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Abstract

Ballast compaction with the Dynamic Track Stabilizer (DTS) is known to improve the lateral track resistance by increasing the stiffness of the ballast. This paper presents two approaches for estimating the strain dependent ballast shear modulus G based on measurements during dynamic track stabilization. One approach uses the Hardin equation with its enhancement for coarse-grained soils to estimate the small strain shear modulus for a given grain size distribution curve. Since different shear strains are induced on an observed sleeper (with fixed location) by a bypassing DTS, the measurement of ballast shear strains with accelerometers on the top and the bottom edge of the ballast layer yield shear modulus degradation curves G/G_max. It is shown for data from a regular maintenance operation, that these shear modulus degradation curves are in accordance with previous research. For the second approach, a SDOF model for the track-ballast interaction under harmonic loading (caused by the dynamic track stabilizer) is developed. Influence lines are used to estimate the mass, the mass moment of inertia and the geometry of an equivalent machine foundation (with spring and dashpot coefficients according to the Hall analog). An optimization algorithm is used to fit the model response to the measured data from field tests with the DTS. The resulting shear moduli show plausible values for loose and compacted ballast conditions, which proves the potential of the presented method as a basis for a future system for Intelligent Compaction (IC) with the DTS.

Keywords

Railway track track maintenance unit track stiffness train-track-soil model vehicle track interaction vibration dynamics

Introduction

Ballast compaction after tamping can improve certain properties of the ballast bed,¹ affecting some of its most important functions²:

1. Resistance to vertical, lateral and longitudinal forces transmitted by the sleepers to retain track geometry.

2. Providing resilience and energy dissipation for the track.

3. Reduction of high pressure activated in the sleeper-ballast interface for the underlying material.

While the positive influence of ballast compaction on the lateral track resistance has been proven by recent research,^1,3,4 a significant positive influence on the vertical track resistance, and thus on the durability of track maintenance operations regarding track resistance, has not yet been described.⁵ Furthermore, very little is known about the influence of ballast compaction on the resilience and the energy dissipation of railway tracks and the stresses activated in the sleeper-ballast interface.

The resilience of railway tracks (or track stiffness k) is determined by the rail flexural rigidity EI and by the track modulus u.⁶ The track modulus u can be interpreted as stiffness of the rail foundation consisting of fasteners, sleepers, ballast, subballast and subgrade and is considered a measure for track quality, with higher values usually describing a better track foundation.^6,7 The track modulus is governed by the stiffness of the ballast, the subballast and most of all by the stiffness of the subgrade.⁶ Therefore, a variety of methods for measuring the track modulus have been developed in recent decades, but to the knowledge of the authors none of these methods are capable of measuring the crucial stiffness conditions of the individual granular layers.

This paper aims to contribute to the estimation of the stiffness of the granular layers of a railway track by providing a conceptual design for a new method to estimate the shear modulus of the ballast layer. There are several methods for evaluating the stiffness of railway ballast. Some researchers suggest determining parameters of the granular layers by the use of advanced numerical models.^6,8 Berggren⁸ performed an estimation of certain substructure parameters (like ballast bed thickness and subgrade shear wave velocity) using the VibTrain-software to calculate a simulated stiffness response for a typical field test with the Swedish RSMV (Rolling Stiffness Measurement Vehicle)⁹ and machine learning algorithms to fit this output to actual measurements. Li⁶ performed a parametric study to show the influence of the resilient modulus of the granular layers on the overall track stiffness using the software GEOTRACK. However, none of these software based approaches allow for a stiffness estimation of the individual layers based on in situ measurements.

A common method for the ballast condition evaluation is the ground penetrating radar (GPR), which is mainly used to evaluate the thickness of the ballast layer, the moisture content of the ballast, and the fouling index.¹⁰ Rapp and Schuk^11,12 combined continuous measurements from the GPR with local data from soil samples (from rotary core samplers) and from dynamic cone penetrometers to develop an approach to estimate the track modulus. However, this approach also cannot distinguish between the individual stiffnesses of the granular layers.

The individual stiffnesses of these granular layers can either be found by in situ tests or by laboratory tests. Results from measurements with the light dynamic penetrometer (PANDA) and from measurements with the light falling weight deflectometer were performed in a previous study by Lamas-Lopez et al.¹³ on a French railway track. According to their study, both methods showed reproducible results for all layers in case of the PANDA test or only for the ballast layer in case of the light falling weight deflectometer.¹³

In terms of laboratory testing, the stress-strain behavior of ballast is usually analyzed using large-scale triaxial tests.^2,14–21 Selig and Roner² examined the influence of the void ratio and the particle shape on the stress-strain behavior of new ballast. Guldenfels¹⁴ analyzed the influence of ballast fouling on the stress-strain behavior of ballast and showed that ballast stiffness (resilient modulus E_r) decreases with increasing ballast fouling. Extensive research on the (soil-)mechanics of ballasted railway tracks was performed by Indraratna and his team.^15–19 Their laboratory tests indicate that the resilient modulus of ballast depends on the confining pressure,¹⁶ and that the resilient modulus of loose ballast build up with increasing load cycles until 100,000 cycles, after which no further increase of the resilient modulus can be observed.^16,18 Furthermore, Sun¹⁷ described a dependency of the resilient modulus on the loading frequency. After exceeding a confining pressure dependent critical frequency, ballast shows strain softening which implies a lower resilient modulus at higher loading frequencies.¹⁷ All these tests resulted in a recently published method for predicting the ballast resilient modulus based on certain predictor variables.¹⁹ Other recent studies based on triaxial tests have dealt with the influence of the water content on mechanically-fouled ballast (fouling caused by particle breakage)²⁰ or with the determination of the small-strain shear modulus of railway ballast.²¹ None of the described methods for the determination of stiffness properties of railway ballast can provide continuous and soil mechanically meaningful information at the same time.

The presented paper is the first to address the estimation of track ballast stiffness in the context of ballast compaction using the Dynamic Track Stabilizer (DTS), a machine used in track maintenance all over the world. The DTS induces (horizontal) vibrations of the track. These vibrations cause shear deformations of the ballast layer which lead to a rearrangement and an interlocking of the ballast particles.⁶ As a result, some of the settlement from rail traffic is already achieved through compaction. By monitoring a single sleeper with accelerometers, different amplitudes of shear deformation can be measured, as ballast shear deformations increase with decreasing distance between sleeper and DTS. Within this study, additional sensors in the ballast layer as well as results from laboratory tests are used to estimate the shear modulus of ballast during ballast compaction. Furthermore, the measured accelerations are used to outline a possible framework for a continuous stiffness measurement of railway ballast utilizing the DTS.

Shear modulus from on site measurements and field samples

The analysis of the deformation of the ballast layer during dynamic track stabilization aims to calculate the strain dependent shear modulus of ballast for a broad range of shear strains. Such shear modulus degradation curves can serve for the evaluation of other methods for shear modulus estimation, like the novel method described in the second part of this paper or as a reference for other studies in this field (e.g. Barbir²²). The calculated shear modulus degradation curves are compared with the results obtained by other researchers. Based on this comparison, the potential of these in situ measurements is assessed.

