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Cost optimization of a composite floor system using an improved harmony search algorithm
Journal of Constructional Steel Research 66 (2010) 664–669
Contents lists available at ScienceDirect
Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Cost optimization of a composite floor system using an improved harmony search algorithm A. Kaveh a,∗ , A. Shakouri Mahmud Abadi b a
Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran
b
Building and Housing Research Centre, Tehran-14, Iran
article
info
Article history: Received 19 June 2009 Accepted 17 January 2010 Keywords: Structural optimization Composite structures Harmony search algorithm Load and resistance factor design
abstract In this study, cost optimization of a composite floor system is performed utilizing the harmony search algorithm and an improved harmony search algorithm. These algorithms imitate the musical performance process that takes place when a musician searches for a better state of harmony, similar to the optimum design process which looks for the optimum solution. A composite floor system is designed by the LRFDAISC method, using a unit consisting of a reinforced concrete slab and steel beams. The objective function is considered as the cost of the structure, which is minimized subjected to serviceability and strength requirements. Examples of composite floor systems are presented to illustrate the performance of the presented algorithms. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Over the last three decades, the cost optimization of composite structures was mainly considered from the viewpoint of the development and application of different optimization techniques. A cost-based optimization model for the design of composite beams has been developed by Lorenz [1]. An optimum-cost design of partially composite steel beams using LRFD is due to Bhatti [2]. An optimization of composite floors is presented by Shock [3]. Cost optimization of a composite I-beam floor system has been developed by Klanšek and Kravanja [4]. Composite frame design using genetic algorithms can be found in the work of Camp et al. [5]. The design of steel structures generally requires the selection of member sections from a discrete set of practically available sections. This selection should be carried out in such a way that the structure has the minimum weight or cost while the performance of the structure is within the limitations described by the code of practice. In recent years, structural optimization witnessed the emergence of novel and innovative design techniques. These stochastic search techniques make use of ideas taken from nature and do not suffer the discrepancies of mathematical programmingbased optimum design methods. The basic idea behind these techniques is to simulate the natural phenomena. One of the recent additions to these techniques is the harmony search algorithm [6–8]. This approach is based on the musical performance process that takes place when a musician searches for a better state of
∗
Corresponding author. Tel.: +98 21 44202710; fax: +98 21 7720398. E-mail address: [email protected] (A. Kaveh).
0143-974X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2010.01.009
harmony. Jazz improvisation seeks musically pleasing harmony similar to the optimum design process which seeks optimum solutions. The pitch of each musical instrument determines the aesthetic quality, just as the objective function value is determined by the set of values assigned to each decision variable. In the process of musical production a musician selects and brings together a 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. The musician then tunes some of these notes to achieve a better harmony. Similarly it is then checked whether this candidate solution improves the objective function or not, much like finding out whether it is euphonic. This candidate solution is then checked to see whether it satisfies the objective function or not, similar to the process of finding out whether euphonic music is obtained or not. Different applications of the harmony search algorithm in structural optimization problems can be found in the work of Saka [9,10], Saka and Erdal [11], Kaveh and Talatahari [12], and in the recent book of Geem [13]. In this study, the harmony search algorithm is used to determine the cost optimization of a composite floor system, consisting of a reinforced concrete slab and steel beams, with the AISC load and resistance factor design (LRFD). 2. Harmony search algorithm The method consists of five basic steps. The detailed explanation of these steps can be found in Lee and Geem [8]; the steps are summarized in the following: Step 1. The harmony search parameters are initialized.
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669
Step 2. The harmony memory matrix is initialized.
operation prevents stagnation and improves the harmony memory for diversity with a greater chance of reaching the global optimum.
Step 3. A new harmony memory matrix is improvised. Step 4. The harmony memory matrix is updated. Step 5. Steps 3 and 4 are repeated until the termination criterion is satisfied. Step 1. A possible value range for each design variable of the optimum design problem is specified. A pool is constructed by collecting these values together, from which the algorithm selects values for the design variables. Furthermore, the number of solution vectors in harmony memory (HMS), that is the size of the harmony memory matrix, harmony considering rate (HMCR), pitch adjusting rate (PAR) and the maximum number of searches are also selected in this step. Step 2. The harmony memory matrix is initialized. Each row of the harmony memory matrix contains the values of design variables which are randomly selected feasible solutions from the design pool for that particular design variable. Hence, this matrix has n columns, where n is the total number of design variables, and HMS rows, selected in the first step. HMS is similar to the total number of individuals in the population matrix of the genetic algorithm. The harmony memory matrix has the following form:
x 1 ,1 x 1 ,2 ... [H ] = ...
x1,hms−1 x1,hms
x2,1 x2,2
... ...
x2,hms−1 x2,hms
... ... ... ... ... ...
... ... ... ... ... ...
xn−1,1 xn−1,2
xn,1 xn,2
xn−1,hms−1 xn−1,hms
xn,hms−1 xn,hms−1
... ...
... ...
