General Mathematical Model for Analysing the Bending Behaviour of Rectangular Concrete Beams with Steel, Fibre-Reinforced Polymers (FRP) and Hybrid FRP–Steel Reinforcements

Abstract

The design guidelines available in building codes for steel and fibre reinforced polymer (FRP) reinforced concrete (RC) beams have been developed on the basis of empirical models. While these models are successfully used for practical purposes, they require continuous improvements with more experimental data. This paper aims to develop a general mathematical model derived from the intrinsic material properties of concrete and certain reinforcements to analyse the bending behaviour of reinforced concrete beams. The proposed model takes into account the effects of non-linearity and ductility on the real behaviour of concrete under compression as well as the concrete tension stiffening. The model focused on analysing the flexural behaviour of rectangular steel, FRP and hybrid FRP–steel RC beams, using the moment–curvature relationship. A general static equilibrium equation was developed and mathematically solved with precise methods to establish a moment–curvature relationship. The effective flexural stiffness (EFS) is therefore calculated by the slope of the moment–curvature diagram, and then the load–deflection response is immediately deduced according to the loading conditions. The present model results were compared with numerous test data reported by various researchers. The comparisons reveal a good accuracy for predicting the EFS and load–deflection response for either steel, FRP, and hybrid reinforced concrete beams, with an error average less than 10%.

1. Introduction

Since its invention, the combination of concrete and steel has demonstrated great functionality in all types of buildings. Based on more than a century of experience on construction sites and a large database of laboratory tests, the use of steel-reinforced concrete (RC) is currently subject to well-established building codes. Nevertheless, the corrosion of steel bars in cracked concrete greatly reduces stiffness and load capacity. This has a significant effect on the durability of structures and their maintenance and rehabilitation costs. Therefore, research has focused on finding other combinations of concrete with reinforcement materials that are more resistant to corrosion, such as FRP.

Due to their excellent performance, especially in corrosion resistance and a higher strength, FRP composites are good alternatives to conventional reinforcing steel [1,2,3,4]. In fact, FRP RC has been introduced to the construction industry through numerous projects in Europe, North America and Japan (ACI Committee 440, 1996). Moreover, several design guidelines for FRP RC structural members have been produced based on existing models initially developed for steel RC and adapted with empirical factors through experimental test results.

Previous experimental studies observed a significant loss in the flexural stiffness of FRP RC beams compared to those reinforced with steel. This variance in flexural stiffness was justified by the difference between the mechanical properties of FRP and steel [5]. Therefore, it was concluded that the deflection control may become as important as the flexural strength for the design of FRP-reinforced concrete structures. This is why researchers have made a multitude of revisions to FRP RC design models over the past three decades to accurately estimate their flexural stiffness.

Different approaches have been proposed in the literature for estimating the flexural stiffness of steel RC beams. Effective flexural stiffness is the most popular approach used by structural engineers to design flexural concrete members. This approach was originally introduced by Branson [6] for calculating the effective moment of inertia ${\mathrm{I}}_{\mathrm{e}}$ of steel RC member. The Branson equation frequently used in North America was adopted by many building codes for its good estimate of the deflection of steel RC beams. This equation has been re-evaluated by numerous studies for the design of FRP RC beams. Indeed, many studies have shown that the Branson equation overestimates the effective moment of inertia for FRP RC beams [7]. These studies also observed a good tendency between the results obtained by the Branson equation and experimental tests. Therefore, it was concluded that a proper calibration of this equation allows the same accuracy for FRP RC beams as that of steel RC beams. Numerous proposals were made to calibrate the Branson equation by introducing empirical factors [8,9]. These empirical factors were later revised by incorporating the elasticity modulus and the reinforcement ratio of FRP on the stiffness of RC beams, as proven in several investigations [10]. This revision is adopted by ISIS, 2001 [11] and ACI-440, 2003 [12].

Another equation derived from the tension–stiffening relationship was found in European code [13] that was proposed to accurately predict ${\mathrm{I}}_{\mathrm{e}}$ for FRP RC beams [14]. This equation is simplified [15] and revised to account for different types of loading and boundary conditions [16], which were later recommended by ACI440, 2015 [17]. Other experimental studies found that the European model is also inaccurate for estimating the tension–stiffening of FRP RC beams. To improve the precision of the model, a modified expression, including the effects of the elasticity modulus and the reinforcement ratio, was proposed [18].

Thus far, researchers care about the accuracy of the available methods for estimating the effective moment of inertia of RC FRP beams [19,20]. Recently, it was discovered that, in addition to the material and dimensional parameters identified above, the accuracy of estimating (EFS) is influenced by the compressive strength of concrete $\left({\mathrm{f}}_{\mathrm{c}}^{\prime }\right)$ and $\left(\mathrm{d}/\mathrm{h}\right)$ the ratio of the tensile reinforcement depth relative to the height of the beam cross-section. Therefore, another equation was proposed, based on the fit results of existing models and experimental tests in order to precisely predict the effective flexural stiffness (EFS) of FRP RC beams [21]. Furthermore, a new model for making analytical predictions of the bending capacity of concrete elements reinforced with FRP bars was proposed to improve the reliability of the analytical model of the ACI440, 2015 design code by introducing a new empirical coefficient incorporating material parameters [22].

All of these previous modifications mainly aimed to soften the response of the FRP RC beams by reducing the tension–stiffening effect in the Branson equation. This is integrated by introducing empirical correction factors to decrease the ratio of the un-cracked to fully cracked stiffness of the cross-section (${\mathrm{I}}_{\mathrm{g}}/{\mathrm{I}}_{\mathrm{cr}})$ to improve the accuracy of estimating ${\mathrm{I}}_{\mathrm{e}}$ for the FRP RC beams. The researchers’ awareness of material parameters leads to more experimental testing and continuous revisions of empirical factors. The empirical models developed thus far are of great practical importance for the design of RC beams with FRP and steel. However, these models cannot be generalized for all RC beam designs and require continuous adaptation by experimental tests for new cases.

The present study proposes a general mathematical model derived from the intrinsic material properties of concrete and reinforcement. The model aims to analyse the bending behaviour of rectangular concrete beams simply or doubly reinforced with steel, FRP or hybrid FRP–steel. The proposed model could be of great practical use in the building industry by opening up the potential for alternative materials to steel bars in constructions with concrete and avoid the time-consuming process of testing the reliability of models through prototyping and experimental testing. The model describes the flexure-dominated cross-section response of RC beams by plotting the moment–curvature diagram. This is obtained by combining the analytical expression of static equilibrium and the strain compatibility condition, accounting for the tension–stiffening effect in a cracked state of concrete under tension. Subsequently, the effective flexural stiffness variation and the load–deflection response are immediately predicted.

2. Proposed Mathematical Model

The development of the proposed model starts with a cross-section analysis. Subsequently, the calculation of the general static equilibrium equation and the moment balance for FRP, steel and hybrid FRP–steel RC beams are established following the fundamental principles of compatibility and equilibrium. The resolution of the static equilibrium equation is mathematically obtained with precise methods by analysing the deformation processes in the blocks of tensile and compressive stress under increasing curvature. An algorithm is therefore established to generate the moment–curvature relationship using the moment balance equation. Finally, the effective flexural stiffness is immediately deduced, and then the load–deflection response is calculated based on the initial loading conditions of the RC beams.

2.1. Basic Assumptions

The assumptions taken in the model are as follows:

• Plane sections before bending remain plane after bending (Bernoulli’s principle).
• Perfect bond between reinforcing and concrete.
• The beam cross-section does not contain any deformation before loading.
• Shear effect is not considered.

The first hypothesis (a) is basic assumption derived from the general theory of flexural strength of a reinforced concrete cross-section [23] The assumption (b) ensures the full applicability of Bernoulli’s principle to concrete in the vicinity of a crack [24]. The hypothesis (c) assumes a correct implementation and that the stripping of the beam is carried out within the regulatory deadlines. In the assumption (d), shear effect is not considered. This is practically achieved by the use of steel stirrups to confine concrete and enhance the shear strength of beam members.

2.2. Cross-Section Analysis

In this section, the geometric, dimensional and material behaviour laws are presented to characterize the beam cross-section.

2.2.1. The Geometric and Dimensional Model

Figure 1 shows the cross-section of the rectangular RC beam.

In Figure 1, $\left(\mathrm{h};\mathrm{b}\right)$ are the height and the width of the cross-section, respectively; $\left(\overline{\mathrm{y}}\right)$ is the neutral axis depth; and $\left({\mathrm{d}}^{\prime };\mathrm{d}\right)$ are the depths of the area’s centroids of the upper reinforcements and lower reinforcements, respectively, measured from the most compressed layer at $\left(\mathrm{y}=0\right)$. ${ \mathrm{A}}_{\mathrm{Upp}.\mathrm{R}}{ \mathrm{and} \mathrm{A}}_{\mathrm{Low}.\mathrm{R}}$ are the cross-section areas of compression and tension reinforcement, with their elastic modulus: ${\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{ \mathrm{and} \mathrm{E}}_{\mathrm{Low}.\mathrm{R}}$, respectively. Their available strains are ${ \mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}}{ \mathrm{and} \mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}}$.

