Design Optimization of Reinforced Ordinary and High-Strength Concrete Beams with Eurocode2 (EC-2)

1. Introduction

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 accordance with EC-2 [13], the assumptions used at the limit state for the typical reinforced T-beam-cross section are, respectively, illustrated in Figure 1(a)–(c).

In the linear strain diagram of Figure 1b, the symbols ε_s and ε_cu3 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, f_cd is the design value of concrete compressive strength, γ_c is the partial safety factor for concrete and f_ck 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 f_ck in accordance with the following Equations [13]:

with 50 ≤ f_ck ≤ 90 MPa and λ = 0.8,η = 1.0 for f_ck ≤ 50 MPa.

F_c and F_s denote the resultants of internal forces in the HSC section and reinforcing steel, respectively.

The design yield strength of steel reinforcement is f_yd = f_yk/γ_s where f_yk is the characteristic elastic limit of steel and γ_s 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, ε_yd = f_yd/E_s where E_s represents the elasticity modulus for steel.

3. Formulation of the optimization problem

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; M_Ed = 1.35 M_G + 1.5 M_Q = 9 MNm; V_Ed = 1.35 V_G + 1.5 V_Q = 3.1 MN.

w = 0.60MN/ml (the total distribution load (dead load + live load)), δ_lim = L/250 = 0.100 m.

Input data for HSC characteristics:

C70/85; f_ck = 70 MPa; γ_c = 1.5; f_cd = 46.67 MPa; ρ = 0.025 MN/m³; E_cm = 40,743 MPa;

λ = 0.75; η = 0.90; ε_cu3(‰) = 2.7; ε_c3(‰) = 2.4; h_fmin = 0.10 m; f_ctm = 4.6 MPa;

μ_limit = 0.329; α_limit = 0.554 for S500 and C70/85.

Input data for steel characteristics:

S500; f_yk = 500 MPa; γ_s = 1.15; f_yd = f_yk/γ_s = 435 MPa; n = 15;

S400; f_yk = 400 MPa; γ_s = 1.15; f_yd = f_yk/γ_s = 348 MPa; f_yd/f_cd = 9.32 for classes (S500, C70/85);

f_yd/f_cd = 7.46 for classes (S400, C70/85); μ_limit = 0.352; α_limit = 0.6081 for S400 and C70/85;

E_s = 2 × 10⁵ MPa; p_min = 0.26 f_ctm/f_yk = 0.002392; p_max = 4%.

Input data for units cost ratios of construction materials:

C_s/C_c = 40 for HSC concrete;

C_f/C_c = 0.01 for wood formwork;

C_f/C_c = 0.10 for metal formwork;

C_f/C_c = 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
b(m)	1.20	0.86	0.52
bw(m)	0.40	0.28	0.28
h(m)	1.40	1.58	1.56
d(m)	1.26	1.42	1.40
hf(m)	0.15	0.11	0.10
AS(m2)	185x10−4	161x10−4	181 x10−4
α	0.554	0.342	0.554
Gain		22%	47%

Table 2.

Comparison of the optimal solutions with minimum weight and minimum cost design for HSC.

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 = h_imposed; (ii) the effective width of compressive flange is imposed, b = b_imposed; (iii) the reinforcing steel is imposed, A_s = A_simposed; and (4i) the flange depth is imposed, h_f = h_fimposed.

The gain depends on the type of formwork used. We distinguish the wood formwork: C_f/C_c = 0.01 and the steel formwork C_f/C_c = 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

Table 3.

The variation of relative gain with particular conditions imposed such as the HSC T-beam dimensions and reinforcing steel.

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: C_s/C_c = 10; 20; 30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio C_f/C_c = 0.01

The corresponding results are reported in Table 4 and represented in Figure 2.

(S500; C70/85) Cf/Cc = 0.01	Gain (%)
10	33
20	27
30	24
40	22
50	22
60	22
70	23
80	24
90	26
100	27

Table 4.

Variation of relative gain in percentage (%) versus unit cost ratio C_s/C_c for a given cost ratio C_f/C_c = 0.01.

It is shown in Table 4 and Figure 2 that the relative gain decreases for increasing values of the unit cost ratio C_s/C_c, stabilizes around an average value for 40 ≤ C_s/C_c ≤ 60 and then increases significantly beyond this average value for a given cost ratio C_f/C_c = 0.01.

