The solution to the differential equation is dependent upon the boundary conditions of the beam as well as the ..... Advanced Mechanics of Materials 6th Edition.
ON THE RELATIONSHIP BETWEEN LOAD AND DEFLECTION IN RAILROAD TRACK STRUCTURE Sheng Lu, Richard Arnold, Shane Farritor* *Corresponding Author Department of Mechanical Engineering University of Nebraska Lincoln N104 Scott Engineering Center Lincoln, NE 68588-0656
Mahmood Fateh, Gary Carr Federal Railroad Administration Office of Research and Development 1200 New Jersey Avenue SE Washington, DC 20590
ABSTRACT Track Modulus, defined as ratio between the rail deflection and the vertical contact pressure between the rail base and track foundation, is an important parameter in determining track quality and safety. The Winkler model is a widely used mathematical expression that relates track modulus to rail deflection. The Winkler model represents railroad track as an infinitely long beam (rail) on top of a uniform, linear, and elastic foundation. The contact pressure between the rail base and track foundation increases linearly with vertical deflection. However, it is widely accepted that actual track deflection is highly non-linear. Several other models have been used to better represent the behavior of railroad track structure including a model that includes a shear layer and one that uses discrete supports. This paper presents a new model of track deflection where the elastic foundation beneath the rail has a cubic polynomial relationship between applied pressure and vertical deflection. This new cubic model is compared to other models of railroad track structure, including the Winkler, Pasternak, and Discrete Support models, as well as with experimental data. It is shown that the cubic model is a better representation of real track structure. INTRODUCTION Background The relationship between applied loads, track stresses, and track deformations are important factors to be considered in proper track design and maintenance. A representative mathematical model that accurately describes this relationship is desirable. Winkler proposed the use of an elastic beam theory to analyze rail stresses and calculation of a fundamental parameter, called the track modulus, which represents the effects of all the track components under the rail (1). Track Modulus (represented by u in this paper) is defined as the supporting force per unit length of rail per unit rail deflection (2). Track Stiffness (represented by k in this paper) is simply the ratio of applied load to
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resulting vertical deflection. Track stiffness relates load to deflection while track modulus relates a distributed load to deflection. Railway track has several components that all contribute to track modulus including the rail, subgrade, ballast, subballast, ties, and fasteners. The rail directly supports the train wheels and is supported on a tie pad and held in place with fasteners to crossties. The crossties rest on a layer of rock ballast and subballast used to provide drainage. The soil below the subballast is the subgrade. The subgrade resilient modulus and subgrade thickness have the strongest influence on track modulus. These parameters depend upon the physical state of the soil, the stress state of the soil, and the soil type (3, 2). Track modulus increases with increasing subgrade resilient modulus, and decreases with increasing subgrade layer thickness (2). Ballast layer thickness and fastener stiffness are the next most important factors (2, 4). Increasing the thickness of the ballast layer and or increasing fastener stiffness will increase track modulus (5, 2). This effect is caused by the load being spread over a larger area. The system presented in this paper measures the net effective track modulus that includes all these factors. Track modulus is important because it affects track performance and maintenance requirements. Both low track modulus and large variations in track modulus are undesirable. Low track modulus has been shown to cause differential settlement that then increases maintenance needs (6, 7). Large variations in track modulus, such as those often found near bridges and crossings, have been shown to increase dynamic loading (8, 9). Increased dynamic loading reduces the life of the track components resulting in shorter maintenance cycles (9). It has been shown that reducing variations in track modulus at grade (i.e. road) crossings leads to better track performance and less track maintenance (8). Ride quality, as indicated by vertical acceleration, is also strongly dependent on track modulus. The economic constraints of both passenger and freight rail service are moving the industry to higher-speed rail vehicles and the performance of high-speed trains are strongly dependent on track modulus. It has been shown that at high speeds there will be an increase in track deflection caused by larger dynamic forces (10, 11). These forces become significant as rail vehicles reach 50 km/hr (30 mph) (12) and rail deflections increase with higher vehicle speeds up to a critical speed (11). It is suggested that track with a high and consistent modulus will allow for higher train speeds and therefore increased performance and revenue (11).
