Influence of movable bearings performance on the dynamic characteristics of a cable-stayed bridge : insights from seismic monitoring records

Abstract

An investigation on seismic response characteristics and modal parameters of a cable-stayed bridge with movable bearings using vibration records obtained from a seismic monitoring system is presented in this paper. The object of this study is the Tokachi cable-stayed bridge, a single plane prestressed cable-stayed bridge that is equipped with moveable spherical frictional plate (SFP). The bridge seismic responses subjected to various levels of earthquake including the 2003 Tokachi Great earthquake were obtained from a permanent seismic monitoring system. The bridge dynamic characteristics that include frequency and damping were estimated by both the time-invariant and time-variant recursive subspace identification methods. The identification results reveal that damping ratios of the low order modes estimated during the largest earthquake can reach more than 10% of the critical damping, which is higher than what usually assumed in the seismic design code. Supplementary damping caused by friction force at the movable SFP bearings was identified as the main source of the large damping, and it was quantified by a modal-based inverse analysis using the sparse regularization method. Results of the analyses confirm the influence of moveable SFP bearings performance with its Coulomb-friction type stick–slip behavior on the dynamic characteristics of the bridge. The bearings remain stuck at smaller excitation causing higher stiffness of the structure and slip during a larger excitation resulting in stiffness reduction. The sliding process induces friction force between contacting surfaces of the SFP bearings which results in the increase of damping as the excitation level increases. The damping, however, does not increase continuously, but becomes steadier after a certain level of excitation.

Appendix

$${\varvec{u}}_{i} \left( k \right) = \left[ {\begin{array}{*{20}c} {u\left( {k - i + 1} \right)^{T} } & {u\left( {k - i + 2} \right)^{T} \cdots u\left( {k - i} \right)^{T} } & {u\left( k \right)^{T} } \\ \end{array} } \right]^{T}$$

(A1)

$${\varvec{y}}_{i} \left( k \right) = \left[ {\begin{array}{*{20}c} {y\left( {k - i + 1} \right)^{T} } & {y\left( {k - i + 2} \right)^{T} \cdots y\left( {k - i} \right)^{T} } & {y\left( k \right)^{T} } \\ \end{array} } \right]^{T}$$

(A2)

$${\varvec{u}}_{i} \left( {k + 1} \right) = \left[ {\begin{array}{*{20}c} {u\left( {k - i + 2} \right)^{T} } & {u\left( {k - i + 3} \right)^{T} \cdots u\left( {k - i + 1} \right)^{T} } & {u\left( {k + 1} \right)^{T} } \\ \end{array} } \right]^{T}$$

(A3)

$${\varvec{y}}_{i} \left( {k + 1} \right) = \left[ {\begin{array}{*{20}c} {y\left( {k - i + 2} \right)^{T} } & {y\left( {k - i + 3} \right)^{T} \cdots y\left( {k - i + 1} \right)^{T} } & {y\left( {k + 1} \right)^{T} } \\ \end{array} } \right]^{T}$$

(A4)

$$[{\varvec{Y}}_{{\varvec{i}}} \left( {k{|}k - s + 1} \right)] = \left[ {{\varvec{y}}_{i} \left( {k - s + 1} \right) ; \begin{array}{*{20}c} {{\varvec{y}}_{i} \left( {k - s + 2} \right) ; \cdots ;\user2{ y}_{i} \left( {k - 1} \right);} & {{\varvec{y}}_{i} \left( k \right)} \\ \end{array} } \right]$$

(A5)

$$[{\varvec{U}}_{i}^{f} \left( {k{|}k - s + 1} \right)] = \left[ {\begin{array}{*{20}c} {u\left( {k - s - i + 1} \right)} & {u\left( {k - s - i + 2} \right)} & \cdots & {u\left( {k - i + 1} \right)} \\ {u\left( {k - s - i + 2} \right)} & {u\left( {k - s - i + 3} \right)} & \cdots & {u\left( {k - i + 2} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {u\left( {k - s + 1} \right)} & {u\left( {k - s + 2} \right)} & \cdots & {u\left( k \right)} \\ \end{array} } \right]$$

