Optimization of Cylindrical Shells Under Combined Loading Against Brittle Creep Rupture M. ZYCZKOWSKI and M. RYSZ

Politechnika Krakowska(Technical University of Cracow) 31-155 Krakow, ul. Warszawska 24, Poland.

Summary

Optimal structural design of cylindrical shells under overall bending, torsion, tension and internal pressure in creep cond­itions is considered. The material is assumed to be governed by the Norton-Odqvist nonlinear steady creep law. Minimal weight of the shell is the design objective, radius and wall thickness are design variables, and the constraint refers to brittle creep rupture as described by the Kachanov-Sdoburev hypothesis. Elim­ination of circumferential bending in the wall results in a circular profile. The condition of uniform creep strength deter­mines the thickness distribution, whereas the optimal radius is determined numerically.

1. Introductory Remarks

Cylindrical shells work very often in creep conditions, e.g.

metal shells at elevated temperatures (pipelines, elements of

jet engines etc.), or shells made of plastics or concrete at

room temperature. If the loading does not conform to rotational

symmetry, then circular cylindrical shells of constant thick­

ness are not optimal inasmuch as their weight is concerned. In

general, two functions as design variables may then be consider­

ed: current radius r r(¢) and wall thickness g = g(¢), where

rand ¢ denote polar (or cylindrical) coordinates. Sometimes

the areas of possible longitudinal ribs may also serve as design

variables. Such optimization problems under elastic stability

constraints were discussed by Zyczkowski and Kruzelecki [1,2,3].

Optimal structural design in creep conditions was initiated at

the end of the sixties by Reytman [4], Prager [5], Nemirovsky

[6]and Zyczkowski [7], and widely developed at the Technical

University of Cracow; more recent results were summarized by

Zyczkowski in [8]. In contradistinction to static problems of

InelaS!ic Behaviour of Plates and Shells IUTAM Symposium Rio de Janeiro 1985 Editors: L. Bevilacqua,_ R Feijoo and R Valid © Springer, Berlin Heidelberg 1986

386

elastic or plastic optimal design, it introduces a new factor,

namely the factor of time. Most structures working in creep

conditions are designed for a finite life-time, usually deter­

mined by creep rupture or creep buckling; however, in some part­

icula~ problems the stiffness in creep or stress relaxation may

also serve as optimization constraints.

Optimal structural design of shells is very well developed; a

survey by Kruzelecki and Zyczkowski [9] reviews over 600 papers.

Most papers, however, deal with elastic or plastic shells, where­

as optimal design of shells in creep conditions is represented

by very few papers only.

The present paper considers optimal structural design of cylind­

rical shells under fairly general combined loadings which result

in longitudinal homogeneity of the stress state: overall bending

by the moment Mb , overall torsion by Mt , axial force N, and

uniformly distributed normal internal pressure p, Fig. 1. Such

a system of loadings may often be encountered e.g. in pipelines

y

Fig. 1. Cylindrical Shell under Combined Loadings.

387

The paper constitutes a generalization of previous considerations

by the authors, [10,11]: first, torsion is introduced here and

axial and circumferential directions are no longer principal;

second, the creep rupture hypothesis is more general than that

used before, and hence it covers a wider class of materials.

Both these generalizations change the problem, mainly ~licating

it; however, a certain simplification will also appear, since

ordering of principal stresses is easier if shearing stresses

due to torsion take place and division of the shell into separ­

ate zones is no longer necessary.

2. Statement of the Problem

The optimization problem is stated as follows:

(1) Minimal weight of the shell is the design objective. Under

the assumptions of a constant bending moment along the axis and

of a homogeneous material this design objective reduces to the

minimal area of the overall cross section. In most engineering

applications the bending moment is variable along the axis; then

the most stressed section may be considered as decisive.

