# An Effective Beam Width Model of Slab in Thick Wall-Thick Slab Joints Subjected to Lateral Forces
## 4 Results
### Modeling Procedure  
#### 4.3.1 Analytical Model  
The dimension diagram of an individual wall-slab joint is shown in Figure 7a, while Figure 7b presents the deformation diagram when a lateral load named by \(Q\) is subjected to the top of the bearing wall, in which the wall and joint portion are assumed as rigid bodies. \(R\) is the rotational angle at the foot of the bearing wall while \(R_{s}\) represents the rotational angle of the effective beam, and \(L_{s}\) represents half the slab span length. Thus, the rotational stiffness \(K\) of the entire wall-slab joint can be expressed using the lateral load \(Q\) and the rotational angle \(R\) that represents the deformation angle of the strong bearing wall as given by Equation (1). \[K=\frac{Q}{R} \tag{1}\]  
Due to considerations of (i) the force supporting at the slab ends represented by \(Q_{s}\), (ii) the rotational angle at the end of the effective beam noted using \(R_{s}\), and (iii) the connection portion treated as a rigid region, the stiffness of the effective beam \(K_{s}\) can be consequently expressed as given by Equation (2). \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]  
Next, a relationship between \(K\) and \(K_{s}\) can be established using \(Q\), \(R\), \(Q_{s}\), and \(R_{s}\) as shown in Equation (3). \[\frac{K}{K_{s}}=\frac{\frac{Q}{R}}{\frac{Q_{s}}{R_{s}}}=\frac{Q}{Q_{s}}\times \frac{R_{s}}{R} \tag{3}\]  
Because there is a relationship between \(Q\) and \(Q_{s}\) concerning the equilibrium of moment around the joint core as shown in Equation (4), \"\(Q/Q_{s}\)\" can thus be expressed as shown in Equation (5) by rearranging Equation (4). \[Q\cdot 2H=Q_{s}\times 2L_{s} \tag{4}\]  
Figure 7: The effective beam at the wall-slab joints in TWTS structures. \[\frac{Q}{Q_{s}}=\frac{L_{s}}{H} \tag{5}\]  
Additionally, \"\(R/R_{s}\)\" can be expressed using dimensions of wall and slab as given by Equation (6) due to the assumptions that the wall and the connection portion are treated as rigid bodies. It is noted that this equation is theoretically correct if the plastic hinge forms on the slab and is adjacent to the joint core. \[\frac{R}{R_{s}}=\frac{L_{2}}{L_{1}} \tag{6}\]  
Accordingly, by rearranging Equation (3) after substituting in \"\(\frac{Q}{Q_{s}}\)\" and \"\(\frac{R}{R_{s}}\)\" using Equations (5) and (6), respectively, \"\(\frac{K}{R_{s}}\)\" can be expressed by Equation (7) using values of the joint dimensions. \[\frac{K}{K_{s}}=\frac{Q}{Q_{s}}\times\frac{R_{s}}{R}=\frac{L_{s}}{H}\times \frac{L_{1}}{L_{2}}=\frac{L_{s}L_{1}}{HL_{2}} \tag{7}\]  
Thus, the effective beam stiffness \(K_{s}\) can be obtained from \(K\) by rearranging Equation (7) as shown in Equation (8). \[K_{s}=\frac{HL_{2}}{L_{s}L_{1}}K \tag{8}\]  
On the other hand, considering the thickness of the slab is \(D\), \(b_{e}\) can be as well expressed by Equation (9) if the required moment of inertia of area \(I\) is known. \[b_{e}=\frac{12I}{D^{3}} \tag{9}\]  
Additionally, considering the effective beam as a normal beam element in which the flexible length of the beam is represented by \(L_{2}\) (equals to \"\(L_{s}\)-\(L_{1}\)\") as shown in Figure 7a, the rotational angle of the effective beam represented by \(R_{b}\) can be consequently expressed by Equation (10). \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]0  
Thus, assuming the stiffness of the effective beam as \(K_{b}\), \(K_{b}\) can be given by Equation (11) by rearranging Equation (10), where the moment of inertia of area \(I\) has been given by Equation (12). \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]1  
\[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]2  
By substituting Equation (12) for \(I\) into Equation (11), \(K_{b}\) can be expressed by Equation (13). \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]3  
Because \(K_{b}\) and \(K_{s}\) are both representing the stiffness of the effective beam, indicating they should be equal, by setting \(K_{b}\) expressed by Equation (13) to be equal with \(K_{s}\) expressed by Equation (8) as shown in Equation (14), Equation (15) can thus be obtained.