DNA 3954T,ELASTO-PLASTIC BEHAVIOR OF D 3954T

PLATES AND SHELLS

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ELASTO-PLASTIC BEHAVIOR OF PLATES AND SHELLS Technil..LAeport

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19 KEY WORDS (Continue on reverse sidu If necessary and Identify by block number)

Plates PlasticityShells Moment-Curvature Relation

20 ATRACT (Continue on reverse side If necessary and Identify by block number)

For an isotropic solid shell, a yield condition, a strain-hardening rule,and a flow rule have been formulated in terms of the membrane forces andbending moments and the strains and curvatures of the middle surface. Theresults of the proposed theory are -':mpared to the results of the through-the-thickness integration of the stress-strain relations of the material.

j I __________.....____________________FORM

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P REFACE

The help of Professor G.A. Wempner in this work is

mo st gratefully acknowledged. Professor F.L. IDiMacggio,

Dr. 1.8. Sandler and Dr. D. Rubin of fered numerous help1)f ul

suggestions.

4

.................................................

By ... .... . .

GISR~s JGNAALW COIT "EIA D

TABLE OF CONTENTS

Page

PREFACE .. .. .. ......... ..............

LIST OF SYMBOLS .. .. .. .......... ........ 3

I I NITIAL AND LIMIT YIELD SURFACES. .. .. .. ........ 10 .

III PROPOSED YIELD CONDITION AND HARDENING RULE .. .. ..... 14

IV PLOW4 RULE AND TANGENT STIFFNESS .. .. ... ........ 17

V EXAMPLES .. .. .. ........... .......... 20

VI CONCLUSIONS. .. .. . ............ ....... 28 -

REFERENCES .. .. .. . .......... ......... 29

2

LIST OF SYMBOLS

a, b, c parameters in the equation of the imit

yield surface Eq. (13)

h shell thickness (solid shell, Fig 2)

d 1dimensions of a sandwich shell (Fig. 2)

i, j = indices, i = 1, 2; j = 1, 2

e.. strains components of the middle surface

k.. = curvature components of the middle surface

$1 = see Fig. 2

e =column matrix of shell strains (Eq. (22))

e" = plastic part of e

s column matrix of stress res,itants,(Eq. (21))

t time

A expression defined by Eq. (31)

D = elasto-plastic tangent stiffness matrix

E = e astic moduli matrix

IFF0 subsequent, iiitial, and limit yield functions

FLI

s absolute values of the gradiets of the yield

M function F (see Eqs. (18) and (19))

I M resultant stress invariants (see Eqs. (6), (7), (8))1 NM

N ij,Nil,N 2 2 ,N1 2 membrane forces in shell

Mi , M ,2 , = moments in shell

1. = strain hardening parameters

M0 = 00 h 2 /6

2/ - for solid shellsML 0 0 h/4

M 0 M 0 td for ideal sandwich shellsL 0

3

LIST OF SYMBOLS (Continued)

= a parameter in the yield condition

a parameter in the hardening rule

U = a parameter in the flow rule

= Poisson ratio

CF. stress c#eponents130 0 yield stress in unlaxial tensionS.. = strain components

13

4I~

This report discusses the elasto-plastic behavior of

thin plates and shells whose material is an elasto-plastic

solid, linear in the elastic range, obeying Mises' yield

condition, and deforming plastically according to the

Prandtl-Reuss flow rule. The mechanical properties of this

solid are characterized by two elastic constants and the

yield stress in uniaxial tension, a0*

The plate or shell is assumed to be a solid layer, with

thickness h; occasionally, reference will be made to an ideal

sandwich shell (Fig. 1) such that d >> t. The basis assumption

of the plate and shell theory used in this paper are summarized

in the expressions for the strain components, at any point of

the shell in term, of the strain and curvature components of

the middle surface

3 z). = e.. + ki z (iZ i "

ii (1)

; and in the expressions for the membrane forces and the bending

moments (stress resultants)

h /2h/ 2

N.. cr. dz M a .i z dz (2)

-h/2 -h/2

It becomes evident from the above equations that, within the

accepted approximation, there is no distinction between a

plate and a shell, as far as the stress-strain relations are

5

*- 0 4

Q) 0

U) -

x x .1 >

0 0

(0U) 0

4

00+ #40)j 0

xx *~PA

concerned, and the term "shell" will be used to describe

)o I the strncture under consideration.

The strain - normal force and the curvature - moment

relations under the conditions of uniaxial stress (i.e. beam

behavior) are illustrated in Fig. 2. The specific objective

of this work is the development of the relations between the

stress resultants, Nij and Mij, and the strain components,

e. and k.., for a general type of loading.

