Retrospective Theses and Dissertations Iowa State University Capstones, Theses andDissertations

1966

The behavior of a folded plate roof systemFrederick Mitchell GrahamIowa State University

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This dissertation has been

microlihned exactly as received 66*10,421

GRAHAM, Frederick Mitchell, 1921-THE BEHAVIOR OF A FOLDED PLATE ROOF SYSTEM.

Iowa State University of Science and Technology, Ph.D., 1966 Engineering, civil

University Microfilms, Inc., Ann Arbor, Michigan

THE BEHAVIOR OF A FOLDED PLATE F'^OF SYSTEM

by

Frederick Mitchell Graham

A Dissertation Submitted to the

Graduate Faculty in Partial Fulfillment of

The Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Major Subject: Structural Engineering

Approved:

Head of Major Départiront

College

Iowa State University Of Science and Technology

Ames, Iowa

1966

Signature was redacted for privacy.

Signature was redacted for privacy.

Signature was redacted for privacy.

ii

TABLE OF CONTENTS

Page

I. INTRODUCTION AND SUMMARY I

II. REVIEW OF LITERATURE 2

A. Brief Statements on Various Theories 2

B. Primary Theory (No Cross-Sectional Distortion) 3

C. Secondary Theory 5

1. Solution by arbitrarily induced A-values 2. Iterative procedure 3. Particular solution procedure '

D. Present Experimental Status 8

E. Published Literature 9

III. DEVELOPMENT OF THEORY 10

A. Fundamental Concepts and Assumptions 10

B. Primary Theory 12

C. Secondary Theory 23

1. Iterative procedure 2. Particular solution procedure

D. Development of Secondary Theory 26

E. Additional Analytical Refinements 36

F. Buckling Analysis of Edge Plate 37

G. Manipulative Techniques for Computations 40

IV. THEORETICAL INVESTIGATION 41

A. Criteria for Parameter Choice 41

B. Pilot Model Considerations and The Analysis _ 42 of a Typical Test Model

C. Dimensionless Parameter - 60

D. Check System for Theoretical Calculations 61

iii

E. Stress and Moment Determination at Gage Location 64

F. Beam Analysis 65

V. EXPERIMENTAL INVESTIGATION 66

A. Loading System , .. 66

B. Deflection Measurement 74

C. Stress and Moment Measurement 74

D. Property Tests of Model Material (Aluminum llOO-H-14) 89

E. Effect of Gage and Adhesive Thickness on Moment 90 Determination

F. Testing Procedure 92

G. Typical Set of Model Data and Data Reduction 92

VI. TEST RESULTS - INTERPRETATIONS AND OBSERVATIONS 116

A. CTjj - values 116

B. - values 120

C. my - values 120

D. Experimental .Observations on Buckling Behavior 127

E. Final Comments 128

VII. SELECTED REFERENCES 140

VIII. ACKNOWLEDGEMENTS 141

1

I. INTRODUCTION AND SUMMARY

The popularity of folded plate roof structures has prompted a study

to be made of an 8 plate simply supported roof system. This study vas made

with 8 aluminum models of varying plate thickness, span length and roof

slope. The models were Loaded until pronounced yielding had taken place

and concurrently a continuous record of strains and deflections at strategic

locations was kept.

The object was to discern the correlation between the ordinary folded

plate theory and the experimental observations when the various parameters

were varied. The anticipated information to be gained was the effect of

certain parameters on the theoretical prediction of stresses and deflec­

tions, the behavior of the structural system at ultimate loads, the buck­

ling behavior of the edge plates and ultimately, how these factors could

be incorporated into the design procedure.

The problem proved to be a very interesting one both from the theoreti­

cal and experimental standpoint and in conclusion answered several questions

and provided an insight into others.

2

II. REVIEW OF LITERATURE

A. Brief Statements on Various Theories

It is the thinking of the writer that the various folded plate

theories can be broadly grouped into two categories - the Engineering

Theory of Elasticity and the Mathematical Theory of Elasticity. Theories

employing the Engineering Theory of Elasticity fall into the majority

group that make the familiar assumption regarding planar distribution of

flexural, torsional and axial strains with their consequent planar dis­

tribution of stresses. Of course this assumption is only made in regards

to individual eleménts of the structure and not to the structure as a

whole.

On the other hand the Mathematical Theory of Elasticity utilizes the

general equations of equilibrium and the equation of strain compatibility.

The problem is then executed without the usual simplifying assumptions.

Insofar as the vast majority of folded plate theories fall into the

category of Engineering Elasticity, the following brief statements will

emphasize this group of theories.

Since all of the Engineering Elasticity theories are based on the same

fundamental assumptions one would expect to obtain the same conclusions

from all. As an analogy to this situation, the common beam can be analyzed

by slope deflection, moment distribution, conjugate beam, strain energy etc.

with the consequence of identical results - the only stipulation being

that the calculations be carried out to the same degree of refinement.

The degree of refinement encompasses such considerations as accuracy of

computation, inclusion or exclusion of shearing strains and axial strains,

3

temperature compensation, the importance of secondary stresses, non-linear

material properties etc. Likewise the folded plate theories can be

expected to yield identical results with identical refinements.

B. Primary Theory (No Cross-Sectional Distortion)

This theory has at times been referred to as the membrane theory but

in light of the familiarity that American engineers have with the primary

and secondary analysis of structures, it is thought that this terminology

would be very appropriate. The first Primary Theory published was by

G. Ehlers and H. Craemer (3,4) in Germany in 1930, and later the theory

was further developed by E. Gruber (7) and others, mostly German. Insofar

as all of their theories made the same simplifying assumption of no rela­

tive deflection between the ridges, the only difference between theories

was in the manner of computing the unknown stresses and moments.

In 1947 G. Winter and M. Pel (14) made the first contribution to

American Literature in the form of a more streamlined version of the same

theory. It is this version of the Primary Theory that will be briefly

outlined in the subsequent statements.

The paper make no claims as to the originality of the basic theory

but does take credit for the development of the distribution system that

eliminates the necessity of solving several simultaneous equations.

A folded plate structure is formed when several flat plates are

joined at the edges and are supported by end dlaphrams. This, of course,

is the simplest case because it Is possible to have intermediate dlaphrams

and have continuous spans. In the development of the Primary Theory the

following plate configuration was used by Winter and Pel:

4

g

A

/ V EKlti

RAN"5.\I&?«.>»<£ STRIP

The basic theory assumes that since an individual plate is long and

thin it is capable of only two modes of structural participation which

are designated as slab action and plate action e.g.

It also assumes that the structure is homogeneous, elastic and that

the lines of intersection between the individual plates (the ridges) do '

not,undergo any relative displacement. The moments and stresses in the

structure are found by the following procedure:

the structure and analyzed as a continuous beam loaded with the

deadload and superimposed live load and supported at the ridge

lines that are assumed to be unyielding. This approach is

Justified on the premise that all slabs have similar longi­

tudinal load distribution with the consequence that any other

unit strip loading would be similar in form.

1. A transverse strip of unit width is cut from the mid-span of

5

2. The reactions of this transverse strip of slab form the mid-

ordinate of the ridge line-loads and are resolved into components

parallel to the plates, thusly forming the plate loads.

3. The total individual plate load is computed by adding the

component contributions from the boundary ridges of the plate

and is then used to compute the longitudinal plate stresses,

assuming each plate to be a long, thin, independently acting deep

beam that is supported by the end diaphragms.

4. Now observing that the common edges of adjacent plates do not '

have equal stresses, proceed to establish continuity by employ­

ment of the stress distribution process.

5. The structure is now designed on the basis of the transverse

slab moments and the longitudinal plate stresses.

C. Secondary Theory

In compliance with common American terminology the Secondary Theory

considers the moments and stresses existing by virtue of primary deforma­

tions. In this respect we are primarily concerned with the effect that

relative deflection of adjacent ridges has on transverse moments and longi­

tudinal stresses e.g.

-t

1. Solution by arbitrarily induced A-values

This technique vas introduced by 1 Gaafar (5) in 1954 and constitutes

one of the first practical solutions to the secondary problem to appear in

American Literature. Mr. Gaafar's procedure is briefly given in the

following statements:

(a) Determine the primary transverse moments and longitudinal stress­

es in compliance with the Winter and Pel procedure and compute

the plate deflections (6) that result.

(b) Arbitrarily induce a relative deflection between any two

ridges. This will create a translatlonal fixed end slab moment

that can be distributed. The distributed moments are used to

compute ridge loads that can as before be resolved into plate

loads. These will be in terms of A^. This operation is repeat­

ed for the A-value of each individual plate.

(c) The A induced plate stresses can now be computed in terms of the

various A-values and used to determine the plate deflections (6).

These deflections when combined with the deflections from step

(a) form the individual total plate B-values i.e. 6^ = f

(primary stresses, A^, A^)

(d) A set of simultaneous equations can now be formulated by realiz­

ing that any plate-A(A^), is a manifestation of the B-values

existing at its boundary ridges. Consequently for n plates

bounded by ridges at both edges, there will in general be n

equations in the nature of A^ " g 6^) which from

above reduce to A^ " (primary stresses, Aj^, A3, A^).

Solving the^equations simultaneously for A-values constitutes the

. • - ' solution from which all transverse moments and longitudinal (plate)

; ' ' . ' . ' ~ . < stresses can be determined. ]

'• • • s " H. Simpson (11°) has used a -^stem very^ similar in principle to that

/ A 4, ( of Gaafar where he has arbitrarily induced a rotation ( ) in each

successive plate. Each arbitrary rotation case gives rise to a trans-

lational fixed end moment that when distributed and used to compute the

ridge and consequent plate loads, enable one to determine the plate de­

flections (6) that are uniquely associated with this arbitrary rotation.

Since an arbitrary numerical rotation was induced and not an unknown

A-value, it is necessary to multiply each case by an unknown factor (K)

which is essentially a way of saying we are going to use K-amount of this

particular case for superposition purposes. Simultaneous equations are

now formulated according to the following statements.

Define K such that (T) = K ( ) existing arbitrary

Then: ( ), = f 6 _ - values, (6 - values), ^ ^arbitrary — pJ^i^^ry case 1

K„ (6 - values), K (B - values) ^ case 2 case n _

A set of n simultaneous equations are formed and solved for K-values

which in turn dictate what portion of each case shall be superimposed on

the primary case to yield final results.

2. The iterative procedure

After computing the primary stresses and the consequent G-values, we

can proceed to determine the change in transverse slab moment and therefore

8

the change in ridge loading. These load changes can be used to repeat the V

process which will n turn result in another change in ridge leading. ' '

This process works very well with certain structural configurations

but with others thé convergence is slow^if not divergent, as is the case

• 1 of plates intersecting at very small angles.

3. Particular solution procedure

This technique of solution was devised by D. Yitzhaki (15) and is

based on the principle that any structure that is indeterminate to the n^h

degree can be analysed by superimposing n particular solutions in combina­

tions such that the given problem conditions will be satisfied. The

procedure is very orderly, very general and is adaptable to a wide variety

of conditions. It also lends itself to tabularization and the inclusion

of additional refinements that are sometimes dictated by certain structural

configurations. It is this process of analysis that will be exhibited in

the theoretical development in this dissertation.

There have been many other approaches to folded plate theory in the

literature, the inclusion of which is thought to be prohibitive. In the

opinion of the writer the fundamental principles of all of the theories

are included in this brief survey.

D. Present Experimental Status

At the time the writer established his initial proposal there was

very little experimental evidence to substantiate any of the theories.

I. Gaafar seems to have made the first published contribution (5) to the

experimental side of the ledger when he tested a five-plate aluminum

model with concentrated live loads.

