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Prediction of the largest peak nonlinear seismic response of asymmetric buildings under bi-directional excitation using pushover analyses

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Abstract

A simplified procedure is proposed to predict the largest peak seismic response of an asymmetric building to horizontal bi-directional ground motion, acting at an arbitrary angle of incidence. The main characteristics of the proposed procedure is as follows. (1) The properties of two independent equivalent single-degree-of-freedom models are determined according to the principal direction of the first modal response in each nonlinear stage, rather than according to the fixed axis based on the mode shape in the elastic stage; the principal direction of the first modal response in each nonlinear stage is determined based on pushover analysis results. (2) The bi-directional horizontal seismic input is simulated as identical spectra of the two horizontal components, and the contribution of each modal response is directly estimated based on the unidirectional response in the principal direction of each. (3) The drift demand at each frame is determined based on four pushover analyses considering the combination of bi-directional excitations. In the numerical example, nonlinear time-history analyses of six four-story torsionally stiff (TS) asymmetric buildings are carried out considering various directions of seismic inputs, and these results are compared with the predicted results. The results show that the proposed procedure satisfactorily predicts the largest peak response displacement at the flexible-side frame of a TS asymmetric building.

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Acknowledgments

The author thanks the two anonymous reviewers who provided considerable help in improving the content and text of the original manuscript.

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Correspondence to Kenji Fujii.

Appendices

Appendix 1: Formulation of the equivalent SDOF model considering bi-directional excitations

Considering a set of orthogonal \(\xi \)- and \(\zeta \)-axes in the X–Y plane with an angle \(\psi \) as shown in Fig. 18, the equation of motion for an \(N\)-story asymmetric frame building model can be written as Eq. (25).

$$\begin{aligned} \mathbf{M}\ddot{\mathbf{d}}(t)+\mathbf{C}\dot{\mathbf{d}}(t)+\mathbf{f}_\mathbf{R} (t)=-\mathbf{M}\left\{ {\varvec{\upalpha }_{\varvec{\xi }} a_{g\xi } (t)+\varvec{\upalpha }_{\varvec{\zeta }} a_{g\zeta } (t)} \right\} , \end{aligned}$$
(25)

where

$$\begin{aligned} \varvec{\upalpha }_{\varvec{\xi }}&= \left\{ {\cos \psi } \quad \cdots \quad {\cos \psi } \quad {-\sin \psi } \quad \cdots \quad {-\sin \psi } \quad 0 \quad \cdots \quad 0 \right\} ^{\mathbf{T}},\end{aligned}$$
(26)
$$\begin{aligned} \varvec{\upalpha }_{\varvec{\zeta }}&= \left\{ {\sin \psi } \quad \cdots \quad {\sin \psi } \quad {\cos \psi } \quad \cdots \quad {\cos \psi } \quad 0 \quad \cdots \quad 0 \right\} ^{\mathbf{T}}. \end{aligned}$$
(27)

In Eq. (25), C is the damping matrix, and \(a_{g\xi }(\hbox {t})\) and \(a_{g\zeta }(\hbox {t})\) are the ground accelerations of the \(\xi \)- and \(\zeta \)-axis components, respectively. Let the U-axis be the principal axis of the first modal response, while the V-axis is orthogonal to the U-axis. The tangent of \(\psi _{1}\), the angle of incidence of the U-axis with respect to the X-axis, is determined from Eq. (28).

$$\begin{aligned} \tan \psi _1 =-\sum \limits _j {m_j \phi _{Yj1}}\Big /\sum \limits _j {m_j \phi _{Xj1}} \end{aligned}$$
(28)

Assume that the building oscillates predominantly in the first mode under U-directional (unidirectional) excitation, and predominantly in the second mode under V-directional excitation. Under bi-directional excitation, it is assumed that displacement \(\mathbf{d}(t)\) and the restoring force \(\mathbf{f}_\mathbf{R } (t)\) can be written in the form of Eqs. (29) and (30), respectively, even if the building oscillates beyond the elastic range.

