Interactive reliability-based structural optimization

Compurers & Swucrures Vol. 19, No. 4, pp. 559-563, 1984 Printed in the U.S.A.

0045-7949/l%.+ $3.00 + .m Pergamon Press Ltd.

INTERACTIVE RELIABILITY-BASED OPTIMIZATION

STRUCTURAL

DAN M. ~RANGOPOL~

Department of Civil Engineering, University of Colorado at Boulder, CO 80309, U.S.A.

(Received 15 August 1983; received for publication 26 October 1983)

AhstracG-In the field of structural optimization the interactive computer-aided design (CAD) system becomes central as a tool for searching and evaluating alternatives to obtain the best design among given choices. Along these lines, this paper presents a new interactive reliability-based CAD optimization procedure for plastic framed structures. Both loads and strengths are assumed to be random variables. Load correlations and resistance correlations are also considered. The procedure to obtain the best design is based on the minimization of the total expected weight for a specified probability of failure of the structure.

INTRODUCTION

Structural optimization over the last two decades has successfully concerned itself with computer-aided de- signing of civil engineering systems. Most optimization studies to date report on how an engineering system should be designed in a non-statistical deterministic fashion. Recent developments in structural reliability with simultaneous rapid growth of computing power and speed of interaction with the computer provide the necessary conditions for the applica- tion of probabilistic concepts in the optimum design of complete structures. One of the main advantages of the probabilistic methodology is that it permits to smooth out the inconsistencies in reliability inherent in the deterministic optimization of structures.

This paper presents a reliability-based optimum interactive approach to the design of plastic framed structures. In this approach, the correlation between any two random loads and the correlation between any two random plastic moments is accounted for by using a technique which incorporates the .eflect of the statistical dependence between any two collapse modes. The optimum reliability-based design of a framed structure against plastic collapse consists in proportioning its members for a specified overall failure probability such that the structure meets minimum weight requirement. Minimization of weight for a specified reliability ensures an optimum distribution of safety amongst different failure modes. The optimization problem is a nonlinear constrained minimization. A computer program was prepared for solving this problem. The nonlinear programming techniques used in this program are the feasible direction method and the penalty function method. Both of these methods are utilized in many optimization problems that involve systematic searches for the best solution at the boundary between overdesign and un- derdesign.

In all reliability-based optimization problems it is necessary to subject the results to sensitivity analysis in order to determine the influence of the input statistical parameters, including distribution functions and coefficients of correlation, on optimum solutions. In this final detailing stage, where engineering judgement

TAssoeiate Professor.

and experience are required, the graphical interaction provides the user with the means to define potential solutions quickly, perform an evaluation of these solutions, choose the best solution and display the results.

The program has been written in Fortran and it was implemented at the University of Liege Computing Center. Recently, there have been considerable im- provements in system reliability bounds[l]. It should be noted that these new close bounds are considered in a new version of the program which is in its early stage of implementation at the University of Colorado at Boulder. The central computing facility at the Univer- sity of Colorado is equipped with the CDC Cyber 172 mainframe system. The present configuration has a core memory of 262 k 60 bit words, 20 peripheral and 2 central processing units. The system is supported by an extended core storage of 500 k words, 6 tape drives and approx. 2 billion discharges of disk storage. The mainframe is used for both batch and interactive processing. In addition a VAX 1 l/780 has been installed recently with 4 mbytes central memory and 512 mbytes of disk storage primarily for instructional computing. Moreover, a CALCOMP 1051 plotter is available for graphic illustration of the computational results.

