Res Eng Design (2006) 17:73–84 DOI 10.1007/s00163-006-0020-8

ORIGINAL PAPER

Practical design optimization of truss structures using the genetic algorithms A. Dominguez Æ I. Stiharu Æ R. Sedaghati

Received: 28 September 2005 / Accepted: 18 May 2006 / Published online: 10 August 2006
Springer-Verlag London Limited 2006

Abstract Truss structures are widely employed in the industrialized world. They appear as bridges, towers, pylons, roof supports, building exoskeletons or high technology light space structures. This paper investigates the simultaneous size, geometry and topology optimization of real life large truss structures using genetic algorithms (GAs) as optimizer and finite element method as analyzer. In general, the large truss structures are constructed for practical reasons from the duplication of some basic structures called bays. Thus, the final optimum design may be reached by optimizing the characteristics of the basic bays instead of optimizing the whole structure. Both single and multiobjective functions based on the mass of the structure and the maximum nodal displacement have been considered as the cost functions. In order to obtain realistic optimal designs, the cross-sectional areas have been extracted from the standard profiles according to AISC codes and practical conditions are imposed on the bays. The design optimization problem is also constrained by the maximum stress, maximum slenderness ratio and the maximum and minimum cross-sectional area of the truss members. To accommodate all these constraints, two different penalty functions are considered. The first penalty function considers the normalization of violated constraints with respect to the allowable stress or slenderness ratio. The second penalty function is a constant function which is used to penalize the violations of the

A. Dominguez Æ I. Stiharu Æ R. Sedaghati (&) Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd., West Montreal, Quebec, Canada H3G 1M8 e-mail: [email protected]

slenderness ratio. Three illustrative examples of realistic planar and space truss structures have been optimized to demonstrate the effectiveness of the proposed methodology. However, other criteria such as cost and/ or manufacturability could be quantified and included in the optimization formulation.

1 Introduction The objective of this work is to present an efficient and practical approach based on the genetic algorithms (GAs) capable of design optimization of realistic truss structures. Generally, the design optimization of truss structures generates the design parameters with limited practical application (Galante 1996; Deb and Gulati 2001; Prendes et al. 2005; Erbatur et al. 2000). For instance, the optimum cross-sectional areas of the truss members may not be found on the market, thus imposing high manufacturing cost. Even if the optimization process considers only profiles from the market, the applicability of the final designs is limited due to involvement of numerous types of profiles and the challenge associated with the assembly of such a structure. Also, generally the optimum design implies many changes of cross-sectional areas at the location of joints, which is not practically recommended. In addition, when simultaneous optimization of the size and topology of the structure is considered, the optimum structure may include numerous types of elements from different material and with geometrical characteristics, which is not feasible from the manufacturing point of view. A comprehensive review of the literature

123

74

in the area of discrete structural optimization is provided in Shea et al. (1997) and Arora (2002). Although the term ‘real life truss structures’ implies a broad kind of structures, here the term is adopted to describe those very common structures with similar bays and the relative members between bays with same profile. In this study, it has been considered that the realistic large truss structures are fabricated from the arrangement of the single bays. Thus, the objective is the optimization of the parameters of the bay. Based on this realistic consideration, the solution space is drastically reduced and the optimum result can be found efficiently and accurately. Furthermore due to this arrangement, the final optimum design will require few different standard elements, which can be easily sourced and managed. Thus, a final optimum design could be fabricated with low manufacturing cost. Although different softwares such as SOL2000, GAOS and MOSES have been developed to optimize discrete section sizes, they do not include the modularity property of structures. The optimum design of practical structures may include many aspects such as manufacturing cost, energy, cost of the material, assembling, mass, stress and displacement which may be the objective of a very extensive cost analysis. In practical applications, the general design will decide upon the criteria of optimization as well on the weighting factors. These criteria are usually based on the destination of the product. In this paper, a methodology that takes advantage of the condition of real structures which are built from basic modules is proposed. Further, the final optimum designs involve few different profiles and all the elements in same position of the bay are fabricated from the same profile as it can be observed in many real life structures. It is clear that the optimum selection of the profile for each element of the structure can produce lighter structures than the proposed methodology; however, the implementation of such final designs is not practical in many cases. Truss structures consist of flexible truss members under axial forces only and are pin-connected at joints. Due to the light weight and easy assembling, these structures play an important role in the economy and have numerous applications such as bridges, towers, cranes, roof supports, building skeletons, space deployable structures, etc. Structural design optimization can be realized in three broad categories (Kirsh 1989; Hafka and Grandhi 1986) namely as size, topology and geometry optimization. The cross sectional optimization, also known as size optimization, assumes that the elements, nodes, connectivity and locations are fixed while it searches

