-- ." ...

·7.!?~ -'"'':.

C~Vll ENGINEERING STUDjES S7RUCTU RAL RESEARCH SERl ES NO. 321

DISCRETE ANALYSIS OF

CONTiNUOUS FOLDED PLATES -' :..: .. :... :> .r·: _.' .: - ... ~' .. , , .... _ _ r' _~_

.- ... - . - ... . ...... ....., ~.- ,

• - • ~. 0..., ,... ...

- -' -'-~ :". ~~. ~-.. ~:~::. ~ .• ~- "~: ;:~ .. ~~ "::" ~ -. ... ~- . -. '_._-- ..... _.-: ... :.':...

~.'. ,-.. ~ ....... -. ,_....... .., :"""-- ~: .. :'

I by

A. KARIM CONRADO and

W. C. SCHNOBRlCH

A Report on a Resea;-c~

Program Carr:ed O'..JT

National Science Foundation

Gra!l7 No. GK-538

URBANA, iLLiNOiS

DISCRETE ANALYSIS OF CONTINUOUS FOLDED PLATES

bY A. Karim Conrado

and

W. Q. S~hnobrich

University of Illinois Urbana, I 1 1 i no i s

March, 1967

ACKNOWLEDGMENT

This report was prepared as a doctoral dissertation by

Mro AD Karim Conradoo The work was under the general supervision

of Oro Ao R, Robinson and the immediate superv~sion of Dr, W, C.

Schnobrich. The research has been partially cupported by the

National Science Foundation under Research Grant N$F-GK 538.

The authors wish to express thei r thanks to the staff of

the Department of Computer Science for thei r cooperation in the use

of the IBM 7094-1401 computer system (supported partially by a

grant from the National Science Foundation under Grant NSF-GP700).

- iii-

TABLE OF CONTENTS

AOKNOWLEDgMENTS ..... . . . ~ . LIST OF TABLES. ~ .... , . LIST OF FIGURES . . . . . . . . . . • , 0 ••••••

INTBOPUCTION .... p 0 0 • 1 • 0 •

1.1. 1.2. 1·3. 1.4.

General Remarks. Previous Work. . Object and Scope . Nomenclature ~.

. ~

" • • ~ • • " II

. , . , . . ~. . . • • f

IYIETHOD OF ANALYSIS. . .

2.1. 2.2. 2~3· 2.4. 2~5·

Introductory Remarks . . r

Assumptions and ~imitations ... r •••••

Coordinate $ystems . . . . . Descriptiop of the Discrete Syst~m Mode~ • . Sign C9nvention. . . . ~

Displacements~ ..

~ . . . y •

II •• l'

2·5·1. 2·5·2. 2·5·3· 2.5.4,.

External Forces. , ..

2·5·5·

Interpal Forqes .• , ~

Ecc~ptricity of Stiffeners Angl~ of Incl:Lnation o·f ELate. .

. ,

2~6. Equilibrium Equations. • , ~ • • • 0 • • e •

2.6,1. 2.6.2. 2.6.3.

U.:Equation . V Equation .• W :E;:quation

• II! • " •• 0 •• , 0

.. 2.7. Bo~ndary Forces ... ,0 G "

I

2.7.1, In-Plane Force Ny' ..• 2·7·2~ Longitudinal Shear rorce Nyx .. 2.7.3. Transverse Mo~~nt My .. , . 2.7.4. Transverse Reaction By .•..• Description of Method of Solution ...

NUMERICA~ RESUL~S . . • . .

. . . . ," ,

General Remarks. . . . . . ... Si~ply~Supported Tro~gh-Shaped Folded Plate .. Two-Span Contin~ous V rolded Plate . . . . ~ . • . .

-iv.,..

Page

iii vi vii

1

1

3 5 5

8

8 8

10 10 l2 12 13 13 13 l3 14

l4 16 16

21

21

22 23 24

25

32

32 32 33

TABLE OF CONTENTS (Continued)

3.4. Two-Span Continuous Saw-Tooth Folded Plate.

3.4.1. Comparison of Transverse Moments :vr 3.4.2. Comparison of In-Plane Shears Nxy ·

y

3·4.3· Comparison of Forces Ny .. 3. 4.4. Comparison of Longitudinal Stresses

CONI':::;LUSIONS AND RECOMMENDATIONS FDR FUTURE STUDY. .

4.1. 4.2.

Conclusions. . . . . . ... Recommendations for Future Study

LIST OF REFERENCES. .

TABLES.

FIGURES

APPENDIX A. OPERATORS FOR INTERIOR POINTS.

APPENDIX B. EQUILIBRIUM EQUATIONS AND OPERATORS FOR THE TRANSVERSE STIFFENERS .....

APPENDIX C.

B.l. B.2. B·3· B~4.

B·5·

B.6.

General Remarks ........ . Strain-Displacement Relations .. . Axial Forces and V Operators . . Bending Moments and W Operators. Determination of Eccentricity and Depth of Equivalent Beam . . Coupling of Linear Equations due to Eccentricity of Stiffeners .

DESCRIPTION OF COMPUTER PROGRAM.

Nx

v

Page

35

36 37 38 39

41

41 42

43

45

52

81

81 82 83 85

87

88

91

Table

1

2

3

4

5

6

7

LIST OF TABLES

Right-Hand Side of Equilibrium Equations for Unit Displacements at the Boundaries. No Stiffener Present.

Terms to be Added to Right-Hand Side of Equilibrium Equations for Unit Boundary Displacements When a Rectangular Stiffener ~s Present ......... .

Solution of Simply Supported Folded Pl~te. Problem 1 ................ .

Properties and Loading of Two-Span Continuous V Folded Plate. Problem 2. ~ ...... p'

Relative Distribution of Total Vertioal Reaction Between End and Center Supports. Problem 2. I:f 0 •

Solution of Two-Span Continuous S?w-~Qoth Folded Plate. Problem 3. . . . . . ...

Values of N Over Central Supports. Problem 3 .. x

-vi-

Page

45

46

47

48

49

50

51

Figure

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

LIST OF FIGURES

Some Common Folded Plate Cross-Sections .

Global and Local Systems of Coordinates ,

Discrete System Model . . ,

Contributing Areas and Location of Displacement Points.

Sign Convention for Displacements ,

Positive Bending Moments and Normal Forces. D

Positive Twisting Moments and Shear Forces.

Identification of Interior Points on Plate ..

In-Plane Forces Acting on U Bar (J,n)

In-Plane Forces Acting on V Bar (m,K)

Transverse Forces Acting on W Point (J)K) .

Moments and Transverse Shears Acting on U Bar (J)n) ..

Displacements and Rotations of Bar (J,n).

Forces and Displacements at Boundary.

Identification of Points Near Edge of Plate . D •

In-Plane Forces Acting Upon Longitudinal Bar n) at Edge . . . . 0

Transverse Forces on W Point at Edge.

Local and Global Displacements at Edge of Plate

Forces Acting Upon Longitudinal Joints of Plate

N Values of Midspan. x

Problem 1. . ...

Variation of M at Mi.dspan 0

y Problem 1 ..

Transverse Displacements at Midspan. Problem 1 .

-vii

Page

52

53

53

54

55

55

55

56

56

57

57

57

58

59

60

60

61

62

62

63

64

64

Figure

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

LIST OF FIGURES (Continued)

N at Simple Sunport. Problem 1 D x:y 1:;'

N at Free Edge. Problem 2-DB .. x

N at Foldo Problem 2-DB x ooooa~oooqOD

M Along Free Edge. Problem 2-DB 0 0 x

M at Center of Each Span. y Problem 2-DBo 0 • •

Variation of N Across Several Sections of Folded PI t P b l X 2. TV··,) a e. ro ...Lem ··"J_t;. • 0 0 o. • Q 0 0

Nand N Values 0 Problem 2 . x xy

Variation of M Along First Valley. Problem 3. y

Variation of M Along Central Ridge. Problem 3 . y

Moment in Transverse Stiffener. Problem 3 ..

Transverse Variation of N xy Problem 3-BO. .

Axial Forces in Stiffener. Pro'blem 3 .

Variation of N~l Along Valley Line B .

Transverse Variation of N 0 Problem 3-NB . x

Distribution of Longitudinal Stresses in Central Plate 0 Problem 3-CP 0 0 •

B-1 Dimensions of Actual and Equivalent Stiffener .

B=2 Identification of Points Along Stiffener. 0

B-3 V Displacements at Point (m) K) 0

B=4 Element of Bent Stiffener . 0 0

viii

Page

65

66

66

67

67

68

69

70

70

71

72

73

73

74

75

90

90

INTRODUCTION

1010 General Remarks

Much has been written in recent years on the subject of folded

plate structures. The majority of this literature has dealt with pris-

matic folded plates of single span and simple supports.

A prismatic folded plate structure consists of a series of

rectangular plates connected along their longitudinal edges and supported

transversely by two or more diaphragms or frames. The main field of

application of this type of construction is for structures where rela-

tively large spans are required such as hangars, auditoriums and

industr:i.al bulldings. Some of the most commonly used cross sections

are shown in Figo 10

The material quantities in folded plate construction are

slightly higher than the quantities required when singly :or. doubly

curved shells are used. Offsetting this increase in material usage,

easier formwork and concreting frequently make the application of

folded plates more economical than curved shell systems. In addition,

the corrugations formed in the folded plate system may be used for

ducts and utility troughs. An excellent discussion of the economics

and practical aspects of folded plate construction is given by Whitney)

(23) -~ Anderson and Birnbaum. \

Longitudinally continuous folded plates are usually stiffened

by means of transverse frames at the column lines. These frames prevent

~~

Numbers in parentheses refer to entries in the List of References.

-1-

2

distortion of the cross-section of the folded plate, thus eliminating

the high transverse bending moments that would be present at the

colUc'1ln lines if no frame existed there. In addition, the plates

transmi.t the external loads to the frames which in turn are supported

by the columns. When no frames are used) the loads are transmitted

directly from the plate to the column supports resulting in high stress

concentrations in that zone. Simple-span folded plates are sometimes

provided with transverse frames spaced at intervals along the span

ei.ther to induce beam-like behavior or to serve as anchors for tie-rods.

