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The effects on the strength of reinforced concrete (RC) beams with fiber-reinforced polymer were investigated experimentally and numerically in a precracked condition [10], showing an improvement in the ultimate load capacity of the member. Cosenza [11] developed a finite element analysis of reinforced concrete elements in the cracked state that provides an accurate methodology for evaluating the stiffness matrix and load vector. The significance of the moment of inertia in the cracked cross-section in the analysis of structural elements for service load was demonstrated in [12] by experimental testing of RC beams with glass fiber-reinforced polymer and steel-reinforced beams with different ratios of gross to cracked cross-section moment of inertia.

The control of deformation of reinforced concrete elements is of utmost importance due to the current requirements regarding excessive cracking and deformation [13,14]. In addition, there has been an increased use of elements with great height and slenderness, which requires a more rigorous deformation control. Structural codes used in Portugal for the control of deformations in reinforced concrete structures, namely Regulamento de Estruturas de Betão Armado e Pré-Esforçado (REBAP [15]) and Eurocode 2 (EC2 [16]), provide two types of limit states: ultimate limit state and service limit state. Although failure to meet a design criterion for deformation does not compromise the safety of structures at the failure level, it is necessary to ensure excellent behavior of structures under service loads. Furthermore, good behavior avoids inconvenience to users due to the poor esthetics of structures associated with excessive cracking, which is the main objective of the Portuguese structural code.

Equations (6) and (7) present the constitutive model for concrete used in this work, which is taken from EC2. Figure 5a shows the momentum-curvature curve for reinforced concrete, where point 1 denotes the occurrence of cracks. Point 2 is the point at which the reinforcement reaches the yield stress, and point 3 is the failure of the structure, with straight line I representing the uncracked state and straight line II representing the cracked state.

(a) Momentum-curvature curve of reinforced concrete, adapted from [21], (b) Illustrative graphic of creep, adapted from [22]. (1) non-crack mode, (2) transition mode, (3) crack mode.

Cracking in a particular cross-section is said to occur when the most stressed fiber reaches the minimum strength and shows up as a crack in the concrete. Therefore, the tensile stress passes into the steel and causes a sudden rupture of its structural stiffness [21], as can be seen in Figure 5a. Figure 6a presents a simply supported beam with distributed load. MI is considered to be below the cracking moment (Mcr), and MII is superior to this momentum. Figure 6b,c also show the stresses of concrete and steel for moments I and II (where state I is uncracked and state II is cracked). Therefore, the beam stiffness is variable and depends on whether it is uncracked or cracked. To calculate the deformations of a simply supported beam, it is necessary to make an integral by the Principle of Virtual Works (PVW), and considering all these limitations, an analytical calculation is difficult. Then, it is necessary to use computational methods.

The cracking moment is defined in Equation (10), where fctm is the average value of the tensile strength of the concrete, Ic is the inertia of the concrete, and z is the distance between the neutral axis and the most stressed tensile fiber. 1e1e36bf2d