123

Res Eng Design (2006) 17:73–84

for the optimum shape (cross-sectional area) of the elements. Topology optimization addresses the issues that size optimization ignores; it is concerned with the number and connectivity of the truss members and joints. On the other side, the geometry optimization looks for the optimum shape of the whole structure. In the present work, the size, topology and geometry optimization of the single bays with which the real life engineering truss structures are constructed has been addressed. The application of the optimization techniques in the field of structural engineering optimization dates back to at least 1904, when Michell (1904) presented the optimal discrete truss structures now called Michell structures. Heyman (1956) optimized frame structures using the linear programming technique. Since then, extensive research has been conducted in the structural optimization field with limited industrial applications. GAs was first applied by Goldberg and Samtani (1986) to optimize structures for the minimum weight under maximum stress constraints. Galante (1996) then applied the same technique to optimize real-life truss structures. The ‘rebirth technique’ was then proposed to improve the results of the traditional GA. A two-phase method based on the GAs is proposed by Rajeev and Krishnamoorthy (1997) to perform size and topology optimization of truss structures. The result of the optimization process is highly influenced by the penalty functions applied to accommodate the constraints. Yang and Soh (2002) applied genetic programming to optimize truss structures using commercial profiles. The squared of the normalized values with respect to the maximum permissible was used as a penalty function. Sandgren and Cameron (2002) used a hybrid method to solve the multicriteria optimization and applied the standard deviation as a penalty function. An adaptive penalty function is proposed by Nanakorn and Meesomklim (2001). Coello (2000) has also carried out a comprehensive survey of multicriteria optimization techniques. The structural optimization problem has generally been recognized as a constrained nonconvex problem (Anthony et al. 2000). Additionally, the discrete variables to define the possible profiles to construct the structures convert it to nonsmooth problem. Therefore, it is necessary to employ a robust optimization method capable of handling these types of problems while providing global optimum solution. GAs have been used as the optimizer due to its capability to catch solutions close to the global optimum and operating with discrete variables. Also, GAs can easily be applied to multiobjective functions.

Res Eng Design (2006) 17:73–84

75

In this paper first the optimization problem has been discussed in detail. Further, the GA methodology is briefly explained. Finally, three illustrative examples have been investigated under different conditions in order to demonstrate the capability of the proposed methodology.

2 Optimization problem The optimization can be defined (Osyczka 1985) as the problem of finding a vector of decision variables, which satisfies the constraints and optimizes a vector function whose elements represent the objective functions. These functions form a mathematical description of performance criteria, which are usually in conflict with each other. Hence, the term ‘optimize’ means finding such a solution, which would give the values of the entire set of objective functions acceptable to the designer. Formally, the optimization problem is stated as: find the vector x
¼ ½x
1 ; x
2 ; . . . ; x
n  which will satisfy the m inequality constraints: gi ðxÞ>0;

i ¼ 1; 2; . . . m;

ð1Þ

the p equality constraints: hi ðxÞ ¼ 0;

i ¼ 1; 2; . . . p;

ð2Þ

and optimize the vector function: f ðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; . . . ; fn ðxÞT

ð3Þ

where x ¼ ½x1 ; x2 ; . . . ; xn  is the vector of decision variables and the optimal solution is denoted by the vector x
: In the case of n = 1, the problem is called single-objective optimization, otherwise it is known as multicriteria or multiobjective optimization. In this work both single and multicriteria objective functions are investigated. For the latter, the objective is to minimize both the mass and the maximum nodal displacement and for the former the goal is just minimization of the mass. Thus, the objective functions may be defined as: f 1 ðxÞ ¼ qLA

ð4Þ

f 2 ðxÞ ¼ maxðU mag Þ

ð5Þ

subject to: r6rallow

ð6Þ

k6kallow

ð7Þ

Amin >A>Amax

ð8Þ

where q ¼ ½q1 ; q2 ; . . . ; qk  is the mass density vector; L ¼ ½L1 ; L2 ; . . . ; Lk  is the length vector; A ¼ ½A1 ; A2 ; . . . Ak  is the vector of cross-sectional areas; Amin and Amax are the vector of lower and upper bound of cross-sectional areas and k is the number of elements. Another constraint regarding the geometry characteristics of the bays or the number of bays will also be specified for each particular problem. The vectors r and k contents the stress and the slenderness ratio of each element and their respective allowable values are defined in the vectors rallow and kallow ; respectively. U mag ¼ ½Umag1 ; Umag2 ; . . . ; Umagj  is the vector of nodal displacement of the j nodes. It is noted that nodal displacement vector can be calculated from the equilibrium equation as: U ¼ K 1 f

ð9Þ

where K and f are the system stiffness matrix and the nodal force vector and can be obtained by the finite element method formulation and analysis. To guard against buckling, the stability constraint is considered as part of the design optimization problem. In this study in order to have a sense of real-life design practice, the American Institute of Steel Construction (AISC) codes (AISC 1989) have been adopted for the relative design specifications. Thus, the allowable tension stress is considered to be 0.6r y; the allowable member slenderness ratio is specified to be 300 for tension members and 200 for compression members; to guard against buckling the allowable compression stress rbi of member i may be determined from the following equations according to AISC formulation: rbi ¼

rbi ¼

12p2 E 23k2i

for ki[C




 k2 1  2Ci 2 ry

5 3

i i þ 3k 8C  8C3

k3

for ki\C

ð10Þ

ð11Þ

wherepffiffiffiffiffiffiffiffiffiffiffiffiffi k i = Li /ri is the slenderness ratio, C ¼ p 2E=ry ; and r y, Li, and ri are the yield stress, the length and the radius of gyration of the cross-section of the member i, respectively, and the superscript b is related with buckling. To convert the constrained problem to unconstrained problem, a penalty function is introduced to the objective function. Penalty functions penalize

