Buckling and post-buckling behavior of a large aspect ...

BUCKLING AND POST-BUCKLING BEHAVIOR OF

A LARGE ASPECT RATIO ORTHOTROPIC PLATEUNDER COMBINED LOADINGS.

Fred Lewis Ames

^AVAL POSTGBADUATB SCHOW

ioHTEBK, CAT IF. 93940

BUCKLING AND POST-BUCKLING BEHAVIOR

OF A LARGE ASPECT RATIO ORTHOTROPIC PLATE

UNDER COMBINED LOADINGS

FRED LEWIS MIES

LIEUTENANT, UNITED STATES COAST GUARD

B.S./ United States Coast Guard Academy

(1968)

SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE

DEGREE OF OCEAN ENGINEER

AND THE DEGREE OF

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June, 1973

LIBRARYNAVAL POSTGRADUATE SCHOODMONTEREY, CALIF. 93940

BUCKLING AND POST-BUCKLING BEHAVIOUR

OF A LARGE ASPECT RATIO ORTHOTROPIC PLATE

UNDER COMBINED LOADINGS

Fred L. AmesLieutenant, United States Coast Guard

Submitted to the Departments of Ocean Engineering and Me-chanicc'il Engineering on 11 May 1973 in partial fullfillmentof the requirements for the degrees of Ocean Engineer andMaster of Science in Mechanical Engineering.

ABSTRACT

Orthotropic plate theory has been increasingly used tomodel plate-stiffener combinations typical of those whichare used in ship hulls. The behaviour of a thin plate withlarge deflections is described by two nonlinear partialdifferential equations of equilibrium and compatibility.The orthotropic form of these equations is derived andsolved for a large aspect ratio plate with long edges simplysupported and short edges fixed as boundary conditions.Buckling and post-buckling regions are investigated undercombined loadings of lateral pressure, inplane edge com-pression and edge shear. Results are presented for virtualaspect ratios 1/1.5 to 1/6 and both isotropic and ortho-tropic plate properties in the form of design and behaviourcharts.

Thesis Supervisor: Alaa E. Mansour

Title: Associate Professor of Ocean Engineering

Thesis Reader: Thomas J. Lardner

Title: Associate Professor of Mechanical Engineering

Acknowledgement

The author would like to express his appreciation to

Professor Alaa Mansour for his support and guidance in this

research. Gratitude is extended to Commander Joseph L.

Coburn,, U. S. Coast Guard Headquarters Research and Develop-

ment Branch, for providing the funding of the necessary

computer services.

Special thanks is given to my wife Holly for her con-

tinued understanding and long hours spent deciphering and

typing the rough draft. In addition, the fine typing of

the final draft by Ms. Cathy Bayer is greatly appreciated.

Table of Contents

Abstract 2

Acknowledgement 3

Table of Contents 4

List of Figures 7

List of Tables 9

Nomenclature 10

1. Introduction 14

2. Formulation of the Problem 17

2.1 Orthotropic Material Properties 17

2.2 The Rectangular Orthotropic Plate with

Large Deflection 17

2.3 Bending and Membrane Stresses 21

2.4 Load Boundary Conditions 22

2.5 Support Conditions 23

3. Theoretical Analysis 24

3.1 Analysis Procedure 24

3.2 Solution 25

3.3 Bending and Membrane Stresses 29

3.4 Effective Width 33

3.5 Principal Stresses 35

4. Numerical Solution. 37

4.1 Analytic Solution 37

4.2 Computer Solution 39

5. Results 41

5.1 Design Charts 41

5.1.1 Charts of Deflection 41

5.1.2 Charts of Effective Width 41

5.1.3 Charts of Bending Moment 41

5.2 Behaviour Charts 4 2

5.2.2 Charts of Maximum and Minimum

Total Principal Stresses 42

6. Discussion of Results 43

6.1 Orthotropic Properties 43

6.2 Comparison with Existing Solutions 43

6.3 Design Charts 44

6.3.2 Charts of Effective Width 45

6.3.3 Charts of Bending Moment 4 6

6.4 Behaviour Charts 4 6

6.5 Examples Demonstrating Use of the Charts. ... 49

6.5.1 Design Example 49

6.5.2 Behaviour Example 50

7. Conclusions and Recommendations 52

8. References 54

9. Appendices 101

Appendix A. Details of Derivation of Basic

Equations of Large Deflections 102

Appendix B. Effective Width 115

Appendix C. Details of Solution 118

Appendix D. Computer Programs 126

List of Figures

Figure Title Page

1 Illustration of Coordinate System ggand Inplane Edge Loadings

Design Charts of Deflection vs. InplaneLoad N*

2,3 p = 0.667 67

8,9 P = 0.50 70

14,15 P = 0.25 73

20,21 P = 0.167 76

Design Charts of Effective Width vs.Inplane Load N*^ X

4,5 P = 0.667 68

10,11 P = 0.50 71

16,17 P = 0.25 74

22,23 P = 0.167 77

Design Charts of Bending Moment vs.Inplane Load N*^ X

6,7 P = 0.667 69

12,13 P = 0.50 72

18,19 P = 0.25 75

24,25 P = 0.167 78

Behaviour Charts of Deflection

26,27 p = 0.667, N* = 79

36,37 p = 0.80, N* = 84

46,47 P = 0.667 89

Figure Title Page

Behaviour Charts of Maximum andMinimum Total Principal Stresses

28-31 p = 0.667, N* = 0, X = 80,81

32-35 p = 0.667, N* = 0, y = 82,83

38-41 p = 0.80, N* = 0, X = 85,86

42-45 p = 0.80, N* = 0, y = 87,88

48-51 p = 0.667, X = 90,91

52-55 p = 0.667, y = 92,93

56-60 Deflection and Principal Stresses, 94,95,96Top and Bottom of Plate

61-63 Design Charts of Deflection, Effective 97,98Width and Bending Moment vs. InplaneLoad N*, p = 1.0

64 Plate Deflection Comparison with Levy 99Solution [5]

65 Effective Width Comparison with Levy 100Solution [5]

List of Tables

Table Title Page

1 Coefficients C 56pq

2 Rigidity and Compliance Coefficients 60

3 Integrals 61

4 Computer Symbols 65

NOMENCLATURE

y= ordinate in long direction

X = ordinate in short direction

a = plate length in x-direction

b = plate breadth in y-direction

3 = — = aspect ratio

u, V, w = displacements of a point in x-, y-, and z-

directions respectively

e , e / e = middle-plane strainsX y xy ^

D , D = flexural rigidity of orthotropic plate in x-

or y-directions respectively

D = effective torsional rigidity of orthotropic

V , V = Poisson's ratio of orthotropic plate in x- or

y-directions respectively

E , E = modulus of elasticity in x- or y-directionsX y

respectively

G = modulus of elasticity in shear

N # N / N = middle-plane loads per unit length

M , M = bending moment in orthotropic plate acting

around a line perpendicular to x- or y-axis

respectively, per unit width

M = twisting moment in orthotropic plate

b = nondimensional deflection coefficientmn

Airy's stress function

h = plate thickness, orthotropic

xy Gh X y y x

2D =Dv +Dv +4Cxy X y y X

C = 9hl12

e = i2^(i-v v„)1,2

P , PX y

= total loads in x- and y-directions respectively

= uniform lateral load, per unit area of plate

= constant inplane shear load, per unit length

P = KY _ virtual aspect ratio of orthotropic plateX

n = xy _

Id dvl X !

^= torsion coefficient of orthotropic plate

J JM X y

N*X 2t^

IT D= nondimensional in-plane load x-direction

N b^_ yN* = —

= nondimensional in-plane load

y-direction

qb"* . .

Q* = —^ = nondimensional lateral load7T'*hD

SabS* = = nondimensional edge shear load

TT^Dxy

b , a = effective width, x- or y-directionse e r j:

respectively

o , o ', o* , a* = membrane stresses in x- and y-

directions respectively; nondimensional

T ; T* = shear stress in xy plane; nondimen-xy xy J t- '

sional

^bx' ^by' ^bx' ^by ^ bending stresses in x- and y-

directions respectively; nondimensional

T, '"^h

~ twisting stress in xy plane; non-

dimensional

o. , a. , T^ = total stressestx ty txy

a* , ai / tJ = nondimensional total stresses

ai'f of^' "^t't~ nondimensional total stresses

cr, « ; a? p = maximum and minimum principal stresses,

plane stress; nondimensional

a r a = edge membrane stress in the x- or y-e ^e

direction respectively

M*, M* = nondimensional benclinq momentsX y -^

per unit length

M* ,M* = nondimensional bending moments pero ^o

unit length due to curvature in x-

direction only (M* ) and y-directiono

only (M* )

C ,6 , A / X , Y = coefficientspq pq pq q q

a ; a* = Von Mises combined stress; non-o o

dimensional

1,1 = moments of inertia of the stiffenersX y

with effective plating in the x- and

y-directions respectively

1,1 = moments of inertia of the effectivepx' py

plating alone in the x- and y-

directions respectively

S , S = spacings of the stiffeners extending

in the x- and y-directions respec-

tively

h , h = equivalent thickness of the plate

and the stiffeners (diffused) in the

x- and y-directions respectively

h = thickness of plate aloneP

1. Introduction

Of major concern to the naval engineer is the reaction

of ship bottom plates to a combination of inservice loadings.

In gene::al, the plate-stiffener combinations will be sub-

jected to inplane compression or tension edge loads due to

ship hogging or sagging, lateral hydrostatic loads and in-

plane edge shear loads. With increasing compression and

shear loads, a critical state will be reached where buckling

will occur. Plate deflection will exceed plate thickness,

however, the plate may still carry considerable loads. High

slenderness ratio (b/h) plates are encountered in modern

longitudinally framed ship designs. The yielding load may

be considerably higher than the buckling load and investi-

gation of the plate-stiffener residual strength in the post-

buckling region is of great importance.

Orthotropic plate theory has been increasingly applied

to ship structures. With this analysis, the plate-stiffener

combination is modeled as an equivalent flat plate with

elastic properties that are different in two perpendicular

directions. Schade [12, 13] and Mansour [9, 10]*, among

others, have worked with this concept.

The small deflection theory is useful only in the pre-

buckling region where deflections of a plate are small in

*Numbers in brackets designate references in Chapter 8.

comparison with its thickness, ^.he assumption of no deforma-

tion in the middle plane of the plate is made in this case.

w 1If deflections are not small, r- > :r, the strain in the middle

plane is no longer negligible and must be considered. Von

Karman's equations describe the behavior of an isotropic

plate under combined lateral and inplane loads. The non-

linear equations describe the behavior of the plate in both

small and large deflection regions.

Considerable application has been made of von Karman's

equations. Levy [4, 6] investigated square and rectangular

plates simply supported and subjected to both inplane edge

loading and norroal pressure. Coan [3] included the effects

of small initial curvature for this case. Levy [5], using

two fixed and two simply supported ends, solved the equations

with edge loading only for an aspect ratio of 4 , and studied

pure shear for the simply supported square plate [7, 8].

The square plate with variations of the boundary conditions,

subjected to edge loading only, was studied by Yamaki [17]

Payer [11] applied von Karman's equations to deep web frames

and included uniform edge shear in addition to compressive

edge stress and normal pressure. Shultz [15] investigated

wide plates having aspect ratios of 3 = 1.5 to 8 for con-

ditions of a transversely framed ship. All of these

analyses apply only to isotropic plates. Mansour [9, 10]

extended von Karman's equations to consider slightly rec-

tangular orthotropic plates under various boundary, loading

and initial deflection conditions.

A large aspect ratio orthotropic plate with the shoD't

edges fixed and the long edges simply supported is considered

in this paper. The plate-stiffener combination is subjected

to loads of inplane edge compression, uniform edge shear and

uniform pressure normal to the plate. Figure 1 illustrates

the loadings and coordinate system used.

The orthotropic form of von Karman's equations are

solved similar to the theoretical analysis of Mansour [9, 10].

The IBM System 37 model 165 is employed to produce the

numerical results. "Design" charts of deflection, effective

width and bending moment are given for orthotropic plate

virtual aspect ratios of 1/1.5 to 1/6 with inplane edge

compressive and lateral loadings. In addition, "behaviour"

plots of plate centerline deflection and total principal

stresses are given for virtual aspect ratios of 1/1.5 and

1/1.25 for various combinations of inplane edge compression,

edge shear and lateral loadings.

