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Optimal Weight Design of Steel Structures Using Adaptive

Simulated Annealing Algorithm

Jun S. Bae1, a

, Young S. Cho1, b

, and Seong U. Hong1, c

1 Dept. of Architectural Engineering, Hanyang University, Korea

a forum0407@hanmail.net,

b ycho@hanyang.ac.kr,

c abssnhong@empal.com

ABSTRACT

Structural optimization is widely adopted in the design of structures with the development of computer

aided design (CAD) and computer technique recently. By applying the structural optimization in the

last decades, designers have gained the design scheme of structures more feasibly and easily.

In this paper, an optimum design of one 30-story high rise building is performed considering

material nonlinearity. Based on finite element analysis and adaptive simulated annealing (ASA)

algorithm, the minimum weight of structure is derived under constraints of allowable normal stress,

allowable shear stress, and local buckling in web and flange. All H-section members are categorized to

5 groups and 9 load combinations were applied into the procedure of structural analysis. A fast

convergence on the optimal solution in design space is obtained from the data iterations between finite

element analysis and ASA algorithm.

This method can achieve the global optimum point in the limited iterations without requirement

of continuity and differentiability of functions. Including analysis of nonlinear material, the efficiency

of the methods is significantly illustrated using a simplified practical design.

KEYWORDS: Structural optimization, Steel structures, Adaptive simulated annealing algorithm,

Global optimum

1. INTRODUCTION

In conventional design, structural parameters of members in buildings are based on the assumption by

the designer’s experience, intuition, and originality. Therefore the conventional trial-and-error method

is generally highly iterative and very time-consuming due to multiple characteristics of tall buildings

consisting of thousands of structural members.

However, optimum design which is based on the types of structures, load combination, material,

codes and various constraints sets up the computational models, and by using the theory of

optimization, obtains the optimal solution of model under the specific objective function. Structural

optimization is widely adopted in the design of structures with the development of computer aided

design (CAD) and computer technique recently.

In this paper, an optimum design of one 30-story high rise building is performed considering

material nonlinearity. Based on finite element analysis and adaptive simulated annealing (ASA)

algorithm, the minimum weight of structure is derived under constraints of allowable normal stress,

allowable shear stress, and local buckling in web and flange. All I-section members are categorized to

5 groups and 9 load combinations were applied into the procedure of structural analysis. A fast

convergence on the optimal solution in design space is obtained from the data iterations between finite

element analysis and ASA algorithm.

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This method can achieve the global optimum point in the limited iterations without requirement

of continuity and differentiability of functions. Including analysis of nonlinear material, the efficiency

of the methods is significantly illustrated using a simplified practical design.

2. BACKGROUNDS

2.1 Description of optimization

Generally, optimal design of structural system is the design method for deciding variables satisfy all of

constraint conditions including minimizing objective function when given all of design constants and

loads. The optimal design with static or dynamic constraints can be stated as following mathematical

formation based on the approach by Cho et al. (2006):

Minimize )(xF (1)

..tS ),,2,1(

),,2,1()(

nixxx

mkgxg

U

ii

L

i

U

kk

Λ

Λ

=≤≤

=≤ (2)

where )(xF represents the objective function considering the minimum weight, costs or ductility in

optimal design of structures. ..tS is “subjected to”, which means the set of problem constraints.

Normally, these constraints represent state variables of structural response of each loading case:

stresses, internal forces, and displacements, natural frequencies, buckling loads. )(xgk is the

thk constraint function. ),,2,1( nixi Λ= means variables of structural design. On this occasion,

constraints have to satisfy their upper limit and variables also have to exist between their lower and upper limit.

2.2 The theory of Simulated Annealing

Simulated annealing (SA) is so named because of its analogy to the process of physical annealing with solids, in which a crystalline solid is heated and then allowed to cool very slowly until it achieves its most regular possible crystal lattice configuration, and thus is free of crystal defects. If the cooling

schedule is sufficiently slow, the final configuration results in a solid with such superior structural integrity. SA establishes the connection between this type of thermo-dynamic behaviour and the search for global minima for a discrete optimization problem. SA is a local search algorithm (meta-

heuristic) capable of escaping from local optima. The key feature of simulated annealing is that it provides a means to escape local optima by allowing hill-climbing moves in hopes of finding a global optimum shown in Figure 1based on the approach by Aarts et al. (1989).