Test track in Retz (section A)

The estimation of the shear modulus of ballast based on dynamic measurements inside the ballast layer is performed using data from an Austrian track section close to the town of Retz (referred to as section A). This section was originally equipped with a pressure cell and an accelerometer at the bottom of both, the ballast layer and the subballast layer during formation rehabilitation in July 2020 in order to collect data for future sub-ballast optimization.²³ The final tamping and associated ballast compaction took place one year after the completion of the formation rehabilitation. During this maintenance operation, the accelerations of the bypassing DTS and the sleeper above the sensors inside the granular layers as well as the accelerations and the pressure at the bottom of the ballast layer and at the bottom of the subballast layer were monitored (in vertical, longitudinal and in lateral direction). The average operating speed of the DTS during the passage of the section was 700 m h⁻¹. Additionally, the phase of the horizontal excitation force caused by the DTS was measured using inductive proximity sensors and the position of the DTS was tracked using a GPS-antenna. A ballast sample was taken on site and analyzed in the scope of a research project between the company Plasser & Theurer, Autrian Federal Railways (ÖBB), and Graz University of Technology. The track section and the measurement setup are shown in Figure 1.

Figure 1.

Measurement setup at section A. The track consists of UIC 60 rails, pandrol e-clip fasteners and concrete sleepers (L2) with Under Sleeper Pads (USP) with a spacing of 60 cm.

Small strain shear modulus from grain size distribution curve

Ballast may be considered as a coarse grained soil with uniform grain size distribution. Although the stress-strain relations for soils are highly non-linear, for small (shear-) strains, the material behavior of soils under cyclic loading can be described with linear elastic constitutive equations.²⁴ For sand, shear strains γ can be considered small if γ < 0.005 %.²⁴ Recent research indicates that the small shear strain behavior of ballast differs for initial loading and subsequent load cycles.²¹ For initial loading, shear strains smaller than γ < = 0.002 − 0.003 % may be considered as small shear strains. For subsequent load cycles, the small shear strain behavior is also valid for higher strain levels of γ < = 0.02 − 0.03 %.²¹ Several empirical formulas for calculating the small strain shear modulus of soils based on their grain size distribution curve have been described in the literature (an overview is given by Wichtmann²⁴). The most common formula is an empirical equation developed by Hardin.²⁵ The formula is based on laboratory tests with uniform sands and may overestimate the small strain shear modulus of well-graded soils. Therefore, adjusted versions of the original formula exist.²⁴ Another limitation of the original Hardin equation is that it is not applicable to coarse-grained soils.²⁶ This limitation can be circumvented with the introduction of an additional scalar function $f (D)$ as an extension of the (generalized) Hardin equation.²⁶ The restriction to uniform grain size distributions is not important for clean ballast with an approximately uniform grain size distribution curve, but may be important for older (fouled) ballast with a broader grain size distribution curve (e.g., due to particle breakage during maintenance operations or due to infiltration of finer grains from the ballast surface). However, since the ballast sample taken from the measured track section has a low level of fouling and thus a uniform grain size distribution curve, the generalized Hardin equation²⁶ with the enhancement for coarse-grained soils is used in this study, assuming a rotational symmetric stress state: $G_{max} = \frac{{O C R}^{k} f (D)}{0.3 + 0.7 e^{2}} \frac{S}{2 (1 + ν)} p_{atm}^{1 - n} {(σ_{v} σ_{h})}^{n / 2}$ (1)

Where OCR is the overconsolidation ratio, k is a constant factor, e is the void ratio, $f (D)$ is a scalar function, S is the stress coefficient, ν is the elastic Poisson’s ratio, p_atm = 100 kPa is the atmospheric pressure, n = 0.5 is an elastic constant that is the same for all types of soil, and σ_v and σ_h are the vertical and the horizontal in situ stresses.

The scalar function for the extension to gravelly soils is given by²⁶: $f (D) = 1 + \frac{f_{0}}{1 + f_{0} \frac{n_{s}}{30} \frac{p}{p_{atm}}}$ (2)

The factor f₀ depends on the grain size distribution curve, n_s is a shape factor that quantifies the particle shape and p is the mean confining pressure (p = σ_h for a rotational symmetric stress state). For clean ballast with an angular particle shape n_s = 25. Hardin²⁶ provides recommendations for the calculation of f₀ based on laboratory test results from large scale resonant column tests: $f_{0} = f_{0}^{max} \frac{α \frac{(D_{5} - D_{Sand}) γ_{w}}{p_{atm}}}{1 + α \frac{(D_{5} - D_{Sand}) γ_{w}}{p_{atm}}}$ (3)

According to Hardin,²⁶ the constants $f_{0}^{max}$ , α and D_Sand can be estimated with $f_{0}^{max} = 1.6, α = 2.8$ , and D_Sand = 1 mm. D₅ is the diameter at 5 % passing on the grain size distribution curve.

Shear modulus degradation curve

The (secant) shear modulus depends on the shear strains. The dependence can be described by a (hyperbolic) constitutive equation, connecting shear stresses and shear deformations, which was originally formulated by Hardin and Drnevich²⁷: $\frac{G}{G_{max}} = \frac{1}{1 + (\frac{γ}{γ_{r}}) \{1 + a \exp [- b (\frac{γ}{γ_{r}})]\}}$ (4)

In equation (4), a and b are curve-fitting parameters. The stress dependency is considered by the reference shear strain γ_r = τ_max/G_max using the shear strength τ_max, which depends on the stress state. Thus alternative equations to equation (4) exist, where the confining and the atmospheric pressure are used directly instead of the reference shear strain and the shear strength.²⁶ Wichtmann²⁸ developed several enhancements of equation (4) to consider the dependence of the modulus degradation on the uniformity of the grain size distribution curve. In the present study, the following equation is used to calculate the modulus degradation curve²⁸: $\frac{G}{G_{max}} = \frac{1}{1 + \frac{γ}{\sqrt{p / p_{atm}}} \{1 + a \exp [- \frac{γ}{\sqrt{p / p_{atm}}}]\}}$ (5)

The curve-fitting parameter a was found from correlations with the uniformity coefficient C_u and is $1093.7 + 1955.3 \log (C_{u})$ .²⁸

The calculation of a modulus degradation curve based on the measured pressure and shear strains using equation (5), requires an estimation of the small strain shear modulus G_max. Equation (1) to equation (3) allow the calculation of G_max of the ballast based on a given grain size distribution curve. The grain size distribution curve of the ballast from track section A was originally determined for a research project between Plasser & Theurer, Austrian Federal Railways, and Graz University of Technology²⁹ and is shown in Figure 2. Equation (1) is governed by the Poisson’s ratio ν and the void ratio e. Different authors estimate different values for the Poisson’s ratio.¹⁴ According to Guldenfels¹⁴ the Poisson’s ratio ν is in the range of [0.3, 0.4], for clean or moderately fouled ballast (mass of ”fines” with grain size $\leq 22.4 mm$ is less than 50 % per weight). The ballast found at track section A was rather clean (see Figure 2) and therefore a Poisson’s ratio of ν = 0.33 is assumed in this study. The void ratio depends on the dry density ρ_d and the grain density ρ_s. The ballast used at the analyzed site consisted of crushed granite, which has an average grain density of ρ_s = 2700 kg m⁻³. The dry density of crushed limestone ballast with different levels of compaction was studied by Thom and Brown.³⁰ Their results were also discussed by Indraratna³¹: ρ_d varies between ρ_d = 1300 kg m⁻³ for uncompacted and uniformally graded ballast and ρ_d = 2100 kg m⁻³ for compacted ballast with a broader grain size distribution curve.³¹ Thus a void ratio in the range of $e = [0.350, 1.077]$ can be expected. Other governing factors of equation (1) are the OCR (which is one for a coarse-grained material like ballast), the stress coefficient S, which can be assumed to be in the range of $S = [1200,1700]$ ²⁶ defined by σ_h and σ_v.

