Proceedings of the Fourth International Conference on Mathematical and Computational Applications

June 11-13, 2013. Manisa, Turkey, pp.221-230

OPTIMAL DESIGN OF STEEL TRUSSES USING STOCHASTIC SEARCH

TECHNIQUES

Serdar Carbas1 and Oguzhan Hasancebi

2

1 Middle East Technical University, Department of Engineering Sciences,

06800, Ankara, Turkey, 2 Middle East Technical University, Department of Civil Engineering,

06800, Ankara, Turkey

carbas@metu.edu.tr

Abstract- The aim of this study is to present the efficiency of stochastic search

techniques and their applications to the optimum design of steel truss structures.

Suitable profiles for each element in a truss have to be selected from a given selection of

standard profiles. Stochastic particle swarm optimizer, harmony search optimization and

genetic algorithms are used as a solution method. These stochastic optimization

algorithms are widely used and suitable methods for hard combinatorial problems. The

comparison of algorithms is done using example problems. The optimum design

problems of these structures are formulated considering the design limitations imposed

by ASD-AISC (Allowable Stress Design Code of American Institute of Steel

Institution).

Key Words- Stochastic Search Techniques; Particle Swarm Optimizer; Harmony

Search Optimization; Genetic Algorithms

1. INTRODUCTION

Optimization has been expanding in all directions at an astonishing rate during the

last few decades. New algorithms and theoretical techniques have been developed; the

diffusion into other disciplines has proceeded at a rapid pace. At the same time, one of

the most striking trends in optimization is the constantly increasing emphasis on the

interdisciplinary nature of the field [1]. There are great advantages to recognizing or

formulating an engineering problem as an optimization problem. The most basic

advantage is that the problem can then be solved, very reliably and efficiently, using

stochastic optimization methods. There are also theoretical or conceptual advantages of

formulating an engineering problem as an optimization problem [2]. Development of

optimization algorithms made it possible for engineers to analyze a structure in much

more detail. Stochastic search techniques based algorithms, which are borrowing their

characteristics from biological methodologies and other natural phenomena, are widely

used in structural engineering to obtain the optimal designs of complex structures [3].

The main idea behind the skeleton of these algorithms is that a structure is analyzed by

dividing it into structural element groups and solving the equations of mechanics for

each of those element groups separately. With the increase of the research and

applications based on stochastic search techniques [4,5], constraint handling used in

metaheuristic computation techniques has been a hot topic in both academic and

engineering fields [6-10].

222 S. Carbas and O. Hasancebi

min max

min max

min max

1,2,...,

1,2,...,

0 1,2,...,

1,2,...,

i

i

b

i i

i

i m

i n

i nc

A A A i ng

1

( ) . .n

i i iiW x A L

In this study, the performance of the optimum structural design algorithms based

on particle swarm optimizer [11], harmony search optimization [12] and simple genetic

algorithms [13] is examined on the optimum design of real size plane trusses. For this

purpose, a simple truss structure and a shallow truss structure are designed using each of

the mentioned algorithms and their performances are compared.

2. OPTIMUM DESIGN PROCEDURE OF STEEL TRUSSES

The general formulation of the weight minimization problem for a truss structure is

formulated as in Eqns. (1) and (2).

minimize (1)

subject to

(2)

where W({x}) = weight of the structure; n = number of members making up the

structure; m= number of nodes; nc = number of elements subjected to compression; ng

= number of groups (number of design variables); الi = material density of member i; Li

= length of member i; Ai = cross-sectional area of member i chosen between Amin and

Amax; min = lower bound and max = upper bound; ζi and δi = stress and nodal

deflection, respectively; b

i =allowable buckling stress in member i when it is subjected

to compression. The design limitations of the optimization problem are imposed

according to ASD-AISC [14].

3. STOCHASTIC SEARCH TECHNIQUES

The stochastic search techniques inspire from the nature such as survival of the

fittest, immune system or cooling of molten metals through annealing to develop a

numerical optimization algorithm. These methods are offbeat stochastic search and

optimization methods used to find the optimum solution of combinatorial optimization

problems. Having not needed to the gradient information of the objective function and

constraints, they have been come in to vogue nowadays. The particle swarm optimizer,

harmony search optimization, and simple genetic algorithms are selected among others

from literature to be used as solution tools in this study. As can be understood from their

names each technique mimics one particular phenomenon that exists in the nature [15].