123

76

Res Eng Design (2006) 17:73–84

unfeasible solutions by reducing their objective function values in proportion to the violation. In other words, the penalty function is applied whenever the constraints are violated. Different types of penalty functions have been proposed to handle the constraints. In deterministic methods such as the sequential unconstrained minimization technique (SUMT), the quadratic penalty function is frequently used in order to keep the continuity of the gradients (Pardalos and Rosen 1987). Similar functions in combination with GAs have been applied by Galante (1996) and Goldberg and Samtani (1986). In these functions the constraint violation is squared and penalized by a scalar. Simple constants or normalized stresses are used by Deb and Gulati (2001) to penalize the cost function. Sandgren and Cameron (2002) employed the difference and standard deviation between the allowable and the nominal stress and displacement. Nanakorn and Meesomklin (2001) used an adaptive penalty function, which is able to adjust itself during the evolution. In this work two different types of penalty functions are considered. The first type of penalty function considers the normalization of violated constraints with respect to the allowable stress or slenderness ratio. The second type of penalty function is a constant function, which is used to penalize the violations of the slenderness ratio. Considering the above, the penalty functions may be defined as:

 p1 ðxi Þ ¼  p2 ðxi Þ ¼

 p3 ðxi Þ ¼

cp1
ððrðxi Þ  rmax ðxÞÞ=rmax ðxi Þ for jrðxÞj[jrmax ðxi Þj 0 otherwise cp2
ððkðxi Þ  kmax ðxi ÞÞ=kmax ðxi Þ for kðxi Þ > kmax ðxÞ 0 otherwise

cp3 0

for kðxi Þ > kmax ðxi Þ otherwise

 ð12Þ

 ð13Þ

3 Genetic algorithms

 ð14Þ

where cp1, cp2 and cp3 are constants. The normalization of the violations allows having large penalty values at the initial stage when the solution is far from the optimum point and small values when the solution is close to the optimum point. The penalty functions in Eqs. 12 and 13 will be used to penalize the size of the members and Eq. 14 will be applied

123

to modify the number of bays. The coefficients cp1, cp2 and cp3 are used to control the magnitude of the penalty values. For the case of discrete design variables, the penalty functions required to be rounded since the possible cross-sectional areas of the commercial profiles are sized in an arranged order and each cross-sectional area is defined by an integer which is represented in binary form. In this work, the proposed methodology considers that the whole truss structures are fabricated from the multiplication of some fundamental bays, which is mainly the case in many real-life truss structures. Thus, the design optimization of entire structures is reduced to the design optimization of the bays. This will enable us to perform size, topology and geometric optimization of large real-life truss structures with relatively few design variables and reduced computational effort. From the optimization point of view, this practical arrangement will cause the space solution to be reduced drastically and also cause the convergence to the optimum solution to be faster and more accurate. Since the optimized bays are made of truss elements with commercially available profiles, the fabrication cost would be significantly reduced. Moreover, the optimized bays can be practically used to assemble the whole structures easily, thus reducing the assembling cost drastically and avoiding joint connection problems.

Due to efficiency of GAs in capturing the global optimization solution, structural optimization problem studied in this research mainly utilizes the GAs as optimizer. The form of direction is based on Darwin’s ‘survival of the fittest’ theories (Goldberg 1989; Holland 1975). In GAs an initial population is created randomly or heuristically. Each element of the initial population represents a design and is called chromosome that are typically strings of binary bits. Each bit is

Res Eng Design (2006) 17:73–84

called genes. Genes occur at different locations or loci of the chromosomes, and take on certain values which are called alleles. These sets of chromosomes evolve over generations to get new and hopefully better designs. In biological science, the term genotype refers to the overall genetic makeup of an individual and is analogous to a structure in structural design. Also, the external characteristics are referred with the phenoype that is analogous to an actual parameter set such as design parameters. The design variables are codified in binary form and arranged in only one chromosome or string. Thus, length of the chromosome is sum of the bits need to represent each design variable and each chromosome represents the whole structure. Four differences separate GAs from more conventional optimization techniques (Goldberg and Samtani 1986): (1) direct manipulation of a coding, (2) search from a population, not a single point, (3) search via sampling, a blind search, (4) search using stochastic operators, nondeterministic rules. The implementation of the GAs usually involves the following cycle: (1) evaluate the fitness of all of the individuals in the population; (2) create a new population by performing operations such as reproduction, crossover and mutation on the individuals whose fitness has just been measured; (3) discard the old population and iterate using the new population. The operations of the second step, which represents the heart of GA, are briefly described. Reproduction is a process in which individual strings are copied according to their objective values, f (biologist call this function the fitness function). Intuitively, one can think of the function f as some measure of profit, utility or fitness that we want to maximize. Copying string according to their fitness values means that strings with a higher value have a higher probability of contributing one or more offspring in the next generation. This operator, of course, is an artificial version of natural selection, a Darwinian survival of the fittest among string creatures. In natural populations, fitness is determined by a creature’s ability to survive predators, pestilence and the other obstacles to adulthood and subsequent reproduction. In our unabashedly artificial setting, the objective function is the final arbiter of the string-creature’s life or death. Many reasonable algorithms exist to enable the individuals with the higher fitness values have the higher chance to be selected into the mating pool. The most widely used technique is the proportional fitness selection or roulette-wheel selection (Goldberg 1989) that can be represented by the following equation