The methods of analysis of prismatic folded plates may be

divided into four categories~

(a) Beam method.

(b) Folded plate theory neglecting relative joint

displacements.

(c) Folded plate theory considering relative joi.nt

displacements.

(d) Elasticity method.

In the beam method it is assumed that no distortion of the

cross-section of the folded plate structure takes place. As a result,

the longitudinal stresses on a section cut transversely through the

folded plate system are considered to vary linearly with the vertical

d.istance from the centroidal axis just as they would in a beam. Further­

more, the in~plane shear forces are therefore parabolically distributed.

Finally, transverse moments are computed from e~uilibrium using the

incremental change in shear plus the vertical loadso This method is

justifiably used fre~uently for long-span folded plate structures 0

3

Method (b) differs from the Beam Method in that the linear

variation of longitudinal stresses is assumed to exist over the

individual plates rather than over the whole section.

Method (b) disregards the transverse moment and longitudinal

stresses caused by the relative displacement of the longitudinal joints.

This effect is taken into consideration in method (c). In 1963) the

Task Committee on Folded Plate Construction (Committee on Masonry and

Reinforced Concrete) of the American Society of Civil Engineers

recommended the use of method (c) for the design of simply-supported

folded plate structures. (13)

The elasticity solution as developed by Goldberg and Leve(7)

makes use of the classical theory of plates for loads perpendicular to

the plane of the plate and the plane stress theory of elasticity for

loads in the plane of the plate. Useful as it is) the elasticity method

has some shortcomings) namely: (1) It applies only to folded plates

simply-supported at the ends by flexible diaphragm$. (2) It cannot

handle folded plates with tie-rods or transverse stiffening ribs.

(3) It cannot handle folded plates with intermediate column supports.

The advent of the high-speed electronic digital computer makes

it possi,ble to analyze numerically the cases listed above as being

excluded from the Goldberg and Leve elasticity method.

1020 Previous Work

First papers on folded plate theory appeared in the technical

literature beginning in 19300 A theory based on a linear variation of

longidudinal stresses in each plate was proposed by Ehlers(5) in that

4

year. This was an improvement over the beam analysis which considers

a linear variation of longitudinal stresses across the entire structure.

Ehler1s theory considered the structure to be hinged along the longi­

tudinal joints. In the same year Craemer(3) proposed a method that

considered the transverse moments due to continuity at the joints.

Winter and Pei(24) introduced) for the first time in the United States)

the folded plate theory neglecting relative joint displacements.

In 1932 Gruening(9) proposed a theory that takes into account

the relative displacement of the joints of the fo~ded plat~. Gaafar(6)

and. Yitzhaki(25) modified and further developed this theory.

Werfel(22) first proposed a theory which included all possible

stress resultants. Later Goldberg and Leve(7) developed the so-called

elasticity solution. De Fries-Skene and Scordelis(4)18) presented a

di.rect stiffness method in matrix form for both the ordinary(18) and

the Goldberg and Leve elasticity(4) methods. Reviews of several design

theories of folded-plate structures have been given by Traum(21) and.

more recently by ;Powell. (14)

Experimental and analytical studies by Ali(l) and paulson(12)

showed that the nQmber and spacing of transverse diaphragms has a sig-

nificant effect on the transverse distribution of longitudinal stresses.

Goldberg) Gutzwiller and Lee(8) used a finite difference technique to

study continuous folded plates. The effect of transverse stiffeners on

the behavior of closed prismatic structures was studied by Craemer. (2)

5

1030 Object and Scope

The objective of this study is to develop a numerical method

for the elastic analysis of transversely stiffened prismatic folded

plates either simply-supported or continuous in the lD_ngitudinal

direction. The method must be able to handle folded plates with tie-

rod.s) supporting columns) concentrated loads and transverse stiffening

ribs that have an eccentricity relative to the plane of the plate.

The representative problems shown are compared with existing

solutions whenever possible.

1.4. Nomenclature

The symbols used in this work are defined where they first

appear 0 For convenience they are summarized below~

A a square matrix of coefficients

B a column matrix

b width of stiffening beam

D

d.

d

E

e

F

G

h

2 12(1-v )

flexural stiffness of plate

depth of stiffening beam

d J3 = separation of bars of equivalent beam

Young's modulus

eccentricity of beam relative to plane of the plate

a column matrix of boundary forces

shear modulus of elasticity

thickness of plate

K

L ,L x y

MX(J,K),My(J,K)

m

N N ) x(J,K)' y(J,K

r

t

u ,v J,n m,K

XJ ,Y K,ZJ K ,n m, ,

Yxy(m,n)

6

a stiffness matrix

grid lengths in x- and y-directions respectively

plate moments about the y.., and x.,.axes respectively

at extensional node J,K

plate twist;i.ng moment at shear node m,n

an influence matrix (Section 2.8)

plate moment per unit length

plate membrane forces in the x- and y-directions

respectively at extensional node J,K

in-plane shear force at shear-node m,n

load per unit area in z-direction

transverse shear in x- and y-directions at locations

JJn and m,K respectively

load per unit area in x-direction

reaction at edge at location J,K

load per unit area in y-di.rection

separation of bars of framework

plate displacements in x- and y-directions at

locations J,n and m,K respectively

plate displacement in z-direction at location J',K

total external loads in the x-) y-, and z-directions

at locations J,n, m,K, and J,K respectively

rotation of bar of framework

twist of surface at point m,n

column matrix of boundary displacements

7

column matrix of interior displacements

T B Ex(J)K))EX(J,K) extensional strain in x-direction at level of top and

bottom bars of framework respectively

Ex(J,K))E,y(J)K) extensional strains at mid-depth of plate in

x- and y-directions respectively

plate displacement in the X-direction

v Poisson's ratio

plate displacement in the Z-direction

extens~onal stresses in x- and y-directions

respectively at extensional node J)K

in-plane shear stress at mid-depth of plate at

shear-node m)n

cp angle of inclination of the plate with the

horizontal

curvature of plate in the x-direction at location J)K

curvature of plate in the y-direction at location J,K

rotation of plate about X-axis

METHOD OF ANALYSIS

2.10 Introductory Remarks

In the method of analysis for prismatic folded plates pre-

sented in this work the individual plates are solved by means of a

discrete system model developed by Schnobrich. (17) The details of the

model are presented in Section 2.4.

A stiffness method of solution is used for the analysis of

the structure as a whole. The description of this method as applied

to folded plates is given in Section 2.8.

202. Assumptions and Limitations

The assumptions made relating the flexural action of the

plate are those of the classical small-deflection theory of plates.

For the in-plane or membrane action of the plate the assumptions are

those of the plane-stress theory of elasticity.

The transverse stiffeners are assumed to resist bending about

an axis parallel to the longitudinal span of the plate and to be com-

pletely flexible when bent about the weak axis. Torsional resistance

of the stiffener is neglected. This omission of the torsional sti.ffness

may be j~stified on the grounds that stiffeners are usually located at

or near locations where the twist of the surface is zero. The stiffener

and the plate are assumed to work monolithically, therefore they are

assumed to have the same strain at their points of junction.

-8-

9

For bending of the plate) the follow;Lng partial differential

equation must be satisfied~(20)

(1)

where W is the deflection of the plate in the z-direction, D is the

flexural stiffness of the plate and p is the load per unit area acting

perpendicular to the plane of the plate.

The in-plane actions must satisfY simultaneously the following

partial differential equations~(19)

Eh [a~ (l-v) d~ + (l+V) 02V J 0 --+ ~

+ q 2 dx2 2 2 . 2

(l-v ) dy

Eh [o2V + (l-v) d2

V (l+V) o~j 0 . --- + dXdY

+ r 2 2ly2 2 dx2 2

(l-v )

where U and V are displacements in the x- and y-directions respectively)

E is YoungVs modulus of elasticity) V is Poisson's ratio) h is the

thickness of the plate and q and r are the loads per unit area in the

X- and y-directions, respectivelyo

Within the domain of the small-deflection theory) the normal

deflection of the plate is governed by Eqo (1). This equation is in-

dependent of Eqs 0 (2) and (3) which are in turn coupledo Di.scretization

of the plate results in one system of linear equations invo;:Lving 'W dis-

placements only and another system involving U and V displacements only 0

This situation is altered only when the plate is stiffened with eccentri.c

ribs 0 In this latter case the entire system of linear equations i.s

10

coupledo Uncoupling of the equations implies that the equations may

be solved as two separate sets of simultaneous equations 0 With a given

storage capacity of the computer, much finer grids can be used if the

uncoupled systems are solved separately 0

2030 Coordinate Systems

For the analysis of the structure a global orthogonal frame

of reference (X,Y,Z) is established as shown in Fig. 20 The X and. Y

axes lie in a horizontal plane, the X axis running parallel, and the Y

axis perpendicular to the fold lines. The Z axis points downwards.

In addi.tion to this global system, local systems of orthogonal

coordinates are defined for each plate. The local frame of reference

for plate n i.s denoted as (x ;y J Z ), Fig, 20 The x and y axes lie n n n n n

in the plane of the plate) the x axis being parallel and the y axis n n

perpendicular to the fold lines. The z axis points in a direction n

such that the triad (x ;y ,z) is a right-handed systemo n n n

204. Description of the Discrete System Model

A discrete system model(17) will be used for the analysis of

stresses and deformat:ions of the structure 0 The model was originally

developed for the analysis of cylindrical shells but it is equally

applicable to plates provided the curvature is made zeroo The model

consists of a grid of straight J weightless J ri.gid bars arranged as

shown in Fig. 30 The network is made of two bar-joint systems called

the primary and secondary systems 0 They are arranged in two layer3 a2

in sandwich construction.

11

At the intersection of the bars of the primary system there

are extensional elements capable of developing extensional forces and

bending moments in both the longitudinal and transverse directions 0

The bars of the secondary system are connected to the midpoints of the

bars of the primary system and are fastened to deformable shear nod.es.