3.1 Particle Swarm Optimizer (PSO)

The particle swarm optimization method is one of the stochastic random search

methods that is developed by Eberhart and Kennedy [16], inspired by social behavior of

bird flocking or fish schooling. This behavior is concerned with grouping by social

forces that depend on both the memory of each individual as well as the knowledge

gained by the swarm. The phenomenon behind this behavior is called swarm

Optimal Design of Steel Trusses Using Stochastic Search Techniques 223

intelligence. The steps of the algorithm are outlined in the following as given in [17,

18]:

Step 1. Initializing Particles: A swarm consists of a predefined number of particles

referred to as swarm size. Each particle incorporates two sets of components; a position

(design) vector and a velocity vector. The position vector retains the values (positions)

of design variables, while the velocity vector is used to vary these positions during the

search.

Step 2. Evaluating Particles: All particles are analyzed, and their objective function

values are calculated.

Step 3. Updating the Particles’ Best and the Global Best: A particle‟s best position

(the best design with minimum objective function) thus far is referred to as particle‟s

best and is stored separately for each particle in a vector. On the other hand, the best

feasible position located by any particle since the beginning of the process is called the

global best position, and it is stored in a vector. At the current iteration, both the

particles‟ bests and the global best are updated.

Step 4. Updating a Particle’s Velocity Vector: The velocity vector of each particle is

updated considering the particle‟s current position, the particle‟s best position and

global best position.

During numerical applications to structural optimization problems, it has been found

that all the particles in the swarm are eventually dragged to the position identified by the

global best. When this happens, the current and best positions of all the particles

become identical to the global best, resulting in almost zero velocity vectors. Since

particles cannot fly any more in such a case, the search gets stuck in a very poor design

point. A reformulation is therefore considered in the present study, where an additional

velocity term is defined and added as in Eqns. (3-4) to give each particle a random

move in certain directions in the close neighborhood of its current position.

t

Nr

t

IBrc

t

IGrcwvv

s

i

k

i

k

i

k

i

k

ik

i

k

i

3

)()(

22

)()(

11

)()1( (3)

1 1/ 2 or 0 1/ 2i d dif r N if r N (4)

In Eqns. (3-4), r3 is a random number between 0 and 1; Ns is the number of steel

sections in the profile list; and i is 0-1 Heaviside function implemented by sampling a

random number r between 0 and 1, Nd is design variables. Probabilistically, Eqn. (3)

implies that in every two iterations only one design variable in a particle is changed to a

new position with random velocity term. The reformulated equation has been observed

to eliminate the aforementioned drawback and greatly improve the efficiency of the

technique.

Step 5. Updating a Particle’s Position Vector: Next, the position vector of each

particle is updated with the updated velocity vector, which is rounded to nearest integer

value for discrete variables.

Step 6. Termination: The steps 2 through 5 are repeated in the same way for a

predefined number of iterations.

224 S. Carbas and O. Hasancebi

3.2 Harmony Search Optimization (HSO) The harmony search method is first introduced by Geem and Kim [19]. This

presented stochastic technique is based on the musical performance process that takes

place when a musician searches for a better state of harmony [20]. In the process of

musical production, a musician chooses and brings together number of different notes

from the whole notes and then plays these with a musical instrument to find out whether

it gives a pleasing harmony. Similarly, a candidate solution is generated in the optimum

design process by modifying some of the decision variables to find optimum solution.

Step 1. Initialization of Harmony Memory Matrix: A harmony memory matrix is

generated and initialized at first. It incorporates a specified number of solutions referred

to as harmony size.

Step 2. Evaluation of Harmony Memory Matrix: Solutions are then analyzed, and

their objective function values are calculated.

Step 3. Improvizing a New Harmony: A new harmony is improvised (generated) by

selecting each design variable from either harmony memory or the entire discrete set.

The probability that a design variable is selected from the harmony memory is

controlled by a parameter called harmony memory considering rate ( hmcr ). If a design

variable attains its value from harmony memory, it is checked whether this value should

be pitch-adjusted or not (par).

