Effects of circular cutout on the plastic buckling of ring-stiffened cylindrical shell

CHINA SHIP SCIENTIFIC RESEARCH

CENTER

Effects

of

Circular

Cutout on the Plastic

Buckling of Ringstiffened Cylindrical Shell

Xu Binghan Vlan Zhengquan

CSSRC Report

June 1995

English version 95003

P. 0. BOX 116, WUXI, JIANGSU

Contents

page

Absttact i

Introduction i

Plastic buckling analysis of plate and shell 2

Finite element formulation 6

Numerical results 9

Conclusions io

ABSTRACT

The buckling behaviour of elastic-plastic shell structure under compressive loading is the results of a complex Interaction between geometrical and material nonlinearities. lt has received a great deal of attention in the past. but the

progress in this field has been slow due to the major odstacle of the "plastic buckling paradox" . in present paper, a modified method of the conven-tional incremental theory of plasticity is developed based on the exact linea-rization of the simplest flow theory for analyzing the plastic buckling behaviour of elastic-plastic shell structure. The validity and accuracy of the present method are verified by comparison with model tests and ordinary numerical

ana-lysis. The effects of circular cutout on the plastic buckling load of ring-stiffened cylindrical shell subjected to external hydrostatic pressure are

investigated by nonlinear finite element analysis in detail.

1. INTRODUCTION

The problem of cutout in shells is inevitable in the design and construction of nuclear pressure vessels, pressure hulls of submarines and aerospace structures The current design of shell with openings is only based on the general requirements for the reinforcement of such openings by the strength criteria and there Is no rational method existing for predicting the effects of cutout on the buckling load of a shell subjected to different types of loading. For the buckling of axially compressed cylindrical shell with cutouts,Tennyson El] and Starnes [2]

investigated the effects of unreinforced circular cutouts on the buckling behaviour of circular cylindrical shells by experiments. Almroth & Brogan [3] developed a two-dimensional finite difference scheme(STAGS) for the stability of elastic cylindrical shells with rectangular cutouts. Bushnell & Melter [4] presented the nonlinear finite element (BOSOR) results of several ring-stiffened

steel cylindrical shells with or without reinforced circular cutouts, which had been tested by the Chicago Bridge & Iron Company and the Los Alamos National Laboratory. lt is found from these experimental and numerical results that the

large reduction in the critical buckling loads of axially compressed cylindrical

shell occurred even for relatively small cutout, in the case of buckling of cylindrical shell with cutout subjected to external hydrostatic pressure, an important result of the nonlinear finite element analysis and celluloïd tests which was presented by Xu & an [5] shows that the effects of cutouts with small reinforcement on the buckling load of cylindrical shell can be neglected in elastic range. But the reduction in critical buckling lead of cylindrical

shell with cutout may be obviously taken place in elastic-plastic regions. Xu et al [61 published a rational approximate method to estimate the collapse load of ring-stiffened cylindrical shell with reinforced cutout subjected to externat

pressure based on the experiments which were carried out at China Ship Scientific Research Center. Until now there is no publication which indicates the effect of cutouts on the plastic buckling load of ring-stiffened cylindrical shell subjected to external hydrostatic pressure theoretically. So it is necessary to carry out the elastic-plastic analysis for predicting the effects of cutouts on the buckling load of cylindrical shell subjected to external pressure.

In contrast to the elastic buckling of shell structure,which has been inve-stigated extensively and a through understanding of its behaviour has been achieved, the progress of plastic buckling of shell is still very slow, although it has received a great deal of attention in past. lt was discovered that the plastic buckling loads of shell and plate calculated by the Simplest smooth yield surface flow theory was consistently larger than that buckling loads obtained by

tests. on the other hand, calculation based on the corresponding deformation theory, which is less accurate in the physical meaning comparing with the simplest flow theory, provided reasonably good agreement with test results. This results iii what has been termed the "plastic buckling paradox" and making a major obstacle to further progress in the plastic buckling analysis of plate and

shell structre [71.