(2) As the design variables we consider two functions describ­

ing a cylindrical shell (Zyczkowski and Gajewski [12]): middle

surface of a cylindrical, not necessarily circular shell is desc­

ribed by the function r = r(~), and wall thickness - by the func­

tion g=g(~). Moreover, in the case of a shell reinforced by long­

itudinal ribs, located at extreme fibres and convenient to carry

large bending moment, we should introduce two parameters Al and

A2 corresponding to optimal areas of those ribs regarded as con­

centrated; however, the present paper will not discuss any rib

reinforcement, thus leaving prevailing bending beyond consider­

ation.

(3) The optimization constraint refers to creep rupture of the

shell under a given system of loadings: ~,Mt,N and p. In part­

icular, the Kachanov-Sdobyrev hypothesis of brittle creep rupture

in its scalar form is adopted [13,14]. According to that hypo­

thesis, a measure of material continuity during the damaging

process, W, is governed by the evolution equation:

(2.1)

388

where Os denotes Sdobyrev's reduced stress,

(2.2)

01 and 0e are the algebraically maximal principal stress, and

the effective stress (stress intensity) ,'respectively, finally

R, v and 0 are material constants, 0 , 0 ~ 1. Leckie and Hay­

hurst [15] found that damaging process of some materials is bet­

ter descr-ibed by 01 (e.g. copper), and of others - by 0e (e.g.

steel and aluminium alloys). Hence the combination (2.2) is

sufficiently general as to cover a fairly broad class of mater­

ials.

If a steady creep is considered and 0 .. = const(t), without l]

redistribution of stresses due to elastic effects, geometry

changes etc., we may integrate (2.1) in a general form. Making

use of the initial condition ~(O) = 1 (perfect continuity, no

deterioration) and of the condition of full local deterioration

at the point under consideration ~(tR) = 0, we obtain for creep

rupture time tR the following "local" formula

1 (2.3)

It is supposed that in optimal structures designed for a given

creep rupture time t R, this value should be reached, if possible,

simultaneously at all points of the body (a structure of uniform

creep strength). Denoting the relevant stress in (2.3) by 0 SR '

introducing a certain safety factor for stresses, j, we obtain

the following condition of uniform creep strength

1 def

° o constr (r ,0) . (2.4)

It should be noted that the condition of uniform creep strength

is, in general, neither a necessary nor a sufficient optimality

condition. It may not be necessary either if geometry changes

are taken into account (Swisterski, Wroblewski and Zyczkowski

[16]), or if other constraints are introduced; it may not be

sufficient if it does not result in a unique solution. In the

case under consideration geometry changes are disregarded and

no other constraints are introduced, and hence we regard (2.4)

as a necessary condition; on the other hand, it is not suffic~

ient here and further optimization will be performed.

389

As was mentioned above, other constraints, in particular creep

buckling constraints for the shell, will not be considered in

the present paper. Such an approach may be justified if the

wall thickness is not too small; this case takes place if press­

ure and axial tension are predominant in comparison to torsion

and bending.

(4) As a first step towards optimal design we look for elimin­

ation of bending states in the shell, in particular for elimin­

ation of circumferential bending. Indeed, any bending of the

wall results in transversally nonhomogeneous state of stress

and (2.4) cannot be satisfied at any point of the shell. Subst­

ituting into the general equilibrium equations of the engineer­

ing theory of shells (Wlassow [17], p.201) all the moments and

shearing forces equal to zero, we obtain

and hence

0,

const k = L , 2 n cP

0, (2.5)

const, (2.6)

where k 2 (CP) = l/p(CP) denotes the circumferential curvature of

the shell ncp = 0cp(CP)g(CP) is the circumferential membrane force,

and the internal,pressure p was assumed to be constant (self­

weight of the medium inside the shell being disregarded). So,

it turns out that the necessary conditions of the membrane state

result here in a circular cylindrical shell, though this shape

was not assumed a priori. Hence we reduce design variables in

the optimization problem under consideration to one function of

one variable g = g(CP) and to one parameter p = r = const. More­

over, the design variable g(CP) and the stress 0cp(CP) are inter­

related by

pr, (2.7)

390

resulting from (2.6).

