A simplified method for calculating load distribution and rail deflections in track,incorporating the influence of sleeper stiffness

Abstract

When performing railway track stiffness analyses, in current standards sleepers are often regarded as a rigid member. For flexible sleeper materials like timber or polymers, this may lead to underestimating rail deflections up to a factor two and overestimating rail seat loads up to 20%. Calculations incorporating sleeper bending can currently be performed analytically by a two-layer beam-on-elastic-foundation calculation, or by finite element analyses, but a simple approach does not yet exist. This article introduces a simple calculation method to establish rail deflections, track stiffness and rail seat loads, incorporating the effects of both bending and shear stiffness of the sleeper, applicable to new or tamped track. A sleeper flexibility factor f_S is introduced as a deformation multiplication factor compared with a rigid sleeper. Validation against current calculation models shows a deformation accuracy within 6% and a load accuracy within 2%. When a track is not maintained, a gap will develop under the sleeper at the rail seat. The size of this gap correlates to the stiffness of the sleeper. Eventually a uniform load distribution can develop under the sleeper. A calculation method is introduced to estimate the gap and the track behaviour at uniform load distribution. When for flexible sleepers the track deflection due to the gap can develop outside of the desired range, timely track maintenance or monitoring is advised.

Keywords

beam on elastic foundation composite sleeper polymer sleeper railway sleeper track stiffness

Introduction

Track stiffness – defined as the wheel load required to produce a unit displacement of the rail at the location of the load – is an important parameter of railway track and has been investigated extensively (Berggren, 2009; Berggren and Saussine, 2009; Powrie and Le Pen, 2016; Wehbi, 2017). However, the role of railway sleepers in the establishment of track stiffness has mostly been simplified, which can lead to errors.

The wheel load of a train is distributed over a number of sleepers. The extent of this distribution depends on the relation between the bending stiffness of the rail and the stiffness of the rail seat support. The rail seat support stiffness again depends on the bending stiffness of the sleeper and the stiffness of railpad, ballast and subgrade.

In current standards, for example, EN13230 (EN 13230-6:2017, 2017), the sleeper is mostly regarded as a rigid member, simplifying the calculation significantly. For stiff sleeper materials, like concrete, this simplification is justified. For timber or polymer sleepers, this simplification can create serious errors in the calculation of load distribution, rail deflections, and track stiffness. The higher the sleeper flexibility, the wider the wheel load distribution over the sleepers and the lower the sleeper bending moments. Track deflections will however increase and ballast compression will localise, which may increase ballast settlement. For timber sleepers, there is a long history of experience, but with the introduction of polymer sleepers, the occurring sleeper stiffness range expands, giving a need to incorporate sleeper bending in the calculations to determine rail seat loads, rail deflections and track stiffness.

Track stiffness analyses incorporating sleeper bending stiffness can currently be performed by an analytical two-layer beam-on-elastic-foundation calculation, in which both sleeper and rail are regarded as a beam-on-elastic-foundation. This method is rather complex and also neglects shear deformations in the sleeper. Since sleepers are relatively thick and short beams and since timber and many polymer sleeper constructions are anisotropic, shear deformations can play a significant role.

Calculations can currently also be performed by finite element analyses. This is normally beyond the possibilities for a first assessment of sleeper suitability. In addition, verified finite element models are mostly not available for others than its makers, whereas analytical calculation methods, once verified, are available for everyone by publication of the calculation method.

To overcome shortcomings in current available analysis tools, this article investigates three load cases, applicable for newly laid track, tamped track and worn track.

For newly laid or tamped track, a simple one-step calculation method is introduced to establish rail deflections, track stiffness and rail seat loads, incorporating bending and shear stiffness of the sleeper. To validate this calculation method, outcomes are compared with analytical two-layer beam-on-elastic-foundation solutions and finite element analyses.

For worn track, when maintenance is not performed, eventually a uniform load distribution will develop under the sleeper, associated with a gap at the rail seat which closes every time a train passes. The size of this gap at uniform load distribution depends on sleeper stiffness. Formulas are presented to estimate the magnitude of this gap under the sleeper and its relation to track stiffness parameters.

Current calculation models

First, the current calculation methodology is discussed, considering both the rail and the sleeper as deforming bodies.

The Winkler foundation

In the 19th century, Winkler (1867) analysed a beam on an elastic foundation by assuming that the foundation compression is proportional to the pressure exerted on it. Winkler calculated the pressure and bending moment distribution for two loads at a distance apart which later proved also applicable for a single load on a continuous beam on an elastic foundation and as such for rails on longitudinal bearers. Zimmermann (1888) calculated several situations for beams on an elastic foundation, including lateral sleepers, assuming that the sleeper load is a given value. In addition, he attempted to analyse the distribution of forces between rail and subsequent lateral sleepers. The rail is in this case not continuously supported, but resting on discrete supports. Schwedler (1889) analysed the track with the beam-on-elastic-foundation theory and stated that the ballast reaction is only proportional to the applied load after a certain compression. This toe-effect can be correlated to the establishment of full contact between ballast particles (Talbot, 1980). Timoshenko (1925) showed that also for lateral sleepers the assumption of a continuous elastic support under the rail is a good approximation, if there are at least four sleepers in a wave length of the deflection curve, which is normally the case (Hetényi, 1946). In 1921, Hayashi (1921) published a book dedicated to the subject, contributing with an uplift analysis of the beam from the foundation. Hetényi (1946) published solutions for a variety of beam-on-elastic-foundation cases.

