The modelling of railway subgrade strengthening foundation on weak soils

3.1 Applied model of the train–track interaction

The physical model is shown in Figure 3 and is composed of equation (1). The model consists of the Bernoulli–Euler beam resting on a linear foundation with viscous damping. The symmetry of the track in the cross section and continuous support are assumed. The foundation is a single layer with stiffness for each support and damping parameters representing the rail pad, the ballast, and the top layer of the subgrade. The wheel suspension consists of a linear spring and a viscous damping element. The contact force between the wheel and the rail is modelled as a so-called Hertzian spring.

Figure 3

Vehicle-track model.

The equations are solved using the finite difference method (FDM). In previous research [30], the FDM solutions were compared with other analytical and numerical models (FEM) and good agreement was found. The novelty in these equations is the variability along the track with regard to support stiffness, damping coefficient, mass of the track beam, and the bending stiffness of the beam.

where x  – coordinate along the track, x 1 and x 2  – positions of the wheels, w w  – vertical displacement of the bogie (no rotations), w k 1 and w k 2  – vertical displacements of the wheels, w  – rail deflections w 1 = w ( x 1 ) and w 2 = w ( x 2 ) , m n  – mass of ½ of locomotive body at the bogie torsion pin, m  – mass of one boogie, m k  – mass of one wheel set, c  – damping coefficient of suspension, k  – primary suspension stiffness, k H  – Herzian spring stiffness, m s ( x )  – unit mas (mass of one rail plus mass of ½ sleeper per 1 m of the track), E  – Young’s modulus of rail steel, I ( x )  – moment of inertia of the rail (which may vary along the track), U ( x )  – stiffness parameter of the foundation, c t ( x )  – viscous damping parameter of the rail support, N  – longitudinal force in the rail, P i = k H ( w k i − w i )  – wheel–rail interaction force i = 1 , 2 , d  – axle spacing in a bogie, δ  – Dirac delta, which in numerical calculations is replaced with a continuous load applied on a very short section of the rail.

3.2 Validation of the train–track model

An example of validation of the model using measurements of rail deflections on a line in service in Krakow is shown in Figure 4. Rail deflections were measured using a geodesic method under a ballast-carrying wagon TMS. The parameters of the wagon are presented in Table 1 and the results of track stiffness measurements are shown in Table 2. The results of validation are shown in Figure 5.

Figure 4

Geodesic measurements of rail deflections under a ballast-carrying wagon.

Figure 5

Calculated rail deflection lines in the transition zone compared with the measurements of rail deflections (denoted by dots): (a) north side of the object and (b) south side of the object.

As a result of the measurements, the maximum deflections under axles were obtained, which were then corrected by “separation” of the second axle using an iterative procedure. This was performed using a model of a beam on an elastic foundation. In the procedure, two axles were assumed and the load magnitude corresponded to the axles given in Table 1.

Using this classical method, the static stiffness of the track was estimated with an accuracy of 0.1 mm. The error grows when the non-uniformity of the track stiffness is significant. In the case of dynamic deflection measurements as obtained in ref. [28] using the magnetic induction method, the values were similar.

The results of the comparison between the measured deflections (denoted by dots) and the calculated deflection lines are shown in Figure 5a and b. Due to the fact that the speed limit on the line was 80 km/h, the authors used this speed in the numerical calculations. The damping coefficient of the track c t ( x ) is assumed as zero due to the small influence of the speed. All other parameters are shown in Table 2 and the track structure 60E1/SB/PS-94 is shown in the Figure 5.

3.3 Data for the train–track modelling

For the purpose of the present calculations, locomotive TRAXX was used (Figure 4), the mechanical parameters of which are given in Table 3. A good parameter study of various trains is given in the literature [31]. In this context, it is sufficient that this locomotive has similar characteristics to that which was run on a section of a French line in which the stresses and the deflections were measured [23]. Therefore, the obtained stress fields, when compared to those measured, may be treated as a realistic estimation of the load on the subgrade (Figure 6).