In this study, it was considered that:

• $\left({\mathrm{A}}_{\mathrm{Upp}.\mathrm{R}}=0\right)$: For the simply reinforced cross-section.
• $\left({\mathrm{A}}_{\mathrm{Upp}.\mathrm{R}}\ne 0\right)$: For the doubly reinforced cross-section.

2.2.2. Mechanical Properties of Materials

In order to design reinforced concrete beams, several studies have characterized the mechanical behaviour of the materials constituting the beam (concrete and reinforcing material) under compressive and tensile stresses.

(a)

Mechanical behaviour of concrete under compression

Many models developed for the mechanical behaviour characterization of concrete under compression emphasize stress–strain curves, such as the bilinear diagram, the rectangular equivalent stress diagram, and the parabola–rectangular diagram. In this study, the latter was chosen to account for the non-linearity and ductile effects on the confined concrete real behaviour under compression [25]. The diagram in Figure 2 can be expressed by using Equation (1) for concrete with a compressive strength ${\mathrm{f}}_{\mathrm{co}.\mathrm{c}}$ of less than 50 Mpa:

$${\mathsf{\sigma }}_{\mathrm{co}.\mathrm{c}}=\{\begin{array}{c}{\mathrm{f}}_{\mathrm{co}.\mathrm{c}}\left[1-{\left(1-\frac{{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}}}{{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}}\right)}^2\right]; 0\le {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}}\le {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\\ {\mathrm{f}}_{\mathrm{co}.\mathrm{c}} ; {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\le {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}}\le {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{ult}} \end{array}$$

(1)

where ${\mathsf{\sigma }}_{\mathrm{co}.\mathrm{c}} ;{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}}$ are the available stress and the available strain of concrete under compression, respectively; ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$ is the compressive strain of concrete at the peak compressive stress and ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}=-0.2\%$;${ \mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{ult}}$ is the ultimate compressive strain of concrete and ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{ult}}=-0.35\%$.

(b)

Mechanical behaviour of concrete in tension

The mechanical behaviour of concrete in tension is assumed to be linearly elastic and perfectly brittle. This behaviour may be characterized by the stress–strain diagram, as shown in Figure 3 and Equations (2)–(4):

$${\mathsf{\sigma }}_{\mathrm{co}.\mathrm{t}}=\{\begin{array}{c}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}} {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}}; {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}}\le {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}\\ 0; {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}}>{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}\end{array}.$$

(2)

$$\mathbf{1.1}{ \mathrm{E}}_{\mathrm{co}.\mathrm{t}}=1.1{\mathrm{E}}_{\mathrm{co}}$$

(3)

$$\mathbf{1.2}{ \mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}=1.5 {10}^{-0}$$

(4)

where${ \mathsf{\sigma }}_{\mathrm{co}.\mathrm{t}}$; ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}}$ are the uniaxial available tensile stress and available tensile strain of concrete. ${\mathrm{E}}_{\mathrm{co}.\mathrm{t}}$ is the tangent modulus of elasticity of concrete given by Equation (3) [26]. ${\mathrm{E}}_{\mathrm{co}}$ is the secant modulus of concrete in compression, and ${ \mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$: is the tensile strain corresponding to the tensile failure stress ${\mathrm{f}}_{\mathrm{co}.\mathrm{t}}$ when the crack initiates in concrete under tension. In this study, concrete tensile strain softening is neglected and the tensile stress of concrete when the available tensile strain is over ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$ is abruptly reduced to zero.

(c)

Mechanical behaviour of unidirectional FRP

FRP is considered to behave linearly and elastically up to failure under uniaxial loading [27,28,29]. The stress–strain curve can be described by Hooke’s law, as shown in Equation (5) and the diagram in Figure 4:

$${\mathsf{\sigma }}_{\mathrm{FRP}}=\{\begin{array}{c}{\mathrm{E}}_{\mathrm{FRP}} {\mathsf{\epsilon }}_{\mathrm{FRP}} ; {\mathsf{\epsilon }}_{\mathrm{FRP}-\mathrm{Lt}}\le {\mathsf{\epsilon }}_{\mathrm{FRP}}\le {\mathsf{\epsilon }}_{\mathrm{FRP}-\mathrm{Lc} }\\ 0 ; {\mathsf{\epsilon }}_{\mathrm{FRP}}\langle {\mathsf{\epsilon }}_{\mathrm{FRP}-\mathrm{Lt} }\mathrm{or} {-}_{\mathrm{FRP}}\rangle {\mathsf{\epsilon }}_{\mathrm{FRP}-\mathrm{Lc}} \end{array}.$$

(5)

where ${\mathsf{\sigma }}_{\mathrm{FRP}};{\mathsf{\epsilon }}_{\mathrm{FRP}}$ are the available stress and its corresponding available strain in FRP, respectively; ${\mathrm{E}}_{\mathrm{FRP}}$ is the elastic modulus of FRP; ${\mathsf{\sigma }}_{\mathrm{FRP}-\mathrm{LT}};{\mathsf{\epsilon }}_{\mathrm{FRP}-\mathrm{Lt}}$ are the ultimate tensile and the tensile ultimate strain of FRP, respectively; and ${\mathsf{\sigma }}_{\mathrm{FRP}.\mathrm{Lc}};{\mathsf{\epsilon }}_{\mathrm{FRP}.\mathrm{Lc}}$ are the ultimate compressive stress and the ultimate compressive strain of FRP.

(d)

Mechanical behaviour of steel

Steel is assumed to be elastic and perfectly plastic (Equations (6) and (7); Figure 5) [30]. In this paper, the work hardening of steel is not needed for its large plastic strain:

$${\mathsf{\sigma }}_{\mathrm{s}}=\{\begin{array}{c}{\mathrm{E}}_{\mathrm{s}}{\mathsf{\epsilon }}_{\mathrm{s}} ;-{\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{el}}\le {\mathsf{\epsilon }}_{\mathrm{s}}\le {\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{el}}\\ {\mathrm{f}}_{\mathrm{s}.\mathrm{el}} ; {\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{el}}\le {\mathsf{\epsilon }}_{\mathrm{s}}\le {\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{pl}}\\ -{\mathrm{f}}_{\mathrm{s}.\mathrm{el}} ;-{\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{pl}}\le {\mathsf{\epsilon }}_{\mathrm{s}}\le -{\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{pl}} \end{array}$$

(6)

$${\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{el}}=2{\% \mathrm{and} \mathsf{\epsilon }}_{\mathrm{s}.\mathrm{pl}}=10\% \mathrm{or} 20\%$$

(7)

where ${\mathrm{E}}_{\mathrm{s}}$ is the Young’s modulus of steel;${ \mathsf{\sigma }}_{\mathrm{s}}{ \mathrm{and} \mathsf{\epsilon }}_{\mathrm{s}}$ are the available stress and available strain in steel, respectively. ${\mathrm{f}}_{\mathrm{s}.\mathrm{el}}$; ${\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{el}}$ are the yield stress and the yield strain of steel, respectively; and ${\mathsf{\epsilon }}_{\mathrm{s}.\mathrm{pl}}$ is the ultimate plastic strain of steel.

2.3. Compatibility Condition

Figure 6 shows an infinitesimal element of length$\mathrm{dx}$, taken at the mid-span of an RC beam subjected to simple bending. According to the Bernoulli’s principle, the strain distribution over the depth $\left(\mathrm{y}\right)$ of the cross-section is linear. By considering the small strain assumption, the equation of compatibility can be stated as follows:

$$\mathsf{\phi }=\mathsf{\epsilon }\left(\mathrm{y}\right)/\left(\overline{\mathrm{y}}-\mathrm{y}\right), \mathrm{with} \overline{\mathrm{y}}ϵ\left[\mathrm{d}\prime ,\mathrm{d}\right],\mathrm{y}ϵ\left[0,\mathrm{h}\right]$$

(8)

where $\mathsf{\phi }$ is the curvature of the cross-section; $\mathsf{\epsilon }\left(\mathrm{y}\right)\ne 0$ is the strain corresponding to the deformed layer at$\mathrm{y}\ne \overline{\mathrm{y}}$.

