OPTIMIZATION OF SUSTAINABLE OFFICE BUILDINGS IN STEEL USING GENETIC ALGORITHMS
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1 Nordic Steel Construction Conference 2012 Hotel Bristol, Oslo, Norway 5-7 September 2012 OPTIMIZATION OF SUSTAINABLE OFFICE BUILDINGS IN STEEL USING GENETIC ALGORITHMS Martin Mensinger 1 a, Li Huang 2 b, Phoenix Zhang 3 c, Heidrun Hogger 4 d a,b,c,d Technische Universität München, Lehrstuhl für Metallbau of authors 1, 2, 3 and 4 Abstract: Constructions of composite steel and concrete slabs for office buildings contain primary and secondary beam systems and a composite slab. The arrangement of the beams is affected by a lot of geometrical boundary conditions such as room geometry, position of the intermediate support, maximum construction height, etc. The design of the composite floor is complicated and highly iterative depending on the design parameters. Genetic algorithm is developed for optimization of the composite floor. The object function considered is global warming potential of the floor, or the primary energy of the construction. The object function is minimized subjected to serviceability limit state (SLS) and ultimate limit state (ULS). 1 Introduction Constructions of composite steel and concrete slabs for office buildings contain primary and secondary beam systems and a composite slab. The arrangement of the beams is affected by a lot of geometrical boundary conditions such as room geometry, position of the intermediate support, maximum construction height, etc. For the planning engineer or the architect it is important to know which constructions are best suited for the building they are planning according to ecological criteria like global warming potential. In practice, a composite beam is designed in a trial-and-error process [1] to select the structure geometry parameters and cross section types, etc. The design of composite beams is complicated and highly iterative. Since the value of a design parameter affects other values, all design parameters cannot be found simultaneously [1]. A genetic algorithm (GA) is implemented for the optimization of the composite floor. As a first step, a program was written that calculates not only the important design rules for the construction but also includes ecological criteria to search for the best solution. This program starts with a typical rectangular floor plan, which consists of a composite floor slab supported by, and composite with, composite beams that span between columns. The primary beam could only be in longitudinal or cross direction. The object function considered is the global warming potential (GWP) of the floor, or the primary energy of the construction, or the cost of the composite floor. The object function is minimized subjected to serviceability limit state (SLS) and ultimate limit state (ULS).
2 2 Nordic Steel Construction Conference Model formulation The primary purpose of model formulation is to formulate a robust optimization model for composite floor. To this end, the present model is formulated in two major steps: 1) to determine the major decision variables affecting the design of composite floors; 2) to formulate the objective of optimization of composite floor in a robust optimization model (see also [2]). 2.1 Optimization objectives The present optimization model is formulated in order to provide the optimization of the energy or global warming potential (GWP) or cost of composite floor. The model is also designed to quantify and measure the impact of various decision variables that affect the optimization of the composite floor. It incorporated the following objective equation: (1) Where is the volume of the concrete, are the mass of the reinforcement, the profiled sheeting and the steel section respectively, is the area of steel section. are the weighting factors listed in Table 1 ( is set to be 0, that means the influence of the area of steel section is not yet considered). Table 1: Weighting factors of objective functions Objective a 1 a 2 a 3 a 4 a 5 Function Energy 1222 MJ/m³ 13,4 MJ/kg 21 MJ/kg 12,3 MJ/kg 0 GWP 237 kg CO 2 - Äqv./m³ 0,874 kg CO 2 -Äqv./kg 1,41 kg CO 2 - Äqv./kg 0,80 kg CO 2 - Äqv./kg Decision variables The decision variables are those who may have an impact on the global warming potential or energy or even cost optimization of composite floors. In order to reduce the complexity of the optimization model, the present model combines the structure variables into three variables called: (1) column position decision variable, (2) column number decision variable, and (3) secondary beam number decision variable; combines the decision variable related to concrete slab into two variables: (1) slab type decision variable and (2) slab thickness decision variable; and combines the decision variables related to the steel section into a single variable called cross section decision variable. 3 Model optimization For the optimization of the composite floors, a genetic algorithm is implemented. Genetic algorithms are a class of probabilistic optimization algorithms [3], which are inspired by the biological evolution process. The general procedural of GA is shown in Fig.1. Initially, a group of individuals (population) is created randomly, and then some individuals are selected as parents, thereafter by recombination and mutation, offspring are created. At the end of one generation, a group of new individuals is generated by survivor selections.