The yield conditions for the simplified behavior in

bending have been proposed in several previous investigations;

a critical evaluation and comparison of the existing yield

surfaces can be found in a paper by M. Robinson (Ref. [1]). In

order to take into accornt the actual moment - curvature

relation, M.A. Crisfield (Ref. [2]), applies the ideas of

isotropic strain hardening of the I-eory of plasticity to the

elasto-plastic behavior of shells. '[he approach reported in

this paper is a continuation and exLension of the above

earlier works.

The fact that the stress resultants N and M.. of the

classical shell theory are not sufficient to describe the

A state of stress has been recognized by G. Wempner. In Ref. [3],

1 he introduced certain higher - order moments which, together

with the classical stress resultants, form the dynamic variables

of the problem. In Wempner's most recert work, described in

a private communication to this author, the stress distribution

through the thickness of the shell is expanded in te.ins of

,!7

M

SIMPLIFIEDN

No NL EXC

e0e kk

Fig. 2 Normal force vs. strain and moment vs. curvature in

uniaxial stress, i.e. beam behavior.

(1 8

Legendre polynomials, with the coefficients of this expansion

being the additional dynamical variables.

It is, of course, possible (and it has been done in

many investigations of elasto-plastic shells) to avoid

completely the problem of the shell constitutive equations.

In the "through-the-thickness-integration" approach, for

given increments of eij and kij, the increments of strains

.. (z) are determined with Eqs. (1); then, from appropriate4j

Aconstitutive equations of the material, the increments of

stresses ..(z) are computed; finally, the increments of shell2j

forces and moment, are determined by numerical evaluation of

the integrals in Eqs. (2). Although workable and accurate,

the procedure requires sometimes prohibitively large storage

capacity of the computer.

, 9

II INITIAL AND LIMIT YIELD SURFACES

The initial yield condition, or the initial yield surface

in the stress space, can be easily derived from our basic

assumptions. The stress components at the top and bottom

surfaces of a solid shell are

.. 6m..C . (3)ii h 2

where the plus sign applies to the top and the

minus sign to the bottom of the shell. Substitution of

expression (3) into Mises' yield condition

1( 2 + 2 0 +3 22 11 22 11 22 312) (4)

Go0

results in

F = I + I + 2 I = 1 (5)

where1N2 + N2 N N + 3 NI 2

(6)

N N2 11 22 11 22 12~ 60

I M 2 2 M 12 2 + 3 M2) (7)M o2(11 + 2 2 11 22 +3

I t MI + N M - 1 N - 1 N2 + 3 NIM (8)NM M 0N 1 1 11 22 22 2 11 22 2 22 11 12 12

- d 2d N = oh M 0 = U0h2/6 (9)0 0 0 0

When the expression (5) is used, the top or the bottom of

the shell should be considered in order to have the larger,

i.e., positive, value for + 2 I . This is assured

10

by writing the initial yield condition of a solid shell as

F 1N + I M + 2 11 NM 1 1 (0

In r' case of an ideal sandwich shell the stresses are

computed from

N.. M.= ... + -_..1ij 2t - dt

and an identical argument leads to the condition (10) except

that N0 and .0 are now

No = 0 2t , M =0 td (11)0 0 0

The physical meaning of the quantities N0 and M0 is

as in Fig. 2. The corresponding strains are e0 and k0,

respectively. For a solid shell

00 o2e0 k0 = '|- (12)

0 E0 Eli

No direct derivation, of the type used above, is possible

for the limit surface. Instead, approaches which are essentially

surface-fitting procedures have proven to be useful. Suppose

that the limit surface is represented by a linear equation in

IN , IM , and INM

F a I + b I + c I = 1 (13)L N it NM

In principle, the parameters a, b, and c should be

determined in such a way as to minimize th*- difference between

the surface (13) and the yield surlace determined by some more

precise calculations. For practical purposes, however, the

11i

following argument results in sufficiently accurate values

of a, b, and c. By considering the case of membrane forces

only, we find that with a = I Eq. (13) becomes the exact

2/2limit condition. Similarly with b = M0 , Eq. (13) will

produce the exact results for the cases of bending moments

only. It is now necessary to consider one more loading case,

with known exact solution, in order to find the value of c.

Such a case may be taken as corresponding to the maximum

value of INM* Upon reflection, it becomes evident that it

is Ni= N2 2 , MI M2 2 , N1 2 = 0, and M1 2 = 0.