A non-dimensional formulation of Equation (8) is introduced by putting: ${\mathrm{y}}_{\mathrm{h}}=\mathrm{y}/\mathrm{h}$; ${\overline{\mathrm{y}}}_{\mathrm{h}}= \overline{\mathrm{y}}/\mathrm{h}$;${ \mathrm{d}}_{\mathrm{h}}^{\prime }= \mathrm{d}\prime /\mathrm{h}$; ${\mathrm{d}}_{\mathrm{h}}=\mathrm{d}/\mathrm{h}$; ${\mathsf{\phi }}_{\mathrm{h}}=\mathrm{h}\mathsf{\phi }$; ${\mathsf{\epsilon }}_{\mathrm{h}}\left({\mathrm{y}}_{\mathrm{h}}\right)=\mathsf{\epsilon }\left(\mathrm{y}\right)$.

The non-dimensional compatibility equation is given as below:

$${\mathsf{\phi }}_{\mathrm{h}}={\mathsf{\epsilon }}_{\mathrm{h}}\left({\mathrm{y}}_{\mathrm{h}}\right)/\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{y}}_{\mathrm{h}}\right),{ \mathrm{with} \overline{\mathrm{y}}}_{\mathrm{h}}ϵ\left[{\mathrm{d}}_{\mathrm{h}}^{\prime },{\mathrm{d}}_{\mathrm{h}}\right],{\mathrm{y}}_{\mathrm{h}}ϵ\left[0,1\right]$$

(9)

2.4. Equilibrium Condition

Figure 7 shows the distribution of stress and strain across the depth of the beam cross-section under the balanced condition, in which ${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}{ \mathrm{and} \mathrm{F}}_{\mathrm{co}.\mathrm{t}}$ are resultants of the compressive and tensile stresses in the concrete, respectively; ${\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}{ \mathrm{and} \mathrm{F}}_{\mathrm{Low}.\mathrm{R}}$ are resultants of compression and tension reinforcement, respectively.

(a)

Static equilibrium equation

Establishing static equilibrium equation requires the calculation of stress resultants acting on a beam cross-section:

•

Calculation of ${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}$:

${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}$may be expressed as:

$${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}=\mathrm{bh}{\int }_0^{{\overline{\mathrm{y}}}_{\mathrm{h}}}{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{c}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{dy}}_{\mathrm{h}} .$$

(10)

This integral can be calculated as:

 ○

Case1 $\left(\mathsf{\epsilon }\left({\mathrm{y}}_{\mathrm{h}}=0\right)\le {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right):$ The strain in the extreme compression fibre is less than ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$ (Figure 8).

 ○

Case2: $\left(\mathsf{\epsilon }\left({\mathrm{y}}_{\mathrm{h}}=0\right)>{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right):$ when the strain in the extreme compression fibre is greater than ${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$ (Figure 9).

Introducing (${\mathrm{u}}_{\mathrm{h}}\in \left[0,{\overline{\mathrm{y}}}_{\mathrm{h}}\right]$) into Equation (10), ${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}$can be expressed as:

$${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}=\mathrm{bh}{\int }_0^{{\mathrm{u}}_{\mathrm{h}}}{\mathrm{f}}_{\mathrm{co}.\mathrm{c}}{\mathrm{dy}}_{\mathrm{h}}+\mathrm{bh}{\int }_{{\mathrm{u}}_{\mathrm{h}}}^{{\overline{\mathrm{y}}}_{\mathrm{h}}}{\mathrm{f}}_{\mathrm{co}.\mathrm{c}}\left[1-{\left(1-\frac{\mathsf{\epsilon }\left({\mathrm{y}}_{\mathrm{h}}\right)}{{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}}\right)}^2\right]{\mathrm{dy}}_{\mathrm{h}}$$

(11)

where:

$${\mathrm{u}}_{\mathrm{h}}=\{\begin{array}{c}0 ; \mathrm{for} \mathrm{case}1 \\ {\overline{\mathrm{y}}}_{\mathrm{h}}-\frac{{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}}{\mathsf{\phi }} ; \mathrm{for} \mathrm{case}2 \end{array}$$

(12)

This equation can also be written in a simplified form in Equation (13) by introducing a Boolean parameter${ \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$:

$${\mathrm{u}}_{\mathrm{h}}={\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-\frac{{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}}{{\mathsf{\phi }}_{\mathrm{h}}}\right) ; {\mathrm{with} \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}=\{\begin{array}{c}0;\mathrm{for} \mathrm{case}1 \\ 1;\mathrm{for} \mathrm{case}2 \end{array}$$

(13)

By substituting ${\mathsf{\phi }}_{\mathrm{h}}$ with its expression given in Equation (9) and then integrating Equation (11) the following is obtained:

$${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}=\mathsf{\alpha }\mathrm{bh}\left[\frac{-1}{3}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\mathsf{\phi }}_{\mathrm{h}}^2{\overline{\mathrm{y}}}_{\mathrm{h}}^3+{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\phi }}_{\mathrm{h}}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}^2+{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\overline{\mathrm{y}}}_{\mathrm{h}}-\frac{1}{3}{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left({\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3/{\mathsf{\phi }}_{\mathrm{h}}\right)\right]$$

(14)

$\mathrm{with} \mathsf{\alpha }={\mathrm{f}}_{\mathrm{co}.\mathrm{c}}/{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2$

• Calculation of ${\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}$ and${ \mathrm{F}}_{\mathrm{Low}.\mathrm{R}}$:

${\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}$ and ${ \mathrm{F}}_{\mathrm{Low}.\mathrm{R}}$: are expressed by the following equations:

$${\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}=\{\begin{array}{c}{\mathrm{A}}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}} ; 0\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}} \\ {\mathrm{A}}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}} ; {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{ult}}\end{array}$$

(15)

$${\mathrm{F}}_{\mathrm{Low}.\mathrm{R}}=\{\begin{array}{c}{\mathrm{A}}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}} ; 0\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}} \\ {\mathrm{A}}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}} ; {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{ult}}\end{array}$$

(16)

where, ${\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{ \mathrm{and} \mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{ult}}$ are the yield and ultimate compressive strains of reinforcement placed in the upper side, respectively; ${\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{ \mathrm{and} \mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{ult}}$ are the yield and ultimate tensile strains of reinforcement placed in the upper side, respectively.

For the case of concrete reinforced with FRP, the ultimate strain is taken as equal to the yield strain, since FRP is assumed to behave as a perfectly brittle material.

Similar to Equation (13), these equations can be simplified in Equations (17) and (18) by introducing Boolean parameters ${\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{ \mathrm{and} \mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}$ and applying the non-dimensional compatibility equation, Equation (9):

$${\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}={\mathrm{bh}\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right){\mathsf{\phi }}_{\mathrm{h}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{d}}_{\mathrm{h}}^{\prime }\right)+{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld} }\right)$$

(17)

$${\mathrm{F}}_{\mathrm{Low}.\mathrm{R}}={\mathrm{bh}\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}\left(\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right){\mathsf{\phi }}_{\mathrm{h}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{d}}_{\mathrm{h}}\right)+{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right)$$

(18)

in which:

$${\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}=\{\begin{array}{c}0 ; 0\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}} \\ 1 ; {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{ult}} \end{array}$$

$${\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}=\{\begin{array}{c}0 ; 0\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}} \\ 1 ; {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}}\le {\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{ult}} \end{array}$$

${\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}=\left({\mathrm{A}}_{\mathrm{Upp}.\mathrm{R}}/\mathrm{bh}\right)$and${ \mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}=\left({\mathrm{A}}_{\mathrm{Low}.\mathrm{R}}/\mathrm{bh}\right)$are ratios of upper and lower reinforcements, respectively.

•

Calculation of ${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}$:

${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}$is expressed as:

$${ \mathrm{F}}_{\mathrm{co}.\mathrm{t}}=\mathrm{bh}{\int }_{{\overline{\mathrm{y}}}_{\mathrm{h}}}^1{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{t}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{dy}}_{\mathrm{h}}$$

(19)

To take into account the concrete tension–stiffening, there are two cases for calculating the resultant of the tensile stresses in concrete:

 ○

Case1: ${\mathsf{\epsilon }}_{\mathrm{h}}\left({\mathrm{y}}_{\mathrm{h}}=1\right)<{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$ un-cracked state of tensile concrete when the strain in the extreme tension fibre is less than${\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$

 ○

Case2: $\mathsf{\epsilon }\left({\mathrm{y}}_{\mathrm{h}}=1\right)\ge {\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$ cracked state of tensile concrete when the strain in the extreme tension fibre is greater than or equal to ${ \mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$. In this case, $\exists { \mathrm{y}}_{\mathrm{h}.\mathrm{cr}}\in \left[{\overline{\mathrm{y}}}_{\mathrm{h}},1\right] \mathrm{such} \mathrm{as} \mathsf{\epsilon }\left({\mathrm{y}}_{\mathrm{h}.\mathrm{cr}}\right)={\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}$(Figure 10). Let ${\mathsf{\chi }}_{\mathrm{h}}$ be the non-dimensional length of the crack in the tensile concrete, such as${ \mathrm{y}}_{\mathrm{h}.\mathrm{cr}}=1-{\chi }_{\mathrm{h}}$.

