Numerical optimum design of elastic annular plates with respect to buckling

0045-7949184 s3.w+.oo

Compulers & Sfr!~fures Vol.18.No.2,pp.369-378. 1984 Printed inGreat Britain.

Pergamon Press Ltd.

NUMERICAL OPTIMUM DESIGN OF ELASTIC ANNULAR PLATES WITH RESPECT TO BUCKLINGt K.

RZEGOCI~~SKA-PEEECH and

Techiiical University

WASZCZYSZYN

31-155Krakdw. Poland 1982;received for publication

(Receired Abstract-Annular plates of divided into N segments

Z.

1983)

volume, under uniform radial edge compression

plate is

stress are assumed. types of objective functions are explored: fixed number design for equals max (inf(P’1). method is applied to compute the distribution

value of critical such a number axial forces distribution internal

state and the k

buckling I,

NOTATION

Main physical and dimensionless quantities are listed in the following: a, b external and internal radii of the plate /3 = b/a ratio of the radii D = Eh’/l?(l - v’) bending stiffness E Young’s modulus cp rotation angle of the normal to the plate middle plane dimensionless thickness of the plate ,y = h/h ho comparative thickness number of circumferential halfwaves h = a/ho slenderness of the plate 1 m, = 6Mr/(Eho-1. dimensionless moments mt = 6M,/(Eh2), mr,= 6M,,/(Ehd) n, = NAEho). 1 dimensionless axial forces nt = N,/(Ehn) I p = P/EL) dimensionless intensity of radial uniform compression PO, Ph uniform compression applied to the edge of radius r = a, b radial and modified shear forces respectively

1

(r,

t. 2) cylindrical coordinates um0, allowable and effective stresses respectively p= r/a dimensionless radius of the middle plane radial and transverse displacements respectively 1. INTRODUCTION

The history of the problem of optimum design of columns made of linear-elastic material is quite long. The simplest case of a column under constant axial compression was formulated by Lagrange between 1770 and 1773 and was then developed and analysed by many other authors [ 11. There are only few papers related to optimum design

tThis paper is based on an investigation sponsored by the Polish Basic Problem PW 05.12.

computed

plate of variable thickness. The optimal means of Rosenbrock’s method with design problems

of circular plates with respect to their buckling. In [2] an optimal shape of circular and annular plates was evaluated under the assumption of homogeneous stress distribution before buckling. The Pontryagin maximum principle and optimal control methods were used in [3] but computations were carried out only for the simplest cases of circular plates. The results of one-parameter optimum design of annular plates were reported in [4]. In the presented paper the optimal distribution of the thickness of elastic, annular plates is considered. Plates are under uniform radial compression applied to one of the edge contour. Essential difficulty of optimum design of such a structure lies in the lack of appropriate formulae for stresses and critical load which cannot be expressed explicitly for plates of variable thickness. Thus, both the objective function and stress constraints are to be assumed in the implicit form. In the paper a numerical approach is proposed. The shooting method is applied to analyse the prebuckling state and to compute the basic critical load. Nonlinear programming is used for the evaluation of a finite number of the plate thicknesses treated as control variables. The following basic assumptions are adopted: (1) The plate is ideal (without imperfections) and thin, i.e. on the basis of the Kirchhoff hypotheses all considerations are referred to the middle plane. (2) The linear theory of plates is valid. (3) Material of the plate is ideally elastic, isotropic and homogeneous of coefficients E, v and allowable stress a,. (4) The plate is divided into N number of annular segments. The middle plane of the segment k is bounded by circles of radii rk = a&. and rk+l = aA+1 respectively, as it is shown in Fig. 1. The thickness h(p) = x(p)hO is assumed to be constant or varies linearly within every segment k: X=

xk+l ~

A+1

-Xk

-

pk

7

for 7 E [a, Tk+l].

(1)

(5) The plate is subject to external uniform radial compression P (Fig. 2) which produces axisymmetric plane stress up to the buckling in the Euler sense. In the

370

K. RZEC~I~KA-PEFXH and Z. WASZCZYSZYN The primary problem of the optimum design V + P is considered, i.e. for a fixed volume of the plate V(x) = const.

(6)

the maximum value of the basic critical load parameter P is to be calculated. Two definitions for such an objective function are used: (I) Optimum design at the fixed number of circumferential buckling half-waves j: P = max \P’(x)l, x E x, j = const.

(7)

(II) Unimodal optimum design P = max (inflP’(x)lf, x E x, j = 0, 1. i

.