ASSUMPTIONS

The basic assumptions adopted herein for the structural reliability optimization of frames are: (a) the members are prismatic and straight; (b) shear defor- mation is neglected; (c) the loads and plastic moments are assumed to be independent random variables; (d) the statistical dependence among loads and the statistical dependence among plastic moments is accounted for through the coefficients of correlation between loads and between plastic moments; (e) the necessary statistical information is assumed to be available; (f) the failure of a framed structure by collapse is defined as in the simple bending plastic theory of structures (an indeterminate plastic frame structure fails by collapse when enough plastic hinges make possible the occurrence of a plastic mechanism); (g) local fracture, instability and other possible causes of failure are avoided; (h) the topology and geometry of the frame are given; and (i) the design variables are the cross- sectional dimensions of the structural members.

559

560 D.M. FRANCOPOL

PROBLEM FORMULATION

In general, the reliability-based optimization problem in the automated design of plastic frames consists in the optimization of sizes of members for a specified overall failure probability of the structure. This problem can be stated as follows:

minimize W = W(M)

subject to Pf= PJM) 5 P;.

The design is defined by the vector

(1)

(2)

M={M,,M, ,..., M,JT (3)

of the mean values of the plastic moments capacities (member strengths) at all critical sections, the objective function W is the total structural weight (or cost) as a function of M, Pf = P,(M) is the probability of system failure and P; is the specified allowable failure probability of the frame.

An alternative approach minimizes the probability of collapse for a fixed weight (or cost). However, the approach generally adopted in the design process is to minimize the weight (or cost) for a specified probability of failure of the structure.

In mathematical terminology, the optimization problem given by eqns (1) and (2) is a constrained minimization. The objective function W(M) is a linear (deterministic) function of the mean values of plastic moment capacities M and the reliability constraint (2) is nonlinear. A good minimization procedure is needed because no explicit function for the overall failure probability P/exists without evaluating numerical integrations.

OPTIMIZATION PROGRAM

In developing this program care has been taken to make the input as simple as possible and to keep the storage requirements at a minimum. The engineering modeling, which means identification, description and enumeration of the n various failure modes of a structure is generated by the computer (a failure mode is defined as the minimum set of plastic hinges to form a collapse mechanism in the structure). Probabilistic calculations to determine individual mode failure probabilities P(Fi), i = 1,2,. . . , n, are then automatically computed for the following probability distributions: normal, log-normal, gamma, extreme value Type I (Gumbel), extreme value Type II (Frechet) and extreme value Type III (Weibull). Additional distribution functions may be added if desired.

The identification of failure mode probabilities leads logically to a combination problem to determine overall probability of failure of the structure. The probability of system failure P, can be expressed in terms of the modal failure events Fi, i=l,2,... , n, as follows[2-4]:

pr=p ( > iJ, Fl

where the symbol U signifies the union of the events. The total failure probability may be difficult to calculate in practice. However, it can be evaluated in terms of upper and lower bounds. These bounds on

system reliabilities include correlation between the failure modes. The bounds are given solely by the probabilities of the single mode failure events F,, F2, . . , F, and their pairwise intersections F, n F2, F2 fl F,, . . . , F,, _ , fl F,,. The total failure probability (4) is bounded as follows[l]:

P,2 P(F,)+ i max 0; P(F,)-‘2 P(4flE;) i=2 j=l

(5)

PII i P(F,) - i max P(F, fl F,). i=2 Jc’

(6) i=l

Upper and lower bounds on the true probability of system failure P, can thus be generated. The bounds (5) and (6) are very narrow (almost coincide) for all of these correlation coefficients between failure modes not larger than about 0.6 provided the total failure probability Pf is smaller than about 10e3, which is generally the case in usual structural plastic design problems. Without restriction on the mag- nitude of the failure mode correlation coefficients, very narrow bounds can be constructed on basis of (5) and (6) using Ditlevsen’s method of conditional bounding[l]. This method requires only a single numerical integration. Very close upper and lower bounds on the true probability of system failure P,, can thus be generated. Ditlevsen’s method is used to evaluate P,. For problems with many failure modes, this method is very efficient if the Vanmarcke’s concept of failure mode decomposition is also considered [2].