In present paper, the difference between the deformation theory and the simplest flow theory is discussed and the "plastic buckling paradox" which causes the discrepancy in predicting buckling loads of shell and plate has been analyzed. A modified numerical method of conventional incremental theory of plasticity is developed based on the exact linearization of the simplest flow theory of plasticity. The results obtained by the present method are quite reasonable in analyzing the plastic buckling of plate and shell structure. For computational efficiency, the generalized plastic yielding criterion is repre-sented by the resultant stress and strain of plate and shell. The plastic flow rule which is consistently defined in each element is discretized by assuming the plastic deformation occurs only at the yielding nodes[8]. The elastic-plastic element stiffness matrix considering material hardening effects is ob-tained by eliminating the additional internal generalized degree of freedom. So

the difficulty to the numerical integrations through the thickness as well as over the area of each element can be discarded, as it usually have to be perfo-rmed at each loading step in the conventional finite element analysis. The

ef-fects of circular cutout on the plastic buckling load of ring-stiffened cylin-drical shell subjected to external pressure are investigated in detail.

2. PLASTIC BUCKLING ANALYSIS OF PLATE AND SHELL

The buckling analysis of elastic-plastic structure is concerned with the equilibrium of force in the deformed configuration of structure.For the nonlinear plastic buckling of plate and shell, except the elastic-plastic material linearities the phenomenon of buckling itself arises also from geometrical

-2--liiearities of the small strain with large rotation. The relative motion of the consecutive configuration is larger and the additional plastic strain due to the variation of the principal directions of stress deviator between the updated reference configuration and current configuration is no longer too small to be neglected.

in this paper, the two dimensional elastic-plasticity with nonlinear

iso-tropic hardening and Von Mises yield equation is adopted for plate and shell problem. Let be vectors containing the components of the stress and

strain tensor respectively.

{ }=

{u,»,,a}

(i)

{}1r r,-"}'

I .xx' _)y

The yield surface f=f({o}.{e})=O can he characterized by means of the

so-called equivalent stress and plastic strain

{a}'[si{cr}

¡P=JdJ'

The deformation theory emphasizes on a one to one relations of stress and

strain. {e}

[C}{o-}+C5[S]cr}

- E - E

E

where E is Yonug's

i

-v

O

V

i

O O

02(I+v)

moduie, V

uniaxial stress-strain curve.

(3).

Poisson's ratio,

E

the secant module of the

1 -0.5 0

[SJ

-0.5

1

0

(2)

0

0

3

where the function

UÇ')

is defined by the isotropic hardening law in terms

of

This implicates that the principal direction of plastic strain coincide with the principal diretion of stress deviator. Although the assumption is less

acceptable in the case of nonproportional loading but the additional plastic due to the variation of principal direction of stress deviator can be estimated.

1d

{de}

=

([C]+C[S]){da}+(C, -Cs)[S]{aì-r

(4)

a

c

E

E

where

E is

the tangent module of the uniaxial stress -- strain curve.

The simplest flow theory which can consider the effects of loading history and describe the relation between the plastic strain rate and current state of stress is more physically acceptable than the deformation theory.

{}=[c]{r}+Â1s]{a}

(s)

where

₂

is the plastic Lagrange multiplier, which can be obtained by the plastic consistency condition.

Nevertheless in the conventional structurai analysis the additional plastic strain increment due to the iotation of the principal axes of stress is always

regarded as too small to be neglected in each incremental step.

de}

_{= [C]{da}+c[SJ{o}

(6)

It results In that the increment of plastic strain reaching the bifurcation stress along with the normal direction of the plastic yèilding surface in the updated reference configuration. in fact owing to the relative rotation of-

the

principal axes of stress, the plastic strain increment should be determined corresponding to the normal direction of yieldeng surface between the updated reference configuration and the current configuration. See Fig. 1.

Using the configuration C,, as reference configuration, the structural vari-able field in the configuration C,, Is assumed to be known and the elastic predi-ction of the stress in the configuration C,,1 can be taken as

= {a} +[D]({a]

-{e'}

(7)

[D]=[CJ'

If the material Is yielded and the stress is beyond the elastic limit, the plastic strain as a function of the intermediate state of stress should be calculated by integrating the simplest flow theory of plasticity over the consecutive configuration C,, and C,, with the generalized midpoint rule.

