The torsional response of plan-asymmetric structures in the inelastic range becomes quite different from that in the linear elastic range. In the linear elastic range, the rotation of the asymmetric structure increases in proportion to the level of the lateral load applied, but if any of the lateral force resisting walls start to yield, there is change in the overall torsional stiffness of the structure, which leads to a change in the eccentricity value as well. This again affects the torsional moment induced in the asymmetric structure, and thus the rotation of the structure increases in a nonlinear fashion in the inelastic range.
To accurately assess the inelastic torsional response of the structure, we derived the generalized form of the inelastic lateral displacement-rotation relation of the structure by following a step by step procedure, and we represent this relation in the lateral displacement-rotation coordinate system in this section. According to the authors’ knowledge, this coordinate system has never been introduced before and is very convenient to describe the inelastic torsional behavior of the asymmetric structure. In addition, we discuss the effects of transverse wall stiffness on the yielding procedure of asymmetric structures.
3.1. Torsional Response of Asymmetric Structures in the D-R Coordinate System
In this section, the generalized form of the displacement-rotation relation of plan-asymmetric structures is derived by assuming that the structure is in the linear elastic range. From this relation, we can derive the inelastic displacement-rotation relation of plan-asymmetric structures.
Figure 1 illustrates a representative asymmetric wall structure subjected to a lateral force in the
y-direction (
V) as well as its deformed shape. In the figure, Wall X
i and Wall Y
i are the
i-th lateral load resisting walls in the
x- and
y-directions, respectively. The
x-coordinate of Wall Y
i and
y-coordinate of Wall X
i with respect to the center of rigidity (CR) in the undeformed configuration are denoted as
ryi and
rxi, respectively. Similarly, the same quantities with respect to the center of mass (CM) are expressed as
dyi and
dxi, respectively.
The lateral force (
V) and torsional moment (
T) of the system can be calculated by Equations (15) and (16), respectively:
Here,
Ky and
KT are the lateral stiffness in the
y-direction and torsional stiffness of the asymmetric system, respectively. In addition, Δ and
ϕ are the lateral displacement and rotation, respectively, and they are estimated at the center of mass. The counterclockwise direction of
ϕ is assumed to be positive. In the derivation of Equation (16), we assumed that the diaphragm has an infinite rigidity, and the out-of-plane deformation of the shear wall is ignored. If no external torsional moment is applied, the resulting torsional moment of the asymmetric system can be determined by multiplying the lateral force and the eccentricity in the
x-direction between the centers of mass and rigidity (
ex), as given below:
The eccentricity in the
x-direction can be computed by
where
kyi is the lateral stiffness of the
i-th wall in the
y-direction (Wall Y
i).
By substituting Equations (15) and (16) into Equation (17), the rotation can be expressed in terms of the lateral displacement as
The lateral and torsional stiffness of the asymmetric structure can be estimated by
where
kxi represents the lateral stiffness of the
i-th wall in the
x-direction. By inserting Equations (20) and (21) into Equation (19), the relationship between the lateral displacement and rotation can be obtained as follows:
As shown in
Figure 2, this relationship can be represented by a solid line with an arrow on the lateral displacement (Δ) and rotation (
ϕ) coordinate system, which is denoted as the D-R coordinate system in this paper. This indicates that a linear relation holds between the lateral displacement and the rotation as long as no wall component yields. The yield condition for Wall Y
1 can be stated as
where
uy is the yield displacement of Wall Y
1. The figure shows that the yielding of Wall Y
1 occurs at the intersection of Equations (22) and (23). As will be discussed later, this coordinate system is very convenient to describe the inelastic torsional behavior of plan-asymmetric structures.
3.2. Inelastic Torsional Response Assessment Procedure
In this section, we discuss a procedure to assess the inelastic torsional response of plan-asymmetric wall structures. If any wall components yield during the application of lateral force, the linear relationship between the lateral displacement and rotation is no longer valid, and thus it must be modified by considering the reduced stiffness of the wall component that has yielded.
Figure 3 illustrates a simple model problem of plan-asymmetric wall structures to explain the assessment procedure of the inelastic torsional response. This indicates that the structure has two lateral force resisting walls in the
x- and
y-directions, respectively, and the lateral force is applied in the
y-direction. The two walls in the
y-direction (Walls Y
1 and Y
2) have elastic lateral stiffnesses of
k and
αk (
α > 1), respectively, and thus Wall Y
2 is stiffer than Wall Y
1. The two walls in the
x-direction have the same elastic lateral stiffness
βk. Here,
β can be of any value, but it may affect the inelastic behavior of the asymmetric structure after yielding of Walls Y
1 and Y
2, as will be discussed later. For simplicity, the out-of-plane deformation of the walls is ignored.
The post yielding behavior of the two lateral force resisting walls Y
1 and Y
2 is defined as a bilinear function, as shown in
Figure 4. They have the same yield displacement (
), and their post-yield stiffness is
γ times the elastic stiffness (
). In contrast, we assumed that the other two walls in the transverse direction (Walls X
1 and X
2) remain in the linear elastic range even after the two lateral force resisting walls yield. The inelastic lateral-torsional response of this structure can be obtained by following the step-by-step procedure described below.
(1) Linear elastic stage
In the linear elastic stage, the lateral displacement and rotation of the asymmetric structure can be computed by Equation (19), where the lateral and torsional stiffnesses and the static eccentricity are estimated by Equations (24)–(26):
In the above equations, the superscript 0 indicates the linear elastic stage. Finally, Equation (19) can be rewritten as
where
and
denote the lateral displacement and rotation of the asymmetric structure in the linear elastic stage, respectively.