The stress distribution in a section is then as shown

in Fig. 3, and the maximum of I corresponds to fl h/2/3.NM

A simple computation results in

3 ' 4ML /9M0 2 =2/s ML/9M0

IN L 0 M INM0

Equation (13) will be satisfied with the above values if

c = M0/M L v"

This form of F is identical with Iliushin's, (Ref. [41)L

yield condition obtained from different considerations.

For an ideal sandwich shell, the functions F0 and FL

coincide, since there is no distinction between initial

yield and limit state.

12

V I I ? .. . . .

I/A

* Li

00 It_______ N

Fig. 3 Stress distribution corresponding

to maximum I (with n = h/2 F )4NM

13

III PROPOSED YIELD CONDITION AND HARDENING RULE

The following yield condition is proposed to describe

the "subsequent" yield surfaces as the loading path moves

from the initial yield surface towards the limit surface

F I + I* + t IINM =14)N m (14)

where

* 1 *2 +2

M M [(Mll - M) + (M2 2 - M2 2 )(1

(15)

(M, -m MI (m22 - M2?) + 3(M 1 2 - MI2

The quantities M.., which will be referred to as "hardening

parameters", are defined by the following

If F = 1 and DN i + -MiMij > 0:

2

dMj = ( L) k 0 2 dk (16)FM

If F < 0 or DN +3 -M.. 3 0:~3N. ij mi ~ M. -j 0

d M.. = 0 (17)

The symbels F s and F,- are defined as

2 2 F 2Fs = [(NO -- F + (NO aN22(N1 1 22 12

(18)

2 aF 2 DF 2 1/2+F ) + (M 0 -) + (M 1 2

+111) + (Ml <

14

2 2 1/2

.=) + (M 0 -) + (M 0 .- ) ] (19)1122 1+M2

F is evidently the absolute value of the vector grad F, inS

a dimensionless formulation; FM is the part of grad F which

A corresponds to the bending moments only.

'TThe function F which appears in Eq. (16) is defined byL

Eq. (13). With the values for a, b, and c as determined in

the preceding section and with ML/M O 3/2 (for solid shells),

it reads

FL =I + 9IM + 2 IINMI (20)L N 9 3/-3

The above formulation contains two parameters, a and 3.

The parameter a should be variable. When the loading path is

still in the initial elastic range, its value should be

a= 2 to assure correct predictions of the instant of first

yielding. As the loading path approaches the limit surface,

the value of at should approach the value of c in Eq. (13).

It appears however, that sufficiently close approximations

to the exact results can be obtained with a constant value

aof X. Here, 2 has been used, i.e. the limit surface

is reproduced correctly while an error is accepted in the

initial yield surface.

The parameter 0 controls the moment - curvature relation

in the plastic range. Again, a constant value of 2, has

been found reasonably satisfactory for solid shells.

15

i I w- ., . , . .,. .. . ,. - . -- ,.. . . .

The hardening law represented by Eq. (16) is neither

isotropic nor kinematic. Its choice is motivated solely by

the fact that it reproduces fairly closely the actual

behavior of a solid shell in the plastic range. It reproduces

also the lowered yield point ("Bauschinger effect") which

4I manifests itself if the bending moment is reversed, the shell

fj unloaded and then loaded in opposite direction.

iJ116

U 16

IV FLOW RULE AND TANGENT STIFFNESS

To complete the formulation of the behavior of elasto-

plastic shells, it is necessary to state the elastic law

and the flow rule. For this purpose, the stress resultants

and the strain components of the shell will be represented

by 6 x 1 column matrices

s = { NIll N22, NI12' Mll M22' M12 1 (21)

e = j ell, e2 2 , 2e1 2 , kll, k2 2 , 2k 1 2 } (22)

The following elastic law is assumed

s = E (e - e" ) (23)

where the elastic matrix E is the usual shell stiffness

matrix relating the membrane forces N.. and the membrane

strai s e i , and the moments M ij and the curvatures ki

(its size: 6 x 6).

The associated flow rule is assumed for the plastic

strain rates:

as

lif F T(24)

F = and (-) s > 0;

e = 0

if ,FT . (25)

F <l i or (--)s < 0.s)

The symbol DF/3s stands for the column matrix

DF a F DF DF 3F DF DF

11 22 ,N12 Mll 'M22 '1M12

17

4 4

and superscript T indicates the trauspose.

The parameter X in Eq. (24) can be eliminated with the

aid of the condition F = 1, or

F = (F-) s + (-F,) ;* = 0 (27)

where

s* 0 0 0, 0, M1 , M2 2, M 2 } (28)

Fa F IF

- = { 0, 0, 0, 0 ' (29)

11 M2 2 M1 2

both being column matrices. Following a, by now, routine 44

procedure, one obtains

~FT e,DF, E e

i . .. . (30)T F)T T(s) E-- s A

where1 M0 F 2 .

