The design and analysis of reinforced concrete members subjected to shear may be performed taking into consideration different strategies reported in the literature, among several others (ASCE-ACE Committee 445 on Shear and Torsion 1998; Hernández-Díaz and Gil-Martín 2012; Jeong and Kim 2014; Mofidi and Chaallal 2014). One of the most widely known is the so-called Modified Compression Field Theory (MCFT) (Vecchio and Collins 1986). In the MCFT, the stress–strain relationship for the steel reinforcement is assumed to be elastic-perfectly plastic, being the Young’s modulus constant up to the yield strength (*f*
_{
y
}) and then zero upon yielding at the crack location. To allow new increments of shear force, MCFT introduces the notion of local shear stress, and as a consequence, requires the check of equilibrium conditions for local shear stresses at the crack location in order to ensure that the steel stress does not exceed the steel yield strength.

A few years ago, Gil-Martín et al. (2009) proposed a new steel constitutive model leading to the Refined Compression Field Theory (RCFT). In the line of a few other shear theories, such as the Rotating Angle-Softened Truss Model (RA-STM, Belarbi and Hsu 1994), the RCFT proposes a stress–strain relationship for the reinforcing bars stiffened by concrete (“embedded bar model”); the novelty is that the embedded bar stress–strain relationship is obtained imposing equilibrium on the tension stiffening effect; so new formulation for the steel model would no longer be needed (compared with RA-STM) and the crack check can be avoided (compared with MCFT). According to Gil-Martín et al. (2009), from equilibrium of forces between a cracked section and a generis section (see Fig. 1a), the RCFT predicts the average stress (along the bar between cracks) of an embedded bar as a function of the average strain (i.e., measured on certain length including several cracks):

$$\begin{aligned} \sigma_{s,av} &= \left\{ {\begin{array}{*{20}c}
{f_{y\,} - \,\frac{{A_{c} }}{{A_{s} }}\frac{{f_{ct} }}{{1 + \sqrt
{3.6M\varepsilon_{s} } }}} & \quad{if\,\,\varepsilon_{s} \ge
\varepsilon_{{max} } } \\ {E_{s} \varepsilon_{s} } &
\quad{if\,\,\varepsilon_{s} < \varepsilon_{{max} } } \\
\end{array} } \right. \hfill \\ {with}&\hfill\\
\varepsilon_{{max} } \, &= \,\frac{{f_{y\,} }}{{E_{s} }}\,\, -
\,\,\frac{{\frac{{f_{ct} }}{{1 + \sqrt
{3.6M\,\varepsilon_{{max} } } }}}}{{E_{s} A_{s} }}A_{c} \hfill
\\ M &= \,\frac{{A_{c} }}{{\sum {\phi \pi } }}\, \hfill \\
\end{aligned} $$

(1)

where *σ*
_{
s,av
} is the average tensile stress in steel (for longitudinal or transverse reinforcement), *A*
_{
s
} is the cross section of steel bar (longitudinal or transverse), *ε*
_{
s
} is the average tensile strain in steel and concrete, *f*
_{
ct
} is the tensile strength of concrete, *E*
_{
s
} is the elastic modulus of reinforcement, *ε*
_{
max
} is the apparent yield strain (or average tensile strain when first yielding occurs at the crack location, Fig. 1) and *A*
_{
c
} is the area of concrete bonded to the bar. Technical codes (e.g., EHE 2008) usually define *A*
_{
c
} as a value equal to the rectangular area (tributary to and surrounding the bar) over a distance not exceeding 7.5Ø from the center of the bar (Fig. 1b), and Ø is the diameter of the bar. Hereafter, we refer to *A*
_{
c
} as the prescribed tension stiffening area. In Eq. (1) the embedded bar stress–strain relationship is established for the concrete tension stiffening model proposed by Bentz (2005).

Numerical results obtained from RCFT for different tested specimens (Gil-Martín et al. 2009; Palermo et al. 2013) show a better fitting of the experimental results, in particular near the peak point in the shear response curve, where the MCFT significantly deviates from the experimental evidences. Nevertheless, it can be proved that when the prescribed tension stiffening area (*A*
_{
c
}) is adopted, for certain specimens (specifically those with high values of the ratio *f*
_{
ct
}/*ρ*, being *ρ* the reinforcement ratio) it is not possible to obtain a positive real solution for the apparent yield strain (*ε*
_{
max
}) defined in Eq. (1). If all terms in the expression of the apparent yield strain are moved to the right hand side, the following function *G* is obtained:

$$ \varepsilon_{{max} } \, = \,\frac{{f_{y\,} }}{{E_{s} }}\,\, - \,\,\frac{{\frac{{f_{ct} }}{{1 + \sqrt {3.6M\,\varepsilon_{{max} } } }}}}{{E_{s} A_{s} }}A_{c}^{\prime} \to G[f_{ct} ,\varepsilon_{{max} } ] = \left( {\varepsilon_{y} - \varepsilon_{{max} } } \right) - \frac{{\frac{{A_{c}^{\prime} f_{ct} }}{{1 + \sqrt {3.6M\varepsilon_{max} } }}}}{{E_{s} A_{s} }} $$

(2)

where *ε*
_{
y
} is the strain corresponding to *f*
_{
y
} (i.e., *ε*
_{
y
} = *f*
_{
y
}/*E*
_{
s
}). A new variable, \( A_{c}^{\prime} \), has been introduced in Eq. (2) in order to discuss the solvability of this equation in terms of the ratio \( A_{c}^{\prime} /A_{c}^{{}} \). To illustrate the effect of the tension stiffening area in the solvability of the RCFT model, the top longitudinal reinforcement of a beam (specimen H75/2) tested in shear by Cladera in 2002 has been considered. The top longitudinal bar diameter is 8 mm, the side cover is 25 mm, the prescribed tension stiffening area (according to EHE) is 9025 mm^{2}, the tensile strength of concrete (*f*
_{
ct
}) is 4.5 MPa and the yield stress of steel (*f*
_{
y
}) is 530 MPa. The function *G* [*f*
_{
ct
}, *ε*
_{
max
}] has been represented in Fig. 2 for different values of \( A_{c}^{\prime} \), resulting in a set of curves. For this specimen, the apparent yield strain (*ε*
_{
max
}) corresponds to the intersection points of these curves with the abscissa *f*
_{
ct
} = 4.50 MPa.

The interval adopted for the strength *f*
_{
ct
} in Fig. 2 coincides with the range established for this parameter by EC-2 (2002). It can be seen that, for high values of \( A_{c}^{\prime} \) (like \( A_{c}^{\prime} = 0.7A_{c} \) and \( A_{c}^{\prime} = 0.9A_{c} \)), the curve *G* [*f*
_{
ct
}, *ε*
_{
max
}] = 0 presents a knee that breaks the bijection between *f*
_{
ct
} and *ε*
_{
max
} (*problem of uniqueness*), or even, no solutions exists for *ε*
_{
max
} (*problem of existence*), as it occurs by taking \( A_{c}^{\prime} = A_{c} \) in this specimen. In relation to the problem of uniqueness, by continuity and taking into account that *ε*
_{
max
} = *ε*
_{
y
} when \( A_{c}^{\prime} = 0 \) (cf. Gil-Martín et al. 2009), the actual value of *ε*
_{
max
} is that of the solution closest to the yield strain *ε*
_{
y
}, that is, the greatest one of the two positive real solutions of Eq. (2). However, the absence of solution in the steel constitutive model proposed by RCFT indicates that the equilibrium of internal forces along the cracked member (see Fig. 1) is not verified, and therefore, the stress–strain relationship for the steel must be corrected.