Adaptation of spherical and cylindrical vessels to variable internal pressure and temperature

Pergamon

lnt. J. Mech. Sci. Vol. 37, No. 7, pp. 783-792, 1995

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 12020 7403/95 $9.50 + 0.00

0020-7403(95) 00009-7 ADAPTATION OF SPHERICAL AND CYLINDRICAL VESSELS TO VARIABLE INTERNAL PRESSURE AND TEMPERATURE PHAM DUC CHINH Institute of Mechanics, Vien Co hoc, 224 Doi Can, Hanoi, Vietnam

(Received 27 October 1992; and in revised form 28 October 1994) Abstract--The kinematic theorem is applied to solve some problems of shakedown of spherical and cylindrical vessels subjected to variable internal pressure and temperature, taking into account the temperature dependence of the yield stress.

NOTATION

a,b inner and outer radii of a hollow sphere (cylinder) r t

O(r,O 0°(0 e(O e~(r, t), e~(r, t)

~e(r, t), ~; (r, t) O'r(0 ) = o'r° " [1 - - 9 ( 0 ) ]

g(0) E; v o~

radial coordinate time parameter temperature field internal temperature internal pressure plastic radial and circumferential strain rates plastic radial and circumferential strain increments over a cycle elastic radial and circumferential stresses yield stress a material function of temperature (e.g. g(O) = fl' 0, fl = const) Young's modulus and Poisson's ratio thermal expansion coefficient

1. I N T R O D U C T I O N

The problem of shakedown of pressure vessels under the combined actions of variable internal pressure and temperature has attracted much attention in the literature recently I-1-6]. Most of the studies dealt with thin cylinders subjected to a constant pressure and a variable temperature field. The approximate scheme of thin shell stress resultant theory was used and various boundary conditions were examined. Only a few dealt with thick shell problems. In Ref. [7] a radial nozzle in a spherical vessel was analysed, which indicated some important aspects of plastic deformation and particularly of the shakedown problem, which the thin shell stress resultant theory failed to account for. Plastic deformation may vary across the wall of vessels and tends to be localized. In Ref. [4] shakedown of a closed hollow cylinder under both variable pressure and temperature field was considered. The temperature dependence of the yield stress was taken into account. Instead of giving the shakedown domain, the author succeeded only in establishing a so-called perfect incremental collapse curve. However, from the kinematic theorem one could say only that the curve lies outside the shakedown domain. Later in this work we shall see that this curve really goes along a part of the shakedown boundary. One approach in shakedown analysis is based on the lower bound static theorem [4, 7, 8]. The difficult task one has to perform there is to construct a set of self-equilibrated stress fields. An alternative approach comes from the upper bound kinematic theorem. Although the kinematical variables are usually subjected to fewer constraints than the statical ones, the problem which arises is how to deal with compatible plastic strain rate cycles. To overcome this difficulty, a specific form of the plastic strain rate field is presumed so that the time integrals can be integrated. The result obtained subsequently is called the perfect incremental collapse criterion [1, 2, 4, 6, 9]. Specifically the method yields an upper bound on the unshakedown load. 783

784

P . D . Chinh

In this paper, the difficulty of application of the kinematic theorem is overcome for the particular problems considered. The shakedown domain as well as possible mechanisms of unshakedown are determined. The temperature dependence of the yield stress has been taken into account, while the weak influence of the temperature on the elastic constants and viscous effects are disregarded. 2. HOLLOW SPHERE Consider a hollow sphere of inner and outer radii a and b subjected to internal pressure P and temperature 0,, which may vary arbitrarily and independently (but slowly so that dynamic effects can be disregarded) within the limits:

0 <~ PL <~ P(t) <<.Pu,

OL <<.Oa(t) <<.Or.

(1)

The pressure and temperature on the outer surface of the sphere are assumed to be zero. The Tresca yield condition for our spherically symmetric problem assumes the form:

where a+ and a, are the circumferential and radial stresses; yield stress ar depends on the temperature 0 as av = a ° ' [ 1 - g(0)],

(2)

g(0) is an increasing function of temperature; g(O) < 1; g(0) = 0; a ° is the yield stress at the reference temperature 0 = 0. The material is supposed to be plastically incompressible, so that the radial and circumferential plastic strain rates for the spherically symmetric problem are related by e~ = - 2e~. A compatible plastic strain increment should have the form: C

ev = - 2 e , ~ = - -

a~
r 3~

c=const.

