Structural Optimization 16, 226-229 (~) Springer-Verlag 1998

B r i e f N o t e

Opt im al heads

des ign of a hor izonta l circular tank wi th e l l ipsoidal

K. M a g n u c k i

Institute of Technology, Pedagogical University of Zielona Gdra, al. Wojska Polskiego 69, PL-65-625 Zielona G6ra, Poland

A b s t r a c t A horizontal circular tank, supported at both ends, is loaded by internal or external pressure. In the design process of such structures the proper choice of basic dimensions to ensure minimal mass may cause a problem. In this paper the optimal radii, lengths and wall thicknesses of a series of tanks of given capacity have been defined. The results of numerical analysis are presented in the form of diagrams.

1 I n t r o d u c t i o n

Shells may be optimized by parametric or variational shap- ing. A detailed classification of optimization problems, to- gether with many examples,, particularly focused on varia- tional shaping, is given by Zyczkowski (1990). For practi- cal solutions of shell optimization computer methods are re- quired. Ringertz (1992) presented numerical methods for op- timization of nonlinear shell structures. The methods were illustrated by two examples of parametric optimization re- lated to a cylindrical panel with a rib and a cylindrical panel with a circular hole. Zhou and Haftka (1995) presented a gen- eralization of continuum optimality criteria methods (COC). They developed multiple displacement constraints and mul- tiple load conditions. Kru~elecki (1997) determined optimal dimensions of a barrel-shaped cylindrical shell, loaded by an axial force and external pressure. For solution purposes the use was made of the concept of uniform stability of the shell. Magnucki and Szyc (1996) determined the optimal, rectan- gle delimited shape of the cross-section of a cylindrical shell loaded by constant internal pressure.

The paper considers a horizontal circular tank with el- lipsoidal heads, loaded by internal or external pressure (Fig. 1). The tank, of required capacity Vo, may be designed in many ways, e.g. as a short tank of big diameter, or as a long one, of small diameter. The optimization problem is, in this case, of parametric type and resolves into determination of its radius a, length L, wall thicknesses t 1 and t 2 , giving the least mass of the tank. Wilby (1977) similarly formulated the problem of reasonable choice of the basic dimensions of a vertical cylindrical tank of circular cross-secti0n.

2 S t r e n g t h cons t r a in t

A vertical cylindrical tank of circular cross-section (Fig. 1) is subject to two different loads. The first load is an internal

pressure Pi, being the sum of the hydrostatic pressure of a liquid having mass density Pro, contained in the tank, and an additional uniform pressure Po. The second load is constant external pressure Pext.

Fig. 1. Circular cylindrycal tank with ellipsoidal heads

In the first load case the tank walls are subject to inter- nal normal pressure, provided that the dead weight of the structure is neglected

Pint = P(~) = PO + ~'ma(1 - - cos!a), (1)

where 7m = g *pm is the specific weight of the medium, and g = 9.81 m/s 2.

Making use of the membrane theory of shells, discussed in detail by Fliigge (1973) or recently by Farshad (1992), longitudinal az and circumferential a~ stresses of the cylin- drical tank wall can be determined. Maximal values of these stresses occur in the lower part of the central cross-section and amount to

= 2t - - + + ' 7ma

rma2( PO) (2)

where A = L/a is a dimensionless parameter of the cylinder length.

The effective (ttuber-Mises) stress is

~rre d = - ~z(r~ + o -2

4t 2 + 2 7 m a ] ' (3)

The strength condition %ed -< C~allow gives the required thickness of cylindrical part of the tank, as

_ 4aallow ~ + - 1 , (4)

where C~allo w is the allowable stress. Tank supports may cause important increase of the stress

due to bending. However, a suitable shape of the support may reduce its unfavourable effect. An exemplary solution of such a support, obtained by means of FEM, was obtained by Maguncki et hi. (1997). In the region of contact between the ellipsoidal head and the cylindrical part of the tank a local stress concentration arises, as discussed by Spence and Tooth (1994). On the grounds of the theory of boundary dis- placements numerical analyses and experimental verification of stress concentration occuring in this region were carried out by Magnucki et al. (1994). The least concentration of reduced stress in the tank of dimension proportion b/a = 0.5 occurs when the cylinder to ellipsoid thickness ratio satisfies the condition

e

Xl~ = X~-' (5)

where x I = t2 / t l , x 2 = a/ t2 , c = 0.84, a = 0.0572. The minimal value of stress concentration coefficient is

