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Reliability-Based Optimal Design of Truss Structures Using Particle Swarm Optimization C. K. Dimou1 and V. K. Koumousis2 Abstract: In this work, the particle swarm optimization method is employed for the reliability-based optimal design of statically determinate truss structures. Particle swarm optimization is inspired by the social behavior of flocks 共swarms兲 of birds and insects 共particles兲. Every particle’s position represents a specific design. The algorithm searches the design space by adjusting the trajectories of the particles that comprise the swarm. These particles are attracted toward the positions of both their personal best solution and the best solution of the swarm in a stochastic manner. In typical structural optimization problems, safety is dealt with in a yes/no manner fulfilling the set of requirements imposed by codes of practice. Considering uncertainty for the problem parameters offers a measure to quantify safety. This measure provides a rational basis for the estimation of the reliability of the components and of the entire system. Incorporating the reliability into the structural optimization framework one can seek a reliability-based optimal design. For the problems examined herein, the reliability indexes of the structural elements are obtained from analytical expressions. The structure is subsequently analyzed as a series system of correlated elements and the Ditlevsen bounds are used for the calculation of its reliability index. The uncertainrandom parameters considered in this work are the load, the yield-critical stress; and the cross sections of the elements. The considered design variables of the optimization problem are the cross-sectional areas of the groups, which control the size of the truss, and the heights and lengths that control the shape of the truss. The results of the optimization are presented for a 25-bar truss and a 30-bar arch and the robustness of the optimization scheme is discussed. DOI: 10.1061/共ASCE兲0887-3801共2009兲23:2共100兲 CE Database subject headings: Particles; Optimization; Reliability; Trusses.

Introduction Particle swarm optimization 共PSO兲 is a population-based stochastic optimization technique suitable for global optimization with no need for direct evaluation of gradients. The method, introduced by Kennedy and Eberhart 共1995兲, mimics the social behavior of flocks 共swarms兲 of birds and insects 共particles兲 and satisfies the five axioms of swarm intelligence, namely, proximity, quality, diverse response, stability, and adaptability 共Millonas 1994兲, which are essential for this particular class of optimization algorithms. The algorithm searches the design space 共DS兲 by adjusting the trajectories of individuals, called “particles,” viewed as moving points in the DS. These particles are attracted toward the positions of both their personal best solution and the best solution of the swarm in a stochastic manner 共Clerc and Kennedy 2002兲. A discrete 共binary兲 version of the algorithm was subsequently introduced by Kennedy and Eberhart 共1997兲. In binary PSO 共BPSO兲 the term chromosome defines a candidate design. Each chromo1 Postdoctoral Researcher, Institute of Structural Analysis and Aseismic Research, National Technical Univ. of Athens, Zografou Campus, 15780, Iroon Polytexneiou 9, Athens, Greece. E-mail: ckdimou@central. ntua.gr 2 Professor, Institute of Structural Analysis and Aseismic Research, National Technical Univ. of Athens, Zografou Campus, 15780, Iroon Polytexneiou 9, Athens, Greece. E-mail: [email protected] Note. Discussion open until August 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this paper was submitted for review and possible publication on April 8, 2008; approved on September 12, 2008. This paper is part of the Journal of Computing in Civil Engineering, Vol. 23, No. 2, March 1, 2009. ©ASCE, ISSN 0887-3801/2009/2-100–109/$25.00.

some contains the set of parameters which define a candidate solution of the problem in the DS. The main difference of the BPSO, as compared to the standard PSO, is in the use of the information contained in the velocity vector. In PSO the velocity vector is used to calculate the particle’s future position. In BPSO the velocity is used to calculate the probability of the bits of the chromosome to set either to 0 or 1. Thus, velocity is interpreted as a probability or entrapment threshold. As this threshold increases the bit flip probability decreases and the bit value is locked to 0 or 1 depending on the sign of the velocity parameter. PSO and BPSO have been used as optimization tools in a wide range of applications that include various structural optimization problems. Fourie and Groenwold 共2001, 2002兲 use PSO for topology, size, and shape optimization of structures. Schutte and Groenwold 共2003兲 implement PSO in the size optimization of two- and three-dimensional trusses. Venter and SobieszczanskiSobieski 共2004兲 use an enhanced version of PSO in the aerodynamic optimization of an aircraft wing. Parsopoulos et al. 共2004兲 implement a variant of PSO suitable for multiobjective optimization. Elegbede 共2005兲 uses PSO for solving the classical reliability analysis problem. Zhang et al. 共2006兲 implement PSO in the resource-constrained project scheduling problem. Yang 共2007兲 uses multiobjective PSO to obtain the Pareto front of the complete time-cost profile of a set of acceptable project durations. Kumar and Reddy 共2007兲 use PSO to derive reservoir operation policies. Dimopoulos 共2007兲 examines the performance of PSO against the standard genetic algorithm 共SGA兲 in both mathematical and engineering problems. BPSO is used mainly in problems of combinatorial nature. Franken and Engelbrecht 共2005兲 examine the influence of the BPSO parameters on the Knights cover problem. Gao et al. 共2007兲 propose a branch and bound BPSO for