Since it is not always practical to solve the optimization problem, represented by eqns (1) and (2), by a single optimization method, the proposed solution is implemented using a double-phase interactive optimization technique. Two optimization methods are used in this procedure: the feasible direction method of Zoutendijk and the penalty function method [5-lo]. For problems of moderate size and complexity the penalty function method, in which constrained optimization is solved using techniques of uncon- strained minimization, is usually very efficient. The penalty function methods are not as efficient as the feasible direction methods for large size and high complexity system optimization problems.

Using as input an acceptable design and interaction, the computer is first used to determine the optimization algorithm which will be the most appro- priate in function of the size and complexity of the problem. However, in the optimization procedure proposed in this study, it is the author’s experience that for structures with up to about 3 design variables and about 30 failure modes the penalty function method is very efficient. The feasible direction method is generally recommended for more complex problems. The discussion in the following will be based on the use of this last method. In determining the direction vector {D} which changes the initial design in the minimum weight direction until the reliability constraint (2) is met, both conditions of feasibility (a move in that direction does not cause constraint violation) and usability (a move in that direction results in a reduction of the objective function W) must be satisfied. Then, a move is made in a direction which continues to reduce the objective

Interactive reliability-based structural optimization 561

function without leaving the feasible domain. A graphical interaction program is used to observe the convergence of the step-by-step optimization technique of the feasible direction method with the purpose of (a) choosing the steepest descent direction and (b) eliminating unnecessary cycles near the optimum. With this program it is possible to plot on the screen the objective function against the number of iterations. The procedure is continued until the optimum is reached or when the movement in the design domain produces no change in the value of the objective function. Since this optimum point may be a local minimum of the objective function, the design procedure should be started from a number of different initial points. A graphical structure display program can be used to interactively build the starting structural frame and to modify this starting structure at any later stage in the optimization process. A high degree of confidence in the optimum solution is achieved when the same design point is determined from several starting points.

The final structure chosen is not necessarily the one with the least weight[l l-151. In all reliability optimization problems it is necessary to subject the theoretical optimum solution (the structure with the least weight) to sensitivity analysis in order to determine the influence of the input statistical parameters, on optimum design variables and objective functions [16]. This type of study provides a means for evaluating the results obtained by optimization. In this final step of the optimization process a graphical interaction program can be used for sensitivity analysis. The sensitivity of the theoretical optimum structure to: (a) the type of load distribution; (b) the type of plastic moment distribution; (c) the coefficient of variation of load; (d) the coefficient of variation of plastic moment; (e) the coefficient of correlation between loads; (f) the coefficient of correlation between plastic moment capacities; and (g) the specified allowable failure probability, is interactively exam- ined in order to obtain a practical optimum design. This practical optimum structure could be a structure with slightly higher weight and/or slightly smaller probability of failure than the theoretical solution, but still considered better because of other engineering reasons (e.g. the dominant mode of failure for the practical optimum solution is not so danger- ous for the loss of human lives than that of the theoretical solution).

NUMERICAL EXAMPLE

Several optimization examples and sensitivity studies have been studied. One of them will be demon- strated here.

The example selected for illustrating the proposed method of interactive reliability-based optimization is the two-story one-bay frame shown in Fig. 1.

A deterministic collapse analysis using linear programming would show sixty possible failure modes and four simultaneous active failure modes at the optimum[l7]. The following input data have been assumed for the optimum reliability-based design of the example frame.

(a) The loads P,, P2, H,, H2 and the plastic moments M,, M2, M, are normally distributed with

VW, vu43

) = V(M,) = 1 z ;($) . . = 0.20 and F(M,

(b) The mean values of the loads are PI = 0.5 P2 = 154 kN and Hi = 0.5 HZ = 10.7 kN.

M2

Ml Ml 6m

I I

Ir --L 1, 4m ,I 4m 4

Fig. 1. Example frame.