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An improved mathematical understanding of the relationship between loads and deflections will lead to better track design and increased safety. Problem Definition: A Beam on an Elastic Foundation (BOEF) Model of Track Structure The BOEF model describes a point load applied to an infinite Bernoulli beam on an infinite elastic foundation. Figure 1 shows a free load and deflection diagram of the rail under a one-wheel load (Figure 1, top). Here, the rail is considered as a continuously supported beam where x represents the distance along the beam and w(x) represents the vertical beam deflection. The approximation that the rail is continuously supported improves as the cross-tie spacing decreases and as the rail bending stiffness increases (i.e. modulus of elasticity and second moment of area). The applied load, P, is assumed to be a point load and creates a distributed load on top of the rail, p(x), 0+
where P = ∫ − p( x)dx . The supporting structure supports the bottom of the rail with a reaction distributed force, q(x). 0
In real track the supporting structure consists of tie plates, fasteners, cross-ties, ballast, etc. In the Winkler model this supporting structure is an infinite elastic medium. The difference in the vertical distributed force applied to the beam (q(x) and p(x)) causes curvature in the beam as given by the following differential equation:
EI
d 4w = q( x) − p( x) , or dx 4
EI
(1)
d 4w + p ( x ) = q ( x) dx 4
The solution to the differential equation is dependent upon the boundary conditions of the beam as well as the loading conditions. A free body diagram that shows sections of the beam is shown in Figure 2. Here it can be seen that one half the applied load the boundary condition for a concentrated applied load, P, must be supported by the foundation reaction distributed force, q(x), on each half of the infinite beam, or:
∫
∞
0
p( x)dx =
P 2
(2a)
In addition, symmetry and the stiffness of the beam demand that the slope of the beam be zero at the point of loading.
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dw dx
=0
(2b)
x =0
The above differential equation and boundary conditions can now be set up and solved in different ways to represent various track behavior. Four such solutions are defined in Section 3. FIELD MEASUREMENTS OF TRACK MODULUS Figure 3 shows the experimental results of the track responses under various applied loads. Rail deflection was measured at given locations using linear variable differential transformers (LVDTs) as a short, slow moving train of known weight passed. The axles of the train weighed 150600 N (33850 lbf), 60230 N (13540 lbf), and 30650 N (6890 lbf). The LVDTs were mounted to steel rods (about 1m (3ft)) driven into the subgrade to provide a stable reference. The LVDTs then measured the vertical motion of the flange relative to the steel rod. The results from four LVDTs are shown in Figure 3. Here the LVDTs were placed at 1m (3ft) increments along the track (x=1m, 2m, 3m, 4m). These measurements, along with many others dating back to the Talbot Report (13) clearly indicate that the vertical rail deflections are not linearly proportional to the wheel loads. It is also important to note that the “degree” on non-linearity can change dramatically over very short distances along the track. Note the deflection of the track under the 30650 N (6890 lbf) load doubled over a distance of one meter. This non-linearity and variability greatly complicates determining and modeling track structure. Several methods have been developed for calculating modulus with each method assuming a different definition of track modulus that approximate the non-linear behavior of real track. Consider the definitions of track modulus represented in Figure 4 and described in the following sections. Beam On an Elastic Foundation (BOEF) Method The most straightforward method to estimate track modulus at a given track location is to simply measure the vertical deflection at the point (w(0)=wo) of an applied known load, P. This is a measurement of the track stiffness, k, but this measurement can be related to track modulus, u, using the BOEF model and assuming that the relationship between rail supporting load p(x) and deflection w(x) is linear and elastic (i.e. p(x)=uw(x) as in 2, 1). These assumptions lead to the Winkler model as described in Section 3.1. The resulting track modulus is given by:
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1
1 ⎛ 1 ⎞3 ⎛ P u = ⎜ ⎟ ⎜⎜ 4 ⎝ EI ⎠ ⎝ w0
4
⎞3 ⎟⎟ ⎠
(3)
