Critical Correlation of Bidirectional Horizontal Ground Motions

Abstract

The ground motion is a realization in space and simultaneous consideration of multiple components of ground motion is realistic and inevitable in the reliable design of structures . It is often assumed practically that there exists a set of principal axes in the ground motions. It is well recognized in the literature that the principal axes are functions of time and change their directions during the ground shaking. In the current structural design practice, the effect of the multi-component ground motions is often taken into account by use of the SRSS method (square root of the sum of the squares) or the CQC3 method (extended Complete Quadratic Combination rule).

Appendices

Appendix 1: Computation of Coherence Function and Transformation of PSD Matrices

Let $\ddot{u}_{g1}$ and $\ddot{u}_{g2}$ denote the ground-motion accelerations along the building structural axes X ₁ and X ₂, respectively. Under the 2DGM along the principal axes of ground motions in the P–W model, $\ddot{u}_{g1}$ and $\ddot{u}_{g2}$ are described by

$$ \left\{ {\begin{array}{*{20}c} {\ddot{u}_{g1} } \\ {\ddot{u}_{g2} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\ddot{u}_{z1} } \\ {\ddot{u}_{z2} } \\ \end{array} } \right\}, $$

(11.21)

where $\ddot{u}_{z1}$ and $\ddot{u}_{z2}$ are the ground-motion accelerations along the principal axes of ground motions. θ denotes the angle between two sets of horizontal axes (= incident angle).

Let ${\mathbf{S}}_{{\ddot{Z}\ddot{Z}}} (\omega)$ denote the auto PSD matrix of the components along the principal axes of ground motions. Then the PSD matrix, consisting of S ₁₁, S ₂₂, S ₁₂, S ₂₁, of the components along the building structural axes may be described as

$$ {\mathbf{S}}_{{\ddot{X}\ddot{X}}} (\omega) = \left[{\begin{array}{*{20}c} {\cos \theta} & {\sin \theta} \\ {- \sin \theta} & {\cos \theta} \\ \end{array}} \right]{\mathbf{S}}_{{\ddot{Z}\ddot{Z}}} (\omega)\left[{\begin{array}{*{20}c} {\cos \theta} & {- \sin \theta} \\ {\sin \theta} & {\cos \theta} \\ \end{array}} \right] $$

(11.22)

$$ {\mathbf{S}}_{{\ddot{Z}\ddot{Z}}} (\omega) = \left[{\begin{array}{*{20}c} {S_{{\ddot{Z}_{1} \ddot{Z}_{1}}} (\omega)} & 0 \\ 0 & {S_{{\ddot{Z}_{2} \ddot{Z}_{2}}} (\omega)} \\ \end{array}} \right] $$

(11.23)

The coherence function between the components of ground motions along the building structural axes is defined by

$$ \rho_{12} = \frac{{E\left[{\ddot{u}_{g1} \ddot{u}_{g2}} \right]}}{{\sqrt {E\left[{\ddot{u}_{g1}^{2}} \right]E\left[{\ddot{u}_{g2}^{2}} \right]}}}, $$

(11.24)

where E[ · ] denotes the ensemble mean. It is assumed in the P–W model that there is no correlation between the 2DGM along the principal axes of ground motions (i.e. $E[\ddot{u}_{z1} \ddot{u}_{z2}] = 0$). Let γ _org denote the ratio of the auto PSD functions $S_{{\ddot{Z}_{2} \ddot{Z}_{2}}} (\omega)/S_{{\ddot{Z}_{1} \ddot{Z}_{1}}} (\omega)$ along the principal axes of ground motions. Substitution of $\ddot{u}_{g1}$ and $\ddot{u}_{g2}$ in Eq. (11.21) into Eq. (11.24) and some manipulations provide

$$ \rho_{12} (\gamma_{\text{org}},\theta) = \frac{{(1 - \gamma_{\text{org}})\sin 2\theta}}{{\sqrt {(1 + \gamma_{\text{org}})^{2} - (1 - \gamma_{\text{org}})^{2} \cos^{2} 2\theta}}} $$

(11.25)

Appendix 2: Horizontal Stiffness of Frame

Let u ₁ and ϕ _AB denote the horizontal displacement of the upper node in the frame and the angle of member rotation of column, respectively. When the horizontal force is denoted by P ₁, the horizontal stiffness of the plane frame can be expressed as

$$ k_{1} = P_{1}/u_{1} = P_{1}/(H \cdot \phi_{AB}) $$

(11.26)

