Appropriate locations of fixed bearings of continuous beams considering rail-bridge thermal interaction

Abstract

Due to the rail-bridge thermal interaction, the high additional axial force in continuously welded rails on continuous bridges may lead to rail buckling or breaking. However, there is little research on the influence of the location of the fixed bearing of continuous beam on the additional force of rail. In order to study the influence of bridge bearing arrangement on the additional longitudinal force of CWR, the thermal interaction model is established for rail, and simple and continuous beams considering nonlinear stiffness and the methods are proposed to determine the locations of fixed bearings of continuous beams corresponding to the maximum additional forces in rail reaching minimum values. Multiple continuous beams with several different lengths and simple beams with three types of bearing arrangements are taken into account to find the effect laws of the locations of the fixed bearings of continuous beams on the maximum additional forces in rail. The results show that as long as the same number of continuous beams, the ratios of the distances of adjacent two fixed bearings to the distance between the two fixed bearings of the simple beams neighbour to the first and last continuous beams respectively are approximately equal to each other. Furthermore the appropriate locations of the fixed bearings of continuous beams are recommended. The results can guide designing the location of the fixed bearing of continuous railway bridge while reducing the additional axial force in continuously welded rails due to bridge thermal effect.

Introduction

A larger number of continuously welded rail (CWR) track are laid on railway bridges. The high additional axial force in CWR due to the rail-bridge longitudinal interaction caused by the temperature change of bridge may lead to rail buckling or breaking. Thus, the problem of the rail-bridge thermal interaction has been gained wide attention. In the early stage, the analytical methods were mainly used. Frýba^1,2 and Esveld ³ put forward basic differential equation and analytic solutions of the additional thermal force of CWR on simply or continuously supported bridges considering linear stiffness.

Then, the numerical or field measurement methods were mainly employed by many scholars. For example, Larsson and Karoumi ⁴ established a finite element model to predict the thermal effect in a hollow concrete section. Kašpárek et al. ⁵ and Carvalho et al. ⁶ monitored the rail-bridge interaction under daily temperature cycle for a long time, and proposed a new method to determine the given rail safety and buckling temperatures by using ANASY finite element software. Mirza et al.^7,8 used the finite element software ABAQUS to study the mechanical properties of rail-bridge systems under the action of thermal load and different levels of seismic load. Zakeri et al.^9,10 analysed the lateral resistance of frictional concrete sleepers and proposed the new definition for variations in neutral and float temperatures in CWR. Mosayebi et al. ¹¹ studied the effects of continuous or discrete supports, V-shaped rail irregularity and geometrical stiffness on the vehicle-track dynamic interaction. Jiang ¹² analysed the transmitting ways of the rail-bridge longitudinal interaction. Chen et al. ¹³ studied the influence of several factors on the additional longitudinal force of CWR on continuous beams with linear stiffness. Ruge and Birk ¹⁴ and Ruge et al. ¹⁵ investigated the longitudinal forces of CWR due to rail-bridge interaction considering nonlinear characteristic. Okelo and Olabimtan ¹⁶ analysed the nonlinear rail-structure interaction of an elevated skewed steel guideway by using the commercial software GT STRUDL. Zhang et al.^17,18 obtained the fastener resistance parameter in the laboratory and studied the rail-bridge interaction of a long-span bridge. Alfred et al.^19,20 used the monitoring-based nonlinear finite element modeling to research the rail-bridge structure interaction. Yun et al.^21,22 measured and investigated the response of the rail-bridge interaction caused by the temperature change. De Backer et al. ²³ used finite element method to research the application limits of CWR on temporary bridge decks. Lou et al. ²⁴ proposed a method to determine the locations of the rail expansion regulator and the fixed bearing of the continuous beam, corresponding to the maximum additional forces of rail reaching minimum values. Yan et al. ²⁵ studied the distribution of the longitudinal force of CWR on suspension railway bridge with length exceeding 1000 m. Wenner et al. ²⁶ analysed the long-term measured data of rail stress and displacement in recent 2 years and compared to those of the model calculation results. Lou et al. ²⁷ investigated the influences of nonlinear stiffness, span number of bridge, constant longitudinal restoring force, bridge bearing arrangement, and span length of bridge on the additional longitudinal stress and displacement of rail for multi-span simply supported bridges. Ramos et al. ²⁸ studied the bridge length limits due to track-structure interaction in continuous girder prestressed concrete bridges. Based on the bilinear resistance model, Dai et al. ²⁹ proposed an analytical algorithm to analyse the track-bridge interaction of long-span steel bridges under thermal action.

