Probability-based optimization of truss structures

Computers & Srrucrurrs Vol. 32. No. I. pp. 87-91. 1989 Printed in Great Britain.

0

PROBABILITY-BASED OPTIMIZATION TRUSS STRUCTURES S. F.

@X5-7949/89 S3.W + 0.00 1989 Pergamon Press plc

OF

J&WIAK

Institute for Road and Bridge Design, Warsaw Technical University, al. Armii Ludowej 16, 00-637 Warsaw, Poland (Receiued 15 March 1988) Abstract-The paper deals with the probability-based optimization of truss structures. The mean value of the structural mass is taken as the objective function and the constraints concern the capacity of structural elements. The random character of structural parameters is considered by estimating the value of the coefficient of variation of element strength. Hazard functions are also introduced to estimate global structural safety and to compare designs. The probability-based problem is converted into an equivalent deterministic nonlinear programming problem which can be solved by the application of mathematical programming library. The numerical results of the plane truss optimization are presented to provide insight into the basic features of the approach.

INTRODUCrION

many national standards [2,3] and recommendations of international standardization organizations [4,5] have statistical and probabilistic basis. Probabilistic analysis is mainly used for calibration of parameters remaining traditional formats, but there is also a tendency of more direct introduction of probability concepts and methods to engineering design. Introducing probability concepts into structural designs requires developing suitable optimization techniques which enable solution of design problems formulated on the assumption that structural parameters exhibit statistical regularity. Optimization of structures with random parameters was the subject of works by Moses [6], Rao [I, Davidson, Felton and Hart [8], Frangopol[9], Ishikawa [lo], and complete review of the literature concerning probability-based optimization of structures is given in a paper by Brandt [ll]. The paper presents the solution of the probability-based truss optimization problem and extends previously reported formulations [12, 131. The objective function is formulated as the mean value of the mass of the structure and the constraints concern the capacity of the structural elements taking into account random character of structural parameters. Hazard functions are also introduced to estimate global structural safety and to compare designs. The numerical results of the plane truss optimization are presented to provide insight into the basic features of the approach and to illustrate the effects.

Despite the generally recognized nondeterministic character of parameters defining engineering structures, probability methods were applied in structural design on a relatively limited scale. Over a period of time, codified parameters, factors of safety or load and resistance factors were selected or adjusted largely due to experience or intuition rather than to consistent probabilistic analysis. The situation has changed in the last few years; continuous progress in structural mechanics and in the building industry has stimulated the search for consistent and mathematically correct solutions of structural safety problems. These can be achieved by taking advantage of probabilistic methods which can be used to handle the random character of structural parameters as well as uncertainties arising in the formulation of design problems (e.g. poor mathematical modeling). The application of probability methods in structural design has become possible thanks to the development of suitable mathematical methods and new facilities for data collection and processing. Many of the limitations in computational capabilities which at first restrained the application of probabilistic analysis are now being overcome. An important impact on the development of probability-based methods for structural design is also due to the necessity of performing exact safety and reliability analysis for special-purpose engineering structures (e.g. off-shore structures, chemical plants, nuclear power plants). The consequences of potential accidents in nuclear or chemical plants are so serious that detailed analysis of structural safety must be performed. Numerous works have been devoted to the problem of nuclear power plant safety; many of them are collected in conference transactions (e.g. [I]). All this results in even wider introduction of probabilistic concepts in structural design. At present

FORMULATIONOF THE PROBABILITY-BASED STRUCI’URALOPTIMIZATIONPROBLEMS In probability-based structural design and optimization, performance requirements [ 141are stated in the form:

(1) 87

S.

88

F.

where g(X) is the limit state condition, X is a set of design variables (material strength, dimensions, geometry etc.), P, denotes the probability of attaining by the structure its limit state, and Pf is a target limit state probability specified by regulatory authority. Formulation (1) is not suitable for performing routine calculations that are used for practical design and most codes and recommendations use safety checking conditions of the general form. Design resistance R, > Design action S,,

(2)

in which R,, S, must be determined so that the probability of attaining a limit state by the structure designed according to the rule (2) is equal to that set by the inequality (1). In the present paper the constraints concern the capacity of the structural elements and the optimization problem is stated as follows: minimize mass of the structure subjected to P,(X, Y) = P&(X,