The tensile concrete contribution in the cracked zone area is neglected; therefore, ${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}$can be expressed as:

$${ \mathrm{F}}_{\mathrm{co}.\mathrm{t}}=\mathrm{bh}{\int }_{{\overline{\mathrm{y}}}_{\mathrm{h}}}^{1-{\mathsf{\chi }}_{\mathrm{h}}}{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{t}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{dy}}_{\mathrm{h}}+\stackrel{=0}{\overbrace{\mathrm{bh}{\int }_{1-{\mathsf{\chi }}_{\mathrm{h}}}^1{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{t}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{dy}}_{\mathrm{h}}}}=\mathrm{bh}{\int }_{{\overline{\mathrm{y}}}_{\mathrm{h}}}^{1-{\mathsf{\chi }}_{\mathrm{h}}}{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{t}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{dy}}_{\mathrm{h}}$$

(20)

By applying the non-dimensional compatibility condition of Equation (9) and integrating it into Equation (20), ${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}$is obtained as follows:

$${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}=-0.5{\mathrm{bhE}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\phi }}_{\mathrm{h}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-\left(1-{\mathsf{\chi }}_{\mathrm{h}}\right)\right){ }^2$$

(21)

Once the flexural crack starts in the extreme-tension fibre (case of the cracked state of tensile concrete), the non-dimensional curvature under static loading may be expressed as:

$${\mathsf{\phi }}_{\mathrm{h}}={\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}/\left({\overline{\mathrm{y}}}_{\mathrm{h}}-\left(1-{\mathsf{\chi }}_{\mathrm{h}}\right)\right)$$

(22)

This gives:

$$1-{\mathsf{\chi }}_{\mathrm{h}}={ \overline{\mathrm{y}}}_{\mathrm{h}}-\left({\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}/{\mathsf{\phi }}_{\mathrm{h}}\right)$$

(23)

In order to integrate the case of un-cracked state of the tensile concrete in the last equation, a Boolean parameter ${\mathsf{\beta }}_{\mathrm{cr}}$ is introduced as follows:

$$1-{\mathsf{\chi }}_{\mathrm{h}}=\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right)+{\mathsf{\beta }}_{\mathrm{cr}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-\frac{{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}}{{\mathsf{\phi }}_{\mathrm{h}}}\right)$$

(24)

with: ${\mathsf{\beta }}_{\mathrm{cr}}=\{\begin{array}{c} 0 \mathrm{for} {\mathsf{\chi }}_{\mathrm{h}}=0 \\ 1 \mathrm{for} {\mathsf{\chi }}_{\mathrm{h}}\ge 0 \end{array}$

By substituting Equation (24) in Equation (21), the following equation is obtained:

$${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}=-\frac{0.5}{{\mathsf{\phi }}_{\mathrm{h}}}{\mathrm{bhE}}_{\mathrm{co}.\mathrm{t}}\left[\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathsf{\phi }}_{\mathrm{h}}{}^2{\overline{\mathrm{y}}}_{\mathrm{h}}{}^2-2\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathsf{\phi }}_{\mathrm{h}}{}^2{ \overline{\mathrm{y}}}_{\mathrm{h}}-\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathsf{\phi }}_{\mathrm{h}}{}^2+{\mathsf{\beta }}_{\mathrm{cr}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^2\right]$$

(25)

Now, the general equilibrium equation of internal forces on the cross-section for steel, FRP and hybrid FRP–steel RC beam be expressed as follows:

$${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}+{\mathrm{F}}_{\mathrm{Upp}.\mathrm{r}}+{\mathrm{F}}_{\mathrm{Low}.\mathrm{r}}+{\mathrm{F}}_{\mathrm{co}.\mathrm{t} }=\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}{}^3+\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}{}^2+\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=0$$

(26)

in which:

$$\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\frac{1}{3}\mathsf{\alpha }\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\mathsf{\phi }}_{\mathrm{h}}{}^3$$

(27)

$$\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=\left({\mathsf{\alpha }\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right)-0.5\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\right){\mathsf{\phi }}_{\mathrm{h}}{}^2$$

(28)

$$\begin{array}{l}\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=\left[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right)+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right)+\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\right]{\mathsf{\phi }}_{\mathrm{h}}{}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{ \mathsf{\alpha }\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\phi }}_{\mathrm{h}}\end{array}$$

(29)

$$\begin{array}{l}\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\left[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}^{\prime }\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right)+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathrm{d}}_h\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right)+0.5\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\right]{\mathsf{\phi }}_{\mathrm{h}}{}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}]{\mathsf{\phi }}_{\mathrm{h}}-\frac{1}{3}{\mathsf{\alpha }\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-0.5{\mathsf{\beta }}_{\mathrm{cr}}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^2\end{array}$$

(30)

(b)

Balance of moments

The moment equilibrium condition is expressed as follows:

$$\mathrm{M}={\mathrm{M}}_{\mathrm{co}.\mathrm{c}}+{\mathrm{M}}_{\mathrm{Upp}.\mathrm{R}}+{\mathrm{M}}_{\mathrm{Low}.\mathrm{R}}+{\mathrm{M}}_{\mathrm{co}.\mathrm{t}}$$

(31)

where M is the acting moment in the cross-section, and ${\mathrm{M}}_{\mathrm{co}.\mathrm{c}};{\mathrm{M}}_{\mathrm{Upp}.\mathrm{R}};{\mathrm{M}}_{\mathrm{Low}.\mathrm{R}};{\mathrm{M}}_{\mathrm{co}.\mathrm{t}}$ are the moments of the resulting internal forces of compressed concrete, compressive reinforcement. tensile reinforcement and tensile concrete, respectively. These variables are as follows:

$${\mathrm{M}}_{\mathrm{co}.\mathrm{c}}={\mathrm{hF}}_{\mathrm{co}.\mathrm{c}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}\right)$$

(32)

$${\mathrm{M}}_{\mathrm{Upp}.\mathrm{R}}={\mathrm{hF}}_{\mathrm{Upp}.\mathrm{R}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{d}}_{\mathrm{h}}^{\prime }\right)$$

(33)

$${\mathrm{M}}_{\mathrm{Low}.\mathrm{R}}={\mathrm{hF}}_{\mathrm{Low}.\mathrm{R}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{d}}_{\mathrm{h}}\right)$$

(34)

$${\mathrm{M}}_{\mathrm{co}.\mathrm{t}}={\mathrm{hF}}_{\mathrm{co}.\mathrm{t}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{y}}_{\mathrm{h}.\cot }\right)$$

(35)

Additionally, ${\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}};{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{t}}$ are centroids corresponding to the resultants of the internal forces of compressive and tensile concrete.

As in ${\mathrm{F}}_{\mathrm{co}.\mathrm{c}}+{\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}+{\mathrm{F}}_{\mathrm{Low}.\mathrm{R}}+{\mathrm{F}}_{\mathrm{co}.\mathrm{t} }=0$, the acting moment can be expressed as:

$$\mathrm{M}=-\mathrm{h}\left({\mathrm{F}}_{\mathrm{co}.\mathrm{c}}{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}+{\mathrm{F}}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}^{\prime }+{\mathrm{F}}_{\mathrm{Low}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}+{\mathrm{F}}_{\mathrm{co}.\mathrm{t}}{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{t}}\right)$$

(36)

•

Calculation of ${\mathrm{hF}}_{\mathrm{co}.\mathrm{c}}{ \overline{\mathrm{y}}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}$:

${\mathrm{hF}}_{\mathrm{co}.\mathrm{c}}{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}$ can be expressed as:

$$\begin{array}{l}{ \mathrm{hF}}_{\mathrm{co}.\mathrm{c}}{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}={\mathrm{bh}}^2\underset{0}{\overset{{ \overline{\mathrm{y}}}_{\mathrm{h}}}{\int }}{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{c}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{y}}_{\mathrm{h}}{\mathrm{dy}}_{\mathrm{h}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ={\mathrm{bh}}^2{\int }_0^{{\mathrm{u}}_{\mathrm{h}}}{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{c}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{y}}_{\mathrm{h}}{\mathrm{dy}}_{\mathrm{h}}+{\mathrm{bh}}^2{\int }_{{\mathrm{u}}_{\mathrm{h}}}^{{ \overline{\mathrm{y}}}_{\mathrm{h}}}{\mathsf{\sigma }}_{\mathrm{co}.\mathrm{c}}\left({\mathrm{y}}_{\mathrm{h}}\right){\mathrm{y}}_{\mathrm{h}}{\mathrm{dy}}_{\mathrm{h}}\end{array}$$

(37)

