Effect of dowel joints on dynamic behavior of train–discrete floating slab track system

Abstract

Dowel joints are installed at the end of floating slabs and can effectively reduce the discontinuity of deformation between adjacent floating slabs. The shear spring model had been proposed for modeling the dowel joints of the floating slab track, while the constraint effect of dowel joints on the bending/rotation has not been considered. For a more comprehensive investigation of dowel joints, two idealized dowel models are introduced here, that is, the shear spring–dashpot model and the bending spring–dashpot model, respectively. The effects of these two models on the vibration of train–floating slab track system are analyzed by numerical examples. It is concluded that both shear dowel model and bending dowel model can reduce the dynamic responses of the train–floating slab track system. In detail, the shear dowel model can effectively reduce the displacement difference between adjacent floating slabs but can hardly decrease the displacement amplitudes of the rail and slabs, while the bending dowel model can effectively decrease the displacement amplitudes but has little influence on reducing the displacement difference. In addition, the spring stiffness of these two dowel models has significant influence on the vibration performance of the train–floating slab track system, while the effect of damping coefficients can almost be neglected.

Introduction

Floating slab track (FST) is widely used to reduce the vibration or noise induced by passing trains, especially in areas with poor foundation conditions or high vibration isolation requirements.^1–4 The heavy floating slabs are rested on high-elastic supports to reduce the dynamic responses due to train/rail interaction, such as rubber bearings and steel springs. The floating slabs can be either continuous if they are casted in situ or discontinuous if discrete precast sections are used and mounted end to end. For a discrete FST subjected to a moving train, a parametric excitation may occur due to the discontinuity of slabs, which increases the train–track vibration and, therefore, has negative influence on the train–track system. ⁵ In this case, the dowel bars or plates are usually installed between adjacent floating slabs to reduce the stiffness discontinuity of slabs and improve the overall dynamic performance of the train-FST system.

The vibration problems of an FST system have been widely investigated based on the theoretical methods^6–8 and experimental methods.^9,10 Cui and Chew ⁴ discussed the force transmission of the Singapore FST system using an analytical method. Hussein and Hunt^5,11 modeled the discontinuous and continuous FST systems as infinite beam systems and investigated the dynamic performance of FST subjected to oscillating moving loads. The vehicle–track impact at the connection between the FST and ballasted track was studied by Li and Wu. ¹² Gupta and Degrande ¹³ established a coupled model for the FST and tunnel–soil system, and the vibration isolation efficiency of continuous and discontinuous FST systems was discussed.

As mentioned before, dowel joints are usually installed at the end of discrete floating slabs to reduce the vibration and noise induced by the slab discontinuity. However, the studies on the effect of dowel joints in a discrete FST system are rare. Chung et al. ¹⁴ tested four floating slabs with different dowel joints, and the effects of dowel joints on the load transfer efficiency of the FST were studied. A theoretical model which considered the dowel joints as shear springs was also calculated and compared to the experimental results. Hussein and Costa ¹⁵ also used the shear spring model for the connection between adjacent slabs and investigated the effect of shear spring stiffness on the dynamic responses of the FST. A more complicated shear spring–damper element was adopted by Xu et al.,^16,17 and the influence of slab length on the dynamic performance of train-FST system was studied, as well as the environmental vibration induced by the subway track. It is noted that the aforementioned work considered only the vertical constraint of dowel joints on the discrete FST, while the restraint effect of dowel joints on the bending or rotation at slab ends is not taken into account.

In this article, two dowel models are introduced to analyze different effects of dowel joints on the train-FST vibration, that is, the shear spring–dashpot model and the bending spring–dashpot model. The dynamic responses of the train-FST system with different dowel models are calculated based on the modal superposition method and Newmark-β method. The effects of two dowel models on the vibration reduction of the train-FST system are analyzed by numerical examples.

Modeling of train–track system with dowel joints

Consider a train-FST system with dowel joints. A quarter model of a carriage with four wheel axles and two bodies is used here, as shown in Figure 1. The mass for a quarter of the carriage is $m_{\mathrm{c}}$ , the mass for half a bogie is $m_{\mathrm{b}}$ , and the mass for a single wheel axle is $m_{\mathrm{w}}$ . The wheel and bogie are connected by a spring–dashpot element with the spring stiffness and damping coefficient being $K_1$ and $C_1$ , respectively, while the bogie and carriage are connected by the spring $K_2$ and damper $C_2$ . The interaction between the wheel and rail is represented by vertical force $F_{\mathrm{wr}}$ .

Figure 1.

Model of a quarter train.

