail adequatliL serv1Ce1•iH'E.;1,' •• cr~ft_ Structural .Joint_s,~ which contains 3) To• salvage a "fatl~ue d.ilmaged." iiJ;i'.q,- 'numerous examples. It also' accounts for

·ture 'where_ fatigue:- damage' has··~-~;i..· ,fastener. !t()l<!_,c_le~ranqe,,()J:' :"slop" .. and .... . . accum111aJ;irig _too rapidly ih a par'-''' loadings· into ·:1:11e p;Lastiil: rar;ge o( · the . ticularc'vehicle or group" of vehic'les ·Joints.. The report 1S over 187 pages and

will enaple 1;1!e reader to' obtAin'· •;,:, good. un­derstanding ()f the subject and 1t·s jnany ap­plications_, and is __ available'·at• _some· 11 b- ' raries. A copy,,with correcfions' and _add•. ed comments, is .. also 'available':··rrom•· w; F.' Mccombs, 2106 Siesta Dr., Dallas,TX 7522~,· $19.95, plus $3:.00 shipping. •·The reader is encouraged, to. consult the' report'*"

c) Reinforcement' for iidd.i ti,;llai st1'i'f;;~ss ptirpcises wh1Ch must include a consl.d.:; _. eration'of possible fatigue fit" liin'i'~ , tat.ion, due•· to' the fnstallat1on. ·· _ .. ,: , --. •;~·:. · .. ·;·:1/1·}·' .,,-,"C·'.•V;'.i .'')'i; c.;:·Jr-0>'1.',"' . "

d) Any. member •atti!l:ched t'o an ~:l'{a1i';;i load"­ed base structure will act as a local

,. !' '; -, t· : '', .. , ,, '

Q - : : 2 2 z_ z d:;eJ_: . . . . . s DOUBLER

' ,,., . "''

Q - ~--:::vs:::. .s .

SPLICE

0 ~----~-----~O O ~-------LOAD IN MEMBER D

i ~~11 I I I I

FASTENER LOADS

WAD IN MEMBER D

I_ .. -•.-./~. ;Fi 8 .. . i;. ll1

<~I I I I I

FASTENER IJJADS

'* Being the only such comprehensive report available, it should be included in any structures library.

APPENDIX B JOGGLED. FLANGES AND. RIGID JOINT TRUSS BUCKLING ANALYSIS

L/O

K j

D/t 2

Fig.C?.45 Cr1~pl1ng Stress {F00 ) Reduction Factor for Aluminum Flanges

Ca1cu1ating the Critica1 (Buck1ing) Va1ue· for a Rigid.Joint Truss's Loading 'I'he critical t'bUckl1ngJ .Loading ·ror the

truss-{all members deforming 1n bending) is round _by a sticc'esS1ve trial·- pr~cedure as follows.

1. For the &p'J'11ed loading determine the member lOiids assumtng ~in joints •. Factor the aPlJlled loads ·to cause a 1011.d or about 1. 6 tl.mes the 'buckling load in the rtrst m~mbe:r that would fall, llssumlng pl.n joints. Then determine the COF and OF for all members, .tor rigid" joints. 2......_A?"PlY a ~nit cou~le at the joint~ N, whose members have the largest compression loads .and carry out the moment d1.str1 "bu.-tion procedure un­t11· all. joints have been balanced and. the· 'balan­

' c1ng moments carr1ed back to the Jo1nt N. U,Do not reDalance N but again rebalance all other Joints (1n the same se~uenee as before} carrying the ba.1anc1ng moments back to N. !±..&....Ele'Peat unt11 .the.moments carried 'back to N are negl1g1ble. usually 2-) sets or 'balancing. 5, Ir the sum or the moments carried back to N, ~PIN' is"'-·l.O the .. load1ng is'less __ than .c.r1t1cal if more than 1.0 it ls above the critical load. 6. Re~eat the above with another loading, suc-9Css1ve1y, unt11ECOMlf"l-.0 .. (or, sa1, be'..:ween .99 and 1.01) and that 1s the cr1t1cal load.1hg.

The erocedure is 11.lustrated below for the truss or F1g.A11.92 ~here, after several succes­sive trials. it is round. that for an a'Pi:illed load o! 1-2,)77 lbs ECOMc=-t.OlJ, so that is,essl!lnti-. all-Y,the critica1 load (tor a load of 12,J60 lbs .I1COl!C"1. 00 )-.

As discussed 1n·Art.tl.15b ~nd Table Alt.6, .the falling load Will be less than the bUckllng 1.oad·beeause of beam-column failure. UsualJ1 it will be at ~bout 90% or the 'bUckl1ng load, 'btlt only a· successive trials analysis as in Art.

·A1t.15a will give the M.s.. ·

The COF and. SC value's used to obtain. SF and 'OF :v"a.lues were' ca1culated..us1ng the formulas: on p.1.

Gusset ~lates can cons1ders.bl.y increase the critical Loading and the falling lo~d1ng. The analyses are as befon, but the S.C and COF Talues are difterene. These are available i:n aer.14-and -1n ... The Analysis or .. Structure_s" 1 Hoff, N ,J., _<Ln<l_on P· 612.-.IS. . · · ·

Assuming t1in Joints to determine. the ax­ial loads is ~uite reasonable, although the Joint second9.rY bending moments do. change these slightly. Actually the as­sumµt1on 1s conserTat1ve, the. final ben~ ding moments always being smaller lsee "Moment D1str1bu"tion·.cere,Ja.mes 1'!"., Van Nostrand Co. ) . In '?abti 111 ... 4 cross-sections are square. At the end of item 6 1 ·p.4- add1"'I'hen see Art.A5.J1a: .