Y) < S,(X, WI
(3)

where X = vector of random variables, Y = vector of decision variables, R,(X, Y) = strength (capacity) of structural elements. S,(X, Y) = action, and J = number of constraints. Inequality (3) denotes that probability of attaining R, < S, must be smaller than or equal to specified value p,. The objective function (structural mass) must be formulated in such a manner that random character of the parameters could be considered. In the paper the mean value of the structural mass W is taken as the objective function; W = W(X)

0,

d(-P)=p,

(7)

where 4 (. . .) = Laplace function. In the present analysis it is assumed that random variables follow lognormal distribution and that random character of parameters influencing the capacity of structural members (material strength, geometrical and shape imperfections, model uncertainties) is considered by estimating value of the coefficient of variation vR. STRENGTH OF THE COMPRESSION MEMBERS

Stability analysis of the compression members presents difficulties even in the case of deterministic formulation of the problem. In the present paper the capacity of the compression truss members have been estimated taking advantage of results presented by Murzewski [15]. The capacity of the compression members can be expressed as:

R,, = min(R,,, &I,

(8)

where R,,,, R,, are independent random variables (R,,, = plastic capacity, R,, = elastic capacity). Solution of eqn (8) (estimation of R,,) is simpler when assuming for R,, and R,, Weibull probability distribution. Carrying out transformations as shown in [I 51, the strength of the compression member R, is given by the formula

In R - /I[vi + v$‘* - In S 2 0,

where fi,, = plastic strength (for Weibull distribution), w = Weibull coefficient of variation, I= A/J.,, I = slenderness ratio, 1, = characteristic slenderness A,= n$%G, The coefficient of variation is given by: w(i)

(5)

where i?, S= mean values of R and S, bR, rrs = standard deviations of R and S. If the distributions of R and S are lognormal, the inequality (3) can be transformed into:

where I?, S = medians cients of variation.

Symbol p denotes here the reliability index which is generally accepted as a quantitative measure of safety of structures. The relation between the probability of failure p eqn (1) and the reliability index fi has the form

(4)

where X = vector of mean values of random variables. The constraint (3) can be converted into an equivalent deterministic inequality by carrying out transformations as shown in numerous references e.g. [7. 81. If R and S follow normal distribution the corresponding deterministic inequality has the form: K- b[u’, + a;]‘12 - sa

J&WIAK

(6)

of R and S, vR, vs = coeffi-

=

; W+--1 +x*

An>*

-

l+h-?I+l+X*

u’*= w:-(wi + w;y2,

w;12

1’ 10

(10)

where wg = w(0)

coefficient of variation for X = 0,

w(1)

coefficient of variation for X = 1.

w, =

wE= lim w(A) coefficient of variation for 1+ cc. I-cc

Probability-based

The Weibull parameters can be transformed normal parameters:

89

optimization of truss structures

to log-

I ;

3

5

7

9

II

13

4

5

8

lo

l2

I4

I5

I7 i &ci

l6

$ I8

cd I r-,YY*

-F--x

a x 3.00= 24.00 m

+

Fig. I. Plane truss.

where i?,, = strength of the compression member (for lognormal distribution), 1?,, = lognormal plastic strength, &, = lognormal elastic strength, vgr= lognormal coefficient of variation. INTRODUCING HAZARD FUNCTIONS

The constraints in the form (1) or (3) can protect the structure against specified mode of failure but cannot give any insight in global structural safety. The necessity of developing global safety measure for a structure is clearly evident in the case of considering dual problem of minimizing weight (mass) and maximizing structural safety. The problem of global safety measure for complex structures is concerned with so called level III probabilistic methods [ 161which introduce a safety measure called level class ratio l/k; l/k = R*h(R*) = S*h(S*),

(13)

where k = safety class index, k(R*), h(S*) = risks of exceeding the limit values of strength R* and loading S*:

h(R)=

f(R) 1 _

F(R)’

h(S) =$$,

(14)

where f = density function, F = distribution function. The hazard ratio for a structure with n independent admissible collapse modes can be estimated as [16]: R* i h,(R*) = l/k, i-l

(15)

and corresponding formulae in the case of lognormal and Weibull distribution or random variable are:

&i/g=

l/k;

i *tfi) T = l/k; i-l I

for Weibull distribution,

(16)

for lognormal distribution

(17)

where ti = In m, q(t)/@(t), q(t)-Gauss tion.

