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Real-Time Measurement of Track 

Modulus from a Moving Car 

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Real-Time Measurement of Track Modulus from a Moving Car 

6. AUTHOR(S) 

Shane Farritor

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13. ABSTRACT

The economic constraints of passenger and freight railroad traffic are moving the industry to higher speed rail vehicles 

and higher axle loads. The dynamics of high-speed trains and the high static loads produce high stresses in the rail 

components. Track performance under these conditions is strongly related to the vertical track modulus—the 

relationship between the rail deflection and the vertical applied load. Both low track modulus and large variations in

track modulus increase dynamic loading leading to increased maintenance needs and reduced ride quality. Currently, 

no widely accepted method to measure vertical track modulus from a rail car traveling at significant train speed exists. 

This report presents a method to measure vertical track modulus from a moving rail car. The method measures the

shape of the rail deflection under the moving car and uses a mathematical track model to estimate modulus. The report

presents theoretical, simulation, and experimental results.

14. SUBJECT TERMS 15. NUMBER OF 

PAGES

Track modulus, rail car, high-speed trains 37 

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ii 
Table of Contents

Executive Summary........................................................................................................................1 

1 Technical Investigation............................................................................................................2 

1.1 Introduction ..................................................................................................................... 3 

1.2 Background and Previous Work...................................................................................... 4 

1.2.1 Trackside Measurement of Track Modulus................................................................. 5 

1.2.2 Onboard Track Modulus Measurement....................................................................... 6 

1.3 Models of Rail Deflection ............................................................................................... 7 

1.3.1 The Winkler Model ..................................................................................................... 7

1.3.2 Discrete Support Model............................................................................................... 8 

1.3.3 Comparison of Winkler and Discrete Models........................................................... 12 

1.4 Modeling and Simulation .............................................................................................. 14 

1.4.1 Dynamic Railcar Model ............................................................................................ 14 

1.4.2 Sensor Model............................................................................................................. 17 

1.5 Trackside Measurements............................................................................................... 19

1.6 System Testing .............................................................................................................. 20 

1.6.1 Field Test Arrangement............................................................................................. 21 

1.6.2 Test Site A–System Verification ............................................................................... 21 

1.6.3 Test Site B–Usefulness of The Discrete Model......................................................... 23 

1.6.4 Test Site C–Effects of Speed..................................................................................... 24 

2 Summary and Conclusions.................................................................................................... 26

3 References............................................................................................................................. 27 

Abbreviations & Acronyms........................................................................................................... 29 

iii 
List of Figures 

Figure 1. Measurement System...................................................................................................... 4 

Figure 2. Relative Rail Displacement under a Railcar ................................................................... 8 

Figure 3. Discrete Model and Free Body Diagram ........................................................................ 9 

Figure 4. Middle Segment of Discrete Tie Model........................................................................ 10 

Figure 5. Comparison of Winkler and Discrete Models............................................................... 13 

Figure 6. Railcar Free Body Diagrams......................................................................................... 15 

Figure 7. Sensor System Model.................................................................................................... 17 

Figure 8. Sensor System Geometry .............................................................................................. 18 

Figure 9. Track Modulus (u) versus Sensor Measurement (d)..................................................... 19 

Figure 10. Trackside LVDT Measurements................................................................................. 20 

Figure 11. Comparison of Model and Trackside Data ................................................................. 20 

Figure 12. Typical Test Images.................................................................................................... 21 

Figure 13. Site A: Trackside, Simulation, and Field ................................................................... 22 

Figure 14. Track Geometry at Test Site A.................................................................................... 23

Figure 15. Mile Post 31 East: Trackside Data, Discrete Model Simulation, and Sensor 

Measurements........................................................................................................................ 24 

Figure 16. Effects of Speed–Test Site C ...................................................................................... 25 

Figure 17. Variation between Rails–Test Site C .......................................................................... 25 

iv 
ACKNOWLEDGEMENTS 

This report presents the results from a 4-year project to develop a system to measure 

Vertical Track Modulus (VTM) from a moving railcar. The work completed during this 

project has significantly advanced the state of the art and understanding of VTM data 

collection and interpretation in the railroad environment. The follow-on phases will 

implement the techniques presented in this report to create robust automated data 

collection and interpretation and to further investigate safety-compromising VTM track 

conditions. 