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solving integer separable concave programming problems. Monteiro and Kosugi 共2007兲 use BPSO for processing hyperspectral imagery data. Reliability-based optimal design 共RBOD兲 has attracted the attention of the many researchers 共Mc Donald and Mahadevan 2008兲. In RBOD, part of the constraints and/or the objective function is of probabilistic nature. Reliability analysis provides a rational basis to quantify safety. A structure is considered safe when it will not fail under foreseeable demands and it is unlikely to fail under extraordinary demands or circumstances 共Elms 1999兲. In reliability analysis, the yes/no 共acceptable/nonacceptable兲 concept of current codes of practice with regard to the requirements of a design is substituted by a probability measure for a particular component of the structure to behave in an unacceptable manner 共failure兲. The main advantage of RBOD with respect to typical structural optimization processes based on the requirements of structural codes and regulations 共Level 1 methods兲 is its ability to overcome the problem of optimal designs with reduced reliability which often occurs in the case of typical optimization. Due to its high computational cost, RBOD has become practical with the advancement of computing hardware. Natarajan and Santhakumar 共1995兲 use the branch and bound method for the reliability analysis of transmission line towers. Ang and De Leon 共1997兲 address the problem of optimum reliability indexes to design, or retrofit, existing structures with the minimum expected life-cycle cost. They state that optimal designs depend upon the statistical characteristics of the acting loads, therefore, no deterministic design code is capable of providing guidelines that would result in economic and safe designs, equally suited for zones with different load characteristics. Royset et al. 共2001兲 implement a semiinfinite optimization approach to perform a decoupled RBOD. Thampan and Krishnamoorthy 共2001兲 couple SGA with a modified branch and bound method to minimize the overall expected cost of the structure. Wen 共2001a,b兲 demonstrates that redundancy factors related directly to the acceptable risk are required to obtain designs of the same level of safety and economy. Wen also notes that current U.S. design codes result in inconsistencies among designs in different regions, with regard to their reliability/redundancy factors. Dimou and Koumousis 共2003兲 implemented a competitive GA algorithm for the RBOD of truss structures. Youn and Choi 共2004兲 address the problem of maximizing the safety of a vehicle on side impacts, whereas the weight of the safety cell is kept below a certain threshold value. In their work, the reliability level of a candidate design is estimated using the hybrid mean value method coupled with a response surface method to obtain approximations of the fail-safe boundaries of the constraints under examination. Foye et al. 共2006兲 develop the framework for an objective assessment of optimal resistance factors used in the ultimate state design of swallow foundation design with the aid of reliability analysis. Royset et al. 共2006兲 develop a gradient-based optimization algorithm for the RBOD and implement it in a series of typical structural problems with multiple failure modes. Foley et al. 共2007a,b兲 present a state of- the-art overview of performance-based methodologies in the design of steel moment resisting frames subject to seismic loading incorporating risk in structural design. They successfully implement a multiobjective optimization scheme coupling an evolutionary GA with radial fitness and balance fitness functions for the probabilistic performance-based design of both single story and multistory frames. The design objectives considered are the cost of the structure and the confidence levels associated with meeting collapse prevention and immediate occupancy performances.

In this work, the reliability-based optimal design 共RBOD兲 of two planar statically determinate trusses—a 25-bar truss and a 30-bar arch—is presented. The BPSO is employed for the derivation of the optimal designs. The design variables 共DVs兲 considered are the cross-sectional areas Ai of the groups that control the size of the truss, and the heights h j and lengths lk that control the shape of the truss. For the derivation of the reliability indexes the examined problems are modeled as series systems of partially correlated groups of elements. In these groups the elements are considered as fully correlated. The random parameters considered are the load, the yield-critical stress, and the cross sections of the elements. Numerical results regarding the optimal designs and the effects of the BPSO parameters are presented.

Continuous and Binary Particle Swarm Optimization In PSO, for each particle of the population, a given position reflects a candidate design in the DS. The position of the particle at time step t is advanced using the information of the velocity vector, where for simplicity the time step is considered as unity. The velocity vector at the subsequent time is a function of: 共1兲 the current speed of the particle; 共2兲 the position of the best solution found by the particle; and 共3兲 the position of the current best solution found by the swarm. For a given DV the position and velocity of a particle of the swarm, at time step t + 1, as a function of its position and velocity at time step t are given as 共Clerc and Kennedy 2002兲 vi,j;t+1 = wvi,j;t + c1r1共pi,j − xi兲 + c2r2共gi − xi兲,

兩v j;t+1兩艋 vmax 共1兲

xi,j;t+1 = xi,j;t + vi,j;t+1

共2兲

where vi,j;t = velocity component of the jth particle for the ith DV at time step t; w = inertia coefficient weighting the influence of the current speed; c1 = cognitive parameter weighting the influence of the best solution found by the particle; c2 = social parameter weighting the influence of the best solution found by the swarm; r1 and r2 = uniformly distributed random values in the range 关0,1兴; pi,j = ith coordinate of the best solution encountered by the jth particle until time step t; gi = ith coordinate of the best solution encountered by the swarm until time step t; vmax = maximum velocity of the particle; and xi,j;t = coordinate of the jth particle for the ith DV at time step t. Typical values for the parameters of Eq. 共1兲 are w ⬇ 1 and c1 = c2 ⬇ 2 共Eberhart and Shi 2001兲. As E关ri兴 = 0.5 共E关·兴 = expected average兲, when w = 1 and c1 = c2 = 2, inertia cognition and sociality are equally important in determining the velocity vector. The variation of ri directly introduces randomness with regard to the weight of cognition and sociality and indirectly affects the relative importance of inertia. Analyses of the influence of the basic parameters in the dynamics of motion in the DS can be found in the work of Eberhart and Shi 共2001兲, Trelea 共2003兲, and Liu et al. 共2007兲. The performance of the optimization process depends heavily on the velocity vector and thus, numerous variants for Eq. 共1兲 have been proposed. The velocity vector can be expressed also without the inertial part, as in Venter and Sobieszczanski-Sobieski 共2003兲, to accelerate the particle trajectory away from infeasible areas of the DS. Time varying w can be used as in Shi and Eberhart 共1998兲, Fourie and Groenwold 共2002兲, and Venter and Sobieszczanski-Sobieski 共2003, 2004兲, to improve performance. The maximum velocity vmax is of primary importance. Eberhart

