Improved synthesis capability for “T” ring-stiffened cylindrical shells under hydrostatic pressure

Computers& S~r~cturcs, Vol. 6, pp. 339-343. PergamonPress 1976. Printed in Great Britain

IMPROVED SYNTHESIS CAPABILITY FOR “T” RING-STIFFENED CYLINDRICAL SHELLS UNDER HYDROSTATIC PRESSURE-l3 M. PAPPASB Department of Mechanical Engineering, Newark CoUege of Engineering, New Jersey Institute of Technology, 323 High Street, Newark, NJ 07102,U.S.A. (Receiued 30November 1975) Abstract-The new “Direct Search-Feasible Direction” (DSFD) nonlinear mathematical programming optimization algorithm is applied to the design of stiffened submersible shells. An automated design capability for this problem (SBSHL6) is described whereii the program will generate the least weight design by locating the optimal, or near optimal, values of skin thickness, web thickness and height, flange thickness and width, and stiffener spacing given the design parameters such as shell size, immersion pressure, shell eccentricity, materials properties, and minimum natural frequency. Constraint equations contrdl, general, panel (between stiffener), web, and flange instability, skin and stsener yielding, and minimum natural frequency. The DSFD procedure appears capable of reliably locating optimal designs whereas earlier attempts in investigations using other optimization methods, including the popular SUMT procedure, failed to provide optimal solutions to the same problem. This and an earlier detailed comparison study strongly suggest that SUMT is not a reliable procedure for structural optimization while DSFD seems to provide reasonably reliable performance. Designs generated by SBSHL6 are presented and compared with those of the earlier studies. The results of a series of synthesis runs from widely separated starting points are also presented. The designs developed by SBSHL6 are substantially lighter than those reported earlier. The multipath runs for each set of parameters studied all converged to similar designs of essentially identical weights demonstrating program reliability.

1. INTRODUCTION

(MP) procedures represent one of the most versatile and powerful methods available for treating optimal design problems[ll. Such procedures have been widely applied to a variety of structural problems including trusses, beams and shells [2,31. Where the problem is linear in the MP sense, as in certain truss problems, available linear MP methods guarantee a global optimum in a finite number of steps[4]. Unfortunately, most structural optimization problems are nonlinear. The nonlinear MP problem is quite formidable and none of the available procedures can guarantee a solution except under certain restrictions[5]. The difficulties are greatly compounded when constraints are used, as they always are in structural problems. Most available nonlinear optimization procedures appear rather unreliable[6]. Yet, the majority of investigators applying these methods fail to discuss this reliability problem or offer any indication that they have taken steps to determine that the designs they offer as optimal are indeed optimal. This writer has found that it is unrealistic to assume that merely because one has applied a widely used nonlinear MP procedure to a given problem that it will produce optimal solutions. There must be confirmation studies to determine if the selected scheme works on the particular problem to which it has been appliedi’l, 81. For example, when the DSDA procedure, a method that Mathematical

programming

proved equal or superior in reliability to all the schemes tested by Eason and FentonlQ 91 (it solved 9 out of 10 of Eason and Fenton’s test problems), was applied to the optimal shell problem described in Ref. [81 it failed to reliably locate the optimum when all the variables were considered as independent (the six variable problem). Special techniques such as variable coupling and problem subdivision had to be employed to obtain solution reliability. Bronowicki et al. [ 101in their extension of Ref. [81using the popular, but seemingly relatively unreliableI61, SUMT procedure[ll] (it solved only five of ten of Eason and Fenton’s test problems) apparently encounter similar difficulties. The designs originally presented by Bronowicki et al. as optimal for their type I problem were found by the author to be non-optimal using a modified version of the shell design program used in Refs. [ 12-141. This paper extends the work of Refs. [8,10,12] by applying a new nonlinear, direct search-feasible direction (DSFD), optimization procedure of apparently superior speed and reliability[l5] to the submersible, ringstiffened, cylindrical shell, design problem. Thus, it provides an additional comparison between the DSFD, SUMT and DSDA procedures on a difficult design problem. The use of the effective DSFD algorithm allows the development of an automated optimal design capability for submersible, cylindrical, ring-stiffened shells that can generate optimal designs to the original six variable problem with a reasonable degree of reliability.