Figure 2.

Grain size distribution curve for ballast sample taken at the track section A.²⁹

In soil mechanics, σ_h is often calculated by factorizing σ_v using an earth pressure coefficient K_i. For the estimation of in situ stresses usually K_i = K₀ = 1 − sin(ϕ) is assumed. Where ϕ is the internal friction angle of the granular material (soil). However, in railroad engineering different definitions of the friction angle of ballast exist.^31–33 The present paper refers to a definition according to the classic Mohr-Coulomb failure criterion. Estaire and Santana³³ published test results from direct shear tests with dense (compacted) and loose ballast (dropped). For these tests a friction angle in the range of $ϕ = [42.0 °, 53.0 °]$ was determined, assuming a cohesionless soil (c = 0).³³ Based on this friction angle range, the earth pressure coefficients for the determination of the horizontal in situ stresses are between K₀ = 0.371 for a friction angle of 42° and K₀ = 0.181 for a friction angle of 53°.

The vertical stresses are obtained by in situ measurements within the ballast layer. The measured vertical stresses are shown in Figure 3 for a distance of ±2 m between the measured section and the DTS during the maintenance operation. Whereas the vertical stresses at the bottom of the ballast layer are known, the stresses at the sleeper-ballast interface were not measured due to the original research purpose²³ and thus the vertical stress distribution in the ballast layer is unknown. However, as the distribution of the vertical and horizontal stresses in the ballast layer is crucial for the application of equation (5), the stress in the ballast-sleeper interface has to be estimated. The vertical stress on the sleeper-ballast interface can be estimated by calculating the load carried by one sleeper. Therefore, an assumption of the load distribution among the activated sleepers is necessary. The number of activated sleepers depends on the stiffnesses of track, ballast and subgrade.³⁴ Nevertheless, measurements have indicated that a single load on a railway track is mainly supported by approximately five sleepers (the loaded sleeper and four adjacent sleepers),³⁵ which was regularly assumed in previous studies on railway engineering (e.g. Klugar³⁶). For simplicity, the load carried by a single sleeper is estimated using this traditional assumption of the distribution of support of a vertical load. The loaded sleeper bears 50 % of the load P, the two sleepers next to the loaded sleeper 20 % and the outer sleepers 5 %. The load P comprises the load from the hydraulic cylinders and the weight of the DTS. During the observed maintenance operation the pressure within the hydraulic cylinders was kept constant at 70 bar which is equal to 106.5 kN according to the operation manual of the DTS.³⁷ The weight of the DTS is approximately 24.10 kN. This load is distributed by two axles and thus the load of one wheel is $\hat{P} = P / 4 = (106,5 + 24,1) / 4 = 32.65 kN$ . For the situation of a DTS straight above one sleeper, the expected loading of one sleeper is: $\tilde{P} = 2 (0.33 \hat{P} + 0.27 \hat{P}) = 39.2 kN$ (6)

Figure 3.

Measured vertical pressure at the bottom of the ballast layer at site A, with detailed plot of the pressure signal for a DTS located right above the monitored sleeper. The mean pressure is shown as dash-dotted line.

In equation (6), the loading of the individual sleepers was assumed to depend linearly on the distance of the DTS axles to the sleepers.

Given a uniform stress distribution, the vertical stress on the sleeper-ballast interface is approximately $\tilde{P} / A_{s} = 39.2 / 0.73 = 53.7 kPa$ . A_s is the area of the sleeper-ballast interface of a single sleeper. The mean measured vertical pressure for one DTS straight above the sleeper is σ_v,bottom = 69.1 kPa (see Figure 3). The calculated stress at the sleeper-ballast interface due to the DTS loading is lower than the measured stress at the bottom of the ballast layer. This difference can be explained by the dead load of the track and the ballast layer itself. The contact stress due to the dead load of the track with a sleeper spacing of 60 cm is caused by the dead load of a sleeper (W_s = 2.91 kN) and of the rails (W_r = 2 ⋅ 60 ⋅ 0, 6 = 0.72 kN). $σ_{v,t} = \frac{W_{s} + W_{r}}{A_{s}} = 5 kPa$ (7)

The stress at the sleeper ballast-interface, taking into account the dead load of the track, is therefore σ_v,top = 53, 7 + 5= 58.7 kPa. The vertical load due to the dead load of a d_b = 40 cm thick ballast layer with an assumed specific weight of γ_b = 19 kN m⁻³ is σ_v,b = 7.6 kPa. This gives a calculated vertical stress of 66.3 kPa, which is only 4 % less than the measured stress at the bottom of the ballast layer. Consequently, a mean vertical pressure of the ballast layer is used in this research: ${\bar{σ}}_{v} = \frac{σ_{v,top} + σ_{v,bottom}}{2} = σ_{v,bottom} - \frac{γ_{b} d_{b}}{2}$ (8)

Another important aspect for the calculation of the shear modulus degradation curve according to equation (5) is the calculation of the shear strains. The displacements (required for this calculation) were determined from the measured accelerations of the sleeper and ballast using an adapted approach of the method proposed by Hofmann,³⁸ who analyzed the effects of high-pass filtering on the results of the double integration of the accelerations using the trapezoidal rule to estimate the displacements. The difference between the lateral displacements of the sleeper (with mean value ${\bar{y}}_{s}$ of the two 3-days sleeper sensors) and the horizontal displacements measured at the bottom of the ballast layer (y_b) is calculated by: $γ = \frac{{\bar{y}}_{s} - y_{b}}{d_{b}}$ (9)

The shear strains calculated in this manner comprise deformation components caused by different frequencies. Only shear strains caused by the governing frequency (excitation frequency of the DTS) are considered to allow for a simplified mechanical analysis of the shear deformations. Hence, the signal created with equation (9) is analyzed in the frequency domain using a FFT which yields the amplitude and phase of the shear wave (with the governing frequency) induced by the DTS. Equation (9) assumes a linear shear deformation of the ballast layer. Consequently, this approach implies the wavelength of the governing wave to be much longer than the thickness of the ballast layer. The shear wave length λ_s (see Figure 1) can be estimated using equations derived for an elastic half space model³⁹: $λ_{s} = \frac{G^{0.5}}{ρ^{0.5} f_{s}}$ (10)

Assuming G = 20 MPa, ρ = 1600 kg m⁻³ and f_s = 20 Hz yields λ_s ≈ 3.5 m which is much larger than the ballast layer thickness of d_b ≈ 0.4 m and thus the assumption of linear shear strains is permissible.