Based on the non-dimensional compatibility equation, the calculation of the integral can be expressed a$\mathrm{s}$:

$$\begin{array}{l}{\mathrm{hF}}_{\mathrm{co}.\mathrm{c}}{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}=\mathsf{\alpha }\mathrm{bh}(-\frac{1}{12}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\mathsf{\phi }}_{\mathrm{h}}^2{ \overline{\mathrm{y}}}_{\mathrm{h}}^4+\frac{1}{3}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\mathsf{\phi }}_{\mathrm{h}}{ \overline{\mathrm{y}}}_{\mathrm{h}}^3\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left(\frac{1}{2}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2{ \overline{\mathrm{y}}}_{\mathrm{h}}^2-\left(1/3\right)({\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3/{\mathsf{\phi }}_{\mathrm{h}}){ \overline{\mathrm{y}}}_{\mathrm{h}}-\left(1/6\right)({\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^4/{\mathsf{\phi }}_{\mathrm{h}}^2)\right))\end{array}$$

(38)

•

Calculation of ${\mathrm{hF}}_{\mathrm{Upp}.\mathrm{R}}{ {\mathrm{d}}^{\prime } \mathrm{and} \mathrm{hF}}_{\mathrm{Low}.\mathrm{R}}\mathrm{d}$:

$${\mathrm{hF}}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}^{\prime }={\mathrm{bh}\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right){\mathsf{\phi }}_{\mathrm{h}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{d}}_{\mathrm{h}}^{\prime }\right)+{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right){\mathrm{d}}_{\mathrm{h}}^{\prime }$$

(39)

$${\mathrm{hF}}_{\mathrm{Low}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}={\mathrm{bh}\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}\left(\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right){\mathsf{\phi }}_{\mathrm{h}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{d}}_{\mathrm{h}}\right)+{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right){\mathrm{d}}_{\mathrm{h}}$$

(40)

•

Calculation of ${\mathrm{F}}_{\mathrm{co}.\mathrm{t}}{\mathrm{y}}_{\mathrm{co}.\mathrm{t}}$:

$$\begin{array}{l}{\mathrm{hF}}_{\mathrm{co}.\mathrm{c}}{\mathrm{y}}_{\mathrm{h}.\mathrm{co}.\mathrm{c}}={\mathrm{bhE}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\phi }}_{\mathrm{h}}{\displaystyle \underset{{\overline{\mathrm{y}}}_{\mathrm{h}}}{\overset{1-{\mathsf{\chi }}_{\mathrm{h}}}{\int }}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{y}}_{\mathrm{h}}\right){\mathrm{y}}_{\mathrm{h}}\mathrm{d}{\mathrm{y}}_{\mathrm{h}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-\left(1/6\right){\mathrm{bhE}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\phi }}_{\mathrm{h}}\left[{\overline{\mathrm{y}}}_{\mathrm{h}}{}^3-3{\left(1-{\mathsf{\chi }}_{\mathrm{h}}\right)}^2{ \overline{\mathrm{y}}}_{\mathrm{h}}+3{\left(1-{\mathsf{\chi }}_{\mathrm{h}}\right)}^3\right]\end{array}$$

(41)

The moment equilibrium equation can be written as:

$$\mathrm{M}={\mathrm{bh}}^2\left({\sum }_{\mathrm{i}=0}^4{\mathrm{M}}_{\mathrm{i}}{\mathsf{\phi }}_{\mathrm{h}}{}^{\mathrm{i}}/{\mathsf{\phi }}_{\mathrm{h}}{}^2\right)$$

(42)

in which:

$${\mathrm{M}}_0=-\frac{1}{6}{\mathsf{\alpha }\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^4+\frac{1}{2}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{c}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^3$$

(43)

$${\mathrm{M}}_1=\left(-\frac{1}{3}{\mathsf{\alpha }\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3+2{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{c}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^2\right){ \overline{\mathrm{y}}}_{\mathrm{h}}$$

(44)

$$\begin{array}{l}{\mathrm{M}}_2=\left(\frac{1}{2}{\mathsf{\alpha }\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2-\frac{5}{2}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{c}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathrm{d}}_{\mathrm{h}}^{\prime }\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathrm{d}}_{\mathrm{h}}\end{array}$$

(45)

$$\begin{array}{l}{\mathrm{M}}_3=\left(\frac{1}{3}{\mathsf{\alpha }\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right)+\frac{5}{6}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}^3\\ \hspace{1em}\hspace{1em}\hspace{1em}+\left({\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right){\mathrm{d}}_{\mathrm{h}}^{\prime }+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right){\mathrm{d}}_{\mathrm{h}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}\\ \hspace{1em}\hspace{1em}\hspace{1em}-({\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right){\mathrm{d}}_{\mathrm{h}}^{\prime 2}\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right){\mathrm{d}}_{\mathrm{h}}^2)\end{array}$$

(46)

$${\mathrm{M}}_4=-\left(1/12\right)\mathsf{\alpha }\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){ \overline{\mathrm{y}}}_{\mathrm{h}}^4$$

(47)

2.5. Moment–Curvature Relationship

The moment–curvature response is an essential tool for understanding the flexure behaviour of the cross-section of an RC beam. To predict the moment–curvature diagram, the static equilibrium equations have to be solved; this requires knowledge of the deformation state.

a.

Deformation state:

Based on the compatibility condition and stress–strain distribution, the deformation state can be controlled by only two parameters: the non-dimensional curvature ${\mathsf{\phi }}_{\mathrm{h}}$ and strain ${\mathsf{\epsilon }}_{\mathrm{h}}\left({\mathrm{y}}_{\mathrm{h}}\right)$ at any given non-dimensional depth${ \mathrm{y}}_{\mathrm{h}}\ne { \overline{\mathrm{y}}}_{\mathrm{h}}$. The evolution of these parameters defines several regions of the deformation state, which are characterized by the Boolean factors mentioned above. These regions are delimited by non-dimensional curvature functions of strain limits versus the neutral axis depth of the tensile and compressive stress blocks as shown in Figure 11 and Figure 12. These figures show that the limit non-dimensional functions can be ordered with increasing curvature as follows:

•

For the tensile stress block:

$${\mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)<{\mathsf{\phi }}_{\mathrm{h}.\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)<{\mathsf{\phi }}_{\mathrm{h}.\mathrm{Low}.\mathrm{R}.\mathrm{ult}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)$$

(48)

•

For the compressive stress block:

$${ \mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)<\left(\begin{array}{c}\begin{array}{c}{\mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{c}.\mathrm{ult}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)\le {\mathsf{\phi }}_{\mathrm{h}.\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)\\ \mathrm{or}\end{array}\\ {\mathsf{\phi }}_{\mathrm{h}.\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)\le {\mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{c}.\mathrm{ult}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)\end{array}\right)$$

(49)

b.

Deformation processes and possible flexural failure modes:

Based on these observations, the following conclusions may be drawn:

○

The concrete in the tensile stress block cracks before the tensed reinforcements yield.

○

The concrete in the compressive stress block reaches the strain at the peak stress before the upper reinforcements yield.

○

The possible flexural failure for the (FRP/steel) RC beams under perfect bond assumption between reinforcements and concrete can be divided into two modes:

■

Mode1: Tension failure when the reinforcement strain reaches the ultimate value.

■

Mode2: Compression failure when the concrete strain reaches the ultimate value. When using steel rebars in the compressive block, mode2 can be divided into two cases depending on whether the steel yields or not, whereas this mode occurs before the compression reinforcement ultimate strain (either with steel or FRP rebars) is reached.

Note that balanced failure occurs when both mode1 and mode2 occur at the same time.

These conclusions can be summarized by the following diagram (Figure 13).

c.

Equilibrium equation solving

This section aims to extract the explicit function of the neutral axis position ${\overline{\mathrm{y}}}_{\mathrm{h}}$ by solving the equilibrium static equation, Equation (26), using a parametric study of the Boolean factors ${\mathsf{\beta }}_{\mathrm{cr}};{ \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$. The definition interval of the root function is determined by analysing the deformation processes in blocks of tensile and compressive stress under increasing curvature.