2. Formulation of the Problem

2. 1 Orthotropic Material Properties

An orthotropic material has different elastic proper-

ties in two perpendicular directions. For plane stress in

the xy plane, the stress-strain relations are

^x = l-v^'v ^S."^ ^y^y^

X y -^-^

Eo = -. ^— (e + V e ) (1)y i~^x^y y X X

xy 'xy

from energy symetry

E V = E VX y y X

which gives 4 independent elastic constants.

2. 2 The Rectangular Orthotropic Plate v/ith Large Deflection

(Figure 1)

w 1For thin plates with large deflection, r- > y, a satis-

factory approximate theory makes the following assumptions:

1. Points initially on the normal to the middle plane

of the plate remain on the normal after bending.

(Disregard of shear deformation.)

2. The normal stresses in the direction transverse to

the plate can be disregarded. (Plane stress where

a = 0.)z

3. Hooke's Law relating stress-strain applies.

For the large deflection theory, deformation in the

middle plane of the plate must be considered. These strain

components therefore include the effect of deflection and

are approximately

X ~ ax 2^dx'

e ~ 9X + 1(9^)2 (2)y 3y 2^3y' ^"^^

^ 9u 9v; 3w ^ 9w^xy ~

8y 9x 9x * ay

Equilibrium of forces and moments on an element of a

plate produces the equilibrium equations

aN aNz + -^ =

a^M a^M a^M^ - 2. ^y +

ax^ ^^3^ ay^

_(5 + N ^+ 2N ^-t- N ^) (4)Xg^2 xyaxay y ^^^^

Introducing the familiar Airy's stress function F which

satisfies equations (3)

N = l!Z; N = ^ ; N = -v^ (5)

^ 8y^ ^ 8x2xy 8x8y

From substitution of large deflection strains (2) into

the stress-strain relations (1) and using the definition of

bending and twisting moments, the moment-curvature relations

for an orthotropic material are derived

M = -D (^ + V ^)^ ax^ ^ 8y2

M = -D (^ + V i^) (6)y ^\y' ^ 8x2

M = 2CB^w

xy 8x8y

where the rigidity coefficients are defined as

E h^D -X - 12(l-v^Vy)

D = JL

y - 12(l-v V )J ^ X y

Substitution of equations (6) and equations (5) into equation

(4) , the "equilibrium" equation is obtained as

D i^:^ + 2D 9'^ -f D i>

Q + •*2 • — Z • + •

rv 2 ^2 3x3y 8x8y ^2 ^2

To obtain the "compatibility" equation, the strain equations

(2) are first differentiated and combined to eliminate dis-

placements u and V

^'^x _ ^'^xy ^^'^y

_( a^w^2 _ 3lw ^ aH/^g^

3y2 axay g^2 axay ^^2 9^2

Differentiation of equations (1) , substitution into equation

(8) along with (5) and using the equilibrium equations (3)

the "compatibility" equation is obtained as

J i!l ^. 2J 9'^ + J 3^""ax** ""âxây^ ây"*

a^w.j a^w a^w O)= (~:;^)axay .^2ax^ ay 2

J =^ ; J = ^

X E„h ' y E^h

2J = -prr- - V J - V Jxy Gh X y y x

Equations (7) and (9) are fourth-order non-linear

partial-differential equations that describe both small and

large deflections of an orthotropic plate. This includes the

buckling and post-buckling behavior of the stiffened plate.

Substitution of the isotropic material properties

EE =E =E;v =v =v; G = Tm—\X y X y 2(l+v)

result in the familiar von Karman's equations.

Solution of these equations with 16 boundary conditions

results in the two functions F and w. The boundary con-

ditions are eight support conditions and eight edge load/

displacement conditions.

2 . 3 Bending and Membrane Stress

The membrane stresses are determined from equations (5)

X 1 8^Fas N.. -. ^ 2

X h h 3^2

a =!z 1 alF(10)

^ h ^ ax^

xy ^ _ 1 9^Fxy h h 9x8y

Noting that the maximum normal stress acts on those sections

parallel to the xz or yz planes, and using equations (1)

(2) and (6) the bending and shear stresses are obtained as

bx Cl-v V ) DX y X

^ X y,l y

The maximum bending and shear stresses occuring at z = ±y

M M Ma, = ±6— ; a, = ±6-^ ; x, = +6-^^ (11)bx ^2 'by ^2 ' bxy ^ ^2

where M and M and M are given in equations (6) . Thex y xy ^ ^

total stresses are the sums

a. = a ± a,tx X bx

"ty = "y * "by <12)

txy xy bxy

2. 4 Load Boundary Conditions

The following most general conditions state that the

edges are subjected to an inplane average compressive load

per unit length in the y-direction and x-direction of

magnitudes N and N respectively. Additionally, all edges

are subjected to a constant inplane shear load per unit

length of magnitude S. Referring to Figure 1,

at X = ±2

N = S , —-^ = -Sxy ' dxdy

Inplane load resultant P is (13a)

^x = : ,. H ^^ = "^x^-b/2 9y

at y = ±2

N = S , TT^—^ = -Syx ' dxdy

Inplane load resultant P is (13b)

P = / ^-^ dx = -N ay -a/2 8X2 y

2.5 Support Conditions

Edges simply supported at x = t-y

Deflection equals zero: w =

External moments equal zero: (14a)

9^+ V ^ =Bx^ ^ 8y2

Edges clamped at y = ± y

Deflection equals zero: w =

Slope equals zero: |— = (14b)

3. Theoretical Analysis

3.1 Analysis Procedure

The solution method used is identical to that of

Mansour [9, 10] which is an extension of Levy [4], Coan [3]

and Yamaki [17]. Briefly, the outline of the procedure is:

1. Express the deflection of the plate satisfying the

support boundary conditions, in a double trigono-

metric series choosing only a finite number of

terms, w will be a function of unJcnown nondimen-

sional coefficients bmn

2. Substitution of this expression into the "compati-

bility" equation (9) results in a fourth order

partial-differential equation with the Airy stress

function F and quadratic functions of the coeffi-

cient b expressed as coefficients Cmn ^ pq

3. A solution for the stress function is assumed which

satisfies equation (9) and the load boundary con-

ditions. F will be expressed as functions of

unknown coefficients 6 which in turn are functionspq

of coefficients Cpq

4. To determine the unknown coefficients b , Galerkin'smn

Method is applied to the "equilibrium" equation (7)

with substitution for w and F by their appropriate

expressions. Using the orthogonality properties of

the trigonometric functions, a set of simultaneous,

non-linear algebraic equations involving cubic

products of the coefficients b results.^ mn

3. 2 Solution

The deflection of the plate surface can be expressed in

the form

w = h E Z b f (x) g (y) (15)m n mn m ^n -^ ^^-'^

Satisfying the boundary conditions (14) / the deflection terms

take the form

J. I V miT , _ pf (x) = COS—X ; m=l, 3, 5....m^ ' a '

/ \ / T \ n+1 , 2mT T o ->

gj^(y) = (-1) + cos-^-y ; n = 1, 2, 3

Differentiation of expression (15) and substitution into

the "compatibility" equation (9)

J i!£ + 2j^"^

+ J i^

= h'ai;

b„ (!!^) (2niL)sin^x sinSgly}^m n mn a b a b

- h={E E b„r (-!)"+!+ cos2gi:y]{!HL)acosi^x}m n mn b a a

•{E Z b (-T—)^ cos—-X cos-r—y}m n mn b a b

{Z Z E E 4b b. .mnii sm—^x sm-r—-y2, 2 . . mn 11 -^ a b -^

a b m n 1 3-^

. ilT . 2111 „., , 2-2 IHTT• Sin—X sm-^—y - Z E Z E 4b b. .m^i'^cos—^x

a b ^ ^ ^ . j mn 13 J

r/ , V n+1

2n7T , iiT 2i7r , ,^,.• I (-1) + cos-^y] cos—^x cos-^y} (16)

m, i = 1/ 3/ 5....

which can be expressed in the form

J i!£ + 2J -A!IL + J i!Z

EEC cos^^i^^x cos^^y (17)h^TT** V ^ A^ 2077 2qTrEEC cos—^£—^x cos-^—

a2j32 P q pq a b

P/ q ~ 0/ 1/ 2....

where C are quadratic functions of the nondimensionalPSl

coefficients bmn

A particular solution for (17) is assumed in the form

F = h^ E Z ({) cos^^x cos^y (18)p p q pq a b -^

substitution of (18) into equation (17) defines coefficient

6 aspq

D 2 /-I

^ = P3(19)

The stress function F to satisfy the load boundary conditions

(13) and equation (9) is

N NF = - -^x^ - -~Y^ - Sxy + F (20)

The method of B.G. Galerkin is applied to the "equilib-

rium" equation (7) to determine the unknown coefficients b^ mn

Galerkin 's condition requires that the following equation be

satisfied by all functions f (x) g (y)

a/2 b/2 ^ It ^ ! ^j,

/ / [D 1^ + 2D _AJ!L_ + d ^ "^

^ax"* ^^9x^8y^ ^ay**

8^F 9^w_^^ a^F 9^^ 9^F 9 ^w, ,^^." ^ " 77 • 7T "" ^91^ * 93^ - 77 * 7T^ ^^^^

9y 9x -^ -^ 9x 9y

f^(x) g^(y) dxdy =

where f (x) g (y) is given be equation (15) . Equation (21)

may also be obtained by applying the principle of virtual

Substitution of w and F, expressions (15) and (20)

respectively, into equation (21) and using the orthogonality

properties of the trigonometric functions, there results

Y TT^D y IT D^ y -^ X

Zf g b^^{2(-l)^"^^r'* + [rS2p2n(2s) 2r2+ p'*(2s)'*]6j

eJ n f g rn'^ ' - . .- .-h ,,v--y - h x--/ j u^^g

2.r2(-l)^-'^^)} + i; I°f E 4in(i)^A (-1) s+l^(-l!^) b2r5 m r n s tt' mr v tt^d

^^^ X ^ xy

+ I Z Z 2mn(i) ^A (-I- + -!_) §£_ (_SafcL.) bm r , IT mr n+s n-s GJ 2rs ninn+s y 7T D^ xy

+ Z E E E b {2(-l)^'^^n^rM(j) ^ + (}) + <^ ]m n r s mn ^m+r ^m-r ^r-m2 fH 2 'J^ 2 '^

+ 2(-l)^"^-'-m^s^[(}) ^ + (j) + (}) ]^m+r ^m-r ^r-m^/S <),S -),S

+ (ms-nr)^I4) ^^ +4> +4> •-(|) +cl) ]^m+r , ^m-r ^m-r ^r-m ^r-m—2"fH+s —2—/ n-s —J—/S-n —2~>n-s —2~> s-n

+ (ms+nr)^I(}), +

, + <^ +<t> ]>m+r ^m+r ^m-r ^r-m

—J-,n-s —2—/ s-n —2—/ n+s —2~7n+s

r s ej.. rfT _it,

3r+l ^ Q -,it= Z Z (.l)^+-2-^ :^ (-2^—) (22)

m, r = 1, 3/ 5....

where 6. . is the Kronecker delta, (|) . . = if i < or

j < 0, = if n = s and-" ' n-s

A = —,— if —^r- oddmr m+r 2

A = if —rr- oddmr iti-r 2

Equation (21) is identical, except for the addition of the

two shear and two N load terms, to that of Mansour [9] case

(i) and, if isotropic material properties are substituted,

reduces to Yamaki [17] case Illb for zero initial curvature.

Substitution for coefficients cf) in equation (22)

results in a set of simultaneous, non-linear algebraic

equations involving cubic products of the coefficients b .

Solution of these equations gives the coefficients b and^ -^ mn

hence defines the functions w and F.

3 . 3 Bending and Membrane Stresses

The membrane stresses are now determined from equations

(10) , (18) and (20) . These are

NX T_ r. V X /2q7Tx 2 2pTr 2q7T

a = - -u hZZd) (-r— ) cos-^x cos-g—

X h p q ^pq b a b ^

NY 1- V V J.

/2p7T. 2 2p7r 2qTra = - -tf

h E Z d) {-^—-) cos--^-—

cos-^—

y h p q ^pq a a b -^

_ S u V V A /2p'n'. ,2qTT. . 2p7T . 2qTr„ ,^-.