$$\begin{aligned} \mathbf{d}(t)&= \Gamma _{1U} \varvec{\upvarphi }_\mathbf{1} D_{1U}^{*}(t)+\Gamma _{2V} \varvec{\upvarphi }_\mathbf{2} D_{2V}^{*}(t),\end{aligned}$$
(29)
$$\begin{aligned} \mathbf{f}_\mathbf{R} (t)&= \mathbf{M}\left\{ {\Gamma _{1U} \varvec{\upvarphi }_\mathbf{1} A_{1U}^{*}(t)+\Gamma _{2V} \varvec{\upvarphi }_\mathbf{2} A_{2V}^{*}(t)} \right\} ,\end{aligned}$$
(30)
$$\begin{aligned} \Gamma _{1U}&= \frac{\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{U}}{\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M\phi }_\mathbf{1}}, \quad \Gamma _{2V} =\frac{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{V}}{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upvarphi }_\mathbf{2}}, \end{aligned}$$
(31)

where

$$\begin{aligned} \varvec{\upalpha }_{\mathbf{U}}&= \left\{ {\cos \psi _{1}} \quad \cdots \quad {\cos \psi _{1}} \quad {-\sin \psi _{1}} \quad \cdots \quad {-\sin \psi _{1}} \quad 0 \quad \cdots \quad 0 \right\} ^{\mathbf{T}},\end{aligned}$$
(32)
$$\begin{aligned} \varvec{\upalpha }_{\mathbf{V}}&= \left\{ {\sin \psi _{1}} \quad \cdots \quad {\sin \psi _{1}} \quad {\cos \psi } \quad \cdots \quad {\cos \psi _{1}} \quad 0 \quad \cdots \quad 0 \right\} ^{\mathbf{T}}, \end{aligned}$$
(33)

It is also assumed that Eqs. (29) and (30) are still valid in the nonlinear stage if the change in mode shape is properly taken into account, and the U-axis is determined based on the first mode vector in the nonlinear stage. By substituting Eqs. (29) and (30) into Eqs. (25) and (34) is obtained:

$$\begin{aligned}&\mathbf{M}\left\{ {\Gamma _{1U} \varvec{\upvarphi }_\mathbf{1} \ddot{D}_{1U}^{*}(t)+\Gamma _{2V} \varvec{\upvarphi }_\mathbf{2} \ddot{D}_{2V}^{*}(t)} \right\} +\mathbf{C}\left\{ {\Gamma _{1U} \varvec{\upvarphi }_\mathbf{1} \dot{D}_{1U}^{*}(t)+\Gamma _{2V} \varvec{\upvarphi }_\mathbf{2} \dot{D}_{2V}^{*}(t)} \right\} \nonumber \\&\qquad +\mathbf{M}\left\{ {\Gamma _{1U} \varvec{\upvarphi }_\mathbf{1} A_{1U}^{*}(t)+\Gamma _{2V} \varvec{\upvarphi }_\mathbf{2} A_{2V}^{*}(t)} \right\} \nonumber \\&\quad =-\mathbf{M}\left\{ {\varvec{\upalpha }_{\varvec{\xi }} a_{g\xi } (t)+\varvec{\upalpha }_{\varvec{\zeta }} a_{g\zeta } (t)} \right\} \end{aligned}$$
(34)

By multiplying \(\Gamma _{1U} \varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\) from the left side of Eq. (34) and considering Eqs. (35) through (38), the equation of motion for the equivalent SDOF model representing the first modal response is obtained as Eq. (39).

$$\begin{aligned}&M_{1U}^{*}=\Gamma _{1U}^{2}\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upvarphi }_\mathbf{1}, \quad C_{1U}^{*}=\Gamma _{1U}^{2}\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{C}\varvec{\upvarphi }_\mathbf{1},\end{aligned}$$
(35)
$$\begin{aligned}&\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upvarphi }_\mathbf{2} =0,\quad \varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{C}\varvec{\upvarphi }_\mathbf{2} \approx 0,\end{aligned}$$
(36)
$$\begin{aligned}&\cos \Delta \psi =\cos (\psi -\psi _{1})=\frac{\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_{\varvec{\xi }}}{\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{U}},\end{aligned}$$
(37)
$$\begin{aligned}&\sin \Delta \psi =\sin (\psi -\psi _{1})=\frac{\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_{\varvec{\zeta }}}{\varvec{\upvarphi }_\mathbf{1}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{U}},\end{aligned}$$
(38)
$$\begin{aligned}&\ddot{D}_{1U}^{*}(t)+\frac{C_{1U}^{*}}{M_{1U}^{*}}\dot{D}_{1U}^{*}(t)+A_{1U}^{*}(t)=-\left\{ {a_{g\xi } (t)\cos \Delta \psi +a_{g\zeta } (t)\sin \Delta \psi } \right\} . \end{aligned}$$
(39)

