study of folded plate structure
TRANSCRIPT
Lehigh UniversityLehigh Preserve
Theses and Dissertations
1963
Study of folded plate structurePierre ChevinLehigh University
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Recommended CitationChevin, Pierre, "Study of folded plate structure" (1963). Theses and Dissertations. 3147.https://preserve.lehigh.edu/etd/3147
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StDDY OF A FOLDED PU.TE SD.UCTUIE
by
Pierre Chevin ! -
• . ,,iJ ...
A Thesis
Presented to the Grad~te Faculty
of Lehigh University
in Candidacy for the Degree of
Master of Science
Lehigh University
1963
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This thesis is accepted and approved in partial
·fulfillment of the requirements for the degree.of Master of
Science in Civil Engineering.
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A C K ~, .Q. fl L E, D. G M E N T §
This thesis is an attempt to verify experimentally the theory
relative to the Folded Plate Structures. All the tests were performed
at Lehigh University in the Models Laboratory of Fritz Engineering
Laboratory from September 1961 to May 1962.
The author is indebted to Professor Eney, Bead of the
Department of Civil Engineering, and Fritz Engineering Laboratory, who
was Thesis Supervisor and to Professor Le-Wu Lu, whose guidance was
greatly appreciated.
' Thanks go to I. J. Taylor, Instruments Associate who helped ,
with instrumentation and strain gage application necessary for the test.
Sincere appreciation must b~ .. exten4ed to Mr. E. Zanoni, who ,f -- . . .,,h . ,).
worked on the same model in 1960 and 1961 and whose experience was very
helpful.
Mrs. D. Fielding typed the manuscript and her sympathy and '\,
comprehension are gratefully acknowledged . .
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TABLE OF CONTENTS·
ABSTRACT
.1. INTRODUCTION
A. GENERAL ASSUMPTIONS
B. DEFINITIONS
2. DESCRIPTION OF THE STRUCTURE
,.
3. ANALYSIS OF STRUCTURE
... ,_ \ ...
Page, (
1
2
:4:
s:
A. OUTLINE OF METHOD ·7
B. COMBINATION OF ELEMENTARY LOADINGS 9·
C. COMPLETE ANALYSIS FOR THE ELEMENTARY LOADINGS 16
D. ·RIDGE STRESSES WHEN THE MODEL IS SUPPORTED BY 33
INTERMEDIATE COLUMNS AT MIDPOINT
E~ RIDGE STRESSES DUE TO THE TIES
·4. DESCRIPTION OF THE HlDEL
40
47
-A. MATERIAI,
B. THE M:>DEL I TS ELF
C. SUPPORTS
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5. . TESTS ON THE K>DEL
·,: . ·::.•
A. PRELIMCNARY TESTS
B. ACTUAL TESTS OF THE LOADED K>DEL . ~-
C. THE ADOPTED SOLUT"ION
D. STRAINS AT RIDGES, FROM THE STRAINS AT
SR-4 LOCATIONS
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T A B L E O F C O N T E N T S {continued
6. DISCUSSION OF THE RESULTS AND CONCLUSIONS
A. DISCUSSION
B. CONCLUSIONS
7. SUGGESTIONS FOR FURTHER TESTING
A. EFFECT OF DIAPHRAGMS
B. PROPOSED TEST '
Page
62
62
64
66
66
66
C. STRESSES DUE TO A DIFFERENTIAL SETTLEMENT 67 '
OF THE SUPPORTS
D. MEASURE OF DEFLECTIONS
8. APPENDIX
RIDGE STRESSES FOR THE ELEMENTARY LOADINGS
ACTUAL RIDGE STRAINS (TESTS DATA)
6:7
6-S.
69 ·- 7'6
77 - 88
FINAL STRESSES (THEORETICAL AND EXPERIMENTAL) 89: - 100
REFERENCES lQl . '
VITA i 102.:
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LIST OF_TABL'ES
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Table j
1 ACTUAL LOADING VERSUS ELEMENTARY LOADINGS
2 PLATE LOADS
3 MOMENTS DUE ro PLATE LOADS
4 NON-CORRECTED STRESSES FOR l'lm ELEMENTARY
LOADINGS
5 DEVLECTION OF PLATES
MOMENTS AT RIDGE 4
:j .,~~
RIDGE" LOADS EQUIVAJ,ENT 'lU THE POINTS
{".. DISPLACEMENT
8 STRESS CORRECTION COEFFICIENTS
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THE STRUCTURE
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POINTS SUPPORTS
COLUMN SUPPORT ~
'IYPICAL TIE
THE ''LOADING TREE"
ATTACHMENT OF THE LOADING TREE
STRAIN READING EQUIPMgNT '(.
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ABSTRACT
David Yitzhaki developed a simplified theory for the
analysis of the Folded Plate .Structures.
~theory.
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The purpose of this thesis is to verify the validity of his
A simple structure was chosen and analyzed for several cases ~, \~.
of loading under different conditions of supports. In this manner
the stresses at the joints of the structure were determined.
A model was then built in plexiglas and the strains were
measured at the joints. After determination of the Modulus of
Elasticity of the material the stresses were computed.
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Finally the theoretical and experimental results were compared.
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I 1-. ~ N T_R O D 1! C T I O N
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The use of the folded plate structures has considerably in-
creased in the past decade mainly for the simplicity of their con-..,
struction. This type of structure is particularly well adapted to the
large buildiJi,gs, where a vast free space is needed since no columns are
1'f necessary tor supporting the roof. ..
·-The main obstacle to a wider use has been the complex-ity of
the theoretical analysis of the prismatic structures. In 1958 Yitzhaki
prepared a simplified method which can be used by any designing engineer.
(Ref. No. 1). This approach will be followed here for determining '), ... ) . .,
theoretically the stresses induced in the ridges of a folded plate roof. '
The object of this study is to verify experimentally the
assumptions connnonly used in the analysis of folded plate structures.
These general assumptions have been defined in detail by Boward
Simpson (Ref. No. 2). They are as follows: / .
A. GENERAL ASSUMPTIONS
"1. The material is homogeneou~ ,_ uncracked, ~n(.f •
elastic. ~
2. Longitudinal edge joints are fully· monolithic
and continuous; there is no rel.ative rotation
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or translation of two adjoining plates at their
common boundary.
.3. The principles of sup'erposition hold; that is,
the structure may be analyzed separately for
the effects of its redundants and various
external loadings and the results added.
4. Individual plates possess negligible torsional
resistance. Torsional stresses du·e to twisting
of the plates can be ignored.
-5. The function of the supporting diaphragms or
bents is to supply the end reactions for the
plate action and for the longitudinal slab
action. They are assumed incapable of providing
restraint against rotation of the ends of the
plates in their O'Wll. planes, but may provide some
end res tr'ain t for longitudinal slab bending.
..
The following additional assumptions can be made if
each plate is relatively long compared with its width -
( length-to-width ratio of, say, three or more):
6. Longitudinal strain due to plate action varies
linearly across the width of e~ch plate ('~lane
sections remain plane"). '!he rate of change of
strain with respect to width ordinarily will
differ from plate to plate, however, from which
- it can be inferred that there will be some
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· relative displacement of the joints of a cross-section.
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7. Longitudinal slab action can be neglected; that
is slab bending carries loads applied to the
surface of a plate to the longitudinal edges
only, as in a one-way slab system."
)
These assumptions will be more clearly understood, if a few ,
terms are more fully ~plained.
B. DEFINITIONS
··;-
"The individual planar elements com.prising the
structure will be termed ''plates."
The "length" of a plate is its dimension between
the diaphragms or ribs; its ''width" is the perpendicular
-· distance .between its two longitudinal edges.
In conformance with the terminology introduced by
Winter and Pei,(3) '~lab'' action will be used to refer
~-
to bending ··of the individual plates out of· their planes,
and ''plate" action to bending in their planes. (The
latter has been termed "disk effect" by Craemer ( 4) ,
and 1\nembrane" action by Ashdown. (S)"
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i . 2. D E S C R I P T I Q N O F T H E S TR UC T UR E
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of a fic'tftious structure has been built because a ,,~
'1-e model
struct~~ ~£ these L.-,'
proportions is not used in an actual building. Gener-
ally the buildings are flatter. The dimensions of the model were deter
mined in such a way that the theoretical analysis is as simple as possible.
The structure (Fig. 1) is composed of five identical plates;
(length L, width d, and thickness tare the same for the five elements). '
The angle between two adjacent plates is constant around the structure.
L = 36 in.
·d: • 5 in.
t ~ 0 .125 in.
A·= d x t = 0.625 in. 2
t d2 l s •
12 = 0.521 in.3
t d3 I= 12 = 1.302 in. 4
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The model is closed at its ex1tremi ties by a diaphragm, the
thickness of which is t (theoretically the thickness of the diaphragms
is without importance since the diaphragms are assumed (Assumption No. 5)
to behave as simple supports:
..
The six ridges are referred as O, 2, 4, 6,·a, 10
The five plates are referred as 1, 3, 5, 7, 9
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= 36 in. = 5 in. = 0.125 in.
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:3 • A N A L Y S . l S O F S T R U C T U R E
A. OUTLINE OF METHOD
'nte purpose of this analysis· is to determine the stresses at·tbe ·
Joints of the structures (including the free edges).
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4 6
2 s ,.
0 10 ·O··
For any type of loading:
1. Consider first the joints 2, 4, 6, and 8 as being fixed
and determine the reactions of each support, the t:'"i
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structures being continuous:
0 10 = free = end of ·c.ant·ileve:rs·
· 2, 4, 6, 8 = fixed
The only loading to be analyzed are vertical ones.
Under these conditions 2 and 8 serve as hinges.
0 10 free
2 8 hinges
4 6 ,j .
fixed
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2. Resolve thes·e reactions into forces· acting in the ~planes of··, ,
of the plates.
J. Consider each plate as a beam carrying the plate forces.
Determine moments. · For each plate write the equations .. ··-··~
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of equilibrium. By consideration of compatibil.~ty ',.
define the stresses existing at the edges.. These
are the initial stresses (or stresses without
corrections).
4. Allow the joints to deflect. Determine the
displacements at the joints corresponding to
5.
the state of stresses e.atablished before. The
procedure is as follows: Define -
a. The deflection of each plate in its plane
b. The deflection of each joint (by using
the Williot-Mohr diagram)
c. The relative displacement of two con-
secutive joints .'lo
Determine the moments induced by these
displacements at each joint. These moments
will induce· lateral bending in each plate.
According to the assumptions made, there should
be no lateral bending, only a longitudinal one.
The correction will be obtained by "suppressing"
this bending,._
:"6. Determine the forces acting at the ridges which
will prevent these bendings from :occ·urring. This
is the secondary loading (also called normal
loading) •
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From there on, repeat the procedure used before,
starting at step 1 in order to det·ermine the
stresses acting at the ridges, due to the secondary
loading which are the correction stresses.
The final stresses are simply the respective
s1mnnations of initial and correction stresses.
B. COMBINATION OF El.gENTARY LOADINGS
The model will be tested for various types .. of loading. For each
of these cases a theoretical solution has to be provided in order to
check the validity of the theory. Instead of developing a complete
solution as outlined on page 7 for each type of condition, it is much
simpler to derive the solutions for a limited number of elementary i \
.... loadings, which will be combined with the appropriate coefficients 1.n ,
order to match the real loading situation.
The method of Yi tzhaki (Ref. 1), based on the combination of
symmetrical and antisymmetrical loadings, is used here. -...
1. Elementary loadings for cases of external vertical
loads, uniformly distributed along the length of
the structure.
As the load is to be converted into ridge-loads, only
ridge loading cases need be analyzed:
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U S 0 U S I
USO= Uniform Symmetrical Outside 'Line Load
US I= Uniform Symmetrical lnaide Line Load ··--
ridge line loads ~
100 -.100 100 100 i t
UAO U A I
U AO= Uniform Antisynunetrical Outside Line Load
U A I= Uniform Antisynunetrical Inside Line Load
Any case of vertical, uniformly distributed load may be obtained
by the combination of these four elementary loadings.
2. Elementary Loadings for center line reactions
The model will be tested for different conditions of
supports.
a. Simply supported
b. reposing on two middle columns
c. With ties placed at the top of the vertical
walls
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1000 T S
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No antisynnnetrical case is needed since the force carried by the
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3. Combination co.efficients:· for the loads actually
applied to the s true ture.
100 100 100·
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:(:l:). (3)
( l) = Cen·terft·ne loa.dl,1'g_· !:.'I.
(2.)\ Rid'ge 4 line loading
(-3...) C.e..nter line:_-Plate 3 line·· loading
Analysis of _c-as_e (1): . ·• .-
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p
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l
In terms of the elementary· loadiTI:gs ..
(1) = (-0.106065 (USO) + 0.60606S(t1Sl)}r~o . ·- ...
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:·~'a.J.::y$:.:Ls: .o:f case (2} :.: . .-. •·.. . . •' - . ·-:: ... '
p
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(Z) .·;;: (0. 5 (lJS:t) + 0, 5 {UAI)) iriO . . . . . -- ·- .. . ··, :. ·: .. ··
• ,•' :MO •
.Analys:i.s of case (3):
O.SP 0.5P 0.5P 0. 5P
-- +
•• {.i·
-13 ·-··",
.. -14 l. •• ,, _ ... ..t. ... ,- .............. . .. . . . ,,, .. -.
' '' , .. ,... , .... , • '•''1>""~ ..• . . ......... "':'" •. ,.' ·~·. .. ,,,., . ,,,.,,,.. .... . ....... :
., ...... ' .- .. [ . . . . • ··.· -.·-:··-·. >-·~ -.. --:. -.. ·--·.--~- .. ·. .. . . . ·-··· ·--· w.---· - -------.-- __________ ,,l_ ---·-·-···--'·-::·----.. ---·--=- - ....... __ ··-·--- . -- ... ---· ~ .- . . . ....._ ... - -- ~-~ -.--
Syrmnetrical factors: .. -;
-0. SP O.SP M = 0 .125 P
I l ~l. )~ t t t t ., ..
0.25P 0.025P
' t t 0.225P 0.275P 0.225P
O.SP 0.SP
~ t . . M:;:: O.r7314P
< t t t 0 .25P 0.0417P 0.10059P 0.10059P 0.0417~:
t t
0.2083P 0~06P 0.3506P- 0.2083P "'
(3) = (0.225 (US()) + 0.2}.5 {t1ST) +. Q.2083 (UAO) + 0.3506 (UAI)) l~O ·. '
. ' . . .
\
.. . ...
·,... ·:. ·\ . ·•
< .. ~
: "·-,-.,. .
, -·- _.._ , • ' · • •. • -- -.jt.' • .:.--- ' - - : , ' ·.: • · •• _ '_.,.,:....,.:. ..... ·;_!.!;_~.,.....,.,. _ _._~.., . .,.- ..... ·~~- -..::-""...;.____-.. =·-- ·- , :. ............... _,"C.'C't-;-· .._ .... _____ .._ ___ ...... · ... • ··.-.- 1 ~".' • .,, -=~-· ---...=,··_,. ___ -~----,, .. ~-.,;a;.~=-11..:··~ ... .:.::.-- ... ...,,_~~J;:-..:l..~~-:1--·--':,.~·~1.-=::-...:;;.=u..~.~-.;,..;;~;;.;..-,.;.;;.,·.•-~-·:~.· .. ·• -~~:_ ____ ._.. _ _. _ ___...__.;. •. ,...,._..,.__.1. .... ......::....---.-::...;...;....:~1~~_.____:..s...::~~_.;...-~,..;..!"-:"'""~;,..J~..-W:.--·•~ .. _ ~ ---------------------v- - - -~~ ___ ....__ ----.- ------ ----- ~---r ,. --- ...- - -- ,,.~- .,-------- ........,,. - • ' · .... ,t 'O . . . . ""'·'
Ii I l
i I ! ~ t @ fi ~ t •! Ci E I
I I I I J ill
I I
.q,
<.9
•
, , ..
• .
TABLE 1 · ACTUAL LOADING VERSUS. ELEMENTARY LOADINGS
u s 0 u s I u A 0 UA I
'
·fl) ; ' .. '·: - 0 .106065 + 0. 606065 0 0
(2) 0 + 0.5 0 + 0.5 '.
(3) + 0.225 + 0.275 + 0.2083 + 0. 3506
,1J'
All these values are for the l of the Structure. Stresses must
be ,computed for the 1/4 points. This will be done by multiplying the
individual cases by the proper ·coef!icients. Stresses are directly pro-~,., "( ' l
portional to the moments:
A
I ! ' ! ' t ' ' ~ ~ . ~ ~ i • ~ ~ ~ '6
I ·Uniform load.
At ' ?" ~ I ~~"" At 1/4 .·. . M = 3/4 110 points · ·
B
' Concentrated load A 0
At <i M Mo --
At 1/4 M 1/2 M0
--points ,.
C I· i J. * i l J l + t i Sinusoidal distribution 4 0 I
(correction laad) /'I;
At ~ M - Mc -1 At 1/4 M - \[2' Mc -
po·ints
11)
for A·
.•·
rrl/4 points= 3/4 ao + 1/t/T <1c
d.1/4 points= l/2 ~o + l/v'f' ac ·: for B
-15
"'·;._-.,•--'•·····-~--~•1,- ,, . ., .. .,_.,,_, ____ .... __ .......__, -·----•·n~-.•--------------...-,-·__.,,,_,_..,_ ______ .. ,,, ,-, .. .-..-- ·-··t. •, • · . .,._._ .... .,,... .. - · -· ·--:t·· ... .__,...,. •-·-" --- ~ .· -:....:...--..... _,..· -,..,.'. _;, ....... '•-. · '• :•-+• · .. • • -•· -· ·-- --- ,_-' .. _,., '"".: _ .• • · _---i,,..-,_~---• :~·•_ •. ·~· .. --i.,~ .. _-~; .. _.,.... _ '"~~--u,,.. ., ........ ~ -..- ,..._,.,_ .............. .-... -..;,-,t,~ ... , .a-~·-·>--- __ ._.,. ... ,. .. ,,.,., .. , .. ,,,,. .. . ... ~.... . ... - " . . . ' -. . ..
f ··4 I ~l ·,:Ii
..; . ,,:, . ' ,.,.. . ·.,. ·'-
,.- .. ,;
., . - - ~·- _... . . .. - - - :., __ - - . - '
C. - COMPLETE ANALYSIS FOR ·rHE ELEMENTARY LOADINGS
!ABLE 2 PLATE LOADS 1. Plate forces
Plate :/11 Plate fl3 Plate/JS Plate 17 Plate 19 100 100
5 7
i
1 9 us .Q
100 100 ·
~ ' US I
U A I
. ,
C S
CA
t1000 + 1000 f 100 tlOO ·
J 100 r .\·
N S
241. 421 ' t '
NA
;r.
100 ~
_,
100 ~
_/
0
141.421/
1000 f
1000 t 0:
100 J, 141. 421/ .
0 0
:_Q . :. -0 .
o 14L421 \
o .
;Q :o
a 141.421 '\
100 ! 341. 421/ 482. 84:t-° 341.421,
100 i
0
100 f
0
10.00 t
tooo i
100 i
100 t
'\,--16
'4,
_ .__.__ _ _., _ ____ _ -·- _ , _ ·-,. _ .. .._., .... __ . _ ... , .. " .... - ... ·. -:..;.~.._.. --.-.... . _.,J ··~ - .. b . ~· •• -~~{' --.. ... _,"":"_'_'="'" 1 IS .,_.,~,- -:-',,....-· .... •.:..~~•: ... ---.,-~ •• - ........ .,.. ...... _._. .... ---~•·-:_ ··-··~--·, _ _.__.,·_._:,.. .. -~-.-..... ~ .......... -:---·....., --::-: .. _, ._•1111"1'~~,-.-n:;t,....:_r .... ,""'.',~, ..:..-.~~Hr: ,,_ , m.-.,i,~,~..a·~~,ff ....... "l,:;i,~.;.-....;:,:,..,c,·.,:-.n -~- _ ,. . ~ ,,
•' \ '
I f
~17 ···-··~ ___ .. __ ., ..... ,1 .... ~- ·-·· ....... ~... . .
-~ .. ···-~-~-. .-,.~·-··.-""' .. -•.. -.. ~- .. ; - ... -.-~·-_ . . -· . .--.·-···: .- -·' ·, ·.... .... . '
TABLE 2 PLATE LOADS (continued):·
Plate /11 Plate 113 Plate 115 Plate ./17 Plate 1$
0 141.421/ . 0 141.421, 0
T~S
."1·
\_
.,
' ;
.. ,. ..... -...1. ..... ·.
--·······---········-·-··-·--·.~. ~--·--·--· .-··-·-·-·--,,-.-- _ ....• _..-,-._--· ' . ' ------·- 4 0 - ·--- - ,_ 0 ._ ..... ~ 40 ... -- -·
...
\
-
2.. Moments in Plates Due to Plate Loads
The plates are considered as simply supported beams
bending in their plates.
(~) USO, USI, UAO and UAI are uniform loadings.
(~) NS and NA are ''Normal loading" (this is defined, p. 1~)
(r) CS, CA, TS are concentrated loadings.
(a,)
...
J
,. ·--:•
i
= f X (L - :~) 2
(() . Met
M
(+) if p (4-)
' . ·,
(~)
PL =-4
PX --- 2
,- .
j \'' •
-~-: .: .. 1 , ..
.2 = 90
9 .8.7
= 90 12
i ~ s n -9 .87 L f
._,
-()·
-18 · -·
J
-.~. ,_.,., "1 ~ '.:._·-;_,, ' ', • ",--, •, ., ' ' 1' : ,• > - , , • ' •
I ·- -19 .. j -·· ,,.
....• ,,, .,
r··· .. TABLE 3 MOMENTS-DUE TO PLATE LOADS
K10 M30 M50 M70 Mgo
us 0 + 16,200 <
16,200 -
us"-- ·+ 22(~ .. 910 + 22,910
+ 16,200 U A O + 16,200
U·A I ·+ 22 :9·:io - 32,400 + 22,910 ' ..