The extreme-fiber stress at the top of the column under one-directional horizontal input may be expressed by

$$ \sigma_{BA}^{1} \left(t \right) = \{6EI_{b}/(Z_{c} L_{1})\} \theta_{B} $$

(11.27)

From the moment equilibrium around the node B, the angle of rotation of the node B can be expressed by

$$ \theta_{B} = 3\phi_{AB}/[2 + 3({{I_{b}} \mathord{\left/{\vphantom {{I_{b}} {I_{c}}}} \right. \kern-\nulldelimiterspace} {I_{c}}}) \cdot ({H \mathord{\left/{\vphantom {H {L_{1}}}} \right. \kern-\nulldelimiterspace} {L_{1}}})] $$

(11.28)

Equation (11.28) and the equation of story equilibrium provide

$$ \phi_{AB} = \frac{{P_{1} H^{2} \{2 + 3({{I_{b}} \mathord{\left/{\vphantom {{I_{b}} {I_{c}}}} \right. \kern-\nulldelimiterspace} {I_{c}}}) \cdot ({H \mathord{\left/{\vphantom {H L}} \right. \kern-\nulldelimiterspace} L}_{1})\}}}{{12EI_{c} \{1 + 6({{I_{b}} \mathord{\left/{\vphantom {{I_{b}} {I_{c}}}} \right. \kern-\nulldelimiterspace} {I_{c}}}) \cdot ({H \mathord{\left/{\vphantom {H L}} \right. \kern-\nulldelimiterspace} L}_{1})\}}} $$

(11.29)

Then the story stiffness can be expressed by

$$ k_{1} = \frac{{12EI_{c} \{1 + 6({{I_{b}} \mathord{\left/{\vphantom {{I_{b}} {I_{c}}}} \right. \kern-\nulldelimiterspace} {I_{c}}}) \cdot ({H \mathord{\left/{\vphantom {H {L_{1}}}} \right. \kern-\nulldelimiterspace} {L_{1}}})\}}}{{H^{3} \{2 + 3({{I_{b}} \mathord{\left/{\vphantom {{I_{b}} {I_{c}}}} \right. \kern-\nulldelimiterspace} {I_{c}}}) \cdot ({H \mathord{\left/{\vphantom {H L}} \right. \kern-\nulldelimiterspace} L}_{1})\}}} $$

(11.30)

Appendix 3: Stochastic Response 1

The auto-correlation function of $\sigma_{BA}^{1} (t)$ can be expressed by

$$ E\left[ {\sigma_{BA}^{1} (t_{1} )\sigma_{BA}^{1} (t_{2} )} \right] \hfill \\ = A_{{\sigma 1}}^{2} \int\limits_{0}^{{t_{1} }} {\int\limits_{0}^{{t_{2} }} {\left[ {c_{1} (\tau_{1} )c_{1} (\tau_{2} )g_{1} (t_{1} - \tau_{1} )g_{1} (t_{2} - \tau_{2} )E\left[ {w_{1} (\tau_{1} )w_{1} (\tau_{2} )} \right]} \right]d\tau_{1} d\tau_{2} } } $$

(11.31)

The auto-correlation function of w ₁(t) can be described in terms of the auto PSD function S ₁₁(ω) by

$$ E\left[{w_{1} (\tau_{1})w_{1} (\tau_{2})} \right] = \int\limits_{- \infty}^{\infty} {S_{11} (\omega)e^{{{\text{i}}\omega (\tau_{1} - \tau_{2})}} d\omega} $$

(11.32)