According to the theory of structural dynamics, the governing equation of a quarter train can be written as follows

$${\mathbf{M}}_{\mathrm{tr}}{\stackrel{\mathrm{··}}{\mathbf{y}}}_{\mathrm{tr}}+{\mathbf{C}}_{\mathrm{tr}}{\stackrel{.}{\mathbf{y}}}_{\mathrm{tr}}+{\mathbf{K}}_{\mathrm{tr}}{\mathbf{y}}_{\mathrm{tr}}={\mathbf{f}}_{\mathrm{tr}}$$

(1)

where ${\mathbf{M}}_{\mathrm{tr}}$ , ${\mathbf{C}}_{\mathrm{tr}}$ , and ${\mathbf{K}}_{\mathrm{tr}}$ are 3×3 vehicle mass matrix, damping matrix, and stiffness matrix, respectively; ${\mathbf{y}}_{\mathrm{tr}}$ , ${\stackrel{·}{\mathbf{y}}}_{\mathrm{tr}}$ , and ${\stackrel{··}{\mathbf{y}}}_{\mathrm{tr}}$ are the displacement, velocity, and acceleration vectors of the train, respectively; and ${\mathbf{f}}_{\mathrm{tr}}$ is the external force vector applied on the vehicle, including the weight of the vehicle and the wheel/rail interaction force $F_{\mathrm{wr}}$ . For simplicity, an elastic wheel/rail relation is adopted here as follows

$$F_{\mathrm{wr}}\left(t\right)=K_{wr}\left(y_w\left(t\right)-\epsilon \left(x_w\right)y_r\left(x_w,t\right)\right)$$

(2)

where $K_{\mathrm{wr}}$ is the wheel/rail interaction stiffness; $y_w(t)$ and $y_r(x_w,t)$ are the vertical displacements of the wheel and rail, respectively; $x_w$ is the position of the wheel at arbitrary time t; and $\epsilon (x_w)$ equals one if the wheel and rail are contacted, otherwise $\epsilon (x_w)$ equals zero.

The rail is modeled as a long simply supported Timoshenko beam with length $L_{\mathrm{r}}=250\mathrm{m}$ , while the floating slab is modeled as a free-free Timoshenko beam with length $L_{\mathrm{s}}=25\mathrm{m}$ . The rail pads are represented by discrete spring–dashpots with stiffness $K_{\mathrm{rs}}$ and damping coefficient $C_{\mathrm{rs}}$ , so are the slab bearings with stiffness $K_{\mathrm{st}}$ and damping coefficient $C_{\mathrm{st}}$ . The rail pads and slab bearings are laid with distance $L_{\mathrm{rs}}=0.625\mathrm{m}$ and $L_{\mathrm{st}}=1.25\mathrm{m}$ , respectively. The total number of floating slabs considered in this model is $n_{\mathrm{s}}=10$ , while the number of rail pads and slab bearings per one slab are $n_{\mathrm{rs}}=40$ and $n_{\mathrm{st}}=20$ , respectively. The dowel joints are installed between adjacent slabs to reduce the discontinuity, as shown in Figure 2.

Figure 2.

Model of a discrete floating slab track with dowel joints.

Based on the Timoshenko beam theory, the equation of the rail is given by

$$\{\begin{array}{c}\kappa A_{\mathrm{r}}{G}_{\mathrm{r}}(\frac{{\partial }^2{y}_{\mathrm{r}}(x,t)}{\partial x^2}-\frac{\partial {\varphi }_{\mathrm{r}}(x,t)}{\partial x})={\rho }_{\mathrm{r}}{A}_{\mathrm{r}}\frac{{\partial }^2{y}_{\mathrm{r}}(x,t)}{\partial t^2}-F_{\mathrm{r}}(x,t)\hfill \\ \multicolumn{1}{c}{E_{\mathrm{r}}{I}_{\mathrm{r}}\frac{{\partial }^2{\varphi }_{\mathrm{r}}(x,t)}{\partial x^2}+\kappa A_{\mathrm{r}}{G}_{\mathrm{r}}(\frac{\partial y_{\mathrm{r}}(x,t)}{\partial x}-{\varphi }_{\mathrm{r}}(x,t))={\rho }_{\mathrm{r}}{I}_{\mathrm{r}}\frac{{\partial }^2{\varphi }_{\mathrm{r}}(x,t)}{\partial t^2}}\end{array}$$

(3)

where $y_{\mathrm{r}}(x,t)$ is the vertical displacement of the rail; ${\varphi }_{\mathrm{r}}(x,t)$ is the rotational angle; $E_{\mathrm{r}}{I}_{\mathrm{r}}$ and $\kappa A_{\mathrm{r}}{G}_{\mathrm{r}}$ are the bending stiffness and shear stiffness, respectively; ${\rho }_{\mathrm{r}}$ is the mass density; $A_{\mathrm{r}}$ is the cross-sectional area; and $I_{\mathrm{r}}$ is the moment of inertia of the cross section.

The external force of the rail $F_{\mathrm{r}}(x,t)$ is expressed as follows

$$\begin{array}{l}{F}_{\mathrm{r}}\left(x,t\right)=\delta \left(x-x_{\mathrm{w}}\right)F_{\mathrm{wr}}-{\displaystyle \sum _{j=1}^{n_s}{\displaystyle \sum _{i=1}^{n_{rs}}\delta \left(x-x_{ij}\right)}}\\ \left\{K_{\mathrm{rs}}\left[y_{\mathrm{r}}\left(x_{ij},t\right)-y_{\mathrm{s}j}\left(x_{\mathrm{s}ij},t\right)\right]+C_{\mathrm{rs}}\left[{\stackrel{·}{y}}_{\mathrm{r}}\left(x_{ij},t\right)-{\stackrel{·}{y}}_{\mathrm{s}j}\left(x_{\mathrm{s}ij},t\right)\right]\right\}\end{array}$$

(4)

where $\delta (x)$ is the Dirac delta function; $x_{\mathrm{w}}$ is the position of the wheel at arbitrary time t; $n_{\mathrm{s}}=10$ is the number of slabs considered in the present model; $n_{\mathrm{rs}}=40$ is the number of rail pads installed on one single slab; $K_{\mathrm{rs}}$ and $C_{\mathrm{rs}}$ are the stiffness coefficient and damping coefficient of rail pads, respectively; $x_{\mathrm{ij}}$ is the position of the ith rail pad on the jth floating slab, while $x_{\mathrm{s}\mathrm{ij}}=x_{\mathrm{ij}}-(j-1)L_{\mathrm{s}}$ is the corresponding position in the local coordinate of the jth slab; $y_{\mathrm{r}}$ is the vertical displacements of the rail; and $y_{\mathrm{s}j}$ is the vertical displacements of the jth slab.

