Definition of design variables.
This chapter presents a method for minimizing separately the cost and weight of reinforced ordinary and high-strength concrete (HSC) T-beams at the limit state according to Eurocode2 (EC-2). The first objective function includes the costs of concrete, steel, and formwork, and the second objective function deals with the weight of the T-beam. All the constraints functions are set to meet the design requirements of Eurocode2 and current practices rules. The optimization process is developed through the use of the generalized reduced gradient (GRG) algorithm. Two example problems are considered in order to illustrate the applicability of the proposed design model and solution methodology. It is concluded that this approach is economically more effective compared to conventional design methods used by designers and engineers and can be extended to deal with other sections without major alterations.
- cost and weight minimization
- reinforced ordinary and high-strength concrete beams
- Eurocode2 (EC-2)
- nonlinear optimization
Structural elements with T-shaped sections are frequently used in industrial construction. They are used for repeated and large structures because they are cost effective when using the optimum cost design model which is of great value for designers and engineers. Compression reinforcement is not often required when designing the T-beams sections. One of the great advantages of T-beams sections is the economy in the amount of steel needed for reinforcement. The objective function is usually simplified to represent the weight, disregarding the costs of shaping and the construction details. However, the economy aspects in terms of costs and gain achieved should be the area where scope exists for extending the research works [1, 2, 3, 4].
Recent developments in the technology of materials have led to the use of the high-strength concrete (HSC); this is mainly due to its efficiency and economy. The reduction in the quantities of construction materials has enabled both a gain in weight reduction and in the foundation’s cost. HSC has a high compressive strength in the range of 55–90 MPa; it not only has the advantage of reducing member size and story height, but also the volume of concrete and the area of formwork. In terms of the amount of steel reinforcement, there is a substantial difference between the normal-strength concrete structures compared to high-strength concrete structures [5, 6]. In this chapter, not only does it presents the minimum weight design but it presents a detailed objective function that considers the ratio cost not the absolute cost with sensitivity analysis of this cost ratio as well. It considers both shaping and material costs. The generalized reduced gradient (GRG) method is used to solve nonlinear programming problems. It is a very reliable and robust algorithm; also, various numerical methods have been used in engineering optimization [7, 8, 9, 10, 11, 12].
This work shows a method for minimizing separately the cost and weight of reinforced ordinary and high-strength concrete (HSC) T-beams at the limit state according to Eurocode2 (EC-2). The first objective function includes the costs of concrete, steel and formwork, whereas the second objective function represents the weight of the T-beam; all the constraints functions are set to meet the ultimate strength and serviceability requirements of Eurocode2 and current practices rules. The optimization process is developed through the use of the generalized reduced gradient algorithm. Two example problems are considered in order to illustrate the applicability of the proposed design model and solution methodology. It is concluded that this approach is economically more effective compared to conventional design methods applied by designers and engineers and can be extended to deal with other sections without major alterations.
2. Limit state design of reinforced concrete T-section under bending
In the linear strain diagram of Figure 1b, the symbols εs and εcu3 designate steel strain and the ultimate strain for the rectangular stress distribution compressive concrete design stress–strain relation. The parameter α represents the relative depth of the compressive concrete zone and the plastic neutral axis is located at the distance αd from the upper fiber for the ultimate limit state design, and x is the depth of elastic neutral axis for serviceability limit state design. In the assumed uniformly distributed stress diagram of Figure 1c, fcd is the design value of concrete compressive strength, γc is the partial safety factor for concrete and fck is the characteristic compressive cylinder strength of ordinary or HSC at 28 days. In accordance with EC-2, the possibility of working with rectangular stress distribution is offered. This requires the introduction of a factor λ for the depth of the compression zone and a factor η for the design strength. The λ and η factors are both linearly dependent on the characteristic strength fck in accordance with the following Equations :
with 50 ≤ fck ≤ 90 MPa and λ = 0.8,η = 1.0 for fck ≤ 50 MPa.
Fc and Fs denote the resultants of internal forces in the HSC section and reinforcing steel, respectively.