The Winkler foundation can be interpreted as a system of independent vertical springs. The model assumes that the deformations appear only in the loaded zone and that outside this zone the deformations are zero. This discontinuous displacement field is the main disadvantage of the Winkler model. Two-parameter soil models such as the Pasternak model restore the continuity of the elastic foundation by introducing a second parameter (Dobromir, 2012). Another shortcoming of the Winkler model is the assumed linearity of the ballast spring support. Ballast in reality behaves as a non-linear spring which influences the track response (Sadeghi, 1997). Furthermore, the wheel loads in the Winkler model are considered quasi-static and a factor is added to compensate for dynamic effects and for the large variances that can exist in longitudinal or transversal support due to differential settlement or differential vertical stiffness of the ballast (Talbot, 1980; Zakeri and Sadeghi, 2007). Despite these limitations, the beam-on-elastic-foundation theory using a Winkler foundation has proven to perform in good agreement with reality provided that the used parameters are determined in a similar setting (Birmann and Rubin, 1969; Essenburg, 1962; Hetényi, 1946; Manalo et al., 2012; Sadeghi, 1997; Sadeghi and Barati, 2010; Shokrieh and Rahmat, 2007; Timoshenko, 1925; Zakeri and Sadeghi, 2007).

The calculation of the rail seat loads and rail deflections according to the Winkler model have the form, respectively, as per equations (1) and (4). The rail seat load on the sleeper can be determined with standard track formula (EN 13230-6:2017, 2017; Esveld, 2001; ISO 12856-1:2019, 2019)

P = \frac{1}{2} QS λ_{R} \sum η_{i}

(1)

In which S is the sleeper centre-to-centre spacing, Q is the dynamic wheel load and λ_R is the characteristic of the rail according to

λ_{R} = \sqrt[4]{\frac{c_{T}}{4 E_{R} I_{R} S}} with \frac{1}{c_{T}} = \frac{1}{c_{P}} + \frac{1}{c_{S}} + \frac{1}{c_{B}}

(2)

The total support stiffness c_T can be calculated from c_P (the stiffness of the railpad), c_S (the stiffness of a baseplate/rail on the sleeper), and c_B (the support stiffness of half of a sleeper in the ballast). E_R and I_R are, respectively, the bending modulus and the moment of inertia of the rail. The compression stiffness of the rail and the indentation of the ballast into the sleeper are neglected, since these are much higher than the other stiffnesses and thus their influence is small. In equation (1), η_i is defined as the influence of the ith wheel at a distance x_i from the sleeper according to

η_{i} = e^{- λ_{R} x_{i}} [\cos (λ_{R} x_{i}) + \sin (λ_{R} x_{i})]

(3)

For a wheel directly above the sleeper, the value η = 1. This case gives the highest rail seat load, but adjacent wheels can create additional loading on the sleeper, which can be calculated with equation (3).

With the calculated rail seat load, the rail deflection y can be calculated (Esveld, 2001; Powrie and Le Pen, 2016) with

y = \frac{P}{c_{T}}

(4)

The stiffer the support, the higher the load on the sleeper directly under the wheel and vice versa. This means that sleeper loads and consequent stresses depend on the stiffness of the system railpad–sleeper-ballast. A sleeper with a lower bending or compression stiffness will experience lower loads.

Two layer beam-on-elastic-foundation

Standards like EN13230 (EN 13230-6:2017, 2017) consider the rail as a beam-on-elastic-foundation, but simplify the load distribution under the sleeper as being uniform (Birmann and Rubin, 1969; EN 13230-6:2017, 2017; Rahrovani, 2016; Rutten, 1915). Numerous alternative load distribution patterns under the sleeper exist, but mostly as a worst case in determining ultimate sleeper bending moments, not for deformation analyses. The assumption of a uniform load distribution under the sleeper is adequate for a stiff sleeper, such as a concrete sleeper. For a more flexible sleeper, such as a timber or a polymer sleeper, also the sleeper itself should be considered as a beam-on-elastic-foundation, otherwise rail seat loads will be overestimated and rail deflections will be underestimated (Rahrovani, 2016; Sadeghi, n.d.; Sadeghi and Youldashkhan, 2005; Winkler, 1875). The load distribution under the sleeper is then not uniform but parabolic as shown in Figure 1.

Figure 1.

Stress distribution in track.

Incorporation of the sleeper stiffness in the calculation creates a two-layer beam-on-elastic-foundation analysis (One could even say a three layer analysis, since the baseplate could be considered as a beam-on-elastic-foundation, where the foundation is the compression of the sleeper. In this article, this third layer is not considered and is assumed to be covered by a measured value of sleeper compression stiffness).

The complex part of this two-layer analysis is that the foundation stiffness c_B in equation (2) depends not only on ballast stiffness and the sleeper-ballast contact area but also on sleeper bending stiffness. Since a simple formula for C_B is now more difficult to obtain, a multiple step analysis can be employed:

Assume an arbitrary rail seat load on the sleeper;

Calculate the rail seat displacement;

Calculate the support stiffness c_B of the sleeper;

Calculate the total support stiffness c_T and rail seat load P with equation (1);

Calculate the actual rail deflection y with equation (4).