(1)
where xi,j is the value of the ith design variable in the jth randomly selected feasible solution. These candidate designs are sorted such that the objective function value corresponding to the first solution vector is the minimum. In other words, the feasible solutions in the harmony memory matrix are sorted in descending order according to their objective function value. It is worthwhile mentioning that not only the feasible designs are inserted into harmony memory matrix, but those designs having a small infeasibility are also included in this matrix with a penalty on their objective function. Step 3. In generating a new harmony matrix, the new value of the ith design variable can be chosen from any discrete value within the range of ith column of the harmony memory matrix with the probability of HMCR which varies between 0 and 1. In other words, the new value of xi can be one of the discrete values of the vector {xi,1 , xi,2 , . . . , xi,hms }T with the probability of HMCR. The same is applied to all other design variables. In the random selection, the new value of the ith design variable can also be chosen randomly from the entire pool with the probability of 1 − HMCR. That is, w xne = i
{xi,1 , xi,2 , . . . , xi,hms }T {x1 , x2 , . . . , xns }T
with probability HMCR (2) with probability (1 − HMCR)
where ns is the total number of values for the design variables in the pool. If the new value of the design variable is selected among those of harmony memory matrix, this value is then checked to see whether it should be pitch adjusted. This operation uses the pitch adjustment parameter PAR that sets the rate of adjustment for the pitch chosen from the harmony memory matrix as follows: w Is xne to be pitch-adjusted? i
Yes No
with probability of PAR with probability of (1 − PAR).
665
(3)
w Supposing that the new pitch-adjustment decision for xne came i new out to be yes from the test and if the value selected for xi from the harmony memory is the kth element in the general discrete set, w then the neighboring value k + 1 or k − 1 is taken for new xne . This i
Step 4. After selecting the new values for each design variable, the objective function value is calculated for the new harmony vector. If this value is better than the worst harmony vector in the harmony matrix, it is then included in the matrix while the worst one is taken out of the matrix. The harmony memory matrix is then sorted in descending order by the objective function value. Step 5. Steps 3 and 4 are repeated until the termination criterion, which is the pre-selected maximum number of cycles, is reached. This number is selected large enough such that within this number of design cycles no further improvement is observed in the objective function. 3. Improved harmony search algorithm The parameters HMCR and PAR in Step 3 help the algorithm to find globally and locally improved solutions, respectively. PAR in the harmony search (HS) algorithm is an important parameter for fine-tuning of the optimized solution vectors, and it can be potentially useful in adjusting the convergence rate of the HS to an optimal solution. Therefore, fine adjustment of this parameter is of great importance. The classic HS algorithm employs a fixed value for PAR. In the HS method, the PAR value is adjusted in the initialization step and it cannot be altered during subsequent generations. The main drawback of this method appears to be in the number of iterations the algorithm needs to find an optimal solution. Small PAR values can lead to poor performance of the algorithm and a considerable increase in the number of iterations required for finding the optimum solution. The main difference between the improved harmony search (IHS) algorithm, developed by [14] and the classic HS method is in the way of adjusting the PAR parameter. In order to improve the performance of the HS algorithm and to eliminate the drawbacks encountered with the fixed values of PAR, the IHS algorithm uses a variable PAR in the improvisation step. This parameter changes dynamically with the generation number as PAR(gn) = PARmin +
PARmax − PARmin NI
× gn
(4)
where PAR is the pitch adjusting rate for each generation; PARmin is the minimum pitch adjusting rate; PARmax is the maximum pitch adjusting rate; NI is the number of solution vector generations; and gn is the generation number. In this study, we have used the improved harmony search developed in Ref. [14], where the effects of this improvement on different mathematical functions and optimization problems are also illustrated. In this paper, we have considered simple application of the IHS algorithm in our specific engineering problem. 4. Objective function By minimizing a suitable cost function one can reach an optimum solution for a composite floor. According to [4], the percentages of different costs of a composite floor are as shown in Fig. 1. It can be seen that the power cost is very little and we can ignore it. The labour cost in a composite beam is almost permanent; therefore it is not necessary to include it in the objective function. The optimal design of a composite floor system is proposed to be determined by the minimum of the costs of concrete, steel beams and shear studs. The objective function can be expressed as follows: Min Q = Ws × L × N × Cs + Wc × Cc + Ns × Cst
(5)
Wc = L × W × tc × ρ.
(6)
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A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669
Slab breadth Span
Composite beam
Slab thickness
Fig. 1. The distribution of the manufacturing costs of a composite floor system [4].
By replacing Eq. (6) in Eq. (5) and considering Q¯ = Q /L × Cs , we have Min Q¯ = Ws × N + W × tc × ρ ×
Cc Cs
+
Ns L
×
Cst Cs
,
(7)
Steel sections
subject to
δ/δu ≤ 1
(8)
Mu /(φb Mn ) ≤ 1
(9)
Vu /(φv Vn ) ≤ 1
(10)
where Ws is the weight of the steel beam in length units, L is the length of the beam, N is the total number of steel beams in the composite floor, Cs is the cost of the steel beam in weight units, Wc is the total weight of concrete, Cc is the cost of concrete in weight units, W is the length of the bay, tc is the thickness of the concrete slab, ρ is the density of concrete, Ns is the total number of studs and Cst is the cost of each stud. δ in is the maximum displacement of the steel beam and δu is its upper bound. φb is the resistance factor for flexure, which is given as 0.9, Mn is the nominal moment strength and Mu is the factored service load moment for a steel beam. φv represents the resistance factor for shear, given as 0.9, Vn is the nominal strength in shear and Vu is the factored service load shear for a steel beam. The details for obtaining the nominal moment strength and nominal shear strength of a steel beam according to LRFD are given in the Appendix. 5. Optimum design process The harmony search algorithm initiates the design process by selecting random values for the steel beam spacing, the beam size and the concrete slab thickness. The algorithm tries to find the best value for each design variable to minimize the objective function. The design process consists of six steps, as follows. Step 1. Select the values of the harmony memory parameters (HM, HMCR, PARmin and PARmax ). Step 2. The harmony memory matrix is initialized (values for beam spacing, beam size and concrete slab thickness are chosen). Step 3. Check whether the newly selected design vector should be pitch-adjusted. Step 4. With the values selected for the beam spacing, beam size and concrete slab thickness, the algorithm designs a composite floor according to AISC-LRFD. Step 5. Calculate the objective function value for the newly selected design vector. If this value is better than the worst harmony vector in the harmony matrix, it is then included in the matrix, while the worst one is taken out of the matrix. The harmony memory matrix is then sorted in descending order by the objective function value. Step 6. Repeat Steps 2 and 6 are until the pre-selected maximum number of iterations is reached.
