Cause of Rail Corrugation on Ladder Sleeper Track

SONG Qifeng; CHEN Guangxiong; DONG Bingjie; ZHANG Juncai; FENG Xiaohang

doi:10.3969/j.issn.0258-2724.20230573

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Journal of Southwest Jiaotong University > 2024 >

LEI Ming, ZHANG Bingqiang, LIU Hai, HUANG Zhibin. Theoretical Method for Calculating Rail Deformation of Ballastless Railway Caused by Tunnel Undercrossing Based on Dual Beam Model[J]. Journal of Southwest Jiaotong University, 2024, 59(3): 646-652, 669. doi: 10.3969/j.issn.0258-2724.20230033

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SONG Qifeng, CHEN Guangxiong, DONG Bingjie, ZHANG Juncai, FENG Xiaohang. Cause of Rail Corrugation on Ladder Sleeper Track[J]. Journal of Southwest Jiaotong University. doi: 10.3969/j.issn.0258-2724.20230573

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Cause of Rail Corrugation on Ladder Sleeper Track

doi: 10.3969/j.issn.0258-2724.20230573

School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China

Received Date: 27 Oct 2023
Rev Recd Date: 29 Dec 2023

Available Online: 18 Jul 2024

Abstract

Abstract

To study the formation mechanism of inner rail corrugation on the ladder sleeper track in the small-radius curve, a finite element model of the leading wheelset–ladder sleeper track system was developed based on the theory that self-excited frictional vibration triggered by saturated wheel-rail creep force causes rail corrugation. In this model, solid elements were used to model the fastening system. Complex eigenvalue analysis and transient dynamic analysis were applied to solve the motion stability and dynamic time-domain response of the wheel-rail system, respectively. Furthermore, the effects of the parameters of the cushioning pad and ladder sleeper structure on the self-excited frictional vibration of the wheel-rail system were studied. The results show that the self-excited frictional vibration with a frequency of 150 Hz caused by the saturated wheel-rail creep force is the cause of inner rail corrugation on the ladder sleeper track in the small-radius curve section. The predicted corrugation wavelength is 69 mm, which agrees well with the measured wavelength. Parameter sensitivity analysis shows that increasing the damping of the lateral cushioning pad and laying ladder sleepers with a spacing of 1.25 m between lateral steel pipes can suppress rail corrugation on the ladder sleeper track to a certain extent.
- rail corrugation,
- ladder sleeper,
- self-excited frictional vibration,
- complex eigenvalue analysis

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在城市地铁和城际铁路修建中，经常会遇到新建隧道近距离下穿既有铁路的情况^[1-2]. 新建隧道开挖施工不可避免会引起铁路路基沉降^[3-4]，诱发轨道产生几何变形，加剧轮轨动力相互作用，进而影响轨道结构服役性能和列车乘坐舒适性，严重时将造成列车脱轨和铁路停运^[5-6]. 因此，研究隧道下穿施工诱发上覆既有铁路轨面变形具有重要的工程意义.

目前，国内外学者采用理论方法分析新建隧道施工对邻近既有结构的影响时，通常采用两阶段分析方法：第1阶段，忽略既有结构的影响，计算新建隧道施工引起邻近结构所处位置土体的附加应力或自由变形；第2阶段，将既有结构视为连续地基上梁模型，并将土体附加应力或自由变形视为外部作用，应用数值方法^[7-8]或简化方法等^[9-10]分析邻近既有结构的变形和内力的变化. 两阶段分析方法由于使用简单，被广泛用于分析隧道施工对邻近管线^[11-16]、上方铁路^[17]、和邻近建筑物^[9,18]的影响.

同时，针对路基沉降引起铁路线路轨道变形开展了一定的研究：邹春华等^[19]、付龙龙等^[20]分别采用室内试验和数值方法对余弦型路基沉降与有砟轨道变形之间的互相关系进行了研究，并且，邹春华等^[21]基于弹性点支承梁模型提出了有砟铁路路基沉降引起轨面变形的计算模型；考虑无砟道床为板状多层结构，张乾等^[22]采用梁-板模型分别模拟钢轨及无砟道床，建立了无砟轨道-路基-土体的三维空间有限元模型，对地面沉降对无砟轨道系统平顺性的影响进行了分析，并指出支承层和路基表层间沉降差较大易出现离缝问题；郭宇等^[23]、Jiang等^[24]分别对双块式和整体板式无砟轨道路基沉降引起轨面变形几何特征与路基沉降波长、幅值之间的对应关系进行了研究.

