OPTIMUM COST DESIGN OF R. C. ONE WAY SLABS

In this research , the formulation of optimum cost design for one way reinforced concrete slabs is presented , since it is useful and have a widespread usage among practicing engineering and applied to realistic structures subjected to the actual constraints of commonly used design codes such as the American concrete instituted code (ACI,2005). The formulation contains minimizing an objective function that represent the cost of steel reinforcement and the cost of concrete , which is subjected to many constraints containing : flexural constraints , serviceability constraints and deflection constraints that illustrated with details in this paper . the optimum solution is calculated using the lagrangian multipliers method , and a visual basic computer software were developed to find the optimum solution .


Introduction
The goal of optimization is to find the values of the variables in the process that yield the best value of the performance criterion . A trade -off usually exist between capital and operating costs . The described factors -process or model of the performance criterionconstitute the optimization problem .
The standard form of a mathematical programming problem is Find n X X X X . . 2 1 Which minimize Z = f ( x ) Subject to g i ( x ) < 0 , where i = 1 , 2 , . . . . . , n . n : no. of design variables .
In the above statement X is a vector of n design variables , Z = f ( x ) is the objective function and g i are design constraints .
Objective function is defined as a function of the design variables , the value of which provides the basis of choice between alternate acceptable designs . The objective may be minimization of weight , cost or stress concentration factor or it may be maximization of efficiency . In structural design the objective function is usually weight or cost minimization .
The constraints of a problem are the restriction which are to be satisfied to make a design acceptable , and in structural design may be classified into : 1-behavior constraints which are related to design variables implicitly and usually structural analysis is necessary to evaluate them like the limitation on stresses , displacement and stability requirements . 2-Side constraints ( Geometric constraints ) which termed as a specified limitations in explicit form on a design variable or a relationship among a group of variables like the codal provisions of minimum or maximum values on a design variable , in other words , they limit the range of acceptable designs in the problem [ 2 ] . Some of structural optimization researches deal with minimization of the weight of a structure , and others deal with minimizing the cost of structure which included many terms such as : the cost of concrete , the reinforcing steel , fiber , prestressing steel , form work , shear steal ….etc. This paper deals with minimizing the steel reinforcement cost and the concrete cost of atypical one way slab using the lagrangian multipliers method .
Olhoof [ 3 ] determined the thickness of a simply supported rectangular plate , in which the fundamental frequency of transverse vibration is an optimal value , Khot , et al. [ 4 ] presented a method based on the optimality criteria for designing minimum weight fiber reinforced structure with stress displacement constraints.
Minimization of weight of a stiffened conical and cylindrical shells was carried out by Rao and reddy [ 5 ] considering practical constraints including natural frequency , they minimized the weight by appropriately selecting the shell thickness and spacing of rings and stringers using the interior penalty function method .
Brown [ 6 ] presented an i terative method for minimum cost selecti on of the thickness of simply supported uniformly -loaded one way slab using only the flexural constraints of the ACI -Code ( " Building " 1971 ) , the cost function includes only the cost of concrete and the cost of reinforcing steel . The author reports cost saving of up to 17 % . Chou [ 7 ] uses the lagrange multiplayer method for minimum cost design of a simply reinforced T -Beam using the ACI Building Code -1971 , the writer defines only two design variables : effective depth and area of steel reinforcement . The cost function includes the cost of concrete and the cost of reinforcing steel , in the formulation , its assumed that the neutral axis is located inside the flange of the T -section , the author reports a cost reduction up to 14 % of the cost of the beam with maximum steel ratio . Gunaratnam and Sirakumaran ( 1978 ) [ 8 ] presented minimum cost design of reinforced concrete slab satisfying the limit states requirements of the British Code -1972 for members having uniform , triangular or parabolic moment distribution using a combination of the lagrange multiplier and graphical methods . Their cost function includes only the cost of concrete and the cost of reinf orcing steel . They present curve f or optimum design parameters as a function of the thickness of the slab . They point out the significant influence of the serviceability limit state of deflection on the optimum design parameters .

Objective Function
The objective function can be expressed as :- Where :-C t : The total material cost . C s : The steel reinforcement cost / unit volume . C c : The concrete cost / unit volume . V s : Volume of steel . V c : Volume of concrete . Equation ( 2 ) can be rewritten as :- Where :r : is the cost ratio of the cost of a unit volume of steel to a unit volume of concrete (C s / C c ). V s = 1 * 1 * As V c = 1 * 1 * h h : Total depth of the section . : Reinforcement ratio Where :- Since b = unit width ( 1 m ) equation ( 4 ) can be C t = C c * h *( * r + 1 ) (5)

Constraint Functions
The constraint functions of this problem can be explained as :- Where :x : is the vector of constraint design variables including flexural constraints , serviceability constraints and deflection constraints .
Where :-Mu : The ultimate design moment . Mn : The nominal bending moment . The ultimate design moment is calculated from the external loads as follow :- Where :k : moment coefficient . w : The factored uniformly distributed load . L : The effective span . The nominal bending moment is calculated as follow :- Where :-As : The steel reinforcement . fy : Yield stress of steel reinforcement . d : Concrete cover . The serviceability constraints are presented in terms of limits on the steel reinforcement ratio and the bar spacing .

Vol.18
No.6 December 2010 20 u = 0.003 . 1 : is equal to ( 0.85 ) for concrete strength up to 28 MPa , and for concrete strength greater than 28 MPa 1 shall be reduced at a rate of ( 0.05 ) for each 6.9 MPa but not less than ( 0.65 ) .
Constraint normalization [ 9 ] By using the lagrangian multiplication method to find the optimum solution for the cost of steel and concrete for the given variables ( d , ) , and using equations ( 5 ) and ( 11  ) , The Total objective cost function will be :- In which ( ) is the lagrangian multiplication constant that will be found during the solution with the other independent variables ( d , ) .
Solving equation ( 14 ) by derivation with respect to the independent variables ( h , , ) and equaling the resulting equations to ( 0 ) gives these three equations :-

Numerical Examples
Three examples were solved using these equations and a visual basic computer program ( As seen in appendix A ) developed for finding the results of these examples . First a one way slab with moment equal to 50 kN.m/m , fc -= 20 MPa , fy = 276 MPa and r = 75 , were solved to find the optimum depth and the optimum reinforcement ration that gives the optimum cost for the constant example comparing to the costs that results from changing the depth of the slab or the reinforcement ratio .
As it seen from Table 1 the depth of ( 0.142 m ) and the reinforcing ration of ( 0.01096 ) gives the optimum cost solution even when the depth of the slab is less than the optimum depth that found and Fig. 2 shows the visible region [ 2 ] for this example for the    Fig. 6 and 7 , obviously from these figs. The minimum savings will be when using a minimum values of fcand fy while using a larger values for these variables during the design gives use a better chance to be closer to save more cost .

Conclusion
Optimum cost have been found using the lagrangian multiplication method for the reinforcing steel ratio and the effective depth for a Reinforced concrete one way slabs , and the following conclusions have been found :-