The floating slabs are modeled as a Timoshenko beam with free-free ends. Different dowel materials and joint types have been adopted in FSTs to reduce the slab discontinuity, ¹⁴ for example, the steel dowel bars/plates installed either on the top of the slabs or embedded in the middle of the slabs. In this article, two idealized dowel models are introduced to stimulate different types of dowel joints, that is, the shear spring–dashpot model as shown in Figure 3(a) and the bending spring–dashpot model as shown in Figure 3(b).

Figure 3.

Model of dowel joints at the end of slabs: (a) shear spring–dashpot model and (b) bending spring–dashpot model.

For the dynamic analysis of slabs with the shear dowel joints or bending dowel joints, the governing equation of the jth slab under local coordinates $x_{\mathrm{s}}-y_{\mathrm{s}}$ can be obtained as follows

$$\{\begin{array}{c}\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}(\frac{{\partial }^2{y}_{\mathrm{s}j}(x_{\mathrm{s}},t)}{\partial x_{\mathrm{s}}^2}-\frac{\partial {\varphi }_{\mathrm{s}j}(x_{\mathrm{s}},t)}{\partial x_{\mathrm{s}}})={\rho }_{\mathrm{s}}{A}_{\mathrm{s}}\frac{{\partial }^2{y}_{\mathrm{s}j}(x_{\mathrm{s}},t)}{\partial t^2}-F_{\mathrm{s}j}(x_{\mathrm{s}},t)\hfill \\ \multicolumn{1}{c}{E_{\mathrm{s}}{I}_{\mathrm{s}}\frac{{\partial }^2{\varphi }_{\mathrm{s}j}(x_{\mathrm{s}},t)}{\partial x_{\mathrm{s}}^2}+\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}(\frac{\partial y_{\mathrm{s}j}(x_{\mathrm{s}},t)}{\partial x_{\mathrm{s}}}-{\varphi }_{\mathrm{s}j}(x_{\mathrm{s}},t))={\rho }_{\mathrm{s}}{I}_{\mathrm{s}}\frac{{\partial }^2{\varphi }_{\mathrm{s}j}(x_{\mathrm{s}},t)}{\partial t^2}+M_{\mathrm{s}j}(x_{\mathrm{s}},t)}\end{array}$$

(5)

where $y_{\mathrm{s}j}$ and ${\varphi }_{\mathrm{s}j}$ are the vertical displacement and rotational angle of the jth slab, respectively. The bending stiffness $E_{\mathrm{s}}{I}_{\mathrm{s}}$ , shear stiffness $\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}$ , mass density ${\rho }_{\mathrm{s}}$ , cross-sectional area $A_{\mathrm{s}}$ , and the moment of inertia $I_{\mathrm{s}}$ are constant for all slabs. $F_{\mathrm{s}j}(x_{\mathrm{s}},t)$ and $M_{\mathrm{s}j}(x_{\mathrm{s}},t)$ are the external force and external moment applied on the jth slab, respectively.

The detailed expressions of $F_{\mathrm{s}j}(x_{\mathrm{s}},t)$ and $M_{\mathrm{s}j}(x_{\mathrm{s}},t)$ depend on different types of dowel models.

The shear dowel model

The external moment $M_{\mathrm{s}j}$ is equal to zero in this case. The external force $F_{\mathrm{s}j}$ includes the forces applied by rail pads, slab bearings, and dowel joints, respectively

$$\begin{array}{c}\multicolumn{1}{c}{F_{\mathrm{s}j}(x_{\mathrm{s}},t)=\sum _{i=1}^{n_{\mathrm{rs}}}\delta (x_{\mathrm{s}}-x_{\mathrm{s}\mathrm{ij}})\{K_{\mathrm{rs}}[y_{\mathrm{r}}(x_{\mathrm{ij}},t)-y_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{ij}},t)]+C_{\mathrm{rs}}[{\stackrel{·}{y}}_{\mathrm{r}}(x_{\mathrm{ij}},t)-{\stackrel{·}{y}}_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{ij}},t)]\}}\\ -\sum _{k=1}^{n_{\mathrm{st}}}\delta (x_{\mathrm{s}}-x_{\mathrm{s}\mathrm{kj}})[K_{\mathrm{st}}{y}_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{kj}},t)+C_{\mathrm{st}}{\stackrel{·}{y}}_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{kj}},t)]\hfill \\ +K_{\mathrm{q}}\{\delta (x_{\mathrm{s}}-0)[y_{\mathrm{s}(j-1)}(L_{\mathrm{s}},t)-y_{\mathrm{s}j}(0,t)]-\delta (x_{\mathrm{s}}-L_{\mathrm{s}})[y_{\mathrm{s}j}(L_{\mathrm{s}},t)-y_{\mathrm{s}(j+1)}(0,t)]\}\hfill \\ +C_{\mathrm{q}}\{\delta (x_{\mathrm{s}}-0)[{\stackrel{·}{y}}_{\mathrm{s}(j-1)}(L_{\mathrm{s}},t)-{\stackrel{·}{y}}_{\mathrm{s}j}(0,t)]-\delta (x_{\mathrm{s}}-L_{\mathrm{s}})[{\stackrel{·}{y}}_{\mathrm{s}j}(L_{\mathrm{s}},t)-{\stackrel{·}{y}}_{\mathrm{s}(j+1)}(0,t)]\},\hfill \\ \mathrm{for}j=2,3,\dots ,n_{\mathrm{s}}-1\hfill \end{array}$$