The meml^rane stresses will be represented in the most

general form, the nondimensional form used by Mansour [9]

Using equation (19)

a N*-a. X XX 2u 12(l-v V )

(24)C q^p"* o o

- Z Z ^-^ cos --X cos y- yP ^ 4(p'* + 2Yp2p2q2 + p'^q'*) ^ ^

cr.. N** = iL_ y^

(_zi2l)12(l-v^Vy)

2p7T 2q7Tcos--^—-X COS-——

4(p'* + 2YP^P^q^ + p-^qM

Z E C p^ o opq^ 2p7T 2q7Tp q £-s cos--^—-X cos-——

T* = ^xy ^ _ S*nYxy ^2, 12(l-v V )r_jT_h_, X y

^abJ ^

Z Z £-^ sin—^-—X sm—r—

p q It 2 a b4(2— + 2p^q^ + ^')

P/ <3 - , 1 , 2....

The bending and twisting stresses from equation (11) are

6DQ = + _r^ = ± - [- i!:^: - V

'by= ± _Lm = ± __Z [-

a^wy ay^

bxy- 12C a^w•*"

,2 8x8y

The normal bending stresses may be expressed as functions of

nondimensional bending moments M* and M* where^ x y

MM*X D D

M* + Vx_ "^

^ M*D y

MM*y hD

D XX o

h pnr^[

= - Da^w

X yX . 2 h nj~D^

8x \| X y

M*y. hD

= - Da^w b'

y -N 2 hD^ 9y y

The bending stresses as functions of the nondimensional bending

moments are

6M*(29)

The nondimensional moment due to curvature in the x-direction

only (M* ) and that due to curvature in the y-direction only^o

(M* ) are determined from (15) and (28)^o

„ „ 'n'^m^, m-n r, , . n+1

2n7T ,

E E ^b^^cos-—X [(-1) + cos^j—-y]m n 2 inn a b

Z Z 411 n'^b cos—X cos-T—-ym n mn a b -^

m = 1 , 3 / 5 . , . .

The bending stresses may be expressed as nondimensional bend-

ing stresses in a form compatible to equations (24) and (25)

Using equations (15) and (26) , the nondimensional bending

and twisting stresses are

'bx= 'bx 1 r' r- 1- ii^'n'= ± -TTjr: r- Z E b COS X

2 (1-v V ) m n mn aX y

rm2(-l)^''l+(m2 + ^^!-^) cos^y]3^ ^

by ^2^ 2 (l-\v^) m n mn ""^^a

andX X. D

bxy ^ . Yn ^ _ y x^ " .

"''^r^L!h_, '^-''x^y' 2np^B^

E Z b run sm—r--y sm—^xm n mn b a

m = 1 , 3 , 5 . . . .

Xl ~~ -^Z ^ i -^ • • • m

The nondimensional total stresses are therefore, from (12)

a* = a* ± a^tx X bx

ty "^y -''by (33)

T* = T* X T*txy xy bxy

3.4 Effective Width

The effective width of a rectangular plate which has

buckled is defined as that width of a uniformly stressed

phantom plate of the same thickness stressed to the same

maximum stress and sustaining the same total force as the

real plate. The effective widths a and b , in the y and x

directions respectively are hence defined as

Nao = a -T^ C34a)e ye n

Nho = b -^ (34b)e xe h

where a is the edge membrane stress in the y-directionye ^ ^

(a at X = ± -tt) and a is the edge membrane stress in they 2' xe ^

x-direction (a at y = ± -5-) . Substitution of (25) evaluated

at X = ± p- into (34a) gives

a N*_e = YpM-i)Pc

N* + 3(l-v V )E E 22 cos^yy X y p q,it,^ 222, »»i+\ b"'

(P + 2Yp^p^q'' + p q )

P/ q = 0, 1, 2..,.

!e^ ^2

(-l)P"^ P'(-l)%q

2qTrN* + 3(l-v V ) [Z

^ ^^ C +Z E E3 Icos^yy X y p p2 p,0 Pq=i(pV2Yp^p2q2+p'*qM ^

substitution of (24) evaluated at y = ± j into (34b) gives

b N*e _ Xb

N* + 3(l-v V )Z E ^ ^"-^^ ^ Sq ^2pTTX X y p q ^-^ cos-^^—

(p- + 2Yp'p'q' + p'qM^

p , q = 0/ 1, 2,,..

b N*e _ X

N* + 3(l-v V )[Ei-ii-Go,q + E I EH_] cos^^x

"" X y q g2 p=i^ (p^* +2Yp2p2q2 +p^qM ^

Clearly, the effective widths a and b are not constant over

the y and x-directions respectively. Neglecting the small

amount of change due to the periodic terms in y or x, taking

only the average values into account [15] , the effective

widths take the form

a N*e _ ya

N* + 3(l-v V )E ^^^

Cy X y'p 2 P,o

p = , 1, 2 . . . .

b N*e X

N* + 3(l-v V )Z ^^'

CX X y q „2 o,q

q=0/ 1/ 2....

3.5 Principal Stresses

For the state of plane stress, the maximum and minimum

principal stresses, a, and o^r are given by

- aX i)- + T

Substituting the nondimensional total stresses, equations

(33) , the nondimensional principal total stresses are

a*' + a*'tx ty

r*' - rr*'r* - o'

2 txy (37)

1,21,2

[IL!iL.3

'* o2P'^B

r*' = a*ty ty

txy= txy

. YP'3

4. Numerical Solution

4 . 1 Analytic Solution

Shultz [15] found that eight deflection terms were

sufficient to describe a simply supported plate up to an

aspect ratio of 8. In this solution eight deflection terms

in the y-direction and one in the x-direction were assumed.

From equation (15) with m = 1 and n = 1,2,3. ..8, the

deflection of the plate takes the form

w = h cos—X [b, - (1+cos-T—y) + b, „ (-l+cos-r—y)a 11 b -^ 12 b -^

+ b^3(l+cos^y) + b^^(-l+cos^y) + b^^^ (1+cos^y)

I i_ / T .1277 X , , ,,

,14tt » , , It, 16Tr .+ b^g (-1+cos-^y) + b^-, (1+cos-^—y) + b^^g (-1+cos-g—y)

Similarly, the right hand side of equation (16) is ex-

panded for m, i = 1; n,j = 1,2, 3... 8. The coefficients C

are determined by collecting terms on the right hand side of

equation (16) and matching the coefficients to the series

on the right hand side of equation (17) for p = 0, 1;

q = 0, 1, 2... 16. The coefficients X and Y are listedq q

in Table 1 where the coefficients C are given bypq ^

C = Y + (-l)Px (38)pq q ' ' q

P = 0, 1

q = 0, 1, 2. . . .16

Substitution of coefficients C into equation (22)

using equation (19) and with va, r = 1; n, s = 1,2,3. ...8

produces the following equation after considerable simpli-

fication

N*Z (2s)2N*bT + Z -^, - —- Z E b, {2(~1)^"*"^s ylfS s 4l,s uns l,n

+ ri+2p2Ti(2s)2 + pV2s)'*]6^^} + -^ Z Z b (-D'^'^'^N*Ilo If II o X / ii ^

+ 2(-)^-^ Z Z b, (-1)^"^^S* + (-)^-^ Z b, n(-4- + -^)S*^Tj' 2ns l,n IT 2 ,

l^n n+s n-s'p p n+s

+ Z Z b, {2(-l)^"^^n^[A, C, + 2A C ]

n s l,n l,n l,n o,n o,n

+ 2(-l)^'*"-^s^[A, C^ + 2A C ]^ l,s l,s o,s o,s

+ (s-n)MA^^^^^C^^^^^ + 2A^Jn-s|CoJn-s

+ (s+n)^[2A ^ C , + W«AtI

„|C,I

^i]}^ o,n+s o,n+s 1, n-s l,In-s|

- Z (-1)^"^^ ig* = (39)S TT

n,s = 1,2,3. ...8

where W=1.0ifn7^s

W = 2.0 if n = s

and4<1 - \\^

^^ (p** + 2Yp2p2q2 + q'»p'»)

<t>= A - ^

pq pq pqQJ,

Equation (39) gives a set of eight simultaneous, non-

linear equations for the eight coefficients b^^, ^12' ^13'

^14' ^15' ^16' ^17' ^"""^ ^18-

4 . 2 Computer Solution

Equation (39) is programmed on the IBM System 370 model

165 using Fortran IV level Gl. The subroutine ZEROIN from

the M.I.T. MATHLIB program library is used to solve the

system of simultaneous, non-linear equations. This sub-

program uses an iterative method of solution and convergence

is achieved when the difference between two successive values

-12of b IS less than 0.5x10 .nm

The foregoing analysis is for the completely general

case with all the loadings as indicated in figure 1, however,

the computer solutions were restricted to certain specific

cases.

Basically, two programs were written to calculate the

deflection coefficients. The first program produces curves

of deflection at the center of the plate, effective width

and bending moment in the y-direction at the center of the

fixed supports versus inplane edge compressive load N* with

Q* as a parameter, N* = and S* fixed. Curves of this

nature, for a range of orthotropic parameters y and ri,

result in a set of design charts. The other program permits

investigation of the behaviour for any set of loading con-

ditions with N* = 0. Deflection along the centerline x =

and total principal stresses along the centerlines x =

and y = are plotted.

5. Results

5.1 Design Charts

The design program was used to generate a set of de-

sign charts with N* = S* = 0. Virtual plate aspect ratios

of 1/1.5, 1/2, 1/4 and 1/6 are presented. For each aspect

ratio a complete set of curves for orthotropic material

parameters y = n = 1 • and y = 4.0, n = 0.5 are given.

Description of design charts:

5.1.1 Charts of Deflection

The nondimensional deflection at the center of the

plate, from equation (15) with x = y = 0, is plotted versus

the inplane compressive edge load N* with the lateral load

Q* as a parameter. Figures 2,3,8,9,14,15,20 and 21 give the

deflection.

5.1.2 Charts of Effective Width

The nondimensional effective width given by equation

(36b) is plotted versus the inplane compressive edge load

N* with the lateral load Q* as a parameter. Figures 4,5,10,

11,16,17,22 and 23 give the effective width.

n5.1.3 Charts of Bending Moment

The nondimensional bending moment M* at the middle ofy

the supports y = ± y and x = 0, equation (27) , is plotted

versus the inplane compressive edge load N* with the lateral

load Q* as a parameter. Figures 6,7,12,13,18,19,24 and 25

give the bending moment.

5. 2 Behaviour Charts

The behaviour program was used to investigate a few

general sets of loading conditions with N* = 0. Virtual

plate aspect ratios of 1/1.5 and 1/1.25 are presented for

orthotropic material properties y = ri = 1.0.

Description of behaviour charts:

The nondimensional deflection, from equation (15) , along

the centerline of the plate (x = 0) is plotted versus y/b

with either S* or N* as a parameter and Q* fixed. Figures

26,27,36,37,46, and 47 give the deflection.

5.2.2 Charts of Maximum and Minimum Total Principal Stresses

The nondimensional total principal stresses, equation

(37) , along the centerlines on the bottom surface of the

plate (z= —^) are plotted versus y/b (x = 0) and x/a (y = 0)

with either S* or N* as a parameter and Q* fixed. Figures

28-31, 38-41, and 48-51 give the stresses for x = and

Figures 32-35, 42-45, and 52-55 give the stresses for y = 0.

6. Discussion of Results

Orthotropic Properties

Reference [10] contains a complete discussion of the

rigidity coefficients D , D and D and compliance coef-^ J- X Y xy ^

ficients J , J and J which are used to express the non-x' y xy ^

dimensional parameters and orthotropic material coeffi-

cients. Possible approximate formulas for their calculation

are given in that reference. The two extreme cases of

orthotropic plate coefficients are the isotropic case where

ri = Y = 1 • sri^ the grid (intersecting beams, no plate)

n = and Y = °°« Most plate-stiffener combinations fall

between ri = y = 1*0 ^rid n = 0.5/ y - 4.0; hence the choice

of parameters for the design charts.

Comparison of Figures 2 to 25 as to the nondimensional

load parameters N* and Q* indicates as p decreases the

assigned values of N* decrease and Q* increase. N* and Q*

are indirect functions of the aspect ratio since they are

nondimensionalized by the plate edges a and b respectively.