(a) Concept of SA (b) SA algorithm

Figure 1. Concept of SA and SA algorithm

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2.3 The theory of adaptive simulated annealing Adaptive simulated annealing (ASA) is a variant of SA algorithm in which the algorithm parameters that control temperature schedule and random step selection are automatically adjusted according to

algorithm progress. ASA algorithm is very well suited for solving highly non-linear problems with short running analysis codes, when finding the global optimum is more important than a quick improvement of the design. There have been a number of studies concerning on this issue, such as

Ingber (1996), and Raghu et al. (1996) and Steven et al. (1995).

3. CASE STUDY 3.1 Modelling

The three-dimensional steel structure shown in Figure 2 was studied as the example of optimal weight design using ASA algorithm. The 30-story high rise building is consisted of 1998 nodes and 4980

elements. Parameters of the structure are shown in Table 1.

Figure 2. 30-story steel structure

Table 1. The steel properties

Weight density (N/mm3) 7.698x10

5

Elastic modulus (N/mm2) 2.06x10

5

Poison’s ratio 0.3

Steel SS400

3.2 Assumptions

3.2.1 Weight minimization Construction and maintenance costs of structures increase in proportion to total weights of structural members (i.e. increasing costs caused by construction or maintenance are fixed or very little).

Therefore optimal member section design come to the conclusion that weight minimization.

Table 2. Load combination

Figure 3. I-section

Case 1 1.2D+1.6L

Case 2 1.2D+1.3WX+0.5L

Case 3 1.2D+1.3WY+0.5L

Case 4 1.2D-1.3WX+0.5L

Case 5 1.2D-1.3WY+0.5L

Case 6 1.2D+1.5EX+0.5L

Case 7 1.2D+1.5EY+0.5L

Case 8 1.2D-1.5EX+0.5L

Case 9 1.2D-1.5EY+0.5L

Note) D: dead load, L: live load, W: wind load, E: seismic load, X,Y: directions of loads

3.2.2 Sectional shapes of steel members Sectional shapes of steel members are limited to H-section shown in Figure 3. This assumption is

applied to selection of member section due to preventing variables of structural design from infinitely extending.

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3.2.3 Load combinations In this paper, 9 load combinations were applied to the procedure of structural analysis considering the critical value among those combinations shown in Table 2.

3.2.4 Grouping of the number of members All H-section members are categorized to 5 groups were applied to the procedure of structural analysis

shown in Figure 4 and Table 3. It is an assumption for investigating applicability and efficiency on computer aided optimization (CAO) for structure by reducing groups of the number of members.

Table 3. Initial value of member section

Figure 4. Grouping of the number of members

Group No. Member section

1 H-692x300x13x20

2 H-400x400x40x40

3 H-458x417x30x50

4 H-498x432x45x70

5 H-700x700x70x70

3.3 Computer aided optimization (CAO) using ASA algorithm

Adaptive simulated annealing (ASA) algorithm was used by one of computer aided optimization (CAO) programs iSIGHT to simulate the analysis results from the FEM program. Mathematic

formulations of optimization are applied to optimal weight design of the steel structure as follows;

Minimize ∑=

==m

i

iii LAWxF1

)( ρ (3)

where )(xF represents the objective function considering the minimum weight. iρ means weight

density, iA means the sectional area and iL is the length of the ith element which is constituent m-finite

elements.

Minimize ∑=

+=n

k

k xgRxFxP1

2)]([)()( (4)

Where )(xP represents the penalty function as the evaluation function of ASA algorithm is achieved

by SUMT. Here SUMT represents the sequential unconstrained minimization technique which is to solve repetitively a sequence of the unconstrained problems whose solutions at the limit approach the

minimum of the nonlinear programming problem.

..tS allowablei

allowablei

ττ

σσ

≤

≤ (5)

Here allowableσ = 250MPa , and allowableτ = 100MPa .

Problem constraints are set to the selection of member section considering H-shape and local buckling as follows;

..tS U

fi

L

i tH 2≤ (6)

whereL

iH means the lower limit of the length of web, and U

fit represents the upper limit of the

thickness of flange.

..tS

0/23528

1

0/235124

1

=≤−=

=≤−=

U

fi

i

fi

y

fi

U

wi

i

wi

y

wi

lB

t

fl

lH

t

fl

(7)

Where wil and fil mean each local web and flange buckling.