Using the data from section A, the shear modulus degradation curve is calculated considering the range of the governing parameters listed before. This leads to a set of reasonable degradation curves which are shown in Figure 4. The degradation curves are plotted as gray lines and the curve resulting from the use of ’mean’ parameters (ρ = 1700 kg m⁻³, ϕ = 43° and S = 1500) is highlighted as a black dash-dotted line. In addition, the envelopes of the shear modulus degradation curves defined by the ’stiffest’ and the ’softest’ degradation curves are plotted as bold black lines. The normalized shear modulus degradation curve (for which a normalization based on the pressure was used) for the mean value curve in Figure 4 is shown in Figure 5. The obtained normalized degradation curve is in accordance with the results of Hardin and Kalinski²⁶ (shown in the detailed section of Figure 5).

Figure 4.

Shear modulus degradation curves derived from (implicit) measurements of shear strains and vertical pressures during dynamic track stabilization for various parameter sets.

Figure 5.

Normalized shear modulus degradation curve for a set of ’mean’ parameters. A detailed section is compared with results of Hardin and Kalinski.²⁶

Not many researchers have dealt with the shear modulus of ballast in the context of track maintenance so far. Barbir²² has estimated a shear modulus of 20 MPa as a constant input value for an analytical model of the ballast reaction during tamping operations, which seems a reasonable assumption considering the results shown in Figure 4 and suggest that the shear strains induced by the dynamic loading are similar during tamping and track stabilization operations.

Previous research has shown a frequency dependence of the resilient modulus of ballast.¹⁷ A frequency dependence of ballast behavior during maintenance operations has been observed in the past and is known as ballast fluidization.² Thus it is indicated that the ballast shear modulus also depends on the excitation frequency. However, because only one excitation frequency was used in the described field test or during the regular maintenance operation respectively, the frequency dependence of the ballast shear modulus during dynamic track stabilization is a limitation of this paper and should be further investigated in future research.

Framework for a continuous stiffness measurement with the DTS

The existing methods for estimating ballast stiffness do not provide continuous results along the railway line. This limitation of current measurement procedures could be solved by developing a system for continuous stiffness estimation with the DTS. Similar methods already exist for rollers and are often referred to as Continuous Compaction Control (CCC) or Intelligent Compaction (IC).^40,41 CCC and IC are based on the measurement of the motion behavior of compaction devices, which highly depends on soil stiffness. If process parameters (excitation frequency, eccentricity of imbalance and vertical pressure in case of the DTS) are constant, irregularities of the motion behavior are caused by a variation of the soil stiffness.

Dynamic track stabilization consists of two parts in terms of structural dynamics: the DTS and the track. Recent research⁴² showed that due to the rigid design of the hydraulic cylinders, the machine frame is dynamically excited during dynamic track stabilization. Consequently, the frame needs to be considered in a dynamic analysis of the motion behavior of the DTS, which is thus governed by variant conditions of the ballast (variable subgrade conditions along the track, variable loading history due to variable track geometry) and variant conditions of the machine frame (variable mass due to machine loading, individual frame structure for different machines). Due to the additional set of variant conditions compared to other compaction devices, the proposed study aims to develop a simplified approach to estimate ballast stiffness based only on track movement behavior. The results of the simplified model are used to discuss the potential of IC with the DTS and the challenges for future research.

Test track in Oberwart (section B)

The analyzed track section was part of a test campaign which was conducted in an Austrian Railways test facility (openrail lab, ÖBB) close to the town of Oberwart in October 2019. The following tests were performed on section B using an 09-4X E³ tamping machine with two integrated and synchronized dynamic track stabilizing machines (the two machines vibrate with the same phase), working directly behind each other:

• Initial test run – the DTS passes section B (without tamping) with in situ ballast conditions.

• Tamping operation without track stabilization (DTS switched off).

• First test run – the DTS passes section B (without tamping) after tamping (loose ballast).

• Second test run – the DTS passes section B (without tamping) a second time after tamping (compacted ballast).

The process parameters of the DTS remained constant during the tests with an excitation frequency of f = 33 Hz, a vertical load of 130 kN per DTS and a maximum eccentricity of the imbalances. The operating speed of the tamping machine (and the DTS) was 2000 m h⁻¹ during all test runs, which is typical for the type of tamping machine used. Thus, a comparison between dense and less dense ballast conditions (after tamping) is possible. The measurement setup included three-dimensional accelerometers (again in vertical, longitudinal and lateral direction) and a one-dimensional accelerometer (in vertical direction) mounted on one sleeper, three dimensional accelerometers on the DTS and one-dimensional accelerometers (vertical direction) on the machine frame. Additionally, inductive proximity sensors to measure the phase of the excitation force caused by the DTS, and a GPS antenna to monitor the DTS position were used. The track section and the measurement setup is shown in Figure 6.

Figure 6.

Measurement setup at track section B. The track consists of S 49 rails, pandrol fasteners and classic wooden sleepers (dimensions: 20/24/253 cm) with a spacing of 68 cm.

Lumped-parameter model

In this paper, the track-ballast interaction during dynamic track stabilization is modeled as a Single Degree of Freedom (SDOF) system. There have been attempts to develop similar systems for an efficient analysis of vehicle-track-interaction in the past.⁴³ These models split the track and the vehicle for calculation purposes, allowing for a separate calculation of the track response while the wheel-rail reaction force is separately computed and used as a known input for the track model. Due to the unknown influence of the machine frame on the motion behavior of the DTS, this approach is also used in this paper (see Figure 7).

Figure 7.

Lumped parameter model for the analysis of the interaction between frame, DTS, track and ballast on the left, the reduced model for the track-ballast interaction in the center and the simplified SDOF model after coupling η₁ and y₁ through the instantaneous center of rotation on the right.

In other words, the dynamic behavior of the track (section) is considered as a harmonically loaded machine foundation, with the horizontal harmonic load acting on the top of the rails. This results in a horizontal translational movement and, due to the eccentricity of this force to the center of mass, also in a rotational movement of the track. The use of the law of conservation of momentum and impulse yields a system of two coupled differential equations of motion for the two degree of freedom system shown in the center part of Figure 7: $\begin{aligned} m \ddot{y} + C_{y} (\dot{y} - \dot{η} h_{0}) + K_{y} (y - η h_{0}) = P_{e} \sin (ω t) \\ I_{x x} \ddot{η} + C_{η} \dot{η} + K_{η} η + C_{y} \dot{η} h_{0}^{2} + \dots \\ + K_{y} η h_{0}^{2} - \dot{y} C_{x} h_{0} - y K_{y} h_{0} = \dots \\ = P_{e} (t) (h - h_{0}) + M_{e} \sin (ω t) \end{aligned}$ (11)