The ranges of values used in the design curves were predi-

cated on achieving a certain minimum value of effective

breadth in the calculations.

Comparison with Existing Solutions

The first set of design curves to be produced were

those for the case p=Y=n=1.0, Figures 61,62 and 63,

for N* with Q* as a parameter and S* = N* = 0. This choicey ^ x

of parameters permitted comparison with the solution obtainef

by Mansour [10] . The number of coefficients retained in

this analysis (only one in the x-direction vice two by

Mansour) does not favor the solution for the square plate

case, however, the results of Figures 61, 62 and 63 agree

to within 3% of the figures 1, 2 and 3 of Mansour [10]

The deflection coefficients obtained in reference [10] show

that the one term assumption only neglects terms that are

less than 5 1/2% of the primary deflection term b, ^ which

accounts for the reasonable agreement in this case.

For the loading conditions of the design curves.

Figures 2-2 3, comparison was made with the solution of

Levy [5] . Levy calculated deflection at the center of the

plate and effective width for the isotropic plate of

3=1.5, Q*=S*=0 and the same boundary conditions as

this investigation. Comparison of Levy's results. Figure 3.

and Table 11 of reference [5] , with the curves for Q* =

from Figures 2 and 3 is graphically displayed in Figures 64

and 65. Correlation of the two results is very good,

6.3 Design Charts

The deflection at the center of the plate is always the

maximum deflection since only the first deflection mode is

encountered in these cases. With reference to the curve

Q* = in the deflection charts, the plate remains unde-

flected until the critical compressive load N* is reached.

Increasing N* further buckles the plate and plate deflection

increases rapidly. Hence, intersection of the curve Q* =

and the N* axis gives the lowest buckling load. The non-

linearity of the curves is evident. In the small deflection

w 1range, r- < -x-, a doubling of the lateral load Q* will double

the deflection, however, in the large deflection range the

lateral load effect becomes nonlinear with decreasing incre-

ments of plate deflection for increasing additions of load.

As p goes from 1/1.5 to 1/6 the magnitude of the maxi-

mum deflection of the plate is considerably reduced. The

plate becomes more of a beam supported all around and hence

"stiffer" than the slightly rectangular plate. The effect

on the deflection of increasing Q* as p -^ 1/6 becomes less.

Comparison of buckling loads and the range of N* values

over the four aspect ratios shows that for p = 1/1.5 the

maximum value of N* considered is four times the bucklingX ^

load while it is less than twice the buckling load for

p = 1/6. The reason for this selection of N* values is^ X

that the range where the solution is reliable decreases as

p -> 1/6. Convergence to the correct solution becomes in-

creasingly more difficult. In addition, the 8 term solution

begins to become insufficient for p = 1/6 where the eight

term, t),«, has increased to a value almost 6% of the primary

term, b-.^, for the highest set of N* and Q* considered.

6.3.2 Charts of Effective Width

By definition, the plate is fully effective in carrying

the external compressive edge load until it buckles. With

reference to the effective width figures, b /b = 1 for Q'^ =

until the buckling load is reached. Extensions of the

curves for Q* 7^ at values of N* less than the buckling

load are meaningless and should be ignored. After buckling,

the effectiveness drops off as the plate deflects. Compeiri-

son of different aspect ratios shows that the effective

width decreases more rapidly and takes on significantly

lower values a p -> 1/6.

6.3.3 Charts of Bending Moment

The bending moment M* at the middle of the fixed sup-

ports y = ±-J

is calculated because it is expected to be

large. M* is equal to M* at these points since the curva-

ture along these supports in the x-direction is zero (M* =0)

^ohence equation (28) is plotted. With reference to the

bending moment figures, the bending moment is zero until

the plate deflects. Increasing Q* increases the bending

moment in all cases with the nonlinearity associated with

large deflections.

6.4 Behaviour Charts

Unfortunately, there are no existing solutions to which

the results of shear loading may be compared for this par-

ticular set of boundary conditions. Levy [7,8] and Payer

[11] considered the simply supported isotropic case. How-

ever, the results obtained are consistent with those antici-

pated in the magnitude and shape of the center line

deflection curve and values of principal stresses.

Results are given only for p = 1/1.5 and 1/1.25.

Investigation of the solution revealed that for virtual

aspect ratios greater than about 1/1.75, the plate will have

one symmetric buckle about the diagonal while for smallei*

values of p antisyiranetric buckling shapes are experienced.

The choice of one term in the x-direction limits the solution

to some degree in the former case*, however, it cannot begin

to approximate the true solution in the later case. Hence,

the narrow range of investigation for p.

Additionally, with reference to the nondimensionalizing

of the total stresses, sections 3.3 and 3.5, to present the

results in a concise and meaningful form, the solution is

restricted to the isotropic case where y = t\ = 1.0 . This is

a result of the isotropic aspect ratio 3 appearing expli-

citly in the nondimensional equations. Nondimensionalizing

the bending and membrane stresses to eliminate 3 will result

in a set of incompatable nondimensional stresses which must

be plotted and evaluated separately, thus increasing the

number of charts required for each specific case.

Initially, the total principal stresses at both the top

and bottom {z = ± -y- respectively) of the plate were investi-

gated. Figures 56-60 show the results for p = 0.667, y = n =

1.0, S* = 14 (above critical load) and N* = Q* = 0. Using

the Von Mises yield criteria for the biaxial case, a = 0,

*Note: The assumption of the one team solution in the x-direction will model a symmetric buckle about the platecenterline, not the diagonal.

(a*) 2 = (a*) 2 + (a*) 2 - a* a* (40)

a combined stress may be calculated to evaluate the relative

importance of the total stress field at the top or bottom

of the plate. With reference to Figures 57-60, the table

of (Oq)*^ is as follows:

y=x=0 y= ± ~r x=0 x= ± ^, y=0

Top 12.4 17.7 10.8

Bottom 12.2 19.5 10.8

The combined stresses are slightly greater at the top

than at the bottom for the center of the plate, however, the

magnitude and difference is considerably greater at the

fixed supports with the bottom stresses being of higher

value. By the nature of the upwards (+z) buckling mode,

this is the expected result. Due to the higher combined

stress at y = ± ^, the behaviour charts were plotted only

for the principal stresses at the bottom of the plate

(z = - 2^ •

Figures 26-45 show the effect of increasing shear loads

above the critical load for p = 1/1.5 and 1/1.25. Inplane

edge loads N* = N* = and cases are presented for both Q* =

and a finite value. Increasing S* and/or Q* increases the

deflection and total principal stresses throughout the plate.

Figures 46-55 show the effects of completely general

sets of loading conditions with increasing inplane edge

compressive loads N* above the critical load. Comparison

of the charts for the sub-critical (S* = 6) and super-critical

(S* = 14) shear loads clearly indicate the influence of

shear on the shape of the deflection curve and values of

principal stress. Increased shear increases deflection and

principal stresses. In addition,, the highest maximujn

principal stresses are observed to occur at the fixed

support, y = ±-J,

6. 5 Examples Demonstrating Use of the Charts

The following examples are given to demonstrate the use

of the design and behaviour charts. The approximate formu-

las for calculation of the rigidity and compliance coeffi-

cients are discussed in detail in reference [10] . The

formulas are listed in Table 2.

6.5.1 Design Example

Consider the following orthotropic characteristics and

nondimensional loads for the stiffened plate:

p = 0.5, Y = 4.0, n = 0.5, N* = 4.0, Q* = S* =

From Figure 9

The critical load N = 1.8X 2c a""

The center of the plate deflection

w = 1.64h

where h is the average stiffener's depth plus the

plate thickness.

From Figure 11

The effective width b = 0.505be

Edge membrane stress (equatilons 24,34b and 35b) is

X b^' T^/T 2. 0.505 10.9e e 12(l-v )

^ . 0.73 :!I-^

^e a^Jy

From Figure 13

The bending moment at the middle of the fixed support

M* = -102y

The maximum bending stress at this point (equation 29)

is6M* hD D

a, = ± -^ . -^ =q: 612 -^

^y h^ b^ hb^

6.5.2 Behaviour Example

Consider the following orthotropic characteristics and

nondimensional loads for an isotropic plate:

p = 0.667, Y = n = 1.0, N* = 3.0, Q* = 8.0, S* = 6.0

From Figure 4 6

The maximum deflection at the center of the plate w =

1.74 h

From Figures 48 and 50

The maximum principal stresses occur at the fixed sup-

ports Y = ± J, X = and are

a, = 6.501 ^2b^J

1 oifi'^ha^ =-1.542

and the combined stress

^0 = ^1^ + ^9 - ^1^9 = 54.6 [^^-^]2 ^ ^2 ^ ^2 _ ^ ^ _ rx ^ ^^ iL_] 21^2

7 . Conclusions and Recommendations

The problem of a large aspect ratio orthotropic plate

with the short edges fixed and long edges simply supported

has been modeled for the case of inplane edge compressive

and lateral loads and subcritical edge shear loads. The

design charts provide valuable information of use in estab-

lishing design criteria for the plate-stiffener combination.

From the behaviour Figures 46 to 55, it is evident that

the highest principal stresses occur at the fixed supports

y = ± 2-. The design Figures 2 to 25 do not present informa-

tion to permit the calculation of the membrane stresses at

the fixed supports. Design charts of a / edge membrane^e

stress in the y-direction (denominator of equation (36a) )

should be produced.

As noted in the results, the one term assumption in the

x-direction will not model the deflection of a plate with

edge shear as the predominate loading. As the major point

of this investigation was to model a large aspect ratio

plate, the solution chosen was an inevitable consequence

when the large range of p is considered. It is recommended

that to model a large aspect ratio plate subjected to shear

loading, close attention must be paid to the important

deflection terms that should be included in the solution.

Levy [8] confined himself to a B = 1 . 5 and used 14 selected

terms in the deflection equation.

Additionally, the eight term solution is just suffi-

cient to model the case for the orthotropic plate aspect

ratio of 1/6.

8. References

1. Bleich, F. and L.B. Ramsey, Buckling Strength of MetalStructures , McGraw-Hill, New York, 1952.

2. Bleich, F. and L.B. Ramsey, A Design Manual on theBuckling Strength of Metal Structures , Technicaland Research Bulletin No. 2-2, SNAME, May 197 0.

3. Coan, J.M. , "Large-Deflection Theory for Plates WithSmall Initial Curvature Loaded in Edge Com-pression," Journal of Applied Mechanics, June1951, p. 143.

4. Levy, Samuel, "Bending of Rectangular Plates with LargeDeflections," NACA Report Number 737, 1942.

5. Levy, S. and P. Krupen, "Large-Deflection Theory forEnd Compression of Long Rectangular PlatesRigidly Clamped Along Two Edges," NACA TechnicalNote No. 884, January 1943.

6. Levy, S., D. Goldenberg and G. Zibritosky, "SimplySupported Long Rectangular Plate Under CombinedAxial Load and Normal Pressure," NACA TechnicalNote No. 949, October 1944.

7. Levy, S., K. Fienup and R. Woolley, "Analysis of SquareShear Web Above Buckling Load," NACA TechnicalNote No. 962, February 1945.

8. Levy, S., R. Woolley and J. Cornick, "Analysis of DeepRectangular Shear Web Above Buckling Load,"NACA Technical Note No. 1009, March 1946.

9. Mansour, Alaa, "On the Nonlinear Theory of OrthotropicPlates," Journal of Ship Research, December1971, p. 266.

10. Mansour, Alaa, "Post-buckling Behavior of StiffenedPlates with Small Initial Curvature Under Com-bined Loads," International Shipbuilding Progress,June 1971 and Dept. of NAME, MIT, Report No.70-18, Oct. 1970.

11. Payer, Hans Georg, "Notes on the Buckling and Post-buckling Behavior of Deep Web Frames," MarineTechnology, July 1972, p. 302.

12. Schade, Henry A., "Bending Theory of Ship BottomStructures," Trans. SNAME, vol. 46, 1938, p. 176.

13. Shade, Henry A., "The Orthogonally Stiffened PlateUnder Uniform Lateral Load," Journal of AppliedMechanics, Dec. 1940, p. A-143.