These shear nodes provide the resultant in-plane shear forces and

twisting moments.

The distance between the top and bottom rigid bars) thickness

t) of the gri.d is equal to h/"J3 ) 'VIlhere h is the actual thickness of

the plate 0 This separation insures that the bending response of the

model corresponds to that of the plate. The derivation of this rela-

tionship is given in Section 206030

The stress-strain properties of the material comprising the

nodes of the model are the same as those of the material in the actual

structure 0 Si.nce the nodes are subjected to plane stress) the followi.ng

stress-strain relations are valid for the nodes~(19)

E (E + VE ) a

2 x (l-v ) x Y

E (E + VE ) ( a

( 2· Y l~v ) Y x

E T

2(1+v T or xy xy

where (J ) a and'! are the extensional stresses in the x- and y.-x y xy

directions and the shear stress in the x-y plane respectively. The

corresponding unit strains are Ex' E ) and y 0 y xy

12

As shown in Fig. 4 the U and V displacements are defined at

the center of the 1 and 1 bars of the primary system along the x-x y

and y-directions respectively. These displacements are at mid-depth

of the plate. The W displacements are defined at the intersections of

the primary bars along the z-direction.

Uniformly distributed loads are replaced by concentrated

loads equal in magnitude to the total loads acting in the respective

contributing areas. Non-uniform loads are replaced by statically

equivalent concentrated loads. The loads are called X, Y, and Z; they

act in the direction of the local coordinate axes at the points where

the respective displacements are defined ..

2.5. Sign Convention

In this section the sign convention for the following items

is given: displacements, internal and external forces, eccentricity

of stiffeners and angle of inclination of plateo

As shown in Fig. 5, the U, V, and W displacements are positive

in the positive x J Y J and z -directions respectively. Rotations of n n n

the 1 bars, a J about the x axis are positive if they are clockwise y x n

when viewed from the origin in the positive x direction. Rotations n

of the L bars, a J about the y axis are positive if counterclockwise x y n

when viewed from the origin in the positive y -direction. n

13

2.5.2. External Forces

The X,Y) and Z external forces are positive in the positive

directions of the local coordinates.

Bending moments M and M are positive if they produce tension x y

in the bottom fibers) as shown in Fig. 6. In-plane forces Nand N x y

are positive if they produce tension q Twisting moments Mxy acting on

the positive face are positive if counterclockwise when viewed from

the origin in the positive x direction as shown in Fig. 7. Moments M n yx

on the positive face are positive if clockwise when viewed from the

origin in the positive y -direction. ,In~plane shear forces Nand N n yx xy

acting on the positive faces are positive if they. act in the positive

x and y ~directions respectively. Transverse shears Q and Q acting n n x y

on the positive faces are positive when acting in the positive

z -direction. n

20504. Eccentricity of Stiffeners

The eccentricity of the stiffener is positive if its center

of gravity is located above the middle surface of the slab.

205.50 Angle of Inclination of Plate

The angle of inclination of a plate is positive when measured

from the positive Y axis counterclockwise to the middle plane of the

plate 0

14

206. E~uilibrium Equations

In this section the e~uilibrium e~uations for typical interior

points are derivedo The analysis of any plate problem by means of the

discrete system model re~uires the solution of a system of linear

simultaneous e~4ations. Each e~uation relates the appropriate external

force applied at a po~nt of displacement definition with the displace-

ment at that point and at neighboring displacement points. These

e~uations are called U, V, or W e~uations according to the point and

direction for which e~uilibrium is established. Collectively they are

called e~uilibrium e~uations.

For the derivation of the e~uilibrium equations reference is

made to the =i,dentification scheme shown in Figo 8. The equations are

presented in operator form in Appendix A. The e~uilibrium e~uations

for the st~ffeners are given in Appendix B.

206.10 U E~uation

Consideration of the e~uilibrium of the horizontal forces

acting on the U bar, shown in Fig. 9, givep:

o (5)

where the Nls are the internal forces and X(J,n) is the total external

load acting upon that baro The internal force resultants may be

expressed in terms of the stresses by the following e~uations:

N x(J,K)

N yx(m+l"n)

L ohoO( ) y x J,K

L . h· 'T x yx(m+l,n)

(6)

Substitution of Eqs. (4) into Eqso (6) yields:

EhL

Y2 fE (J K) + VEy (J) K)] (l-v ) LX·"

N yx(m+l"n)

EhL x

The strains at points (J"K) and (m+l"n) are as follows~

U - U J"n+l J"n

L x

v - V m+l"K m"K

L y

Yyx(m+l"n)

u - U J+l)n J)n L +

Y

V -.V m+l)K ; m+l)K-l L

x

15

(8)

Substttutton of Eqs. (8) into Eqs. (7) yields the force-displacement

equations:

EhL [u -u v(Vm+l)K .,. Vm'K)l N

y J,n+\x J"n + (9) x(J"K) 2 L (l-v ) y

and

EhL ~ -u V -V j N

x . J+l,n J,n m+l,K Lx m+,l,K-l (10) yx(m+l)n) 2(1+v) L + y

Upon substitution of the force-displacement relations into E~. (5)"

the"U ~quationl! is obtained:

16

(l+V) 2

+ [V - V - V + V J + (l-V ) X = 0 2 m+l,K m,K m+l,K-l ID,K-l Eh (J,n)

(11)

2.6.2. V Equation

The derivation of the IIV equation ll is similar to the deriva-

tion of the "U equation. II Equilibrium of the forces shown in Fig. 10

gives~

o (12)

Substitution of force-displacement relations into Eq. (12) results in

the !IV equation":

L Y Iv - 2V + V ] Lx m,K+l m,K m,K-l

2.6.3. W Equation

In the z-direction, equilibrivrn of the forces shown in

Fig. 11 gives:

Qx(J,n+l) - Qx(J,n) + ~(m+l,K) - ~(m,K) + Z(J,K) o (14)

where the Q1s are the internal transverse shears and Z(J,K) represents

the external load applied at point (J,K). The transverse shear may be

expressed in terms of t4e bending and twisting moments; e.g.) Q I ) x,J,n

17

may be obtained by taking moments about the right end of the U bar

shown in Fig. 12~

Mx(J,K) - Mx(J,K-l) + Myx(m+l,n) - Myx(m,n) L L x x

Substitution of expressions for the Q1s similar toEq. (15) into

Eq. (14) gives ~

Mx ( J , K+ 1) - 2M x ( J -' K) + Mx (J -' K -1) +M .....:y~(~J...,....+_l'""--' K_)"-!---_2M---'\y:...-(~J~,_..,.K.!-) _+_M~y~( J_-....,.l..,..::.J_K.!.-)

L x

L Y

Myx(m+l,n+l) - Myx(m,n+l) - Myx(m+l)n) + Myx(m,n) + L

x

Mxy(m+l,n+l) - Mxy(m,n+l) - Mxy(m+l,n) . + MXY(rn,n) -z + ., Ly , + ( J , K) o

(16)

The bending and twisting moments in Eq. (16) may be expressed in terms

of the displacements. Only the moments M ( ) and M ) will be x J,Kjx(m+l,n

derived in this section. The derivation of moments M and M proceeds y xy

in a similar fashion~

The moments may be expressed as follows:

[ -(~)(!2.) aB _ aT M L x(J,K) x(J,d x(J,K) 2 2 Y

t h [-r~x(m+l,n) - -rJx(m+l,n)] M - (-) (-) L yx(m+l)n) 2 2 x (18)

where superscripts Band T indicate the top or bottom bars of the frame-

work. Substitution of Eqs. (4) into Eqs. (17) and (18) gives~

M yx(m+l,n)

EhtL -

8(1+~) t~x(m+l)n)

18

(20)

If the strain quantities are replaced by the appropriate displacement

terms given by Eqs. (8), Eqs. (19) and (20) become:

M x(J,K)

EhtL . x - 8(1+v)

+ V

From Fig. 13, it may be seen that:

therefore, since the sines of both angles are equal,

(21)

(22)

(24)

Substitution of expressions for the difference between bottom and top

displa cements, as in Eq. (24), into Eqs. (21) and (22) yields ~

M x(J,K)

M yx(m+l,n)

Eht2 ~-L J... (-W + 2WJ K 4( 2) Lx2 J,K+l , l-V

19

(26)

The moments per unit length are obtained by dividing the moment expres-

sions just derived by Land L , respectively, y x . ,

myx(m+l,n) (28)

Equations (27) and (28) are the equivalents in the model of the usual

moment curvature relations of plate theory:(20)

m x

where D 2

12(1-V )

20

Therefore) for the response of the model to be e~ual to that of the plate

Eh3 Eht2 2

12(1-v ) 2

4(1-v )

from which

t h

(30) = ~3

Substitution of moment expressions similar to E~s. (25) and (26) into

E~o (16) yi~lds the "w e~uationll:

t2

1

+ 4L3Y [-WJ,K+2+4WJ,K+1-6WJ,K+4WJ,K_I-WJ,K_;] x

2 -W -W -W -, + (l-v )

J-l)K-l J+l,K-l J-l)K+lJ Eh Z(J,K) o

A comparison of the e~uilibrium e~uations obtained from the model with

finite difference expressions of E~s. (1)) (2), and (3) will show the

mathematical e~uivalence of the two procedures. If the plate is trans-

versely stiff~ned by beams) the V and W equilibr~um e~uations for the

stiffener must be added to the corresponding e~uations for the plate.

21

2.7. Boundary Forces

Because of the special conditions existing at the boundaries

of a 'p).ate, the formulas for the boundary forces differ from those of

a typical interior point. In this section) the formulas for the four

boundary forces (see Fig. 14a) are derived. They are applicable whether

the longitudinal edge is free or restrained by adjacent plateq. The

points and bars of the framework near a longitudinal edge are identified

by means of the notation shown in Fig. 15.

Compatibility of deformations at the longit~dinal joints of

a folded plate re~uires tha,t all plates meeting at that joint have

the same rotation about the X axis and that the tpree displacements

along the X) Y( and Z axes be e~ual. Therefore) for each panel length

three linear displacements and one rotation must be defined at the edge.

