Subdomain bounding technique for large scale shakedown analysis

Compurers & Stnrcrures Vol. Printed in Great Britain.

30, No.

4, pp. 883-886.

&MS7949188 $3.00 + 0.00 Pergamon Press plc

1988

SUBDOMAIN BOUNDING TECHNIQUE FOR LARGE SCALE SHAKEDOWN ANALYSIS YUEHENG Xut and HAIWAN YANG$ TDepartment of Mechanics, Peking University, Beijing, The People’s Republic of China IDepartment

of Mechanics, Tianjin University, Tianjin, The People’s Republic of China

Abstract-A technique is proposed to handle large scale non-linear mathematical programming shakedown analysis and it is illustrated by an example. The idea is to split the original structure several substructures, each of which can be dealt with independently. Upper and lower bounds of the load factor of shakedown for the original structure can be. obtained from similar load factors for of the substructures. The Melan theorem and a hybrid finite element are used in the formulation discretization. Only elastic-perfectly plastic material is considered.

usually much larger than that which the available MP algorithm can deal with, even if we adopt a piecewise linearized yield surface and to formulate the problem as a linear programming, as in [3]. A technique is proposed here to overcome the above difficulty.

NOTATION limit load factor of shakedown for the original structure yield function, Q(u,~) = f(ui, - $J~~~,~)(u,,- f0~6,~) -uf elastic stress field with load factor 1 = 1 self-equilibrium stress field

SUBDOMAIN BOUNDING THEOREM

the set of first kind of self-equilibrium stress field in U the set of second kind of self-equilibrium stress field in U the Kth subdomain or substructure of the body V the optimum load factor of shakedown for the Kth subdomain V, when P,~is restricted to w(V,) the optimum load factor of shakedown for the Kth subdomain V, when pi, is restricted to iI the vector of nodal displacement, its virtual variation

Definition 1: A stress field pij(x) in the domain U belongs to the first kind of self-equilibrium stress field in the domain V and will be denoted as: Pij(x)Ea(U),

if (i)

INTRODUCTION

(ii)

The main purpose of a shakedown analysis is to find out the limit range of cyclic loading within which the elastoplastic body shakedowns to an elastic state, i.e. there will be no further plastic deformation after several cycles of loading. Now we consider a body subjected to a single parameter load Pi(x, t) = ,l(t )P,(x). In terms of the Melan theorem [ 11, a mathematical programming can be formulated as [l, 21 1,

Subject to

4 [&r;(x) + p,(x)] < 0, V1E[O,i,],

XEV,

pii,, = 0 piin,=

in U

(2)

0nau.

(3)

Definition 2: A stress field pii in the domain U belongs to the second kind of self-equilibrium stress field in the domain U and will be denoted as: Pij(x) E W”),

if (i)

Maximize

for into limit each and

Pi,., =0

inU

(4)

(5)

(1) Theorem: If & is the shakedown limit load factor of the V, as defined by the optimum objective value of(l), and the domain V is divided into n subdomains V,, V2, . . . , V., then

where a;(x) is the elastic solution for A(t) = 1, and pii belongs to the set of self-equilibrium stress field

in the body V and plays the role of variables in the MP formulation. The main difficulty encountered in the MP approach of shakedown analysis is that, when the body is discretized by FEM or by other method, the number of variables in the MP formulation is

min{l,,l,,...

,~,}<1,~min{~,,~,,...,K}

(6)

where 1,) AZ,. . . ,A, and p,, p2,. . . , pn are defined as the optimum objective values of following MP

883

YUEHENGXu and HAIYUAN YANG

884

I

+r

0

Fig. 1

problems which are formulated independently

in each subdomain

maximize 1, P,,EO(Vk) Subject to

4[Aat(x) + pij(x)] G 0, x E Vk

VA E[O, &I,

(7)

THE HYBRID FINITE ELEMENT DISCRETIZATION

The e:(x) and pij(x) in formulation (1) are constructed by the hybrid finite element discretization described here. g:(x) is obtained by the well known procedure of elastic analysis. pii is constructed by the same stress mode in each element as that for a:(x), that is

maximize pk P,,EWVk) Subject to

IPI = [Pl{~l.

4[lo$(X) + p,(x)] < 0, VP E[O,Pkl,

XE Vk.

@I

Proof: Let p,(x) in each subdomain be the solution fields of problems (7) when x is in that domain, then there is 4P~O,(x) + p,,(X)] < 0

for x E V,

The self-equilibrium requirement (3) for the first kind of field and (5) for the second kind of field are satisfied in a variational sense here. From the principle of virtual work the following equation holds for the self-equilibrium field p,-(x):

Pij

i E [O, A,],

where

ib [~ij+ u,,[]da = 0,

L’ which can be transformed

A,=min{L,,&

,...,

into

A,},

thus we have L, > A,. If the body V can shakedown and the corresponding self-equilibrium stress field is rij(x) which is denoted by 7f)(x) in the subdomain V,(k = 1,2,. . . , n), it is obvious that JI

(9)

7!?n & co, U I

pijn,6ui ds = 0

dc

Table Radius Numerical results?

avk

I.

a=5cm,b=lOcm

a =45cm,

1256.3 1238.8 (kg/cm*) 1242.5

p, =

b =50cm

181.2 Wcm*)

Pa=

0.697% -0.707% -0.414%

189.7 -4.452% 0.014%

thus

Relative errors

it follows from definition that

t Different working parameters in SCDD algorithm produce different results. The analytic solution for thick walled tube is [S]:

p(k2.&=1,2

,...,

n), p,=min

thus ~.“~~,=min{~,,~,,...,/l,}.

u,ln-,

bu a

AC,-2

a2 b2

p, = 1247.7 kg/cm2 (a = 5 cm, b = 10 cm) p, = 189.6 kg/cm2 (a = 45 cm, b = 50 cm).