Continuous beams are widely adopted in the railway bridge. Compared with the length of simple beam, that of continuous beam is longer, thus the maximum additional force in rail induced by the temperature change of beam is greater. If the location of the fixed bearing of continuous beam is not appropriate, the high additional axial force induced by the rail-bridge longitudinal thermal interaction may lead to rail buckling or breaking. To the knowledge of the authors, few papers have studied the influence of the location of the fixed bearing of continuous beam on the additional force of rail. Therefore, this paper focuses on the issue about the appropriate location of fixed bearing of continuous beam to reduce the additional axial force in CWR. Compared with the existing literature, the new findings and novelties of this paper are below. (1) The effect of the locations of the fixed bearings of continuous beams on the maximum additional force of rail is investigated considering nonlinear stiffness; (2) The methods are presented to determine the locations of fixed bearings corresponding to the maximum additional forces in rail reaching minimum values; and (3) To reduce the additional axial force in continuously welded rails due to bridge thermal effect, the appropriate locations are recommended for the fixed bearing of continuous railway bridges with the different number.

Longitudinal thermal interaction model of CWR track and foundations

The foundations of a CWR track from left to right are considered as embankment, simply supported bridges, continuously supported bridges, simply supported bridges, and embankment. CWR, simply and continuously supported bridges are modelled as beam elements. Each continuous beam has only one fixed bearing and the fixed bearing is assumed to locate at any position below the beam in the numerical analysis. A schematic planar model of CWR track and foundations longitudinal thermal interaction is shown in Figure 1, in which, $L_{\mathrm{s}}$ and $L_{\mathrm{c}}$ denote the length of simple and continuous beams, respectively, $L_0$ denotes the distance between the two fixed bearings of the simple beams neighbour to the first and last continuous beams respectively, the number of continuous beams from left to right is named as 1, 2, …, n, and n is the number of the last continuous beam, $l_i$ denotes the distance between the left end and the fixed bearing of the i-th continuous beam (hereafter referred to as the i-th fixed bearing), $l_{\mathrm{ls}1}$ denotes the distance between the fixed bearing of simple beam neighbour to the first continuous beam and the first fixed bearing, $l_{n\mathrm{rs}}$ denotes the distance between the fixed bearing of simple beam neighbour to the last continuous beam and its fixed bearing, ${\mathrm{o}}_{\mathrm{c}i}$ denotes the midpoint of the i-th continuous beam, $l_{ii+1}$ denotes the distance adjacent two fixed bearings of continuous beams, ${\mathrm{o}}_0$ denotes the midpoint of $L_0$ , and the letters of SB and CB denote the simple and continuous beams, respectively. Three types of bearing arrangements of simple beams are considered. As shown in Figure 1(a) to (c), the type A denotes that the fixed and movable bearings of simple beams are arranged alternately from left to right, the type B denotes that the fixed and movable bearings of simple beams on the left side are arranged alternately, but they are reversed on the right side, and the type C denotes that the movable and fixed bearings of simple beams on the left side are arranged alternately, but they are reversed on the right side.

Figure 1.

Schematic planar model of CWR track and foundations longitudinal thermal interaction: (a) type A of bearing arrangement of simple beams, where $L_0=L_{\mathrm{s}}+n·L_{\mathrm{c}}$ , (b) type B of bearing arrangement of simple beams, where $L_0=2L_{\mathrm{s}}+n·L_{\mathrm{c}}$ and (c) type C of bearing arrangement of simple beams, where $L_0=n·L_{\mathrm{c}}$ .

The track between rails and bridge is modelled as coupling shear elements with nonlinear stiffness as shown in Figure 2. The rail-beam thermal interaction descriptions can be found in Lou et al. ²⁷ For convenience, the key statements are listed here. The nonlinear characteristic of the difference $u_{\mathrm{D}}$ of the coupling element dependents on the a critical value $\tilde{u}$

$$u_{\mathrm{D}}=\{\begin{array}{cc}{u}_{\mathrm{R}}-u_{\mathrm{B}}& \mathrm{for}\left|u_{\mathrm{R}}-u_{\mathrm{B}}\right|<\tilde{u}\\ \tilde{u}·\mathrm{sign}(u_{\mathrm{R}}-u_{\mathrm{B}})& \mathrm{for}\left|u_{\mathrm{R}}-u_{\mathrm{B}}\right|\ge \tilde{u}\end{array}$$

(1)

Figure 2.

Longitudinal track-bridge coupling element and the nonlinear stiffness law: (a) $u_{\mathrm{R}}>u_{\mathrm{B}}$ , (b) $u_{\mathrm{R}}<u_{\mathrm{B}}$ and (c) nonlinear stiffness law.

where, $u_{\mathrm{R}}$ and $u_{\mathrm{B}}$ denote the longitudinal displacements of the rail and the upper surface of the beam, respectively. It is assumed that the simple and continuous beams are not influenced by the rails during the temperature change of beam and that they can move freely. $u_{\mathrm{B}}$ of simple and continuous beams can be written as

$$u_{\mathrm{B}}={\alpha }_{\mathrm{B}}·\mathrm{\Delta }T·x_{\mathrm{B}}$$

(2)

where, ${\alpha }_{\mathrm{B}}$ denotes the coefficient of thermal extension of the beam, $\mathrm{\Delta }T$ denotes the temperature change with reference to the initial temperature of the beam, and $x_{\mathrm{B}}$ represents the distance between the calculated section and the fixed bearing of beam.