77

Pr ðiÞ ¼

f ðiÞ n P f ði Þ

ð15Þ

j¼1

where i represents the chromosome, Pr the probability of being reproduced and f is the objective function or fitness and the size of the population n. However, the drawback of this technique is that when a very good element emerges in the population, this element will be reproduced many times. This would lead to have clones significantly far from the global optimum. On the other side when the fitness of the population is too uniform and the convergence is slow because the search may become a random walk. To address these shortcomings, fitness scaling is often used. The simplest way to scale fitness is linear scaling (Goldberg and Deb 1991): f 0 ¼ af þ b

ð16Þ

here, the scaled fitness f¢ is weighted by using two coefficients a and b such that the average scaled fitness f¢avg equals to the average of raw fitness favg and the best individual fmax is scaled down to around 2f¢avg. However, fitness scaling does not work for all populations. Another approach is to use a selection with good selection pressure. Selection pressure is the degree to which the better individuals are favored. For instance, Goldberg and Deb (1991) proposed a scheme that randomly chooses a set of individuals from the population and picks the best for reproduction. However, the important genetic information of a good chromosome can be ignored if it is in competition with better elements. In this study a new approach is employed (Dominguez et al. 2003), which considers the Table 1 to determine the number of times that one chromosome is reproduced. For example, if the size of the population n is 200 and the probability of reproduction Pr is 0.2; then the number of chromosomes to be copied (nr = nPr) is 40 elements of the best elements. Thus, the 1st best element is copied six times, the 2nd and 3rd are copied five times, and so on until it is completed 40. Therefore, the probability of reproduction can control the speed of convergence. After reproduction operation, simple crossover may proceed in two steps. First, members of the newly reproduced strings in the mating pool are randomly mated. Second, each pair of strings undergoes crossing over as follows: an integer position k along the string is selected uniformly at random between 1 and the string

123

78

Res Eng Design (2006) 17:73–84

Table 1 Number of times to copy the chromosome nr

Times of copying

2 7 16 30 50 77 ...

2 3,2,2 4,3,3,2,2,2 5,4,4,3,3,3,2,2,2,2 6,5,5,4,4,4,3,3,3,3,2,2,2,2,2 7,6,6,5,5,5,4,4,4,4,3,3,3,3,3,2,2,2,2,2,2 ...

length minus one [1,n – 1]. Two new strings are created by swapping all characters between positions k + 1 and n inclusively. Mutation is a low probability random operation, which may slightly perturb the design represented by the prodigy. The operator works on a bit-by-bit basis and is governed by the probability of mutation. The mutation operation allows each bit to change from 0 to 1 or vice versa. This operator also allows new zones to be explored in the search space. In optimization problem investigated in this study, the design variables are the discrete cross-sectional area, the dimensions of the bay and the number of bays. Thus, the cost function may be continuous and subsequently nondifferentiable function; thus, the gradient-based optimization routines imply some difficulties because of the calculation of the gradients. Considering this, the optimization method should be chosen from the stochastic methods that have the capacity of solving discrete and continuous problems and are able to find the global optimum. As was mentioned before, GAs have been selected due to its capacity to handle discrete design variables and also catch the global optimum point. Also, the penalty functions can be easily incorporated in GAs which is not the case for deterministic methods. In deterministic approaches the penalty methods have often been criticized due to steep ridges that they impose on otherwise smooth problems (Goldberg and Samtani 1986). These ridges can cause difficulty among search techniques, which depend upon a particular shape of local search surface. However, this objection is not relevant to GAs method since they do not depend on continuity or derivative existence for their operation. The GAs are significantly different from the most deterministic design optimization techniques. The GAs work with a coding of the design variables, as opposed to working with the design variables directly. The search is over a set of designs, unlike the traditional algorithms, which start from a single point. Finally, the GAs require only the objective function information (gradient information is not required) and is based on

123

the probabilistic rules. It is important to mention that there are other techniques such as the simulate annealing (SA), evolutionary programming (EP), evolutionary structural optimization (ESO) and Tabu search that do not rely on gradient information and have been used in the structural optimization area.

4 Illustrative examples To demonstrate the accuracy and efficiency of the proposed methodology, three practical 2D and 3D truss structures have been optimized using the abovedescribed GA method. The practical structures have been selected given the fact that the literature provides global optimum for the problems. The objective function is to minimize the weight or minimize both the weight and maximum nodal displacement. The selected examples look at a limited number of aspects related to performance than cost. Besides the constraints in Eqs. 6–8, structures are also guarded against the local buckling by Eqs. 10 and 11. In the problems discussed in this section the material is steel with modulus of elasticity E = 201 · 103 MPa, yield stress r y = 248.8 MPa and density q = 7,851.03 kg/m3. In order to ensure practical designs the possible cross-sectional areas will be selected from the profiles available on the market. Two different groups of selected steel profiles are presented in Table 2. The first group comprises only pipes. The second group contains W, S, HP and L profiles. In both groups the sections are arranged in an ascending order. All the specific properties such as the cross-sectional area, the radius of gyration, the density (mass per unit length) and the moment of inertia of the cross-sections could be obtained from the AISC standards (AISC 1989). In the GAs process a single point crossover with a probability of pc = 0.6 has been used. Consequently, the reproduction operation is conducted (Dominguez et al. 2003) which requires to define a probability of reproduction, pr, which is assumed to be pr = 0.15. Moreover, the probability of mutation, pm, and the initial population are assumed to be 0.005 and 150, respectively. Since the GAs operations are performed independently, the sum of the probabilities of each operation is not conditioned to be one. 4.1 Example 1: optimization of a 22-bar planar truss structure The 22-bar planar truss structure is shown in Fig. 1. The structure is fixed at nodes 1 and 2 and a downward force of 49,050 N has been applied at the last node.

