NONLINEAR ANALYSIS OFAXISYMMETRIC CIRCULAR PLATES

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Authors Reeves, James Willard, 1931-

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REEVES, James Willard, 1931-NONLINEAR ANALYSIS OF AXISYMMETRIC CIRCULAR PLATES.

University of Arizona, Ph.D., 1970 Engineering, civil

University Microfilms, Inc., Ann Arbor, Michigan

NONLINEAR ANALYSIS OP AXISYMMETRIC CIRCULAR PLATES

by

James Wlllard Reeves

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 7 0

THE UNIVERSITY OF ARIZONA.

GRADUATE COLLEGE

I hereby recommend that this dissertation prepared under my

direction by Jamas Wi I lard Reeves

entitled Nonlinear Analysis of Axisymmetric

Circular Plates

be accepted as fulfilling the dissertation requirement of the

degree of Doctor of Philosophy

ho an Director Date /J f ' tation

After inspection of the final copy of the dissertation, the

following members of the Final Examination Committee concur in

its approval and recommend its acceptance:"

/r, tff

T" ts, !%<f

Aug. I5} acq

ft This approval and acceptance is contingent on the candidate's adequate performance and defense of this dissertation at the final oral examination. The inclusion of this sheet bound into the library copy of the dissertation is evidence of satisfactory performance at the final examination.

STATEMENT BY AUTHOR

This dissertation has "been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allow­able without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manu­script in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED

AC KNOWLEDGMENTS

The author is indebted to Dr. Ralph M. Richard,

director of this dissertation, whose guidance, and initial

conception of the merits of the problem., led to the

completion of this investigation.

Appreciation is extended to all of the Civil

Engineering faculty members and to the author's colleagues,

whose instruction, association, and assistance ultimately

have made this dissertation possible. In particular,

sincere thanks are expressed to Wilbur D. Birchler and

Melvin L. Callabresi for many rewarding discussions pertain­

ing to this study.

I am indebted to the National Science Foundation

and to the University of Southwestern Louisiana for

financial support during my graduate program.

The encouragement, support, and unfailing confidence

of my parents, Leon and Julia Reeves, are acknowledged and

deeply'appreciated.

As a token of my deepest gratitude for their love,

patience, and understanding, I proudly dedicate this dis­

sertation to my wife, Clara, and rny children Tracy and

Christine.

iii

TABLE OP CONTENTS

Page

LIST OP ILLUSTRATIONS vl

LIST OP SYMBOLS ix

ABSTRACT xii

1. INTRODUCTION I

Problem Statement 1 Review of Literature . . • 4

2. FORMULATION OP EQUATIONS 12

Nonlinear Plate Equations 12 Equilibrium 12 S t r e s s - S t r a i n . . . . . . . . . . . . . . 1 6 Moment-Deflection 18

3. METHOD OP SOLUTION 22

Introduction 22 Matrix Formulation 22 Element Stiffness Matrix 24

Linear Analysis 24 Nonlinear Analysis 29

Applied Loads 31 Constitutive Relationship 32 Cap Element 37 Mechanics of Solution 40

4. APPLICATIONS AND EXAMPLES 46

Introduction 46 Linearly Elastic Material ' 46

Edge Loaded—Simply Supported Annular Plate 47

Uniformly Loaded—Simply Supported Plate. 50 Nonlinear Material . 52

Uniformly Loaded—Clamped Plate 52 Concentrated Load—Simply Supported

Plate No. 1 ....... 55 Concentrated Load—Simply Supported Plate

No. 2 66

iv

V

TABLE OF CONTENTS—Continued

Page

5. SUMMARY AND CONCLUSIONS 80

REFERENCES . 82

LIST OF ILLUSTRATIONS

Figure Page

1-1. Description of Discrete Model 8

2-1. Infinitesimal Element of a Circular Plate . 13

2-2. Projection of an Infinitesimal Element on the r-z Plane . 14

3-1. Plate Element showing Generalized Displacements and Forces 27

3-2. Nondimensional Stress-Strain Relationships . 33

3-3* Nonlinear Parameter n 34

3-4. Nondimensional Secant Modulus 36

3-5. Circular Cap Element 38

4-1. Results for Inner Edge of an Annular Plate . 48

4-2. Annular Plate Results—Single Load 49

4-3- Results for a Uniformly Loaded, Simply Supported Plate 51

4-4. Load-Deflection Curve for a Uniformly Loaded, Clamped Plate .... 53

4-5. Radial Stresses 56

4-6. Plate Model showing Nodal Circles 57

4-7. Radial Slopes 57

4-8. Lateral Displacement Profiles 58

4-9. In-Plane Displacement Profiles 58

4-10. Membrane Forces 59

4-11. Bending Moments ........ 59

vi

vii

LIST OP ILLUSTRATIONS--Continued

Figure Page

4-12. Top Surface Radial Stresses 60

4-13- Top Surface Circumferential Stresses ... 60

4-l4. Bottom Surface Radial Stresses 61

4-15. Bottom Surface Circumferential Stresses . . 61

4-16. Radial Stress Distribution through the Plate Thickness 62

4-17- Circumferential Stress Distribution through the Plate Thickness 63

4-18. Load-Deflection Curve for a Simply Supported Plate with a Concentrated Load—No .1 64

4-19. Lateral Displacement Profiles— Concentrated Load 67

4-20. Radial Slopes—Concentrated Load 67

4-21. Top Surface Radial Strain over the Support . 68

4-22. Top Surface Circumferential Strain over the Support 68

4-23. Top Surface Circumferential Strain at Station •§• Inch away from Support ... 69

4-24. Top Surface Radial Strain at Station Inch away from Support ... 69

4-25. Top Surface Radial Stress— Concentrated Load 70

4-26. Bottom Surface Radial Stress— Concentrated Load 71

4-27. Top Surface Circumferential Stress— Concentrated Load 72

4-28. Bottom Surface Circumferential Stress— Concentrated Load 73

LIST OP ILLUSTRATIONS—Continued

viii

Figure * Page

4-29- Stress Distribution through the Plate Thickness—Concentrated Load 74

4-30. Load-Deflection Curves for a Simply-Supported Plate with a Concentrated Load—No. 2 ~. . . 76

4-31. Results for a Concentrated Load of 20 lbs 77

4-32. Results for a Concentrated Load of 500- lbs. 78

LIST OP SYMBOLS

Symbol Definition

A equilibrium matrix

a outer radius of the plate

B compatibility matrix

b inner radius of the plate

f1 v2 I . 5- dz - flexural rigidity of the J - 75- 1-u- plate

•cn-,3 D1 ———5— — flexural rigidity of the plate

12 (1-u )

d matrix of nodal displacements

d differentiation with respect to the radial cEr coordinate, r

E . modulus of elasticity of plate material

Es seoant modulus of elasticity

e_ radial normal strain r

e^. circumferential normal strain

f vector of applied forces

h

H P Es

H- _a» _ (i-u2)

ix

plate thickness

initial "boundary of* finite element

final boundary of finite element

B^kB = system oriented stiffness matrix

element stiffness matrix

radial bending moment per unit length

circumferential "bending moment per unit length

radial membrane force per unit length

circumferential membrane force per unit length

nonlinear stress-strain parameter

applied concentrated load

vector of internal forces

shearing force per unit length

intensity of uniform load

/ h A i d z ' " 2"

Poisson's ratio I

membrane radial displacement

normal displacement

coordinate directions

^ = rotation

radius of curvature in the radial direction

xi

radius of curvature in the circumferential direction

effective von Mises stress

uniaxial yield stress of the material

radial normal stress *

radial bending normal stress

radial membrane normal stress

circumferential normal stress

circumferential bending normal stress

circumferential membrane normal stress

ABSTRACT

An annular plate element is presented for analyzing

structural, axisymmetrical, initially flat, circular plates

subject to finite deflections and having nonlinear material

properties. The displacement method of matrix structural

analysis is used.

The element stiffness matrix is formed by utilizing

the differential equations for the annular element. The

equations are integrated across the element. To account

for finite deflections, the equations are uncoupled by

separating membrane and bending effects.

A cap element is used to circumvent the singularity

which exists at the center of the plate. This allows a

solid plate to be simulated and permits the application of

a concentrated load at the center of the plate.