以上相关理论研究中邻近既有结构均被简化为弹性地基上的单层梁模型，不能模拟地下结构与底部地基局部分离现象，不适用于计算新建隧道下穿施工引起既有无砟铁路的轨面变形. 为此，本文将无砟铁路结构视为双层地基梁模型，采用改进的Winkler弹性地基模型模拟轨道板与地基的互相作用，建立考虑轨道板底部局部脱空影响的路基沉降引起无砟铁路轨道变形的理论计算模型，进而采用解析方法推导了隧道下穿施工引起轨道变形计算式，为预测隧道下穿既有无砟铁路引起轨道变形提供理论支持.

1. 问题描述及控制方程

新建隧道下穿引起既有无砟铁路轨道变形的计算模型如所示，以隧道与铁路相交处为坐标原点，铁路线路方向和x轴建立水平坐标系. 图中： $\beta$ 为隧道与铁路的水平交角， ${R_{\text{t}}}$ 为隧道开挖半径， ${{\textit{z}}_{\text{t}}}$ 为隧道中心到铁路路基顶面的竖向距离， ${l_{\text{t}}}$ 为轨道叠合梁底部局部脱空区长度， $s\left( x \right)$ 为隧道斜交下穿引起路基沉降，（0，x_i）为铁路上任意点坐标.

图 1 隧道下穿既有铁路力学分析示意

Figure 1. Mechanical analysis of existing railway caused by tunnel undercrossing

下载: 全尺寸图片幻灯片

本文拟采用两阶段分析方法进行求解，并作如下基本假定：

1）钢轨视为两端自由的上层Euler梁，其竖向变形、弹性模量和截面惯性矩分别为 ${w_{\text{r}}}$ 、 ${E_{\text{r}}}$ 和 ${I_{\text{r}}}$ ；

2）轨道板和其下部支承层简化为下层弹性叠合梁，其竖向变形、弹性模量和截面惯性矩分别为 ${w_{\text{b}}}$ 、 ${E_{\text{b}}}$ 和 ${I_{\text{b}}}$ ；

3）采用改进Winkler地基模拟叠合梁下的路基作用，地基的弹性系数为 ${k_{\text{b}}}$ ；

4）采用线性弹簧模拟扣件系统，钢轨扣件系统单位长度的等效弹性系数为 ${k_{\text{r}}}$ ；

5）隧道施工引起周围土体沉降曲线符合高斯分布.

根据Euler梁挠曲理论，叠合梁变形诱发钢轨（上层梁）变形的控制方程为

${E_{\text{r}}}{I_{\text{r}}}\frac{{{{\rm{d}}^4}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^4}}} = - {k_{\text{r}}}\left[ {{w_{\text{r}}}\left( x \right) - {w_{\text{b}}}\left( x \right)} \right].$

(1)

新建隧道施工前，叠合梁在轨道系统及自重作用下产生均匀沉降. 新建隧道下穿施工引起路基沉降，改变轨道板叠合梁与地基土间的互相作用，导致叠合梁产生变形. 若叠合梁变形大于路基顶面土体沉降，则叠合梁与地基土体间保持接触，两者之间仍然存在互相作用；否则，叠合梁与地基土体将发生分离（即局部脱空），两者互相作用力为0. 轨道板叠合梁（下层梁）变形的控制方程为

$\qquad\qquad\qquad {E_{\text{b}}}{I_{\text{b}}}\dfrac{{{{\rm{d}}^4}{w_{\text{b}}}\left( x \right)}}{{{\rm{d}}{x^4}}} =\left\{ {\begin{array}{*{20}{l}} { {k_{\text{r}}}\left[ {{w_{\text{r}}}\left( x \right) - {w_{\text{b}}}\left( x \right)} \right] + \gamma h},\quad{\left| x \right| \leqslant {l_{\text{t}}}/2}, \\ {{k_{\text{r}}}\left[ {{w_{\text{r}}}\left( x \right) - {w_{\text{b}}}\left( x \right)} \right] + \gamma h - {k_{\text{b}}}\left[ {{w_{\text{b}}}\left( x \right) - s\left( x \right)} \right]},\quad{\left| x \right| > {l_{\text{t}}}/2} , \end{array}} \right.$

(2)

式中： $\gamma$ 、 $h$ 分别为叠合梁重度和高度； $s\left( x \right)$ 计算式^[7,13-14]如式（3）.