As an alternative approach, in this article a simple one-step approach is introduced to calculate the flexural behaviour and force distribution over the rail, incorporating the sleeper bending stiffness, by deriving a simple formula for the sleeper-in-ballast support stiffness c_B.

Support conditions

Three support conditions are considered for this track analysis. For a new ballast bed, the sleeper can be considered as supported by a continuous elastic foundation as shown in Figure 2(a). This support condition leads to a parabolic load distribution under the sleeper with a maximum load under the rail seat area. The shape of the parabolic load distribution depends on the relation between the stiffness of the ballast and the stiffness of the sleeper. For a very stiff sleeper, the load distribution will tend to an approximate uniform load distribution as is shown in load case C.

Figure 2.

Considered sleeper support conditions: (a) new track, (b) tamped track, (c) worn track. The left-hand figures show the spring support situations. These can be translated into the load distributions which are shown in the right-hand figures.

When performing maintenance in track, by tamping the ballast, only the area around the rail seat is tamped, creating a more local support, as can be seen in Figure 2(b) (Sadeghi, n.d.). The situation in Figure 2(a) can be interpreted as a specific case of the situation in Figure 2(b), that is, the situation where L₂ = 0 and L₃ = ¹/₂ L. By solving the generic load case B, also specific load case A will be solved.

Load case C is applicable for worn track and will be further discussed at the end of this article. This load case is characterised by a uniform load distribution under the sleeper, which is caused by a gap between the sleeper and the ballast at the rail seat.

Current calculation new or tamped track

First, load case B is calculated using existing calculation methods (according to Hetényi). This calculation is performed to validate the proposed simplification, and to show what the current method encompasses to determine the sleeper-ballast support stiffness c_B.

The sleeper can be divided into five parts according to Figure 3. When looking at beam-part 4, this beam-part is loaded with rail seat load P on a distance L₄ from the left side, as well as a bending moment M at the left end, the latter due to its connection with beam-part 3. To calculate the displacement at the position where P is acting, first bending moment M has to be calculated, which can be performed by considering that the rotation of the left end of beam-part 4 equals the rotation of the right end of beam-part 3.

Figure 3.

Analysis of partly supported beam-on-elastic-foundation.

Hetényi (1946) gives us the following equations for the rotation (θ) at the left end of beam-part 4 caused by, respectively, the bending moment M and the rail seat load P (rotation clockwise is positive)

θ_{M} = \frac{4 M λ^{3} a}{Cw} with a = \frac{\sinh λ L_{3} \cosh λ L_{3} + \sin λ L_{3} \cos λ L_{3}}{\sin h^{2} λ L_{3} - si n^{2} λ L_{3}}

(5)

\begin{matrix} θ_{P} = \frac{2 P λ^{2} b}{Cw} with b = \frac{\sinh λ L_{3} (\sin λ L_{4} \cosh λ L_{5} - \cos λ L_{4} \sinh λ L_{5}) + \sin λ L_{3} (\sinh λ L_{4} \cos λ L_{5} - \cosh λ L_{4} \sin λ L_{5})}{\sin h^{2} λ L_{3} - si n^{2} λ L_{3}} \end{matrix}

(6)

In above equations, λ is called the characteristic of the sleeper (Hetényi, 1946) and is defined by

λ = \sqrt[4]{\frac{Cw}{4 EI}}

(7)

In which C is the bedding modulus of the ballast and subgrade, w is the width of the sleeper at its bottom side and E and I are, respectively, the bending modulus and the moment of inertia of the sleeper (Throughout this article, symbols without subscript refer to the sleeper). Bending moment M is equal to the bending moment around beam-part 3. For beam-part 3, standard beam formulas give (Gere and Timoshenko, 1997)

θ_{L_{2}} = - \frac{M L_{2}}{2 EI}

(8)

Since the rotation of the right end of beam-part 3 has to equal the left end of beam-part 4, equations (5), (6), and (8) can be combined and reworked to

M = \frac{- 4 P λ^{2} bEI}{Cw L_{2} + 8 λ^{3} aEI}

(9)

Denoting downwards deflections as positive, the ballast compressions due to, respectively, M and P at the rail seat are (Hetényi, 1946)

δ_{M} = \frac{2 M λ^{2} b}{Cw} with b as defined in equation 6

(10)

\begin{matrix} δ_{P} = \frac{P λ c}{Cw} with c = \frac{(\cos h^{2} λ L_{4} + \cos^{2} λ L_{4}) (\sinh λ L_{5} \cosh λ L_{5} - \sin λ L_{5} \cos λ L_{5})}{\sin h^{2} λ L_{3} - \sin^{2} λ L_{3}} + \\ \frac{(\cosh^{2} λ L_{5} + \cos^{2} λ L_{5}) (\sinh λ L_{4} \cosh λ L_{4} - \sin λ L_{4} \cos λ L_{4})}{\sinh^{2} λ L_{3} - \sin^{2} λ L_{3}} \end{matrix}

(11)

Combining equations (9), (10), and (11) gives the total ballast compression at the rail seat

δ_{R} = \frac{P λ}{Cw} [c - b^{2} {(\frac{Cw L_{2}}{8 λ^{3} E I} + a)}^{- 1}]

(12)

The support stiffness of half of a sleeper in the ballast (c_B) is then defined as the rail seat load P divided by the ballast compression at the rail seat δ_R according to

c_{B} = \frac{P}{δ_{R}}

(13)

In combination with equations (1) and (4), the rail seat loads and rail deflections can be calculated. In order to easily analyse track displacements, there is a need to simplify this calculation. Also the current method neglects shear deformations in the sleeper which decreases accuracy.