(3)

Koiter's unshakedown condition [10, 11, 4] for the spherically symmetric problem in the presence of thermal effects, taking into account (2), can be given as

dt where

[aeA'2eg + a°'g(O)'21egl]r2dr >>.

dt

3a3b3P(t) a~ (r, t) = ae~ - a~ - 2(b3 _ a 3 ) r 3 0(r, t) =

O.(t)

a°.21e~lr 2 dr,

,

a/r - a/b 1 -

a/b

(4)

(5) '

a~ and are are the stress response of the sphere to the external load and temperature field on the presumption of its perfect elastic behaviour; the temperature 0 is the result of the thermal equilibrium problem, which is assumed to be independent of the mechanical field (since the process considered is quasistatic); e,~ should satisfy the condition for a compatible plastic strain rate cycle, which in our particular problem has the form (consulting (3)): e~ =

e~ dt = 2r---5 .

(6)

From (4), Koiter's kinematic theorem can be expressed as

dt k 7 1 = Sup eg~t6)

[ae~'2e~, +

a°'g(O)'21e~,l]rZdr ,

dt

tr°" 2le~l r2 dr

(7)

Adaptation of vessels to variable pressure and temperature

785

and with ks > 1 the sphere will shakedown, while at ks < 1 it will not. e~ e (6) means that the supremum should be taken over all fields eg satisfying condition (6). In the special case g(0) = 0, ks has the usual meaning as the shakedown factor, i.e. the external agencies multiplied by it would still keep the body inside or on the boundary of the shakedown domain. Our objective now is to simplify (7), particularly to eliminate the time integrals. First, notice that any e~ ~ (6) can be expressed as A(r, t)'~(r) = A • 2r-~c,

[ ,T A d t = l

eg (r, t) =

ife,~¢O(c¢O)

(8)

:o A(r,t)'g(r)

,

Adt=O

if~g=c=0,

(9)

0

where g(r) is an arbitrary function. Substituting the given expression of eg into (7) we obtain k [ ' = max {I, A },

(10)

where dt I =

Sup ~g,a~(s)

[eX'2e~'A + a°.g(O).2le~.Al]r2dr

f:f: dt

dt = Sup A~(S)

dt

A = Sup ~.A~(9)

(11)

[o'X"A + a°'g(O)'[Al]r-ldr

fff] dt

a°" 2[e~" Alr 2 dr

a °" lAir -1 dr

[aX" ~" A + a °" g(0)" Ig" All dr (12) dt

a°r" [g" AI dr

Let us examine, first, I in (11). The whole process will then be applied similarly to A in (12)• Denote: (rlAI-A S(r) = Jo 2 dt i> O, (13) then from (8), (9) one deduces

fflAl+Ad

t={S;1

2

ifc¢0(t,~¢0) if c = 0 (eg=0).

(14)

(15)

Denote:

U(r, t) = aX(r, t) + a°.g(O, (r, t)), Urn(r) = max U(r, t) = U(r, t,,) t

L(r, t) = - a~(r, t) + a°r'g(O, (r, t)),

L'(r) = m a x L(r, t) = L(r, tt,).

(16)

t

With (13), (14), (16) we can rewrite (11) in the following form: - - + L

[ A f ; A i r _ 1 dr

I = Sup AE(8)

ff

(17) a . (2S + 1) r - 1 dr

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P . D . Chinh

Looking at the time integral in the n u m e r a t o r of (17), one notices ffIU'~+L''A' +A

[A[ 2 - A I dt ~< U " ' ( S +

1) + L " ' S .

On the other hand, taking A(r, t) = (S + 1). 5(t - t.r) - S. 5(t - hr) satisfying (8) we get (remember t,, and tu from (16); 6(t) is the Dirac function):

;;[

u I A I + A + L IAI

;q

dt = U ' . (S + 1) + Z m. S.

