Design of Reinforced Cantilever Retaining Walls using Heuristic Optimization Algorithms

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Procedia Earth and Planetary Science 5 (2012) 32 – 36

Procedia Earth

and Planetary

Science

Procedia Earth and Planetary Science 00 (2011) 000–000

www.elsevier.com/locate/procedia

2012 International Conference on Structural Computation and Geotechnical Mechanics

Design of Reinforced Cantilever Retaining Walls using

Heuristic Optimization Algorithms

Yaoyao Pei

a

_{, Yuanyou Xia, a*}

School of Civil and Engineering, Wuhan University of Techonololy, Wuhan, 430070, China

Abstract

This paper aims at automatic design and cost minimization of reinforced cantilever retaining walls (RCRW). The design requirements and geometrical constraints are imposed as design constraints in the analysis. 9 parameters are selected to define the structure and 25 constraints are established. Three heuristic algorithms, including genetic algorithm (GA), particle swarm optimization (PSO) and simulated annealing (SA) are presented to solve the constrained optimization model. The computation programs have been developed and validated by taking an example design. Results show that heuristic optimization algorithms can be effectively applied to cost minimization design of RCRW. It is found that no single algorithm outperforms other methods. With respect to effectiveness and efficiency, PSO is recommended to be used.

Keywords: Reinforced concrete soil retaining structures; cost minimization; heuristic optimization algorithms; penalty function 1. Introduction

Present design of cantilever retaining wall is far from being optimized. Generally, on the trial and error basis [1], the conventional procedures lead to safe designs, but the cost of cantilever retaining wall is very much depend on the experience of the structural designer. The design optimization of cantilever retaining walls is generally undertaken on the basis of target reliability or cost minimization. Babu[2] proposed an approach for reliability-based design optimization of reinforced concrete cantilever retaining wall. A minimum cost based design of reinforced concrete retaining structures was undertaken by Ceranic et al [3].

Cantilever retaining walls are investigated, being representive of reinforced concrete retaining * Corresponding author: Yaoyao Pei. Tel.: 15872410584.

E-mail address: yaoyao.bae@foxmail.com.

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structures; simulated annealing was applied to the solution. But the design variables merely included the variables, neglecting the variables of reinforcement. A similar research was undertaken by Huang [4] with complex method.

Heuristic optimization algorithms have been widely applied in civil engineering. Bhattacharjya [5] used a hybrid genetic algorithm to design a stable trapezoidal channel section. Six heuristic algorithms were applied to slope stabilities and performances were discussed by Cheng et al [6, 7]. Similar research was undertaken by McCombie [8], with the application of simple genetic algorithm. Three algorithms, including the multi-start global best descent local search, the simulated annealing and the meta-threshold acceptance were used to design reinforced concrete road vaults by Carbonell et al [2].

In this paper, a further attempt is undertaken for automatic design and cost minimization of reinforced cantilever retaining walls. The optimization mode defines the structure by design variables, and objective function is the cost of the structure. Design requirements and geometrical constraints are imposed as design constraints. To solve the mode, three heuristic optimization algorithms, including genetic algorithm (GA), particle swarm optimization (PSO) and simulated annealing (SA) are proposed to obtain the global optimal design variables and cost minimization.

2. Optimization model 2.1. Design variables w B2 B1 B3 d dw pmax A B pmin pA pB Gk Vk Hk qk O 1 2 3 4 1 1 2 2 2 l 1 1-1 3 4 2-2 ey ex a l 2

Fig. 1. Cantilever retaining wall

This analysis includes 9 design variables. As shown in figure 1, five geometrical design variables are the width of the vertical wall stem at the top and bottom (w and B2), length of the toe and heel (B1 and B3), the depth of the toe and heel (d); four design variables of reinforcement consist of the required bar areas for the vertical wall stem and footing (A1, A2, A3), the length of the non-overall-length bar for the vertical wall stem. Notice that the bars for the vertical wall stem are non uniform distribution, two non-overall length ones plus an overall-length one with the equal spacing consisting of a unit.

2.2. Objective function

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of concrete and reinforcement per linear meter. To simplify the problem herein, the cost associated with labour, making, fixing and striping the framework, steel fixing and material loses is neglected. The total cost of the reinforcement consists of the cost of required to resist the tension and compression ultimate forces, and the secondary steel necessary to prevent cracking.

2.3. Constraints and Constraints handling

Generally, the design constraints consist of stability requirements and geometrical constraints. Geometrical constraints are easily acquired and do not discuss in this paper. The stability requirements include overturning check, sliding check, bearing capacity check and normal section check. Penalty technique is used for unfeasible solution. An adaptive function by Brarbus [9] is applied to constraints handing for GA and PSO. Another dynamic penalty function by Joines and Houck [10] is used for SA.

3. Example problem

Problem description: The wall height is 3 m (dw) with uniformly distributed surcharge load pk = 10 kN/m2_{and groundwater is ignored. The backfill is with an effective angle of internal friction, φ, of 34}o

and unit weight, γb, 17 kN/m3. The height difference between the backfill in front and back of the wall is 2.4 m. The permissible bearing pressure is 120 kN/m2_.