Concrete slab Welded stud connector
Fig. 2. Schematic view of a simple composite floor system.
6. Design examples Here we have considered a span, and in a structure this span is repeated to cover a ceiling. Such a span behaves independently, and once we optimize the problem for one span the result will correspond to the entire ceiling. This process can be repeated for spans of different dimensions. Example 1. The considered composite I-beam floor system is 6 m long, subjected to the combined effects of the self-weight and the imposed dead load of 3 kN/m2 and imposed live load of 2 kN/m2 ; the width of the floor is 8 m. The base diameter of the stud is 13 mm and the overall height is 50 mm. The compressive strength of the concrete is 21 MPa, and the yield strength of the steel beam is 240 MPa. For the classic HS algorithm, the parameters for this example are taken as HMS = 30, HMCR = 0.9 and PAR = 0.45. The improved harmony search algorithm parameters are taken as HMS = 30, HMCR = 0.85, PARmin = 0.35 and PARmax = 0.99. Bounds of the design variables are provided in Table 1. A schematic view of the composite floor system for Example 1 is shown in Fig. 2. The output consists of the following: Steel beam spacing = 1600 mm; Concrete slab thickness = 80 mm; Steel beam size = IPE18. The histories of design with and without modification for this example are shown in Fig. 3. It can be seen that the IHS algorithm converges after 210 iterations, while the HS algorithm requires 370 iterations to attain the same result.
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669 Table 1 Bounds of design variables.
Table 2 Compression of the specifications of the IPE18 and the W250X17.9 section.
Bounds of the design variables No. Bounds
Steel beam spacing (mm)
Concrete slab thickness (mm)
Steel beam sizea
1 2
500 300
80 140
1 29
Lower bound Upper bound
667
a The steel beam consists of 29 steel I-beams (IPE12 to 30, INP12 to 30 and IPB12 to 30).
Section properties
IPE18
W250X17.9
Area Depth Width Web thickness Flange thickness Moment of inertia about x-axis
23.9 cm2 18 cm 9.1 cm 0.53 cm 0.80 cm 1940 cm4
22.84 cm2 25.07 cm 10.05 cm 0.48 cm 0.53 cm 2239.32 cm4
Table 3 Improved harmony search parameters used for the sensitivity analysis. Case
HMS
HMCR
PARmin
PARmax
1 2 3 4 5
30 30 30 30 30
0.85 0.9 0.95 0.95 0.85
0.4 0.45 0.3 0.25 0.35
0.85 0.9 0.95 0.95 0.99
Fig. 3. Design histories for Example 1.
Fig. 5. Design histories for cases 1, 2, 3, 4 and 5 of Example 1.
7. Sensitivity analysis for improved harmony search parameters
Fig. 4. Design histories for Example 2.
Example 2. Here, Example 1 is studied by considering 267 AISC cross sections. Other input data are the same as in the previous example. The output consists of the following: Steel beam spacing = 1600 mm; Concrete slab thickness = 80 mm; Steel beam size = (W10X12)W250X17.9. The histories of design with and without modification for this example are shown in Fig. 4. It can be seen that the IHS algorithm after 725 iterations attains an objective function of 169, while for the HS algorithm we have 174.5 after 1000 iterations. Compression of the specifications of the IPE18 section and the W250X17.9 section is provided in Table 2.
The HS algorithm has proven its capability in many optimization problems. Here, we make a sensitivity analysis only to find suitable initial values of the parameters for better performance in our problem. As can be seen from Fig. 5, all the cases have converged and only their rates are different. This is true for many other heuristics as well, and different initial parameters can lead to different convergence rates. In this section, a sensitivity analysis is performed for the improved harmony search parameters involved in Example 1. The results of the sensitivity analyses carried out to determine the appropriate values of the improved harmony search parameters are given in Table 3. The design histories for the four cases are shown in Fig. 5. It is apparent from this figure that the values of 0.95 for HMCR, 0.25 for PARmin and 0.95 for PARmax are the best values for the parameters in Example 1. In this paper, a wide range is considered for the design variables. As an example, the distance between two beams is considered as 500 mm to 3000 mm, while in the previous researched this distance has been considered as constant. In the second example, 267 different sections are considered for design. Naturally these wide ranges for the variables increase the number of iterations. 8. Concluding remarks This paper presents the cost optimization of a composite floor system where the design constraints are implemented as in LRFDAISC rules. The composite floor system consists of a reinforced concrete slab and steel I-beams. The optimization was performed
668
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669
by the recently developed improved harmony search method. This mathematically simple algorithm sets up a harmony search matrix, each row of which consists of randomly selected feasible solutions to the design problem. In every search step, it searches the entire set rather than a local neighborhood of a current solution vector. It needs neither initial starting values for the design variables nor a population of candidate solutions to the design problem. The results obtained show that the improved harmony search method is a powerful and efficient method for finding the optimum solution of structural optimization problems. The main aim of this paper has been to present a simple and efficient algorithm which can be used in practical engineering problems. Such a simple approach can be utilized in many other engineering design problems to reduce the cost of the construction. Acknowledgement The first author is grateful to Iran National Science Foundation for support. Appendix. Load and resistance factor design Load and resistance factor design for composite flexural member consists of the following steps [15].
