## 3 Design Response Spectra for Lebanon
### Design Response Spectra According to IBC  
The _International Building Code_ (IBC) gives the following formulas to build the DRS in terms of horizontal absolute acceleration \(S_{e}(T)\):  
\[S_{e}(T)=S_{DS}*\left[0.4+0.6*\left(\frac{T}{T_{0}}\right)\right]\text{ for }T\leq T_{o} \tag{10}\]  
\[S_{e}(T)=S_{DS}\text{ for }T_{o}<T\leq T_{s} \tag{11}\]  
\[S_{e}(T)=S_{D1}/T\text{ for }T\geq T_{s} \tag{12}\]  
IBC also gives a fourth branch of the spectrum that applies for values of period exceeding 4 s, similarly to UBC. This branch will not be considered here, according to Assumption 3. The values of \(S_{DS}\), \(S_{D1}\)\(T_{0}\) and \(T_{s}\) can be obtained through the formulas:  
\[S_{DS}=\frac{2}{3}*S_{MS} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Site Class**} & \multicolumn{2}{c}{**UBC**} & \multicolumn{2}{c}{**IBC**} \\ \cline{2-5}  & **C\({}_{\mathbf{a}}\)** & **C\({}_{\mathbf{v}}\)** & **S\({}_{\mathbf{1}}\)** & **S\({}_{\mathbf{s}}\)** & **F\({}_{\mathbf{a}}\)** & **F\({}_{\mathbf{v}}\)** \\ \hline A & 0.20 & 0.20 & 0.4 & 1.2 & 0.8 & 0.8 \\ B & 0.25 & 0.25 & 0.4 & 1.2 & 1.0 & 1.0 \\ C & 0.29 & 0.40 & 0.4 & 1.2 & 1.0 & 1.4 \\ D & 0.32 & 0.47 & 0.4 & 1.2 & 1.0 & 1.6 \\ E & 0.35 & 0.74 & 0.4 & 1.2 & 0.9 & 2.4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Values of the parameters for different site classes (according to UBC, IBC and NL135).  
Figure 3: Lebanese horizontal design response spectra as obtained according to American (**a**) UBC and (**b**) IBC codes for the five soil types (A, B, C, D, E) and 5% damping ratio.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Site Class**} & \multicolumn{2}{c}{**Average Properties at 30 m Depth**} \\ \cline{2-5}  & **Shear Wave Velocity** & **Standard Penetration Resistance** & **Undrained Shear Strength** \\ \cline{2-5}  & **v\({}_{\mathbf{x},\mathbf{30}}\) (m/s)** & **N\({}_{\mathbf{SPT}}\) (blows/30 cm)** & **S\({}_{\mathbf{u}}\) (kPa)** \\ \hline A & \textgreater{}1500 & - & - \\ B & 760–1500 & - & - \\ C & 360–760 & \textgreater{}50 & \textgreater{}100 \\ D & 180–360 & 15–50 & 50–100 \\ E & \textless{}180 & \textless{}15 & \textless{}50 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Site classes according to UBC and IBC.  
\[S_{MS}=F_{a}*S_{s} \tag{14}\]  
\[S_{D1}=\frac{2}{3}*S_{M1} \tag{15}\]  
\[S_{M1}=F_{0}*S_{1} \tag{16}\]  
\[T_{0}=0.2\frac{S_{D1}}{S_{Ds}} \tag{17}\]  
\[T_{s}=\frac{S_{D1}}{S_{Ds}} \tag{18}\]  
The values of the parameters \(S_{1}\), \(S_{s}\), \(F_{a}\) and \(F_{v}\) are provided in Table 3, in accordance with NL135, for the five site classes defined in Table 4 (the same classes are referred in fact by IBC and UBC). The DRS obtained through Equations (10)-(18) and Table 3 are displayed in Figure 3b, which are the same for the two main horizontal directions. It is noted that on the third branch of the IBC spectra the design acceleration increases as the mechanical properties of the soil decrease (as PS92). The UBC spectra follow the soft soil amplification rule all over the period range instead. When comparing Figure 3a,b, it can be inferred that, for the Lebanese design, UBC is almost always less conservative than IBC (site classes A, B and C), or similarly conservative (site class D).