The total load is first applied to the plate in one

step. Using the linearly elastic, small deflection solution

as an initial estimate, an iterative technique is employed

to arrive at the final solution.

The Richard equation is used to account for work

hardening of the material. The effective stress is cal­

culated using the von Mises yield criterion, and then is

used in the Richard equation to obtain the secant modulus

xii

xiii

of elasticity. This modulus is determined at discrete

points through the thickness of the plate at the centroid

of each element.

Several numerical examples are given to illustrate

the accuracy of the method.

CHAPTER 1

INTRODUCTION

Problem Statement m

The classical theory of plates which assumes small

deflections is noticeably deficient in describing the

behavior of thin plates. When lateral deflections exceed

one-half of the plate thickness,, the prediction of stresses

and displacements by the classical theory is erroneous

(Timoshenko and Woinowsky-Krieger, 1959)* Since many plates

remain useful for deflections far in excess of this,.a more

general theory is needed.

Von Karman in 1910 presented the fundamental theory

for the analysis of elastic plates with large deflections

(Timoshenko and Woinowsky-Krieger, 1959)• These equations

are nonlinear and include the effects of membrane forces and

the second order effects of lateral displacements in the

associated strain-displacement equations. The solution of

these equations in the general case is unknown.

Large deflections, as used in this paper, mean

deflections which are of the same order as the plate thick­

ness, yet which are small compared to the other plate

dimensions. In studying the case of large deflections,

combined action of membrane stretching and bending must be

X

2

considered. The equations of equilibrium would relate to

an element of the plate in the deflected position and not

to an element of the initially flat plate. Since this

distorted shape is not known beforehand, but is part of the

required solution, nonlinear!ty is introduced. Even for

Hookean material the problem yields a solution which is not

linear in that deflections and stresses are not proportional

to applied loads.

The problem is further complicated if yielding of

the material is included. Attempts to include large deflec­

tions and yielding of the material have generally been

restricted to specialized stress-strain relations such as

rigid-plastic material (Onat and Haythornwaite, 1956$ Cooper

and Shifrin, 195*0 and elastic-perfectly plastic material

(Crose, 1967)•

In certain design problems, such as aerospace

structures, plastic deformation may be tolerated in order

to utilize the materials more efficiently. In fact, plastic

deformation may have the beneficial effect of alleviating

the stress concentrations.

The treatment of problems of large deflections with

full generality represents a task which is at this time

outside the realm of mathematical tools. The availability

of the digital computer, the techniques of finite element,

and a tractable constitutive relationship now make the

3

numerical solution for nonlinear geometry and nonlinear

material possible for special problems.

In this paper the circular plate is divided into a

number of annular rings, called plate elements. The edges

of the rings where the elements are connected are called

nodal circles. Because each element may be considered as a

separate unit, different material properties, thicknesses,

and loadings can be ascribed to the individual elements.

The direct stiffness method is used to establish a

force-displacement relationship between the nodal circle

forces and moments and the corresponding displacements and

rotations. *

Birchler, Callabresi, and Murray (1968) derived the

element stiffness matrix for a linearly elastic annular

element subject to small deflection by integrating the

governing differential equations. A searching.procedure

was used to convert the boundary value problem to an initial

value problem. This same procedure is used here after the

nonlinear plate equations are stepwise linearized.

An Iterative technique is used to handle the

geometric nonlinear!ty. Holding certain terms constant

from the previous iteration and uncoupling the membrane and

bending effects results in a linearization of the nonlinear

governing differential equations. Utilizing the linear,

small deflection solution as an initial estimate of the

4

deflected shape, iteration is continued until the solution

converges within a given desired tolerance.

A constitutive relationship developed by Richard

(1961) is used to solve the material nonlinearity (Goldberg

and Richard, 1963)• This relationship gives the stress as

an explicit function of strain and ensures a smooth transi­

tion from elastic action to plastic action. The degree of

nonlinearity may be adjusted by varying the three parameters

in the Richard equation. Yielding of the material is

governed by the von Mises yield criterion. '

The material respons.e of the element is assumed to

be governed by the variation of effective stress through the

plate thickness at the center of the element. The plate

thickness is divided into a number of layers. The secant

modulus is evaluated at the interface of each layer and at

the upper and lower faces of the plate. These secant moduli

are held constant during each iteration.

Review of Literature

Solutions for moderately large deflections of thin

plates are based on the equations formulated by von Karman.

Numerous authors have presented methods of treating these

nonlinear equations for axially symmetric circular plates.

Way (1934) presents a series solution for uniformly

loaded, elastic circular plates. This results in an "exact"

solution of the differential equations. The differential

5

equations are first written in dimensionless form. Since

the radial membrane stress is a symmetrical function of the

radius, r, this stress can be expanded in a series of even

powers of r, and since the slope Is an antisymmetrical

function of r, it can be expanded in a series of odd powers

of r. These series in dimensionless form are as follows:

= BQ + BgX2 + B^x^ + ...

=VfT(C1x + + C^x^ + )

where x = r/h. Substituting these two relationships into

the governing differential equations results in recurrence

formulas for and Ck. The general solution is now known.

The particular solution results from application of the

boundary conditions. $

The author considers two cases: (1) plates with no

load, with a moment at the edge, and zero tensile force at

the edge; and (2) plates with a uniform load, clamped edge,

and zero radial edge displacement. Central lateral deflec­

tions and stresses are presented for deflections up to

approximately 1.2 times the plate thickness. In attempting

to extend this method to larger deflections, it was found

that convergence was very difficult to obtain. This same

problem confronted McPherson, Ramberg, and Levy (19^2) when

they attempted to extend Way's method.

6

Wempner and Schmidt (1958) present a series

solution of the von Karman equations for elastic, annular

plates, a case for which Way's solution is not applicable.

The method is very similar to Way's method. Series

solutions are assumed for the radial membrane force and

the radial slope. Two recurrence formulas for "the series

coefficients result from substituting the series into the

differential equations. All coefficients are determined in

terms of four arbitrary constants. For the case of un­

restrained radial movement and rotation at the edges

discussed in this paper, two of the arbitrary constants are

eliminated. A numerical solution is accomplished by

estimating and successively correcting the other two'

arbitrary constants until the boundary conditions are satis­

fied to a desired degree of accuracy. Although the only

case discussed is an annular plate with unrestrained radial

movement and rotation at the edges, the authors state that

the method is equally applicable to other edge conditions

and other load conditions. A method for demonstrating

convergence and appraising the error is included. Conver­

gence of the series is dependent on the magnitude of the

ratios of trie inner radius to the outer radius, the plate

thickness to the outer radius, and on the load. In partic­

ular, small values of plate thickness to outer radius can

result in extremely slow convergence. In fact, there is no

assurance that the series will converge for all values of

7

these parameters. Results are presented for deflections up

to approximately 1.4 times the plate thickness. Although

no statement is made in this regard, possibly the method

also has the same convergence problem that occurs in Way's

method when the deflection to thickness ratio exceeds the

range presented.

Several investigators have analyzed plates with

large deflections which do not remain elastic. In all

cases bilinear stress-strain relationships were assumed.

Crose (1967) presents a method for analyzing

nonlinear circular plates which is based on a physical

discretization of the continuum. The models shown in

Figure 1-1, is a lumped parameter model interconnected by

axial and shear springs. It is essentially a physical

analog of the finite difference equations for the continu­

ous plate. The total mass of the solid continuum is

concentrated at the mass points. The procedure is further

simplified by adopting a sandwich configuration. The plate

consists of three layers. The top and bottom layers,

containing all of the mass, are identical and are assumed

to be characterized by a uniform plane stress distribution

across the thickness. Separating these two sheets is a

shear core of constant thickness which is infinitely stiff

with respect to transverse shearing forces, and which

provides no reslstence to flexural and extensional forces.

The idealization of the solid plate to a sandwich plate

(a) Plan View

•W A/WV O

•' "O" •VSAAr Unde formed Position Deformed Position

(b) Elevation View

Figure 1-1. Description of Discrete Model

9

simplifies the problem by eliminating the effect of

yielding through the thickness.

