HLAVACEK, I.; LOV~SEK, J.: Optimal Design of an Elastic Plate. I 377

ZAMM . Z. angew. Math. h'lech. 77 (1997) 5, 377-385

HLAVACEK, I.; LOV~SEK, J.

Optimal Design of an Elastic Plate with Unilateral Elastic Foundation and Rigid Supports, Using the Reissner-Mindlin Plate Model. I: Continuous Problems.

Einige optimale Design-Probleme werden untersucht, in denen das gemeinsame Zustandsproblem eine Variationsunglei- chung mit einem monotonen Operator ist. Die Ungleichung beschreibt eine Reissner-Mindlinsche Platte auf einseitig- elastischer Unterlage und mit einseitigen Pfeilern. Die Dicke der Platte und die Steifheit der Unterlage werden als Design- Variable geuiihlt. Fur alle Probleme wird die Existenz der Losungen bewiesen.

Several optimal design problems with a variational inequality with a monotone operator as common state problem are considered. The inequality represents the bending of a Reissner-Mindlin plate, resting on a unilateral elastic foundation and on some unilateral rigid piers. Both the thickness of the plate and the stiffness of the foundation play the role of design variables. The cost functionals include the intensity of shear stresses, and reactive forces on the piers or the weight of the plate. The solvability of all the problems is proved.

MSC (1991): 49A29, 65N30, 73K40

1. Introduction

We consider five optimal design problems for an elastic plate, the bending of which is described by means of the Reissner-Mindlin model. The plate rests on a unilateral elastic foundation and is supported unilaterally by a finite number of rigid piers. The role of design variables is played by (i) the thickness of the plate, and (ii) the stiffness characteristic of the elastic foundation. The design variables have to belong to a set of Lipschitz-continuous functions. Two cases of boundary conditions are considered, namely those of hard clamped or hard simply supported edges of the plate.

The cost functionals represent: (i) an integral of the intensity of the shear stresses, (ii) a norm of the deflection function, (iii) the compliance, (iv) a resultant of transverse reactive forces on the piers, and (v) the weight of the plate. In the last case, i.e., in the weight minimization problem, we introduce constraints, which express bounds for some mean values of the intensity of the shear stresses.

The state problem is modelled by a variational inequality, where the design variables influence both the coeffi- cients of the nonlinear monotone operator and the set of admissible state functions. On the basis of a general existence theorem for a class of optimization problems with variational inequalities, which was proved by the authors in a pre- vious paper [1], we prove the existence of at least one solution to four of the optimal design problems mentioned above. The last one, i.e. the weight minimization problem, is treated via a penalty method, and the existence of a solution is proved, as well.

In Part I1 of the paper we will propose and study approximate optimal design problems by means of so called mixed-interpolated finite elements.

1. Setting of the problems

Let us assume that the midplane of the plate occupies a given bounded and simply connected domain 52 c R2 with a piecewise smooth boundary. We denote the standard Sobolev function spaces by Hk(52) = Wk,2(.Q), k = 1, 2. Let the norm in H1(Q) be denoted by l l . l l l , and the seminorm by ] . I 1 , i.e.

For brevity, we use the following notations:

U,a = &/ax, ..P = zapa, (u, v ) O = J uv dx, n

the repeated Greek subscript implies the summation within the range (1, 2). Here t denotes the unit tangential vector to the boundary 852.

Assume that the plate occupies the domain

{. = ( X I , x2) t 52, 2 3 E (-t(x), t ( X ) > ) 1

378 Z-4MM. Z. angew. Math. Mech. 77 (1997) 5

where the half-thickness, t E Uid, is defined as

Uid = { t E C(n),l(SZ) (i.e., Lipschitz-continuous function): tmin 5 t(x) 5 t,,, , It,,l 5 C:, a = 1, 2) , tmin, t,,, CA are given positive constants, tInir, < tnla.

The Lipschitz-continuous functions are functions satisfying

/ t ( x ) - t(x’)l 5 Clz - 2’1 for all x, z’ E S Z , where C is a positive constant independent of x, 2’. For instance, all continuous, piecewise smooth functions (such as finite element functions) belong to them.

The assumption enables us to use the Arzeli Theorem at the end of Section 2 below. In addition to that, only “mild” changes of the thickness function (Le., small c“,) are in accordance with the derivation of the plate model from the 3D-elasticity.