The U and W displacement$ at the edge are defined in the usual manner.

As shown in Fig. 14b, the rotation and the V displacement at the edge

are defined at the W points.

In-P~ane Force N y

The in-plane force N at point (J,K) on the edge is expressed y

in terms of the strains in the following manner:

EhL [ ----,,-~- Ey(J)K) + VEX(J)K)J (l-v )

The strains at (J,K) are given in terms of the displacements by the

expressions:

VJ K - V K , m, L (1)

2

u - U J,n+l J,n L

x

22

The y-strain at the ed.ge is given in terms of the displacement at the

edge and at an interior point. Substitution of Eqs. (33) into Eq. (32)

gives N at the edge~ y

N y(J,K) EhL l-2 (V T K-V K) (UJ 'l-UJ )1 x u, m, ,n+, ,n ~-2- L +v ~L (l-v ) Y x J

Longitudinal Shear Force N yx

Longi.tudinal equilibrium of the forces shown in Fig. 16

requires that~

N (_ ) = N ( ) + N yx ,J, n yx m, n . x

The forces in the right-hand side of the equation above may be expressed

in terms of the displacements as follows~

N yx(m,n)

N x(J, K)

EhL y

, 2· 2(1-v )

EhL y.

2 2(1-v )

ru -U_ J,n J,n-l L L x

tU -U J,n+~x Jln

2v (V J ,K-l-Vm,K-lj + (36)

L Y

+ 2v (VJ-, ~~ VmJK)j

23

Substitution of Eqs. (36) into Eq. (35) yields the longitudinal shear

at the edge:

by~

-UJ ,n_l+2U

J ,n.- U J ,n+l 2v(V J, K-l-VmL;-l-V J ,K+Vm, Kj. v 1 l' +

x

Transverse Moment M y

(37)

The transverse moment M at a typical interior point is given y

M Y

Eht21

- 4(1-}) [Xy + VXx]

where Xx and Xy denote the curvatures along the x- and y-directions.

For a point (J)K) at the edge the curvatures are as follows:

WJ K-WJ-l K , . ,

1 ( ...1L) \ 2

1 y

where a(J,K) is the rotation of the plate at point (J,K). Therefore

the expression for the y-bending moment at the edge is:

M y(J,K)

Eht2L tW w 1 x J,K ~y J-l,K - a(J,K)

2(1-V2

)L y

Eht2v -l J-+ 2 -WJ,K+l + 2WJ ,K - WJ,K-l

4(1-V )L x

Transverse Reaction R y

24

(40)

The transverse forces acting at edge point (J,K) are shown

in Fig. 170 E~uilibrium of the forces shown re~uires that the reaction

at the edge of the plate be:

(41)

The shears on the right-hand side of this e~uation may be expressed in

terms of the moments as follows:

~-- ( ) yx m,n - L

Mx(J,K+l) - Mx(J,K) L

x

x

M yx(m,n+l) L x

Substitution of E~so (42) into E~o (41) yields:

-IMY(J)K) - My(J-l,K) Mxy(m,n+l) - Mxy(mJn~ L + L -I

L y Y J

(42)

I.-M ( ) +2M ( ) -M ( \ M ( ) -M ( Ll + L x J,K+l xL~,K x J,K-l) + yx.m,n+lLx

yx m,nJ (4.3 )

25

Equation (43) corresponds to Kirchhoff 1 s expression for the edge

reactiono The second bracket is equal to the rate of change of the

twisting moment along the edge of the plate. (20) Substitution of the

expressions for the moments into Eqo (43) yields the reaction at the

edge~

Eht2 2 (l-v ) {

L l-3WJ K- 4wJ_l K+WJ -2 K ] ~ , , ) - 2a

4~2 L (J,K) l.J y

y

+ 2L1 L [-W J, K+1 +2WJ-, K-W J, K-1 +W J _1,K+1- 2W J -1, K+W J -;L, K-1l_ y x

L

+ 8L~ [WJ,K+2-4WJ,K+1+6WJ,K-4WJ,K_1+WJ,K_~ x

2080 Description of Method of Solution

(44)

The method of solution used belongs to the class variously

known as il stiffness J 11 11 displacement Ii or 11 equilibrium!! methods. The

structure is divi,ded into segments corresponding to the individual

plates; the boundaries of the segments therefore correspond to the

longitudinal joints of the folded plate structure.

The displacements and rotations at the longitudinal joints

of the structure are treated as unknowns. Each plate is first analyzed

assuming that the longitudinal joints are completely fixed. Next, the

di,splacements at the joints are each gi,ven unit values in turn and the

resulting boundary forces computed. Thus) each boundary force may be

expressed as a i!fixed-edge li value plus a linear combination of the

26

unknown displacements at the boundaries. The actual boundary displace-

ments are obtained from the equations of equilibrium of forces and

moments at the joints. Once the boundary displacements are found) the

plate may be analyzed again under the action of external loads and the

known boundary displacements.

The forces at the boundaries of each segment are expressed

in matrix form as follows:

(F} (45)

Where F is the column matrix of the boundary forces)

Nb yx

Nb y

F Rb

(46)

Y

Mb y

and 6 is the col~mn matrix of boundary disvlacements)

~

6 ~ (47) if

b a

In Eqsp (46) and (47) above the superscript b indicates that the forces

and displacements are at the boundary. The number of elements in F

and 6 is equal to the number of boundary-degrees-of-freedom of the

27

plate. Along the boundari~s of a segment there are four forces and

four degrees of freedom for each panel length) Fig. 14. Thus) for a

plate with eight panel lengths in the longitudinal direction there are

sixty-four boundary forces and sixty-four degrees of freedom (thirty-

two on eacp boundary).

The square matrix K in Eq. (45) is a stiffness matrix. The

column matrix Ff

represents the values of the boundary forces when the

boundaries are completely restrained. Thus) Eq. (45) is similar to

the slope-deflection equations used in the theory of plane frames.

In order to obt~in the Ff column matrix it is necessary to

generate a matrix A that relates the interior displacements 5 of the o

fixed plate with the external loads B acting on the plat~ according o

to the relation:

[A](5 } o

(B } o (48)

Equation (48) is in effect the matrix representation of the system of

equilibrium equations derived in Section 2.6. The set of linear

simultaneous algebraic equations given by Eq. (48) may be solved by

means of the Gauss elimination procedure. Once the displacements 5 o

of the fixed plat~ are known, the boundary forc~s Ff are easily computed

by means of the equations presented in Section 2.7.

In order to obtain the matrix K it is necessary to give) in

turn) a unit value to each boundaTY displacement keeping all others

equal to zero. A unit displacement at a boundary affects the equilibrium

equations of neighboring points only. For instance, referr~ng to Table 1)

a unit U displacement at point B will affect the equilibrium equations

28

for points 5) 6) and 8 only. The table shows the right~hand sides of

the e~uilibrium e~uations for unit displacements at the boundaries

for the plate alone. Table 2 shows the additional terms when there

is a transverse stiffener present. To illustrate how the tables

are fanned the term corresponding to point (m) K) in Fig. 15 will be

derived for the case when no stiffener is present. The only force

entering the e~uilibrium e~uation of thip point and involving the

boundary displacement VJ)K is Ny(J)K)) which is given by,Eq. (34). The

term involving VJ)K is:

The term is taken to the

and VJ)K is given a unj,t

2Eh1~

2 (l-y )1 y

right-hand side

value. The net

21 x

L Y

and multiplied by (1 ... y2)

Eh

result is:

rnis term appears in Table 1 in the fifth row under VA. Other terms

in the ~ables are obtained in a similar manner.

When a boundary point is given a unit displacement the

resulting set of linear simultaneous e~uations will have the form:

[A](5.} ~ (E.} l l

where A is exactly the same as in E~. (48) and 5. represents the l

interior displacements of the plate when the ith boundary point is

given a unit displacement.

29

Solution of the set of linear simultaneous equations given

by Eq. (49) y~elds the column matrix 5. from which the boundary forces l

that make up the ith row of stiffness matrix K are obtained.

Thus) tne following sets of linear simultaneous equations

have to be solv~d~

[AJ(5.} 1.

(B, ) l

i 1) ..... ) £

where £ is tpe number of boundary degrees of freedom of the plate. In

the computer prog~am dev~loped advantag~ is taken of the fact that the

(£ + 1) systems of equations represented by Eq. (50) have the same

matrix of coefficients A. In the Gauss elimination procedure the

eli,minatic:m part is performed only once on A keeping (£ + 1) right~hand

vectors. The back substitution operation) however) must be ~one for

each one of the B vectors) or a total of (£ + 1) times~

From th~ solutiop of Eqs. (50) ~he matrix M is obtained~

M

where the 5's are column vectors whose elements are the displacements

of the interior P9ints of the plate. Matrix M is an influence matrix.

Its element m. K located in the ith row and kth column represents the l,

displacement of tbe ith interior point due to a unit displacement at

the kth boundary po~ntD

Once matrix M is known) a set of boundary forces for each

column of M is computed. These boundary forces make up the columns of

the stiffness matrix K.

30

In order to establish equilibrium at the edges of the plate

the forces must be given in terms of displacements in the global system

Of coordinates~ Referr~ng to Fig. 18, the transformation for displace-

ment$ from the local to the global system is given by the following

relations:

t... u

T1 V coscp + W s::i.ncp (52)

~ ;:::: -V sincp + W coscp

ljr ;::::CX

whe;re t...,n, ~ and ljr are the displacements along the X, Y, and Z axes

an~ the rotation about the X axis respectively and cp is the ~ngle of

inclination of the pa;rticu~ar plate relative to the horizontal. With

thE; transformat~ons given b;y Eq. (52) the bm;mdary forces may be

expressed now in terms of the displacements in the global system.

