OPTIMAL DESIGN OF PRESTRESSED CONCRETE HOLLOW CORE SLAB PANELS

The research has prepared programmed mathematical techniques by Visual basic language for analysis, design and calculating the optimization of precast prestress hollow core slab panels. The research deals with the optimization by adopting the modified Hooke-Jeevs method which is considered as a very suitable method especially for problems have many constraints. The formalizing of objective function was discussed according to required purpose. The aim of the study is to discuss three parameters (optimum weight, optimum cost and optimum live load). It is found that the average void percentage ratio regarding the optimum weight is about (50%) whereby the section tends to be in a shape where the voids become less than thickness and width take into consideration that the section is subjected to all the constraints (voids percentage tends to be much more than the regular case), as well as, it is found that the average of void percentage ratio concerning with the optimum cost is about (41%). The research also adopted preparing designable tables which are informative and easy in use practically for different kind of hollow core slab sections, it is found from the prepared maximum live load tables that the deflection restricts the span length not less than (60%), furthermore that adding topping slab (5cm) thickness increase the span length about (16-20) % for thicknesses (15-22) cm.


Introduction
Hollow core slab is a precast prestress concrete member having continuous voids as shown in Figure 1. This structural member had been produced for more than sixty years ago, it is fast started for spreading where they are currently become widely in use for all over the world Figure  2, and as an example of this, approximately 18 million square feet of hollow core slabs are produced annually across Canada. in Different Countries of World (14) .
Hollow core slab is used in any type of building construction regardless the building size, height of building or the function of building. The present study adopt the optimum design of hollow core slab because there are so many variables of this structure and they have many solutions where the designers already search about the best solution (optimum solution).  It was concluded in previous studies that twenty cycles (loops) of expanding loading by Fourier series (Levy's method) (12) will be fair enough to be close to the exact solution as shown in Figure 4. Finally, it's normally in optimum design of hollow core slabs to use (PCI analysis) because the difference among the previous studies is not so high moreover that there is tendency in optimization to fix most of the parameters to see the behavior clearly.
Regarding the design of hollow core slab, it is general similar to the design of prestressed concrete members (6) .All the necessary design equations and requirements have been used as constraints of a problem where flexural strength design, shear design, deflection, stresses and the effect of topping slab have been taken into consideration. All of these equations will be clarified in optimization technique.

General
Most of the design problems and studies have several solutions and the main topic for any design or study is how to obtain the best solution (optimum solution). Finding the optimum solution by classical seeking (try and error) among the variables is considered to be acceptable when less variables are considered while this method becomes invalid when the number of variables is large. The development in computers leads to an increase in the number of optimization methods where there are very large numbers but everyone has a limitation for its use. There is no general method available for solving all optimization problems efficiently.

Method of Optimization
There are many methods of optimization according to the type of problem but in all of the types of optimization methods, the design variables are modified to minimize or maximize the objective function. In (1961) Hook and Jeeves suggested direct-search method for optimization for an objective function without constraints (3), In (1984) Bunday modified this method where the method became to be in use for an objective function with constraints (4) . Hooke-Jeeves pattern search method and its modifying is adopted in present work. It can be briefed below:-1-Suggest an initial value …..Checked with constraints 2-Make the first exploration …..Checked every step with the constraints 3-Make pattern move ….. Checked with constraints 4-Make the second exploration..... Checked every step with the constraints 5-Terminate the process when the step length has been reduced to a small value.

Formulation of the Problem
Hollow core slabs are one of the most structural forms that are widely in use in most of the world countries and any saving in that forms will reflect highly beneficial. As it is mentioned before that optimization is a body of mathematical results and numerical methods for finding maximization or minimization where that is according to the applications or the problem in the field. The present work will study three cases of optimization:-1-minimum weight of hollow core slab 2-minimum cost of hollow core slab 3-maximum allowable live load Construct the model-objective function and its constraints if it is available-is already depending on the purpose of the work. So the studying of minimum weight is considered informative if the field need for that kind of problem and so on for the minimum cost or maximum live load.