C s - 9,000 + 9,000
CA. ... 9,000 - 9,000
:T· $ - 12,728 + 12,728 ,,:
.N S + 13,130 - 18,570 + 18,570 - 13,130
NA + 13,130 - 44,829 + 63,397 - 44,829 + l_J, :130 '; ..
'
.a,,
. ., ..
,. 'Pt . ,,,
fl
~. .., ....
,;;
"'·
. ..... _ ...... "'- .
:if ' .
3. State of Equilibrium of the Structur·e .. ct '
10---------------------i
,,: ·w~ -
..
9 . } Mgo 8 '"'-----:==----t
Ta a--~zd"'===---~
' ) M70 7 6----:=::----; >
T6 6
,,
5
>
T4 <
3
>b
4 .-----!:==------J
, ) M30 2 ...__--===----,
Tz < 2 ----!===:..--...J
1 I
' MlO
0
\, y J
•
Tz ) Tz 'd. .. _,
2··
·a ... 76: •..
(l_f():
·a· -·· ... · 78
0:1_2:
. Exa~p:'~e .. : n·etermina tion of the stresses Plane Projection of ..
the Folded Plate "'
Mso T6 T4 T6 + T4 - d Structure + ..
~54 - -= s z A s 2 ' ' ~
M30 T4 - lltwi T4 + Tz. d + -:C134 - -I z A s 2
By ·compatibility a = er = cr 3:4 54 40
The fore.es .T's are the solut~n of a system of 'simulitaneous equa..ti'ons :
!_ = 4 Tz + r4 25 · ·
~ = Tz + 4_f4 + T6.
A (M50 + M70) - T4 + 4 T6 + TS - -25 ~
(M70 + M90) A - T6 + Ts --25
q -
-20 . . -·-··-·---·, ....... ~ .
I
(
---------------~------:-,-~--:-------------------------------------...... ----., . . I I • '
. .,
.... . ........ ~- .. ···••·. . . :::-. .:. ..
,. t;
\. "~ ... ,~
~ --
............ ....... ~ .... ~.· -·~ . ... -.,,.,.,,., .... •'• ..
I}·.·.
': "-'-' :- •' -~- •' -4. Non-corrected Ridge Stresses
A 2S
A'
M10 + M30 .. ,-.·r, 4 1 0 0 T2
M30 + M50 1 4 1 0 T4 = -
M50 + M70 0 l 4 1. T6 ..
M70 + Mgo 0 :0 1 ,4 Ts
T2 , T4 , r6 , T8 may be expressed in functions of M10 ,
K30 , M50 , M7o , 'and Mg O wlii-c h are known.
I By replacing into the expressions of a, Ts by their
' values in function of Mo 8 , the stresses can be written
as functions of the known moments only.
(160
0100
...
+ 306
- 194
.... ·+ :52. . -
- 14:
. ·+ 4 ..
... 2
- 82
+ 164
- J-5·6
,+··-. - - 6
+ 22
·- 4~4
• .L:54
+· li/+-
2.2:
Where ·A 1 = 348. 33 A
1 - = A' 0.0045933
-..
..;.-.
- . 6·
+ ti.
'• 164
+ 82
+ .2
-,' .4
..... ::52
+ 194
- 306
-21 ·-· . '" -.... ,, ........ .
' ..
Mgo
' , ·-·.-../
\ '
,:·
·'I!" ~
• ~------:~ --,"r"-. ~•"I""~,--~,-,, :::;-;o;,::.,_·;_:,.,.r-·,... ·~.::...~-~ ...... ..,_ •• ,,., ~ ' ,, \ • .,, <•r-~ I•• ' ~-
-22 ••••• io ......... ___ ................... .
' t
. ,: - .,· - . . ~ - .... . '-... • -·- .• . J, . ............ ---1;.· ··--------· ..
This can _be' simplified since ·the particular loadings are
all either symmetrical or antisymmetrical.
.~
.• •· .
000
,A'
000
A' 0 20
040
,, •
For
+ 304
= - 190
+ 38
syounetrical
t6
+ 152
- 114
cases:
f I
MIO = -Mgo
~o = -M 70
M50 - 0 -
0 00 = 0 100
0'20 = 0 so
cr 40 = 0 60
For antis,ymmetrical cases: M10 = Mg0
+ 308 - 88
- 198 + 176
+ 66 -· 198
+ ·22 ' - ·44
+ .154
M10
M30
M50
j/-
000'. ;: -·0100
(1'20· = -(180
.-·,:-----•. . J. . . '
•. .#,-f I._~ /C">"'< • -23
,,. •lt,-4, .. ,• ... · .., ....... ,~
,',•. ...... '"'1 ............. ·•••· .• • "'••-• ---• -· •• -··· + + + - 0 + •-• ••-••• ••••••0 .. +Lo.4 o-.1-L ......... -•+•• .. --'•••Oo+O•+ ··-··-·· ..................... +04---'0 0 ··-
_ ·---••u+ ·' "- _ --,,,.u· - • • •
r
TABLE 4 NON-CORRECTED STRESSES FOR THE ELEMENTARY LOADINGS
Solving for the different cases: ...
~ <S
aoo + 22,621 36,759 -u s 0 0'20 - 14,138
+ 16,966 cr 40 + 2,828
0 ;, ~.
er 00 - 7,998 23,944 +
I :U· .,$. I er 20 + 15,996 27,993 -! ·,
C140 - 11,997 l )
0 /
.. u A 0 croo + ·22 ,919 37,653 -
a 20 - 14,734 ·+ 19,645 0 40 + 4,911
9,822 -"t ..
12,535 0 00 -~
+ 3 7,604 u A I 0 20 + 25,069
- 68,824 :. ;.:-.
0 40 - 43,755 + 87,510
cr · .. 1.2 ;56 7 ' -o.o: + 20,421
:c-s-:: er · + 7,854 ;..; ·-2'0: 9,425 -
0'4:0-. - 1,571 0
12, 732 ~U-"!"
0 00 -+ 20,917 •
CA a- . + 8,185 20 -· ,10 .9:13 2,728 .. · . ' ....
Gt;iO + .5.,-4.56
croo + 4,443 :13. 330 -·
8,887 ... ' ' - TS cr20 -J
+ 1-5 ,5S2 Ji
r
6,665 '
0"40 + ' 0
{
~ I'
:~ ' '
24,817 i
000 + t
49,241 · ~ -~
t NS 0 20 - 24,424 ·:.. ... .-f
~
+ 36,440 0'40 + 12,016
0 •
",
j 000 + 43,103 ' -104,098
.~. ' ~
0 20 - 60,995 •..
+150,592 -~-' 0"40 + 89,597 -17,9 ,194 ~
,.
:-
,,..-,,. .
- '. ·.~·· ·.·: .. ~ .... ,,,,.......,.. . -~-.,,... ........ . .... .,.
...... ,t
These ridge stresses are not the-.final ones. The joints
deflect under the load, inducing moments, the reactions to which,
at the joints, act like additional ridge loads.
These ridge loads are not uniform but vary as the deflected
shape of the joint since they are proportional to the deflection.
It is assumed that these loads vary like a sine curve along the
joints, which is a very clo~e approximation of .the deflection
curve.
~o ''Norma 1 11· load
! ~ = io sin
2 goL :-1rx:: M :s\:i:n
... -- 9 .·87 :L·.
. n.i - Mc1 • - sin ·1.
M
L
:~
-24 -- -- : . •.• ... -
\ ...
ii .
..
I
' I . I
I I
•. J
I I I
I l
I I I I
. ! I
I I
I
' I
! i i.
t r l I t
t f'. R ! .:,. f !·'
--·~ ....... .., ....
,·
!
A
ii
,-~--·' - r--,' •' • --· ' .•• ,-~ ,,, ''"""-~· ' ~
,,.,.(,
-25
0-, 7. ·~.
5. Deflections of Joints
a. Displacement of individual plates.
p L4 Met, L2 1 J • ~ I j i • l
76.8 E I 9.6 E I
! p L3 Met 12 -=' -0 -
48 E I 12 E I
L4 s 90 M(a. L2 i ' i i + • -?>n b - =
97 .4 E I 9 .8 7 E I
2 I a Mq, -- h
Acr A a 2 I 2 I Llcr 1\ - = - h h
)
8 IAcr 12 1 ~a L2 with' a· E s Thus - = X X --
' .. ' - h ·:E I E K h
For uniform loadings .8·'·· = 12 ". Ao
h 9.6
d cr 12 h 12
For concentrated load _g·f.' =
.,
A a . 12 h 9.87
For normal loadings $' --
,:
.- .~ ' ·-.. 1,. '4·--·· -· . - - . -- -
• ... .. ,.
..
-26 --..,.,........... . .. ' .- ...... ;, .
- -::- -'.----------.---... ·----·-·· --.----- ··--·------ ---··· ..... ······ ·-·· ···--· --··-----) -·-.- . --· ... _ .. ·--- -------·-···. -------- - ·-· ·--(
Ridge (K + 2)
. - - . --.. ·--···-·-· . .
!:'
SIGN CONVENTIONSP
~• (2K +
Jidge . !'(K + 1)
l-J A' (2K + 3).(2K. +, .iJ i'is positive if the deflection
occurs from the ridge 2K tow
ards the ridge 2 (K + 1)
~ 2(K + 1)
Ridge 2K
' ' ', / ' ' . '
- - - - -- - .... -- - ~
A' (2K + 3) (2K +· l!J): .A' is positive if directed towards
outside of structure A 1 (2K+ 3)
A '{2K + 3) (2K + 1)
Initial plate position
-----w Final plate position
A '(2K + 3) (2K + 1) = (2I( + 1). 2- (2K + 3)
A' (2K + 3) (2K + 5) = (2It + 3)- (2K + 5) 2
~' (2K + 3), = (2K + 3) "(2K + 1)
, ..... ::. . .. ~~.- ..
- (2K + 3) (2K + 5)
A' (2K + 3) 1is positive if the
rotation is in the clockwise
directiom
A1 = ELi
·i I
' ' I
•
• • • • • ' ,• • .:.. • I• • • • - • ~ ~ -•-
' . .... • _.,:. ___ .......... _, '-# ·-
TABLE 5 DEFLECTION OF PLATES
According to these conventions E fl.
has been computed for each case. _
" ·us o
U: A:O
u A I
:c-s·
-·CA
TS ..
. '
NA
PLATE
1
3
5-.. :
1
:3
.S-
1 3-
=s-
1
3
5
1
3::
5·
l
3 ..
5:
"
3
.5:
].
-3.
5-:
1
3
5
8'
- 992.5
+ 458.1
0
+ 647 .8
- 755.8
0
- 1d16. 6
+ 530.4
- 265.-2
+ 1015.3
- 1858.3
+ 2362.8
+ 441 • 1
- 203 • 6,
0.
+ 4.:5'1 .8 '~ 235. 7
+ 117 • 8
·- 28 7. 9·
+ 335.9 -0
-.- I2.9-3 .... ·i +- 9·56,:. 9:
Cl
-~- 2733. 6
·+ 3954. 5 •.• .....6-1
- 4705.6
X 103
~-
,·.
.~
.. ~ ...
- __ ·A_· -f_: = L,l
A' X
0
-- 2320 _Q:,
·o· :. '- ·_:
+.· :24:28'
·o-
o - 2873. 5
+ 2030.6
+ 8488. 5
- 9981. 6
0
C + 7036
()
·o +:, :12.76 ... --9.
- :9.·0·2 .• 2
0-
:J07<). 0
.0
0
:3,..742·-. ·.5
·o,
0.
..... 18,429.6
+ .. 20,596.2
103
-<:,
!
: ..
·---~--~- -"Of,:,.__.,'.!._ .· _;:_·-:.,,,r.-1...: .... ~-,..._ 'f.., "~----' '.._., .. .-~- ~-,,~~"11<•-f: ··- ~ :c-____ _.,..,_ ....... .,~-.... ,. ;_'A.-'-' ........ "; ;,.hrt,..··· •. ,,., ••.• ·-· .. - .4_,...,.. \i"'t •-:::---~.. -. ~ .. ·.~·:.,,,._ ~-"!'r •·::.: -,l·~-.f -e.-+-,:•-: ;.• --.-._ ~-...... ~~~ , ... ;;,.·:.- ,_--.:. :;.._:-.~~~ . .,"!!,,~ ...,.._..,.--;1...--...,.-p.;-fL, .. ..-..-__,:- ... ,~,u,,,¢.a.-~,..,=--'--.-,~,...,,,,_'(..:,,, ... 7,...,_ ~"~ : .. ::--Ai·..,.,. ,j.·:--- ,,,..;·-.._.-.f.1:..i-.,uc~-•...:!~,.;...,u.t.:."1~,T·:..-::-,~•.._,11......,,1;;:.~.-;.:ii. ... ~ool.• .. •-·-.,- , ·1 .. ... ......... . . •, .. .... . •'•. . . . ..
,'.
(';
' :
1• •• , ............ _ ••.•. ._, < ,,,,.. -~ • ........... --:.· ..... •,.,·, .. •' ,·, ,, ·- .; ...... .• . .
6. Induced Moments . ~:
.,··
{
---
- ·• . -~.
Fix~a End Moment M4
-
rr, .... J .. t~l1:_.:.,.... .... ! ._ •. _ ,.·:..- ..:---y·· .. , .•..• -..· . .,.,i.:,-- J-, 1: ,J""-·-·· .. ~ .. :.-~-,":.-, • .._:._ ... _ .... .,_ .... -- ..... - • ..t•--~ ... __ ........ ,-,_, __ ,.;,: _ .,,_, .. _ ..... ...... 1..J.
•
·- . No ~r·estrai·n:t ________ _
.. 4. 6
5:
.5.11 .. • ,. 5/V2 II
. . : . . . .
·._lrt· ca·· ·s-e__ o-il= · :1. symme.try'
In cas.:~ o·f anti-symmetry
....
:li:o
8
: . __ ... _,' · .. _- .. _·:_ ._ -:·.-. - -....
rr
Und_t{t- ·~l¢t-t-f c-a.l Lo.:ads•
. . I M43 = .J <l2° E .43
M45 ·= 6 -1 E.15 d2
Rz = ff -5
= {[+ 1.. 5 2.5
~ .... _ .. , ,., .. ,,\~--.,.,,.,., ... _ ... ____ ,
• • , •., •,!'•, I'•·,.,•.·•" "'· - ·", •. ,· ',,~ •'·.
-28
.......
.,,.
I
• I
.(
' .,. ' \
. 'l
, ...... ~>-·-"·
1, .•.. t .. 1-.., .•• • ,.
. . ' .............. •'
'·
.'(,..
> I • .. ,:.<
I.
Distributed Moment M4
.; . ~ ..
. ~- .
) I
In case of syuli1let:ry
3/5'2/5 I
I
2/51 3/ 5 I
·M·· ··3· 4 . .. . · .•
I-n cas.e. of· .an·ti.s.ynnnetry
,
1/.} i
= 3/2K__, ~ M I M
451 .43
., 2'/ 3. ,j
1·-
..... : ... .., .• -... ;. 1(
l-.,
-29 l \ . l (
l ....... --............. -... -• ..... ~ ,, ·.-:~
'--· .
. ,,
..... - ... , .. ~ ... <;. ,..,. -?--"' ,..., ~,.,.., ,.· ,, -, -·J ,;..,··, .... ,. ,,-~ ...,-,. ,--~,io .'~rllt.• ,,,r,-~----r-.. 1. -~
,.. . . <' --;--r:!:'.- l ~-~ ---' •.;;. ..• ..,..- • •-~~.,,_......,,1-.. ___ ;-_ ;_...-6:.,.,.,,_, ~•· .. ,.......;- _,..,,.;,,;·..-~ ,_.l,f.,.,., ,-~_,..._;_-~ta-•-..,"",'-...,,~,.-•~-~-·--=..:.-•·'~•-••"""'-... ~- ~ ... ~.,. .. ~,.u.,..,...._., ,µ.j,,', •. .,..._.lo,c;.1,y.•.-Q:;.;.t.t.••·"'·~:-,"'-•ir.f!'T"-... ..... ___ : "".· "--'_ ~-·-r.;:cz,,··x:n,
• - _, - , ...... • ~' ..... ,,, • -'-.l~"". ''. - .. .,., .• • ' ..
__ ,..l . . •· ....
t\
=·it '
"
...
..................... ____ .... _ ....... --.
. ':'. ~(
...... ,, .... . . ...... ,1,, ..•.
. '- ""Ill· ~ ,.,
. ..... : .. ·•
From the Table 5 the values of rn 4 are computed:
US 0
US I
UAO
UAI
cs
CA
TS
NS
NA
I ' .
TABLE 6 MOMENTS AT RIDG~ 4
m4 Fixed End
- 45.3096
+ 47 .4188
- 56.1194 + 79.2055
165. 7706
-389.3420
+ 20.2330
+ 24. 938
- 35.191
+ 2.1 .• 0 6-5()'.
- 73:._091
-359. 93008
+804.48751
·!
m't Distributed·
- 18. 1238
+ 18. 96752
- 63. 8144
+240. 294
+ 8. 0932
+ 28. 3556
+ 8. 42915
- 29. 2364
--sos: ... 1.fsg:
;.:-
/
. ,
. . .
·'
..
..
;~··
( --·-..-··- .......... •-,--·--·' ... _ . --· -- ~, , •• , , .... _.-.., ... ,,,.,"- -·•, ... •-..::r ••••• .. • · ·-·..a•- ·--~---· .,._....,~_..,L,.....---..--.--,,..-.+..-.,,,.,,, _.._ ,, t...-.a~ .. ,r--v•,·"- ·• -···
-:)
j ·~1
;
' j
·'
I l ' I
,, ,, 1 ' ,.
,1 z·
,1
........ ., ' .. ' ' . .. . . . •' . . • j
.... ,.
.• ~~-t"·~··=-;-;,-;>·r~.-.. ;,~1 .. - _- ... , · - •.... , -, ..
/ .. , .. ,, ... " . . ... •' ·,~ '. • '•.- I ,A ...... : ... , •.
- . . . ... ,_. ,., •• '·-· • k- ...... ~..... .,,,
• ,, • J • ,.;..:., .... ~~- ... ,. . ~ . : ,.J
7. ·.Secondary. Load-ings
Loading
US 0
.-1_~ ...
U S I
tl AO
U A I
C S
.c A
TS
NS
NA
They are the reactions to the moments induced at the
joints 4 and 8.
Symmetrical Case: Rz = Rs
R4 = R6
R2 = - R4 = M4
( v;-) = M4
(0.282~~;8).
R2 = - Ra ~
R4 = - R6
R2 = K4 ( vi) 5 =~ (0.28428)
-.
R4 = M4 ( {i ~ . .1.) = M . (2 + '\12.) 5 5 4 - 5
·~.
R4 = M4 (0. 6828428)
TABLE 7 RIDGE LOADS EQUIVALENT ro THE POINTS DISPLACEMENT
' R 2 + 5.12618
5.3648
+ 18.0494
90! 6 7. 9654
2. 2891
- 8.0202
2.3841
+ 8.26930
+ 143.717
' Rt+ - 5.12618
+ 5. 3648
-· 43. 5752
+· 1'64. 0830 ~
+ 2.2891
+ 19. 3624
+ 2.3841
- 8.2693
- 346.963
~s = 100 + 8.2693 = 108.2693
~NA -:~ , .. 100 + 143. 717 • 243 • 717
' R6
-· 5.1-2618
+ 5. 3648
+ 43.5752
- 164.0830
+ 2. 2891
- 19.3624
+ 2.3841
8.26930
. + 348.963
' Ra + 5.12618
. - 5.3648
... 18.0494
+. 6 7. 9654
- 2.2891
-+ 8.0202
~ 2.3841
+ 8.26930
- 143.717
-31
I
I II
~ V ~ I t~"' , ~ - !
< ~ - ": • •• ~~, .. r•., ... ..__:__..,..,._ "- ~·"1· •"""·t1•t::Wl~c~·,,;..--.... ,t. ... ,~r1.. .. ~~·-·"·'-'. ·--- ·-·- 1.._ ..... -., ~ ~ - ....... ..-.._.._._..~~- .;:11·,w,1---..-' -·~- •t.'11- ··~-. ,........,....,;.,.. ... ....;.~·1&----ii ••• ~~ ...... ,.~ ..... --... ··--·.:.. ............ -.,~ .... ':Ir_;, ~-·~~ .... ~-· ,, .......... ". ,.,_,_...,..r, •. ·- _ ......... , ..... .,..._! ... w,.,. • -.-:q.-..,····•'Lt'IF".:r -·a.-~,_ ... ._._ .. -... ;, ..... ·!'\,.,,-.;,;.,.... .... ,~J".• - ·...-, .. :;...... .......... -A.'."""t,"!i'•~•·-~ .... r.·-...,~.~"!"~ -....4 ____ - -~-----· - - ··- - - ~-· !~ - '.
-. V-
I
l ' i ' Ii
~ $ ~ ! .~;
~ r I• i' ~ f ), I ~: .. r
t )j ~
f ~
i i i, ;!
~ f. f,;
~
t ~-fe
I
·-
-....
The initial normal loadings (NS and NA) and their respective
-corrective loadings have- ,the same shape and are in the same ratio.
As a result the computed stresSes crNS and crNA are due to:
A =(R ... R')NS or NA
R is the actual ridge load.
The cod:-ection to be added to the deflections fo.r· the·
other cases will then be: R + L Ci (loi.dinCJ i)
TABLE 8 STRESS CORRECTION COEFFICI:tniTS
Correction Coe·fficients for:
... '
+ 5.12618 tr S 0 108.2693
·u ·s I 5.3648
108.293 •.
18.0494 UA 0 243.717
UA I - 6 7. 9654 243.717
C s 2.2889103 108.2693
1.02017 -C A 243.717
T S - 2.38412 108.2693
::: + :O= •. 04 7 346 NS
=0.049550 NS - ·--
·+· 0.0 740588 - NA -V
-~ :.- 0. 2 788 701 NA
+ 0.21140898 NS --
-- 0. 03290 771 NA - -
0.0220277 NS - --·· l'·
TA None -·Always Synnnetrical
---
..... .. , ~
Cuso
Cus1
CUAO
CUAl
Ccs
CCA
CTS
-32 · . ······-a,),.lo-. ••; .,
"
,..