Equilibrium of forces at the joints of the folded plate may

be established at this stagE;~ Using the superscripts 1 and R to

denote forces at the left and right of each joint, the eq~iliprium

equations are as follows (Fig. 19)~

L FX 0: N1 _ NR 0 yx yx

I Fy 0: Nf{ R N1 1 + (RR + ZR) sincp R

:;::: coscp coscp y y Y

(R1 _ z1) sincp 1 0 -

Y

31

L FZ 0: _NR . sincp R NL sincp L (R -R) R + + R + Z . coscp y Y

(RL zL) L 0

Y coscp

L:~ 0: L MR 0 (53) M y y

-L .,....R where Z and Z are external forces applied at the joints of the

structure~ ~xternal couples could also have been considered in the

moment equilibrium equations. Wherever there is a transverse stiffner

the forces inEq. (53) are to be taken as the sum of the forces in

the plate and in the stiffener.

Solution of the system given by Eq. (53) yields the final

boundary displacements (~) ~} ~J ~) in the global system of coordinates.

Fixity against rotation or translation of points along tne fold lines

may be obtained by elimination of rows and columns in Eqs. (53)~

The final Qounda~ displacements in the local system m~y be

obtained from the displacements in the global system.

Jf the column vector of final boundary displacements in the

local syst~m is called 6 fb , then the final interior d~splacements 5

in each plate are given by

where [M]T is the transpose of M (Eq. 51).

As the final step in the analysis, the interna~ forces and

moments are computed from the known displacements of the Plate.

NUMERICAL RESULTS

3.1. General Remarks

The method of analysis developed in this study is used for

the solution of three example problems:

1. Simply-Supported Trough-Shaped Folded Plate

2. Two-Span Continuous V Folded Plate

3. Two-Span Continuous Saw-Tooth Folded Plate.

The numerical results obtained in examples 1 and 2 are compared with

available solutions. All the structures analyzed are loaded sym-

metrically and have two lines of sJ~~etry; therefore only one ~uarter

of the structure need be analyzed. For the solution of the problems

shown in this study a FORTRAN program for an IBM 7094 digital computer

was written.

3.2. Simply-Supported Trough-Shaped Folded Plate

A three plate) trough-shaped) simply supported folded plate

with uniformly distributed load is considered. The supports are assumed

to be the usual diaphragm support) rigid in their own plane but flexible

normal to that plane. The loading) material properties) cross~section)

* and dimensiops of the structure are given in Tabl~ 3. Two grid sizes

were used: 8 x 5 and 10 x 8. The values of the elasticity solutions

* The two numbers denote the panels in the longitudinal and transverse directions respectively.

-32-

33

calculated to the 7th harmonic are used as a basis for comparison. The

midspan values of longitudinal stresses at the free edge and at the

fold line and the transverse moment at the fold line are given in

Table 3. The transverse distribution of longitudinal stresses at

midspan is shown in Fig. 20. Very good agreement is obtained between

the elasticity solution of Goldberg and Leve and the 8 x 5 grid solution

from the model. The 10 x 8 solution is not shown since it coincides

with the solid line to the scale of the figure. The transverse varia-

tions of M and W displacements, both taken at midspan, are given in y

Figs. 21 and 22. Again good agreement is found with the elasticity

solution. Within plotting accuracy, the same curve for W displacements

was obtained for the elasticity method, the 8 x 5, and the 10 x 8 grid.

Figure 23 shows the shear N at the simple support. The discontinuous xy

line represents extrapolated values since the model defines the shears

at points one half panel away from the support.

It is interesting to note that the longitudinal stresses

obtained by the beam method come within 5% of thevalu~s given by the

elasticity solution. The midspan moment M at the fold lines predicted y

by either of the engineering methods is 33% larger than that given by

both the elasticity solution and the model. This discrepancy is probably

due to the fact that the engineering analyses do not take into account

the contribution of the twisting moments in resisting external loads.

3.30 Two-Span Continuous V Folded Plate

The properties and loading of the inverted V folded plate

used in this example are shown in Table 4. As outlined in this table,

34

three sizes of stiffener supports are considered at the center column

line of the structure 0 The three cases are designated 2-DB (with Deep

Beam) J 2-B (with Beam) J and 2-NB (with No Beam) 0 The two end supports

are assumed to be normal simple support diaphragms.

r:[lhe N values at the free edge and at the fold line for x

structure 2-DB are shown in Figs 0 24 and 250 The discontinuous li.nes

indicate the solution obtained with the model using·a 12 x 5 grido The

solid lines) presented for comparison) were obtained by Goldberg)

« 8) Gutzw111er and Lee' by extrapolation from 12 x 5 and 24 x 10 fini.te

difference solutions 0 At the center support their 12 x 5 solution

deviates farther from the extrapolated values than does the 12 x 5

values obtained with the modelo Near the center of each longitudinal

span there is a slight reversal in the accuracy trend between the

Goldberg solution and the modele

The values of M along the free edge and of M at the center x y

of each long1tudinal span for s"tructure 2-DB are given in Figso 26 and

270 The values obtained with the model are practically identical to

the values given by Goldberg et alo Figure 28 shows the transverse

distribution of N values at several sections of folded plate 2=DB. As x

shown in the fig~re) the distribution of stresses is strongly non-linear.

Moreover) this nonlinearity becomes more pronounced as the beam size is

decreased) as shown in Fig. 29ao Thus) the beam method could in no

way predict these stresseso Other methods of solution) such as those

suggested for short span folded. plate structures) are not applicable

because of the support conditions a Figl1res 29b and. 29c show the in·-plane

shears at the s1mple support and at the center 0 The total in-plane

35

shear) as represented by the integrated or summed value at the simple

end support increases with decreasing beam size. At the center)

although the peak value increases with decreasing beam size) the total

area under the curve decreases. The transverse shear reactions (not

shown) tend to do exactly the opposite. The net result is that the

vertical reaction at the center incr~ases with increasing beam size,

as shown in Table 5. The relative distribution of vertical loads

transferred to the various supports approaches that of a continuous

beam a$ the stiffener size is increased.

3.4. Two-Span Continuous Saw-Tooth Folded Plate

A saw-tooth folded plate continuous over central column

$upports is considered in this example. Five variations of the same

proble~ are studied. The loading, material properties) and geometry

of the structure are shown in Table 6. In addition, an interior plate

with the same loading and dimensions of plate DE in Table 6 is con­

sidered. The structures are designated 3-NB (No Beam)) 3-B (with Beam),

3-BO (with Beam and Poisson's ratio equal to zero), 3-DB (with Deep

Beam), 3-BTR (with Beam and Tie-Rod), and 3-CP (Central Plate). Two

sizes of central stiffeners and two values of Poisson's ratio were used.

The two end supports are assumed to be normal simple support diaphragms.

All structures are supported at the center on points Band D. These

points are completely restrained against vertical and horizontal dis­

placements for all structures except 3-BTR. Structure 3-BTR rests on

roller supports at points Band D. The horizontal movement of these

rollers is assumed to be partially restrained by a steel tie-rod with

an area of 0044 square inches 0

The changes of the significant forces in the folded plate

structure for the different versions of the problem are discussed in

the following sections,

Comparison of Transverse Moments M y

Figures 30 and 31 show the longitudinal variation of the

transverse moment M in the plate along the first valley and the central y

ridge. A substantial reduction in the magnitude of the moment occurs

in the vicinity of the stiffener. The magnitude and longitudinal

variation of M is practically unaffected) however) by a change in y

stiffener depth from 21 to 50 inches (an increase in stiffness of 12.5

times). Similar results were obtained for other longitudinal sections.

The design moment for the central ridge is affected very little by the

introduction of the stiffener. The major portion of the slab along the

ridge must still be designed for a moment of about -0.35 k-ftjfto The

maximw~ moment along the first valley is decreased by about 25% by the

introduction of the stiffenero

Figure 32 shows the moment taken by the stiffener. A change

in Poisson's ratio from 0.18 to 0.0 for the material properties assumed

for the plate elements causes almost no change in stiffener moment.

Notice that a small moment is present at the free edge of the beam for

the case V = 0.18. This moment is balanced by the moment in the central

slab-strip and will shrink to zero as the mesh size is decreased. IThe

37

reason for the existence of this small moment is explained in the

following paragraphs.

At a free edge of the slab the moment boundary condition for

the slab~strip of width L is~ x

For the free end of a beam the condition is:

-EIX Y

o

o

For non-zero values of X these conditions can be satisfied simul­x

taneously only when V = o. The condition imposed at the free end of

the beam for the combined beam and slab-strip is:

o

This condition satisfies the requiremept of statics that the net moment

be equal to zero and will converge to the condition given by Eq. (56)

when the mesh length L is decreased. For the case V = 0 the moment at x

the free edge of the beam is zero) inqependent of the mesh size.