Step 4. Update of Harmony Matrix: After generating the new harmony vector, its

objective function value is calculated. If this value is better (lower) than that of the

worst harmony vector in the harmony memory, it is then included in the matrix while

the worst one is discarded out of the matrix.

Step 5. Termination: The steps 3 and 4 are repeated until a pre-assigned maximum

number of cycle is reached.

3.3 Genetic Algorithms (GAs) Theory of GAs method depends on the principle of Darwin‟s theory of survival

fittest. This can be summarized that any individual animal or plant which succeeds in

reproducing itself is "fit" and will contribute to survival of its species, not just the

"fittest" ones, though some of the population will be better adapted to the circumstances

than others [21, 22]. The fundamentals of this algorithm are outlined below [13].

Step 1. Initial Population: In simple GAs, a design variable is not represented by its

actual design value; instead it is encoded as a binary string of finite length. A coded

design variable consists of zeros and ones, such as 001001, and is referred to as a

substring in GA terminology. If there are design variables in all, then such substrings

are joined together to form a complete string (chromosome or individual) that represents

a potential design to a problem at hand.

Step 2. Encoding: For each individual decoding is carried out to map all substrings to

some integer values representing the sequence numbers of standard steel sections in a

given profile list.

Step 3. Evaluation and Fitness: Once an individual is decoded, analysis is carried out

to obtain its structural response under external loads, and an objective function value is

assigned to it. After all the individuals are evaluated in this manner, each individual is

assigned a fitness score, which indicates the merit of the individual with respect to

overall population. In the present study, an inverse transformation function (Eqn. 5) is

Optimal Design of Steel Trusses Using Stochastic Search Techniques 225

used to calculate the fitness score kf of an individual k such that the objective function

value k of the individual is proportioned to the maximum ( max ) in the population.

k

kf

max (5)

The individuals‟ fitnesses calculated from Eqn. (5) need to be scaled to eliminate the

dominance of highly fit individuals during selection process. Eqn. (6) shows the fitness

scaling function used for this purpose.

ave

avemax

avemaxave

avemax )(

)(

)(

)1(f

ff

fsff

ff

sff cc

kk

(6)

In Eqn. (6), maxf , minf and avef are the maximum, minimum and average fitnesses

in the population, respectively; cs is a real valued scaling factor (typically taken as 2.0);

and kf is the scaled fitness score of the k-th individual. By virtue of fitness scaling, the

ratio of maximum scaled fitness to average scaled fitness is set to a value around cs .

Step 4. Selection and Reproduction: Selection is carried out next, where individuals

of high fitness scores are selected and reproduced, while the least fit ones are eliminated

to create an intermediate population called mating pool. Theoretically, the number of

copies (reproductive trials) which an individual is represented with in the mating pool is

determined in accordance with fitness proportionate selection scheme (Eqn. 7).

avef

f kk

(7)

In Eqn. (7), avef denotes the average scaled fitness of the population; and k

represents the number of reproductive trials allocated to an individual. Eqn. (7) is not

directly used due to possible round-off errors; instead its algorithmic realization is

conducted by the roulette wheel selection approach. In this approach, firstly the

individuals take up slots on a simulated roulette wheel in connection to their scaled

fitnesses. The roulette is then spun times to select and reproduce the individuals for

mating pool.

Step 5. Crossover: The selected and reproduced individuals are mated randomly to

form 2/ pairs, and crossover is separately implemented between each pair according

to a preset probability value called crossover probability. When applied to a pair,

crossover swaps the genetic information between the mating (parent) individuals to

produce two new child individuals. A number of different strategies have been devised

to implement this task. In the present study, a two-point crossover approach is used,

where mating individuals are cut at two randomly selected crossover sites, and a design

exchange is facilitated by swapping either the inner portion falling between the

crossover sites or the two outer portions.

Step 6. Mutation: Mutation is applied on the genes of child individuals by randomly

altering a gene of 1 to 0 or vice versa based on a specified gene-wise mutation

probability.

Step 7. Termination: The child (new) population replaces the parent one and the steps

2 through 6 are repeated until a maximum number of generations.