In Eq. (35), \(M_{1U}^{*}\) and \(C_{1U}^{*}\) are the first modal mass and the first modal damping coefficient, respectively. The ground acceleration component in the U-axis, \(a_{gU}(t)\), is defined as:

$$\begin{aligned} a_{gU} (t)=a_{g\xi } (t)\cos \Delta \psi +a_{g\zeta } (t)\sin \Delta \psi . \end{aligned}$$
(40)

Therefore, Eq. (39) can be rewritten in a simplified form as:

$$\begin{aligned} \ddot{D}_{1U}^{*}(t)+\frac{C_{1U}^{*}}{M_{1U}^{*}}\dot{D}_{1U}^{*}(t)+A_{1U}^{*}(t)=-a_{gU} (t). \end{aligned}$$
(41)

To derive the equation of motion for the equivalent SDOF model representing the second modal response, the tangent of \(\psi _{2}\), the angle of incidence for the principal direction of the second modal response with respect to the X-axis, is formulated as:

$$\begin{aligned} \tan \psi _2 =-{\sum \limits _j {m_j \phi _{Yj2}}}/{\sum \limits _j {m_j \phi _{Xj2}}}. \end{aligned}$$
(42)

By multiplying \(\Gamma _{2V} \varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\) from the left side of Eq. (34) and considering Eqs. (36) and (43), Eq. (44) is obtained.

$$\begin{aligned}&M_{2V}^{*}=\Gamma _{2V}^{2}\varvec{\upvarphi }_2^{\mathbf{T}}\mathbf{M}\varvec{\upvarphi }_2,\quad C_{2V}^{*}=\Gamma _{2V}^{2}\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{C}\varvec{\upvarphi }_\mathbf{2},\end{aligned}$$
(43)
$$\begin{aligned}&\ddot{D}_{2V}^{*}(t)+\frac{C_{2V}^{*}}{M_{2V}^{*}}\dot{D}_{2V}^{*}(t)+A_{2V}^{*}(t)=-\left\{ {\frac{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_{\varvec{\xi }}}{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{V}}a_{g\xi } (t)+\frac{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_{\varvec{\zeta }}}{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{V}}a_{g\zeta } (t)} \right\} . \end{aligned}$$
(44)

In Eq. (43), \(M_{2V}^{*}\) and \(C_{2V}^{*}\) are the second modal mass and the second modal damping coefficient, respectively. It is assumed that the principal directions of the first and second modal responses are mutually close to orthogonal. This assumption can be expressed as:

$$\begin{aligned} \left( {\tan \psi _1} \right) \left( {\tan \psi _2} \right) \approx -1. \end{aligned}$$
(45)

In other words, the principal axis of the second modal response is close to the V-axis. From Eqs. () and (45), Eq. (46) is obtained.

$$\begin{aligned} {\sum \limits _j {m_j \phi _{Yj2}}}/{\sum \limits _j {m_j \phi _{Xj2}}}=-\tan \psi _2 \approx -\frac{1}{\tan \psi _1}=\frac{\cos \psi _1}{\sin \psi _1}. \end{aligned}$$
(46)

Therefore, considering Eq. (47), Eqs. (48) and (49) can be derived.

$$\begin{aligned} \frac{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_{\varvec{\xi }}}{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{V}}&= \frac{\sum \nolimits _j {m_j \phi _{Xj2}} \cos \psi -\sum \nolimits _j {m_j \phi _{Yj2}} \sin \psi }{\sum \nolimits _j {m_j \phi _{Xj2}} \sin \psi _1 +\sum \nolimits _j {m_j \phi _{Yj2}} \cos \psi _1}\approx -\sin (\psi -\psi _{1})=-\sin \Delta \psi ,\end{aligned}$$
(47)
$$\begin{aligned} \frac{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_{\varvec{\xi }}}{\varvec{\upvarphi }_\mathbf{2}^{\mathbf{T}}\mathbf{M}\varvec{\upalpha }_\mathbf{V}}&= \frac{\sum \nolimits _j {m_j \phi _{Xj2}} \sin \psi -\sum \nolimits _j {m_j \phi _{Yj2}} \cos \psi }{\sum \nolimits _j {m_j \phi _{Xj2}} \sin \psi _1 +\sum \nolimits _j {m_j \phi _{Yj2}} \cos \psi _1}\approx -\cos (\psi -\psi _{1})=\cos \Delta \psi ,\nonumber \\ \end{aligned}$$
(48)

Substituting Eqs. (47) and (48) into Eq. (44) and considering Eq. (49), the equation of motion of the equivalent SDOF model representing the second modal response is obtained as Eq. (50).

$$\begin{aligned}&a_{gV} (t)=-a_{g\xi } (t)\sin \Delta \psi +a_{g\zeta } (t)\cos \Delta \psi ,\end{aligned}$$
(49)
$$\begin{aligned}&\ddot{D}_{2V}^{*}(t)+\frac{C_{2V}^{*}}{M_{2V}^{*}}\dot{D}_{2V}^{*} (t)+ A_{2V}^{*}(t)=-{\alpha }_{gV}(t). \end{aligned}$$
(50)

In Eq. (49), \(a_{gV}(t)\) is the ground acceleration component along the V-axis.