BLL

At the bOttom or p.4 add the rollowing1 "If. the.applied. loading is abov~ the critical value. the· successtve c9.r?'Y o"V"er · moments w111 diver~e. To calcu]Ate the

: cri t1cai { buck1ing) lo-id 1nfC value see ff>1 BI (, :1.iu11E!s thv Pe!H' gQv•r, R1P.=id Joint str­ength is usua11y oVer 60% more than is ·

~ ~~~fl 'Predicted bye. 'Pin Joint stress analysts.

C:-~,~OJ~C:::J 'fue (ai'l1ng toad C'or tl-\e tI"U!S 1S 1\,088 1 1 1. cs, failur" occur1ng 1n member CE as a

A '--::]~::===:!::~~~=::::·~·~2~•~:;~·0~27~'r:~~~~~E.. beam-column, j.6" rrom C (M-448 in-lbs). -.~R';lf;. 07) ..:.:o669 ·This is e9.6% or the aeove ouck11np; 1oe.d ----------·-.Ot2J/ZI .0109:::::::::::·Looo£ 13.nd 1s obtained by success1ve tria1s .. us----------- ,0002 ,oooo ~ 1np;. the -procedure o~ i\rt,All.15a. ·

8

T-:-r I,, . ,1!;;

ptt{lj I I

.. . . . ' ... • .. ·· ~~~- ·;~-' r m m1' ¥:'~~; 1· 1 --r-+----.: 4 ~

'I

2

0 I···' I Slif;fness:coeff;cient:=S~1<···1· f~\~j (SF' =' 4SCEI/L)

-·

-2i----:-t-t-t-+"-+-~-+--l--+-~~LI (J It) is/L .•....

'\:I- . . •0:30 ... -4 '

;.

-6 I I I I I I I I I I l . I '•I

~· . - 8 J--f,--, --t--l----t-'--1----1--1----t--l---'--I-"-'

I

-10 ~--'~--'-~-'-~"----'~--'-~-'-~-'---'~--'-~-'---'-' 0 2 3 4

kL "'- L/,/ Fig. 2.4.9. Stiffness coefficients for compression bars with gusset plates.

From author's paper in ASCE Transaction~ lf><?.,'/.O..,,

'·" ~~~th£~ 1.5l~I 1·· ········

1.0 ~ >T. · L' . I I I _;:;

f o.5 I L .... I I . I I /\ - ·.·. ... . •. Carry-over factor:C I .I <eA~'IS: · I I · \

. 0 l . I··.:· . I I · ·· I· I I . ·r , l'~ l 't I .·. :.· " . . . .. I .'. •:<y::;>'L\ \' ..

)> 'ti 'ti

·\;l tJ H ~

tJj

Gl c:: (/] (/]

t'1 8

'ti t< )> 8 t'1

0 )> 8 )>

":! 0 ::<:!

s/L (/] -0.s 1 I I I I I I I I: ' ·r I\~-~\~ o.3o ~

.... ,

1

• .

1

. ~~ 0.20 g -1.0 r--+I 1--'-+·.·.· · 1-+--1 --+-I --l-1 +--ti 1--1-· 1~; \--t=-< LI g:6~ ~

. 0 1 2 3 . -4 ; 5 . 6 t<

"="

kL "' /,/J.· . ., c..i . I 0

. ' . H Fig. 2A, l0" ;;ca~-ry,oveJ factors for c?mpression bars \~ith ~sset plates. ~

' · '' . F}'om author's paper m ASCE Transactions .!!V r'1' (/]

L;,,,r '11~!' "·"'C""""'Y c11J .. ,..,-. &,h .... ,.~,;;.,, b,,..t:..a4~ ~i¥C y,)... ~ S-. 7 .i!or J' r1~S

tJj ~

N

M ~

!11

(/)

E-< z H 0 ,., ... :i' ~ D E-< u D

18 . ·.,'.

'16 I>

14

12

'•

' ' I I .

·P·~P ....._ :- . ..:-:. --~L '-:1•.r •.. /

/ k2= PYEI

j= IJ~, v . . . v /

Stiffness coefficient =SC ./ v v

.I ' ' ' , ' ~

(I)'. / v (SF = ~SCEI/L)

01) /

/ )v;/ ~ 0 Ji,· 10

~ \.) ;;: ~ Ci ~ 8 [ii E-< ~ ..:11' n. E-< [ii. (/). (/)

D t.9

6

4 0

!11

x H Ci z [ii.

Fig. 2.4.l l.

n. n. -.:

, _,..,,,,. ~

~

v v

-v

--~- .. ) ..

·~

1 2

/

/. '\)'.l-y

-<-A' 1,_,/ -? ()· •/

MV l)\lq

;:....---V.\v,. .

... L--- krY\): s .. -. ;

3 4 kL::. L/j

·/ /

v

/ "" v

/

,/

:.......--- .

I 5 6

Stiffn1'ss coefficients for tension bars with gusset plates. From author's paper in ASCE T1·a11sac1io11s"&Yrl

,·;

c

0.7 t---.; :-........ ' .

~ ..._ -.......:. ~ -:.... " ~ :' - ....._

0.6

--........... ~ '-..._

-......:. '~ !'-... 0.5

' ' .......... "

" 0.4

0.3 Carry-over Factor= C

0.2

o. 0 I 2

p 0 . -- _Qi' ~~._.....,

4_•f--__ L -'-l•_H-

!'--....

'

"" I~ !--.._

i

' L --- -.- I .:..,__

k2= P/El ·-

' ':- J J

.

f',._

"" ........

I'-.

~ '-..._

' " " ~ ""--

" " "-

'!'-.....

3 4 kL .,_ J./j

i;;;-

-i-

~

~

~

....._

"

"- . -I'-

' t---. I'--

r-....... '""-.. ..........

I"----..