226 S. Carbas and O. Hasancebi

4. DESIGN EXAMPLES

In order to present the effectiveness of the stochastic search techniques described

above are encoded in three abovementioned optimization algorithms. Two different

design examples, a 46-bar simple truss structure and a 46-bar plane shallow truss

structure, are used to test and compare their numerical performances on optimum design

of real size truss structures. The number of structural analysis is taken as 50,000 for

design examples to make sure that all the stochastic techniques are given the equal

opportunity to reach the global optimum. The following material properties of the steel

are used in all design examples; modulus of elasticity (E) 29000ksi (203893.6MPa)

and yield stress ( yF ) 36ksi (253.1MPa).

4.1. 46-Bar Simple Plane Truss As a first design example, a 46-bar simple plane truss shown in Figure 1 is

selected [23]. This truss has three-span with a total length of 33m. The truss is to be

optimized for minimum weight with the cross-sectional areas of the members being the

design variables. The members of the structure are grouped into 4 independent design

variables considering the upper chords members as first group, the lower chord

members as second group, the braces as third group and the column members as fourth

group. The grouping of members is shown in Figure 1. A single point design load

through x-axis at the mid-point of lower chord as well seven point design loads through

y-axis are applied at lower chord as shown in Figure 1. A discrete set of 137 economical

standard steel sections selected from W-shape profile list based on area and radii of

gyration properties is used to size the variables. The stress and stability limitations of

the members are calculated according to the provisions of ASD-AISC [14]. In addition,

the displacements of all nodes in any direction are restricted to a maximum value of

0.254 cm. The minimum weights reached by each stochastic search technique and

section designations attained for each member group for 46-bar simple truss structure is

given in Table 1. The optimum solution of this problem is reached as 35723.73 lb

(16204.01 kg) by both PSO and HSO. The GAs has obtained the minimum structure

weight as 35815.09 lb (16245.45 kg). Figure 3-(a) displays the variation of feasible best

designs attained by stochastic optimization techniques.

Figure 1. 46- Bar Simple Plane Truss Structure

11 x 3 m = 33 m

4

m

48 kN

48 kN

48 kN

48 kN

48 kN

48

kN

48 kN

48 kN

1

2

3

4

3

3

4

1

2

x

y

Optimal Design of Steel Trusses Using Stochastic Search Techniques 227

Table 1. Optimal designs of 46-bar simple plane truss

Table 2. Nodal coordinates for the left side of 46-bar shallow truss structure

4.2. 46-Bar Shallow Truss

The shallow truss containing 46 elements and 25 nodes is considered as the

second truss example. This structure represents a typical crane or robotic manipulator

[24]. Symmetry is to be enforced so that the 23 design variables define the cross-

sectional areas of all bars of the structure. The node coordinates of the left part of the

truss is demonstrated in Table 2. Figure 2 shows the geometry and the 23 element

groups. The all nodes of the truss in the x and y directions are subjected to the

displacement limits of 0.254cm. The allowable cross-sectional areas in this example are

selected from W-shape profile list with 137 economical standard steel sections. The

structure is to be designed for loads of 13.34, 53.38, and 13.34 kN applied in the

negative y-direction at nodes 7, 13 and 19, respectively. The optimum design weights

obtained by each technique for 46-bar shallow truss and the pipe section designations

attained for each member group are given in Table 3. For this structure, PSO technique

gives the least weight, which is 10599.52 lb (4807.86 kg). This design is considered to

be the optimum solution of this problem. But HSO and GAs methods obtained the

optimum design weights as 10601.58 lb (4808.79 kg) and 10675.79 lb (4842.46 kg),

respectively. Noticeable is that GAs once more has given the heaviest design. The

design history graph is shown in Figure 3-(b).

Figure 2. 46-Bar Shallow Truss Structure

Size

variables

PSO HSO GAs

Ready

Section

Area, in2

(cm2)

Ready

Section

Area, in2

(cm2)

Ready

Section

Area, in2

(cm2)

1 W16x67 19.7 (127.09) W16x67 19.7 (127.09) W12x65 19.1 (123.23)

2 W12x45 13.2 (85.16) W12x45 13.2 (85.16) W12x45 13.2 (85.16)

3 W10x54 15.8 (101.94) W10x54 15.8 (101.94) W12x50 14.7 (94.84)

4 W21x62 18.3 (118.06) W21x62 18.3 (118.06) W12x65 19.1 (123.23)

Weight 35723.73 lb

(16204.01 kg)

35723.73 lb

(16204.01 kg)