Equation (11.31) can then be modified to

$$ \begin{gathered} E\left[ {\sigma_{BA}^{1} (t_{1} )\sigma_{BA}^{1} (t_{2} )} \right] \hfill \\ = \int\limits_{ - \infty }^{\infty } {\left[ {\begin{array}{*{20}c} {{\text{A}}_{\sigma 1} \int\limits_{ 0}^{{{{t}}_{ 1} }} {{\text{c}}_{ 1} (\tau_{ 1} ){\text{g}}_{ 1} ({{t}}_{ 1} { - }\tau_{ 1} )(\cos \omega \tau_{ 1} {\text{ + i}}\sin \omega \tau_{ 1} ){\text{d}}\tau_{ 1} } } \\ { \times {\text{A}}_{\sigma 1} \int\limits_{ 0}^{{{{t}}_{ 2} }} {{\text{c}}_{ 1} (\tau_{ 2} ){\text{g}}_{ 1} ({{t}}_{ 2} { - }\tau_{ 2} )} (\cos \omega \tau_{ 2} {\text{ - i}}\sin \omega \tau_{ 2} ){\text{d}}\tau_{ 2} } \\ \end{array} } \right]} {\text{S}}_{ 1 1} (\omega ){\text{d}}\omega \hfill \\ \end{gathered} $$

(11.33)

By substituting t ₁ = t ₂ = t, τ ₁ = τ ₂ = τ in Eq. (11.33), the mean-squares $E[\sigma_{BA}^{1} (t)^{2}]$ can be derived as

$$ E[\sigma_{BA}^{1} (t)^{2} ] = A_{\sigma 1}^{2} \int\limits_{ - \infty }^{\infty } {\{ B_{c} (t;\omega )^{2} + B_{s} (t;\omega )^{2} \} } S_{11} (\omega )d\omega $$

(11.34)

where

$$ B_{c} (t;\omega ) \equiv \int\limits_{0}^{t} {c_{1} (\tau )g_{1} (t - \tau )\cos \omega \tau d\tau } $$

(11.35a)

$$ B_{s} (t;\omega ) \equiv \int\limits_{0}^{t} {c_{1} (\tau )g_{1} (t - \tau )\sin \omega \tau d\tau } $$

(11.35b)

On the other hand, the component in the direction X ₂ may be transformed into

$$ E\left[ {\sigma_{BA}^{2} (t_{1} )\sigma_{BA}^{2} (t_{2} )} \right]\\ = A_{{\sigma 2}}^{2} \int\limits_{0}^{{t_{1} }} {\int\limits_{0}^{{t_{2} }} {\left[ {c_{2} (\tau_{1} )c_{2} (\tau_{2} )g_{2} (t_{1} - \tau_{1} )g_{2} (t_{2} - \tau_{2} )E\left[ {w_{2} (\tau_{1} )w_{2} (\tau_{2} )} \right]} \right]} \,d\tau_{1} d\tau_{2} } $$

(11.36)

The auto-correlation function of w ₂(t) can be described in terms of the auto PSD function S ₂₂(ω) by

$$ E\left[ {w_{2} (\tau_{1} )w_{2} (\tau_{2} )} \right] = \int\limits_{ - \infty }^{\infty } {S_{22} (\omega )e^{{{\text{i}}\omega (\tau_{1} - \tau_{2} )}} d\omega } $$

(11.37)

The mean-squares $E[\sigma_{BA}^{2} (t)^{2}]$ can be derived as

$$ E[\sigma_{BA}^{2} (t)^{2} ] = A_{{\sigma 2}}^{2} \int\limits_{ - \infty }^{\infty } {\{ C_{c} (t;\omega )^{2} + C_{s} (t;\omega )^{2} \} } S_{22} (\omega )d\omega, $$

(11.38)

where

$$ C_{c} (t;\omega) \equiv \int_{0}^{t} {c_{2} (\tau)g_{2} (t - \tau)\cos \omega \tau d\tau} $$

(11.39a)

$$ C_{s} (t;\omega ) \equiv \int\limits_{0}^{t} {c_{2} (\tau )g_{2} (t - \tau )\sin \omega \tau d\tau } $$

(11.39b)

Appendix 4: Stochastic Response 2

The cross-correlation function of $\sigma_{BA}^{1} (t)$ and $\sigma_{BA}^{2} (t)$ can be expressed as

$$ E\left[ {\sigma_{BA}^{1} (t_{1} )\sigma_{BA}^{2} (t_{2} )} \right] \hfill \\ = A_{\sigma 1} A_{\sigma 2} \int\limits_{0}^{{t_{1} }} {\int\limits_{0}^{{t_{2} }} {\left[ {c_{1} (\tau_{1} )c_{2} (\tau_{2} )g_{1} (t_{1} - \tau_{1} )g_{2} (t_{2} - \tau_{2} )E\left[ {w_{1} (\tau_{1} )w_{2} (\tau_{2} )} \right]} \right]} d\tau_{1} d\tau_{2} } $$