where: u is the track modulus E is the modulus of elasticity of the rail I is the moment of inertia of the rail P is the load applied to the track w0 is the deflection of the rail at the loading point This method only requires a single measurement and it has also been suggested to be the best method for field measurement of track modulus (14). However, as shown in Figure 3, it is clear that this linear approximation has large error for real track. Using a single applied load and a single measurement of deflection does not capture the changes in the load deflection curve present in real track. Deflection Basin Method The Deflection Basin Method uses the vertical equilibrium of the loaded rail and several deflection measurements to more directly estimate track modulus. In this approach, rail deflection caused by a point load(s) is measured at several (ideally infinite) locations along the rail and the entire deflected “area” calculated. This method requires several deflection measurements over the section of track that supports the load(s), which makes it more time consuming (2). Using a force balance this deflected area, or deflection basin, can be shown to be proportional to the integral of the rail deflection (2, 1): ∞
∞
−∞
−∞
P = ∫ q (x ) dx = ∫ uδ (x ) dx = uAδ
where: P is the load on the track q(x) is the vertical supporting force per unit length u is the track modulus δ(x) is the vertical rail deflection Aδ is the deflection basin area
(4)
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(area between the original and deflected rail positions) x is the longitudinal distance along the track These measurements and calculations result in a numerical solution to the BOEF equation given in Equation (1). This solution does include the non-linear behavior of the track, but the measurements are extremely time consuming and only reveal the track modulus at a given point. As shown in Figure 3, these measurements could change dramatically for a point just centimeters away. Heavy-Light Load Method Many have represented the load/defection curve as piecewise linear with a low stiffness at low loads and a much higher stiffness at higher loads (15). This is seen in real track as slack in the rail and can be caused by many things such as the ties not contacting the ballast. As the rail is loaded, a low stiffness is experienced until the tie contacts the ballast resulting in a higher stiffness. This leads to a measurement of track stiffness using two loads, Figure 4, that are ideally both in the “high stiffness” range (e.g. slack is removed) (16, 6, 17). k=
P2 − P1 w2 − w1
(5) where: k is the track stiffness P1 and P2 are the applied loads w1 and w2 are the corresponding deflections Again, a linear assumption is used to then transform the stiffness measurements of the two loads to track modulus (substitute k = P into Equation 3).
The clear difficulty with this measurement is that the real
wo
load/deflection relationship is not piecewise linear and the resulting stiffness varies with the selection of the two loads, P1 and P2. Track Modulus at Characteristic Load It is proposed in this paper that a good definition of track modulus is the variation in supporting distributed force relative to the variation in deflection near the characteristic load for a given track. This characteristic load might be
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defined as the nominal axle load for a given freight line (e.g. 160kN or 286,000/8=36kips). This can be expressed mathematically as the derivative of the pressure deflection curve evaluated at the characteristic load P*:
u* =
∂p ∂w
(6) P
*
where: u is the track modulus p is the supporting force per unit length of rail P* is the characteristic load corresponding for a given rail line To evaluate the derivate at the characteristic load, the load must again be transformed to a distributed load. This can be done with the linear assumptions as described above or with the cubic model given in Section 3.4. This definition of track modulus has been used in field measurements (18). MODELS OF RAIL DEFLECTION The Winkler Model In the Winkler model, the BOEF model described above assumes the distributed supporting force of the track foundation is linearly proportional to the vertical rail deflection (i.e. p(x)=uw(x)). The BOEF differential equation then becomes:
EI
d 4 w( x) + uw( x) = q( x) dx 4
(7)
This model has been shown to be an effective method for determining track modulus (19, 20) and derivations can be found in (12, 21). The vertical deflection of the rail, w, as a function of longitudinal distance along the rail x (referenced from the applied load) is given by: w( x ) = −
where:
Pβ − β ⋅ x e [cos(β x ) + sin (βx )] 2u
(8)
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⎛ u β =⎜ ⎜ 4 EI ⎝
1
⎞4 ⎟ ⎟ ⎠
where: P is the load on the track u is the track modulus E is the modulus of elasticity of the rail I is the moment of inertia of the rail w is the longitudinal distance along the rail When multiple loads are present, the rail deflections caused by each of the loads are superposed (assuming small vertical deflections) (21). A plot of the rail deflection given by the Winkler model over the length of a four-axle coal hopper is shown in Figure 5. The deflection is shown relative to the wheel/rail contact point for five different reasonable values of track modulus (6.89, 13.8, 20.7, 27.6, and 34.5 MPa or 1000, 2000, 3000, 4000, and 5000 lbf/in/in). The model assumes 115 lb rail with an elastic modulus of 206.8 GPa (30,000,000 psi) and an area moment of inertia of 2704 cm4 (64.97 in4). The limitations of the Winkler model are clear given the widely accepted non-linearity of track structure. However, this model is often used because it does