However, recent measurements⁴² suggest that the motion of a track section during dynamic track stabilization can be described as a purely rotational movement around a (temporarily) constant instantaneous center of rotation. This behavior also indicates the existence of an excitation moment M_x. The track section moves in lateral and vertical direction during dynamic track stabilization. Consequently, the instantaneous center of rotation can be calculated as vertical distance from the measurement plane, defined by the position of the accelerometers placed on the sleeper (upper edge of the sleeper): $z_{m} = \frac{e_{s} {\bar{\dot{y}}}_{s}}{({\dot{z}}_{s2} - {\dot{z}}_{s 1})}$ (12)In equation (12), e_s is the horizontal distance of the sleeper sensors and the other variables describe measured velocities in vertical direction (z) or in lateral direction (y). However, equation (12) is not able to guarantee the existence of a (temporarily) constant instantaneous center of rotation. Thus a constraint using the phase angle gained from a FFT of the rotational and the translational terms of equation (12) needs to be formulated. The signal analysis is performed using rectangular time windows of the signals with a length equal to the time needed for 10 imbalance rotations, measured by the inductive proximity sensors. A temporarily constant instantaneous center of rotation exists if the phase difference between the rotatory motion and the translational (horizontal) motion is 0° or 180° for a certain time window. However, a tolerance for this phase difference of 45° is accepted in this research: $| φ {({\bar{\dot{y}}}_{s})}_{f_{s}} - φ {({\dot{z}}_{s2} - {\dot{z}}_{s 1})}_{f_{s}} | \leq 45 °$ (13)

Where $φ {(s i g n a l)}_{f_{s}}$ is the phase angle from a FFT of the input signal at the excitation frequency f_s of the sleeper. In Figure 8, the position of the instantaneous center of rotation and the corresponding phase difference of equation (13) for the old track from track section B and for the modern track from track section A are shown. Whereas the phase difference for the older track is within the feasible domain defined by equation (13), the modern track violates the constraint. The detailed plot of the measured signals for the rotatory (around x-direction) and translational velocities (y-direction) illustrate the delay between these motions for the modern track, while being approximately in phase for the old track. Consequently, the old track with wooden sleepers rotates at the same time as it is laterally displaced. The modern track with concrete sleepers with Under Sleeper Pads (USP) rotates before it is laterally displaced and is approximately horizontal at the peak of its lateral motion. As a result, the motion of an old track can be described by a rotation around a temporarily invariant center of rotation, but the motion of the modern track can not. Therefore, the coupled system from equation (11) can be simplified for old tracks.

Figure 8.

Left: Instantaneous center of rotation for a modern track from section A (a) and an old track from section B (b) with shaded area showing allowed phase differences between rotation and translation. Top right: a detailed plot for the measured displacement and rotation signal for the DTS above the monitored sleeper. Bottom right: sleeper motion in the y-z-plane.

The law of conservation of momentum using the geometry defined in Figure 9 yields: $\begin{align} I_{x x} \ddot{η} = - η \underset{\begin{array}{c} \tilde{K} \end{array}}{\underset{⏟}{[K_{y} (z_{m} - h_{s}) + K_{η}]}} \dots \\ - \dot{η} \underset{\begin{array}{c} \tilde{C} \end{array}}{\underset{⏟}{[C_{y} (z_{m} - h_{s}) + C_{η}]}} \dots \\ + \underset{\begin{array}{c} \tilde{M} \end{array}}{\underset{⏟}{[P_{e} (z_{m} + h_{r}) + M_{e}]}} \sin (ω t) \end{align}$ (14)Rewriting equation (14) to account for the variables introduced by the underbraces gives the well-known equation of motion for an SDOF: $I_{x x} \ddot{η} + \tilde{C} \dot{η} + \tilde{K} η = \tilde{M} \sin (ω t)$ (15)

Figure 9.

Geometry used in equation (14).

In equation (14), K_i and C_i are the ballast stiffness and ballast damping parameter, respectively. These parameters are calculated using the mass-spring-dashpot model by Hall,⁴⁴ which is also discussed by Das.³⁹ The variables h_i and z_i are geometric properties regarding the sleeper (index s) or the rail (index r), M_e and F_e are the excitation moment and the excitation force with angular frequency ω, and I_xx is the mass moment of inertia of the track. An alternative for the estimation of K_i and C_i is the application of the cone model according to Wolf.⁴⁵ The cone model can consider an embedding of the machine foundation, a layered half-space and plastic strains. Both methods are based on the Reissner solution of a circular foundation vibrating on an isotropic and elastic half-space. Therefore, the two methods should lead to similar results for K_i and C_i for the assumed simple case of a rectangular foundation vibrating on an elastic isotropic half-space. Due to its simplicity the Hall⁴⁴ analog is used in this paper.

The mass moment of inertia I_xx from equation (16) is: $I_{x x} = I_{x x, s} + I_{x x, r} + m_{s} z_{s}^{2} + m_{r} z_{r}^{2}$ (16)

Estimation of initially unknown, dynamic parameters

Equation (14) requires some parameters that are initially unknown:

• Components for calculation of the mass moment of inertia I_xx.

– Dynamically activated masses of sleepers m_s and rails m_r.

– Dynamically activated mass moments of inertia of sleepers I_xx,s and rail I_xx,r.

• Geometry of an equivalent rectangular machine foundation.

• Shear modulus G of the ballast.

• Poisson’s ratio ν of the ballast.

• In situ density of the ballast ρ for:

– Spring coefficients K_y and K_η.

– Damping coefficients C_y and C_η.

• Excitation force P_e and excitation moment M_e.

The activated masses and mass moments of inertia can be estimated using an approach similar to the influence lines commonly used in civil engineering. The movements of one equipped sleeper are continuously measured. Thus, the amplitude and the phase of the sleeper reaction (with respect of the phase of the dynamic excitation caused by the imbalance of the DTS) are known for any distance between sleeper and DTS which leads to a continuous influence line for the sleeper reaction. The DTS causes a rotatory and a translational sleeper response. Consequently, there are two different influence lines for the two types of reaction. These influence lines are either connected to the activated masses in case of the lateral accelerations, or connected to the activated mass moments of inertia in case of the angular accelerations.

The use of influence lines to calculate dynamically activated masses and moments of inertia is described in detail below using the example of (laterally activated) track masses. In a first step, the horizontal acceleration amplitudes of the sleeper are calculated using a FFT and only the amplitudes caused by the excitation frequency are considered. The calculated amplitudes for the initial test are shown in Figure 10. These amplitudes are already normalized by dividing by the peak acceleration caused by the preceding DTS. In case there was only one DTS, the influence line for the activated sleeper mass could be generated by simply multiplying the normalized amplitudes with the sleeper masses, considering the position of these masses as lumped masses located at the actual sleeper position. However, at track section B, two DTS were used. Initially, the section of the normalized ampitudes generated by the preceding DTS is flipped at its amplitude peak, in order to create the influence line. Then this procedure is repeated for the section created by the following DTS. A continuous function is generated for both curves by fitting a Gaussian curve. The parameters for these curves are averaged and the generated ’mean curve’ is multiplied by the lumped masses of the sleepers as previously described. The generated curve can be interpreted as the maximum mass that contributes to the measured accelerations of the monitored sleeper. For example, in Figure 10 a maximum of 40 % of the mass of the adjacent sleeper can contribute to the accelerations of the monitored sleeper, when fully excited by the DTS.

Figure 10.

Left: Normalized amplitudes of horizontal accelerations $({\ddot{y}}_{s})$ for the initial test (dash-dotted line) with applied factor to consider the phase difference at different DTS distances (bold line). Shaded areas represent the normalized amplitudes caused by the preceding DTS (dark gray) and the following DTS (light gray). Right: Resulting influence line for the sleeper masses with averaged normalized amplitudes (multiplied with the mass of a single sleeper) caused by the two DTS as a dashed line.