(A6)

$$\left[ {{\varvec{Y}}_{i}^{f} \left( {k{|}k - s + 1} \right)} \right] = \left[ {\begin{array}{*{20}c} {y\left( {k - s - i + 1} \right)} & {y\left( {k - s - i + 2} \right)} & \cdots & {y\left( {k - i + 1} \right)} \\ {y\left( {k - s - i + 2} \right)} & {y\left( {k - s - i + 3} \right)} & \cdots & {y\left( {k - i + 2} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {y\left( {k - s + 1} \right)} & {y\left( {k - s + 2} \right)} & \cdots & {y\left( k \right)} \\ \end{array} } \right]$$

(A7)

$$\hat{\user2{L}}_{{\varvec{0}}} \left( k \right) = {\varvec{L}}_{11} \left( k \right){\varvec{L}}_{11}^{{\varvec{T}}} \left( k \right)$$

(A8)

$$\hat{\user2{L}}_{1} \left( k \right) = {\varvec{L}}_{21} \left( k \right){\varvec{L}}_{11}^{{\varvec{T}}} \left( k \right)$$

(A9)

$$\hat{\user2{L}}_{2} \left( k \right) = {\varvec{L}}_{31} \left( k \right){\varvec{L}}_{11}^{{\varvec{T}}} \left( k \right)$$

(A10)

$$\hat{\user2{L}}_{3} \left( k \right) = {\varvec{L}}_{21} \left( k \right){\varvec{L}}_{21}^{{\varvec{T}}} \left( k \right) + {\varvec{L}}_{22} \left( k \right){\varvec{L}}_{22}^{{\varvec{T}}} \left( k \right)$$

(A11)

$$\hat{\user2{L}}_{4} \left( k \right) = {\varvec{L}}_{31} \left( k \right){\varvec{L}}_{21}^{{\varvec{T}}} \left( k \right) + {\varvec{L}}_{32} \left( k \right){\varvec{L}}_{22}^{{\varvec{T}}} \left( k \right)$$

(A12)

$$\hat{\user2{L}}_{{\varvec{0}}} \left( {k + 1} \right) = \hat{\user2{L}}_{{\varvec{0}}} \left( k \right) + {\varvec{u}}_{i} \left( {k + 1} \right){\varvec{u}}_{i}^{T} \left( {k + 1} \right) - {\varvec{u}}_{i} \left( {k - s + 1} \right){\varvec{u}}_{i}^{T} \left( {k - s + 1} \right)$$

(A13)

$$\hat{\user2{L}}_{1} \left( {k + 1} \right) = \hat{\user2{L}}_{1} \left( k \right) + {\varvec{\varepsilon}}_{{\varvec{i}}} \left( {k + 1} \right){\varvec{u}}_{i}^{T} \left( {k + 1} \right) - {\varvec{\varepsilon}}_{i} \left( {k - s + 1} \right){\varvec{u}}_{i}^{T} \left( {k - s + 1} \right)$$

(A14)

$$\hat{\user2{L}}_{2} \left( {k + 1} \right) = \hat{\user2{L}}_{2} \left( k \right) + {\varvec{y}}_{i} \left( {k + 1} \right){\varvec{u}}_{i}^{T} \left( {k + 1} \right) - {\varvec{y}}_{i} \left( {k - s + 1} \right){\varvec{u}}_{i}^{T} \left( {k - s + 1} \right)$$

(A15)

$$\hat{\user2{L}}_{3} \left( {k + 1} \right) = \hat{\user2{L}}_{3} \left( k \right) + {\varvec{\varepsilon}}_{i} \left( {k + 1} \right){\varvec{\varepsilon}}_{i}^{T} \left( {k + 1} \right) - {\varvec{\varepsilon}}_{i} \left( {k - s + 1} \right){\varvec{\varepsilon}}_{i}^{T} \left( {k - s + 1} \right)$$

(A16)

$$\hat{\user2{L}}_{4} \left( {k + 1} \right) = \hat{\user2{L}}_{4} \left( k \right) + {\varvec{\varepsilon}}_{i} \left( {k + 1} \right){\varvec{\varepsilon}}_{i}^{T} \left( {k + 1} \right) - {\varvec{y}}_{i} \left( {k - s + 1} \right){\varvec{\varepsilon}}_{i}^{T} \left( {k - s + 1} \right)$$

(A17)