(1)Charles Camp, Jifei Li, and Shahram Pezeshk
A design procedure utilizing a genetic algorithm (GA) is developed for discrete optimization of composite structures. This procedure conforms to the load and resistance factor design (LRFD) method. The objective function considered is the cost of the structure. The objective function is minimized subjected to serviceability and strength requirements.
Optimization, genetic algorithms, composite-frames, structural engineering, and load and resistance factor design.
The design of partial-composite steel frames is more challenging than that of steel frames because of the complexity associated with selecting the number connectors between the concrete slab and the steel beam or girder. The variation of number of shear connectors will change the stiffness of a beam, affecting the capacity of the frame.
Much work has been done in the area of composite design using the strength of the concrete floor slab and steel girders acting together by means of shear connectors. Schaffhausen and Wegmuller (1978) compared a 12-story frame design with and without consideration of composite action. Ito and Galambos (1993) presented a formulation for minimum weight design of continuous composite girders based on the American Association of State Highway and Transportation Officials (AASHTO) specifications. The economies of using LRFD in composite floor beams were discussed by Zahn (1987). A cost-based optimization model for design of composite beams was presented by Lorenz (1988). An optimum-cost design of partially-composite steel beams using LRFD was presented by Bhatti (1996).
In this study, a genetic algorithm (GA) is used to design composite steel frames in compliance with the AISC load and resistance factor design (LRFD). The objective is to find the minimum cost of the rigid frame with composite girders which satisfies strength and serviceability requirements.
The GA used in this study is a modified version of a program originally developed by David Carroll at the University of Illinois. The source code for the GA driver is free for public use and is available over the Internet. Carroll's program is a FORTRAN version of a genetic algorithm driver. The GA driver program can be used for a variety of different problems by simply designing an encoding scheme and supplying routines for estimating the fitness of a individual solution. The main advantage of using the GA drive system is modularity and code reuse. New options can be added to the GA portion of the program with little to no modifications to the fitness evaluation routines.
The GA technique is based upon theory of natural adaptation (Holland, 1975). The primary strength of the approach is that information from relatively good solutions is exchanged to create better solutions. The basis of the mechanism is patterns or schema developed in the encoded representation (Goldberg, 1989).
The GA driver initializes a random sample of individual solutions upon initiation of the algorithm. The GA driver uses binary coding for individual solutions in the population. The modified version of the GA driver has two strategies for choosing random pairs for mating: tournament selection with a shuffling technique and a partitioning scheme (Camp et al. 1998). There are several crossover techniques in the modified version of the GA driver: single-, double-, triple-point or uniform. In addition, there is an option to randomly vary the crossover method at each application. Reproduction allows for the generation of either a single child or two children from each set of parent solutions. In addition, there is an concurrent option for an elitist operation that guarantees the survival of the best solution into the next generation. Mutation is handled either at the genotype (jump mutation) or at the phenotype (creep mutation). Additional features include: a niching (sharing) operator and an option for the number of children generated per pair of parents. Each operator is designed to either enhance the convergence properties or to slow the process to ensure adequate exploration of the design search space.
The objective function for the optimization problem is to minimize the total cost. For the design of frames with partially composite beams there is a trade-off between steel beam weight and number of shear connectors for a given set of design requirements. Thus, the primary design variables are the W-Shape for each beam and column in the frame and the number of shear connectors (Ns) for each composite beam. The total costs of a composite frame is the sum of steel beam and column costs and the shear connector costs. Consistent with standard practice, it will be assumed that the cost of connectors is proportional to the cost of the steel, which is called the relative cost. The objective function can be expressed as follows Bhatti (1996):
(1)
where f is the objective function, n is the total number of beams, m is the total number of columns, Ws is the weight of steel (lbs/ft), L is the length of a member (ft), Ns is the total number of connectors per beam, and Csm is the relative cost of connectors to the cost per pound of steel. The relative cost coefficient Csm varies based on the job-size and the geographical region. Typically, the relative cost coefficient ranges between 6 and 12 (Lorenz, 1988)
In this study, seven constraints are applied to composite girder design. These constraints enforce LRFD design specifications for strength (before and after the concrete cures), nominal moment capacity, live load deflection, vibration, and other practical design considerations (limiting the depth of girders). In general, the girder constraints gi may be expressed as:
(2)
where mi is the degree of violation of constraint gi.
The are two constraints on the design of the shear connectors. In general, the connector constraints si may be expressed as:
(3)
where qi is the degree of violation of constraint si.
There are four design constraints for column design. These constraints enforce LRFD requirements on beam-column interaction, shape compactness (for local stability), and frame stability. The column constraints ci may be expressed as:
(4)
where ni is the degree of violation of constraint ci.
There are several penalty function schemes proposed for structural optimization design (Camp et al. 1998). The quadratic penalty function used in this study for composite frames is:
(5)
where F is the penalty factor, gi are the girder constraints, cj are the column constraints, sk are the shear connector constraints.
Having computed a penalty factor, the penalized objective function of a particular solution can be easily determined as:
(6)
Consider the design of a simple composite beam as presented by Bhatti (1996). The problem parameters are: Csm is 10, Fy is 36 ksi, fc' is 3 ksi, the live load is 250 psf, the dead load is 90 psf, the beam span is 40 feet, the beam spacing is 10 feet, a 3 inch metal deck with a 4.5 inch slab, the unit weight of concrete is 145 psf, the maximum damping for vibration is 4%, and the strength of the shear connectors is 26.1 ksi
There are two design variables in this example: the beam size and the number of shear connectors. The member sizes and the number of connectors per beam are determined by the GA in compliance with the AISC LRFD code. In this example, 256 AISC cross-sections are considered.
The design presented in this example is developed using a population of 100 solutions run for 50 generations. A partitioning selection scheme is used where the upper 25% of the population is assigned a 50% probability of selection and the lower 75% of the population shares the remaining 50% probability. Reproduction uses uniform crossover to generating two new solutions and employs an elitist strategy. Exploration of the search space is enhanced by using a jump mutation operator.
Table 1 lists the comparison of the results from the GA with Bhatti's solution. Both methods obtained same beam size; however, the GA design required fewer connectors which results in a lower relative cost.
Table 1. Comparison of Results for Simple Composite Beam.
Analysis |
W-Shapes |
Connectors |
Cost |
GA |
W27x84 |
33 |
3,660 |
Bhatti (1996) |
W27x84 |
35 |
3,724 |
American Institute of Steel Construction (1995). Manual of Steel Construction, Load & Resistance Factor Design, Second Edition.
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Camp, C. V., Pezeshk, S. and Cao, G. (1998). "Optimized Design of Two-Dimensional Structures Using A Genetic Algorithm," Journal of Structural Engineering, ASCE, Vol. 124, No. 5, 551-559.
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Holland, J. H. (1975). Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Michigan.
Ito, M. and Galambos, V. T. (1993). "Minimum-Weight Design of Continuous Composite Girders," Journal of Structural Engineering, ASCE, Vol.119, No.4.
Lorenz, E. R. (1988). "Understanding Composite Beam Design Methods Using LRFD," Engineering Journal, AISC., Vol. 25, No. 2.
Schaffhausen, R. J. and Wegmuller, A. W. (1978). "Multistory Rigid Frames with Composite Girders Under Gravity and Lateral Forces," Engineering Journal, AISC, Vol. ??, No. 2.
Zahn, M. C. (1987). "The Economies of LRFD in Composite Floor Beams," Engineering Journal, AISC, Vol. 24, No. 2.