$s\left( x \right) = \frac{{{\text{π}} R_{\text{t}}^2{V_{\text{t}}}}}{{\sqrt {2{\text{π}} } {i_{\text{s}}}}}\exp \left[ { - 0.5{{\left( {\frac{{x\sin \;\beta }}{{{i_{\text{s}}}}}} \right)}^2}} \right],$

(3)

式中： ${V_{\rm{t}}}$ 为隧道施工引起的平均地层损失比，与施工条件相关； ${i_{\text{s}}}$ 为路基顶面沉降槽宽度半径，采用建议公式 ${i_{\text{s}}} = \kappa {{\textit{z}}_{\rm{t}}}$ 进行计算， $\kappa$ 为地层沉降槽宽度参数，与土体条件相关.

2. 解析方法

将式（1）变形可得

${w_{\text{b}}}\left( x \right) = \frac{{{E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}}}\frac{{{{\rm{d}}^4}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^4}}} + {w_{\text{r}}}\left( x \right).$

(4)

对式（4）求4次导数，可得

$\frac{{{{\rm{d}}^4}{w_{\text{b}}}\left( x \right)}}{{{\rm{d}}{x^4}}} = \frac{{{E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}}}\frac{{{{\rm{d}}^8}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^8}}} + \frac{{{{\rm{d}}^4}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^4}}}.$

(5)

将式（4）、（5）依次代入式（2）中各分段方程，整理后可得关于钢轨变形的控制方程.

1）当 $\left| x \right| \leqslant {l_{\text{t}}}/2$ 时

脱空区上方钢轨的变形控制方程为

$\frac{{{{\rm{d}}^8}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^8}}} + C\frac{{{{\rm{d}}^4}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^4}}} = \frac{{{k_{\text{r}}} \gamma h}}{{{E_{\text{r}}}{I_{\text{r}}} {E_{\text{b}}}{I_{\text{b}}}}},$

(6)

式中： $C = \dfrac{{{k_{\text{r}}}}}{{{E_{\text{b}}}{I_{\text{b}}}}} + \dfrac{{{k_{\text{r}}}}}{{{E_{\text{r}}}{I_{\text{r}}}}}$ .

求解式（6），可得钢轨中间脱空段的变形表达式为

$\begin{split} \\[-10pt] & {w_{\text{r}}}\left( x \right) = {{\rm{e}}^{{\beta _1}x}}\left( {{c_1}\cos\; {\beta _1}x + {c_2}\sin \;{\beta _1}x} \right) + \\ &\quad {{\rm{e}}^{ - {\beta _1}x}}\left( {{c_3}\cos \;{\beta _1}x + {c_4}\sin\; {\beta _1}x} \right) + {c_5} + {c_6}x +\\ &\quad {c_7}{x^2} + {c_8}{x^3} + \frac{{\gamma h}}{{24\left( {{E_{\text{r}}}{I_{\text{r}}} + {E_{\text{b}}}{I_{\text{b}}}} \right)}}{x^4}, \end{split}$

(7)

式中： ${\beta _1} = \sqrt[\uproot{13}{4}]{{\dfrac{{{k_{\text{r}}}}}{{4{E_{\text{b}}}{I_{\text{b}}}}} + \dfrac{{{k_{\text{r}}}}}{{4{E_{\text{r}}}{I_{\text{r}}}}}}}$ ；c₁～c₈为待定参数，可用边界条件求出.

将式（7）代入式（4），可求得中间脱空区上方的轨道板叠合梁变形表达式为

$\begin{split} & {w_{\text{b}}}\left( x \right) = {D_1}\left[ {{{\rm{e}}^{{\beta _1}x}}\left( {{c_1}\cos \;{\beta _1}x + {c_2}\sin \;{\beta _1}x} \right)}+ \right. \\ &\quad \left. { {{\rm{e}}^{ - {\beta _1}x}}\left( {{c_3}\cos \;{\beta _1}x + {c_4}\sin \;{\beta _1}x} \right)} \right] + \\ &\quad {c_5} + {c_6}x + {c_7}{x^2} + {c_8}{x^3} +\\ &\quad \frac{{\gamma h}}{{24\left( {{E_{\text{r}}}{I_{\text{r}}} + {E_{\text{b}}}{I_{\text{b}}}} \right)}}{x^4} + \frac{{\gamma h {E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}\left( {{E_{\text{r}}}{I_{\text{r}}} + {E_{\text{b}}}{I_{\text{b}}}} \right)}}, \end{split}$

(8)

式中： ${D_1} = 1 - 4{\beta _1^4}\dfrac{{{E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}}}$ .