=

{"},, +2[S1{a}

(8)

where,

Ç9 [0,1]

and the plastic parameter A can be determined by plastic yield surface constrains.

An intermediate midpoint value of plastic strain within the increment step is defined by the following linear interpolation

{?}

=(1-ç){e}

n it+L

={e'}+ça[s]{cr}

(9)

For equations (8) and (9) it has to append the elastic stress -- strain relations.

{cr} +[D]({e}_-{e'})

(io)

Substituting equation (9) Into equation (1O),we have

{tT}n*ç

{s

_{}+ ÇQFD1[S]{0i+ç;}

This leads to the foljowing expression

{c.r}

₌

([Il

.3[D](S]_1{OR}

(12)

The equivalent plastic strain may be determined in terms of stress and plastic strain tensor. Defining a plastic work function

4P=(P_P)_{a}t ({gJ'}

_-{}

₎

Il

Çt{a}t

[SJ{cr} CX

(13)

This gives the equivalent plastic strain expression

=

₊

₍₁₄₎

In the special case of Ç=1. the evolution of the stress and strain with the yield surface constrains in the configuration take the form.

{cr}

_{([I]+Â[D][S]y{o}}

e'}

{'}

+A[SJ{or}_n'-t

+1

=t& [S]cr}

L _Jn+I

=

fflf

i =

--HÇ1)= O

Then a nonlinear scalar equation for 2 is obtained by substituting the equa-tion (15e) and (lSd) into equation (15e) and it is easily solved by iteration method.

A modified method of the conventional incremental theory Is developed by differing the equation (15h)

{del' }

=

_dA[Si{

cr} +

2[S]{da}

(16)

where

the plastic Lagrange multiplier

dA

Is determined by the plastic consistency equation (dfO) and the

plastic

parameter ?

is determined by

the plastic yield

surface constraints (fO)

Introducing the plastic potential function for an associated plastic flow rule the yield surface and plastic potential are identical,the alternative expression of the present incremetal theory of plasticity can be formulated as

{d}=d. '

2

2{da}

4cx}

¿'{cr}

3 FINITE ELE?NT FORMULATION

In deformation path of a shell structure from its initial configuration to its final one,

C

and

C1

denote two consecutive intemediate configurations. Using

_C,,

as updated reference configuration, for the faceted representation of

the shell structure the incremental equation of the triangular hybrid stress element of

the elastic plate can be derived by

following the conventinal finite element method _[71.

[K]{Lq}

=

{AF}

{Aß}

=

-6

(17)

(

\ 18a) (1

Sb)

where

(AF}

denotes the incremental column matrix of the force applied to the element node,

sq}

thc incremental dsplacement of the element node and

{M}

the internal stress resultant at the element node. The constant matrix [Hj is associated with tli material property and the trial functions of the increments of

the internal stress resultant and the constant mattix [B] is related with the trial functions of the boundary displacement and surface traction. The tangent stiffness matrix

_R]

is composed of material component and Initial stress and

geometrical component.

[K]

[Kml+IKg]

[Km]

=

[B]'[H][B]

(19)

The part

[K] is a consequence of the incremental

response of the material while the part

[K5]

expresses the influence of the initial stresses in the deformed geometry of the structure.

Because the internal stiess resultant in each element are in equilibrium with the generali7ed nodal force the alternative expression of the incremental equation (18a) can be writen in the following form

IBI'ß[Xg]{q}

=

{AE}

(20)

ihis implies that the material stiffness matrir

_[Km]

is dependent on the material elastic plastic property and the geometrical stiffness matrix

[Ks] is

only function of the mechanical state in reference configuration.