➢

Parametric study of Boolean factors ${\mathsf{\beta }}_{\mathrm{cr}};{ \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$

As shown in Figure 14, Equation (26) can take three polynomial forms depending on the values taken by (${ \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}$ and ${\mathsf{\beta }}_{\mathrm{cr}}$) as given below:

$$\{\begin{array}{c}\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=0 \\ \mathrm{With} { \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}={\mathsf{\beta }}_{\mathrm{cr}}=1\end{array}$$

(50)

$$\{\begin{array}{c}\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=0 \\ \mathrm{With} { \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}=1; {\mathsf{\beta }}_{\mathrm{cr}}=0 \end{array}$$

(51)

$$\{\begin{array}{c}\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=0 \\ \mathrm{With} { \mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}=0\end{array}$$

(52)

These equations are analytically solved as follows:

•

For the first-degree polynomial form in Equation (50), the root function of the neutral axis position ${\overline{\mathrm{y}}}_{\mathrm{h}}$ is given in Equation (53):

$${\overline{\mathrm{y}}}_{\mathrm{h}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right)$$

(53)

•

For the second-degree polynomial form in Equation (51):

At equilibrium, the existence of the solution requires that the value of the discriminant ${\Delta }_2=\mathrm{C}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2-4\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)$ is positive and the roots are in the form given below:

$${\overline{\mathrm{y}}}_{\mathrm{h}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=\frac{\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\pm \sqrt{\mathrm{C}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2-4\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)}}{2\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)}$$

(54)

•

For the third-degree polynomial form in Equation (50):

Equation (52) can be solved by the Cardan method to obtain three explicit functions for neutral axis position versus curvature, these functions can be expressed in their general form as Equation (55):

$${\overline{\mathrm{y}}}_{\mathrm{h}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=2\sqrt{\mathrm{p}/3}\cos \left(\left(1/3\right){\cos }^{-1}\left(\left(3\mathrm{q}/\left(2\mathrm{p}\right)\right)\sqrt{-\left(3/\mathrm{p}\right)}\right)+(2/3)\mathrm{k}\mathsf{\pi }\right)-(\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/3\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right))$$

(55)

with $\mathrm{k} ϵ \left\{0,1,2\right\}$

In which:

$$\mathrm{p}=-\left(\mathrm{B}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2/3\mathrm{A}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2\right)+(\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right))$$

(56)

$$\mathrm{q}=\left(\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/27\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\right)(-(2\mathrm{B}{({\mathsf{\phi }}_{\mathrm{h}})}^2/3\mathrm{A}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2)-9(\mathrm{C}({\mathsf{\phi }}_{\mathrm{h}})/\mathrm{A}({\mathsf{\phi }}_{\mathrm{h}})))+\left(\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\right)$$

(57)

The roots founded in Equations (53)–(55) are continuous functions by continuity of the coefficients $\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right); \mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right); \mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right); \mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)$ [31] and the regions $\mathrm{Ti}\cap \mathrm{Cj} \left(\mathrm{i},\mathrm{j}=1,2,3\right)$ are connected spaces; this implies that Equations (54) and (55) have a unique solution. Consequently, it is concluded that the evolution of the neutral axis position versus curvature can be plotted according to one of possible curves, as depicted in Figure 15. Thus, the curvature in each region $\mathrm{Ti}\cap \mathrm{Cj} \left(\mathrm{i},\mathrm{j}=1,2,3\right)$ can be delimited by $\left[{\mathsf{\phi }}_{\mathrm{h}-\mathrm{ij}-0},{\mathsf{\phi }}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}\right]$, in which ${\mathsf{\phi }}_{\mathrm{h}-\mathrm{ij}-0};{\mathsf{\phi }}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}$ are the initial and boundary conditions of the non-dimensional curvature at region $\mathrm{Ti}\cap \mathrm{Cj}$, respectively. Hence, the unique non-dimensional neutral axis depth is ${\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}$ $\in \left[{\mathrm{d}}_{\mathrm{h}}^{\prime },{\mathrm{d}}_{\mathrm{h}}\right]$, corresponding to [${\mathsf{\phi }}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}$].

➢

The definition interval of the root function:

By considering ${\mathsf{\epsilon }}_{\mathrm{b}}/\left({\overline{\mathrm{y}}}_{\mathrm{h}}-{\mathrm{y}}_{\mathrm{b}}\right)$ as a general form for non-dimensional curvature functions and substituting it into non-dimensional curvature ${\mathsf{\phi }}_{\mathrm{h}}$ in the equilibrium equation. Equation (26) is as follows:

$${\mathrm{A}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}{}^3+{\mathrm{B}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}{}^2+{\mathrm{C}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right){\overline{\mathrm{y}}}_{\mathrm{h}}+{\mathrm{D}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=0$$

(58)

with:

$$\begin{array}{l}{\mathrm{A}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\frac{1}{3}\mathsf{\alpha }\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\mathsf{\epsilon }}_{\mathrm{b}}{}^3+\left({\mathsf{\alpha }\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right)-0.5\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\right){\mathsf{\epsilon }}_{\mathrm{b}}{}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\alpha }\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{b}}-\frac{1}{3}{\mathsf{\alpha }\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3-0.5{\mathsf{\beta }}_{\mathrm{cr}}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^2\end{array}$$

(59)

$$\begin{array}{l}{\mathrm{B}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\left(\mathsf{\alpha }{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}})-0.5(1-{\mathsf{\beta }}_{\mathrm{cr}}\left){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\right){\mathsf{\epsilon }}_{\mathrm{b}}{}^2{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right)+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}]{\mathsf{\epsilon }}_{\mathrm{b}}{}^2-2\mathsf{\alpha }{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{b}}{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}]{\mathsf{\epsilon }}_{\mathrm{b}}+\mathsf{\alpha }{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{3}{2}{\mathsf{\beta }}_{\mathrm{cr}}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}^2{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}\end{array}$$

(60)

$$\begin{array}{l}{\mathrm{C}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\left[{\mathsf{\rho }}_{\mathrm{Upp}\cdot \mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Upp}\cdot \mathrm{R}\cdot \mathrm{yld}}\right)\right({\mathrm{y}}_{\mathrm{h}-\mathrm{b}}+ \mathrm{d}\prime )\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right)\left({\mathrm{y}}_{\mathrm{h}-\mathrm{b}}+\mathrm{d}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(1-{\mathsf{\beta }}_{\mathrm{c}\mathrm{r}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}\left({\mathrm{y}}_{\mathrm{h}-\mathrm{b}}+0.5\right)]{\mathsf{\epsilon }}_{\mathrm{b}}^2+\mathsf{\alpha }{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^2{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{b}}{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}{}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}]{\mathsf{\epsilon }}_{\mathrm{b}}{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}-\mathsf{\alpha }{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{3}{2}{\mathsf{\beta }}_{\mathrm{cr}}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^2{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}{}^2\end{array}$$

(61)

$$\begin{array}{l}{\mathrm{D}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}^{\prime }\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right)+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathrm{d}}_{\mathrm{h}}(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+0.5\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathrm{E}}_{\mathrm{co}.\mathrm{t}}]{\mathsf{\epsilon }}_{\mathrm{b}}{}^2{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+[{\mathsf{\rho }}_{\mathrm{Upp}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Upp}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\rho }}_{\mathrm{Low}.\mathrm{R}}{\mathrm{E}}_{\mathrm{Low}.\mathrm{R}}{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\epsilon }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}]{\mathsf{\epsilon }}_{\mathrm{b}}{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}{}^2+\frac{1}{3}\mathsf{\alpha }{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}^3{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}^3\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+0.5{\mathsf{\beta }}_{\mathrm{cr}}{\mathrm{E}}_{\mathrm{co}.\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{co}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}{}^2{\mathrm{y}}_{\mathrm{h}-\mathrm{b}}^3\end{array}$$

(62)

Similar to Equation (55), ${\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}$ is given as follows:

$$\begin{array}{l}{\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}=2\sqrt{{\mathrm{p}}_{\mathrm{L}}/3}\cos \left(\left(1/3\right){\cos }^{-1}\left(\left(3{\mathrm{q}}_{\mathrm{L}}/\left(2{\mathrm{p}}_{\mathrm{L}}\right)\right)\sqrt{-\left(3/{\mathrm{p}}_{\mathrm{L}}\right)}\right)+(2/3)\mathrm{k}\mathsf{\pi }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-({\mathrm{B}}_{\mathrm{L}}/3{\mathrm{A}}_{\mathrm{L}})\mathrm{With} \mathrm{k} ϵ \left\{0,1,2\right\} \end{array}$$

(63)

in which:

$${\mathrm{p}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=-\left(\mathrm{B}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2/3\mathrm{A}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2\right)+(\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right))$$

(64)

$${\mathrm{q}}_{\mathrm{L}}\left({\mathsf{\phi }}_{\mathrm{h}}\right)=\left(\mathrm{B}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/27\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\right)\left(-\left(2\mathrm{B}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2/3\mathrm{A}{\left({\mathsf{\phi }}_{\mathrm{h}}\right)}^2\right)-9(\mathrm{C}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right))\right)+\left(\mathrm{D}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{A}\left({\mathsf{\phi }}_{\mathrm{h}}\right)\right)$$

(65)

The only admissible value for ${\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}$ is:

$$\{\begin{array}{c}{\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}\in [{\mathrm{d}}_{\mathrm{h}}^{\prime },{\mathrm{d}}_{\mathrm{h}}]\\ \mathrm{and}\\ {\mathsf{\epsilon }}_{\mathrm{b}}/\left({\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}-{\mathrm{y}}_{\mathrm{b}}\right)=\min \left({\mathsf{\phi }}_{\mathrm{h}.\mathrm{ij}.\mathrm{Lt}}\left({\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}\right);{ \mathsf{\phi }}_{\mathrm{h}.\mathrm{ij}.\mathrm{Lc}}\left({\overline{\mathrm{y}}}_{\mathrm{h}-\mathrm{ij}-\mathrm{L}}\right)\right)\end{array}$$