If the absolute value of displacement difference $u_{\mathrm{D}}$ is less than this critical value $\tilde{u}$ , there is a linear elastic relationship between the difference $u_{\mathrm{D}}$ and the longitudinal restoring force q, as shown in Figure 2(c), they can be expressed as

$$q=-c·u_{\mathrm{D}}\mathrm{for}\left|u_{\mathrm{D}}\right|<\tilde{u}$$

(3)

in which, c denotes spring stiffness per unit length in the linear phase, if $u_{\mathrm{D}}>0$ , a force q with negative value acts on the rail, consequently a force q with positive value acts onto the bridge; if $u_{\mathrm{D}}<0$ , a force q with positive value acts on the rail, consequently a force q with negative value acts onto the bridge.

If the absolute value of difference $u_{\mathrm{D}}$ is not less than this critical value $\tilde{u}$ , the rail slips relative to the ballast or concrete strip. The corresponding nonlinear stiffness law shown in Figure 2(c) is applied, and the longitudinal restoring force in the coupling element is a constant $\tilde{q}$ with

$$q=\tilde{q}=-c·\tilde{u}·\mathrm{sign}(u_{\mathrm{R}}-u_{\mathrm{B}})\mathrm{for}|u_{\mathrm{R}}-u_{\mathrm{B}}|\ge \tilde{u}$$

(4)

The track between rails and embankment is modelled as longitudinal spring with nonlinear stiffness. Eqs. (1), (3) and (4) can be used in the rail on embankment only by the value of $u_{\mathrm{B}}$ with zero.

The mechanics equation of CWR track and foundations longitudinal thermal interaction considering nonlinear stiffness can be found in Lou et al., ²⁷ and are omitted here to reduce the length of this paper. The corresponding program for calculating the additional thermal force in rails of the presented model is compiled.

Verification

A CWR track on single continuous beam and two approach embankments are considered. The parameters are listed in Table 1 from Chinese code, ³⁰ except the parameters of $L_{\mathrm{c}}$ and $L_{\mathrm{E}}$ . The spring stiffness per unit length with 4.4 × 10⁶ N/m² of rail foundation is adopted here. The fixed bearing is assumed to located at 30 m away from the left end of the beam. The four parameters of ${\tilde{u}}_{\mathrm{B}}$ , ${\tilde{q}}_{\mathrm{B}}$ , ${\tilde{u}}_{\mathrm{E}}$ and ${\tilde{q}}_{\mathrm{E}}$ listed in Table 1 are not adopted in the example, but used in the Section 5 to reduce the number of Table. Figure 3 plots the distributions of the additional displacement and force in the rail along its length induced by the temperature change of beam, which are obtained by the finite element method and the analysis method, ² respectively. The abscissa indicates the distance between the certain section and the left end of rail. The positive and negative values of the ordinates represent tension and compression, respectively, and the following signs have the same meaning. Figure 3 shows that the two solutions are in good agreement. Therefore, the presented model and the self-compiled program can be regarded as correct.

Table 1.

The parameters of CWR track on continuous beam and embankment.

Notation	Item	Value
E	Modulus of elasticity of rail	2.1 × 1011 N/m2
A	Area of the cross-section of rail	7.745 × 10−3 m2
${\alpha }_{\mathrm{R}}$	Coefficient of thermal extension of rail	1.18 × 10−5/°C
$L_{\mathrm{c}}$	Length of continuous beam	120 m
$\mathrm{\Delta }T$	Temperature change of beam	15°C
${\alpha }_{\mathrm{B}}$	Coefficient of thermal extension of beam	1.0 × 10−5/°C
c	spring stiffness per unit length of rail foundation	4.4 × 106 N/m2
${\tilde{u}}_{\mathrm{B}}$	Critical value of relative displacement on beam	0.5 mm
${\tilde{q}}_{\mathrm{B}}$	Constant longitudinal restoring force corresponds to the critical value ${\tilde{u}}_{\mathrm{B}}$ on beam	8000 N/m
${\tilde{u}}_{\mathrm{E}}$	Critical value of relative displacement on embankment	2.0 mm
${\tilde{q}}_{\mathrm{E}}$	Constant longitudinal restoring force corresponds to the critical value ${\tilde{u}}_{\mathrm{E}}$ on embankment	8800 N/m
$L_{\mathrm{E}}$	Length of CWR track on each side embankment	100 m

Figure 3.