I 5 6

•IL 0.30

0.20

0.13

0.09

0

Fig. 2.4.12. Carry-over factors for tension bars with gllsset plates .. From author's paper in ASCE Tra11sactio11s,"' i!.'r'SV

INDEX Arch Analysis B5

Beam deflections 1, A21 Beam formulas A21 Beam-columns ·

analyses J4, J9 allowable stresses JB approximate formula 1, J? calculating Per J6 carry-over factor J deflections 1, J5 equivalent loads J? formulas 8 1ni tially '!lent J? ina.rgin of. safety JB, J9 numerical analysis J, J4-J6 · 0th.er .consid.erations J? varying EPJ4

Eol t!j, '74 ' BOlt strengths 77, A21 Bent Analysis B7 '.llusii~n.o:s 14 · Bulkhead Analysis Bl. Carry-over factor J' Clamp-up importance 74-75 Columns. . . . . .

buckling .data 23-29 .,B9 design curves 12, A4..:A5 e ffe c t.1ve· modulus 25 end friction effects JO iititially bent 21 instability criteria 15 non-dimensional curves 24 one end free 12 sh.ear effect on buckling 14 stepped 16 ····. .. · torsional buckling 7, 44-46 two-s,,ari 15 ' .· moving (mechanisms) JO· mul tis'?an 14' · ,; numerical anal,ysis 17 varying EI ·17 ·

Combined stresses 11 Crippling 4i44· curved bes.ms 6 Doublers BlO , , · Design check i1st Ai Diagonal tension,

stringer system 51•60 longeron sys telil 60-66 flat web holes 49-50

Etrec.ti ve modulus. of elasticity 25

Faetors of s~fety 8, ?4 Fastener/joint design guides Ali> Fas'tener /joirit design data A10-A21 Fillers, effects on jointS 76· ' Fittings

.design 74 marglns of safety 9, ?4, Al factors of safety 74~"·""·

Fixed end moments 4 Frame stiffness criteria 47

Il

.;

Frame Analysis Bl Flexible lateral.supports 2J, 21';, 29, .B9 !nteraction equations 10 Gusset plate data B12-13' Joint strengths A10 -2t' · Jotnts, orittle and ductile 75 Joggled flanges 42, S II Lugs • oblique loads 74 . Lugs'• transverse lo_ads 74

Margins of ,safety 9,10, JB,39, Al tl'Ue 10 . . apparent 10

Mat~rial propel'ties 8, A6""A9 Moment distribution 1, J, 15 Monocoque shell buckling

• cylindrical shells 67-71 conical shells 72.-77 spherical end ~aps 73

Numel'ical analyses c!llumns 17-21 beam..:columns 34-36

Nuts 7q, Nut/collar tension'strength A21 Operating deTices . B9. Preload torque factors for bolts A2 Pla~tic bending 1, · J·1

residual stressesJJ shear stresses J2

Plastic torsion 1, JS Plate buckling

shear :39 bending, one edge free 40

Principal stresses 11 Proportional limit stress JS

Rib crushing load~ S Riv'et de-sign 50· Rockwell/Brinell hardness data A2 Secondary structure B9 Shear stresses in plastic range J2 Sheet buckling (see plate buckling) Shims (see fillers.l

· Slgn convention Jo 19 Stiffener .minimum, I .. A2 St1ffness coefficient J St1ffness factor J ·

. Stress-strain curves. AJ Successive trial solut1.ons 10, 2), J9

· ~plices BlO Tangent modulus 1; 24, 25 Tension cl1p strengths·77, Al-A2 Thread design/strength 75-76 Thick-web beam analysis 47 Tolerances; allowable 11 Torsional buckling 44-46 Truss analysis 1, 3~5, 811 Tension field (see diagonal tension)

Yield stress bepding.J!l.qdulus JJ

INDEX FOR THEC BRUHN TEXTBOOK

The current (1973) textbook has no Index. Following is an updated index for that edition,

Accelerated Motion of Rigid Airplane· .••.

Aircraft Bolts • . .. Aircraft Nuts . . , . Aircraft Wing Sections -

Types • ·• '. '; · ••••• Aircraft Wing Structure_­

Truss Type; • _. .. • •..• Air Forces on Wing . . . Allowable Stresses· (3.'nd

Interactions) . ··. Anilysts ·of -Fra'rDe with

PiMed Supports·. • . • Angle MethOd . • • • • • • Application of Matrix Methods

to Various Structures Applied Load •.. AxiS of Symmetry • • •

Beaded Webs • • • Beam Design - Special Cases. Beam Fixed End Mori:ientS· by

Method of Area :M:Olnents Beam Rivet Design··~:-.'"~;·. ~ • Beam Shear arid Ben:d_ing

Moment ••• · • .' • ', •••• Beams - Forces at a Section • Beanls • Monlerit Diagl'.llins • Beams with Non-~?J1e1

Fl<µ1ges ••• ;- .· ••••• Beams • Shear and Moment

Diagrams ••• -:· .«· ••• .ae·ams - Statically I>eter·

minate & Indeterminate • Bending and Compression

of Columns •••• , •• Bending Moments • Elastic

Center Method • • ." '. •

A4. 8 Dl. 2' Dl.2

Al9; 1

A2.14 A4.4

Cl1. 36

A9. 16 C7. l

A7. 23 A4.1 A9.4

Cl0.16 D3.10

A7.32 Cl0.8

A5.1 A5.7 A5.-6

Cl!.9

A5.2

A5. l

A18~ 1

A9.1 Bending of Fectangula.r

Plates ....••••• Bending Strength - Basic

Approach. ~ • • • • • • , Bending Strenith • Example

Problems • • • • . ; · ... ; · Seri.ding Strength of Rolind

• AlB.13

Tubes ... -•• ·• ; ~ Bending Strength - Solid

Round Bar • , • • • • • Bending Stresses , , • , BenP,ing StrEiSs'es '" Curved

Beams •.•••••.•• Bendiilg StreSSes'. • Elastic

Range •• ;··-~ ;·: .•.•• Bending Stresses - Non·

homogeneous Sections .• Bending Stresses About

Pri.ticipal Axes • • • . • Bending of Thul Plates • Bolt Bending S_~ength . _. Bolt & Lug Strength Analysis