3 Nordic Steel Construction Conference Representation of Modeling Fig. 1: General procedural of GA Mapping from real world model to the genetic individuals are called representation. In this paper, integer representation is used. The genes value of each chromosome must be within some ranges. The ranges should be determined based on what would be structurally reasonable. As an example, let us consider a composite floor which is 50m long and 15m wide. We use 13 genes for each chromosome to represent the structure arrangement. Gene(1) represents the position of the columns. The columns stand in a row within the range of [x,y], as shown in Fig.2a). Supposing the difference between each possible position is 0.3m, then e.g. gene(1)=2, the position of the columns is =x+0.3m*gene(1) y. a) Position range of the columns b) Position of the columns Fig. 2: Position/position range of the columns Gene(2) represents how many columns are there. As shown in Fig.2b), d is the distance between columns. Then gene(2)=50m/d-1, with 4m d 10m, d=1.25*i, i=1,2,, and 50m/d must be an integer. So gene(2) can be 4, 7 or 9, i.e., there can be 4, 7 or 9columns in the middle of the floor if the length of the composite floor is 50m. Gene(3) represents which kind of slab is used. There are two slab types: Superholorib SHR51 and Cofraplus 60. If gene(3) = 0, Superholorib SHR51 is chosen. Otherwise, gene(3) = 1, which means Cofraplus 60 is used for the floor construction. Gene(4) represents the thickness of slab. Four levels of thickness are generally used for the office building floor, which are 120mm, 140mm, 160mm and 180mm. For example, gene(4) = 140 means the thickness of the slab should be 140mm. Gene(5) represents the cross section type of the secondary beams. Only IPE and HEA will be used for secondary beams. Gene(6) represents the steel grade of the secondary beams. There are three steel grades for choosing, S235, S355 and S460. For example, gene(6) =235 means that the steel grade of the secondary beam is S235.
4 4 Nordic Steel Construction Conference 2012 Gene(7) represents which kind of cross section is chosen for the primary beams. IPE, HEA, HEB and HEM are available for the primary beams. Gene(8) represents the steel grade of the primary beams. The same as gene(6), S235, S355 and S460 are available for choice. Gene(9) represents which construction system is used. In system A (gene(9) = 0), main beams are in cross direction and secondary beams are in longitudinal direction as shown in Fig.3a. In system B (gene(9) = 1), main beams are in longitudinal direction while secondary beams are in cross direction, as in Fig.3b. a) system A b) system B Fig. 3: Construction systems Gene(10) represents the slabs are propped or un-propped during the construction time, while gene(10) = 0 means they are un-propped and gene(10) = 1 means propped. Gene(11) represents the number of the secondary beams. Assume the distance of the secondary beams should be between [2m, 6m]. If the structure system is system A, gene(11)=randominteger between [15m/6m,15m/2m]+1, otherwise, in system B, gene(11)=randominteger between [50m/6m,50m/2m]+1. Gene(12) represents the cross section type of the columns. Gene(13) is the steel grade of the columns. Like gene(6), it could be S235, S355 or S Approach Schemes Two different schemes are used to optimize the composite floor. One is using general genetic algorithm to approach the optimization, the other runs a genetic algorithm within genetic algorithm (GA-within-GA), the inner GA is run only when the outer GA reaches a threshold score Approach 1: GA The computation procedure of the composite floor design is shown in Fig.4 (see also [1]) as a flowchart while the steps are described as follows: 1) Read project and genetic algorithm parameters needed to initialize the search process. The project parameters include: (1) floor length and width, (2) maximum structure limit, (3) range of the column positions, (4) Young s modulus of steel section and concrete, (5) partial factors of steel, concrete and reinforcement, (6) additional unit dead load, (7) unit live load, (8) fitness weighting factors, (9) section properties for 65 commonly used I-shaped sections, (10) profiles of 8 profiled concrete slab, (11) factors for the continuum beam moment calculation. The required genetic algorithm parameters for this initialization phase include: (1) chromosome size, (2) the number of generations, (3) population size, (4) mutation rate, (5) crossover method, (6) crossover rate, (7) selection scheme. 2) Create initial population P 1 in the first generation (g = 1). The population size is S. 3) Check ULS and SLS of each individual i ( i = 1 To S) to find the feasible design solutions.