When the depth of the stiffener is increased from 21 to 50

inches the moment taken by the stiffener changes by a very small amounto

3·4.2. Comparison of In-Plane Shears N , xy

Figure 33 shows the values of t4e in~plane shear forces for

structure 3-BO at the end support and at the center colu..mn line. These

values are extrapolations of the shears from thei.r points of definition

at the center of each mesh panel to the edge line of support. The

curves are essentially a series of parabolas. The curves for the other

structures are similar. From the data presented in the figure) the

load on the span is split between center and end supports in the ratio

of 37% to 63% or in about the same ratio as would be obtained by a beam

approach 0

Comparison of Forces N y

The axial forces in the stiffeners for structures 3-BO and

3-BTR are shown in Fig. 34. The difference between the ordinates of

the solid line immediately to the left and right of points Band D is

due to the presence of horizontal reactions at these points. The intro-

duction of a roller support at points Band D eliminates any horizontal

reaction, therefore bringing the values of the axial forces closer

together. Small tensile forces are present at the ridge lines but the

major portion of the stiffener is subjected to compression. As shown

in Figo 35 the value of N in the slab along the first valley stays at y

a constant value of about O. 5k/ft for most of the span and then dips to

a large compressive value near the center support of the structure 0

The same variation occurs for fold line D. This transition of N from y

tension to compression is independent of the presence of stiffeners for

the problem studied. The tensile force in the tie~rod between D and D? )

on the other side of the line of symmetry, was 1043 Ibs. The tie-rod

between points Band D carried a compressive force of 126 pounds. Pro-

visions may be built into the computer program to disregard compressive

.39

forces arising in tie rods 0 The resulti.ng compressive force in the

present case was deemed small enough not to warrant any modifications

of results since the tie rod would not buckle at such a low stresso

It must be pointed out that at the free end of a stiffener the same

situation occurs forN forces that occurs for transverse moments M 0

y y

The condition established is that the sum of the axial force in the

stiffener and the in-plane force in the central slab-strip must be

zero at the free edge 0 For the case V ~ 0 both the axial force in

the beam and N in the slab are zeroo y

Comparison of Longitudinal Stresses N x

As shown in Fig. 36) the distribution of longitudinal stresses

at the center support and at midspan is almost linear. Table 4 shows

that the main effect of the stiffener is to bring the values of the

tensi.ons and compr~ssions at the edges of the plates closer together 0

The transverse variations of longitudinal stresses in a central plate

are plotteo_ in Figo .370 The stress at the line of symmetry remains

unchanged whereas that at the support increases from ~12.l to -1402 k/fto

The total compressive force) however) remains practically unchanged"

Thus) there exists a stress concentration at the support that can not

be predict~d w~th the coarser grids" This concentration of stress is

due to the fact that the support is assumed to provide a reaction con-

centrated at a pointe Similar concentrations of stress are obtained if

the supports are assumed to provide a reaction distr:Lbuted uniformly

over a short length, as in Ref. (15)0 A more realistic picture would

40

be obta;ined if a very fine grid was used around the column perimeter

since actual columns are neither line nor point supportsp

CONCLUSIONS AND RECOMVlliN~TIONS FOR FU~urtE STUDY

4.10 Conclusions

A stiffness method of analysi.s for folded plate structures

using a discrete system model has been developed 0 By giving unit values

to the boundary displacements of the fixed plate, stiffness matrices for

the plates ,.are generated. Equations of equilibrium yield the final

boundary displacements at the fold lines 0 The method may be used for

both simply-supported and continuous structures. Excellent agreement

was found with other elasticity solutions) even when relatively coarse

grids were used. Rapid variations of forces and moments were success~

fully predicted. Although roller-type diaphragms were used at the end

supports) the method is not limited to this type of end condition.

Rotational and translation~~ fixity of points along the fold lines may

be obtained by eliminatiop of columns and rows in the equations of

equi.librium at the the longitudinal joints. Structures with tie-rods

or with columns placed at points other than the centerline can be

successfully treatedo

For the continuous cases studied, the introduction of trans-

verse stiffeners affects the relative distribution of applied load.

between the supports 0 This effect is more pronounced for plates of

low span-to-width ratio. The transverse distributions of longitudinal

stresses is affected by the presence of transverse stiffeners. Trans-

verse moments M i.n the plates are red.uced considerably in the immediate y

vicinity of the transverse stiffeners but the longitudinal distribution

-41..,

of M is relatively insensitive to large changes in stiffener size. y

The zone of reduced M on both sides of the stiffener is wider for y

exterior plates than for interior ones.

4.20 Becommendations for Future Study

42

The analysis of folded plate structures supported at the ends

by flexible stiffene~s with torsional resistance should be undertaken

in future studies. The $tudy of folded plates with rectangular holes

and of box girders should not present great difficulties. Simply-

supported and oontinuous folded plates having more than two plates

intersecting at a jo~nt can also be studied q

The method of solution should be combined with a mod.el similar

to the one used by Mohraz and Schnobrich(ll) to study the elasto-plastic

behavior Of folded plat~s. Laboratory tests to failure should be con-

ducted on small scale folded plates in order to verify experimentally

the ultimate capacities predicted by the analytical models.

Fipally" by using a model with non-orthogonal coordinates"

the method may be extended to folded triangular plates and to umbrella

and pyramidal roofs.

LIST OF REFERENCES

1. Ali, A. A., "The Analysis of Continuous Hipped Plate Structures, II Ph.D. Thesis, University of Michigan) 1951.

2. Craemer, Ho, "Prismatic Structures With'Transverse Stiffeners,iY Concrete and Constructional Engineering) Vol. 45) No.3) March) 1950.

3. Craemer, Ho, "Theorie der Faltwerke," Beton und Eisen, Vol. 29, 1930.

4. DeFries-Sk,ene, Arnim and Scordelis) A. C., if Direct Stiffness Solution for Folded Plates)" Journal of the Structural Division) ASCE, Vol. 90, No, ST4" Proc. Paper 3994" August" 1964.

5. Ehlers) G.) "f:in :ijeues Konstruktio~sprinz ip, II Bauingenieur) Vol. 9" 1930.

6. Gaafar) I., IIHipped Plate Analysis Considering Joint Displacement" II Transactions) ASCE) Vol. 119) 1954.

7. Goldberg) J. E. and ;Leve) H. L., "Tpeory of Prismatic Folded Pla,te Structures)" International Assoc. for Brid,ge and Structural Engineer~n~? Publications, Vol. 17) 1957.

8. Goldberg) J. E.) Gutzwiller) M. J. and Lee) R. H~) "Experimental and Analytical Studies of Continuous Folded Plates," ASeE Structural Engiqeering Conference" Miami) Florida, February) 1966.

9. Gruening, GO) IIDie Nebenspannungen in Prismatischen Faltwerken JII

Dr. Ing. Thesis, Darmstadt) 1932.

10. Huang, P. C~) "The Analysis of Hipped-P;late Structures with Intermediate DiaphIlagms)" Sc.D. Thesis, University of Michigan, ;l950~

11. l1ohraz, B.) and ScbnobrichJ W. C.) "The Analysis of Sh?llow Shell Structures by a Discrete Element System)" Civil Engineering Studies) Structural Research Ser;Les No. 304, University of Illinois, March) 1966.

12. Paulson) J. M." "The Analysis of Mutiple and Continuous Folded Plate Structures," Ph~D. Thesis, University of Michigan, 1958.

44

13. i!Phase I Report on Folded Plate ConstructioD)ll Report of the Task Committ~e on Folded Plate Construction of the Committee on Masonry and Reinforced Concrete of the Structural Division) Journal of the Structural Divi.sion) ASCE) Vol. 89) No. ST6) Proc. Paper 3741) December) 1963.

14. Powell) Graham He) "Comparison of Simplified Theories for Folded Plates)iI Journal of the Structural Division) ASCE) Vol. 91) No. ST6) Proc. Paper 4552) December) 1965'0

15. Pulmano, Victor A e) and Lee, Seng-Lip, ilprisma tic Shells With Intermediate Columns,l1 Journal of the Structural Division, ASCE, Vol. 91, No. ST6, Proc~ Paper 4578, December, 1965.

16. Reiss) Max) and Yitzhak, Michael, "Analysis of Short Folded Plat~s)1t Journal of t};le Structural Division) ASCE) Vol. 91) No. ST5) Proc. Paper 4516) October) 1965.

17. Schnobrich, W. C., "A Phys ical Analogt.:l.e for the Numerical Analysis of Cylindrical Shells, I! Ph.D. Thesis, University of Illinois, 1962.

18. ,Scordelis) A. C., rIA Matrix Formulation of the Folded Plate Equatic:ms," Journal of the Strl,lctural Division) ASOE) Vol. 86) No. ST10, Proc. Paper 2617, October, 1960.

19. Timoshenko, S., and Goodier, J. N~, Theory of Elasticity, McGraw,Hill Co.) New York, 1951.

20. Timoshenko, S., and Woinowski~Krieger) S.) Theory of Plates and Shells, McGraw-Hill Co.) New York, 1959.

21. Traum,) Eliahu) liThe Design of Fold~d Plates)n Journal of the Structural Division) ABOE, Vol. 85, No. ST8) Proc. Paper 2229) October, 1959.

22. Werfel, As, 1!Die Genaue 'I'heorie der PJ;'ismatischen Faltwerke und ihre Praktische Anwendung) II International Asso~. for Bridge and Structural-Engineering, Publications, Vol. 14, 1954.

23. Wltitney, C. S.) Ander$on, B. G. and, Birnbaum) H.) !!Reinforced Con~rete Folded P1t:-te Construction)1I Journal of the Structural Division) ASCE) Vol. 85) No. ST8) October) 1959.

24. Winter) G. and Pei) M.) "Hipped Plate Construction)" Journal) American Concrete Institute) Vol. 43) January) 1947.

25. Yitzhaki) Do) Prismatic and Cylindrical Sb.ell Roofs) Haifa Science Publishers) Haifa) 1958.

1

~ Edge

2 3 4 5 A x

6 B

7 8

RoHoSo Displacement of

Point UB VA WA aA

1 t

2 -- ~~ --21 1

Y x

2 t 21 -- -- x ----413

y

,3 -- == -- --

r2L 2 j t21

4 L3x + L:Ly x -= -- = --

212 Y .y

~l+v) 21

5 x

2 -1 -- --y

(l-v) 1

6 - .2 =~ == --

2 L Y

7 t

2 -- --

2L 1 --Y x

8 _ O,+'~) -- -.- --2

TAB1E 10 RIGHI'=HAND SIDE OF EQUI1IBRIUM EQTJATIONS FOR uNIT DISPLACE­MENTS AT THE BOUNDARIES 0 NO STIFFENER PRESENT.

45

I

46

1

Edge x

2 3 4 5 A L 6 stiffener

B y

7 8

R.HeSo Displacement of

Point VA WA CiA

1 -- -~ --

2 [ 0 e~ 2

{l-v )bd d- , -- 12 + --

hL3 Y

3 -- - eb.d~1-v2) --hL2

Y

2 4 ( 1-V

2 ) b'" Ii'

e2]

2(1-v2

)bd [2 e

21 4 2bCle(1-v ) - -- + t2 + hL2 hL3 12 hL2

Y Y Y

5 2b-ci( l-v

2) ~'h'/l 2) 2bde (1_v

2) - euc, -v -hL h T.