Let w E HA(&?) be the deflection function (positive upwards), /3 E V, the rotation function, where V, = [HA(&?)]’ for a hard clamped plate, V1 = {p E [H1(52)]’: t . P = 0 on aQ) for a hard simply supported plate. We consider the small strain tensor components

eap = -- 2 z3(Pa,/3 +Pp,,, > ea3 = f (w,a - P a ) , e33 = 0 . (1.1) 1

We restrict ourselves to homogeneous anisotropic elastic materials, so that the stress-strain relations are

for all symmetric matrices (sap) with some positive constant Q, (E,B) is a diagonal matrix with positive entries. The loading of the plate is given by (i) the surface forces f = (0, 0, f)T, f E L2(D);

(ii) the own weight F = (0, 0, - 2 ~ t ) ~ ,

We consider several unilateral inner obstacles-rigid piers as follows. Given mutually disjoint subdomains QT = 0 for i # j and constants a, > tmir,, i = 1, 2 , . . . , N , we define the set of admissible

where w = const 2 0 is a given specific weight.

such that fit c 52, fi:n deflections,

~ ( t ) = {w E H ~ ( Q ) : w ( x ) 2 t ( x ) - up for a.a. x E a:, i = 1,. . . , N ) . (1.4)

Moreover, we consider a unilateral elastic foundation (nonhomogeneous and frictionless) acting on D \ Q*, where N

z=1 52* = u Qt, and having the stiffness characteristic z E Uld, where

U” ad = { z E C(O)J(fi\Q*): 0 5 z(x) 5 z,,, Iz,,I 5 c;, a = 1, 2 ) , and z,,,, C: are given positive constants.

u = (P, w) E V with the norm Let us introduce the notation: V = 6 x H,(52) , V* the dual space, [. , .] the dual pairing between V and V*,

I14 = (IIPIIB + lw17)1’2~

[Al(t)U, 4 = $ s t3cupysPa,pvy.s dx + s teup(w,a - P a ) ( 5 , p - ap) dx >

Let w = (a, 5 ) E V and define operators Al(t ) :V -+ V*, A2(z):V -+ V* for t E Uid, z E Uad by the relations

(1.5) Q n

[ A ~ ( z ) U, W] = J zw-5 d z , n\n*

where w- = min { w , 0) denotes the negative part of w. Moreover, we introduce the functional L ( t ) E V* by means of

[ L ( t ) , .I = (f - 2wt, 5 ) o ’ On the basis of the principle of virtual displacements we may define the

S t a t e P r o b l e m : Find u(e) = (P(e), w(e)) E K x ,X(t) , where e = ( t , z ) E U k x Uid, such that

[(Al(t) + Ad.)) 44, 27 - 4.11 2 [L(t) , 21 - 441 for all w E 6 x X(t) .

HLAVACEK, I.; LOV~SEK, J. : Optimal Design of an Elastic Plate. I 379

We will express statical constraints by means of the second invariant of the stress tensor deviator (intensity of shear stresses),

Assume that 12 plays a decisive role at the extreme fibers ( 2 3 = ft) of the plate or at the midplane ( 2 3 = 0). Inserting the relations (1.1) and realizing that 0,s vanishes for 2 3 = ft due to the symmetry of the stress tensor and oap vanishes for 2 3 = 0, we obtain that

for 2 3 = f t and

4 = IZZ(VW - P ) = 2 (E?l(W,l - PA2 + 42(w,2 - PJ2) for 23 = 0.

We define cost functionals of the following four types:

where 5 is any (fixed) function of HA(52) such that 5 = 1 on 52* a.e. and Z is an extension of the function z by zero on 52 , e = ( t , z ) and u = (P, w). *

Remark 1.1: The functional j , is the well-known compliance, i.e., the work of external forces. The functional j ,

Let us justify the definition of j , in detail. For any continuous w E X(t) , we decompose each subdomain Sz; into represents a resultant of transverse reactive forces on the piers.

the set

S,(W) = ( 2 E 52:: w(2) > t ( 2 ) - a,}

and its complement Jz(w) = Sz; \ Sz(w). Denote

N

Obviously, w = t - ai on Ji(w). We introduce the set 2 = ( 5 E HA(Sz): 5 = 1 a.e. on Sz*}, and obtain

Lemma 1.1: I f u ( e ) = (P, w) is a solution to the State Problem (1.8) and zf w E H2(52), then

j4(e, ~ ( e ) ) = - J ((teap(W,a -P,)),B + f - 2mt) d x , J(4

i.e., it has the same value f o r all 5 E 2.

