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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1–4].

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 [7–12].

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.

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

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

Figure 1. (a) Typical T-beam cross section; (b) strains at ultimate limit state and (c) stresses at ultimate limit state.

<sup>λ</sup> <sup>¼</sup> <sup>0</sup>:<sup>8</sup> � fck � <sup>50</sup>

<sup>μ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � fck � <sup>50</sup>

Fc and Fs denote the resultants of internal forces in the HSC section and reinforcing steel,

The design yield strength of steel reinforcement is fyd = fyk/γ<sup>s</sup> where fyk is the characteristic elastic limit of steel and γ<sup>s</sup> 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

<sup>400</sup> (1)

http://dx.doi.org/10.5772/intechopen.78734

123

<sup>200</sup> (2)

accordance with the following Equations [13]:

represents the elasticity modulus for steel.

respectively.

3.1. Design variables

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

3. Formulation of the optimization problem

The design variables selected for the optimization are presented in Table 1.