The deformation history in the displacement-rotation coordinate system and location of the center of rigidity in the linear elastic range are provided in
Figure 5. As shown in the figure, the displacement-rotation relationship in the linear elastic stage defined by Equation (27) is represented by a line with an arrow.
(2) First yielding point
As the lateral force increases, the wall on the flexible side yields first since (i) it has greater displacement than the one on the stiff side and (ii) the two walls have the same yield displacement. The wall on the flexible side starts to yield if its
y-displacement reaches the yield displacement, and this yield limit can be stated by
Consequently, the first yielding point can be defined as the intersection of the lines represented by Equations (27) and (28), as shown in
Figure 5. This first yielding point (
YP1) can be obtained by inserting Equation (28) into Equation (27), and its coordinates are given by
(3) First inelastic stage
After the yielding of the wall component on the flexible side, its lateral stiffness is reduced to
γ. From this, the lateral and torsional stiffnesses and the eccentricity of the asymmetric structure can be re-estimated as follows.
Here, superscript 1 indicates the first inelastic stage. Similarly, the relationship between the increments of the lateral displacement (
) and rotation (
) after the first yielding can be expressed by
(4) Second yielding point
A second yielding occurs if the lateral displacement of the wall on the stiff side also reaches the yield displacement with increasing lateral load. This limit state can be expressed by
Like the first yielding, the second yielding point can be obtained by finding the intersection of Equations (33) and (34). This procedure is also illustrated in
Figure 6. Thus,
and
can be expressed by
Finally, the coordinates of the second yielding point (
YP2) can be written as
(5) Second inelastic stage (final stage)
As in the case of the wall component on the flexible side, the lateral stiffness of the wall on the stiff side is also reduced to
γk. After yielding of the two wall components in the
y-direction, the modified lateral and torsional stiffnesses and the eccentricity of the asymmetric system can be respectively expressed by.
where the superscript 2 indicates the second inelastic stage. Note that the eccentricity in these equations is the same as the one in the linear elastic stage. This is mainly because the lateral stiffnesses of the two walls in the
y-direction are assumed to be reduced by the same rate after yielding and the transverse wall components in the
x-direction do not contribute to the
x-directional stiffness eccentricity. The transverse walls in the
x-direction still remain in the elastic stage and now offer relatively large torsional stiffness compared to the wall components in the
y-direction. Finally, the relationship between the increase in lateral displacement (
) and rotation (
) at this stage can be given by
and this is plotted in
Figure 7.
Figure 8 schematically shows the typical lateral-torsional responses of asymmetric structures obtained using three different approaches on the D-R coordinate system based on the discussion up to this point. In the figure, the dashed line represents the linear elastic torsional response given by Equation (22), and the dotted line is the amplified linear elastic torsional response, which is adopted in most of the current design codes in terms of the amplified design eccentricity, as already discussed in
Section 2. The solid line represents the inelastic torsional response based on the procedure presented in this section. A comparison of these three curves shows that the amplified linear-elastic response, which is adopted in most of the current design codes, may both underestimate the torsional rotation at the initial yielding stage and overestimate it at the post yielding stage, resulting in an inaccurate estimation of the torsional response over the entire inelastic range.
3.3. Effect of Transverse Wall Stiffness on Yielding Procedure
In the derivation introduced in
Section 3.2, we assumed for simplicity that the two lateral force resisting walls have the same yield displacement. As a result, the wall on the flexible side always yields first, and then the wall on the stiff side. However, wall stiffness and yield displacement are determined mainly by the length of the wall. According to [
24,
25], the stiffness (
k) and yield displacement (
uy) of the lateral force resisting wall can be determined by the following two relationships:
where
lw denotes the wall length. In this case, the stiffness of the transverse wall may affect the overall yielding procedure of asymmetric structures, and this issue is further discussed in this section.
An example problem illustrated in
Figure 9 is considered to explore this issue. This is similar to the model problem shown in
Figure 3 except that its floor plan has a square shape, not a rectangular shape. The lateral stiffness (
kw1), wall length (
lw1) and yield displacement (
uy,w1) of the flexible side wall Y
1 are denoted by
k,
lw and
uy, respectively. The same quantities for the stiff side wall Y
2 (
kw2,
lw2 and
uy,w2, respectively) are given by
where
α is greater than 1 and is the stiffness ratio of the stiff and flexible side walls. Note that the parameters given by Equations (44)–(46) satisfy the conditions stated by (42) and (43). We assumed that the two transverse walls in the
x-direction (Walls X
1 and X
2) have the same elastic lateral stiffness as
βk and never yield.
By using Equations (18)–(21), the relationship between lateral displacement and rotation angle of the example problem can be expressed by
From this equation, the displacement of the two lateral force resisting walls can be derived as
The flexible side wall yields first if the following condition is met:
By substituting Equations (48) and (49) into Equation (50), we obtain
The condition stated by Equation (51) is plotted in
Figure 10. The horizontal and vertical axes of the figure represent the total wall stiffness in the excitation and transverse directions, respectively. The results in the figure indicate that first yielding may occur even in the stiff side wall if the total transverse wall stiffness is located in the region above the curve plotted in the figure. For this reason, some researchers have classified plan-asymmetric structures into torsionally unrestrained (TU) and torsionally restrained (TR) structures to consider this effect [
11,
12,
13]. The inelastic torsional response assessment procedure proposed in this study can properly handle this effect without introducing the classification as in the previous research, but none of the current code provisions like IBC and Eurocode consider this effect.