Another experimental contribution (10) was made by A. C. Scordelis,

I ' p /• •< , Ei L. Croy and I. R. Stubbs iji which a simple-span aluminum folded plate .

model consisting of 3 north light shells was analyzed theoretically and

experimentally using ridge line loads^ Analytical results obtained by

the ordinary folded plate theory, the theory of elasticity and the elemen­

tary beam theory are compared witA experimental results and the validity

of the assumptions used in the analytical methods is examined.

The "Report of a Research Survey Regarding Folded Plate Construction"

conducted by the A.S.C.E. Task Committee on folded plate construction,

discloses the fact that there is currently rather extensive activity, both

theoretical and experimental, taking place but most of t;he results are

unpublished.

E. Published Literature

There is a considerable amount of European published Literature but

the majority of American Literature is in the form of Engineering Society

Papers. There is one book in English that is devoted in its entirety to

folded plate considerations and this is written by David Yitzhaki (15).

It is a very thorough and orderly treatment and is presented with

the option of including many of the refinements that are usually neglected

in the process of simplifying the assumptions.

10

III. DEVELOPMENT OF THEORY \

•>> ^ ^

A. Fundamental Concepts and Assumptions

The Theoretical Development that has been chosen for exposition in

this dissertation can be thought of as being composed of two major parts:

(1). Primary folded plate theory with no cross-sectional distortion.

(2). Secondary theory which takes into account the effect of distortion.

The basic concepts and tools of the method are cdnunonly known to all

engineers but the structural interaction is very complex and consequently

destroys the problem's simplicity. An attempt will hereby be made to

execute an imaginative development that will promote an intuitive feeling

for the structural interaction. All symbols will be introduced as needed.

The heart of the theory lies in the behavior of the most basic struc­

tural element - the individual plate, which can behave as follows:

L ATERAL ÎV.A A C TIOIJ PLATÊ ACTIOM LOI^CIFUOMAI- II-AS ACTION» AL. ACTIOO

The basic theory assumes that since an individual plate is long and

thin, it is capable of only two modes of structural participation, which

are designated as slab action and plate action. The longitudinal slab

11

action is neglected due to the high ratio. If longitudinal slab action

were considered it would immediately be seen that there exists an extreme

flexibility in this respect.

'

Since these plates are assumed to be capable of only two modes of

structural participation, the plates are worthless as individuals but when

connected along their common edges and framed into end diaphragms, a very

rigid structure is formed. These end diaphragms restrict all lateral move­

ments in the end plane but offer no restriction normal to the end plane.

The mechanism of load transfer is as follows: the surface loads are trans­

ferred to the ridges through transverse slab action, each ridge load is

resolved into components parallel to the plane of the plates intersecting

at the ridge and through plate action these components are carried out to

the end supports. This is the general scheme of behavior and the specific

execution of this scheme will now be developed. The assumptions under­

lying the theory development are as follows:

1. There are two modes of structural participation

(a) lateral slab action

(b) plate action

2. Longitudinal slab action and torsional rigidity of the plates

are neglected.

3. The structure is composed of continuous, elastic homogeneous

elements

4. The principles of superposition are applicable.

5. In regards to the individual plate, plane sections remain plane,

i.e., a linear relationship exists between the longitudinal

strains and the distance from an edge of the plate.

12

B. Primary Theory

In the primary analysis of the plate system the assumption is made

that each slab is subjected to a similar longitudinal load distribution

with the consequent effect of producing a set of similar ridge loads.

This fact makes it possible to isolate a typical unit strip of the folded

system, as follows:

R io(.G Lo*»e>t>

^ -P . i-.

K

LATERAL 5LA6

LoKiSITUplMAU PuAiTt

The isolated strip is analogous to the following continuous beam,

with the exception being - for all stiffness computations it is necessary

to use h rather than d.

r 132

A®

<-

'34

4-

•t-

cl4Ç

45

*5-

u6 7

*—*4-dn

R:

Note! The R'-values are primed because they are the initial values resulting from moment distribution

and are correct only if no relative settlement takes place between ridges. They can be thought of as ridge loads, later R" will signify loads required to make the slab system conform to the plate system.

13

These R' values are now resolved into components parallel to the

plates at respective plate intersections, i.e.

IM GtMEEAL;

For a.n)

l^Qgca. 9o uyGo"^

%- /Z f/AJ 90-^33

33

SffJ

_ Cos. ^3

S w oC

•à/

t. = CÏ, R.' Afio fai = G /

a

These C values are coefficients that when multiplied by the ridge

loads, will yield the contribution of the ridge load to the specified plate

load. The next development will involve values, that can be used for

the resolution of a horizontal ridge load (Fig. l)into components 11 to the

plates. Even if these values are seldom needed for horizontal ridge

loads, they will later prove expedient in the calculation of horizontal

ridge movements, if required.

14

IF WE. M &RE L_Y TR A MSL ATE OMR THINK! M G Q.Y ^ IT IÇ. PossisLe

To iORlTE DiKSCTLX

P " Co p « _ Co s (90-g/a) «5' (T • , , ^<3S ..

5/a/

S//V^ a3 ^i»^Q /Q,

' 9 " -

The sign of C may be determined by simply observing statically the

component directions of a plus load i.e.

-+- LOACS

Each individual plate load is arrived at by combining the contribu­

tions from its boundary ridges, e.g.

UJHERÇ. P. = aA

t. " L

1^'$

The computed P2f3 will be the maximum ordinate of the plate load-distribu­

tion which could presumably be distributed in a number of ways - the most

probable of which are :

«fa This is possible when the only loads are those applied directly to the

ridges, such as a possible crane load.

15

il

L : 4

Result of uniform slab loading

L —4

3 3 ,,. - L_ Equivalent sinusoidal loading which

is quite often used to facilitate certain types of calculations.

This Pg 2 - value is now used for computation of M° which is the

maximum moment that would exist if the plate were free to behave indepen­

dently. 2 is now used to compute o g the corresponding free edge

stress, e.g. 2,3

^.3

th^/6

On completion of this step it will be observed that the free edge

plate stresses at common edges are not equal and therefore introduce the

problem of establishing continuity. This is accomplished by a stress

distribution method that is analogous to the Hardy Cross moment distribu­

tion method. The writer develops this method by first demonstrating the

analogy to regular slope deflection equations and with this analogy as a

tool, proceeding to execute other developments.

To initiate the development, use is made of the free bodies of two

adjacent plates, the applied forces being the - values and the conti­

nuity-restoring edge shearing forces:

16

iU

i I j I

© -I-

34-

A/,

JÉL

V i i Ma t S.

@

(2)

©

®

@

•k M

6 5

'W3 <5 34

3 - '2-

ri2, G 3

M4.

Ng

No

NJ '<2.

Note: For clarity in derivation positive 'stress is considered a manifestation of positive forces or positive moments.

This induces simplicity into the equations. Later we will resort to the more normal

® @ . 4" FORC6<> A4D

= Moment caused by plate loads

a 2^, etc = Stress at 3 in plate - 3,4

-comp. + tension convention, with its inherent design advantages.

The longitudinal stress equations can now be expressed in terms of

the designated forces and moments.

23

M, 2.3

N„

^23^23/6 *^23^23

23 («2)( -T )

f h2 ^23 23/6

N. 23

*23^23

(NaX— )

^23^23/6

23

6M: 2 il

*23^23

No 3N

A 23 23 23

3N,

23

23 (21.2 + Nj)

23

or

*23^23

M: 2 , 3

'23

(1)

A set of equations can now be written, utilizing (1) i.e.

17

^sz- + <oîi_ =0 s4

6j/ f iSja = o ^43+ S"~®

©

S Ç + 6 67 =0

from which, in the case of the illustrated configuration, there are 5

eously and substituting the acquired N values back into (1) will yield

the longitudinal stresses existing at the plate edges.

When there are several equations to solve simultaneously the task

becomes rather formidable and the use of an iterative procedure can greatly

ease the solution. If the stress equations are carefully scrutinized it

is seen that there is a strong resemblance to the slope deflection

equations for ordinary beams, thereby making the following analogy possible.

equations and 5 unknowns (N^, I^lving these equations simultan-

18

Beam

r

LoA,t>

"if MBA

©

I

L

GB

»

"ab" [zO^+GgJ 20^+Qb I + F.E.M.^g

M _ m BÂ L ["b-^A] + F.E.M. BA

(D ' Ng,

Analogous Plate Cross-section

M°9 ® DEIJOTE^ foiU'J

' > A OF rj VeCTXSR.

s

0 iVa

<^32,

IZ H

23 4 F "^^S]

M. 2,3

32 A

23

23 L"'3-™2]

23

^23

Moment Distribution Stress Distribution

AD

with corresponding Apply a C.O.F

4N.

2N,

and C.O.F

J

19

I Y A"® (

Stiffness Factor

"AB° 4^ [(2) (I) +0] + 0- ^

K Distribution Factor

AB

ZK

23 A 23

[(2)(1)40] S.F.-K 23

Relative Stiffness 23

D.F. 1/A

2: 1/A

Mb/»»-o

"3) (S) P

Stiffness Factor for hinged end

"ab"^ [^'A'^B]

But a^'O. [28j+8j

® B ° • "

"AB' d VL 3EI L

So for hinge condition

S.F. = the fixed end-S.F.

<3

Ali=/

1 ' : 1

(-)J 1. 1 L

A/,

This condition arises at the center ridge of an anti-symmetrically loaded system, where the common edges are tend­ing in opposition to each other and thereby nulify the stress.

23 A 23

32"°" 4 : "3 1 " -1

23 A 23 M ' 4 i-Â^

23

i.e. 3/4 free edge stiffness factor

" restrained edge stiffness factor

20

As an illustration of the developed Primary Theory the following

example is presented:

Q/ 1 X

-TTt

^ i&iD "fie.

'V/

® "e-i t.

/ /

E.= 3(/ORPSI

S' 10'

o 1

Load on horizontal projection of a 1' unit strip of transverse slab

/oo/6/pT

(Z) ^ ^ 1 I !• \ L

. _ 1 L L

I /d)' .1 1 1 ' ' . " 1

Determine primary slab moments:

(Z) ©

F.EIA. (N-IB^

M' (FFLB)

O I 0.5 O. 5 / O

-33% -835

4416 -f£85

-4lé»

—3o8 +2ofl -(,2-4

==t

21

Determine slab reactions and plate loads

CD

R' (It!)

F ' (it^

é2.4-

Î-

^4.8

h

©

(Statics)

4'57-C> (Plate Load • -R')

1+

Determine the effect of primary ridge loads on the platersystem

0) ®LI

r' —

f-

~ —

/6

c<=50*

F'(lb)

P

-62.4

.1.732^2.00

+625

+ll

+438

I

2+1.732

+108-125 +625

+108 +500

M°(in-lb) +5.82x10^ +27x10

(psi) +242 +211

-625 -876 4-758

-1501

-81x10

-632

+758

(SuAB

( FL.ATE

Operation

'n,n+l cos B

n,n-l

F' X C sin o<.

+ P_„ etc. P23 + ^32

+40x10^11/8 Plfxl2

+1710 M°/B a=l/6th^

Note I The above stresses are consistent with the loads and moments. (2) c

22

But for the actual distribution of these stresses, the (+ ten, -comp.)

sign convention will be utilized.

Stress "Uistribution

© © 0

S.F. 1/5 1/11.58 1/11.58 1/11.58 1/11.58 1/5

D.F. 0.7^.3

C.O. 0.350.15

F.E.S. (aO) +242 -242+211

-159

- 4 "

j+162.

+ 12

+ 1

Z 1+ 22 -243+386

/

Distribute iO +440-189

+ 22 +198+197

Final (a ) + 22 +198

1/ht or 1/A

0.50.5 0.30.7 0 S.F./ZS.F.

0.250.25 0.150.35 Oi 1/2 D.F.

