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Fritz Laboratory Reports Civil and Environmental Engineering

1-1-1960

Limiting deflections in plastic design by limiting l/dratios, C.E. 406 Report, Lehigh University,(February 1960) M.S. thesisD. N. C. Pai.

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Recommended CitationPai., D. N. C., "Limiting deflections in plastic design by limiting l/d ratios, C.E. 406 Report, Lehigh University, (February 1960) M.S.thesis" (1960). Fritz Laboratory Reports. Paper 1723.http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1723

LIMITING DEYLECTIONS IN PLASTIC DESIGN

BY"

LIMITING L!dRATIOS

David H. C. Pai

Submitted to Protessor G. C. Driscoll asfulfillment of the course requirement of C. E.406 "Special Problems in Civil Engineer1ng~

DEPARTMENT OF CIVIL ENGINEERING

FRITZ ENGINEERING LABORATORY

LEHIGH UNIVERSItyBETHLEHEM, PENNSYLVANIA

February. 1960

---------

TULE OF CONTENTS

1

• • • • • • • • i1

1. INTRODUCTION • • '. • • '. • '. 1

2. d\$SUMPTIONS AND FUNDAMENTAL CONCEPTS • .' • 2

METHODS OF CALCULATING DEFLECTIONS •

4. CASE I;: BEAM FIXED AT ONE END, SUPPORTED ATTHE OTHER .. UNIFORMLY DISTRIBUTED LOAD. • • • 5

•5. CASE II. BEAM FIXED AT ONE END SUPPORTED AT 'mE

OTHER ... CONCENTRATED LOAD AT lUD-SPAN. • • 8

6.

8.

CASE III' aEAM FIXED JAT BOTH ENDS .. UNIFORMLYDISTRIBtJTm LOADS.. • • '. •

CASE IVe CONTINUOUS BEAM ... TW'O EQUAL SPANS ...UNIFOmU DISTRIBUTED LOADS" " " •

/

CiASE V:,IN...BASED RECTANGULAR FRAME. UNIFORMLYDISTRIBUTlID VERTICAL LOADS.. .. .. •

•

•

•

•

•

10

12

14

10.

DISCUSSION • •

•

•

•

•

• •

•

•

16

11

11.

12.

NOMENCLA1'URE

REFERENCES "

•

•

•

•

•

•

•

• • •

•

•

• 18

19

FIGURES. .. • • • • .. .. 20

"-- -- -----

ii

ABSTRACT

Defieetion calculations at working load are time oonsuming

in elastic design. Specifioations of the Amerioan Institute of Steel

Construction and other specification" writing bodies have made it

more or less unnecessary to oalculate deflections b.Y limiting length

of span to depth of beam (LId) ratios to 24. And when beams are sub­

ject to shock or dynamic loading, the LId ratio is reduced to 20.

It is proposed, therefore. that a study could show that by

limiting LId ratios. beams designed plastioally would not have a

maximum deflection at working load more than a certain specified

amount. These studies would help write specifications limiting tId

ratios in plastic design.

, "

.1

1• INTRODUCTION--

Plastic analysis and design of steel structures is a good tool

for steel construction. It has not. however, eliminated the necessity

of defleotion calculations. In elastic design, deflection calculations

are tedious and are much to be avoided if possible. Therefore, the

AISC partially circumvented the necessity of defieotion calculations

by specti'ying length of span to depth of beam (LId) ratios. Deflection

calculations in plastic analysis is just as necessar,y as that in elastic

design, although there are approximations whioh do s~.mpli1)r the calcula­

tions to a limited extent. But still, t~e amount of work involved

remains objectionable. It is the purpose of this reporttD investigate

into the feasibility of determining appropriate LId ratios for plastic

design. In this report. five cases will be investigated:

(I.) fA beam fixed at one end and. simply supported at the other

with uniformly distributed loads.

(II) Same beam as (I). but loaded with a concentrated load

at mid-span.

(III) A fixed~ended beam with uniformly distributed load.

(IV) JA two-equal-spaned continuous beam l-lith uniformly

distributed load throughout.

(V) fA pin...based rectangular frame with uniformly distributed<J

vertical loads.

-- ---------------

\" -.'

2. ASSUMPTIONS AND FUNDM1ENTAL CONCEPTS

!Aside from assumptions made in simple plastic theory. CI this

report is based on the followS.ng concepts I

del) The)lIJ;~ relationship is idealized as shown inFig. 1.

(2) As a consequence of assumption ~, eaoh spanretains its nexural rigidity EI for the whole·length between binge sections.