The material in the top and bottom sheets is

assumed to be elastic-perfectly plastic. Yielding is

governed by the von Mises criterion and plastic flow

follows the Prandtl-Reuss flow rule. The author states

that the approach is equally applicable for other inelastic

materials.

The stress-strain equations must be in incremental

form because of the use of the incremental theory of

plasticityj therefore the remainder of the field equations

are converted to incremental form. Load is applied in

increments3 and an iterative technique is utilized. The

linear elastic solution results from the first iteration.

Within each iteration, the linear equations are solved by a

modified Gauss elimination scheme. Load is applied in

small increments until a desired level of loading is

achieved. The efficiency of solution depends on the mag­

nitude of the load increments.

The main restriction on the load increment is made

during plastic flow. If the increment is too large, the

yield criterion will be seriously violated. When this

happens, the load increment is decreased by interpolation

and a smaller increment of load applied. The magnitude of

the load increment is also limited during plastic flow by

certain stability requirements of the nonlinear equations

10

and by certain assumptions made regarding transformation

of displacements.

The primary advantage realized in using the discrete

element formulation is that the resulting equations can be

derived through the use of elementary mechanics and simple

geometry. Also, the equations are physically meaningful

when related to the model.

Three specific problems are presented and discussed.

Two of these are clamped plates with uniformly distributed

lateral loads, and one is a simply supported plate with a

concentrated load at the center.

Naghdi (1952) used the theory of plastic deformation

from the theory of plasticity to develop a method of'

handling strain-hardening materials. The equilibrium

equations are written to include the effects of large

deflections. The amount of strain hardening is related to

the octahedral shear strain and the plate rigidities and

material moduli are adjusted accordingly. Using a tensile

stress-strain curve, an octahedral shear stress-shear

strain curve is constructed according to a procedure which

Naghdi states has been verified experimentally. Various

parameters are then plotted versus octahedral shear strain,

and the variation of the octahedral shear strain is

determined through the thickness of the plate.

An iterative procedure is used. To simulate a

closed plate, a small rigid plug.is assumed to be inserted

11

at the center of the plate. Once fair results are obtained,

the plug is removed and computations are carried out for

several more cycles. Numerical solutions for two load

cases are presented for a simply supported circular plate

with a central concentrated load. Tests were conducted to

verify the analytical results and good agreement was found

for deflections.

Onat and Haythornwaite (1956) combine the solution

for flat plates due to Hopkins and Prager (1953) and the

Tresca yield criterion to develop an analysis for large

deflections of circular plates composed of rigid plastic,

nonstrain-hardening material. Their analysis is compared

to the results from two plate tests and is found to give

poor agreement with the tests.

CHAPTER 2

FORMULATION OF EQUATIONS

Nonlinear Plate Equations m

Equilibrium

Figure 2-1 shows an infinitesimal element of a

circular plate in its undeformed state and in its deformed

state including loads and forces. Since the displacements

may be large, the equations of equilibrium must be satis­

fied in its deformed position (Novozhilov, 1953)-

Figure 2-2 shows*the projection of the deformed

shape on the r-z plane. The load, q, and the forces,

M.J., Nr, N^. and Q, are the forces acting on the element.

The displacement functions v and w determine the deformed

shape of the plate and as such determine the radius of

curvature pr.

Summation of forces in the radial direction, r, for

the element in its deformed state yields

dN Nr(rd0) cos -gp + Ntdrd0 - (Nr + dr)(r + dr)

d© cos dr) = 0 (2-1) ar dr

12

13

9<S

(oQ Uncle formed Position

• Q

Mr+^-rdr r dr

(b) Deformed Position

V

Figure 2-1. Infinitesimal Element of a Circular Plate

14

W

dw/dr\

dr

Figure 2-2. Projection of an Infinitesimal Element on the r-z Plane

15

Expanding the last term in Equation (2-1) gives

dN Nr(rd0) cos + Ntdrd6 - (Nr + dr)(r + dr)

d0(cos cos dr - sin 5^ sin ^ dr) = 0 (2-2) dr dr

By taking the cosines of small angles equal to one and the

sines of small angles equal to the angles this equation

becomes

dN Nrrd9 + Wtdrd9 - (Nr + dr) (r + dr)

de(l - dr) =0 (2-3) dr

Summation of forces in the z direction yields

QrdS + nrrde - (Q + §§ dr) (r + dr)d©

- (Nr + dr)(r + dr)de(|2 + dr)

- qrdrd© = 0 (2-4)

Summation of moments in the r-z plane yields

dM, M^d© - (Mr + dr)(r + dr) d0 + M^drd0

+ Qrdrd0 = 0 (2-5)

16

Dividing "by drd© and neglecting higher order terms,

Equations (2-3), (2-4), and (2-5) become, respectively,

These are the final equilibrium equations for the plate.

The last term in Equation (2-6) results from expanding the

cosine term in Equation (2-1). This term is retained

because it is not a higher order term, and because it may

become large where a plate has a large membrane force and

large curvature.

Stress-Strain

For an elastic, isotropic material with plane stress

conditions, the stress-strain relationship is described by

Hooke's Law. These equations in polar coordinates are

dW ^ - Kt + wr (2-6)

Q + r^ + rNrg» + I^^£ + Nr^ + qr = 0 (a-7)

dM. M r + r ^ - M t - Q r = 0 (2-8)

(2-9)

17

In large deflection analysis* the middle plane stretching

is included, therefore the membrane strains must be included, *

These strains are (Timoshenko and Woinowsky-Kriegerj 1959)

e - dv + .Lydw\g r» r}y 2 V HW 'r dr- 2Vdrj

(2-10)

e. = t r

5y definition

Nr=/| £

h 2" o;dz

(2-11) h

"S-wt=/; h °^dz

Substituting Equations (2-9) and (2-10) into Equation (2-11)

results in

N r = [ £ + i ( ! £ ) 2+ ^ ] H

(2-12)

r ~3r~ T 2 cir'' 1 " n, = t i + ) H

where h

K = f ^ Es dz

7- h

18 In Equations (2-12) the modulus of elasticity* E3 has been

replaced by the secant modulust E . because nonlinear

material is considered in this study. The method of

evaluating E_ will be explained in Chapter 3>

Moment-Defleetion

The definitions of radial and circumferential

moments are

The Kirchhoff-Love hypothesis* which states that for thin

plates a normal to the middle surface remains normal during

deflection, allows the bending strains to be expressed as

er = z:/ Pr and e^ = z/ Equations (2-9) can therefore

be expressed as

h

(2-13)

h

(2-14)

19

The curvatures are defined as (Timoshenko and Woinkowsky-

Krieger, 1959)

_1_ = -d2w

Pr dr2

(2-15)

1 _ 1 dw pt 'r w

Substituting Equations (2-14) and (2-15) into Equation

(2-13) gives

^ [ gw + « g ]

(2-16) vi - i-i r 1 dw . „ d w i

where

h 2 r t v D =/ d -2-rr dz / h l-u^

Again E has replaced E.

The plate rotation, dw/dr, is also expressed as .

Equations (2-6), (2-7), (2-8), (2-12), and (2-16) are com­

bined to form six first order, ordinary, nonlinear differen­

tial equations as follows:

20

Hr = P C0"1?)

N o 5 v = _Z HZ fo ifti dr H atdr; ~ (2-lts)

d/3 _ Mr u/6 lqv "Sr D r~ (2-19)

- a - . a + v ^ . M ( 1 . u ) dr r D r ^ '

Ax v N„ N A* UN A -f* (1-u) [ X(1+U) - -jf ] + r l/0 + —e£

2 „3 (2-20) c iitvt /1~>

dNr _ H(l-u) r V/-, , v Nr , WrMr/^ uWr/3 (2-21)

« L V. k J-"™-; H J ri ~

- q

dr r L rv ' H J D

= -D(l-u) t A (1+u) + ] + Q (a_22)

These six equations will be used to solve for •

the' , forces, Q3 NrJ and MrJ and the displacements,

Wj Vj and /3. After these forces and displacements

have been found, the circumferential force and moment may

be found by using Equations (2-12) and (2-16). By utilizing

Equations (2-17)» (2-18),, and (2-19), and noting that

d^w _ d/?

dr2 dr

21

Equations (2-12) and (2-16) may be expressed as

+ UN3

Mj. = - .(1-u2) + uMj.