P r o o f : At first, we show that

T E - t ( ~ , p ( w , ~ - P,)),p + Zw- - f + 2wt = 0 in 52 \ J(w) a.e. (1.11)

1" Consider a point 20 E S,(w). There is a ball B,(xo) c S,(w) and a nonnegative function rp E C ~ ( B e ( 2 0 ) ) , rp > 0 on a closed ball Bei2 (q), such that

w 2 t - a, + cp in B,(zo) a.e.

Hence for any 6 E C ~ ( B , ~ ~ ( l c o ) ) , we may find an E > 0 such that

w + EE 2 t - a, + p/2 in Be/2(zo)

Consequently, 5 = w + ~4 E X(t) .

a.e.

Substituting v = (P, 5) into the inequality (1.8), we find v - u = (0, E E ; ) , and

s t&ap(w,a - P a ) 6,fl dz 2 (f - 2w4 9 0 . B

Since the same inequality can be derived for -6, we have

J ( t ~ u ~ ( w , a -Pa) 6 , ~ - (f - 2mt) t ) dx = 0 R

for all 6 E Cr(Be/z(zo)). Integrating by parts, we obtain T = 0 in a* \ J (w) .

obtain from (1.8), that T = 0 in 52 \ 8*. 2" Consider a point 20 E Q\Q* . We may find B e ( q ) , and for any 4 E C?(B,(zo)) we set v - u = (0, 3~6) to

380 ZAMM. Z. ancew. Math. Mech. 77 (1997) 5

3" Integrating by parts and using (1.11), we may write

jd(e, u(e))= J T [ d z = T < d x = J T d x . Q J(4 J(4

Remark 1.2: Some results on the regularity of a solution to the obstacle problem (see [3 - pp. 139, 1421) can

These assumptions seem to be sufficient for the justification of the functional j 4 . We cannot prove, however, that justify the conjecture, that w E H2(SZ), provided 52 is convex and t E H2(SZt) for all i = 1 , . . . , N .

they are necessary.

Remark 1.3: Let < be the function from the definition (1.10). Then we have

j 4 ( e , u ( e ) ) = J < d p = f d p > O , I2 J ( U J )

where p is a nonnegative Radon measure, supp p c J ( w ) (see the Riesz-Schwarz Theorem [4]).

(1.8) to obtain that Indeed, let us consider 5 E H;(SZ), 5 2 0 a.e. Then w + E X(t ) , and we may substitute 2) = (p, w + 5) into

s Tcdx 2 0 . n

Consequently, T is a nonnegative distribution and j 4 ( el u( e ) ) 2 0. 0 Now we are able to define the main task of the paper, i.e. the

O p t i m a l D e s i g n P r o b l e m s : Find

et, = arg min ji(e, u(e)) , i E {I, 2, 3, 4}, e E Uad

where u(e) denotes the solution of the State Problem (1.8), and Ij;dd = uid X uid. We also introduce functionals

yl,(e, u) = (meas ~ ~ 1 - l J t2121(v~) dx - crd 2

G,

f o r j = 1, ..., s, and

y,(e, u) = (meas GI)-' Izz(Vw - p) dz - ri G3

(1.12,)

(1.13)

for j = s + 1 , . . . , M , where Gj are given subdomains of SZ and Qd, td are given constants. We define the set of admis- sible design variables,

and the following

W e i g h t M i n i m i z a t i o n P r o b l e m : Find

e* = arg min j5(e) , e e Sd

where j ,(e) = J ot d z .

n

(1.14)

2. Existence of a solution to the Optimal Design Problems

First of all, we consider a general class of optimal control problems. Let U be a Banach space of controls, Uad c U a set of admissible controls. We assume that Uad is compact in u. Let a reflexive Banach space V be given with a norm ((.II, and let V* be its dual with a norm 1 1 . 1 ( * and [. , .] the duality pairing between V and V*.

Definition 2.1: We say that a sequence {K,}, n = 1 , . . . , of convex subsets of V converges to a set K , i.e.,

K = lim Kn ,

if the following conditions are satisfied:

n i m

(i) any w E K is a limit of a sequence {w,} such that v, E K, holds for all n, (ii) if v, E K, and w, - w (weakly) in V , then w E K.