Methods •• , .•• , Bolt Shear, Teri:Sion &

Bending Strengtlis , Boundary Conditions Box Beams Analysis Brazing •• Buckling Coefficient Buckling or Flat Panels With

Dissimilar Faces Buckling of Flat Sheets under

Combined Loads ••••• Buckling of Rectangular

C3.1

C3.4

C3.1 A~3.1

A13. !5

Al3. 13

Al3-, 11

A13.2 AlB."10

Dl.9

Dl. 5

Dl. 3 A24. 8 A22.5 D2.4 CS.1

C12. 25

C5.6

Plates • • • • • • . . . . . AlS. 20 Buckling of Stiffened Flat

Sheets under Longitudinal Compression ....

Buckling under Bending Loads Buck.ling under Shear Loads. Buck.ling under Transverse

Shear .•.•.•

Carry Over Factor Castigliano' s Theorem Centroids - Center of Gravitv.

C6.4 C5.6 cs. 6

CS.)4

All.4 A7. 5 A3.1

Cladding Reduction Factors. Column Analogy Method, , Column Curves .:. Non•

Dimensional ••..•. Column Curves ·• Solution Column End Restraint. Column Formulas •.••. Colunm Strength. • . . • • • Column Strength with Known

End Restraining Moment Combined Axial and Trans­

verse Loads - Genera.i. Action ••••••.••••

Combined Bending and Compression .• ·'· .,. , •

Combined Bending" and· Flexural Shear •

Colnbined Bending and Tension •••••••

Combinei;i Bendl.ilg and_ , Tension or Compression of

CS.5 AlO. l

C2.2 C2. l3 C2. l C4~2

C7.21

c~. 16

A5.21

c4.22

C3.10

C4. 23

Thin Plates ••.•.•••• AlB. 17 Combined Bending & Torsion , C4. 23 Combined Stress _Equations , • Cl. 2 Compat:a.biliiy EquatiOii.l:t , • • A24. 7 Complex Bending - ·

Sym'metrical Section_. • • • • C3. 9 Compressive Buclding Stress

for Flanged _El_em.ents • • Conical ShellS __ • Bui:kiirig

Strength , · , • • , • ' • • • ConStant Shear Flow Webs Constant Shear FioW Webs -

C6.1

CB.22 · A14. 10

Single Cell • 2 Flailge 'Bi!am. A15. 3 Constant Shear Flow Webs -

Single Cell - ·3 F1ani'e Beam. A15. 5 ContinUous Structures -

Curved Members ••••• , All. 31 Continuous Structures •

Variable Moment of Inertia • Cor~. Shear.,~,,_· ••.•. • : • Corr_ection for Claddlng •• Corrugated Core _,Saitdwich

Faillire Modes· • • CoZzone ·Procedure Creep. of Materials Creep· Pattern • • Crippling Stresses

Calculations • Critical Shear Stress • Crystallization Theory Cumulative Damage Theory. Curved-Beams Curved Sheet PanEils -

Buckling Stress .. • .••.• Curved Web SystetilS .-'·. Cut-Out:S in Webs or Skin

Panels •• -_, •• · •••••

Deti.ection Limitations in Plate Analyses. • • • • •

Defl~ctiori.s by Elastic ~eights Deflecuoils by Moment Areas. Deflections for Thermal

Strains ••..•.•• · •..• Deflections by Virtual Work Delta. Wing Example Problem, Desigti for Compression Design Conditions and Design

Weights .•.••.••. Design Flight Requirements

for Airplane , •••.• Design Loads •..•• Design for Tension •..• Differential Equation Or

Deflection Surface • • • Discontinuities •••• Distribution of Loads to

.• ,; '"'-~SheeFPanels; ·•·• Ductility ••••. Dummy Unit Loads • • . Dfnamic Effect of Air Forces.

I2

All. 15 012. 26

C7.4

C12". 27 ca·:2 Bl.8

Bl.12

C7.7 Cll. 16 Cl3.l C13.Z6

AS.8

C9.l Cll. 29

D3.7

A17. 4 A7.27 A7.30

A7.17 A7. 9

A2ll/ C4.2

A5.12

A4. 6 A4. 1 C4.1

A18. 12 A20, 15

A21.2 Bl. 5 AB. 6

A4.13

Effect o! Axial Load on Moment Distribution. •

Effective Sbe"et Widths • 4 , #

Elastic Buckling Streiigth ·a( Flat Sheet in __ Compression.

Elastic - Inelastic Action • • Elastic Lateral _Si1j:ii)ort

Columns • • ; • • , : • • • ElastiC ·Stability of Column • Elastic Strain Energy • • • , Elasticity and Thermo-

elasticity - One~Dimensional Problems ••••••••••

Elasticity and TJiermo­elasticity • Two-Dimensional Equations ~ , • • .. _ . • ••

Electric Arc \Velding • • • • End Biy Effects • • • • • , • End Moments for Continuous

Frimeworks , , • EQ'UitiOns of Sta.ti'c

Equilibrium Equilibrium Equations

Failure of Coluinrl:s by

All. 22 C7.10

C5.1 Bl. 5

C2.17 A17.2 Cl.6

A26.1

A25.1 b2.2

Cll. 23

All.10

A2.l A24.2

Coiiipression •••.•• ~ AlB. 4 Failure Modes in Cui-Ved

Honefcomb Panels. • • C12. 20 Failure of Structures • • Bl..1 Fatigile ~ysis ___ - .Statistical