Introducing the increments of the nodal strain

{As}

which is composed of the middle surface extensional strain

{As0}

and curvature

_{AK}

at the element

node, the increments of displacement and strain at the yeilding element node can be divided Into elastic and plastic parts by the linear decomposition of the strain and displacement increment

{Aq}= {Aq'}+{Lq}

(21)

Then the nodal plastic strain -- displacement matrix is same as the nodal elastic

strain - displacement

matrix, considering the equation (8b) the increments of the nodal strain can defined as:

1AI=*

[SI

b[Sj

b1[S]

a2[S]

8k

a=0.25t2(1_)exp(

₎

8'

exp(--

,) 3

K0

(23)

{A? }

=

_{[Bj{Aq' }}

(22

a)

{Ae"}= [BJ{Aq'}

(22h)

{} = [H]' {Le' }

(22c)

In order to aviod the numerical integrations of the plastic zone expanding over the thickness of the shell plate, basing on the results of Ref [14], [15], an approximate plastic yielding condition that is a direct function of the internal stress resultant, plastic strain and plastic curvature is employed for the whole section of shell plate

_{[141, [15].}

f =

H2()=O

2cr

K

_°Et

where a is the sign function of the coupled term of membrane stress and bending moment In the quadratic form of plastic yield criteron, the matrix EQI is a complete linear expansion for membrane stress and bending moment,

K"

denotes the equivalent plastic curvature, and o is initial yielding stress.

Following the plastic node method of Ueda et al. 181,present authors have proposed a pair of strqss resultant and plastic strain at each plastic yielding element node Instead of the nodal force and plastic nodal displacement. So the difficulty to calculate the transforming relations between the element stress and nodal force can be discarded.

It is postulated that the plastic flow at each nodal point of the element can occur only if the internal stress resultant at: this nodal point satisfies the equation (23) and the sunimtion of the plastic work increment done at the element yielding nodes is equal to the integration of plastic work increment doue ¡n the element domain. The plastic flow rule which is consistently defined in yielded element may he relaxed and expressed only at the yielded node, in fact, when the element stress state tends to constant strain distribution, the element elastic plastic behaviour can be determined by the nodal elastic pias-tic behaviour. Tue increment of plastic strain can he determined based on the prescrit method in equation (17)

{Ls'}= u{u}+2[vJ{Ap}

[V] = 2[Q}' [A][QJ

(24)

{u}

= 21V]{ß}

The equation (22c)can be rewritten as

{Aß} =[H]'({Lte}_{e"})

[H]({Lis}-A2{U}-2[V1{AuJ})

This leading to

= ([H]+ ,%[VJ)'

(25)

As long as the yielding element node i is under loading, the plastic

con-sistency condition must be satisfied at the yielding node.

4f = {ui{ß

K1A2 =

K, =2H2H'(1-21H')'

(26)

The elastic-plastic element stiffness matrix considering material hardening effect can be obtained by eliminating the plastic multiplier ¡n equation (19), (25), (26).

{JÇ]{L\q} ={LF}

{ß} = [HJB1{Ltq}

IKipl=EK,,ui+EKg]

(27)

[Ks,] = [B]'IH1'[B]

[H,,,,

1

= [H 1

- [H I'Eul([K I i(UJ'[If I1[UIY'[Ui'[H i1

4 NUMERICAL RESULTS

A nonlinear finite element program based on the outlined formulation has been developed. The solution algorithm is based on the modified incremental iterative scheme [101 which has been suggested to trace the nonlinear buckling deformation path including the bifurcation and snap-through by simply perpendi-cular decomposing of unbalance force in each iteration step.

The first. example Is a simpy supported square plate under uniformly distri-buted lateral loads, see Fig. 2. The obtained load---deflection curve is in quite agreement with the numerical results given by Ueda & Yao [81. Fig. 3. shows the plastic expanding progress of the yielding node obtained by the present analysis.

lt is seen that the plastic node is expanding along the diagonal of the plate

which coincides with the plastic hinge line predicted by the conventional limit

analys is.

As an illustration of the validity of the modified scheme of the conventional incremental flow theory for the nonlinear buckling behaviour, a simply supported square plate with or without circular cutout under uniform compression in one direction is investigated. The effect of plastic deformation on the nonlinear buckling load of the square plate is shown in Fig.4. lt Is found from the numerical analysis that the distribution of membrane force is not uniform before the plate. buckling occurs due to the effects of boundary condition and geometrical nonlinearity. The real distribution of the membrane force is shown in Fig. 5,which tends to the assumption of effective width iii the buckling plate with

load increased. The effects of cutout on the buckling load of the plate is given in Fig6. In elastic range, the present analysis is in good agreement with the experiment results by Zhang

Fill,

Kumai [121, Schlack [131. The critical buckling load of elastic-plastic plate is analyzed in this paper and hcrethe

plastic buckling paradox" is eliminated. The corresponding spread of plasticity around hole within the plate Is shown in Fig.7.