35815.09 lb

(16245.45 kg)

Node X (m) Y (m) Node X (m) Y (m)

1 0.00 0.00 8 3.92 0.24

2 0.28 0.13 9 4.62 0.61

3 0.25 0.18 10 5.38 0.38

4 1.01 0.05 11 6.07 0.76

5 1.70 0.32 12 6.10 0.46

6 2.46 0.09 13 6.35 0.635

7 3.16 0.47

12.702 m

0.635 m

13.34 kN 13.34 kN

53.38 kN

1

2

3

4

5

6

7

8

9

10

11

12

13 23

22

21

20

19

18

17

16 15

14

13

12

11

10

9

8

7

6

5

4 3

1

2

x

y

228 S. Carbas and O. Hasancebi

10000

12500

15000

17500

20000

0 10000 20000 30000 40000 50000

PSO HSO

GAs

35000

37500

40000

42500

0 10000 20000 30000 40000 50000

PSO HSO

GAs

(a) 46-bar simple plane truss (b) 46-bar shallow truss

Figure 3. The design history graph obtained with stochastic optimization techniques

for (a) 46-bar simple plane truss, (b) 46-bar shallow truss.

5. CONCLUSIONS

The highly promising outcome of this research suggests that the stochastic

numerical optimization techniques that are inspired by imitating biological principals

provide efficient and robust techniques for obtaining the solution of discrete optimum

design problem of steel trusses. These techniques use one of the natural phenomena to

generate a numerical optimization technique that are quite suitable for obtaining the

solution of combinatorial optimum design problems. For example, genetic algorithms

are based on the principle of survival of the fittest while harmony search optimization

imitates the improvisation process of a musician seeking a pleasing harmony. The

design examples show that these techniques are more powerful than the ones based on

mathematical programming methods in solving the discrete optimum design problem of

steel trusses.

Table 3. Optimal designs of 46-bar shallow truss

Size variables

PSO HSO GAs

Ready Section

Area, in2 (cm2)

Ready Section

Area, in2 (cm2)

Ready Section

Area, in2 (cm2)

1 W33x130 38.3 (127.09) W33x130 38.3 (127.09) W30x124 36.5 (127.09)

2 W14x145 42.7 (85.16) W36x150 44.2 (85.16) W24x112 47.7 (85.16) 3 W14x68 20.0 (101.94) W21x68 20.0 (101.94) W18x55 16.2 (101.94)

4 W27x114 33.5 (118.06) W10x112 32.9 (118.06) W30x108 31.7 (118.06)

5 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09) 6 W30x124 36.5 (85.16) W30x124 36.5 (85.16) W27x117 34.4 (85.16)

7 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94)

8 W24x104 30.6 (118.06) W27x102 30.0 (118.06) W18x106 31.2 (118.06) 9 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09)

10 W30x132 38.9 (85.16) W30x132 38.9 (85.16) W12x136 39.9 (85.16)

11 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94) 12 W14x90 26.5 (118.06) W12x87 25.6 (118.06) W12x106 31.1 (118.06)

13 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09)

14 W36x135 39.7 (85.16) W12x136 39.9 (85.16) W30x124 36.5 (85.16) 15 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94)

16 W21x101 29.8 (118.06) W27x102 30.0 (118.06) W27x102 30.0 (118.06)

17 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09) 18 W21x122 35.9 (85.16) W30x124 36.5 (85.16) W24x131 38.5 (85.16)

19 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94)

20 W30x116 34.2 (118.06) W30x116 34.2 (118.06) W33x118 34.7 (118.06) 21 W8x67 19.7 (127.09) W18x65 19.1 (127.09) W18x60 17.6 (127.09)

22 W36x135 39.7 (85.16) W14x145 42.7 (85.16) W14x132 38.8 (85.16)

23 W36x135 39.7 (101.94) W12x136 39.9 (101.94) W36x135 39.7 (101.94)

Weight 10599.52 lb (4807.86 kg) 10601.58 lb (4808.79 kg) 10675.79 lb (4842.46 kg)

Number of Analysis

Bes

t F

easi

ble

Des

ign

(lb

)

Number of Analysis

Bes

t F

easi

ble

Des

ign

(lb

)

Optimal Design of Steel Trusses Using Stochastic Search Techniques 229

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