Before the application of any heuristic algorithms, the random direction search method, complex method (CM), which works well when design variables and constraints are relatively less, is adopted for the solution. All the algorithms were programmed in MATLAB and ran at a personal laptop with an Inter processor Quad Core I3. Figure 2 shows one solution of cost versus No. of trials for CM. The stop criterion is maximum No. of trials of 2×105_{. It is obvious that this solution is far from optimal solution.}

Then more than ten computing with the maximum No. of trials varying from 1×105_{to 1×10}7_{are tried, but}

similar solutions are obtained. Solutions are far from the optimal solution. It is testified that CM cannot be used for this complex problem.

Fig.2. Cost versus No. of trials for CM; (left) Fig.3. Cost versus No. of trials for GA (right)

The stop criteria of these algorithms are all maximum iteration steps, 1×104_{, 1×10}4_{, 3×10}5_{for GA,}

PSO, and SA, respectively, which are obtained by a number of trial-and-error attempts. The final parameters are given as follow: for GA, pop = 100, k1 = 0.5, k2 = 0.9, k3 = 0.03, k4 = 0.1; for PSO, pop =

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100, c1 = 1.4801, c2 = 1.4801, wmin = 0.5, wmax = 0.9; for SA, q = 0. 95, εf = 1e-10. As shown in Figures 2-4, three typical iteration processes of the three algorithms are given. Figure 5 shows the final solution versus No. of computing (local minima are not included). The results indicate that CM is easily entrapped in local minima and cannot convergence whilst the other three approaches are optimal solution approximation. However, SA applied in this paper is also easily to be entrapped in local minima due to the penalty technique used. Figure 6 illustrates the time consuming versus No. of computing. With respect to time consuming, PSO outperform the other two algorithms. For the problem discussed, PSO is still highly recommended and suggested to be used to deal with complex engineering problems due to its effectiveness, efficiency and simple theoretical framework.

Fig. 4. Cost versus No. of trials for PSO; (left) Fig. 5. Cost versus No. of trials for SA(right)

Fig. 6. Minimum costs versus No. of computing (left) Fig. 7. Time consuming versus No. of computing (right)

As Table 1 indicated, the cost of original design double the costs obtained by heuristic algorithms. Note V corresponds to the volume of steel required per linear meter. The origin design needs more steel volume whilst the heuristic algorithms require more length of the footing.

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Table 1. Results comparison of different methods Parameters

Method (m) w1 (m) B1 (m) B2 (m) B3 (m) d (m) l (mmV 3_/m) _(Yuan/m)Cost _---Fs F_---over _(m)e p_(kN/mmax/pmin2₎ Original

Design 0.200 0.250 0.40 1.60 0.250 2.00 6.972e6 1.53e3 1.570 3.280 0.375 107.90/22.67 GA 0.150 0.200 0.320 2.590 0.250 0.960 5.080e6 7.10e2 3.100 3.180 0.450 118.2/8.50 PSO 0.150 0.200 0.430 2.370 0.250 0.702 4.798e6 6.98e2 2.935 2.938 0.457 119.00/5.50

SA 0.150 0.200 0.420 2.380 0.250 0.66 4.737e6 6.95e2 2.944 2.938 0.457 119.00/5.54

4. Conclusion

An analysis of design optimization of reinforced cantilever retaining walls has been attempted and three heuristic algorithms and one random direction search method (CM) are adopted for the solution. Results indicate that

 CM is not suitable for the optimal solution of problems with too many design variables and complex constraints

 GA and PSO can successfully applied to the optimal solution of structural optimization problems with many design variables and complex constraints

 General penalty techniques are not suitable for SA to handle constraints and more effective approach is required

Although this investigation has obtained optimal solutions of the problem considered herein, further work should be done. Additional constrains include total and differential settlement, different distribution s of ground bearing pressures. The cost function should also be more robust as well.

References

[1] Alfonso Carbonell et al. Design of reinforced concrete road vaults by heuristic optimization. Advanced in Engineering

Software, 2011, 42:151-159

[2] G. L. Sivakumar Babu and Munwar Basha. Optimum design of cantilever retaining walls using target reliability approach.

International journal of geomechanics, 2008, 8 (4): 240-252

[3] B. Ceranic, C.Fryer, R.W. Banies. An application of simulated annealing to the optimum design of reinforced concrete retaining structures. Computers and Structures, 2001, 79: 1569-1581

[4] Huang Hansheng. Optimal design of Cantilever Retaining wall. Hubei Water Power, 2002, 46 :10-13

[5] Rajib Kumar Bhattachariya and Mysore G. Satish. Optimal design of a stable trapezoidal channel section using hybrid optimization techniques .Journal of irrigation and drainage engineering, 2007, 133 (4): 323-329

[6] Y. M. Cheng, L. Li, S. C. Chi. Performance studies on six heuristic global optimization methods in the location of critical slip surface. Computers and Geotechnics, 2007, 34: 462-484

[7] Y. M. Cheng, Liang Li, Shi-chun Chi, W. B. Wei. Particle swarm optimization algorithm for the location of the critical non-circular failure surface in two-dimensional slope stability analysis. Computers and Geotechnics, 2007, 34: 92-103

[8] Paul McCombie, Philip Wilkinson. The use of the simple genetic algorithm in finding the critical factor of safety in slope stability analysis. Computers and Geotechincs, 2002, 29: 699-714

[9] Helio J. C. Barbosa, Afonso C. C. Lemonge. A new adaptive penalty scheme for genetic algorithms. Information Sciences, 2003, 156: 215-251

[10] Joines. J., Hourck, C..On the use of non-stationary penalty functions to solve non-linear constrained optimization problems with GAs. Proceedings of the First IEEE International Conference on Evolutionary Computation, 1994, 579-584