(2-23)

CHAPTER 3

METHOD OP SOLUTION

Introduction m

Structural systems which "behave elastically can be

analyzed by either the displacement method or the force

method. The same methods can be used for nonlinear systems

as presented by Richard and Goldberg (1965) and Goldberg

and Richard (1963). Because of the ease of generating the

required equations on the digital computer, the displacement

method is used here. • -

Matrix Formulation

The matrix formulation of the displacement method

is based on the equilibrium, stress-strain, and compati­

bility relationships of a deformable body. From

equilibrium considerations, the vector of applied forces,

f, is related to the vector of internal forces, p, "as

f = Ap (3-1)

where A is known as the equilibrium matrix. Using the

stress-strain relationship, the internal forces, p, are

determined in terms of the internal deformation, v, as

p = kv- (3-2)

22

23

inhere k is called the stress-strain matrix, or the element

stiffness matrix. Maintaining continuity results in the

expression relating external deformationss d, to the internal

deformationsj Vj

v = Bd (3-3)

where B is known as the compatibility matrix. It can be

shown from the theorem of virtual work that the compati­

bility matrix is the transpose of the equilibrium matrix.

Combining the above equations results in

or

or

where

also

The displaced configuration for any applied loading can be

found from the set of simultaneous algebraic equations

represented by Equation (3-4).

f = AkBd (3-4a)

f = BtkBd . (3-4b)

f = Kd (3-^c)

K '= BtkB

p = kBd (3-5)

24

Element Stiffness Matrix

Linear Analysis

Birchler, Callabresi, and Murray (1968) demonstrated

the feasibility of deriving the element stiffness for an

elastic annular element for small deflections by using the

governing differential equations. Because a similar

procedure is used for the nonlinear analysis, a review of

the procedure is presented here for convenience.

Using the infinitesimal element of a circular plate

in its undeformed state, as shown in Figure 2-1, and

assuming small deflections, the equilibrium equations

analogous to Equations (2-6), (2-7)9 and (2-8) are (Timo-

shenko and Woinowsky-Krieger, 1959)

Hr + r ££ " Nt - 0

Q + r^| + qr = ° (3-6)

dM» + r - Mt - Qr - 0

The stress-strain relationship analogous to Equations (2-11) t

is

Nr = H' [ + u 2 ]

Nt = H' I? + u3? 1

(3-7)

where H1 - 3 l-u

25

The moment-deflection relationship analogous to Equations

(2-15) is

^ = "D' + 1 dr

2 w • _ n i r l d w J . « d w i = " D t r ¥ + u ^ ]

(3-8)

•cn-,3 where D' = ——

12(l-u2)

The rotation, /3 s of the plate is expressed as a function

of the displacement w "by the equation

A - dw P ~ ~Sr

The above equations are combined to form six first

orderA ordinary, linear differential equations as follows:

N dv r uv "3? H1 r

_ Vi/3 dr r T51"

= _ £ _ q dr r H

dNr _ H'(1-u2)v wr(1_u)

" > ~

dMr _ D' (1-u2) & M1-*) -ar ^5-^ r +

(3-9)

<Z

26

After solving these equations at discrete points along the

plate, the values of and Mt can "be found at each point

from the equations

= h,(1;"2)V + un t r r

(3-10) « _ _ P' + t r _ r

By definition each column of the element stiffness

matrix is a generalized force vector which deforms the

element in accordance with the differential equations., omit­

ting applied loads, so that all the generalized displace­

ments are zero except for the unit displacement associated

with that column. Based on this definition, a procedure

to determine the element stiffness matrix is developed.

The undeformed plate element, Figure 3-1* has six

generalized displacements and six associated generalized

forces, three of each at each "boundary in the radial direc­

tion. If all six of the initial conditions at node i are

known, the six first order governing differential equations,

Equations (3-9)* can "be solved by a fourth order Runge-

Kutta numerical integrating procedure. Prom the unit

displacement method where all displacements are zero except

one which is unity, three initial and three final general­

ized displacements are known. This means that three initial

conditions, the generalized forces Nr^, and Mr^, are

unknown; and they must be the particular values to insure

27

Wi Wt

Vi

•e 'ft

•~-r

(a) Generalized Displacements

Q, Mri

•

' / J

Q;

Mrj

•*~r

(b) Generalized Forces

Figure 3-1* Plate Element showing Generalized Displacements' and Forces

28

the known values of the final displacements in accordance

with the governing differential equations.

A searching procedure is used to find these initial

forces. Using a set of assumed initial forces (Q^°^ ^ri^

and "the differential equations are solved by numerical

integration to give a set of final end displacements

v(°>, and By incrementing the initial forces and

re-solving for the final displacements, the change in final

displacements due to unit force changes can be determined.

These results may be written as

<P/3j

3% ' S>e± * <9

<5>W1 5vJ o>"ri d "ri '

9/5s

SMri ' o>Mri ' olMri

(3-11)

where the ratios represent partial derivatives. By applying

the chain rule of differential calculus, the following matrix

relationship is obtained.

Awj1 aWj

f * A^ Awj1

d 9 Nri a

f * A^

Avj • =

3 v 1

d<i± •

3 Nri

3 vj o>Mri

< Auri ,

A/Sj

* 3 \ 3Mri

*fi3 3Mri J A^

(3-12)

29

The quantities Aw^, and A/0j are the changes

required in "Wj0^ VJ°^ and sive known final

end displacements. Solving, Equations (3-12) for the quan­

tities AQ^ AN and AMr^ and then adding them to

Nri^ an(* Mri^ respectively will yield a set of initial

forces that corresponds to the known final end displace­

ments. The differential equations are re-solved using the

proper set of initial conditions to obtain the final forces

These final forces Qj, rj3 and ^r.j* al°nS with the initial

forces ^ri3 and Mri* ^orm a column of the element

stiffness matrix. The above procedure is repeated for the

six sets of the generalized displacements, thus generating

the complete 6x6 stiffness matrix for one element.

Nonlinear Analysis

The above procedure for forming the element stiff­

ness matrix is applicable when the governing equations are

linear. For the case of large deflections, the equations

are nonlinear and must be linearized before the element

stiffness matrix can be formed. To accomplish this*, the

equations associated with bending, Equations (2-17)/ (2-19)

(2-20), and (2-22), are separated from the equations associ

ated with stretching of the membrane, Equations (2-18) and

(2-21). All bending terms in membrane equations and all

membrane terms in the bending equations are held constant

for each iteration. The secant modulus, Eg, and therefore

30

H and Da is also held constant for each iteration.

The bending equations become

(3-13)

a/3 = _ ^ (3-14) UF- 15 u ;

* * „* n /3 /?TT * N

H = - § + -F- (i-«) - 4s (1-u) [ T-(1+u)- -r ]

* , /i*.2 *, -,*v2 -+ V^j!+ j^.q (3-15)

^ = . D£l=]il [ .£ (1+u) + ] + Q (3-16)

where the starred terms are held constant. Using Equations

(3-13) through (3-16) instead of Equations (3-9)> four

columns of the element stiffness matrix are determined by

the method described above.

The membrane equations become

N O dv _ r if uv

" 2 (/J J - T (3-lTa)

£ . Jllal I J,l„, ^ ,

where the starred terras are held constant. In trying to

form two additional columns of the element stiffness matrix,

31

Equation (3-17a) presented difficulty because of the (^*)^

terra. This term remains constant regardless of the

boundary conditions, and therefore gives extraneous values

when the unit displacement method is used. To circumvent ^ O

this problem, the (/£> ) term is omitted during the forma­

tion of the stiffness matrix. Equation (3-17&) then becomes

N dv _ r uv /q "5r ~ ~H r* (3-17b)

Using Equations (3-17b) and (3-18) instead of Equations

(3-9)9 two additional columns of the element stiffness

matrix are formed, resulting in the complete 6x6 matrix for

one element for a particular iteration.

The effect of the omitted term is brought into the

problem by considering it as a load term. When the effects

of the applied loads are calculated, as described in the

next section, Equation (3-17a) is used instead of Equation

(3-17b).^

Applied Loads

• The displacement method requires that loads be

applied only at the generalized displacement points. Dis­

tributed loads on an element are handled by calculating

the fixed end forces in a manner similar to that used in

the preceding section to determine a column of the stiff­

ness matrix. The only differences are that in solving for

the fixed end forces all six of the generalized

32

displacements are set to zero, and the differential equa­

tions are solved with the applied load term included.