The design yield strength of steel reinforcement is fyd = fyk/γs where fyk is the characteristic elastic limit of steel and γs is the partial safety factor. In addition, the steel strain is considered unlimited in accordance with the Eurocode2 provisions. In this chapter, for an optimal use of steel, the strain must always be greater or equal to elastic limit strain, εyd = fyd/Es where Es represents the elasticity modulus for steel.
3. Formulation of the optimization problem
3.1. Design variables
The design variables selected for the optimization are presented in Table 1.
|Design variables||Defined variables|
|b||Effective width of compressive flange|
|As||Area of tension reinforcement|
|α||Relative depth of compressive concrete zone|
3.2. Objective functions
3.2.1. Cost function
The objective function to be minimized in the optimization problems is the total cost of construction material per unit length of the beam. This function can be defined as:
Thus, the cost function to be minimized can be written as follows:
3.2.2. Weight function
The weight function to be minimized can be written as follows:
ρ is the density of the reinforced concrete T-beams and W is the unit weight per unit length of the reinforced concrete T- beams.
3.3. Design constraints
(external moment ≤ resisting moment of the cross-section)
(internal force equilibrium)
(minimum steel percentage)
(maximum steel percentage)
Conditions on strain compatibility in steel:
(In the case of Pivot B, optimal use of steel requires that strains in steel must be limited to plastic region at the ultimate limit state (ULS).)
(Compression reinforcement is not required.)
(external shear force ≤ resisting shear force)
μlimit is the limit value of the reduced moment.
θ is the angle between concrete compression struts and the main chord
ν1 is a nondimensional coefficient, ν1 = 0.60(1-fck/250);
z is the lever arm, z = 0.9d;
hfmin is the minimum depth of flange.
3.4. Optimization based on minimum cost design
The optimum cost design of reinforced concrete T-beams under the limit state can be stated as follows:
For given material properties, loading data and constant parameters, find the design variables defined in Table 1 that minimize the cost function defined in Eq. (4) subjected to the design constraints given in Eq. (6) through Eq. (23).
3.5. Optimization based on minimum weight design
3.6. Solution methodology: Generalized reduced gradient method
The objective function Eq. (4), the objective function Eq. (5) and the constraints equations, Eq.(6) through Eq.(23), together form a nonlinear optimization problem. The reasons for the nonlinearity of this optimization problem are essentially due to the expressions of the cross-sectional area, bending moment capacity and other constraints equations. Both the objective function and the constraint functions are nonlinear in terms of the design variables. In order to solve this nonlinear optimization problem, the generalized reduced gradient (GRG) algorithm is used. This algorithm was first developed in late 1960 by Jean Abadie  as an extension of the reduced gradient method and then since has been refined by several other researchers [17, 18]. GRG nonlinear should be selected if any of the equations involving decision variables or constraints is nonlinear.
Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on values and formulas of a spreadsheet model. Version 4.0 and later include LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The Microsoft Office Excel Solver tool uses several algorithms to find optimal solutions. The GRG nonlinear solving method for nonlinear optimization uses the Generalized Reduced Gradient code. The Simplex LP solving method for linear programming uses the Simplex and dual Simplex method. The Evolutionary solving method for non-smooth optimization uses a variety of genetic algorithm and local search methods. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formulae cells whose values are to be constrained (the constraints) and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions and approximates derivatives, using finite differences. The NLP solution engine for the Excel Solver is GRG.
The generalized reduced gradient method is applied as it has the following advantages: (i) the GRG method is widely recognized as an efficient method for solving a relatively wide class of nonlinear optimization problems; (ii) the program can handle up to 200 constraints, which is suitable for reinforced ordinary and HSC beam design optimization problems; and (iii) GRG transforms inequality constraints into equality constraints by introducing slack variables. Hence all the constraints are of equality form. A more detailed description of the GRG method can be found in .
4. Numerical results and discussion
4.1. Design example A for reinforced HSC T-beams
The numerical example A corresponds to a high-strength concrete T-beam belonging to a bridge deck, simply supported at its ends and predesigned in accordance with provisions of EC-2 design code.
The corresponding preassigned parameters are defined as follows:
L = 25 m; MEd = 1.35 MG + 1.5 MQ = 9 MNm; VEd = 1.35 VG + 1.5 VQ = 3.1 MN.
w = 0.60MN/ml (the total distribution load (dead load + live load)), δlim = L/250 = 0.100 m.