... ,-
'.'
..
"
I' ~ · I (. rl •• ,.,_ ....... , ........ ""~" .......... #•' ·:. . • " .. ~ -·· ...• "''~ ............ .,. ........ ,.
' •.• •.•,1111:.1.,._..._., .• .., ••• , •-.-:·:~, 411•,C' !' ,........__ · ..... _:_~.-~QI'~~.~'· ,t : ; . \ ~
.,. ... , ,...,. ...• "'....... .. . . ~
. ··•' ..
D. RIDGE STRESSES WHEN ·rHE H)DEL IS SUPPORTE1) BY INTERMEDIATE
COLUMNS AT MIDPOINT
The joints stresses have been computed for the simply supported
model in the three ·cases of loading (Table rot\ !?age 69).
The joints stresses for the theoretical syunnetric and anti
syunnetric reactions :. only (of 1000 units) are also known (Table 4 ) .
The joints stresses of the model on columns will be determined
by using the principle of ,ruperposi:tion.
Assumptions
a. Define the reactions in each case.
b.
c.
Define the joints stresses due to these
forces.
Add these stresses to the stresses of I
(
the joints when the model is simply
supported for the corresponding case
of loading.
Deflections at centerline of ridges O and 10 are zer,oe
The reactions of the columns are considered as external
forces acting on the sttjlcture •
".
. .
-=-· .:
...... --· '·-•~ -,1,_,-,..·.-a. ....... :.·-'"'-. -·~··.~ •. :. : .... _ • ..,_ __ .J ,.,.,. • .. - • !. - ,.:: - .. :_ .:..·--; . -~·- ;_ -- .-
". -- •,.·.. •• ,o, ..,_ ·,.>" -~ ·.-•• :-,~o.pc. " c • i. -.;,, ·."c.-, • .=.,..,-.~·~···•""'"~'-'--~~=-·· : . ..;.;.~-~·•"·•·•·•..,w···- ~-,~-"~"-'····~<'-· CN.·,• •=~~•""'---~~-,,.,.. ~ --~···------. ,.-,·m~w c. - , . ..,;,_,. ' .• ~ • -·~,-~,......,,..;;_.u~~·~-~··'· .. ·' • .
. ·-·· -- ........ ~ --····· . . ........ -.... --·· ' .. ,
"'
I -
- I ,I
--~-
-~-
1. Reactions due to\ the columns
1 9
If the structure is simply supported deflections ·of the model at the
c.enterline for the free edges O and 10 are:
and
.tt:'
· b. 0
= deflection of plate 1 = 6. 1
-8. .· = deflection of plate 9 = ~ · ··10 9
~ ~
C 0 = Kg (synunetrical load) + KA (antisymmetrical load)
c10 = K8
(synunetrical load) - KA (antisynnnetrical load)
.c0 = Ks x 1000: +).CA x 1000
ClO =K s X 1000 - K x A 1000
I
then the corresponding deflections are • •
60 = Kg (deflection of plate 1 due to 1000) or K$~1 S -
(deflection of ~
1000) or KA~l A + KA plate 1 due to
:i6. ·10 = Kg (deflection of plate 1 due to 1000) or Kg69 S
;),
1000) or KA6g A + KA (deflection of plate 1 due to
.,
•O
-34
·'. .. -- ''!, .. •• - .... •(· .• . .. ~ ... ·i--; - ····-z-·· ... ' ·~:1.'!: ..... !, .,.. -~ ·: - . +-.. - :· • ~--·-,- .... , __ ---·· ._. - ...... ;.,..•" •i,,- .... ±~ .... ..,.._,__, __ •••• - •• ~ •• -, •••• ~-·· •;. ' .i...• • ... , ,; • I. ' ·-;, ;,!'
.1
I \
, .... , .
' . .- .. ,._: ....... --.~---.·-·.~-: .. ··• .,:~:.- .. --· ... - .· _.-.. ··.:-, .... · ...... -··.··.·- ... -~· ........ -~---- -·-. :------~~~-
L i'" '· .
By compatibility: I'
fl. 1 +Ao = o ~ 9 +A10, -= ._o,_,
A 9 = &g A 9' :s + KA A9 A
A = K · 9 S A 1 S - KA A 1 A
• ' !:i 1 A = - fl. 9 A
Ks = A-1 +~. 9
2 ~1 s _ ~ 1 - A 9 - 2A 1 A
t
Are the solutfot;ts of the system "bf equations.-
The reactions are then:_
~ 1 · + & 9:· . ·~- 1 --·- A. 9 + ·· ..... · .... --
-
• _ .2 .. 8 1 S·
81 + ~ 9
2 81 S -
:2 -~-.- A .· .. 1
~l - ~ 9
2 ~ l A
·1000
1000:
•..
The computations will be ma.de using the elementary loadings
(tr'S I, U ~ O, etc.).
The deflections A 1 and A 9 in each case will be obtained by
combining the "elementary deflections" (Table 5 ) . ..
--~·-
··. -35 41
- --~ . ..:.,··;.; ·----- -.. :·, .. ~ .... ·-·-··-·--·-··---· -
' · " •. ·' . ~ · . ·- . ",. · ~ ··-tr~ f": -"i".,;,:'-',, ......... : .. ... •. ,:_ .. ·,- _1. ,,.,-11 ··- ... ·-""•"- ..... .._ -"'=''''.!~--t;1•'1-.... ~J-' .. _., .... _. ,r __ •,:,.<J·"'" , .·- • 1.-, 1 ,.,, _, &.-ui- ,1.-,::i • .-. -· , ~ ,-l.; ""~-··.;;,..-,a..;.•• ~ ... ...i._ •• -· -~..,- ·-·--- ~ .i.....1:.-...;_•1:....· r-• .:.l-l·-~1....1.1~-''~--~-- _.1.- ·--..;.•-~'-{J., ;..-._,m~.-·.:,_..ll.~:...; . .&JT_J.U.:C..L --···- ~~- •• -,..;.,. •• , .i;,t.a .. t 'IS~.--- ~- .,.r;:a'. •-· •. -;. -f ·1 ,,..,__ ~s·1.•,, _ __.~, ·• ' -~,.:.;._. ·--v • • - ,1,,.. ...,- ,.
p· '
1-,,' '•.· •. ·.,.
..
··-····- -····· --· ... ···-- ·-- -- ·--. ·- . ··------ ~-···-------·.--.... -,.--, ... _ ..... ,_ .. _.. ... -....................... ,,.,,.--.~--·-- ..... ~ ...... , ---, .. , .. .,., ... -,.-~,------~--··"-~-~-----·-·-------~--·- -- .. ····---~---~----·-·-····-~------ .... ., ......... -................ , ......... ·--
-· '~ ., . ~~
. " .,,, ,,
- ., . .; .. -·- ~- .;.~ .... -..
Middle Co ltunns
-,_:
Deflection due to load
~ L = - 0.106065 x USO
+ 0.606065 x USI
Loading (1)
~::
t \()
+ (( 0.047346 X 0.106065 ) + ( 0.04955 x 0.606065 )) NS
By sy11w1etry,
\
~ 1 = 6. 9 = + 105. 269
\
t ·.1 •
+ 392. 609
- 45.326
. ·&.,l = ~ 9 =+ 452.552
..
Deflection due to reactions
~ c = - 0.0211409 x KS x NS
+ 1 xKSxCS
By symmetry,
6. cO =~ clO =· + KS x 441.100
+ KS x 27 .337
Llco =flclO •+KS x 468.437 \,
·,
By compatibility•
+ 452.552 = KS X 468.437-
solving _for KS KS = - 0.966089
. ~-··· I
. ........ ··_· -;.;J6. . ~-
·~ 1
~-
. ' .... . . . ~ .
.. •· -~- -·- ·- ·----· -- ... -;-"7""····.-. .~~--= .. · -··.~--: ___ . __ :_:_ ______ .._, _____ ·-----· -· -· .. ..... ~- -...... -·-· .._ .. - ..... •··-- ... - _,_ - -·~. -·=c--.,,;.. .··,._~·- · ... · 1,,, ~-- •.•• . -. . .
.....
}li.ddle Colt1111ns
\.":._;_ /
Deflection due to load
6_ L = + 0.5 x (US!) + 0.5 x ( UAI ) + 0.5 x ( 0.04950 ) X ( NS ) + 0.5 x ( 0.27887) x (NA)
A 1 = + 323.90 + 507.65 - 32.04 - 381.16
A 1 = + 418.35
Deflection due to reactions
Ac=+KSx(CS) +KAx ( cA)
._i.- + KS x (' - 0.021141 ) x ( NS ) + KA x ( + 0.039077) x (NA.)
~ CO = KS + 441.1 KA+ 451.8 KS + 41.0 KA - 68.3
L\ CO = KS ( 482.1 ) + KA ( 383.5 )
By compatibility, 1,
- 418.35·= 482.1 KS+ 383.5 KA . - 165.37 = 482.1 KS - 383.5 KA
solving for KS and KA .! '
r··
Loading (2)
·"·
8. 9 = + 323.90 - 507.65 - 32.04 + 381.16
8 1 = + 165.37
f
~ ClO =KS+ 441.1 KA - 451~8 KS + 41.0 KA + 68.3
~ ClO =KS ( 482. 1 ) - KA ( 383.5)
KS = - 0.60777 KA = - 0.32983
·--~ . '·
·-37
"
• J
,,
i
. .. ,.. ·=·~·-···=· --~-~=====·--= . .---=-=· ===~~--~~--"~--... ·~------~-'" ~~~-----!!!llll!l!l!!!lll!!'!l!!"!!!I ______________ _ .,=, ..... ;:· . .,., .... · .. , .. , . ..__.. . . . . .
.. ' 'lit I •
-38 .. . . . . ~. -- ........ .. ••• d •••••• _ ....... ,..
'',' _,,;
·,..,;... ... . .,,.-.,~.::,l••·••"''"~...,..,.,,.:,,,1:h••·• ,·:. 4 I ,••1 •'°',~-~.,._.....-.. ~-, ~ - -- ---- - -·· .--- ·•·-- ·--. -··----~---------··-- -· ... -- ·.·--·--··-- -
.,
... \
'
Middle-Columna I
I 9
O·
t + t0
Deflection due to load
.. A. L a USO X 0.2125 USI x 0.2875
NS X 002125 X ( - 0.047346 ) NS X 092875 x ( + 0.049550)
UAO x Oo 1875 UAI x O. 3125 NA x 0.1875 x ( - 0.074059) NA x 0.3125 x ( + 0.278870)
,•· .. ,-...
b. 1 = - 210.906 + 186.242 + 13.099 - 18.421 - 190.612 + 317.2tl ~ 37 .959 - 238.225
ll 1 = - 103.583
h-. . ..
L0a41ng (3) .
-~-
A 9 = - 210,~06 + 186.'f42 + 13.099 - 18.421 + 190.612 - 317 .281
9
- 37.959 + 238.225
n 9 = + 43.611
Deflection due to reactions
CO and ClO are same,'ifor the case (2) By compatibility,
103.583 = 482.1 KS+ 383.5 KA. - 43.611 = 482.0 KS - 383.5 KA.
" • ,t
solving for KA and KS
- .:: - ,· .:. - ·.::.
... .. . .... ~ .. --- -
'"
: ... !
KS•.- 0.062198 KA. = 0.1919087
.,
~ ~ • , ,, •. , · : ,1; __ , • • 11 . • ~ , ·• • ·,, • • ·,c • · , , . . ... ' "\ . '·. '
.. ·~~-·,,
-~·
'·'
...
'I '.--·-f>
. .. .
~• .... -
2. Stresses due to the columns . \
Stresses and loads are proportional. The stresses
are simply the algebraic summation of the stresses due
to CS and CA (Table 4, page 23) multiplied by the
proper coefficients.
3. Final Stresses
These are the summation of the stresses determined
.above and of the stresses of the simply supported
structure (pp. 69 and 70).
The final stresses are .. ·tab.ttlat.e·d· ·on pp. 71 :arl'd: .72: .•
-~
. .. '.
'r
-39 'P,.--·-- --- --~---- ------.--- -' '
l
. I
'.
-40 1.---·•"'••r-w-•-.,~·- -~. .. ...... -• •IIP.-,l'to • .•• , •a-.-......,•. f• ---- ....... ~ ...... . --·-··•·· - - .... - ........ ..., _______ ... ~----~-- -·-·------~--- ·- --· ··- ---· ··-•- _____ ..._ r ---- -· --·--- -·--------·- -----·- .. ..... ..,....... ···-· - - - •- -11-...1 - .. -~.: '••~HJUJQl'"I,._.,.,,_,.. ___ ~~ --~ ...,.,,,.
---------------- ---- - - -- ---· .. ··-· -- -.,-- ·-- ---·--- --~-----.. - - ·- ., - . - . -- ·-- ---- ·-- --- - - - . -
E. RIDGE STRESSES DUE TO THE I TIES
As with the columns, the ties are considered as external loads,
and the final stresses will be determined by using the principle of
superposition .
The stresses due to a tie exerting a reaction of 1000 at the
centerline of ridges 2. and 8 are tabulated (Table 4 ) .
The actual ties are located as shown in Fig~i J..
36 --
'~. 5 9 - t ~.s - - ~
--------
u
..
Assumptions_
'the length of the ties remain constant. The ties behave as pa. ·2-=:,-..
simple supports and resist the horizontal components of the plate forces .. ,
in plates 3 and 7.
\
The loa<;lings studied here are all uniformly distributed loads, I \
that is, r3 and F7 remain constant along the length.
Since the ties are not at the centerline, a correlation must be \
established between the effects of forces applied at the middle cross-
sectfon and the effect of the •ctual ties. The stresses are prop-6rtional,
Q.· <;j ~ .~~ '•'·;:;.·· •.• ,. -•-.!;--~.-:.;_· ., .• --.;:•,u-'-"t,;, .. -.:-•. .r;.:.-:-r.~r..,-,.·••_..-.:_ -·-:=-:\.,.:,,.....:,_ .... .,:.,·~.:..•,";. ... - ~-•- .- . ....__·,_µ\.~ ·.-;;,• ,• -.... ~~-f-~ .- .... ~~~-.... ~n, ,•,• -~~-~~-- .~·..:;.;.:..:, i.~i,.,_,.._,.:,:;·~-;~·.,,. ...... ~"'fj-'1.T,J ... l'T,•-,·~r,:,.,;:li:"S.• ~ ..__.__,.._~NT:~"•,.,..........,..."'r.a.-:cru,'<..,'U.~AG•~•·:-::i&D.W1 ,.I._.S.._..P ..... ~,~.•~.._..,,,,_..._._,...,._ • .-.1'!'"_,._ ..... ,..~.t,..,.......,,~~=f~,..,..._"'_'~ •
·, -r
i
um
t·
. -
...... ' ., •• ,,, •• ,,. r.>"- ......... , -( - . ,. ...... ... . .... ·····
to the moment due to the load at the considered section. This correlation
will he established with the moments. (
... ,a 18 ,~.s 9
T T
M = TL -4
M - TL - -4 M <t = 13. 5 T
M = 9 T Ml/4 = 4.5 T Ml/4 = 13.5 TX 4 .x3~3s == 9 T
Actual Mq, = 13.5/9 M in the case of a single G:, t:ie.
., Actual M1;4 = M
The coefficients 1~· 5 and 1 will be used for defining the
stresses at the centerline and at the 1/4 points.
The following computation will be ma.de in function of the case
where the tie is applied at the centerline which has been investigated.
·:~·,
Relations of compatibility ;.
...... ;
2
,.
If the structure is simply supported, the ridge No. 2 deflects
horizontally of l12. and the ridge No.· 8 of 4s·
These horizontal d·eflections are nothing but the deflections of
the, plates No. 3 and No. 9 multiplied by cos 45°.
b Z = ,63 (cos 45°)
Let us call _a H = ( Ll3 - tJ 7) (cos 459.).
The tie does not prevent an absolute lateral displacement but
si.nce its elongation is zero, it prevents any relative deflection of #
ridges No~·2 and No. 8. Replace the tie by 2 synunetrical forces
applied to the structure. These forces create synunetrical lateral
deflecti.ons:"
Llt+A8 =6T
If the model with a tie is loaded at the cros,s--:s.ectloq q·( t:Jie:
tie AT+ ~H = 0, by compatibility~
or T = K x 1000
( A3
.. .
-42
'• ·• u . ·. . . .. . .. · .. ....,,
~--···· •.·~ ·.~__... ~,-~_ ... ~ .. ,;1...,.,l'll~-~,i,.-"!"~ .... c..-:---.--- ': -·· ··'.' . - ·---. _.,.·--~ "'~·~-.----;--··· .. - ... ~---:~ . • ...... . • ~ , ... ..,.~1-e':':...-:r·,,1:!::· _. _- ' t-- --:!-:-.::......:.. .. '" • :.;•J•• • .:,...,-._;,.,..'.""-: ·~.,.:.. ~·:.i,1, __ • ,._;..., __.:. ,-:...:_'.-.:..r-:.!' t-.. - I • ..:.,. ::;...,,-. • •• •~ ••' ..... .,, __ ,-~-~l..i.~-'~:,,r;,::...t....-r~.,·t!~:i.:.a.~.l;MJ("-~t..i.~~~-·~.,o..--...,;..&..:..i.ig .. i,.-,!'l=~~-~-•~.:-:..<•,.._~~._,._.>,lf;'•....._. .....:.-aJ,,-~~ ... ----":'-- ·,:--·--·· ••
. ..,, . .
. \
...... ~,-.......
-43
--- . ------ ------- --~ -----------------~-¥-~~--~---,.
The solution is very simple. The loading has been decomposed
into elementary loadings. An antisynunetrical load will give Li3 - 47 • O,
by definition. All is needed is to consider the deflections of plates
3 and 7 due to the symmetrical elementary loadings which are the only
relative deflections of ridges 2 and 8 .
.A3 S = .OT S
t ~ 3 S = K ~ 3 ( 1000) = 0
K = -
I'
Z.LlJs A 31000
-~--
f. '
-.
·-··,~~·-. ·-··--··-· .. ---·- -.--·-- -~·-..... · ... -· .. '··-~.---~--~~- -· _ ...... ---·_ .... '. --·:.,.;;.,,,-·_ .. · ... -·.--~-----~i:ui~-'.?,L .... ~. ·-~ .. -... ~-"·_-· .. 0:.-· •. ··._..·- .. ~ · __ ·_····k": -·.- ._ .. -
.,._
I·:
,.; .. ~-... "; ..... , ... •., , , .,,;; :...,;.:..,._~- 1i ·.-"': ....... :.:~-· :.
.p ·' .
. . . . -~····-,.,._ .... -,1~,~-.......... , ... _ ...
. .. --··.e ...• ,.. .. , ... , ., .. ·--11•_, . ..,..-, •. _ .. :,·:,.: ... - ..
..
,r
:i ,, ~
l, .,
........,. __ .___ • -·-· -··---- ..... ~4--· -
-44
___ ..• .:.:.... __ ~ ----·-------·---~-----
Ties
.-.
. :: .... :; .-·,
Loading (1) _,
.....
. _ .........
a.
Deflection due to load
= - 0.106065 x USO+ 0.606065 x USI >;...
+ ( 0.004346 x 0.1U606S + 0.04955 x 0.606065)NS
83 =6_7 = + 48.59 - 458.06 + 33. 54
A 3 = ll. 7 = -. 3 7 5 • 9 3
Deflection due to reactions
....
AT = + K ( TS - 0. 0211401 x NS )
A 3 =!::,. 7 = K x 473.56
By compatibility:
375.93 = K X 473.56
Solving for K:
· K = 0. 79284
:"':.,, .·
K'= 13.5 x K = 1.2000 9
·-·· ' • , .••• ' ·• •. •-•·-··-·. 4 • ·• •. •··· .• ,~ _...,~....,, .•• ,0_. .... , .. ,,_• _•, • - ••. , •• ,.-__,, •• • R•"" • _ '-:- ~' -. . • ..... ~,..:~~,J'"I ..,, ,L\.:..;,,i,.,_,1,1 . .:..;\l,.C,1.a~,fo"- ·'"·' • .••. c • ;.•. :1,.:.. •·•
• •. ,. - •.• 1, .... .
,r.
. I
·- , , ..... , . , . ' .......
'
.. "9 .
_, ; .: .. ;.· __ :.., ... - .. · ~ ~--- ··----·--- ---~- _, ___________ ---·------- .. ··--·-· -- ----- -· -.,;., - ...... -. -
Ties
_._ .. ~--------
Deflection due to load
AL = + 0.5 (USI) + 0.5 (0.04955) (NS)
A3 =fl7 ~ - 377.90 + 23. 68
A 3 =A 7 =r - 354. 22
. ,,. ..
Loading (2)
\
The deflection due to reactions is ·the same as for
the previous case
.,
ll.3 =6.7 = K x 473.56
By compatibility
354.22 = K x 473:.5·6:
Solving for K
K = + 0. 748 ------n
I ,',
13.5 X 0.748 = + 1.122 9
, .
-45
- ---· - -- - - .. .
. I}
FR
Ties
¥. -- - ______ ...__ ____ ,. __ ~·- ":_ .. __ , __ ·-- I .;
.... .,..~-··· ., ......... ., ... . .,,, ... ..,..... . ,.~ • • .. • - •" I ' • ,I' • ',• ... ,-.. ,., •.at• I. ,_ .. • • ••• ••::. , ...... ,. "',._.'I . . . . . . ......... .
:. .. ~.-....... . _ :.
Deflection due to load
AL=+ 0.2125(US0 - 0.047346 x NS) + 0.2875(US1 + 0.049550 x NS)
A3 =A7 = + 97.35 - 217.29 - 9.63 + 13.63
A3 =A7 = - 123.94
Loading (3)
The deflection due to reactions is the same as
..•
·fior the previous case
By compatibility
123.94 = K X 473.56 ) .
Solving for K:
·, .. ·.