_ ,g(cyl) 1.12, where (r (cyl) (v'~/2) x 2 is f~r = Ore dmax/ red = red = the effective stress in cylindrical shell. Taking the above into account one can modify (4), therefore the following wall thick- ness of cylindrical part of the tank is assumed:

t2 - - 4~allowTrna2 ~ 8 ( l+~--~ma/p0 "]2 q_ ( ~ _ 1 ) 2 (6)

Considering (5) the head thickness is

t I = ~ x ~ . (7)

The wall thickness of cylindrical part of the tank determined this way corresponds only to the first load case and results from the strength condition.

3 Stabi l i ty cons t ra in t

In the second load case the tank walls are subject only to constant external normal pressure Pext, provided that the dead weight of the structure is neglected. Therefore buckling may occur. A comprehensive description of the stability of shells subjected to various loads is given by Volmir (1967). He drew attention to simplification made in the theory of shells and their effect on critical loads. His general considerations were illustrated with many detailed examples. Making use of his conclusions one may write the following formula for the critical load of a circular cylindrical shell, loaded with external pressure:

PCR = E ( ~t2~3min ; Q ; \ h i n k l t j '

227

@2 _~ n2) 2 n2) 2 1 2 uk 2)

k4 ~2 '

,

where k --= rr~ = ~, n is the number of circular waves, E is Young's modulus, and ~, is Poisson's ratio.

The stability condition of the tank is of the form Pc <_ PCR, where Pc = fb * Pext, and fb is the buckling safety factor. Due to the condition and taking (8) into account, one can derive algebraic equation of the third-order

�9 - ,2x - " 0 : 0 , (9)

where

E k 4 B 2 = - -

Pc ( lk2 + n 2 _ 1) (k 2 + n2) 2 '

B 0 ---- 12 (1 - ~,2) Pc l k2 + n 2 - 1

from which the value of dimensionless parameter x 2 = a / t 2 may be obtained for the assumed value of the external pres- sure Pc. Thus, the wall thickness t 2 is obtained, delimited by the stability condition.

4 Objec t ive func t ion

The tank is a thin-walled structure, consisting of two ellip- soidal heads, characterized by the ratio b/a -= 0.5, and a cylindrical shell (Fig. 1). The capacity Vo of the tank is usu- ally slightly oversized, as for the mathematical description a middle surface, instead of external and internal ones, is taken into account. Hence

Vo = 2VI + V2 =~ra3 ( ~ + )~) , (10)

where V 1 = ~a 3 is the capacity of ellipsoidal head, V 2 = ~ra2L is the capacity of the cylindrical shell.

Therefore, a dimensionless parameter, describing the tank length for given capacity, may be written as

~_yo 2 ~ra 3 3 ' (11)

The mass of the tank

ms -- 2m I + m2 = 27rpsa 2 (~tl -I- At2) , (12)

where m 1 = rc~pstl a2 is the mass of ellipsoidal head, ~ =

1+ 4 - t - l n (v~-t- 2) ~ 1.38017 m 2 -- 2~rpst2aL is the mass of cylindrical shell, and Ps is mass density of the vessel material.

228

Introducing the head thickness (7) one obtains the objec- tive function in the following form:

m s =

(~x~ ) 27ra2t2 ~ e 2 + ~ Ps -~ 2~ra2t2 (1.6431x~ +)~) P s , (13)

where the dimensionless length parameter A is determined by (11), while the wall thickness t 2 is given by (6), when a strength condition prevails, or by the root of (9), when a sta- bility condition is decisive. Thus, the optimization problem is of parametric type and consists in such a choice of the a radius of the cylinder, for which the objective function, i.e. the tank mass, takes the least value.