Fig.X. shows a simply supported circular cylindrical shell subjected to external pressure. The numerical results including the elastic analysis and elastic plastic analysis are presented. lt seems reasonably good for predicting the nonlinear buckling load by comparison with the model test and Von Mises formul a.

The final example is nonlinear buckling analysis of a ring-stiffened elastic plastic cylindrical shell with reinforced cutout subjected to external

pressure. In Fig.9. the parameter R denotes the radius of cylindrical shell, a denots the radius of cutout,t is cylindrical shell thickness,F is the effective section area of the reinforced cylinder [6], 1 is frame spacing, and L is frame

second moment of area.

The reduction in the critical buckling load due to the cutout is obtained at elastic-plastic stage.However, the numerical results show that the influence of cutout on the buckling load can be neglected for F/at>O.6,. From Fig. 9. it is quite evident that for a/R>0. 2 and F/at<0. 15, a 30 percent reduction in the critical buckling load can be expected.

5 CONCLUSIONS

The validity and accuracy of the present method are demonstrated by appli-cations to the buckl,ing of elastic-plastic shell structure. The "plastic buckling paradox" can be eliminated in the plastic buckling analysis of plates and shells by the present modified scheme of the conventional incremental theory

of plasticity. The large reduction in the plastic buckling load occurs for externally pressured ring-stiffened cylindrical shell with reinforced cutout, although the effect of a reinforced cutout on the elastic buckling load of ring-stiffened cylindrical shell subjected to external pressure can he neglected

[51.

REFERENCE

i Tennyson, R. C., The effects of unreinforced circular cutouts on the

buckl-¡ng of circular cylindrical shell under axial compression. Trans. ASME, J.Eng. for Industry, Vol. 90. l963.pp54l-546

2 .Starnes,J.El.,The effects of a circular hole on the buckling of cylindrical shells, Ph. D. ttiesis, 1970, California, lnst.of Technology, Pasadena, Calif. 3 Alsnroth, B. 0., Brogan, F. A. & Marlowe, M. B., Collapse analysis for shells

of general shape. Technical Report AFFDL TR-71-8, Calif.

4 Bushnell, El. _& Meller, E., Elastic-plastic collapse of axially compressed

cylindrical shell: a brief survey with particularapplication to ring-stif-fened cylindrical shells with reinforced openings, Trans. ASME, J. Pressure Vessel Tech., Vol. 106, 1984, pp2-l6

5 Xu, P. H., Van, Z. Q. & Xu, X., Nonlinear buckling analysis of circular

cylindrical shell with circular cutout, Proc. of the lot. Symp. on Marine Structures, shanghai, China, 1991, pp289-294

6 Xu,P. H.,Feì,J.H. & Zhu, B. J., Theory and experiment on shells with cutouts, National Defense Industry Publishing House, Beijing. 1987

7 Horrigmoe, G., Hybrid stress finite element model for nonlinear shell problems,

-mt. J. Humer. Meths. Eng., VoI. 12,1978, ppl8l9-l839

8 Ueda, Y. & Yao,T.,The plastic node methods: A new method of plastic analysis,

Comput. Meths. in Appi. Mech. and Eng., Vol. 34, 1982, pplO89-llO4

9 Wan, Z. Q., Plastic buckling of ring-stiffened cylindrical shell with various

cutout Ph.D. thesis, 1993, China Ship Scientific Research Center, wuxi, China 10 ran, Z. Q., Nonlinear finite element analysis of cylindrical shell subjected

to axial compression and lateral pressure, M. Sc. thesis, 1987. China Ship Scientific Research Center, Wuxi, China

11 Zhang,Y. L., Stability of perforated plates, M. Sc. thesis, 1981, China Ship Scientific Research Center, Wuxi, China

12 Kumai,T. Elastic stability of the square plate with a central circular hole under edgé thrust, Proc. First Japan National Congress AppI. Mech.,1951 13 Schlack,A. L., Experimental critical loads for perforated square plates, Exp.