The generated influence line of the masses (continuous function f(x)) is subsequently multiplied with the normalized amplitudes g(x) caused by the DTS at different locations. Thus the function of the activated mass m(x), with respect to the distance between the preceding DTS and the monitored sleeper, can be calculated using a convolution integral described by equation (17): $m (x) = f (x) * g (x) = \int_{- x 1}^{x 2} f (χ) g (x - χ) d χ$ (17)

Where χ is a local representation of the x-axis for performing the integral. The method outlined is based on the assumption that all corresponding masses vibrate with the same phase with respect to the excitation caused by the DTS. While this is a reasonable assumption for the DTS close to the monitored sleeper, the phases differ for greater distances between the DTS and the monitored sleeper. The normalized amplitudes $| {\ddot{y}}_{s} (x) |$ are multiplied with the cosine of the phase difference between the phase of the signal and the phase for the excitation (the DTS) right above the monitored sleeper, in order to consider the ’out-of-phase’ parts of the masses that are at a further distance from the DTS: $| {\ddot{y}}_{s} (x) | = | {\ddot{y}}_{s} (x) | \cos [φ {({\ddot{y}}_{s} (x))}_{f_{s}} - φ {({\ddot{y}}_{s} (0))}_{f_{s}}]$ (18)

The described method for estimating the activated mass with respect to the distance between DTS and the monitored sleeper is then performed with the reduced normalized amplitudes calculated by equation (18). The resulting amplitudes are shown in Figure 10. Analog estimations were performed for the activated masses of the rails and for the mass moments of inertia of the sleepers and the rails. The activated mass moments of inertia were estimated on the basis of the measured angular accelerations of the monitored sleeper. The activated masses of the track for all tests are shown in Figure 11 and the activated mass moments of inertia of the track for all tests are depicted in Figure 12.

Figure 11.

Left: Normalized and corrected amplitudes of lateral accelerations for all tests. Right: Calculated dynamically activated track masses for a SDOF model.

Figure 12.

Left: Normalized and corrected amplitudes of angular accelerations for all tests. Right: Calculated dynamically activated track mass moments of inertia for a SDOF model.

The foundation geometry of an equivalent machine foundation was also determined with an influence line. The width and the height of the equivalent machine foundation are assumed to be the same as for a single sleeper. Thus the foundation length in longitudinal direction must be estimated. In mass-spring-dashpot models, the governing parameters for the principle behavior of the model are the mass and the spring stiffness. The dashpot coefficient mainly causes a reduced system response close to the natural frequency of the system. Consequently, the geometry was estimated using the corresponding motion quantity for the spring stiffness. The activated length was therefore estimated in the same way as the dynamically activated track masses, except that the normalized displacement amplitudes were used instead of the normalized acceleration amplitudes. The activated foundation length is shown in Figure 13.

Figure 13.

Activated length for the equivalent machine foundation from convolution of EL with normalized lateral displacement amplitudes.

Back calculation of shear modulus from sleeper motion

In the last section, it was shown that some unknown parameters for the application of equation (14) can be estimated using influence lines. However, the shear modulus, the Poisson’s ratio and the density of the ballast as well as the excitation force remain unknown.

A possibility to estimate these uncertain parameters is using optimization algorithms. In this research, a gradient-based optimization algorithm of the MATLAB optimization toolbox (function fmincon) is used to perform a least-squares optimization of the steady-state solution of the differential equation (15). The steady-state solution of a harmonically excited SDOF system is given in equation (19). $η (t) = \frac{\tilde{M}}{\tilde{K}} \frac{\sin (ω t - φ_{m})}{\sqrt{{[1 - {(ω / ω_{n})}^{2}]}^{2} + {[2 ζ (ω / ω_{n})]}^{2}}}$ (19)In equation (19), ω_n denotes the (undamped) circular natural frequency of the SDOF system, ζ denotes the damping ratio and φ_m denotes the mechanical phase angle: $φ_{m} = \tan^{- 1} (\frac{2 ζ (ω / ω_{n})}{1 - {(ω / ω_{n})}^{2}})$ (20)

The stiffness $\tilde{K}$ , the dashpot coefficient $\tilde{C}$ and the excitation moment $\tilde{M}$ are required, in order to obtain a solution through equation (19). The stiffness and the dashpot coefficient are calculated using the estimated masses, mass moments of intertia and foundation geometry from the previous section. According to Hall,^39,44 the stiffness and dashpot coefficients can be calculated from a set of equations. The parameters for a horizontal translation (’sliding’) motion of a rectangular foundation with width B and length L are calculated according to equation (21) and for a rocking motion around an axis at the bottom of the foundation, which is parallel to the foundation length, according to equation (22): $\begin{aligned} K_{y} & = \frac{32 (1 - ν) G}{7 - 8 ν} r_{0} \\ C_{y} & = \frac{18.4 (1 - ν)}{7 - 8 ν} r_{0}^{2} \sqrt{ρ G} \\ ζ_{y} & = \frac{C_{y}}{C_{cr, y}} = \frac{0.288}{\sqrt{B_{y}}} \\ B_{y} & = \frac{7 - 8 ν}{32 (1 - ν)} \frac{m}{ρ r_{0}^{3}} \\ r_{0} & = \sqrt{\frac{B L}{π}} \end{aligned}$ (21) $\begin{aligned} K_{η} & = \frac{8 G}{3 (1 - ν)} r_{0}^{3} \\ C_{η} & = \frac{0.8 \sqrt{G}}{(1 - ν) (1 + B_{η})} r_{0}^{4} \\ ζ_{η} & = \frac{C_{η}}{C_{cr, η}} = \frac{0.15}{\sqrt{B_{η}} (1 + B_{η})} \\ B_{η} & = \frac{3 (1 - ν)}{8} \frac{I_{0, x}}{ρ r_{0}^{5}} \\ r_{0} & = \sqrt[4]{\frac{B^{3} L}{3 π}} \end{aligned}$ (22)

In equation (22) I_0,x is the mass moment of inertia about the x-axis through the foundation base I_0,x = m/3(H² + B²/4). In equation (22), B is the lateral dimension of the wooden sleepers (253 cm), L is the activated foundation length calculated with the influence line and H is the height of a wooden sleeper (20 cm). Further, r₀ is the radius of an equivalent circular foundation, C_cr,i is the critical damping and B_i is the mass ratio. The unknown soil parameters in equation (21) and equation (22) influence the stiffness and the dashpot coefficients required for solving equation (15) with equation (19) only linearly or even appear under a square root. However, while plausible values for Poisson’s ratio and density vary in a rather small range, the shear modulus has a wide range in terms of possible values due to its strain dependence. Thus, a constant value for the Poisson’s ratio (ν = 0.33) and for the density (ρ = 1700 kg m⁻³) are assumed for the optimization. The shear modulus, as well as the excitation force and the excitation moment are computed by performing a least-square optimization: $S = \sum_{i = t_{1}}^{t_{end}} {(η_{i} (t) - η_{i}^{*} (t, G, M_{e}, F_{e}))}^{2} \to m i n .!$ (23)

With the error term S describing the difference between the measured signal η_i and the calculated signal $η_{i}^{*}$ using equation (19) within the chosen time windows. These time windows are the same as those used to calculate the instantaneous center of rotation, starting at t₁ and ending at t_end.