(6)

$$M_{\mathrm{s}j}(x_{\mathrm{s}},t)=0$$

(7)

where $x_{\mathrm{ij}}$ and $x_{\mathrm{s}\mathrm{ij}}$ are the positions of the ith rail pad on the jth floating slab in the global coordinates $x-y$ and the local coordinates $x_{\mathrm{s}}-y_{\mathrm{s}}$ , respectively; $n_{\mathrm{rs}}=40$ and $n_{\mathrm{st}}=20$ are the number of rail pads and slab bearings per one slab, respectively; $x_{\mathrm{s}\mathrm{kj}}$ is the local position of the kth slab bearing on the jth floating slab in the local coordinates $x_{\mathrm{s}}-y_{\mathrm{s}}$ ; $K_{\mathrm{st}}$ and $C_{\mathrm{st}}$ are the stiffness coefficient and damping coefficient of slab bearings, respectively; $K_{\mathrm{q}}$ and $C_{\mathrm{q}}$ are the stiffness coefficient and damping coefficient of shear dowel joints, respectively; $L_{\mathrm{s}}$ is the length of each slab; and $y_{\mathrm{s}(j-1)}$ , $y_{\mathrm{s}j}$ , and $y_{\mathrm{s}(j+1)}$ are the vertical displacements of the (j − 1)th, jth, and (j + 1)th slab, respectively. It is noted that the forces of dowel joints are vanished at the left end of the first slab and at the right end of the n_sth slab.

The bending dowel model

Both external force $F_{\mathrm{s}j}$ and moment $M_{\mathrm{s}j}$ are nonzero, but $F_{\mathrm{s}j}$ only contains the forces of rail pads and slab bearings. The expressions for $F_{\mathrm{s}j}$ and $M_{\mathrm{s}j}$ are given by

$$\begin{array}{c}{F}_{\mathrm{s}j}(x_{\mathrm{s}},t)=\sum _{i=1}^{n_{\mathrm{rs}}}\delta (x_{\mathrm{s}}-x_{\mathrm{s}\mathrm{ij}})\hfill \\ \{K_{\mathrm{rs}}[y_{\mathrm{r}}(x_{\mathrm{ij}},t)-y_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{ij}},t)]+C_{\mathrm{rs}}[{\stackrel{·}{y}}_{\mathrm{r}}(x_{\mathrm{ij}},t)-{\stackrel{·}{y}}_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{ij}},t)]\}\hfill \\ -\sum _{k=1}^{n_{\mathrm{st}}}\delta (x_{\mathrm{s}}-x_{\mathrm{s}\mathrm{kj}})[K_{\mathrm{st}}{y}_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{kj}},t)+C_{\mathrm{st}}{\stackrel{·}{y}}_{\mathrm{s}j}(x_{\mathrm{s}\mathrm{kj}},t)]\hfill \end{array}$$

(8)

$$\begin{array}{c}{M}_{\mathrm{s}j}(x_{\mathrm{s}},t)=K_{\mathrm{m}}\{\delta (x_{\mathrm{s}}-0)[{\frac{\partial y_{\mathrm{s}(j-1)}}{\partial x}|}_{x_{\mathrm{s}}=L_{\mathrm{s}}}-{\frac{\partial y_{\mathrm{s}j}}{\partial x}|}_{x_{\mathrm{s}}=0}]-\delta (x_{\mathrm{s}}-L_{\mathrm{s}})[{\frac{\partial y_{\mathrm{s}j}}{\partial x}|}_{x_{\mathrm{s}}=L_{\mathrm{s}}}-{\frac{\partial y_{\mathrm{s}(j+1)}}{\partial x}|}_{x_{\mathrm{s}}=0}]\}\hfill \\ +C_{\mathrm{m}}\{\delta (x_{\mathrm{s}}-0)[{\frac{\partial {\stackrel{·}{y}}_{\mathrm{s}(j-1)}}{\partial x}|}_{x_{\mathrm{s}}=L_{\mathrm{s}}}-{\frac{\partial {\stackrel{·}{y}}_{\mathrm{s}j}}{\partial x}|}_{x_{\mathrm{s}}=0}]-\delta (x_{\mathrm{s}}-L_{\mathrm{s}})[{\frac{\partial {\stackrel{·}{y}}_{\mathrm{s}j}}{\partial x}|}_{x_{\mathrm{s}}=L_{\mathrm{s}}}-{\frac{\partial {\stackrel{·}{y}}_{\mathrm{s}(j+1)}}{\partial x}|}_{x_{\mathrm{s}}=0}]\},\mathrm{for}j=2,3,\dots ,n_{\mathrm{s}}-1\hfill \end{array}$$

(9)

where $K_{\mathrm{m}}$ and $C_{\mathrm{m}}$ are the stiffness coefficient and damping coefficient of bending dowel joints, respectively.