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Displacements at the top and story drifts as serviceability in structures are also considered as

problem constraints as follows;

..tS

U

kl

k

lkkl

kl

U

hl

k

k

lhhl

dh

d

d

h

d

≤−

=

≤=

−

=

∑

)1(

30

1

30

δδ

δ

(8)

where hd represents displacement at the top and kd means story drift. Here, l represents the

direction of lateral loads (i.e. x or y direction), U

hld = H/500 (H: total height of structure) and

U

kld =h/300 (h: story height).

4. TEST RESULTS

Graphs show the key feature of adaptive simulated annealing is that it provides a means to escape local optima by allowing hill-climbing moves in hopes of finding a global optimum shown in Figure 5through 10.

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

1 1 2 23 34 45 5 6 6 7 78 89 100 11 1 12 2

H1 W 1 tw1 tf1

0.0E +00

1.0E +08

2.0E +08

3.0E +08

4.0E +08

5.0E +08

6.0E +08

7.0E +08

8.0E +08

9.0E +08

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 0 0 1 1 1 1 2 2

s tre ss( σ )

Figure 5. Convergence of group No.1 Figure 6. Convergence of allowable yield stress

0.0E +00

1.0E +06

2.0E +06

3.0E +06

4.0E +06

5.0E +06

6.0E +06

7.0E +06

8.0E +06

9.0E +06

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129

w e ig h t

-0.500

-0.450

-0.400

-0.350

-0.300

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 0 0 1 1 1 1 2 2

L W 1 L W 2 L W 3 L W 4 L W 5

Figure 7. Convergence of objective function Figure 8. Convergence of local web buckling

In graphs x-axis represents the number of iterative times. All of values convergence at 122th

iterations by allowing hill-climbing moves in hopes of finding a global optimum in stable state except for local flange buckling in group No.1 shown in Figure 11. The convergence of local flange buckling

in group No.1 doesn’t satisfy the set of problem constraint about buckling from 96th iterations. Therefore values resulted from 95th iterations satisfied all of constraints can be recognized as optimum values in this case study.

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

0.200

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 0 0 1 1 1 1 2 2

L F2 L F3 L F4 L F5

-0.350

-0.300

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 0 0 1 1 1 1 2 2

LF1

Figure 9. Convergence of local flange buckling from group No.1 to group No.5

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Total weights of structural members by optimizing the structure have fallen off to 24.55% of weights by the initial design shown in Figure 12.

Table 3. optimum value of member section 5758.864

4345.097

0.00

1000.00

2000.00

3000.00

4000.00

5000.00

6000.00

initial v alue optimum value

Group No. Member section

1 H-407x413x10x17

2 H-812x112x10x41

3 H-163x171x41x52

4 H-606x256x26x66

5 H-578x740x45x60

Figure 10. Comparison between initial value and optimum value of total weight of structure (ton).

5. CONCLUSIONS

In this research work, ASA algorithm is applied for optimum design, that is, minimizing the weight of

steel structure. The method which is a variant of ASA algorithm parameters that control temperature schedule and random step selection are automatically adjusted according to algorithm progress is very well suited for solving highly nonlinear problems with short running analysis codes. Based on these

analysis and simulation, the following conclusions are made; (1) Proposed formulas for optimization give satisfaction limits which are not only allowable yield

stress and shear stress but also serviceability of structure through ASA algorithm.

(2) CAO reduced the trial-and-error method and got the optimum value by standardized formulas. Continuous investigation about this topic can bring not only time effect but also cost effect in aspects of future constructions. Furthermore it can bring advancements of performance and

reliability through optimization and have the possibility to reduce CO2 emissions in aspects of steel industries.

ACKNOWLEDGEMENTS

This research was financially supported by the Sustainable Building Research Center of Hanyang

University which was supported by the SRC/ERC program of MOST (Grant#R11-2005-056-01004-0).

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Aarts, E. H. L., and Korst, J., 1997. Local search in combinatorial optimization. John Wiely & Sons,

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Ingber, L., 1996. Adaptive simulated annealing (ASA): lesson learned. Control and cybernetics, Vol. 25, pp. 33-54.

Raghu, K. C.,Mrinal, K. S., and Paul, L. S., 1996. “2-D resistivity inversion using spline

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