The track parameters are as follows:

• 60 E1 rail;

• static stiffness of rail pad k b = 32 kN/mm;

• twin-block sleeper U-41 (Figure 7) – length = 2.4 m, support area = 0.437 m², sleeper mass = 225 kg, and sleeper spacing = 60 cm, thus its support stiffness k _s = 68 kN/mm (half of the sleeper);

• ballast layer = 45 cm;

• the subgrade modulus is 75 MPa, next using the assumed stress distribution angle of 22° [33,34] and the sleeper dimensions, the stress area for the uniform distribution of stresses is obtained as 0.761 m², which with the stiffness modulus C = 150 MN/m² [35] yields the substitute stiffness for the foundation k sub = 0.761 × 150 MN/m² = 114 kN/mm;

• stiffness of the three layers is determined according to ref. [36], k track = 1 / ( 1 / k b + 1 / k s + 1 / k sub ) = 18.28 kN/mm;

• if this stiffness is distributed uniformly for a sleeper spacing of 0.6 m, the foundation stiffness parameter is U = k track / 0.6 = 30.5 MPa.

Figure 6

Twin-block sleeper U-41.

Figure 7

Calculated deflection lines for the track as the first stage of the model. The data are given in Table 4, and the speed is 200 km/h. Deflection under 1st axle is 1.74 mm, deflection under 2nd axle is 1.88 mm, and in the case of one axle, the deflection is 1.81 mm.

Figure 8

Vertical displacement map (Ty), [mm] – the sleeper deflection here is 1.03 mm (which is the sum of ballast deflection of 0.44 mm and subgrade deflection of 0.59 mm at the level of 95 cm below the running surface of the rail).

Figure 9

Vertical stress distribution [kPa] – maximum value of 30 kPa at the level of 95 cm below the running surface of the rail.

The mass of the rail including half of the sleeper per one meter of the track is m = 60 kg + 225 kg/2 × 1/0.6 m = 248 kg/m. The parameters assumed for calculations are shown in Table 4. Generally, the parameters correspond to those provided in the literature [37,38].

On the basis of the calculations of the deflection lines, the reaction per half sleeper was determined as 31.5 kN, and next the stresses at the bottom of the ballast was determined assuming the stress distribution angle in the ballast as 22° [34,39]. Thus, they are equal to 34 kPa. These stresses further spread in the subgrade, and at the level of 95 cm below the running surface of the rail, they reach a value of 23 kPa. They are 30% larger than those measured in previous research [23]. This may be attributed to the larger stress distribution angle assumed in the latter work.

3.4 Numerical model of the ballast-subgrade system and its validation

In order to make sure that the stress levels in the numerical model of the subgrade are correctly determined, the upper part of the subgrade is first modelled. For this purpose, a 2D FE model was developed in program MIDAS GTS NX. The calculations were performed according to the guidelines given in the Eurocode [33] assuming the sleeper loads as the reaction forces obtained from the numerical train–track model in Section 3.2. The soil parameters are shown in Table 5. The results of the validation of this model and the numerical train–track model are summarised in Table 6 (Figures 8 and 9).

Additionally, in order to compare the stress values discussed above, the laboratory tests described in the literature [34,39] are quoted. There, the ballast was carefully compacted and its depth was 32 cm. At the bottom, special measurement plates were put underneath the ballast layer which measured the stresses while the load wads were applied (statically and dynamically). As can be seen in Figure 10, the distribution of the stresses is not uniform.

Figure 10

Distribution of the vertical (normal) stresses underneath the ballast layer [39].

The vertical load was correlated to the possible reaction per sleeper in service conditions. It was equal to 41 kN per fastening system. The measured stresses were around 30 kPa in the area directly under the fastening system (area 1 in Figure 10).

