In the continuous beam model on elastic supports, the stiffness of the rail supporting represents the elastic characteristic of the supporting (Fig. 1).

Supporting stiffness is defined as the ratio between force and supporting deflection.

(1)

Where:

F is the force applied on the top surface of the rail supporting (kN)

y is the deflection of rail supporting under the effect of force (mm).

In this study, the supporting stiffness is considered the in-plane stiffness of the track. Modeling in-plane stiffness of the track is the stiffness total of the rail pad, sleeper, and substructure (Fig. 2).

Where:

k_total is a coefficient of quasi elasticity (in-plane stiffness of the track) of the rail supporting (kN/mm);

k_i is the stiffness of each layer (kN/mm);

k_{rail pad} is the stiffness of resilient rail pads in the track structure (kN/mm);

k_sleeper is the stiffness of sleeper (kN/mm);

k_substructure is the stiffness of substructure (kN/mm);

For prestressed concrete sleepers, k_sleeper can be approximated = ∞, stiffness of resilient rail pad is determined in the laboratory. Stiffness of substructure according to the following formula:

(2)

Where:

C₀ is the substructure modulus (kN/mm³), and C₀ is determined from the field test.

l is sleeper length (mm)

b is sleeper width (mm)

Coefficient of quasi elasticity:

(3)

The track structure can be seen as a system consisting of rails that are elastically supported by means of rail pads by sleepers spaced at a fixed distance. The sleeper is supported by an elastic substructure consisting of ballast plus roadbed.

The experiments were implemented on the substructure in turnout with a full-scale model (Fig. 3).

Fig. (1). Beam model on elastic supporting.

Fig. (2). Model of the total in-plane stiffness of the track.

Fig. (3). Components of track.

There are some methods that may be used to measure the in-plane stiffness of the track. The measurements are made at discrete intervals while standstill and the measurements are continuous while moving. Standstill measurements have been more widely used, often for research purposes.

In this study, the measurements with discrete intervals while standstill were done. The authors research and produce an equipment system for measuring the modulus of the substructure under sleepers (Figs. 4 and 5). The loading is set through the jack and rack system. This force is transmitted to the substructure through the press plate. The dimensions of the pressing plate are 230 mm x 230 mm. Two strain gauges are installed to measure displacement relative to the applied load. Carry out continuous loading and record the corresponding displacements through strain gauges.

Fig. (4). Equipment system to measure modulus of substructure.

Fig. (5). Test at the field.

Fig. (6). Displacement of substructure.

The survey of the elastic modulus of the substructure was carried out at 10 different locations at the turnout (From location 1(LC1) to location 10(LC10)). Preload is applied to reduce the effect of gaps and slips. When the load-displacement relationship graph is stable, data is collected. The load versus displacement curves under test load for the substructure are given in Fig. (6).

The displacement graph at positions is different due to moisture of the substructure, and the impact load of the previous trains. In some cases, low substructure stiffness values can also result from fouled or dirty ballast that prevents adequate support for the train load. This problem occurs especially when the ballast and roadbed deteriorate in the presence of water due to train-induced repeated loading. The resulting migration of fines into the ballast and subsequent formation of wet spots can lead to a reduction in the stiffness of the track support system.

Based on the relationship between load and displacement, we can determine the elastic modulus of the substructure as shown in Table 1.

The average substructure modulus is 1.05 N/mm³. We calculate for a sleeper with dimensions of length 1800 mm and width 230 mm. Based on Formula 2, the stiffness of the substructure such as in Table 2. Average of the substructure stiffness is 216.73 kN/mm. Substructure stiffness is very useful for investigations related to the bearing capacity of the track.

High stiffness leads to faster deterioration of the track and its components due to higher dynamic loads. In such cases, a method that may be used to reduce the stiffness is the installation of rail pads under the rail foot. The stiffness of the rail pad is also an important parameter in determining the overall stiffness of the track.

In order to determine the stiffness of the rail pads, in this study they were determined through laboratory tests (Fig. 7). Rail pads were made of rubber with a thickness of 10mm.

Fig. (7). Test of the rail pad.

The test was done with 5 specimens of the rail pad. The results of the rail pad’s stiffness are shown in Table 3 and the rail pad’s corresponding displacement curves are shown in Figs. (8-12). The average rail pad’s stiffness is 33.50 kN/mm.

Fig. (8). The load-displacement curve of the rail pad’s specimen 1.

Fig. (9). The load-displacement curve of the rail pad’s specimen 2.

Fig. (10). The load-displacement curve of the rail pad’s specimen 3.

Fig. (11). The load-displacement curve of the rail pad’s specimen 4.

Fig. (12). The load-displacement curve of the rail pad’s specimen 5.

Total track stiffness is important for track maintenance and design. Track stiffness determination provides useful specifications for the evaluation of track system safety and performance. The overall stiffness of the track is determined by the stiffness of the track's components. In this study, the track stiffness is determined through the stiffness of the rail pad and the stiffness of the substructure.

According to Formula 3, the in-plane stiffness of the track is shown in Table 4. The average in-plane stiffness of the track is 28.73 kN/mm. The results are consistent with references [48] and recommended that rail supporting static stiffness (for the track using concrete sleeper and rubber rail pad) should be taken as 22~42 kN/mm.

Fig. (13). Discretization of rail.

Fig. (14). An element of rail.

When we consider the rail as a beam on elastic supports. Using the finite element method to discrete the rails into non-overlapping elements and satisfy the strain continuity and force balance conditions (Fig. 13).

Considering a rail element as shown in Fig. (14).

We have the element's node displacement vector:

(4)

Since in direction v_i, v_i+1 has an elastic supporting of stiffness k, so we have an element stiffness matrix:

(5)

Moving the load to the node. Node force of element:

(6)

Set up the overall hardness matrix K*. Solving the system of equations:

(7)

Displacement at nodes:

(8)

Substituting into the equation:

(9)

We have:

(10)

Determining the forces on rail supports R_i (Fig. 15):

Fig. (15). Forces on rail supporting.

(11)

Using the P60 rail with the characteristics shown in Table 5. A section of rail with a length of 50m is divided into 80 elements. The length of each element is 0.625m. The distance between rail supportings is 0.625m. The locomotive has 4 axles with an axle load of 14 tons/axle. Calculating the force transmitted on the rail supporting with the values of in-plane stiffness of track in Table 4. Forces acting on the rail supports such as in Fig. (16). The value of the maximum force acting on the rail supporting is determined at the position under the wheel, and is shown in Table 6.

Fig. (16). Chart of forces acting on the rail supports.

These results show that the stiffness of the rail support is proportional to the value of the force transmitted to the rail support. High stiffness increases the track's bearing capacity under the effect of train dynamic load. However, this will also cause large dynamic forces transmitted to the substructure. Therefore, research to choose the appropriate stiffness to create comfort for passengers and increase the life of the structure is necessary.

The maximum bearing force on a (single) discrete rail support due to the wheel load is:

F_max = DAF.F_mean = 44.22 kN

Contact pressure between rail and sleeper:

Where:

F₀ is the total pre-tensioning force of fastening on rail support (kN), F₀ = 20 kN

A_rs is the area of rail pad (cm²), A_rs = 550 cm².