O sA = (l -F ) - - (31) AL k 0F2

Substitution of Eq. (24) into Eq. (23) yields

sE( - ) (32)

or, with ) as in Eq. (30),

s D

where the elasto-plastic tangent stiffness D is given by

DFT EF

D = E[l -(~ ~ F.T T (3 4)

--) E - ( A-

It sould be noted that the stress-strain relation,

Eq. (33) i of rate type. A proper numerical procedure

18

should be uised in evaluating Eq. (33) for finite increments

Iof strin A'.. eadsrs hsrqieet, of.

course, in force for any flow type theory of plasticity.

'-i

.14

V EXAMPLES

In order to test the present theory, the effect of some

typical loading histories has been applied to the following106

shell: thickness = 1 in, Young modulus = 29 x 10 psi,

Poisson ratio = 0.3, yield stress in uniaxial tension,' 3

0 = 30 x 10 psi.

Figure 4 shows moment - curvature relation for bending

in one plane; the curvature kll increases from zero to

-2 -20.4 x 10 then decreases to -0.4 x 10 and increases again

-2to 0.4 x 10 The results of the present theory are shown

with continuous line. For the same strain history, the

computations have been performed with the use of through-the-

thickness integration (trapezoidal integration, 21 points

through h); the corresponding results are shown with broken

line in Fig. 4.

Figure 5 contairs a similar ca,.e of bending in one plane,

except that the maximum and minimum values of the curvature

-2k are double of those in Fig. 4, i.e. + 0.8 x 10 and

- 0.8 x 10 - 2 , respectively. It is evident that further in-

crease of maximum and minimum of k would not bring any new11

aspects of the m'oment - curvature relation, since both curves

approach the values Ml1 = ML for larger 1k,11

The interaction of membrane forces and bending momEnts

is obviously important in various problems of shell analysis.

Figures 6 through 9 show the effect of interaction between Ni1

and M1 1 . The loading histories in these figures are: first,

20

0

I IA

(n

-. 04 -03 0;', -.01 -000 ;01 -02 -03 -04

Fig. 4 Moment ve,'sus cur"-ture for bending in one

plane; k1 # 0, K2, k1 0, e1 e2 e 0;

continuous line: p-esent theory, broken line: through-

the-thickness in!' -,ration

21

ED

4-LI

NU)

COII

08 -. 6 -0 0-00 0 0 0< a C

Fi.5 Mmn eru uvtr orbnigi n

pln; k 0 k k 0

11 ~ ~~~ 22 1 2 1

Fig.nou 5lioet vresust curvatre, frorkendling; inroneh

the-thickness integration.

'~ 22

404

(D

0)

CD

N

O

Il)

01 -

Fi.6 Mmn esscrvtr o edn n

e- e -k 1022 12 22 1

-. 0 -. G -04 .02 -.0 .0 .0 .0230

Ii

- V

,4.

V)

, - .06 - ,04 -02 -00 .102 ,04 006 .0

N 01

Fig. 7. Moment versus curvature for bending and

extension in one plane; kl 0 , el= 0.50 e0

e22 1 e2 =22 = 12=0

.24

C,,)

L?

I

- -.

CD,.

I //

N

Ln.

-. 9 -.oS -.04 -.02 -00 .o2 .o4 . .o

< X1O"

Fig. 8 Moment versus curvature for bending and

extension in one plane; kl 0, el 1 0.75 e0 ,

Se22 e e2 k 22 k12 0

25

-- - -_-_-

CD

M11Vi)

SNIIf

08 -CO -006 0 -0 -4nO 0

X1)-

Fig 9Aoetvru uvtr o edn n

exenio inoepae I1 0 1 10 ~

e) 22 e1 2 k1

(D -26

the strain component ell is increased to ell = 0.25 e

(Fig. 6), el 0.50 e0 (Fig. 7), ell = 0.75 e0 (Fig. 8)

and ell = 1.00 e0 (Fig. 9). Then, with ell kept constant,

the curvature k is varied through the cycle from 0 to

-2 -2+ 0.8 x 10 and to - 0.8 x 10 -

. In Figs. 6 through 9,

the results of the present theory are shown in continuous

line, with the results of through-the-thickness integration

in broken line.

j

27

VI CONCLUSIONS

A theory of elasto-plastic behavior of plates and shells

has been presented in terms of the membrane forces and moments

and the strains and curvatures of the middle surface. The

structure of this theory is analogous to the classical theories

of plasticity of solids. It consists of a yield condition,

a strain hardening rule, and a flow rule. The concept of the

yield surface, as known in the classical plasticity, exists

here in the stress space of points (N N N M M ,k

1412, M21).