2）当 $\left| x \right| > {l_{\text{t}}}/2$ 时

脱空区外侧钢轨变形控制方程为

$\begin{split} & \frac{{{{\rm{d}}^8}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^8}}} + E\frac{{{{\rm{d}}^4}{w_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^4}}} + F {w_{\text{r}}}\left( x \right) = \\ &\quad F s\left( x \right) + \frac{{{k_{\text{r}}} \gamma h}}{{{E_{\text{r}}}{I_{\text{r}}} {E_{\text{b}}}{I_{\text{b}}}}}, \end{split}$

(9)

式中： $E = \dfrac{{{k_{\text{r}}}\left( {{E_{\text{r}}}{I_{\text{r}}} + {E_{\text{b}}}{I_{\text{b}}}} \right)}}{{{E_{\text{r}}}{I_{\text{r}}} {E_{\text{b}}}{I_{\text{b}}}}} + \dfrac{{{k_{\text{b}}}}}{{{E_{\text{b}}}{I_{\text{b}}}}}$ ， $F = \dfrac{{{k_{\text{r}}}{k_{\text{b}}}}}{{{E_{\text{r}}}{I_{\text{r}}} {E_{\text{b}}}{I_{\text{b}}}}}$ .

求解式（9），并结合边界条件 ${\left. {{w_{\text{r}}}} \right|_{x \to \infty }} = 0$ ，可得脱空区外侧钢轨变形表达式为

$\begin{split} & {w_{\text{r}}}\left( x \right) = {\bar w_{\text{r}}}\left( x \right) + \frac{{\gamma h}}{{{k_{\text{b}}}}} + {{\rm{e}}^{ - {\beta _2}x}}\left( {{c_9}\cos\; {\beta _2}x + {c_{10}}\sin \;{\beta _2}x} \right) + \\ &\quad {{\rm{e}}^{ - {\beta _3}x}}\left( {{c_{1{\text{1}}}}\cos \;{\beta _3}x + {c_{1{\text{2}}}}\sin \;{\beta _3}x} \right), \end{split}$

(10)

式中：c₉～c₁₂为待定系数，可由边界条件确定. ${\bar w_{\text{r}}}\left( x \right)$ 为路基沉降部分对应的方程特解，可采用展开级数法进行求解， $\beta _2$ 、 $\beta _3$ 取值如式（11）.

$\qquad\qquad\qquad \left\{\begin{array}{l} \beta _2=\sqrt[\uproot{13}{{4}}]{{\dfrac{{E - \sqrt {{E^2} - 4F} }}{8}}},\;{\beta _3} = \sqrt[\uproot{13}{{4}}]{{\dfrac{{E + \sqrt {{E^2} - 4F} }}{8}}},\quad\;{E^2} - 4F > 0,\\ {\beta _2} = \dfrac{{\sqrt {2\sqrt[4]{F} - \sqrt {2\sqrt F } - E} }}{2},\;{\beta _3} = \dfrac{{\sqrt {2\sqrt[4]{F} + \sqrt {2\sqrt F } - E} }}{2},\quad\;{E^2} - 4F < 0. \end{array}\right.$

(11)

将 $s\left( x \right)$ 采用三角函数进行展开，记为

$s\left( x \right) = {a_0} + \sum\limits_{n = 1}^{\infty } {\left( {{a_n}\cos {\frac{{n{\text{π}} x}}{{{l_{\text{s}}}}}} } \right)},$

(12)

式中： ${a_0} = \dfrac{1}{{{l_{\text{s}}}}}\displaystyle\int\limits_0^{ + {{{l_{\text{s}}}} / 2}} {s\left( x \right){\rm{d}}x}$ ， ${a_n} = \dfrac{2}{{{l_{\text{s}}}}}\displaystyle\int\limits_0^{ + {{{l_{\text{s}}}} / 2}} {s\left( x \right)\cos {\frac{{n{\text{π}} x}}{{{l_{\text{s}}}}}}{\rm{d}}x}$ ， ${l_{\text{s}}}$ 为线路计算长度，n为三角级数项的编号.