2 hL 'U Y oJ

Y

6 -- -- --

7 -- -- --

8 -- -- --~ i

TABLE 2 e TERMS TO BE ADDED TO RIGHT-HAND SIDE OF EQUILIBRIUM EQUATIONS FOR UNIT BOUNDARY DISPLACEMENTS WHEN A RECTANGULAR STIFFENER IS PRESENT.

~

A

Forces at

Midspan

N at A x

(K/ft)*

N !-3 t R

rt)·y'*

M at B y

(K-ft/ft)***

100 Ib I ftl

~---151

E = 3 )( lOG ps i y = 0,18 Span = 60 I

8 x 5 Grid 10 x 8 Grid

20.55 21.08

-10.28 -10.54

-4.162 -4.184

* Beam method: 20.48 K/ft. ** Beam method: -10.24 K/ft.

*** Engineering method: -5.625 K-ft/ft. **** Highest harmonic taken is 7.

47

Elasticity****

21.52

-4.236

TABLE 3. SOLUTION OF STh1PLY stJPPORTED FOLDED PLA.TE. PROBLEM 1.

structure Designation

2-DB

2-B

2-NB

I '

Dead load: !

1.414 ib/tt

i E :: 10.6 x 10 psi 'V :: 0.3 e :: O.OD!

Beam Size

8" x 24"

No Beam

., 6'

Conditions of Support at

Center of Structure

Supported at points A, B, and C

Supported a t A and C

Supported at A'and C

TARLE 4. PROPERTIES AND LOADING OF TWO -SEA.N CONTINUOUS V FOLDED PlATE. P.ROBLEM 2.

48

49

Percentage of Total Load

Case

At End Supports At Center Supports

4803 5107 No Beam

42·3 57·7 With Beam

40·9 59·1 With Stiff Beam

at Center ;

-r-

37·5 62·5 Beall Method of Analysis

.. ~ , ~

TABLE 50 RELATIVE DISTRIBUTION OF TOTAL VERTICAL REACTION BETWEEN END AND CEl\fTER SUPPORTS 0 PROBLEM 2.

p

f. 6

8

81 .).

r-- 40' ·1·

Problem Grid Beam Designation Size Size V

3-NB l2x6 No· Beam 0.18

3-B l2x6 8"x2l" 0.18

3-BO l2x6 8"x2l" 0.00

3-llB l2x6 8"x50 ii 0.18

3-BTR l2x6 8"x2l" 0.00

3-CP l6x6 No Beam 0.18

16 I

Tie-Rod

No

No

No

No

Yes

No

E

I

8.J 0

.1 '

It Wo :: 37.5 Ib/tt WL :: 30,0 Ibltt! h :: 3,0"

E :: 3 X 10' Ib/ind! e :: 0,0"

Comments

Vertical and horizontal displacements at points Band D are restrained.

" " Ii Ii

fI rr

Roller supports at points Band D permit horizontal movements only.

Rotations and horizontal displacements restrained along fold lines D and E. Vertical displacements unrestrained everywhere except at point D.

TABLE 6. SOLUTION OF TWO -SPAN CONTINUOUS SAW -TOOTH FOLDED PLATE. PROBLEM 3.

Structure

3-NB

3-B

3-DB

TABLE 70

N in 1b/ft at location~ x

A B C

10867 -13265 10697

10602 -12376 11334

10709 -12140 11569

"

VALUES OF N OVER CENTRAL SUPPORTS 0 x

D

=12142

-11918

-11743

PROBLEM 30

51

E

10557

11024

11325

(a) Saw - tooth

(b) Trough

(c) Barrel

(d) Northlight

FIG. I SOME COMMON FOLDED PLATE CROSS-SECTIONS.

53

z

FIG. 2. G

x Shear Nodes

1 Ll

J t

t

SECTION A-A

G. 3.

Contribution Area for Perpend icu lor Loads

Cont I' i bution Area for udinal Loads

FI

54

Contribution Area for Transverse Loads

55

w

G. 5. S N C o

x

FIG. 6.

Z n

, 7 ISTI

K+2

n+2

K'" I

n+1

t K

Lx n

L K-I

n -I

K-2 m-I m m+1 m+ 2

J-2 J-I J J + I J+ 2

~Ly~

FIG. 8. IFle OF I lOR POI

-Ny!(m,n) x (J, n)

G. u ( J, n) .

57

Nxy (m, n)

G. I . I -. P (m, K)

FIG. I Ie T CT (J, K).

FIG. 12.

t 'IN J, 1(-1

FIG. 13. 01 ROTATIONS OF (J, n)

59

(a) Bound ary Forces

w

(0) Boundary Displacements

FIG. I . o

60

T K+I

Lx n+1

l K

n ~ Edge of Plate

K-i

-m I m J-2 J-I J

~Ly~

G. I IFI S N EDGE OF PTE.

Nyx (m, 1'1 )

G. 16. IN- NE u ( J II n ) E

Qy ( m, K)

Qx.( J, n)

/ /

/ /

R y (J, K)

/ /

/ /

OX (J, n+l)

FIG. 17. TRANSVERSE FORCES ON W POINT AT EDGE.

61

FIG. 18. LOCAL GE

-L l

~

G. I . T E.

-lit l

/':1" /

S

62

Resultant

EMENTS

,.....

0 I

o o

o In

I

u

o

c: 0

:::ll

0 en

"CI >'.~

~C) u :;:It) (I)

0 )(

WOO

o o

E ~ -.0 0 .... n.

C 0 Q. en "0

::i! ... 0

@I)

~ :J

0 >

)(

Z

0 (\J

(!)

I.J...

o o N

4000

2000

o

I b - ft / ft

0.2

0.1

o

ft

-4236

_ ............... Elasticity Solution - -- 8 x 5 Grid

FIG. 21. Variation of My at Midspan. Problem I.

/ /

I I

I I

I I

/ I

I

/ /

I I

I /

~7 I

/

.--.----~-

64

-1150

I ct I

/ FIG. 22. Transverse Displacements at Midspan. Problem I.

l I

~ A -_ ...... Elasticity Solution

--- 8 x 5 Grid

o A B c

Si Support

G. 51 su T. P LE l .

-8 Reference 8 (Extrapolated)

-2

8 0

FIG. -10

-8

-6

-4

-2

10

120

-_ ...... 12 x 5 Grid

0.1 0.2 0.3 0.4

Reference a ( Extrapolated)

--- 12 x 5 Grid

O.! 0.2 0.3 0.4 !. L

FIG. 25. D.

66

0.5

2-

\

0.5

2 .... DB.

+-"0-....... +-"0-

f .a

+­"0-.......

-5

-4

-3

-2

-I

0

2

3 0

FIG.

-2

+- -I "0-

I .c

Reference 8 --- 12 x 5 Grid

26.

0.1 0.2

X

2-

8 12 x 5 Grid

67

0.3 0.4 0.5

:Ii

L

EE

Fold Line

Simple Support ct

Free Edge

-4.88 -7.04 + 0.41 + 8.23

3.87 5.98 -6.87

A B c o

FIG. 28. VARIATION OF Nx ACROSS SEVERAL SECTIONS OF FOLDED PLATE. PROBLEM 2- DB

0\ CD

8.23 1.28 A

B

-5.45

Deep Beam

--- 8eom

-- """'"",,,.,., No Be a m

-6.87 -8.81

( a) N x Va I ues at Center Support

-4

-2

0.2 0.4 0.6 0.8 1.0 y/o

( b) Nxy E Su

,p1"''''''''' ""'~

0 0 0.2 0.4 0.6 0.8 1.0

'I/o ( c)

FIG. 29. Nx N 2.

-12.48

- 2.5

- 2.0

+- -1.5 '+-, +-'+-

I -1.0 ~

- 0.5

0 0 0,1

3 -NB

---- 3-B _ ...... - 3-0B

0,2 )it

L

0.3 0.4 0.5

FIG. 30. VARIATION OF My ALONG FIRST VAllEY. PROBlE 3.

-0.4

-0.3

+-'+-.......... -0.2 +-'+-

3 - NB \\ I ~

1J -0.1 3-B 3 -DB

0.1 0.2 0.3 0.4 0.5

x

L

FIG. 31. VARIATION OF ALONG CENTRAL RIDGE. PROBlE 3 .

70

-..... I

oX

20 I

lOr-A

I ~~ -10

-20

-30~ \ -40

-50

8

/

-49.6

ct A C EI

V'0

10.4

I " 0 C '\J7

3 - 80

--_ ..... 3-8

~

Go 32. MOMENT IN TRANSVE,RSE STIFFENER. PROBLEM 3 s

<t

~0.3

;j

72

3.0

2.0

1.0

-..... A 8 C E ........ 0

~

-3.0

lo) At Center of Structure

3.0

2.0

1.0

- 0 ..... ........ ~

-1.0

-2.0

-3.0 (b) At End Support

FIG. 33. T E ION OF . PROBlE 3-80.

en a.

5

~ -5

-10

-15

-20

3 -80

................. _- 3 -8TR

FIG. 34. XI L IN STIFFENER. P 3.

I.O".........--...........,.----.,..-----r----.....,......--~

-1.0

+-...... ..........

~ -2.0

-3.0

-4.0

-5.0 0

FIG. 35. VARI

3 -8

_ ..... _-- 3 -08

0.1 0.2

OF Ny

x L

0.3

NG

0.4

\ \ \ \ I \ \

0.5

EY LINE B.

73

-

<t I

-15r- I -13.2

-12.1 ~ -lor \ / \ B D

-5 ... ~

" A '" E .:Jit. 0

5

lOt- V " " 10.6 10.9 10.7

I

(a) At Center of Structure I i

-lOr- I -6.4 -6.5 -6.6

I- '" ~ -5

+-~

" ~ 5

t- 'V' 'W'

6.9 6.2

(b) At Center of Each Span

FIG. 3 . NSVERSE ION x . P 3 ..... NBo -...J +:-

10.5

75

O -12.1 ,-.......----......... ---14.2

3 - NB (12 x 6 Grid)

.......... _- 3 -CP (16 x 6 Grid)

E

FIG. 37. DISTRIBUTION OF LONGITUDI STRESS 3-CP. IN CENTR PLATE. PROBLE

APPENDIX A. OPERATORS FOR INTERIOR POINTS

For convenience the operators for internal forces and the

equilibrium equation patterns are summarized in this appendix.