18)

2. BASICEQUATIONS

(A) Prebuckling, plane state In the prebuckling state two following assumptions are of importance: (i) No initial deflections occur, Le. W,(r) = 0, (ii) Axial symmetry of the plate and load implies displacements V(r) = W(r) = 0 and tangent force N,,(r) = 0 for r E [b. a]. The set of equations of the state (A) can be easily derived in the dimensionless form. It is completed by two ordinary differential equations:

Fig. 1.

1 u’= _“U[ 1-v” n,

n:=-

1

P[

u x--(l-v)& P

P

,

@a)

X

and by an algebraic equation: Fig. 2.

II, =

xtt

WI,.

P

prebuckling state the effective stress flc = .\/ar2 - up* + ali

(2)

is used as an exersion factor of the plate.

(l-tY,)tf,t(Yi(tI,i+P~)~Of01.i~~~~~

F~~~ff~a~i~ff of the optimum des~gffproblem

The dimensionless thickneises Xk are used as compot:nts of the vector x E R m the control space x C R . x={x:gi(x)50,

i=G}

The boundary conditions are formulated for three cases: (i) free edge, (ii} clamped edge, (iii) uniform compression pi is applied to external (i = a) or to internal edge (i = b) of the annular plate. All these boundary conditions are embraced in the following formulae:

(3)

(IO)

Combinations of 0, i values of the coefficients OL.,& corresponding to the above mentioned boundary conditions are set down in Table 1. Values (Y,, (Y~E (0. I) enable us to formulate mixed types (elastic fastening) of boundary conditions.

where gi(x) are combined of two types of constrains: (B) ~~ck~jng, bending state

(i) Geometrical constrains Xkmin5

Xk

s

k max.

k=1,N,

(4)

(ii) Stress constrains with respect to the effective stress (2): u,&) 5 u,. k = 1,N.

(5)

Three assumptions and their implications are worth emphasising (i) Modified shear force Vr=Q+$

(11)

371

Numerical optimum design of elastic annular plates with respect to buckling Table 1. Supporting

Coeff ic isnts

of external edse

da

pa

I

I

--___q

L

I

__--..._&

i

0

’

1

I

is used not only in bound~y conditions but also in equations related to the inside of the plate under consideration. (ii) Deflection W = W(r, t) is assumed to be a function separated with respect to independent geometrical variables: W(r, t) = W’(r) co5 jt,

(Qa)

where j is the number of circumferential halfwaves. The displacement (12a) implies the following distribution of other internal forces:

I

0

I

1

I

J

i

where the following algebraic relations are valid:

m,,=-2(1+

ix’

(14’3

w

(

p-q

f

-

Various boundary conditions of the state (I?) are shown in Table 2 and related to 0.1 values of the coefficients pa, /3b,ya, yb in the relations:

t1- pi)+% +

pi&[

=

0,

(I- “f)Cpi+ ~#n,i = 0 for i = a, b.

V,(r, t) = V:(r) cos jt, M,tr, t) = h&j(r) cos jt, M,(r, t) = M:‘(r) cos jt,

(W M,(r, t) = M”,(r) sin jr, Qt(r, t) = Q:(r) sin jr. (iii) At the moment of buckling axisymmetric dis~bution of axial forces is valid: N,(r, t) = N?(r), N,(r, t) = N:(r).

(13)

The set of differential equations of the state (B) can be transformed into the following dimensionless form: U”Z - lo* (0’= - $ (m, - Yrn,), ‘2 u:=

*2

-'t+m*-?nn)t+vn, P 6AP

m:= -~(rn,-m,)+6hv.-~m,+6hpn,, Wa)

3. METHODOF SOLUTION

For the plates of variable thickness it is impossible to derive explicit formulae for the objective function p as well as for dist~bution of stresses inside the plate. That is the reason why both states (A) and (B) are analyzed numerically, The set of differential equations (9a) formulates a two-point BVP of range r = 1. This set can easily be solved by means of the shooting method[51. In the algorithm the Runge-Kutta IV order method has been used to integrate nurne~~aily eqns (9a). In boundary conditions (IO) the unit uniform compression fl= 1 is assumed at the loaded edge of the plate. Solution of eqns (9) gives the axial forces ri,(p,; x), ri,(p,; x) in the integration nodes I = 1, . . . L. These values are stored in arrays and are used in the analysis of the buckling state (B). Equations (14a) formulate a two-point BVP of range r = 2. They are also solved by the shooting method, modified in [6] for computation of lower eigenvalues of eqns (14a). In the modification, a critical value of the load parameter pCris computed as one of the free initial values of the shooting procedure. The lowest critical load