14. Schade, Henry A., "The Effective Breadth of StiffenedPlating Under Bending Loads," SNAME, vol. 59,1951, p. 403.

15. Schultz, H.-G., "Post-buckling Behavior of Wide ShipPlating," Report No. NA-64-3, Department ofNaval Architecture, University of Californiaat Berkeley, February 1964.

16. Timoshenko, S. and S. Woinowsky-Krieger , Theory ofPlates and Shells , McGraw-Hill, New York, 1959.

17. Yamaki, Naboru, "Postbuckling Behavior of RectangularPlates with Small Initial Curvature Loaded inEdge Compression," Journal of Applied Mechanics,September 1959, p. 407.

18. Kantorovich, L.V. and V.I. Krylov, Approximate Methodsof Higher Analysis , P. Noordhoff LTD, Holland,1964.

TABLE 1 - COEFFICIENTS Cpq

C = Y + (-D^Xpq q q

p = 0,1

q = 0,1,2. . .16

where X and Y are given byq q ^ -^

X^ = b?T +4b?^ + 9b2 +16b2 + 25b^2p + 3 6b,2^ + 49b2 + 64b2o 11 12 13 14 15 16 17 18

Y = -Xo o

X, = 4bTTbT^ + 12b, ^b,, + 24b,^b, .+ 40bT.b, £.+ 60b, ^.b^^1 11 12 12 13 13 14 14 15 15 16

-"bi2bi3 - 25b^3b^4 - 41b^^b^5 - eib^^b^g - 85b^gb^, - llSb^^b^^g

^2 =-"^il

"^'^ll'^lS

-^ "''12''14 + 30bj_3b^5 + ^Sb^^bj^g

+ 70b^5b^,+ 96b^gb^g

Y^ = Sb^^'-l'll^ "^12 - "=13 - ^14 - ^15 + "^le - \l + \b^

-^il -"''ll'=13

- 34bi3b^5 - 52b^4b^5- 74b^5b^,

- "°'>16''l8

^3 = -*^l''l2" ^\l\4 -^ 20b^2''l5 * 36b^3b^g + 56bj^^b^^

+ BOb^jbj^g

Y3 = l«b^3'-hl ^ "^la- ^3 +^4 -^5 - ?16 -"=17 ^^s'

+ 64b^^b^g

Y, = ^''•^14^-^11 "- ^12 -^13 •" ^14 - ^15 -^ ^16 - ^17 " 2^18^

-lOb^TbT- - 26bTTbTt. - Ab^ ^ - 40bT^b, ^ - SSb^-bT-11 13 11 15 12 12 16 13 17

Y5 = 50b^5(-b3^^ + b^2 - t-is + 1^14 - "15 ^''le

- '^17 * \s'>

X, = -10b, ,b,, + 14b, ,b,, - 16b, ^b,. + 32b,,b,„ - 9b11 15 11 17 12 14 12 18 '13

Yg = 72b^g(-b^^ + b^2 - "=13 ^ \4 - hs ^ ^6 - '^17 " "^is'

h = -l^b^^b^g H- 16b^^b^g - 20bj^2'=^5 - 24b^3b^^

^7 = SS''17(-^1 -^ h2 - ^3 ^ h4 - ^5 ^ ^6 " ''n "^'='lB>

-37b^jLb^g - 65b^^b^3 - 29b^2''l5 " 25bi3b^4

Xg = -14b^^b^, - 24b^2b^g - 30b^3b^5 - ISb^^

Yg = 128b^g(-b^^ + b^2- - ^3 " ^4 - ^5 ^ ^6 " ^7 "^ ^8^

-50b, ,b,_ - 40b, ^bT^ - 34bT.b,c.- 16b,^.11 17 12 16 13 15 14

Xj =-16b,^b^g - 28b^2b^^ - 36b,3b^g - 40b^,b^5

Yg =-65bj^^b^g - 53b^2'=^^ - 45b3_3b^g - 41b3^^b^5

^10 =-««h2^8 - ^^\3\l - 52bi4'=16 - "bJs

^11 =-^2'î3''i8 - ^^û^n - ô'îsîe

^11 =-"'"l3''l8 - "'"I4''l7 - "'^is'^ie

X^2 =-64bi4h8 - ^°h5^7 " ^Sb^^

Y^2 =-80b^4b,g - 74b,5b,, - 36bjg

X^3 =-B0b,5b^g - 84b^gb^^

"13 =-8«'=i5'=i8 - ^^he^n

^14 -^^ê'îs - ""Ît

"l4 =-"°^6'^18 - "l^iv

^^15 ° ^^^''l7''l8

"l5 = -113^7^8

^16 = '^'Ks

^16 " ^16

TABLE 2 - RIGIDITY AND COMPLIANCE COEFFICIENTS

1» I s '

_Z XI s

n = xy

JD D . X yX y ^ ^

1 1px pyI I

J -y Eh

xy1 , 1+vE ^ h

V.- -]

Y = xy(1 + V) ^

h hX y - V-

h hX y

J JX y

h =2h h

X yh + hX y

TABLE 3 - INTEGRALS

b/2 a/2

/ / COS—X COS—X axdy = -rr m = r^ a ^ 8

= m 7^ r

b/2 a/2 -

/ ; COS—^x COS—^x cos-^r—-y dxdy =^ ^ b ^ ^

b/2 a/2r r IHTT rTT 2n7r - , ./ / COS—^x cos—^x cos , y dxdy =

^ ^ b ^^

b/2 a/2 ^ _

/ ; cos—^x cos—X cos-T—-y cos-^r—-y dxdy =v-7rj m=r, n=sQ Q

a a b-^ b-^-'ie '

= mf^r or Ht^s

b/2 a/2 ^ ^r f n\7T 2piT rTT 2q7T _, , -.

/ / cos—X COS—^-—

cos—X COS—c^y dxdy =^/^ a a a b

b/2 a/2 ^ ^ ^/ / COS—^x cos—7—y dxdy =

b/2 a/2^^ ^j^ ^

/ / COS—X dxdy = ;^---(-l) r odd^ ^ 2r7T

= otherwise

/ / cos—^x COS—£--x COS—^x COS—g—y cos-r-—y dxdy^ ^ ^ b ^ b ^

ab , ^= -^i m+r = 2p, q = s

= Yo'l^^-^i ~ 2pf q=s, m > r

ab 1I ^= •jjM^"!^! = 2p, q=s, r > m

= otherwise

b/2 a/2 ^^ 2p7T . rir 2nTr 2qTr , ,

/ / cos—X cos X cos—X cos-4r-Y cos—5—y dxdy

ab, ^= j2 ' in + r = 2p,q = n

= jy; [m - r| = 2p, q = n, m > r

abI I ^= jy^ [r - m] = 2p, q = n, r > m

= otherwise

b/2 a/2 - o o/ / sm—^x sm-^-—X cos—^x sm-^r-^y sm-^y dxdy

^ ^ ^ ^ ^

ab, ^= -s-T ' m + r=2p,n = q

abI I o= r^ > |m - r[ = 2p, n=q, m > r

= -T?' I

•'^ - m] = 2p, n=q, r > m

= otherwise

, , ITITT 2p7T riT 2n7T 2q7T 2stt -3 -,

/ / cos—^x COS X COS—X COS—r-'-y COS—c-^y cos ,-y dxdy

r. r. a a a b. b-' b-' ^

; m + r = 2p, n + s = q

m + r = 2p,|n - s|= q, n > s

m + r = 2p, I

- n|= q, s > n

= -^) |m - r|= 2p, n + s = q, in>r

= -^J |m - r|= 2p,|n - s|= q, m>r, n>s

= -^t |m - r|= 2p,|s - n|= q, m>r, s>n

= -^1 |r - in|= 2p, n + s = q, r>m

ab•64 ) [r - in[= 2p,|n - s = q, r>m, n>s

~i |r - m|= 2p, |s - n = q, r>m, s>n

= otherwise

b/2 a/2

/ /. mTT . 2p7T riT . 2n7T . 2qTr 2stt j ,sm—^x sin--E—-X cos—^x s.in—?—-Y sm—^y cos-t-—y dxdy

a a a b-^ b-' b-'-'

«= -^ ) m + r = 2p, n + s = q

64' m + r = 2p, n - s = q/ n > s

ab-64' m + r = 2p, s - n = qf s > n

ab~ 64' m - r = 2p, n + s = q/ m > r

ab" 64' m - r = 2p, n - s = q. in>r, n>s

ab="64 ' m - r = 2p, s - n = q/ m>r, s>n

ab-64' r - m = 2p, n + s = q/ r > m

ab=~64' r - m = 2p, n - s = q/ r>m, n>s

ab" 64' r - m = 2p, s - n = q/ r>m, s>n

= otherwi.se

b/2 a/2

/ / sin—^x cos—X sm—7—y dxdy

odd , n odd

odd, n odd

otherwise

, ^ ^' '. infT rir .. 2nTT 2s7t _ ,

/ / sm—X cos—X sm—r—-y cos-r—y dxdya a D D

ab ^ 1 , , 1 . 1 . m+r ,, . ,j= (-T—) (—r— + ) » -TT- odd, n+s odd

o 2 m+r n+s n-s '' 2 '

ab , 1 , , 1 , 1 V m-r ,, , ,

=( ) (—;— + ) . —TT— odd, n+s odd

o„2 m-r n+s n-s' '' 2 '

= otherwise

Table 4' - Computer Symbols

Fortran Symbol Variable

B(N) b^ ^l,n

GAMMA Y

QSTAR Q *

NXSTAR N*

NYSTAR N*

SSTAR S*

W/H w/h

AE/A a^/a

BE/B b^/b

MYSTAR M*

SIGMA STAR a* or a*

CPQ Cpq

APQ A^^pq

RANG Max. value of N* or S*

RANGQ Max. value of Q*

-9kk *-

i r t t i t I *

Figure 1

J . ^ —

^^2.4 - J^^

/^^°*1.6 - yY//

y^X / / P = 0.667

y^y / / Y = 1.0

0.8 - y^ / / n = 1.0

y^ 1 Q* = 0,4,8,12

/\J m \J ""

1 1 1 1

0.0 3.0 6.0 9.0 12.0

0.0 3.0 6.0

9.0 12.0

Figures 2 & 3

0.85 -

0.70 -

0.55 -

0.70 -

0.55 -

7.0 .5

9.5 12.0

Figures 4 & 5

6920 -,

-100 -

-160 -

-110 -

-170 -

0.0 3.0

p = 0..667

y = 1.

n = 1..0

Q* = 0, 4,8, 12

9.0 12.0

Figures 6 & 7

J . z -

2.4 - ^y^/^^^Q*

1.6 - y^A/yy// P " °-^°

yyyy y = i-o

y//// n = 1.0

0.8 - yy^/ / Q* = 0,5,10,15

(^ — /tillJ » u —

0.0 2.0

4.0 6.0 8.0

/^^^Q*

y/y/ y = 4.0

y\/// n = 1.0

y^X// Q* = 0,5,10,150.8 -

0.0 ~1 1 1 1

6.0 8.0

Figures 8 & 9

1.02 -I

0.34 -

0.66 -

0.48 -

0.66 -

0.48 -

1.0 2.8

6.4 8.2

Figures 10 & 11

^\J —

P = n. RO

^^^"^^^\ \ Y = 1.0

^^^^i;;\^\\. n = 1.0

-60 - ^\!;XN\v Q* = 0,5,10,15

-140 -

\^X\/Q*

-220 _ ^\\

-300 - ^1 1 1 1

-150 _

-310 -|

N* Figures 12 & 13

1.60 -1

1.20 -

0.80 -

0.40 -

0.45 -

0.0 0.6 1.2 1.8

N*X Figures 14 & 15

1.02 n

0.84 -

0.66 -

0.48 -

0.80 -

0.60 -

0.40 -

0.20 -t

0.9 1.3

2.1 2.5

Figures 16 & 17

-160 -

-250 -

0.0 0.6 1.2 1.8 2.4

-160 -

-280 -

0.0 0.6 1.2

1.8 2.4

Figures 18 & 19

1.60 -1

1.20 -

0.0 0.5 1.0

1.5 2.0

Figures 20 & 21

0.80 -

0.40 -

0.80 -

0.60 -

0,50,100,150

1.50 1.75

0,50,100, 150

1.75 2.00

Figures 22 & 23

J u w —

p = 0.167

\\\/^*

450 - Y = 4.0

n = 1.0

Q* = 0,50,100,150 .