-211-632

+68-162"

+632+1710 -1710] Just carry-1 ^ I over

-162 + 377 distribute later

+24-24 12

'+i-2 - 1

+ 57

T 4

+11^-820

-351j+351

+457+1710 -1272

+376-876 0

-46fi|-469

-468

+833+834 -1272}

J834 -12721

The analysis of the given structure would now be complete if the

implied assumptions were true, i.e. the validity of the slab moment distri­

bution process is dependent upon a structural behavior that is limited to

ridge rotations with no accompanying relative ridge deflections.

This immediately suggests that an investigation of relative ridge

deflections be made and that the findings be utilized to effect a correc­

tion in the primary transverse slab moments and the primary plate stresses.

23

C. Secondary Theory

The secondary phase of folded plate theory involving the analysis

and disposition of these movements has at times been designated as the

Bending-Theory. Here the use of the term secondary in conjunction with

the lateral moments and longitudinal plate stresses induced by deflection

phenomena is thought to be quite consistent with terminology common to

the American Engineering profession.

1. Iterative procedure

Hence the iterative approach to secondary analysis will consist of

the determination of (a) Ridge deflections that are consistent with primary

plate stresses, (b) Transverse slab moments that are induced by ,a ,

(c) Additional ridge loads induced by ^ , (d) Longitudinal plate stresses

induced c. , (e) Repeat a,b,c and d The structural behavior of

the system might be such as to obviate the secondary operations beyond

step (a), the implication being that the distortion in structural config­

uration is too slight to produce any secondary effects e.g.-all ridges

undergo the same vertical deflection.

On the other hand there might be a substantial inducement of slab

moments without a consequent substantial inducement of ridge loads - the

implication being that observation of step (c) dictates that operations

cease.

Contrary to this structural characteristic of rapid convergence, there

are certain structural configurations that are very sensitive to secondary

effects, a condition favoring slow convergence or even divergence. When

confronted with this situation the iterative technique must be abandoned

in favor of some exact procedure employing a set of simultaneous equations.

24

2. Particular solution procedure

The technique herein presented is founded on the basic premise that

if the true loading, consisting of N concentrated loads, is known and if

a set of N particular solutions can be found, each set consisting of N

arbitrary loads then the true solution will exist in the form of a super­

position of the particular solutions. _

Example;

Basic system

The load for which it is desired to find associated longitudinal stresses and slab moments.

Particular load system (a)

A loading for which there is a uniquely associated set of o-values and m-values.

Particular load system (b)

A loading for which there is a ITnlquely associated set of a-values and m-values.

Particular load system (c)

A loading for which there is a Uniquely associated set of a-values and m-values.

The systems are superimposed using a-amount of system (a), b-amount

of system (b), etc., which results in the following set of equations:

aF* + bFg + cF® - F^

25

apf + bFÎ' + cF^ = F, 4 4 4 4

aF® + bFg + cFg = F^

The coefficients a, b and c can now be determined and they will

dictate the % of each corresponding element of the various particular load

systems that must be superimposed to construct the desired basic system. ••

The philosophy of this method is somewhat analogous to the well

known "General Method of Structural Analysis" or the "Maxwell Method"

wherein a solution is assumed that satisfies the statics of the problem

and subsequently the geometry is corrected by means of a superposition of

individual effect solutions. The particular solution technique does not,

however, involve the individual-effect solutions in a strict sense insofar

as it is impossible to apply the conventional unit load to a specified

ridge without simultaneously introducing other secondary ridge loads during

the execution of the computation procedure. Therefore a particular load

system can be constructed by adding these secondary effects to the arbi­

trarily assumed primary ridge loads. It is somewhat analogous to saying

that if the holding forces in a bent subject to sideway were superimposed

on the original actuating forces, the result would be a system of loads

that would be uniquely associated with the computed moments and the

observed geometry. -

The basic system loads directly related to the superposition procedure

are not, however, formed by this type of superposition. They are, in fact,

the holding forces that exist by consequence of the primary deflection

phenomena. We herein are essentially determining the effect of the absence

26

of these holding forces when we superimpose the particular systems in the

manner specified above.

D. Development of Secondary Theory

1. Determination.of ridge deflections that are consistent with the

primary plate stresses:

The deflection of a plate in its own plane could normally be calculat­

ed on the basis of loads and moments but the known primary ridge stresses

suggest a method utilizing angle changes, i.e. double integration - y" =

^ = nn-i t-^tpno^h * where (j) is the slope or conjugate beam - beam with a

load of

unit length Ad)

The choice will naturally be determined by computa-unit length

tional expedience.

The angle change per unit of length is a function of the stress diff­

erential existing between the two edges of a plate. The foregoing ideas

are illustrated in the following cases:

Concentrated load case: (conjugate beam)

a( - si - csl

Uniform load case: (conjugate beam)

a^

27

Sinusoidal load case: (double integration)

1

•-•À . ©

o

The individual plate deflections are used to calculate vertical

ridge deflections by making use of the already known C-values coupled

with energy principles. Thusly,

^23 Force II to plate 2,3 caused by a unit vertical ridge load.

&2 can also be determined by the following williot geometry

Work of 1^^ = work of its components

\jork op 1 — work, op lis cofapo'nse'^t')

0

0 o las

2. Determination of secondary moments induced by relative ridge deflec­

tions!

This step consists of finding the fixed-end moments resulting from

relative ridge deflections i.e., plate rotations,as follows, and distribut­

ing these moments.

'ùtlr. "I [£-£].IF

rf-^rp.r -

M»IA

The i%alameii»c^ of T<t*se MOMCiXTt voikv.

28

3. Determination of secondary ridge loads induced by the secondary

moments (m"):

Just use horizontal projection of the system as follows:

% m.

J

i

a? —j" 134-

II n " _ nlVWI, -ml-Tnl

^ da ^34-

Ridge Load = -R

Rg as shown is the ridge reaction required to hold the slabs in a position that Is consistent with plate movements

4. Determination of secondary longitudinal plate stresses induced by

secondary ridge loads

This is accomplished in identical manner as were stresses in the

primary system.

5. Repeat 1, 2, 3, 4 and 5 until scheme converges.

6. An alternate to the above procedure would be to follow step 3 with

the establishment of the various particular load systems and thereby

directly determine the effect of the secondary ridge loads.

It is obvious that the secondary effects for this rather peculiar

loading condition are quite pronounced which suggests that convergence

may be slow. With this In mind a solution will be sought utilizing the

29

example ûf seco^oa^y

100

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op&rktio a

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bâisi Ù.F.

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30

particular load technique. The philosophy of the arbitrary selection of

a particular loading is arrived at by observing that the R"-values are

functions of the ridge moments and thereby form a balanced set of forces

(ZR" = 0). In light of this fact it is expedient to assume arbitrary load­

ings in the form of mutually balanced forces that can be thought of as

being manifestations of arbitrarily assumed ridge moments, e.g.

looo FT

" "i" " , f-vauu.^

Choosing thusly will result in a set of particular loadings equal in

number to the number of redundant ridge moments rather than the number of

R"-values. In the case at hand the ridge moment at 3 is the only

redundant quantity, so one abritrary loading will suffice. The particu­

lar load will be assumed as 1000 » distributed sinusoidally in the

longitudinal direction. The reason for the sinusoidal variation lies

in the fact that the secondary effects that are being studied are func­

tions of the elastic curves of the ridges. Of course the elastic curves

of uniformly loaded members are not sine curves but the difference is

negligible. Therefore, in light of the ease with which normal functions

can be manipulated, all particular solutions will be constructed on the

basis of normal variation of loads.

The significance of the calculations (Page 31) lies in the fact

that the F-values constitute a set of actual loads that are uniquely

associated with the o', 6^ and m" values of this particular system (a).

Now insofar as our problem originated from the fact that the r"-values

31

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32

of the basic load system are fictitious holding forces and ve are seeking

the effect of their absence, we are lead to the following indicial

equation:

aF + r" = 0

Particular Basic

a (-287) + (-326) = 0

326 ® -287 ~

Superposition of the basic and particular systems now yield the

following solutions:

0 , . + aa ' basic

, xtitï i 11 particular \ —^ parabolic sinusoidal

gvegv + a6v , , ^<ttttrn^ + ^ttttttx basic particular^ ^ very close sinusoidal

' to sinusoidal

i m=m + m" . + am" m

basic' "* basic' particularj:^uniform ' very close ' sinusoidal ni+ >-rt11 ( i itt>^

to sinusoidal

Results

Max. mid-span values

1272 basic

a (psi)

0.2625 1.384 0.2554 basic

-0.212 -0.041 0.212

0^ (in) +0.05 +0.214

basic

basic

part.

m (1| )

33

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36

) E. Additional Analytical Refinements

The original assumptions used for the development of the theory

neglected the torsional stiffness of the individual plate as being

insignificant and in the majority of cases it seems to be. But in event

the structural configuration makes torsional stiffness important, the

particular solution technique is well adapted for these additional

corrections.

The additional corrections will exist in tWe form of a secondary

ridge load (R^ ) due to^ torsion that is to be added to the Rj^^^^^^-values

and the Rpgj.j.-values respectively.

The torsional ridge loads are a result of asking oneself what distri­

buted edge forces are associated with a specified twisted configuration

of a thin plate. The development stems from the consideration of the

following twisted plate configuration:

slatics

Geometry

The problem is to derive an expression for R^ in terms of 6 or

—J-. (Use analogy to twisting of thin bar)

^ ^ 37

v;

Geometry Statics

sin dl^y = dx d ,

t

^ = Y cos ~ Relationship between Statics and Geometry

d^O ù£)^ . Jtx ^y'^x de M^y _ = . _ _ r s z n - ^ d 9 = ^ , _ = ^

^ ' 150" dx *

„ . d^O Jt^ A5^ . Jtx Equate — = — = - . — sin -

dx L

.-. 4 sin ÏÏ - ZSÊ» • sin If L d 3L d ^

2 3 (R^ ) = 2 To add refinement to the analysis, this

3L d value is merely superimposed on r" existing by virtue of secondary moment.

Further refinement in analysis could be had by considering the longi­

tudinal slab stiffness but this value has been shown to be insignificant

throughout the literature.

F. Buckling Analysis of Edge Plate

It is anticipated that certain structural configurations will cause

buckling of the edge plates to be a significant consideration and therefore

the writer is moved to develop a rational approach to the prediction of

this phenomenon.

The first step in the analysis of the edge plate is to ascertain the

distribution of normal and shearing stresses throughout the plate. This

is done as follows:

38

ih

X j dy dy

* 4- Stress €«I at a wi'i t

f 4- 4g^ <ix J cJx

iy dx «ùx

L &6

© =. lyx.-,- .

^y

4/ g)x

hquiuis^ium çqo atioivi<^

sv - si + i -

=<j»e»j ei -

nci^/^l. tggts oftglfttjc^j 4c<«s5 (latc

coge_ platte^

® = Gl[iH-«</)l |=^Ty^=^«[y+^y^L^°4

yy/ « i^'^' s -^^1 /fso (jso

= <$]ky = ^-^[u+g^y^lS^wcp^ 0j/ "x ox ax li -c j jl l )

— 4-ljl g lj|j

4-0

M -o

c OMPIUMCs = < 0 d *=^41 sinj 6

^ =-s^ [^+%y9B

YXY = <^o + X y IR

39 .'f k

Observation of these equations leads to the conclusion that and

Tjjy are of minor comparative magnitude. Therefore will be the predomi­

nating factor in the consideration of buckling.

Observation of the buckled configuration of a simulated edge plate

model yields the following results:

i i i 4 4 i edge. pupttfc-

In light of this configuration it is thought that a conservative

estimate of the critical compressive edge stress can be made by isolating

a strip of edge plate that is Cgho long and has a width equal to the part

in compression, i.e.