The relative gains can be determined for various values of unit cost ratios: C_f/C_c = 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 C_s/C_c = 40.

The corresponding results are reported in Table 5 and presented in Figure 3.

(S500; C70/85) Cs/Cc = 40 Cf/Cc	Gain (%)
0.01	22
0.02	21
0.03	21
0.04	20
0.05	20
0.06	19
0.07	19
0.08	19
0.09	19
0.10	18

Table 5.

Variation of relative gain in percentage (%) versus unit cost ratio C_f/C_c for a given cost ratio C_s/C_c = 40.

From Table 5 and Figure 3, the gain decreases monotonically with the increase of unit cost ratio C_f/C_c for a given cost ratio C_s/C_c = 40.

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; M_Ed = 5MNm; V_Ed = 1.1MN; w = 0.043MN/ml; δ_lim = L/250 = 0.080 m.

Input data for ordinary concrete characteristics:

C20/25; f_ck = 20 MPa; γ_c = 1.5;f_cd = 11.33 MPa; ρ = 0.025MN/m³; E_cm = 30,000 MPa;

λ = 0.80; η =1.00; ε_cu3(‰) = 2; ε_c3(‰) = 3.5; h_fmin = 0.15 m; f_ctm = 2.20 MPa; n = 15;

μ_limit = 0.372; α_limit = 0.6167 for S500 and C20/25.

μ_limit = 0.392; α_limit = 0.6680 for S400; and C20/25.

Input data for steel characteristics:

S400; f_yk = 400 MPa; γ_s = 1.15; f_yd = f_yk/γ_s = 348 MPa;

E_s = 2 × 10⁵ MPa; p_min = 0.26 f_ctm/f_yk = 0.00143; p_max = 4%;

f_yd/f_cd = 30.71 for classes (S400, C20/25);

f_yd/f_cd = 38.39 for classes (S500, C20/25).

Input data for units cost ratios of construction materials:

C_s/C_c = 30 for ordinary concrete.

C_f/C_c = 0.10 for metal formwork.

C_f/C_c = 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
b(m)	1.20	1.30	1.25
bw(m)	0.40	0.28	0.29
h(m)	1.60	1.57	1.60
d(m)	1.44	1.41	1.44
hf(m)	0.14	0.17	0.16
AS(m2)	125 × 10−4	123 × 10−4	122 × 10−4
α	0.668	0.668	0.668
C	1.171		1.0281
Gain		23%	14%

Table 6.

Comparison of the optimal solutions with minimum weight and minimum cost design.

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 A_s = 120 × 10⁻⁴ m²; (iii) imposed web width b_W = 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

Table 7.

Variation of relative gain with particular conditions imposed such as the T-beam dimensions, reinforcing steel and weight.

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: C_s/C_c = 10; 20; 30; 40; 50; 60; 70; 80; 90; 100 for a given unit cost ratio C_f/C_c = 0.01

The corresponding results are reported in Table 8 and presented graphically in Figure 4.

(S400; C20/25) Cf/Cc = 0.01	Gain (%)
10	18
20	16
30	14
40	13
50	12
60	12
70	12
80	11
90	11
100	11

Table 8.

Variation of relative gain in percentage (%) versus unit cost ratio C_s/C_c for a given cost ratio C_f/C_c = 0.01.

It is shown in Table 8 and Figure 4 that the relative gain decreases for increasing values of the unit cost ratio C_s/C_c for a given value of C_f/C_c = 0.01.

The relative gains can be determined for various values of the unit cost ratios: C_f/C_c = 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 C_s/C_c = 30.

The corresponding results are reported in Table 9 and illustrated graphically in Figure 5.

(S400; C20/25) Cs/Cc = 30 Cf/Cc	Gain (%)
0.01	14
0.02	14
0.03	13
0.04	13
0.05	13
0.06	12
0.07	12
0.08	12
0.09	12
0.1	12

Table 9.

Variation of relative gain in percentage (%) versus unit cost ratio C_f/C_c for C_s/C_c = 30.

From Table 9 and Figure 5, the gain decreases monotonically with the increase of unit cost ratio C_f/C_c for a given value of C_s/C_c = 30.

5. Conclusions

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.

Appendix