5 N u m e r i c a l r e s u l t s

The numerical analysis was applied to the series of steel tanks of capacities V 0 = 25, 50, 100,200,300 m 3. In the first load case the tanks are filled with water of specific weight 7m = 9.81 kN/m 3 and, additionally, subject to constant internal pressure P0 = 2.5 MPa. In the second load case they are affected only by constant external pressure Pc = fbPext = 0.1 MPa. The following properties of steel have been assumed: Young's modulus E = 2.05,105 MPa, Poisson's ratio u = 0.3, and allowable stress aallow = 330 MPa or gallow = 250 MPa.

m S

[kg]

30.10I

29-1 ff

28.1 0 ~

27.10 ~

1.2 1.3 1.4 1.5] 1.6 1.7 118i 1.9 a[m~ o,I ~1"

"r ",r-

Fig. 2. Graph of the objective function

The objective function (13) of every considered tank was of similar shape. For example, Fig. 2 shows the objective function of the tank of capacity V 0 -- 200 m 3. The minimal value of the objective function, i.e. the mass of the structure, is achieved when the wall thickness t 2 of the cylindrical shell simultaneously approaches both strength and stability con- ditions. For the case limited only by the strength condition, the objective function has a minimum (M2) , which corre- sponds to the least function value. Optimal dimensions, the radius aop t and thickness t2opt, corresponding to point M 1 for assumed series of tanks are shown in Figs. 3 and 4, in the form of diagrams. The dashed lines represent an optimal solution for the tank made of steel of less allowable stress

~allow = 250 MPa.

1

Fig .

0 20 30 50 100 200 300 500 Vo[m 3]

3. Optimal radius aopt - - capacity 110 graph

t2opt

17

16

15

14

13

12

11

10

9

8

7

J / . , l ~ ""

J/ / /

, , / t /

�9 J f f /

10 20 30 50 100 200 300 500 Vo[m 3]

Fig. 4. Optimal thickness t2opt - capacity V0 graph

6 C o n c l u s i o n s

Optimal solutions achieved for a cylindrical pressure tank are characterized by almost constant dimensionless length parameters ~ = L / a and z 2 : a / t 2, For the other tanks of the series, made of the steel of allowable stress equal to crallo w -- 330 MPa, these parameters are of the following values A -- 9.36,9.37,9.44,9.50,9.56 and z 2 = 130.0,129.8,129.4,129.0,128.6. The presented solution for

the parametric opt imizat ion allows reasonable acceptance of the dimensions of pressure tanks.

R e f e r e n c e s

Farshad, M. 1992: Design and analysis of shell structures. Dor- drecht: Kluwer

Fliigge, W. 1973i Stresses in shells. Berlin, Heidelberg, New York: Springer

Kru~elecki, J. 1997: On optimal barrel-shaped shells subjected to combined axial and radial compression. In: Gutkowski, W.; MrSz, Z. (eds.) Proc. WCSMO-P, Second World Congress on Structural and Multidiseiplinary Optimization, Vol. I, pp. 467-472. Polish Academy of Sciences

Magnucki, K.; Szyc, W. 1996: Optimal design of a cylindrical shell loaded by internal pressure. Struct. Optim. 11,263-266

Magnucki, K.; Szyc, W.; Stasiewicz, P. 1997: Selection of design parameters of a cylindrical pressure vessel together with its sup- port. Silesian Technical University in Gliwice, Proe. XXXVI-th

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Syrup. on Modelling in Mechanics, Vol 4, pp. 211-216

Magnucki, K.; Kaczyfiski, R.; Walczak, M. 1994: Minimization of stress concentration in pressurized cylindrical vessel (in Polish). Zagadnienia Eksploatacji Maszyn, Polish Academy of Sciences, 2, 321-331

Ringertz, U.T. 1992: Numerical methods for optimization of non- linear shell structures. Struct. Optim. 4, 193-198

Spence J., Tooth, A.S. 1994: Pressure vessels design concepts and principles. London: Chapman & Hall

Volmir, A.S. 1967: Stability of deformations structures (in Ru- sian). Moscow: Izdatielstvo Nauka

Wilby, C.A. 1977: Optimization of design of circular tanks. Proc. Inst. Civil Engrs, Part 2, 63, 921-924

Zhou, M.; Haftka, R.T. 1995: A comparison of optimality criteria mathods for stress and displacement constraints. Comp. Meth. Appl. Mech. Engrg. 124, 253-271

Zyczkowski, M. (ed.) 1990: Structural optimization under stability and vibration constraints. CISM Udine, Vienna: Springer

Received Jan. i, 1998 Revised manuscript received April P2, 1998