HLAVACEK, I.; LOV~SEK, J. : Optimal Design of an Elastic Plate. I 381

Theorem 2.1: Assume that a system { K ( e ) } , e E Uad, of convex subsets of V is given such that

{e, -+ e in U, erL E Ud} + K ( e ) = 11-00 lirn K(e,) ; (2.1)

a system { A ( e ) } , e E Uadr of operators A(e ) : V -+ V* is given, satisfying the following conditions:

[A(e ) u - A ( e ) v, u - v] > 0 (2.2)

forall u, v E v u f v , and for alle E Ud; lim [A(e) (u + s(v - u)), w] = [A(e) u, w]

s-o+

for all u, v, w E V and all e E Uad;

{v E v llvll I c> =+ 114.) vlI* I c1

for all e E uad , where o ( e ) E K ( e ) such that

is independent of e; there exists a function r : [O, co) -+ IR and for all e E Uad, there exists

lirn r ( s ) = +co, n - t o o

with C independent of e and

for all u E V Let a functional f E V* and a continuous operator B : U -+ V* be given. For any e E Uad consider the following variational inequality: u (e ) E K ( e ) ,

[ A ( e ) u (e ) , v - u(e ) ] 2 [f + Be, v - u(e)] for all v E K ( e ) .

Let a functional j : U x V -+ IR be given such that

{en + e in U and u, - v (weakly) in V} * lim inf j (enl v,) 2 j ( e , v ) . n-o3

Then (i) there exists a unique solution u (e ) of the problem (2.7) for any e E U d ; (ii) there exists at least one solution of the following optimal control problem: Find

eo = arg min j ( e , u ( e ) ) . e c U d

P r o o f of (i) follows from [2, chapt. 2 , Theorem 8.21. The assertion (ii) has been proven in [ l , Theorem 1.11.

We will apply Theorem 2.1 to the proof of the existence of solutions to the Optimal Design Problems (1.12i)1 i = 1, 2 , 3, 4. Let us verify the assumptions of the Theorem 2.1, defining

u:= c(G) x c(G\Q*) , A ( e ) := Al(t) + A ~ ( z ) ,

v := V, K ( e ) := V, x X ( t ) , e := (t, z ) , Ud := u i d x Us.

Lemma 2.1: For any t E qd the set X ( t ) , defined in (1.4), is a closed and convex subset of HA(sZ), and

{t , + t in c(G), t, E u:,> + X ( t ) = lim X(t , ) . n-cc

P roof : The closedness follows from the Lebesgue Theorem. The convexity is immediate.

Let t , E u i d , t , + t in c(G). There exists a V, E co(G), such that 0 5 V, 5 1 in $2, and V, = 1 for all II: E Q*. For any 7u E X ( t ) , we construct a sequence

w, = w + V, Iltn - ill,.

IIWn - wIl1 = JIG7 - tll, 11V,l11 -+ 0 .

Then w, E HA(52), w, 2 t - ai + (t, - t ) = t , - ai holds for &.a. J: E Q$, so that w, E X(t,) . Moreover,

Next, let w, E X(t,), w, - w (weakly) in H:(Q). Using the Rellich Theorem, we have w, -+ w in L2(Q) ; wn 2 t , - ai a.e. in 52:. From the Lebesgue Theorem, w 1 t - a, follows a.e. in a$, so that w E X ( t ) .

Corollary: From Lemma2.1 and the closedness of b5 in [H'(52)]2, at follows that the system { K ( e ) } , e E Uad,

satisfies (2.1).

382 ZAMM . Z. angew. Math. Mech. 77 (1997) 5

Lemma 2.2: The system of operators { A ( e ) } , e E uad, satisfies the assumptions (2.2) till (2.6).

P r o o f : From [5-Lemma 1.31 we have

for all u, w E V. Since

(u- - b - ) (a - b) 2 (a- - b-) ' ,

we may write

(2.9)

(2.9a)

(2.10)

The sum of (2.9) and (2.10) yields that

[ A ( e ) u - A ( e ) w, u - w] 2 CIIu - w1/' (2.11)

holds for all u, w E V, with C independent of e E Uad. Consequently, (2.2) is fulfilled. Since (2.9a) implies that

(2.12)

(2.13)

is readily seen, with C independent o f t E ULd. As a consequence of (2.12) and (2.13), the condition (2.3) is verified. Since Al( t ) 8 = Az(z) 8 = 8, we can set u = 8 in the estimate

l lA(4 u - A ( e ) wIl* I C1lIu - 41 , (2.14)

where Cl is independent of e E Uad, to verify (2.4).

deed, then (2.5) follows from (2.11) and (2.14), where we set u = 8. Since there exists a k E X ( t ) for all t E U:d, we can set u ( e ) = (0, k ) and ~ ( s ) = Cs - C1 Ilu(e)ll in (2.5). In-

The following estimates are easy to obtain:

(2.15)

(2.16)

(2.17) and (2.6) follows.