Input data for HSC characteristics:
C70/85; fck = 70 MPa; γc = 1.5; fcd = 46.67 MPa; ρ = 0.025 MN/m3; Ecm = 40,743 MPa;
λ = 0.75; η = 0.90; εcu3(‰) = 2.7; εc3(‰) = 2.4; hfmin = 0.10 m; fctm = 4.6 MPa;
μlimit = 0.329; αlimit = 0.554 for S500 and C70/85.
Input data for steel characteristics:
S500; fyk = 500 MPa; γs = 1.15; fyd = fyk/γs = 435 MPa; n = 15;
S400; fyk = 400 MPa; γs = 1.15; fyd = fyk/γs = 348 MPa; fyd/fcd = 9.32 for classes (S500, C70/85);
fyd/fcd = 7.46 for classes (S400, C70/85); μlimit = 0.352; αlimit = 0.6081 for S400 and C70/85;
Es = 2 × 105 MPa; pmin = 0.26 fctm/fyk = 0.002392; pmax = 4%.
Input data for units cost ratios of construction materials:
Cs/Cc = 40 for HSC concrete;
Cf/Cc = 0.01 for wood formwork;
Cf/Cc = 0.10 for metal formwork;
Cf/Cc = 0.00 in the case of the cost of the formwork is negligible.
4.1.1. Comparison between the minimum cost design and the minimum weight design of HSC T-beams
The vector of design variables including the geometric dimensions of the T-beam cross-section and the area of tension reinforcement as obtained from the standard design approach solution and the optimal cost design solution using the proposed approach is shown in Table 2.
|Design variables vector.||Initial design||Optimal solution with minimum cost|
(S500, C70/85), Cs/Cc = 40, Cf/Cc = 0.01 wood formwork
|Optimal solution with minimum weight|
The optimal solutions using the minimum cost design and the minimum weight design are shown in Table 2.
It is shown from Table 2 that the gain and optimum values for minimum cost design and for minimum weight design are different.
From the above results, it is clearly shown that significant cost saving of the order of 47% can be obtained using the proposed minimum weight design formulation and 22% through the use of minimum cost-design approach.
4.1.2. Parametric study
In this section, the optimal solution is obtained according to practical consideration: (i) the total depth is imposed, h = himposed; (ii) the effective width of compressive flange is imposed, b = bimposed; (iii) the reinforcing steel is imposed, As = Asimposed; and (4i) the flange depth is imposed, hf = hfimposed.
The gain depends on the type of formwork used. We distinguish the wood formwork: Cf/Cc = 0.01 and the steel formwork Cf/Cc = 0.10.
Further practical requirements can also be implemented, such as esthetic, architectural and limited authorized templates. The optimal solutions obtained using the particular conditions imposed are shown in Table 3.
|Optimal solution with.||Gain (%)|
|Classes(S500, C70/85); Cs/Cc = 40; Cf/Cc = 0.01wood formwork||22|
|Classes(S500, C70/85); Cs/Cc = 40; Cf/Cc = 0.10 steel formwork||19|
|Classes(S500,C70/85) and Cf/Cc = 0 the cost of the formwork is negligible||23|
|Classes(S400,C70/85); Cs/Cc = 40; Cf/Cc = 0.01wood formwork||08|
|Imposed height h = 1.70 m; S500 and C70/85||21|
|Imposed width b = 1.00 m; S500 and C70/85||22|
|Imposed reinforcement As ≤ 0.0150 m2; S500 and C70/85||22|
|Imposed flange depth hf = 0.10 m; S500 andC70/85||22|
From the above results, it is clearly seen that a significant cost saving between 08% and 23% can be obtained by using this parametric study.
4.1.3. Sensitivity analysis
The relative gains can be determined for various values of unit-cost ratios: Cs/Cc = 10; 20; 30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio Cf/Cc = 0.01
|(S500; C70/85) Cf/Cc = 0.01||Gain (%)|
It is shown in Table 4 and Figure 2 that the relative gain decreases for increasing values of the unit cost ratio Cs/Cc, stabilizes around an average value for 40 ≤ Cs/Cc ≤ 60 and then increases significantly beyond this average value for a given cost ratio Cf/Cc = 0.01.