路基沉降对应的轨道变形特解为

${\bar w_{\text{r}}}\left( x \right) = {a_0} + \sum\limits_{n = 1}^{\infty } {{\dfrac{{F\cos \left( {\dfrac{{n{\text{π}} x}}{{{l_{\text{s}}}}}} \right)}}{{{{\left( {\dfrac{{n{\text{π}} }}{{{l_{\text{s}}}}}} \right)}^8} + E{{\left( {\dfrac{{n{\text{π}} }}{{{l_{\text{s}}}}}} \right)}^4} + F}}} }.$

(13)

同理，将式（10）代入式（4），可求得脱空区外侧轨道板叠合梁变形表达式为

$\begin{split} & {w_{\text{b}}}\left( x \right) = {\bar w_{\text{r}}}\left( x \right) + \frac{{{E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}}}\frac{{{{\rm{d}}^4}{{\bar w}_{\text{r}}}\left( x \right)}}{{{\rm{d}}{x^4}}} + \frac{{\gamma h}}{{{k_{\text{b}}}}} +\\ &\quad {D_2}{{\rm{e}}^{ - {\beta _2}x}}\left( {{c_9}\cos \;{\beta _2}x + {c_{10}}\sin \;{\beta _2}x} \right) +\\ &\quad {D_3}{{\rm{e}}^{ - {\beta _3}x}}\left( {{c_{11}}\cos \;{\beta _3}x + {c_{12}}\sin \;{\beta _3}x} \right), \end{split}$

(14)

式中： ${D_{\text{2}}} = 1 - 4\dfrac{{{E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}}}{\beta _{\text{2}}^4}$ ， ${D_{\text{3}}} = 1 - 4\dfrac{{{E_{\text{r}}}{I_{\text{r}}}}}{{{k_{\text{r}}}}}{\beta _{\text{3}}^4}$ .

根据Euler梁变形理论，可求得钢轨和轨道板叠合梁的转角 ${\theta _j}$ 、弯矩 ${M_j}$ 和剪力 ${Q_j}$ （ $j$ =r，b分别表示钢轨和叠合梁）：

$\left\{ {\begin{array}{*{20}{l}} {{\theta _j} = \dfrac{{{\rm{d}}{w_j}}}{{{\rm{d}}x}}}, \\ {{M_j} = {{ - }}{E_j}{I_j}\dfrac{{{{\rm{d}}^2}{w_j}}}{{{\rm{d}}{x^2}}}}, \\ {{Q_j} = {{ - }}{E_j}{I_j}\dfrac{{{{\rm{d}}^3}{w_j}}}{{{\rm{d}}{x^3}}}}. \end{array}} \right.$

(15)

1）由于模型对称性，当 $x = 0$ 时，钢轨和叠合梁的转角与剪力都为 ${\text{0}}$ ，即

$\left\{ {\begin{array}{*{20}{c}} {{{\left. {\dfrac{{{\rm{d}}{w_j}}}{{{\rm{d}}x}}} \right|}_{x = 0}} = 0}, \\ {{{\left. {\dfrac{{{{\rm{d}}^3}{w_j}}}{{{\rm{d}}{x^3}}}} \right|}_{x = 0}} = 0} . \end{array}} \right.$

(16)

2）当 $x = {l_{\rm{t}}}/2$ 时，钢轨和叠合梁的位移、转角、弯矩与剪力都保持连续，即

$\left\{ {\begin{array}{*{20}{l}} {{{\left. {{w_j}} \right|}_{{x^ - } = {{{l_{\text{t}}}} / 2}}} = {{\left. {{w_j}} \right|}_{{x^ + } = {{{l_{\text{t}}}} / 2}}}}, \\ {{{\left. {\dfrac{{{\rm{d}}{w_j}}}{{{\rm{d}}x}}} \right|}_{{x^ - } = {{{l_{\text{t}}}} / 2}}} = {{\left. {\dfrac{{{\rm{d}}{w_j}}}{{{\rm{d}}x}}} \right|}_{{x^ + } = {{{l_{\text{t}}}} / 2}}}} ,\\ {{{\left. {\dfrac{{{{\rm{d}}^2}{w_j}}}{{{\rm{d}}{x^2}}}} \right|}_{{x^ - } = {{{l_{\text{t}}}} / 2}}} = {{\left. {\dfrac{{{{\rm{d}}^2}{w_j}}}{{{\rm{d}}{x^2}}}} \right|}_{{x^ + } = {{{l_{\text{t}}}} / 2}}}}, \\ {{{\left. {\dfrac{{{{\rm{d}}^3}{w_j}}}{{{\rm{d}}{x^3}}}} \right|}_{{x^ - } = {{{l_{\text{t}}}} / 2}}} = {{\left. {\dfrac{{{{\rm{d}}^3}{w_j}}}{{{\rm{d}}{x^3}}}} \right|}_{{x^ + } = {{{l_{\text{t}}}} / 2}}}} . \end{array}} \right.$