An examination of the test cases presented in this paper

indicates that the accuracy of the results of the present

theory will probably be acceptable in a large number of en-

gineering applications. .

There exists always the possibility of furth2r optimization

of the accuracy of this theory. It can be achieved by adjusting

the values of the parameters a and , by introducing integer

or fractional powers of the terms (1 - F L ) and F is/FM in the

strain hardening ruie (Eq. 16) etc. Finally, some thought Lshould be given to comparing the predictions of approximate

computations not to some other theoretical results (even if

they are "exact") but to realistic experimental data.

28

J __ 28

REFERENCES

[1] M. Robinson, "A Comparison of Yield Surfaces for Thin

Shells", Int. J. Mech. Sci., Vol. 13, pp. 345-354,

1971.

[2] M.A..Crisfield, "On an Approximate Yield Criterion

for Thin Steel S'hells", Report 658, Transport and Road

Research Laboratory, Crowthorne, Berkshire, U.K.,

1974.

[3] G. Wempner, "Nonlinear Theory of Shells", presented at

ASCE National Environmental Engineering Meeting, New

York, 1973, Meeting Preprint 2095.

(4] A.A. Iliushin, Plastichnost', Gostekhizdat, Moscow,

1948, French translation: Plasticite, Eyrolles, Paris

1956.

to29

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32I__

DEPARTMENT OF DEFENSE CONTRACTORS DEPARTMENT OF DEFENSE CONTRACTORS (Continued)

Aerospace Corporation Physics International CompanyATTN. Prom N. Mathur ATTN: Doe. Con. for Fred M. Sanuer

2 cY ATTN: Tech. Info. Services ATTN: Doc. Con. for Larry A. BehrmannATTN: Doe. Con. for Robert Swift

Agbablan Assoc.aes ATTN: Doe. Con. for Dennis OrphalATTN: Al. Agbabian ATTN: Doe. Con. for Tech. Lib.

ATTN: Doec. Con. for Charles GodfreyApplied Theory, Inc.2 cy ATTN: John G. Trullo R & D Associates

ATTN: J. G. Lewis

Bat'lic Memorial In'tltute ATTN: William B. Wright, Jr.ATTN: R. W. Khangesmith ATTN: Tech. Lib.

ATTN. Henry CooperThe BDM Corporation ATTN: Cyrus P. Knowles

ATTN: A. Lavagnino ATTN: Harold L. Brode

The BDM Corporation Science Applications, Inc.ATTN: Richard llengley ATTN: Tech. Lib.

The Boeing Company Science Applications, Inc.ATTN: R, If. Carlson ATTN: I. A. Shunk

California Research & Technology, Inc. Southwest Research InstituteATTN: Tech. Lib. ATTN: A. B. \Venzel

ATTN Ke.KroenhgenStanford Research InstituteGeneral American Transportation Corporation ATTN: Burt R. Gasten

General American Research Division ATTN: George R. AbrahamsonATTN: G. L. Neldhardt

Systems, Science & Software, Inc.General Electric Company ATTN: Tech. Lib.TEMPO-Center for Advanced Studies ATTN: Donald R. Grine

ATTN: uASIAC ATTN: Robert T. Ai!cnATTN, Thomas D. finty

liT Research InstituteATTN: Tech. Lib. Terra Tek, Inc.

ATTN: Sidney GreenInstitute for Defense Analyses

ATTN: IDA, Llbrarian, Ruth S. Smith TRW Systems Group 'ATTN: Tech. Info. Cir., S-1930

Kaman AvidyneDivision of Kaman Sciences Corporation TRW Systems Group

ATTN: E. S. Criscione San Bernardino OperationsATTN: Norman P. Hobbs ATTN: Greg lialcherATTN: Tech. Lib.

Weldlinger Assoc. Consulting EngineersKaman Sciences Corporation ATTN: Melvin L. Baroi

ATTN: Gunning Butler, Jr. ATTN: Dr. M. P. BienlekATTN: Library ATTN: Dr. J. R. Funaro

Lockheed Missiles & Space Co., Inc. Weidlinger Assoc. Consulting EngineersATTN: Tech. Lib. ATTN: J. Isenberg

Martin Marietta AerospaceOrlando Division

ATTN: Al Cowan

Merritt Cases, IncorporatedATTN: Tech. Lib.ATTN: J. L. Merritt

Nathan M. NewmarkConsulting Engineering Services

ATTN: Nathan M. Newmark

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