PRELIMINARY DESIGN OF CONCRETE STRUCTURES USING GENETIC ALGORITHMS AND SPREADSHEETS

    PRELIMINARY DESIGN OF CONCRETE STRUCTURES USING GENETIC ALGORITHMS AND SPREADSHEETS J. Kong*, City University of Hong Kong, SAR China T. C. Kwok,...
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PRELIMINARY DESIGN OF CONCRETE STRUCTURES USING GENETIC ALGORITHMS AND SPREADSHEETS J. Kong*, City University of Hong Kong, SAR China T. C. Kwok, City University of Hong Kong, SAR China

31st Conference on OUR WORLD IN CONCRETE & STRUCTURES: 16 - 17 August 2006, Singapore

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31st Conference on OUR WORLD IN CONCRETE & STRUCTURES: 16 – 17 August 2006, Singapore

PRELIMINARY DESIGN OF CONCRETE STRUCTURES USING GENETIC ALGORITHMS AND SPREADSHEETS J. Kong*, City University of Hong Kong, SAR China T. C. Kwok, City University of Hong Kong, SAR China

Abstract This paper presents the application of the genetic algorithm to the design of some common concrete structures in Hong Kong, namely, tall reinforced concrete building frameworks, long-span concrete portal frames and prestressed concrete beams or bridges using spreadsheet. Cross-sectional dimensions of member groups are treated as discrete design variables and the method aims to minimize the weight of the structure subject to various constraints like multiple inter-storey deflection constraints for tall building frameworks, various stress limitations under serviceability limit state for fully prestressed concrete beams or bending and shear strength under ultimate limit states for portal frames. These constraints are treated as penalty function during the evolutionary process. In addition to the usual selection, cross-over (2-point) and mutation operators of a genetic algorithm, various special operators are introduced and their effectiveness in accelerating the convergence of the algorithm and the minimal weight of the structure are investigated. To verify the effectiveness of the method, present results are compared with those obtained using optimality criteria method for tall building frame solutions. The results presented herein are only preliminary; further development of the method is currently in progress. Keywords: preliminary structural design, genetic algorithm, spreadsheet.

1. Introduction It is obvious that the construction cost of a concrete structure, depends on, among other factors, the volume of concrete, cost of formwork, the tonnage of steel and the foundation. In many places like Hong Kong, where foundation cost contributes a very significant portion of the total cost and the steel cost is relatively insignificant, engineers are very often trying to reduce the weight of concrete structures (not necessarily an absolute minimum) with an appropriate and construct-able geometric configuration during the preliminary design stage that hopefully leads to a more cost-effective structure before getting into the detail design stage. In the conventional design practice of a reinforced or prestressed concrete plane frame or a continuous bridge, for example, a trial geometric profile with a uniform or non-uniform cross-sectional depth is first selected, based on