Solution of train–track vibration equation

Based on the modal superposition method, the displacement and rotational angle of the rail can be assumed as

$$\{\begin{array}{c}{y}_{\mathrm{r}}(x,t)=\sum _{p=1}^{N_{\mathrm{r}}}{Y}_{\mathrm{r}p}\left(x\right)q_{\mathrm{r}p}\left(t\right)\hfill \\ \multicolumn{1}{c}{{\varphi }_{\mathrm{r}}(x,t)=\sum _{p=1}^{N_{\mathrm{r}}}{\mathrm{\Phi }}_{\mathrm{r}p}\left(x\right)q_{\mathrm{r}p}\left(t\right)}\end{array}$$

(10)

where $Y_{\mathrm{r}p}(x)$ and ${\mathrm{\Phi }}_{\mathrm{r}p}(x)$ are the pth mode shape functions of the displacement and rotational angle of the rail, respectively; $q_{\mathrm{r}p}(t)$ is the generalized function of the rail; and $N_r$ is the number of modes considered for the rail.

For a simply supported Timoshenko beam, the mode shape functions of the rail are given as

$$\{\begin{array}{c}{Y}_{\mathrm{r}p}\left(x\right)=\sin {\lambda }_{p}x\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{\mathrm{r}p}\left(x\right)=g_p\cos {\lambda }_{p}x}\end{array}$$

(11)

in which the wavenumber ${\lambda }_p=j\pi /L_{\mathrm{r}}$ can be determined by the boundary conditions of the rail; $g_p=(\kappa A_{\mathrm{r}}{G}_{\mathrm{r}}{\lambda }_p^2-{\rho }_{\mathrm{r}}{A}_{\mathrm{r}}{\omega }_{\mathrm{r}p}^2)/\kappa A_{\mathrm{r}}{G}_{\mathrm{r}}{\lambda }_p$ is the amplitude coefficient; and the pth natural frequency of the rail is expressed as

$${\omega }_{\mathrm{r}p}=\frac{[\kappa G_{\mathrm{r}}{A}_{\mathrm{r}}/I_{\mathrm{r}}+{\lambda }_{\mathrm{r}}^2(E_{\mathrm{r}}+\kappa G_{\mathrm{r}})-\sqrt{{(\kappa G_{\mathrm{r}}{A}_{\mathrm{r}}/I_{\mathrm{r}}+{\lambda }_{\mathrm{r}}^2(E_{\mathrm{r}}+\kappa G_{\mathrm{r}}))}^2+4\kappa G_{\mathrm{r}}{E}_{\mathrm{s}}{\lambda }_{\mathrm{r}}^4}]}{2{\rho }_{\mathrm{r}}}$$

(12)

The displacement and rotational angle of the jth slab can also be obtained similarly as

$$\{\begin{array}{c}{y}_{\mathrm{s}j}(x_{\mathrm{s}},t)=\sum _{p=1}^{N_{\mathrm{s}}}{Y}_{\mathrm{s}p}\left(x\right)q_{\mathrm{s}p}\left(t\right)\hfill \\ \multicolumn{1}{c}{{\varphi }_{\mathrm{s}j}(x_{\mathrm{s}},t)=\sum _{p=1}^{N_{\mathrm{s}}}{\mathrm{\Phi }}_{\mathrm{s}p}\left(x\right)q_{\mathrm{s}p}\left(t\right)}\end{array}$$

(13)

The mode shape functions for the free-free Timoshenko beam can be written as

$$\begin{array}{c}{Y}_{\mathrm{s}1}\left(x\right)=1,{\mathrm{\Phi }}_{\mathrm{s}1}\left(x\right)=0,\mathrm{for}p=1\hfill \\ Y_{\mathrm{s}2}\left(x\right)=\sqrt{3}(1-\frac{2x}{L_{\mathrm{s}}}),{\mathrm{\Phi }}_{\mathrm{s}2}\left(x\right)=0,\mathrm{for}p=2\hfill \\ \{\begin{array}{c}{Y}_{\mathrm{s}p}\left(x\right)=c_{1p}\cos {\lambda }_{1p}x+s_{1p}\sin {\lambda }_{1p}x+c_{2p}\cosh {\lambda }_{2p}x+s_{2p}\sinh {\lambda }_{2p}x\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{\mathrm{s}p}\left(x\right)=-g_{1p}{c}_{1p}\sin {\lambda }_{1p}x+g_{1p}{s}_{1p}\cos {\lambda }_{1p}x+g_{2p}{c}_{2p}\sinh {\lambda }_{2p}x+g_{2p}{s}_{2p}\cosh {\lambda }_{2p}}\end{array}\hfill \end{array}$$