Vhis work was partially supported by the Officeof Naval Research under Contract Number ONR-OOOM-71A-012_. SPresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, D.C., 29-31 March 1976. OAssociate Professor of Mechanical Engineering, Newark CoUege of Engineering, New Jersey Institute of Technology, Newark, NJ 07102,U.S.A.

2. OPTIMIZATION The

PROCEDURE

optimization problem

The structural optimization problem may be stated as follows: Find those values Zi of the real design variables xi such that

339

f(&)=minf(xi)

i=1,2...1

(I)

M. PAPPAS

340

subject to the constraints gj(xi)~O

j=i,Z...J

(2)

and X,’c: xi s x,“.

(3)

The “objective” or “merit” function f(xi) may represent the weight or cost of the st~cture or any quan~tative characteristic that is to be optimized. The constraints gi(Xi)usually control behavior such as stress and buckling or geometric relationships. “Side” or regional constraints of the form of eqns (3) limit the variable range. Although these constraints are a special form of eqns (2), they are separated here because they are treated somewhat differently than those of (2). The general mathematical programming strategy is to start from some arbitrary initial design xi”and then on the basis of the local properties of the objective and constraint functions move to an improved design xi’ such that f(xil) < f(xr@,L The process is then repeated untit no further improvement can be made. The best design located is usually called the optimum. The basic op~imaZsearch The “pattern” search of Hooke and Jeevesff61 modified to substantially improve its performance[l7] is used here as the basic optimal search. Since the basic search scheme is formulated to treat unconstrained opti~zation problems, the constrained problem of (Q-o-) is t~nsfo~ed to the form: Find

The justification for the use of this penalty function form is discussed in detail in Ref. 1177. Violations of eqns (3) are handled by prohibiting moves that violate any of these equations. The original Hooke and Jeeves pattern search procedure is modified so that instead of taking the local exploration steps in the direction of the coordinate axes they are taken in, and normal to. the direction of the previous pattern move( 171. The secondary search

Unfortunately, the pattern search, even with the rotating coordinate improvement, is not particularly reliable ]6,17]. Thus, some method is needed to determine if the point of pattern search termination is optimal and if it is not, to determine a direction in which to restart the search. Pappas and Amba-Raol71 and Siddal[lli] apply a secondary search at points of pattern search termination. Unfo~unately. the earlier secondary search strategies although more powerful locally than the pattern search do not provide a rigorous procedure for confirming optimality. The direction finding procedure of Zoutendijk [191can, however, be used to confirm the optimality of a point of pattern search termination and determine a direction in which to restart the pattern search if the point is not optimal. The direction finding problem can be stated: Given the set s,, fmd the set Sothat results in a max ff

(111

for which F(b) = min F(X)

(4) u>o

(12)

(si~~~~(x,)+~
(13)

where

- (s,)‘Vgj(x, ) + W,o 5 0 j E f, Here P(z) is the largest of the penalties pJ(xi) which are of the form (6)

(141

- 1 5 sk 5 0 k E K,,

(151

OS sk I 1 k E K,‘.

(16)

Here (si)’ indicates the transpose of vector s,, WI is a weighting parameter, the set .L contains the active constraints .g,(x#l gi > - ez, then Ai

=21f(~i)-j(Xi tAXt)l/[g/(Xi)-gj(Xi

+AxOI

(8)

where AX, = ervjb

)/IVj(XtI/.

(9)

The quantity el is an arbitrary small real number representing the size of the step, Axi, taken in the direction of the gradient to f(x,). It may be seen that a penalty is used only if a constraint is violated. If the violation of any of these constraints is greater than an arbitrarily specified amount ez, then A, is not calculated from eqn (8) but is made arbitrarily large. That is, when gi(xi)<-e2 A, = I(

1

where K is an arbitrary large number.