Remark: For the displacements in vertical direction at the rail seat a difference is made between the

Ballast compression: the displacement at the bottom side of the sleeper, determined by sleeper bending and ballast compression

Rail seat displacement: the displacement at the top side of the sleeper, additionally incorporating sleeper compression.

Rail deflection: the displacement at the rail level, incorporating the rail seat displacement and the railpad compression.

Simplified calculation model

A simplification is proposed, based on the common geometric proportions of a sleeper. To enable this simplification, the dimensionless variable λL₃ is introduced. This variable simplifies the analysis of the sleeper deformations.

λL₃

In the current calculations, the length L₃ (see Figure 2) is the governing length for the sleeper performance. Combining L₃ with the sleeper characteristic λ according to equation (7) gives a dimensionless variable λL₃ representing the relative sleeper flexibility. A long and flexible sleeper on a stiff ballast gives an upper value for λL₃ of approximately 6; see Table 1 for the considered parameters. The lowest considered value for λL₃ is 0.5 for a short and stiff concrete sleeper on a flexible ballast. So relative sleeper flexibility λL₃ can vary from 0.5 to 6 within the considered values according to Table 1. Also for narrow gauge or broad gauge track, this λL₃ range can be calculated to cover the expected range of sleepers.

Table 1.

Considered ultimate values and consequent λL₃ variable and flexibility factor (f_S) for the most stiff sleeper/flexible ballast combinations and for the most flexible sleeper/stiff ballast combinations for different sleeper materials.

		Sleeper dimensions(w × h) (mm)	Sleeper bending modulus E (GPa)	Ballast bedding modulus C (N/mm³)	Sleeper supported length L₃ (mm)	λL₃ (–)	f_S (–)
Concrete	Stiff	300 × 250	38	0.04	700	0.5	1
	Flexible	250 × 200	38	0.25	1500	1.9	1.2
Timber	Stiff	275 × 160	20	0.04	700	0.8	1
	Flexible	250 × 150	6	0.25	1500	3.7	1.9
Polymer	Stiff	275 × 160	20	0.04	700	0.8	1
	Flexible	250 × 150	1	0.25	1500	5.8	2.9

A graph of the ballast compression at the rail seat as a function of the relative stiffness of the sleeper λL₃ can be found in Figure 4. In this graph, the separate displacements caused by the bending moment M and the force P according to equations (10) and (11) are given, as well as the combined total displacement. On the vertical axis, the relative ballast compression at the rail seat δ_RR has been plotted, which is defined as

δ_{RR} = \frac{δ_{R} Cw L_{3}}{P}

(14)

In which δ_R is the ballast compression at the rail seat. The physical meaning of the relative ballast compression at the rail seat δ_RR is that the relative ballast compression is set to unity for a rigid sleeper, since in that case δ_R = P/(CwL₃). A relative ballast compression value of one – the compression a rigid sleeper would have – is therefore the minimum possible relative ballast compression. The interesting effect of using these dimensionless variables on both axes is that for every stiffness property of sleeper or ballast, this graph is identical.

Figure 4.

Relative ballast compression at point of load application as a function of the relative stiffness of the sleeper, caused by bending moment M and force P, as well as the combination (load case A, sleeper length 2600 mm, standard gauge).

The graph shows that for stiff sleepers (λL₃ < 1), the relative ballast compression tends to go to the value one. For flexible sleepers (λL₃ > 3), the relative ballast compression tends to ½λL₃. The stiffness range that is to be expected for sleepers (λL₃ between 0.5 and 6) covers both of these areas and the transition zone in between.

For the expected sleeper stiffness range, Figure 4 also shows that the displacement can become up to three times the minimal displacement. That is the influence a sleeper can have on the system deformation. For concrete sleepers, λL₃ values are usually below two, meaning that the sleeper flexibility can be neglected. For higher λL₃ values (more flexible sleepers), the influence becomes significant.

Introduction of sleeper flexibility factor f_S

An approximation of the ballast compression at the rail seat can be derived from Figure 4 by curve-fitting the transition zone with help of the formula $\sqrt[n]{1 + x^{n}}$ and can be defined as

δ_{R} = \frac{P}{Cw L_{3}} \sqrt[4]{1 + {(\frac{1}{2} λ L_{3})}^{4}}

(15)

In which the part before the 4th root is the ballast compression that occurs for a rigid sleeper, so this compression is only due to the ballast. The 4th root part is the additional compression that occurs due to the bending of the sleeper. Denoting this root as the sleeper flexibility factor f_S, equation (15) can be written as

δ_{R} = \frac{P f_{S}}{Cw L_{3}}

(16)

With

f_{S} = \sqrt[4]{1 + {(\frac{1}{2} λ L_{3})}^{4}} or : f_{S} = \sqrt[4]{1 + \frac{Cw}{64 EI} {L_{3}}^{4}}

(17)

The last derivation with help of equation (7). The sleeper flexibility factor f_S equals by approximation the relative ballast compression at the rail seat (δ_RR) as shown on the y-axis of Figure 4.

