# 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
\[R_{s}=\frac{L_{s}}{L_{2}}\times R \tag{25}\]  
The slab stiffness can be expressed in two ways: (i) \(Q_{s}\) divided by \(R_{s}\) shown in Equation (2), and (ii) \(3 EI\) divided by squared slab flexural length \(L_{2}\) as given by Equation (26), respectively. By equalling these two formulas, \(Q_{s}\) can thus be expressed as shown in Equation (27). Figure 10:0  
Figure 10:1  
Then, by substituting Equation (25) for \(R_{s}\) into the Equation (27), \(Q_{s}\) can accordingly be expressed by \(R\) as shown in Equation (28). Figure 10:2  
Next, substituting Equation (28) for \(Q_{s}\) into Equation (23), and considering \(L_{2}\) equals \"\(L_{s}-L_{1}\)\", a relationship between \(Q\) and \(R\) can be established as given by Equation (29). Figure 10:3  
Using this formula, the criterion value \(Q_{0}\) can consequently be expressed as given by Equation (30). Figure 10:4  
Figure 10:5 Calculation diagram. Thus, a correction coefficient \(\rho\), representing the ratio between the unknown value \(Q\) for a target wall-slab joint and the criterion value \(Q_{0}\), can be established using Equation (29) divided by Equation (30) as given by Equation (31). Figure 10:6  
Because the initial stiffness is defined as 0.05% inter-story drift ratio in this study, \"\(R=R_{0}=0.05\)%\" can be substituted into Equation (31). Thus, Equation (31) is simplified to Equation (32) by eliminating \(R\) and \(R_{0}\). Generally, the applied lateral force changes and the stiffness reduction needs to be considered as the deformation process, which leads to a reduction of the effective beam width, while when the deformation is small enough, the deformation change and the stiffness reduction can be ignored so that the stiffness can be seen as a constant value. Thus, the applied lateral force can be consequently considered stable as a constant. Based on these considerations, it is noted that the \(R\) and \(R_{0}\) are set as 1/2000 rad for the initial stage in this work because the deformation angle is relatively small enough at the initial stage. Figure 10:7  
Additionally, the moment of inertia of area \(I\) and \(I_{0}\) in Equation (32) are assumed by 3.20 times the wall thickness for joints with both-side slabs and 2.60 times the wall thickness for joints with one-side slab according to measured results from the experimental data as shown in Table 2 previously. Thus, the unknown value \(Q\) can be estimated using the criterion value \(Q_{0}\) and the correction coefficient \(\rho\) is given by Equation (33). Figure 10:8  
As a result, an estimated value of the lateral force at the initial stage obtained using Equations (32) and (33) can thus be applied to Equation (18) for predicting the effective beam width for the initial stiffness. This analytical-based, effective beam width estimation is derived for practical designing the wall-slab joints in TWTS structures. The proposed effective beam-width model provides a relatively accurate and convenient estimation for determining the slab design by limiting the minimum value on the width. Additionally, using this proposed model, it can as well model the entire wall-slab joints as a wall-beam frame, which leads to a simplified two-dimension frame analysis by substituting the slab components using the proposed effective beam.