G(t)-Mills function, $(t) = function, 9(t)-Laplace func-

NUMERICAL EXAMPLE

The example concerns minimum mass design of plane truss presented in Fig. 1. The loading has been assumed according to the Polish Standard (PN-85/S-10030). The element cross-sectional areas are listed in Fig. 2. The cross-sectional areas are assigned to truss elements as shown in Table 1. Decision variables yi, i = 1,2,. . . ,9 have been selected as follows: variables y, , y, concern cross-sectional dimensions as shown in Fig. 2, variables y,-y, concern joint coordinates as shown in Fig. 3. The general shape of the truss is initially prescribed and results from architectural and functional

Table 1. Assignment of cross-sectional areas to truss elements Element no. I

2 3 4 5 6 7 8 9 IO II 12 13 14 I5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Joint no. 1 3 3 5 5 7 7 9 9 11 II 13 13 I5 15 17 1 2 2 4 1 4 3 4 4 6 3 6 5 6 6 8 5 8 I 8 8 10 7 10 9 10 10 12 10 I1 11 12 12 14 12 13 13 14 14 16 14 15 15 16 16 18 16 17 17 18

Cross-section no. I I I I 1 f

1

2 3 4 2 2 4 5 6 4 10 7 8 4 9 8 4 7 8 10 6 4 8 3 4 8

S. F.

90 (a)

J&WIAK

(b)

such that the mass of the structure w = Y5 aXY)f,(Y) i=I is minimized with following constraints P[ai > RI Q ~1 P]aylm,

for i = I, 2,

(18) imposed

. ,33

> RI 6 P,

(19) (20)

150 for compression members, 1, < J-

L

6

520

Y2

-#

i 200 for tension members

a,,, < 1isoo,

(22)

where y = material density, a,(Y) = cross-sectional area of ith element, l,(Y) = length of ith element, ui = stress in ith element, a: = stress due to characteristic loads in ith element, ml, = fatigue coefficient calculated according to the formula stated in Polish Standard PN-82/S-10052, Izi= slenderness ratio for ith element, and u,,, = maximum deflection. Constraints (19-22) have been assumed according to performance and safety requirements stated in Polish Standard [17]. Probabilistic inequalities (19-20) can be transformed to the form

14

u=il

In R - flvR- In bi 2 0, In R - flvRIn ai/rnzm 2 0,

Fig. 2. Element cross-sections. (A) Chord cross-sections. (B) Cross-sections of verticals and diagonals. requirements. To assure this the following analysis has been performed: first, a second order function (a parabola) has been defined which could approximate the shape of the bottom chord of the truss. The decision variables are transformed in such a way that ,r-_r9 denote the distance between actual joint coordinate and the curve representing the function (Fig. 3) while the values of decision variables could change within a limited range. To simplify analysis it is assumed that random variable is element strength R E _4-(l?, vR) only. Random character of material strength, geometry, dimensions, model uncertainity etc., are considered by proper estimation of the value of VR. The optimization problem is formulated as follows: find values of the decision variables yi, i = 1,2, . . . ,9

16

Fig. 3. Choice of the decision variables.

(21)

i = 1,2,. . ,33

(23) (24)

where R mediana of the element strength [R = i?,,, for tensional elements, R = II,* for compression members and I$, is estimated according to formula (1111,VR= lognormal variation coefficient considering random character of structural parameters and model imperfections. The deterministic programming problem was solved by the application of the Mathematical Programming Library [18], using Powell’s method. The main purpose of the numerical analysis was to compare results obtained using the probabilistic approach presented in the paper with the results obtained on the assumption that the problem is deterministic. At first, the structure was optimized using formulas, requirements and design values specified in design codes (deterministic solution IV,). Then the structure was designed using the present probability-based approach. The following values were adopted in the numerical analysis: /3 = 2, R = 300 MPa, .??= 200 GPa (for the calculation of the compression strength), v0= 8%, vE= 8%, Av = 20%. The values of variation coefficient adopted in the analysis have been chosen in accordance with the values suggested by other authors [15]. For the assumed values, optimum values of the decision variables and minimum mass were calculated ( W,,).The value of W,, is 4.8% smaller than the value W,, obtained for deterministic problem. Numerical analysis concerned also the influence of the magnitude of the variation coefficients