Therefore, b [Urn'( S + 1 ) + L ~ . S J r - l dr

(18)

I = Sup- a

fi

S~>0

'a°r.(2S + 1)r -1 dr

Denote: g(r) = a ° ' 2 S ( r ) r -1,

X =

'

W = max ,

f

Oa

Urn(r) + Lm(r) 2o.0

b .i

u = ( r w ) + L'(rw) 2a ° ,

then (18) can be rewritten as

I=

Sup ~, x~(19)

f]

(19)

Sdr,

~.(Um+L=).(2a~,)-ldr+

(20)

£,

Urn.r-1 dr (21)

X + a °" In (b/a)

It is easy to see that W'X

I ~< Sup x ~o

+

U'~.r -1 dr

X + a °" ln(b/a)

F u r t h e r m o r e putting S = X" 6 (r - rw) satisfying (19) into the expression after Sup in (21), taking into account (20), we deduce W.X

I = Sup x ~o

+

£,

U'~.r - i dr

(22)

X + a °" ln(b/a)

The right-hand expression in (22) depends monotonically on X s [0, + m), so the supremum is attained at X = 0; or X = + oe, thus

= Max f M a x a~(r, t) - aX(r, t') + aO" [o(O(r, t)) + 0(0, (r, t'))] V.,,,,,' 2a° '

x

fl

max [a~ (r, t) + a~,. 0 (0, (r, .t))] • r -1 dr a °" In (b/a)

}

"

(23)

Adaptation of vessels to variable pressure and temperature

787

Analogously one derives

[0.X(r, t) - 0-X(r, t')] . g + 0.° . [ g(O (r, t)) + 9(0, (r, t'))]" Igl

A = Sup g,,,,,,'

20.°'lgl

(24)

= Max 0.~(r, t) -- 0.~(r, t') + 0.°" [9(O(r, t)) + 9(0, (r, t'))] >,\ ~,,,,, 20.o \ F r o m (10), (23), (24) one obtains k~-1 = max {I, A } = I = max {Ip, A },

(25)

where

/p ~

fi

'max [0.X(r, t) + 0.O" g(0, (r, t))]" r - 1 dr t

0.o. In (b/a)

(26)

F r o m (8), (9), (11), (12) and (25) one notices that on the part of the shakedown boundary, where 1 = ks = I = Ip > A, the sphere ~¢ill fail because of incremental collapse (e~ # 0), while on the other part, where 1 = ks = I = A, either of incremental or alternating (g # 0) collapse modes could take place. Looking at (1), (2), (5) and taking into account the fact that g(O) generally is an increasing function of temperature, one can see max 0(r, t) = 0," a/r a / -b 1 -

a/b

maxg(0, (r, t)) = g(i(r)),

_ if(r),

max if(r) = 0,,

maxg(if(r)) = g(0.),

t

r

3a3b3p, max [a~ (r, t) + a°.g(O (r, t))] - 2(b3 _ aa)r 3 + a °. g(ff(r)), M a x {aX(r, t) - aX (r, t') + a°.[g(O(r, t)) + g(O, (r, t'))] } r, t, t'

=maxl

,.t.t" 12(b 3 - a3)r 3 [P(t) - P(t')] + ao. [9(O(r, t)) + 9(O(r, t'))]

= max

{

3a3ba

2(b3 _ aa)r 3

(P,

}

P,)+20.°.O(ff(r))}

3b 3 - 2(b 3 - a a) (P. - P~) + 2a°r.g(0,). N o w substituting (5) into (24)-(26) and taking into account the last formulae, we get finally: k2 x = max {Ip, A),

(27)

where p.

Ip = 20.Oln(b/a ) + "

a/r - a/b

9 (if(r))- r - 1 dr

ln(b/a)

3b 3 (p. _ Pz) A = 4-~ ~y + o(O,).