L GL 4

\r//7777 77/77/1 K D +

t -p 0' tb'6u-r.0m op soq.e.

a-riow s wdic. ai-re. COFTO« L ,2 ei

Now using an analogous Euler column condition i.e. P^r " » l

the following analogy, is realized:

0.5 c^h^t jt^ehot^

2 2 2 12(1- )0%

40

ff- Critical^:

6(l- )cv

Investigation of several different conditions should bear out this trend and constant

G. Manipulative Techniques for Computations

The example shown in the development was not involved enough to

warrant any short cut procedures but for a non-symmetric loading of a

symmetric structural configuration consisting of many folds it behooves

one to seek out all computational expediences. One of these expediences

is familiar to all structural engineers and involved the concept of

reduced stiffness factors when pinned-end conditions are known to exist.

Another is the complete carry-over process before balancing moments or

stresses. The third, which is most uniquely advantageous relative to the

problem at hand is the resolution of the problem into symmetrical and

anti-symmetrical parts (Fig. 2) each entailing the use of just one half

of the structure.

This type of solution yields results for both symmetric and non-

symmetric loading if needed and even though it requires two separate

solutions involving one half of the structure the computations are much

less in number than one solution involving the whole structure. This is

the technique used -to analyze the models used in conjunction with this

dissertation.

i i

a,

uh i ulu

Example :

w p

SlIUL i ^ * _

§ ^ w±â-

fit>. £ Statical- AIJTI-

/ > ' 41

IV. THEORETICAL INVESTIGATION /

A. Criteria for Parameter Choice

In regards to the theoretical investigation the first problem to arise

is - what model size and shape should be investigated. In the case of the

writer this problem was resolved by asking the question - what are the

parameters of structural configuration existing in practice? With this in

mind, a definite configuration was chosen

This configuration was chosen because of its simplicity in form and its,

nevertheless, inherent complexity in folded plate interaction. It was

thought that this number of folds would be just enough to substantiate the

trend from folded plate interaction at the edges, to beam action in the

interior without such a large number of folds that computation and instru­

mentation would become an unnecessarily formidable task.

The construction periodicals were then combed to obtain a representa­

tive list of existing ~ and ratios. A careful study of this list

influenced the choice of the following set of parameters as being the set

encompassing the range of values most commonly encountered in the field.

Table 1. Model characteristics • k

Model No. L h

H h

t h

t •

, L h

1 4 1/4 1/45 0.0888" 16" 4"

2 4 1/4 1/20.9 0.1915 -16" 4"

3 4 5/8 1/45 0.0888 16" 4"

4 4 5/8 1/20.9 0.1915 16" 4"

5 8 1/4 1/45 0.0888 32" 4"

6 8 1/4 1/20.9 0.1915 32" 4"

7 8 5/8 1/45 0.0888 32" 4"

8 8 5/8 1/20.9 0.1915 32" 4"

/ - 42

V

B. Pilot Model Considerations and The

Analysis of a Typical Test Model

Before becoming too deeply involved in something that may prove to be

a lost cause it behooves one to engage in a sort of preliminary test pro­

gram in order to more or less obtain a preview of coming attractions. The

function of these crude preliminaries is to establish prematurely whether

our ideas are going to produce results that are in the realm of feasibility.

To accomplish this a pilot model was designed, built, theoretically

analyzed and experimentally tested. The model was fabricated from a 0.030"

thick sheet of aluminum which was subsequently folded into the chosen model

configuration. The model was then loaded with a roving single concentrated

load at the midspan of successive ridges and valleys and the resulting de­

flections were observed. By and large the agreement between the theoretical

and experimental deflections was very good, with the consequence of a green

light for pushing more deeply into the program. A typical set of pilot

model calculations are given on the following pages. This set is based on

the concentrated load being at point C.

r The computed B'^-values could be slightly improved by iterative opera­

tion, i.e. load the structure with the reversed set of holding forces

(R"-values). But insofar as very little would be gained by this additional

step, the operations were terminated at this point, theoretical and experi^

mental data were obtained for loading at points other than C - in fact the

experimental agreement is much better for loading at D but the C-case was

presented because it reflects a folded plate characteristic that will be

discussed later.

' r

43

Also included in the theoretical investigation is a typical set of

model calculations using the particular solution technique. Model No. 6

was chosen for this typical example.

Given:

A o '/

loo

Concentrated load @ midspan

H = 1.812" t = 0.05" p = 26.9° E => 10.1 (10*) psi

h = 4.00" L = 31.25" =<= 53.8°

Determine Z- values

Zab = = 0.0333 in4 — Zab

= (0-05) (16) = 0.1333 in* 'bc

Determine C^-values

= cos 6 , 0.893 vab sin 0.808

=-1.105 (All others ±

Determine slab moment distribution constants,synna. system

b

D.F.

Anti-symm system

A

g zs

o.4^g[O.S7& 0.5 [o.5 O i /

D.F. O'L I 0M8\0.57& O.S7Z\0.4Z8 ! O

(Moment [Distr. :onst.

V

44 > -

Determine plate stress distribution constants

A B C [) Symm. system i i ^ o o.CLl 0.333 0,5 0.5 0.5" 0.5" ii. o C/Stress

Distr. Const.

Anti-sjmm. ^ C D EL

system^^ q.sfp^s o.57£to.4g8 7^0

Determine plate deflection factors

Uniform loading

Plate 31.25'

AB i2Eh (12)(10.1)(10*)(2)

Plate 31.25'

BC (12)(10.1)(10°)(4)

4.03 (lO-S in/psi

2.015 (10"*) in/psi

Sinusoidal loading

plateab 31.25'

Plate

(9.87)(10.1)(10*)(2)

31.25%

4.89 (10'*) in/psi

BC (9.87)(10.1)(10°)(4)

= 2.44 (10"*) in/psi

Determine slab fixed end moment factors

49.7 AB'

Slab : 5v

FEM

45

oop l.

1.812." V

-sycmm. soi-u-^ioivj

h

g 6.9°

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-3 4371437 -304 X/Ô^ - ojddz 4-0.027 -Ù.Z04-

k"

4-ÛÛÛO8 -0.0005 -0.ÛOS4. -V0.0ÛS4+0.04,4-7 -ù'û(/l7rÙJi(fi7

4-cx0008 -0.00%, 4-0.073/ - 0./294

46

PI L O T fMoOE. L - , A/oti - Syma^. s>ouu-r i-oaJ

^ 1, loo' L=3|. "

.@ ©A- ©Ar. <5/W F' ( t W 1 o + /oo • o 6 C/ i -ilO'i + l.lOS +/./&S-/-/OS -/./cs -t/JO^ i-UC6\

(lb) : O o -fiio.'B-uo^S o o oi p (lb)' O • + 1 ( 0 . 5 . - I I O . ' S O 1

O : -tsa> -SG3 o ; ^°(PSi) ! O ^ L470 -G^70 O '\

O . P . / I Û 0 . 6 6 7 0 . 3 5 3 0 . 5 Û . . S 0.^7Z 0.4£S 10 C.O.r, ! ' C.33S 0./67 0.2? 0.&5 0.286 p.2/4

:0 O +UTÔ -C>47Û-C47Ù +C47Ù O 0

1-/-^ .. . --J— 1 +1 r<30<S3 O 4-6254 -5/?2é-455S 4-(,686 O |C> -+41'?/ +434 - 45^^ -Zfi,2A-hzSCZ^

(fsO h.2.062 +417/ -/W +£S62, O A^'(psi ! - 6&S4- + 9/65 : —7SS4- J.S^Z

^Azb i 4-0 3 3 . 0 I S Q , 0 / ' S < 0 . 0 / 2 Ix/gT*"

( w ) ( w )

+;?5.aO -hl&.4C:> . + IS.53) -+ S.77 lx/3' ( w ) ( w )

+J7.S'S fZû.4û +^0.40+17.49 ' +17.49+4.3S +6.3S-W@ ( w ) ( w ) + 48. J S +37.89 +J3.87 O ( w ) ( w )

i + l O » S < o + I 4 - ' C ) ^ + 2 ^ 8 ^ ! - + S / 6 + 6 9 7 + / / S 6 Ix/cf

O-FT o i / Q.dZ^o.S7iL 0.^7^0.^£S /10 Co. FT O !Q.S O.'S/i 0,264 o.£S6o.Z.I4- <a5!

+ 51^ -S-/5+^97 -<^97 +118G -//86 p—• /^5? 1 /o-,, 5^93 "'"^

;

! i-^.' 1 + 5-15 -.75'2 ivSSO -S-/4+S93 -''^6 i-g/S +a.7% ! -365 +t34-'-473 i o + 1 5 ; + I S f / ^ O + / 2 0 o k / o " '

m" Q +<0.0/5 +OJSLO O c'4^^:) \-Ô.ÛÔ4t-^0.Cù4^l-Ù.OZ9 +<3.0a+0.03S -(K>5J^+M«

R " C % ^ —0.0042, —û.cù-^S +-Ô'Ô42. 0

47

When the symmetrical and anti-symmetrical solutions are superimposed

the résultant applied loadUbecomes 260j^^. It therefore becomes necessary

to divide all results by 2. This could have beéi( avoided by using a /

for each loading to start with but 100 was used' because of numerical

simplicity. . Results :

A B C D F G H I J

-2108 +4215 -5118 +3310 -1655 +3310 -5118 +4215 -2108

°200 -2083 44171 -4992 +2862 0 -2862 +4992 -4171 +2083

-4191 +8386 -10,110 +6172 -1655 +448

vO CM r-

4 1 +44 -25

"lOO' ^2095 44193 -5055 +3086 -827 +224 -63 +22 ^(psi)

48.95 39.55 29.81 22.10 29.81 39.55 48.95

eV

® 200 48.24 37.89 23.87 0 -23.87 -37.89 -48.24

97.20 77.44 53.68 22.10 5.94 1.66 0.70

«v ° inn u-/./',48.6 38.72 26.84 11.05 2.97 0.83 0.35 (inxlO"^)

Following are the complete pilot model results:

48

V

Load at

Dial at

B

C

D

F

G

H

I

C

Test Theo

37 49

35 ' 39

25 27

9.5 11

3.8 3

1.4 0.8

0.7 0.4

I

Test Theo

16 16.7

25 26

33 34

25 25

11 10

4.2 2.9

2.7 1.2

Test Theo

5.2 4.5

11 11

25 26

37 35

24

11

26

11

6.5 4.5

Test Theo

2

4

12

1.2

2.9

10

27 25

37 34

27 . 26

18 v 17

H

Test Theo

1

2

4

10

24

38

42

0.4

0.8

3

11

27

39

49

Model Characteristics

E = *' f " i' 5 ° iï L = 32.00in, h = 4.00^", t = 0.191%

© •

0

3.876' ^ = 14«4s

0 = 35.94,°

Compute Z values

= 0.506 in3

«12 = = 0.127 in^

Compute C values

for ûifltctio/j

C21 = = -2.00 (all other C^ » ± 2

49

Compute C values'

c" =;SÏn p _ô.517 (all otheir c" = -0.517 sinoc

bet&rmi^e slab moment distribution constants

, t \

Symmetrical system:

D . F . O T 1 0.428 to.572 oTsTo.J 0 1

Anti-symmetrical system:

' w k j l I

0 1 0.428 0.572 0.572 0.428 1 0

Determine plate stress distribution constants

Symmetrical system:

1 —

1 0 0.67 1 0.33 0.5 0.5 0.572 0.428

Anti-symmetrical system:

0.67 0.33 0.5 0.5 0.572 1 0.428

Determine plate deflection factors

Factors for uniform loading:

12, 9.6eh 32'

9.6x10x2 5.33x10'* in^/lb

Plate 32'

9.6x10^x4 2.67x10"* in^/lb

50

Factors for sinusoidal loading

Plate 1. 32''

1-2, 9.87Eh 9.87x10^x2 = 5.18x10-6 ifi^/lb

Plate 32'

23, 9.87x10 x4

,2.6.0x10" in /lb

Determine slab fixed end moment factors

F.E.M, 6E*I

/ \

d — 4

10^x0.19^

2(l-(j) 4)(3.876)

2.49x10 X

Determine F.E.N, for symmetrical case

1_ @ @

t ^ O 9 » I ' ! <y <y v o* ^ \ «t v v _ \ — —

r-FEM = 10 X = -18.8 in-lb

FEM = 10 %: 3-876 = -12.5 in-lb 12

Determine F.E.M. for anti-symmetrical case

10

r i ' ' 1 0 f t 0 n\ 0 k 1 3) i >

' 1 f -/s8|f/2.ff -zg-sj-z .s +/ .9i-/g.s

VI "2 O

V <

:5 u j <

o|

2

j u) Ù O $

kl 0 li

tl j (f)

1 0

F 4 (y ui a-

0

tû

0

h <

p

CÀ uj (k

0-h u

ol

12

(©

©-

II

J w a o

—bf (G)^ (\

ri

h

o

2r <3

«0 »ô

CO

'] 5 iv)

\4-

\d n oo

g rô

« qo

s X

o

S3 U) i »o O "V

O

i 5 s

10

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O'*^!