either past experience and previous records or simple rules of thumb like the span-to-depth ratio. Structural analyses are then carried out to check whether all stress/strength and/or deflection constraints are satisfied under various load combinations, taking into account prestressing effects, if necessary. The process is repeated until a feasible and lighter structure is obtained. To improve the design, different preliminary geometric profile can be tried out and the most cost-effective one is then determined for subsequent detail design. Unlike other conventional optimization techniques, the genetic algorithm is a heuristic method that imitates the natural evolutionary process and it is relatively easy for practicing engineers to understand, although its practical applications in structural design is still very limited. The method is particularly suited to the class of combinatorial optimization problems with discrete variables, to which many structural design problems belong, and applications have been developed in previous studies. In respect of structural design, this paper presents a study of the application of genetic algorithm to the finding of a minimum-weight concrete structure by varying its geometric profile or member sizes during the preliminary design stage. Together with the popular and powerful tool of spreadsheet, the GA method can be demonstrated to be a practical tool for engineers to use for routine analysis and design of structures. The work reported herein only represents a preliminary study of applying the genetic algorithm to the design of three common concrete structures in Hong Kong, namely, tall reinforced concrete building frameworks, long-span concrete portal frames and prestressed concrete beams or bridges. In these problems, cross-sectional dimensions of member groups are treated as discrete design variables and the method aims to minimize the weight of the structure subject to various constraints like multiple inter-storey deflection constraints for tall building frameworks, various stress limitations under serviceability limit state for fully prestressed concrete beams or bending and shear strength under ultimate limit states for portal frames. These constraints are treated as penalty function during the evolutionary process. In addition to the usual selection, cross-over (2-point) and mutation operators of a genetic algorithm, various special operators are introduced and their effectiveness in accelerating the convergence of the algorithm and the minimal weight of the structure are investigated. To verify the effectiveness of the method, present results are compared with those obtained using optimality criteria method for tall building frame solutions. Applications to the said problems are described in the following sections. 2. Tall building plane frames In the design of tall building frameworks, one of the most important tasks for the engineer is to design an efficient structural layout for resisting the high lateral wind load and, at the same time, satisfying both the architectural and building services requirements. Once such a layout is determined, member sizes should be minimized, particularly for vertical members of the framework, so that usable floor space can be maximized. This is of particular commercial importance in areas like Hong Kong where the price per unit floor area is very high. In addition, minimizing the weight of the superstructure also leads to a significant saving in foundation cost, particularly in areas with poor soil conditions or newly reclaimed lands which are commonly encountered in Hong Kong. To facilitate the process of finding the optimal member sizes for a given structural framework, this paper presents the application of the genetic algorithm to the design of tall reinforced concrete buildings under lateral loads. Cross-sectional dimensions of member groups are treated as discrete design variables and the method aims to minimize the weight of the structure subject to multiple inter-storey deflection constraints, based on the Hong Kong Code of Practice [1]. These constraints are treated as penalty functions during the evolutionary process. Initial trial runs using the conventional selection, cross-over (2-point) and mutation operators of a genetic algorithm indicated that the evolution process was slow and it is not always guaranteed that member sizes in the lower storeys would have larger or equal member sizes than the upper storeys. As such, an additional member size constraint is introduced into the algorithm. In addition, a special accelerating operator, together with a pre-selection scheme and elitist strategy are adopted and their effectiveness in accelerating the convergence of the algorithm and minimizing weight of the structures is investigated; details of formulation are given in [2]. To demonstrate the aforesaid procedure, a two-bay, 12-storey high plane frame as shown is Figure 1 is studied. The columns are divided into 12 groups. Beam sizes and column width are fixed. Genetic algorithm is applied to the plane frame. The usual selection, cross-over and mutation operators are adopted; in this study, the two-point crossover operator is chosen because it has been demonstrated to be more stable, reliable and efficient. In addition, to increase the performance of the simple genetic algorithm, elitist strategy is also used in conjunction with the pre-selection

concept. For the elitist strategy, the best chromosome is retained and copied into the next generation, thus increasing the speed of domination of a population of highly fit individuals, as reported in many studies. For other chromosomes, according to the pre-selection concept, an offspring replaces its parent if the offspring’s fitness exceeds that of the inferior parent. Initial runs, based on the aforesaid algorithm, with 10 and 30 chromosomes were carried out for the plane frame in Figure 1. Each chromosome has a binary string length of 60, comprising 12 design variables (i.e. 12 groups of column depth) and each variable being represented by a binary number of 5 digits. Results of the minimum weight and average weight of each generation are summarized in Table 1, based on a cross-over probability of 80% and mutation probability of 0.01. It can be observed that there is no significant difference between the minimum weight obtained using 10 or 30 chromosomes, although the latter reaches the same value earlier than the former. 10 6