Res Eng Design (2006) 17:73–84

79

Table 2 Selected standard steel profiles from the manual AISC (AISC 1989) Group I

Group II

SP1/2 EP1/2 SP3/4 EP 3/4 SP 1/2 EP 1 SP 11/4 SP 11/2 EP 11/4 SP 2 EP 11/2 EP 2 SP 21/2 SP 3 EP 21/2 SP 31/2 EP 3 SP 4 EP 31/2 SP 5 EP 4 SP 6 EP 5 SP 8 EP 6 SP 10 EP 8 SP 12 DEP 6 EP 10 EP 12 DEP 8

L 1X1X1/8, L 1-1/2X1-1/2X1/4 L 2X2X1/4, L 2-1/2X2-1/2X5/16 L 3X2X1/2, L 3X2-1/2X3/8 L 3X3X1/2, L 3-1/2X3-1/2X3/8 L 3-1/2X2-1/2X1/2, W 8X10 L 4X3X1/2, L 6X3-1/2X3/8 L 5X3X1/2, L 5X3-1/2X1/2 L 4X4X5/8, L 5X5X1/2 W 12X19, L 6X6X1/2 L 6X4X5/8, L 7X4X5/8 L 9X4X9/16, W 14X26 W 10X30, W 16X31 W 14X34, HP 8X36 L 8X4X1, W 10X39 S 12X40.8, HP 10X42 L 8X6X1, W 18X46 W 12X50, L 8X8X1 W 14X53, S 18X54.7 W 21X57, W 16X57 HP 13X60, W 24X62 W 8X67, W 18X71 S 20X75, W 12X79 W 14X82, S 20X86 W 16X89, W 21X93 W 10X100, W 24X103 W 14X109, W 18X119 W 27X129, W 21X132 W 14X145, W 30X146 W 18X158, W 27X161 W 33X169, W 24X176 W 30X191, W 33X201 W 36X210, W 36X230

SP, ES and DEP are standard pipe, extra strong and double extra strong, respectively

The objective function is to minimize the mass of the structure under stress and slenderness ratio constraints. The allowable compressive stress has been determined using Eqs. 10 and 11 according to the AISC standards. The possible cross-sectional profiles are selected from the elements of Group I in Table 2. Three different cases have been investigated in this problem. Case I is a size optimization in which the geometry and topology of the structure are assumed to be fixed and the design variables are only the cross-

sectional area of the members. A minimum mass of 243.1 kg has been obtained for this case. The optimum results for cross-sectional areas are shown in Table 3. It is noted that all constraints have been satisfied in the optimum point. This problem has also been investigated by Erbatur et al. (2000) and the mass of 248.8 kg has been reported using GAs. The true optimum mass is recorded as 238.1 kg which is obtained by evaluating all the possible combination and the optimum profiles are provided in fifth column of Table 3. It can be realized that a slightly better optimum solution has been obtained in this study. It is also interesting to note that in this case only six optimum profiles are matched with the true optimum profiles; while 17 optimum profiles recorded by Erbatur et al. are matched with true optimum result. However, the optimum mass in this study turns out to be lower than that of Erbatur et al. (2000). This may be due to the fact that structural optimization problems are highly nonunimodal. In the case II, the number of bays and the height of the structure are also introduced as design parameters. Increased number of bays reduces the stresses and the buckling effect, however it increases the weight of the structure. It is interesting to note that the minimum mass of 236.7 kg has been obtained in this case, which is about 1.4 kg lower than the true optimum result for the original problem. This is due to the fact that in this study the topology was modified since the number of bays was included as design variable. Also the maximum nodal displacement was found to be 16% lower than that of case I. The optimum height and the number of bays of the structure have found to be 2.095 m and 3, respectively. Knowing the total length of structure and the number of bays, the length of the individual bay is automatically determined. It is noted that no constraint violation exists at the optimum point. The iteration history of the evolutionary process for cases I– III is shown in Fig. 2. Improvement of the chromosomes through the process can be realized from this figure. In the trial III, a single bay has been optimized in order to find the optimum solution of the whole

Fig. 1 a Basic bay of a statically determinate 22-bar truss; b complete structure

a)

b)

123

80

Res Eng Design (2006) 17:73–84

Table 3 Optimum results for 22-bar planar truss structure Member 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Mass (kg)

Stress, r Case I ( · 108 Pa) –1.1558 1.4827 –0.7372 1.2984 –1.0804 1.4669 –0.7297 1.3568 –1.0344 1.4526 –0.7231 1.1645 –1.0138 1.4405 –0.7176 1.4334 –1.0704 1.4307 –0.7132 1.3188 –0.6482 1.4894

rallow Case I ( · 108 Pa) –1.2576 1.4928 –0.8977 1.4928 –1.3037 1.4928 –0.8977 1.4928 –1.2631 1.4928 –0.8977 1.4928 –1.1894 1.4928 –0.8977 1.4928 –1.0995 1.4928 –0.8977 1.4928 –0.7828 1.4928

k Case I 51.8027 182.2689 100.0510 50.9096 44.0710 182.2689 100.0510 62.3602 50.9096 182.2689 100.0510 77.0955 62.3602 182.2689 100.0510 94.7915 75.0383 182.2689 100.0510 183.9723 112.7006 241.8310