«18

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l"

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+

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_Ô in

Ô

ip

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ill

m iô

m M

s

d

é os: U 4 cl \i)

lil à u) 5

52

jsï ? r fo f

\ t [

^ Y i

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+ vî

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54

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N- i eo

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CIS

m

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56

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po

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59

Indlcial equations for model No. 6

af3a + bpgb + basic = °

aF4a + bF^y + cF^g + 4 basic " °

af5a + bfgb + cfse + r"g ^asic = °

Substituting values from operation tables gives;

676a + 88.89b - 56.9c - 281.2 = 0

-330a - 219.67b - 127c + 196.7 = 0

.40.5a + 232.23b +• 310c - 86.4 = 0

The solution of which is:

a = 0.418, b = 0.135, c - 0.232

Compilation of Results;

i5zx-5

? > u <

m' -18.8 -10.80 : -12.98 -12.26 Ui

> u < a -67,072 +51,970 -42,303 +39,915 -39,657

Vj cë b" +1.7740 +0.9432 +0.8644 +0.8504 i/^ <

DÛ c U; C/7

ra" 0 -494 +100 -67

u < s a ' +39,350 -10,330 -5,770 -796 +8,970 JS y r

<?

6v }x0.418 -0.5375 +0.0021 -0.0250 -0.1017 JS y r

<? m" j 0 +364 -41.7 -74.4

a < s 0 ' ^ -1,032 +2,060 -128 -1,547 +775

y H-

•£ Ui 6V J^xO.135 40.0435 +0.0040 -0.0194 -0.0241 ar

dî ln % m" J 0 -7.65 -9.5 -1.1

60

i / 1 +442 -885 +2,434 +663 -5,090 6 § s" a ôV ^xO.232 -0.0310 -0.0264 +0.0207 +0.0597

m" j 0 -26 -0.75 • +48.7

J < 2

a -28,312 +42,815 -45,767 +38,235 -35,000 J < 2 +1.249 +0.9229 +0.8407 +0.7843

ll m -18.8 -174 +35. -106

58% Corr.

to-' basic system

r 315%

C. Dimensionless Parameter-

There could conceivably be times when it would be desirable to carry

out the folded plate solution to some predetermined degree of accuracy.

In fact it could be that the basic solution alone would give answers to

the desired degree of precision. In order to obtain a measure of the

importance of the particular solution corrections the dimensionless para­

meter ^ was developed. It was developed by making a step by step check

of the operation table and thereby establishing the proportionality

existing between r" and the system parameters (l, t, h and h) - also

between F and the system parameters and then dividing the r" - propor­

tionality by the F -. proportionality. The result of this operation was

X' <ïï>'

\ '

From this it is seen that ^ is a measure of the ratio of superfluous bI'

to the activating force f'. It was therefore thought that would also

be a measure of the correction to the basic system afforded by the parti­

cular solutions.

61

To confirm this reasoning a plot (Fig. 3) was made showing the

relationship existing between ^ and -i at first valley C and F ^ '

between ^ and the percentage correction to the basic edge beam at point ii

A. As seen from the plotj X vs. —f— is linear as anticipated and the F

percentage correction to the basic system increases at a decreasing rate

as A increases. In fact the trend is toward a percentage correction

that is asymtotic to some limiting correction value. The significance of

all of this is that in knowing A for this particular configuration we also

have an idea of the magnitude of particular corrections to the basic

system and therefore can predetermine their importance.

D. Check System for Theoretical Calculations

After the execution of a complete set of model calculations it is

comforting to have a check system to insure the correctness of our mani­

pulative operations. No check will tell us that we have used the wrong

geometry, loads, or material properties but once we have selected these

quantities, the operational correctness will be reflected in the degree

to which the statics and geometry of the problem have been simultaneously

satisfied. With this in mind a system based on slope deflection was

devised for checking the compatibility of the m and 6 values.

Starting with the well known slope deflection equation

^ab = [Ϯa + ^ab where:

m J noje M is

AiJAUOGOUS

TO lU

62

ago

<340-

300-

/ao-

/20-

80-

vs A"r SUPERFLUOUS f'^ actuating force.3

vs CORRECTlOis l TO g_og.e &e=ai^ / strq,5s

corî^tctvoni to ed^ s.ea,fy/\ r5:$$

/éo

A6.1) A FU'^CTlOi^i op \

63

The following relationship was derived:

(2CAB " ^BA) ~ (2CAO ~ CQA) 3EI 2hd "^OA

- asy

This has direct application to the following real situation;

Comparing the final values obtained in the folded plate solution

to those determined by using this equation gives assurance that the mani­

pulations involving M - values and 5-values stand a good chance of being

correct.

Another check to make is based on the longitudinal stresses and is

carried out by the following operation:

64

and

/y C/s

o

v

y

y ^

V\Z"=TOTM- LOA.D otl modfc-l-

E. Stress and Moment Determination at Gage Location

Inasmuch as it vas thought to be unwise to place gages directly in

the sharp folds of the structure the need arises for the computation of

the theoretical stress and moment at the point of experimental measurement

because the folded plate theory gives the stresses and moments at the

ridges. The following reduction formulas are based on the already stated

stress and moment patterns.

basic La

particular

y rrn>v. ^ ^<ftttnrittr^

m

Parabolic

basic

L

Sinusoidal

ri asic

Zm particular

m. y

= M i i i i i i m m /rtt' i i rn">x, + i i ttttv^

h- ^ Uniform Almost Sinusoidal

I

_ L ».

Sinusoidal

\ uuittl^»"

a

65

m basic at any

gage

™ = m. ± . at mid-span i h i-*-j

gage

at any gage

-H

Am^^j -

[h 2 2-y

1 -h 2

1 -

Ih t?

\ l l

iwd^

|wd2

lh\ ' •

^basic

at mid-span gage

F. Beam Analysis

- For comparison purposes the folded plate system was assumed to behave

as one huge simple beam and stresses were thereby determined. The most

important phase of this endeavor was probably the moment of inertia computa­

tion technique which hinges upon the following chosen element of plate

system as being basic to the complete solution.

°ï2 t\

COS 2 + th^ sinf ^

The remaining operations in determining the moment of inertia a,ad locating

the neutral axis of the whole system are quite involved from a manipulative

standpoint but follow from routine statics. The final step in the solution

involved the application of the flexure formula.

66

V. EXPERIMENTAL INVESTIGATION

A. Loading System

Before the design of the experimental set up could be initiated the

question of what type of loading was to be administered and what quantities

were to be measured, had to be answered. The design was strongly influenced

by the pilot test observations, wherein the rather thin model was subjected

to a concentrated ridge load. The question was - were these realistic

conditions? Even With this rather crude setup the correlation was excel­

lent, but the outcome was probably influenced by the favorable conditions

of thin plates having smaller secondary effects and the load, being applied

at the ridge, not having to undergo the physical process of finding its

way to a ridge as would a uniform surface load. In other words when we

apply the forces at the ridges, they are already "there" - but when we

apply the loads on the surface we have to "assume" that they get to the

ridges. Therefore the former case is circumventing a part of the assump­

tions and thereby stands a better chance of producing effects that are

closer to those theoretically predicted.

With this background thinking, the decision was made to use a uniform

surface load which was simulated by a pattern of discrete loading pads.

The pads were fabricated by splitting 1 in. wood dowel stock in half,

gluing a layer of form rubber on the flat side, and then drilling holes

for the wire load hangers as shown:

To /\cc0<v»0 DATE-

Di; 'peR&^-r fSool=

SLoç»e.

67

The pads were uniformly distributed on the surface in a pattern such

that a pad was centered on every 2-in. x 4-in. area. The load hangers then

formed the upper part of a whiffle-tree mechanism, Fig. 4, that served as

the transition between the concentrated applied load and the simulated

uniform surface load. The load application mechanism consisted of a 12-ton

hydraulic jack, enclosed in a frame designed for transmitting à vertical

load to the structure. Fig. 5. The load measuring system originally con-

sisted of a 3-range (low, medium, high) system composed of 3 pressure gages

that were tapped into the fluid reservois of the jack. Fig. 6. As a con­

sequence of the internal ram friction, difficulty was encountered with this

measuring system and calibration of the system yielded plots with the

following characteristics:

Judicious use of these calibration curves could conceivably have

resulted in reliable load determinations were it not for the fact that the

nature of the test procedure dictates that the load be sustained at each

increment for a substantial period of time. During the actual model test­

ing operation there would be an element of uncertainty as to where on the

calibration curve we would find our true load.

With this outcome the decision was made to use the pressure gages as

approximate load indicators and incorporate a more dependable load

losal plot, ra

IRA

friçt1û<v1

trut- l oand (e>y macki/oe.)

<^acl<^ 1 es r at i ojsi cui^n/êt

Fig. 4. Whiffle-Tree Mechanism

# -, •• • f

Fig. 5. Jack Assembly

Fig. 6. Experimental Set-up Showing

Pressure Gages as Load-Measuring

Device

74

measuring device in series with the jack system. Subsequently two SR-4

strain gage load cells (low range 0—>— 5000^^, high range 0—^-20,000^^)

were designed by mounting SR-4 gages.on high strength aluminum rods.

Swivel end-connectors were designed for the rods to eliminate as much load

eccentricity as possible. Four gages were used per load cell - 2 diametri­

cally opposite longitudinal gages and 2 diametrically opposite transverse

gages. This arrangement corrected for eccentricity, compensated for temp­

erature and increased the sensitivity approximately 30%, Fig. 7. This

high capacity loading system was designed to enable plastic behavior obser­

vations to be made.

B. Deflection Measurement

1 " For deflection measurement,Yôôô dial indicators were pulled with

fine wire to alleviate transverse contact effects that sometimes arise

when the structure has a deflection component that is perpendicular to the

axis of the dial indicator stem. The fact that there is no measurable

error induced by this pulling system was confirmed by placing two indi­

cators in series - with a 12-in. spacing, and observing any differential

dial movement between them.

C. Stress and Moment Measurement

The next step in the experimental program involves the measurement of

stresses and moments. Care was taken to select strategic gage locations

what would reflect a representative behavior of the whole structure with­

out employing an excessive number of SR-4 gages. As it was, each of the

8 models used a total of eighteen.90°-rosettes which required 36 gage

Fig. 7. Small Load Cell (0-^5000

76

\

77

readings per load interval. It was thought that the gages would reflect

a truer strain picture if they were not placed right in the sharp folds

of the plates, so the location in the majority of cases was 0.25-in. from

the folds. The gages used (AX-5-1) had a 7/16-in. gage length which

obviously caused them to still be awfully close to the folds. There were

essentially 4 gages at each location (90°-rosettes, top and bottom) pur­

posely placed to facilitate the determination of membrane stresses and

slab moments. Figs. 8 to 15.