K. RZE(XKIASK,~-PEIWHand Z.

372

WASZCZYSZVN

Table 2. Supporting

___ f:

of external edge

Coefficients AL I

..a-

r: 0

0

0

1

1

0

1

1

i I

_r--

I I+--

; I

---

I 1

Supporting

of

internal

Coefficients s h 0

1

is evaluated easily if the initial value in the iteration procedure is p”’ = 0. As in the analysis of the state (A) The Runge-Kutta IV order method with the same L nodes of integration is used to compute the state (B). The computation is performed for a fixed member j of the circumferential halfwaves in the problem I of the optimum design (see formula 7). If the problem II is considered then, according to (8). the state (B) is computed for increasing number j=O. I, . . up to the number of half-waves j, which corresponds to the first minimal value of the critical load p = pcJj*; x). The field of axial forces n, = PA,, n, = pA, is used for computation of effective stresses in every numerical integration node I in order to check the condition (5) of the stress constrain. The above mentioned part of the algorithm has been performed for the fixed number of segments k and the vector x of control variables. This part is sketched in Fig. 3 as a general flow-chart of the subroutine FLJNMOD (see [71). The maximum value of the load parameter p and a corresponding distribution of the plate material (values of components of the vector x satisfying condition (6)) are computed on the basis of the Rosenbrock method. An internal penalty function has been used to satisfy geometrical and stress constrains, (4) and (5) correspondingly. Two annular segments of constant thickness are assumed at the beginning of the computation. The optimal values x,, x2. xXare adopted as initial values for the next step where the number of segments is doubled.

(i)

4. NUMERICAL EXAMPLES design at fixed j Critical loads have been computed for axisymmetric

Subroutine

1

Ophwn

Fig. 3

FUNMOD

373

Numerical optimum design of elastic annular plates with respect to buckling

buckling of plates (j = 0). Plates with free initial edge and with either clamped or simple supported external edge have been analysed. The following data have been assumed: ho= O.Olm, a = l.OOm, b = 0.5~~

E = 2.06 x lO”N/m*,

v = 0.3, a, = 2.94 x 108N/m2,

&in = 0.2, Xmax= 3.0.

(16)

In Tabte 3 results of computations for the clamped plate of continuous change of thickness are shown assuming only geometrical constrains. The computations have been brought up to the number of segments N = 16.

For higher numbers N the number of iteration is low since the distribution of plate material does not change significantly. In the last column of Tables the CPU time is pointed out for iterations needed for doubled number of segments, e.g. for N = 8 the common CPU time equals 281 sec. Critical load decreases if the stress constrain is considered as it is shown in Table 4. Comparison of results for the segment numbers N = 2, 8 is presented in Fig. 4 where also distribution of axial forces at N = 8 is drawn. Optimal shape of the simple supported plate is independent of the stress constrain. The reason is the lower value of critical load, as shown in Table 5. The most interesting fact is that the materia1 “concentrates”

Table 3. I IL&l-

j = 0,

p = 0,s

min= 0.2

1

lumber of

xm*= 3,o

P cr

5egments

Number

[103H/re 1

2

-1179,952

4

-1231,486

I,49974 I I,43926

0,52351

CPU

of

IS

iteration6

time

[El

1.00570

0,20002

50

116

I,33910 0,20004

I.00970

14

94

I,41430 l,339ll l.09203 0.76663

?,P907 I,25684 l,@969 0,64507

2

67

0.44264

0.36772

I

16

I.43933

-1232,335

I,36411 I,17442 0,88815 %52351

0,20023

0.28073

Table 4. jz0

I i

!Z&L__ “3

Number of segments

16

I

v

p=

Xmin = 0,2, OE?=

OS5

rmex = 3,o

294,3.10GN/m2

P cr

Nuclber of iteretions

z.

I 103WmI

-940,629

1*2799 1,2052 I,1120

1.2519 I.1815 1,0773

1.2296 1,1469 -I,‘3424

%9819 0,80?4 OF4199

0.9230 0,6SYY o,26jo

0.8614 0.5189

1

CPU time

IS]

54

K. RZEGOCI~~SKA-PEZECH and Z. WASZCZYSZYN

374 j=O

--a

p --J

x min

ZZ

100

(J=

= c,2

?L max = 3,Q

Xmin = C,?

5cmax = 390

0.5

D = 294 9 3 * 106N/m2 a -.-

-.__._.