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and Shi 共2001兲 note that high values of vmax reduce the algorithm’s capability to locate the global optimum especially in nonsmooth multimodal objective functions, whereas low values of vmax reduce the convergence speed of the optimization algorithm and may trap the algorithm into local optima. Suggested values for the vmax vary from 10 to 20% of the range of the DVs, as in Eberhart and Shi 共2001兲, to 40–50%, as in Fourie and Groenwold 共2002兲 and Schutte et al. 共2003兲. Higher values of the vmax parameter allow the particles to move faster in the DS, improving the rate of convergence, but also increase the probability of a particle missing 共overpassing兲 the true optimum design. Shi and Eberhart 共1998兲 use the constriction parameter ␹ to bind the particle velocity to avoid erratic behavior and preserve the capability of the algorithm to locate the global optimum. Another measure to avoid premature convergence of the algorithm is the implementation of the craziness operator, as in Fourie and Groenwold 共2002兲, Schutte and Groenwold 共2003兲, and Venter and SobieszczanskiSobieski 共2004兲. In BPSO, the velocity vector of Eq. 共1兲 is used to derive the probability of the ith bit of the chromosome that represents the individual design, to be equal to 1. An extended analysis on the behavior of BPSO and its derivations can be found in Clerc 共2007兲. The value of the ith bit is given as 共Kennedy and Eberhart 1997兲 i xk+1 =

再

i 1, rk+1 艋 sig共vik兲 i 0, rk+1 ⬎ sig共vik兲

冎

共3兲

where sig共vik兲 = 1 / 关1 + exp共−vik兲兴; sigmoid function sig共vik兲 varies i in the range of 关0,1兴; and rk+1 = random value uniformly distributed in the range of 关0,1兴. The vmax acts as an entrapment probability of the bit in either value of 0 or 1. Kennedy and Eberhart 共1997兲 recommend clamping of the vmax in the range of 关−4 , 4兴 to avoid saturation of the sigmoid function. Franken and Engelbrecht 共2005兲 support this statement, suggesting for the vmax values over 3.0 stating that small vmax slows convergence as the entrapment probability is low and random bit flips do not allow the population to converge to the optimum. For the c1 and c2, Franken and Engelbrecht 共2005兲 suggest values around unity for problems of small complexity. As the complexity of the problem increases, larger c2 values are preferable to avoid premature convergence. Herein, the effects of the BPSO parameters are investigated with respect to the problems addressed. Specifically for the vmax, an extensive parametric study is performed to verify if the notion of optimal mutation probability of GAs 共Hesser and Männer 1990兲 can also be applied to BPSO.

Reliability-Based Optimal Design BPSO as well as other evolutionary methods such as GAs and their variants, as compared to classical mathematical programming methods, are suitable for computationally intensive and mathematically hard problems, the main reason being that they need only zero and no first order information, i.e., sensitivity, evaluations. RBOD is a computationally intensive and mathematically hard problem even in its simplest form. As such it is employed in this analysis, the main goal of which is to reveal the performance characteristics of BPSO to RBOD problems of a particular class. For a given statically determinate plane truss structure loaded with a specific load pattern, controlled by one parameter P, the

objective is to minimize the sum of the construction cost and the cost of potential structural failure 共Wen 2001a兲. This favors designs of increased reliability in the case of equal construction cost. The DVs are the cross-sectional areas Ai of the member groups 共taken from a list of available tubular cross sections兲, the heights h j and lengths lk that control the size and shape of the truss. The constraints consider the failure probabilities of both the elements and the structure. The optimization problem is formulated as follows:

共4兲 subject to

gn共Ai,h j,lk兲 =

gs共Ai,h j,lk兲 =

P f,n − 1.0 艋 0 Pn,lim

P f,s − 1.0 艋 0 Ps,lim

共5兲

where Nt = number of truss elements; Vm = volume of the mth element; Cmat and Cfail = structural cost per unit volume and the cost of potential structural failure, respectively; P f,n and Pn,lim = failure probabilities of the nth element and its maximum failure probability, respectively; and P f,s and Ps,lim = failure probability and its limit value for the entire structure. Note that no construction constraints are considered. The failure probability for an element is defined as the probability of the ratio of resistance over action obtaining values below unity 关P f,n = Pr共Rn / Sn 艋 1兲兴. For the problems examined, every member force Sn is given as the load acting on the structure multiplied by an appropriate coefficient for each element 关Sn = f n共h j , lk兲P兴. For statically determinate structures f n共h j , lk兲 is only a function of the geometry of the design and the load pattern. For the tensile and compressive-buckling resistance the expressions listed in Eurocode 3 共EC3兲 are employed. The tensile resistance is given as the cross-sectional area multiplied by the tensile yield stress 共Rn = ␴yAn兲. The compressive resistance is given as the cross-sectional area multiplied by the reduced critical stress in compression given as the compressive yield stress multiplied by a reduction factor ␹ accounting for buckling, based on the provisions of EC3 共Rn = ␹n␴yAn兲. The random variables 共RVs兲 of the problem are the load P, the yield stress of the material ␴y, the cross-section areas A j of all groups of bars, and the reduction factors ␹ accounting for buckling for all groups of bars. All RVs are considered as log-normally distributed; thus, the problem of calculating the failure probability of an element has an analytical solution 共Dimou and Koumousis 2003兲. The average and coefficient of variation 共COV兲 of the RVs are given in Table 1. Notice that the statistical variability of the cross-sectional areas is considered constant for the whole list of available cross sections. The truss elements form four groups, the lower and upper chords, the vertical struts, and the diagonal members. In every group the elements are considered as fully correlated; thus, the failure probability of the group is given from the maximum of the failure probabilities of the members of the group. The structural failure probability is obtained as the average of the Ditlevsen bounds of the corresponding series system of correlated groups following the provisions of Christensen and Murotsu 共1986兲 and Christensen and Baker 共1982兲. The versatility of these algorithms can be enhanced further by considering numerous conflicting objectives, e.g., cost, reliability,



Table 1. RVs for the Problems Examined 25-bar truss Random variable Load 共kN兲 Yield stress 共MPa兲 Cross section 共cm2兲 Buckling factor ␹

30-bar arch

E 关·兴

COV 共%兲

E 关·兴

COV 共%兲

30.0 275.0 Variable Calculated

12.5 7.0 10.0 10.0

30.0 275.0 Variable Calculated

25.0 7.0 4.0 5.0

Fig. 1. Twenty-five bar truss 共load and DVs兲

uniformity of design, robustness, etc. Multiobjective optimization is expected to be even more beneficial as it provides the engineer with a complete set of nondominated designs 共Zitzler 1999; Parsopoulos et al. 2004兲.

Examined Problems Two examples which cover the structural behavior of a wide range of statically determinate truss structures—a simply supported 25-bar truss 共Dimou and Koumousis 2003兲共Fig. 1兲 and a 30-bar 3-hinge arch 共Dimou and Koumousis 2004兲共Fig. 2兲—are examined. The DVs are the cross-sectional areas Ai of the groups 共taken from a list of available tubular cross sections兲, the heights h j, and lengths lk that control the size and shape of the truss. The relevant data are presented in Table 2. For the 25-bar truss, the total length is L = 10 m. The number of DVs for this problem is equal to 13. Four DVs control the size of the truss selecting the cross-section Ai of each group. Seven DVs are used to determine the h j of the truss and two DVs are

Fig. 2. Thirty bar arch 共load and DVs兲

used for the span lengths lk that control the shape of the truss. For the DVs controlling the Ai and h j, 4 bits are used to describe each DV, whereas for the lengths, 3 bits are used. The number of bits needed to fully describe a design is equal to 共4 + 7兲 ⫻ 4 + 2 ⫻ 3 = 50 thus, the multiplicity of the DS enumerates 250 共=1 , 125, 899, 906, 842, 624兲 designs. A candidate design for the 25-bar truss is described as follows:

0011兩0101兩0010兩1111兩0000兩1111兩0001兩0100兩0010兩1110兩0000兩000兩110 ⇓ Cross sections

0011

0101

0010

1111

L. chord

U chord

Strut

Diagonal

⌽42.7⫻3.2

⌽60.1⫻3.2

⌽33.7⫻3.2

⌽152.4⫻4.0

↓

↓

↓

↓

Heights hj

兩

Lengths lk

0000

1111

0001

0100

0010

1110 ↓

0000 000 ↓ 兩 ↓

110

0.25 m

h2=h1+0.50 m

h3=h2+0.025 m

h4=h3+0.10 m

h5=h4+0.050 m

h6=h5+0.275 m

h7=h6 2.00 m

3.50 m

↓

↓

↓

↓

↓

↓

whereas for the lk, 5 bits are used. The number of bits needed to fully describe a design is equal to 共4 + 4兲 ⫻ 4 + 3 ⫻ 5 = 51 and the multiplicity of the DS enumerates 251 共=2 , 251, 799, 813, 685, 248兲 designs. A candidate design for the 30-bar arch is described as follows:

For the 30-bar arch, the total length is L = 13 m and the load participation factors 兵ai其 depend on the shape of the structure. The number of DVs for the 30-bar arch is equal to 11 共four control the size of the truss and seven control its shape兲. For the DVs controlling the Ai and h j, 4 bits are used to fully describe each DV,