Distribution of the additional force and displacement in the rail: (a) additional force of rail and (b) additional displacement of rail.

Procedures to determine the appropriate locations of the fixed bearings of continuous beams

The maximum additional tension and compression forces of rail vary with the location of the fixed bearing of continuous beam. When the maximum additional tension and compression forces of rail on two continuous beams reaches their minimum values, the steps to determine the locations of fixed bearings of two continuous beams are as follows.

1. Firstly, the second fixed bearing is located at the left end of beam.

2. The first fixed bearing is also located at the left end of beam. Then, the additional force of rail along its length is calculated and their maximum values can be obtained. It should be noted that the maximum and minimum values of compression force refer to the absolute values.

3. The location of the second fixed bearing remains unchanged. The location of the first fixed bearing is changed from left to right until arriving the right end of the beam, and the changing interval may be adopted the length of beam element.

4. The maximum additional tension and compression forces of rail corresponding to each location of the first fixed bearing are obtained and their relationship curves are plotted.

5. The minimum values in the above curves are just those of the maximum additional tension and compression forces of rail while the second fixed bearing is located at the left end of the second continuous beam.

6. Then, the location of the second fixed bearing is changed from the left to right until arriving the right end of the beam, and the changing interval may be also adopted the one length of beam element. The minimum values of the maximum additional tension and compression forces of rail corresponding to each location of the second fixed bearing are gained by repeating the steps (2)-(4).

7. The curves of the minimum values mentioned in the step (6) with the location of the second fixed bearing are drawn. The abscissas corresponding to the minimum values in the curves are the locations of the second fixed bearing to be determined.

8. Finally, the location of the second fixed bearing determined above remains unchanged. Using the steps (3) and (4), the curves of the maximum additional tension and compression forces of rail with the location of the first fixed bearing can be obtained. The abscissas corresponding to the minimum value in the curves is the locations of the first fixed bearing to be determined.

The procedures to determine the appropriate locations of the fixed bearings of three continuous beams are similar to above those of two continuous beams, only with the following modifications. The second fixed bearing in the above procedures is replaced by the third fixed bearing, then the new step, i.e., the second fixed bearing is located at the point ${\mathrm{O}}_0$ with unchanged, is added as the step (1).

Results

The influence of the locations of the fixed bearings of continuous beams on the maximum additional force of rail, and their locations corresponding to the maximum additional force of rail reaching the minimum value are researched in this section. The foundations of a CWR track are the same as the described in Figure 1. The total number of simple beams is 10 and they are evenly distributed on two sides of continuous beams. The length with 32 m of simple beam is adopted. The 3 types of bearing arrangements of simple beams are considered as shown in Figure 1(a) to (c). Except for the parameter c, the other parameters in Table 1 are adopted in this section. In order to make the obtained representative laws, the continuous beams with different numbers and lengths are considered. The continuous beams with total number of 1, 2, 3, 4 and 5 are used, and the length of $L_{\mathrm{c}}$ with 120, 135, 150, 165 and 180 m are adopted. The following results are based on the temperature rise with 15°C of beam. If the changed temperature is −15°C, then the tension and compression in the results will be reversed.

Results of one continuous beam

Table 2 lists the minimum values of the maximum additional forces of rail and the corresponding location of the fixed bearing of one continuous beam, as well as the maximum additional force of rail corresponding to the fixed bearing located at the midpoint ${\mathrm{o}}_{\mathrm{c}1}$ . Table 2 shows that (1) For the same length of continuous beam, the bearing arrangements of simple beams have the influence on the values of $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ , and according to the values of $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ in descending order, they are respectively B, A, C, and C, A, B. Accordingly, the type A of bearing arrangement is recommended. (2) While the maximum additional compression and tension forces of rail reach their minimum values, there are $\frac{l_{\mathrm{ls}1,\mathrm{c}}}{L_0}=\frac{l_{1\mathrm{rs},\mathrm{c}}}{L_0}=0.5$ , and $\frac{l_{\mathrm{ls}1,\mathrm{t}}}{L_0}=\frac{l_{1\mathrm{rs},\mathrm{t}}}{L_0}=0.5$ , and this means the fixed bearing is located at the point of ${\mathrm{O}}_0$ . (3) For the type of A, the values of $\frac{F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}-min{F}_{\mathrm{maxc}}}{min{F}_{\mathrm{maxc}}}\times 100\%$ are 14.79% to 24.80% for the beam with different lengths, which show that the difference between $F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}$ and $min{F}_{\mathrm{maxc}}$ is obvious, and this indicates the fixed bearing located at the midpoint ${\mathrm{o}}_{\mathrm{c}1}$ is not appropriate.

Table 2.

The minimum values of the maximum additional forces of rail and the corresponding location of the fixed bearing of one continuous beam, as well as the maximum additional force of rail corresponding to the fixed bearing located at the midpoint ${\mathrm{o}}_{\mathrm{c}1}$ .