7nn — ^1 \j\) ""

1 1 1 I

0.0 0.5 1.0

1.5 2.0

Figures 24 & 25

p = 0.667 Y = n = 1.0 N * =

Figures 26 & 27

p = 0.667 y = n = 1.0 N* =

S* = 13,14,16,20

Figures 28 & 29

p = 0.667 Y = n = 1.0 N* =

Figures 30 & 31

p = 0.667 Y = n = 1.0 N * -

O —r-

CEt—

Q* = 8

S* = 13,14,16,20

Figures 32 & 33

p = 0.667 Y = n = 1.0 N* =

S* = 13,14,16,20

01cr(—in

-.33 ,15 .18

X/R.35

Q* = 8

S* = 13,14,16,20

-.bU1^.33

X/R.52

Figures 34 & 35

p = 0.80 Y = n = 1.0 N* =

Figures 36 & 37

p = 0.80 Y = n = 1.0 N* =

S* = 12,12.5,14,16

Figures 38 & 39

p = 0.80 Y = n = 1.0 N* =

Figures 40 & 41

p = 0.80 Y = n = 1.0 N* =

S* = 12,12.5,14,16

7- .50

Figures 42 & 43

p = 0.80 Y = n = 1.0 N* =

ccacI—en

= 12,12.5,14,16

CCCE}~in

33 -.151^

.01 .1! .52

Q* = 4

S* = 12,12.5,14,16

X/R.18 .35 .52

Figures 44 & 45

p = 0.667 Y = n 1.0 N* =y

Figures 46 & 47

p = 0.667 Y n = 1.0 N* =y

Figures 48 & 49

p = 0.667 Y = ri = 1.0 N* =y

I -.50

Figures 50 & 51

p = 0.667 Y = n = 1.0 N*y

Figures 52 & 53

1,2,3,4

qI 1.50 -.33 -.16 .01

X/R.35

Figures 54 & 55

p = 0.667 Y = n = 1.0 Q* = N* =

S* = 14

Figure 56

p = 0.667 Y=n=1.0 Q*=N*=0

Figures 57 & 51

p = 0.667 Y = n = 1.0 Q* = N* =

Figures 59 & 60

2.7 - ^^1.8 - y;^0.9

-^^.^ y^ / Q* = 0,2,4,6

0.0 /1 1 1 1

5.0 10.0 15.0 20.0

Y = n = 1.0

0,2,4,6

15.0 20.0

Figures 61 & 62

1.02 -I

0.89 -

0.76 -

0.63 -

4.0 8.0

p==Y = n = i.o

Q* == 0,2,4,6

12.0 16.0 20.0

Figure 63

•HPL4

•5C X U

Cn•HP4

9 . Appendices

APPENDIX A

Details of Derivation of Basic Equations of

Large Deflections

Definition of External Forces and Moments

Shearing force per unit length parallel to x axis

Q = / T dz^ -h/2

Shearing force per unit length parallel to y axis

Q = / T dzy -h/2 i'^

Bending moments per unit length of sections of a plate per-

pendicular to X and y axis respectively

h/2M = / zcr dz^

-h/2 ^

h/2M = / za dzy -h/2 y

Twisting moment per unit length of section of a plate per-

pendicular to X and y axis respectively

h/2M = -/ ZT dz

h/2M = -/ ZT dzyx

.h/2 y^

Strain - Displacement Relations

Consideration of strain in the middle of the plate

u + —-dxdX

For large deflections, the effect of deflection on the

strains has to be included in the strain-displacement re-

lations. The accompanying figure shows a typical elongation

of a linear element of length dx due to displacement in the

X and z directions from bending.

Elongation = dl - 7r-<3x + -x-Cr— ) ^dx^ 8x 2 9x

The strain in the x direction is therefore

^x ~ 9x 2^dx'(a)

The strain in the y direction is likewise

^y ~8y 2^9y^ (b)

The shearing strain due to displacements u and v is

dvdy 9x

It can be shown that the shearing strain due to displacement

9w ^ 9v£

The total shear strain in the middle plane is

"Yxy ay 8x 8x * 3y

For small angles the displacements u and v can be represented

by the change in displacement w as

u ~ - z -—9x

V ~ - z ^^—ay

Substitution into (a) , (b) and (c) gives

e3^2 2 ax

+ ^(ffr)' (2r

E„ =^ - 2

9^w_^ i(9W) 2

y .„2 2 ay'

-9^ 9^^ + 9^ 9W'xy axay ax ay

*Numbered equations correspond to those used in the maintext.

Equilibrium of Forces and Moments

1. Lateral load q only

yx 9y -^

*. M +8Mxy

xy 9xdx

M +—idyy ay

^Q + -^xX 9x

Q + —^<ayy ay

The above diagrams are superimposed to make one loading

condition. For equilibrium in the z-direction:

8Q 3QX V ""

-dxdy + -5-^ydx + q dxdy =3x ^ 3y

9Q 8Q+ -^ = - q (a)8x 9y

Moment equilibrium about the x-axis

EM =0XX

8M 8M 3Q-^^^xdy 9^y^^ + Q dxdy + -^^^ydxdy + qdxdyO (dy) =

Neglecting the last tv7o terms since they are of higher order

8M 3M_^EZ _ .^ + Q =0 (b)3x 9y y

Moment equilibrium about the y-axis and similarly neglecting

small terms

EM =0yy

9M 9My^ ^ _^ - Q =9y 9x ^x

2. Forces in the middle plane of the plate (Membrane Stresses)

The forces acting in the middle of the plate may have a

considerable effect on the bending of the plate and must be

considered.

I*—dx—

9N*^ N + -^x

N —X

^"xy '~TxN +-^r^^X

8NX* N + -T-^x

8N-N ._. + -T^^yyx 8y

y dy ^

Assuming no body forces or tangential forces

^Asxdy + -g^ydx =

8N 9N-r-^ydx + ^^^^dxdy =dy dx

and from symmetry noting N = N"^ ^ xy yx

8N 8nX

+ -^ =ax 9y

_Ji + -ii:^ =8y 8x

Equations (3) are independent of equations (a) , (b) and (c)

and can be treated separately.

Projecting normal forces N onto the z axis and taking

into account the bending of the plate and resulting small

angles

-N dy|^ + (N + -2idx) (|^ + A!w^x)dyX -^ 8x x 8x 8x ^2

Similarly projecting normal forces N onto the z axis

-N dx|^ + (N + ^y) i^ + 9i^y)dxy ay y ay ^ ay ^^2

Neglecting higher than second order terms gives

Si 2 aN .iv-T

a w, , , X aw, , ,,.

^xTl^^^^ -^ -aF • 33?^^^^ ^^^oX

^2 aN ^XT a w , J , y aw, J , .N, dydx + -^ • TT—dydx (e)y9y2 ^ ay ay ^

Treating the projections of the shearing forces onto the

z axis the same way

(1) for N with slope of deflection surface in the yxy ^

direction on the two opposite sides of the element as

aw -1 aw, a^w ,

ir— and "5— + Tc—;^— dxay ay axay

xy9x8y -^ dx dy -^

(2) for N with slope of deflection surface in the xyx ^

direction on the two opposite sides of the element as

9w J 9w , S^w ,

ri 9^ ^ ^ j^ yx 3W, ,

Va^s^^^"" " "ly • ^^^"^

combining (1) and (2) gives

^2 9N . 9N -

01.T 9 w n -, . xy 9w, J , xy 9w, , ,jr»2N -r——-dxdy + —r—^ -r—dxdy + —r—^ tt—dxdy (f)xy8x9y -^ 9x 8y -* 3y 9x -^

Differentiation of (b) with respect to y and of (c) with

respect to x and substitution into (a) eliminates shearing

forces Q and QX y

a^M a^M a^M 32mX

_^yx ^ y xy = _ -

3^2 9x8y 3^a 9x9y

Noting that M = - M , adding expressions (d) , (e) and (f

to the load qdxdy originally defined and making use of (3)

S^M g^M a^MX _ 2

xy ^ ^ y^

9x^ ^^^^ 3y2

= -(5 + N ^ + 2N 1^ + N li^) (4)

^3x^xyaxsy y^^^

Moment - Curvature Relations

Substitution of equations (2) for e , e and y in equations

CD gives*

^ ^"^xV 3x^ 2 3x Ygyz 2 y 3y

y r 9^W,1,^W^2 9^W

,1 ,9V7. 2t / v

^rr= "TilTrTr-I'Z + o-C-^;) - zv + o-v (^^] (a)

Y 1-V V r. 2 ^ oV X^ 2 ^ X dX^ X y 9y ^ dx

^ = -2zg|-4- + G |^.|^xy 9x3y 9x 9y

Taking the moments as defined before and integrating

h/2M = / ZQ dz^ -h/2 ^

h/2M = / za dzy -h/2 y

h/2M, ,^ = -/ ZT dz^y

-h/2 ^y

Even functions will drop out over the integration interval

leaving the same expressions for moments M , M and M as^ ^ X y xy

for a plate undergoing pure bending and only small deflec-

tions. The result is

*Numbers in parenthesis correspond to equations in the maintext.

M = -D (^ + V ^) (6)

M = 2cr^xy 8x3y

where the rigicity coefficients D and D are defined as^ -^ X y

E h3D EX - 12(l-v V )X y

E h3D - ^-~

y - 12(l-v V )jt""x y

CZ ^^'12

"Equilibrium" Equation

Substitution of equations (6) into (4) gives

3""+ 2D 3"" + D 3'"

âx* ''îx^n^ ây"

= 5 + N 2i^ + 2N i^ + N 2iwXj^, xyaxay y^ya

Substitution of the stress function defined by equations (5)

D i!^ + 2D -i-^ + D ^

-, d^F d^\7 ^d^F d^\7 ^ d^F a^w ,_,n + • — 2 • + • i7 J

"Compatibility" Equation

The forces N , N , and N in the middle plane of theX y xy ^

plate depend on the strain due to bending as well as the

external forces applied in the xy plane.

Again assuming no body forces and requiring load q is

perpendicular, equations (3) apply for equilibrium in the

middle xy plane

-y-+ _^=Sy 8x

The corresponding strain components are those of equations (2)

X 3x 2^9x^

_ 8v . 1 ,8w.

" 2^87^ ^2)

^xy By 8x 8x * 8y

Differentiating equations (2) and combining to eliminate

u and V results in

Using Hooke's Law to relate strain and forces N

N Nx_ _ y_

^x hE ^y hEX -^ y

S hE X hE ^^'•^ y X

NY = -^^'xy hG

Differentiation of equations (a) and substitution into

equation (8) with use of equation (3)

3^N 9^N a^N- ^ + 2J . ^y + J ^X3^2 xy 8x3y y ^^2

29 w X 2 9 w 9^w

^^^y 9x^ 9y^

T - 1 T - 1

x E,,h ' y E h

xy Gh X y y x

Substitution of the stress function defined by equations (5)

APPENDIX B

EFFECTIVE WIDTH

Effective Width

f? < -

Figure a Figure b

Consider the simple case of a rectangular plate simply

supported on all edges where the loaded edge remains straight

in the plane of the plate at all times. If the load is below

the buckling load, the stresses will be distributed evenly

as shown in figure a. For post-buckling loads the center

of the plate will exhibit less compressive strain than the

edges because of large deflections of the center of the

plate. The stress distribution is shown in figure b. The

effective width relates the maximum stress o uniformlymax

distributed along a phantom plate sustaining the same total

load as the real plate, thicknesses being the same.