The optimization problem described in equation (23) has multiple local minima and is not sufficiently defined to obtain a unique solution. As a result, the resulting shear modulus degradation curve highly depends on the initial values selected for the shear modulus, the excitation force, and the excitation moment. Therefore, these initial values, as well as the boundary conditions for the optimization, must be chosen carefully. The chosen initial values and boundary conditions for the unknown parameters G, M_e and F_e are listed in Table 1. The initial shear modulus was estimated based on the calculation results from track section A (see Figure 4). The initial excitation force was estimated to be 20 % of the imbalance excitation of the DTS at 33 Hz excitation, which is F_imb = meω² = 123 kN. No reasonable estimation method could be identified for the excitation moment. Therefore, it was estimated at a relatively low level of 10 kN m, taking into account that the rotational degree of freedom should be suppressed by the vertical loading of the hydraulic cylinders. These initial values were used only for the first time window analyzed. For the optimization of the subsequent time windows, the results of the previous time windows were set as initial values. The lower bound was primarily used to avoid negative values for the parameter and the values were therefore chosen close to 0. The upper bound for G was set at reasonable limits from a soil mechanical perspective. The upper bound of the excitation force was estimated based on the maximum excitation force caused by the rotating imbalances at the given excitation frequency. The excitation moment could not be reasonably estimated, which is why the upper bound was chosen at a rather high value.

Table 1.

Boundaries and initial values for optimization algorithm used to estimate G, M_e and F_e.

Value	Lower bound	Upper bound	Initial value
G (MPa)	1	300	50
M_e (kN)	0	1000	10
F_e (kN)	10	200	25

The back-calculated shear modulus, excitation force and excitation moment are shown in Figure 14. The shear modulus estimation is highest for the initial conditions and lower after tamping. For the additional test, approximately the same shear modulus is calculated as for the first test after tamping, which means, that no further compaction was achieved with the second pass of the DTS. Figure 14 also shows the mechanical phase angle and the damping ratio of the SDOF model. Both, the mechanical phase angle and the damping ratio are comparatively high in terms of structural dynamics. However, other researchers have reported similar results when modeling the soil-machine interaction during compaction processes as SDOF systems. For example Nagy,⁴⁶ calculated a damping ratio of ζ = 0.5 with a mechanical phase angle of φ_m = 120° at the observed resonance frequency of a deep vibro compaction test.

Figure 14.

Top left: back calculated shear modulus for the SDOF model with respect to DTS distance. Bottom left: mechanical phase angle for the SDOF model. Top right: back calculated excitation force and moment for the SDOF model with respect to DTS distance Bottom right: Damping ratio for the SDOF model.

Conclusion

The objective of the present research was to estimate ballast stiffness using the DTS. Based on measurements on a modern track section (section A) and an old track section (section B), two different approaches for determining ballast shear modulus degradation curves were described. The following conclusions are drawn from the evaluation of these approaches:

(1) The presented measurement setup of the modern track is suitable to determine the shear strains and the vertical stresses of the ballast layer during dynamic track stabilization. It was shown, that the measured shear strains and vertical stresses can be used to calculate reasonable (normalized) shear modulus degradation curves. The curves obtained are consistent with results of other researchers and could be used for material characterization in the future. Long-term monitoring of maintenance work on equipped sections (similar to that described in this paper) could map changes in the normalized shear modulus degradation curve and thus provide information on ballast fouling.

(2) In this paper, a simplified model (rotational SDOF system) was presented to describe the motion of old tracks during dynamic track stabilization. For this SDOF system, the mass moment of inertia, the spring and dashpot coefficients and the excitation moment are initially unknown. It was demonstrated that the mass moment of inertia can be estimated using influence lines based on the measured angular and translational accelerations at the measurement cross-section in combination with convolution integrals. The spring and dashpot coefficients of the SDOF model were linked to soil parameters using simple and well-established physical models originally developed for vibrating foundations. The geometry of such an equivalent machine foundation can be calculated by another influence line obtained from displacements (numerically integrated accelerations) and another convolution integral. It was not possible to reasonably estimate the excitation moment in advance.

(3) The developed SDOF system could serve as the basis for a possible system for Intelligent Compaction (IC) for the Dynamic Track Stabilizer (DTS). It was shown that by fitting the steady-state solution of the SDOF system with the measured signal for short time windows (defined by the time required for 10 imbalance rotations), shear modulus degradation curves can be estimated using an optimization algorithm. The calculated shear modulus degradation curves showed higher values for the first test than for the following tests after tamping. These results confirm the expected loosening due to the tamping process.

However, the proposed method is not yet suitable for an IC system at the current state of development. It is limited to older and rather stiff tracks where an instantaneous center of rotation exists. Furthermore, a reduction of the number of unknown input parameters is essential to obtain a unique solution. Future research should therefore aim to design and use a dynamically decoupled DTS, to allow for the calculation of the rail reaction forces (in lateral and vertical direction) as an input parameter of the model. Consequently, only the shear modulus would remain unknown and an analytical solution for the proposed system would be possible – at least for the time window where the DTS is located right above the monitored track section. The shear modulus calculated for this time window may be utilized as a constraint for identifying a suitable shear modulus degradation curve. Overall, further research with the DTS seems promising and could lead to measurement systems that provide unique as well as continuous information on ballast behavior, thus contributing to the optimization of track maintenance (i.e., maintenance cycles, selection of required maintenance measures, etc.).

Footnotes

Acknowledgements

The authors acknowledge Plasser & Theurer for funding the proposed research and for providing test machines and personnel as well as for organizing test tracks. Furthermore,the authors acknowledge the contribution of the former Master’s student Franziskus Paech.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

ORCID iDs

Manuel Dafert

Johannes Pistrol

References

Feurig

. Experimentelle und theoretische Untersuchungen zur Optimierung des Dynamischen Gleisstabilisators (DGS) im Hinblick auf eine Verbesserung der Gleislagestabilität [dissertation]. München: TU München, 2020.

Selig

Waters

. Track Geotechnology and substructure Management. Repr ed. London: Telford, 1994.

Khatibi

Esmaeili

Mohammadzadeh

. DEM analysis of railway track lateral resistance. Soils Found. 2017; 57(4): 587–602.

Mohammadzadeh

Esmaeili

Khatibi

. A new field investigation on the lateral and longitudinal resistance of ballasted track. Proc Inst Mech Eng F: Rail Rapid Transit. 2018; 232(8): 2138–2148.

UIC(ORE) . Frage D 117: Optimale Anpassung des klassischen Oberbaus an den künftigen Verkehr – Bericht Nr. 20: Die Effektivität dynamischer Gleisstabilisierung unter Verwendung des DTS, 1981. Technical report, UIC, Utrecht, NL.

Hyslip

Sussmann

, et al. Railway Geotechnics. Boca Raton: CRC Press, 2015.

Tong

Liu

Yousefian

, et al. Track vertical stiffness –Value, measurement methods, effective parameters and challenges: a review. Transportation Geotechnics 2022; 37: 100833.

Berggren

Kaynia

Dehlbom

. Identification of substructure properties of railway tracks by dynamic stiffness measurements and simulations. J Sound Vib. 2010; 329(19): 3999–4016.