The effective width of the concrete slab is the sum of the effective widths for each side of the beam centerline, each of which should not exceed: (1) one-eighth of the beam span, center-to-center of supports; (2) one-half the distance to the centerline of the adjacent beam; or (3) the distance to the edge of the slab.
tw
s > 3.76
E Fyf
,
(A.8)
Mn should be determined from the superposition of elastic stresses, considering the effects of shoring, for the limit state of yielding (yield moment). A.5. Shear connectors (1) Load transfer for positive moment. The entire horizontal shear at the interface between the steel beam and the concrete slab should be assumed to be transferred by shear connectors, except for concrete-encased beams as defined in Section I3.3 of Ref. [15]. For composite action with concrete subject to flexural compression, the total horizontal shear force, V 0 , between the point of maximum positive moment and the point of zero moment should be taken as the lowest value according to the limit states of concrete crushing, tensile yielding of the steel section, or strength of the shear connectors. (a) Concrete crushing: (A.9)
(b) Tensile yielding of the steel section: V 0 = Fy As .
(A.10)
(c) Strength of shear connectors: V0 =
X
Qn
(A.11)
where Ac = area of concrete slab within the effective width.
A.2. Shear strength of beam
As = area of steel cross section.
P
Vu ≤ φ v Vn
(A.1)
req’d Vn = Vu /φv ≤ Vn .
(A.2)
The nominal shear strength, Vn , of unstiffened or stiffened webs, according to the limit states of shear yielding and shear buckling, is Vn = 0.6Fy Aw Cv .
(A.3)
s For webs of rolled I-shaped members with h/tw ≤ 2.24
φv = 1.00 and Cv = 1.0.
E Fy
(A.4) (A.5)
When temporary shores are not used during construction, the steel section alone should have adequate strength to support all loads applied prior to the concrete attaining 75% of its specified strength fc0 . A.4. Positive flexural strength of composite beams with shear connectors The design positive flexural strength, φb Mn , should be determined for the limit state of yielding as follows:
φb = 0.90.
(A.6)
For
s = 3.76
E Fyf
,
Qn = sum of the nominal strengths of shear connectors between the point of maximum positive moment and the point of zero moment. (2) Strength of stud shear connectors. The nominal strength of one stud shear connector embedded in the solid concrete or in the composite slab is Qn = 0.5Asc (fc0 Ec )1/2 ≤ Asc Fu .
A.3. Strength during construction
tw
h
V 0 = 0.85fc0 Ac .
A.1. Effective width
h
Mn should be determined from the plastic stress distribution on the composite section for the limit state of yielding (plastic moment). For
(A.7)
(A.12)
(3) Required number of shear connectors. The number of shear connectors required between the section of maximum bending moment, positive or negative, and the adjacent section of zero moment should be equal to the horizontal shear force divided by the nominal strength of one shear connector. (4) Shear connector placement and spacing. Shear connectors required on each side of the point of maximum bending moment, positive or negative, should be distributed uniformly between that point and the adjacent points of zero moment, unless otherwise specified. However, the number of shear connectors placed between any concentrated load and the nearest point of zero moment should be sufficient to develop the maximum moment required at the concentrated load point. Shear connectors should have at least 1 in (25 mm) of lateral concrete cover. The diameter of studs should not be greater than 2.5 times the thickness of the flange to which they are welded, unless located over the web. The minimum center-to-center spacing of stud connectors should be six diameters along the longitudinal axis of the supporting composite beam and four diameters transverse to the longitudinal axis of the supporting composite beam. The maximum center-to-center spacing of shear connectors should not exceed eight times the total slab thickness nor 36 in (see Table 4).
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669 Table 4 Summery of AISC-LRFD specification for composite beam. Summery of AISC-LRFD specification for composite beam Section
Item
Summery
I3.1
Effective width on each side of beam (Lesser of the 3 values)
b = Beam length/8 (L/8) b = Beam spacing/2 (s/2) b = Distance to edge of slab
I5.1
Material
I5.2
Horizontal shear force (Lesser of the 3 values)
Hs > 4ds (Minimum stud height) = 0.85fc0 Ac =A s Fy P = Qn
I5.3
Strength of stud
Qn = 0.5Asc (fc0 Ec )1/2 ≤ Asc Fu
I5.6
Shear connector placement and spacing
= 6 ds Longitudinal = 4 ds Transverse
References [1] Lorenz ER. Understanding composite beam design methods using LRFD. Engineering Journal, AISC 1988;25(2):35–8. [2] Bhatti MA. Optimum cost design of partially composite steel beams using LRFD. Engineering Journal 1996;33(1):18–29. [3] Shock BT. automated design of steel wide flange beam floor framing systems using genetic algorithms. M.S. thesis. Milwaukee (WI): Marquette University; 2003.
