To Study the Performance of SLP and Pattern Search using MATLAB

GRD Journals- Global Research and Development Journal for Engineering | Volume 2 | Issue 11 | October 2017

To Study the Performance of SLP and Pattern Search using MATLAB Roopa A. K Assistant Professor Department of School of Environmental & Civil Engineering K.L.E Technological University

Dr. S. S. Bhavikatti Emeritus Fellow Department of School of Environmental & Civil Engineering K.L.E Technological University

Abstract The responsibility of structural engineer is to produce safe and economical designs. The primary responsibility of producing safe design is now achieved easily because of Codes of practice have standardizes the design procedure software are available to carry out analysis and design without committing any numerical mistakes. Optimization algorithms increasing popular in engineering design activities, because a designer or decision maker can derive the best way for maximum benefits from available sources. In many software optimization techniques have been developed such as MATLAB,NAG,MOSEK, NMATH etc In this paper, study had been carried out for performance of sequential linear programming and pattern search of optimization technique using matlab and extend to the structural optimization of one-way slab.

Keywords- Optimization, Sequential Linear Programming, Pattern Search, One Way Slab

I. INTRODUCTION Since 1960s, various formulations for optimization of problems in many fields, such as structural, chemical, industrial and mechanical engineering, economics, optimal control and others have been developed. Optimization is used to obtain the best solution amongst all the available solutions taking into account all the necessary aspects of a system or a problem. The optimization involves choosing of the design variables in such a way that the objective of the design is achieved, subject to the satisfaction of constraints. The structural optimization problem is always one of the minimizing or maximizing a certain specific characteristic of structural system like cost, weight, performance capability of the system depends on the problem.

In this paper optimization for one way slab with simply supported end condition is carried out in order to reduce cost per square meter. Parametric study is also done for different span ratio. The procedure carried out for analysis and design as per IS:4562000[6]. Sequential linear programming and pattern search method is used for optimization. Results obtained for both optimization method is compared.

II. LINEAR PROGRAMMING The LP consists linearising the constraints and the objective function in the neighborhood of a design vector and solving the resulting linear programming problem to get a new design vector. The linearization and solution of linear programming problem is continued in a sequence till optimum is reached.

Sequential Linear Programming was originally proposed by Griffith and Stewart. The method uses linear programming as a search technique. A starting point is selected, and the nonlinear model and constraints are linearized about this point to obtain a linear problem which can be solved by the Simplex Method. The point from the linear programming solution can be used as a new point to linearize the nonlinear problem [12].

III. POWEL’S METHOD Pattern search (PS) is a family of numerical optimization methods that do not require the gradient of the problem to be optimized.

Hence PS can be used on functions that are not continuous or differentiable. The pattern search finds for the minimum along the direction s(i) = x(i )-x(i-n), where x(i) is the point obtained at the end of ‘n’ univariate steps and x(i-n) is the starting point before taking the n univariate steps and s(i) is the pattern direction. In the pattern search method, two points are created for pattern movement. A set of search direction is taken iteratively to work with pattern search. Starting from one point the next point can be found by walking along the search direction. Exploratory move consists in moving from one temporary point in one direction to another temporary point in next direction, till exploration reached in all n directions are made. For exploration a small fixed step length is taken to decide whether to move in positive direction or negative direction or no move in the coordinate direction selected after exploring all n direction the temporary point obtained be Y 1n.

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To Study the Performance of SLP and Pattern Search using MATLAB (GRDJE/ Volume 2 / Issue 11 / 005)

A. Powel’s Method In this method, starting from a point X1 minimization is made in the coordinate direction 1 and then in the coordinate direction 2.

After completing search in all n coordinate direction the point X 2 obtained. From point X2 onwards the search is in the coordinate direction 2, 3, …..n and pattern S1where S1 = X2 – X1. After reaching the design point X3, One more search direction is given up in favour of pattern direction S2 = X3 – X2. The iterative procedure of search is continued till all coordinate are replaced by pattern search S1 to Sn. Even at this stage, if optimum is not reached, whole procedure is restarted by searching in all coordinate directions .

Fig.1 shows typical search procedure by Powel’s method in a two-dimensional problem. After finding optimum step lengths in coordinate directions S1 optimum step length in this direction gives X2. From this point search is in the direction coordinate 2 and s1. This search locates point X3. Then S2 = X3 – X2. From this point search is in S2 and S1 to get point X4. Now both pattern directions have been explored. Hence from X4 the search is in the coordination direction 1 and 2 to get new pattern search directions.