5 Nordic Steel Construction Conference ) Compute the fitness value for each individual in generation g, while the individuals with feasible design solutions are reset to better fitness values. 5) Create a child population C g using selection, crossover and mutation operators. 6) Combine the child population C g and parent population P g to form a new combined population N g with a size of 2S. This combined population allows good solutions of the initial parent population to pass on to the following generation in order to avoid the loss of good solutions of the initial parent population once they are found. 7) Sort the new combined population N g based on their fitness scores. 8) Keep the top S individuals from the combined population N g to form the population P g+1 of the next generation. This population is then returned to step3 for generating a new iterative process, which continues until the specified number of generations is completed. These computational steps were implemented using a C++ computer program based on GAlib (Matthew s genetic algorithms library) [4]. Fig. 4: Computation flow chart of the first approach Approach 1: GA-within-GA The outer GA tries to arrange the composite floor, while the inner GA is used to choose the beams and columns. It doesn t try to run the inner GA until the outer GA reaches some threshold. The computation procedure is shown in Fig.5 as a flow chart while the steps are described as follows: 1) Read project and genetic algorithm parameters needed to initialize the search process. The project parameters are the same as chapter ) Create initial population P₁ in the first generation (g₁ = 1) of outer GA. The population size is S₁.
6 6 Nordic Steel Construction Conference ) Set fitness1(i)=0 for each individual i ( i = 1 to S₁) in population P₁. Then the inner GA for the individual I starts: 1. Read in genetic algorithm parameters needed to initialize the inner GA: (1) chromosome size, (2) the number of generation, (3) population size, (4) mutation rate, (5) crossover method, (6) crossover rate, (7) selection method. 2. Create initial population P₂ in the first generation (g₂ = 1) of inner GA. The population size is S₂. 3. Check the composite beam design for each individual in population P₂ to give the feasible solutions better fitness score. 4. Compute fitnss2 for each individual of population P₂ in generation g₂. 5. Create a child population C2 g2 using selection, crossover and mutation operators for the inner GA. 6. Combine the child population C2 g2 and parent population P2 g2 to form a new combined population N2 g2 with a size of 2S₂. 7. Sort the new combined population N2 g2 based on their fitness scores 8. Keep the top S₂ individuals from the combined population N2 g2 to form the population P2 g2+1 of the next generation. This population is then returned to step iii for generating a new iterative process, which continues until the termination generation is accessed. Then the best fitness2 is passed to the fitness2(i) for individual I in the outer GA. 4) Set fitness value for each individual i: fitness1(i) = fitness2(i) for i = 1 to S₁. 5) Create a child population C1 g1 using selection, crossover and mutation operators for the outer GA. 6) Combine the child population C1 g1 and parent population P1 g1 to form a new combined population N1 g1 with a size of 2S₁. 7) Sort the new combined population N1 g1 based on their fitness scores. 8) Keep the top S₁ individuals from the combined population N1 g1 to form the population P1 g1+1 of the next generation. This population is then returned to step 3 for generating a new iterative process, which continues until the specified number of generations is completed. 4 Illustrative examples and parametric study 4.1 Illustrative examples Although the following example is rather simple, it is chosen in order to demonstrate the efficiency of the GA used [5]. A composite floor plan 50m(l) 15m(w) thickness composed of I-shaped primary and secondary beams is optimized, and the columns between the floors are 3m in height, see Fig.6. The dead load is 3 kn m 2, live load is 2 kn m 2, concrete unit weight is kn m 2. Table 2 shows all the chosen properties of the GA used. Two cases were considered, with the same formulae for fitness function: (2) but different factors a i. v c is the volume of the concrete; m R, m p, m q are the mass of reinforcement, profiles sheeting and steel section, respectively; A a is the area of steel section. In the first example, the fitness function is for energy, and the factors of the fitness function are in the second row of Table 1. In the second example, the fitness function is for global warming potential (GWP), and the corresponding factors are in the third row of Table 1.
