Equivalent Lateral Forces for Design of Multistory Sliding Base Structures

Abstract

In Chap. 4, the peak superstructure responses of single-story sliding base (SB) structures subjected to three-component earthquake excitation were studied, and simplified equations for estimating the peak superstructure responses were also developed. For multistory SB structures, these equations may still be used to determine the total earthquake force acting on the superstructure; however, they are not sufficient to determine the earthquake forces acting on each story of a multistory structure. For this reason, this chapter focuses on the development of equivalent lateral forces for the design of multistory SB structures.

5.1 Model Descriptions

In Chap. 4, the parameter mass ratio, $\alpha$, was introduced. This parameter is defined as the ratio between the superstructure mass and the overall mass of the SB structure. The response of the superstructure is heavily influenced by the parameter $\alpha$, as demonstrated in Chap. 4. The mass ratio of a multistory SB structure presented in Fig. 5.1 can be calculated by

$$ \alpha = \frac{{\sum\nolimits_{i = 1}^{N} {m_{i} } }}{{\sum\nolimits_{i = 1}^{N} {m_{i} } + m_{b} }} $$

(5.1)

Fig. 5.1

Schematic plot of a multistory SB structure

in which m_i is the mass of the ith floor; m_b represents the mass of the sliding base; and N corresponds to the story number.

Based on practical applications, a maximum of five stories can be considered as the upper limit for the story number of SB structures. For a building with N ≤ 5, it is reasonable to consider that the mass of every floor is equal. Thus, Eq. (5.1) becomes

$$ \alpha = \frac{Nm}{{Nm + m_{b} }} $$

(5.2)

in which m represents the mass of each floor. In practical applications, the value of $\alpha$ should be at least 0.5 as m is usually equal to or greater than m_b. Hence, the value of $\alpha$ can be considered not less than 0.5.

It is assumed that the stiffness of each story is equal. The parametric study in Chap. 4 demonstrates that the natural period ratio between the two orthogonal horizontal directions has a negligible impact on the superstructure responses. Therefore, it is assumed that the story stiffness in both the x and y directions is identical and represented as k. Given these assumptions, the fundamental period, T₁, of the superstructure is expressed as follows:

$$ T_{1} = 2\pi \sqrt{\frac{m}{Ck}} $$

(5.3)

in which the values of coefficient C are equal to 0.382, 0.198, 0.121, and 0.081 for N = 2, 3, 4, and 5, respectively. As Chap. 4 demonstrates, the superstructure responses in single-story SB structures remain basically unchanged when the superstructure period falls within the typical range, and it is feasible to conservatively represent the responses of potential SB structures with the response at the period that generally yields the highest superstructure responses. The truth of this result persists in multistory SB structures. The computed results based on T₁ = 0.3 s are used in the subsequent analyses as maximum superstructure responses occur mostly at T₁ close to 0.3 s.

The construction of the damping matrix involves the utilization of Rayleigh damping. For N = 2, the damping ratios for the first and second modes are taken as 5%, while for N = 3 and 4, the damping ratios for the first and third modes are 5%, and for N = 5, the damping ratios for the first and fourth modes are taken as 5%.

The static and dynamic friction coefficients are assumed to be the same as the difference between them was found to have little effect on the superstructure responses. In accord with Chap. 4, the maximum value of $a_{gx0} /\mu g$ is taken as 20, where a_gx0 is the peak value of the x component of the ground acceleration, $\mu$ is the friction coefficient, and g is the gravity acceleration.

It was found that the variance between the static and dynamic friction coefficients had minimal impact on the superstructure responses, thus they are assumed to be equal. In accordance with Chap. 4, an upper limit of 20 is placed on the value of $a_{gx0} /\mu g$.

