# Seismic Response of a Three-Dimensional Asymmetric Multi-Storey Reinforced Concrete Structure
## 3 Finite Element Modeling of the SMART 2013 RC Structure
### Failure Criter Ion for Concrete  
For the simulation of the behavior of concrete materials in multiaxial stress states, the Willam-Warnke failure criterion [29] was implemented in ANSYS with a SOLID65 element (ANSYS Mechanical APDL Release 14.5, ANSYS Inc., Canonsburg, PA, USA, 2012), which is a three-dimensional hexahedral element that can be used to model concrete. Both cracking and crushing failure modes can be accounted for by using the failure criterion. In the concrete element, cracking occurs when the principal tensile stress lies outside the failure surface. On the other hand, crushing occurs when all of the principal stresses are compressive and lie outside the failure surface. The Willam-Warnke failure criterion prescribes the portion of the failure curve in deviatoric plane as part of an elliptic curve. The cylindrical von [PERSON] model and the conical Drucker-Prager model are all special cases of the Willam-Warnke failure criterion. The three-parameter Willam-Warnke failure surface can be expressed as:  
\[f(\sigma_{m},\ \tau_{m},\ \theta)=\frac{1}{\rho}\frac{\sigma_{m}}{f_{c}^{ \prime}}+\frac{1}{r(\theta)}\frac{\tau_{m}}{f_{c}^{\prime}}-1=0 \tag{3}\]  
where \(f_{c}^{\prime}\) is the uniaxial compressive strength of concrete, while \(\sigma_{m}\) and \(\tau_{m}\) denote the mean normal and mean shear stresses, respectively. The variable \(\theta\) denotes the angle of similarity. They can be represented as:  
\[\sigma_{m}=\frac{1}{3}I_{1},\tau_{m}=\sqrt{\frac{2}{5}}\sqrt{J_{2}},\theta= \frac{1}{3}\cos^{-1}\left(\frac{3\sqrt{3}I_{3}}{2\frac{3}{2}}\right) \tag{4}\]  
where \(I_{1}\) is the first invariant of the stress tensor, while \(J_{2}\) and \(J_{3}\) are the second and third invariants of the deviatoric part of the stress tensor, respectively. In Equation (3), \(r(\theta)\) is the radial coordinate of the failure surface on the deviatoric plane, which can be expressed as the following:  
\[r(\theta)=\frac{2r_{c}\big{(}r_{c}^{2}-r_{t}^{2}\big{)}\cos\theta+r_{c}(2r_{t }-r_{c})\big{[}4\big{(}r_{c}^{2}-r_{t}^{2}\big{)}\cos^{2}\theta+5r_{t}^{2}-4r _{t}r_{c}\big{]}^{\frac{1}{2}}}{4\big{(}r_{c}^{2}-r_{t}^{2}\big{)}\cos^{2} \theta+\big{(}r_{c}-2r_{t}\big{)}^{2}} \tag{5}\]  
The parameters \(\rho\), \(r_{c}\), and \(r_{t}\) in Equations (3) and (5) are the three parameters of the Willam-Warnke failure surface model. The parameters \(r_{c}\), and \(r_{t}\) are the magnitude of the position vectors at locations \(\theta=0\) and \(\pi/3\) on the deviatoric plane, respectively, as shown in Figure 7. The three model parameters can be expressed as:  
\[\rho=\frac{\overline{f}_{bc}^{\prime}\overline{f_{t}}}{\overline{f}_{bc}^{ \prime}-\overline{f_{t}}} \tag{6}\]  
\[r_{c}=\sqrt{\frac{6}{5}}\ \ \frac{\overline{f}_{bc}^{\prime}\overline{f_{t}}}{ \overline{3f}_{bc}^{\prime}\overline{f_{t}}+\overline{f}_{bc}^{\prime}- \overline{f_{t}}} \tag{7}\]  
Figure 6: Strain–stress curves for concrete and steel reinforcement.  
\[r_{t}=\sqrt{\frac{6}{5}}\ \ \frac{\overline{f}^{\prime}_{hc}\overline{f}_{t}}{2 \overline{f}^{\prime}_{bc}+\overline{f}_{t}} \tag{8}\]  
In Equations (6)-(8), \(\overline{f}_{t}\) and \(\overline{f}^{\prime}_{bc}\) are the strength ratios, which are represented as:  
\[\overline{f}_{t}=\frac{f_{t}}{f^{\prime}_{c}},\overline{f}^{\prime}_{bc}=\ \frac{f^{\prime}_{bc}}{f^{\prime}_{c}} \tag{9}\]  
where \(f_{t}\) and \(f^{\prime}_{bc}\) are the uniaxial tensile strength and the equal biaxial compressive strength of concrete, respectively. Figure 8 shows the trace of the three-parameter Willam-Warnke yield surface in the \(\sigma_{1}\)-\(\sigma_{2}\) principal coordinate plane for \(\overline{f}_{t}=0.1\) and \(\overline{f}^{\prime}_{bc}=1.16\).