Forces equal to the fixed end forces but opposite

in direction are applied at the appropriate nodes. After

the element forces and displacements are found., these

applied forces are removed.

Constitutive Relationship

A three parameter, nonlinear constitutive relation­

ship developed "by Richard (1961) is used to give the stress

as an explicit function of the strain (Goldberg and Richard,

19633 Richard and Callabresi, 1967s Richard and Blacklock,

1968). This relationship is

r E? I" Vn - (3-19> [ 1+N] where (J*is the stress, E is Young's modulus, e is the

strain, e„ is the strain which is equal to CV/E where CP ' o ^ ,0 o

is the.maximum uniaxial stress, and n is a parameter defin­

ing the shape of the nonlinear stress-strain relationship. 1

Figure 3-2 is a nondimensional plot of this formula for

various values of n, and Figure 3-3 shows how the three

material parameters, E, CF", and n, may be obtained from a

uniaxial stress-strain curve. It may be noted that this

single-valued expression represents essentially the inverse

of the Ramberg-Osgood equation; however, in Equation (3-19)

0.8

0.6

0.4

0.2 %

0.4 0.8 t.a 2.0 2.4 2.8

e/e,

Figure 3-2. Nondimensional Stress-Strain Relationships

Figure 3-3. Nonlinear Parameter n

35

the stress approaches the plastic stress asymptotically

with increasing strain.

In the studies made, in this dissertation^ yielding

of the material will be governed by the von Mises yield

criterion^ which for plane stress is

= (o 2 + o 2 - + 3r|y)4 (3-20)

For an axisymmetric circular plate^ the effective stress

becomes

OJ = (OJ2 + of - (3-21)

In order to determine the stress distribution in

the element once the strains and displacements are known,

the secant modulus of elasticity is used. From Equation

(3-19)i the secant modulus Is

E E = s

[ 1 + TT—175 (3-22) ]

This equation represents an exact solution for uniaxial

stress elements because for this case

0^ - Ee (3-23)

Figure 3-^ shows a nondimensional plot of these functions.

Equation (3-22) ensures a smooth^ univalent stress-

strain relationship. A given stress-strain curve can be

approximated by choosing appropriate values for the

so 20

0.4

0.2

0.4 2.4 0.8 A2 LQ 2.0 i

Figure 3-4. Nondiraensional Secant Modulus 00 o\

37

parameters Ea 0^, and n. It should be pointed out that

stresses resulting from this development are only approxi-

mate values, just as are the stresses found by all other

investigators allowing plastic action of the material;

however, they should give a close approximation of the true

stresses.

Cap Element

For a closed axisyimnetrlc circular plate the

governing differential equations become singular at the

axis of symmetry. One of the advantages of the finite

element method is that this singularity can be removed by

using a cap element with four degrees of freedom.

The displacement functions for the circular caps

shown in Figures 3-5a and 3~5b are presented by Tiraoshenko

and Woinowsky-Krieger (1959)• Superimposing these two

plates results in the plate shown in Figure 3-5c, which

shows the generalized forces and displacements associated

with bending. The membrane generalized force and displace­

ment is shown in Figure 3-5d. The membrane force-

displacement relationship is given by Den Hartog (1952)'.

The cap radius will be limited in size so that

small deflection theory will apply. The material is assumed

to be nonlinear. For these conditions the bending and

membrane effects are uncoupled. By allowing the four

generalized displacements to take on unit values one at a

Mm, fij (C) G

I 1/ 0

Mrj)/3j

W

a

ft) rQ/, Wj.

T— a.-Wj

Mrj3/3j

(c)

Qj,wJ Qij- v/f

\

1 Qj,Wjf

(oO .

I

4 Nrj, Vj

Figure 3-5- Circular Cap Element

39

time, the element stiffness matrix is formed by columns.

This matrix is -p. * - -

f "

167td

" a2 0 0

167td

a2 0

87td a wi

•h 0 0 0 0 0 0 i* vi

Mri = 0 0 0 0 0 0

• 8d

" a3 0 0 8d

a3 0

" a2

< >

Wd

NJ 0 0 0 0 T a 0 VJ

Mrj 4D

l" ? 0 0 4D

a2 0 (3+u)d

a

6 E_z2 where

I h 1-u •J~ o dz and T = /

e ^ s

h 2

jr=uj dz

Because of the singularity existing at the center,

Mri and Wi rema^n undefined and are given the value of

zero for convenience.

The fixed end forces are found by solving for the

forces due to the applied loads while making all generalized

displacements equal to zero. For a uniformly distributed

lateral load, q, the fixed end forces are

40 "• - • «•

rrr^a.

Mi • 0

«ri — .0

• <

.3ga |

Hd 0

p

"rd m * • «

and remain undefined and are given the value of

zero for convenience. The membrane force, Nj, is zero

because the bending and membrane effects are uncoupled for

small deflection theory; therefore, no membrane force

results from a lateral l<3ad.

Mechanics of Solution

To initiate the solution the plate is divided into

annular elements. These elements are made small where high

stress and strain gradients occur which is where radial \

membrane forces or radial slopes are changing rapidly. The

material properties, dimensions, and loads for each- element

are established. If the plate is closed at the center, a

cap element is used.

The elastic, small deflection solution is used as

an initial estimate of the deflected shape. The element

stiffness matrix is generated for this case by using

Equations (3-9) and the procedure presented above. A fourth

41

order Runga Kutta numerical Integration scheme (Hildebrand,

1956) is used to integrate the equations across the element.

The fixed end forces are calculated by the same procedure.

Using Equation (3-4b), the element stiffness matrices are

transformed to the global coordinate system, and by super­

imposing these matrices, the global stiffness matrix, K, is

formed. The global displacements, d, are now found by

solving Equation (3-4c) by the Gauss elimination method.

The element forces and displacements are obtained from

Equations (3-3) ar*d (3-5)-

In the radial direction, the membrane forces, the

slopes, the membrane displacements, and the moments are

retained for use in the next iteration. Each quantity is

averaged at the two nodes of each element with the averages

being represented by the starred terms in Equations (3-I3)

through (3-18). These values then remain constant through­

out the next iteration.

A new element stiffness matrix is formed for each

iteration from Equation (3-13) through (3-16) and Equations

(3-17b) and (3-18). This matrix changes for each iteration

due to the effect of the starred terms in these equations.

The procedure is repeated until the solution converges to

within a given tolerance.

During iteration the solution oscillates and con­

verges 'slowly even for slightly nonlinear problems. In

order to accelerate convergence a "damping factor" is

• 42

applied to the membrane fixed end forces of each element

except the cap element. These are the most rapidly changing

of the fixed end forces, and therefore they exert the most

influence on the modification of the element stiffness

matrix. Values of the damping factor between 0.0 and 1,0

were tested, and best results were obtained for a damping

factor of 0.6. If the value gets too large, oscillations

of the solution occur; and if it gets too small, the

equations become overdamped and require many iterations to

achieve a solution.

When the oscillations become small, indicating that

a solution is near, the damping factor should be removed

so that the plate may seek its true deflected shape un­

hampered. But when this happens, oscillations will again

occur even though they are now small. A procedure was used

during these final iterations which does not artificially

dampen the equations and yet does control the oscillations.

After several iterations, when the solution has stabilized

somewhat, the damping factor is discarded and the fixed end

forces are calculated for each iteration by averaging the

most recent three values, one newly calculated value and

two values from the previous two iterations. It was found

that this procedure could effectively begin after five

iterations. There is no mathematically rigorous Justifica­

tion for using the damping factor or the averaging

procedure, but convergence is eaqpedited when they are used.

43

When lateral deflections exceed approximately two

plate thicknesses, the solution may diverge very quickly

even though damping factors are used. This happens "because

the initial small deflection solution results in deflections

which are much too large, and when the element stiffness

matrices are adjusted to the rapidly changing deflected

shapes which occur during the first few Iterations, the

solution may diverge. In order to furnish sufficient time

for the adjustment of the element stiffness matrices, a

limit is placed on the allowable change in the membrane

fixed end forces. Good results were obtained by using a

maximum change of 100$ of the membrane fixed end forces.