True optimum (Erbatur et al. 2000)

Using GAs (Erbatur et al. 2000)

Case I

Case II

SP 3 1/2 EP 1 SP 3 SP 2 SP 3 1/2 EP 1 SP 2 1/2 SP 2 SP 3 EP 1 EP 2 SP 2 SP 2 1/2 EP 1 SP 1 1/2 SP 2 SP 2 EP 1 EP 3/4 SP 2 SP1 1/2 EP 1 238.1

SP 3 1/2 EP 1 SP 3 SP 2 SP 3 1/2 EP1 1/2 SP 2 1/2 SP 2 SP 3 EP 1 SP 2 1/2 SP 2 SP 2 1/2 EP1 1/4 SP 1 1/2 SP 2 SP 2 SP 1 SP 1 1/4 SP 2 SP 1 1/2 EP 1 1/2 248.8

EP 3 SP 1 1/4 SP 2 SP 3 SP 3 1/2 SP 1 1/4 SP 2 SP 2 1/2 SP 3 SP 1 1/4 SP 2 EP 2 SP 2 1/2 SP 1 1/4 SP 2 SP 1 1/2 SO 2 SP 1 1/4 SP 2 EP 3/4 EP 1 1/4 EP 1 243.1

SP 4 SP 2 SP 2 EP 2 EP 3 SP 2 SP 2 SP 1 1/2 SP 2 1/2 SP 2

structure. The objective is to minimize the mass of total structure through the optimization of the parameters of an individual bay. The total structure will then be assembled through the multiplication of this single bay. The design variables are the member’s cross-sectional areas, the height and the number of bays. Knowing the length of the total structure, the length of each bay could be automatically determined. These design variables are sufficient to define the size, topology and geometry of the whole structure. The minimum mass of

Fig. 2 Genetic history of the objective function for size, topology and geometry optimization of 22-bar plane truss for cases I–III

123

236.7

293 kg has been obtained for this case. The optimum height and length of the bay was found to be 1.69 m and 3 m, respectively. The optimum cross-sectional profiles for elements 1, 2, 3 and 4 for the individual bay shown in Fig. 1a have been found to be SP-3-1/2, SP-1/ 2, SP-1/2, EP-1-1/4, respectively, and none of the constraints have been violated. It is noted that although the optimum mass for this case is slightly higher than those in cases I and II, the less number of profiles required to assemble the structure and thus enable an easy fabrication, makes this approach extremely cost effective and practical. The comparison between these three cases is summarized in Table 4. It is clear that case III is computationally significantly more efficient than the other two cases. Typical optimization of truss structures that find the optimum cross sectional area or profile for each element allows a more uniform distribution of the forces. One of the intuitive optimality criteria methods is the fully stressed design (FSD) which considers that ‘‘the optimum design of each member of the structure that is not at its minimum gage is fully stressed under at least one of the design load conditions’’ (Haftka and Gu¨rdal 1992). However, under this concept the final design leads to have many different profiles for optimum structure which is not practical in most of the cases. For the case of proposed methodology it was observed that the distribution of forces is not as uniform as case I; however, the final design implies few profiles as it

Res Eng Design (2006) 17:73–84

81

Table 4 Comparison between different optimization cases

Objective function (kg) Time (s) Analyzed structures Space solution Bits of chromosome Number of bays Number of elements Iterations Dispmax (m) Height (m)

Case I

Case II

Case III

243.1 345 13,554 1.298e33 110 5 22 100 0.0409 2.0

236.7 451 16,937 6.646e35 119 3 10 100 0.0343 2.065

293.0 91 3564 5.368e8 29 3 10 29 0.0297 1.69

happens in many real truss structures which is our objective. 4.2 Example 2: optimization of a single lacing space truss structure In this section, the proposed methodology has been applied to optimize a single-lacing truss structure as shown in Fig. 3. These kinds of structures are usually used to construct various types of cranes (Fig. 4). Practically, the whole structure shown in Fig. 3b can easily be fabricated by using a basic bay shown in Fig. 3a. The basic bay has nine elements. Taking advantage of the symmetry in the structure, it has been assume that the elements 2 and 3, 4 and 5 and 7 and 8 are similar. Thus, six design variables can define the size (cross-sectional area) of the profiles used for these elements. Geometry and topology of the structure can also be defined by using the width, the height and the number of bays as design variables. As illustrated in Fig. 3b, the structure is fixed from one side and the forces of 78,480 N in Z-direction, 7,848 N in Y-direction and 3,924 N in X-direction are applied at the last two nodes on the other side of structure. The lateral X Fig. 3 a Description of the basic bay of a single lacing space truss structure; b cantilever truss structure with five bays