0110兩0001兩0111兩0000兩0000兩1111兩0001兩0011兩00011兩10000兩01011 ⇓ Cross sections

0110

0001

0111

0000

L. chord

U chord

Strut

Diagonal

⌽76.1⫻3.2

⌽26.7⫻2.9

⌽82.5⫻3.2

⌽21.3⫻2.8

↓

↓

↓

↓

Heights hj

兩

Lengths lk

0000

1111

0001

0011

h1=1.000 m

h2=h1+0.775 m

h3=h2−0.150 m

h4=h3−0.300 m

↓

↓

↓

↓

兩

00011 10000 01011 ↓

↓

↓

3.15 m

3.30 m

2.55 m



Table 2. Cross Sections 共CS兲共⌽ = d and t Are in mm兲, Heights 共h j兲, and Lengths 共lk兲 Considered CS name ⌽21.3⫻ 2.8 ⌽26.7⫻ 2.9 ⌽33.7⫻ 3.2 ⌽42.7⫻ 3.2

d

t

CS name

d

t

CS name

d

t

CS Name

d

t

21.3 26.7 33.7 42.7

2.8 2.9 3.2 3.2

⌽48.4⫻ 3.2 ⌽60.1⫻ 3.2 ⌽76.1⫻ 3.2 ⌽82.5⫻ 3.2

48.4 60.1 76.1 82.5

3.2 3.2 3.2 3.2

⌽88.9⫻ 3.2 ⌽101.6⫻ 3.6 ⌽108.0⫻ 3.6 ⌽114.3⫻ 3.6

88.9 101.6 108.0 114.3

3.2 3.6 3.6 3.6

⌽127.0⫻ 4.0 ⌽133.0⫻ 4.0 ⌽139.7⫻ 4.0 ⌽152.4⫻ 4.0

127.0 133.0 139.7 152.4

4.0 4.0 4.0 4.0

25-bar truss h1 h2 h3 h4 h5 h6 h7 L 1, L 2

0.25– 0.50 m with step= 0.05 m 0.50– 1.50 m with step= 0.10 m 0.00– 0.30 m with step= 0.025 m add to h1 0.30– 0.50 m with step= 0.050 m add to h1 Values as in h2, add to h2 Values as in h2, add to h3 0.00– 0.30 m with step= 0.025 m Values as in h5, add to h5 Values as in h5, add to h6 From 2.00 to 3.75 m with step= 0.25 m

Coding and decoding the binary expressions of all particles is performed at every time step to evaluate the performance of every candidate design as determined by the optimization problem of Eqs. 共4兲 and 共5兲. The ratio of costs is set to Cfail / Cmat = 200,000, Pn,lim = 10−6, and Pstr,lim = 5 ⫻ 10−6. These values correspond to reliability indexes of ␤n,lim = 4.753 and ␤str,lim = 4.417, respectively. The ␤str,lim is taken equal to the proposed reliability index in the draft of the Probabilistic Model Code 共JCSS 2001兲 for structures “with small relative cost of safety measure and moderate consequences of failure.” The selection of more stringent limits for the elements is dictated by the type of structure examined. A statically determinate structure has no mechanism to redistribute the acting stresses after failure of any of its elements. The probability of failure of the entire structure is greater than or at least equal to the probability of failure of its most likely to fail element 共Christensen and Murotsu 1986; Christensen and Baker 1982兲, thus leading to a more stringent limit.

BPSO Parameters The BPSO parameters affect the capacity of the optimization algorithm to explore the DS and exploit areas of interest. They should be selected appropriately to avoid undesirable behavior such as bit “locking,” erratic bit flipping, etc. For the swarm size 共the number of particles that comprise the swarm兲 the following values are considered: Ns = 兵50, 60, 70, 80, 90, 100其 creating a pool of solutions ranging from 100 to 200% of the length of the chromosome that describes the design which is considered adequate. The following combinations of the cognitive and social parameter are considered: 兵关c1 , c2兴其 = 兵关0.0, 4.0兴 , 关0.5, 3.5兴,关1.0, 3.0兴, 关1.5, 2.5兴 , 关2.0, 2.0兴 , 关2.5, 1.5兴 , 关3.0, 1.0兴 , 关3.5, 0.5兴 , 关4.0, 0.0兴其, in accordance with the recommendations of Franken and Engelbrecht 共2005兲. Limits with regard to the value of w are considered as a function of c1, c2, and vmax. After some manipulation, the upper and lower limit of w, for any bit of the chromosome, results as follows:

30-bar arch

h2

From 1.00 to 1.40 m with step= 0.025 m From 1.40 to 2.15 m with step= 0.050 m 0.00– 0.775 m with step= 0.025 m, add to h1

h3 h4 L1 L2 L3

−0.175– 0.60 m with step= 0.025 m, add to h1 −0.375– 0.40 m with step= 0.025 m, add to h1 3.00– 4.55 m with step= 0.05 m 2.50– 4.05 m with step= 0.05 m 2.00– 3.55 m with step= 0.05 m

h1

1−

储pi,j − xi储r1c1 + 储gi − xi储r2c2 艋w⬍1 vmax +

储pi,j − xi储r1c1 + 储gi − xi储r2c2 vmax

共6兲

The upper limit ensures no bit value “locking” either to 0 or 1. The lower limit ensures that the velocity of the particle can reach the corresponding limits. The Euclidian metric of 储g-x储 and 储p-x储 is either equal to 0 or 1. Values of w ⬍ 兵1 + 共c1 + c2兲 / vmax其 ensure no bit locking and for w ⬍ 兵1 + 共c1 + c2兲 / 共2vmax兲其 this condition is also satisfied in the average sense 共as E关rc1,2兴 = 0.5c1,2兲. For w ⬎ 兵1 − 共共c1 + c2兲 / vmax兲其 the limit values of velocity can be reached. For w兵1 − 共共c1 + c2兲 / 共2vmax兲兲其 this condition is also satisfied in the average sense. Based on the results of Eq. 共6兲 for w the following values are considered: 兵w其 = 兵0.5, 0.6, 0.7, 0.8, 0.9, 1.0其. These values remain constant during the optimization process. For the range of w considered no bit locking is expected as w 艋 1. For w = 0.5, vmax 艋 8 共the limit for c1 + c2 = 4兲 ensures that the corresponding limits can eventually be reached. In total, 324 combinations of Ns, w, c1, and c2 are considered. For the basic study vmax = 4.0 for a threshold probability of approximately 1 / k, where k = length of the chromosome resulting in one bit flip per chromosome per time step. For valid statistics in each combination of parameters 共runs兲, 30 analyses with a different initial random seed are performed, with a total duration of 200 time steps.