/._

-+=?I

-

P = -938,7.103N/m

I

---

P = -1232,1.103N/1n

I

-.-

-._-._,+.-

-

p = -940,6,~03r?[m

.-

.-.-.

.-.-

..-_-,

-.-.-

I

.-.j_

i

P = -94,8.103~/~ -.-.-.

I

-

-.__._,_,

3

I

P=-‘06’ ;

Fig. 4.

in an opposite direction to that in the clamped plate. In the case of simple supporting of external edge the plate is thicker at the internal edge contrary to the case of the clamped plate. The results shown in Fig. 4 are in agreement with simple evaluations given in (41. The results of computations for the clamped plate with constant thickness of segments are given in Table 6. Criticaf load is slightly lower than in the case of linear varrying thickness, as it has been shown in Table 4. (ii) Unimodal optimum design

Computations have been carried out for the plate of clamped external edge, free internal edge and data[lO]. Results obtained for the linear change of thickness are given in Table 7. Depending on number of segments N the basic critical loads are found for the number of circumferential halfwaves corresponding to symmetric (j = 0) or antisymmetric (j = 1) buckling.

If the plate is of constant thickness within segments then j = 3 (see Table 8) and the optimal shape related to the corresponding buckling mode is different than that in the case of continuous variation of thickness (Fig. 5). The optimum design of the clamped plate evaluated on the basis of criterion (7) for the fixed number j = 0 gives the critical load parameter 93% higher than for the plate of constant thickness, But on the other hand such a plate would buckle asymmetrically (jif 0) for lower loads. The more realistic criterion (8) leads to the optimal shape of the plate which is safe for buckling at any number j of circumferential halfwaves. In this case the lowest critical load corresponds to j = 1 (or to j = 3 for constant thickness of segments) and it is only 13% higher than for the constant thickness plate. Dependence of critical load on assumed volume of the clamped plate is shown in Fig. 6. The lowest curve is drawn for plates of the constant thickness ho=

315

Numerical optimum design of elastic annular plates with respect to buckling

x0 x O.Olm. The upper curve is traced for an optimal shape under assumption of the axissymetric buckling (j = 0). The middle curve, which is rather close to the curve of x = const., corresponds to different number of buckling halfwaves (j = 1 or 2) according to the criterion (8). It is interesting to point out that application of criterion (8) can lead in fact to the bimodal solution. For instance if ho = 0.08m then P,, = 267.51x lo3 N/m for j = 1 and P,, = 267.56x lo3 N/m for j = 2. If the fixed volume corresponds to ho = 0.06m then PC, = (104.3507, 104.3508)x lo3 N/m for j = 2, 1 accordingly. The number j of adjoint halfwaves is shown in Fig. 6. 5.CONCLUSIONS On the basis of numerical computations the following conclusions can be drawn: (1) If the number of buckling halfwaves j is fixed then

material distribution in the optimal plate depends on boundary conditions. In the case of free internal edge and simple supported external contour the optimal plate is thicker at the internal edge than at the external one. Opposite distribution of the plate thickness is found for the clamped external edge (Fig. 4). (2) Optimum design at the fixed number of halfwaves j can give nearly double critical load than that obtained for a plate of constant thickness, but the “optimal” plate can buckle for loads significantly lower for another j, (3) Optimum design for the criterion (8) max (inflP’I) gives critical loads only about 10% higher than it the constant thickness but it makes the optimal plate safe against buckling at any number of halfwaves. (4) The optimal plate computed on the basis of criterion (8) can correspond to bimodal solution for the neighbour number of buckling halfwaves j.

Table 5. j P -7

q

0,

J3= 0,5

4-A

'

cab=294,3.106N/m2

I I Number

x 4

2

of iterations

I

-94,829

CPU time

[ B1

1

I

-

0,37638

0.82982

2.41758

2

27

4

-106,653

0.20157 1935574

0,68821 2,99885

0.85624

4

43

a

-106,814

0,20129 0,776J37 1.35922

0,45227

0,67746

1

32

0,85687 2,17922

1 IO9745 2.99901

0,20129 0.56486 0,77887 0,97716 1.35922

0,32658 0.67746 0,81687 1.09745 I,76922 2,49444

0.45237 0,72716 0.85687 1.22833 2,17922

1

40

16

-106,815

2,56911

Table 6.

p

!