5.2 Peak Base Shear

After obtaining the displacement history by performing response history analysis, the equivalent static forces in the x-direction, ${\mathbf{F}}_{x} = \left[ {F_{x1} ,F_{x2} , \ldots ,F_{xN} } \right]$ (F_xi is the force acting on the ith floor) for a multistory SB structure can be determined using the following equation (Chopra, 2001)

$$ {\mathbf{F}}_{x} \left( t \right) = {\mathbf{k}}_{x} {\mathbf{u}}_{rx} \left( t \right) $$

(5.4)

The peak base shear, V_bx, can subsequently be computed by

$$ V_{bx} = \mathop {\max }\limits_{t} \left| {{\mathbf{1}}^{T} {\mathbf{k}}_{x} {\mathbf{u}}_{rx} \left( t \right)} \right| $$

(5.5)

Because of the close relationship between the peak base shear and the mass of the superstructure, as well as the friction coefficient, the normalized peak base shear, $\overline{V}_{bx}$, is introduced as follows

$$ \overline{V}_{bx} = \frac{{V_{bx} }}{Nmg\mu } = \frac{{\mathop {\max }\limits_{t} \left| {{\mathbf{1}}^{T} {\mathbf{k}}_{x} {\mathbf{u}}_{rx} \left( t \right)} \right|}}{Nmg\mu } $$

(5.6)

For a single-story SB structure, Eq. (5.6) is simplified to

$$ \overline{V}_{bx} = \frac{{\mathop {\max }\limits_{t} \left| {ku_{rx} \left( t \right)} \right|}}{mg\mu } = \frac{{\omega_{x}^{2} \times \max \left( {\left| {u_{rx} (t)} \right|} \right)}}{\mu g} = \frac{{A_{x} }}{\mu g} $$

(5.7)

As indicated by Eq. (5.7), the normalized peak base shear, $\overline{V}_{bx}$ is equivalent to the normalized peak pseudoacceleration, $A_{x} /\mu g$, for single-story SB structures. The normalized peak pseudoacceleration has been thoroughly examined in Chap. 4. Hence, with certain modifications, the equations created for predicting $A_{x} /\mu g$ could be employed to predict $\overline{V}_{bx}$ for multistory SB structures.

By replacing $A_{x} /\mu g$ in Eqs. (4.10) and (4.11) with $\overline{V}_{bx}$, the following equations are yielded:

$$ \overline{V}_{bx} = \frac{{\beta_{1} \left( {a_{gx0} /\mu g} \right)^{{\beta_{2} }} }}{{\left( {a_{gx0} /\mu g} \right)^{{\beta_{2} }} + \beta_{3} }} $$

(5.8)

and

$$ \delta_{{\overline{V}_{bx} }} = \gamma_{1} \exp \left( { - \gamma_{2} \left( {a_{gx0} /\mu g} \right)} \right) + \gamma_{3} $$

(5.9)

in which $\delta_{{\overline{V}_{bx} }}$ represents the coefficient of variation (COVs) of $\overline{V}_{bx}$. In order to verify if Eqs. (5.8) and (5.9) are applicable to multistory SB structures, the mean values and COVs of $\overline{V}_{bx}$ of structures with varying story number N are compared in Fig. 5.2a, b, respectively, using the 180 non-pulse-like ground motion records for site classes C and D. As can be seen in Fig. 5.2a, when $a_{gx0} /\mu g$ is given, the mean value of $\overline{V}_{bx}$ decreases and ultimately approaches a constant value as N increases. In order to account for this impact, the introduction of a reduction factor, denoted as $\gamma_{N}$, is necessary. This factor is calculated by taking the ratio of the mean $\overline{V}_{bx}$ of a structure that has N stories, to the mean $\overline{V}_{bx}$ of a single-story structure that has the same $\alpha$ under the same $a_{gx0} /\mu g$. The analyzed data suggest that the influence of $\alpha$ and $a_{gx0} /\mu g$ on $\gamma_{N}$ is minimal. The nonlinear regression analyses yielded the following formula for calculating $\gamma_{N}$:

$$ \gamma_{N} = 0.25e^{ - 0.65N} + 0.86 $$

(5.10)