(3) ~nlimited rotation is possible at hinge sections.. at a moment value of M :::: ~.lt 1

--~------~----_. __ .~~---,----------~-~See pp. 25. 60, 98 of Reference 1 ..

3. METHODS OF CALCULATING DEFLECTIONS

Def'lebtion oalculations in plastio design fall into two types:

(1) The magnit.ude of deflection at ultimate load I this

is sometimes desirable because the load factor 01'

safety does not guarantee absolute~ against over­

load on rare ocoasions.

(2) 'lhfil magnitude of deflection at working load t this

is of value since a great majority of structures

td1l function at working load most ot the time.

In th~ rsport interest is oentered exclusivoly around the

second type or deflection calculations. i.e. magnitude of deflections

at working load. J\ sufficiently aocurate approximation 1-TUl be used

to oalculate deflections at working load. This consists or a plastic

analysis to obtain the ultimate load. which is then divided by the

load factor (1.8S) to reduce the loading down to working load. For

the types of structures to be investigated. the structures are all in

the elastio range at working load. Theretore. elastic deflection

equationo t-dll be used to calculate defleotions.

The scheme tor the investigation here is to oxpress LId as a

function of SIt (the ratio of deflection to span length.). In

generalt

---------

-4

£' =It ~r (p & Ware interchangeable)

PL c X ~

t:=K (}t ~)

1. =KX .fz . Zd (L!d)L E r

or I 1:1 ra uil - . 2.£ (L!d) ••••• (1)

where K and X are the tactors which depend on the loading and the

geometry of the stNcture, (;"'Y is the propertis ot ASTM tA-7 SteelE

the material assumed in this report, ¥-= 2t, f is the shape factor

which governs how severely the beam will detleet.

. --...

-5

4. CASEl

BEAM F~ AT ONE END. SUPPORTED AT OTHER - UNIFORMLY. DISTRIBUTED LOADS'

STRUCTURE400)

LCMDIN(1

MECHANISM

MOlmtTDIAGRAM

f-- -----'-J...=-----'--f

Following the procedure outlined in the previous chapter. we

have

We = lrli

Wfi./ L c "p fi [ 1 + 1 +

~c [~+ X:x ]"p ••••• (2)

l~ =11.73 ~ (by trial and error)

::e...L •

F = 1.85 ••••• (:3>

--=~.-=-- ~~= =e-. ---~.~__

'.

.6

Since the stxucture is elast.ic at working load, the defiect10n is

then given as

185 EI(Ref. 2) ...... (4)

substitutiYlg the value of t-Tw in Eq. :3 into Eq. Ito, we have

de 6.3.5 ~ L3,L 185 EI

But Mp ::: Uy Z

. [::: 6.35QyZ

(V/185 EI). . ,.,..............I.

Since Oy::: 33 kat and E ::: 30 x 103 kai, vIe have:

zt2I

Divide through by L and multiply the right hand side of the equa­

tion by dId we have

f· ::: 3.78 x 10-5 Zd.-I

(Lid) I' ---~- -- --[

\ '

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

or ~ ::: 3.78 x 10.5 (2f) (LId) •• • •• (5)

Using f/L::: 1/360 as an arbitrary value, computations are made tor

the lightest w1de-~lange ot each group in the AISO handbook starting

with the 12" members through the 36 11 shapes. The result is plotted

on a LId va S Ix. curve. (See Figure 2) It is observed that all

---- =~------------ -- ----

',- '

' .. _...-

-7

curves tall within a narrow boundary. The oritical member having

a limiting LId:: 33.0 for a [/L = 1/360. Incidentally, the ratio

d/L = 1/360 is specified by the AISO as the maximum recommended

live load deflection for members Sl1pportingplastered cellings.

Therefore,- tor this particular type of beam and loading plastically

designed, the LId ratio will be 33 which is considerably greater than

a reoommended 24 in elastic design.

5. CASE II

BEAM FIXED AT ONE END, SUPPORTED AT THE OTHER .­

CONCENTRATED LOAD A.T MID-SPAN

STRUCTUREtAND

LOADING

MECHANISM

MOMENTDIAGRAM

p

j~);if

.j.I J;i 1i • If- -i,L

~9

I... ~ ~I

Prooedure here is simUar to that of Case II

••..• (6)

----i

1.