Bearing arrangements and length of each continuous beam $L_{\mathrm{c}}$ (m)	Bearing arrangements	120	135	150	165	180
$min{F}_{\mathrm{maxc}}$ (×105 N)	A	−1.767	−1.974	−2.174	−2.368	−2.555
	B	−2.213	−2.400	−2.581	−2.760	−2.934
	C	−1.277	−1.511	−1.733	−1.945	−2.148
$l_{\mathrm{ls}1,\mathrm{c}}$ / $l_{1\mathrm{rs},\mathrm{c}}$ (m)	A	76/76	83.5/83.5	91/91	98.5/98.5	106/106
	B	92/92	99.5/99.5	107/107	114.5/114.5	122/122
	C	60/60	67.5/67.5	75/75	82.5/82.5	90/90
$\frac{l_{\mathrm{ls}1,\mathrm{c}}}{L_0}$ / $\frac{l_{1\mathrm{rs},\mathrm{c}}}{L_0}$	A	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5
	B	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5
	C	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5
$F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}$ (×105 N)	A	−2.204	−2.395	−2579	−2.759	−2.933
	B	−2.213	−2.400	−2581	−2.760	−2.934
	C	−1.277	−1511	−1.733	−1.945	−2.148
$\frac{F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}-min{F}_{\mathrm{maxc}}}{min{F}_{\mathrm{maxc}}}\times 100\%$	A	24.73%	21.33%	18.63%	16.51%	14.79%
	B	0	0	0	0	0
	C	0	0	0	0	0
$min{F}_{\mathrm{maxt}}$ (×105 N)	A	2.167	2.269	2.335	2.376	2.402
	B	1.987	2.157	2.268	2.337	2.379
	C	2.288	2.343	2.379	2.402	2.417
$l_{\mathrm{ls}1,\mathrm{t}}$ / $l_{1\mathrm{rs},\mathrm{t}}$ (m)	A	76/76	83.5/83.5	91/91	98.5/98.5	106/106
	B	92/92	99.5/99.5	107/107	114.5/114.5	122/122
	C	60/60	67.5/67.5	75/75	82.5/82.5	90/90
$\frac{l_{\mathrm{ls}1,\mathrm{t}}}{L_0}$ / $\frac{l_{1\mathrm{rs},\mathrm{t}}}{L_0}$	A	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5
	B	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5
	C	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5	0.5/0.5
$F_{\mathrm{maxt},f{b}_{\mathrm{mid}}}$ (×105 N)	A	2.174	2.273	2.337	2.377	2.403
	B	1.987	2.157	2.268	2.337	2.379
	C	2.288	2.343	2.379	2.402	2.417
$\frac{F_{\mathrm{maxt},f{b}_{\mathrm{mid}}}-min{F}_{\mathrm{maxt}}}{min{F}_{\mathrm{maxt}}}\times 100\%$	A	0.32%	0.18%	0.09%	0.04%	0.04%
	B	0	0	0	0	0
	C	0	0	0	0	0

where, $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ denote the minimum value of maximum additional compression and tension forces of rail, respectively. Their definitions are as follows. The maximum additional compression and tension forces of rail vary with the location of the fixed bearing, the minimum value among all maximum additional compression forces is called $min{F}_{\mathrm{maxc}}$ , and the minimum value among all maximum additional tension forces is called $min{F}_{\mathrm{maxt}}$ . $l_{\mathrm{ls}1,\mathrm{c}}$ and $l_{\mathrm{ls}1,\mathrm{t}}$ denote the distance between the location of fixed bearing of simple beam neighbour to the first continuous beam and the locations of the fixed bearing corresponding to $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ , respectively. $l_{n\mathrm{rs},\mathrm{c}}$ and $l_{n\mathrm{rs},\mathrm{t}}$ denote the distance between the locations of fixed bearing of simple beam neighbour to the last continuous beam and the locations of the fixed bearing corresponding to $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ , respectively. $l_{ii+1,\mathrm{c}}$ and $l_{ii+1,\mathrm{t}}$ denote the distance between the locations of the two fixed bearings of the i-th and i+1-th continuous beams corresponding to $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ , respectively, i = 1, 2, …, n-1. $F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}$ and $F_{\mathrm{maxt},f{b}_{\mathrm{mid}}}$ denote the maximum additional compression and tension forces of rail corresponding to the fixed bearing located at the midpoint continuous beam, respectively.

Figure 4 draws the curves of the maximum additional tension and compression forces of rail with the location of the fixed bearing of one continuous beam with $L_{\mathrm{c}}$  = 120 m under the three types of A, B, and C, in which, the abscissa represents the location of the fixed bearing away from the left end of continuous beam. The laws can be found from Figure 4 as follows.