Subdomain

bounding

technique

for large scale shakedown

analysis

885

or in a discretized form: ., p,,n,bu, ds = 0.

(10)

= 11

k=l

?q

If 6ui is interpolated by its values in the nodes of FE mesh and the au, in each node is independent, then eqns (9) and (10) provide the first kind of self-equilibrium field:

If Sui is independent in each inner node of the domain and constitutes a virtual rigid motion on boundary, then eqns (9) and (10) provide the second kind of self-equilibrium field: P~,(x)ER(V). This is illustrated by an axisymmetric problem as follows. If 6ui is independent in each nodes, we will obtain a linear constraint equation from eqn (10)

I Fig. 2.

I’ a,,

...

Now, if we replace all the rows corresponding to the virtual displacements of boundary nodes in z direction in eqn (10) by one row which is the sum of these rows, then a linear constraint equation follows from (11) [b,,

.‘.

bl”l

Vl

set the reference load to be P” = 0.32 kg/cm*, then 1, = 2404.5, p4 = 2513.5, and we find that, with the load factor A(r) = 14, the subdomains V, , V, and V3 are still in the elastic range

r”l

I: b,,

Equations (9) and (12) will give a self-equilibrium field of the second kind, because the unique degree of freedom of rigid motion in axisymmetric problems is the dislocation along the z direction. If formulation (1) is not large scale mathematical programming, we can solve it directly by the available MP algorithm. Numerical tests are given for two thick-walled tubes of different sizes and compared with the analytic solution (see Fig. 1 and Table 1). Throughout this paper, we have adopted the axisymmetric hybrid stress element AXH9C [4] for our numerical implementation. The yield function 4 is averaged at each element, as was done by Huang and Konig [5]. The general nonlinear programing code SCDD [6] is applied here. A pressure vessel is analysed here by the subdomain bounding technique. The vessel is shown in Fig. 2; the section under consideration for numerical analysis is divided into four subdomains, with 29 elements in all, as shown in Fig. 3. In order to calculate the elastic stress for A(t) = 1, P = PO, we

Fig. 3.

YUEHENG

886

Xu and HAIYUANYANG

Thus, we have PI 2 1, 2

P4

Let the shakedown limit load factor to be ;i, = $(A, + p,); we obtain the shakedown limit load P, = 1, PC = 786.8 kg/cm*, and the error bound due to the uncertainty is less than

It should be noted here that the su~omain bounding relation (6) is valid analytically, irrespective of the numerical method used to obtain A,, A,, . ..1 1n, PI>&, . * 3gn and 5, so long as 1, is obtained by the same numerical method rather than by the subdomaining technique. Of five numerical results for two test tubes in Table 1, two are slightly higher than the exact solutions, which should be explained by the fact that the very simple algorithm SCDD is not sophisticated enough to guarantee the results to be totally independent of the working parameters in SCDD. Thus in our later application of SCDD, a minor adjustment is made, which is reported in detail in 171,

REFERENCES

1. J. B. Martin,

Thus, the bounding

is successful for this problem.

2.

3. CONCLUDING

REMARKS

The bounding technique proposed in this paper provides a possible means to deal with large and complex structures and the numerical results obtained here are satisfactory. However, more practice is needed before this technique can be successfully applied to other types of structures besides the shelllike structures where the self-equilibrium stress fiefd pu(x) is weakly interacting among connected subdomains. The self-equilibrium stress field is constructed here by eqn (91, together with constraints (IO), and the requirement of completeness is satisfied only approximately; therefore, the numerical results tend to be conservative.

4.

5.

6. 7. 8.

Plasticity: Fundamentals and General Results. MIT Press, Cambridge, MA (1975). L. M. Kachanov, Foundation of the Theory of Plasticity. North-Holland (1971). L. Corradi and A, Zavelani, A linear programming approach to shakedown analysis of struct&& Cornput. Meth. Mech. Enana 3. 37-53 (1974). R. L. Spilker, imiroved hydrid stress axisymmetric elements including behavior for nearly incompressible materials. fnr. J. Nurner. Meth. Engng 17, 483-501 (1981). N. D. Huang and J. A. Konig, A finite element formulation for shakedown problems using a yield criteria of the mean. Comput. Meth. appi. Mech, Engg 8, 179-192 (1976). Y. Wan et al., A Collection of the Programs in Common use for Optimization. Worker Press, China (1983) (in Chinese). Y. Xu, The hybrid FEM for the shakedown analysis of axis~met~c pressure vessel. Master degree thesis, Tianjin University (1985) (in Chinese). W. Prager and P. G. Hodge, Theory of Perfectly Plastic So/i& John Wiley, New York (1951).