In summary of Section 5, it may be stated that the validation, comparative calculations (Table 6), and tests [39] indicate that the stresses determined using the models developed in Section 3 are a good representation of the track behaviour. The subgrade deflections shown in Table 6 and obtained using a numerical ballast-subgrade model (0.59 mm) are convergent with the results of measurements in a previous research [22] for a similar locomotive. The comparative calculations and the validations are proof that the model parameters and their structures are correct and may be justly used for the subgrade and its foundation modelling in Sections 4 and 5.

5.1 Embankment prior to strengthening

On the basis of the calculations, it is concluded that the maximum settlement will reach 278 cm, which is unacceptable. This is due to the fact that the foundation of the embankment is very weak, so the self-weight of the soil mass above the foundation is dominating the behaviour of the structure.

5.2 Embankment after strengthening with palisades on both sides

The calculations of this case showed that the maximum vertical settlements reach 267 cm, which is not a significant improvement in relation to the previous case. Such behaviour of the structure indicates that the main component of the settlement is the foundation – the whole embankment settles as approximately one block.

5.3 Embankment after strengthening with palisades on both sides and piles placed in the centre

In this case, the calculated maximum settlement was 3.0 cm. This result is satisfactory from the point of view of the regulations. To be complete, the calculation should also include checking the gradient of settlements along the embankment to identify whether it exceeds admissible values. This is omitted here as it is irrelevant for the purpose of the article.

5.4 Embankment with weakened soils due to train-induced vibration

In this case, the structure was identical to that given in Section 5.2, but it was additionally weakened by the influence of vibration that propagates in the soils. The parameters of the soils in this case will be reduced following the findings of the research performed by Cracow University of Technology during test runs of the Pendolino train [40].

Research concerning vibration propagation into the subgrade and the environment has been the subject of a great number of papers. The analyses were made from the point of view of the transmissibility of vibration through soils, strain waves, or vibration mitigation [41]. A summary of research is given in the literature [42]. Natural frequencies of soil layers and resonance phenomena have also been studied [43,44]. However, the problem of the influence of vibration on the soil-bearing capacity of railway subgrades had not yet been fully analysed; therefore, this was undertaken by the authors. The question arises of how a speed increase on a line – which means higher accelerations and amplitudes of vibration – influences subgrade settlements and the formation of track unevenness.

Some data concerning this issue is provided in the literature. As a result of the vibration, the internal friction angle is reduced, which means a loss of strength. In the case of cohesive soils, the cohesion coefficient is also reduced. In the literature [45] it was noted that accelerations of soil particles at some critical level change the state of the soil, which leads to plasticisation and an abrupt loss of strength. The author tested ground samples (granular soils) under various stresses, amplitudes, and frequencies of vibration. For example, as a result of vibration action in the order of 140–250 Hz, the soil particles exhibited compaction in the order of 20% in relation to the sample height. This compaction remained after the vibration action was stopped, but the shear strength was restored to its original value. The total strength loss of granular soil (coarse sand) during vibration action was determined at 13% at a vertical acceleration of 0.1 g, and reached 22% at an acceleration of 0.145 g. The particle displacement amplitude was in the range of ( 1.0 − 1.5 ) × 10 − 3 mm . The reduction in the friction angle was found to be at a level of 10%.

In the literature [46], a vibration loss coefficient was proposed. This coefficient was proposed to be a measure of the influence of vibrations on soil strength. This would result in the change in the oedometric compression modulus of the soil.

In the present calculations, the above strength loss was assumed in the oedometric moduli of the non-cohesive soils (which is 13%) and the friction angle was reduced by 10%, since such an order of magnitude was found in the quoted research. The reduced parameters are shown in Table 8. It has been assumed in these calculations that this loss in the internal friction angle is independent of the normal stress in the soil. At present, the authors have left out the problem of the possible plasticisation of soils in the foundation of the embankment.

5.4.1 Research into vibration propagation during tests with the Pendolino train

The measurements of vibration propagation were made by a team mainly comprised of University researchers in 2014 [40]. The measurements were made on the main line at various train speeds and with a top speed of 293 km/h (speed record on the Polish railway network). Curves of the vibration decay are shown in Figure 15.