The support stiffness of the sleeper in the ballast (c_B) can then be calculated with help of equation (16) to

c_{B} = \frac{Cw L_{3}}{f_{S}}

(18)

The load that is acting on one sleeper and the rail deflections can now be calculated with this support stiffness and equations (1) and (4).

The implications of this flexibility factor f_S are interesting. It represents the contribution of the sleeper bending stiffness to the ballast compression at the rail seat. When f_S equals one, the sleeper is infinitely stiff (A factor f_S lower than one is not possible). It therefore gives the limitations of what can be achieved with sleeper stiffness. If the bedding modulus of the ballast is low, it is not possible to correct that with the sleeper stiffness if f_S is already almost one. A solution then has to be found in the bedding modulus of the ballast, not in the sleeper stiffness. An f_S-value of two means that the ballast compression at the rail seat is doubled compared with a rigid sleeper. Typical values of f_S can be found in Table 1 for the chosen range of λL₃ values. Since the polymer sleeper flexibility range is wide, it is important to understand the consequence on track behaviour. However, larger deformations also lead to lower rail seat loads and a better load distribution over the sleepers. The whole track calculation therefore needs to be made to understand all implications.

Including shear deformation

The beams in current calculation models and also in the proposed simplification are all considered Euler–Bernoulli beams: deformations due to rotational bending are considered, but deformations due to shear are not. For the rails and for other beams with a high span-to-height ratio – 16 is often chosen as the minimum value for a neglectable influence – this simplification is valid, but sleepers are relative thick and sleeper materials can be anisotropic. Wood is anisotropic and many polymer sleeper compositions are anisotropic. Shear deformations can be more pronounced due to this anisotropy and using a Timoshenko beam for the sleeper, incorporating shear deformations, improves the calculation accuracy (Timoshenko, 1921, 1925). The current analytical beam-on-elastic-foundation theories do not take shear deformation into consideration and therefore have limited applicability to sleepers.

Essenburg (1962) analysed Timoshenko beams on an elastic foundation, from which the displacement at the position of loading for an endless beam can be derived according to Table 2, as compared with an Euler–Bernoulli beam. AGκ in this equation is the transverse shear stiffness of the sleeper with shear modulus G, cross section A and shear coefficient κ. For a rectangular beam, κ = 5/6 and for an I-beam, κ ≈ web area /total cross-sectional area (Cowper, 1966).

Table 2.

Displacements of a beam-on-elastic-foundation directly under a load for an endless Euler–Bernoulli beam (Winkler, 1875) and a Timoshenko beam (Essenburg, 1962).

Euler–Bernoullibeam	Timoshenko beam
$δ = \frac{P λ}{2 Cw}$	$δ = \frac{P λ}{2 Cw} (2 β - \frac{1}{β}), with β = \sqrt{1 + \frac{\sqrt{EICw}}{2 AG κ}}$

The effects of shear deformations can be incorporated in proposed simplifications by substituting the sleeper bending modulus E by an apparent bending sleeper modulus E_A in equation (17). E_A can be calculated from the real modulus E– which applies for long spans – and the shear modulus G according to

E_{A} = \frac{E}{{(2 β - \frac{1}{β})}^{4}}

(19)

With β according to Table 2. E and G can be determined by bending tests at two different spans or can be taken from literature. As an alternative, the apparent modulus E_A of the sleeper can be determined directly by a short span bending test with a span that is appropriate for the beam-on-elastic foundation situation. This will however always be an estimate, since the shear and bending stresses in a three- or four-point bending test will never match the beam-on-elastic-foundation situation.

Discussion of simplified model

Validation of the simplified calculation model (excluding shear deformations) is performed by comparing with the analytic Hetényi calculations as previously presented. Validation of the simplified model including shear deformations is performed by comparing with finite element analyses. Furthermore, geometrical limitations of the simplified model are discussed as well as the necessity of incorporating sleeper bending in track deformation analysis.

Model validation excluding shear deformations

Figure 5 shows the comparison of the simplified equations with the Hetényi solution for the situation excluding shear deformations. The accuracy of the simplification is within 6% of Hetényi’s equations. Derived sleeper bending moments and rail seat loads on basis of these displacements are within 2% accurate compared with the full two-layer beam-on-elastic-foundation calculations for both fully or partly supported sleepers.

Figure 5.

Relative ballast compression as a function of relative stiffness. Left: load case A, Right: load case B.

The factor L₄/L₃ in Figure 5 is a measure for rail position on the sleeper according to Figure 2. The range 0.5–0.65 represents the most common sleeper lengths. A fully supported short sleeper with a length of 2300 mm in standard track has a L₄/L₃ value of around 0.65 and is the considered the upper boundary value. For a sleeper of 3000 mm length, the L₄/L₃ value is around 0.5 for a fully supported sleeper and represents the lower boundary. For the situation after tamping (load case B), the load will be more or less in the centre of the beam part L₃, so L₄/L₃≈ 0.5. Also for narrow gauge or broad gauge track the mentioned L₄/L₃ range can be calculated to cover the expected range of sleepers.