1. The maximum additional tension and compression forces of rail vary with the location of fixed bearing of continuous beam, and they are distributed symmetrically with the respect to the point of ${\mathrm{O}}_0$ , as mentioned in Section 2, the point ${\mathrm{O}}_0$ is the midpoint of $L_0$ . It should be pointed out that for the types B and C of bearing arrangement, the point ${\mathrm{O}}_0$ is same as the midpoint ${\mathrm{O}}_{\mathrm{c}1}$ , while for the type A, the symmetrical point ${\mathrm{O}}_0$ is not same as the midpoint ${\mathrm{O}}_{\mathrm{c}1}$ .

2. The location of the fixed bearing of continuous beam has little influence on the maximum additional tension force of rail, but has a very significant influence on the maximum additional compression force of rail. As shown in Figure 4(b), the maximum and minimum values of the maximum additional tension force of rail are 2.115 × 10⁵ N and 1.987 × 10⁵ N, their difference is only 6.44%, however, those of the maximum additional compression force of rail are −3.591 × 10⁵ N and −2.213 × 10⁵ N, their difference is 62.27%. Figure 4(c) shows that those of the maximum additional tension force of rail are 2.394 × 10⁵ N and 2.388 × 10⁵ N, their difference is only 4.63%, however, those of the maximum additional compression force of rail are −2.888 × 10⁵ N and −1.277 × 10⁵ N, their difference is 126.16%.

Figure 4.

The curves of the maximum additional tension and compression forces of rail with the location of fixed bearing of one continuous beam with $L_{\mathrm{c}}$  = 120 m: (a) type A of bearing arrangement, (b) type B of bearing arrangement and (c) type C of bearing arrangement.

Results of multiple continuous beams

Symmetrical law

Two continuous beams with $L_{\mathrm{c}}$  = 150 m and the type B of bearing arrangement of simple beams are considered as an example. The symmetrical law will be found for the locations of fixed bearings corresponding to the maximum additional tension and compression forces of rail reaching their minimum values by the procedures in the Section 4. Figure 5 plots the curves of the minimum values of the maximum additional forces of rail with the location of the second fixed bearing, in which, the abscissa represents the location of the second fixed bearing away from the left end of beam. Figure 5 (a) draws the compression and tension forces, and Figure 5(b) supplements the tension force due to the unclear change of the ordinate of the tension force in Figure 5(a). Figure 5 shows that the locations of the second fixed bearing to be determined are 69.5 and 111 m, respectively. This means that the distances between the locations of the second fixed bearing and the location of the point of ${\mathrm{O}}_0$ are 69.5 and 111 m, respectively.

Figure 5.

The curves of the minimum values of the maximum additional forces of rail with the location of the second fixed bearing for the case of two continuous beams with $L_{\mathrm{c}}$  = 150 m and the type B: (a) compression and tension forces and (b) tension force.

According to the step (8) in the Section 4, their locations remain unchanged, then the curves of the maximum additional compression and tension forces of rail with the location of the first fixed bearing can be obtained and plotted in Figure 6(a) and (b), respectively. Figure 6(a) and (b) show that the locations of the first fixed bearing to be determined are 80.5 and 39 m, respectively. This means that the distances obtained between the first fixed bearing and the point of ${\mathrm{O}}_0$ are also 69.5 and 111 m, respectively. In addition, it can be found that the minimum values of compression and tension forces in Figure 5 are equal to those in Figure 6, respectively. The figures for continuous beams with the other lengths, the other type of bearing arrangement of simple beams and the number of three are omitted to reduce the length of this paper. It can be concluded that the locations of the first and the second fixed bearings are both symmetrical with respect to the point ${\mathrm{O}}_0$ while the maximum additional compression and tension forces reach their minimum values. Furthermore, for the continuous beams with the number of three, the locations of the first and the third fixed bearings are symmetrical with respect to the point ${\mathrm{O}}_0$ while the maximum additional compression and tension forces reach their minimum values.

Figure 6.

The curves of the maximum additional forces of rail with the location of the first fixed bearing for the case of two continuous beams with $L_{\mathrm{c}}$  = 150 m and the type B, in which, the values of $l_2$  = 69.5 and 111 m remains unchanged for (a) and (b), respectively: (a) compression force and (b) tension force.

Modified procedures for determining the locations of fixed bearings

Based on the above symmetrical law of the fixed bearings, the procedures mentioned in the Section 4 for determining the locations of fixed bearings of two continuous beams can be modified as follows. The modified procedures can significantly improve the calculation efficiency due to the reduction of the number of cycles in the calculation process.

1. The first and second fixed bearings are, respectively, located at the left and right ends of the beam for the types B and C. But for the type A, the second fixed bearing is located at the distance of $L_{\mathrm{s}}$ away from the right end of the beam. Then the maximum values of the tension and compression forces of rail are obtained.