The relative gains can be determined for various values of unit cost ratios: Cf/Cc = 0.01; 0.02; 0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.08; 0.09; 0.10 for a given unit cost ratio Cs/Cc = 40.
|(S500; C70/85) Cs/Cc = 40 Cf/Cc||Gain (%)|
4.2. Design example B for reinforced ordinary concrete T-beams
The numerical example B corresponds to a concrete T-beam belonging to a pedestrian deck, simply supported at its ends and predesigned in accordance with the provisions of EC-2 design code.
The preassigned parameters are defined as follows:
L = 20 m; MEd = 5MNm; VEd = 1.1MN; w = 0.043MN/ml; δlim = L/250 = 0.080 m.
Input data for ordinary concrete characteristics:
C20/25; fck = 20 MPa; γc = 1.5;fcd = 11.33 MPa; ρ = 0.025MN/m3; Ecm = 30,000 MPa;
λ = 0.80; η =1.00; εcu3(‰) = 2; εc3(‰) = 3.5; hfmin = 0.15 m; fctm = 2.20 MPa; n = 15;
μlimit = 0.372; αlimit = 0.6167 for S500 and C20/25.
μlimit = 0.392; αlimit = 0.6680 for S400; and C20/25.
Input data for steel characteristics:
S400; fyk = 400 MPa; γs = 1.15; fyd = fyk/γs = 348 MPa;
Es = 2 × 105 MPa; pmin = 0.26 fctm/fyk = 0.00143; pmax = 4%;
fyd/fcd = 30.71 for classes (S400, C20/25);
fyd/fcd = 38.39 for classes (S500, C20/25).
Input data for units cost ratios of construction materials:
Cs/Cc = 30 for ordinary concrete.
Cf/Cc = 0.10 for metal formwork.
Cf/Cc = 0.01 for wood formwork.
4.2.1. Comparison between the minimum cost design and the minimum weight design of ordinary concrete T-beams
The optimal solutions using the minimum weight design and the minimum cost design are shown in Table 6.
|Design variables vector||Initial design, C20/25 & S400||Optimal solution with minimum weight, C20/25 & S400||Optimal solution with minimum cost, C20/25 & S400|
|AS(m2)||125 × 10−4||123 × 10−4||122 × 10−4|
It is shown in Table 6 that the gain and the optimum values for minimum weight design and for minimum cost design are different.
From the above results, it is clearly shown that a significant cost saving of the order of 23% can be obtained using the proposed minimum weight design formulation and 14% through the use of the minimum cost design approach.
4.2.2. Parametric study
In this section, the optimal solution is obtained through the considerations: (i) one of the dimensions of HSC T-section is imposed, h = 1.50 m; (ii) the imposed reinforcing steel As = 120 × 10−4 m2; (iii) imposed web width bW = 0.30 m; and (iv) imposed relative depth of compressive concrete zone α = 0.6000
Further practical requirements can also be implemented, such as esthetic, architectural and limited authorized template.
The optimal solutions obtained using the particular conditions imposed are shown in Table 7.
|Optimal solution with||Gain (%)|
|fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01 wood formwork, C20/25 & S400||14|
|fyd/fcd = 38.39; Cs/Cc = 30; Cf/Cc = 0.01wood formwork, C20/25 & S500||09|
|fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.00; C20/25 & S400||15|
|Imposed web with bw = 0.30 m; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400||13|
|Imposed reinforcementAs ≤ 0.0120 m2; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400||14|
|Imposed height h = 1.50 m; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400||11|
|Imposed relative depthα = 0.600; fyd/fcd = 30.71; Cs/Cc = 30; Cf/Cc = 0.01; C20/25 & S400||14|
From the above results, it is clearly seen that a significant cost saving between 09 and 15% can be obtained by using this parametric study.