(14)

where $g_{1p}=(\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}{\lambda }_{1p}^2-{\rho }_{\mathrm{s}}{A}_{\mathrm{s}}{\omega }_{\mathrm{s}p}^2)/\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}{\lambda }_{1p}$ and $g_{2p}=(\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}{\lambda }_{2p}^2+{\rho }_{\mathrm{s}}{A}_{\mathrm{s}}{\omega }_{\mathrm{s}p}^2)/\kappa A_{\mathrm{s}}{G}_{\mathrm{s}}{\lambda }_{2p}$ ; ${\omega }_{\mathrm{s}p}$ is the pth natural frequency of the slab; $c_{1p}$ , $s_{1p}$ , $c_{2p}$ , and $s_{2p}$ are modal coefficients. The natural frequency ${\omega }_{\mathrm{s}p}$ and modal coefficients can be determined by the boundary conditions of the slab. The wavenumbers ${\lambda }_{1p}$ and ${\lambda }_{2p}$ are given as

$${\lambda }_{1,2p}=\sqrt{\frac{1}{2E_{\mathrm{s}}{I}_{\mathrm{s}}}[{\omega }_{\mathrm{s}p}^2{\rho }_{\mathrm{s}}(\frac{E_{\mathrm{s}}{I}_{\mathrm{s}}}{\kappa G_{\mathrm{s}}}+I_{\mathrm{s}})\mp \sqrt{{\omega }_{\mathrm{s}p}^4{\rho }_{\mathrm{s}}^2{(\frac{E_{\mathrm{s}}{I}_{\mathrm{s}}}{\kappa G_{\mathrm{s}}}-I_{\mathrm{s}})}^2+4E_{\mathrm{s}}{I}_{\mathrm{s}}{\rho }_{\mathrm{s}}{A}_{\mathrm{s}}{\omega }_{\mathrm{s}p}^2}]}$$

(15)

Substituting equations (10) and (13) into equations (3) and (5), the partial differential equations of the rail and floating slabs can then be turned into the ordinary differential equations. Combining with equation (1), the coupled equations for the train-FST system can be obtained. The dynamic responses of the train-FST system are finally solved based on the modal superposition method and the Newmark-β method.

Numerical results and discussions

In this section, the dynamic responses of FST with dowel joints under a moving train are calculated based on the aforementioned theoretical model. According to the properties of B-type metro train in China subway and FST system, the parameters adopted for the simulation are listed in Table 1. The train speed is assumed to be 20 m/s.

Table 1.

Parameters of train and floating slab track.

Parameter	Units	Value
Mass of 1/4 carriage $m_{\mathrm{c}}$	kg	10,220
Mass of 1/2 bogie $m_{\mathrm{b}}$	kg	1520
Mass of a wheel axle $m_{\mathrm{w}}$	kg	1760
Stiffness of primary suspension $K_1$	MN/m	2.36
Damping of primary suspension $C_1$	kN s/m	80
Stiffness of secondary suspension $K_2$	MN/m	0.4
Damping of secondary suspension $C_2$	kN s/m	50
Young’s modulus of rail $E_{\mathrm{r}}$	GPa	210
Shear modulus of rail $G_{\mathrm{r}}$	GPa	81
Density of rail ${\rho }_{\mathrm{r}}$	kg/m3	7900
Cross-sectional area of rail $A_{\mathrm{r}}$	m2	1.53 × 10−2
Moment of inertia of rail $I_{\mathrm{r}}$	m4	5.92 × 10−5
Stiffness of wheel/rail interaction $K_{\mathrm{wr}}$	MN/m	1200
Stiffness of rail pad $K_{\mathrm{rs}}$	MN/m	80
Damping of rail pad $C_{\mathrm{rs}}$	kN s/m	150
Young’s modulus of slab	GPa	30
Shear modulus of slab $G_{\mathrm{s}}$	GPa	12.5
Density of slab ${\rho }_{\mathrm{s}}$	kg/m3	2500
Cross-sectional area of slab $A_{\mathrm{s}}$	m2	1.92
Moment of inertia of slab $I_{\mathrm{s}}$	m4	5.76 × 10−2
Stiffness of slab bearing $K_{\mathrm{st}}$	MN/m	10
Damping of slab bearing $C_{\mathrm{st}}$	kN s/m	150

Effect of dowel joints on train–track vibrations

Three different cases are considered here to investigate the effects of dowel joints on the dynamic responses of the train-FST system, that is, the floating slabs with no dowel joints, with shear dowel model and with bending dowel model, respectively. The effects of shear dowel model and bending dowel model on reducing the vibration performance of train–track system are analyzed through numerical results. Here, the stiffness of the shear dowel model is taken as 100 MN/m, while the stiffness of the bending dowel model is 100 MN m/rad. The damping coefficients for these two dowel models are assumed to be zero. A more detailed parametric analysis on stiffness and damping coefficients of dowel models can be found in the next subsection.

Figures 4 and 5 show the vertical displacements of the rail and floating slabs under a moving train, respectively. A total of three different cases are discussed here, that is, the floating slabs with no dowel joints, with shear dowel model, and with bending dowel model, respectively. It can be seen that both shear dowel model and bending dowel model can decrease the amplitudes of rail displacement and slab displacement, and the bending dowel model is more effective compared to the shear dowel model. Besides, it is found that the bending dowel model can barely reduce the displacement difference between adjacent floating slabs, while the shear dowel model can effectively smooth the displacement curve compared to the case of no dowel joints.