Figure 15

Vibration decay as measured by sensors placed on a line perpendicular to the track, the maximum distance is 128 m and the curves represent various trains that passed the site – the curve for the Pendolino is indicated by an arrow [40]. Other lines are obtained for lower speeds starting at 60 km/h and for other types of trains, which is less important for the present article.

The results of the research show that there is a linear relationship between the speed of the train and the accelerations of sleepers. These accelerations propagate to the ballast and are approximately 10 times smaller next to the track (but still on the top of the subgrade). They decay very quickly at the foot of the embankment, and at some distance from the embankment, they became indistinguishable with regard to the train type. The order of magnitude of the accelerations is 80–90 cm/s² (approximately 0.1 g) and at the foot of the embankment (which was about 6 m high), it is 5–20 cm/s² (about 0.02 g). These results justify use of the aforementioned rates of strength losses of soils because they correspond to the range of amplitudes and frequencies in the quoted paper [45].

5.4.2 Model of vibration decay in soils

For the purpose of the present article, the authors use a simplified model of vibration decay in soils with the assumption that the foundation is an elastic half-hemisphere [44]. This simplified model is as follows:

(2) u ( r , f ) = p ( 1 − ν ) 2 π G r e − Dr ⁎ ,

where, u ( r , f )  – amplitude of vibration (mm), r  – distance from the source (m), f  – frequency of vibration (Hz), p  – vertical service load (reaction per half sleeper) – Table 8, ν  – Poisson ratio of the soil – Table 8, G  – shear modulus of the soil – Table 8, D  – damping coefficient of the soil – Table 8, r ⁎ = ω r / v s  – dimensionless coefficient of the distance from the source of vibration, ω  – angular frequency of vibration – Table 8, v s  – speed of vibration-wave propagation in the soil – Table 8.

Examples of vibration decay plots are shown in Figure 16. The data for the calculation are presented in Table 8. Assuming small strain theory and taking the linear relationship between the compressibility modulus and the shear modulus G = E / 2 ( 1 + ν ) , which for layers Ia and Ib (Table 7) means that E belongs to interval 12,000–18,000 kPa, one obtains the average shear modulus G = 6,000 kPa. The Poisson ratio was assumed as average for gravel, coarse sand, and fine sand which is ν = 0.225 [47]. In the calculations that follow, two layers were considered for the purposes of comparison: layer Ia with E = 18,000 kPa and layer IVa with E = 29,700 kPa. These values correspond to shear moduli G = 7,500 kPa and G = 12,500 kPa, respectively. The damping coefficient of the soils was assumed as being constant at D = 0.05 and the shear wave propagation was v s = 270 m/s [44] for gravel and sands.

Figure 16

Dependence of the vibration amplitude on the distance from the source (sleeper bottom) on the basis of ref. [44].

For the following calculations, the speed of the train was 250 km/h (69 m/s) as in Figure 15. The decay curve is similar to that obtained for a classic train with a locomotive at a speed of 160–200 km/h. The load was treated as the reaction force underneath the sleeper (per rail) (denoted by p = 32.1 kN as in Table 6 and equal to 3,270 kg). The frequency was calculated from the sleeper excitation as the speed of the train/sleeper spacing, which gives 69 m / s / 0.6 m = 116 Hz, and the angular frequency was 727 Hz.

In Figure 16, we can observe that the vibration amplitude at the bottom of the sleeper (distance close to zero) is within the range of about 0.3–0.5 mm, which is a realistic estimation and this is convergent with the calculated sleeper deflections in the track model and the FEM soil model (Table 6).

5.4.3 Calculations of the embankment settlements with soils weakened due to vibrations

The last calculation example is comparable to that in Section 5.2; however, this time the soils were weakened as shown in Table 8. The results are presented in Figure 17.

Figure 17

Vertical displacement map of weakened soils with the weakening effect.

The results of the calculations shown above do not at this point indicate a significant difference in the settlements between the weakened and not weakened cases. However, some effect can be seen in the form of a difference of around 10 mm.