Probability-based

91

optimization of truss structures

Table 2. Influence of the magnitude of Av on the structural mass vo

0.08

2 w [tl

0.05 0.08 9.98 0.2187

k

0.08 0.10 0.08 IO.32 0.1976

0.068

: w 1tl k

0.068 0.200 1I .48 0.1670

0.080

0.113

0.150

0.080 0.200

0.200 0.113 11.95 0.1703

0.200 0.150 12.26 0.1854

1I .60 0.1623

on the optimal solution. The results are presented in Tables 2 and 3. In this case, as the influence of the

variation coefficients on structural mass is rather small, the parameter k has been introduced to assure equal safety of the compared designs. The equal value of k for different designs provides equal safety of structures and assures that reducing the mass does not result in reducing structural safety. The safety class index k proved a very efficient measure in comparing structural designs as it is more sensitive to the change of structural parameters than the overall structural mass. CONCLUDING

0.15 0.08 11.16 0.1702

0.13 0.08 10.64 0.1836

Table 3. Influence of the magnitude of v. and vE on the structural mass

vo

0.08

0.08

REMARKS

Although structural optimization techniques are mainly devoted to solve deterministic problems, the developing of techniques which can be used to handle probability-based design problems cannot be neglected. The paper presents a solution of the truss optimization problem taking into account the random character of structural parameters. Although the approach seems to be too complicated for standard usage, it can be recommended for the design of individual, special-purpose structures or for the verification of design results obtained using altemative approaches.

0.08 0.20 0.08 11.60 0.1626

0.08 0.25 0.08 12.26 0.1521

2. A. S. Nowak and N. C. Lind, Practical bridge code calibration. J. Slrucr. Div. AXE ST12, 2497-2510 (1979). 3. NBS/US Dept. of Comm., Deuelopment ofa Probability Based Load Criterion for American National Standard A58, National Bureau of Standards, Special Pub]. 577

(1980). 4. NKB, Recommendations for loading and safety regulations for structural design. The Nordic Committee on Building Regulations, Report 36 (1978). 5. ISO/TC 98, General principles on reliability for structures. Warsaw (1981). (Revision of IS0 2394-1973.) 6. F. Moses, Structural system reliability and optimization. Comput. Struct. 3, 283-290 (1977). 7. S. S. Rao, Structural optimization by chance constrained programming technique. Compur. Struct. 12, 777-781 (1980). 8. J. W. Davidson,

9.

10.

II. 12.

13. 14. 15.

16.

L. P. Felton and G. C. Hart, On reliability-based structural optimization for earthquakes. Comput. Strucr. 12, 99-105 (1980). D. Frangopol and J. Rondal, Optimum probability design of plastic structures. Eng. Opt. 3, 17-25 (1977). N. Ishikava, T. Mihara and M. Iizuka, Reliability analysis of large scaled structures by optimization technique. Transaction of the 9th SMiRT Conference, Vol. M. pp. 79-90, Balkena (1987). A. Brandt, S. Jendo and W. Marks, Probabilistic approach to reliability-based optimum structural design. Engng Trans. 32, 57-74 (1984). S. Joiwiak, Minimum weight design of structures with random parameters. Comput. Struct. 23, 481485 (1986). S. J6iwiak, Optimum Design of Structures with Random Parameters. WTU Press (1986). H. Hwang, B. Elhngwood, M. Shinozuka and M. Reich, Probability-based criteria for nuclear plant structures. J. Struct. Engng 113, 925-942 (1987). J. Murzewski and M. Gwoidi, Strengths of compression members in level 2 and 3 probability-based analysis. Proceedings Conference on Probabilistic Problems in Structural Mechanics, Gdansk (1985). (In Polish.) J. Murzewski, A. Miynarczyk, Risk summation rule in probabilistic interaction problems. Proceedings Conference on Probabilistic Problems in Structural Mechanics,

REFERENCES

I. F. H. Witmann (ed.), Structural reliability, probabilistic safety assessment. Transactions of 9th International Conference on Structural Mechanics in Reactor Technology (SMiRT), Lausanne 1987. Balkena, Rotterdam

(1987).

Gdarisk (1985). (In Polish.) 17. PN-82/S-10052 Polish Standard. Bridge structures. steel bridges, design. Wyd. Normalizacyyne Warszawa (1983). (In Polish.) 18. T. Kreglewski, T. Rogowski and A. Ruszczynski, Optimization in FORTRAN, PWN, Warszawa (1984). (In Polish.)