,

if(r) = 0~ . 1 - a/b

(28)

A peculiarity of our spherically symmetric problem is that the elastic constants E, v and the thermal expansion coefficient ~ do not appear in the equations determining the shakedown domain (27), (28) (the situation is partly similar to that of plastic limit analysis where E, v could be disregarded). This occurs because of the simple forms of elastic and thermal fields (5). Once a specific material is chosen and O(0) is given, from (27), (28) one can determine the

788

P.D. Chinh

shakedown domain for a particular problem in the space of coordinates P,, P~ and 0,:

k,(P., Pt, O) >i 1. In the special case g (0) - 0 and Pz = 0, the equation k, = 1 yields the unshakedown load for the simple problem of adaptation of a hollow sphere subjected to variable internal pressure 0 ~< P(t) <<.P,:

P,, = min{2a°.ln(b/a),

~ay4o.(1

_

a3/b3)}.

For comparison, the instantaneous plastic collapse pressure is P~ = 2o-o. In (b/a). Note that: g(O(r)).r- 1 dr

max g(#(r)),

<

In (b/a)

r-1 dr

= g(O.),

In (b/a) b3

so in the case of a sufficiently thick sphere ( ~ i> ~ 1) and Pl = 0, from (28) one finds out A > Ip and, from (27), k[ 1 = A. The expression of the unshakedown load would be very simple (k~ = 1): 4 ( b 3 - - a 3 ) o"°"

p. -

[1 - g(O.)].

3b 3

3. C L O S E D

HOLLOW

CYLINDER

Following Ref. [4], consider a hollow cylinder of inner and outer radii a and b subjected to quasistatic internal pressure P and temperature 0a (consult (1), (2)). The cylinder is long and its ends are closed with rigid diaphragms. Koiter's kinematic theorem takes the form: dt

k[ 1 = Sup

[a~,"e~ + a °" g (0)" [e,~[J r dr

f:£ dt

e~e(30)

a°'le~lr dr

where

e 2aZb2p °X=°;-~'-(b2_a2)r2

EaZ°~Oa

~ 2 b2

2(l_v)(b2_a2)Lr~

b2-a2

7

a~i~)_l '

(29)

ln(b/r) 0 = O, ln(b/a)' a~ and a e are the circumferential and radial stress response of the cylinder to the external load and temperature field on the presumption of its perfect elastic behaviour, and 0 is the result of the respective thermal equilibrium problem. Note that the expression of a] in (29) here is more involved than that of (5) because of the axial load effect in the case of a closed cylinder. The Tresca yield condition is used. The plastic strain rates and compatible plastic strain increments for our axisymmetric problem are supposed to be: erv =

eg =

- - e~, ~

c

e,~ dt = ~ .

(30)

The procedure used in the previous section is applied successfully here, which leads us to the same formulae (25), (26), (24). The argument after (26) is also valid here. It is interesting to note that the perfect incremental collapse curve constructed by Konig in Ref. [4] coincides with the curve Ip = 1, which in fact is a part of the shakedown boundary. To derive it, Konig suggested a plastic deformation rate of the special form (8) with additional condition A(r, t)~> 0, which he called the perfect incremental collapse mode. In fact, this is only

Adaptation of vessels to variable pressure and temperature

789

a special kind of incremental collapse if we understand incremental collapse as the m o d e with steady-increasing plastic deformations after each cycle (e~ ~ 0), while the increase of plastic deformations during each cycle is not required to be m o n o t o n o u s . As we have already noted, ratcheting could h a p p e n even at 1 = k71 = A > Ip. In the case of absence of thermal effects: 0 - 0a - g(0) --- 0, from (25), (26), (24), (29) we get;

p, k~-1 = M a x

(P,-- P,)b2~

0.01-n-(b/a)' 0.°(b2 - aZ)J"

Given Pz = 0, the equation ks = 1 yields the same u n s h a k e d o w n pressure as that obtained by the static a p p r o a c h (see Ref. [4]): P, = Min {0.°ln

(b/a), o-°(1

-

a2/b2)}.