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[4] Klanšek U, Kravanja S. Cost optimization of composite I beam floor system. American Journal of Applied Sciences 2007;5(1):7–17. [5] Camp C, Li J, Pezeshk S. Composite frame design using a genetic algorithm. Department of Civil Engineering, Memphis (TN 38152): The University of Memphis; 1999. [6] Geem ZW, Kim JH, Loganathan GV. A new heuristic optimization algorithm: Harmony search. Simulation 2001;76:60–8. [7] Lee KS, Geem ZW. A new structural optimization method based on harmony search algorithm. Computers and Structures 2004;82:781–98. [8] Lee KS, Geem ZW. A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice. Computer Methods in Applied Mechanical Engineering 2005;194(36–38):3902–33. [9] Saka MP. Optimum geometry design of geodesic domes using harmony search algorithm. Advances in Structural Engineering 2007;10(6):595–606. [10] Saka MP. Optimum design of steel sway frames to BS5950 using harmony search algorithm. Journal of Constructional Steel Research 2009;65(1): 36–43. [11] Saka MP, Erdal F. Harmony search based algorithm for the optimum design of grillage systems to LRFD-AISC. Structural Multidisciplinary Optimization 2009;38:25–41. [12] Kaveh A, Talatahari S. Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Computers and Structures 2009;87:267–83. [13] Geem ZW. Harmony search algorithms for structural design. Springer Verlag; 2009. [14] Mahdavi M, Fesanghary M, Damangir E. An improved harmony search algorithm for solving optimization problems. Applied Mathematics and Computation 2007;188:1567–79. [15] AISC. LRFD specification for structural steel buildings. Chicago (IL): American Institute of Steel Construction; 2005.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Cost optimization of a composite floor system using an improved harmony search algorithm A. Kaveh a,∗ , A. Shakouri Mahmud Abadi b a
Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran
b
Building and Housing Research Centre, Tehran-14, Iran
article
info
Article history: Received 19 June 2009 Accepted 17 January 2010 Keywords: Structural optimization Composite structures Harmony search algorithm Load and resistance factor design
abstract In this study, cost optimization of a composite floor system is performed utilizing the harmony search algorithm and an improved harmony search algorithm. These algorithms imitate the musical performance process that takes place when a musician searches for a better state of harmony, similar to the optimum design process which looks for the optimum solution. A composite floor system is designed by the LRFDAISC method, using a unit consisting of a reinforced concrete slab and steel beams. The objective function is considered as the cost of the structure, which is minimized subjected to serviceability and strength requirements. Examples of composite floor systems are presented to illustrate the performance of the presented algorithms. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Over the last three decades, the cost optimization of composite structures was mainly considered from the viewpoint of the development and application of different optimization techniques. A cost-based optimization model for the design of composite beams has been developed by Lorenz [1]. An optimum-cost design of partially composite steel beams using LRFD is due to Bhatti [2]. An optimization of composite floors is presented by Shock [3]. Cost optimization of a composite I-beam floor system has been developed by Klanšek and Kravanja [4]. Composite frame design using genetic algorithms can be found in the work of Camp et al. [5]. The design of steel structures generally requires the selection of member sections from a discrete set of practically available sections. This selection should be carried out in such a way that the structure has the minimum weight or cost while the performance of the structure is within the limitations described by the code of practice. In recent years, structural optimization witnessed the emergence of novel and innovative design techniques. These stochastic search techniques make use of ideas taken from nature and do not suffer the discrepancies of mathematical programmingbased optimum design methods. The basic idea behind these techniques is to simulate the natural phenomena. One of the recent additions to these techniques is the harmony search algorithm [6–8]. This approach is based on the musical performance process that takes place when a musician searches for a better state of
∗
Corresponding author. Tel.: +98 21 44202710; fax: +98 21 7720398. E-mail address: [email protected] (A. Kaveh).
0143-974X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2010.01.009
harmony. Jazz improvisation seeks musically pleasing harmony similar to the optimum design process which seeks optimum solutions. The pitch of each musical instrument determines the aesthetic quality, just as the objective function value is determined by the set of values assigned to each decision variable. In the process of musical production a musician selects and brings together a 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. The musician then tunes some of these notes to achieve a better harmony. Similarly it is then checked whether this candidate solution improves the objective function or not, much like finding out whether it is euphonic. This candidate solution is then checked to see whether it satisfies the objective function or not, similar to the process of finding out whether euphonic music is obtained or not. Different applications of the harmony search algorithm in structural optimization problems can be found in the work of Saka [9,10], Saka and Erdal [11], Kaveh and Talatahari [12], and in the recent book of Geem [13]. In this study, the harmony search algorithm is used to determine the cost optimization of a composite floor system, consisting of a reinforced concrete slab and steel beams, with the AISC load and resistance factor design (LRFD). 2. Harmony search algorithm The method consists of five basic steps. The detailed explanation of these steps can be found in Lee and Geem [8]; the steps are summarized in the following: Step 1. The harmony search parameters are initialized.
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669
Step 2. The harmony memory matrix is initialized.
operation prevents stagnation and improves the harmony memory for diversity with a greater chance of reaching the global optimum.
Step 3. A new harmony memory matrix is improvised. Step 4. The harmony memory matrix is updated. Step 5. Steps 3 and 4 are repeated until the termination criterion is satisfied. Step 1. A possible value range for each design variable of the optimum design problem is specified. A pool is constructed by collecting these values together, from which the algorithm selects values for the design variables. Furthermore, the number of solution vectors in harmony memory (HMS), that is the size of the harmony memory matrix, harmony considering rate (HMCR), pitch adjusting rate (PAR) and the maximum number of searches are also selected in this step. Step 2. The harmony memory matrix is initialized. Each row of the harmony memory matrix contains the values of design variables which are randomly selected feasible solutions from the design pool for that particular design variable. Hence, this matrix has n columns, where n is the total number of design variables, and HMS rows, selected in the first step. HMS is similar to the total number of individuals in the population matrix of the genetic algorithm. The harmony memory matrix has the following form:
x 1 ,1 x 1 ,2 ... [H ] = ...
x1,hms−1 x1,hms
x2,1 x2,2
... ...
x2,hms−1 x2,hms
... ... ... ... ... ...