Another technique for managing highly nonlinear

problems is to use a combination of the incremental load

and iteration procedures. In this method of solution, the

load is applied in increments and iteration is continued at

each increment until convergence occurs. The load in­

crements are applied until the maximum value of load is

reached. This method is particularly useful where

solutions are desired for intermediate load cases. The

straight iteration procedure, where the total load is

applied in one step, is usually faster than the combination

method.

For the first seven iterations the material is kept

linearly elastic. During this period the nonlinearity is

due to geometry only. By introducing the secant modulus of

44

elasticity as given in Equation (3-22), the material is

then allowed to become nonlinear. It is assumed that the

material response throughout each element is governed by

the variation of effective stress through the thickness at

the centroid of the element. The secant modulus will

therefore be a function of the z spatial coordinate only.

These moduli are calculated at discrete points through the

plate thickness, and are held constant throughout each

iteration. Because of the singularity existing at the

center of the plate, these calculations cannot be carried

out for the cap element; therefore, it is assigned the same

moduli as the adjacent element.

Experience has shown that convergence is slowed if

the material is permitted to be nonlinear too early in the

solution. If -the nonlinear geometry is treated first and

then material nonlinearity is introduced, convergence is

rapid.

In addition to all generalized forces and displace­

ments the solution consists of stresses and strains

calculated through the plate thickness at the centroid of

each element. The strains result directly from integrating

the differential equations across the plate element. Com­

bining the membrane strains and the bending strains, the

total strains are

45

" S + *<&>* " dr

= X . 2/dwx r r Mir'

The stresses are calculated from the following equations

= .^(er + uet> *

E <JT = — (e. + ue ) * 1—VI * r

where these are a modified form of Hooke's equations in

polar coordinates^ Equations (2-9)•

CHAPTER 4

APPLICATIONS AND EXAMPLES

Introduction

It is of interest to verify the accuracy of this

analysis "by comparing the results with other analyses and

with experimental data. For the case of large deflections

and linear material some "exact" solutions are available

for certain particular problems. No "exact" mathematical

development exists for the case of large deflections and

nonlinear material. Although the primary concern of this

study is with nonlinear material behavior, the linear

material analysis is verified to insure correct initial

phases of the more general problem.

Linearly Elastic Material

Two problems in this section were chosen to provide

a comparison of the method with two different plate* struc­

tures % one a solid plate which would utilize the cap

element, and the other an annular plate. In both problems

results for forces or stresses in addition to deflections

are presented. This furnishes an additional check on the

method.

46

47

Edge Loaded—Simply Supported Annular Plate

Figures 4-1 and 4-2 contain graphs reproduced from

a report by Wempner and Schmidt (1958). The dimensionless

quantities shown on these graphs are defined as follows:

- _ (1-u2)P q 2*7TEha

(3.-u2)Ht Nt M r

12(l-u2)M.b Mt = 5—— z EhJ

where P is equal to the total applied load. Theoretical

results are presented for deflections to approximately 1.5

plate thicknesses. The linear exact solution* in which

small deflections are assumed, is shown in Figure 4-1. A

comparison with the large deflection analysis clearly shows

the importance of the membrane effect on the plate

deflection.

The results from the analysis contained herein are

also shown in Figures 4-1 and 4-2. The plate used had

inner and outer radii of 10 and 20 inches respectively,- a

Poisson's ratio of 0.3, a thickness of 1 inch, and a Young's

modulus of 10,000,000 psi. Ten elements, each 1 inch wide,

were chosen. It is noted that excellent correlation is

shown between the two analyses.

48

— Redrawn from Wempner and Schmidf (05&)

o Calculcrfeof Resu/fs

0.8

vy/h 16

± ± J.

Figure 4-1,

/ Z 3 4 5

T7t x 10s and MtxlO

Results for Inner Edge of an Annular Plate

I

49

—. Redrawn from Wempner and Schmidt (1358)

o Calculated Results

-6

M,

0.2

OA

W

/a /7 /£ £0 /s it IB IS to r - Inches

Figure 4-2. Annular Plate Results—Single Load

50

Uniformly Loaded—Simply Supported Plate

Theoretical results obtained by Federhofer and *

Egger (Timoshenko and Woinowsky, 1959) for a simply sup­

ported circular plate with immovable edges are shown in

Figure 4-3- This'figure contains graphs of the central

deflection, the maximum normal stress at the center, and

the maximum normal stress in the radial and circumferential

directions at the edge of the plate. Results are presented

for deflections up to approximately five plate thicknesses.

The points denoted by circles were obtained from the

analysis presented herein.- The agreement between the two

methods is excellent.

To check this problem, a 10 inch diameter plate was

assumed with a thickness of 1 inch, a Poisson's ratio of

0.25* a Young's modulus of 10,000,000 psi, and nodal circles

of radii 0.1, 0.5* 1.0, 2.0, 3-0, and 4.0 inches. The

combination incremental load and iteration technique was

used successfully on this problem. Linear material action

was assumed in this problem, and this resulted in a- maximum

stress at the center of 97a400 psi at a qa^/Eh^" value of

300. For most plate materials, the normal stress could not

attain a value nearly this high; therefore, the plate

material would become nonlinear and the results would be

considerably different from that shown in Figure 4-3.

51

— Redrawn from Tt'moshenko o Ca feu/a fed Results

24

Uj 20

too so 300 25

a* Eh

Figure 4-3. Results for a Uniformly Loaded, Simply Supported Plate

52

Nonlinear Material

From the tests of circular plates with large de­

flections available in the literature, three have been

selected here to provide a comparative study with the

method contained herein. In all cases the tests apparently

were carefully performed, the plates underwent large elasto

plastic deformations, and additional data were presented

other than just the deflections. Exact agreement with

experimental work cannot be expected, but sufficient agree­

ment is shown to Justify the assumptions and procedures of

the nonlinear material analysis.

Also presented is an analysis which includes the

effects of large deflections and nonlinear material.

Uniformly Loaded—Clamped Plate

McPherson, Ramberg, and Levy (1942) tested several

•aluminum-alloy and stainless-steel plates which were

uniformly loaded and clamped at the outer edge. Material

stress-strain curves are presented. Corrections were

applied to test results for errors due to initial tension

(or compression) in the plate during clamping, rotation of

the clamping rings, contraction of the clamping rings due

to membrane tension in the plate, and deviations from flat­

ness at no load.

. Figure 4-4 shows results from tests of a 5-0 inch

diameter, .0300 inch thick, 17S-RT aluminum-alloy plate.

53

240 280 200 40. 120 80 O aa* Wf

Linear Theory Nadai Equation Test Results (SAcPherson, Ram bergi Levy, I94l) Calu fated Results

Figure 4-4. Load-Deflection Curve for a Uniformly Loaded,•Clamped Plate

54

The tests were conducted to deflections of 4.7 plate thick­

nesses. The test curve duplicates Way's "exact".solution

up to a deflection of 1.2 plate thicknesses, which was the

maximum value presented by Way (1934). The solid squares

represent points where a permanent set at the center

exceeded 0.01 inches. Also drawn on Figure 4-4 is the

theoretical curve representing Nadai1s method. This method

differs from Way's "exact" solution in that Nadai solved

the nonlinear differential equation by assuming a convenient

but not quite uniform pressure distribution, whereas Way

used a uniform pressure. The points denoted by circles

were obtained from the method presented in this dissertation

These points fall between the results obtained from tests

and from Nadai's method. The elementary theory, assuming

small deflections, is also shown on Figure 4-4, thereby

illustrating the importance of membrane effect on plate

deflections.

Tensile properties were determined on coupons cut

both longitudinally and transversely from the sheet from *

which the plates had been cut. Although the material proved

to be slightly anisotropic., isotropy was assumed in this

study. The modulus of elasticity was 10,300,000 psi, the

yield stress was taken to be 44,000 psi, and Poisson's ratio

and the parameter n were set at 0.3 and 4 respectively. To

define the finite elements, nodal circles were choden as

shown in Figure 4-6.