In the general case, there is no simple analytical integration of (25), (24), (26) with (29) and numerical calculations have to be done. Assume P~ = 0~ = 0; 9(0) = fl" 0; fl = const > 0. We have:

A

=

1 ~2P, b 2_ [-Eo~[Oa(t)--Oa(t')]( 20-0 [b e _ a2 + m a x - - - - - - ~-~ -- -227 2bz -,,,, L2(1 v)(b -)

b2-a2~ ~n~/~J

a

-

+ao[0o(t)+0o(c)]]} - 2 a ° [ b 2 - a 2 + 2(1 -

v)(b2 - a 2) 2b2

In (b--~

+ flo-°0" "

The last expression is obtained because usually for metals: B0-° < ca~.

Ip follows the procedure presented in Ref. [4]: 1{ [[ln(b/p)] 2 Eota2 Ip-aOln(b/a) P" +0"" fla ° ~ +2(l_v)~ 2-a

Calculation of

.((b 2 --a2)ln(b/p) \a21n(b/a)

z)

b2 +

~5 l)]},

where

P F o r example, taking

b/a =

= b( 21n(b/a!~U2.(1 \b2/a 2 - 1,,1 _

flu°r(1 Ec~

3

v)/.

2 and material constants a p p r o p r i a t e for mild steel:

a°r=2.45x10 sPa, fl=2xl0-3deg

E=2.06x1011Pa, -1,

v=0.3

a=2xlO-Sdeg -1.

Calculations from the last formulae yield: k~-1 = m a x

{Ip, A } =

m a x {1.44 ft, + 0.093 Ou, 1.33 ft, + 0.50~ },

(31)

where

fi. = P./a °, O. = OuEo(a°r. The s h a k e d o w n d o m a i n is given in Fig. 1. O n the b o u n d a r y Ip = 1 (which coincides with Konig's curve in Ref. [4]), we have incremental collapse, while on A = 1, b o t h incremental and alternating plasticity m o d e s are possible. If the t e m p e r a t u r e dependence of the yield stress is neglected, i.e. fl = O, we get a greater coefficient k~: k~-~ = m a x {Ip, A} = m a x {1.44P, + 0.0890,, 1.33/5, + 0.440,}.

790

P . D . Chinh

I1

10.o

5.0

0.0

'1

0.0

0.4

0.8 u

Fig. 1. The shakedown domain for a thick-walled cylinder (see (31)).

Some authors studied thin-walled cylinders with constant pressure. A linear throughthickness temperature distribution is assumed. Thermal radial stress as well as the temperature dependence of the yield stress are neglected (see, e.g. Ref. [3]). Following the request of the reviewer of this paper, we apply the approximations of the thin shell theory to our formulae (24), (25), (26), (29) to deduce a simple expression for the equation determinating the shakedown boundary for the closed thin cylinder considered in Ref. [3]. Assume: P=const,

0~<0a~<0u,

g(0)=0.

We have (tr~ and tr~ are the thermoelastic results of the thin-shell problem): 2a2b 2 Ect tr~ = tr,~ -- tr~ = (b 2 _ a2)r 2 • e + 1 ~

r - (a + b)/2 0a, b- a

2a2b 2 E~ r - (a + b)/2 0u, J(b 2 - - a2)r 2 P + l ~ v b- a maxa~(r,t) = ) 2a2b 2 ~(b2 _ a2)r 2. P, r < (a + b)/2.

r

/> (a +

b)/2

Now (26), (24) are calculated readily:

(1 lP= [a°ln(b/a)]-l" I P + 2 ( i - -E~t --v)

ab--a + b ' l n a 2b + bJ~" On1 , (32)

E~ A - 4(1 - v)tr° 0~. As in Ref. [3], take _ ~ = 20. Denote: 19.5 P /~ = a--~-r" '

E~ /~ = 2(1 - v)a~," 0u.

Finally (25), (32) yield: k~-1 = max {Ip, A} = max{/~ + 0.244ffu, 0.5ff~}.

(33)

Equation k, = 1 determines the shakedown boundary given in Fig. 2. Note that there are differences between our derivation of (33) and the approach in Ref. [3]. In the latter, the finite element analysis is applied to a particular loading process and the von Mises yield condition is taken. Our result is obtained directly from the shakedown kinematic theorem without referring to any particular loading process, and the Tresca yield condition is taken. Despite the differences, the ratcheting curve in Ref. [3] seems to approach the part Ip = 1 of the shakedown boundary constructed in Fig. 2.