provide a clear closed form solution to the relationship between load and deflection in track structure. Discrete Support Model A second model assumes a similar linear relationship between the rail support and deflection, but uses discrete springs to provide the rail support forces rather than the infinite elastic medium used in the Winkler model. The discrete support model is similar to the Winkler model when the ties are uniformly spaced, have uniform stiffness, and the rail is long. The discrete springs represent support at the crossties and the single applied load represents one railcar wheel and is fully described in Norman (22). The discrete support model is useful because track modulus can vary from tie to tie (as in Figure 3). The proposed model also only considers finite lengths of rail and a finite number of ties, Figure 6. To reduce the model’s
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computational requirements, the rail is assumed to extend beyond the ties and is fixed at a (large) distance from the last tie. This ensures the boundary conditions are well defined (the rail is flat, far away, or w(∞)=0 and w’(∞)=0) and the rail shape is continuous, Figure 6 top. The deflection in each of the springs (i.e. the rail deflection) can be determined by first solving for the forces in each of the springs using energy methods and the free body diagrams in Figure 6. The principles of stationary potential energy and Castigliano’s theorem on deflections are applied (21). These methods require small displacements and linear elastic behavior. The number of equations needed to determine the forces in the springs is equal to the number of springs (i.e. spring forces are the unknowns). The moment and shear force in the cantilevered sections of the model (Figure 6(A) and (C)) can now be calculated. Static equilibrium requires the moment, for Section A, to be:
M 1 = M A + V A x1 , and M 2 = M C + VC x2
(9)
Similar equations can be written for the sections of beam between each of the discrete supports. This leads to N+4 equations where N is the number of discrete supports used in the model.
Now, the total system energy can be
written as the sum of the energy stored in the bending beam (sheer energy is negligible) and the energy stored in the springs: U TOTAL = U Beam + U Springs = ∑ ∫
Mi F2 dx + ∑ i 2 EI 2 ki
(10)
Where Mi is the bending moment in each segment of the beam and E and I are the sectional properties. The bending energy in each segment is summed. Also, Fi is the force in each support spring and ki is the stiffness of each spring. Castigliano’s theorem can now be used to create equations needed to solve for the unknown spring forces and boundary conditions (moment and shear forces): ∂U ∂U ∂U = = = 0, ∂Fi ∂M A ∂M B
and
∂U ∂U = =0 ∂VA ∂VB
(11)
With these relationships, a set of N+4 equations and N+4 unknowns can be developed by substituting the moment expressions into Equation (12). These expressions can be written in matrix form: (12)
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MF = P where: P is the load vector M is a N+4 x N+4 matrix of the external forces F is a vector of the spring forces and reaction forces and moments The spring forces lead directly to spring displacements and the details can be found in Norman (22). The discrete support model gives results similar to the Winkler model for similar inputs. However, the discrete model has the additional ability to represent non-uniform track. Figure 7(B) compares the deflections from the two models for uniform modulus and a single applied load. The continuous line represents the Winkler model and the boxes indicate the tie locations in the discrete model. The track modulus used in the Winkler model was 20.7 MPa (3000 lbf/in./in.) and the corresponding tie stiffness was 10.5 MN/m (60000 lbf/in.). Track modulus is equated to tie stiffness by dividing by the tie spacing (ties spacing of 50.8 cm (20”)). A single point load of 157 kN (35750 lbf) was applied over the center tie. The deflection predicted by both models is very similar with a maximum variation of 6.47%. The clear advantage of the discrete support model is that it can represent non-uniform track. In Figure 7(C) the stiffness of the 3rd tie from the left end has been decreased by 50% (to 5.25 MN/m or 30000 lbf/in.). In Figure 7(D), the stiffness of the 3rd tie has been increased by 100% (to 21.0 MN/m or 120000 lbf/in.). The track deflection with a single soft tie (Figure 7(C)) is no longer symmetric about the loading point. The rail is deflected more on the left side of the load where the soft tie is located. The maximum deflection of the rail was also slightly increased (by approximately 0.1219 mm (0.0048 in.)). Figure 7(D) shows the rail deflection where the stiffness of the 3rd tie has been doubled to 21.0 MN/m (120000 lbf/in.). The discrete model shows that the deflection near the stiff tie and the maximum deflection have both decreased (by approximately 0.1829 mm (0.0072 in.)). The results from these examples show that 1) the two models give similar results for similar inputs, and 2) the deflection curve can be affected by a single tie.