K = 0.2617
K'~ 13.5 x K = 0.3925 9 '
-46 ... t ••
. ~ ... .. ......... ... ..,,...: .,.., ., . . ..... .. ' . ,., ... , ••••• " •• > • ....
---------- ________...._. __ -...- ------~-------. _ __..._ ________________________________________ ...... _ _....
I ! I
I t
·'·~1 ._, __ ---·-· ·--~ - ......... , .. --··----- ...... ~ .... _'I\" ....... .... :...J" .. • •.
. ()
4. :n· __ E S C R I P T I O N O F T H E M O D E L
A. MATERIAL
Methy1 methacrylate (plexiglas) has been selected as the model
material. Plexiglas presents some disadvantages: the most si\gnificant
is that the physical properties might be slightly different in the
same sheet and between different sheets. Also one must benextremely .
careful when placing the strain gages and when reading them. The
problem of bonding the gages has been solved by using the Eastman 10
Cement, a special agent manufactured by Eastman Chemical Products, Inc.
The procedure for reading the SR-4 gages will be explained in ,detail
later.
Plexiglas has been c·hosen because of its very low modulus of
.~las:.t:i:ci·ty - approximately !/30th of E for steel. This allows a large
deformation for a relative.ly small load and makes the testing_ very
simple.
B. THE MODEL ITSELF
It is composed o·f five- identical plates, a diaphragm cut in
the same sheet as the plate with the structure closed at each end. ,
The bottom of the model is open. The model is represented in Fig.2.
The details of the construction are shown also.
1·" .
. '• . ~ . . . .. --·- -··-~---·~----·· ... --.... ~--- .... - ....... _ ...... -... ', ~-
I '
.ft:.
.-
t,
r
n tj '1 [,
11 .11 I\
T j: :•
t r [! fl
11
t '' ,,
il ~;
_,
ll
fi I ,,
j tr
i~ . ii
I ~
' !· 1· ,. '
J. I·
~.
!
L !
i· ,. ~ ~ j
i
J . rl
i\ j'
~-
-~
-~
·.· _..__,_
:~
Di.a.J)hl:'agml
- --._;,
CONNECTION
, ,,.,
PLATE -
__ ,,. ;
.. ·;; .. .,. ..
.. .. -,
.,,. ./ _,,,._
,
DIAPHRAGM
., ,,,
.. .,, .,··
....... . : ,,. .,.
Fig.
-~-· ,,,
·"' .;, :
2
/ .,, .
. .,.- .
.;:, . ,,_
:.,, _,, .. . ., .
/.
THE MODEL
/. .,...._ ,,,_ ..
--~ /.
-, . .,
: ..... ,.,
-~-
·/
.-,.~
;,.,_.1
/· ~~···
~ ;-'~ .... -: :...,,·~:·- - -- -·,_ . ,., , .,. :, , ., -~ . , ,, ,, ,., . ,,-,., .
,/-..... I
..-: ·' '" I
. .,._ . ;,:·
;,,.
I _I ... lJ: ·.,· . .I:,·· ·I. I . ·1J JJ·· 1 . ..
. ·,-~ ~ - ~. ..... - -- - .-.:~. ,;._ ... ..-_ - - -
.
TYPICAL JOINT
\ ; r l
I {
I I
j t·· I .< ;
l
I I
. I I
- I
.. •
I
I i
i ,
I . .i:,,. co
~Jl~~l!'E!"'-.Y,O•~~, - - • ., • l '1,---,i •J...,"..'•"'-.. t- ;. ;.A'"'"-;;o1,.l.:._,:•~f-tL,d1,,""~•=-', _,•,.:'t'""'-"',,' .. : .. ~,,•('...,',•.' ....... .,.,.1-.--l;:~ .... -':_.~-•~;;t,~', :..-- '"".,-'!,,.,,,
.... --,--""!"--., . . ~ . ... . ' ..
-49 - ---~------------··-----·~~--------------- ·------------ -- ·-·---------·
.. ~ - - .. -- - . - . :- ""': -
· This model has been built for Lehigh University by a pro
fessional model maker. The total cost is around $80.00, most of the
cost being due to the labor expended.
C. SUPPOR'l'S
1. Model Simply Supported - Four aluminum balls provide
point supports, one at eaci/Corner. /.
' been made in the diaphragm in order
place.
Smal 1 notches have
to keep the balls in
In the s·imply supported case, two supports at one end
provide a free rotation of the structure but prevent any
displacement. The supports at the other end are free to
displace in a horizontal plane. Ball bearings allow
' these supports to behave\as rollers (Fig.~).
2. Model on Middle Colunms - The columns should prevent
any vertical (upward as well as downward)
displacements. At the centerline, 1/8 of
' and hor~ontal
an inch 'way
from the free edge a small hole was drilled in each side
panel. A small brass rod fixed at its ends passes through
the hole and acts as a hil}.ged support (se~ Fig.4)•
The four corner supports are mounted on rollers as mentioned
previously. (
3. Ties - A brass rod (diameter = 1/16 in.) acta as a tie,
which is not supposed to elongate or contract. Holes were
V
i I
i I
I I
I ' ! I
I l
-- - ________..J .. -
. '
' ... _ .. __ ··--~-~~---- -··-. ·~----· ·. ~--- ..... •"'- ~---•L-......___•-•• --~.--,-~.---.. ----,.-•~~-·-•
.. .,.
I
. - 50 -~ -.~ .. ' ,,_;_~ . - :.~· .a.._....:., .. \ •,, ..... \. ~ - . .. .
·-· t . -------·------------------------ ·-----------~-_: ___________ .... __ .. ______ _ -----"-----•-----------
I , - .
"
'
( ) I I
If' .
.
.'
'.
Simple support roller action
~-----+ ----
Fig. 3
·. . . ••
,
Ir washer rod
,,
• •
; POINTS SUPPORTS
-.
·•
Simple support no roller action
1~811 of st ucture
.... t •
,, ·~''\ • I I I . , ' ,.
'
,,,:
_ _..,. ____ .~.:---i" ·_: : ... - - - - - - - - - -- ...... ---...;· .. -~, .---'
COLUMN SUPPORT
-~~-· ,.;:,,,_-_~_-: .. ~,:,...;,, :'--- ~".: .. :'•'-.,;.,;,,,,._•~-·-·· ··-·.~·,.;·:· .. -;---·:•·: .;..'..~j~-... ···~.
t,
..
..• . .... . ,. : ' ....... . ··: -
;,
·., ~( ..
l
..
,,. . ................. ,,,,..,, ....... ' " .. .. ·, .... . . . ..... . ••.,, .,
made at the top of the edge panels. The ties are passed
through these holes and four nuts, one on each side of
each wall, prevent any relative motion of these panels
(Fig. 5) •
Fig. 5 : TYPICAL TIE
:.~ .
•'9•
...
I ,, ,.,..
.... ., ,:;.. :,:
,~ ... . . =..:-..:..:.;.::,x:.~~:....._..;:;..._ .... _:... __ _:~.--· ._......--- -~-v'.'=-.:.~::--:--.::--~ ... -- -··- ~::-:r= - -C'.'%'l!· .• -"· •• .c._-.n;J.'.::.::.....:.,:r..,:.i:.~=C-::0~ .... ~....:,~=:::..'C.: _.;.J.....;...s... c..;.: -:--..• ;,,..~ ::;.~~.::... .. ---~~i;i;.: ;:.:::.-;:c:..?o - .:..;-·.:..r-~Jo;:;-_r:~~.u.~.:,-:,.-~_.;:: • .;~~-::~~-~~ -~ .... ·.1-···•.,..:.,.~ ..... - ... >J_:.,,. . ,··?-..
,_....~·
• .. ' - -.-, .... u,,1.·,,,." • 'I:-.'
·It/,
-51 '· .1... ..
···-. ., ..... ,._,_ .... ·-····~-"' ..
...... . '-'
l I
~ . -
.... .• ~ L .... • .......... I,... -~.-,.. .•.W ... .... ~ ....... ""
t
·\.
--- ----- --- - - - ·---·-···--
5. T E S T S O N T H E M O D E L
A. PRELIMINARY TESTS
E. Zanoni(l) ran. a few tests in 1962 in order to establish the
physical properties of the plexiglas and to verify that the SR-4 gages
were behaving satisfactorily.
1. Test of a tensile coupon for determining
the Modulus of Elasticity of the plexiglas,
. the Po'isson 's ratio, and the creep.
2.. Test of an I shaped beam in flexure for
determining the Modulus of Elasticity
and the creep.
3. Test of the I beam as a column for
determining the Modulus of Elasticity
and the creep.
These tests have showed an excellent behavior of the strain \._.._..-,
gages. The Modulus of Elasticity was in the range between 450,000 and
" 480,000 psi; the Poisson's ratio around 0.37.
From ~11 these tests only the test on the tensile coupon was
repeated in order to check the value of the Modulus of Elasticity. ,In
one year it had increased to 500,000 psi. This value will be used for
the computation of the stresses. (See rr go t• lot).
-- ~s2 . 'J.~ •. ,. !'
..
. ~ ....
------~~, ·- ~ .. ---·---~ . ' . - ' ...,.._, .... ...,.,...._. ' -lf!'l'UI,---:-.... ,_...__._,,._. __ ·--··,,_,. __ ,.__.. _ .... ~~ , _. _- ~"': ,i •:, ;~}; ~ • ...- 'S ·• I,. -:1. . .·· ,,. "'- 1 .- • .
..
.• w.. ~ ...
'···. -53 . .. ' .... . ... . . .. ,...... . " ......
The rate of creep had not changed. The variation of elongation .
of the plexiglas under load varied in function of time as an exponential.
The amount of creep was very small after 3 or 4 minutes. (For safety,
an interval of 10 minutes between application of load and reading of the
SR-4 gage.)
B. ACTUAL ,TES !§(oF THE LOADED !l>DEL .,
The joint stresses were measured at the centerline points and at
the quarter-line points of the model for the different cases of loads,
supports, and ties which have been theoretically investigated.
1. Loading System - A uniformly distributed line load
is needed for each case. This is obtained by using
what Scordelis ( 7) calls a "loading tree" (Fig. 6 ) •
·~. " .
In this _case the system divides one single load,
applied at ,the centerline, into 16 e}ual loads equally
spaced along the length of the model. Considering
the stiffeners of each plate, this is a sufficient
approximation of a true uniformly distributed load.
The tree is attached to fhe structure by means of
nylon loops passed into holes drilled in the plexiglas I
and retained by aknot (see Fig. 7) .,- The nylon line can
resist a tensile force of 20 lbs. In the three cases
. of-loading the holes have been drilled vertically in the
model.
2. Magnitude of t~e Loads - Readings are taken for three
values of the single load applied at the centerline and
·-
•
l l 1 ~ ;;._ I(
i }.
~ ; i: 'l
l
-'
.\
I I
I I
\ . Fig. 6 ·: THE "LOADING TREE"
'··
longitudinal cross-section of :s-t-ruc:t·ur:e '· !
. " ;· (
' ' {
r. I
equivalent uri.i.forni line··-load: t. p = ·p1:·36 I
I r I
l. .I-- I 1 ' ~ I I .. ~ I I·
:• ~ : . -·~"' ~"' '~ ~ ~ ~ ~ ~ ~ ~ ~" r--,.. ""' "r-,.... ~ I'--.. "" I"'-. "' I"-. "'-... I'-... ................... r---.. ~ .........
r-.. "'-... "' ~ ~"" ~ ........ i' ....
;V· . 'I> '-J ~ Kl k, ..
~ 1....-- ~ 1..-, V '· -.v ... . .
V V. '
V : . . . .. . - . ' . . -
•'
. . . . . f: l' -.. . .. '· 1 :/ :
: ·- .. . . . - -· . ' . . . . - . v .- -. . . . -· . . .-..
V - . :...
v - ..
- . --· ... .. V .... -- -/ . - ... - .
.... . " . · ... · . ·-' .
. . ~ .. . . .. - - ..
. .
··-·
" - ....... -. . . . . . . .. .. . ....
I I
,
r--.J (..... ~ j[..... f..J [.....
.. ..... ... ·1 .1 : '
.. . - -
-~ . . . .. . .. . . - . - -
... • ;
. . . .
;
' .. - . . . -
"
. . . - .. -· .., ..
..
p ·~: T-ota.l. LO'a·d
... ~
hi v V
/
./
/
L,,"
I/
L,,"
V
·~ - - . .. . . --· .. • -·-·
"
r l
\ /: I !,• ,·.
[.
I I I
!: '· i ;·
! : \
I I I
I I
I I
,.
r I
• ' '
.
-· } .
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!
I ·,u, +:"'
.,
~ ~.._. ... --, ' . ' '
. -- ' -- . --- . -··-- .. • • • ·. ' .. 'I
-55 ' ...•.
.. - ' .....
.~- ~--··· -- . --- --------·-·- ~ ' .. ' ··-~-----:-----·-·r-•----·.-----~• ··- --·-·· •. ·.,._·-· •• .:...,. --------._.. . .,:.,.'-:. .. _..: ,...:..:..:...:.:. ...
•
,\
-.
-·- ...----- -- -·~ -, ·"-' -~ ,
Loadings :
(3) (2) (1)
Location of loops,in lines along the structure
.. ~. '.
' ' . ' :
_,
.,.
knot Fig. 7 : ATTACHMENT OF
·tHE LOADING TREE
I ·.·· .I
nylon line ( 20 lbs. test)
- ·- .. _:. ··-
..... ,. . "..j,:......;.· -· • ' - ·=- C=:.;,.::...:..~e.:..: 7· .... . -- ·- ···,;"J;,--,- ....... --,r- ·- ---..:-~~-.. -:e---·. - ·- -·-··--·-=-..:-:·~-f ..,,e,. ".
,C •• I '• •.,,
---- - .. ~---------·· ---- -
)
"
.. ',';•
... - --~-. -. -;·-~ -:-·-:-'" -~··· ·. -·. •' -. - ·-- - r- - - - --· - --."'- - -- - ~---- . -· -.-··"' - - --· ---~ :-- . tJ .. ~.: - ; - - -- ...... - - - - ·--- ----------·
distributed by the "treet': \
O, 20, and 40 lbs. The
computations -of the strains are made for the uniform
load
3.
_P _ 20 100 L - 36 = 180 lb.fin.
Locations of the SR-4 Gages ~Fig.1)
I I I
8~
' II
Io f I
-~-
The two sections are identical and so are the strain \_ .. fl
ga-ge locations. There are 2 gages at each point in order to
check the bending of the plates.
.;.
Locations 1 to 10 refer to the centerline, locations 11 :·to
to 20 to the q~rter-line. All together there are 40 SR-4
gages.
4. Reading of the Strain Gages - The material of the model
(plexiglas) poses a few problems relative to the readings ···, ..
of the SR-4 gages.
(a) Plexiglas creeps.
(b) A temperature compensating gage must be
provided.
(c) Plexiglas is a poor heat conductor. As a
result ~the heat input, due to the warming up
:~,
J4. • ,• • "' 0 •M ...... 0 -·-·~· ~- -. oO •• ... -.-~... :- ,, • ~
l .,p
-56
. . =:-·.:r==,,:" .. ""-'~c;_,.=_.-_._. z-.c..,-_·.-=nt-:cs=_;_-;,..-.-.•,c;.·"-";:;oco~..::-,., ::·~'--'-',, .c,sc-1'__.:..;~, .. , ~,·.·:=. c=-:.c.o;ce.:_;~~-ccc-.;c.:•,w·c.. ··,>·f.• · ,., .•~, .....,., =~":!\ ,.,:,-,~·• ·.,,r=-,;-;-·.,·.T"<'.""' '"'""~'''·=·c:1,.:-~-:--;_-:,~ .. :-.==.-a..:.,_ c...:.. :,;,.,• ·.;:;.,,....=.t..'"='' ="'1•·-'· "°''....fi;.;·n',c Cl~•l "-*·-<o..,., ... ~, :.,·.w.·.a,.=c--,..._:;r;:,===::....-r.-• .i~~ ,.:. ···- :.., '''·';'
"
-
·-- -· --- --·------~. ·-··~. - '.-.· .. - .. - ... ·,··,- - ... · •,· . - _, ___ . ___ .' ___ ._:.·~ ...•• :· ••.• -- ·._·_·,4 .. ··_-· .. ·-. ·-•. · .. ·- -··~·+.--_-· '. -_ .... ·- --··- .:------· ·--. -~---. - .---~--:-.·- .. · ... -· -~:. --. ·-.. ·---:-------. --:. -:--~-~-~ ... -----
"I'
.··
of the gage itself, is not immediately
dissipa,ted as is ~he case for, ~etals,
but contributes to an additional deformation
of the model and of course modifies the
readings. This ''heat illput effect" is
shown in evidence by the fact that when a
gage is modified by the loading, after the
creep has occurred (10 minutes) the indicator
does not remain at a fixed position, but
varies slightly with time, always showing a
positive .. increment (elongation) whether the
gage is under tension or compres~ion. This
last point indicates clearly that this
elongation cannot be attributed to creep.
Obviously this phenomenon is valid for the
temperature compensating gages. -~·
C. THE ADOPTED SOLUTION
1. Instrumentation -
(a) A strain gage indicator, l:S _us·:~d .. ~
(b) A clock is needed.
(c) The 'measuring gage" plugs are connected to
a shift-box relayed to the 40 gages of the
model.
(d) . The "compensating gage" plugs are connected to
another shift-box relayed to 2 identical
'
./
' '
-57
- --· ._:::.-:::::::'7:"...:.. 1·.:.~-,-- ~rr---;;- ~,. . ,r.'Tl~~~';"7, -~r-·..:....-:;r,:L-..,.~ ... ,lr __ .::-~":""..;...-:-r .1.·:_ ·.-:::.;..·1 ... 'L;-:::t.--.·::rr-:::l'.~..::-..r·: -_•:-:-r-:---:_~ ,. • ::..C-:r---r.:.:"':~~1...:.."::"~-.: -::.::.- -:..:.:..'"':.r-=:+=,."::: ·;..·_-::.::::.:·~ :_--:==-=;-::. -·.:.:...·:1:-:-l"::""'-.:.:r::"C--.:i1~~- ~-- _1•-,"'!:.--=;~. ·-.~..::'!'...'."!/t:..:~'lf'"..::"'"~--=.1.1...:::."!..-.r-.;..,......-:--::··~..:--..-.-:..:.t..;.c.--;,~r..J;..-.:•,;:,..::. .: ... :~ . .1:.:.-:....1:.1 .. ~ ·.:...-:--_;.:~--":!::...:~t<.::..:..i.;-,.:.~;;.-'.:.:::.::_...;:;:::_.W,'z_:-:. .. ~~u..'.:x--~ ... tt...~.i~r.l..:...._=-.-~~,.::-_:-__:_::,;,.::::..:.:_ __ _ --:::. .. ;.
---···' _.,.' (•
. I
.. -,. :,:·:'
Ii J I A-f:il1 L•L• -., • •••• ;;;.: 8 w•:s• r•••- :: :,; -.!rrZ!i=!:::::;: tl MWt•:tlJAP ,Q l;
-:-.M .... 1----- -- - -·- -- -~ - --- - M __ .._ - --·~ -- --· • • -- • ••-.:- -
it compensating gages (I) and (2).
2 • Procedure - ( Fag . 8 )
(a) After applying the load, allow 10 minutes for
the creep.
'(b) The box Bl is switched on measuring gage 1,
the box B2, on compensating gage 1.
(c) Balance the indicator. Keep it in balance
f,' for 30 seconds .
( d) At "30 seconds " switch Bl to 2 and B2 to 2.
(e) Record reading of gage 1 (The indicator has·
not been changed).
(f) Balance indicator for gage 2.
(g) At 30 se·conds switch Bl to 3 and :82 to 1 ... • e t-:c. •·• ~
In this manner each gage is warmed up for·· 30
seconds and tafttn off afterwards. The heat input is the f
same for the measuring gages and the compensating gages.
Th~- ab,lute reading contains the effec.t of heat, but
this is cancelled when the difference of reading is
calculated. (It is as·sumed that 30 seconds is enough
for the cooling of the compensating gages.)
Under these conditions, one full ·test for a defined
, loading and given boundary conditions will Dequire 30
minutes; 10 minutes for creep, 20.minutes for reading
and recording the 40 gages .
: ' .
t
-58 ', I . . . 1
' .
.·,~·
·- -,, ,=,:o;:.:,,,.=.,=..-=c:-.-c.·=-··c-c-,-;:·-~.~:·=1!-.... ·=·-:.,=:•. ·: ... ;· . .-:f~ .. ,, ......... ~ .,.. ..-,,··=-s·,.-.. ' '-::---.-~. . .. "'" •c-,,,,·,;;,· .. , ,.-.-. "'· :;.:.-:,,:~;.;~:~.,"'"''· ,:-::,:...==="'·'' .::c"::'.C.:::r-==c . .:!!..._",·""·""":, .. ~t:, .. :;::i...e-::r ="-:.:-..;;.\C.'.::. ,. u.,,:C .. c·,:c.: .... c....-::,: ~:·.,·,=~~-'--~-=,=~,==--"·':f-'-'-" ·=-'-·""·-·="'"-"'-c-'c.,:,:;c.=·"'..:==-~-;.:-~=· =--=· .. .- -==-..:..,c..., .. : . ; . ·r· .