(17)

where e3 is small arbitrary positive constraint defining “activity”, and K,-K,’ constitute the active upper and lower side constraints, respectively. Zoutendijk[l9] shows that if the solution s, is a null vector, then the point is a local optimum and if not then the direction of s, is the best feasible direction. Equations (1l&(16) constitute a linear pro~amming problem with the variables si and o. Equation (I 11is the objective function and the remaining equations the constraints. The solution si can be obtained reliably and efficiently using any suitable linear pro~amming method such as the simplex procedure. The combined strategy

The details of the overall DSFD procedure are given in Ref. [15]. Essentially, the search is started from an arbitrary starting point. The point need not be confined to the feasible regions as in most other procedures. The rotating pattern search is then applied until it fails to generate any design improvement. This point is checked

341

“T” ring-stiffenedcylindricalshellsunder hydrostaticpressure for optimality by solving Zoutendijk’s direction finding problem. If the point is not optimal, the pattern search is restarted. This procedure is repeated until optimality is confhmed or a predefined level of convergence is achieved. The procedure has been demonstrated to be reasonably fast and reliable[H]. A code CADOP2 based on this method was the only one of the twenty codes studied that solved all of Eason and Fenton’s test problems[6,15]. Furthermore, it was faster than any of the reasonably reliable codes (a code solving at least seven of the ten problems). By comparison, the very popular SUMT procedure solved only half the test problems and required an average of twelve times greater CPU time than CADOP2 on the problems both codes solved. This new, superior, numerical optimization method was thus chosen to treat the difficult six variable submersible shell design problem that had failed to yield to the earlier DSDA and SUMT procedures in an effort to produce an improved automated design capability for such problems. 3. TRR SHELL DESIGNPROBLEM

This paper describes a procedure for the automated optimal design of a hydrostatically loaded, ring-stiffened, circular, cylindrical shell. It extends the work of Refs. [8,10,12] by coupling the problems described in these works with an optimization algorithm that is capable of reliably locating optimal designs where all the design variables are independent. This was not possible with the DSDA and SUMT optimization procedures used earlier on these problems. The objective function and constraints are as follows:

f(xi)= wo

has no real substance. For this reason, these problems were not considered here. The Type I problem of Ref. 1101 which is essentially identical to that of t121except that a minimum in uacuo natural frequency constraint is employed is treated here to allow comparison of the DSFD procedure with the SUMT algorithm used in HOI. The constraints used control: = gross buckling gz = shell (between stiffener) buckling g, = shell yielding g4= stiffener yielding g5= stiffener flange buckling g6= stiffener web buckling g7 = minimum natural frequency. gl

The equations used for g3 and g, are eqns (4~(17) of Ref. [8] and those used for gs and 86 are essentially eqns (ll), (20)and(21) of Ref. [lo]. Reference [lo] uses the same equations as [8] for the shell and stiffener yield constraints g3 and g, except that [lo] ignores the effect of eccentricity on stiffener stress. The basic behavior prediction equations for constraints g,, gz and g, are adapted from Ref. [12] which uses a procedure described in [21]. Gross buckling control is achieved by stating that g, =

P:, -F&P Pl,

(b) Ring and skin detail

Fig. 1. Shelldesign variables.

(20)

where n is the number of circumferential waves and m the number of axial half-waves. The equation for p I, and the method for finding the minimum buckling load p F, are given in Ref. [12]. The equation for the shell buckling constraint is similar to that for gross buckling except that x5 the distance between stiffeners replaces I,, the overall shell length in eqn (2) of Ref. [12]. The minimum natural frequency constraint is written as 01 -&Ii” g1= ->O 01

(21)

where wtin is the specified minimum frequency and O, the lowest natural frequency of the structure. o, is found from ol=mino(n,m)

segment cross section

(19)

where FS, is the factor of safety for gross buckling and p$ = minp,,(n, m)

(18)

where the six design variables xi are given in Fig. 1. WDis the weight of the shell divided by the weight of liquid displaced. The objective function is identical to that used in Ref. [8]. Reference [lo] uses a simplified but somewhat less accurate form for the weight/displacement ratio. Reference [lo] formulates two problems (Types II and III) involving the maximization of the separation of the two lowest in uacuo natural frequencies. In uacuo dynamic behavior, however, bears little resemblance to the behavior of a shell submersed in water[20]. It is felt, therefore, that an optimization problem where the objective function is based on such inaccurate estimates