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Sheared Layer Model A third solution of the BOEF model adds a shear layer to the uniform elastic rail foundation. In this model, known as the Pasternak foundation, vertical displacement of one section of the elastic foundation will result in displacement of neighboring sections of the elastic foundation (e.g. a mattress where the springs are tied together). This distinction is most prevalent when the beam has low bending stiffness (i.e. low EI). Here, the supporting distributed load, p(x) is given by:
p( x) = −G p
d 2w +upw dx 2
(13)
where up is a track modulus and Gp is a shear modulus. Substituting into Equation (1) gives the following governing differential equation:
EI
d 4w d 2w − G +upw = q p dx 4 dx 2
(14)
The solution (from Kerr****) for a single applied load P acting at x=0, is
w( x) =
P β 2 −α x [κ cos(κ x ) + α sin(κ x )],−∞ < x < ∞ e 2u p ακ
(15)
where:
β2 =
up 4EI
; α,κ = ± β 2 ±
Gp
(16)
4EI
The resulting relationship between applied load, P, and deflection, wo, is still linear as in the Winkler Model, Figure 8. Here Gp=60GPa(8,702,400psi), I=3663cm4(88in4), E=206.8GPa (30,000,000psi) are used as parameters. However, the effective stiffness of the Pasternak model is higher because more of the elastic foundation is involved in producing reaction supporting pressure. The difference between the Pasternak model and the Winkler model are more evident when either the beam is not stiff (low EI) or the shear modulus is high. Figure 9 shows the correlation between the deflections of the two models, for an identical beam under identical loads, with two shear modulus values. Again, as the shear modulus is increased more of the elastic foundation produces supporting pressure resulting in both a stiffer track and an altered shape. The very significant difficulty in using the model is in identifying an appropriate value of the shear modulus.
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Nonlinear Cubic Model The limitation with all the previous models is that each uses some form of linear elastic behavior to represent the supporting pressure. Field tests conducted by the ASCE-AREA Special Committee on Stresses in Railroad Track (13) clearly showed that the vertical rail deflections were not linearly proportional to the wheel loads. An extensive experimental study conducted by Zarembski and Choros (14) also clearly documented this nonlinear response. Here, a new model is proposed that represents the relationship between vertical rail deflection and the rail support distributed load as a cubic polynomial. To define this relationship the experimental results of Zarembski and Choros (14) are plotted in Figure 10 along with a cubic polynomial curve fit. The polynomial fit is excellent (R2=0.9987). Using a cubic polynomial has several advantages. First, it clearly captures the behavior of real track (Figure 10) in that it provides for low stiffness at low loads and higher stiffness at higher loads. Also, negative displacement of the track (track lift) does not result in significant downward forces being applied to the rail. Unlike the previous models, the cubic polynomial represents the fact that if the track rises slightly, the ballast does not pull the track down. Here, the supporting distributed load p(x) has a cubic relationship between p(x) and w(x):
p( x) = u1 w + u 3 w 3
(17)
Note, that symmetry about the applied load requires the second order term to vanish. Substituting into the BOEF model gives the following differential equation.
EI
d 4w + u1 w + u 3 w 3 = q 4 dx
(18)
Equation (19) is a nonlinear differential equation and a closed form analytical solution is not straightforward. One analytical approximation based on the Cunningham’s method can be found in McVey (23). However, a numerical solution for this Boundary Value Problem (BVP) can be obtained. The BVP can be written in state space notation as:
w′ = func ( w, x)
(19)
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⎡ w( x) ⎤ ⎢ ⎥ ∂ ⎢ w′( x) ⎥ w′ = = func( w, x) ∂x ⎢ w′′( x) ⎥ ⎢ ⎥ ⎣ w′′′( x)⎦
(20)
w′( x) ⎤ ⎡ w( x) ⎤ ⎡ ⎢ ⎥ ⎢ ⎥ w′′( x) ⎥ ∂ ⎢ w′( x) ⎥ ⎢ =⎢ ′ ′ ′ ⎥ ( ) w x ∂x ⎢ w′′( x) ⎥ ⎢ ⎢ ⎥ − 1 u w( x) + u w 3 ( x) ⎥ ′ ′ ′ ⎥ ( ) w x 3 ⎣ ⎦ ⎢⎣ EI 1 ⎦
(21)
Given equation (19) the BVP becomes:
(
)
As the name implies, the fourth order BVP described above requires the value of four boundary conditions, here:
w( x) | x =∞ = 0 w( x) | x =−∞ = 0 w′( x) | x =0 = 0
(22)
w( x) | x =0 = wo Now, since the BVP can have more than one correct solution, an initial “guess” for the last boundary condition that will cause the solution to converge to the expected solution. In this work, the initial guess is provided by the Winkler model evaluated at x=0 and u=u3 given by:
Pβ w(0 ) = wo = − 2u 3
⎛ u where : β = ⎜ 3 ⎜ 4 EI ⎝
1
⎞4 ⎟ ⎟ ⎠
(23)
The mechanics of this problem also requires the solution be found subject to the additional constraint given by the free body diagram in Figure 2 by:
P ∫ (u w + u w )dx = 2 ∞
0
3
1
3
(24)
The unique solution that satisfies all these constraints will give the rail deflection. Any number of numerical techniques can be used to solve this well posed BVP. In this work the “bvp4c” function in Matlab (24) was used.