(17)

根据上述推导的钢轨和轨道板叠合梁变形计算式，利用边界条件（式（16）、（17））可建立起一个关于待定系数的12 × 12阶线性方程组，编制MATLAB程序可求出待定系数. 回代入式（7）、（10）即为隧道下穿施工诱发无砟铁路钢轨变形. 由于脱空区长度 ${l_{\rm{t}}}$ 未知，需要采用多次迭代法进行计算.

3. 数值验证

郭宇等^[23]基于通用有限元软件ABAQUS，对余弦型路基沉降引起双块式无砟轨道的几何变形规律进行了模拟分析. 模型中采用空间梁单元模拟钢轨，采用弹簧－阻尼器模拟扣件系统，道床板、支承层和路基结构均采用实体单元模拟，结构材料参数见表1. 在轨枕与混凝土道床板、道床板与支承层、支承层与路基表面均进行接触设置，并充分考虑轨道结构和路基之间可能出现离缝甚至空吊现象.

表 1 无砟轨道材料参数

Table 1. Parameters of ballastless rail material

结构	弹性模量/Pa	泊松比	说明
钢轨	2.10 × 10¹¹	0.30	T60 轨
道床板	3.25 × 10¹⁰	0.17	C40 混凝土
支承层	2.55 × 10¹⁰	0.17	C20 混凝土
路基	180	0.25	密度 300 kg/m³

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为验证本文理论方法的正确性，采用本文理论方法对文献[23]中不同路基沉降幅值和波长条件下的轨面变形进行分析，模型中钢轨抗弯刚度为6.624 MN·m²，叠合梁抗弯刚度为1.225 GN·m²，扣件刚度为30 MN·m⁻¹，路基弹性系数为200 MN·m⁻¹. 本文计算结果与文献[23]的分析数据对比如图2所示，图2（a）中路基沉降波长取20 m，幅值分别取5 mm和20 mm；图2（b）中路基沉降幅值10 mm，波长分别取10 m和40 m.

图 2 本文结果与文献[23]数据的对比

Figure 2. Comparisons between results of this study and data from literature [23]

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从图2中可看出：当轨道结构未产生脱空（5 mm/20 m、10 mm/40 m）时，理论解析计算结果与数值模拟结果几乎一致；当轨道结构产生脱空（20 mm/20 m、10 mm/10 m）时，考虑轨道结构脱空的理论解析计算结果与数值模拟结果基本一致，小于未考虑轨道结构脱空的理论计算结果；当路基沉降为10 mm/10 m时，未考虑轨道结构脱空的理论方法求解出的钢轨变形最大值（7.9 mm）约为考虑轨道结构脱空的理论计算结果（2.3 mm）的3.4倍. 分析表明，当路基不均匀沉降引起无砟轨道结构产生离缝或空吊时，采用本文提出的理论方法仍能较为准确计算出轨面各点处变形值.

4. 参数分析

为控制新建隧道下穿既有铁路引起轨道变形，实际工程中通常采用减小隧道施工引起周围地层损失率或选择不同穿越角度和隧道埋深等措施. 为此，本节将针对上述3个参数对隧道下穿施工诱发轨面变形的影响进行分析. 算例中：开挖半径 ${R_{\text{t}}}$ =3.1 m，地层损失率 ${V_{\text{t}}}$ =0.5%，隧道埋深 ${{\textit{z}}_{\text{t}}}$ =6 m，地层沉降槽宽度参数 $\kappa$ =0.5，隧道与铁路线路垂直相交；无砟铁路结构参数取值与验证算例相同.