(66)

with:

$${\mathsf{\phi }}_{\mathrm{h}.\mathrm{ij}.\mathrm{Lt}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)=\left(1-{\mathsf{\beta }}_{\mathrm{cr}}\right){\mathsf{\phi }}_{\mathrm{h}.\mathrm{t}.\mathrm{L}.\mathrm{cr}}+{\mathsf{\beta }}_{\mathrm{cr}}\left(1-{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}\right){\mathsf{\phi }}_{\mathrm{h}.\mathrm{Low}.\mathrm{R}.\mathrm{yld}}+{\mathsf{\beta }}_{\mathrm{Low}.\mathrm{R}.\mathrm{yld}}{\mathsf{\phi }}_{\mathrm{h}.\mathrm{Low}.\mathrm{R}.\mathrm{ult}}$$

(67)

$$\begin{array}{l}{\mathsf{\phi }}_{\mathrm{h}.\mathrm{ij}.\mathrm{Lc}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)=\left(1-{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\right){\mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{c}.\mathrm{pk}}+{\mathsf{\beta }}_{\mathrm{co}.\mathrm{c}.\mathrm{pk}}\left(1-{\mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\right)\min \left({\mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{c}.\mathrm{ult}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right).{\mathsf{\phi }}_{\mathrm{h}.\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{ \mathsf{\beta }}_{\mathrm{Upp}.\mathrm{R}.\mathrm{yld}}{\mathsf{\phi }}_{\mathrm{h}.\mathrm{co}.\mathrm{c}.\mathrm{ult}}\left({\overline{\mathrm{y}}}_{\mathrm{h}}\right)\end{array}$$

(68)

The valid root function for a region $\mathrm{Ti}\cap \mathrm{Cj}$ must satisfy:

$${\overline{\mathrm{y}}}_{\mathrm{h}} \left({\mathsf{\phi }}_{\mathrm{h}.\mathrm{ij}.\mathrm{L}}\right)={ \overline{\mathrm{y}}}_{\mathrm{h}.\mathrm{ij}.\mathrm{L}}$$

(69)

Finally, an algorithm was written to generate the moment–curvature relation under increasing curvature for a rectangular RC beam cross-section, as shown in Figure 16.

After obtaining the diagram$\left(\mathrm{M}-{\mathsf{\phi }}_{\mathrm{h}}\right)$, the calculation of ${\left(\mathrm{EI}\right)}_{\mathrm{e}}$ is immediately deduced, as shown in Equation (69):

$${\left(\mathrm{EI}\right)}_{\mathrm{e}}=\left(\mathrm{M}\left({\mathsf{\phi }}_{\mathrm{h}}\right)/\mathrm{h}.{\mathsf{\phi }}_{\mathrm{h}}\right)$$

(70)

Consequently, the deflection at mid-span can be estimated according to the initial condition of loading.

3. Results and Discussion

A large number of rectangular RC beams with FRP, steel and hybrid FRP–steel reinforcements obtained from different experimental investigations were used to validate the proposed model. Fifty-seven samples collected from 11 experimental studies were chosen with various geometric and material characteristics.

Table 1. Geometric and material variability of selected beam samples.

	b (mm)	h (mm)	d (mm)	ρLow.R%	Elow.R (GPa)	σLow.R.yld (Mpa)	fco.c (Mpa)
Min	120	150	125	0.36	35.6	465	21.27
Max	500	300	285	3.6	200	1988	46.52

shows the variability of the chosen beams in terms of width, height, reinforcement ratio, modulus of elasticity, reinforcement tensile strength, and concrete compressive strength.

Table 2. Test data parameters for four glass FRP RC beams.

No	Ref	Beam	b (mm)	h (mm)	d′ (mm)	d (mm)	fco.c (Mpa)	Lower Reinforcement
								Bar Type	ρLow.R	ELow.R	σLow.R.yld
									(%)	(Gpa)	(Mpa)
1	Zheng He (GFRP) [32]	G30W-A	150	300	-	285	21.27	GFRP	0.50	52.00	1.230
2	Zheng He (GFRP) [32]	G30W-B	150	300	-	285	27.31	GFRP	0.70	52.00	1.230
3	Zheng He (GFRP) [32]	G40W-A	150	300	-	285	27.20	GFRP	0.50	52.00	1.230
4	Zheng He (GFRP) [32]	G40W-A	150	300	-	285	33.07	GFRP	0.70	52.00	1.230

provides test data parameters for four rectangular glass FRP RC beams to compare moment–curvature curves. The values at the crack initiation and in the ultimate state of moments and curvatures from the proposed model and the test data are listed in

Table 3. Accuracy comparison of present model with experimental data for moment and curvature results.

No	Mcr (Cracking Moment) (kN·m)		ϕcr (Cracking Curvature) (10−6/mm)		Mu (Ultimate Moment) (kN·m)			ϕu (Ultimate Curvature) (10 × 10−6/mm)
	Exp	Proposed Model	Exp	Proposed Model	Exp	Proposed Model	Proposed Model Error	Exp	Proposed Model	Proposed Model Error
1	8.80	4.52	1.35	0.54	47.30	43.11	9%	62.67	63.66	2%
2	8.20	5.64	1.35	0.57	59.60	57.00	4%	60.56	63.61	2%
3	7.90	5.56	1.50	0.57	46.60	49.98	7%	70.21	70.96	1%
4	7.50	6.20	1.45	0.58	66.80	65.20	2%	69.80	70.35	1%

, from which a good prediction of the flexural capacity of all beams was observed. A scatter at the early moment was noted. This could be justified by the difficulty of the experimental tests to control the sudden variation of the flexural stiffness during the initiation of cracking in the tensile concrete. Figure 17 shows a good tendency of moment–curvature diagrams for all beams. This result confirms that the proposed model can also accurately predict the effective flexural stiffness.

Table 4. Test data parameters for six rectangular FRP, steel and hybrid FRP–steel RC beams.

No	Ref	Beam	b (mm)	h (mm)	d′ (mm)	d (mm)	fco.c	Upper Reinforcement				Lower Reinforcement
								Bar Type	ρLow.R	EUpp.R	σUpp.R	Bar Type	ρLow.R	Elow.R	σlow.R
							(Mpa)		(%)	(Gpa)	(Mpa)		(%)	(Gpa)	(Mpa)
1	Aiello and Ombres [33]	B1	150	200	25	175	45.70	Steel	0.36	200.00	465	Steel	0.75	200.00	465
2	Alsayed [34]	SERIE A	200	210	46	160	31.00	Steel	0.07	200.00	553	Steel	1.10	200.00	553
3	Rafai and Nadjai [35]	BRC1	120	200	30	160	46.52	Steel	0.42	201.00	530	CFRP	0.56	135.90	1.675
4	Nakano et al. [36]	RC-A4	200	300	-	240	29.43	-	-	-	-	AFRP	1.45	56.00	1.265
5	Wang, Land et al. [37]	S-14-1	120	250	29	188	42.60	CFRP	0.19	106.40	1.628	Steel	0.86	200.00	500
6	Wang, Land et al. [37]	S-16-1	120	250	29	187	42.60	CFRP	0.19	106.40	1.628	Steel	1.34	200.00	500

provides test data parameters for six rectangular beams: two steel doubly RC beams, one Aramid FRP simply RC beam and two hybrid steel–carbon FRP RC beams. The comparison between the results of the present model and experiment data reveals a good agreement for the prediction, with an average accuracy equal to 5% for the load bearing capacity and 7% for deflection at mid-span, as shown in

Table 5. Accuracy comparison of present model with experimental data for load capacity and deflection at mid-span.

No	Load Capacity (kN)			Deflection at Mid Span (mm)
	Exp	Proposed Model	Proposed Model Error	Exp	Proposed Model	Proposed Model Error
1	50.5	48.4	4%	70.1	74.5	6%
2	56.8	56.6	0%	27.8	31.0	11%
3	88.9	82.7	7%	36.0	29.2	19%
4	151.7	160.3	6%	28.5	28.6	0%
5	81.3	75.0	8%	17,50	17.3	1%
6	95.0	97.4	3%	17.2	18.2	6%
		Average	5%		Average	7%

. A good tendency was observed and justified by the precise accuracy of the effective flexural stiffness obtained by the present model as shown in Figure 18, Figure 19 and Figure 20. Figure 18b show some difference in the load–deformation response of the model compared to the test data at failure stage. However, load bearing capacity and deflection limit have been accurately estimated.