... ... ... ... ... ...
xn−1,1 xn−1,2
xn,1 xn,2
xn−1,hms−1 xn−1,hms
xn,hms−1 xn,hms−1
... ...
... ...
(1)
where xi,j is the value of the ith design variable in the jth randomly selected feasible solution. These candidate designs are sorted such that the objective function value corresponding to the first solution vector is the minimum. In other words, the feasible solutions in the harmony memory matrix are sorted in descending order according to their objective function value. It is worthwhile mentioning that not only the feasible designs are inserted into harmony memory matrix, but those designs having a small infeasibility are also included in this matrix with a penalty on their objective function. Step 3. In generating a new harmony matrix, the new value of the ith design variable can be chosen from any discrete value within the range of ith column of the harmony memory matrix with the probability of HMCR which varies between 0 and 1. In other words, the new value of xi can be one of the discrete values of the vector {xi,1 , xi,2 , . . . , xi,hms }T with the probability of HMCR. The same is applied to all other design variables. In the random selection, the new value of the ith design variable can also be chosen randomly from the entire pool with the probability of 1 − HMCR. That is, w xne = i
{xi,1 , xi,2 , . . . , xi,hms }T {x1 , x2 , . . . , xns }T
with probability HMCR (2) with probability (1 − HMCR)
where ns is the total number of values for the design variables in the pool. If the new value of the design variable is selected among those of harmony memory matrix, this value is then checked to see whether it should be pitch adjusted. This operation uses the pitch adjustment parameter PAR that sets the rate of adjustment for the pitch chosen from the harmony memory matrix as follows: w Is xne to be pitch-adjusted? i
Yes No
with probability of PAR with probability of (1 − PAR).
665
(3)
w Supposing that the new pitch-adjustment decision for xne came i new out to be yes from the test and if the value selected for xi from the harmony memory is the kth element in the general discrete set, w then the neighboring value k + 1 or k − 1 is taken for new xne . This i
Step 4. After selecting the new values for each design variable, the objective function value is calculated for the new harmony vector. If this value is better than the worst harmony vector in the harmony matrix, it is then included in the matrix while the worst one is taken out of the matrix. The harmony memory matrix is then sorted in descending order by the objective function value. Step 5. Steps 3 and 4 are repeated until the termination criterion, which is the pre-selected maximum number of cycles, is reached. This number is selected large enough such that within this number of design cycles no further improvement is observed in the objective function. 3. Improved harmony search algorithm The parameters HMCR and PAR in Step 3 help the algorithm to find globally and locally improved solutions, respectively. PAR in the harmony search (HS) algorithm is an important parameter for fine-tuning of the optimized solution vectors, and it can be potentially useful in adjusting the convergence rate of the HS to an optimal solution. Therefore, fine adjustment of this parameter is of great importance. The classic HS algorithm employs a fixed value for PAR. In the HS method, the PAR value is adjusted in the initialization step and it cannot be altered during subsequent generations. The main drawback of this method appears to be in the number of iterations the algorithm needs to find an optimal solution. Small PAR values can lead to poor performance of the algorithm and a considerable increase in the number of iterations required for finding the optimum solution. The main difference between the improved harmony search (IHS) algorithm, developed by [14] and the classic HS method is in the way of adjusting the PAR parameter. In order to improve the performance of the HS algorithm and to eliminate the drawbacks encountered with the fixed values of PAR, the IHS algorithm uses a variable PAR in the improvisation step. This parameter changes dynamically with the generation number as PAR(gn) = PARmin +
PARmax − PARmin NI
× gn
(4)
where PAR is the pitch adjusting rate for each generation; PARmin is the minimum pitch adjusting rate; PARmax is the maximum pitch adjusting rate; NI is the number of solution vector generations; and gn is the generation number. In this study, we have used the improved harmony search developed in Ref. [14], where the effects of this improvement on different mathematical functions and optimization problems are also illustrated. In this paper, we have considered simple application of the IHS algorithm in our specific engineering problem. 4. Objective function By minimizing a suitable cost function one can reach an optimum solution for a composite floor. According to [4], the percentages of different costs of a composite floor are as shown in Fig. 1. It can be seen that the power cost is very little and we can ignore it. The labour cost in a composite beam is almost permanent; therefore it is not necessary to include it in the objective function. The optimal design of a composite floor system is proposed to be determined by the minimum of the costs of concrete, steel beams and shear studs. The objective function can be expressed as follows: Min Q = Ws × L × N × Cs + Wc × Cc + Ns × Cst
(5)
Wc = L × W × tc × ρ.
(6)
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A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669
Slab breadth Span
Composite beam
Slab thickness
Fig. 1. The distribution of the manufacturing costs of a composite floor system [4].