-9

Since the structure is still elastic,

[ = .009371 .pt,3EI

(From p. 370, Ref. 2)

substituting Fq. 6 into Fq. 7,

r ; ~L20= 9.317 x 10- x 3.24 :E

EI

d = 3.02 x 10.2 Uy· ZL2E I

d=3.33 x 10-; ZL2T

~/L = 3.33 x 10·; . 2t (tId)

..... (7)

,; •• ,; (8)

After the LId V8 [It is plotted, (See figUre 3) it is observed that

the critical LId ratio for a itt = 1/360 is 40 which is even better

than Case I.

6. CASE III

BEAM FIXED AT BOTH ENDS - DISTRIBUTED LOADS

..10

STRUCTURE!AND

.LOADING

1l1ECHANISM

MOMENTDIAGIW'1

Similarly

We = Wi

(J e

~ ze• % ·1·

W L 0. L ='2 v 2'

W\1::: 16 ~

1m:::: ~ = 8.65 !!e1'" L

The structure is elastic at working load,

.11

(p. 371, Ref. 2)

substituting Eq. 9 into Eq. 10 we have;

••• ,; (10)

d = 8.6.5 or

ZL2-I

dlL == 2.lf8 x 10~.5 . 2f. (LId) ..... (11)

The critical LId ratio for a d/t =1/360 is 48.3 (See Figure 4)

which is greater yet. This 15 logioal since there is more continuity

in this beam than in previous one.

,~~---I

-12

7. CASE IV

CONTINUOUS BEAM .. TWO EQUAL SPANS - UNIFORMLY DISTRIBUTED LOADS

STRUCTUREAND

LOADING /:1.

MECHANISM

MOMENTDIAGRAM

~ an elastic ana~sls. the expression for deflection is found

to be J = wx (L.3 + .3LX2 + 213). Maximum'deflection occurs at48EI

X = 0.4215L; thus giving the expression d =.005416 wt3 ••• (12)max.. EI

Advantage will be taken of the symmetry of the structure in

calculating for the ultimate load. Consider the lett span:

We = Wi

t-b~L (1-X) e = ~O

with X =.4215L

2L( T - 1)

~ = 11.65 ~L

,", .

• .• (13)

. Since the maxi.mu.m defleotion 1s

-1)

6 tilL)= .00,5416 -. the value ofEI

•••• (14)

Ww will be substituted into the above equation giving the maximum

defleotion at working load as d= .00,5416 (6.)0 .~') .ii or

f = 3.78 x 10"',5 2f (LId)

'l'he LId ratio for this oase has a critioal value. of 3J. whioh

is identioal to that in Case I. (See Figure S)

o

.14

8. CASE V

PIN-BASED REOTANGULAR FRAME - UNIFORMLY DISTRIBUTED VERTIOAL LOAD

5frudUrecme;!

loacl,/).7

/!fomenT"

iJ/4frt:7/'YJ

, '.,

T ~::=:::::::::'-_.===lTI

r J. . -I

Bj?1B

JAn elastic analysis was carried out for a three span continuous

beam with side spans 41.. and side span loads 0( w. The deflection

at the center line of the main span is given as /

By letting d • the side span l~d factor equab zero, a denection

expression for all pin.based rectangular frames with vertical loads

is obtained. Thus.

.1.5

••.• (15)

is the deflection expression for a rectangular frame with variable

column height ~I- •

As for the plastic anelysis to obtain the ultimate load, similar

procedure will be followed noting that hinges at the knees form simu1-

taneously. Thus the ultimate load is

W ::: 16u

. and ••.• (16)

have

Substituting this value into the expression for deflection, we

or

J _8.6j ~L2 I ( a 2 6 £P ), -3840+2 ).2 f I 20lJ '"' +3 jJ +9

'j. =2.48 x 10"5 (2f) J,fd (~~d2 ~Ji9L3+2,4 r;r- ••.• (1'7)

This equation shows that the severity of defleo"tion 1s directly pro­

portional to the column height and the magnitude of the shape faotor.

In figure 6, graphs of SIt vs tid tor several factors of colunln

heifJht <J3') have been plotted. The graphs show that the critical

L/q rat:los are 31., 24, 21 and 19. for a {3 value of t. i, 3/4 and 1

respectively.