Fig. 18
figure 18

Plan of the asymmetric buildings and corresponding equivalent SDOF model. a Plan of asymmetric building structure. b Equivalent SDOF model (first mode)

As described in Sect. 2, it is assumed that the spectra of the two horizontal ground motion components are identical. From this assumption, the relationship for the response acceleration spectra of the U- and V-components \(S_{AU}(T)\) and \(S_{AV}(T)\) is expressed as:

$$\begin{aligned} S_{AU} (T)=S_{AV} (T)=S_{A\xi } (T)=S_{A\zeta } (T). \end{aligned}$$
(51)

In Eq. (51), \(S_{A\xi }(T)\) and \(S_{A\zeta }(T)\) are the response acceleration spectra of the \(\xi \)- and \(\zeta \)- components, respectively. Therefore, the same response spectra are used to predict the peak response of the first and second mode.

Appendix 2: The flow of “the displacement-based mode-adaptive pushover analysis”

In the pushover analysis, which is referred to as the “displacement-based mode-adaptive pushover analysis”, the following assumptions are made.

  1. (1)

    One-component model, with two nonlinear flexural springs at both ends and one nonlinear shear spring in the middle of the line element, is applied to all members. The envelope curve for each nonlinear spring of all members is symmetric over the positive and negative ranges.

  2. (2)

    The equivalent stiffness of each nonlinear spring can be defined by its secant stiffness at the peak deformation previously derived in the calculation.

  3. (3)

    The first mode shape at each loading stage \({}_\mathbf{n}\varvec{\upvarphi }_\mathbf{1}\) can be determined based on the equivalent stiffness.

  4. (4)

    The deformation shape imposed on a model is similar to the first mode shape obtained in (2) and (3).

Figure 19 shows a flow chart of the displacement-based mode-adaptive pushover analysis procedure applied in this paper. The main difference between the present analysis and the pushover analysis proposed by Antoniou and Pinho (2004) is that in the present analysis the secant stiffness of each element is used to determine the mode shape and the displacement shape (not the displacement increment) at each nonlinear stage. Antoniou and Pinho use the tangent stiffness of each element to determine the mode shape and the displacement increment in their analysis.

Fig. 19
figure 19

Flow chart for the displacement-based mode-adaptive pushover analysis procedure

Appendix 3: Formulation of the torsional index based on mode shape

The \(k\)th equivalent modal mass with respect to the \(k\)th principal direction of the modal response \(M_{k}^{*}\) is expressed as Eq. (52). Assuming from Eq. (51) that the \(k\)th mode is purely translational \((\phi _{\Theta jk} = 0),\) the \(k\)th equivalent modal mass ignoring the rotational component \(M_{kT}^{*}\) can be expressed as Eq. (53).

$$\begin{aligned} M_k^{*}&= \frac{\left( {\sum _j {m_j \phi _{Xjk}}}\right) ^{2}+\left( {\sum _j {m_j \phi _{Yjk}}} \right) ^{2}}{\sum _j {m_j \phi _{Xjk}^{2}} +\sum _j {m_j \phi _{Yjk}^{2}} +\sum _j {I_j\phi _{\Theta jk}^{2}}},\end{aligned}$$
(52)
$$\begin{aligned} M_{kT}^{*}&= \frac{\left( {\sum _j {m_j \phi _{Xjk}}}\right) ^{2}+\left( {\sum _j {m_j \phi _{Yjk}}} \right) ^{2}}{\sum _j {m_j \phi _{Xjk}^{2}} +\sum _j {m_j \phi _{Yjk}^{2}}}. \end{aligned}$$
(53)

From Eqs. (52) and (53), the ratio \((M_{kT}^{*}/M_{k}^{*})\) is obtained as:

$$\begin{aligned} \frac{M_{kT}^{*}}{M_k^{*}}=\frac{\sum _j {m_j \phi _{Xjk}^{2}} +\sum _j {m_j \phi _{Yjk}^{2}}}{\sum _j {m_j \phi _{Xjk}^{2}} +\sum _j {m_j \phi _{Yjk}^{2}} +\sum _j {I_j \phi _{\Theta jk}^{2}}}. \end{aligned}$$
(54)