Hetényi notices that 0.55 is the ideal value for a fully supported sleeper, where bending of the sleeper does not lead to any rotation at the position where the rail is attached, so also not to any lateral movement of the rail head (Hetényi, 1946).

Model validation including shear deformations

When including shear deformations into the simplified calculation, it can no longer be compared to the Hetényi solution, since shear deformations are not incorporated in the Hetényi model. Validation is therefore performed with finite element analyses (see Figure 6).

Figure 6.

FEA model (top) and one of the results (bottom).

Around 100 calculations have been performed using a Nastran solver and a Femap v.11.0 post- and preprocessor. 3D orthotropic solid element have been used to incorporate all E-modulus and G-modulus ranges as defined for a 250 × 150 × 2600 mm sleeper in a standard gauge track. The sleeper consists of 27,840 solid elements with an average size of 15 mm. Under the sleeper CBUSH Spring/damper elements have been used without damping and with a vertical stiffness of 0.1 N/mm per mm² surface. The rail seat load is set at 60 kN.

Performing the calculations shows that the calculated deformations of the simplification incorporating shear deformations vary no more than 6% from the finite element analyses for any G/E value as per Figure 7.

Figure 7.

Comparison of ballast compression for a fully supported sleeper (left) and a partly supported sleeper with L₂ = 400 (right) by the simplified calculation including shear effects (SIM) with finite element analyses (FEA) for different shear modulus (G) and bending modulus (E) values for a 250 × 150 × 2600 mm sleeper in standard gauge track.

Limitations

As mentioned, L₄/L₃ values will normally lie in the range of 0.5–0.65. Figure 8 gives the relative ballast compression at the rail seat (δ_RR) as a function of L₄/L₃ and shows that for L₄/L₃-values between 0.2 and 0.8, the relative displacement is more or less independent from L₄/L₃. The simplified calculation does not incorporate the L₄/L₃ relation and is therefore only applicable for the horizontal area in Figure 8. It can be calculated that L₄/L₃ values have to lie between 0.4 and 0.65 to stay within mentioned margin of 6% for the simplified calculation method. This means that the simplified calculation model can be used as long as the sleeper length (L) is between 1.5 and 2.5 times the centre-to-centre distance of the rails (L₁). For regular sleeper, this is practically always the case. For standard gauge track this would include sleepers ranging roughly from 2300 to 3800 mm length and for narrow gauge sleepers from 1750 to 2850 mm length.

Figure 8.

Relative ballast compression at the rail seat as a function of the relative position of the rail seat loads.

Necessity of incorporating sleeper bending

When calculating rail seat loads and rail deflections with and without incorporating sleeper bending, it shows the necessity of incorporating sleeper bending (see Table 3). For each sleeper material, the most flexible sleeper is taken, in the most rigid surrounding (assumed sleeper and ballast properties according to Table 1), giving a worst-case situation. Table 3 shows that, for concrete sleepers, the effect of sleeper bending is limited. The worst-case situation can be found for the polymer sleepers, where rail seat loads can reduce over 20% by including sleeper deformations (P reduces to 78% of the value excluding sleeper bending) and rail deflections can double by including sleeper deformations (y increases to 214% of the value excluding sleeper bending).

Table 3.

Influence of sleeper flexibility on the rail seat load and deflection for a 54E1 rail, 600 mm sleeper spacing and sleeper/ballast parameters as per Table 1. The left material columns exclude sleeper bending and compression; in the right material columns, it is included. The right two columns show the influence of railpad stiffness for a concrete sleeper.

		Influence of including sleeper flexibility						Influence of railpad stiffness
		Concrete		Timber		Polymer		Concrete
Variable	Unit	Excl.	Incl.	Excl.	Incl.	Excl.	Incl.	Stiff railpad	Flexible railpad
c_P	kN/mm	200	200	500	500	500	500	1000	80
c_S	kN/mm	10000	10000	250	250	750	750	10000	10000
f_S	–	1	1.2	1	1.9	1	2.9	1	1
c_B	kN/mm	25	20.8	25	13.2	25	8.6	25	25
c_T	kN/mm	22.2	18.8	21.7	12.2	23.1	8.4	24.3	19.0
P/Q	–	0.351	0.337	0.350	0.303	0.355	0.275	0.360	0.338
P / P excl.	%	= 100%	96%	= 100%	87%	= 100%	78%	= 100%	94%
y / y excl.	%	= 100%	113%	= 100%	154%	= 100%	214%	= 100%	120%

The support stiffness of the baseplate/rail on the sleeper c_S is a function of the sleeper compression stiffness and can be determined by dynamic testing. For concrete sleepers, this value is so high that it is normally neglected in the calculation (5–10 MN/mm). The concrete sleeper compression rigidity has to be compensated in the flexibility of the railpad. Timber sleepers can be calculated to have a c_S value in the range of 75–250 kN/mm and polymer sleepers in the range of 75–750 kN/mm. Bold figures indicate the chosen variable range, other values result from this choice.