-16

9. DISCUSSION

From this investigation, it is observed that LId ratios could

be specified in plastic design in order to circumvent the necessity

of deflection calculations. Before discussing suitable LId values,

it should be borne in mind that the investigation here is limited

to the particular cases of frequently occurring structures only. Also,

a SIL =1/)60 represents the dene~tion limitation 1n the elastic

AISC specUioations due to live load. Further. the structural shape

assumed in this report is the standard American wide-flange made of

ASTM A.7 stael-Keeping tha above conditions in mind one may safely

say that the critical LId ratio for a plastically designed beam is

)) as compared with ,24 in an ~lastically designed one. !As for a

rectan~r trame wi,th uniformly distributed vertical loads, the

~/d ratios vary accor4ing to the Qolumn height, 1. e., the taller the

columns, the more defieotion (see Figure 6). Therefore, the general

deri.ation in case V ser~es ets a guide to obtain limiting Lid ratios

for any factor of column height to span length.

iAs was mentioned previously, this report assumes the wide-flange

as the structural shape. The values of 2f or Zd/I. varies from 2.1)

to 2.)2 (see table in back of report) and these .alues establish the

boundaries for the ['It vs tId curves. For other structural shapes

such as the I beams, the 2f values are higher, being between 2.28

and 2.46. Therefore. there will be a corresponding reduction in limit­

ing Lid ratios. However, the prooedure for obtaining critical Lidratios remains the same regardless of the type of structural shapes

used.

-~=_._. -----

-17

. 10. ACKNOWLEDGEMENTS

The author is deeply indebted to Professor G. C.

Driscoll, supervisor of this study. His most valuable

suggestions andcttticisms are sincerely appreciated.

\ \ ' , .,Professor W. J.' Eney is the director of Fritz

Engineering Labor.atory and Head of the Department of

Civil Engineering.

-18

11. NOMENCLATURE

SYMBOLS

d depth of beam

e external when used as a sUbscript

E Young's modulus

load factor of safety = 1'.85

means internal when used as a subscript,

length of span

plastic hinge moment =6"'y zconcentrated load

f

F

.1, i

L

l~

P

Shape factor, t =z/s 2£ =Zd/I

s

w

wZ

E

V-

a-;f

section modulus

means ultimate when used as a SUbscript

weight per unit length, or "working" when used as a subscript

total uniform load

ful~ plastic secti.on modulus

strain

stress

yield stress of steel

vertical deflection

slope of deflection curve

-~-,------

2.

4.

..19

12. REFERENCES

Beedle, L. S. PLASTIC DESIGN OF STEEL FRAMES.John Wiley & Sons, .1ew York,1958. pp. 184-204. .

Steel Construotion AMERICAN INSTITUTE OF STEELC011STRUCTION, NeTtI York. 5thFdition, 1951.

Driscoll, G. C., Jr. ROTATION CAPACITY OF A THREESPAN CONTINUOUS BEAM, FritzLabora.tory Report No. 268.2,

·Lehigh.Universitr. 1957.

Drif,lcoll; G. C.. Jr.. ROTATION CAPACITY REQUIREMENTSFOR BEAMS AND FRAlJI.ES OF STRUCTURALSTEEL, Dissertation, Lehig~

University, 1957. PP. 186, 196.

rY

t

tv(

t

--~- e(a) IDEALIZED STRESS-STRAIN CURVE

/IL- . . _

--_. c;zf

(b) IDEALIZED MOMENT-CURVATURE CURVE

FIGURE 1

CURVE OF LId VB 6/L FOR CASE I

LId50

40

I.. -L--~

30

20

10

0lL.... L- l...- ..L- -L- -.L,,-_--'-_

1/2000 1/1000 .: 1/666 1/500 1/400 1:/360

----~ .6/L

FIGURE 2

CURVE OF L/d VB O/L FOR CASE n

rt.....: -----)

.. L/d 40!I!I!

30

20

10

OIL-__~L-__~.L-__--:-..l...- :L----_-:-L-_~---:--

1/2000 1/f-000 1/666 1/500 1/400 1/360

-------- J/L

FIGURE 3

CURVE OF LId VB d/L FOR CASE III

o L-.__----,L- ~----Ll ---1-1 -:---LI__LI:---

1/2000 1/1000 1/666 1/500 1/400 1/360

20

30 l.I

10

I

IL ·1I

50 ~

L/dIII

I40 ~

----~ ~/L

FIGURE 4

CURVE OF tid vs lit FOR CASE IV

50

L/dr-i- -+--L --1

40

1/10001/2000o J<:...-~__-'--__~-'----'-'-_--'-'-'--'--L-I --'--I_~'_'___Il___ _'_I_

1/666 1/500 1/400 1/360

20

10

30

'-FIGURE 5

40

CURVES OF LId vs oil FOR CASE V

TC3i

1

(3_1--;r

L/d30

20

10

o1/2000

//.J-- -­/-'/-2

;J=!

1/1000 1/(;,66

aILFIGURE 6

1/500I L

-'1/400 1/360