In Eq. (54), the ratio \((M_{kT}^{*}/M_{k}^{*})\) is the reduction ratio of the \(k\)th equivalent modal mass resulting from the rotational component; if the \(k\)th mode is purely translational, the ratio \((M_{kT}^{*}/M_{k}^{*})\) is unity, while if it is a purely torsional mode, the ratio \((M_{kT}^{*}/M_{k}^{*})\) is zero. Equation (54) can be rewritten as Eq. (55), considering the torsional index of the \(k\)th mode, \(R_{\rho k}\), defined by Eq. (56), which is identical to Eq. (22).

$$\begin{aligned}&\displaystyle \frac{M_{kT}^{*}}{M_k^{*}}=\frac{1}{1+R_{\rho k}^{2}},&\end{aligned}$$
(55)
$$\begin{aligned}&\displaystyle R_{\rho k} =\sqrt{\sum \limits _j {I_j \phi _{\Theta jk}^{2}}\Big /\left( \sum \limits _j {m_j \phi _{Xjk}^{2}} +\sum \limits _j {m_j \phi _{Yjk}^{2}}\right) }.&\end{aligned}$$
(56)

From Eq. (55), it can be seen that the ratio \((M_{kT}^{*}/M_{k}^{*})\) is unity when \(R_{\rho k}\) is zero (purely translational), while \((M_{kT}^{*}/ M_{k}^{*})\) is close to zero when \(R_{\rho k}\) is significantly large. Therefore, the terms “predominantly translational” and “predominantly torsional” can be defined by the value of \(R_{\rho k}\); when \(R_{\rho k} < 1,\) the mode is “predominantly translational” and when \(R_{\rho k}> 1,\) the mode is “predominantly torsional”. Note that the index \(R_{\rho k}\) can be used for both single-story and multi-story irregular buildings.

Appendix 4: Classification of structural systems as torsionally stiff (TS) or torsionally flexible (TF)

In general, the classification of structural systems as TS or TF systems is based on the ratio of uncoupled torsional to lateral frequencies \(\Omega _{\theta }\) of the corresponding torsionally balanced system (e.g., Hejal and Chopra 1987). However, here the classification is made based on the first and the second modes because the ratio \(\Omega _{\theta }\) can be rigorously evaluated only for one-story asymmetric buildings (and multi-story asymmetric buildings that satisfy certain conditions). In other words, the classification is made using the torsional indices of the first and second modes, \(R_{\rho 1}\) and \(R_{\rho 2}\).

Bosco et al. (2013) proposed a method to evaluate the static eccentricity and the ratio \(\Omega _{\theta }\) of multi-story asymmetric buildings from static analyses. In this appendix, their method is applied to the six four-story building models investigated in this article and the ratio \(\Omega _{\theta }\) in each story is evaluated.

The evaluated results are shown in Table 4. The ratio of uncoupled torsional to lateral frequencies in the X- and Y-direction, \(\Omega _{\theta X}\) and \(\Omega _{\theta Y}\), respectively, are larger than 1 for all the stories in the four building models (Models A1, A2, B2, and B3). Therefore, from the classification based on \(\Omega _{\theta }\), these four building models are classified as TS systems in both directions. However, in the case of Models B1 and B4, the classification based on \(\Omega _{\theta }\) is difficult because in the first story \(\Omega _{\theta X}\) and \(\Omega _{\theta Y}\) are smaller than 1, while in the upper two stories \(\Omega _{\theta X}\) and \(\Omega _{\theta Y}\) are larger than 1. Conversely, the classification based on \(R_{\rho 1}\) and \(R_{\rho 2}\) is quite clear, as shown in Fig. 8; for all building models,\(R_{\rho 1}\) and \(R_{\rho 2}\) are smaller than 1. Therefore, the author believes that for a multi-story building the classification of the structural system as a TS or TF system should be made based on the mode shape of the first and second modes.

Table 4 Ratio \(\Omega _{\theta }\) in each story of building models evaluated based on Bosco et al. (2013)

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Fujii, K. Prediction of the largest peak nonlinear seismic response of asymmetric buildings under bi-directional excitation using pushover analyses. Bull Earthquake Eng 12, 909–938 (2014). https://doi.org/10.1007/s10518-013-9557-x

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