I l ,)
"
-59
' . • <.•f .-·, •• • II•• ' ,, .Jt. I ' • ,.r•_, .. _,1o1t,9\,"', I ,.. • ..... ··' ~ '~
1 •,.•~J._·;, •• • •' . ~ ., . (. , . •· --------....,,.......~-----~- ~--·-----"- -· ··-- ·-·-----~-·~
-
··);:(g-~ 8 i STRAIN READING EgUIPMENT
to the 40 AS SR-4 gages of the model
l
j, ••
1 2
r
· ..• :
switch box' B2
A5 SR-4 compensating gages
switch box Bl
measuring gage
compensating gage
clbck
strain indicator
. . ~ ·- ·.;._' •• · . .: .. :~---.-~-.·.~~_:, _ _:._:_ •• _ ....... :.....u.1-~ .... ~~-• .I.~ll,_J!_,.;,."-·o.L.·.,_;,_____...L,.~~_.::.,;.' ' .. ::: ..... : _:.J';:':-= .• ··<" -r:-:~.:-.~:=-'"'.-.,::r:-._;,,_.._ ~..._._...,..,:t-.!C..! ... :::._.'!.....;.\-~--- ,_:_.------'--=-~'~.'..!l, •.• •. ·-~.,r-r:.:,~ ,~1~·~11.':~L!l.'..t.L C:_.'., . ._ .. -J1,,~·-.. ~ :.t.>--G. ~~~ ~"::!!J:.:. .-~tn-~~1'1:Y.11.,<lu.r~ •• \'. ·~J • .;J,-"\..'-"'•'e'IU.J;:l.." .. T"l'Jo!~J.:;;:a:Li...'1,',;"l'l,;;.:'_.-.:; ...... ~ .... ,.u::r.~·.,·_:in;c":-.':.11.)..' ~-.~~J.r:.1d.:'L.l!:l.:...."l.,J/1-----=.. ;c:.a.:.,,~~-- ... '\:C'"..t.....:_~ .. .J.'.., . .:!.\.l ;,;.::..::.:ll.'.lt...x.r··.:r..:::::..·:-:-_-;-, ..:...·:_-~·____..r;:_ ;_ : ... -:. .:._::. ... _;;.i:... ~..:!"!" 111_'!:..,,-·.:.·.· -~\ L~,•-... •.;.: --.. ...._
. • ....... ,,.,. ·-·-,, ..... ·•, ··,·: ··., ~ ,,.:.,-,, •;.,.-.t>·- -., ••. ,., ·,· -,_•'"•"1: .-;;,.•·, •. ,•, :.,r.,•, ., , .•. ,.,
I -
.r
)/
"/·'
I
'·
•
. ' . . -· ' . ·. . '. . ' ' ' ·- .. :··:~:""--:.~· .. :"~·:···-.•·· ~-"' -. .
·,. -:r:•
· · 4 t ·"
l
A
.. ,. /'
,..._
I ~ .u
-
u
Twelve different cases. The average results of these tests.·
for Cl and 1/4 points are recorded pps. 77 to 88.
\ D. S!RA:INS AT RIDGES, FROM THE STRAINS AT SR-4 LOCATIONS
I
A linear variation across the width of each plate has been
assumed (assumption No. 6, p. :>)
5
-~ .. -- .. ;.
*4 5R4
b -
.._ ..... o.s
I C l C. -
-60
.. . , .. ., . ' ·•··. •·,'
- -·' ~ -··· .
---· ----. ··- -·- -- - ----... - ·. --··,·- . ~- -· - ... ~~ --- . ------- ·--'. .. --··-- -----... ---·· --
B ' B --
Let us consider the plates 3 and 5: . ,..
a' b' b" c 1 are known.
A" B' B" C' must be determined.
From figure on pag_e 60 .
+ B" 2
b" - Cf
4
a" - b'
a.nd
a" - b' = 5 - 0.5 - 0.5
-------=slope of the strain line 4
·A" = a" +
B' :·;:: :.b·' +
B." - b": + -
c• f - c + ...
a" - b" 4
b • - a" 4
.b·'' C ' 4
C ' - b" 4
(+ 0:~ 5).·
(+ 0:.:5):
..
( 0 , •. 5.)
.( 0 •. 5:)
These joint strains and the corresponding joint stresses have
been computed; they are tabulated (pp. 89 to 100).
-61·
.. ;_.
I
I
I
I
I
l ·
-62 .. .. .. ... . '· .... ~ .... _.,~ ~·· '..... "" "' 1; . ••• • .6'
, .. -. --- ........... _,,,. _______ _ ' • I - - • - - •-- - ,.,. • -· - • • '• • . ., ..... ; . ---- - ---------- - --- ---- --~=&.---- .., _ __:_ ~--;-·-·---- ---- --· ... _____ ,:__ _______ --- -~ - -- -- ----- ·- ------- --------- ------ --~ - . ·- -- ----·-·- - . -- - -·· --· --
6. D I S C U S S I O N O F T H E R E S -U L T S
AND CONCLUSIONS
A. DISCUSSION
1. Preliminary Tes ts
The determination of the Modulus of Elasticity, using a small 1/\
'
tensile coupon, might not be quite accurate. E. Zanoni suggested that
b:"~fore building the model, the Modulus of Elasticity should be defined
for each plate. Each plate would have a length of 56 in. (instead
of 36 in.) and would be used itself as a tensile coupon. Afterwards
it. would be cut at the exact length. If the values of E vary from
one plate to another, these different values would be used for con
verting the strains into stresses .. This would give more accurate
values.
Yet this would not take into account the effect of the joints
since the gluing might affect the value of the Modulus of Elasticity .
.....
,---'--:
,,,. ... ' .... , ~. . . . ' .. ,',. } '.!"''··· ..... ,..,
·J
2'. Comparison of ,the Experimental and Theoretical Stresses
(See Tables on pp. 89 to 100)
\
(a) Model simply supported
ct points
1/4 point.a
(b) Mode·t,. on mi;.ddl.e,. colunms
ct polnt~=
-l/4 ,pp:i.:nts·.,
(c) Model simply supported with ties
CL points
·t/4 'po . .±nts'.
(d) Model on middle columns with ties
(i points
1/4 _po-int~
Loading~(!)
excellent correlation
excellent correlation
good correlation at ridge 4 and: ·6
Load!ng (2)
good correlation
good ·correlation
go:od correlation
good good correlation correlation at ridge 4 and 6: at ridge 4 and 6
no correlation even signs are different
no correlation
no correlation
-
p·oor correlation signs respected
poor correlation
poor correlation
poor correlation
Loading (3)
good correlation
good correlation
good correlation at r-idge 4 and 6.
good correlation at ridge·4 and .6
satisfactory correlation
similar variation of stresses
satisfactory correlation
satisfa~tory correlation
-63
'" '
;,• .... - ,.,
~·--_./. .. -~-
' . .
. .,,. . .,, . •. "' . ··.:- , .... "'"' . .. ,,.. .. , ·.. '\ ,.. . :·.· .. ·- .... ~ ... .,. .. _. , . .-~ ......... ~ .....
.... . . . .. ' '•·
' .
..,. " -- . .. .. ... ~ . ..... ,, .. -•-•'if ..... ~..-~-··.'.~.·:···~--~~- ····',"'!""••• ' ". ,..... : ''." ... , '·~ .. ..
B. CONCLUSIONS
1. The results are quite satisfactory as long as uniformly distributed
loads are appli~d to the structur~.
In these cases the experimental stresses are very close to the
stresses computed theoretically.
2. The results at the quarter points are good.
This would indicate that the effect of the diaphragm is not
e·o .be considered at these points.
3. However, in the cases of concentrated loads the theory does not
seem to be valid.
The distribution of the stresses has to be investigated for ....
these cases. The moments in the plates are likely not as they are
considered in the present form of the theory. A knowledge of the )
actual plate deflections would be appreciable in order to modify
the analysis. ...
'
·-~ ·.
·1
. .
• : d
-64
., •"!!'
-··-~ _ .:_;,-;:_ .. ;:: ;.. .. ;...-~:::;_;_::_-:,.:.=.::, ... ;. .... :..;.;-.;.-;..~-~:..:;.·._·r·~c-"l:-'"..;:.'.~.Z:.c...:;~·=,..~:.:;-.:-=-~-·.;".::.CT-·..;·.i~~~--=-~,--..-...r.'lt:~~~~tl=-":--:"':-~::;_r ___ -'!::'.·--r~ ... ~-::L""):1r-.....!".:.'.;:.:.:.' • .:..=. . ..: • .l:-.t . ..:-=.s~.,:..=J~-::C...'.J'"--.".---,...:::...:..1~--..r.:....:.•t:;,:.1:1;:.11.":'t'.1_. .•• ~-;::::--..,::;:-:~-:.1,;J.a.-;,--~.r.7;,;..1,,·:,-...,-.-~:-e<J1.~ti -~..::.:;:.:"~~-:::. .. ,.;..~.r.1r.~o,Q,.,~~::.. .. :.>m:.: •. "CJo.;...;-..,--.:_:--..,.,-::---,.:,.;,.=.\~~-•=-·.._c• • ....::: 1:.:?::Xr--:r.:r::"'.-~~.i.•~~ •• - ..... ,J.~~J.:;·_-.:..· ·.-. ··
--.. .. . ... ~- ·.·,•. ;,;- .~,,~,h.,.:r; ',lit ~ ,< ~· 1'!.~~' ' 1 , • .' L"'-~,r :11• ,,f"'d • .,,,,,ht'J.).;_\,"..,!.. .. , ~.n. i,,l,.-d/ 1,.. "'.;:._ .;..':. l,1'.. JIJ.~ .. ,,..,:;.:
, .,, - ... ... , ', ... -65 ..... '
I
C<r,,_,..._,;.,. • • . . .'"··' . ,• ... • I •.· ' "'• •
'. . ..... , ...... , ' . •.. . .
,t ...... , .... -~ ........... ' ....... ---------- --- --·- - .. - - -- - ------- --·- ... ·---- ---· -- - -:-- - .. -- ... -
- --· ·-- _.. - ·- ... --- --·-- ·- ·-· ·,,·. - -------·-,·--- _;_;._ - ....... ~ ... ! ___ .;::. ::: ••. -:~.: .. ,:... ~:_; '.:- . .,.
I ' ,'
3. Possible Causes of Errors
Unsatisfactory results in the case of ''model on ·middle
supports" might be explained by the fact that the beams (steel 1 in.
x 1 in.) supporting the model under each of its free edges are not
·sufficiently rigid relatively· to the rigidity of the model. If this
is the case, a certain amount of deflection is allowed at the middle
supports. The theory assumes zero deflections at these points. Since·,
the model does not behave as a be·aD.J, -t;·lle i.rtvestigation of the effects
of partial deflection:s on the joint s,t:re:ss:es is rather involved and
will not be made here. However, before proceeding to further testing,
"thJs .investigation should be ma.de. Stiffer beams should eventually
b·e used.
In a general manner, the. s:trains obtat.rted :in ·t.,hes e tests
-~re of small magnitude, usuall}r less than 100 micro ''/". The accuracy
+ . of the reading is of the or.der o._f - 5 micro ''/''. When the strains y ·-
are below 20 rnic.ro '-•/:''.,- thi.s gives a high percentage of inexact! tude.
In the: f1Jture, higher loads should be applied. The total load
·here was 40 lb_s -., A load of 80 1 bs . ·c.~uJd. be used without danger to
the model.
-·~ . ,
. .....
:·•.
--c:,.:_-,·· ----•·-----.... ,~-,_,.,_·;·..:=.-,.:-c,.J'--"::,-".:._i'-=-->..:.-"'...-=---c...._=-...==..,.,~'--'-'•...___"--===•,--, -·,c-a.r-ad!:.a.:::'.....,.:.:""-=.:::..e-~-:-='"'.:._;~-__,,,.---=-=j;:---,:::,~-;:,_•=~--".•~L~°":4a.;.C:..:""-'-;t:U.a."~...:;.:"=·=-""'----t='~~~· __::·'--f\.:_ ... ·~w'-·.,~_-.:..,.;;::,_~~-~.·~_ ....... :JUA.:21..:.i.01·-~~-lo~~ ....... lt.U;p.,,.~>1"11~~~U~~-.-;.·,·._1.·.:; ..• - ". = ,,
• • •.. .· ,<.
. -
....
-66 . . .. (I. • .· ' ,,..-:&;' • . '. ,, ...
••• •••• '·' ..................... ,. ... ......,_ ... ..£ .. ·'4\i.' ...... -, •..• r'.4i:•,.i • ,"4~•· ............ , •• i ,.- 1•.-,1 ~ • >'•'••••••,• •. • •' ',·~" • I •
---------·---
.,
7. S U G G E S T I O N S F O R F U R T H E R T E S T I N G
A. EFFECT OF DIAPHRAGMS
I The assumptions at the base of this theoretical behavior of a
folded plate structure neglects the effect of the diagrams. A diagram
is considered as a simple support which does not induce any restraint.
M:>st likely this is not true. When the structure is· .loaded, the
diaphragm deforms ind:u~t·ng res train ts .
B. PROPOSED TES1~-
· Place ·strain :ga·ges a,roltt)d: ·the edges of the diaphragm (foil
g_a:g_·'=!-~ would: gl:ve .mos-t acc:.u·rate results because ·of their small si.ze.) ..
·p1ace closely spaced fo-1.1. ga:·g¢:$·· to.w:~rds ·the encl$.· Pit. t.:he
·P.a.ne·ls .•
.. :Load the structure and take readtlJ.gs. Towards the ends the
:cfe:fo.rmat:ion, if the diaphragms have no e'ffect, should be negl-igible. ·
If the readings are not· zeros, place a stiffer diaphragm
(by gluing). Load the structure and take the readings.
(
By repeating this procedure it would 'b\ possible to determine
the effect of the diaphragms on the stresses at the ends of the
structure and eventually to modify the assumptions, regarding the ~
diaphragms in consequence.
i
.I
r-...-. -,~. I • • •• rNr ,, .- I -
.~ ' ..
,,.
. .. •., ...... ............. , .. . .. ,... • Y'I'• ~ ,. " ',. .... •• .• •• • ...-• ..... "I' • • '"' , ••
.( .. ,_ -
C. STRESSES DUE 'ID A DIFFERENTIAL SETTLEMENT OF THE SUPPORTS
., The model could be either simply supported or reposing on
intermediate columns.
The difference in settlement co.ul.d be provided by usi.n_~:.
screws at the supports, which could ·b~ ,c:idJ.usted at different tevels· ..
D. MEASURE OF DEFLECTIONS
It is assumed that a plate deflects only in its· plane:-., ·Fo:r
the, external panels, which are free edges, this is obvious.ly· not: t-ru-'e \
:and it is mos-t li:kely no--t true for the other panels, es'1?¢:·c1.·a1l,y :in
·the c·ase of non;..s:yrtnnettical loading.
A. knowle4·ge of the exact defle.c.t:fot1s, wou·td help in defining
these deformation·s i.n a theoretical w.'a,y· and.: :on .this basis a correction
for stress e_s: could ·be· .proyldecf ..
,• ":
~-
(. }
'
•..• . ' ... ~· ' ...... -· ~ .. ,--, '':", . _,,_ .- .• , ... _ ......... - '"';;,. ·--: c-;..·-·.,·---.·- __ ,... ::i.x---·•···=::c.-_-:r- ---..;:.;.:..4r.::··· ...... =..,,_...:····--v=··-,,-':.-!.:-..,....f~ ~.~~--.:.:;;:-: - .....,.:: -~,~~~~;-,~~~=·.,-.,..~.;:z:t..-.:=.,...~.:t::J;L...-::-=..--:-r~-;:c-~~-~~ - . '
rl • •. -r
..
·11
' -68 . .. , . ,. ... ,,v .. ~-. ..... ., ...... . ...... .... ~ ... , ............ . .........., ...... I
,.\,,,,,,.'•""1'°D ... , .. ~ ....,lllf ,•,•;•
. ...
., .
8. APPENDIX
:-,.,..
••
•I ·, .-•
I'
;:_ . .-- : .. ~ ·-.. - ----··-·n·.
·····-·-,l.... - -- -·-
'
•.
I '·. I :
'
' . I
1
' I
I I ' i
'
I
' I.
~-.
..
. -·- ...:•.:-.,.......... ·"'·' -".~ ... - -,·-·· .. - _, ...... ,. -----···· ·--- ----- ···-· . ,... --~ ... , ..... ---,--~ .. -· -- - ........ ~ -··· -·~ ... , ....... '• .. ·- . ··-··-~ -..----------... -·---- ··--· . ~- ----------·- ---------~
.. ,./ .
. ~'
.',
;\ . I
·C EiiTER LI lJ.E ..IJI.D(f E 3TR~S ES {J"~~r·:L_) t :ODJ~L f3·I l ::J>L:Y :SJTP:P:OlTTED
Loadi.ng· .. Elem:. loadi.ngs. ·c) .4 6 8 lo - 2 3~·9 1 .·4·99 - 2·42
( l) USO US:[ NS
- 4 847 ~ 9 695 - 7 221. SYl-ll·IETRICltL 870 - 856 .421
_. 6 ·376 1_0 ;338 - 7 092 - 7 092 p
:us:1 -· 3 999 7 998 - 5 998 -. 5 998 . (·2)
U:~l ..... 6 267 12 534 -21 877 21 877 ·:trs 615 - 605 2C?8' 2Q8 , . ,;
l'lA. 6 010 - 8 505 12. 4·93 -12 493 ,, . ' .
~ 3611-:1 11 4·22 --15 08.4 3 .68lt-
lJSO .5: ·0°0 - 181 :636 636 . . ') : ..
USI 2: 199 3°9 ·3 299 3 299 . ' ~ . ·- . :,I · ... - -UAO 1.- 774 .. .3 06:9 1 023 1 023 -· -' UAI 4. 3qt-"" ·e3 789 -15 340 15 3ltO ('3) :--· ., )
NS - 264 260 - 128 - 128 liS 338 ~i 161~ 164 -lTA 665 l 382 1 382 ~: -!TA 4· 214 5 ()64 8 760 ... ' 8 760 - ,
6 893: 1 842 .- a 566 4 312 ... , .,
.
10 338 .. 6 376
7 998 -12 534 - 605
8 505
3 361+
- ~ ·1s1 399
~ 069 789 -260
~~f --5 964
448
- ~. 999 6 267
615' - 6 010
- 3 227
5 090 2 199 -
~0 4 774 4 395
- 264 338
-665
4214
- 963
:'t· ..
.•. i
. . .. . -~---~ . . ·: )_:
.. ·,:i
l .. . J ..
,. " -~ '.·· . • • • r
. - .
,i .. .
., :i .J ..
r ·• tf :.: .
. .•· f -~-
• . .. . ' j: •. t :
I : 0\ ' "" ~ 't
. ~ ~... --- . ... ~.
' I
I
i
! I t I
,,
i .
i
. '
; .
• I
!
·,
\.
(i):
. (3)
E:lem. .l·og__c1.i~g.s.
USO USI NS
USO. USI U .. t\I UAI NS !'JS IJA IfA
.; ~-
.. ..-./'
f.
.•
~·-
TJIE.O·R·E.l'l C·AL. V.A:L~UES
J;::on)~.X, ·SI{iJ?.LX ;,:1UP:P:OrtteEr>;
.r ()
... 1 751{-· ]l .. 124.: -.. ·18.1. - 3 6 3 5 7 .. 2_7·:·1. ... 5 '4_2>.59.· .• _ ·2_ ..• 615 -~ •' .. :f3Q5 '8
7= ?90 - 4 774
- l 000 --·~ 998 - 4 4-9$ .~ 4·4qg 5 998 ~ 3 000. ... 4: 700 9 400 -16' 407 :16 407 - 9 400 4 700
·435 - 1~28 211 211 - 428 435 4 250 - 6 014 8 834 ~- '8: 8311~ 6 014 - 4 250
... 3 015 8 956 ... :1·1 '86.0' I\ . .
3. ~-86 2 184 - 2 115.
8 817 - 2 3·86 - 1 649 3 2'?9 - 2
3 580 - 2 302 - 3 206 6 592 -11 /
- 187 184· -239 - 235
~- 470 665 -·l+ 216 6 ·2. 979 -.: .
:5 .€113 ·1 601 - 7·
477 474 - 2 767 505 11
90 -116 ·977 -1.93 - 6
.:493 3
477 474 767 505
90 116 977 193
·551
- 2 386 3 299 2 302
3 817 -- l 649 . - 3 580
- 6 592 184. -
3 295 187 239 470
- 235 - 665
4 216 - 2 979
123 - 573
.! . I·· . 3;
:• ., •· ..
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l • I ,': '
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,..t'..;._::·:·,-~--.. __ 5,!r,"'f!QZ-llftft ... ----~~·-·=·=····"'!'r.l'. -'=""--· --· ·""""""'·-.·-·---·-···-·· .... _ ---------,-------------------------...--...... -
'-
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Elem.. loa:dil7tg.s .. •
$8
cs CA
:.c.s .C.A
·-,,,.,
:tHEOR.Ett·CAL -V.'ALlJES. -.. - .··. ·;-·. ·. . -., ... ·._ '.-. . ... ·:
- 6:. 3··7·6
ll ::63t._
)- 258
- 3: 61~·1
7 3].9 3 7:3.'1
7 :1*0·9
:2-·
.. 1:0. 33'8
~.: ·? 08·9
.3: .2.4.9·
-· ·4 460· ,. ~
:-.,. .2 038:
:4 ·924
,,
4
... 7: 092
l _2·7'2
.... ·.5 7·20
8.0(). .. -_ :: . ,_-
- 7 .. 3
:--1.1+ 3:57
p. 89:3:
,7··4 .. ·a
··1· ·s··4·.:·2·.·. ·.·'' :_._. _-- ~ 9 :56..6
8::2 -.... : .-,?
·O ·1··:.6·.2···.·. : !"'"". i: ..
3: 9..8·2:
.4.56 ·1 ·::t8.0: ....
-:'42 -
. 0 ~.· .. ·2· .... ·.-6.-.:. -:~ '';/ /
-~.
-6:
S:Ylil·fETR:IC· ~~L.· .. · - - . . . . . ~ .. -- .
..
3· ::68:1+
·80:"0 '7:3:
3 364 - 3 227· :· '·
·.·.·4·_ · ,.:.,:5··7 ) · .. :
4 312
82 42
- >+ 460 7 319 2 03·s: - 3 7,31
/
942 ,361
448 - 963
456 - 749 - 1 180 2 162
276 ~50
. ,.,
... . . ,-- -··-~ - ~ .,._
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GAGE_ :fl .TE.ST 1il ~,~T 4iZ TEST 111 TEST 112 AVGE STRAIN
l ..... : 15:: - 30 - 60 - 60 - 41.25 ·2 - . =·100:. 100 90 .90 95.00 3, .... ·90· 110 65 70· 83. 75 ·4. ,"'!" :·10 - 75 -- 50 - 65 -- 50.00 ,5 ... .. 80 80 - ·to - 75 - 76.25 6 :7Cl 80 7·5 85
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ll --- . :2:5 -· 35 - 35 . ·- :25· -~ :10:. 00 ·-· ··--- -
. -
12: .-so 80: '6'5 .• 5-s·: 10~·00 13·, :so 7~: 70 70:: ·73 -- ... 75 14· 45 . s·s. 35· =40 4-3-.7~ - - --· ·-. -..