Fig. 5.2

Statistical values of $\overline{V}_{bx}$: a mean values; b COVs

Therefore, the mean value of $\overline{V}_{bx}$ for an N-story SB structure can be calculated by the following equation:

$$ \overline{V}_{bx} = \left( {0.25e^{ - 0.65N} + 0.86} \right)\frac{{\beta_{1} \left( {{{a_{gx0} } / {\mu g}}} \right)^{{\beta_{2} }} }}{{\left( {{{a_{gx0} } / {\mu g}}} \right)^{{\beta_{2} }} + \beta_{3} }} $$

(5.11)

The values of the coefficients $\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ are provided in Table 4.1. As depicted in Fig. 5.2b, The COVs of $\overline{V}_{bx}$ are not significantly affected by the story number N. Hence, Eq. (5.9) can be used for multistory SB structures without requirement of modifications. The values of $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ are provided in Table 4.2.

5.3 Equivalent Lateral Force Distribution

The response histories of V_xi/(Nmg) for a three-story SB structure (with $\alpha = 0.8$ and $\mu = 0.1$) and corresponding FB structure under the Mammoth Lakes record (a_gx0 = 0.39g) from the 1980 Mammoth Lakes earthquake are presented in Fig. 5.3, where V_xi is the story shear of the ith story. In the case of the FB structure, the peak story shears for various stories occur simultaneously. However, for the SB structure, the peak shear time for each story varies. Hence, it is not feasible to employ the distribution of equivalent static forces at the peak base shear to ascertain the peak shear of other stories.

Fig. 5.3

Response histories of each story shear of a three-story SB structure (with $\alpha = 0.8$ and $\mu = 0.1$) and b three-story FB structure subjected to the Mammoth Lakes record from the 1980 Mammoth Lakes earthquake

Peak story shears are the response quantities that need to be used in design. Therefore, the equivalent lateral forces were calculated through the following process: (1) Identify the peak story shear of each story for every ground motion; (2) Organize the peak story shears for each story from the 180 ground motion records in ascending order; (3) Determine equivalent lateral forces using the peak story shears at the corresponding percentile. The equivalent lateral force of the ith floor is represented as $F_{xi}^{e}$. The distributions of ${{F_{xi}^{e} } /{F_{xN}^{e} }}$ for various percentiles are illustrated in Fig. 5.4 when $\alpha = 0.8$. As shown in Fig. 5.4, the selected percentile does not have a significant impact on the distribution of ${{F_{xi}^{e} } / {F_{xN}^{e} }}$. Accordingly, the distribution of ${{F_{xi}^{e} } / {F_{xN}^{e} }}$ for the 50th percentile is selected for the subsequent analyses.

Fig. 5.4

Distributions of ${{F_{xi}^{e} } / {F_{xN}^{e} }}$ under different percentiles ($\alpha = 0.8$): a ${{a}}_{{{{gx}}0}} /\mu {{g}} = 6$; b $a_{{{{gx}}0}} /\mu {{g}} = 12$

5.3.1 Parametric Study

Figure 5.5 depicts the data points (${{F_{xi}^{e} } / {F_{xN}^{e} }}$, i/N) corresponding to different N in the same plot. As shown in this figure, the trend is almost the same for the data points (${{F_{xi}^{e} } / {F_{xN}^{e} }}$, i/N) for different N. Therefore, the same relationship between ${{F_{xi}^{e} } / {F_{xN}^{e} }}$ and i/N can be used for different N.