Remark 2.1: We have proved that the system { A ( e ) } , e E Uad, is uniformly strictly monotone (2.14) and uni- formly Lipschitz continuous (2.14).

Lemma 2.3: The cost functionals j , ( e , u), i = 1, 2, 3, 4, satisfy the condition (2.8).

P r o o f : It is not difficult to realize that j1 is convex in V for any e E U d , being a sum of quadratic functionals. Since it is also continuously differentiable, it is weakly lower semicontinuous. We may write

j ~ ( ~ , u n ) = j i ( e i ~ n ) + J n ,

where

I J ~ I I C I I ~ ; - t211c IIPnII? . If en + e in U and un - u weakly in V, then

IIt2,-t211C+0, P n - P i n & , IIPnIIl I C ,

so that J, -+ 0. We have

lim inf j 1 ( ~ , u,) 2 lim inf j1 ( e , u,) + lim J , 2 j l ( e , u) . n-cc n+cc n-+m

2 For j , the verification of (2.8) is immediate, since llwlll is a convex and continuous functional on HA(f2).

HLAVACEK. I.: LOV~SEK, J.: Optimal Design of an Elastic Plate. I 383

For j , we may write

j 3 ( e n , 'tt",) = [L(t) , %I + Jn , (J,( = 2/w(t, - t , w,)oI I Cllt, - tll, IwnI1 -+ 0 1

so that

limj3(en, u,) = lim [L( t ) , u , ] = [L( t ) , u] = j3(e1 u ) .

j,(e,, 21,) = j4(e, urn) + Mrn , 1 ~ n I = J ( t n - t ) EaB(wn,a -Pna) 5 , ~ dz + (20(tn - t ) + (2 , - 4 wi, C)ol

For j , we take a fixed 5 E 2 in (1.10) and write

I* I W n - tll, (Iwnl1 + IlPnllo + 1) + 112, - 211c llwnllo) -+ 0 1

since the norms IJw,/I1 and \lPnllo are bounded. Consequently, we have

liminfj*(e,, u,)>liminfjB(e, u,).

The weak convergence of w, and the Rellich Theorem yield that w, -+ w in L2(52). Consequently, we have

so that

l imjl(el u,) = &(e, u ) .

Let us define B: U -+ V* by means of

W e ) , 4 = -2(wt, 510 '

Then it is easy to see that B is continuous in U.

respectively.

thus obtain the following

Making use of the Arzel& Theorem, we can verify that both qd and U& are compact in C(n) and in C(fi \ Q*),

As a consequence of Lemmas 2.1, 2.2, and 2.3, we conclude that the assumptions of Theorem 2.1 are fulfilled. We

Theorem 2.2: There exists a unique solution of the State Problem (1.8) for any e E Uad. The Optimal Design Problem (1.12%), i E (1, 2, 3, 4) has at least one solution.

3. Existence of a minimum weight design

The existence of a solution to the problem (1.13) will be proved by means of a penalization method. Let us consider the penalized cost functional

M

j= 1 Jde, 4 = j 5 ( 4 + f l c [v,(e, 4 I + 7

where F is an arbitrary positive parameter. We define the

P e n a l i z e d O p t i m i z a t i o n P r o b l e m : Find

e, = arg min JE( e, u( e ) ) , e t wad

where u(e) is the solution of the State Problem (1.8).

Proposition 3.1: Let e, -+ e in U , e, E Uad, as n -+ co. Then

114%) - 4e)ll + 0 . P roof : For brevity, let us denote u, = u(e,) . We may follow the proof of [l, Theorem 1.11 to derive the weak

convergence

u,- u(e) in V. ( 3 4

384 ZAMM Z. angew. Math. Mech. 77 (1997) 5

As in [l, (1.18)] we derive that

lim [A(%) u,, u, - 74 = 0 .