(a)Shell

20

(22)

where o is given by eqns (31)-(333 of Ref. [213. The II and m producing the minimum frequency are located in exactly the same fashion as for the minimum buckling load. The minimum natural frequencies of both the entire shell and the segment of the shell between stiffeners are determined and the lowest value is used in eqn (22). The frequency equations are derived using the smeared stiffener orthotropic shell approach also used in Ref. [12]. The effects of stiffener placement (interior or exterior) and torsional rigidity are considered but imperfection sensitivity is ignored. Experimental results indicate that such equations produce reasonably accurate results for hydrostatically loaded stiffened shells where the stiffener

342

M. PAPAS

spacing is small compared to the buckling wave length as is the case for the range of parameters studied in this paper. Imperfections do not play a major role in such shells [22,23]. 4. RESULTS

A computer program called SBSHL6 coupling the DSFD optimization procedure and problem described above was used to repeat some of the shell design studies of Refs. [8, lo] in order to evaluate the performance of SBSHM on this problem and to allow a comparison of the results generated with those of Refs. 18,lo]. All studies use the following design parameters unless otherwise indicated; shell midplane radius R = 5.029 m (198 in.); shell length L, = 15.09m (594in.) shell eccentricity of zero; immersion depth 304.8 m (1000ft); specific weight of immersion fluid (sea water) yW= 1.0256g/cm2 (0.282lblin3), specific weight of shell material A, = 7.733 g/cm3 (0.282lb/in.‘) Young’s modulus 20.68 x lo6 N/cm” (30 X lo6 psi), Poisson’s ratio p = 0.30 and allowable yield stress of 41,360 N/cm’ (60,000psi) for the shell and stiffener material. Factors of safety of 2 are used for all buckling constraints and a minimum natural frequency of zero is specified. Ail shells use interior stiffeners. Table 1 demonstrates the reliability of the new optimization procedure. The starting points, in mm, used for this study were: (Xi)I = (0, 0, 0, 0, 0,O) (xi): = (12.7, 12.7, 12.7,127,254, 127) (xi)3= (25.4,25.4,25.4,254,508,2543 (x1)1= (38.1,38.1,38.1,381,762,381). It may be seen that a series of runs from widely separated starting points converged to similar designs with weights that are within 0.15% of the lowest. This behavior is indicative of reliable optimization algorithm performance and the absence of local optima. The designs generated by SBSHM are substantially lighter than those given in Ref. [S] due primarily to the Table

I, Convergence by SBSHL6 to an optimal configuration using widely separated starting Ref.

wD weight/ displacement x1,

akin

x2,

web

X3' X4' X5* X6'

91' 42' 43' 94' 45' 46'

ratio

[El

0.1357

thicknem,

mm

thickness,

flange

mm

thickness,

(in) (in)

mm

flange

width,

stiffener

mm

(in)

spacing,

height,

mm

mm

(in)

(in)

Starting Point 2

0.10303

6.7158 CO.26441

6.6751 13.2628)

6 7 5 i0 (O.Zh5'1)

6.700s (O.26381

33.609 (1.3232)

27.277 (1.0739)

12.893 (0.5076)

15.133 (0.59581

125.308 (4.933)

25.293 10.9958)

30.322 Il.19381

67.114 (2.6683)

55.515 (2.18641

580.29 (22.846)

405.79 (15.9763)

407.67 (16.050)

412.41 (16.2391

463.07 (17.1681

292.68 (11.5238)

296.93 (11.690)

295.14 (11.777)

0.622

0.000

cl.000

buckling

0.003

0.4hO

0.438

0.004

0.000

0.000

yield

0.339

0.088

0.088

buckling

0.003 *

1.000

0.993

0.041

0.001

web

buckling

*web

buckling

controlled

by

0.10312 28.067 11.1050)