It should be noted that the necessary conditions of membrane state (2.5) are not the sufficient ones. For example, the circ­umferential bending effects in the wall under similar loadings were studied in the plastic range by Mrowiec and Zyczkowski [18,

19]. In the present paper, however, we neglect these effects and assume the membrane state.

(5) Equations of state are assumed as the Norton-Odqvist const­itutive equations for an incompressible body:

(2.8)

~ii = 0, (2.9)

where eij and Sij denote deviatoric strain rates and deviatoric stresses respect1vely, Ee and 0e are the effective strain rate and the effective stress as described by the Huber-Mises-Hencky hypothesis, K and n stand for material constants, and in the last equation the summation convention holds. In the case of plane stress under consideration we have

(2.10)

(2.11)

where the usual engineering notation for stresses and strain rates has been introduced.

(6) Finally, the optimization problem is stated as follows.

We minimize the overall cross-sectional area

A 2r

TI/2

~ g(~)d~ +

-TI/2

min (2.12)

under the constraint for stresses (2.4), and under the integral

constraints

N

n/2

2r S cr z (<jJ ) g (<jJ ) d<jJ

-n/2

n/2

S Tz<jJ(<jJ)g(<jJ)d<jJ

-n/2

n/2

const

const,

2r J ',($) (r s1n$ - yo)g($)d$

-n/2

391

(2.13)

(2.14)

const, (2.15)

where Yo denotes the coordinate of the centre of gravity of the

unsymmetric cross-section,

n/2 n/2

r S g(,)s1n,d,/ S g($)d$. (2.16)

-n/2 -n/2

with the equations of state (2.8), (2.9), and the remaining

fundamental equations of continuous media (equilibrium, compat­

ibility) .

3. Stress and Strain Rate Distribution

The distribution of shearing stresses TZ<jJ follows directly from

equilibrium equations. Hydrodynamic analogy yields

TZ<jJ(<jJ)g(<jJ) = const(<jJ) = c. (3.1)

Substituting (3.1) into (2.14) we calculate C and express shear­

ing stresses in terms of the twisting moment:

Mt --2-2nr g

(3.2)

392

now the constraint (2.14) disappears. Further, making use of

(2.7) we eliminate the thickness g($) and rewrite (2.12)-(2.16)

in the form: TI/2

A 2pr 2 S 1 d$ -+ min, 0$

(3.3)

-TI/2

TI/2

N 2pr 2 J °z 0$

d$ const (3.4)

-TI/2 TI/2

TI 12 J sin$ d$ 3j" -TI/2 0$ ~ 2pr 0; (sin$ - TI/2 )d$ const (3.5)

J 1 d$ -TI/2 0$

-TI/2

M;)reover, the formula for 'z$' (3.2) , turns into

, z$ Mt

0$ (3.6) 2TIpr3

Compatibility equations make it possible to determine the dist­

ribution ox axial strain rates ~z. Using Cartesian coordinates

we may write

(3.7)

in the problem under consideration the strains do not depend

on z, hence two derivatives in (3.7) vanish, and ~z mustbe lin­

ear function of y. Returning to cylindrical coordinates we write

393

this function in the form

K r sin<j> + ~o' (3.8)

where K denotes the curvature rate and E is the rate of elong­o ation of axis of the cylinder

roidal axis, shifted by Yo). hypothesis, but, in fact, it

(not coinciding here with the cent-

Eq. (3.8) resembles Bernoulli's

exceeds that hypothesis: plane

sections not necessarily remain plane, warping may occur.

Now, the rema~n~ng four unknowns 0z' 0<j>' E<j>' and Yz<j> may befamd from the Norton-Odqvist equations (2.8), (2.9), and the condition

of uniform creep strength (2.4). First we eliminate f from (2.9)

performing contraction of the deviators and making use of (2.8):

Using the last equation as joining ~z and Sz we obtain n-l

(3.9)

K r sin<j> + ~o ~(O~ + O~ - oz0<j> + 3m20~}--2-(20z - 0<j>)'

(3.l0)

where the dimensionless parameter:

m (3.11)

Jo~ns the loadings and the design variable r. The rema~n~ng

two independent equations (3.9) determine E<j> and YZ<j>' but they will not be used effectively. On the other hand, (2.4) with