Therefore

a a = aae max

Where o is the average edge stress. For this particular case

h- ^ and

aj^ax - Oy^

Where a is the edge membrane stress in the y-directionye ^ -^

(a at X = ± -^r) . Substitution into the above relation givesy 2 ^

the result

Na a = a -T^ (34a)e ye h

Similarly, for loads in the x-direction

Nho = h — (34b)e xe h

APPENDIX C

Details of Solution

1. Determination of Coefficients Cpq

Substitute deflection expression (37) into the right

hand side of equation (16) . The right hand side of equation

(16) becomes*

Term #1

= hljLL{^sin'^x[b,sin^Y + 2b^sin4?y + 3b,sin^y2V.2 al b-^ 2 b-^ 3 b-^

+ 4b.sin-T-y + Sbr-sm-r—-Y + 6b-.sin-v—

4 b-' 5 b-^ 6 b"^

. -ji_ • 1417 , „, . 16Tr ^^

}+ 7b^sin^-y + 8bgSin-^y]

Term #2

- 14cos —xlb, + b,cos-r—y - b^ + b^cos-r—y + b-2, 2 al 1 b-' 2 2 b'' 3

+ b-cos-T—y - b- + b.cos-r—y + b,. + bcCos—r—y - b^3 b-^ 4 4 b-' 5 5 b"* 6

. ,_ 12iT , , , 1471 , . , 1611 , r, 217+ bgcos-^y + b_ + b-cos-^y - bg + bgCos-^yj • [b^cos-^y

, A-L. 47T , rt, 67r , , ^, Sir , ^_, IOtt+4b^cos-T-y + 9b-cos-T—y + 16b.cos-^y + 25b_cos-^—

+36bgCos^y + 49b^cos^y + 64bgCos^y] } (a)

*Note: since m = 1 for all cases, for convenience b, isrepresented by b .

Each of the two terms of equation (a) is multiplied out and

like terms are collected in the form cos ^ y for

q = 0,1, 2... 16 using the function product relations

4sinasin3 = 2 cos(a-3) - 2cosCa+B)

4cosacos3 = 2 cos(a-3) + 2cos(a+3)

The first and second terms produce coefficients 2X and 2y^q q

respectively. Using the function relationships

. 2 71" 1 1 27Tsm —X = -^ - -sK^os—

a 2 2a9 7T 1,1 271"COS^—X = 77 + oCOS ^Xa 2 2a

there results

First terra:

„ 2qiT „ 2qiT 27TX cos-^y - X cos-^-y cos—^x

q b "^

q b "^ a

Second term:

Y cos—^y + Y cos-^y cos—^-x

q b -^ q b -^ a

Comparison with the right hand side of equation (17) , the

coefficients C arepq

C = Y + X0,q q q

C, = Y - Xifq q q

C = Y + (-l)Pxpq q q

p = 0/1

q = 0,1,2 16

2 . Application of the Method of' B.' G. Galerkin to Equation

Method of B.G, Galerkin (from reference [18] )

It is required to determine the solution of the equation

L(w) =

where L is a differential operator in two variables, which

solution satisfies homogeneous boundary conditions.

The approximate solution sought is in the formi

w(x,y) = Z h^ F^(x,y)n=l

where F (x,y) (n = l,2...i) is a system of functions chosen

to satisfy the boundary conditions and b are undetermined

coefficients. Consider the functions F (x,y) to be linearly

independent over a given region (no one of the functions can

be expressed as a linear combination of the others) . For

w(x,y) to be the exact solution of the given equation, it is

necessary that

L(w) =0

If L(w) is continuous, this requires that the expression

L(w) is orthogonal to all functions of the system F (x,y)

(n = l,2.,.i). With i constants, b. , bp...b., i conditions

of orthogonality can be satisfied.

Requiring that the two functions are orthogonal over a

region produces the following system of equations

//L(w)F (x,y)dxdyR

= f/Li I b^F^(x,y))F^(x,y)dxdy =

(n = 1,2. . .i)

from which the coefficients b„ can be determined. Thesen

coefficients will define the solution w(x,y)

The method of B.G. Galerkin can also be obtained from

the principle of virtual work.

Application of the method of Galerkin to equation (7)

requires that the following equation be satisfied by all

functions f (x)g (y)

a/2 b/2 r,k^, •y'*^^cjit„

/ / [D ^^ + 2D _LJ^_ 4- D ^ ""

""dx' ""^dK^dy' ^dy If

8^F . a_^^ 2 S^^ 3^w _ 8_^ 8^w ,

~ ^ ~^ 2 \ 2 9x8y 8x8y ~

^ 2 .> 29y dx ^ ^ 8x^ ^y"-

X f^(x)g^(y)dxdy = (21)

Substitution of w and F, expressions (15) and (20)

respectively

a/2 b/2/ / Z 5^"^^ ^ S^"^^ ^TTiT.^^"^^ + cos-v—y] cos—-Xn n rs xinna mn d a

+ 2 D , h ZZ (-1-—) (— ) b cos—r—y cos—^xxy mn b a mn b -^ a

+ h D Z Z (-T-— ) b cos—J-—y cos—^x - qymn bmn b-^ a ^

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2q7T miT 2piT• cos- V y cos—X cos-^E-—

b -^ a a

r; v. V V /2n7T. 2i. 2n'n" mTr- N h Z Z (-T—) ^b^^cos-^r—y cos—-xymn b mn b a

3 ,2nTT» 2 /2p7r. 2 J. i_ 2nTr 2q7T mTr 2p'n"- h Z Z {-^r—){-^^-)(b b_ cos-v--y cos—^y cos—^x cos—^-—

m n b a ^pq mn b -^ b -^ a a

— , „ „ /iHTT. 2i r / T \ n+1

2nTr , mTT- N h Z Z (—) "^b [(-1) + cos^r—'ylcos—^xxmna mn b-' a

r^T^-L. r^ r- /2nTr. ,m7T», . 2n7T . mTT- 2Sh Z Z (-y— (—-)b^^sin^r—y sm—-xm n b a mn id a

, -, 3 „ „ „ „ ,2mT. /mTTv /2pTT» ,2qTT. . , . 2nTT . 2qTT+ 2h'^ Z Z Z Z (—r—) (— )

(—^^-) (-^) li) b_ sin—r—y sm-^ymnpq baab ^pq mn b -^ b -^

. mTT . 2pTT T Tf T \ S+ 1 .2STT , ^"^ j j rt

• sm—X Sin—'=—x} •[ (-1) + cos—r—-yjcos—x dxdy =

a a Dam,r = 1,3,5....

The orthogonality properties of the trigonometric

functions must be used to evaluate the integral of equation

(21) . By employing the function relations

cosA + cosB = 2coS2-(A + B)coSy(A - B)

sinA + sinB = 2sin2-(A + B)coSy(A - B)

The trigonometric products may be expressed in typical forms

a/2r miT r-rr J/ cos—x cos—X dxQ a a

1 a/2= -r/ I2cos— (m+r)x + cos— (m-r)x + cos— (r-m) x] dx4q a a a

/ sm—X cos—X dx« a a

= x/ [2sin-(m+r)x + sin-(m-r)x - sin-(r-m) x] dx4 ^ a a a

where integration is carried out for all m and r. By

combining terms that integrate to similar values, these

integrals are equivalent to

=r^ [2cos-|m+rlx + 2cos-|m-r|x + 2cos-I r-mlx] dx4|^ a' ' a' I a' '

1 a/2x/ I2sin-|m+r|x + 2sin-lm-rlx - 2sin— I r-mlx] dx4q a' ' a' I a' i

||where the second two terms in each expression integrate to

zero if r > m or m > r respectively. Manipulation of the

trigonometric products on the right hand side of equation

(21) into these typical relations permits the reduction of

the equation into the convenient form of equation (22)

Table 3 lists all the specific integrals from equation (21)

with their respective values.

APPENDIX D

COMPUTER PROGRAMS

The computer program to solve the simultaneous equations

(39) is written in Fortran IV Level Gl for the IBM system

37 0. The program consists of the main and three subprograms

MATHLIB subroutine ZEROIN, and subroutines EVAL and CPQ.

Table 4 lists the important computer symbols used.

ZEROIN is a Fortran IV subprogram, from the M.I.T.

mathematical library, which computes the solution vector

b of a set of N simultaneous, nonlinear equations

NZF (b) = using double-precision arithmetic, b is obtained

by an iterative method beginning with estimated values of

the solution vector and iteration is performed as

k k k kwith the vector D = J 'FCb ) where F (b ) represents the

k kvector of function values at the point b and J represents

kan approximation to the Jacobian at b . Convergence is

tested with the expression

|b^+l - b^|2 ^ 0.5 X 10-i2|b^+l|2

Subroutine EVAL generates the eight simultaneous

equations and calculates the vector of function values for

a given solution vector b. EVAL is called repeatedly by

subroutine ZEROIN.

Subroutine CPQ is in turn called by subroutine EVAL,

with a solution vector b, to calculate the coefficients Cpq

that are used in the simultaneous equations.

For the starting values of the solution vector b, the

main program uses a well guessed set of values set in the

program. If the lateral load is not zero (Q* 7^ 0) (see

note 1) or the present loading conditions did not change

significantly from the last set, the previously computed

solution vector b is used for the starting values.

If ZEROIN cannot achieve convergence in 50 iterations,

the main program alters the initial set of starting values

and returns them to ZEROIN. Normally convergence is obtained

within the first 50 iterations, however, never more that a

second set of values is required.

Part 1 - Design Charts

The main program varies the nondimensional loads Q*

and N* or S* for a particular set of plate parameters.

Tables of nondimensional deflection coefficients, nondimen-

sional deflection at the center of the plate, nondimensional

effective width and nondimensional bending moment (y-

direction) at the middle of the supports (y = ± b/2) are

outputs. Additionally, CALCOMP plots of the last three

tables are produced.

Part 2 - Behaviour Programs

The main program reads the input parameters for 4 sets

of loading conditions on a particular plate and outputs

CALCOMP plots of nondimensional plate deflection along the

centerline (x = 0) and nondimensional total principal

stresses along the centerlines (x = and y = 0) . These

programs are completely general and can be used with any

set of loading conditions.

The design programs must be used with care. With

reference to equations (36a) and (36b) , the effective width

is only defined for N* 7^ or N* 7^ respectively. Hence/y X

variation of S* with N* = will give a value of zero for

the effective width. The design programs are written so

that no computation or plot of effective width will be

made for any variation of S* with N* fixed. Additionally,

deflection at the center of the plate and the y-direction

bending moment may not be significant parameters in this

case. It is recommended that the "behaviour" program be

used if variation of the shear load, S*, is desired.

Program Descriptions

Main Program 1 - Design (N* = 0)

Input one card per set of complete design curves set up as

follows:

Column Parameter

1-10 RHO

11-20 GAiyiMA

21-30 ETA

31-40 *NXSTAR (Enter Negative Valueto vary)

41-50 *SSTAR (Enter negative valueto vary)

51-60 RANG

61-70 RANGQ

*Note: Either N* or S* may be varied, but not both in the

same run.

Output

1. Tables

a. bl,n

b. r- at y = X

c. b (only if N* is varied)e -^ X

d. M* at y = ± fe, X =y ^

2. CALCOMP Plots

Tables b, c and d above,

Values

N* or S* = to RANG in 10 increments of RANG/20 and

Q* = 0, RANGQ/6, RANGQ/3 , RANGQ

Termination

A blank card as the last data card is required,

Main Program 2 - Design (N* = 0)

Input one card per set of complete design curves set up as

follows:

Column Parameter

*NYSTAR (Enter negative valueto vary)

*SSTAR (Enter negative valueto vary)

*Note: Either N* or S* may be varied, but not both in the

same run.

Output

1. Tables

a. b,1/n

2. CALCOMP Plots

wa. r-aty = x =

b. a (only if N* is varied)e -^

c. Mj«^ at y = ±J,

3. Values

N* or S* = to RANG in 10 increments of RANG/20 and

Q* = 0, RANGQ/6, RANGQ/3 , RANGQ

Termination

A blank card as the last data card is required.

Main Program 3 - Behaviour

Input four car<

Co]Lumn

11-•20

31--40

41--50

51--60

61--70

Parameter

GAMMA smust be same all 4 cards

NSTAR (see note 2)

*Note: If TB other than zero, stresses at top of plate will

be computed. TB = zero defaults to bottom of plate.

Output

1. Tables

a. b,1/n

b. T-aty = x =

c. a? and ai at y = x =

2. CALCOMP Plots

a. g at X = for -I

< ^ < -f I

b. a* and a* at x = for - -^ < ^ < -f i12 2 a 2

c. a* and a* at y = for - -^ < - < + ^12 2 a 2

Termination

A blank card as the fifth card is required.

Subroutine EVAL

Two subroutine programs EVAL - one for N* = , the

other for N* = 0. The appropriate EVAL program must be

used with the corresponding main programs.

Subroutine ZEROIN & CPQ

Completely general subprograms. Use with all combina-

tions of main programs and EVAL.