'15 - 55 -~: ,5.(l ... 55. ... 45 -- '51---.25 - - .
16 -- 6'0: -- ·75: ... 45 - 5.5 - S-8 7-5 - ·-·· - .
17 ~- 55 - .50 - 40 - ·3-5 -· 45 .oo ·.--.•.
18 .. 9(l" 80 70 70 77.50 19 :ao, 90 70 .80 80.00 20 30· 30 50 35"
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Note t Strains are in micro "/" for a load P = 20 .lbs
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RIDGE 114 LINE LOADING
MODEL SIMPLY SUPPORTED
'.OU.TSIDE GAGES 1 INSIDE GAGES
.. -··
GAGE ff TEST 111
1. ,;;. :2.
.3
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5:
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10, -- - ~.
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5
20
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TEST 112
- 5
85:
85
- 160
- 175
10
·30
35::
:25:
- 45:
25
90 70
·at.· 110: ·~ 100:
2·0.:
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35
20
- '35:
,•
TEST 111
- 45
70
35
- 155,
... :145
-. s: 10
·25=
i.s.: _, 45:
~ ·40
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15
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~ 20
TEST 112
- 35 :as: 50.
... 15()
- 145'
.. ,5
25
25
20
-. 25·
- -~·5 ·60:
40
- 9:5
~ llS: :20
3.0:
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., ~' -, .. . . .. - . . ··1
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AVGE STRAIN
- 17.50
81.25
62.50
- 152.50
- 155.00
3. 75
23. 75
30.00
21.a5
- 40.00
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4.8 • 75:
- 103 75 •
- 110 00 •
12 • 50
25 .oo 25 • Oll
18 • 75
·- :23 75. •
I
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Note t Strains are in micro "/ 11 fo_r a loa P = 20 lbs
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CENTER LINE PLATE 113 LINE LOADING
MODEL SIMPLY SUPPORTED
OUTSIDE GAGES INSIDE GAGES
TEST 111 TES:r lk2 -~ TEST 1fol
- 5
- .60·
... ao - ·tao -.. 17'5:
·.- JP ·- 20,
- 55
-~- 65 -· .BQ
B.5_
30:
5
- 7S - 75
20·
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- 140
- 140. .,
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- :au 4~ 4.0
5:
5.:
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5 . ·'
50
75
159
145
40
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40
55
140
135
20
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- :65
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35:
30 .5_
CJ:
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l\VGE STRAIN
17.50
- 45 .00
- 63 75 •
- 152. 50
- 148. 75
- 25 .oo ·- 13. 75. '.~ so:·t>.O: . t .. ~ 55 00 ·. . .. - 68 ._·75.
lJ.-~':25
-o .• oo ,- 72.50
- 71.25
27.50
33.75 '
s.·oo· 125 ... ..
·.-. ll .·25 ...... ·
Note : Strains are in micro '' / 11 for a loa·d.-. p·· ,=· 20 lb·s··.
·'·
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..
CENTER LINE LOADING
MODEL ON MIDDLE COLUMNS
,..
OUTSIDE GAGES INSIDE GAGES
TEST f/1 TEST /12
l -. 9:5: .. 75· - 40 - 20
.2 · 9.5: :_$.0: ·9.0: ao· 3 its.: :.95: 70 :5:0· 4 ..... '.6$: .. .-65 -· :s.s .... 6.'5 5 - 85 -· ·a~, ~ -8·5. lllllit :6Q.'
:6. - 6'5· - 60: -· 6'5 .-.; ,o,
7 -: .. 50 -: 55, ·- :60 -: 40· .
:S: 75 85 ·6:0 " ·'
:65
9: 70 90 ·70 -ao.: 10 ·- l.-.60 - 165 l.0.0 85
11 ._ ... 45 - 10· ' .: _: ~ ·~:5:. - :)'().
12 :ao.:: 80:: :.65 55
13 :so :6'5. :.7Q: ·:65: ......
J:4 - 45 -·, '
6'0 - :3s·: -·· .:3.5:
.15. - 70 - :·50 - 50 .--·· ., s·o 16 - 60 -~ 55· - .5:0 -~:. 45.
17' ' .1·:. - 5.d - ·-40 - 3'0:: - :~·o 1.8- :65 6:5 5$: '60
19 1.0: 7.:5: so: :p'.5:
.20 .... 10 -· s·. .. .. so .. 5.0' ~': -··
. ..
AVGE STRAIN
- 47 50 •
·86·, .• ·~2~
.82:. 50:
-~ .·62' .• ,o: - 78 .. 75
·--~ 6s<.·oo . ' '
=- 51. 25
71 ... z~ 7:7.:5.0
~- lQ, •. 00
- 3 7'. 5()
65 .oo 70 .00
43 75 - •
- 55 .00
- 52 .• 50 " :3_7 50 - . '
'61 • 25
65 .oo 28 • 75
Note : Strains are in micro 11/ 11 fo.:r :a load P = 20 lb:$
.• · ( ; ·. ~
.. ~. i.:
-80-······- . - ... ,,. ·-· . ~. . - .
' ..
. ..
'(!),
I .t .·• \• !
j i) r ·~ . ' '
.;;. ·, /. . • •• _!_, • . •·•·••• . •.• .... . .... ' ,. .. .,, .... ,
-.~ ........ .
~:
'_:,
I j I
I t
,.
l
~ t ~
f ~ • 1 ~
l f ~ ; • f ~ . • I ~ -~.
ij " C ~ • t .! I
z i
I I I
~i ~l . I.
1-· .
2: ..
:3.
4 ;5.
6
7
:s 9' -
l_().'
11
l.Z
l:l
14:
15·
i:P 17·
18·
19
20
-~ ........ - ·•
' ... ~ . ,, , "- .
- RIDGE /JJ. LINE LOADING
MODEL ON MIDDLE COLUMNS
OUTSIDE GAGES · INSIDE GAGES
TEST 111 TEST /12 TEST /fol
- 105 - ioo ·a:s:· 1:25: ..
65 65 . .. - -- .· 70 ·75:
'7():· :55 4;5:. .40:
·- ·1.40 - 140. ~ .. 125 __ , ·13·()
;~. ··145 - t40 - 1.20 - J.:30: :1.5·
..
5 15 .. 20· ..
:•
.30 45 30 :.30
.20 .. 40· 25 ... ":25
\.o· 3.o.: 25 -~:o·
-· : :s:o: ·- 60 35 5 , 3Jl 4Q: -· 40 -· 35
115_: 75 .. 55 5:5
65 50 :50 .,,5
- 95 - 100 - 9:5 ·- ·9.5
- 95 - 95 - 95 :-· 1.o.:5 - ,.._
10 25 :20: 15
25 :35 25:· 25:
15 :35 lp' 25:
:l.5 ··zs 25:: ::3.s
- :25 -· ·20 ~.: 10 ·- 10··
•' . ,
..
AVGE STRAIN
·--~ 1 • 25
:-6:3 • 75:
5:·2·.5,0
·- 143 75 •
;·-- 1·43 7.:5 •
13: .. 75
43. 75
22 .• '50 .. 2-J' .. '75 ,.. .. .. .
- -25 00 •
·-· 1:. ·25:
75 • 00
55 • 00
~- 96 • 25
- 9 7 50 •
1''7 • 5-Cl
z1·. s:o·:
22 • 50.
·25 •:00
- 16 25 ••
Note : Strains are in micro "/" f:ot a lb·ad p·· = 20 lbs
-81
f
.. .
·--•~••·-c••---~-~~-----------------------~------,---,--~e,-.,,--~-,-----,.·
-82 "_ ...... ,--~·:--· -,~·-;::...I __ ·"'~-.:~---,.-:~:~;~:.:··-~~~-.. --~-~--.. ~ "':.:.r_ .. ~::: .. ·:-~-.---,..~:-::, ··rr:w:.:_-:"~.··.r~--~~~: .. ·:~~--·::· ..... --- --~.:~· ·-·-··:·-·· ~-:··"-_- .. - ... _.._ ...... _.,. ~--r .. -~ •,•o<l•", ... :.a~ --~--~ ... ; 't•" • • -· ........... ~ ... - ~-,, ... _..\,,,- --~ -~·-- .. ~- .. •• '·.· .. ·• "', , .. ~d··.-~~-- .. ~·~1_:/".:~.-l.-~ .. ~~·-,_· ... ·1".-~.'~, .... ~l;,: ..... ;"_"·-~· ··'· .. u:,.:-:..::~1--. '""'-::1··::-~·-· '°' ~~::'-~~t:_-~-~-.!~
. ~ ,.
., •·
.. ,
:CENTER LINE PLATE 11.3/ t~I:NE: 'toAD:tNG
MODEL ON MIDDLE COLUMNS ·
OUTSIDE GAGES INSIDE GA~ES
GAGE 1J TEST 1/1. TEST 112- TEST /fol AVGE STRAIN
-·
1 .20: a:s.: 75 6s: 61 • 25 ·2 5:: . 10. ·2.0 :10 11 •. -2.5. .- ,•
10' -3 -~ lS 10 10 ~: ·6.:25 1-·
: __
·-.
,i···· ..•
·95 10·8 75 ..
4 125 100.· :_~ l;l5 - - ·-: . -.· ·- • 5 - 120 - ilO - 105 '!19 :90_ - 106- .·2-5 . . .... 6 ) 15 -35 10 35. :-23. 75
'
:7 40 .50 15 .·45 '37. 50 a· - 15 - 25 ·- 20. ;~:· 5: ~- _16~ ·25
-:.,,.-;·
,/.'~
I ~l - 30 -· 20 ~ rs. 0.: -· i.6:. :25 I to 125· 120 .'30 'llS: .2$. .00 ; .. - -: ..,. ;-_~ .
I ~!
·1.1 ~- . so· '7::5.. ·15 15 46 • ·2-s 7 .. ..· .1' ;§
t 15· t 2· ·10 0: 10 8 :75: '.? • ~ ~
·10. ~,
13 5 15 0 :s·: •. po ·.• , : .. •· .·.~ ~:. -· .... { -i ~
{ :as ·7-,5 :75 a:2.~ "
!1 14· -~- - - - 95 -· 5.0 ,,
" .• , ts ;8:5 80 _:ao: 7'.0 78 7.5. ,Q.
}i ---· -- - ~-· - . . .. i s l-6 25 :2:5 :20: 20· 2-2 50 .. : ~~'
·, . •. . ~
:' ~:
.25: f ·_17 30 45: 1s: ·28 ..• 75 i, t ~ ~: ,. .ta.: 20 10 10 15 8 75· ,. ;f, - - '.~ -' . . ,, :i "',1
19 :10 ·.20· ~~
10 10. 12. 50 .~' . : - ~- ~- - ·-·r f ;rto· 15 25 .-10' 15 00 :1 10· -~ - ·- .. ··'9 - ....
i •
'
Note : Strains are 'in micro "lu for a load P = 20 lbs · ... •,
·-·
'· ~ ,.
CENTER LINE LOADING .,
. -MODEL SIMPLY SUPPORTED ( WITH TIES )
OUTSIDE GAGES INSIDE GAGES
TEST 111 TEST /fol AVGE STRAIN .-~.. .,,t
l ,45· :40, .. 45: 40 43. 75 .. . ,
:2 ·- 10·. :4:5. 15 10 12.50 3· ·o: - :10 :5 - 5 - 2.50 ..
4: ~- 40 5,· - 15 - lO - 15 .oo 5 - 30 ~ 20' - 30 ·- :3_Q - 27 50 .. • ·9· - 35 - 10' - 30 - 15: ~: 22. 50 7 - 30 - '20' ·- :3Q ·- 20· - 25 .00 8. - 5 l'O ·o: 10: .3:. 75 9· 0 15 0 ' .10. ~. 25
10: i:3.,5" ,:S:'O .45 .55 46. 25
1.l. 15 :20 ::2-~.: ·4.0 25 00 .. . .. L2 .25:
•. ! J.o· .. ·'2.0: ·15: 2·2 • 5.0
.l~· :1.0: 10 5 10 a·. 75 /,·,
1:4· ·- 15: - .25· - 10. -. J5:·' ·.-.• 16 25 • .15 .. 20 - 15 :~ 10· - 30 - 18 75 • 16 -· 10: - ::20: - 5 ·-· :20 - 13 75 • 17: -: ..
.2'5 ·- :20 - :5 ·-··· 20.' - 17 .so 18 ·:20. '15 :1.5 20· :1·7 ..50: 19 15 2:0. 20 .20 18 .• 75 . -. ,• .·
20 :25= 15: 3:5. ·2·0 ·:t3 75 . l .. Note t Strafn~ .. ~-r:., i;IJ.:, i;ni¢.-ro· :J.f/\.,· .for ,a loa~·l'. ·P = ::2:o ·tbs
:\ .
..
.I
.. ······•·· ". - ·----·- -·-. . . - ..... -· .. -. . .. , ... - ..... . . . . - ··-· ------ . -- ~·- -·-
.... ,..
I
i I I
l ! j l ! l
~ I . • •" ' ... , •" '"• ' • ' • • • , •.•• · •••
. '- . .
..&·•-·· .•• '-"--. ~ - - ·--· - .. ·-,,_ ..... \ '.. ' - ., .. '. . . -~ ........... __ . ' - .. ·1, . ,,,
,,
RIDGE f}lt LINE LOADING
·,;. MODEL SIMPLY SUPPORTED ( WITH TIES) . .,_.
GAGE 1/
1
2
,J:
4
.5.:
·6:
··t ·a·
9
:10
1·1
12
13
14
15
:1 .. 6·
17
18
19
20
. =:OUTSIDE .. GAGES
TEST 4fol
--·
~-
-·
70
·2$
.:i;o:
'.9.:5:
9·.=.5:
55.
60
'1.5'
25.
25
45
45
15
- 80
·- 8:5
4(l
5:0
- 10:
-· -20::
ct
--
-· -·-
7Q
:4'5:·
l()
105
105
55
65-
20·
25 ;2:5
·60
40·
3-5.
~- ao. . .
80: -- ~.
•45
.5-5·
~- IQ:
·-· 10:
·5
N t • St i i i "/ o e. rans are nm cro -11
·~--
'• .. , ....
INSIDE .GAGES
TEST /fol
--
--··
45:
·25:
5
100
95
40= .• . 45
15
l5 ..
'35:
:20=
25 2·0:
-· 80:
- 10:5 .ao .. 40.
- 15 , __ :10
,:25
--
....
-·
50:
25·
20 :105
9::.5
s.·s 55
20
20:
5~
20:
.35:
30·
·-· .80 .
-~- ·75_
45
40
- 10
... :zo 40
AVGE STRAIN
--
-~:
-
58 • 75
30 .oo 11 • 25
101. 25
97 .50
51 • 25:
S-·6: .. ·25 . ...........
17:.50
23-:. 75:
35 =.Otl .... · ...
'36 25 ... :3J~-. 25
:2s ·oo .· .... -~ ·so .:oo ~- 86. 25
40.00
46.25
- ll.45
- 15:-.• 00
17.50
..
for a load P = 20 lbs
·'\...,,
• • I
.-84 . I
' ............. ----1
:~.
J
-85 . .
.,.. ... ... ... - . '". ___ .,.., .. ,. . ................ -··_, ___ _..~, ....... ,.,.,.,:._.....,-- .. ,....,.,- ....... _,:~~I! • ___ ,_ .... , .... ~ .. __..t....;----• -· ................... '" .• - .• - 'I.,.. .- " I---•• ~ .,-\ ...... ··- ... • • .. • .. • _,. ... ... ' .... - .. -- • • • ~,-,.. • ..,· - --r,.:.:...:.~~.,-.. .... ,"\.,.~-u.L.-'-~..: ·- . ·-~~~ .•~-~,,_:. -~~---..:.~~"--'' -~ :. :, ___ ' -I
• • • ' • ' .o f ~ ~ o' ' .. ·,.''· ....... , ' 'l ~ ,~ ~ ' :~, ,. • ' 'NI I'' QI.-....-.• , ,, •• f'... ' 'I/IN ~r- :. • •. •,;•:"'·1 '.~•· " ' ,-~:,,;,. ..,~.. - :~.
·1 •.
.. i• ,:. ••.
·:·
' i
GAGE 4#
l
:2·.
3::
4.
:5
6
7
.a:
.'9
lQ
. ll.
·.i..z
13
14 15 . ·',
16
17 18
19
20
\.. ·, . ~., ..... .. I
CENTER LINE PLATE fi3 LINE LOADING
MODEL SIMPLY SUPPORTED ( WITH TIES)
OUTSI.DE. GAGES INSIDE GAGES ·,,
TEST }fa.:2 TEST 111 .TES.T .. /p2: , . AVGE STRAIN
,115
:0
0.
~. '85:
-. 5:5..
·:~:
-i!· -
-
"'!".
-·-
35
75·
.25
20. 2·0:
.ss . ·xs 35 ·a.o a.o·
:5 ..
20:
·15 . .. •
10
:60:
1.2·5
2·5
... ·5
- 70
- 75
--
75
60 :~o: 10:
:2.0:
10,5
. 25:
5
55
60
so 50
20
0
10
90·:
~o:: :.Q
- .55.
- ·45
.55
65 15
:"20·
.15
40
- 25
- 15
- 85
- :6'0 .Q:
25:
- 15:
.... 5:
- lQ:
-·-·
too·. ·2.0
.S·
·.- 70
~ 65 ..
:45
so· .20
-s ·30
Si.5:
:25
0:
- .45'
- 50;
45·
.. 40·
20.:
5·
5.
..
107 .50
16·25 ...•. ·- :2· .... ~:o· - 1.o·::.00:
.... 6().00·
52 .... 50 .... :.62.5.0
' .•.. :22 •. $0
·13 .. 75 . . . .. .
6_'3 .=:·75' . ... , .
.2 .•. so - 3. 75
- 66. 25
- 62.50
25~0(l
3·3:.,]5::
z ... so: =- 2.5.Q.:
11. ·2s·. ....•. ' . .
Note : Strains are in micro "/ 11 fot·· ij: lg.ad. P = 20 lbs
••
i'
• ' r " •• ·•"':·
-- . - ... . ;•· - - • - :• • .- ... < !.... .
..... ··- - : ·• .•. :. ' - - . • ~ .... '!'. \" , :. • . - .• . . - • • \ _.,. ,, .. •, .,a _ ,-. , ~. ~~-"'--- ,. ;._,___.l.h .. 4-·-~-- -•· ,;_,··• ·- •• •· •·• • -~ ~ ,,,.,,_ ·, '·
•• -···-~-· J~ • '.~ •• --~ -~ •
• ~ - •.• '1 •' -
:·•··
----·~ • -. .... ~ ....... ·-- ---- . •t
' ) i
i
,, ....... '•
..... ~---, ' ··- ... '
• • .•.•. ..-.. .... ,.,. .• ~tu
•.. , .,, J•
GAGE 11
1.
2
3.
-.4:
.5.'
~
7
:8
·9:
:10: .': ....
.11
1._2_.
13--14: ·1:5:
16
17'
ta·. l9.
.~:.,;,,,
. ---'-.. . . --·.. -·-- ··--·· .
.,,.-,. ··.-· ·- .. ,, ' • ••-•· ••• • I ! .........
CENTER LINE LOADING
MODEL ON MIDDLE COLUMNS ( WITH TIES ):
OUTSIDE GAGES
TEST /fol
10
2:5'
5
- 15
- 20
- 15
- 20
5·
15:
2·~:.
·s· .. ..
.10:
0::
- 15
- :20.:
-· 10:
- 20
xs :to.
5:
TEST /12
10 / I ••
0
10
- 30
- 20
- 15
- 15. is:
:15·
.95.
10
20
10
- 20
- 15
- 15
• 30
20
20
5
INSIDE GAGES
TEST /fol
~. '30
5
:o - . 10
~- 15
..... · 20·
~- 20·
:·20
:o - 20
...
·.2()
1.5
10.
10
- 20
- 15
0
:2.0:
15 ·15
TEST /fo2
-. 30'
:Q:
o,
·- ·20 -~. ·15
.. .15
.. 10 10:
l.0
- ''.4():
3:0.
.5:
o,
- 20
- 15
.. 15
- 20
lO.
2·S:
-86 --• _ .,.,.. • .,. ___ __.._. ................. _ .. -,,1.-<.,· .. -L...~1>,, --- , ___ .., .. _ ·- ,, •. ,,_ ....... , ,: .. c .. .• c· ,1' • • ~~
• . . "1 '. ' ·, '' . . . ~ ..... . _ ... _.,.,.. .. ·, .... . - ····-··--· . .., ... -,.. . ....... ,..,- .. ,,,.._
.•.•.•• 1
AV~ STRAIN
- 10.00
7.50
1.25
• 18.75
- 17.50
- 16.25
-. 16.25
12.50
10.00
15.00
16.25
12.50
5.00
- 16.25
- 17.50
- 13.75
11.50
!6.25;
16.25
12.50
.... ,,. .......... ~,
=:
I ' ' ' i
:I
j '
., ' }
., J ,i ,,
{i
-~
" l ~ <~ ~~ il
J J !J 'I
ij 1 j i
Note ~ Strains are in micro "/" for a l<>ad. P = 20 lbs
. - .. .. ,, .. . " ,, ; .•I. ·' •· .. '
f
I I
l _ __ _ __ . __ -_ _ __ _ __ .h ·-- __ _ --. _ . __ r
I j ••
,_. ---•-• .... ,_.__.,__.,,_...,,--,-•·:::•·~•'• ... -·,1-r,11 ,~-"'-... , 1 ... ,,._,_, ... _,,_,.<1,-.,.L__.,,~,_,.-~~--"• .. -.· •·1 ' ' ... ~ , .. ., .... ~ . ' ...... - . ... ...... " .. -. ... " . --- . - -·- -··- -- ·- ...... -------. ··--·-·- . -- --------.. ._ .. ,..