7 Nordic Steel Construction Conference Fig. 5: Computation flow chart of GA-within-GA Fig. 6: A bird's view of composite floor plan. Table 2: Properties of the GA Population Size N: 2000 Generation M: 2000 Probability of mutation pm: 0,05 Probability of Crossover cm: 1,0
8 8 Nordic Steel Construction Conference Different approach schemas Schema 1: One GA In One GA, the composite floor will be optimized only by genetic algorithm. The following factors will be optimized as genes: 1) the position and number of the columns, 2) the slab type and thickness, 3) the cross section type and steel grade of the secondary beams, 4) the cross section type and steel grade of the primary beams, 5) the cross section type and steel grade of the columns. With selection, crossover and mutation, these gene strings will be optimized generation by generation. The optimization processing (population size = 100) is shown in Fig.7. Fig. 7: Convergence of One GA The horizontal axis is the generations from 0 to 3000 while the vertical axis is energy fitness value. We can see, from the Fig.7, in the 1st generation the optimized fitness value is MJ, thereafter it went down dramatically to MJ in the 59th generation, which is half of the fitness value in the 1st generation. As the generation grows the fitness value became smaller and smaller, and finally it converged to MJ in the 1756th generation. The whole optimization process took 0.807sec Schema 2: GA-in-GA In the schema 2, the outer genetic algorithm optimizes the geometry of the composite floor, like the position of the columns, the number of the columns and secondary beams, while the inner genetic algorithm optimizes the profiles for every geometry in the outer genetic algorithm loop, like the cross section types and steel grades of secondary beams, primary beams and columns. Fig.8 shows a optimization processing of this schema with the following genetic algorithm parameters: population size of inner GA = 100, generation of inner GA =100; population size of outer GA = 100, and it terminates according to the convergence. Fig. 8: Convergence of GA-in-GA It shows in Fig.8 that the optimized energy fitness value was MJ in the 1st generation, and then it dropped to MJ since the 4th generation, and finally converged to MJ in the 18th generation. The optimization process took sec, which is much
9 Nordic Steel Construction Conference longer than the processing time of schema 1. However, the schema 2 converges very fast, which we can also see from Fig.8, it converged before the 20th generation while the schema 1 converged after 2000 generations. The One GA is the least time-consuming approach schema in all these three methods, and it's also a straight-way optimization which is easier to understand. But this approach convergences a little slower regarding to the generations. The GA-in-GA convergences much faster regarding to the generations, but took longer time for the computing. The GA-in-GA is very useful when we want to know the most optimized value: normally, run this approach twice with 100 population and 100 generations, if the optimized fitness values are the same, then we can be "sure" that they are the most optimized ones. The tests in the following parts of this chapter will only run with One GA approach because it takes shorter time. 4.3 The best gene strings By executing the GA, some optimal solutions were found for the energy fitness function. These solutions are given in Table 3. Position of Columns Table 3: Optimal solutions Geometry Slab Secondary Beam Primary Beam Column Number of System Columns A/B Number of Secondary Beams Slab Type Slab Thickness Cross Section It is interesting to note that the construction system is always A and the slab type is always cofrasta, this leads to the conclusion that the system A and cofrasta slab are more environment-friendly. The structure in the 1st row of the above table is shown in the Fig.9, which has 11 columns in the longitudinal direction of the