Numerical Results In Fig. 3 and 4, eight near optimal designs for the 25-bar truss and the 30-bar arch are presented. These designs are obtained from the last time step of the run that produced the best design and demonstrate the capacity of the algorithm to produce a plethora of good feasible near optimal designs. This is quite important in structural optimization, where very rarely we strive for the mathematical optimum which, due to limitations in the formulation of the problem, by no means can be considered as the one and only superior solution of the real problem. All the proposed designs are



Fig. 3. Optimal designs for the 25-bar truss 关monetary units共MU兲兴—grid size 1.0 m

symmetric about the vertical axis at midspan. The optimal designs of the 25-bar truss use the ⌽42.7⫻ 3.2 cross section for the lower chord. For the upper chord, either the ⌽76.1⫻ 3.2 or the ⌽82.5 ⫻ 3.2 cross sections are used. Cross sections of ⌽33.7⫻ 3.2 and ⌽42.7⫻ 3.2 are used for the struts and diagonals. The inclination of the diagonals increases when moving toward the midspan and the height at midspan to span ratio varies between 12.5 and 20%. The optimal designs of the 30-bar arch use the ⌽82.5⫻ 3.2 or the ⌽88.9⫻ 3.2 cross sections for the lower chord, in combination with the ⌽60.1⫻ 3.2 or the ⌽76.1⫻ 3.2 cross sections for the upper chord. For the struts and diagonals, cross sections varying from ⌽21.3⫻ 2.8 to ⌽42.7⫻ 3.2 are used. The height at midspan to span ratio varies from 18 up to 24%. In Table 3, the reliability indexes for the member groups and the structure corresponding to the optimum designs of Figs. 3 and

4 are displayed. For the 25-bar truss the “least” reliable elements 共elements exhibiting reliability index values closer to the limit and thus more likely to fail under extraordinary demands or circumstances兲 are observed in the upper chord and the diagonals. For the 30-bar arch such elements are located in the upper chord. In general, as the objective value increases the structural reliability index also increases. The cost of the eighth in rank nearoptimal design is about 10% more than the cost of the optimal design, but its failure probability is more than an order of magnitude less. This is the additional information the designer can gain from the RBOD analysis which in simple terms quantifies the safety of a proposed design at the components and group-system level. For the 25-bar truss, an increase of ␤str,lim = 4.417 to ␤str,lim = 4.753 关which corresponds to a failure probability of 10−6 and is the set as the target failure probability of structures “with

Fig. 4. Optimal designs for the 30-bar arch 共MU兲—grid size 1.0 m JOURNAL OF COMPUTING IN CIVIL ENGINEERING © ASCE / MARCH/APRIL 2009 / 105


Table 3. Reliability Indexes for the Optimal Designs for the Elements of the Lower Chord 共␤lc兲, the Upper Chord 共␤uc兲, the Struts 共␤s兲, Diagonals 共␤d兲, and the Structure 共␤str兲 — 25-Bar Truss and 30-Bar Arch 25-bar truss Obj. 150.94 152.04 155.69 158.01 160.09 162.00 164.01 167.09

30-bar arch

␤lc

␤uc

␤s

␤d

␤str

Obj.

␤lc

␤uc

␤s

␤d

␤str

5.614 5.566 ⬎10 ⬎10 5.614 6.456 4.867 6.275

4.782 4.767 4.779 4.801 4.782 5.012 5.028 4.972

4.809 5.185 4.809 4.851 6.294 5.273 5.649 5.335

4.777 4.805 5.117 5.812 4.777 4.855 4.962 5.719

4.661 4.685 4.700 4.736 4.689 4.806 4.782 4.957

202.90 204.01 207.09 209.89 211.04 213.25 215.70 222.45

4.925 4.812 4.848 4.829 4.956 4.959 4.967 5.234

4.786 4.829 4.783 4.839 4.854 4.954 5.139 5.769

⬎10 6.601 ⬎10 ⬎10 ⬎10 ⬎10 7.470 5.626

7.060 7.263 7.651 7.417 7.508 ⬎10 ⬎10 6.744

4.774 4.790 4.766 4.806 4.839 4.931 4.957 5.230

small relative cost of safety measure and large consequences of failure” 共JCSS 2001兲兴 results to an increase of the objective of the optimal design by 7%. This does not apply for the 30-bar arch, as all designs exhibit a reliability index ␤str,lim above 4.753. This is due to the characteristics of the considered set of candidate cross sections and can happen easily in every day design practice.