Fig. 1: Powel’s method

IV. METHODOLOGY FOR OPTIMIZATION OF ONE WAY SLAB A. Objective Function The procedure in MATLAB involves Write an M-file for the objective function and constraints as per IS:456-2000. Optimization is done using the MATLAB function fmincon and pattern search. Result is compared for both techniques. The objective function of present work is to minimize the material cost per unit slab area while satisfying the constraints. It must be the function of design variables. The objective function can be written as, f(x) = Vc+ R*Vs Where, f(x) = Objective function, which is to be minimized Vc = Volume of concrete in m3 Vs = Volume of steel in cum R = Cost ratio, i.e ratio of cost of 1 cum of steel to cost of 1 cum of concrete B. Design Variables An important first step in the formulation of an optimization problem is to identify the design variables. In this problem the following design variables are considered: X1= Effective depth of slab in mm (d) X2 = Area of steel for tension reinforcement in mm2 (Ast) C. Constraints In many practical problems, the design variables cannot be chosen arbitrarily; rather, they have to satisfy certain specified functional and other requirements which collectively called design constraints. The design constraints which are considered in this optimization model are listed below.

1) 2) 3)

Moment in the slab -1< 0 Limiting moment Minimum depth of the slab -1< 0 Depth provided Minimum Ast requried for the slab Ast provided

1< 0

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To Study the Performance of SLP and Pattern Search using MATLAB (GRDJE/ Volume 2 / Issue 11 / 005)

4) 5) 6)

Ast provided -1< 0 Maximum Ast required Total deflection in slab -1< 0 Permissible deflection Shear stress in the slab -1< 0 Shear resisted by the slab

Table 1: Optimum results for different initial points for panel 8 x 4m load = 5kN/m2 and M20 & Fe415 Linear Programming Pattern Search Trail no Starting point in mm Optimum d, Ast in mm Iteration Optimum d, Ast in mm Iteration 1 300, 384 98.733, 940.745 9 98.733 , 940.827 8 2 150, 204 98.733, 940.745 14 98.733, 959.03 8 3 100, 144 98.733, 940.745 22 98.733, 954.92 8 4 50, 84 98.733, 940.744 9 98.733, 957.657 8 Table 2: Optimum depth for different span ration with live load = 5kN/m2 and M20 & Fe415 Linear Programming Pattern search Panel size Ly/lx d in mm Ast in mm2 Total cost per m2 d in mm Ast in mm2 Total cost per m2 8x4 2 98.73 940.74 532.53 98.97 957.65 536.67 8x3.6 2.2 87.45 833.25 479.89 87.45 844.01 485.87 8x3.3 2.4 79.20 754.69 441.37 79.20 762.06 442.15 8x3 2.6 71.14 677.88 403.73 73.14 682.71 410.28 8x 2.8 2.8 65.87 627.636 379.08 66.87 631.16 391.52 8x2.6 3 60.67 578.141 354.14 62.67 380.64 380.18 Table 3: Influence of cost ratio for panel 8 x 4m load = 5kN/m2 and M20 & Fe415 Cost ratio d in mm Ast in mm2 Cost per m2 60 98.733 940.745 511.44 70 98.733 940.745 522.852 80 98.733 940.745 534.23 90 99.733 940.745 545.622 100 99.733 940.745 557.075

V. RESULTS Before starting the actual optimization we need to check the optimizer point by taking different initial points. From Table-1 we can see that after taking different initial points, optimization went smoothly for LP and pattern search method till optimum point reached. Even though in trial 4 starting pointing is taken in infeasible region which violated the constraints, still it gives optimum point in feasible region. Both optimization methods continued and it went smoothly and selects the global optima. From the Table2 observation can be, LP works smoothly for all condition and patter search fails in giving corresponding Ast for required depth.

Hence for structural optimization pattern search fails in selecting global optimum. The depth obtained from the optimization is depth required for balanced section. But if we impose the side constraints 100 mm for minimum depth of slab as per IS456: 2000, then for all cases depth will be 100 and corresponding Ast will be 926.19 mm2 and the section will be under reinforced. Tabel-3 and Fig-1 we can see that increasing in cost ratio will increase the objective function but there is no increase in the depth. Hence for different cost ratio optimum depth will be a depth required for balanced section.

VI. CONCLUSIONS The objective of this study is to compare the optimization methods for minimum cost of One way slab. Parametric study with respect to different panel size and cost ratio has been carried out. The results of optimum design for both methods have been compared and conclusions drawn are as follows.

1) It is possible to obtain the global optimum point for both methods.

2) Depth obtained from optimization from is depth required for balanced section. Therefore we can say that for optimum depth is nothing but depth satisfying balanced section.

3) The optimum cost design of slab is bounded by number of constraints as it contains the design variables so problem is known as fully constrained design.

4) Cost ratio has impact on depth of slab. As rate of steel increases, cost ratio goes on increases so resulting section will be under reinforced section.

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