Adaptation of vessels to variable pressure and temperature

791

u 4-

3-

Ip=~ 1

1-

SD = 0

0.0

0.5

t.O

Fig. 2. The shakedown domain for a thin-walled cylinder (see (33)).

4. CONCLUSIONS

Some difficult spherically symmetric and axisymmetric problems of shakedown of a hollow sphere and closed hollow cylinder under simple thermomechanical loading conditions have been resolved in almost explicit forms: (27), (28) (for the sphere) or (25), (26), (24) with (29) (for the cylinder). The only remaining work to be accomplished is to calculate an integral in (28) or (26) and perform some operations of maximization once the parameters of a particular problem are given. The general form of dependence of the yield stress on temperature (2) is allowed. The shakedown domain and possible modes of inadaptation on the shakedown boundary have been determined. A similar approach has been applied in Ref. [12] to study the shakedown of bar structures under cycles of loads and temperature, where we succeeded in transforming Koiter's inadaptation condition to a more applicable form, similar to that of (25), (26), (24). The method can be used to solve the problem for the compound sphere (cylinder), which is composed of spherical (cylindrical) shells made from different materials. The results could also be extended to the quasiperiodic dynamic loading case (consult Ref. [13]), i.e. when the dynamic load amplitudes and frequencies vary slowly with time. The formulae (25), (24) and (26) are still valid--the only change is that the quasiperiodic dynamic loading program takes the place of the quasistatic loading program ~r~(r, t) in (5) and (29) (the quasistatic temperature field O(r, t) is kept unchanged). However, one should keep in mind that in the dynamic case the influence of the coupled effects of the thermomechanical field and the thermal conductivity may sometimes become significant and cannot be neglected as has been done in our quasistatic scheme. Acknowledgements--This work is supported by the Program of Basic Research in Natural Science.

REFERENCES 1. D. A. Gokhfeld and O. F. Cherniavski, Limit Analysis of Structures at Thermal Cycling. North-Holland, Amsterdam (1980). 2. A. R. S. Ponter and S. Karadeniz, A linear program under bound approach to the shakedown limit of thin shells subjected to variable thermal loading. J. Strain Analysis 19, 221 (1984). 3. T. H. Hyde, B. B. Sahari and J. J. Webster, The effect of axial loading and axial restraint on the thermal ratchetting of thin tubes, lnt. J. Mech. Sci. 27, 679 (1985). 4. J. A. Konig, Shakedown of Elastic-Plastic Structures. Elsevier, Amsterdam (1987). 5. J. Bree, Plastic deformation of a closed tube due to interaction of pressure stresses and cyclic thermal stresses. Int. J. Mech. Sci. 31, 865 (1989). 6. M. Robinson, The application of stress resultant theory for incremental collapse of thin tubes. Int. J. Mech. Sci. 33, 805 (1991). 7. F.A. Leckie and R. K. Penny, Shakedown loads for radial nozzles in spherical pressure vessels. Int. J. Solids Structures 3, 743 (1967).

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P.D. Chinh

8. D. Weichert and J. Gross-Weege, The numerical assessment of elastic-plastic sheets under variable mechanical and thermal loads using a simplified two-surface yield condition. Int. J. Mech. Sci. 30, 757 (1988). 9. A. Sawczuk, Evaluation of upper bounds to shakedown loads for shells. J. Mech. Phys. Solids 17, 291 (1969). 10. W. T. Koiter, General theorems for elastic-plastic solids, in Pro#ress in Solid Mechanics (Edited by I. N. Sneddon and R. Hill), p. 165, North-Holland, Amsterdam (1960). 11. V. I. Rozenblum, On analysis of Shakedown of uneven heated elastic-plastic bodies. P M T F 5, 98 (1965). 12. D. C. Pham, Shakedown of bars subjected to cycles of loads and temperature. Int. J. Solids Structures 30, 1173 (1993). 13. D. C. Pham, Extended shakedown theorems for elastic plastic bodies under quasiperiodic dynamic loading. Proc. R. Soc. Lond. A439, 649 (1992).