Parametric Studies As the main objective of this work is to investigate the performance of BPSO in a class of simple RBOD problems, a set of

Fig. 5. Evolution of the objective function—25-bar truss

parametric studies is performed concerning the parameters of the optimization algorithm. Effects of BPSO Parameters w, c1, and c2 on the Robustness of the Optimization Scheme In Fig. 5, the evolution of the objective function of the best solution of the swarm 共seed 15兲, together with the average objective value of the best solution over the 30 analyses, for the 25-bar truss with vmax = 4.0 is presented. In addition, the evolution of this objective ⫾2 SD is presented, which provides an estimate of the scatter of results. The best solution for the 25-bar truss is found for 兵Ns , w , c1 , c2其 = 兵70, 1.0, 2.0, 2.0其. The evolution of the average value shows that the algorithm locates areas where good solutions reside early in the optimization process as illustrated in Fig. 6, where the best designs found by the swarm at different time steps are presented. The algorithm fine-tunes these solutions in a slower fashion, which is typical in all evolutionary algorithms. The history of the objective of the best solution for the 25-bar truss 共Seed 15, in Fig. 5兲 shows a sharp drop at Time Step 10 followed by numerous small improvements of the design. The best design at Time Step 1 共Fig. 6兲, exhibits a cost of 285.14 Monetary Units 共MU兲 and violates the constraint for the upper chord elements by 31% with an estimated failure probability of 1.31⫻ 10−6 共which corresponds to a reliability index ␤ = 4.698兲. At Time Step 5, the best design of the swarm exhibits a cost of 248.49 MU and both its shape and choice of cross sections differ considerably from the best design of Time Step 1. At Time Step 10, the best design

Fig. 6. Evolution of the swarm best design for the 25-bar truss 共Ns = 70, w = 1.0, c1 = 2.0, c2 = 2.0兲—grid size 1.0 m 106 / JOURNAL OF COMPUTING IN CIVIL ENGINEERING © ASCE / MARCH/APRIL 2009


and c1, the average value of all the corresponding objectives of the optimal designs found after an analysis, and the objective of “the worst optimal” solution 共maximum value兲 for the problems examined, with regard to w and c1, are presented, respectively. The average and standard deviation 共statistics兲 are calculated over the remaining parameters of the BPSO over all analyses and runs. Thus, for w, the best, worst, average, and standard deviation of the optimal designs are obtained from a set of 1,620 values 共6 swarm sizes by 9 combinations of c1 and c2 by 30 analyses with different initial random seed兲. Similarly, for c1, the corresponding best, worst, average, and standard deviation of the optimal designs are obtained from a set of 1,080 values 共6 swarm sizes by 6 values of w by 30 analyses with different initial random seed兲. Values of c1 in the range of 关1.5–2.0兴 and w in the range of 关0.9–1.0兴 are recommended. The suggested range for c1 is provided under the assumption of c1 + c2 = 4.0, in accordance with the recommendations of Franken and Engelbrecht 共2005兲. For the 30-bar arch, deterioration of performance is observed as c1 increases with regard to the average objective. However, performance is not affected considerably with regard to the objective of the best design of the swarm.

Fig. 7. Evolution of the objective function—30-bar arch

exhibits a cost of 183.09 MU although it violates the constraint for the upper chord elements by 12%. At Time Step 20, the best design of the swarm establishes a shape which is practically indistinguishable from the shape of the best optimal design with a cost of 176.33 MU. From this point forward the evolution process fine-tunes the good optimal designs until at Time Step 146 the best overall design of the swarm with a cost of 150.94 MU is discovered. In Fig. 7, the same set of objective function histories as in Fig. 5 are presented for the 30-bar arch. The best solution for this problem is found for 兵Ns , w , c1 , c2其 = 兵70, 0.9, 1.0, 3.0其. The history of the objective exhibits stagnation at the early stages, then two sharp drops at Time Step 38 and Time Step 61 followed by relatively minor improvements of the design till convergence. In Tables 4 and 5, the objective value of the best design 共minimum value兲 over all analyses and runs for a particular value of w

Effect of Maximum Velocity vmax on the Robustness of the Optimization Scheme Finally, the effect of vmax on the robustness of the BPSO is examined. The vmax as a function of the bit-flip probability 关Pr共BF兲兴, using Eq. 共3兲 is given as vmax = − ln关− 共A − 1兲/A兴

共7兲

where A = 1 − Pr共BF兲.

Table 4. Analysis Results with respect to the Inertia Parameter 共w兲 w

1.0

25-bar truss Min 150.939 174.755 E 关·兴 SD 14.047 Max 227.379 30-bar arch Min 203.911 224.977 E 关·兴 SD 13.658 Max 277.983 Note: Bold indicated best values.

0.9

0.8

0.7

0.6

0.5

153.198 169.466 6.876 189.782

153.246 172.224 6.668 188.249

156.018 175.734 7.437 195.295

157.703 178.779 8.407 200.506

155.888 181.553 9.023 200.344

203.867 220.260 5.458 234.261

205.240 223.887 6.877 244.792

206.023 228.199 8.564 250.502

213.189 233.246 10.121 259.201

212.758 237.117 11.132 259.209

Table 5. Analysis Results with respect to the Cognitive Parameter 共c1兲 0.0 0.5 c1 25-bar truss Min 153.343 153.668 174.305 174.077 E 关·兴 SD 11.112 10.753 Max 253.096 242.326 30-bar arch Min 205.240 205.086 224.857 225.155 E 关·兴 SD 9.705 9.462 Mac 281.056 280.395 Note: Bold indicated best values.