Minimum Weight of Hollow Core Slab
The first important thing for any optimization is creating the objective function which is already related to the specific study that is wanted in researching. Here the objective function is very simple and it's clarified below:-Weight = width * thickness * (1 -percentage voids ratio) * density The other important thing for any optimization is the constraints that are limiting or restricting the objective function. Here the design constraints are all the equations that deal with flexure strength, shear, deflections, stresses and crack control. These variables will be discussed one by one.

Flexural Constraints
Below, the equations that are taken into consideration during the optimization process. The moment capacity of section is compared with that applied from the load:-The notations in equations have been listed in notation list .
Concerning the flexural constraints, the moment capacity of section should be not less than the moment due to the applied load, taking into consideration the steel ratio (steel index) and ( ).

Shear Constraints
The shear capacity of section is the less of the results of the two equations ( ) and (  )   ) The shear that is applied from the load:-Shear capacity of section should be not less than the shear that is applied from the loads (6) . ACI code (9.5.4.1) permitted to use the moment of inertia of the gross concrete section for class (U) flexural member. For comparison with ACI code, when non-structural elements are attached to the slabs, the portion of deflection after erection will be equal to (Change in camber + final defection +initial live load) where change in camber equal to (final cambererection camber) (6) . Comparison with Table 9.5(b) of the ACI Code is used during the process of optimization.

Stresses Constraints
Stresses in concrete immediately after prestress transfer (before time-dependent prestress loss) at the end of support are:  The permissible compression stresses under the prestress plus total load shall not exceed (0.6 ). The prestress member shall be classified as class (U), class (T), class (C) based on (ftwhich is meaning the tensile strength concrete) as briefed below:-Class U: Class T: Class C: Hollow core slabs are normally designed to be uncracked section under full service load (class U) (6) so there is no need for requirements of crack control. Finally, the dependent design variables that are used in this problem is the (width, thickness of hollow core slab, diameter of void), the other variables are used as a given value. Briefing of minimum weight problem is plotted in Table 2.

Minimum Cost of Hollow Core Slab
The same procedure has been done but for calculating the minimum cost where the equations of constraints are similar to those mentioned previously. The objective function and constraints are briefed as mentioned in Table 3.  Table 3 Optimization-Minimum Cost.

Design variables
Aps H D N

Method of optimization
Modified Hooke-Jeeves

Maximum Allowable Live Load
According to the mathematical point of view, optimization is a maximum or minimum value where it may be obtained after a searching among some of overlapping phases (governing equations) that are related to the problem. Here, the maximum allowable live load is added to the optimization idea because the problem deals with the maximization. The optimized body can be expressed as below: Objective function: Max L.L = Min ( L.L "flexure" , L.L "shear" , L.L " deflection" , L.L " stresses" ) Where L.L = live load which was obtained by using the same equations that are mentioned previously in article (3.3.1). Constraints:-Maximum live load should be not less than zero and from the practical view, the live load under (0.5 kN/m 2 ) is considered non informative. Briefing of maximum live load problem is plotted in Table 4:- Table 4 Optimization-Maximum Live Load.

Design variables
Span length, reinforcement

Method of optimization
Classical searching

Application
As it is mentioned before and according to the field requirements, the optimum weight, cost and live load can be found as shown below:-For a given data:-  Just for a theoretical study, the tendency of optimization is decreasing along reducing the void diameter. When the diameter is taken as variable the optimum case can be shown in Table 5. Briefly, the diameter of voids and its number tend to be maximum as possible to get the minimum weight.
-Concerning with optimum cost, it can be obtained by finding (optimum area steel, optimum thickness, optimum void diameter, optimum number of voids) as shown in Figures(8,9,10 and 11). It is clear from the above figures that the optimum area of steel and thickness vary regularly among different cases of lengths and live loads while there is an overlapping among the optimum diameter and number of voids due to the direct relation between them. The above charts can be used for finding the optimum design variables to get the optimum cost where these charts are produced for the most common lengths and live loads in the field. -Regarding the maximum live load and according to Table 4, the maximum live load is clarified below. For a given data: The effect of topping slab will be taken into consideration where it will be plotted with the (charts without topping) to clarify the behavior and to make visible comparison. The properties of topping slab are as shown below:-Topping slab thickness = 5 cm , = 35 MPa,Other data: will be mentioned later on. * Note: Dashed curves related to the hollow core slabs without topping So the maximum allowable live load for any given data (In case of hollow core slabs, thickness = 15 cm, number of voids = 8 and diameter of void = 105mm) can be found by using the minimum of the curves that shown in Figures (12, 13, 14 and 15). Unique curve cannot be plotted because of the varying of the governing phases during changing the length and area of steel. Unique table is prepared where it covers all the phases (flexure, shear, deflection and stresses), so the maximum live load for different cases of length and prestressed reinforcement is clarified in Table 6 and 7. Different tables have been produced for different sections of hollow core slab to be available for any future studies or designs.