(a)

and Y forces could stand for the inertia forces of the load and wind loads, respectively. The direction of the diagonal member E1 is conditioned to change alternatively from one bay to the adjacent bay. Two different cases have been investigated. In case I, the objective is to minimize the mass of the whole structure through optimization of an individual bay. However in case II, the objective is to simultaneously minimize the mass and maximum nodal displacement. The weighing factor of 20,000 has been considered for the displacement component in order to provide the same preference for both displacement and mass. The possible profiles are extracted from the group II of Table 2. The iteration history for the average and optimum solution of the evolutionary process in both cases I and II is shown in Fig. 5. A minimum mass of 1,677 and 2,724 kg have been obtained for cases I and II, respectively. The maximum nodal displacement for cases I and II are 56 and 49 mm, respectively. The final optimum design for both cases recommends five bays. The final design for the cross-sectional areas (type of profiles), the height of structure and the number of bays for both cases is given in Table 5. From Fig. 5 it is observed that when two objective criteria are in conflict, as is the case of the displacement and mass, the curve of the objective function zigzags due to the penalizations. Furthermore, it is noted that a small reduction for the maximum displacement requires larger profiles. Examination of Table 5 reveals a violation of the allowable stress for each case. This problem can be solved by imposed higher stress penalty coefficient cp1 in Eq. 12 which can lead to heavier structures. A more practical solution is reinforcing just that member with a stronger profile. It should be noted that although with proper weighting coefficient, the same importance for maximum displacement and mass

(b)

123

82

Res Eng Design (2006) 17:73–84 Table 5 Results for different cases of optimization of a singlelacing space truss structure

Objective Mass (kg) Max. disp (m) Stress violations k violations Time (s) Analyzed structure Profiles E1 E2, E3 E4, E5 E6 E7, E8 E9 Final design Fig. 4 A tower crane constructed with the single lacing space truss structures

is considered, the output optimum mass and maximum displacement are not necessarily varying with the same proportion. In other words increase of 62.4% in mass of structure does not produce 62.4% reduction in deflection because the optimized structure in this case (case II) has different configuration and profile with respect to case I. 4.3 Example 3: optimization of a double-layer grid space truss structure A double-layer grid structure has been shown in Fig. 6. This type of structure is commonly used as roof supports as illustrated in Fig. 7. The basic unit shown in Fig. 6a has been used to construct the whole structure shown in Fig. 6b.

Fig. 5 Genetic history of the objective function for size, topology and geometry optimization of examples 2 and 3

123

Case I

Case II

Mass 1,677 0.056 E3 of 14.9% 0 1,015 31,417

Mass + disp. 2,724 0.049 E2 of 51% 0 1,052 31,670

L 3X2-1/2-3/8 W 10X30 4 L 2-1/2X2-1/2X5/16 L 1-1/2X1-1/2X1/4 L 3X3X1/2 4 L 2-1/2X2-1/2X5/16 5 bays, width = 1.1984 m height = 2.668 m

L 3X2-1/2X3/8 W 10X30 L 3X2-1/2X3/8 L 1-1/2X1-1/2X1/4 L 3-1/2X3-1/2X3/8 W 14X26 5 bays, width = 1.706 m height = 2.888 m

The structure is assumed to have a length of 9 m in both X and Y directions with equal number of bays on both sides. The objective is to minimize the mass of the whole structure through the optimization of the parameters of the individual bay. The design variables are the cross-sectional areas of the members in a basic bay as shown in Fig. 6a, which are selected from Group I in Table 2. The height of the structure and the number of bays in X or Y directions are evaluated after the optimization is completed. The optimum results have been provided in Table 6. A minimum mass of 1,767 kg has been obtained. The optimum structure consists of six bays in each X and Y directions with total of 409 elements and 98 nodes (287 DOF). It is noted that the final design is a practical structure with only ten types of elements, which are commercially available. This will naturally cause drastic reduction in assembly and fabrication cost of the whole structure. It should be noted that stress constraints have been violated in some members in final design. Although this violation is not significant, it can be easily remedied using stronger elements for those members. This is possible since the violations and the number of elements violating the stress constraint are small. Although it is desirable to have more supports to prevent buckling of the members at the supports, in this example it was considered a critical condition with only four supports as it can be observed in Fig. 6. However, this is not a limitation for the proposed methodology. Also, it should be noted that the methodology could be applied when there is an offset grid between the lower and the respective upper nodes

Res Eng Design (2006) 17:73–84 Fig. 6 a A single bay is considered to construct the double-layer grid 3D truss structure; b schematic representation of the structure with 4 · 6 bays

83

(a)

which is the case of Fig. 7. For this case the basic bay should be modified to include that offset grid. Furthermore, since the analysis of the structures is made using FEM and considering the whole structure, any support and load can be applied at any position. It is convenient to mention that the optimization of truss structures using the proposed methodology could yield larger mass values compared with the case when each element is optimized; but these optimized designs have limited practical application because they could imply many different profiles. Furthermore, when the optimum design of the structure requires changes in the parameters between the bays such as tapered structures, the modularity concept will not be effective.

(b)

5 Conclusions Structural problems are most of the time solved as direct problems. The designer who is facing a problem as any of the above presented in this paper is tempted to produce a configurational structure and work towards accurate evaluating of the cross-section of each member of the structure. Such an approach significantly limits the capability of producing a topology optimized design. The proposed method based on GAs optimization techniques enables the designer to find the optimal solution from a large map of possible designs to achieve an optimal solution for the inverse problem. Table 6 Result for the optimum design of the double-layer grid truss Optimization process Variables Chromosome (bits) Load Penalty functions Penalty coefficients Results Mass (kg) Profiles

Stress violations (element and associated bay)