55

Figure 4-5 shows elastic radial stresses calculated

from Nadai's theory and radial stresses for nonlinear

material calculated from the method presented herein. The

stresses from Nadai's theory and Way's "exact" theory are

in very close agreement up to w/h = 1.2, which is the

maximum value presented by Way. The effect of nonlinear

material on the stresses is clearly discernible.

Results from three load cases calculated by the

method contained herein are presented in Figures 4-7 through

4-17. Figures 4-7, 4-8 and 4-9 contain the displacements;

Figures 4-10 and 4-11 show, the forces and moments; Figures

4-12 through•4-15 exhibit the surface stresses across the

plate; and Figures 4-16 and 4-17 represent the normal

stress distribution through the plate thickness.

Concentrated Load—Simply Supported Plate No. 1

Cooper and Shifrin (1954) tested several simply

supported mild-steel plates loaded well into the plastic

range. The load was applied to the plate specimens through

a 0.6 inch diameter rod. Stress-strain curves for the plate

material are presented. Although the material was slightly

anisotropic, isotropy was assumed.

The solid line in Figure 4-18 was redrawn from a

report by Cooper and Shifrin (1954). It represents the

test results of a circular plate .104 inches thick and

17.3125 inches in diameter, supported on a 17-Inch diameter

ring..

56

too

Elastic Stresses from Nadai Ca leu fated Stresses

Top oz of Support

Bottom op zs at Center—

lop op at Center

Bottom op at Support

Figure 4-5 • Badlal Stresses

57

Diameter =• S Inches

Figure 4-6. Plate Model showing Nodal Circles

Tfc~ = 280

t.O L5

r - Inches • *

Figure 4-7. Radial Slopes

58

JOO

280

/.& 2.0 o.s 2.S O t.o

r - Inches Figure 4-8. Lateral Displacement Profiles

t.S

AO

JOO

§ O K

> -O.S

-I.O t.o 2.S 0.5 O

r — inches

Figure 4-9. In-Plane Displacement Profiles

v

7

6 mo

5

A too 3

2

/

O t.5 2.S 0.5 1.0 O r - hches

Figure 4-10. Membrane Forces

280

too -2

2.5 1.5 t.O 2.0 O r-Inches

Figure 4-11. Beading Moments

Figure 4-12.

r — inches

Top Surface Radial Stresses

z

28Q

/ tao

too

o

t.5

r - Inches J.o o o.s

Figure 4-13. Top Surface Circumferential Stresses

I

61

3

2

/

1 ° J

5 -2 *

bv -3

-4

-S

-G

/SO

too

I 0.5

± _L » 2.0 /.0 /.ff

r - Inches Figure 4-14. Bottom Surface Radial Stresses

2.5

8. I

*

'o

X

b*

3

2

I

O

-/

-2

^230

-

-

• 1 1 • ' O.S 20 /.O /.S

r - inches Figure 4-15. Bottom Surface Circumferential Stresses

2.5

I

62

_ > 1 1 ti o o

j\\ ,00- M ISO M

2S0—^ "^\\ 280—^ t -¥• C^XiO - pS!

(a) at r « ojs in.

4 2 O 2 4 crp x ID'*— psi

(b) a£ at r~i.Z5in.

x

IOO—N J U loo—^

180—s /77 180—v

280

/

f j / 2 8 0 - J

j. x

07 x iO'* - psi

(c) of at r = 2.35 in.

i I i i L I 4 2 O Z 4 6

x to"* - psi

(d) 07 at r~ 2.50 in.

Figure 4-16. Radial Stress Distribution through the Plate Thickness

63

/OO——^ 1 180 ^ A[\ 280—/ 1ft

L I L 3 2.

i O / -+

J I 2 3

Of * 10 -pst

(a) of at r-o. tsin.

too ^ 1 180 ^

280

I 1 1 1 1 1 1 3 2 2 3

Of x 10-+- psi

(b) 07 at r- t.zsin.

2 3

Of X tO - pst

(c) of at r=2.35 in. (d) of at r = 2.50 in.

Figure 4-17. Circumferential Stress Distribution through the Plate Thickness

1.6

S /.2 *??•

I

^ OA

0.4

Redrawn from Cooper and Shifrin (1954)

—o~ Calculated Results

' 0 O.J 0.2. 0.3

Central Deflection - Inches i

Figure 4-18. Load-Deflection Curve for a Simply Supported Plate with a Concentrated Load—No. 1.

ON 4=-

65

A problem simulating the above test was considered

in the analysis. The results of this analysis are also

shown in Figure 4-18. The boundary was assumed to be simply

supported with unrestrained in-plane movement at the edge.

The loading was assumed to be a concentrated load applied

at the center. Poisson's Ratio was taken as 0.3, Young's

modulus as 30,000,000 psi, the yield stress as 40,000 psi,

and n as 8. These material constants match those of the

test specimen.

The plate as tested was not perfectly flat and

required some initial load.before coming completely down

on the supporting ring. An analysis was made to determine

the effect of an initial load on the test results. Within

the early elastic range of behavior it is assumed that the

test should compare favorably with an elastic analysis.

The elastic analysis indicates that a load of 52 pounds

corresponds to a central deflection of 0.066 inches. A

test point at this same deflection shows a corresponding

load of 170 pounds. The initial load was therefore 118

pounds. Crose (1967) calculated the initial load as 124

pounds. All calculated results have been adjusted to take

into account the initial load.

Figure 4-18 shows a comparison of the load-deflection

results from calculations and the test. The calculated

values are higher than the test values over the load range.

Calculated lateral displacements are presented in Figure

66

4-19- An indication of the yielding occurring in the

material near the center is evident. Figures 4-20 through

4-24 show a comparison between tests and calculations for

radial slopes and strains. The test results are shown as

a range of values to indicate the spread of test data

points. The slight difference between the curves is in a

direction consistent with the difference in the load-

deflection curves in Figure 4-18.*

Calculated stresses at the top and bottom surfaces

are presented in Figures 4-25 through 4-28. Stresses for

two load cases have been redrawn from Crose (1967). The

stresses resulting from the two methods have the same

general shapes however,, no close correlation can be

expected because Crose assumed a sandwich plate with elastic-

perfectly plastic material. Stress distribution through

the plate thickness is presented in Figure 4-29.

Concentrated Load—Simply Supported Plate No. 2

Using the theory of plastic deformation from the •

theory of plasticity, Waghdi (1952) analyzed a simply *

supported, 24S-T aluminum plate for two load cases. Several

plate specimens, made of the same material, were tested and

a comparison was made between the theoretically determined

and the experimentally obtained values of deflection.

A 10 inch diameter plate with a thickness of 0.065

inches was chosen. Young's modulus was 10,000,000 psi and

67

3 4 5 6 7

r - inches Figure 4-19. Lateral

Displacement Profiles—Concentrated Load

2<>r

<o

/<s -

1z -I

a

TesJ- Results -from Cooper and Shifrin (1354)

Calculated Results *

Slope A 10 * — Inches/inch Figure 4-20. Radial Slopes—Concentrated Load

I

68

15

I to

Q. 5

saws

4#1'

WMMTesf Results from Cooper and Sh/fr/n (IS54)

Calculated Results

x 10 — Inches/Inch (Tens/on)

Figure 4-21. Top Surface Radial Strain over the Support

esf Results from Cooper and Shifrin (1954)

Calculated Results

20

et a IO* — Inches/inch (>Compression)

Figure 4-22. Top Surface Circumferential Strain over the Support

69

<o

I

A5

to

* c o.- IffifE Results from Cooper

and Shifrin (1354) Calculated Results

a 12. /<3 .20

ef x 10* - inches/inch (Compression)

Figure 4-23. Top Surface Circumferential Strain at Station % Inch away from Support

/PP 7es"/" Results -from Cooper M0

and Shifrin (1354) Calculated Results J0? >

&r x 10s- Inches/Inch (Tension) »

Figure 4-24. Top Surface Radial Strain at Station •§• Inch away from Support

so

40

30

20

\

-JO

From Crose (/9G7) Calculated Results

-so O.G AO o.z r/a

Figure 4-25. Top Surface Radial Stress—Concentrated Load

ri

SO I-

40

- From Crose (1SG7) Calculated Results

30

20 4 /

X V

10

-feoj

\ \

V

<9

0.2 0.4 O.G

r/a 0.8 AO

Figure 4-26. Bottom Surface Radial Stress—Concentrated Load

%

72

20

—From Crose (OGl) —Calculated Results

3/9

-to

tots \

30

to 0.8 0.2 O

Figure 4-27- Top Surface Circumferential Stress—Concentrated Load

I

73

40

30

20

-to

From Crose. (t3G 7) Calculated Results

0.8 /.o o

Figure 4-28. Bottom Surface Circumferential Stress—Concentrated Load

lb

V \ \ v \ . \ > \ J \V x* J \\j5\ \ Vs /] \ //

y /

1 1 1 1 1 1 L I 30 20 10 O iO 20 30 40 SO

cj: - ksi crp. at r- 0.5 in.