12

6

6

12

6

5

11

5

5

11

5

4

10

4

4

10

4

3

9

3

3

9

3

2

8

2

2

8

2

1

7

1

1

7

1

20 20 20 20 20 36m

20 20 20 20 20 20

5m

5m

Figure 1: A 12 storey high plan frame. Beam and column width = 150mm. Beam depth = 325 (11 &12F), 525 (9 & 10F), 625 (7 & 8F), 700 (5 & 6F), 750 (3 & 4F), 725mm (1 & 2F). Column depth varies randomly between 300mm and 1075mm. Young’s modulus = 2.29x107 kN/m2. 3 Density=24kN/m . Table 1: Column Depth and minimum weight of frame using 10 and 30 chromosomes Column Depth Column 10 30 Group Chromosome Chromosome 1 900 850 2 700 800 3 700 800 4 525 600 5 325 400 6 325 350 7 975 725 8 900 725 9 800 700 10 700 700 11 700 500 12 350 400 Min Weight (kN) 508.5 507.96 The aforesaid application was implemented on a PC (Pentinum III with 800 MHz) and was done

using Excel. The computing time for such a simple framework, with 10 chromosomes and 100 generations took about 20 mins. For more complicated framework and larger number of constraints, there is no doubt that the computing time will be increased tremendously. As such, it would be highly desirable, from practical point of views, to speed up the genetic algorithm. To this end, a special accelerating operator is introduced and incorporated into the SGA. In the conventional structural design practice, analysis for a framework is carried out and the deflection limits are subsequently checked. Should the limit for a particular floor is found exceeded, the corresponding structural member sizes are increased. Conversely, if the interstorey drifts is found to be much lower (the extent of which can be defined by the designer) than the limit for a particular floor, it is up to the designer to decide whether the corresponding member sizes shall be reduced or remain unchanged. This simple and conventional practice can be incorporated into the SGA (details of which is given in [2]); the essence is that the member sizes of each chromosome (framework) shall be given an opportunity to enlarge or reduce or simply remain unchanged. In this example, the probability of having member enlargement, reduction or being unchanged are defined by the user as 0.3, 0.3 and 0.4 respectively. As such, if a chromosome is randomly allocated a number of 0.25, for example, then sizes of this framework will be reduced for members of those floors which deflections are less than 80% (user-defined) of the limit.

Table 2: Weight and size of frames

Column Group 1 2 3 4 5 6 7 8 9 10 11 12 Min Weight (kN) with accelerator Min Weight (kN) using SGA

Column Depth obtained using different number of chromosomes and with accelerator 30 50 10 100 725 700 700 625 700 650 700 600 475 575 625 475 400 525 625 475 325 500 500 400 300 350 350 400 675 800 700 700 675 725 700 675 675 550 625 675 675 525 575 600 650 425 500 550 500 350 350 300 488.5

472.3

508.5

507.96

475.5

466.9

Ref[3] using OC 675 600 550 475 400 300 800 800 725 675 600 500 491.5

The plane frame in Figure 1 is reanalyzed using the genetic algorithm with the accelerating operator. Cross-over probability of 0.8 and mutation probability of 0.01 are adopted. Comparison of the weight of the structure is summarized in Figure 2. It is obvious that, using the same number of chromosomes as the previous SGA, the weight of the structure flattens out with less than 40 generations. The minimum weight results using the genetic algorithm are compared with those obtained using the optimality criterion method [3]. Good agreement can be observed between the two sets of results, although the GA result is about 3% lower than that of the OC method.

Weight of Structure 640

620

600

Average (10 chromosomes with acceleration) Minimum (10 chromosomes with acceleration) Average (30 chromosomes with acceleration) Minimum (30 chromosomes with aceleration) Average (10 chromosomes)

W eight (kN)

580

560

540 Minimum (10 chromosomes) 520

Average (30 chromosomes) Minimum (30 chromosomes)