Operators for forces: x

0 W point I X U point

D. V point I ~ Y

v

v

-76-

N _ Eh yx - 2{1+V)

N Eh

:::: 2(1+\1) x:y

Eh3 M ;:;:;

x I2(1-v2)

Eh3 M ==

I2(1-V2) y

~ -A4 0

-AS

I-A1O

-

~ A U Eh 3 + 2(1+v}

-

1 A4

... A 7

A9

-A1

-A 11

-

Eh u + 2(1+v}

... A8 W

-A,] .... v

w

77 -

.....

o v

-

-

.....

0 V

-AI A ....

-

X

y

A5 -A 5

M Eh3 0 = 2 yx

12(1-v )

-A 5 A5

o

-A6 A6

Equilibrium equations:

-G 3

-0 3

6

78 x

Ly w

w

79

,A 1 -B Bl 1

®-B 2

,B2 ~ B2 U + X V + (l-v l - _

S Bl -B

Eh X - 0

1

.. ~ Al x

B4

Ly , -B

1 BI

f:::. U + V + (1-V2 l - _ Eh Y - 0

Bl -B ~ I

B4

Where

L ~ Al ==-1I.. All ==L

L A6 == L L x x x

L A12 == 2(A10 + All)

A2 == V A == J... 7 L2

x L

AS x V

==y;- AS == L y y

A4 == 1.0 A9 == 2(A7 + AS)

~l-vl L

A = x

5 L A10 == 2' y L

Y

G1

= 6G5

+ 6G6 + 4G4

G2 = 2G4 + 4G5

G3 = 2G4 + 4G6

t 2

G4 = - 2L 1 x y

t2

L x

- 4L3 . y

L x

L y

B _ (l-V) . Ly

4 - 2 L x

80

APPENDIX B 0 EQUILIBRIUM EQUATIONS AND OPERATORS FOR TEE TRANSVERSE STIFFENERS

B.l. General Remarks

Force-displacement relations and the corresponding operator

expressions associated with eccentric stiffeners are derived in this

appendix. Because the stiffeners are assumed to be completely flexible

in torsion) the operators will involve V and W displacements only.

The V and W operators obtained for the stiffeners must be added to

the appropriate ope~ators for the plat~ itself in order to obtain the

complete equilibrium expressions for the plate-stiffenerstructure~

Only stiffeners of rectangular cross-section are considered.

The actual stiffener has a width b and depth d. This stiffener has

an eccentricity e relative to the middle plane of the plate. The

equivalent stiffener be.;;tm u,sed in the model has dimer,Lsions width b)

depth d, and eccentricity ~I (Fig. B.l). For the equivalent and

aotual beams to h~ve identical response it is necessary that e 1 be

equal to e and, d to be equal to d/ .J3 .

In the following derivation the s superscript placed to the

left of the symbol denotes that the quantity to which it is appended

applies to the stiffener. The right-hand superscript indicates whether

the term applies to the top or the bottom of the equivalent beam, thus

S T EY(J,K) means the strain in the top fiber of the equivalent beam at

the point (J, K) .

-81-

82

B.2. Strain~Displacement Relations

Referring to Fig. B.2)the top and bottom strains at location

(J)K) of tne equivalent beam ar~:

(B.l) S~ SVI:

(m+l)K) ~ (m,K) .. , ·L··· .

Y

T4e displacements at th~ top and bottom may be expressed as

follows (Fig. B.3):

(B.2)

and since

then

(Bo 4)

Similarly,

S~ (m+l)K)

V( ) _ (! _ =.1) d (W ) :m+l,K,2 d . L (J+l)K) - W(J)K)

Y

S T 1 e i

V . - V + (-2 + =-) (m+l)K) - (m+l,K) d

lB.5)

Substitution of Eqs. B.4 and B.o5 into Eqs. B.l relates the strains at

the nodes to tne plate displacements by the following equations~

B.3. Axial Forces and V Operators

~he for~e resultapts at top and bottom of the equivalent

beam are found from the stra,insby

Ebd 2

Sl\fB Ebd y(J,K) = ~

therefore) the total axial force in the beam at location (J)K) is~

Ebd 2

Substitution of Eq. Bo6 into Eq. B.8 gives:

84

Similarly,

With the coptribution of the beam 1 s own weight neglected)

the e~uilibrium e~uation of axial forces acting on bar (m,K) may be

written as follows;

o (B.ll)

Upon $ub$tttution of E~s. B.9 and, B~lO into E~. B.ll, the e~uilibrium

eQuation for the beam can be written as:

o

2 Multipl~ing this expresqion by (l~~ ) qnd putt~ng it in operator form)

the following set of line-operators are obtained:

[ I

-c , where

Cl I~. I

C4 $

(bd ) (1_v2) hL

Y

-C 2

@

§j

Cl

IJ I b .. V +

.,.C C -3 L~

W $ $.

C4 3C3

0 (B.13)

(B .14)

B.4. Bending Moments and W Operators

The moment at location ,K) in the stiffener) defined about

the mid-depth of the slab" is as follows ~

This moment may be expressed in terms of strains by substituting the

appropri~te strains for the stress terms within the brackets~

8M

Ebdd y(J,K) = -4-

Ebde! --2 [

s B 8 T l Ey(J,K) + EY(J,K~

(B.l6)

Finally with the strain-displacement relations of Eq. B.6, Eq. B.16 can

be written as~

_ Ebde i (V _ V .) L . m+l"K m,K

y

The transverse shear forces at points (m)K) and (m+l,K) are given by;

s Qy(m+l,K)

Transverse equilibrium of forces at point (J,K) requires that

( 8' ,B.l )

86

(B.19)

Substitution of Eqs. B.18 into B.19 yields:

o (B.20)

The W equilibrium equation for the beam is obtained by further substitu-

tion of the expre$sions for moments into Eq. B.20;

(B.2l)

2 Multiplying this expressj,on by (li~ ) and converting it into operator

form) the following set of line operators are obtained:

[I C3

-C4 C4 -C

1 J V

3 .6 £, ~ 6 6

C5 -4c 6c

5 -4c

C5j 2 5 5 + $: $ @) $ $ W+

(l-v .) S 0 Eh Z(J,K)

(B~22)

where C3

and C4 are given by Eq. B.14 and

Bo 5 0 Detennination of Eccentricity and Depth of Equivalent Beam

R~ferring to F~g. B.~the qtrain at the level of fiber AB is:

dV EAB = ely (B , 24)

because V displacements are defined at the level of this fiber. The

strain at a point a distance S above fiber AB is related to the dis-

placements by:

1 Since R ::;

The force in an elemental area b ~ d~ is:

dV dN = E 0 b (r--, dy

The total rorce in the cross~section is given by

Integration of this expression yields

(B.25)

(B.26)

(B.27)

88

N Ebd (dV + e . d2w) dy dy2

(B.28)

For Eqo B.9 to be the discrete representation of Eq. B.28 it is

necessary that e' be equal to e.

Similarly) the total moment about the mi~ft~:Q.~pth 1.J)f 'the ,slab

is:

M (B.29)

Integration of this expression yields:

d\f. d~ ~2 2l M = ~Ebde • dY' - Ebd . dy2 L12 + e 1

For Eq. B.17 to be equivalent to Eq. B.30 it is necessary that

d=d/J3I.

B.6. Coupling of Linear Equations due to Eccentricity of Stiffeners.

The operat'Jrs given by EqQ B.13 and Eq. B.22 must be added

to the V and W operators obtained for the plate alone. Notice that in

the general case the V operator involves W displacements and vice

versa. This means that the simultaneous equations will be coupled. If

the eccentricity e of the stiffener is zero Eq. B.14 will involve v

displacements only and Eq. B.22 w displacements only. In this case

the U and V equations will be coupled and the W equations will be

independent.

d/2

d

d/2

(0) Actual Stiffener

Ly

-d/2

d/2 - _~~ _-_--·17- h'....,....,...'lII"""7""".....,....,.....,....,...~Il""""'P""~~~...,....,...'7"""I!l

(b) Equivalent Stiffener

FIG. B .... 1. DIMENSIONS OF ACTUAL AND EQU NT STIF NER.

(m-I, K) (J-i, K) ( m) K )

I I I

(m+I"K)l

I ~/2 + e')

( J , K)

---+--- ---+---+ ~/2-e')

I· ·1

1Ft STI

J d/2

+-e l

- -+-t-d/2 - e I

t S B

V (m, K)

FIG. B-3. V DISPLACE ENTS POINT (m II K).

R

I

FIG. 8--4. ELE ENT BENT STIFFE R.

APPEN"DIX C. DESCRIPrION OF COMPUTER PROGRAM

A brief description of the FORTRAN program developed for

this study is given in this appendix, The program uses the so-called

"ping-pong technique ll and consists of ten core-loads as follows:

- 1. Reads all information concerning the geometry~~ rna terl.0..i:';;

properties and loading of the structure. Echo-prints

all information.

2. Generates for each type of plate one force-displacement

relation for each displacement point. Next, two column

vectors for unit perpendicular and in-plane loads are

generated. Finally, it generates one column vector for

unit movements of each boundary degree-of-freedom.

3. Solves the equations generated in core-load 2 by Gauss

elimination procedure.

4. Gene~ates a force matrix for each type of plate. Forces

are expressed as a ilfixed i1 value plus a linear combina­

tion of the movements at the edges in the local system

of coordinates.

5. Generates a force matrix for each plate. Forces are

expressed in terms of edge-movements in the global system

of coordinates.

6. Sets up equilibrium equations at the longitudinal edges

of the folded plate.

7. Solves the equations generated in core-load 6.

-91-

8. Transforms the solution obtained in cere-load 7 back to

local coordinates.

9. Solves for fi.nal displacements in each plate 0

10. Computes forces) moments. Prints deflections) forces

and moments.

92