·,
~ .•.. • a(,,1 ··- • · ···• ·"'•• ~ · ..• ,, ...... ~~--=-:.~:, .. ··:;.,;;;.,,;--:-.·-:= ··-::·· ..... ~ • · .... -.-:-.:,.',._. ·· · ·-·· ...
l
""~''"· ,. , •••o- ~ ,• ••-•--M•#' ,··"'/'-: _,.. __ ,.,.. ........ ,
' ' . ' ,j
. . ..• .. ,, ....
l
i t 1
" l
~ :<
;I
1
~ !
. --I . . . . «- •. - -- --- .. ,
J ,,, . I,'
I .
., •. ..:. .... : •• ·.o.:: ~· •••.••. ', ........... ~ ........ .
. . .. •. . ...
1
'2':=
3·
4
.s. 6
7
a· •'
9
10.
1·1
12
1·3
14 ·1.5:
1-6 '
17
18
19
2~
. ... . ,-. :,, . ·~·~ ........ _...* ... ., ............. , .•. 4'1 ............. _ ............ .
. i .. ::_·.
ft
RIDGE 1F~ LINE LOADING
MODEL ON MIDDLE COLUMNS ( WITH TIES )
OUTSIDE GAGES
TEST #1
100
10
20
- :120
-- :110 I-, .. • •
~-·
---·
-
5:5
:'5''.$
._·5_'
10:
:45
5:<l ::ZO.
70:
75:,
65
55:
5
20
5
... ~ ~.. '.ic-i~··· ,:·.:.,. ·;:
TEST _.1!2
95.
:.5:5
:20
-· :75
- _90
7.0
:as
- 10
- .30
,35
.. 4_.5:
.:45:
20·
- 80:
- 75.
:60:
65:
:- ts.·
-· 10
- 10
-
INSIDE GAGES
TEST 1ft 1 ·'
0
·:25
:5
- :9':5
-· 10:s:
5:5-
50·
10
- 10
25
is:
,35:
15 . ' '
-·· 6S
-- 80 ..
-so· ·35.-
:2:-s
·- ,20
;25
TEST 1fo2
~s -··· <-:.
25
45. :35
- ·100 ...... : .8·-s·
:(l.5
15-
-· 15 -~- 15
J:5.
l5, ,,
30
3:5i
-· 85
- ·ao.:
:5.-()_:
:60:
40
... to 40
... , ..... - .:- ............ -~····;· ....... -....... -. .. .
' .. ,. .. •. ·:·;• ~. "•': ·-· . . "
AVGE STRAIN
55.00
33.75
20.00
- 97 .so - 97.50
61.25
66.25
- s.oo - 20.00
26.25
30.00
40.00
22.50
- 75.00
- 77.50
56.25
51.25
17.50
- 15.00
17.50
.. ·
Note :· Strains are in micro II/" .for a load P =i.~. 20 lbs
-~... -:-. ' . . .. '
• .. ~ ·.,; •• -·- ••• ,;.. ' ..... # •
...... . . - r .. ., _,.,_ -• ·• ... _ • .., . • • - - . --· .,.... .• - ·-
·, ., ..
--· - 'r _.. ._.,., -· ""' -~., • , '•·• - '-••. ., . ·-· .... ~ . --··.' ,_ - ' .
'
) '
~,}y L '',
• 7. > .::._._.. • ::. ••• -·- . ~:-- --:··:-···: ·-. _,. ___ ~ -·- -...
. . .. '
'. --.~ ......... , --.· .. ~ ... -•· .... -- .. ··--···
i !
1
l l ...... -·· '.. ' . ' ~., ................... ,_ , .. -...-••4-.-.•. ·-·· .................... _...,_., .... _,,_ .. ___ ...... .
I . ,
I i j I
:I I I \
... :,·;
,.
'
._.._..___..,.,.. .• , .• _;:, ~,,.;_ • , ,· .. ·.,..'.-, •.• :.·.:.:, ''.., ,I · ,._. .'--• •- . ..,. __ ••• _'.:_,,,_, .• : ·-··~--·:' •. .,;,_',. , .. ,;_'._;: --~·._' • · - s.'.'. .. ·.:. •. " . ,, .•... ·,,,: .;:,···· ; .. ·._ ' .. ....... _ ... __ ~ ...
·····•· -... ,., •... : , .. · .or,'i1•~'" .,.-:.,..,;.~:.~····
... .......... •, 1• ' •. •. •. ,,.., .. ,· •' ,. ·•r • ,• " • ~ .. .,-
. ··---~ .. ··- ~--.... ' . .•. , • ... ...... . .......... u .... .
' .
' .
CENTER LINE PLATE 1F3 LINE LOADING
MODEL ON MIDDLE COLUMNS ( WITH TIES )
OUTSIDE GAGES INSIDE GAGES
:/}2: . GAGE II TEST /fol TES'!' . 1;2 TEST /fol TEST A.VGE . STRAIH
1 so: :So .85. 110 88. 75 2 ·10'. .010: -15. 20 13. 75 ·3: lO - -lO: :o 0 o.oo 4 _, ·so - :90· :- :65 - 70 -- ·73. 75 5 ~ ...
!ie .. 70: 85 70.: 65 72.50 -~ - - -·6· 55: 40 -~,5 40 47.50 '· ·.
7 70.: ·s.o 45 50 53.75 8 ·20· °() :.10 5 B.75 ·9· :s ·- =2-5 ' to: .. 5 - 3. 75
.... . · ..
:10.· ·--· 85 - :1"5 a·s: 90 3.75 '\ -'~
:so: . . .. ll ·BO.: '30 15 :41.25 "~ ..
12· :.30 :ts .3_() 0 23.75 .. 1.3·. 0 - 5, 0: - 5: - 2.50
14· - 55 - 6.0 ~ :45 - 55: - 53. 75 :15 .. 80 ·- 65 ... .S.5· .. 55 - 63. 75 16 35 40 :3:~ 30 35.00 17 35 35 .. •.
;3:S. 40 36.25 18 0 0 5 0 1.25 19 ~ :1.tl 91!"" .io:· 0 .o. ;"9_: s.oo
·10: 5; S·: 1.25 20 5 .....
··-· --
Note t Strains are _in mi_c:ro '''/i", ;
·fo:r a. load P = 20- lbs . . .: ' . . . . . . .
t·
---· . .-..
/
··~ . .
. ----···-~- ------ .,.. __ , .. _..... - - .. ,., '. - . . ' .. . . . ., . . . ,._ - ' ·- •· ~ ... --· ,_ . .,._.. ··~ -~- --·- .. ,,_ -· .. •·' -- ..
II
I
. I !
' ' I . . -· -- ..... _ .. -~\. .... -- ..... -~. -. - . . I l I
.. .,~ .... ~ • < "V, . o, -:.·-•':.•:...,-. , . ...- .. -,. ... -... .. .. .1:.. ••I · • • ~• ·: • · ... ~,a..,., .. "' •
'•' ... ,
-1'"
CENTER
POINTS
QUARTER
POINTS
•
I •
• :.,,,. ~········ ,,.,,, •.. ·.1 .... ·--. • • - .....
CENTER LINE LOADING
MODEL SIMPLY SUPPORTED
RIDGE 1fo A.YGE STRAIN
0
2:
4
6::
8.
'10·
:o
4:
6
:s
·10 •',
- 57.28
106.25
-- 72.16
··- ·7s .• 13
115. 62
·- 57 .34
- 42.50
87.66
~ 58 .J:8 .....
- ·60.06
93.67
- 40. 78
-..
·.· .....
.• - ' . , . ""-'llo;,-
-
-"' -
-
EXPER. STRESS
28.64
53;13
36.08
37 .57
57.81
28.62
- 21.25
43.83
-·- 29.19
- 30.03
46.84
- 20.39
-
--
-
THEOR. STRESS
.35.42
57. 4_3
39.40
39.40
57.43
35.42
- 2.6 .. 52:
43.28'
- 29.64
- 29.64
43.28
- .26. 52
i--. -89 '
. . .• .. .. .., . .......... ' . ······-., ,· ...... ; ·- ' ~ ·····>.~.1••1·, ... \,\--# ....... _ ........... .....
\
Note : .Str·a,ins .. are in,1' / 0 .,:.s·tresses in psi , for a load P = 20 lbs
- .... .... . ·- -- .- - .. .. ·- ..,.. - -·- ,.
' .. ... _.. '~··· .... .. _._,.. .._., .... ·- ,,... ... ,, '°'-I' , .. •• • ,,, .... - ...... . .............. "" .. '" . .... . .. . ..
.l -·- ·•• •-•• -· ••• n--•• -----------·--" ,., . .,._ ·----•• o- ·• - ~--••' ••,--.,---~-, •• -a..•, .. -., .... -•-~-• -, •• L, • • " :----·--~-.... - ·•••.-----------• , .... •--~·-•--"• •• , •. ,_,_,_.,. __ . ' ·•• ,, '
(
..,,~ '· . . ' ·-•-------- ..__. ___ , • ·-•· (• , , , 1 ,.,_ • ,rr • •· .~ ., ........ .._ ... . ,. ........... .. . . ··-··· ., .
. .. . ':...":,.~ .. , ,.
CENTER
POINTS
i i i I ' I ;
·Qtr.1\RTER
. POINT·$;
RIDGE #'J., LINE LOADING
MODEL SIMPLY SUPPORTED
RIDGE 11 AVGE STRAIN
......
.O' - 29.84
-2: 91.48 .,_
4 - 178. 87
6 .. 24.06
a 26.55
ro - 41.06
o: --~- 16.5,6
:2: 64. 64
4 -- 124.06
.. 6· 26.81
.8 23.28
10 .. _ :2& .• 5,6·.
-
-
~·-·
-
-
-
EXPER. STRESS
14.92
45.74
88.4~
12'~-03
13·.2'.8:
.. . .. ..
20 · .:Q::J: . . ....
8.28
32.32
62.03
13.41
11. 64
..13 • ."ZB-
-
·-
--
-
-.
.'!"9
. ... '
THEOR. STRESS
20.23-.
63.46
83.80
20.47
18 69· . .
.l7 •. 9.3
17 .75 . -. .. . . .
49 • 76
65 • 89
18.26
·12 .• :13
:11 .. _:75 . . ..
Note t Strains are t.n. micro "/ 11 , Stresses in pe_i' ,: for a load ~ = 20 lbs
··.-.·.:
'.., . -.:. ,. ;:; - '': ••. ... a
\
-90 1
•·
.· ....
.. . ...... , ... ,. ._, ____ .,, .. ······ .... • ---~:.··. •• ; ·- I 4• - •• ___ _, -~-- ···-- • '•h' • 0 O O ' ••• ---·--- -· •, •• •·- • ••• .,,_, __ ••.•• \ O •· ...... ;" > • ..... • • ~- ............. -........ •• __ -~--... ,- - .. ' O ~~ .- ..... • ..... - ---····-.. •-..,Jo.,· ... _, ... O •
r-· .
'- -
- - .. -------·- - - -- - .. ----- - ······-·- ---• • - _,,uo ,,.-,--~.-. ·--- ,04 . ..-~,-~--, ••• oO -••,•- ·•- '• ··•,-••, ·.•·•·•---~ ..... -,:·-~~M-- •-•• • -· .- ... ~··,_:~-·.: ,..~ .. ,,.~.--~,:.~' .. :.;;,~;.;.:_;:;.,:;.:;.~.::.;;:.:•,~'• ,. ,•,: 0•,,,.~,\-' -•,,.., •oo ••• .. w. 1., ............. ~ ......... _. '. --····· . . ,,., -- .. _ --
- •• 1- ••• ·----··---.A•.·'• .... . .. , .. , ....... •· _,,. ... ~ ....... ,.. .
,·
)
,,,... ..........
CENTER
POINTS
Q.UARTER'.
POI.NTS··
--.,-, •.. • .H
....... _ -----· -...... _,__ ..• ~- - .. .
. CENTER LINE PLATE. 1/3 LINE LOADING
MODEL SIMPLY SUPPORTED
0.
2
4:
•·a· .. · .. ·
10
0:
2:
'.4•
6
8:·
10·
AVGE·STRAIN
·r~· .
8:9: 83 ......•.
3.98
- 110.05 -,;
·46.85
J •. os
~: 12 ... ·.66 .· ........ .
68.91
6 .95
- a·2.sa
38.59
o. 71
... 12.50
EXPER. STRESS
44.92
1.99
·- 55.02
23.43
1.53
- 6.:33
3.4 •.. 4'6
:3·.48
- 41.29
19.29
O. 36:
~! 6· :25 ......
.c
.L_~.,....___,, .. ..
THEOR. STRESS
38.29
10.23
- .5J .•. :L4
.23·.'9.~·.
·,2.;48.
- :5., 35
27.85'
8.8'9
- 41.62
}~. 72.
0,::6~
-· J-.•. ;1.s
Note! Strains are in micro "/11 ,Stresses in psi, for a load P = 20 lbs •
.,
~:,.-., .. ;.,
.· -· •.- -· .... -, '.~. :: : . .. . ~ :- • .,. ,! •.• - .• ''. -
................. ·•·
, .,.,.., -• .... -, , ., , • •~•••a .. ; ' • • • - •
-91 .. ··-·· ... • ............... "
······•···· .
i
:
I . . ~ r---.. ·-· -- ···--···~-·-----······ --, .. ·--~ .... ---·" .,., - .· .. , ... -.. -- ... . ........ -·- ---- ... ·- ........ _ ·--· .... --- ·-· ··----· .. -9·2 .,JI~ --·- .,... '·- - •, _.,..,_ ___ ---· -- .- -·
"•,V1.,,. ··•111.,;,,,,.;.1_.(; ........ •l'<kM . ._... .. , •. , .... , ....... -. •• L .... _ , ............. ,...~-~ ......... .-.,.-~ ...... ·-.. ---~- - •• ~ . . ... ~-' .._..... .. ....~.,~·--·--'·-... -,•-·•' ··- ........ ... . ... . .. ·,.;;;.; . '....... ..... .,.,-, ~ ,, . .. ...
. ;,:
·····~~ .......... :·::··::-·~ ··.,,.;··.,n
·'
..
...
.,...
CENTER
POINTS
QUARTER
'POINTS
.......
CENTER LINE LOADING
MODEL ON MIDDLE COLUMNS
Q'
2·
4 ..
.. 6
8
·to
0:
:2: . .
10.
r
i,o·:
AVGE STRAIN
- ·6'lf,. 2:2. .. -~'
101.80
- s·o .s"L. • •
.. 6'6:.-.:66 - . . ... ·a7·.-so . . . ..
.. ~ 20 .• 94
- ·4.6.J3.7
10.·34
...... 5:4 .. 64 . . · .
. - 61-79 . .. .
"
'. ~ '·
:~l
-
--
-·
EXPER. STRESS
32. 11
50.90
40.27
33 • 33
·4~. 75:
10.47
- 2J.·.4.4
35.17·
3:3. 9.1
-· -18'.16 .,,
-~
-·-
,: , .;.,_!',
THEOR. STRESS
29 • 18
18.05
31 .• 7-7
"31 . .. ~ 77:
. .
:1.a .• o:.s:
29: .. .18 .. . ~. .
5 :·2-2 .. , .. , ..... 43 .• i68
26.39
26.39
ZJ.68
5.22 f'
Note t Strains are in micro "/", Stresses in ps.i ,. for a load P = 20 lbs •
. ...
. ,._,.:,..._~t·· __ ·;,.-....:.:..:...:,:..,,·-......... :.;.· _.,-.:·--. ..•. '· ,.. - . , ........ _,,._ ··--.--·~·-. -.- ___ >·.··--.- ··-.·.,. . -··--··-·--:··~-~~--·-:·~-
. . . ,. '. -·· . .---· .-··-'· .. ·- - ... , ..... ,_ - ·- .. . ·-·~-- -----···· ... -. .:
er:
.. . ...
• ,-,,, ,;., ,,.1
......... --.·-·_ ·· ___ ··. -·-·-------··· ...
•• ••• • •-••••• •• •"- 0 • .- •• - --...... --- ••- ~ ·-- .. ,.., __ • ·• .4 ,. ·-•·--- ,••fl:,. II '411'-· ,, .,1, __ .,. ....... ,-~ .... ··--~""'4• """' -·•1•, • ..__ .• ,.~__. .. • "' __ .. - ...... °''' ........ I ~ _.. __ • - -• -•·---• _... .• ;... ........ , • -· ..... , .... - •• -· --··- 0• ... ~_;_ ... •--···- ,,,,,·;• • 1••,• --0 ,,...,._ -----.. •••• ... •·, •• .... - ... ...:.-r -0.;, ...... ,_ ···-"'"' • '"·• Y- ........... --•-·""- •••~• ••' '-'•• , ... -~ .. ,~,.-- "" ,.,,., • •• ...... ....... . .,, .. \.a_... . .. .... ·• ..... . . .. . .. .... -
CENTER
POINTS
·Q-UARTER
POINTS ~·
I
~'l.'1111' .... r.~, .. ,,. ,,, , , ......... 11.t,.;rt"-'•..,.., .. ,~.,"'J.\'".·'''""·"'···
.. . . .. ' .. ~' . ...• . ... . •···-·-· ... -· ,.
. . ., ........... --... -,-............. ~ .. -- .... • ...... ,._.,._ .... .
RIDGE #4 LINE LOADING
MODEL ON MIDDLE COLUMNS
..,_ ··-···-
..
RIDGE 11 AVGE STRAIN EXPER. STRESS
THEOR. STRESS
0 ·- 9.37 1!19'' 4.,,.6::8: 4116 .. 2 74.46 :37 · .2t3 . . .. 23.85
-4: - 165. 74 ··-· =:82 .•. 87· - 79. 76 .. . . .
6 40. 73 20.37 25=· .. ·3:2. . .
. ... .
.g·, 28.00 14.00 s·.2·3
.fO-. - 31.09 - 15.55 2.00
:.\·,-:
=o ·- 10. 78 - _.5·.:. 3.9 13.04
2· 79.22 3_9:.61 ...•... 32.82
. _4. - 113.51 ·- ~:~.:.7,5 . - .65.16
6' 30.00· .rs.:qq:. 21.62
8 26.64 13.32 s.oo :10 - 21.4i - 10. 70 - 1.61
Note: Strains are in micro "/11 ,Stresses in psi, for a load P = 20 lbs •
(
b
·1 · '.\ I _ ____....,__. -------.- •·•"'• --,-..--.:..._..;,~ ~ .. ~ .... : ., •. .;;,.·--·-··-~· --~.-. - . ~·'!,.·-.. ---.·-. ~-+..I~ ...... :--:--. • '. -. ··-·--:-.,·-!' /:.· -:---::· -- ·' . .' ·-:.-. ,· . : : -~-·:"":"-~': - ·'"!'" ..... _· -. • .. - '--~-~-------"::"'~---. . -: .- .. ~ ...... ~-":'"!""-~-~··-.--.. ~~·--. '':.": ""'~·-------.... , ___ - - ' , . . . ·. ,:
' I
- -.-. ··-·- --- --·-, ..... - ~··.•· - -·,c·, . ; ;- -~---- ___ ....,._ ----=--,. ~·-- - _, -···· ._. •• ' .. -.,1 ,, ,, .. .., .. - •. • . ' H' .-:-! •. __:_,..::.:.:. .. ·.:·.,. O -~, •._:;_;._~, N.::.. • .'~ •••• ....:._• ••-·N,O oo. • • ·-· ,_..a. ... -•• ... ,. "''"'! ··- ··- 00
0
,,• ;;'"')' - -:- - ON :··· --.- • •o - • • --·
.....
'' 't,IIIIIIL•••·-·' ., • .,..., .......... ,. '~. - ............ ...,_..,...'l'•t._-.. ,,.,• ,._ ................. ---- , ............. ,_,,. •. ,_,,,, .... .,.... ........ , .. -1,N-<>--,.- •-··' • -·-.... ,---. ............ --4,-• -•••• 0 •·••~• • • -------- •• --·•-,••-•••-,.•_'-_ , ... _ .. _,. ••• ' ,,o .. •• ONO ...... , .... •• o,f,,,_,_,_,. __ , •.-••.----•~•••-••·•-'"-.IN••-··-·--·-------·--....... -,.•.~-···' .. 1. .. ' •• ~- •' -. • .. lo' • ......... _. ...... ,, :• w•
,-.. •.
~
I I/
. '•., ...... , _ .. ,..... .. I, • .••• • ...... .,, • ...
I.
I ~94 .
• • ' ~ -· t\-_,. . '. . • .....,._. • •. -:-- ...... .._ __ ~ .·1.-(.· .. , .,.., ___ -- • .. • • - .... , .. : .......... - ...... " ,. ,,-···loll,......... , ··- ·• ....... - ... -.. \A'- ...... , .............. - - "·' . ' .......... -. . ..................... --..~--- -.. -~ .. •· .... .
CENTER
POINTS
QUARTER
POINTS
'' ' '~,
.... ·- ,. .• • .. ......... ,.. ............ , •.t1.•.'
\· .CENTER LINE PLATE 113 LINE LOADING
MODEL ON MIDDLE COLUMNS
AVGE STRAIN
·.·o·· .. .. · . ·6·7:.so . . :2 • 6.06
4. - 122.23
:6.-, 42.11
:s - 19 .13
10: - 25.97
..
0: 5:0 .~t4.
'2' .. . :4 .. ·3·s· ..... :4. :.-. 9·1. 80
·.6 .. 34.30 .__;-~·
8 - 12.81
l() - 15··.Jl
-r
-~
·.-
·-
-
-
EXPER. STRESS
33. 75
3.03
61.12
21.06
·9 56 .. ~ .
:12:· .. ·98, '.' .. .