NOTE 1

If the lateral load is equal to zero (Q* = 0) a possible

but trivial solution is that all the coefficients b =0.mn

Care must be exercised to avoid using b =0 for starting^ mn ^

values in this case. Additionally, under certain circum-

stances the solution readily converges to zero rather than

the appropriate non-trivial solution. The programm.er should

be aware of this possibility.

NOTE 2

The behaviour program may be used with either N* or N*

as a loading, but not both. The main program must be used

with the appropriate subroutine EVAL.

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<^. C*—' rO>— •-H>-'C:LL,>-l-l-_l>-l-l-H- I-

LL<:u_oo<LLCOoDa:•-.O 000 Oi_)OCL3

c>«: •-

** CO00 CD:z —

2z (NJ r-< -t000f-t fNj rj "NJ

II • *>—

' c^ c c- iri ix: iii

<-l Lb LU LUr<^ |_ (_ H-

»—• 1—1 I—

o Qt CC a:Q 3: :s 3

in>t o in i:>

in a a^ o

•k V, V. «^ ^ Xw V CM - ro f-

z X • >- • ^ •«

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u_ r^ »> o a •- •~< QCc •

cc: UU1-

00 U- » .c:^•> z

K •• cc IT • < X U.I

^ • < .~4 •» \~ r^ oLU II h- a. X LL •• <—l-H t/> w, o m v» LL'

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o •• (M * «• rO X •k 00o X

eg (M .-J

LU • r-•»

X OOUJ

z •» • cr < U- o »• on 00c ro O m. \- » a: •> 00 m>-• • LL 9^ KTi m < h- LU cc

K •£: •• X II1- 1—

o u. •• o K (/^ oc H- ooLU •i>

< - 00 •_J •» ^ s: • •• •k

U- II Vf • LU SI •• X -J XUJ < m O <. X h- <C enc

1—'Si

o •> Cl —

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•> X r-l —

q: •> 00 • c l~t/) •> rsj •k •V. l~ m z X CJ2Z X »H m • o LL • > f^ o LUUJ rsj fNJ (NJ CO z O xO «« •k z .12: *^ •• • — •—1 •"^ U- X » < li-

i_i V. m vC Ou cc "^^ (_: Z •• LTN in. ^^ LUo 'v. • u_ - m ^ fc-H c •> «^ f* >v. z oz • vO •« ^ •- —

o *—

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^m. (Ni u X CO .-< ^ •-^ —

ai X in •« <r cc LU LU«i« ro ^ •—

OC z cc CD LL C t— CO • X r-l LL o f-r>- • *^ a. «. m « >— OO • r<^

r^. r\) CC s^ •• <1- o o V, c •^ •» •. •» CC * *> ». r-< •k m rO •• -J_J r\j —1 m u X X X X «. •k • . LL » *• » X C-a •« u_ r-l i- CO «-> rg o X X 7-i c r-J o X X in

U. C) C 00 m c m (^ —

CO LT. » • .—

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Uj < < Ci. < < <• o < < < < < < fO < <t— -J 1— J- 51 o s: s: 2: r-4 5: s:. s: 5" s: s t-4 s: ^ LL G-Z -J c£ OC a. cr: (X CD cc cc cc qE QC a: CD c*: QC O O Qo < oc o C' • O c C - o o c c o o m c O . h_ Zo o rs. LL u.

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u_ U- LL LL u_ LLr-«

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o o o o o O o <^ in —

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f\J fSJ r\j M fM ro <\J (\J rg fNJ f<? ro

SUBROUTINE EVAL 1

(N* = 0)Y

KQ. 0**^ << >t Z)•> •*. 00 a

^»i?- UJ LU

^ «»,

s: »-» c 1—

00f-i vt X Q rsCJ Ct: Z c

> •> H- * *—

LU^ 5: •» C? z

h- »^ '•^ Z <\~ vt + c h-«» »• «-» rvj >—

_J3 C/0 4-, oo ID^ CL 1— # -< Z

00 z: •» ex:^•^ »—

•J' •—1 »-> <: c UJ 00_j r- K *^ ccUJ » — oo Q. D- z fMc 00 \~ 00 J —1 *•> O- •• •« UJ «a sr '->. a: I -J

Q. <M ^0 < (X CD LU •^

< <f 1—

y- *^ rD 00•» • *— 1/5 * 3 X 00

-t t— z: LL or < c cc zU •> 1-^ •« *> 2: 00 »•

X -J — a: X lyo 1- 1—4 1-^

cc 00 2: •» < < < z «i« Ci. h-•« a: •> u_ 2: (- < CM •^ <

< rvj »• 3 00 V UJ ^- 1'"^ 13

2: Q- H- f- 00 •^ 2 00 00 C C?c •» t/1 00 >z ». „>« • Q ^ 3 z • Q UJ

tm, >_• < »> z. 00 CO < • > r-i r;i C^ •

r> »- H- •^ CNJ u_ •—1 •-' >— f\J O C7^ a. U 1 vJ- LULU a: 00 •k c:. LU + • r- a 't> -^ h-

cQ m • < ^ 00 •» •• n|- i-( o^ u. UJ * < >^ <w q; <: h- U-' CL 00 2: < •^ CO > • QC 1- 1—4 _J-J •— >: 00 z 2: v c in U_ »—

•—

< LU Q. X 3< Q 2: 1/5 Ul. t—

2: O- m • 1— w J— H "Xs cc> f—

< « 00 CM ci: ^ _J fMLU X Ct rvl 00 f—

"^ ~» <j < in + K . .' 00 fc' <: >4- cc < ¥••• < *> U- (_. ». a *^» fVJ cc —

— UJ r^>d- * if 1— C < LJ ^

U-i 2 c H- •• •k LU' rj < •—

cr^ cc » —1 Ci. • < <: c 00 X \— CO 00z -. to X 00 • 21 CL ^ II in < .-^ •.^ UJ -+

is- K X c? CiT 00 c ». *—

rv u_»—

cr Z. < u. Q- a: —

c II u_ LL' Qi < c LU q: 'r. '"V z z •—

C c c Q c •^

»- a: a: e «:• >;? UJ • 1—

II D r^ X * c -*it < II • • • • • .~.

z> < LU 00 CO CO 00 CO UJ 2: •» 1—1 IVJ f—

— z LU G OC- • Q C 00 ~> -» 00 QK i; * £1- •h K _J c Cl • _ II 1—1 w—l »—

h- • II C CVj c • • Q UJ II II II II II _ •D^ I/) UJ _j _J _j _j _i C, jr »— < ro _J —

p-< -¥ >~ <r r^ v* • •«v • CO —

• H- >-4 ..« u. rgcc 2 ^- <j < <: < <3. 5; G K II < I—

z _j II c- (\1 < CM 1 1fV < »-* rvj m -r f^ II It

Z> ^:- UJ UJ u^ LU LU c a Cl < I—

«^ *1* c 3) a X II II II II 11 II a: c «^ ^ •M> «^ oo fM00 Cl H-

a: q: a. (X CC < C a t- ll Q u_ < cc < M Q UJ > UJ Q 1- t- K h- h- u. 00<; 1—4 •J Zz -H < UJ^ hM •—

00o LUo *:

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«-» r 00 —

to Ji z: COu. rvj -J 5:•> 1- UJ 2:o 00 c u.o it, + N^»«^ ;* —cyex. X Q •< a: • (-«

+ (M +o rvj « too »- 00 Cl• 00 Ol •z.

<NJ M- 2: u.+ M « v.

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* + •-H u_ o LL to 1•i« »- 1- _ —• -^ »^

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lO K CCi o •«— U- U_ LU >.' <_) CM cr- »- — -« CCU. * » ~' —

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c * ^- •» 4- h- in »—

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s: • •—

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II-* (/^ 00 — c LU 5: CC — q: II

1— ZD IT. < ~- 2: II • 1- ZD CC l; —(NJ •-• o II 2r Z + U- 1 U- 00 Ui ^ or II UJ II V- LU II t-^ LU II K- cr: II

-.4 00 z II 1-«^ 1 Q- w ^ II ^ II a. • • U_' u- • .«^ II y- ..» • • >_ C — II UJ •«— Q- • k—

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z — mK -J < o o 2 c. 2: 5r z II IX. LL K UJ U_ w Z u LL LL c <~> Q u_ LL »-I/) 00 u h- Q Li. 2 U- 2 u. 5: —1 ^—4 00

c *-H 1- K-

h- Q —1 >-H

1— c -^ 1—4

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IS en oo cc ,t c LL M ^« —

r>- X *-» cc <•-*

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oo 00 a. + o + X Q £X +00 «^ ^^ >^ •

1• t U. 2 .». ^ • _ »— • < »»

s: • c? G r^ f\j o Qr *•• Uj UL -4- oc: »—

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K II U- < < < o o U u_ o O o w 2 r^ <^ O X w •^ — c UJ 2a: 3 »—

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h- o i~ h- LU h- H- o H- 1- H- 3 o CX UJr3 > U.

c «^ UJ rg »—

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u. r-4 00 u.

SUBROUTINE EVAL 2

(N* = 0)X

oo KQ. ..-. << -t r)•> * 00 o

<». •» Uj UJ<^ «» O

51 o •—( 00•-4 vt X Q Do o oc Z C

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h- O «~ >-» Z <h- o ^ + c H•" »^ ^w f\J »—

_JjC 00 <-; •j^; oo 3^ a. h- * t—

5:00 z *• c:: ,» o 1—

2 r-t <-. <r r> UJ 00_J u r- h- •?' q: a:UJ » ^ oo a O- z <Q 00 h- oo •it

•-H t-» CL •> ^ o UJ 00

o. z — CC -T~ ^ Q 00a (M <'-' o < CX cc; UJ ->;

< <}• •—

y- .^ 3 oo r\j

•> LJ *> •_• 00 a C ZD ¥. «/>

vd- H- z U- O < c * za » r-l •• •» a. I/) O oX _l U —

o: 5: to h- X »«4

a: 00 s: •k 00 < < < Z — a: 1-•> CC » u. ST ~» 1- o < <\J "V <

»-^ < (NJ » -^ Z) 00 ¥^ UJ >-

1 < z>z. a t- K- 00 O *. z o oo oo D K- oo •• o^< oo ^ ^ -o ^ Q ^ => Z • UJ UJ

.an. I—

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t— (M CJ tr a: u 1 r\j OiU UJ en 00 •• u rO UJ -f • r- C o V l~

CC UJ •» < •• 00 •k ^ ^ t—

cr U- UJ Q ^ < < .-«

w oc: < 1- UJ c 00 z: < y •• CO O > • »^ *— ^^ -J C\J f-l

_J •-I •SL oo •« 2 o o >. a o un U_ a »—

1—1 t-«4 U- Qu 3 * "f

< o SL 00 Q u. o »—

s: C c ro • K •«— CL ^< "v o « 00

> < » •• •> ^ 00 < .^ « LT* >- r-* < * *V rvi CC _J CM *-» ^UJ > O QC M 00 l~-J 1—^ o ^ o o < LH + (- o o V <X. <»- < f 00 •i-

•» < ^ u. c: <_; •» CL "N. CM a: ^-* <-» Uj c -^ o J. K c o >» u, :('

LU Z o - O » »> UJ c < r—

c^ Ljl *, 1—

Cl • ;i t C OO X CO 00 V «<«

z -- 00 X oo «^ 2 o a: •^II o IS\ < o —

<-« lU vj-• <f X z^ CC Q ^ /—

o O o u. o o-< a: z < U- c O- S •• t-A Q II u u_. CC V. t _> O CL H V. z 1—1 c c C c n* C cK a: QC c o < a; • »—

II Z3 o X fe O V\ o o < II • t • • *— r-\ • •3<i UJ CO CO CO CO CO UJ 2 » o —

M .—

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z _J II o (M <I rv 1 1< •—1 CM n ^ II II CM II

3 Z UJ UJ UJ UJ UJ o o Q. < —

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LUOO •^»llh- -^ • •--c-«I— z^;z—'-• (Nj'^z^-f-t— (N

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SUBROUTINE ZEROIN

LU oCSJ

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NM •> X•> ^m. ^-4

CO o CCa. <N \~LU •> <•• O so <NQC ..~> ^^ -J» M U. <l

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Buckling and post-buckling behavior of

3 2768 001 89863 8DUDLEY KNOX LIBRARY

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