Figure 4.

Vertical displacement of the rail under a moving train in three cases.

Figure 5.

Vertical displacement of floating slabs under a moving train in three cases.

Figure 6 shows the acceleration of the carriage in three different cases. It can be seen from Figure 6 that the acceleration amplitudes in three cases are 0.0588, 0.0432, and 0.0492 m/s², respectively. Both shear and bending dowel models can reduce vertical acceleration of the train compared to the case with no dowel joints. In addition, the FST system with shear dowel model has a smooth acceleration curve near the end of slabs ( $x=125\mathrm{m}$ in Figure 6), while the FST with bending dowel model has an abrupt fluctuation at the same position. This phenomenon is consistent with the conclusion obtained from Figures 4 and 5 that the bending dowel model cannot effectively reduce displacement differences near slab ends and thus causes an impact on the train-FST system.

Figure 6.

Carriage acceleration in three cases.

The fastener forces and slab bearing forces at the different positions of the sixth floating slab are depicted in Figures 7 and 8, respectively. It can be seen that the fastener force and bearing force near slab ends are reduced obviously in the FST system with shear dowel model, especially for the tensile force of the fastener (negative value in Figure 7(a)). However, both shear and bending dowel models have little effect on the forces of the fastener and slab bearing in the middle of the slab, as depicted in Figures 7(b) and 8(b).

Figure 7.

Fastener forces at different locations of the sixth floating slab in three cases: (a) fastener force at the left end of the sixth slab and (b) fastener force in the middle of the sixth slab.

Figure 8.

Slab bearing forces at different locations of the sixth floating slab in three cases: (a) slab bearing force at the left end of the sixth slab and (b) slab bearing force in the middle of the sixth slab.

Parametric study of dowel models on train–track dynamics

The influence of dowel parameters on the dynamic performance of the train-FST system is further investigated in this subsection. Different stiffness coefficients and damping coefficients for the shear dowel model and the bending dowel model are considered, and their effects on the vibration performance of train–track system are discussed.

Parametric study of shear dowel model

The effect of shear damping coefficient on the dynamic responses of train–track system is analyzed first. A total of three different damping coefficients are adopted here, that is, $C_{\mathrm{q}}=0,1\times 10^3,\mathrm{and}1\times 10^5\mathrm{N}\mathrm{s}/\mathrm{m}$ , respectively. It can be concluded that the damping coefficient $C_{\mathrm{q}}$ of the shear dowel model has little effect on the rail displacement, as shown in Figure 9. A similar conclusion can be obtained for other dynamic responses of the train–track system, but the numerical results are not listed in detail for the concision of this article.

Figure 9.

Rail displacement under different damping coefficients $C_{\mathrm{q}}$ of shear dowel model.

The influence of shear stiffness coefficient $K_{\mathrm{q}}$ on the vibration responses of train–track system is also discussed. As depicted in Figures 10 and 11, both rail displacement and slab displacement decrease with increasing stiffness $K_{\mathrm{q}}$ and tend to be stable for $K_{\mathrm{q}}$ greater than $1\times 10^8\mathrm{N}/\mathrm{m}$ . However, the reduction rates for the rail displacement and slab displacement are only 6.1% and 6.3%, respectively, which means the shear dowel model has limited effect on reducing the track displacements. It is also found that the displacement curves in Figures 10 and 11 become smoother for an increasing stiffness $K_{\mathrm{q}}$ , and no obvious displacement differences can be observed between adjacent slabs.

Figure 10.

Rail displacement under different shear stiffness $K_{\mathrm{q}}$ of shear dowel model.

Figure 11.

Slab displacement under different shear stiffness $K_{\mathrm{q}}$ of shear dowel model.

Figure 12 shows the curves of the carriage acceleration under different shear spring stiffness $K_{\mathrm{q}}$ . The fastener force and slab bearing force at the left end of the sixth slab are shown in Figures 13 and 14, respectively. It is noted that the carriage acceleration and fastener force will decrease with increasing $K_{\mathrm{q}}$ at first but then increase when $K_{\mathrm{q}}$ is too large. Therefore, a reasonable large spring stiffness $K_{\mathrm{q}}$ of the shear dowel model should be designed for a better dynamic performance of train–track system.

Figure 12.

Carriage acceleration under different shear stiffness $K_{\mathrm{q}}$ of shear dowel model.

Figure 13.

Fastener force at the left end of the sixth slab under different shear stiffness $K_{\mathrm{q}}$ of shear dowel model.

Figure 14.

Slab bearing force at the left end of the sixth slab under different shear stiffness $K_{\mathrm{q}}$ of shear dowel model.

Parametric study of bending dowel model

A similar parametric study has been conducted for a floating slab track with bending dowel joints. As can be seen in Figure 15, the damping coefficient $C_{\mathrm{m}}$ has little effect on rail displacement, which is similar to the damping $C_{\mathrm{q}}$ in the shear dowel model.

Figure 15.

Rail displacement under different damping coefficients $C_{\mathrm{m}}$ of bending dowel model.