Fig. 5.5

Comparison of the relationships between ${{F_{xi}^{e} } / {F_{xN}^{e} }}$ and i/N for different N: a $\alpha = 0.5$; b $\alpha = 0.8$

The distributions of $F_{xi}^{e} /F_{xN}^{e}$ for various $\alpha$ and $a_{gx0} /\mu g$ are shown Figs. 5.6 and 5.7. The value of $F_{xi}^{e} /F_{xN}^{e}$ for i < N decreases with an increase in $\alpha$ or $a_{gx0} /\mu g$; and as $a_{gx0} /\mu g$ increases, the distribution of $F_{xi}^{e} /F_{xN}^{e}$ becomes fixed. When $a_{gx0} /\mu g$ exceeds a specific value, the corresponding lateral forces at the lower levels invert direction for $\alpha \ge 0.7$. These results means that the increase of $\alpha$ or $a_{gx0} /\mu g$ results in the concentration of equivalent lateral forces at the upper floors.

Fig. 5.6

Distributions of $F_{xi}^{e} /F_{xN}^{e}$ for a $\alpha = 0.5$; b $\alpha = 0.7$; c $\alpha = 0.8$; d $\alpha = 0.9$ ($N = 4$)

Fig. 5.7

Distributions of $F_{xi}^{e} /F_{xN}^{e}$ for a $a_{gx0} /\mu g = 2$; b $a_{gx0} /\mu g = 6$; c $a_{gx0} /\mu g = 12$; and d $a_{gx0} /\mu g = 20$ ($N = 4$)

5.3.2 Simplified Equations

Referring to the shapes of the $i/N - F_{xi}^{e} /F_{xN}^{e}$ curves depicted in Figs. 5.6 and 5.7, the distribution of equivalent lateral forces was modeled using the following equation:

$$ F_{xi}^{e} /F_{xN}^{e} = c(i/N)^{2} + (1 - c)(i/N) $$

(5.12)

The regression coefficient c in Eq. (5.12) is dependent on $a_{gx0} /\mu g$ and $\alpha$. Equation (5.12) satisfies the boundary conditions that when i = 0, $F_{xi}^{e} /F_{xN}^{e} = 0$, and when i = N, $F_{xi}^{e} /F_{xN}^{e} = 1$. The values of c attained from nonlinear regression analyses for varying values of $a_{gx0} /\mu g$ and $\alpha$ are presented in Fig. 5.8. The value of c increases and approaches a constant as $a_{gx0} /\mu g$ increases for a given $\alpha$, and it also increases with an increase in $\alpha$. These trends are in agreement with the effects of $a_{gx0} /\mu g$ and $\alpha$ on the distributions of $F_{xi}^{e} /F_{xN}^{e}$, which has been discussed in the previous section. Through nonlinear regression analyses, the following equation was developed for calculating the value of c:

$$ c = \left( { - \,4.3\alpha + 0.29} \right)e^{{ - 0.68\left( {a_{gx0} /\mu g} \right)}} + 3.4\alpha - 0.95 $$

(5.13)

Fig. 5.8

Values of c for different values of $a_{gx0} /\mu g$ and $\alpha$

As depicted in Figs. 5.6 and 5.7, the distributions of $F_{xi}^{e} /F_{xN}^{e}$ determined using Eqs. (5.12) and (5.13) demonstrate good agreement with those that were computed based on response history analyses.

5.4 Conclusions

In this chapter, the equivalent lateral forces for the design of multistory SB structures are studied. The increase in story number N tends to reduce the mean value of the normalized peak base shear but has little effect on the coefficient of variation. The peak story shears of a multistory SB structure under an earthquake ground motion occur at varying times for each story. Based on this truth, the equivalent lateral forces can be calculated by utilizing the peak story shears that correspond to the same percentile. Based on the computed results, it can be inferred that the distribution of equivalent lateral forces is generally not influenced by the percentile chosen. The distributions of equivalent lateral forces for various N follow a similar pattern. Additionally, with an increase in the normalized PGA and mass ratio, the equivalent lateral forces tend to concentrate at the upper floors. Using Eqs. (5.9), (5.11), (5.12), and (5.13), the equivalent lateral forces required for the design of multistory SB structures can be determined.