1.0

1.5

2.0

2.5

3.0

3.5

4.0

154.939 173.427 9.327 232.151

153.246 173.109 8.670 231.137

150.939 173.385 8.094 206.962

151.418 173.804 8.612 209.072

154.774 175.178 9.093 205.350

155.714 177.815 8.950 206.919

160.956 183.665 9.048 216.421

203.867 225.349 9.507 278.329

204.480 225.832 9.781 278.243

207.179 226.143 9.520 278.986

203.970 226.742 10.294 266.143

203.911 228.513 11.695 274.226

206.480 231.004 12.401 272.018

205.892 237.934 11.370 275.272



Table 6. Analysis Results for Different Values of vmax vmax

5.6

4.7

25-bar truss 175.616 175.682 E 关·兴 Min 151.228 151.557 Max 304.474 245.424 30-bar arch 228.108 228.040 E 关·兴 Min 203.970 203.905 Max 282.492 280.393 Note: Bold indicated best values.

4.0

3.3

2.3

175.418 150.939 253.096

175.722 152.504 244.287

175.740 152.163 304.459

227.947 203.867 281.056

227.879 203.934 286.812

227.910 202.898 286.703

Conclusions

The probability to obtain x bit flips on average per particle per time step is Pr共BF兲 ⬇ x / k, where k is the chromosome length. Values of vmax permitting an average 0.2, 0.5, 1.0, 2.0, and 5.0 bit flips per particle per time step are considered. For k ⬇ 50 and x = 兵0.2, 0.5, 1.0, 2.0, 5.0其, Pr共BF兲 = 兵0.40, 1.00, 2.00, 4.00, 10.00其 corresponding via Eq. 共7兲 to vmax = 兵5.6, 4.7, 4.0, 3.3, 2.3其. As vmax ⬍ 8 the limit values can be reached for all w. In Table 6, the variation of the objective value of the best design, the average value of the objective of the optimal designs, and the maximum objective value of the worst optimal design, over all runs and analyses, with regard to vmax, is presented. For the 25-bar truss, the best behavior is observed for one bit flip per particle per time step. For the 30-bar arch, the variation of vmax does not affect considerably the performance of the algorithm. For low values of vmax, the capacity of the algorithm to refine the optimum solution increases and the scatter of the objective of the best design found by the swarm also increases. Although this situation is desirable, as diversity increases the robustness of stochastic optimization algorithms 共Dimou 2004兲, the same effect can also be obtained with a local optimizer as in Mishra 共2006兲 and Charalampakis and Koumousis 共2008兲, or with the craziness operator as in Fourie and Groenwold 共2002兲, Schutte and Groenwold 共2003兲, and Venter and Sobieszczankski-Sobieski 共2004兲 without suffering the added computational cost due to slow convergence. In Table 7, the objective of the best design with regard to vmax and Ns for the problems examined is presented. In six out Table 7. Objective Value for Best Design 共Minimum Objective兲 for Different Values of vmax and Ns 25-bar truss Ns / vmax 50 60 70 80 90 100

5.6

4.7

151.228 152.717 152.939 151.833 152.175 152.109

153.040 151.973 151.557 154.469 151.898 154.561

4.0 155.648 151.418 150.939 151.838 153.343 153.198

3.3 155.729 152.611 155.072 155.325 152.609 152.504

5.6

4.7

50 205.948 60 207.665 70 205.909 80 203.970 90 207.863 100 204.314 Note: Bold indicates best

204.055 205.178 208.215 203.905 204.006 204.157 values with

4.0

3.3

The BPSO is implemented in the shape and size RBOD of a 25-bar truss and a 30-bar arch. These structures define two different types of structural systems carrying vertical loads representing two different patterns of structural behavior. The BPSO produces good optimal solutions improving the best solutions reported in the literature 共Dimou and Koumousis 2003, 2004兲 and outperforms the results of the standard genetic algorithm for the problems examined. The overall best results of the BPSO are observed for c1 = 关1.5– 2.0兴 and w = 关0.9– 1.0兴 and for vmax values permitting one bit flip per particle, per time step. This represents a range of parameters for which accurate and computationally efficient results are obtained. The BPSO emerges as a valuable tool for structural optimization in the field of RBOD and is proved quite competent in fully exploiting the regions of optimum solutions, providing numerous near optimal designs. In this respect it offers a plethora of alternative candidate designs. In addition, evaluation of the optimal solutions reveals that a considerable increase of reliability can be achieved with a relatively small increase of the cost of the structure. This fulfills the premises of RBOD and justifies the role of both reliability analysis and optimization as important additional considerations in the design of structures.

Acknowledgments This work was supported 75% by the European Union— European Social Fund, 25% by the Greek Ministry of Development—General Secretariat for Research and Technology and by private funds under Measure 8.3 of the Operational Program for Competitiveness—3rd Community Support Program.

2.3 152.163 153.301 152.693 152.676 154.823 154.677

30-bar arch Ns / vmax

of twelve combinations, the best design is found for vmax = 4.0; thus, the notion of “optimum” mutation probability derived from the area of GAs is also applicable in BPSO. Therefore statistically speaking, a vmax value that results in one bit flip per particle per time step is recommended as a good value when BPSO is implemented in optimization, for this particular type of problem. In other words, it is expected that values of vmax resulting in one bit flip per particle per time step will maximize 共in a statistical sense兲 the robustness of the optimization algorithm.

2.3

205.240 207.344 204.138 204.648 205.432 204.928 203.867 204.486 204.529 204.179 204.035 202.898 203.911 203.934 204.225 203.970 206.617 203.985 regards to vmax for various Ns.

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