Figures 16 and 17 show the displacements of the rail and floating slab, respectively, under different stiffness coefficients $K_{\mathrm{m}}$ of the bending dowel model. Both rail and slab displacements decrease obviously with increasing $K_{\mathrm{m}}$ . The reduction rates for the rail displacement and slab displacement are 18.51% and 28.15%, respectively, which are much larger than those of the shear dowel model. However, an increasing displacement difference can be observed in the displacement curves. It indicates that the bending dowel model can effectively reduce the displacement amplitudes but may enlarge the displacement difference near slab ends.

Figure 16.

Rail displacement under different bending stiffness $K_{\mathrm{m}}$ of bending dowel model.

Figure 17.

Slab displacement under different bending stiffness $K_{\mathrm{m}}$ of bending dowel model.

Figure 18 shows the variation of the carriage acceleration under different bending spring stiffness $K_{\mathrm{m}}$ . The acceleration decreases with increasing $K_{\mathrm{m}}$ , but the location of the maximum acceleration will gradually be closer to the end of slabs. The fastener force will increase slightly as the bending spring stiffness increases, while the force of slab bearing will keep decreasing, as shown in Figures 19 and 20, respectively.

Figure 18.

Carriage acceleration under different bending stiffness $K_{\mathrm{m}}$ of bending dowel model.

Figure 19.

Fastener force at the left end of the sixth slab under different bending stiffness $K_{\mathrm{m}}$ of bending dowel model.

Figure 20.

Slab bearing force at the left end of the sixth slab under different bending stiffness $K_{\mathrm{m}}$ of bending dowel model.

Effect of a mixed dowel model on train–track dynamics

By referring to the research of Chung et al., ¹⁴ we consider a dowel joint constructed by 20 steel dowel bars with each diameter of 25 mm, as shown in Figure 21(a). For such a dowel joint, both shear stiffness and bending stiffness may exist and need to be considered in the train–track system. Here, we introduced a mixed dowel model as shown in Figure 21(b).

Figure 21.

Floating slab with dowel bars: (a) cross section of floating slab and (b) a mixed model for the dowel joint.

The shear stiffness $K_{\mathrm{q}}$ of the dowel joint is calculated as suggested by Chung et al. ¹⁴

$$K_{\mathrm{q}}=\frac{1}{\frac{1}{{\beta }^3\mathrm{EI}}+\frac{d_j^3}{12\mathrm{EI}}}$$

(16)

where $E=206\mathrm{GPa}$ is Young’s modulus of the steel dowel bar, I is the moment of inertia of the dowel bar, and $d_j=2\mathrm{cm}$ is the gap between two adjacent floating slabs. $\beta$ is defined as $\beta =[d_b{K}_b/4\mathrm{EI}{]}^{1/4}$ , in which $d_b$ is the diameter of dowel bar in millimeter and $K_b$ is the concrete bearing stiffness in megapascal per millimeter. The definition of $K_b$ is given by

$$K_b=127c_1\sqrt{f_c}{\left(\frac{1}{d_b}\right)}^{2/3}$$

(17)

in which $f_c=30\mathrm{MPa}$ is the concrete compression strength of slabs, and $c_1$ is a coefficient which ranges from 0.6 to 1.0 and depends on the bar spacing.

The shear stiffness so obtained is $K_{\mathrm{q}}=2\times 10^8\mathrm{N}/\mathrm{m}$ , while the bending stiffness $K_{\mathrm{m}}$ is taken approximately based on the theory of structural mechanics for a fixed-fixed dowel bar, that is, $K_{\mathrm{m}}=2\times 10^6\mathrm{N}\mathrm{m}/\mathrm{rad}$ (Figure 22).

Figure 22.

Fixed-fixed beam model: (a) vertical deformation and (b) rotational deformation.

The rail displacement and slab displacement so calculated are shown in Figures 23 and 24, respectively. It is found that the mixed dowel model can effectively reduce the displacement differences near slab ends and also decrease the displacement amplitudes. Since the shear stiffness is much larger than the bending stiffness in this example, the results obtained here are mainly dominated by the shear stiffness and are, therefore, very similar to those obtained in case of the shear dowel model.

Figure 23.

Rail displacement under a moving train.

Figure 24.

Slab displacement under a moving train.

Conclusion

In this article, a two-dimensional model for a train-FST with dowel joints is developed, and the influence of dowel joints on the vibration performance of train–track system is investigated. A total of two idealized dowel models are introduced to investigate the restriction effect of different dowel joints on the train–track coupling vibrations, that is, the shear spring–dashpot model and the bending spring–dashpot model. The parametric analysis of the shear and bending dowel models is also conducted. The numerical results are calculated using the modal superposition method and Newmark-β method, and the following conclusions are obtained:

1. Both shear dowel model and bending dowel model can reduce the vibration of the train–track system and can thus improve the dynamic performance of the train–track system. For these two dowel models, the damping coefficient has little effect on the vibration responses of the train–track system, while the spring stiffness has significant effect on the train–track coupling vibrations.

2. The shear dowel model can effectively reduce the displacement difference near the end of floating slabs but can barely reduce the displacement amplitudes of the rail and slabs. The spring stiffness of the shear dowel model should not be taken too large since the carriage acceleration and fastener force may be magnified.

3. The effect of bending dowel model on the dynamic performance of train-FST system is just opposite. The two dowel models have complementary effects on improving the dynamic responses of train-FST system.