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As the cubic model closely represents the deflection test data for the whole range of wheel loads, the accuracy of the linear analysis depends on the magnitude of the test load. Because the cubic spring is initially softer than the one in the Winkler model, the rail must deflect more before the base can pick up the full load. This means that the distributed load will be spread over a wider span than it is for the linear model as shown in Figure 11. Meanwhile, the deflection at the contact point for the cubic model is slightly larger than the one for the Winkler model when the applied load is relatively large. Track Modulus at Characteristic Load using the Cubic Model Finally, the track modulus at characteristic load can be calculated:
u* =
∂p ∂w
= P
*
(
∂ u1 w + u 3 w 3 ∂w
)
= u1 + 3u 3 w 2 P
P*
(25)
*
This definition of track modulus is compared to the Winkler model for a given measurement of load of 151240 N (34kips) and displacement of 0.254cm (0.1”) in Figure 12. In this Figure the load deflection curve is plotted from the experimental data of Zarembski and Choros (14) shown in Figure 10. It is clear that for single data points at higher loads the Winkler model will always underestimate the actual track modulus (Figure 12). The Winkler model will also poorly represent changes in deflection with respect to changes in load at these higher values. It is also clear from these data that any two choices of loads (as in the Heavy-Light load definition of track modulus) will give a different value of track modulus. CONCLUSIONS Due to the widely accepted non-linearity of track response, the linear Winkler model obviously has its inadequacy. Other models like the Pasternak model and the discrete model attempt to modify the Winkler model to develop models that could more accurately describe an actual track foundation’s behavior under various applied loads, but they are still based on the linear assumption. The heavy-light load method does provide a better approximation to the nonlinear behavior, but there are still some discrepancies between the piecewise linear approximation and the real continuous nonlinear track behavior. The cubic model clearly captures the behavior of real track in that it provides for low stiffness at low loads and higher stiffness at higher loads. It represents the real track structure under the whole range of wheel loads.
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Combined with the proposed definition of track modulus at characteristic load, the cubic model can sensitively demonstrate the changes in deflection with respect to the changes in load at higher values, which the linear Winkler model will poorly represent. ACKNOWLEDGEMENTS This work is supported under a grant from the Federal Railroad Administration. The authors would specifically like to thank Mahmood Fateh and Gary Carr with FRA and William GeMeiner of the UPRR. We would also like to thank BNSF and UPRR for track access and logistical support.
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Figure 1: Free Body Diagram of the Rail
Figure 2: Boundary Condition of the Rail
Figure 3: Deflection of Track Under Three Loads
Figure 4: Various Representations of Track Modulus
Figure 5: Relative Rail Displacement Under a Railcar
Figure 6: Discrete Model and Free Body Diagram
Figure 7: Comparison of Winkler and Discrete Models
Figure 8: Stiffness of Winkler and Pasternak Models
Figure 9: Comparison of Winkler and Pasternak Models
Figure 10: Experimental Data and Curve Fitting
Figure 11: Comparison of Cubic and Winkler Models
Figure 12: Modulus Calculations in Winkler and Cubic Model
LIST OF FIGURES Figure 1: Free Body Diagram of the Rail Figure 2: Boundary Condition of the Rail Figure 3: Deflection of Track Under Three Loads Figure 4: Various Representations of Track Modulus Figure 5: Relative Rail Displacement Under a Railcar Figure 6: Discrete Model and Free Body Diagram Figure 7: Comparison of Winkler and Discrete Models Figure 8: Stiffness of Winkler and Pasternak Models Figure 9: Comparison of Winkler and Pasternak Models Figure 10: Experimental Data and Curve Fitting Figure 11: Comparison of Cubic and Winkler Models Figure 12: Modulus Calculations in Winkler and Cubic Model