(1.0591)

buckling

flange

0.10113 28.082 (I.10561

2h.226 CC.95381

panel

stiffener

Starting POLnt 4

28.128 11.Oi41

gross

sitin yield

starting Point 3

Point\

28.092 (1.1060)

9.9772

(in)

Starting Point 1

0.10306

26.901

(0.3928)

web

elimination of an unrealistic stiffener buckling constraim used in [8] but not used here or in /IO. 121. The shell buckling equations used here are considerably more complex than those of 181.This added complexity in addition to the increase in problem dimensionality and difficulty resulting from the uncoupling of the stiffener variables greatly increases the computational effort required for solution. The program SBSHL6 typically requires about 100 times more CPU time than that of [S]. Thus, the added sophistication is obtained at substantially increased cost. Still, if a reasonable starting point is specified, the cost of solution is not excessive (normally less than Smin CPU time on an IBM 360-165). The buckling equations used in [X]can unfortunately lead to invalid designs for certain ranges of parameter values [ 121 since the interior buckling minimum tm 3 I) is not considered. This is also true of the procedure of Ref. [IO] where m >6 is ignored. Thus, this added complexity is justified. Table 2 demonstrates the superiority of DSFD over the SUMT procedure used in [lo] which failed to locate the minimum[l3, 141producing designs substantially heavier than those presented here. The shell parameters utilized in [lo]are used for this study. They are identical to those given above except that CL= OX y, = 8.225&m’ (0.30 lb/in.‘), tirnln= 12Hz, and factors of safety equal to unity are used for the buckling constraints. The design of Ref. {IO]is obviously not optimal since an optimal design should converge to the minimum frequency constraint. Neither is the design a local optimum. This design was used as a starting point for a synthesis run using SBSHL6. The search immediately located a better nearby design and moved to a design essentially identical to that presented in Table 2. Thus, the design of Kef. \ 101 does not appear to be a local (~ptimt~rn. The design presented here as generated by SBSHLh used the starting point (x?) = (254, 25.4, 25.4, 254, 508, 254). The design of Ref. 1141 cannot be considered as representative of the performance of the procedure of [lo] since it used a near optimal starting point. The performance of SUMT here, coupled with the evidence developed in an earlier comparison study[6], strongly suggests that SUMT simply cannot cope with this six variable shell design problem.

settinq

x6 = 18 X3.

406.311 (15.9961

343

” T” ring-stiffenedcylindricalshellsunderhydrostaticpressure

Table2. Comparisonof designsdevelopedby SBSHMwiththoseof Refs.[lo, 141 Ref.

h'~, weight/ displacement Xl' x2,

skin web

thickness,

x4,

flange

width,

x5,

stiffener

x6'

web

0.11274

0.10746 29.522 (1.1623)

(in)

9.563 (0.3765)

6.0274 (0.2373)

5.1054 (0.2010)

mm

11.951 (0.4705)

7.8003 (0.3071)

4.6533 (0.1832)

(in)

nun (in)

spacing,

height,

mm

SBSHL6

30.622 (1.2056)

(in)

mm

flange

X3'

[141

0.13317 mm

thickness,

Ref.

30.754 (1.2108)

ratio

thickness,

I101

mm

(in)

(in)

150.80 (5.937)

448.66 (17.664)

263.22 (10.363)

853.49 (33.602)

766.32 (30.170)

424.51 (16.713)

279.91 (11.020)

239.04 19.411)

497.56 (19.589)

gl,

gross

buckling

Not

given

Not

given

0.234

g2,

panel

buckling

Not

given

Not

given

0.742

g3,

skin

g4,

stiffener flange

95' g6, web

g7, min. %'

SZ

W2'

SZ

Not

given

Not

given

0.000

yield

Not

given

Not

given

0.081

buckling

Not

given

Not

given

0.039

Not

given

Not

given

0.000

Not

given

Not

given

yield

buckling nat.

freq.

