Nonsmooth analysis of the impact between successive skew bridge-segments

Abstract

Skew bridges with in-deck joints belong to the most common types of existing bridges worldwide. Empirical evidence from past earthquakes indicates that such, multi-segment, skew bridges often rotate in the horizontal plane, increasing the chances of deck unseating. The present paper studies the oblique in-deck impact between successive bridge segments, which triggers this peculiar rotation mechanism. The analysis employs a nonsmooth rigid body approach and utilizes set-valued force laws. A key feature of this approach is the linear complementarity problem (LCP) which encapsulates all physically feasible post-impact states. The LCP results in pertinent closed-form solutions which capture each of these states, and clarifies the conditions under which each post-impact state appears. In this context, a rational method to avoid the singularities arising from dependent constraints is coined. The results confirm theoretically the observed tendency of skew (bridge deck) segments to bind in their obtuse corners and rotate in such a way that the skew angle increases. Further, the study offers equations which describe the contact kinematics between two adjacent skew planar rigid bodies. The same equations can be used to treat successively as many pairs of skew bridge-segments as necessary.

Appendix

The determinate |G _NN | is

$$ \vert\mathbf{G}_{NN} \vert= \frac{ ( r_{11}r_{21} - r_{12}r_{22} )^{2}}{I_{1}I_{2}} + \biggl[ \frac{ ( r_{11} - r_{12} )^{2}}{I_{1}} + \frac{ ( r_{21} - r_{22} )^{2}}{I_{2}} \biggr] \biggl( \frac{1}{m_{1}} + \frac{1}{m_{2}} \biggr) $$

(58)

The inverse of matrix $( \mathbf{G}_{NN} + \mathbf{G}_{NT}\bar{\bar{\boldsymbol{\mu}}} )$ and its determinate D ₁ can be written as

$$ \begin{aligned} &\bigl( \mathbf{G}_{NN} + \mathbf{G}_{NT}\bar{\bar{\boldsymbol{\mu}}} \bigr)^{ - 1} \\ &{\quad = \frac{1}{D_{1}} \left( \begin{array}{c@{\quad}c} \frac{1}{m_{1}} + \frac{r_{12}^{2} + \mu r_{12}r_{T1}}{I_{1}} + \frac{1}{m_{2}} + \frac{r_{22}^{2} + \mu r_{22}r_{T2}}{I_{2}} & - ( \frac{1}{m_{1}} + \frac{r_{11}r_{12} + \mu r_{11}r_{T1}}{I_{1}} + \frac{1}{m_{2}} + \frac{r_{21}r_{22} + \mu r_{21}r_{T2}}{I_{2}} ) \\ - ( 1 + \frac{r_{11}r_{12} + \mu r_{12}r_{T1}}{I_{1}} + \frac{1}{m_{2}} + \frac{r_{21}r_{22} + \mu r_{22}r_{T2}}{I_{2}} ) & \frac{1}{m_{1}} + \frac{r_{11}^{2} + \mu r_{11}r_{T1}}{I_{1}} + \frac{1}{m_{2}} + \frac{r_{21}^{2} + \mu r_{21}r_{T2}}{I_{2}} \end{array} \right)} \\ &D_{1} = \vert\mathbf{G}_{NN} + \mathbf{G}_{NT} \bar{\bar{\boldsymbol{\mu}}} \vert = \frac{m_{1} + m_{2}}{m_{1}m_{2}} \biggl[ \frac{ ( r_{11} - r_{12} )^{2}}{I_{1}} + \frac{ ( r_{21} - r_{22} )^{2}}{I_{2}} \biggr] \\ & \hphantom{D_{1} =} {}+ \frac{r_{12}r_{21} - r_{11}r_{22}}{I_{1}I_{2}} \bigl[ \mu r_{T1} ( r_{21} - r_{22} ) + \mu r_{T2} ( r_{12} - r_{11} ) + r_{12}r_{21} - r_{11}r_{22} \bigr] \end{aligned} $$

(59)

Similarly for the inverse of matrix $( \mathbf{G}_{NN} - \mathbf{G}_{NT}\bar{\bar{\boldsymbol{\mu}}} )$ and its determinate D ₂, we have

$$ \begin{aligned} &\bigl( \mathbf{G}_{NN} - \mathbf{G}_{NT}\bar{\bar{\boldsymbol{\mu}}} \bigr)^{ - 1} \\ &\quad{= \frac{1}{D_{2}} \left( \begin{array}{c@{\quad}c} \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{r_{12}^{2} - \mu r_{12}r_{T1}}{I_{1}} + \frac{r_{22}^{2} - \mu r_{22}r_{T2}}{I_{2}} & - ( \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{r_{11}r_{12} - \mu r_{11}r_{T1}}{I_{1}} + \frac{r_{21}r_{22} - \mu r_{21}r_{T2}}{I_{2}} ) \\ - ( \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{r_{11}r_{12} - \mu r_{12}r_{T1}}{I_{1}} + \frac{r_{21}r_{22} - \mu r_{22}r_{T2}}{I_{2}} ) & \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{r_{11}^{2} - \mu r_{11}r_{T1}}{I_{1}} + \frac{r_{21}^{2} - \mu r_{21}r_{T2}}{I_{2}} \end{array} \right)} \\ &D_{2} = \vert\mathbf{G}_{NN} - \mathbf{G}_{NT} \bar{\bar{\boldsymbol{\mu}}} \vert= \frac{m_{1} + m_{2}}{m_{1}m_{2}} \biggl[ \frac{ ( r_{11} - r_{12} )^{2}}{I_{1}} + \frac{ ( r_{21} - r_{22} )^{2}}{I_{2}} \biggr] \\ &\hphantom{D_{2} =} {}+ \frac{r_{12}r_{21} - r_{11}r_{22}}{I_{1}I_{2}}\big[ \mu r_{T1} ( r_{22} - r_{21} ) + \mu r_{T2} ( r_{11} - r_{12} ) + r_{12}r_{21} - r_{11}r_{22}\big] \end{aligned} $$

(60)

For μ=0 (59) and (60) reduce to (26) and (58) of frictionless collisions.

Finally, the product $\mathbf{G}_{TN}\mathbf{G}_{NN}^{ - 1}\mathbf{G}_{NT}$ returns the scalar value:

$$ \begin{aligned}[b] &\mathbf{G}_{TN}\mathbf{G}_{NN}^{ - 1} \mathbf{G}_{NT} \\ &\quad= \frac{1}{\vert \mathbf{G}_{NN} \vert} \biggl[ \biggl( \frac{1}{m_{1}} + \frac{1}{m_{2}} \biggr) \biggl( \frac{r_{11} - r_{12}}{I_{1}}r_{T1} + \frac{r_{21} - r_{22}}{I_{2}}r_{T2} \biggr)^{2} + \biggl( \frac{r_{T1}^{2}}{I_{1}} + \frac{r_{T2}^{2}}{I_{2}} \biggr)\frac{ ( r_{11}r_{22} - r_{12}r_{21} )^{2}}{I_{1}I_{2}} \biggr] \end{aligned} $$

(61)