The desired quantities in these tests were plate membrane stresses

(CTX) and slab moments (MY) which are the primary quantities predicted by

X the ordinary folded plate theory; g.

dottof/s

The other quantities (m^ and Oy) are assumed by the theory to be

insignificant, which is an assumption requiring experimental confirmation

or refutation. Since any linear strain pattern occuring across the plate

thickness can be resolved into a membrane strain (average strain, or strain

at mid-plane of the plate) and a moment strain (Ac from top to bottom of

plate) as shown in the following illustration -

1 % . 1

l==s'

" ©otto/^

/ / / /

A

-h

•Ave te AS =

78

o Te rsù "k • L_ O'Neal Tu D 1

lop \/ielu)®(d"^e'^si6'^^ a re mot hor%.. proj,^ "ZZl- —

v-f

0.25 o.zs

Fig . 3 Gage amd diau Placement (^modti- 1

79

Gagse TR.A.K1 SVSR. 'âE. • \_ O fo O 1 T-u O I/vj <4.1—

A & I J

(31 0%: g) b (c ]0 • ^

0 .0 h

<9 h

Rear . cag.e une

m\d- s?afj

fsoisit

Emd

tor vievuy div\e./xisioi\^s a.pe mot hoe.2., proj, ) r6.a,r,

i

G;

H

bs. o,zs"

4"

h -H 0.25' o.e-s

@

^ . 0.2s

r

8'

feoiviv

fit, 9 gage. awn placetva^mt ci^ooe.v_^)

80

• L O'vl GI'tmok»»

G,

fiù». 10 gage, dia,'v_ pca.ceme^t ( mopni- 5 )

81

i ^ ^ A. frO s • L 0 IADi ^ I—

@

0

M ID-SPAN

Top VLELWXÙLCV/TENSLOMÇ. ARE AJOT HORG. proje^ r%^__ /

Rear^ G Ate LWE.

mio-spakj

0.2S

0.2S" O.ZS o.zs

front

F'g. II G a6s a AUD dial PlAC^imç/OT C A-I oCse.L_ 4-^

82

G A G £ o T r V " c . * Lo-v^GaVT ru cs'i'O /X

A e>

mio-spai^^

(51

Rear gaqe liaiti

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It follows that the membrane-stress equation

<5^ f=< + «yl resolves itself into ^^ave ° pXave^^yayl] l-M -+ 1-j4 *— —'

It also follows that the slab moment equation

k = d-^^2 . 2 + ^ — .a y S

Et'

12(1^^)

The solution of these experimental equations for stresses arid moments

would have necessitated the employment of a multiplicity of arithmetic

operations were it not for the switching unit that was conceived and de­

signed expressly for this operation. This unit was designed to be used in

conjunction with a regular SR-4 strain indicator. Fig. 16, and operates

according to the following schematic:

R

ix" gj a c3&-?

L/VTEl

MOW / MEN\

L, u switch

0

du^a^^y g\(j,er ç.l'ra./hj

XiviDicftTOR. rlctr- V Schematic of gage hook-up ^ * K-c-ui-y

For membrane readings the switch is flipped to membrane to automati­

cally complete the following circuit;

Schematic for membrane stress

Fig. 16. Switching Unit

88

89

For moment-readings the switch is flipped to moment to automatically

complete the following circuit: L,

-ll

( ^ / / / / / / f

Schematic for moment reading

This technique reduces the strain readings to a form directly appli­

cable to the membrane and moment equations and with several thousand read­

ings to make any reduction in tedium is a labor economy.

D. Property Tests of Model Material (Aluminum 1100-H-14)

The experimental stresses and moments are accurate only to the degree

of our knowledge of the material properties. The 2 obviously significant

properties are E and as seen from inspection of the experimental equa­

tions. In order to discern these characteristics, the 3 following basic

approaches were used on strips taken from the actual plates that were to

later be formed into the test models:

(1) Three simply supported beams on edge

1-iM J -j

o.ossS " "—n

o.iqis-^ o.

P

^7

I2.3S

Result: E = 10(10) psi

90

(2) One simply supported beam - flat

0.0888

?

w- 3 o

Result: e' was 11.0(10) psi, which indicates that Poisson's

Ratio is closer to 0.30 than 0.33 or possibly that the strip

wasn't wide enough to develop complete plate behavior.

(3) IWo axial tension tests

ir

s)

El" IoOq) PSÎ

= isjooopsi

J" O, /2.S5

When models are loaded into the plastic zone this knowledge of the

proportional limit makes it possible to distinguish between non-linear

relationships caused by material properties and those caused by secondary

geometry effects.

E. Effect of Gage and Adhesive Thickness

On Moment Determination

It was thought wise, in light of the rather thin plates, that a check

be made to ascertain the relative importance of correcting for any error

that might be introduced due to the gage wires not being on the surface,

but at a distance from the surface equal to the thickness of the adhesive

layer plus approximately % the gage.thickness. The scheme for making this

91

check is based on the fact that deflections are just manifestations of

strains. Consequently if we know the deflection, and the strain pattern,

an expression can be derived in the form e - f(6), which enables one to

predict strain without using the modulus of elasticity.

The procedure was essentially the conjugate beam method whereby a beam

is loaded with a known pattern of angle changes. Ihe present case was ^

p % ,«• executed as follows

0.0888

,^phe

6c 70.0444-

}0-04U Jl-

= 0.110'

6t& , 2

Then; e s fll-375\ (6)(0.0888)(0.110) , , .-6

v 12 y ^24)2

But G = 105<<^"/in Error = 8,8%

The implication of these results is that the gage wires are probably

located a distance from the surface equal to 8.8% of t/2, i.e.

8 . 8 100

(0.044) = 0.00392"

a value very closely approximated by taking the micrometer thickness of

the eaee^alone and adding 1/1000" for adhesive thickness. Actually in

92

order to arrive at this adhesive thickness the gage was measured before

attaching it to the member - the member thickness was measured before, and

the combined thickness of gage plus member was measured after. It follows

that the experimental stress formula must include an adjustment in the

measured curvature, i.e. t' rather than t must be used for ^ a asy ax t' ,

a n d — r ^ . O b v i o u s l y w h e n t h e p l a t e t h i c k n e s s i n c r e a s e s , , t h i s ôy^ f

correlation rapidly becomes insignificant but when the thickness is.de­

creased the error can easily become 10% or 20% depending on the gage type

and adhesive.

F. Testing Procedure

The actual testing procedure consisted or applying approximately 10

to 15 load increments with the jack and simultaneously recording the strain

and deflection readings at all the designated locations. All of the models

were loaded into the plastic zone, usually to the limit of the model's

capacity or in the case of excessive deflections, to the limit of the

allowable distortion in the whiffle tree mechanism.

G. Typical Set of Model Data and Data Reduction

Insofar as model No. 5 exhibits some rather interesting behavioral

patterns it has herein been chosen for sample presentation. The compiled

results from this model and all of the remaining models are found in Table

3. (Page 108-115)

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('«/) — —

^jâs £>„. = -asi

6% = u,zs^4co+-'&4j|-—/^ylzo p^'

ttlv), 0,0069^-jzsg 4. \sj =- -^.-so

7/1 5^4; "" q' 17

AI

IN -U»

inj

C?00 &00 /ooo

107

0VS7 0.2.4S

I O C soo

Table 3 Compilation of experimental and theoretical results

Model No. 1,200 lb

Expr. Theor. . v Beam %E

Expr. Theor. , \ %E (""y)

Expr. (m^)

Expr. The or. Beam ^ ^ %E

A 6 G D ' E F

-5,540 +5,430 -4,590 +4,920 -34 -5,150 -7,550 +6,780 -5,440 +5,174 +42 -5,080

+313 +4,680 -5,500 44,680 ' -416 -5,500 +36 +25 +19 +5 XX -1

-0.24 -4.59 -1.84 -1.56 +1.34 -1.85 -3.96 -3.62 -2.03 +1.75 -2.41

. • • • . -14 +97 +30 +30 +30

-0.08 -0.84 -0.33 -0.06 +0.51 -0.19

0.046 0.032 0.031 0.036 • • • . 0.065 0.035 0.032 0.031 . . . 0.031 0.031 0.031 0.031 . . . +41 +9 +3 -14

Model No. 2 4,000 lb

Expr.

%E

Expr.

(-y)

Expr. (m^)

Expr. The or. /.V\ Beam > 7oE

-8,000 +7,760 -7,020 +7,370 +191 -7,080 •10,700 +10,220 -8,550 +7,950 +150 -7,650

+464 +7,020 -8,260 +7,020 -623 -8,260 +34 +32 +22 4« -22 +8

40.33 -14.60 -15.85 +3.68 +9.38 -5.58 • • •. -14.44 -30.30 -3.12 +6.08 -11.22

-1 +91 XX -35 +100

+14.38 +7.92 +2.74 +9.25 +10.34 +5.58

0.068 0.042 0.041 . . . 0.043 0.096 0.053 0.049 . . . 0.048 0.047 0.047 0.047 0.047

441 +26 +20 +12

Ox(psi), my ( ), mx ( ) , 5v (in), Number under Model No. is load

at which tabular values are evaluated..

All errors to nearest 1%

+ Error signifies that absolute value of Theor-absolute value of Expr.

XX Indicates the data obtained was erratic

109

g h i j k l

• • • -6,000 -4,060 -3,700 • • • -7,550 -5,660 -3,820

+313 +235 -4,120 +26 440 +3

• • • -0.84 -0.11 -1.67 • • • -2.34

440

-0.59 +0.28 -0.29

0.039 0.041 0.055 0.032 0.035 0.065 0.031 0.031 0.031 -18 -15 +18

• • • -8,030 -5,660 -5,250

• • • -10,700 -8,080 -5,760 e # # 4464 +348 -6,200

+33 +43 +10

+1.14 +0.57 -4.45 -10.08

+127

+17.60 +9.68 +4.45

0.045 0.046 0.078 0.049 0.053 0.096 • • •

0.047 0.047 0.047 +9 +15 +23

Table 3 Continued

Model No . 3 2000 lb A B C D . E - F

Expr. -6,170 +5,325 -3,580 +3,470 490 -3,330 Theor.

(<^x) -5,150 44,530 -3,580 +3,430 +20 -3,405

Beam (<^x) +334 +3,146 -3,710 +3,146 • -279 -3,710 7oE -17 . -15 0 -1 -78 +2

Expr. -0.62 -9.20 -1.83 -2.64 +1.16 -2.08 Theor.

