# An Effective Beam Width Model of Slab in Thick Wall-Thick Slab Joints Subjected to Lateral Forces
## 4 Results
### Modeling Procedure
#### 4.3.2 Model Processing Using Experimental Tests Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & **The applied** & **Initial Rotational** & **Effective Beam** & **The Ratio of the Effective** \\  & **Lateral Load \(Q_{0}\)** & **Stiffness \(K_{b0}\)** & **Width \(b_{e0}\)** & **Beam Width Divided** \\  & **(kN)** & **(kN/rad)** & **(mm)** & **by the Thickness** \\  & & & & \(b_{e}/t_{w}\) \\ \hline
1 & 24.10 & 48,200.00 & 563.86 & 3.20 \\
2 & 24.10 & 48,200.00 & 558.61 & 3.20 \\
3 & 18.60 & 37,200.00 & 446.37 & 2.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Application experimental test results to the proposed model. Figure 10: Loading arrangement [6,7]. slab joints. Additionally, \(A_{w0}\) represents the criterion value of the total cross-sectional area of the bearing wall, \(l_{w0}\) represents the criterion value of the length of the bearing wall which equals 500 mm, and \(t_{w0}\) is the criterion value of the thickness of the bearing wall which equals 175 mm. Thus, the criterion lateral force \(Q_{0}\) can be expressed using the criterion average shear stress \(\widetilde{\tau}_{0}\), multiplying the criterion value of the total cross-sectional area of the bearing wall \(A_{w0}\) (equaling to the product of \(l_{w0}\) and \(t_{w0}\)) as shown in Equation (19):  
\[Q_{0}=\widetilde{\tau}_{0}\times A_{w0}=\widetilde{\tau}_{0}\cdot l_{w0}\times t _{w0} \tag{19}\]  
where,  
\(\widetilde{\tau}_{0}\): Criterion value of average shear stress distributed over the cross-sectional plane of the bearing wall. \(A_{w0}\): Criterion value of the total cross-sectional area of the bearing wall. \(l_{w0}\): Criterion value of the length of the bearing wall. \(t_{w0}\): Criterion value of the thickness of the bearing wall. Similarly, for a target wall-slab joint, the lateral force \(Q\) applied at the 0.05% inter-story drift ratio can be expressed using its average shear stress \(\widetilde{\tau}\) which is distributing over its bearing wall cross-section plane, multiplying its total cross-sectional area of the bearing wall \(A_{w}\) (equaling to the product of \(l_{w}\) and \(t_{w}\)) as shown in Equation (20):  
\[Q=\widetilde{\tau}\times A_{w}=\widetilde{\tau}\times l_{w}\times t_{w} \tag{20}\]  
where,  
\(\widetilde{\tau}\): Average shear stress distributed over the cross-sectional plane of the bearing wall. \(A_{w}\): Total cross-sectional area of the bearing wall. \(l_{w}\): Length of the bearing wall. \(t_{w}\): Thickness of the bearing wall. Thus, as given by Equation (21), a correction coefficient \(\rho\) can be established by \(Q\) shown in Equation (20) divided by \(Q_{0}\) shown in Equation (19). It is noted that only if the dimensions of the bearing wall is designed identical to that in [PERSON] et al. 's models, the correction coefficient \(\rho\) is thus as well the correction coefficient between the average shear stress \(\widetilde{\tau}\) and the criterion average shear stress \(\widetilde{\tau}_{0}\). \[\rho=\frac{Q}{Q_{0}}=\frac{\widetilde{\tau}\times l_{w}\times t_{w}}{\widetilde {\tau}_{0}\times l_{w0}\times t_{w0}} \tag{21}\]  
As shown in Figure 11, the supporting force applied to the slab end can be defined as \(Q_{s}\) and has a relationship with \(Q\) as given by Equation (22). \[Q\times 2H=Q_{s}\times 2L_{s} \tag{22}\]  
Concerning the geometric relationship, the rotational angle at the slab end \(R_{s}\) can be expressed by \(R_{b}\) plus the rotational angle of the bearing wall \(R\), as given by Equation (23). \[R_{s}=R_{b}+R \tag{23}\]  
Assuming the displacement of the end of the rigid connection area is \(\delta\), \(\delta\) can be expressed by multiplying the rotational angle and the flexural length for both sides as shown in Equation (24) and Figure 11, where the definition of flexural length \(L_{1}\) and \(L_{2}\) are explained in Figure 7.  
\[\delta=R_{b}\times L_{2}=R\times L_{1} \tag{24}\]Next, by substituting Equation (24) for \(R_{b}\) into Equation (23), \(R_{s}\) can thus be expressed using the rotational angle \(R\) and joint dimensions as given by Equation (25).