- Open Access

# Computing the Refined Compression Field Theory

- A. M. Hernández-Díaz
^{1}Email author and - M. D. García-Román
^{2}

**10**:140

https://doi.org/10.1007/s40069-016-0140-0

© The Author(s) 2016

**Received:**3 December 2015**Accepted:**31 March 2016**Published:**19 May 2016

## Abstract

In recent years, some modifications were introduced in the stress–strain relationship of the steel in order to develop a more efficient shear model for reinforced concrete members. The last contribution in this sense corresponding to the Refined Compression Field Theory (RCFT, 2009); this theory proposed a steel constitutive model that has account the tension stiffening area prescribed by technical codes, what simplifies all the design process. However, under certain design conditions supported by such codes, the RCFT model does not provide a real (non-complex) solution for the steel yield strain when the prescribed tension stiffening area is considered; then the load-strain response cannot be computed. In this technical note, the tension stiffening area is fixed in order to guarantee the application of the embedded steel constitutive model for all the standard design range.

## Keywords

- reinforced concrete
- compression field theories
- steel constitutive model
- tension stiffening area
- solvability

## 1 Introduction

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.

*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).

*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:

*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.

## 2 Fixing the Tension Stiffening Area

*f*

_{ ct }is a parameter, and abusing the notation, denote also by

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)] the above mentioned bivariate function

*G*[

*f*

_{ ct },

*ε*

_{ max }]. In Fig. 3a the function

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)] has been represented for three values of \( A_{c}^{\prime} \); this figure shows that the equation

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)] = 0 ceases to have positive real solutions when the value of \( A_{c}^{\prime} \) is greater than the one that makes the graphic of

*G*tangent to the positive part of the abscise axis. This turns out to happen when both functions

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)] and \( G^{\prime } \) [

*ε*

_{ max }, \( A_{c}^{\prime} \)] vanish simultaneously, where:

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)]. Solving the system of equations {

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)] = 0, \( G^{\prime } \) [

*ε*

_{ max }, \( A_{c}^{\prime} \)] = 0} in the unknowns

*ε*

_{ max }and \( A_{c}^{\prime} \)

_{,}the positive real solution for the apparent yield strain is

*G*[

*ε*

_{ max }, \( A_{c}^{\prime} \)] = 0 has, at least, a positive real solution. Let us denote

*λ*the factor:

In a general sense, the factor *λ* represents the greatest portion of the tension stiffening area which may be taken in order to preserve the internal equilibrium of forces, in such a way that as concrete participation increases, the steel stress diminishes. According to this consideration, the effect of tension stiffening area over the embedded steel behavior model is illustrated in Fig. 3b for the case of specimen H75/2.

*λ*, a widely validated shear test (Abersman and Conte 1973),

*apud*(Collins and Mitchell 1991; Hernández Montes and Gil-Martín 2014), has been considered; to this aim, the load-strain curve of the tested specimen has been predicted using the RCFT under two assumptions: (1) neglecting the contribution of the concrete tension stiffening area surrounding the reinforcement bars (i.e., \( A_{c}^{\prime} \) ≈ 0), and (2) considering the limit value of the tension stiffening area established by the coefficient λ. These two curves have been illustrated in Fig. 4 together the experimental shear response obtained in (Abersman and Conte 1973). As shown, the experimental results lie about halfway the shear curve corresponding to the assumption (1) and the limit curve corresponding to the coefficient λ; in particular, for a low-intermediate range of the shear strain, the limit tension stiffening area proposed in this work coincides approximately with the experimental response. In this sense, the factor λ establishes a feasible (solvable) search domain for the experimental adjustment of RCFT model using metaheuristic methods (cf. Hernández-Díaz 2012), what improves the computational effectiveness of this process.

*ε*

_{ y }) at any crack location. Once more, the apparent yield strain (

*ε*

_{ max }) may be obtained from internal equilibrium; in this case, the above-defined function

*G*adopts the following expression (see Eq. (13) at Carbonell-Márquez et al. 2014):

*n*=

*E*

_{ s }/

*E*

_{ c }and

*ρ*

_{ eff }is the effective reinforcement ratio (\( \rho_{eff} = A_{s} /A^{\prime }_{c} \)). Equation (5) represents a monotonically decreasing function over the whole strain domain; therefore, in this case the value of \( A^{\prime }_{c} \) must be only constrained in order to avoid values of

*ε*

_{ max }lower than the average strain

*ε*

_{ ct }corresponding to the concrete tensile strength (

*f*

_{ ct }), then the expression of coefficient λ is given by:

## 3 Conclusions

A practical relationship is obtained between the reinforcement area (*A*
_{
s
}), the tensile concrete strength (*f*
_{
ct
}) and the yield stress (*f*
_{
y
}) for a given value of the tension stiffening area (*A*
_{
c
}). These four parameters are not independent, and three of them constrain the fourth one in order to preserve the internal equilibrium of a cracked member. Therefore, such relation must be satisfied in order to make operative the embedded steel constitutive model for every reinforced concrete section. Finally, this result is extensive to every structural system involving cracked embedded reinforcing bars.

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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