Dtstribution '. • ": • : •••• · £i3~q, Fatigue and Fiil-Safe Design • : Cl.3,3S Fatigue of Materials Bl. l4 Fatigue S-N Curves • C1s. 'r.3 Fillers.· • • • • • ... • D3. s · Fittllig Design • -••. _. Dl: 1 Fixed End MOments }a.~l. a: .. -Fixed 'End MOlnent:S- Due to

Support Deflecti.ODs -. • Fixity Coefficients. • • • ~Design.,.~·-__ ;, .... ··.·,.~ Flange Design Stresses • Flange Discontinu'ities. , Flange 'Loads • • • • ._. • • • Flange Strength (Crippling) ... Flat Sheet Web with Vertical

Stiffeners • • • • • • Flexural Shear Flow

Dtstribution ., :'·_':_:·'. • ,,• Flexural Shear Flow -

Symmetrical Beam Section flexur~ S~ear ... ~~ress •..•.• ",. Flight Structures - Required

Stren.gth ••• _ .~- •.•. : •.•.•• Forces .on Airplane ·in Flight • Formulation of Plane Stress

Problem ••••• FraIIies with Jouit

Displacements • Frames with One Axis or

Symmetry •••••• Frame's· with Unknown

Joint Deflections •.• Frames with Unknown

Joirit Displacement Fuselage • Balance Diagram Fuselage - BasiC Structure • Fuselage ~ Example Problem

Solutions • • · • ; • , , • • • • Fusel3.ge Frames • • • • • • • Fuselage Shear·s·and Moments Fuselage Shears and Momerits

for Landing Conditions · • • • Fuselage Stress MethodS • • Fuselage ~ IDtirnate Bending

Strength • • • . • ••

Gas Welding •• General Organization or

Aircraft Co. • • • • • • • General Types of Loading. Gerard Method Gust Load Factors Gust Loads •••

All.9 •. · J!2. l ,._Cl0.1 .. CI0.2

ClO. 7 ci.1. a

..... ~~0.,4-:

C10.1 A15.24

Al4.5 A~4, .. l

C1.1 A4:4

A25.5

A12.B

Al0.2

All.17

A12.8 AS.13 A20.1

A20.9 A21. 17

A5.12

A5.18 A20.3

A20.6

D2. l

Al. 1 B1.1 C7.-2 A4.6

Cl3. 2'7

INDEX FOB. THE BB.URN TEXTBOOK - Continued

Honeycomb Flat Panel Failure Modes •

Imp~t Loading •• ,- , ·: Impact, Testing Methods Inelastic Buckling , · -~ _·: , lnelasUc Buckling Strength

Cl2.8

Bl. i5 Bl.15 AlB.,6

of Flat Sheet in Comprel!ision C5. 3 · Inelast_ic Bucklint of'Thin

Sheets ,· •• , •• , • , ••• Al8. 23 lnerlia'·Forces·~ ·• ·; ... ··• • • -~4-2 Inertia Loads Due to'AngUJ.~:·

Acceleration •• ~-<<";·'-,' }·'~ · r·\\s.·1a Inertia_ Loil.ds nue 'fu 'Unit

ioti, 'cioo in. lbs. Pitching Moment, •••• · '

Initial Stresses • . Internal Shear Flow Sys.terns Inter-.Rivet Bucklini SireSs.

Joggled Memt>e:rs • • •• · JdhnsoJi:..Euler Equation. Joints - Method of ••

Landiili Gear Unit:S - ;,.-Calculating Reactions & Loads on Members , • • ·

. Landing Impact !.<)ads. '. • Large I:J'eflectionS iii Plates.

. Linlit:L1iads , • , • , -, "'; ·, , Load'· Factors • • . -~; . .. . . • Loaded ContuiUOUS ··Beanf With :Y~eicifug SupP<J1 ~ _ ~ ~, •. , -·-~

I,o~e;.On Type SysteDi:.·:·' '~.: .•

~eu:ver Loads ••• ··/.' :·"~ Mass Moments of Inertia , • • J4atrU:es • Element Stiffness··', MairiX Methods in nenf!Cii.Ons MatriX Methods ·;Stress·''·

PTob.l€n1s •• , , ;·--,;··-;: ·>. Maxirniim Shear Sfress'es··' · ·

for sii:D.ple Cross:-Sections . _. Meml;)I-ane Action lii 'Tiiirl'";

Plat_~s·: .••• ~ •• ··:·~":· .. 1'

Memb_~;tn,e AnaloID: ...... Membrane Equ~ti.orls 'of · Equil~.~rium · • ·, · •· , • •

Metallic Materials •.•.•. Method_ of Displacements • • Method Of Joints • Trusses •• Method of Mom,(ID~ ·:~ 'i'fUs.S_e~ MethO.d" Of SbearS~·~:TJ'usS_es:'· ~ Methods' ·of ColuniI:J·Fa.uure::· ~ Modulus of RuP:tu:r't{' .. ;~.. • .'.·:·· • ModuluS of RuJJ¥e Stre_s.s. _ • Modultis Theory • • ·• : .: • · • Mohr',s. ~irc1e·"~'': .: •• · ·:,. --~ Moment 'Distribution Method Molllent of Inertia • Striifiit))

of-Columns • ; •. '. .. • ... , ·;···;·:c. Moments for Combinations

of·Vii.rious Load Systems_ •• Moments ,of Inertia .: A

0

iri]1ali{' Momen'ts of- Inertia • ··

Centroids • , , , ·:··.·._ :: ..... Monocoque Circula,t: Cylinders

Buckling undl:!i External Pressure .·: .:·.::~ -._,_~ • ,• •••

Monocc:ique CircUlar Cylinders Buckliiig under Pure Bending

Monocoque Circular .. _ Cylinders Problems fd~:._Fin_dirlg

· Buckling strenith · ·" _; Monr;icoque Cylinde_i-S.':

Buckling under Axial Compression ••• ,

Monocoque Cylinders • Buckling under Axial Load an..,~~tf~ernaj,,_Pres,~ure...,

A5·.:i9·' AB.1.3

A6.·6. C7.12

D3.4 · c7,.~.~2 ... A2.-10.