(c) The statistical dependence between any two random loads, X and Y, is accounted for by using the correlation coefficient p(X, Y) as a measure of this dependence. More specifically, the correlation coefficient is a measure of the linear dependence between two random loads. In the following we assume: completely dependent horizontal loads p(H,,HJ = 1, independent vertical loads p (P,, PJ = 0, and statistical independence between horizontal and vertical loads: p(H,, P,) = P (HI, Pz) = P (Hz, PI) = P (Hz, P2> = 0.

Tables 1 and 2 summarize typical reliability-based optimization results of the frame shown in Fig. 1. The computation of the optimum solution was carried out by the proposed CAD system. The six collapse modes which dominate in contributing to the overall failure probability of the frame are indicated in Fig. 2.

Table 1 together with Table 2 provide information for a number of interesting sensitivity studies. These comparisons may be summarized as follows: (a) the influence of the specified overall failure probability of the frame P; on the optimum design solution (Table 1); (b) the influence of the degree of correlation

/ n r---7

coefficients of variation V(P,) = V(P,) = 0.15, Fig. 2. Dominant failure modes.

562 D. M. FRANGWOL

Table 1. Reliability optimum design results of two-story frame shown in Fig. 1

66.1 347 148.2

68.0 324 138.5

75.1 340 147.9

t The corresponding collapse mode is shown in parentheses according to Fig. 2.

Table 2. Comparison of collapse mode failure probability for different reliability optimum designs of two-story frame shown in Fig. 1.

Correl. Spec. between

probab. of Individual collapse mode failure probability at the optimum

plastic failure moments Mode of failure (see Fig. 2)

p; P(Mi, Mj) (aIt (b) (4 or (4 Cd 0-l

0 0.48.10-3 0.24.10-3 0.51.10-’ 0.83.10-4 0.17.10-4 10-s 1 0.25.10-3 0.28.10-’ 0.97.10-4 0.76.10-4 0.21.10-4

0 0.54.10-4 0.18.10-4 0.11.10-6 0.89.10-’ 0.10.10-5 10-4 1 0.28.10-4 0.30.10-4 0.86.10-’ 0.67.10-5 0.20.10-5

0 0.66.10-’ 0.12.10-5 0.16.10-’ 0.69.10m6 0.10.10-6 10-s 1 0.28.10-5 0.31.10-’ 0.82.10-6 0.68.10-6 0.16.10-6

t The corresponding collapse mode is shown in parentheses according to Fig. 2.

between plastic moments p (Mi, M,) on the optimum design solution (Table 1); and (c) the influence of P; and p (M,, Mj) on the relative contribution of each individual collapse mode to the overall failure probability of the frame (Table 2).

As shown in Table 2, only one mode dominates in reliability-based plastic design. A further conclusion is that the same collapse mode is not found to dominate at the optimum when the correlation between plastic moments is taken or is not taken into account.

The evolution of the reliability-based optimization process of the frame, for completely independent plastic moments, is illustrated in Fig. 3 by three isomerit contours. All points on one contour corre- spond to solutions which have a probability value equal to the specified overall failure probability of the frame, P; = 10m3, and equal structural weight. Also shown is the optimum solution. The dominant mode of failure for this optimum solution is the mode a (see Fig. 2); its contribution to the overall failure probability is 48%.

Constant failure probability contours are automatically generated by the computer. These contours are of practical interest in comparing alternative (isosafety and/or isoweight) design solutions because the contribution of the failure modes to the overall

failure probability of the frame may be different from one point to another on the same contour. This is particularly important when the consequences of occurrence of one failure mode (e.g. the combined

kNm fM2

0 100 200 300 kNm

Fig. 3. Isomerit contours and optimum solution.

Interactive reliability-base d structural optimization

mechanisms b, c or d in Fig. 2) greatly exceeds the consequences of occurrence of another failure mode (e.g. the beam mechanisms a, e or f in Fig. 2), as is generally the case in many structural engineering systems.