2·5. .47
2.19'
:45.90'
17.15
6.40
7 .15
-
-
-
-
\
THEOR. STRESS
2.2.12
19.32
52.92
2.3 •. 27
·1 ::5;3 . ... ;
·2 ..• :.SQ
20.09
19.97
41.07
18. 75
0.94
0.49
NOTt··: Strains are in micro "/ 11 ,Stresses in psi , for a load P = 20 lbs •
... ·-- , .... ,
.. .fj
····-· -.
. 'S
..... ·- ·~··--·. ·"": ,,~ ..... _--,.._:·~· .. --::-··:-·· .
" (to.- •••. • .... , ... ~ •.. .......... ...... •
··•·· ·---- - ·--.. -...... ,,.,.. ___ ~· ·-··· .. ·---·· -·-.. ·---·- · • •. ····- ·-· ·-:- -- ·.~-:-.. • ~ ....... --~ ... --····· ..... !~:-'":::.".:-:.:·, ..... ·:-': •. ·., .. ~ --~------- -- ·- ·-.. - --· . ···----·-.. -C.,14~.t- ............. •r ~ ....... •··'• •.. . . •. ----,·-·· .tlllJl!!'lr,1..1.., .. r .. , .- _ ;,r.: .. : ... , .. -.!.", .... _ ............. ···- ..... ~ .............. , ···········- ...... _ . n·-- ..... -..... ~ ........... , .. __ ._ ~~-.------ .. -o·u-.......... , ............. ,, .. ···'· ....• ,;.·
t' -95 ·--'-•- --"--• "-·- • ----· - ------- --••• -- ••- --- ----,- -·--- -·-..,.-• -..---~· ,..__ , I
. .,. ..... "#. ~ .... ~ • .,~, ...... , .... "'.--- ................... ey1,.,... ... ' --·--.·. -- ' -· . -,-.· ........ - "-.• - ••. , •. .........,;4 .... ---.- • - · ....... - ' ......... ---.. ·-.• _.·~··; .......... _.-.-_-.·~--.• ~·.----~---- --··".-~ ·~~ -~.-~--. :...--.-:-··~-:--~--- -- :-- __ .. --: l_. ......................... ~.~!IU..--·--:-~ - , ... ~~----.._-----------.... -----.. -~ . . ' ..
CENTER
POINTS
QUARTER
POINTS
•
·1' ,, •
CENTER LINE LOADING
MODEL SIMPLY SUPPORTED 1
( WITH TIES )
AVGE STRAIN
:o 47.66 . '<,
2· 4. 76
·4 ~ 21.34
6: ·- 25.28
:8.: 5.74
10:: . . - · .. -. 48.25
·o ·.· ·. 25 • 31
.2: 17.03
\ 4: - 19 37 •
:6: - 17.50
8 20-- 00' . --• .-· ..
10 24-.~_.37
--
-· .-.--.
EXPER. STRESS.
·2·3·. 83 . - ..•... 2.38
10."62
12.64
2.87
24.12
12. 65
8.51
9 6:8· ..•.... -
:.8' • .75
·1.0 •. 00
12. 18
-·
:,
-
-
·-'.
THEOR. STRESS
2.42
4.91
6 49 -· 6.49
4.91
2-42: .. . It ·. . .
:5·. 23": ..
,2._.·47
o.rJ7
O 5:7 :· . :.·_·- ...
2· •. 41
5: 2-3' ........
Note : Strains are in micro "/ 11 ,Stresses in psi , for a load P = 20 lbs •
. .,
..
.·''-'·-:>.-··.· ·'
·-~
~-"; ........ /'.· ·'
"7" .• ·_._ ••. -
I •
.,
I,
. '
1-............... :-:::-:.:.:.:.:: ... :~: -·--· ::.: ... -·:· .... ·:' --·;: ... _ ....... ---·- ...... - . : ................ __ , __ .:.:.::.:. .... ~:.::::.: . .:. ... -- _;:~ ___ ..:.::_:_:; ___ .-:... ..... . ... , .... .. ..-.---·-·'·· . ·---··-· ' '-•·-···------------·--··~---~----•-·A•- ..... •._.,... ___ - ·-·-·· -·-·-···-'"-·--- -·--•••-·· ' ... ,,. ....... ~ .. ,,\., ..... ...,. __ .. .,..,... • . .,.~- -
........ (
-- --· ---··----·--'"-·-----------'"--·-· ........................... _____ ··-·-·-·~·-····--·· _,_ ..
__ •. .. ·-,. -----'---~ ... - .. ,-,_...-. - .. !,._J,~.t!t...;v,.;,: ..• - .... ·--· -..... - f - --.-.-,.,...!:.~.-~ .. ~ ---u~'t!°!~~:~·~-·~.-.._._•'!,• . .__-. ~~~:·'"':...""":.:.,.----~-- - --- -·-·· l!'.'r!~·~-!·: • -··· ·- - -· • .. ~ - .. .. -·--
.....
r,
CENTER
POINTS
·-
:QUARTER
POINTS
.. .,:
RIDGE #4 LINE LOADING
{: ''-
MODEL SIMPLY SUPPORTED ( WITH rr~,~ )
·arn·cm 11 :·: .... ···.
0
2
4
6;
8
10
:o.
:·2
..
·4.·. · ...
6
10.·
AVGE STRAIN
.62.34
26.48
- 116.32
.6._7 • .-6'Q
- ·25 .66
42.34
•.; 91' .• 5.7·
54--61-. .. . .. .•
- 18·:,,7.5
·11 •. so
-
-
EXPER. STRESS
.31.-17
13.24
58.16
33.80
1-2 •. 83 .. .• ~·· .
21 ... 1;.
18.12
18.59
- 48.78
2.t .30 ... ·- ·9·· .• :·37·
s. 7.$ . . ...
-
-
-
THEOR. STRESS
10.87
4. 70
40.60
6~.-6·9_
I
40.066'.
13.17
13.35
8.00
- 23.17
,.
·- ·4_5 .i62, \
Note t Strains are in micro "/ 11 ,Stresses in psi , for a load P = 20 lbs •
...... _ _:._.. ... ·,.--,---···-·· -•·-- ------- - -· -·- _____ .......... -··" ... --· .. --,-· ··--------·--- .. ____ ···-··--·-·· . -r·'· ... -- ---,- -··- .: ... _._:.__ --·~-- -·-····- _·.- .... .,. - ·- -------·--·-
..
-i_.;.
...... , ...... • -··• ··-········----··---•·-~---•--·-" -· - -· ·-·- ·---- - • ., .• _ -·- ------ ---- .•• --·--~--·- ·-•~- ··----·· - ··--------·-•·· ·- ----- __ ..._ _ _. -~ • ., ,... . 'I- .,,.1, ... ·• •••• ...... ,. •• .,~~" ,~-. ... ,.. .,.._..,., .• ,(,.. ~ .,_,. ___ ,,.,. ... ,_,., .. ..-~ ............. ~~~ ,, • '.,,. ,_.,.,.., ....... I,._ ........ -,.-.......... , --· ,·l- .. .,. •, '<,._ J .,_ • ·• ' -~,!? •. .,. •. •· .... ., ..... ~,.~- ... ,.,,.-,~.., • ' -· --- - •
-96 ,.t;,.. • - - . ·- - - .
. .,,,,..-
f?'
f .' ,:~
-·-·--·--···- .. --... .. .
- - ---· .. --- -·· ·---"·-~- .
·,.
CENTER
POINTS
- - . . -- - . . -• . ·- --·- - . . - - -
CENTER LINE PLATE 113 LINE· ·LOADING
MODEL SIMPLY SUPPORTED ( WITH TIES )
RIDGE ·t;
Lr
0
,2
:4
.·(i
=a· . .
10
AVGE STRAIN
128· .:91 . . .. . .. .. . .
.5 .• 3.9
. .;.. . 76.2:~.
67· .. 03: •. . .
15 .. ··15 ... 2:2., •.. 1-9
-
EXPER. STRESS
64.45
2.69
38. 12
34.51
7.57
11.04
,-.-.. ' ' .,, ' ' .,_ -""·'...- ~
- -- -~ ..... ·- --- ·- ···---~ .... - -- .. ,--. ·------ - -- - .. - .
-
..... , . ··-··
THEOR. STRESS
49.67
10.32
38.03
39.07
18 •.. 06
: .. ,
5·. ·53 ... • ••
-97
~.
'!
;o. 71.40 35 .• 70· :~l-8.3.8· 'I !
~ ..
. ~ .... - - --- --- - -- . - ... -. --_· .. ~----
QUARTER
POINTS
::2
·4
6
·8
10
~
·-
-
0.54
73.75
36.79
2.80
12.96
~'.
..
-
0 .• 21
.36. 87
18.39
1.40
6.48
- 11.31
- 26.69
34.67
- 19.52
7.34
Note! Strains are in micro"/" ,Stresses in psi , for a load P = 20 lbs •
'·
-~, .... -·. -_ - . ~ "-·.·· ~----·:··--"--;··-i.:.·"''':'·--,.····:.--::-. ·-· ,....,,,-:--~~·-.-,-.·-'i. .... . - r - --- . . - - --···-·•"" . ... ---~;.. ..... ~ _ _. __ ..... ..... .: .• -...~ ... :.~·-·~ ......... ~-..: ... -.,:. • ..:..... ••........• _. ·-· .; •. ,; .... :;-._. ',-: .. · .. -·----:--"-·••--:------=~ ... _;.,- - ...:. . ..:---- ___ .._ -., .... - .,., ·~·-·-·- ... _:''"'--•·--:.-:---·- ·---·
l .. . .. ... ······ l \, .,,....,. ._,.,..,.~ ....... ,-~.,,....-.-.,.,__.- __ , ____ •• ,...--~-·•---.. ~••••••,.,._., t<t,, ,.,,.,., _,..._ • ..,_ •·••• __ .... __..- .. ,,,~,.,,u,, .. ,.. "'"''' ,.'f.,,,,..,., .. ,,l,_,_u., ... ,. ..... ,,.......,_, - •· > ~, -·------ ---··- ............. ~--... • ",,..,. ,-,.-.,.,..,._,..,.,-.. -•"'-•-··-•'"·'•--•-•••·,,_,__,,.,~•-••- - ,-., .. ,- .. ,,.,, _ _..._.,..,~-·,'4·• ,.,I o•L-••-• .,_..,.,"" ·-· h''"·•••·•-··------ ·•·- , • •-•-,•~---··"•••··"·'"·"''' •'·••-• •·--••·•••·•'""'' .... ,~,..r_._, l ,·
l
j (
l
--.~ . .: _ _,.__•;,,:.-- • .;;..>, ............. ' _ _.,,._·, ... --. . ........ -~--- , __ , .... _.......,__. ____ . ·-· . - .. ~- '--·"••·•··--·---·---...- . ·- .. --· ... ___ ; ...
;,
. ' . - ... ,_.. - ....... ·-·-.... . ... -98 i>
-~-
CENTER
POINTS
QUARTER
POINTS
. . . ' .. ~
CENTER LINE LOADING
MODEL ON MIDDLE COLUMNS ( WITH TIES )
RIDGE fl
:()
2:
4
6
s:-.
10
0
2
·4
'6
:a ·1·n :· ·.-:~u·
·'
;"""
:al!'
·-
~
-
12.19
6.72
19.45
17 .96
12.68
15. 62
9.84
18.44
17.50
18.54
12.03
-
-
-
~
-
EXPER. STRESS
6
6.09
3.36
9. 72
8.98
.34
7.81
8 36 ...•. ,.
4.92
9.22
B. 75
9.37
6.01
..
-
-
. ... . ..
THEOR. STRESS
59.71
44.29
14.06
14.06
44 ... :2~)' . . .. '
5"9. 7:l·
:26 .• 79
- 17.19
3.82
3~82
- 17.19
Note : Strains are in micro "/u ,Stresses in psi, for a load P = 20 lbs.
'7-- ·,
l.' ,_,
;._ ·--·~- ·- - ·_ ··- .. , --· .: ...
~•••,._, ·- ••'· --R••-•··•-.. ·••'·• ·"''"" i••-•"'..,.'''·:•·•., .. ;~•,•,I_,,.,,,_,. ... _.,,,..,.., __ ,._ fUI'"' .. ,,.,. __ ,,..,.,..., .. ,, •. ..., •• _ ..... ~.·-"'"., •. ,... ~~.._._.,..,...,~ .. ,_..,._,..,.,..,..-, ..... ..,,_._._.._....._ _ _._~.,_:._ .... ,,. ... ,,._,::.:..·::;:~ •• :.:: •. ..,.:·:·.,,,..,,,. ..,:...::::~::,::=-.:,:·:-: .. ~.-~,:,,.:.:.~ ... -....,.-----.. ,,,.,,.,_.., N~lw .·-...,.,- ,..,,_,.., . .,,.._..,..,.,._.,_,,i... ,.-...,..,.,..,..-.le.. .... -,..,,_,....,.~-----·._ _____
I 'i, •
- : ·' 1
ii ,. - .. :..:
·- :. .. ,.. .. .
... :·"1'. .;.i ·~, ,:
·) -,,1
' ':J
I ;"1'
i
l. I \
i
' ,,
~ . .I
I
!
.,
1 • l l
! I
i I 1 I I i a
I l
. l .. 1
I'
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' ~" '...', ;·t
:;, :j
l j
I I '
i I
'1 . I . )
I
I i
,.,,. .• .
CENTER
POINTS
:.QUARTER
POINTS
;, ....
-· . --- -·-----., -- --·- . ----- .... - - w • ' • _.,.,
..... w ..... - _ ....... --.. -..· , •• ,.,,, • • • • ...... ·-·:· .... •• w •
w .... _•
.,'., .. , ............... ,
"''"····>- ~-.1, .. ( •
· · RIDGE /14 LINE LOADING
MODEL ON MIDDLE COLUMNS ( WITH TIES)
RIDGE #
,0.:
'2
4
6.
a.
10:
0
4
. 6
:8:
10
AVGE STRAIN
57.66
32.89
- 114.69
78.00
- 19.84
32.03
·28 75 ... ,~.
-38.09
- 90.70
65.54
- 14.84
21.56
-
-
EXPER. STRESS
26.83
16.44
57.34
39.00
9.92
16.01
I4·'_, .. 3·._.7··· . .•... 19:.()4_. . ' ..
- ,45 ... ·3·5 ......
·- 7 .42
10.78
..............
--
-
: .. ,, .,. ... •\ ' .... ' ..
THEOR. STRESS
72.25
31.39
36.57
68.51
·s3.52
33.10
43.14·
- 2.~ •. 9~·
- -2·2._45
64.33
- 52. 75
28.49
Note : Straias are in micro "/11 . Stresses in psi , for a load P = 20 lbs. '
-99 " ,1,1 ... •, .. ,,,._:- .·•
I
- - - ___________________ __.,___._ ____ ~--.--·---- ----- --· ----------- --·-----~-
CENTER
POINTS
"
QUARTER
.POIN~tS
·: ......... ,#•,·•·· ,·- -:·.
, .. ., .. . ' .. ' ··~--··-................ ,,,,. . ~ .......... ;•.•.•'":... .._ :.... .. .,, . • ••• . ................... _.- ... - .... ....-. .. t:t ... ..
CENTER LINE PLATE 113 LINE LOADING
MODEL ON MIDDLE COLUMNS ( WITH TIES)
RIDGE ffo AVGE STRAIN ·
o·: ,.9e.,,_:12:
'.f ·6~·ao
4 ~ .. 8'5' ·203 .... 6/" 60.92
8-: ·- 0.77
10 4.69 . ;' - '. .
o· 43.44
·2· 12. 75
·4 •.. .-. - 68. 14
,6. 43.98
8 - 4.29
10 ,_ 0 .78·: ...•...
-
=-
~~-
-·-
EXPER. STRESS
49.06
3.40
42.61
30.46
0.38
2.34
21.72
6.37
34.07
21. 99
2.14
0.39
--
-
-·
-
-
•.• "'·,,, ........ I,
THEOR. STRESS
33.00
1.23
37 .81
38.37
.22 .•. (18'.
:12.-35
J"o·· 61 . . . ... i:.·:23
·26 13 . .. , ... · ... J:a,~;6·:a
2·1 .•. 14
l0:_.:03
t ' \.
Note: Strains are in micro"/,,, Stresses in psi, for a load P = 2o lbs •
· .
.....
'>.· ·.~. ••)•., ....
. -·.··:··- ,_ . ., •,• •• •,: ,, ,:,~• .. ," ,• 0 ,· 1".•••~I ! -.,. ,, H.0 ~-·-~ > • . •. ,; .. "· ;;, .. :.:----.. ·.· .... ,; -~-"-,-.--·"· .. -·----~-.. _, .... ,,..... --~-· . . -'--·-...:.:.---·-'··"'-'-··-
. -100.
~
. r •.
.I ! '
I ,
i,. ••_:_:~• •• ,,. .. - ....._.._. • .,.._ ~ ... -:,.,. • '·-~.~.:::_'_ ___ ·, .. .:._ __ ~ .......... - ... ··- ,_ .... -.-,-... ---.... - ... - ... - .. ..----..--. ~---.,. ........ ....__, __ _. ... _.,,. .. ___ ""' ..... -- .... --... ~··••··-------~-·- ..... ,, .......... -... ... -·--·· f'W"."'lW- - , .
. - ·-·; -----:.s-.. .:.-· --· ·--... . I' . t .
' i
·--·----........... ~.::.::._-,,-•_ ........... -..... _..,_ .... ,, ......... - .. ~,.. ........... -... .......... --RO --· .. -----·- - '·r ' i
i ' I
l I
I
I
I
b
·r t -;···,,~., , ::. ; ! : ..;: 1 f<li' I;_ H MS : V 1 ,..-- -,-,--~ ' .. --II
··----· .. ----·---·---·-----•--' - -----·---------- - . --··· -- ·- - -- - - - -· -- -- -------- ----- - -..... -···-- ---~ - --- - .. ~-··--·- ·101 · ... ,. ---1 ""1· '' .. ··...:
--- --------~- ... --... , .. __ ·~--- -- ---.. ,,
• ....... .-. ... ._. ...... ~·- ... . ..... .,.., ... .... ·~4~·-,.~
'
.. ,• .,. :, . .,..,.,,-,.,.,.,, . -~ .. _
'' I ,&, ,.,,1•••• •• •- •·•·• ......... -~ .•. , ..... ' .... ' ' . . ·- ..... " ......... "~··, ...... ~~·-,,.,..~ -·· •.. -........
ll E F E R 2 N C E S
I. Yitzhaki, D. "PRISMATIC AND CYI..INDRICAL SHELL ROOFS", Haifa Science Publishers, Haifa, Israel, 1958
Simpson, H. ''DESIGN OF FOLDED PLATE ROOFS", Proc. ASCE Paper 1508, January 1958
. (-··
3.. Winter, G. and Pei, M. . ''HIPPED PLATE CONSTRUCTION'', Journal AMerican Concrete Institute, Proc. Vol. 403, January 1947
4;!1 Craemer, H.
5.
''DESIGN OF PRISMATIC SHELLS", Journal American Concrete Institute, Proc. Vol. 49, February 1953
Ashdown, A. J. "THE DESIGN OF PRISMATIC STRUCTURE", ' Concrete Publications Ltd., London 1951
Zanoni, E. A. ''MODEL STUDIES OF A FOLDED PLATE STRUCTURE", Fritz Engineering Laboratory, Lehigh University, 1962
Scordelis, A. C., Croy,~. L. and Stubbs, I. R. "EXPERIMENTAI. AND ANALYTICAL STUDY OF A FOLDED PLATE", Journal of the Structural Division, Proc. ASCE, December 1961
.. \"
-~-
_. -· ;,,:
·~···. - ·····: ·--···· ;·,··-· .·. ·- ...
,.
"• • .... ,. ·; •---:-r•,·-•-.. '". •··-- .---· -,;-- _ . ..., ,· -·--.--------• .... - . ·--~· - ' . - --•- .. ·. ,,,-..,_,. __ - ~---, . .-s.~,- ' - . i.. - ~- -.. ;. .•.•
',. I.. '., ~~····--·· ---·-----.-- ------- ·-, --~-"~ ....... --·- .... ----.--......... -----------------___ ... _________ -----·------ ... _____ _....,. _______ ..,..... .. _~ ~ ............ --~ ...... ll;"'" ----- ' - .. -----:-::- ·---~ _ ....... -.... •• ·-:---· --J..'•----...... , .... ,_ ... ~ .. -· -~~·f'"-~ - -·---;_-~-_-... _ .. _.,._~~-:.~t~~"""- -
- ., -- - .--- - .-------~~--~----- -· --- ·---- ---- ---- -···-· ----·-------- -·------· -------- .--- --- - -. --10-2--·· . . .. ~-- -·· ,- __ ........,_____ ---·'II;.:,- -- ....... - -.. •.,~' . . .... . , .. , .... ,. .. ····, ·-·
..... _______ ...... . ·- ·"·· .fJII/IIIIIIJt,. .. .•.. -
...
.\ ........ .i\S:· •'..... .'. ..: •.• -~.:.;: ·-.... ···,
~· .... .,,.. ,ri,..t(,....• ......
•:
- ·-··· ......... . ······•· .......... __. ........... , .. ·- ..•. .
VI TA
The author was born in Mezieres (France) February 10, 1939.
Be was graduated from the Lycee Chanzy in Charleville (France) in
June 1957. He then attended the Institut Catholique d' Arts et
Metiers in Lille (France), from October 1957 to June 1961. There he
received the Diplome d' Ingenieur I. C. A. M. in June 1961. He
came as a Fulbright Student to Lehigh University, Bethlehem, Pa.
Later he was awarded a Research Assistantship in Civil Engineering
at Fritz Engineering Laborator~. He began studies for a Master's ..
Degree in September 1961.
. .
. •;_S: .·. ''iii
'·
:1 ..
• .
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----:- ::-.·.- ....
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. .. -..,,,- ... • ...... ,... _________ , __ ~.-- -·---. -· - • ··10- ......... -' •· ............ ,. •.•. •. ~' }"~ ........... - . ...... -..... ---·-·· --··· . ·-·····" .
I·
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