(2.2) and (3.6), yield

i[o + VcOz -2 2 2 1 0<j> + . 0<j>} + 4m 0<j> + 2 z

+ (1- 8) /02 2 2 2 (3.l2) + 0<j> - 00+ 3m 0<j> ° . I z z <j> 0

The solution of the system of equations (3.l0) and (3.l2) with

respect to 0z and 0<j> would enable us to determine the stress distribution in terms of the coordinate <j> and of the parameters

394

K, to Then we could use (3.4) and (3.5) to evaluate K and Eo

and perform minimization of (3.3) as a function of the design

variable r.

4. Change of Variables

An analytical solution of (3.10) and (3.12) with respect to Oz

and o~ seems impossible. However, the left-hand side of (3.12)

is homogeneous of the first degree in stresses and an essential

simplification will be obtained by introducing instead of o~, 0z

two new, dimensionless unknowns s, s by the formulae

o~ = 00s sins, (4.1)

These formulae resemble the Nadai-Sokolovsky parametrization of

the Huber-Mises-Hencky yield condition expressed in principal

stresses, but they are used here in some other sense since TZ~

is also present and s is not proportional to the stress intensity.

In the problem under consideration we have 0 < s < TI, because

o~ must be positive. Now, in view of the mentioned homogeneity

of (3.12), this equation may be solved with respect to s = s(s):

s

and (3.10) yields

Kr sin~ + EO

.:!!:.) + sin s + 3

] +

(4.2)

(4.3)

where s(s) is given by (4.2), and the following dimensionless

parameters have been introduced

K E 0 (4.4) K S OnK 0 onK

0 0

So, we have reduced two equations (3.10) , (3.12) to only one

395

equation (4.3) with the unknown ~ = ~(~). This equation cannot be solved for ~, but it can easily be solved with respect to

~ = ~(~), and this inverse function will be used in further calc­ulations:

~ = - £

arc sin ________ ~o f(~)

- (4.5) K r

where f(~) stands for the right-hand side of (4;3) with substit­

uted (4.2).

Now, the integrals (3.3)- (3.5) may be rewritten with the integ­

ration variable changed into ~, and so the optimization problem will be presented in an effective form. Moreover, we replace the dimensional design variable r by a dimensionless one, m, substituting, in view of (3.11),

31M;" r = v'~.

Finally, we look for a minimum of the integral

-a

under the constraints

n

~ J~2~~1~~ d~ d~ + min m2t3 s(~)sin~ d~

1 =m-m

~l

~2

\ sin(~ -

J sin~

~l

2::.) 3 d~ d~

d~

(4.6)

(4.7)

const

(4.8)

396

1;2

rUb (l;l,1;2,m) 1 { sin(, - })

[ sin<p (0 -mb m sinl;

1;2 1;1 [Sin<!> (1;) d<p dl;

I; s(l;)sinl; dl; ] d. (4.9) 1 dl; dl; const,

1;2

~(I;~Sinl; d<p dl; dl;

1;1

where s (1;) is given by (4.2), d<P/dl; is to be calculated from (4. 5) and

the ~ionless quantities a, n, rob are defined as follows

-a -n 2/3

J:...(~) N 2 M ' m p t

n M~· t

(4.10)

-The limits of integration 1;1,1;2 may be expressed in terms of £0

and K by solving (4.5) with substituted <p = -n/2 and <p = n/2,

respectively. However, this is not necessary: Eo and K should be

determined from (4.8), (4.9) in terms of n and ron' so we may sim­

ply solve (numerically) (4.8), (4.9) with respect to 1;1 and 1;2

instead of Eo and K. Finally, after 1;1 and 1;2 have been evalua­ted, we look for a minimum of (4.7) as of a function of one var­

iable m (m is hidden in z (I;) and <P( 1;) as well).