J

--

! I

?I

Number of segment 8

j = 0,

f3

rmin =0*2

q

0,5

7=

100

X,,=3,0

O.= 294, 3.106N/m2

P cr

Number

x

of

[103N/ml

iterations

4

-934,538

I,23354 0.40351

I.09706

I.05486

8

-934,755

: *z;;

I,23324

1.05419

I,23365 I.09732 I, 09726 I,05418 0.40308

CPU time

[Sl

17

157

1 SO9734 0,40307

3

163

I,23327 1,09732 1,OWS 1 SO5417 0.40305

2

0:4W5 16

-934,786

I.23371 1.23326 I,09724 I.05477 0.40322 0,40300

195

316

K. RZEGCUBSKA-PEEECH and Z. WASZCZYSZYN

A= 100 x

P -'

min

= 0,2

p =0.5 cmax

= 3.0

o- = 294 *. 3 10%/m2 a.

I cc

_

-.-

P .-.-.

._.___.-.-.-.-.

-_._.-_c.-

CP

I

= -519,9.103N/m *.-

j =I

I

P CI = -540,7.103N/m j = 0

_.__.~.

I -.+_

I

‘\

2,o . B I,5 2 “0

v 1,O. 0,5

”

I -

-

-._

P

.~.~

cr

__.__

= -497,2.103N/m

j=3 P cr

I---

I

-.-t

1

= -518,1.103N/. ___.__

j=3

&-1

Numerical optimum design of elastic annular plates with respect to buckling

311

Table 7. p= 1

_T+L

"7

Number of segments

cr

I

iIh=

qin = 0.2, KS=

100

Xmax=

3.0

106N/m2

294,3.

P

Number of iterations

x I 103iym]

'

CPU

t imo [s1

2

-483,1225

1

1.02656

I,00938

0,93415

4

321

4

-519,W3

1

I,13945 0,75911

I.03369 I.31339

0.95918

2

258

8

-540.7048

0

I.13945 0,99644 0,70911

I,11157 0,95918 0,98625

1.03369 0,85914

1

318

I,13945 1.07263 0,99m 0.90916 0.70911 I,21880

I,12551 I,03369 0.97781 0.85914 0,79768 I.56772

I.11157 1.01506 0,95918 0.78412 0,98625

1

610

16

P

x

0,5,

-541,3244

0

I,45136

-L

c

.103N/, m

x min =

0,2

%max = 2,0 2000 1800 1600 1400 1200 1000 800 600 400

0.2

0,4

0,6

0.8

I,0

500

250

166.7

125

100

1.2

I,4

83.3

7194

Fig. 6.

I,6 62.5

1.8

A

= ho/O, 01

h = a/ho

378

Table 8.

I p --

p = 0,s

x

il “1

min

CB

=

2 =

100,

xmx=3,0

0*2

= 294,3.106N/m2

x 4

-497,758

3

yo20

1,oooo

1,1C@O

9.3000

2

76

1,OOOO 1,1000

2

86

3000

1

258

,

8

16

-578,146

-535,266

3

3

?* 2000 1,OOOO 0.5* I,2000

I,4000 I,0000 1,1000 0,5444

0,201l

1,loOO

0.37386 I,2000

l,oooo l,oooo 1,1000 0,5444

(5) Piece-wise approximation of continuous dist~bution of the plate thickness x(p) gives satisfactory results for 8 segments of linearly variable thickness. Further increasing of number of segments practically does not influencethe results of computations. REFgKJINCEs 1. Formulations of Optimum Design of Structural Elements (Edited by A. M. Brandt), pp. 161-170.PWN, Warsaw (1978), (in Polish). 2. A. Gajewski, Calculation of elastic stability of circular plates with variable thickness by an inverse method. Bull. Acad. Pofon. Sci., Sir. Sci. Tech. 145) 303-312(1966).

I,

l,oooo 1,woo 1,1000

0,37y3

3, W. 8. Grineyv and A. P. Filippov, Optimum de&

4.

5.

6.

7.

of circular plates in stability problems. Strait. &fech. Rasch. Soorush. 2, K-21(1977)(in Russian). A. Strzelczyk, Critical bucking loads of annular plates with variable thickness. ZA&f&f68 T&&T156 (1980). Z. Waszczyszyn, Modification of numerical methods for analysis of elastic-plastic structures. Symp. France-Polonais ‘Probllmes des Rhioologieet de M&xnique des Sofs’, Nice 1974,PWN, Warsaw 1977,pp. 465-475. Z. Waszczyszyn, Critical load of elastic annular plates for asymmetric buckling. Arch. Bud. Maszpn 23(l), 79-93 (1976) (in Polish). K. Rzegocinska-PeYech. Numerical optimum design of elastic columns and annular plates with respect to buckling. Dr’s Thesis, Cracow TU (1980)(in Polish).