A2.23 Cl3.2.6 Ai'i.'6_ A4.i A4.5

c~~-t-4·f. ,_ " . •l·~.i1. ,.

Ct3.;i:t A3.2

A23.S .:A'i; I-8

'AiLili.

A14.5 ..

Ai7-.~·5'· A6.3

Ai_6 ••. i\ ..::~· A2.10, A2.11 A2.12 c2:\. C3.3

C4.15

~f:·~ .' }."iL'i'

·',,\}•

c2·.:·5 A5.22 A3:'a ..

A3.l

ca. 11

CB. 7'

c8. i7

CB. I

CB.3

NACA ~ethod • • • , • Needham Method ••• Neutral Axis Location Neutral Axis Method •

Cll. 14 C7. i'

Al3. 1 A13 •. 3

Octahedral Shear Stress Theory, •.. • ~ ••.•••

Para!1e1 Axis Theore:~ •

Phjsical Action of Wing Section

Plane Strain • • • •· • · • Plane Stress, • • • • • Plate Bending Analysis • Plate Bending E9:uatiphs~ Poisso'n's Ra~o ~·:,;"'; '.: Practical -Wihg SeCtioh''

Application. • • • • , PressUre Vessels - ...

Applications • • • , .' : Pressurized Cabin,~tr~~5:

Analysis ••••••••. Principal Axes •· • , ,,.• • Pr.incipal Strains • • ~ •• Pi'inciple·of Superposition Product of Inertia •

Radius_oi Grration • r-·-·. ~ • ; .. Ramberg-Osgo.qd_ ;Equation •.• Redundant Prdb1em'Den'ecuon

Calcu!a.tions bf Matrix Methods •... ~-····· .•

REidUndant R~~t~o.~.~.,·P,r.:, .. ~e'<1:5t Work •· •.• _._· ,• '-;·' , ,.

Redilndant Stress Cal.Culations' R~dun~ant stress·es'1by' ·Least

\YO.r-k.' •..• · -:,..;, : .. >.-~ ... :.c,,~ .. • Redundant S~q~11re;~ .. n.th·' ~emt>ers: ~!1l?j.~~,~~?,}B Loadings • • " •. ·'. .~ :~ ._' ... ,.

Relation - She~~·ana,Beilding-. Moment ••••• "~ '.--' •• Rei;;traint Produc·ed 'by 'Lips

and Bulbs • • • • , ,. • • • Rib Loads from .,

DiscOrltinuifiefi· ;·'. . . ··:' . . Rib -: . MUltiple Sb'.'UJg_er .Bea~. Rib - Single Ce:U ·._Be<i.nl".;.~-' .:.':' Rib"'·--rti,ree strmget:'Beam·. Rivet ·Design • • •• Rivet Lbads • • • • • RiVet€"d Connections Rivet! 'in Tension • ,

SandW.ich Construction and oesigri ••• · ••• -•.• <~-.

Saii.dw.i.ch Structural,_ · ... P?-6pertieS'• ~ . .'; •• ·•

SandWich Structures Design'. Secant Modulus ••• · •• ,· • Secondary Bending Mo~ent:S

iri·Trusses •• ~ ..... • ;'_, ., Sectio_n of Maximum Bending

Moment · ••.•• Section Properties . ~ ... Sliear C"enter, , .' . - . · ~ • Sht?ar C.ent_er. ~0Cati.9p.._ :-:.

Neutral Axis Method:~.· Shear Centei ·of Sing1·e.··cel1 -

Three Flange Beahi . • • • , Shear Center oi Single_Cell"'

Two Flange B'e·am ;- ~~-. . ~ • Shear Ciips ..... '_ ••. • • -~ .. ' • Shear Flow in Cellular Bea.ins shear Flow-· M:ultipfe"ceus.· • Shear Flow in Tapered Sheet

Panel ......... : · She<i.r'Lag Influences ••• Shear Loads • • • • . • • ·• Shear Stresses & Shear

Center • Beam Sections • Shear Stresses •

Unsymmetrical Beam Sections •••••••

Shearing Stresses from Principal Stresses . • •

Shearing Stresses • Right Angles •• -••.•.•.•.

I)

Cl.8

A3. l A3.9

A19. 11 A25. 'I. A25. l A17.3 A."11.1

'"')~~; .. 'I

A19. 2~

A16;2

Af~. ~ A3;10. c;,.s A8.1 A3.9

A3. l C2.2·

A.~.,27

A8.2 A.8.27

AB.3

A5.4

ci;·5

Ani·f~' A2'~~- 6 A21~ .. 'l.

C1L18 c1'1~:7 rii'~-:1~· D1~·2s '{!-'''<·

· cild ·

· ci2· .. ·4 c12;'33 ·

Bl,5.

All. 28

A5.'41

A3.2 Al4.1

Al4~,--~~ Al5.·'6:

Al_s:·.4,' ?53. 1

A.15. 24 A15.16

A15.27·' A19:'24 Cli.'5

A14. 6

Al4. 6

CL 1

Cl.1

Sheet-Stiffener Panels • Falling strength ••••

Sheet_.Wrinkling Failure. Single Bolt Fitting. • • • Single Cell Beam -

C'Z.15 C7.15 Dl.4

Symmetrical about One Axis. A15. 1 Single. Cell • Multiple Flange •

Orie Axis of Symm~try. • • • A15. 7 Single, Cell • Unsymriletrical .­

Mu!iiple Flange •. ,'..:;- , • • • A15. 8 Single Spar - Cantqe~er . __ . . .. y.'i~-". Metal .cOVeT'Eid .• '. ;~,)~: J!..i9· •. ~~

Slope Deflection - Hinged End. A12. 3 Slope: penection Method ... ,.. ,Al~. l Spot y.'elding. . • • • • • . • D2. '7 Static Compres;;ion Stre_ss-,

Str:iin l1l:i.$r3.m. ·• _. ~-'' ~ . Static.-TeiuHon Stress·­

StrS.in Diagram • • • • • Statically Determinate

Coplanar Structures and Loadings_ •••••••.••

statiCally· Dete-rminat~ and hi.determinate __ Sti;uctur!'!S ·

Staticilly Indete!rmiila.te:. , .. , Fr.imes -- Joiilt:,ROq.iiOn ·;

Statically Indeterminate .Pi~~eni • :' . •. :~:. •,:_ •,,:-· .-· .. _.