2. Both the locations of the first and second fixed bearings are changed towards the point ${\mathrm{O}}_0$ at the same time until the second fixed bearing reaches the left end of the beam. The maximum additional tension and compression forces of rail for each location of the first fixed bearing are gained and the relationship curves are plotted. The abscissas corresponding to the minimum value in the curves are the locations of the first fixed bearing to be determined. The location of the second fixed bearing can be obtained by using the symmetrical law.

Similarly, if the last paragraph of Section 4 is modified to the above procedure, it can be used to find the appropriate locations of the fixed bearings of three continuous beams. Again, the two continuous beams with $L_{\mathrm{c}}$  = 150 m and the type B of bearing arrangement of simple beams are taken as an example by using the modified procedures. The relationship curves have been plotted in Figure 7. Figure 7 shows that the minimum values of the maximum additional compression and tension forces are −2.715 × 10⁵ N and 2.199 × 10⁵ N, and the corresponding abscissas are 80.5 and 39 m, respectively, and they are equal to those in Figure 6. This illustrates that the minimum values of the maximum additional compression and tension forces and corresponding locations of fixed bearings by the modified procedures are the same as those by the procedures mentioned in the Section 4.

Figure 7.

The curves of the maximum additional forces of rail with the location of the first fixed bearing for the case of two continuous beams with $L_{\mathrm{c}}$  = 150 m and the type B: (a) compression and tension forces and (b) tension force.

In addition, Figure 7 shows that the tension forces are between 2.199 × 10⁵ N and 2.433 × 10⁵ N, with the difference of 10.64%, however, the compression forces are between −2.715 × 10⁵ N and −5.175 × 10⁵ N, with the difference of 90.61%. This further explains that the location of the fixed bearing of continuous beam has little influence on the maximum additional tension force of rail, but has a very significant influence on the maximum additional compression force of rail.

Minimum values of the maximum additional forces of rail and corresponding locations of the fixed bearings of multiple continuous beams

On the basis of the modified procedures, and the laws mentioned above, the values of $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ , and corresponding the locations of fixed bearings for the continuous beams with the number of 2–5, as well as the values of $F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}$ and $F_{\mathrm{maxt},f{b}_{\mathrm{mid}}}$ are obtained and listed in Tables A1 to A4, respectively. The following laws can be concluded from Tables A1 to A4.

1. For the same number and length of continuous beams, the bearing arrangements of simple beams have the influence on the values of $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ , and according to the values of $min{F}_{\mathrm{maxc}}$ and $min{F}_{\mathrm{maxt}}$ in descending order, they are respectively B, A, C, and C, A, B. Thus, the type A of bearing arrangement of simple beams is recommended.

2. Each fixed bearing located at the midpoint ${\mathrm{O}}_{\mathrm{c}i}$ cannot ensure the maximum additional tension and compression forces of rail reaching their minimum values. The value of $\frac{F_{\mathrm{maxc},f{b}_{\mathrm{mid}}}-min{F}_{\mathrm{maxc}}}{min{F}_{\mathrm{maxc}}}$ is generally greater than that of $\frac{F_{\mathrm{maxt},f{b}_{\mathrm{mid}}}-min{F}_{\mathrm{maxt}}}{min{F}_{\mathrm{maxt}}}$ ,which means the influence of the location of the fixed bearing on the maximum additional compression force of rail is greater than that on the maximum additional tension force of rail.

3. With the same number of continuous beams, for the same length of continuous beam and the same type of bearing arrangement of simple beams, there are $l_{\mathrm{ls}1,\mathrm{c}}=l_{n\mathrm{rs},\mathrm{c}}$ , $l_{ii+1,\mathrm{c}}=l_{\mathrm{jj}+1,\mathrm{c}}$ , $l_{\mathrm{ls}1,\mathrm{t}}=l_{n\mathrm{rs},\mathrm{t}}$ , $l_{ii+1,\mathrm{t}}=l_{\mathrm{jj}+1,\mathrm{t}}$ in which, i, j=1, 2, …, n-1, and n is the number of continuous beams. For example, Table A3 shows that there are $l_{\mathrm{ls}1,\mathrm{c}}=l_{4\mathrm{rs},\mathrm{c}}=$ 131.5 m, $l_{12,\mathrm{c}}=l_{23,\mathrm{c}}=l_{34,\mathrm{c}}=$ 163 m for $L_{\mathrm{c}}$ =180 m and the type A. This means that the locations of fixed bearing are symmetrical with respect to the midpoint of ${\mathrm{o}}_0$ .