Parameters such as the railpad stiffness are kept constant in this analysis. Changes in railpad stiffness values have an influence in the track stiffness, but their influence is limited (Kaewunruen and Reminnikov, 2008; Oregui, 2015). When, for example, ranging the railpad stiffness between 80 and 1000 kN/mm, Table 3 shows that for a concrete sleeper the effects on rail seat load and deflection are comparable in magnitude to the errors when neglecting sleeper bending. For timber and polymer sleepers, the error in neglecting sleeper bending is larger.

Worn track

When a track is in use, due to the loads acting on the ballast, the ballast will settle. At the rail seat, where the ballast stresses are the highest, settlement will be the highest. This results in movement and/or compaction of the ballast with levelling of the ballast stresses as a result, until the support condition of Figure 2, load case C is reached (Sadeghi, n.d., 2010; Schwedler, 1889). In this situation, the loads on the ballast are uniform over the sleeper length and thus further ballast settlement will occur evenly. It is noted by Schwedler (1889) and Sadeghi and Youldashkhan (2005) that this load situation is accompanied by a gap under the sleeper at the rail seat area, which closes every time a train passes. The ultimate support condition according to load case C disregards other influences on the ballast and only considers static influences. When the sleepers are dynamically loaded, the difference of dynamic amplification in the rail seat area versus the centre area of the sleeper may lead to further settlement, creating support conditions where the centre loads are even higher than the rail seat loads, as indicated for example in the ORE reports (Birmann and Rubin, 1969). Occurrence of this situation depends on the reaction of the sleeper to dynamic loading and on the bending stiffness of the sleeper (Zakeri and Sadeghi, 2007).

For worn track, load case C should be analysed, which is a straightforward calculation. However, the applicability of load case C is depending on sleeper stiffness, which needs further consideration.

Figure 9 shows the schematic representation of displacement lines for a sleeper. The green line δ₀ in Figure 9 shows the displacement line for a newly laid sleeper on a continuous ballast support. The associated vertical ballast stresses in this initial situation are shown as q_C,0 and q_R,0 for the centre of the sleeper and the rail seat area, respectively. The triangles depict the linear relation between the ballast compression and the ballast stress.

Figure 9.

Schematic displacement situation for worn track, with a gap under the rail seat.

The displacement situation for worn track is shown by the blue line δ₁. If in this situation a uniform load distribution under the sleeper exists, the ballast stresses q_C,1 and q_R,1 have become equal and have the value 2P/L. This support condition is associated with a gap under the sleeper, as depicted by the red line ‘gap’ in Figure 9. When excluding differential settlements in longitudinal track direction – such as hanging sleepers – the sleeper centre will stay supported.

For the uniform load distribution of load case C, the difference of deflection between the centre of the sleeper and the rail seat (δ_C₁-δ_R₁) is equal to the gap (g) and can be calculated for a Timoshenko beam with standard deflection formulas (Gere and Timoshenko, 1997; Young, 1989)

g = \frac{P α}{L} with α = {L_{1}}^{2} (\frac{12 L L_{1} - 6 L^{2} - {L_{1}}^{2}}{192 EI} + \frac{1}{4 AG κ})

(20)

The right hand fracture of equation (20) represents the shear deformation of the sleeper. Again, instead of using the real bending modulus E in combination with the shear modulus G, the apparent bending modulus E_A of the sleeper can be estimated by a short span bending test and can be used instead of E, in which case the right hand fracture of equation (20) can be omitted.

The ballast compression δ_R₁ at the rail seat for a rail seat load P becomes

δ_{R 1} = \frac{2 P}{CwL} + g

(21)

This leads to a support stiffness of the sleeper in the ballast (c_B) of

c_{B} = \frac{L}{\frac{2}{Cw} + α}

(22)

When there is a maximum acceptable gap under de rail seat defined as a function of the track maintenance policy, it depends on the sleeper bending stiffness whether a uniform load distribution under the sleeper will ever be reached. As an example, when assuming a maximum gap of 5 mm (with P=50 kN, w=250 mm, h = 150 mm, L = 2600 mm, standard gauge), a uniform load distribution is only reached for sleepers with a bending modulus of at least 2.5 GPa (disregarding shear deformation of the sleeper). For flexible sleepers, therefore the uniform load distribution is less likely to occur, since due to maintenance the gap will not be allowed to grow until uniform load distribution. For rigid sleepers, the uniform load distribution can be reached and even exceeded, due to the higher dynamic forces and specifically the difference between the dynamic forces at the rail seat and in the centre of the sleeper.

Example gap analysis

Powrie and Le Pen (2016) advise a track modulus of 10–50 N/mm². The track modulus is defined as the relation between the load per metre transferred between rail and sleeper and the displacement at that point (Esveld, 2001; Powrie and Le Pen, 2016). An example analysis can be found in Figure 10, which shows that for load case A and B, the whole sleeper stiffness range as defined (modulus of 1 GPa and up) will give an adequate track modulus when assuming a 0.1 N/mm³ bedding modulus. For load case C (worn track), the track modulus eventually extends outside of the desired range for sleepers with a modulus below 2 GPa if timely maintenance is not performed.