4.1 隧道埋深的影响

为分析隧道埋深对下穿施工引起钢轨变形的影响，采用本文理论方法分别计算不同隧道埋深（ ${{\textit{z}}}_{\text{t}}$ =2，4，6，10，20，40 m）条件下轨道的变形，其中轨道中点变形值及轨道板叠合梁脱空区宽度随隧道埋深的变化曲线如图3所示.

图 3 隧道埋深的影响

Figure 3. Influence of buried depth of new tunnel

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从图3中可看出，随着隧道埋深的逐步增大，轨道中点变形值及叠合梁底部脱空区宽度均先增大而后减小，当隧道埋深为6 m时达到峰值. 这是由于随着隧道埋深的增加，隧道施工引起路基顶部的沉降槽宽度将逐渐增大，但沉降槽最大沉降值却逐渐减小. 当新建隧道埋深较小时，隧道施工引起路基顶部的沉降槽的宽度较小，轨道叠合梁底部脱空区宽度和最大变形值较小；随着隧道埋深的增大，路基顶部沉降槽的宽度将增大，导致叠合梁底部脱空区宽度和钢轨中心点变形逐渐增大；当隧道埋深增大到一定值时，路基顶部的沉降逐渐平缓，导致叠合梁脱空区宽度逐渐减小，直至随路基协同变形. 由此可见，选择适当的隧道埋深穿越既有铁路可以有效控制隧道下穿施工引起上方轨道变形.

4.2 隧道施工引起地层损失率的影响

当隧道施工中地层损失率 ${V_{\text{t}}}$ 分别取0.25%、0.50%、1.00%、1.50%、2.00%和2.50%时，隧道下穿施工诱发钢轨中点变形值及叠合梁底部脱空区宽度变化如图4所示. 由图可知：随着隧道施工过程中周围地层损失率的逐渐增大，钢轨中点变形值及叠合梁底部脱空区宽度均逐渐增大；当隧道施工中地层损失率从0.25%增大到2.50%时，钢轨中点变形值和叠合梁脱空区宽度分别从9.7 mm、4 m增大到48.6 mm和13 m，依次增长了约4.00倍、2.25倍. 由此可见，当减小隧道施工中地层损失率是控制隧道下穿施工引起钢轨变形的有效措施之一.

图 4 隧道施工引起地层损失率的影响

Figure 4. Influence of ground loss rate caused by tunnel undercrossing construction

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4.3 隧道与铁路水平夹角的影响

当新建隧道与铁路间的水平夹角 $\;\beta$ 分别取15°、30°、45°、60°、75° 和90° 时，新建隧道下穿施工诱发轨道中点变形值及叠合梁底部脱空区宽度变化如图5所示. 由图可知：随着隧道与铁路线路间水平夹角的逐渐增大，钢轨中点变形值逐渐减小，而叠合梁底部脱空区宽度逐渐先增大、后基本保持不变. 这是由于当新建隧道与既有铁路线路间的水平夹角较小时，隧道施工引起上方铁路路基顶部沉降较为平缓，轨道叠合梁与路基之间保持跟随变形；随着隧道与铁路线路间水平夹角的增大，路基顶部沉降槽宽度逐渐变窄，轨道叠合梁与路基间产生变形差，轨道叠合梁底部将产生局部脱空，轨道中点变形将减小. 由此可见，选择较大角度穿越既有铁路有利于控制隧道下穿施工引起轨道的变形.

图 5 地铁线路与隧道水平夹角的影响

Figure 5. Influence of intersection angle between between railway and tunnel

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5. 小　结

1）将无砟铁路轨道结构简化为双层地基梁，采用无拉力弹性地基模拟轨道板叠合梁与路基的互相作用，并根据叠合梁与路基间的变形差判别两者的接触状态，建立隧道下穿施工引起无砟铁路轨道变形的控制方程.

2）将双层梁分为中间脱空段和两端接地段共三部分，采用解析方法分别推导出上、下层梁（钢轨和轨道板叠合梁）变形的通解和特解表达式，并结合边界条件和分段处连续条件求出变形表达式中的待定系数.

3）随着隧道埋深的增大，轨道最大变形值及叠合梁底部脱空区宽度均呈先增大而后减小趋势. 随着隧道施工中周围地层损失率的增大，钢轨最大变形值及叠合梁底部脱空区宽度均逐渐增大. 随着隧道与铁路线路间水平夹角的增大，钢轨最大变形值逐渐减小，而叠合梁底部脱空区宽度逐渐增大、后保持不变.

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