4. As shown in Table A1, while the maximum additional compression and tension forces of rail reach their minimum values, the locations of fixed bearings of two continuous beams are not same. Most of the values of $\frac{F_{\mathrm{maxc},f{b}_{\mathrm{mmt}}}-min{F}_{\mathrm{maxc}}}{min{F}_{\mathrm{maxc}}}\times 100\%$ are greater than 50%, however, most of the values of $\frac{F_{\mathrm{maxt},f{b}_{\mathrm{mmc}}}-min{F}_{\mathrm{maxt}}}{min{F}_{\mathrm{maxt}}}\times 100\%$ are smaller than 2%. Therefore, it is not appropriate to arrange the fixed bearings of continuous beams at the position where the maximum additional tension force reaches the minimum value, and it should be arranged at the position where the maximum additional compression force reaches the minimum value.

5. With the same number of continuous beams, for the different lengths of continuous beams and the different types of bearing arrangement of simple beams, the distances of adjacent two fixed bearings (including the fixed bearings of simple beams neighbour to the first and last continuous beams) are not equal to each other, however, the ratios of them to $L_0$ are approximately equal to each other. For example, Table A3 shows that the value of $l_{\mathrm{ls}1,\mathrm{c}}$ is 90.25 m for $L_{\mathrm{c}}$ =120 m and type A, and that value of $l_{\mathrm{ls}1,\mathrm{c}}$ is 116 m for $L_{\mathrm{c}}$ =150 m and type B. The values of $l_{\mathrm{ls}1,\mathrm{c}}$ are not equal obviously. However, the corresponding ratios of $\frac{l_{\mathrm{ls}1,\mathrm{c}}}{L_0}$ are 0.1763 and 0.1747, respectively, with only 0.92% difference, which can be considered as approximately equal. Again, Table A4 shows that the value of $l_{12,\mathrm{c}}$ is 112.25 m for $L_{\mathrm{c}}$ =120 m and type A, and that of $l_{12,\mathrm{c}}$ is 133.5 m for $L_{\mathrm{c}}$ =150 m and type C. Similarly, the values of $l_{12,\mathrm{c}}$ are not equal obviously. But, the corresponding ratios of $\frac{l_{12,\mathrm{c}}}{L_0}$ are 0.1776 and 0.1780, respectively, with only 0.23% difference, which can be considered as approximately equal. The law that the values of ratio are approximately equal to each other, and it can be helpful for the arrangement design of the fixed bearings of continuous beams with other length to reduce the maximum additional force of rail.

6. For the same length of continuous beam and the same type of bearing arrangement of simple beams, the distances of neighbour two fixed bearings increase with the number of continuous beams. For example, the values of $l_{\mathrm{ls}1,\mathrm{c}}$ are 84.25, 88, 90.25 and 91.5 m, and those of $l_{12,\mathrm{c}}$ are 103.5, 108, 110.5 and 112.25 m, respectively, for two, three, four, and five continuous beams with $L_{\mathrm{c}}$ =120 m and type A. This can be helpful for the arrangement design of the fixed bearings of continuous beams with the number more than five.

Conclusion

The longitudinal thermal interaction model of CWR track, simple and continuous beams considering nonlinear stiffness is established. Then, two methods are proposed to determine the locations of fixed bearings of continuous beams corresponding to the maximum additional tension and compression forces of rail reaching their minimum values. Furthermore, multiple continuous beams with several lengths and simple beams with three types of bearing arrangements are considered to find the effect laws of the locations of the fixed bearings of continuous beams on the maximum additional forces in rail. The obtained results can guide designing the location of the fixed bearing of continuous railway bridge to reduce the additional axial force in CWR induced by the bridge thermal effect. The conclusions are as follows.

1. With the temperature rise of beam, the location of the fixed bearing of continuous beam has little influence on the maximum additional tension force of rail, but has a very significant influence on the maximum additional compression force of rail. The law will be reversed with the temperature falling of beam.

2. The appropriate locations are recommended for the fixed bearing of continuous railway bridges with the different number considering rail-bridge thermal interaction. The results can guide designing the location of the fixed bearing of continuous railway bridge while reducing the additional axial force in continuously welded rails.

3. For two continuous beams, it is not appropriate to arrange the fixed bearings of continuous beams according to the locations while the maximum additional tension force reaches the minimum value, and it should be arranged according to the locations while the maximum additional compression force reaches the minimum value.

4. While the maximum additional tension and compression forces of rail reach their respective minimum values, the locations of the fixed bearings of continuous beams are both symmetrical with respect to the point ${\mathrm{O}}_0$ . Each fixed bearing located at the midpoint of continuous beam cannot ensure the maximum additional tension and compression forces of rail reaching their minimum values.

5. As long as the same number of continuous beams, for the different lengths of continuous beams and the different types of bearing arrangement of simple beams, the distances of adjacent two fixed bearings (including the fixed bearings of simple beams neighbour to the first and last continuous beams) are not equal to each other, however, the ratios of them to $L_0$ are approximately equal to each other. This can be helpful for the arrangement design of the fixed bearings of continuous beams with other length to reduce the maximum additional force of rail.