HZ

5. CONCLUSION

The DSFD procedure seems capable of generating

essentially optimal solutions to the six variable submersible circular ring-stiffened shell design problem. A problem on which earlier investigations using the SUMT and DSDA procedures encountered considerable difficulty in achieving even near optimum designs. Thus, as in an earlier study, the DSFD procedure was found to be more effective than DSDA and SUMT. The use of DSFD thus allows the development of an automated optimal design capability for such shells which seems superior to those now in existence. It seems superior to that of [lo] since it apparently can reliably generate optimal or nearly optimal solutions while the latter apparently cannot. It is preferable to that of [8] since it uses more advanced shell buckling equations and is preferable to [8] and [12] since it allows complete uncoupling of the stiffener design variables.

0.000

28.12

12.03

12.00

49.39

22.30

20.50

9. M. Pappas, Performance of the “Direct Search Design Algorithm” as a mathematical programming procedure. AIAA J. 13, 827-829 (1975).

10. A. J. Bronowicki, R. B. Nelson, L. P. Felton and L. A. Schmit, Optimum design of ring stiffened cylindrical shells, UCLAENG-7417, UCLA. School of Entineerinn and Anolied Irm ~~ Science (1974).Published in revised ?orm: A-IAA J. 11. A. V. Fiaco and C. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968). 12. M. Pappas and A. Allentuch, Pressure hull optimization using general instability equation admitting more than one lingitudinal buckling half-wave. J. Ship Res. 19, 18-20 (1975). 13. L. P. Felton, Private communication (24 May 1974). 14. L. P. Felton, Private communication (26 July -1974). 15. M. Pappas and J. Y. Moradi, An improved direct search mathematical programming algorithm. j. Engrg. Indust 97B, 1305-1310 (1975).

16. R. Hooke and I. A. Jeeves, Direct search solution of numerical and statistical problems. J. Comput. Mach. 8,212-229(l%l). 17. M. Pappas, Use of direct search in automated optimal design. REFFXENCES J. Engrg. Indust. 95B, 395-401(1972). 1. C. Y. Sheu and W. Prager, Recent developmentsin optimal 18. J. N. Siddall, OPTISEP designer optimization subroutines structural design. Appl. Me&an. Reo. 21, 985-992(1968). ME/DSN/REPl. Faculty of Engrg., MacMaster Univ, Hamil2. L. A. Schmit and R. L. Fox, An integrated approach to ton, Ontario, Canada (1971). structural synthesis and analysis. AlAA 1.3,1104-1112 (1%5). 19. G. Zoutendijk, Methods of Feasible Directions. Elsevier, 3. T. P. Kicher, Structural synthesis of integrally stiffened Amsterdam (1960). cylinders. J. Spacecraft Rockets 5, 62-67 (1968). 20. P. R. Paslay, R. B. Tatge, R. J. Wemick, E. K. Walsh and D. F. 4. G. B. Dantzig, Linear Programming and Extensions. PrinceMuster, Vibration characteristics of a submerged rington University, Princeton, New Jersey (1%3). stiffened cylindrical shell of tinite length. J. Accoust. Sot. Am. 5. D. J. Wilde, Optimum Seeking Mefhods. Prentice-Hall, 46(Part 2), 701-710(1%9). Englewood Cliffs, New Jersey (1964). 21. J. A. McElman, M. M. Mickulas and M. Stein, Static and 6. E. D. Eason and R. G. Fenton, A comparison of numerical dynamic effects of eccentric stiffening of plates and cylindrioptimization methods for engineering design, J. Engrg. cal shells. AIAA J. 4, 887-894 (1966). Indust., Trans. ASME %B, 196-201(1974). 22. A. B. Burns, Optimum stiffened cylinders for combined axial 7. M. Pappas and C. L. Amba-Rao, A direct search algorithm for compression and internal or external pressure. J. Spacecraft automated optimal structural design. AIAA J. 9, 387-393 Rockets 5, 690-699 (1%8). (1971). 23. B. 0. Almroth, A. B. Burns and E. V. Pittner, Design criteria 8. M. Pappas and A. Allentuch, Optimal design of submersible, for axially loaded cylindrical shells. Proc. AIAA /ASME 11th frame stiffened, circular, cylindrical hulls. J. Ship Res. 17, Structures, Structural Dynamics and Materials Conf. (1970). 201216 (1973).