Stepped Column,:~ .. S~enITTP Stiffeii.ed Cyfiii~r.i,C~.~;, .. · 11 , .:'·

Structures • Ultimite

B1~4

Bl.2

A2. 7

A2.4

Al2. 'l

AS.1 C.2.14

strerigth : ·"~ .... ·._~-:-,,-,.:. .... Ca.a SUffn.ess &: Carry:-over , ...

Factors for C:JJ.ryed lde?nbers Aff..30 Stifin_~ss Factor~:,;.:.;:-_·.,'..'~.",' ..... Ali. 4' Strain • DiSplac:ement

Relat.ions • • • • « . Strain-Energy~ .. ~ .. :.~! •.• Strain. Energy of Plates Due

to Edge Con:ipi'esstt;>~ ~ .. ,

A24.5 A7.1

Ben~ing ••••••.•••• , Al8. 19 Stritn Energy in Ptn-e Bending

of Plates • , • ; • • • • • • • AlB. 12 Streatnline Ttl~Uie;_ . .:. . .streng:th ..• .,: C4.12 Strength Checkihg 3.nd "' · · .. ,· ..

Design - Prob~~ms ,• ,, ••

C4.22 Cll.15

Strength of RoWid. Tubes· ' under ·_combine~, LQadings •

stress_ Analysis FoZ.InuI2.s ... · Stresii'AnalYsis'of Thin.Skin. -

MultiPle stnn,g~r. ,qaptue.ver Wing .•..•• ,',·_-~ ••••.• A19.10

Stress Concentration Factors. C13.2. Stress ,ntstribritiOi:i ·& Angie ·

of TwiSt for 2-Cell Thin· Wall Closed Section • .• • • ~~., 7

Stress·Strain Cur:ve .... ~ , ... Bl."7 Stress-Strain Relations • '· ., ..... • A.24~ 6 Stre;sse,s_ around Panel,Cut(iut •. A22.1 str_esSes··m Up'rlghts, .. :;,._. •.• :,::::~- c~~.17 Strl.nier Systems,in'·n~onai ·

Tension . .. : ".· ·:·· . ._ ... , ·• ';'-~ ·: • Cll. 32 StrilCfural DE!sigri Philosophy • Cl. 6 Structural Fittings".''.'.;.:'.·!,:.. • A.2. 2 Structural Skin Panel_Details.. 03.12 StrU.ci:UI:es with Curved

MembE!rs •••• :·. ~ · .••• All. 29 Successive Approximation

Method for Multiple Cell Beams ••••• , ••

Symbols for. Rea.Cting Fitting Units • • • • , •

Symmetrica Sections: ".' External Shear Loads' •

• ~-~5. 24

Tangent Modulus . • • . Tangent-Modulus Theory Taxi Loads •.•.••• Tension Clips ••• ·, • " •• Tension- Field Beam Action. • Tension- Field Beam Formulas Theorem of Castlgliano . . • Theorem of Complementary

· Energy. , •••••• ~ ••

A2. 3

A14~2

Bl. 5 Al8.B

Cl3.Zb 03.2"

Cll. 1 Cll. 2 A7.5

A7. 5

' ,.1

'

INDEX FOR THE BRUHN TEXTBOOK - Continued

Theorem of Least \lt·ork AS.2 Torsional Strength of Rowid Unit Analysis for Fuselage Tht:orerus o! Yirtual Work and Tubes ........... C4.17 Shears and Moments. • • AS.15

Minimum. l'otentlal Energy A7.!\ Torsional Stresses in Unsymmetrical Frame A9.2 · Thermai. .oenecttons by Multiple-Cell Thin-Walled Unsymmeb:'ical Frames or

Matrix Methods AS.39 Tubes .......... A6.8 Rings . ........... Al0.4 Thermal Stresses • • . • . AB.14 Transmission of Power by Unsymmetrical Frames using Thermal Stresses . . . . • AB. 33 Cylindrical Shaft. • • • , • A6.2 Principal Axes. • •••• A9.18 Thermoelasticity - Three- Triaxial Stresses . , , . . . Cl. 5 Unsymmetrical Structures A9.13

Dimensional Equations. , A24.1 Truss Deflection by Method Thin Walled Shells A16. 5 of Elastic Weights • A7.33 Velocity - Load Factor Three Cell - Multipl~ Fi~g~ Truss Structures A2.9 Diagram •••••••• . A4.7

Beam - Symmetrical about Trusses with Double One Axis •••••••••• AlS. 15 Redundancv •••• _ AS.10 Wagner Equations • • • • • • • Cll.4

Three Flange - Single Cell Tr11sses With Multiple Web Bending & Shear stresses Cl0.5 Wing •••••••••• , • A19.5 Redundancy. AB.11 Web Design ......... Cll. 18

To!sion - C_ircular Sections. A6.1 Trus~es with 5inQ"l~ Web Splices • • • • • • • • , CI0.10

Torsion - Eflect of .!!.Ina Hl?dundancy • • , , • AH.7 Web Strength. Stable Webs • CIC, 5

