# On the Modeling and Analysis of Brittle Failure in Existing R/C Structures Due to Seismic Loads
## Appendix A
The first step towards the calculation of the individual strength terms of an R/C column is the determination of the normalized depth of the compression zone, \(\xi=x/d\), at the ultimate limit state, which is estimated for a specified axial load ratio \(\
u=N/(A_{c}\cdot f_{c})\) from the following interpolations:  
\[\text{if }v_{bal}<\ v<\ v_{max},\ \xi=\xi_{bal,u}+\ (1-\xi_{bal,u})\cdot\frac{v-v_{ bal}}{v_{max}-v_{bal}} \tag{1}\]  
\[\text{if }v_{min}<\ v\leq\ v_{bal},\ \xi=\delta_{2}+\ (\xi_{bal,u}-\delta_{2}) \cdot\frac{v-v_{min}}{v_{bal}-v_{min}} \tag{2}\]  
The values of \(\
u_{min}\), \(\
u_{bal}\), and \(\
u_{max}\) correspond to characteristic values of the depth of compression zone \(x\), for \(\epsilon_{cu}=0.005\), and are given as follows:  
\[\text{for }x=d_{2}\Rightarrow\xi_{2}=\delta_{2},\ \frac{v_{min}}{A_{c}\cdot f _{c}}=0.72\cdot\delta_{2}-\frac{f_{y}}{(1-\alpha)\cdot f_{c}}\cdot[\rho_{s1}+ \rho_{v}\cdot(1-2\cdot\delta_{2})] \tag{3}\]  
\[\text{for }x=x_{bal}\Rightarrow\xi_{bal,u}=0.64,\ \frac{v_{bal}}{A_{c}\cdot f _{c}}=\ (\rho_{s2}-\rho_{s1})\cdot\frac{f_{y}}{f_{c}}+0.462+0.275\cdot\rho_{v} \cdot\frac{f_{y}}{f_{c}} \tag{4}\]  
\[\text{for }x=d\Rightarrow\xi_{d}=1,\ \frac{v_{max}}{A_{c}\cdot f_{c}}=0.85^{2}+ \frac{f_{y}}{f_{c}}.\ (\rho_{s2}+0.8\cdot\rho_{v}) \tag{5}\]  
With the interpolated value of \(\xi\), the following strength terms are now determined:  
\[V_{flex}=\ \left[\rho_{\ell,tot}\cdot\frac{f_{y}}{f_{c}}\cdot(1-0.4\cdot\xi)\ +\ v\cdot\left(\frac{h}{d}-0.8\cdot\xi\right)\right]\cdot\frac{b\cdot d^{2} \cdot f_{c}}{H_{cl}} \tag{6}\]  
Exhaustion of shear strength:  
\[\text{If }\ v<\ 0.10:V_{v}=A_{tr}\cdot f_{st}\cdot\frac{d\cdot(1-0.4\cdot\xi)} {s}\cdot\cot\theta_{v} \tag{7}\]  
\[\text{If }\ v\ \geq\ 0.10:V_{v}=v\cdot b\cdot d\cdot f_{c}\cdot\tan\alpha+A_{tr} \cdot f_{st}\cdot\frac{d\cdot(1-0.4\cdot\xi)}{s}\cdot\cot\theta_{v} \tag{8}\]  
Anchorage failure of longitudinal reinforcement:  
\[V_{a}=\ \left[\rho_{\ell,tot}\cdot\frac{\min\left\{\ \frac{4\cdot L_{tr}f_{b}}{D_{b}}+ \alpha_{hook}\cdot 50\cdot f_{b}\ \ ;\ \ f_{y}\ \right\}}{f_{c}}.\ (1-0.4\cdot\xi)\ +\ v\cdot \left(\frac{h}{d}-0.8\cdot\xi\right)\right]\cdot\frac{b\cdot d^{2}\cdot f_{c} }{H_{cl}} \tag{9}\]  
\[V_{lap}=\ \frac{\left[\ \min\left\{\ \left(\ \begin{array}{c}\mu_{fr}\cdot L_{lap} \cdot\left[\frac{A_{tr}}{s}\cdot f_{st}+\alpha_{b}\cdot(b-N_{b}\cdot D_{b}) \cdot f_{t}\right]+\right)\\ \\[\frac{\text{Shear capacity of joints:}}{\text{Unreinforced or lightly reinforced joints}}:V_{j}=\gamma_{j}\cdot 0.5\cdot\sqrt{f_{c}} \cdot\sqrt{1+\frac{v_{j}\cdot f_{c}}{0.5\cdot\sqrt{f_{c}}}}\cdot\frac{b_{j}\cdot d \cdot d_{beam}}{H_{cl}}\] (A11)  
\[\text{if }v_{min}<\ v\leq\ v_{bal},\ \xi=\delta_{2}+\ (\xi_{bal,u}-\delta_{2}) \cdot\frac{v-v_{min}}{v_{bal}-v_{min}} \tag{2}\]0 (A12)  
In the above expressions, the following terms appear:  
* \(\rho_{\ell,tot}=A_{s,tot}/(b\cdot d)\) is the total longitudinal reinforcement ratio of a column with external dimensions \(h\times b\),
* \(A_{s,tot}\) is the total area of the longitudinal reinforcement at the column's critical section,
* \(d\) is the column effective depth,
* \(f_{y}\) is the longitudinal reinforcement yield stress,
* \(f_{c}\) is the concrete compressive strength,
* \(\xi\) (\(=x/d\)) is the normalized depth of compression zone,
* \(v\) is the axial load ratio acting on the cross-section (\(N_{g+0.3q}/(b\cdot d\cdot f_{c})\)),
* \(H_{cl}\) is the column's deformable length,
* \(\tan\alpha=(h/d\ -0.8\cdot\xi)\cdot d/H_{cl}\), where \(a\) (\(\leq\)\(\theta_{v}\)) is the angle of inclination of the diagonal strut created between the centroids of the compression zones at the top and bottom column cross-sections of the column. This represents the strut forming by the axial load acting on the column according to [PERSON] et al. (2014),
* \(\theta_{v}=\{45^{\circ}\) when \(v\) < 0.10, 30\({}^{\circ}\) when \(v\) \(\geq\) 0.25, whereas for \(0.10\leq v\) < 0.25 \(\theta_{v}\) is calculated from linear interpolation\(\}\) is the angle of sliding plane.