The effective flexural stiffness generated from the model algorithm was compared with those obtained from the data test, ACI440-R-15 [17] and (M.M Aliasghar and A Khaloo [21]) formulations. The last model was chosen for its best fit with a large updated experimental database. Hereafter, the EFS ratio $\left({\left(\mathrm{EI}\right)}_{\mathrm{e}}/{\mathrm{E}}_{\mathrm{c}}{\mathrm{I}}_{\mathrm{g}}\right)$was compared to give a good reading of the results (

Table 6. (EFS) Accuracy comparison of present model with experimental data, M.M Aliasghar and A Khaloo (2021) and (ACI 440.1R-15) formulations for FRP RC beams (the background colour shows the level of disperience).

Ref	No	Ref	b (mm)	h (mm)	d (mm)	ρLow.R%	Elow, R (GPa)	σLow.R.yld (Mpa)	fco.c (Mpa)	Eie/EcIg (Exp)	Eie/EcIg (Proposed Model)	Proposed Model Error %	Eie/EcIg M.M Aliasghar and A Khaloo (2021),	M.M Aliasghar and A Khaloo (2021), Disperience %	Eie/EcIg (ACI440,1R-15)	(ACI440,1R-15) Disperience %
[34]	1	g3	200	260	211	1.2	43.4	886	31.3	0.083	0.074	11%	0.090	8%	0.141	70%
	2	g4	200	300	248	1.15	35.6	700	40.7	0.067	0.061	9%	0.071	6%	0.108	61%
	3	g5	200	250	198	2.87	35.6	700	40.7	0.114	0.115	1%	0.126	11%	0.203	78%
[8]	4	ISO1, ISO2	200	300	250	1.1	45	690	43	0.080	0.074	8%	0.082	3%	0.128	60%
[38]	5	NS-1a	229	286	226	1.1	40.3	690	36.3	0.059	0.059	0%	0.069	17%	0.107	81%
	6	NS-1b	229	286	226	1.1	40.3	690	36.3	0.060	0.059	2%	0.069	15%	0.107	78%
	7	NS-1c	229	286	226	1.1	40.3	690	36.3	0.059	0.059	0%	0.069	17%	0.107	81%
	8	NS-2a	229	286	226	1.66	40.3	690	36.3	0.080	0.083	4%	0.094	18%	0.151	89%
	9	NS-2b	229	286	226	1.66	40.3	690	36.3	0.086	0.083	3%	0.094	9%	0.151	76%
	10	NS-2c	229	286	226	1.66	40.3	690	36.3	0.088	0.083	6%	0.094	7%	0.151	72%
	11	NS-3a	254	286	224	2.05	40.3	690	36.3	0.099	0.095	4%	0.108	9%	0.175	77%
	12	NS-3b	254	286	224	2.05	40.3	690	36.3	0.089	0.095	7%	0.108	21%	0.175	97%
	13	NS-3c	254	286	224	2.05	40.3	690	36.3	0.097	0.095	2%	0.108	11%	0.175	80%
	14	NS-4a	229	286	224	2.27	40.3	690	36.3	0.102	0.103	1%	0.117	15%	0.189	85%
	15	NS-4b	229	286	224	2.27	40.3	690	36.3	0.100	0.103	3%	0.117	17%	0.189	89%
	16	NS-4c	229	286	224	2.27	40.3	690	36.3	0.107	0.103	4%	0.117	9%	0.189	77%
	17	NL-1a	254	184	140	0.71	40.3	690	40.3	0.040	0.036	10%	0.044	10%	0.062	55%
	18	NL-1a	254	184	140	0.71	40.3	690	40.3	0.040	0.036	10%	0.044	10%	0.062	55%
	19	NL-1a	254	184	140	0.71	40.3	690	40.3	0.038	0.036	5%	0.044	16%	0.062	63%
	20	NL-2a	305	184	138	0.94	40.3	690	40.3	0.051	0.044	14%	0.051	0%	0.077	51%
	21	NL-2b	305	184	138	0.94	40.3	690	40.3	0.050	0.044	12%	0.051	2%	0.077	54%
	22	NL-2c	305	184	138	0.94	40.3	690	40.3	0.048	0.044	8%	0.051	6%	0.077	60%
	23	NL-3a	241	184	138	1.19	40.3	690	40.3	0.055	0.054	2%	0.060	9%	0.094	71%
	24	NL-3b	241	184	138	1.19	40.3	690	40.3	0.058	0.054	7%	0.060	3%	0.094	62%
	25	NL-3c	241	184	138	1.19	40.3	690	40.3	0.055	0.054	2%	0.060	9%	0.094	71%
	26	NL-4a	203	184	138	1.41	40.3	690	40.3	0.072	0.061	15%	0.068	6%	0.109	51%
	27	NL-4b	203	184	138	1.41	40.3	690	40.3	0.071	0.062	13%	0.068	4%	0.109	54%
	28	NL-4c	203	184	138	1.41	40.3	690	40.3	0.075	0.062	17%	0.068	9%	0.109	45%
[39]	29	G1-8	200	300	236	2.2	40	617	39.1	0.112	0.101	10%	0.111	1%	0.139	24%
	30	G1-6	200	300	236	1.6	40	617	39.1	0.078	0.078	0%	0.086	10%	0.147	88%
	31	G2-8	200	300	236	1.9	36	747	39.1	0.096	0.082	15%	0.091	5%	0.114	19%
	32	G2-6	200	300	236	1.4	36	747	39.1	0.083	0.064	23%	0.072	13%	0.233	181%
	33	C1-8	200	300	229	1.2	114	1506	39.3	0.132	0.130	2%	0.148	12%	0.187	42%
	34	C1-6	200	300	229	0.9	114	1506	39.3	0.105	0.104	1%	0.115	10%	0.160	52%
	35	C1-4B	200	300	244	0.6	114	1506	40.4	0.084	0.091	8%	0.100	19%	0.134	60%
	36	C1-4	200	300	229	0.6	114	1506	40.4	0.084	0.075	11%	0.082	2%	0.226	169%
	37	C2-8	200	300	229	1.1	122	1988	40.8	0.118	0.127	8%	0.143	21%	0.177	50%
	38	C2-6	200	300	229	0.8	122	1988	40.8	0.097	0.099	2%	0.109	12%	0.146	51%
	39	C2-4b	200	300	244	0.5	122	1988	39.9	0.091	0.082	10%	0.092	1%	0.122	34%
	40	C2-4	200	300	229	0.5	122	1988	39.9	0.087	0.068	22%	0.076	13%	0.139	60%
[40]	41	40#3-1-S	100	150	126	1.13	55.6	1764	40	0.113	0.093	18%	0.104	8%	0.165	46%
	42	40#4-1-S	100	150	125	2.03	48.6	1605	40	0.162	0.131	19%	0.143	12%	0.229	41%
	43	I,6	500	250	195	0.63	42	692	35	0.045	0.037	18%	0.049	9%	0.067	49%
	44	II,5	500	250	195	1.52	42	692	35	0.076	0.077	1%	0.089	17%	0.142	87%
	45	L,4	500	250	195	0.47	147	1970	35	0.103	0.082	20%	0.094	9%	0.152	48%
[35]	46	BRC-1	120	200	165	0.7	136	1676	42.6	0.112	0.123	10%	0.132	18%	0.212	89%
	47	BRC-2	120	200	165	0.7	136	1676	41.7	0.121	0.123	2%	0.133	10%	0.214	77%
											Average	8%	Average	10%	Average	68%

). A background colour is used to show the level of disperience: green for errors less than 10%; yellow for errors between 10% and 20%; orange for errors between 20% and 30%; and red for errors over 30%. Figure 21 illustrate the EFS prediction disperience comparison of the present model with Aliasghar and ACI440.1R.15 models based on Table 6. The comparison reveals a satisfactory accuracy for predicting the (EFS) for an FRP RC beam with an average precision equal to 8%.

4. Conclusions

The proposed model was applied to 57 rectangular FRP, and steel and hybrid FRP–steel RC beams were collected from 11 different experiment investigations. Moment–curvature curves, the effective flexural stiffness and the load–deflection response generated from the model were compared with those obtained from the test data, and the following concluding remarks can be drawn:

• The predicted moment–curvature response for glass FRP RC beams showed a good tendency with those obtained by experimental tests.
• The present model adequately predicts the load bearing capacity of FRP, steel and hybrid FRP–steel RC beams.
• The prediction of the load–deflection response by the present model for FRP, steel and hybrid FRP–steel RC beams showed good agreement with the experimental results.
• The current model can adequately predict the effective flexural stiffness for FRP, steel and hybrid FRP–steel RC beams compared to other recent models.
• The proposed model has some differences regarding test data, especially at the first loads before the initiation of cracking in the tension concrete. Nevertheless, the model showed a phenomenon of sudden change in bending stiffness caused by the initiation of cracks in the concrete in tension observed by several test data.