Conclusions
The following conclusions can be drawn: 1-Precast / prestressed concrete institute (PCI) uses the coefficients that are related to the beam analysis in the analysis of hollow core slabs while the present study found by using Levy's method for the analysis of isotropic plate :a-Twenty cycles in Fourier expansion is enough to be near the exact value. b-The average percentage ratio differences between Levy and (PCI) results are about (0.6%, 23%, 2.7%) for moment, shear and deflection respectively. Taking into consideration that it is normal to use (PCI) coefficients in optimum design of hollow core slab. 2-The average percentage voids ratio to get minimum weight is about (50%) where the minimum weight of hollow core panel is obtained by depending on the void diameter. The thickness of hollow core panel will be a little bit larger than the diameter (for just satisfying the practical and geometrical consideration which is equal to 2.75 cm in each face).From other side the width will be larger than the "diameter of void multiplying by the number of voids" where minimum distance between two voids is 2.75 cm. In addition to that, it is recommended to use width less than (1.2m) in spans less than (5 m) to get minimum weight. 3-It is found from optimum cost of hollow core slabs that:a-General charts can be used for finding the optimum design variables to get the optimum cost. b-The average percentage voids ratio is about (41%) where the diameter of void tend to be less than the thickness by a little bit distance c-In General, thickness, area of prestressed steel and diameter of void tend toward increasing along increasing the length and live load while the number of voids are decreased. 4-Modified Hooke-Jeevs method is considered very suitable method for the problems that have large number of constraints where it is very easy for programming and for connecting the constraints with the problem. From other side the method is not able to move along the constraint and converges on the first point on the constraint it locates as the solution so searching along the initial variable has to be done to avoid that problem. 5-Concern finding the maximum live load, three main points are recorded:a-Many tables for available productions have been prepared to be informative for any work or study. The tables have been covered all the requirements (flexure, shear, deflection, stresses). b-The governing equation for the last three rows for all the tables of max live load is the deflection, from other side the deflection is restricting the span length to be not less than (60%) for any table of any section of hollow core slab. c-Adding topping slab (5cm) increases the span lengths in the tables as it is briefed below:- f pe = Compressive stress in concrete at extreme fiber where external loads cause tension due to the effective prestress only f ps = Stress in prestressed reinforcement at nominal strength f pu = Specified tensile strength of prestressing steel fr = Modulus of rupture of concrete hf = Depth from the face of hollow core slab down to top level of voids Moment of inertia of hollow core slab l.l = Live load L = length of hollow core slab Nominal flexure strength Factored design moment Cracking moment Pi = Initial prestress force after jacking losses Effective prestress force after all losses Ratio of prestressing reinforcement S b = Elastic section modulus V ci = Nominal shear strength of concrete in a shear-flexure failure mode V cw = Nominal shear strength of concrete in a web shear failure mode V d = shear due to unfactored self weight Vp = Vertical component of effective prestress force yb = Distance from neutral axis to extreme bottom fiber yt = Used as either distance to top fiber or tension fiber from neutral axis w = Weight per meter length Wu= design load q = weight per meter length for a plate ACI strength reduction factor Factor for type of prestressing strand Factor depend on the concrete type Prestressing reinforcement index