Figure 10.

Track modulus as a function of sleeper bending stiffness for a standard gauge 54E1 rail track, C = 0.1 N/mm³, S = 600 mm, c_S = 300 kN/mm, c_P = 100 kN/mm, sleeper 260 × 160 × 2600 mm.

Conclusion

The assumption of a uniform load distribution under a sleeper is an adequate assumption for a stiff sleeper, such as a concrete sleeper, but for a more flexible sleeper, such as a timber or a polymer sleeper, this assumption can lead to overestimating loads more than 20% and underestimating rail deflections by a factor two. The sleeper itself should therefore, next to the rail, be considered as a beam-on-elastic-foundation, giving a parabolic load distribution under the sleeper, for which a simplified calculation method is suggested.

For sleeper lengths between 1.5 and 2.5 times the centre-to-centre distance of the rails, ballast compression at the rail seat and consequent rail deflections can be calculated by using a sleeper flexibility factor f_S, as defined in Table 4. The factor f_S determines the rail seat ballast compression due to sleeper bending, as multiplication factor to the ballast compression when the sleeper is considered rigid and can in praxis have a value between one and three. Rail deflections calculated with this simplification are within 6% accurate and rail seat loads within 2%, compared with current analytical methods.

Table 4.

Flow of equations, completed with general formula’s (in blue) (EN 13230-6:2017, 2017; Esveld, 2001; ISO 12856-1:2019, 2019).

Load case A/B		Load case C

(L2 = 0 & L3 = ½L for load case A)
$β = \sqrt{1 + \frac{\sqrt{EICw}}{2 AG κ}}$	Or estimate E_A by testing at appropriate span	$α = {L_{1}}^{2} (\frac{12 L L_{1} - 6 L^{2} - {L_{1}}^{2}}{192 EI} + \frac{1}{4 AG κ})$ or estimated by: $α = \frac{{L_{1}}^{2} (12 L L_{1} - 6 L^{2} - {L_{1}}^{2})}{192 E_{A} I}$ E_A by testing at appropriate span
$E_{A} = \frac{E}{{(2 β - \frac{1}{β})}^{4}}$	Or estimate E_A by testing at appropriate span
$f_{S} = \sqrt[4]{1 + \frac{Cw}{64 E_{A} I} {L_{3}}^{4}}$
$c_{B} = \frac{Cw L_{3}}{f_{S}}$			$c_{B} = \frac{L_{C}}{2 / Cw + α}$
total support stiffness $c_{T} = {(\frac{1}{c_{P}} + \frac{1}{c_{S}} + \frac{1}{c_{B}})}^{- 1}$
$r a i l s e a t l o a d P = \frac{1}{2} QS λ_{R} \sum^{η_{i}}$ with $λ_{R} = \sqrt[4]{\frac{c_{T}}{4 E_{R} I_{R} S}}$ and $η_{i} = e^{- λ_{R} x_{i}} [\cos (λ_{R} x_{i}) + \sin (λ_{R} x_{i})]$
$r a i l d e f l e c t i o n y = \frac{P}{c_{T}}$ , for load case C: $gap g = \frac{P α}{L}$
$t r a c k s t i f f n e s s = \frac{Q}{y}$	$t r a c k m o d u l u s = \frac{c_{T}}{S}$	$r a i l s e a t b a l l a s t s t r e s s = \frac{C P}{c_{B}}$

Since sleepers are relative thick and sleeper materials can be anisotropic, using a Timoshenko beam for the sleeper analysis improves accuracy compared with an Euler–Bernoulli beam. By transforming the sleeper bending and shear modulus into an apparent bending modulus, the shear deformations of the sleeper can be taken into account in the simplified calculation method. With this modification, calculation outcomes when compared with finite element analyses are within mentioned accuracies, also expanding the simplified calculation method beyond the current analytical methods by including beams with high shear deformations.

Summarising the set of proposed equations as derived, completed with the standard track formula, gives the calculation flow for system stiffness, rail seat loads and rail deflections as given in Table 4.

When a track is not maintained, eventually a uniform load distribution will develop under the sleeper, which is associated with a gap under the sleeper at the rail seat, which closes on train passage. The gap belonging to a uniform load distribution is depending on the bending stiffness of the sleeper, for which formulas are given in Table 4. When the consequent track modulus for flexible sleepers due to this larger gap can develop beyond desired track behaviour, track maintenance is required in time and monitoring of the gap is advised.

Footnotes

Appendix 1 Acknowledgements

The execution of the finite element calculation series has been performed by Van Dijk FEM Engineering.

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 iD

Aran van Belkom

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A simplified method for calculating load distribution and rail deflections in track,incorporating the influence of sleeper stiffness

Abstract

Keywords

Introduction

Current calculation models

The Winkler foundation

Two layer beam-on-elastic-foundation

Support conditions

Current calculation new or tamped track

Simplified calculation model

λL3

Introduction of sleeper flexibility factor fS

Including shear deformation

Discussion of simplified model

Model validation excluding shear deformations

Model validation including shear deformations

Limitations

Necessity of incorporating sleeper bending

Worn track

Example gap analysis

Conclusion

Footnotes

Appendix 1

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

References

λL₃

Introduction of sleeper flexibility factor f_S