An analysis model for the temperature and residual stress of tunnel liner exposed to fire

Abstract

The temperature and stress analysis of tunnel liner is the basis of the damage assessment of the tunnel, and it is also have a great significance to tunnel fire protection design. In this study, a thermo-mechanical coupling model is derived to study the temperature and stresses of tunnel liner under the RABT fire curve. In contrast to consideration the effects of flame impingement on the heated surface only, the heat transfer coefficient (HTC) of the heated surface of tunnel liner is considered in the proposed model. The applicability of theoretical method is verified by comparing with the fire tests. According to maximum temperature experienced and material degradation, the residual stress of tunnel liner after fire is discussed, which could provide the basic for the damage assessment after fire. Contributions of the HTC of tunnel liner on the temperature and stresses were quantitatively described. This theoretical model explains the temperature and residual stress evolution of tunnel liner under fire when considering the effect of HTC, which provides a theoretical basis for the tunnel fire proofing.

Appendix 1

$$\Theta_{n} \left( \tau \right) = \sum\limits_{k = 1}^{m} {\left[ {\frac{1}{(m - k)!}\mathop {\lim }\limits_{{s \to s_{1} }} \frac{{d^{k - 1} }}{{ds^{k - 1} }}\left[ {\left( {s - s_{1} } \right)^{m} \frac{M(s)}{{W(s)}}} \right]e^{s\tau } \tau^{m - k} } \right]} {\kern 1pt} + \sum\limits_{k = m + 1}^{K} {\frac{{M(s_{k} )}}{{\left[ {{{dW(s)} \mathord{\left/ {\vphantom {{dW(s)} {ds}}} \right. \kern-\nulldelimiterspace} {ds}}} \right]_{{s = s_{k} }} }}e^{{s_{k} \tau }} } ,$$

${\text{where}}\,M\left( s \right) = s^{{2}} \cdot \left( {d_{11} d_{22} - d_{12} d_{21} } \right);$$M\left( s \right)\,\, = \,\,\left( {sd_{{{1}2}} \left( s \right)H_{2} \Phi_{b} - d_{22} (s)H_{1} \frac{{\Phi_{a} }}{{\tau_{1} }}} \right)g_{n} (s) + \left( {d_{21} (s)H_{1} \frac{{\Phi_{a} }}{{\tau_{1} }} - sd_{11} (s)H_{2} \Phi_{b} } \right)j_{n} (s);$$d_{{{1}1}} \left( s \right)\,{ = }\,\sum\limits_{n = 1}^{\infty } {\left[ {g_{n} \left( s \right)n\left( {R - 1} \right)^{n - 1} } \right]} - H_{1} \sum\limits_{n = 0}^{\infty } {\left[ {g_{n} \left( s \right)\left( {R - 1} \right)^{n} } \right]} ;$

$d_{12} \left( s \right)\,{ = }\,\sum\limits_{n = 1}^{\infty } {\left[ {j_{n} \left( s \right)n\left( {R - 1} \right)^{n - 1} } \right]} - H_{1} \sum\limits_{n = 0}^{\infty } {\left[ {j_{n} \left( s \right)\left( {R - 1} \right)^{n} } \right]} ;$

$d_{21} \left( s \right)\,{ = }\,\sum\limits_{n = 1}^{\infty } {\left[ {g_{n} \left( s \right)n\left( {R - 1} \right)^{n - 1} } \right]} + H_{2} \sum\limits_{n = 0}^{\infty } {\left[ {g_{n} \left( s \right)\left( {R - 1} \right)^{n} } \right]} ;$

$d_{22} \left( s \right)\,{ = }\,\sum\limits_{n = 1}^{\infty } {\left[ {j_{n} \left( s \right)n\left( {R - 1} \right)^{n - 1} } \right]} + H_{2} \sum\limits_{n = 0}^{\infty } {\left[ {j_{n} \left( s \right)\left( {R - 1} \right)^{n} } \right]} .$