Berggren

. Railway track stiffness - dynamic measurements and evaluation for efficient maintenance. Stockholm: Royal Institute of Technology (KTH), 2009.

10.

Wang

Liu

Jing

, et al. State-of-the-Art review of ground penetrating radar (GPR) applications for railway ballast Inspection. Sensors 2022; 22(7): 2450.

11.

Schuk

Rapp

Martin

, et al. Determining the track condition using soil properties - part 1. IOP Conf Ser Mater Sci Eng 2019; 615(1): 012054.

12.

Schuk

Rapp

Martin

, et al. Determining the track condition using soil properties - part 2. IOP Conf Ser Mater Sci Eng. 2019; 615(1): 012055.

13.

Lamas-Lopez

Cui

Costa D׳Aguiar

, et al. Geotechnical auscultation of a French conventional railway track-bed for maintenance purposes. Soils Found. 2016; 56(2): 240–250.

14.

Guldenfels

. Die Alterung von Bahnschotter aus bodenmechanischer Sicht. Zürich: VDF Hochschulverlag, 1996.

15.

Indraratna

Ionescu

Salim , et al. Stress-strain and degradation behaviour of railway ballast under static and dynamic loading, based on large-scale triaxial testing. In Proceedings of the 15th International Conference on Soil Mechanics and Geotechnical Engineering. Istanbul, vol 3, pp. 2093–2096.

16.

Indraratna

Lackenby

Christie

. Effect of confining pressure on the degradation of ballast under cyclic loading. Geotechnique 2005; 55: 325–328.

17.

Sun

Indraratna

Ngo

. Effect of increase in load and frequency on the resiliency of railway ballast. Geotechnique 2018; 69: 1–26.

18.

Indraratna

Ngo

Rujikiatkamjorn

. Performance of ballast influenced by deformation and degradation: laboratory testing and numerical modeling. Int J GeoMech. 2020; 20(1): 04019138.

19.

Indraratna

Armaghani

Gomes Correia

, et al. Prediction of resilient modulus of ballast under cyclic loading using machine learning techniques. Transportation Geotechnics 2023; 38: 100895.

20.

Qian

Tutumluer

Hashash

, et al. Triaxial testing of new and degraded ballast under dry and wet conditions. Transportation Geotechnics 2022; 34: 100744.

21.

Altuhafi

Coop

. The small-strain stiffness of a railway ballast. Géotechnique. 2023; 3: 1–12.

22.

Barbir

. Development of condition-based tamping process in railway enginering [dissertation]. Vienna: TU Wien, 2022.

23.

Kopf

Stern

. Tragfähigkeitsmessungen am Unterbau der Eisenbahn. Eisenbahntechnische Rundschau (ETR). 2022; 71(6): 83–87.

24.

Wichtmann

Triantafyllidis

. Influence of the grain-size distribution curve of Quartz sand on the small strain shear modulus G_max. J Geotech Geoenviron Eng. 2009; 135(10): 1404–1418.

25.

Hardin

Black

. Sand stiffness under various triaxial stresses. J Soil Mech Found Div. 1966; 92(2): 27–42.

26.

Hardin

Kalinski

. Estimating the shear modulus of gravelly soils. J Geotech Geoenviron Eng. 2005; 131(7): 867–875.

27.

Hardin

Drnevich

. Shear modulus and damping in soils: design equations and curves. J Soil Mech Found Div. 1972; 98(7): 667–692.

28.

Wichtmann

Triantafyllidis

. Effect of uniformity coefficient on G/G_max and damping ratio of uniform to well-graded Quartz sands. J Geotech Geoenviron Eng. 2013; 139(1): 59–72.

29.

Offenbacher

Landgraf

. BETamp project team meeting. Technical report, 2021.

30.

Thom

Brown

. The effect of grading and density on the mechanical properties of a crushed dolomitic limestone. In Proceedings of the 14th Australian Road Research Board Conference. Canberra, vol 14, pp. 94–100.

31.

Indraratna

Salim

Rujikiatkamjorm

. Advanced rail Geotechnology - ballasted track. London: CRC Press, 2011.

32.

Esmaeili

Pourrashnoo

. Experimental investigation of shear strength parameters of ballast encased with geogrid. Construct Build Mater. 2022; 335: 127491.

33.

Estaire

Santana

. Large direct shear tests performed with fresh ballast. In: Stark

(eds). Railroad Ballast Testing and Properties, ASTM STP1605. West Conshohocken, PA: ASTM International, pp. 134–151.

34.

Esveld

. Modern railway track. 2. ed. Zaltbommel: MRT-Productions, 2001.

35.

Liu

Lei

Rose

, et al. Pressure measurements at the Tie-ballast interface in railroad tracks using granular material pressure cells. In Proceedings of the Joint Rail Conference 2017 – JRC. Philadelphia, PA, 2017.

36.

Klugar

. Der statische Querverschiebewiderstand des belasteten Gleises. Eisenbahntechnische Rundschau (ETR). 1976; 25(4): 211–216.

37.

Plasser und Theurer . Die Technologie des dynamischen Gleisstabilisierens. Allgemeine Einführung - Funktionsprinzip -Arbeitsanleitung V 1.0 2010.

38.

Hofmann

. Numerische Integration von Beschleunigungssignalen. IMW – Institutsmitteilungen 2013; 38: 103–114.

39.

Das

Ramana

. Principles of soil dynamics. 2nd ed. Stamford, CT: Cengage Learning, 2011.

40.

Winter

. Continuous compaction control in the UK: history, current state and future prognosis. Proceedings of the Institution of Civil Engineers - Geotechnical Engineering 2020; 173(4): 348–358.

41.

Pistrol

Adam

. Fundamentals of roller integrated compaction control for oscillatory rollers and comparison with conventional testing methods. Transportation Geotechnics 2018; 17: 75–84.

42.

Dafert

. Erkenntnisse aus Messungen am dynamischen Gleisstabilisator. Vienna: TU Wien, 2019.

43.

Ahlbeck

Meacham

Prause

. The development of analytical models for railroad track dynamics. In: Kerr

(ed). Railroad Track Mechanics and Technology. Pergamon, 1978, pp. 239–263.

44.

Hall

JRJ

. Coupled rocking and sliding Oscillations of rigid circular footings. In International Symposium on Wave Propagation and Dynamic Properties of Earth Materials. Albuquerque, New Mexico, pp. 139–148.

45.

Wolf

. Foundation vibration analysis using simple physical models. 1st ed. NJ, USA: PTR Prentice Hall, 1994.

46.

Nagy

Pistrol

Kopf

, et al. Integrated compaction control based on the motion behavior of a deep vibrator. Transportation Geotechnics 2021; 28: 100539.

Ballast stiffness estimation based on measurements during dynamic track stabilization

Abstract

Keywords

Introduction

Shear modulus from on site measurements and field samples

Test track in Retz (section A)

Small strain shear modulus from grain size distribution curve

Shear modulus degradation curve

Framework for a continuous stiffness measurement with the DTS

Test track in Oberwart (section B)

Lumped-parameter model

Estimation of initially unknown, dynamic parameters

Back calculation of shear modulus from sleeper motion

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iDs

References