Fig. 7 Typical application of the structure in roof supports (Courtesy of Luis Palome´, Constructor)

k violations Max. disp. (m) Bay

12: high, number of bays and ten for profiles 59 50 kg/m2 The same as the trial II Cp1 = 6.5, cp2 = 8, cp3 = 1 1,767 E1 = SP 3/4, E2 = SP 3/4, E3 = SP 3/4 E4 = EP 1, E5 = SP 3/4, E6 = SP 3/4, E7 = SP 1/2, E8 = SP 1-1/4, E9 = SP 1-1/4, E10 = SP 1-1/4 7 bay (1,1) = 11% 2 bay (2,1) = 2% 2 bay (2,1) = 1.7% 6 bay (6,1) = 4.6% 0 0.0469 Six per side and high of 1.079 m

123

84

Most practical large structures have been constructed from the duplication of a basic unit or bay in order to significantly reduce the fabrication and assembling cost. It has been shown that the design parameters of the basic bay could be used to perform the size, geometry and topology optimization of these kinds of structures. This will not only reduce the computational time drastically, but will also generate a realistic and practical optimum structure. In practical design problems usually the size of the members is selected from the standard profiles. This issue has also been incorporated in the proposed approach with introducing two groups of standard profiles. Practical designs constraints are imposed to the basic bay, thus the final optimum designs are realistic. In order to identify the global optimum the GAs algorithms has been employed as an optimizer. The finite element analysis has also been used to generate the analysis part of the design optimization problem. Genetic algorithms have proved to be a reliable optimizer for discrete and multiparametric problems and have the capability to apply different constrains and penalty functions without any concern regarding the continuity and differentially of the objective function. Although the GAs have proved to be an excellent tool to find the optimum designs, the experience of the designer plays an important role to define some parameters of the optimization process. The design knowledge is essential to define the upper and lower limits for cross-sectional areas of the elements, the number, the length, the height, the width of the bays or even the selection of the set of possible profiles. Acknowledgements Support by Natural Science and Engineering Research Council (NSERC) of Canada and Universidad Autonoma de Queretaro UAQ are acknowledged.

References AISC (1989) Manual of steel construction: allowable stress design. American Institute of Steel Construction (AISC), 9th edn. Chicago Anthony DK, Elliot SJ, Keane AK (2000) Robustness of optimal design solutions to reduce vibration transmission in a lightweight 2-D structure. J Sound Vib 229(3):505–528 Arora JS (2002) Methods for discrete variable structural optimization, in recent advances in optimal structural design. In: Burns S (ed) ASCE Press, pp 1–40 Chan E (1997) Optimal design of buildings structures using genetic algorithms. Ph.D. Dissertation, California Institute of Technology, Pasadena, California

123

Res Eng Design (2006) 17:73–84 Coello CA (2000) An updated survey of GA-based multiobjective optimization. ACM Comput Surv 32(2):109–143 Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37:447–465 Dominguez A, Sedaghati R, Stiharu I (2003) Topology optimization of adaptive structures using optimal placement and number of actuators. In: 6th CanSmart meeting, Montreal, pp 177–186 Erbatur F, Hasanc¸ebi O, Tu¨tu¨ncu¨ I, Kilic¸ H (2000) Optimal design of planar and space structures with genetic algorithms. Comput Struct 75:209–224 Galante M (1996) Genetic algorithms as an approach to optimize real-world trusses. Int J Numer Methods Eng 39:361–382 Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading Goldberg DE, Deb K (1991) A comparative analysis of selection schemes used in genetic algorithms. In: Foundations of genetic algorithms, San Mateo, CA, pp 69–93 Goldberg DE, Samtani MP (1986) Engineering optimization via genetic algorithm. In: Proceedings of the ninth conference on electronic computation. ASCE, New York, pp 471–482 Hafka RT, Grandhi RV (1986) Structural shape optimization—a survey. Comp Meth Mech Eng 57:91–106 Haftka RT, Gu¨rdal Z (1992) Elements of structural optimization. Kluwer, Dordrecht Heyman J (1956) Design of beams and frames for minimum material consumption. Quart Appl Math 8:373–381 Holland JH (1975) Adaptation in natural and artificial systems. The University of Michigan Press, Ann Arbor Kirsh U (1989) Optimal topologies of structures. Appl Mech Rev 42(8):223–238 Michell AGM (1904) The limits of economy in frame structures. Philos Mag 8(47):589–597 Nanakorn P, Meesomklin K (2001) An adaptive penalty function in genetic algorithms for structural design optimization. Comput Struct 79:2527–2539 Osyczka A (1985) Multicriteria optimization for engineering design. In: Gero JS (ed) Design optimisation. Academic, New York, pp 193–227 Pardalos PM, Rosen JB (1987) Constrained global optimization: algorithms and applications. Springer, Berlin Heidelberg New York Prendes MB, Bello A, Coz JJ (2005) A modified elitist genetic algorithm applied to the design optimization of complex steel structures. J Constr Steel Res 61:265–280 Rajeev S, Krishnamoorthy CS (1997) Genetic algorithms-based methodologies for design optimization of trusses. J Struct Eng 123(3):350–358 Sandgren E, Cameron TM (2002) Robust design optimization of structures through consideration of variation. Comput Struct 80:1605–1613 Shea K, Cagan J, Fenves SJ (1997) A shape annealing approach to optimal truss design with dynamic grouping of members. ASME J Mech Des 119(3):388–394 Yang Y, Soh CK (2002) Automated optimum design of structures using genetic programming. Comput Struct 80:1537– 1546