500

480

460 0

10

20

30

40

50

60

70

80

90

100

Generation

Figure 2: Comparison of Results between GA with acceleration and SGA 3. Prestressed concrete beams Prestressed concrete beams and frames are widely used structural elements of buildings and bridges. Prestressed concrete box girder, in particular, is one of the most commonly used structural forms for medium-span highway or railway bridges due to its superior load distribution characteristics, aesthetic appearance, and the relative ease of construction. In practice, structural design of a prestressed concrete bridge usually starts with some trial cross-sections, based on either past experience, or some simple rules-of-thumb like the span-to-depth ratio, and by taking into account the method and sequence of construction. A trial tendon profile is then used, taking into account the stressing sequence and construction method. Structural analyses are subsequently carried out to determine various force and moment envelopes and to check whether the stress, deflection and/or strength constraints are satisfied under various load combinations, during construction and upon completion. This process is repeated until a feasible design is obtained. To improve the design, different designs can be tried out and the most cost-effective one, not necessarily the optimal, is then determined for subsequent detail design. In this work, a study of the application of genetic algorithm to the preliminary design of prestressed concrete box girders with continuity over several spans is being conducted. The objective is to minimize the weight of the whole structure subject to various stress constraints with design variables including cross-sectional dimensions, individual span length of the continuous beam, the size and number of prestressing tendons and geometry of tendon profiles. Data sets are constructed a priori for the said variables, by taking into account the construction practices, designer’s experience and practical ranges so that a finite and countable number of arrangements are pre-defined for each design variable. Genetic algorithm is then used to search for the optimum combination of arrangements from these data sets. The method was implemented on a spreadsheet using VBA because of its familiarity, availability and ease of programming for practicing engineers. A four-span beam is used to illustrate its application and effectiveness; details of formulation can be found in [4]. To demonstrate the application of GA, a four-span continuous, cast-in-situ, box girder is considered, which cross-section is given in Figure 5. The total length of the bridge is 140m and it is assumed to

be symmetrical about the middle support. It is intended to obtain a “good” preliminary design before a detailed design is started. Of particular interest here includes the depth of the section, the ratio of the ultimate and penultimate span lengths, the approximate tendon profile and the total prestressing force required (i.e. the number of prestressing strands required.). The bridge deck is divided into two notional lanes, carrying HA traffic load, in the form of a uniformly distributed line load which magnitude is related to the loaded length, and a HB point load of 1000kN. For this type of bridges, usually the combination of dead load and traffic live loads at the completed construction stage is critical in determining the over-all geometry of the bridge. Multiple lane factors and load factors, including those for dead load and superimposed dead load are all considered in accordance with the local practice [5]. A total of 17 load cases are generated and moment envelopes at the middle of each span and the intermediate supports are considered for the limiting concrete stress constraints. Design variables of the section depth, penultimate and ultimate span lengths and range of prestressing forces are given in Table 3. With reference to Figure 3, the tendon profile is assumed with L1 varies from 0.3 to 0.45 of span 1 and d1 = 0.05 or 0.1 L1 and d2 and d3 = 0.05 or 0.1 L2. The lowest point of tendon for the two internal spans is at the middle of the respective span. The minimum and maximum eccentricities of the tendons are taken as 80% of the distance of the neutral axis from the bottom and top surface respectively. Variation of tendon force due to friction loss along the profile is explicitly calcu-lated together with a further 20% of losses due to shrinkage, creep, relaxation, draw-in and elastic shortening. By applying GA with 30 chromosomes for 50 generations, with 80% probability of crossover, 1% probability of mutation and penalty factors K=1000 and α=0.1, the final weight of the structure so obtained is 15638kN. This minimum weight structure corresponds to a deck of depth 1.52m (i.e. a span/depth ratio of about 25), with maximum prestressing force 27676.8kN used, and ultimate and penultimate span lengths of 32.5 and 37.5m (i.e. a ratio of about 0.85). The tendon profile for the minimum weight structure corresponds to L1 = 0.4 of span 1 and d1 = 0.1 L1, d2 and d3 = 0.1 L2. The variation trends of the minimum weight and average weight of the whole population are given in Figure 4. Due to the limited space, other parametric studies and results will be published elsewhere and presented in the conference. d3 d4

d1 d2

d5 d6

Li hi

Span 2

Span 1

Span 4

Span 3

Figure 3: Tendon Profile of a 4-span bridge 18000 17500 Average weight

16500 16000 15500

Minimum weight

15000

49

46

43

40

37

34

31

28

25

22

19

16

13

10

7

4

14500 1

weight (kN)