(my) • • • -5.18 -3.14 -3.06 +2.30 -3.01

%E (my) • • « -44 +72 +16 +98 +45

Expr. K) -1.87 -3.07 -0.70 -0.88 -0.27 -0.89

Expr. 0.023 0.011 0.012 0.011 Theor. (6V) 0.017 0.009 0.008 0.008 Beam

(6V) 0.008 0.008 0.008 0.008 %E '• • • -26 -18 -33 -27

Model No. 4 12,000 lb

Expr. Theor, . . Beam %E

Expr. Theor. , v %e ('"y)

Expr. (m^)

Expr. Theor. ,gv\ Beam %E

-14,300 +12,720 -10,180 +9,600 +382 -8,850 -13,900 +12,500 -10,040 49,550 +75 • -9,400

+576 +8,675 -10,220 4^,675 -770 -10,220 -3 -2 -1 -1 -83 +6

-0.90 -34.80 -23.53 -9.85 +10.50 -13.80 -31.90 -30.90 -15.92 +14.25 -19.62

-8 +31 +61 +36 +42

+3.57 -5.61 -5.61 -1.07 +2.60 -2.68

0.054 0.033 0.028 0.024 . . # 0.048 0.026 0.024 0.023

0.023 0.023 0.023 0.023 -11 -21 -14 -4

Ill

g h i j . k l /

-5,510 -4,370 -2,460 -5,150 -3,860 -2,555

+334 +250 -2,780 • • • -6 -12 H

-0.37 -0.42 -2.10 -2.99

• * # +42

• • • -1.11 -1.25 -0.85

0.011 0.012 0.019 0.008 0.009 0.017 0.008 0.008 0.008

-27 -25 -10

-14,250 -10,660 -6,910 -13,900 -10,400 -7,050

+576 +432 -7,660 -2 -2 +2

0 0 -14.03 -19.00

+35

0 0 -3.77

0.025 0.028 0.051 0.024 0.026 0.048 0.023 0.023 0.023

-4 -7 -6

Table 3 Continued

Model l)o. 5 1200 lb A B

Expr.

cx)

%E

Expr.

("y)

Expr. (m^)

Expr. Theor. ,gv\ Beam %E

-5,930 +9,550 -9,900 +9,300 +214 -9,020 11,120 . +12,500 -11,470 +10,200 +418 -9,330

+626 +9,360 -11,000 +9,360 -832 -11,000 +87 +31 +16 +10 +95 +3

0 -3.29 -5.73 +2.39 +0.93 -3.00 -2.57 -10.56 +0.85 +0.64 -3.56

• • • -22 +64 -64 -31 +19

+1.70 +0.19 -0.96 +1.83 +1.00 -0.14

0.300 0.221 0.221 0.215 0.457 0.278 0.252 0.239 0.248 0.248 0.248 0.248

+52 +26 +14 +11

Model No. 6 2000 lb

Expr. -2,620 +6,230 -7,170 +6,520 +236 -6,460 Theor. -4,880 +8,340 -9,030 +7,560 +374 -6,810 Beam +250 +6,800 -8,050 +6,800 -623 -8,050 %E +87 +34 +26 +16 +58 +5

Expr. -3.66 -18.80 +9.68 -0.17 -12.40 Theor. -5.82 -36.20 +6.83 -3.87 -20.80 %E \™y/ +59 +93 -29 XX +68

Expr. K) +10.26 +8.60 +1.75 ' +11.45 +7.20 +3.13

Expr. 0.193 0.155 0.146 0.143 Theor. /•rVn 0.286 0.213 0.194 0.181 Beam \P J 0.186 0.186 0.186 0.186 %E +48 +37 +33 +27

113

g h i j k l

• • K -5,840 -3,120 -4,120

.• • • -11,120 -5,100 -4,150 +626 +274 -4,810 +90 +63 +1

• • • 0 -0.10 -1.50 -2.04

• • • +36

• • • +2.02 +0.39 -0.17

0.228 0.239 0.342 • • •

0.252 0.278 0.457 0.248 0.248 0.248 +10 +16 +34

-2,860 -1,700 -3,250 -4,880 -2,580 -3,040

+250 +103 -3,530 +71 452 -7

0 -3.96 -9.00 +127

+11.20 +4.99 +2.44

0.159 0.170 0.206 *1 * * 0.194 0.213 0.286 • • • 0.186 0.186 0.186 • • • • • • +22 +25 +39 # «l #

Table 3 Continued •Ur

v

Model No. 7 2000 lb A B c D E F

Expr.

Theor.

Beam

7,E

-8,020

-9,280 +668

+16

+8,130

46,780

+6,292 +8 '

-7,000

-7,310

-7,420 +4

+7,000

+6,860

+6,292 -2

- • +75 +108 -558

+t4

-6,520

-6,640

-7,420

+2

Expr.

Theor.

%E (my)

-0.22 -3.38

-2.85

-16

-4.29 -5.42

'+26

-0.37 -0.76 +106

40.69

+1.24 +80

-1.82

-2.10 +15

Expr. K) 0 -0.76 -1.43 -0.09 0 -0.67

Expr.

Theor.

Beam

%E

(6V)

0.120

0.134

0.067 +12

0.075

0.075 • 0.067

0

0.073

0.068

0.067 -7

• • • 0.071 0.066 0.067

-7

Model No. 8

6000 lb

Expr.

Theor. ( \ Beam ^

7oE

Expr. Theor. , \

Expr. (m^)

Expr.

Th«

Be: 7JE

Theor. /_V\ Beam ® >

-9,980 +10,900 -9,850 +8,970 +372 -8,250

-9,810 +11,440 -10,700 +9,450 +396 -8,650

+576 +8,675 -10,220 +8,675 -770 -10,220

-2 +5 +9 +5 +6 +5

+0.22 -8.75 -28.4 +3.74 -1.30 -16.15

-10.57 -46.00 44.33 +2.28 -15.80

+21 +62 +16 XX -2

46.00 +3.61 -6.78 +4.51 +2.10 -3.00

0.155 0.108 0.094 0.085

0.166 0.103 0.093 0.088

0.092 0.092 0.092 0.092

+7 -5 -1 44

115

G .H . I J K L

-7,420 -2,900 -2,970 -9,280 -4,120 -2,910

+668 +292 -3,250 +25 +38 . -2

• • • • « « -0.17 -0.22 r i . 28

• • • -1.71 .* ' # # • # +33

-0.06 -0.38 -0.43

0.075 0.077 0.117 0.068 0.075 0.134 0.067 0.067 > 0.067 . . .

-9 -3 +15

-4,530 -3,610 Faulty -4,560 -3,880 Gage +252 -4,480

# # # +1 +7

• • • -11.96

• • • • • • -8.75

-27

0 -3.10

0.088 0.104 0.157 • # . «

0.093 0.103 0.166

0.092 0.092 0.092

+6 -1 +6

116

VI. TEST RESULTS - INTERPRETATIONS AND OBSERVATIONS

, . '

Ai a - values ,

\ -After a critical inspection of Table 3 it is seen that the theoretical

correlation of the a -values is in general, best for the steep-pitched *

configurations and worst fgr the shallow. The length and thickness have

a less pronounced effect on the accuracy of stress prediction but still

proved to be influential. The shorter model configurations tended to have

better correlation than the longer ones and the thicker configurations tend

to have better overall correlation than the thinner ones.

As was stated before, this model configuration in general has proved

to be an extremely interesting one because of its extreme sensitivity to

subtle parameter changes and secondary effects due to the changes in geome­

try caused by deflections. There even seems to be a hidden thrust type of

interaction between the valley folds that doesn't show up in the theory

because the theory presupposes all loads to be transferred to the end sup­

ports through the action of shear in the plane of the plates.

*.

The deceiving aspect of this general configuration lies in the seem­

ingly innocent form of repeated ridges and valleys with its anticipated

simple behavior. First of all, in regards to shallow models the plate

thickness is a significant % of the total model depth, whereas our theory

assumes the model material to be concentrated on a working line represented

by the mid-thickness of the plates. In essence we are saying, as in the

case of models 1, 2, 5 and 6, that the model has a total depth of 1-in.

when in reality the model is 1.0888-in. or 1.1915-in. in total depth. This

fact helps to explain why, when we get away from the outside edge of the

117

model, the beam method is at times even closer to the experimental than the

folded plate theory. This is because the beam method utilizes a moment of

inertia that includes the entire model depth. However, the beam method is

dangerous to use because of the gross errors in predicted stress at the

boundaries.

Conceivably another deterrent to good correlation lies in the fact

V V that in the shallow, long cases the difference between 6^ and 5^ can very

easily create an increased effective depth of the overall configuration In

the magnitude of 10 to 20 per cent, i.e.

F

Increase in effective depth

This effect, coupled with the foregoing thickness effect, probably

accounts for the fact that with shallow configurations all of the experi­

mental values tend to be considerably less than the theory predicts -

whereas one would logically assume that some values would be greater than

and some less than predicted due to the fact that both the a -values *expr.

and the "^xtheo must form the same magnitude of internal resisting

moment.

Another very elusive effect which seems to have entered into the

picture is what could be called a side thrust phenomenon due to deflection

geometry. The behavior of the pilot model first gave clue to the existence

of this behaviorism. If a comparison is made of the correlation of 5g due

to a concentrated load at the fold C and 5% due to a concentrated load at •— d

118

fold D, it will immediately be seen that when the load is at D the correla­

tion is much closer than when the load is at C. For both loading conditions

the ££)^ between B and C is approximately the same, which dispels any notion

that the transverse slab effects could be the explanation. Careful con­

sideration uncovered the fact that the basic difference between the two

loading conditions, in reference to their effect on 6g is that when the load

is at C, plate BC is being deflected by a system of forces having a very

large resultant in the plane of the plate, i.e.

co / /TP//

When the load, is at D, there is no resultant force in the plate BC in

the direction of p„ other than those that will be induced by lateral slab

action. The plate BC is hereby bent in its plane by the effect of stresses

being fed into it at its edge at C. The question is - what is the signifi­

cance of this? If we observe that after loading, p^g can be thought of as

acting on a beam that has been deflected normal to the plane of loading, it

will Immediately be seen that for the plate BC to be in equilibrium there

must be an additional set of edge forces (Q) distributed along the edges

B and C, e.g.

119

G MO SiAf 90 RTS

qnr 77777/

MiD-'îpA^Ni ùeFLÇcTeo PosiT/o'vl

For the concentrated load câse these edge forces will tend to be

distributed in such a manner as to nulify the torsion that would tend to

exist at each section along the plate, i.e.

it nx

QdxH =lpggdz, where z%A sin j-

/.QdxH -ipçg|A cos

J r> I ItA JtX

zooe.

^ -0

As shown above these Q-values are holding forces, so their effect on

the structure is reversed, thereby creating the so-called side thrust

phenomenon. If instead of a concentrated force in the plane of the plate

we had a distributed force p = p^^ sin then the shearing force at any

longitudinal section would equal p„„ cos and from the foregoing

analysis, QdxH = p^^ cos ~ cos ~ dx or Q = p^^ — cos nx . L L

JtX

L A

'CB H 2 jtx L

It is believed that this kick-out or side-thrust force is partially

responsible for the greatly reduced edge beam stresses in the shallow con­

figurations and by consequence the reduction of 6^-values. Of course there

are other factors that contribute to the error In stress prediction but

120

they are readily mentioned in the literature and have already been

mentioned in a previous section of this dissertation. -,

B. 6^ - values

All of the aforementioned items in regards to the - values are also

applicable to the correlation of 6 - values because deflections are just

manifestations of the stresses and strains. There is however a very impor­

tant additional consideration to be made, especially in regards to the

short models. This involves the significance of the shear deflection.

Models 3 and 4 are glaring examples of this phenomenon because in these

models, the aforementioned elements that have in other models produced

conservative results, have here,been overpowered by shear effects that

cause experimental 6^ - values to actually exceed the theoretical. In the

case of the short, shallow models (1 and 2) the shear effect is present but

is not prédominent over the other deflection-reducing effects.

With the interaction of all of these effects it is seen that even

though we get the best o correlation with the short, steep models we get * v -

the best 6 correlation with the long, steep models.

C. "y " values

The correlation of my - values has proven to be a rather unpredictable

affair. As was originally anticipated, the surface loading with its allied

difficulties has introduced some perplexing but nevertheless interesting

problems. If ridge loading had been used, all my moments would have been

a manifestation of the - values and consequently would have yielded

correlations of the same order as 6^, but with the surface loading the Oy -

values become a function of the manner in which the loads are distributed

121

on the surface as well as ù£)^. In other words the discrete loading system

here used, is seen by the total structure as a continuous and uniform

loading system, with the consequence that the - values.and the & -

values respond correspondingly to the system. On the other hand the m^

moments are localized functions of this discrete loading system and corres­

pondingly proved to be very sensitive to their location relative to the

actual loading pad.