4.2.3. Sensitivity analysis
The relative gains can be determined for various values of the unit cost ratios: Cs/Cc = 10; 20; 30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio Cf/Cc = 0.01
|(S400; C20/25) Cf/Cc = 0.01||Gain (%)|
The relative gains can be determined for various values of the unit cost ratios: Cf/Cc = 0.01; 0.02; 0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.08; 0.09; 0.10 for a given unit cost ratio Cs/Cc = 30.
|(S400; C20/25) Cs/Cc = 30 Cf/Cc||Gain (%)|
The following important conclusions are drawn on the basis of this chapter:
The problem formulation of the optimal cost design of reinforced concrete T-beams can be cast into a nonlinear programming problem; the numerical solution is efficiently determined using the GRG method in a space of only a few variables representing the concrete cross-section dimensions.
The space of feasible design solutions and the optimal solutions can be obtained from a reduced number of independent design variables.
The optimal values of the design variables are only affected by the relative cost values of the objective function and not by the absolute cost values.
The optimal solutions are found to be insensitive to changes in the shear constraint. Shear constraint is not usually critical in the optimal design of reinforced concrete T-beams under bending and thus can be excluded from problem formulation.
The observations of optimal solution results reveal that the use of optimization based on the optimum cost design concept may lead to substantial savings in the amount of construction materials to be used in comparison to classical design solutions of reinforced concrete T-beams.
The objective function and the constraints considered in this chapter are illustrative in nature. This approach based on nonlinear mathematical programming can be easily extended to other sections commonly used in structural design. More sophisticated objectives and considerations can be readily accommodated by suitable modifications of the optimal cost design model.
In this chapter, we have included the additional cost of formwork which makes a significant contribution to the total costs. This integration is important for an economical approach to design and manufacture.
The suggested methodology for optimum cost design is effective and more economical compared to the classical methods. The results of the analysis show that the optimization process presented herein is effective and its application appears feasible.
The comparison of optimal solutions for minimum cost and minimum weight shows that the construction cost affects significantly the optimal sizes. Not only do we use the mass but the cost as objective function as well which contains the material and construction provision costs. The difference is caused by construction details costs.
|C20/25||Class of ordinary concrete|
|C70/85||Class of HSC|
|S400||Grade of steel|
|S500||Grade of steel|
|fck||Characteristic compressive cylinder strength of ordinary or HSC at 28 days|
|fctm||Tensile strength of concrete|
|fcd||Design value of concrete compressive strength|
|γc||Partial safety factor for concrete|
|η||Design strength factor|
|λ||Compressive zone depth factor|
|εc3||Strain at the maximum stress for the rectangular stress distribution compressive concrete|
|εcu3||Ultimate strain for the rectangular stress distribution compressive concrete design stress–strain relation|
|fyk||Characteristic elastic limit for steel reinforcement|
|γs||Partial safety factor for steel|
|fyd||Design yield strength of steel reinforcement|
|εyd||Elastic limit strain|
|Es||Young’s elastic modulus of steel|
|Ecm||Modulus of elasticity of concrete|
|pmin||Minimum steel percentage|
|pmax||Maximum steel percentage|
|αlimit||Limit value of relative depth of compressive concrete zone|
|μlimit||Limit value of reduced moment|
|w||The total distribution load (dead load+ live load)|
|VG||Maximum design shears under dead loads|
|VQ||Maximum design shears under live loads|
|VRd,max||Maximum resistant shear force|
|VEd||Ultimate shear force|
|MRd, max||Maximum resisting moment|
|MEd||Ultimate bending moment|
|MG||Maximum design moments under dead loads|
|MQ||Maximum design moments under live loads|
|Fs||Resultant tensile internal force for steel|
|Fc||Resultant compressive internal force for HSC|
|n||Ratio of the modulus of elasticity of steel to that of concrete|
|b||Effective width of compressive flange|
|ds||Effective cover of reinforcement|
|As||Area of reinforcing steel|
|hfmin||Minimum depth of flange|
|δw||The mid-span deflection of simply supported beam under distribution load w|
|θ||Angle between concrete compression struts and the main chord|
|ν1||A nondimensional coefficient; ν1 = 0.60(1-fck/250)|
|z||Lever arm, z = 0.9d|
|ρ||Density of the reinforced concrete T-beams|
|W||Unit weight per unit length of the reinforced concrete T- beams|
|C0/L||Total cost per unit length of T-beam|
|Cs||Unit cost of reinforcing steel|
|Cc||Unit cost of concrete|
|Cf||Unit cost of formwork|