# 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
By rearranging Equation (15), the effective beam width \(b_{e}\) is consequently expressed by Equation (16) using the joint stiffness \(K\), component dimensions, and material features of the wall-slab joint. \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]4  
\[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]5\[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]6  
Additionally, because \(K\) can also be expressed by \"\(\frac{Q}{R}\)\", where \(Q\) represents the lateral force applied to the top of the wall while \(R\) represents the rotational angle at the wall foot, Equation (16) can be rearranged to Equation (17). \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]7  
As a result, Equation (17) can be considered as the effective beam model to calculate the stiffness of an individual wall-slab joint under a lateral load \(Q\) representing an earthquake excitation. It is noted that parts of \"\(\frac{4}{ED^{3}}\)\", \"\(\frac{L_{2}^{3}}{L_{s}L_{1}}\)\", and \"\(H\)\" can be obtained from model dimensions and material features, such as wall length, slab thickness, Young's modulus of concrete, and others which can be obtained directly from design details. Only the \(Q\) and \(R\) which represent the applied lateral force and the rotational deformation angle of the joint are needed carefully discussed for applying this model in practical design. Firstly, \(R\) can be taken by the value of 0.05% due to the initial stiffness defined as the secant stiffness at 0.05% inter-story drift ratio (corresponding to 1/2000 rad inter-story deformation angle according to WRC Standard [3]). Therefore, Equation (17) can thus be rearranged to Equation (18) for predicting the effective beam width for the initial stiffness. \[K_{s}=\frac{Q_{s}}{R_{s}} \tag{2}\]8  
Next, for the applied lateral force \(Q\), if the average shear stress distributing over its bearing wall cross-section plane of wall-slab joints when the required deformation value which represents the initial stage reaches, the value of \(Q\) for Equation (18) can consequently be obtained by multiplying the average shear stress and its total cross-sectional area of the bearing wall. However, it is hard to accurately estimate the average shear stress distributed over the bearing wall, due to the shear stress of concrete at a very small deformation angle which is not stable nor identical for different situations. Therefore, to accurately estimate the value of force \(Q\) applied to the bearing wall when the defined initial stage reaches, results of an experimental cyclic lateral loading test of thick wall-thick slab joints in TWTS structures conducted at Yamaguchi University in 2016 [6,7] are employed in this work. The measured lateral force applied on the top of the bearing wall at the inter-story drift ratio of 0.05% for specimens in their tests is used to provide a reasonable estimate of \(Q\) at 0.05% inter-story drift ratio. These values from their experimental results are employed as criterion values and a correction coefficient is defined. Thus, by applying the criterion value and correction coefficient, an estimation of the crucial factor \(Q\) for the effective beam width of initial stiffness prediction can be obtained. Detailed information of their experimental tests related to this work and the model processing using their resultant data are explained in the following sections.