on the relationship between load and ......lu et al. 3 an improved mathematical understanding of the...
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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
Lu et al. 2
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).
Lu et al. 3
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),
where ∫+
−=
0
0)( dxxpP . The supporting structure supports the bottom of the rail with a reaction distributed force, q(x).
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:
)()(4
4
xpxqdx
wdEI −= , or
)()(4
4
xqxpdx
wdEI =+
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 2
)( Pdxxp
In addition, symmetry and the stiffness of the beam demand that the slope of the beam be zero at the point of
loading.
(1)
(2a)
Lu et al. 4
00
==xdx
dw
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:
(2b)
Lu et al. 5
34
0
31
141
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
wP
EIu
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):
( ) ( )∫ ∫∞
∞−
∞
∞−=== δδ uAdxxudxxqP
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
(3)
(4)
Lu et al. 6
(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).
12
12
wwPPk
−−
=
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 ow
Pk = into Equation 3). The clear difficulty with this measurement is that the real
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
(5)
Lu et al. 7
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*:
*
*
Pwpu
∂∂
=
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:
)()()(4
4
xqxuwdx
xwdEI =+
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:
( ) ( ) ( )[ ]xxeu
Pxw x βββ β sincos2
+−= ⋅−
where:
(6)
(7)
(8)
Lu et al. 8
41
4 ⎟⎟⎠
⎞⎜⎜⎝
⎛=
EIuβ
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
Lu et al. 9
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:
11 xVMM AA += , and 22 xVMM CC +=
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:
∑∑∫ +=+=i
iiSpringsBeamTOTAL k
FdxEI
MUUU22
2
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):
0 and ,0 =∂∂
=∂∂
=∂∂
=∂∂
=∂∂
BABAi VU
VU
MU
MU
FU
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:
(9)
(10)
(11)
(12)
Lu et al. 10
PMF =
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.
Lu et al. 11
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:
wudx
wdGxp pp +−=2
2
)(
where up is a track modulus and Gp is a shear modulus. Substituting into Equation (1) gives the following governing
differential equation:
qwudx
wdGdx
wdEI pp =+−2
2
4
4
The solution (from Kerr****) for a single applied load P acting at x=0, is
[ ] ∞<<−∞+= − xxxeuPxw x
p
,)sin()cos(2
)(2
κακκακβ α
where:
EIu p
42 =β ;
EIGp
4, 2 ±±= βκα
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.
(13)
(14)
(15)
(16)
Lu et al. 12
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):
331)( wuwuxp +=
Note, that symmetry about the applied load requires the second order term to vanish. Substituting into the BOEF
model gives the following differential equation.
qwuwudx
wdEI =++ 3314
4
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:
),( xwfuncw =′
(17)
(18)
(19)
Lu et al. 13
),(
)()()()(
xwfunc
xwxwxwxw
xw =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′′′′′
∂∂
=′
Given equation (19) the BVP becomes:
( )⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+−
′′′′′′
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′′′′′
∂∂
)()(1)()()(
)()()()(
331 xwuxwu
EI
xwxwxw
xwxwxwxw
x
As the name implies, the fourth order BVP described above requires the value of four boundary conditions, here:
ox
x
x
x
wxwxwxwxw
==′==
=
=
−∞=
∞=
0
0
|)(0|)(0|)(
0|)(
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:
( )41
3
3 4 : where
20 ⎟
⎟⎠
⎞⎜⎜⎝
⎛=−==
EIu
uPww o ββ
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:
( )∫∞
=+0
331 2
Pdxwuwu
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.
(20)
(21)
(22)
(23)
(24)
Lu et al. 14
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:
( )*
**
231
331* 3
PPP
wuuw
wuwuwpu +=
∂+∂
=∂∂
=
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.
(25)
Lu et al. 15
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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Lu et al. 16
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Lu et al. 18
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