By replacing Eq. (6) in Eq. (5) and considering Q¯ = Q /L × Cs , we have Min Q¯ = Ws × N + W × tc × ρ ×
Cc Cs
+
Ns L
×
Cst Cs
,
(7)
Steel sections
subject to
δ/δu ≤ 1
(8)
Mu /(φb Mn ) ≤ 1
(9)
Vu /(φv Vn ) ≤ 1
(10)
where Ws is the weight of the steel beam in length units, L is the length of the beam, N is the total number of steel beams in the composite floor, Cs is the cost of the steel beam in weight units, Wc is the total weight of concrete, Cc is the cost of concrete in weight units, W is the length of the bay, tc is the thickness of the concrete slab, ρ is the density of concrete, Ns is the total number of studs and Cst is the cost of each stud. δ in is the maximum displacement of the steel beam and δu is its upper bound. φb is the resistance factor for flexure, which is given as 0.9, Mn is the nominal moment strength and Mu is the factored service load moment for a steel beam. φv represents the resistance factor for shear, given as 0.9, Vn is the nominal strength in shear and Vu is the factored service load shear for a steel beam. The details for obtaining the nominal moment strength and nominal shear strength of a steel beam according to LRFD are given in the Appendix. 5. Optimum design process The harmony search algorithm initiates the design process by selecting random values for the steel beam spacing, the beam size and the concrete slab thickness. The algorithm tries to find the best value for each design variable to minimize the objective function. The design process consists of six steps, as follows. Step 1. Select the values of the harmony memory parameters (HM, HMCR, PARmin and PARmax ). Step 2. The harmony memory matrix is initialized (values for beam spacing, beam size and concrete slab thickness are chosen). Step 3. Check whether the newly selected design vector should be pitch-adjusted. Step 4. With the values selected for the beam spacing, beam size and concrete slab thickness, the algorithm designs a composite floor according to AISC-LRFD. Step 5. Calculate the objective function value for the newly selected design vector. If this value is better than the worst harmony vector in the harmony matrix, it is then included in the matrix, while the worst one is taken out of the matrix. The harmony memory matrix is then sorted in descending order by the objective function value. Step 6. Repeat Steps 2 and 6 are until the pre-selected maximum number of iterations is reached.
Concrete slab Welded stud connector
Fig. 2. Schematic view of a simple composite floor system.
6. Design examples Here we have considered a span, and in a structure this span is repeated to cover a ceiling. Such a span behaves independently, and once we optimize the problem for one span the result will correspond to the entire ceiling. This process can be repeated for spans of different dimensions. Example 1. The considered composite I-beam floor system is 6 m long, subjected to the combined effects of the self-weight and the imposed dead load of 3 kN/m2 and imposed live load of 2 kN/m2 ; the width of the floor is 8 m. The base diameter of the stud is 13 mm and the overall height is 50 mm. The compressive strength of the concrete is 21 MPa, and the yield strength of the steel beam is 240 MPa. For the classic HS algorithm, the parameters for this example are taken as HMS = 30, HMCR = 0.9 and PAR = 0.45. The improved harmony search algorithm parameters are taken as HMS = 30, HMCR = 0.85, PARmin = 0.35 and PARmax = 0.99. Bounds of the design variables are provided in Table 1. A schematic view of the composite floor system for Example 1 is shown in Fig. 2. The output consists of the following: Steel beam spacing = 1600 mm; Concrete slab thickness = 80 mm; Steel beam size = IPE18. The histories of design with and without modification for this example are shown in Fig. 3. It can be seen that the IHS algorithm converges after 210 iterations, while the HS algorithm requires 370 iterations to attain the same result.
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669 Table 1 Bounds of design variables.
Table 2 Compression of the specifications of the IPE18 and the W250X17.9 section.
Bounds of the design variables No. Bounds
Steel beam spacing (mm)
Concrete slab thickness (mm)
Steel beam sizea
1 2
500 300
80 140
1 29
Lower bound Upper bound
667
a The steel beam consists of 29 steel I-beams (IPE12 to 30, INP12 to 30 and IPB12 to 30).
Section properties
IPE18
W250X17.9
Area Depth Width Web thickness Flange thickness Moment of inertia about x-axis
23.9 cm2 18 cm 9.1 cm 0.53 cm 0.80 cm 1940 cm4
22.84 cm2 25.07 cm 10.05 cm 0.48 cm 0.53 cm 2239.32 cm4
Table 3 Improved harmony search parameters used for the sensitivity analysis. Case
HMS
HMCR
PARmin
PARmax
1 2 3 4 5
30 30 30 30 30
0.85 0.9 0.95 0.95 0.85
0.4 0.45 0.3 0.25 0.35
0.85 0.9 0.95 0.95 0.99
Fig. 3. Design histories for Example 1.
Fig. 5. Design histories for cases 1, 2, 3, 4 and 5 of Example 1.
7. Sensitivity analysis for improved harmony search parameters
Fig. 4. Design histories for Example 2.
Example 2. Here, Example 1 is studied by considering 267 AISC cross sections. Other input data are the same as in the previous example. The output consists of the following: Steel beam spacing = 1600 mm; Concrete slab thickness = 80 mm; Steel beam size = (W10X12)W250X17.9. The histories of design with and without modification for this example are shown in Fig. 4. It can be seen that the IHS algorithm after 725 iterations attains an objective function of 169, while for the HS algorithm we have 174.5 after 1000 iterations. Compression of the specifications of the IPE18 section and the W250X17.9 section is provided in Table 2.