40 30 20 20 30 40 SO CTf - ksi

(c) 07 a-f" r-o.sh tn.

L I _L J. J 30 20 IO O /O 20 30 40 SO

oy. - ksi (b) crp. at r~2.4in.

• V \ \ \ \w\s\® \te\(P \

1 1 1 1 1 1 1 1 J 30 20 10

(d) Of

O to 20 30 40 so Ot - ksi at r= 2.4 in.

Figure 4-29- Stress Distribution through the Plate Thickness—Concentrated Load

75

Poisson's ratio was taken as 0.3. The concentrated load

was applied to the center of the plate by means of a one

inch diameter steel rod. The two load cases were:

(1) Load P = 20 pounds for the plate to undergo

large elastic deformation only.

(2) Load P = 500 pounds for the plate to exhibit

large elastoplastic deformations.

The plate was analyzed with the method contained

herein using a yield stress of 69*000 pSi and an n of 2.

Figure 4-30 contains load-deflection curves from this

analysis, from the test, and from Naghdi's analysis. Both

theoretical methods show good agreement with the measured

deflections.

Figures 4-31 and 4-32 contain graphs of forces and

moments for the two load cases as calculated by the two

theoretical analyses. Poor agreement between the methods

is evident. Using the method described herein, it was

found that for load case (1), where P = 20 pounds, the

plate material remained almost perfectly elastic. The

maximum change in the modulus of elasticity was only 1.2#

near'the center. Because the method derived in this dis­

sertation has been shown to give excellent agreement with *

"exact" analyses for the case of linear material, and

because it also gives excellent agreement with the test

data for load case (1) in Figure 4-30, it is concluded that

the results of this method in Figure 4-31 are very close to

I

76

QJ -Q 0

1 *

Experimental Points from Naghcti 0352)

— Theoretical Results -from Naghdi (1352)

Calculated Results

\\p= soo

r — Inches

.Figure 4-30. Load-Deflection Curves for a Simply Supported Plate with a Concentrated Load—No. 2

\ f

77

to

8

"S

I

GO

40

20

8 :Q

I *

-20

-40

-GO

Theoretical Results from Naghdi (195Z)

Calculated Results

x JL 2 3

r - Inches

Figure 4-31. Results for a Concentrated Load of 20 lbs.

\

78

—Theoretical Results from Nacfhdi (!B5Z)

—Calculated Results so

40

M

30

20

/O 1000

, SOO

r — Inches

Figure *}— 32. Results for a Concentrated Load of 500 lbs.

79

the true values. The bending moments from Naghdi1s method

then are too large and the membrane forces are generally

too small throughout the plate.

CHAPTER 5

SUMMARY AND CONCLUSIONS

A numerical technique has been presented to analyze

thin structural circular plates possessing nonlinear

material properties without the usual restrictive assump­

tion of small deflections. The material is assumed to be

isotropic and homogeneous. The plate and its loading are

taken to be axisymmetric, thereby limiting the problem to

one space variable; however, the plate thickness, the

material properties, and the loading can vary in the radial

direction.

The method is based on finite element procedures.

The governing differential equations are derived directly

from the- plate element for large deflection geometry. The

element stiffness matrix is formed by utilizing the step­

wise linearized, differential equations, a fourth order

Runge-Kutta integration scheme, and a searching procedure

which establishes the appropriate boundary conditions in

accordance with the unit displacement method.

The nonlinear material properties are established

through the use of a three parameter stress-strain relation­

ship and a yield criterion. An iterative technique is used

to handle the geometric and material nonlinearity. Within

80

81

each Iteration the differential equations are stepwise

linearized and the material properties are held constant.

The load may be applied in one step or in increments if

intermediate solutions are desired.

Because circular plates are widely used, and because

they can support much larger loads when large deflections

and nonlinear material are permitted, the method of

analysis which was developed herein is applicable to many

important engineering structures.

Prom comparison with theoretically "exact" solu­

tions, the elastic results were shown to be valid. The

nonlinear material solutions, having no "exact" mathematical

development, were shown to compare favorably to experimental

results from several investigators. On the basis of the

studies made in this dissertation, a feasible method for

obtaining solutions to circular plates subject to large

deflections and nonlinear material has been developed and

presented.

REFERENCES

Birchler, W. D., M. L. Callabresi, and J. E. Murray, "Finite Elements Models for Shells of Revolutions, EES, Series Report No. 21, Engineering Experiment Station College of Engineering, The University of Arizona, 1968.

Cooper, R. M., and G. A. Shifrin, "An Experiment on Circular Plates in the Plastic Range," Proceedings of the Second U.S. National Congress of Applied Mechanics, Ann Arbor, ±95^, pp. 527-534.

Crose, J. G., "A Large Deflection Analysis Method for Transversely Loaded Elastic-Perfectly Plastic Circular Plates," Doctoral Dissertation, University of Illinois, 1967..

Den Hartog, J. P., Advanced Strength of Materials, McGraw-Hill Book Company, Inc., New York, ±952.

Goldberg, J. E., and R. M. Richard, "Analysis of Nonlinear Structures," Journal of the Structural Division, ASCE, Vol. 893 No. ST4, Proc. Paper 3fc>o4, August, 1963, pp. 333-351.

Hildebrand, F. B. Introduction to Numerical Analysis, McGraw-Hill Book Company, inc., .New York, 1956.

Hopkins, H. G., and W. Prager, "On the Load-Carrying Capacities of Circular Plates," Journal of the Mechanics ajid Physics of Solids, Vol. 2, No. 1, October, 1953* PP« 1-13.

McPherson, A. E., W. Ramberg, and S. Levy, "Normal-Pressure Tests of Circular Plates with Clamped Edges," NACA, Technical Report 7^-8, 19^2.

Naghdi, P. M., "Bending of Elastoplastlc Circular Plates with Large Deflections," Journal of Applied Mechanics, ASME, Vol. 19, No. 3, September, 1952, pp. 293-300.

Novozhilov, V. V., Foundations of the Nonlinear Theory of Elasticity, Grayiocit Press, Rochester, N.Y., J.953*

82 *

83

Onat, E. T. and R. M. Haythornwaite, "The Load-Carrying Capacity of Circular Plates at Large Deflection,11 Journal of Applied Mechanics., ASME, Vol. 23, March, 1956, pp. 49-55.

Richard, R. M., "A Study of Structural Systems Having Conservative Nonlinear Elements, Doctoral Disser­tation, Purdue University, 196I.

Richard, R. M., and J. E. Goldberg, "Analysis of Nonlinear Structures: Force Method, Journal of the Structural Division, ASCE, Vol. 91* No. ST6, Proc. .Paper 4^53, December, 1965* PP- 33-48.

Richard, R. M., and M. L. Callabresi, "Nonlinear Finite Element Analysis," Engineering Experiment Station Report No. 13, The University of Arizona, Tucson, Arizona, 1967 •

Richard, R. M. and J. R. Blacklock, "Finite Element Analysis of Inelastic Structures," AIAA Journal, Vol. 7* March, 1969* PP« 432-438.

Timoshenko, S., and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill Book Company, inc., New York, 1959•

Way,.S., "Bending of Circular Plates with Large Deflection," Transactions3 ASME, Vol. 56, 1934, pp. 627-636.

Wempner, G. A., and R. Schmidt, "Large Symmetric Deflec­tions of Annular Plates," Journal of Applied Mechanics, ASME, December, 1958* pp. 449-452.