# An Effective Beam Width Model of Slab in Thick Wall-Thick Slab Joints Subjected to Lateral Forces
## 2 Background
### Effective Beam Width Method (EBWM)  
The Effective Beam Width Method (EBWM) concept was initially developed to simplify an analytical procedure for calculating the rotational stiffness of column-slab joints in reinforced concrete flat-plate structure, which is one type of reinforced concrete (RC)  
Figure 4: Relation of initial stiffness and maximum inter-story deformation angle. structure where the slabs are supported on the columns directly without any beams. In this method, the effective beam width is defined as the width of the floor slab that essentially acts as a beam. By assuming the effective beam, the effects of bending, shearing, and torsional behaviors of the slab are incorporated. Only the effective beam portion of the slab is considered when determining the strength and stiffness of the joint. As a result, the stiffness of the original slab could be greater than that of the effective beam because not all of it contributes to resisting loading, and thus the resultant stiffness captured by the effective beam is relatively lower. However, due to the shearing and torsional behaviors of the slab being relatively ignorable compared to the flexural behavior, the effective beam can be considered reasonable to represent the structural behavior of the original slab. As shown in Figure 5, usually, an effective beam width factor, which is the ratio of the effective beam width to the full slab width, is used to represent the scope of the effective beam by substituting the slab with the effective beam width. [PERSON] and [PERSON] [8] proposed an investigation on the EBWM concept for the first time in the early 1960s. It was pointed out that parts of plate elements such as slabs and walls could be considered counting as effective to cooperate with frames to form resisting systems against external forces. They described static experimental studies to investigate earthquake resistance of flat slabs and plates, and as well indicated that the concept of effective width was suitable for the practical design of flat slabs or plates for the estimation of its stiffness that are effective for resisting external excitations. [PERSON] [9] modeled a three-dimensional system as a two-dimensional frame of a conventional column and an equivalent beam by multiplying an effective beam width factor and the original full width of the slab using the EBWM concept. This effort made the equivalent beam rotate identical to the deformation angle as the original slab, based on which it could be considered that the equivalent system has the same elastic rotational stiffness as the original system. [PERSON] and [PERSON] [10] as well applied the EBWM concept in a column-slab joint. In their work, they pointed out that when the column underwent a rotational angle, part of the slab connected to the column that had the same width as that of the column would produce the same rotational angle, while the rotational angle of other parts of the slab located at a longer distance from the column core would be varying to much smaller. If the width of the slab was sufficiently wider, the rotational angle at the edge of the slab would be possible to turn to 0. [PERSON] et al. [11; 12; 13] also applied the EBWM concept to nonlinear seismic analysis. Based on the results of an experimental study on 40 interior column-slab joints and 41 exterior column-slab joints in flat-plate structures, they proposed a modified calculation approach  
Figure 5: Definition of EBWM. for modeling the effective beam width factor. They indicated that the effective beam width factor could be expressed as a function of column and slab aspect ratios and calculated using elastic solutions. [PERSON] and [PERSON] [14] also evaluated the EBWM concept with experimental and numerical results and consequently indicated that the EBWM concept could provide reasonable and suitable predictions for the estimation of stiffness within the elastic range. They suggested that the stiffness of the column-slab joints was independently affected by the slab full-width and the column depth.