The HS algorithm has proven its capability in many optimization problems. Here, we make a sensitivity analysis only to find suitable initial values of the parameters for better performance in our problem. As can be seen from Fig. 5, all the cases have converged and only their rates are different. This is true for many other heuristics as well, and different initial parameters can lead to different convergence rates. In this section, a sensitivity analysis is performed for the improved harmony search parameters involved in Example 1. The results of the sensitivity analyses carried out to determine the appropriate values of the improved harmony search parameters are given in Table 3. The design histories for the four cases are shown in Fig. 5. It is apparent from this figure that the values of 0.95 for HMCR, 0.25 for PARmin and 0.95 for PARmax are the best values for the parameters in Example 1. In this paper, a wide range is considered for the design variables. As an example, the distance between two beams is considered as 500 mm to 3000 mm, while in the previous researched this distance has been considered as constant. In the second example, 267 different sections are considered for design. Naturally these wide ranges for the variables increase the number of iterations. 8. Concluding remarks This paper presents the cost optimization of a composite floor system where the design constraints are implemented as in LRFDAISC rules. The composite floor system consists of a reinforced concrete slab and steel I-beams. The optimization was performed
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by the recently developed improved harmony search method. This mathematically simple algorithm sets up a harmony search matrix, each row of which consists of randomly selected feasible solutions to the design problem. In every search step, it searches the entire set rather than a local neighborhood of a current solution vector. It needs neither initial starting values for the design variables nor a population of candidate solutions to the design problem. The results obtained show that the improved harmony search method is a powerful and efficient method for finding the optimum solution of structural optimization problems. The main aim of this paper has been to present a simple and efficient algorithm which can be used in practical engineering problems. Such a simple approach can be utilized in many other engineering design problems to reduce the cost of the construction. Acknowledgement The first author is grateful to Iran National Science Foundation for support. Appendix. Load and resistance factor design Load and resistance factor design for composite flexural member consists of the following steps [15].
The effective width of the concrete slab is the sum of the effective widths for each side of the beam centerline, each of which should not exceed: (1) one-eighth of the beam span, center-to-center of supports; (2) one-half the distance to the centerline of the adjacent beam; or (3) the distance to the edge of the slab.
tw
s > 3.76
E Fyf
,
(A.8)
Mn should be determined from the superposition of elastic stresses, considering the effects of shoring, for the limit state of yielding (yield moment). A.5. Shear connectors (1) Load transfer for positive moment. The entire horizontal shear at the interface between the steel beam and the concrete slab should be assumed to be transferred by shear connectors, except for concrete-encased beams as defined in Section I3.3 of Ref. [15]. For composite action with concrete subject to flexural compression, the total horizontal shear force, V 0 , between the point of maximum positive moment and the point of zero moment should be taken as the lowest value according to the limit states of concrete crushing, tensile yielding of the steel section, or strength of the shear connectors. (a) Concrete crushing: (A.9)
(b) Tensile yielding of the steel section: V 0 = Fy As .
(A.10)
(c) Strength of shear connectors: V0 =
X
Qn
(A.11)
where Ac = area of concrete slab within the effective width.
A.2. Shear strength of beam
As = area of steel cross section.
P
Vu ≤ φ v Vn
(A.1)
req’d Vn = Vu /φv ≤ Vn .
(A.2)
The nominal shear strength, Vn , of unstiffened or stiffened webs, according to the limit states of shear yielding and shear buckling, is Vn = 0.6Fy Aw Cv .
(A.3)
s For webs of rolled I-shaped members with h/tw ≤ 2.24
φv = 1.00 and Cv = 1.0.
E Fy
(A.4) (A.5)
When temporary shores are not used during construction, the steel section alone should have adequate strength to support all loads applied prior to the concrete attaining 75% of its specified strength fc0 . A.4. Positive flexural strength of composite beams with shear connectors The design positive flexural strength, φb Mn , should be determined for the limit state of yielding as follows:
φb = 0.90.
(A.6)
For
s = 3.76
E Fyf
,
Qn = sum of the nominal strengths of shear connectors between the point of maximum positive moment and the point of zero moment. (2) Strength of stud shear connectors. The nominal strength of one stud shear connector embedded in the solid concrete or in the composite slab is Qn = 0.5Asc (fc0 Ec )1/2 ≤ Asc Fu .
A.3. Strength during construction
tw
h
V 0 = 0.85fc0 Ac .
A.1. Effective width
h
Mn should be determined from the plastic stress distribution on the composite section for the limit state of yielding (plastic moment). For
(A.7)
(A.12)
(3) Required number of shear connectors. The number of shear connectors required between the section of maximum bending moment, positive or negative, and the adjacent section of zero moment should be equal to the horizontal shear force divided by the nominal strength of one shear connector. (4) Shear connector placement and spacing. Shear connectors required on each side of the point of maximum bending moment, positive or negative, should be distributed uniformly between that point and the adjacent points of zero moment, unless otherwise specified. However, the number of shear connectors placed between any concentrated load and the nearest point of zero moment should be sufficient to develop the maximum moment required at the concentrated load point. Shear connectors should have at least 1 in (25 mm) of lateral concrete cover. The diameter of studs should not be greater than 2.5 times the thickness of the flange to which they are welded, unless located over the web. The minimum center-to-center spacing of stud connectors should be six diameters along the longitudinal axis of the supporting composite beam and four diameters transverse to the longitudinal axis of the supporting composite beam. The maximum center-to-center spacing of shear connectors should not exceed eight times the total slab thickness nor 36 in (see Table 4).
A. Kaveh, A. Shakouri Mahmud Abadi / Journal of Constructional Steel Research 66 (2010) 664–669 Table 4 Summery of AISC-LRFD specification for composite beam. Summery of AISC-LRFD specification for composite beam Section
Item
Summery
I3.1
Effective width on each side of beam (Lesser of the 3 values)
b = Beam length/8 (L/8) b = Beam spacing/2 (s/2) b = Distance to edge of slab
I5.1
Material
I5.2
Horizontal shear force (Lesser of the 3 values)
Hs > 4ds (Minimum stud height) = 0.85fc0 Ac =A s Fy P = Qn
I5.3
Strength of stud
Qn = 0.5Asc (fc0 Ec )1/2 ≤ Asc Fu
I5.6
Shear connector placement and spacing
= 6 ds Longitudinal = 4 ds Transverse
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