4. D. M. Frangopol, Probabilistic concepts in structural optimization-theory and applications (in Romanian), Ed. Romanian Acad. 38(SCMA 4 and 5), 519-535 and 695-715 (1979).

CONCLUSIONS

The use of deterministic concepts in optimization of structures may give design solutions which have either an unacceptably low or an unnecessarily high level of reliability. Both these consequences result in an inefficient use of resources. The CAD system described in this paper, based on combining probabilistic techniques, optimization methods, graphical display programs and interaction, provides a power- ful tool to obtain a practical optimum design of plastic frames. As indicated above, recent advances in computer techniques for design have reached the stage where selection between isoweight and/or isosafety alternative solutions can be automatically carried out. In order to reach more significant levels of structural reliability-based optimization it is necessary to compare optimized isosafety structures of different topology, material and geometry. Further research is needed to achieve this potential.

5. F. Moses, R. Fox and G. G. Goble, Mathematical programming applications in structural design, SM Study No. 5. Computer Aided Engineering. University of Waterloo Press (i970). - -

6. F. Moses, Design for reliability-concepts and applications. Optimum Structural Design (Edited by Gallagher and Zienkiewicz), pp. 241-265. Wiley, New York (1973).

7. A. B. Templeman, Optimization concepts and techniques in structural design. Proc. 10th Congress IABSE. Introductory Report, pp. 41-60. Tokyo (1976).

8. F. Moses. Svstem and geometrical optimization for linear and nonlinear structural behavior. Proc. 10th Congress IABSE. Introductory Report, pp. 61-75. Tokyo (1976).

9. U. Kirsch, Optimum Structural Design. McGraw-Hill, New York (1981).

10. D. M. Frangopol, Reliability analysis and optimization of plastic structures. Proc. 4th Int. Conf. on Applications of Statistics and Probability in Soil and Structural En- gineering, pp. 1271-1288. Florence (1983).

11. D. M. Frangopol and J. Rondal, Optimum probability- based design of plastic structures. Engng Optimization 3, 17-25 (1977).

Acknowledgements-The author would like to express his gratitude to Ch. Massonet, Professor and Chairman of the Department of Strength of Materials and Stability of Struc- tures, University of Liege, Belgium for his guidance throughout the investigation. Some of the work reported in this paper was presented at the 3-5 April 1984, 6th Inter- national Conference on Computers in Design Engineering, held at Brighton, England.

12. D. M. Frangopol, Discrete-optimum probability solution of plastic structures. Advances in Civil Engineering through Engineering Mechanics, ASCE, pp. 444-447. New York (1977).

13. E. Somekh and U. Kirsch, Interactive optimal design of truss structures. Computer-Aided Design 13, 253-259 (1981).

14. D. M. Frangopol and J. Rondal, Considerations on optimum combination of safety and economy. Proc. 10th Congress IABSE. Final Report, pp. 45-58. Tokyo (1976).

RJWERENCES

0. Ditlevsen, System reliability bounding by condition- ing. Proc. ASCE., J. Engng Mech. Div. lOlI(EMS), 708-718 (1982). E. H. Vanmarcke, Matrix formulation of reliability analysis and reliability based-design. Comput. Structures 3, 757-770 (1971). F. Moses, Structural system reliability and optimization. Comput. Structures I, 283-290 (1977).

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15. D. M. Frangopol, Statistical properties of the limit equilibrium for plastic structures, Presented as a con- tributed paper at the VII Int. Congress on Mathematical Physics, Boulder, l-10 August (1983).

16. S. R. Parimi and M. Z. Cohn, Optimal solutions in probabilistic structural design. J. Met. Appliq. 2, 47-92 (1978).

17. M. 2. Cohn, S. K. Ghosh and S. R. Parimi, Unified approach to theory of plastic structures. Proc. ASCE., J. Engng Mech. Div. 98(EM5), 1133-l 159 (1972).

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Interactive reliability-based structural optimization