# Nonlinear Finite Element Analysis Formulation for Shear in Reinforced Concrete Beams
## 4 Two-Dimensional Realization of Beam Shear
### Concrete Material  
There are many constitutive models to simulate the concrete behavior [17; 18; 19], and those models are widely used in the finite element analysis. In all analyses presented in this paper, the uniaxial behavior of concrete in compression is obtained by applying the modified Kent and Park model [20] to account for the confining effects provided by stirrups. That uniaxial behavior is scaled by using the compression softening relation suggested by [PERSON] and [PERSON] [21]. In addition, other concrete models with confinement effect [16; 17; 18] can be interchangeably used. To describe tension stiffening, the shear cracking stress of concrete is calculated by using the ACI expression \(0.17\sqrt{f_{c}^{\prime}}\), where \(f_{c}^{\prime}\) is in MPa [22]. The post-cracking envelope is obtained by using the equation proposed by [PERSON] and [PERSON] [22]. Figure 3 illustrates the uniaxial behavior of concrete material. It is assumed that this uniaxial behavior discussed thus far describes the orthotropic concrete material in the principal directions. The composition of material tangent stiffness for an orthotropic material is discussed as follows.  
Figure 3: Behavior of concrete material.  
For an orthotropic material such as concrete, by neglecting the Poisson's ratio, the material tangent in the principal directions can be written as:  
\[\mathbf{d}_{\mathrm{P}}=\left[\begin{array}{ccc}E_{1}&0&0\\ 0&E_{2}&0\\ 0&0&G\end{array}\right] \tag{10}\]  
where, \(E_{1}\) and \(E_{2}\) are the Young's modulus in each principal direction, respectively, which can be calculated by using the ACI expression \(4700\sqrt{f_{c}^{\prime}}\), and \(G\) is the shear modulus. The latter can be determined by:  
\[G=\frac{f_{1}-f_{2}}{2(\varepsilon_{1}-\varepsilon_{2})} \tag{11}\]  
where, \(f_{1}\) and \(f_{2}\) are the principal stresses, and \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the principal strains. Equation (11) satisfies the condition of invariance under the principal axis rotation [23]. The orthotropic material stiffness should be transformed to the current local coordinate system of the element by using the following relation:  
\[\mathbf{\hat{d}}=\mathbf{T}^{\mathrm{T}}\cdot\mathbf{d}_{\mathrm{P}}\cdot \mathbf{T} \tag{12}\]  
where the transformation matrix is defined as:  
\[\mathbf{T}=\left[\begin{array}{ccc}\cos^{2}\theta&\sin^{2}\theta&\cos \theta\sin\theta\\ \sin^{2}\theta&\cos^{2}\theta&-\cos\theta\sin\theta\\ -2\cos\theta\sin\theta&2\cos\theta\sin\theta&\cos^{2}\theta-\sin^{2}\theta \end{array}\right] \tag{13}\]  
The failure criterion of concrete material can be found in the modified compression field theory (MCFT) formulation [7], where the two-dimensional plane reinforced concrete response was simplified with two uni-axial stress-strain relations in the two principal directions. In the principal directions, the compression failure occurs at the compression strain of \(-0.003\). The steel was considered as smeared to the corresponding concrete layers, and the confining effects were also reflected, as described in Figure 3.