(11.40)

The cross-correlation function of w ₁(t) and w ₂(t) can be described in terms of the cross PSD function S ₁₂(ω) by

$$ E\left[ {w_{1} (\tau_{1} )w_{2} (\tau_{2} )} \right] = \int\limits_{ - \infty }^{\infty } {S_{12} (\omega )e^{{{\text{i}}\omega (\tau_{1} - \tau_{2} )}} d\omega } $$

(11.41)

Let us introduce the definition of the cross PSD function $S_{12} (\omega) = C_{12} (\omega) + {\text{i}}Q_{12} (\omega)$.

Then Eq. (11.41) can be expressed by

$$ E[w_{1} (\tau_{1} )w_{2} (\tau_{2} )] = \int\limits_{ - \infty }^{\infty } {\{ C_{12} (\omega ) + {\text{i}}Q_{12} (\omega )\} e^{{{\text{i}}\omega (\tau_{1} - \tau_{2} )}} d\omega } $$

(11.42)

The cross-term can be modified into

$$ \begin{gathered} E\left[{\sigma_{BA}^{1} (t)\sigma_{BA}^{2} (t)} \right] \hfill \\ = A_{\sigma 1} A_{\sigma 2} \int\limits_{- \infty}^{\infty} {\left[\begin{gathered} \int\limits_{0}^{t} {c_{1} (\tau)g_{1} (t - \tau)(\cos \omega \tau + {\text{i}}\sin \omega \tau)d\tau} \hfill \\ \times \int\limits_{0}^{t} {c_{2} (\tau)g_{2} (t - \tau)(\cos \omega \tau - {\text{i}}\sin \omega \tau)d\tau} \left\{{C_{12} (\omega) + {\text{i}}Q_{12} (\omega)} \right\} \hfill \\ \end{gathered} \right]} \,d\omega \hfill \\ = A_{\sigma 1} A_{\sigma 2} \int\limits_{- \infty}^{\infty} {\left[{\left\{{B_{c} (t;\omega) + {\text{i}}B_{s} (t;\omega)} \right\}\left\{{C_{c} (t;\omega) - {\text{i}}C_{s} (t;\omega)} \right\}\left\{{C_{12} (\omega) + {\text{i}}Q_{12} (\omega)} \right\}} \right]} \,d\omega \hfill \\ \end{gathered} $$

(11.43)

Another cross-correlation function $E[\sigma_{BA}^{2} (t_{1})\sigma_{BA}^{1} (t_{2})]$ may be described by

$$ E[\sigma_{BA}^{2} (t)\sigma_{BA}^{1} (t)] \hfill \\ = A_{\sigma 2} A_{\sigma 1} \int\limits_{ - \infty }^{\infty } {\left[ {\{ C_{c} (t;\omega ) + {\text{i}}C_{s} (t;\omega )\} \{ B_{c} (t;\omega ) - {\text{i}}B_{s} (t;\omega )\} \{ C_{12} (\omega ) - {\text{i}}Q_{12} (\omega )\} } \right]} \,d\omega $$

(11.44)

By combining both cross terms, the corresponding term can be expressed finally by

$$ \begin{gathered} E[\sigma_{BA}^{1} (t)\sigma_{BA}^{2} (t)] + E[\sigma_{BA}^{2} (t)\sigma_{BA}^{1} (t)] \hfill \\ = 2A_{\sigma 1} A_{\sigma 2} \text{Re} \left[{\int\limits_{- \infty}^{\infty} {\left[{\{B_{c} (t;\omega) + {\text{i}}B_{s} (t;\omega)\} \{C_{c} (t;\omega) - {\text{i}}C_{s} (t;\omega)\} \{C_{12} (\omega) + {\text{i}}Q_{12} (\omega)\}} \right]} \,d\omega} \right] \hfill \\ = 2A_{\sigma 1} A_{\sigma 2} \int\limits_{- \infty}^{\infty} {\{f_{1} (t;\omega)C_{12} (\omega) + f_{2} (t;\omega)Q_{12} (\omega)\} d\omega} \hfill \\ \end{gathered} $$

(11.45)