This conclusion was reached after a lengthy search for the cause ff

the discrepancy in the moment correlation and was confirmed by what the

writer has chosen to call a post mortem on model No. 3. The purpose of

the post mortem was to establish the variation of m - values as a function y

of their position relative to the actual load application pad. This

operation was accomplished by cutting a representative sample from model

No. 3 in such a way as to form a very basic folded plate structure with a

span short enough to preclude longitudinal action and thereby accentuate

lateral slab bending. The test section, Fig. 17, consisted of 1 fold plus

2 edge plates and had a span of 8-in. A variety of loadings were initiated

to establish the effect of the load being on the plate where m^ - values

were being measured and the effect of the load being on the plate adjacent

to the one in which my - values are being measured.

The reason for this was to discern the Saint-Venant effect of the

moment being transferred across a ridge. It was thought that the chosen

structural element, by being stripped of the usual folded plate interaction

would yield a direct Insight on the relationship between the experimental

and theoretical moments. The results of the post mortem are shown in Figs.

20 to 23.

Fig. 17. Post-Mortem Setup

124

0.2,s

iz.':

r / J • - TR ' I

L _U

k

1.75" , I.7S'' H O.ZS"

A - ,t-.

IS r ,, l./l

d 0" E

'7V;-i 177/1

i j_

/"_

/"

t??;

É 1 B

A

I.. /'Ll

|:>V

-âl-

; l ' d

p7/

/ « "

iLi

T

r

IL r n

F| 6, 18 G> A. C.E_ LO C A T I O /O CAIOT HOR%Z.O/OTA L. PRoirC T lO/vT)

2 ll> Zs^k

^ 3,/22 3./ZZ

fi6,14 loati pattern) 4-

125

ip

iheo.n/auues j

I

i/) 0 \p

I

0

fs

i

m i ® 'i

v%/| i L /I /J

—y-

m 00 v;

i v

%

? y

all vali>6i

(x vo x

i

b

(v

rv

d +

? 3 4aû

f\g,.20 vh^ fob. load parre^m mû^

t^to. values

00 o

i.

Ï '

\d d

I _5r

i

jt

n!

q

vfl

fo d /

fx

m e:

//. ,%/{

M

load/=-/^5''^

"Pad

y/^ B fig». 21 l oab pattera] (oû. 12.,

~r+ieo, ^ v

m 00 All Values C

"d" cO q + v //

o -#-

I r - ZiS c

)'#

~ . o I

(%/>/ &

di

m l<z/1

i

i /a l ôawcy -/2,5

/pad

f16..22 fofe load 0\tterk^ ^

o-r-

tii eo, values -f-

ri

cJ

r iv-.i

d +

5

i ji

f « i m r ' E

/.' / .

in i

d -""^3 I

r"

i_yj

r: I 6

t,

m #

Lôad/ = { Z S /p/\c»

m i •/:•!

w l -'• '

r/'C-'/l

I rwà.zï yuNJ foa load fatter/vj /o^ 4

I

127

Referring to Table 3 one will observe that the moment correlation at

the first valley. (B) is consistently fairly good and the experimental

moment is greater than the theoretical moment. The reason for this lies

in the fact that this valley moment is predominently a function of the

cantilevered load on the edge plate and therefore doesn't largely depend

on the load acting directly upon the plate containing the location B.

If the post mortem information is extrapolated to effect a change in

the my - values given in Table 3 an across-the-board improvement will be

seen in the moment correlation. Qualitatively the results are very con­

clusive - quantitatively this extrapolation process leaves questions of

degree unanswered and subsequently suggests that more extensive investiga­

tions are in order.

Another factor to be reckoned with in the m^ correlation is the fact

that the folded plate theory does not account for the influence of on

m^. This in itself constitutes quite an omission insofar as the - values

at times are greater than the m^ - values.

D. Experimental Observations on Buckling Behavior

All of the models tested buckled either elastically or plastically in

the edge plate. In regards to the elastic buckling an attempt was made in

the theoretical development to approximate the critical buckling stress in

t h e e d g e p l a t e w i t h a n e q u a t i o n o f t h e f o r m c r . = — — - . I n a n

6(1-4

attempt to derive some quantitative conclusions from the test data a plot

of vs ^Xniembrane made for point A of each model, assuming that

buckling could be detected by observing the stress level at which the moment

128

strains increase faster than membrane strains. Careful scrutiny of these

plots revealed a strong tendency for all of the edge plates, regardless of

the thickness, to buckle at an average stress level of around 5500 psi,

Figs. 24 to 28. Of course the thin plates, exhibited a much sharper break In

linearity than the thick plates but nevertheless there were two thick

plates where a break was observed at this stress level. If these limited

buckling observations have any significance it means that

2 2 2 2 a = • 1—T , where = a constant, K

The. K . . (5500M8/9?ft? .

E 10

If more significance is attached to the more distinctive behavior of the 2 2

2 JT t E thin plates, Cq can be evaluated, e.g. C = ' , s— "

(6) (55% (8/9) (4)° Co = ' inhering an effective length of

buckle to be approximately 5-in. which is very close to the observed

geometry. Additional variation of parameters will be necessary to validate

this approximate formula as a design tool but it is definitely indicative

of a trend.

E. Final Comments

It goes without saying, that as usual, more needs to be done. As evi­

denced by the current literature, quite a lot has been done on the mathe­

matical model and not enough on the real model. It is the writers opinion

that in the research endeavor at hand, a definite insight into the behavior

of a folded plate system has been gained. In future research there are

129

^\x.c Xkmo, OP to6E Pw\TE_

Mo CitL I

moivien^t vi msmbka/ug.

406

e.

/ÛOO éco tr

FIG>.2.4 &ucVLLii\)Q» O

130

B u e K L I M G E ç f l A V i O i î . O f LûG E PL A -TE_

m o d ^ u s

strain! ms (^&n/|^sraa3l ^t^am

/ooo ffoo aoo

f i g ^ 5 b u c k l i n q . o b^f e r . v a - t \ o n l s

h-

5 a-

al o a

Lui

II. o

o

> < % ui

v9

v o =5 cq

=£ af

tu 2

(i3

4 1

s: ui 4

1 sr

•3 -2.

v* 9

J -2 ai < 0 ol o . A-

h 2 iij

g vO

o o 00

i/l

7

t-

5 m

flo

0 j

2 ' j

o 3

tû

cj

a

Ll.

o o

gr"") Ooo/

qfivâl^ "acvv^^way si\ r^ivai^ lî^-ano]^

s" "o^x 13aoiaj

390] ig'^oifnvt^ag^ ^cvi")'>i :)ing*

zzz

133

£)ac KL-'I W^ hamlofl of EDGE. fuate~

Mobtu 1^^ G

M OME.rù% ^TR»Alf>^ Mλ M6>R/^i^E ÇTRA/<^

/OOD

l. oc lo

&00 /OOO 600 400

fiû,.2,ô buckucnictt oïiî)e^yat\o/os»

Fig. 29. Edge Plate Buckling

%•

Fig. 30. Colapse by Slab Buckling

3 7

137

138

obviously certain pitfalls to avoid, such as those relative to gage

placement. The placement of gages directly on the ridges, with such

spacing as to average out local disturbances should be investigated. This

will also eliminate the necessity for including surface load influence in

the theoretical calculations for moment, as is required when the point is

not directly on the ridge. In addition to. this there would "probably be a

distinct improvement in load continuity effect if the longitudinally dis­

crete pads were replaced by longitudinally continuous pads.

An obvious out to these problems is to use just ridge loads, with the

subsequent elimination of surface effects, but the question still looms in

the mind of the writer relative to the correctness of the assumption that

the surface loads are transferred to the ridges by a mechanism analogous to

a continuous beam system. The results show conclusively that this trend is

present but the matter of degree of precision of the assumption needs fur­

ther attention. The writer would highly indorse a very basic research pro­

gram that would delve into the actual validity of the fundamental assump­

tions. This would have to be initiated on structural systems so simple as

to preclude the question of theoretical correctness of stress existing at

points of interest. With such a simple system we would hopefully be able

to isolate the various fundamental assumptions.

The hope of applying plastic or ultimate strength design to this

particular folded plate configuration is rather remote because the plastic

behavior of this system is quite frequently not a logical extension of the

elastic behavior but on the contrary contains reversals of trend as evi­

denced by observing the test plots from model No. 5. It is therefore very

dangerous to assume as we do in plastic design, that constant stress

139

levels are attained by successive elements of the structure with the

consequent formation of a mechanism. \

Finally in regards to the much sought after simplified design pro­

cedure for office practice, the writer is firmly convinced from the experi­

mental results that there is much inherent neglected strength in this

system which tends to reduce, in fact, the corrections made to the basic

theoretical solution. This fact tends toward improving the validity of

stresses predicted by the beam method.

Therefore a sensible design approach could be devised whereby the

disturbance-producing edge plate is initially ignored or in essence, where­

by we consider our structure to be of infinite lateral extent, and then

replace the edge plate, assuming its stresses to be some predetermined

proportion of the interior plate stresses.

140

vii. selected references

1. Ashdown, A. J., The Design of Prismatic Structures. London,

Concrete Publications LTD. 1951.

' •

2. Born, J., Faltwerke, Ithr Theorie und Anwendung. étuttgart,

K. Wittwer Verlag. 1954.

3. Craemer, H., Theorie der Faltwerke. Beton und Eisen 29:276. 1930.

4. Ehlers, G., Die Spannungsermittlung in F.laechentragwerken. Beton

und Eisen 29:281. 1930. *

5. Gaafar, I., Hipped Plate Analysis Considering Joint Displacements.

American Society of Civil Engineers Transactions 119:743-784. 1954.

6. Goldberg, J. E. and Leve, H. L., Theory of Prismatic Folded Plate Structures. Proceedings of the International Association for Bridge and Structural Engineers 87:59. 1957.

7. Gruber, E., Berechnung Prismatischer Scheibenwerke. Proceedings of the International Association for Bridge and Structural Engineers.

1932.

8. Gruber, E., The Exact Membrane Theory of Prismatical Structures

Composed of Thin Plates. Proceedings of the International Association

for Bridge and Structural Engineers. Vol. II. 1951.

9. Ketchum, M. S., Design and Construction of a Folded Plate Roof Struc­

ture. Journal of the American Concrete Institute Proc. 51:449.

Jan. 1955.

10. Scordelis, A. C., Croy, E. L., and Stubbs, I. R., Experimental and

Analytical Study of a Folded Plate. American Society of Civil Engineers Proceedings 87, No. ST8. Dec. 1961.

11. Simpson, H., Design of Folded Plate Roofs. American Society of Civil

Engineers Proceedings. Paper 1508. Jan. 1958.

12. Timoshenko, S. and Gere, J., Theory of Elastic Stability. 2nd

Edition. New York, New York, McGraw-Hill. 1959.

13. Timoshenko,. S and Woinowsky-Krieger, S., Theory of Plates and Shells. 2nd Edition. New York, New York, McGraw-Hill. 1959.

14. Winter, G and Pei, M., Hipped Plate Construction. Journal of

American Concrete Institute 43:505. 1947.

15. Yitzhaki, D., Prismatic and Cylindrical Shell Roofs. Haifa, Israel, Haifa Science Publishers. 1958.

141

VIII. ACKNOWLEDGEMENTS

•The writer is deeply grateful to his committee for their willing­

ness to help whenever confronted, to Dr. Carl E. Ekberg, Jr., Head of

Civil Engineering Department, who originally suggested the field of

investigation and to the National Science Foundation whose fellowship

created the opportunity to divert from his established routine and

further his education.