5. Numerical Examples

The system of two equations (4.8), (4.9) is solved with respect

to 1;1,1;2 by Newton's procedure for subsequent values of m and then a numerical minimization of the function a = a(m) is perform­

ed. However, some complications appear when calculating numeric­

ally the integrals: they are improper, since d<P/dl; increases inf­

initely for <p = -n/2 and <p = n/2 i.e. for I; = 1;1 and I; = 1;2. Accuracy of most numerical "procedures is then poor, but those

singulari tie's may be removed by integration per partes of the

type

(5.1)

Y

I

I I

I I I

I1b

Fig. 2.

!J

Optimal shape and stress distribution for 0 anov-Huber-Mises), n=4, IDb=5, n=O.

397

5z Y" 60

0.5 6z.cp. 60

o {Kach-

398

!J

- - F' CfJ2 =?'~

Fig. 3.

'-- , ""'"

o

Optimal shape and stress distribution for 0 anov-Sdobyrev) n=4, mb=S, n=O.

0,5

O. S (Kach-

fJ

Fig. 4. Optimal shape and stress distribution for 0 anov-Galileo), n=4, mb=S, n=O.

399

0,5

1 (Kach-

400

where u i stand for the relevant integrands in (4.7)-(4.9).

The resulting optimal shapes and the relevant stress diagrams

are shown in Figures 2 (8 = 0, Kachanov-Huber-Mises), 2(8 = 0.5,

Kachanov-Sdobyrev), and 4(8 = 1, Kachanov-Galileo).

6. Final Remarks

The shapes shown in Figures 2-4 are optimal, but they may be

difficult in manufacturing. Much easier, but less effective

optimization may be achieved by applying constant wall thickness

reinforced by two longitudinal ribs to carry overall bending of

the shell. This problem of purely parametric optimization will

be discussed separately.

References

1. Zyczkowski, M., Kruze1ecki, J.: Optimal design of shells with respect to their stability, Proc. lUTAM Symp. Optim­ization in Structural Design, ed. by A. Sawczuk and Z. Mroz, Springer 1975, 229-247.

2. Kruze1ecki, J.: Optimization of shells under combined load­ings via the concept of uniform stability, Optimization of Distributed Parameter Systems, ed, by E.J. Haug and J.Cea, Sijthoff and Noordhoff 1982, 929-950.

3. Kruze1ecki, J., Zyczkowski, M.: Optimal design of an elas­tic cylindrical shell under overall bending with torsion. Solid Mech. Arch. 9 (1984), 269-306.

4. Reytman, M.l.: On the theory of optimal design of struct­ures made of plastics with time effects taken into account (in Russian). Mekhanika Po1imerov (1967), 2, 357-360.

5. Prager,W.: Optimal structural design for given stiffness in stationary creep. Z. Angew. Math. Physik 19 (1968), 252-256.

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8. Zyczkowski, M.: Recent results on optimal design in creep conditions. Proc. Euromech. Co11. 164, Optimization Meth-

401

ods in Structural Design, ed. by H. Eschenauer and N. Olhoff Bibliographisches Institut 1983, 444-449 .

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10. Zyczkowski, M., Rysz, M.: Optimal design of a thin-walled pipeline cross-section in creep conditions. Mechanics of Inelastic Media and Structures, ed. by o. Mahrenholtz and A. Sawczuk, PWN 1982, 329-339.

11. Rysz, M.: Optimal rib-reinforcement of a thin-walled pipe­line with respect to creep rupture. J. Pipelines (1985), in print.

12. Zyczkowski, M., Gajewski, A.: Optimal structural design under stability constraints. Proc. IUTAM Symp. Collapse -Buckling of Structures, ed. by J.M.T. Thompson and G.W. Hunt, Cambridge Univ. PRess 1983, 299-332.

13. Kachanov, L.M.: On the rupture time in creep conditions (in Russian). Izv. AN SSSR, Otd. Tekhn. Nauk (1958), 8, 26-31; (1960),5,88-92.

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16. Swisterski, W., Wroblewski, A., Zyczkowski, M.: Geometric­ally non-linear eccentrically compressed columns of uniform creep strength versus optimal columns. Int. J. Non-linear Mech. 18 (1983), 287-296.

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