Restraint •••••••• A6. l6 Tubing t)eSign Fac·ts .... C4.5 Webs with Round Lightening

Torsion - Non-circular Two-Dimensional Problems. A26. 5 Holes ......... ClO. 17

Sections ........ A6.3 Two-Cell Multiple Flange Wing Analysis Problems Al9.2 Torsion Open Sections A6.4 Beam - One .AXi.S of Wing Arrangements • , , Al9.I Torsion of Thin-Walled Symmetry ........ AlS.11 Wing Effective Section , Al9.12

Cylinder havi.ng Closed Type Type of Wing Ribs • • • • • A21.1 Wing Internal Stresses • A23. <t Stiffeners .......... A6.15 Wing Shear and Bending

Torsion Thin Walled Sections , AS. 5 Ultimate Strength ln Combined Analysts •••••••• Al9.14 Torsional Moments - Beams , AS,9 Bending & Flexural Shear • , C4.25 Wing Shear and Bending

Torsional Modulus of Rupture. C4.17 Ultimate Strength in Combined Moments •••••••• AS.9 Torsional Shear Flow in Compression, Bending, Wing - Shear Lag • • • , • A19. 25

Multiple Cell Beams by Flexural Shear & Torsion • , C4.26 Wing Shears and Moments AS.10 Method of Successive tn.timate Strength in Combined Wing Stif!n.ess Matrix ••••• A23...a3· Corrections • • • • • • , A6.10 Compression, Bending & Wing Strength Requirements Al9.5

Torsional Shear Stresses in Torsion ........... C4.24 Wing Stress Analysts Methods Al9.5

Multiple-Cell Thin-Y:'ail Ultimate Strength in Combined Wing - Ultimate Strength • • Al9.11

Closed Section - Distributio~ AS~ 7 Tension, Torsion and Work of Structures Group, Al.2 !Jlternal Pressure p in psi. c4:2s

Ui:iif.qrm Stress Condition • , Cl.1 Y Stiffened.Sheet Panels C7.20

ADDITIONAL TOPICS IN THE BRUHN TEXTBOOK

Beams of Mult1span •••••••••• A11,S Beam-Columns,,, ••••• •••,,,., A5.21 Beam-Column Example,,,,.,,,, C4,22 Beam-ColUllUl Por:mulas,,.,,,,, A5,2J Beam End Bay Etfeots., •••••• c11.1J Euokl1ng ot Composite Shapes C6,1

Centroid Formulas,,,, •• ,, •• , Column, Elastic Supports,,,. Columns, Stepped,,, ••••••••• Columns, Tapered,.,., •• ,,••• Curved Bea.ms,'·',,.,,,,,,,,,,

Diagonal Tension, Shell

AJ,2 c2,9

C2,10 C2,10 J;IJ,10

Analys i11, , , , , , , , , , , , , , , , • , C11, 29 Dummy Unit Load Method.,,,,, A?.9 Effective Sheet Widths •••••• A20,J Elastic Weights ••••••••••••• A7.27

Faotora of Safety,,,•••••••• Di,i Fatigue Problem Area11,,,,,,,c13,27 Fatigue Residual Stress

Ef"reot, , , , • , , , , • , • , , , , , , , ,"Cl J, lJ Fatigue Testing Machines,,,, C13,7 Fillers, Joint Strength

Effects, ... ,,,,,,,•·•••••.•• D3.5.

Pr8..me Stiffness c1'1ter1"a, ••• i:::9,11 Flanges, Bent.,••••••••••••• D3.11 Flange Supports,,,,,,,,,.,,, D3.11 Flexural Shear Stress 1n the

l'la.stie Range,,,,,,,.,,,,, C3,10

Inertia Forces,,,,.,,,, ''I',. Interaction Curves,,,,,,,,,, Interaction Curves,,,,,,,.,,

Loads, Stat1c,,,.,,,,,,,,,,, Loads 1 Dynamic, , , , , , , , • , , •• , Lug Strength.",, , , , , , , , • , , , , ,

A4,2 c4,22 cs.a

A4,12 A4,12 Dl.5

Material Properties,., •• ,,,, Bl,10 Multiapan Beam Anal1sis,,,,, All,S

Nuts., ••••••••• ,, •• , ••••• ." ••

i'lastlc Bending,,,.,.,,,.,~, l'la.te Buckling, •• ,,,,,, •• ,,, l'lastlc Torsion, •••••••••• ,, l'roduct of Inertla,.,,,,,,,, i'ropert1ea of Sect1ons,, ••••

01.2

C3.1 cs.1

C4,20 AJ,9 AJ,1

Recessed Shear Panels,,,,,,, ~J,13

~dundant Structure, Deflection Caleu1at1ons,,, A8,9

Bi vet/Joint Edge Distance.,. Dl,18 Rivet/Joint Shear Strength .. Dl,20 Bivet/Joint Tension Strength Dt,28

Sheet Buokllng,,., •• ,, , , , , , , Shims, See Pillers,,,,,,,,,, Spot~elds.,,,, •.•,,,, o.• •••••• Statics,,,••••, ...... ,,,••,, Structural Detail Design

Guides,.,,,,,,,,,,,,,,,,,, Structures Department

Organization,,,.,,,,,, •• ,,

cs.1 DJ,5

DJ,13 A2.1

D3,14

Al.1

Tension Field Shell AnalyaisC11,29

Virtual Loads,,., ••••••••••• A7,9 Virtual Work,,,,,,,,,,,,,,,, A7,14 v-n Dia.gram.s., •••••• ,, ••• , , , A4,8

'Welding,,,,•••••••••,,,,•••• 02,1 Wr1nkl1ng Failure or·shee~ •• c7,15

ADDITIONAL TOPICS ARE ALSO IN THE SUPPLEMENT'S INDEX, PAGE Il

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