Mech., Vol.8, No.2, 1968

14 Eidsheim, 0. M. & Larsen. P. K., Nonlinear analysis of elastoplastic shells by hybrid stress finite elements, Comput. Meths. in Appl. Mech. and Eng.,Vol.

34, 1982, pp989-1018

15 Crisfield,M. A.,Finite element analysis for combined material and geomerical nonllnearitie.s. Nonlineat Finite Element Analysis in Structure Mechanics, Proc.of the Europe-US workshop. Ruhr-Universtat Bochum Germany, 1980,

pp325-i)

E=U6OOO. O MPa

v=O..3

u=274.O

MPa

bJ2OU.0

mm

mm

H'=OO

(Jf t

1

/

Fig.1 ii-plane represtatlon deviatoric st.re

ard plastic strain

1J( In & inn

--Lowrr hound

Upper hound

Preient

nnflIY$19 I

0.05

0 1 0. 15

0 2

0.25

.:1

o. :s

0.1

-f-Fig.2 Load deflection ciive of plate urxiEr uniform presst.re

-q=.tlO

rftl'a

'k

/

b

Fi.3 Plastic zor

of plate ncr u1ffm

esire

E=205000.0 MPa

V=Q.3

o=274.0

M.Pa.

H'=O. 01E

_/(J,

b

"SIE

t

Fig.4 Nkinhinear ajckling stress 01

scjjar-e plate

13

-MPa

L:\

h

N

q=.3.Si MPa

q=4.22 Mfa

Elastic

\

\

dding

Plastic tjci<lirìg

4. 0

o

[3.

q=l. lt MI'a

{j3.9O lUPa

0.0

2.0

3.0

AA

rA r r

ArA

rA Arr rrr

0111d56.O MPa

L

1.0

0.8

0. 6

0. 1

0. 2

0. 0

I'

, ,/ /

VAAAVA A VArA rA rA

ArArÀrArArArArA

YA VA YA YA YA YA

AUYAYAYAYAYAYA

AYArAVAUUVAU

rArr.rrArrArrA

3=2.O

O.=163.O MPa

02

Fig.5 Plastic.zor

of plate uider tnlfcr caîqresslm

0. 4

Fig.t3 Effect of oJtout on the txJd(ling

_{ot square plaie}

-

14

-0. (i

0. 8

L £

_{'. ;}

.

A

AA

AVAYAVV

'AVAYÀYA

AYAYAYAVAYA

VA VA VArAVA VA VA

4

VA VArAVA VA

AYA

YAYA VA VAYA VA

VAVVVAV r VVA

12.O

u,=173.O MPa

o

No

q o

-\

o o o

15.O

N

N

N

o

Zhang

Schiack

b. O

7.0

6. 0 5. 0

4. 0

3.0

2.0

1.0

[

Fig.7 Plastic zone aro&nd hole whon Uie Lxjrjding occtrs

?

VV

44Ip4

h

VA

AVFAVA

/

1! /

¡

//

I.

V

/

I

tuses

I

CSS1Ctest

Elastic

Plastic

alysis

analysis

/

J

¡

0. 0

1.0

2 0

3.0

4.0

5.0

6.0

7.0

8.0

W ( nuzi)

Fig.8 Plastic budding

o

cyllixirical shell

_{jbject.ed to extErnal}

_Jr'e

15

-.IFa

mm

uim

mm

E=I9G000. O

v=O. 3

R=r1366. O

t= 9.8

L= 217.0

0.8

0.4

- 16

\\

_\

_N

-___

=o. a

\\

N

___ 4=o.

-

=o. i

n t

t

14.0

JRt

't

J_-70

1

0. 0

0. 1

0. 2

0.3

0.1

_0.5

II.

Fig.J effect or wtout. on the

_{Iastic buckling}

_of

_{ring-stjffj cylfrftj}