17000

No. of generations

Figure 4. Variation of average and minimum weight for the 4-span bridge

wtop ttop

troot

ttip Lcant tbot

d

tweb

wbot

Figure 5. Box girder cross-section and the design variables

Table 3: Data for the 4-span bridge Overall Width of top slab External wing thickness Internal wing thickness Top slab thickness Section Depth (with increment of 0.02m; total 64 discrete values) Web Thickness Bottom slab thickness Bottom slab width Wing width Ultimate span (with increment of 0.5m; total 16 combination of span lengths for the bridge) Penultimate span (with increment of 0.5m) Range of prestressing force (total 32 discrete values using various number of VSL prestressing strands)

9.7m 0.25m 0.35m 0.25m 1.20 to 2.46m 0.45m 0.2m 4.5m 2.3m 27.5 to 35.0m 35 to 42.5m 8481.6 kN to 27676.8 kN

4. Reinforced concrete portal frame Unlike previous studies in optimization, the amount of reinforcement is specified based on the designer’s experience and it is not taken into account as a design variable in this study. To start with, a simple concrete frame is considered. The geometric profile is represented in a discrete manner. The structure is subject to various strength constraints with design variables including cross-sectional dimensions of beams along the span and along the height of columns. Individual span length can also be included as design variables, if continuous frames are considered. Data sets are constructed a priori for the said variables, by taking into account the construction practices, designer’s experience and practical ranges so that a finite and countable number of arrangements are pre-defined for each design variable. Genetic algorithm is then used to search for the optimum combination of arrangements from these data sets using penalty factors in conjunction with the usual cross-over and mutation operators. The method was implemented on spreadsheets using VBA. An example of RC frame is given in Figure 6.

Figure 6. Variation of (a) final optimized geometric profile of an RC frame under vertical and lateral load (b) total fitness and (c) best individual fitness 5. CONCLUSION This paper presents the application of the genetic algorithm to the design of reinforced/ prestressed concrete structures using spreadsheet. The problem is formulated as a combinatorial optimization: discrete data sets are constructed a priori for the major design variables in the practical range, taking into account relevant local construction practices and designers’ experience. The method is intended to help designers to find a “good” preliminary design prior to carrying out the detailed design. The method has been implemented on a spreadsheet using VBA which is familiar to most practical engineers. The converged results of the given examples are consistent with local experience. The work will also be extended to include consideration of prestressed concrete frames with continuous spans. REFRERENCES [1] Code of Practice for the structural use of concrete (1987), Building Authority, Hong Kong. [2] Kong, J. (2004) "Design of Tall Reinforced Concrete Buildings using genetic algorithms” (CDRom) Proceedings of the Sixth World Congress on Computational Mechanics, Beijing, China September 5-10, 2004, p.539, Vol.II of Conference Abstract. [3]Chan,C.M. and Sun,S. (1997) Optimal Drift Design of Tall Reinforced Concrete Building Framework, in D.M. Frangopol, F.Y. Cheng eds. Advances in Structural Optimization. Proc. 1st USJapan Joint seminar on structural optimization, Chicago, Illinois, pp. 31-42. [4] Kong, J. (2005) Preliminary Design of Prestressed Concrete Continuous Bridges Using Genetic Algorithms and Spreadsheets” Proceedings of the Third International Structural Engineering and Construction Conference, ISEC-03, Sept.20-23, 2005, Shunan, Japan, pp. 571-576, Vol. 2 [5] Structures Design Manual for Highways and Railways, High-ways Department, The Government of HKSAR. 1997 Edition.

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