Previous stress solutions for cold region tunnels (CRT) are basically based on the assumption of full bond contact between the lining and surrounding rock (SR). Whereas, in actual engineering practices, the lining and SR are in frictional contact condition. Based on the complex variable function theory and optimization method, the stress solution of CRT is established under the frictional contact condition of lining and SR, and the solution considers the transversely isotropic frost heave deformation of SR and the support delay. The Coulomb friction law is applied to describe the frictional contact between SR and lining, and the frictional contact condition is solved by the optimization method regarding the state of lining with the least relative sliding amount on the contact surface as its real working state. The stress solution is validated with numerical simulation. The stress distributions of lining and SR under three contact conditions, namely, the full bond contact condition, the full slip contact condition and frictional contact condition, are analyzed and compared. Results show that when the frictional coefficient fr take the values of 1.0 and 0.0 under the frictional contact condition, the stress distributions solved by the optimization method are consistent with the results directly derived from solutions under the full bond and full slip contact conditions, respectively. This illustrates the validity of the optimization method in solving the frictional contact condition. Under these three contact conditions, the radial stresses of lining and frozen SR increase with radius while the circumferential stresses decrease with radius, and the influence of frictional contact condition on circumferential stresses in lining and frozen SR is more significant. Under different frictional coefficients, the radial contact stress at the lining and SR interface decreases with increasing angle, and the radial stress at 45° keeps as a constant. When the frictional coefficient fr increases, the tangential contact stress at the interface increases. Besides, the tangential stress and relative displacement at the contact surface are always maximum at 45°.

中文翻译：

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支护延迟条件下衬砌与围岩摩擦接触条件下寒冷地区隧道复杂功能应力求解


以前寒冷地区隧道 （CRT） 的应力解决方案基本上基于衬砌和围岩 （SR） 之间完全粘合接触的假设。然而，在实际工程实践中，衬里和 SR 处于摩擦接触状态。基于复变量函数理论和优化方法，建立了衬砌与SR摩擦接触条件下CRT的应力解，并考虑了SR的横向各向同性冻胀变形和支护延迟。采用库仑摩擦定律来描述 SR 与衬板之间的摩擦接触，并以接触面上相对滑动量最小的衬板状态作为其实际工作状态，采用优化方法求解摩擦接触条件。应力解通过数值模拟进行验证。分析比较了全粘结接触条件、全滑移接触条件和摩擦接触条件3种接触条件下衬板和SR的应力分布。结果表明，在摩擦接触条件下，当摩擦系数 fr 取 1.0 和 0.0 时，优化方法求解的应力分布分别与全粘结和全滑移接触条件下的解直接得出的结果一致。这说明了优化方法在求解摩擦接触条件方面的有效性。在这 3 种接触条件下，衬板和冻结 SR 的径向应力随半径的增加而减小，而圆周应力随半径的减小，摩擦接触条件对衬板和冻结 SR 的圆周应力的影响更为显著。 在不同摩擦系数下，衬板和 SR 界面处的径向接触应力随角度的增加而减小，45° 处的径向应力保持不变。当摩擦系数 fr 增加时，界面处的切向接触应力增加。此外，接触表面的切向应力和相对位移在 45° 处始终最大。