基于弹性地基曲梁理论的盾构隧道管片分析方法

黄伟明,王金昌,徐日庆,杨仲轩,徐荣桥

Structural analysis of shield tunnel lining using theory of curved beam resting on elastic foundation

Wei-ming HUANG,Jin-chang WANG,Ri-qing XU,Zhong-xuan YANG,Rong-qiao XU

表 B.1 各荷载相关积分

Tab.B.1

${\bar q_j}$ 关于双曲余弦函数的积分系数 关于双曲正弦函数的积分系数 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;{{\bar q}_1} \\ ({90^{\circ}} \leqslant \xi \leqslant {270^{\circ}}) \\ \end{array} $ $\begin{array}{l} {B_{z,\;{\rm ch},i,1}}{\rm{ = }}{\lambda _i}(\lambda _i^2 + 4),\;{\beta _1} = - {\lambda _i}\sin \;(2\theta ), \\ {\beta _2} = {\lambda _i}\sin \;(2{\theta _0}),\;{\beta _3} = - \left[ {2 + \lambda _i^2{{\cos }^2}{\theta _0}} \right] \\ \end{array} $ $\begin{array}{l} {B_{z,\;{\rm sh},i,1}} = {\lambda _i}(\lambda _i^2 + 4),\;{\beta _1} = 2 + \lambda _i^2{\cos ^2}\;\theta , \\ {\beta _2} = - \left[ {2 + \lambda _i^2{{\cos }^2}\;{\theta _0}} \right],\;{\beta _3} = {\lambda _i}\sin \;(2{\theta _0}), \\ \end{array} $ $\begin{array}{l} {B_{s,\;{\rm ch},i,1}}{\rm{ = }}2(\lambda _i^2 + 4),\;{\beta _1} = - 2\cos \;(2\theta ), \\ {\beta _2} = {\rm{2}}\cos \;(2{\theta _0}),\;{\beta _3} = {\lambda _i}\sin \;(2{\theta _0}) \\ \end{array} $ $\begin{array}{l} {B_{s,\;{\rm sh},i,1}} = 2(\lambda _i^2 + 4),\;{\beta _1} = - {\lambda _i}\sin \;(2\theta ), \\ {\beta _2} = {\lambda _i}{\rm{sin}}\;(2{\theta _0}),\;{\beta _3} = 2\cos \;(2{\theta _0}) \\ \end{array} $ $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;{{\bar q}_2} \\ ({{\rm{0}}^{\circ}} \leqslant \xi \leqslant {\rm{9}}{{\rm{0}}^{\circ}}, \\ {\rm{ 27}}{{\rm{0}}^{\circ}} \leqslant \xi \leqslant {\rm{36}}{{\rm{0}}^{\circ}} ) \\ \end{array} $ ${\eta _{z,\;{\rm ch},i,2}} = {\eta _{z,\;{\rm ch},i,1}},\;{B_{z,\;{\rm ch},i,2}} = {B_{z,\;{\rm ch},i,1}}$ ${\eta _{z,\;{\rm sh},i,2}} = {\eta _{z,\;{\rm sh},i,1}},\;{B_{z,\;{\rm sh},i,2}} = {B_{z,\;{\rm sh},i,1}}$ ${\eta _{s,\;{\rm ch},i,2}} = {\eta _{s,\;{\rm ch},i,1}},\;{B_{s,\;{\rm ch},i,2}} = {B_{s,\;{\rm ch},i,1}}$ ${\eta _{s,\;{\rm sh},i,2}} = {\eta _{s,\;{\rm sh},i,1}},\;{B_{s,\;{\rm sh},i,2}} = {B_{s,\;{\rm sh},i,1}}$ $\begin{array}{l} \;\;\;\;\;\;\;\;{{\bar q}_3} \\ ({0^{\circ}} \leqslant \xi \leqslant {360^{\circ}}) \\ \end{array} $ $\begin{array}{l} {B_{z,\;{\rm ch},i,3}} = {\lambda _i}(\lambda _i^2 + 4),\;{\beta _1} = {\lambda _i}\sin \;(2\theta ), \\ {\beta _2} = - {\lambda _i}\sin \;(2{\theta _0}),\;{\beta _3} = - [2 + \lambda _i^2{\sin ^2} \;{{\theta _0}}] \\ \end{array} $ $\begin{array}{l} {B_{z,\;{\rm sh},i,3}} = {\lambda _i}(\lambda _i^2 + 4),\;{\beta _1} = 2 + \lambda _i^2{\sin ^2}\;\theta , \\ {\beta _2} = - \left[ {2 + \lambda _i^2{{\sin }^2}\;{\theta _0}} \right],\;{\beta _3} = - {\lambda _i}\sin \;(2{\theta _0}) \\ \end{array} $ ${\eta _{s,\;{\rm ch},i,3}} = - {\eta _{s,\;{\rm ch},i,1}},\;{B_{s,\;{\rm ch},i,3}} = {B_{s,\;{\rm ch},i,1}}$ ${\eta _{s,\;{\rm sh},i,3}} = - {\eta _{s,\;{\rm sh},i,1}},\;{B_{s,\;{\rm sh},i,3}} = {B_{s,\;{\rm sh},i,1}}$ $\begin{array}{l} \;\;\;\;\;\;\;{{\bar q}_4} \\ ({0^{\circ}} \leqslant \xi \leqslant {360^{\circ}} ) \\ \end{array} $ $\begin{array}{l} {f_{z,\;{\rm ch},i,4}} = {{\bar q}_4}\sum\limits_{k = 1}^3 {{{{\eta _{z,\;{\rm ch},i,4,k}}}}/{{{B_{z,\;{\rm ch},i,4,k}}}}} \\ {\eta _{{\rm{z}},\;{\rm ch},i,4,k}}{\rm{ = }}{\beta _{k,1}} + {\beta _{k,2}}\cosh\;[{\lambda _i}(\theta - {\theta _0})] +\\ \quad\quad\quad\quad {\beta _{k,3}}\sinh\;[{\lambda _i}(\theta - {\theta _0})]\;;\\ {B_{z,\;{\rm ch},i,4,1}} = 4{\lambda _i}(\lambda _i^2 + 4),\;{B_{z,\;{\rm ch},i,4,2}} = 8(\lambda _i^2 + 1),\\ {B_{z,\;{\rm ch},i,4,3}} = 8(\lambda _i^2 + 9),\;{\beta _{1,1}} = {\rm{2}}{\lambda _i}\sin \;(2\theta ),\\ {\beta _{1,2}} = - 2{\lambda _i}\sin \;(2{\theta _0}),\\ {\beta _{1,3}} = - \left[ {2\lambda _i^2{{\sin }^2}{\theta _0} + 4} \right],\\ {\beta _{2,1}} = - \sin \;\theta ,\;{\beta _{2,2}} = \sin \;{\theta _0},\\ {\beta _{2,3}} = - {\lambda _i}\cos \;{\theta _0},\;{\beta _{3,1}} = {\rm{3sin\;(3}}\theta ),\\ {\beta _{3,2}} = - {\rm{3sin\;(3}}{\theta _0}),\;{\beta _{3,3}} = {\lambda _i}\cos \;\left( {3{\theta _0}} \right) \end{array}$ $\begin{array}{l} {f_{z,\;{\rm sh},i,4}} = {{\bar q}_4}\sum\limits_{k = 1}^2 {{{{\eta _{z,\;{\rm {sh}},i,4,k}}}}/{{{B_{z.{\rm{sh}},i,4,k}}}}} \\ {\eta _{z,\;{\rm {sh}},i,4,k}} = {\beta _{k,1}} + {\beta _{k,2}}\cosh\;[{\lambda _i}(\theta - {\theta _0})] +\\ \quad\quad\quad\quad\;{\beta _{k,3}}\sinh\;[{\lambda _i}(\theta - {\theta _0})];\\ {B_{z,\;{\rm sh},i,4,1}} = 4{\lambda _i}(\lambda _i^2 + 4),\;{B_{z,\;{\rm sh},i,4,2}} = 2(\lambda _i^2 + 1)(\lambda _i^2 + 9)\\ {\beta _{1,1}} = 4 + 2\lambda _i^2{\sin ^2}\theta ,\;{\beta _{1,2}} = - [4 + 2\lambda _i^2{\sin ^2}{\theta _0}],\\ {\beta _{1,3}} = - 2{\lambda _i}\sin \;(2{\theta _0}),\\ {\beta _{2,1}} = 2{\lambda _i}{\cos ^3}\; \theta + {\lambda _i}\left( {\lambda _i^2 + 3} \right)\cos \; \theta {\sin ^2}\; \theta ,\\ {\beta _{2,2}} = - \left[ {2{\lambda _i}{{\cos }^3} \;{{\theta _0}} + {\lambda _i}\left( {\lambda _i^2 + 3} \right)\cos \;{{\theta _0}}{{\sin }^2} \;{{\theta _0}}} \right],\\ {\beta _{2,3}} = \left[ {\left( {\lambda _i^2 + 3} \right){{\sin }^3} \;{{\theta _0}} - 2\lambda _i^2\sin \;{{\theta _0}}{{\cos }^2} \;{{\theta _0}}} \right] \end{array}$ $\begin{array}{l} {f_{s,\;{\rm ch},i,4}} = {{\bar q}_4}\sum\limits_{k = 1}^3 {{{{\eta _{s,\;{\rm ch},i,4,k}}}}/{{{B_{s,\;{\rm ch},i,4,k}}}}} \\ {\eta _{s,\;{\rm ch},i,4,k}} = {\beta _{k,1}} + {\beta _{k,2}}\cosh\; \left[ {{\lambda _i}\left( {\theta - {\theta _0}} \right)} \right] + \\ \quad\quad\quad\quad \;\; {\beta _{k,3}}\sinh \left[ {{\lambda _i}\left( {\theta - {\theta _0}} \right)} \right]; \\ {B_{s,\;{\rm ch},i,4,1}} = 4(\lambda _i^2 + 4),\;{B_{s,\;{\rm ch},i,4,2}}{\rm{ = }}8(\lambda _i^2 + 1), \\ {B_{s,\;{\rm ch},i,4,3}} = 8(\lambda _i^2 + 9),\;{\beta _{1,1}} = 2\cos \left( {2\theta } \right), \\ {\beta _{1,2}} = - 2\cos \left( {2{\theta _0}} \right),\;{\beta _{1,3}} = - {\lambda _i}\sin \left( {2{\theta _0}} \right), \\ {\beta _{2,1}} = \cos \; \theta ,\;{\beta _{2,2}} = - \cos \;{{\theta _0}}, \\ {\beta _{2,3}} = - {\lambda _i}\sin \;{{\theta _0}},\;{\beta _{3,1}} = 3\cos \left( {3\theta } \right), \\ {\beta _{3,2}} = - 3\cos \left( {3{\theta _0}} \right),\;{\beta _{3,3}} = - {\lambda _i}\sin \left( {3{\theta _0}} \right) \\ \end{array} $ $\begin{array}{l} {f_{s,\;{\rm sh},i,4}} = {{\bar q}_4}\sum\limits_{k = 1}^3 {{{{\eta _{s,\;{\rm sh},i,4,k}}}}/{{{B_{s,\;{\rm sh},i,4,k}}}}} \\ {\eta _{s,\;{\rm sh},i,4,k}}{\rm{ = }}{\beta _{k,1}} + {\beta _{k,2}}\cosh\;[{\lambda _i}(\theta - {\theta _0})] + \\ \quad\quad\quad\quad {\beta _{k,3}}\sinh\;[{\lambda _i}(\theta - {\theta _0})]; \\ {B_{s,\;{\rm sh},i,4,1}} = 4(\lambda _i^2 + 4),\;{B_{s,\;{\rm sh},i,4,2}} = 8(\lambda _i^2 + 1), \\ {B_{s,\;{\rm sh},i,4,3}}{\rm{ = }}8(\lambda _i^2 + 9),\;{\beta _{1,1}} = {\lambda _i}\sin \;(2\theta ), \\ {\beta _{1,2}} = - {\lambda _i}\sin \;(2{\theta _0}),\;{\beta _{1,3}} = - 2\cos \;(2{\theta _0}), \\ {\beta _{2,1}} = {\lambda _i}\sin \;\theta ,\;{\beta _{2,2}} = - {\lambda _i}\sin \;{\theta _0}, \\ {\beta _{2,3}} = - \cos \;{\theta _0},\;{\beta _{3,1}} = {\lambda _i}\sin \left( {3\theta } \right), \\ {\beta _{3,2}} = - {\lambda _i}\sin \left( {3{\theta _0}} \right),\;{\beta _{3,3}} = - 3\cos \left( {3{\theta _0}} \right) \\ \end{array} $ $\begin{array}{l} \;\;\;\;\;\;\;{{\bar q}_5} \\ ({0^{\circ}} \leqslant \xi \leqslant {360^{\circ}}) \\ \end{array} $ $\begin{array}{l} {B_{z,\;{\rm ch},i,5}} = \lambda _i^2 + 1,\;{\beta _1} = - \sin \;\theta , \\ {\beta _2} = \sin \;{\theta _0},\;{\beta _3} = - {\lambda _i}\cos \;{\theta _0} \\ \end{array} $ $\begin{array}{l} {B_{z,\;{\rm sh},i,5}}{\rm{ = }}\lambda _i^2 + 1,\;{\beta _1} = - {\lambda _i}\cos \;\theta , \\ {\beta _2} = {\lambda _i}\cos \;{\theta _0},\;{\beta _3} = - \sin \;{\theta _0} \\ \end{array} $ $\begin{array}{l} {B_{s,\;{\rm ch},i,5}}{\rm{ = }}\lambda _i^2 + 1,\;{\beta _1} = \cos \;\theta , \\ {\beta _2} = - \cos \;{\theta _0},\;{\beta _3} = - {\lambda _i}\sin \;{\theta _0} \\ \end{array} $ $\begin{array}{l} {B_{s,\;{\rm sh},i,5}}{\rm{ = }}\lambda _i^2 + 1,\;{\beta _1} = {\lambda _i}\sin \;\theta , \\ {\beta _2} = - {\lambda _i}\sin \;{\theta _0},\;{\beta _3} = - \cos \;{\theta _0} \\ \end{array} $