Next, we have (3.4)

[A(e,) u, u, - u] = [ A ( e ) u, u, - u] + [A(%) u - A ( e ) u, u, - u] -+ 0 , (3.5)

using (3.2) arid (2.6). Inserting (3.4) and (3.5) into (3.3), we are led to the strong convergence u, --+ u ( e ) in M. Lemma 3.1: Let e, -+ e in U, e, E Uad, as n --+ 00. Then

rc lim - 00 [ W j k , 4 % ) ) l + = [w,& 44)1+,

[W&, u(en)l+ - IVj(en+))l+l

j = 1,. . . , M

P r o o f : We may write

I (meas Gj)-' J (tLax Ih(VP,) - I21(VP)/ + It: - t21 IIa(VP)I + 1122(Vwn -P,) - 122(Vw - P I / ) dz , GJ

where P, = @(en), wlZ = w(e,). The latter integral has the upper bound

C{llP, - Plll (IIPnll1 + IIPII1) + Ilti - t211c + (IltJW -PI10 + IIVw, -P,llo) (IIV(wn - w)llo + IIP, -Pllo)} 7

which tends to zero by virtue of Proposition 3.1.

Proposition 3.2: The Penalized Optimization Problem (3.1) has a solution for any E > 0.

P r o o f : The functionals jS(e) and [ y , ( e , u ( e ) ) ] + are continuous in u a d on the basis of Lemma 3.1. Since the set

Theorem 3.1: Assume that Sad # 8. Let { e E } , E -+ O+, be a sequence of solutions to the Penalized Optimization Problems (3.1), {u (e , ) } the sequence of solutions of the State Problem (1.8).

Then there exists a subsequence { e:), E -+ O+, and e* E S a d , such that

U,d is compact in U, there exists a minimizer in Uad.

eg 3 e* in U , llu(eg) - u(e*)11 3 0 , (3.61, (3.7)

where e* is a solution of the weight minimization problem (1.14).

P r o o f : Since Uad is compact in U , there exists a subsequence such that (3.6) holds with e* E Uad. Then (3.7)

Let us show that e* is a solution of (1.14). By definition, we have

follows from Proposition 3.1.

M

j = 1 j,(ea) + ( ~ l C [vJ (eE , u(ea))I+ 5 js (e )

for any e E Sad . Consequently, nr

3 = 1 c [WJ(eE, +dl+ I c i5 (e ) .

c lW,(e*, u(e*))l' 5 0

Passing to the limit with E 3 O+ and using Lemma 3.1, we obtain that

A1

1'1

and e* E Sxl follows. From (3.8) we obtain

j 5 ( G ) I j 5 ( 4

j d e * ) I j 5 ( 4

for all e E Sd. Passing to the limit with E 3 Of, and using (3.6) and the continuity of j , in U , we arrive at the inequality

for all e E &a. Hence e* is a solution of (1.14).

Corollary 3.1: If S a d # 8, there exzsts at least one solution of the weight minimization problem (1.14).

P r o o f : follows from Proposition 3.1 and Theorem 3.1.

Acknowledgement: This research was supported by a Grant No 201/94/1067 of the Grant Agency of the Czech Republic.

HLAVACEK, I.; LOV~SEK, J.: Optimal Design of an Elastic Plate. I 385

References

1 HLAVACEK, I.; LOV~SEK, J.: Optimal design of an elastic or elasto-plastic beam with unilateral elastic foundation and rigid sup-

2 LIONS, J. L.: Quelques mkthodes de r6solution des problemes aux limites nonlin6aires. Dunod, Paris 1969. 3 RODRIGUES, J. F. : Obstacle problems in mathematical physics. North-Holland, Amsterdam 1987. 4 SCHWARZ, L.: Thhorie des distributions. Hermann, Paris 1966. 5 HLAVACEK, I. : Reissner-Mindlin model for plates of variable thickness. Solution by mixed-interpolated elements. Appl. Math. 41

ports. Z. angew. Math. Mech. 72 (1992), 29-43.

(1996). 57-78.

Received May 29, 1995, revised February 28, 1996, accepted June 27, 1996

Addresses: Ing. IVAN HLAVLCEK, Dr. Sc., Matematickf hstav AV CR, Zit& 25, CZ-11567 Praha 1, Czech Republic; Doc. Ing. Dr. JAN LOV%EK, Dr. Sc., Stavebni fakulta SVST, Radlinskkho 11, SK-81368 Bratislava, Slovak Republic

25 Z. angew. Math. hlech.. Bd 77, H 5