structure, wherein the main beams are in the cross direction and the secondary beam are in the longitudinal direction. The secondary beams use IPE220 as cross section and S355 as steel grade, the primary beams use IPE550 as cross section and S460 as steel grade, and the columns use HEA280 as cross section and S460 as steel grade. The slab is cofrasta and 120mm thick. Steel Grade Cross Section Steel Grade Cross Section Steel Grade Energy Fitness Value 4.9m 11 A 6 cofrasta 120mm IPE220 S355 IPE550 S460 HEA280 S m 11 A 6 cofrasta 160mm IPE140 S355 IPE550 S460 HEA280 S m 11 A 6 cofrasta 120mm IPE220 S235 IPE550 S460 HEB240 S m 11 A 7 cofrasta 160mm IPE140 S355 IPE550 S460 HEA280 S m 11 A 7 cofrasta 160mm IPE140 S355 IPE550 S460 HEB240 S m 11 A 7 cofrasta 160mm IPE140 S460 IPE550 S460 HEA300 S m 11 A 6 cofrasta 120mm IPE220 S235 HEA360 S460 HEA280 S m 11 A 6 cofrasta 120mm IPE220 S460 HEA360 S460 IPE450 S m 11 A 7 cofrasta 160mm IPE140 S460 IPE550 S460 HEA320 S m 11 A 6 cofrasta 120mm IPE220 S355 HEA360 S460 HEB240 S m 11 A 6 cofrasta 120mm IPE220 S2355 IPE600 S355 IPE270 S m 11 A 6 cofrasta 120mm IPE220 S460 HEB450 S460 HEB300 S Fig. 9: An example structure of good solutions A further test for the first solution in Table 3 was made to found out the "control factor" in the optimization process. The results are shown in Table 4, where all the values should be smaller
10 10 Nordic Steel Construction Conference 2012 or equal to 1 according to the optimal criteria. From Table 4, it can be seen that the deflection is the key element which controls the optimization, while the sheer force of the primary beam has the least control of the optimization. 7 Conclusions Table 4: A further test for the optimized solution Secondary beam Med/Mrd Ved/Vrd Deflection/(beam_lengeth/350) 0, , , Primary Beam Med-neg/Mrd_neg Ved/Vrd Med-pos/Mrd_pos 0, , , Column F_Ed_m/N_b_y_Rd F_Ed_e/N_b_y_Rd+k_yy*M_Ed_y_col/M_b_y_Rd F_Ed_e/N_b_z_Rd+k_zy*M_Ed_y_col/M_b_y_Rd 0, , , These experiments demonstrate that the GA is able to optimize composite structures [5]. The proposed schemas enable structural designers to generate and evaluate optimal/near-optimal design solutions [1].To accomplish this, the schemas incorporate: (1) a design module that performs the design of composite floors [1]; (2) a GWP/Energy module that computes the global warming potential or the energy consumption of composite floors; (3) an optimization module that searches for and identifies optimal/near-optimal design alternatives [1]. Several parts of this work are interesting starting points for further research or improvement: (1) there is a need to develop a module for visualizing the optimal progress. (2) the length of gene strings could be flexible, i.e., the variables of design problem can be selected by users. (3) it is necessary to calculate more exactly in the design module. For example, the columns in different floors have different loads, and the edge columns and middle columns also have different loads. The edge beams and the middle beams use different cross sections and steel grades. (4) multi-objective optimization should be incorporated into GA. References [1] Ahmed B. Senouci, Mohammed S. Al-Ansari, Cost optimization of composite beams using genetic algorithms, Advances in Engineering Software 40 (2009), [2] Arnaud Lemaire, Example: simply supported secondary composite beam, access steel, [3] Siddhartha K. Shakya, Probabilistic model building Genetic Algorithm (PMBGA): A survey, Technical Report, Computational Intelligence Group, School of computing, The Robert Gordon University, Aberdeen, Scotland, UK. pp.1,, [4] GAlib, Example: simply supported secondary composite beam, Massachusetts Institute of Technology and Matthew Wall, [5] J. L. Marcelin, P. Trompette, R. Dornberger, Optimization of composite beam structures using a genetic algorithm, Structural and Multidisciplinary Optimization, Volume 9, Numbers 3-4, , DOI: /BF , Springer, 1995.
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