This work was performed at the University of Nebraska-Lincoln under the direction of 

Dr. Shane Farritor. The authors are grateful to Dwight Clark, Bill GeMeiner, and Mike 

Boos of the Union Pacific Railroad (UPRR); John Leeper, Mike Oliver, and Hank Lees 

of the Burlington Northern Santa Fe Railroad (BNSF); and Rich Kotan of Omaha Public 

Power District (OPPD). These people served on an Industrial Advisory Board that 

contributed to this research throughout the project. 

Mr. Mahmood Fateh of the Federal Railroad Administration (FRA) served as the 

technical monitor for this project and Dr. Magdy El-Sibaie of FRA contributed to the 

project. UPRR, BNSF, and OPPD provided access to revenue service track, as well as

provided operational support including motive power, train crews, and logistics. UPRR 

donated a caboose that was refurbished to serve as the crew platform during testing. 

UPRR also donated a hopper car, and BNSF loaned a tank car to serve as a heavy axle 

load vehicle for the testing. The authors are also grateful to Dr. Ted Sussmann of the 

U.S. Department of Transportation Volpe Center for his technical input and guidance, as 

well as useful comments on the draft versions of this report. 

v 
vi 
EXECUTIVE SUMMARY 

Track quality is a major factor in railroad safety, and one accepted indicator of track quality is 

the vertical track modulus. Track modulus is defined as the coefficient of proportionality 

between the vertical rail deflection and the vertical contact pressure between the rail base and 

track foundation. Railway track has several components that all contribute to track stiffness, 

including the rail, subgrade, ballast, subballast, ties, and fasteners. 

Track modulus is important because it affects track safety, track performance and track 

maintenance. 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. Large variations in track modulus, such as those often found near bridges 

and crossings, have been shown to increase dynamic loading. Increased dynamic loading 

reduces the life of the track components, resulting in shorter maintenance cycles. 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. Ride quality, as indicated by vertical 

acceleration, is also strongly dependent on track modulus. 

The economic constraints of 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 is suggested that track with a high and consistent modulus will allow for 

higher train speeds and therefore increased performance and revenue. 

This research presents the design, analysis, and testing of an on-board, non-contact track 

modulus measurement system. On-board measurements are difficult because no stable 

reference frame exists. The system presented here measures vertical rail displacement relative 

to the wheel/rail contact point, and these relative measurements are combined with an 

analytical model of the track structure and the measured vehicle loads (weight and dynamic 

forces) to estimate the modulus. 

The sensor system consists of a digital vision system and two line lasers. The camera and line 

lasers are rigidly attached to a bracket mounted to the sideframe (structural member that 

connects axles) of a truck (two-axle assembly). These lasers are projected at an acute angle 

(~30 ̊). They cross and create curves (because of the curved profile of the rail head) across 

the surface of the rail. On softer track the rail will rise relative to the wheel/rail contact point, 

and the laser lines as observed by the camera will move closer together. Conversely, the 

distance between the lasers will increase on stiffer track. The minimum distance between 

these lines, d, can be related mathematically to the track modulus. Using line lasers allows 

the system to compensate for lateral movement of the rail relative to the camera and for 

changes in rail profile. 

This research presents the mathematical models used to estimate track modulus from the laser 

measurements. The system is evaluated using computer simulations and field testing. Field 

testing was conducted at several locations on Class III, IV, and V tracks. Independent 

trackside measurements of rail deflection are used to confirm the on-board measurements. 

1 
2 
1 TECHNICAL INVESTIGATION 

1.1 Introduction 

Track quality is a major factor in railroad safety, and one accepted indicator of track quality is 

the vertical track modulus. Track modulus is defined as the coefficient of proportionality 

between the vertical rail deflection and the vertical contact pressure between the rail base and 

track foundation (Cai, et al., 1994); it can be restated as the supporting force per unit length of 

rail, per unit rail deflection (Selig and Li, 1994). 

Railway track has several components that all contribute to track stiffness, including the rail, 

subgrade, ballast, subballast, ties, and fasteners. The rail directly supports the train wheels 

and is supported on a rail 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 (Li and Selig, 1994; Selig and Li, 1994). Track modulus increases with 

increasing subgrade resilient modulus and decreases with increasing subgrade layer thickness 

(Selig and Li, 1994). Ballast layer thickness and fastener stiffness are the next most important 

factors (Selig and Li, 1994; Li and Selig, 1998). Increasing the thickness of the ballast layer 

and/or increasing fastener stiffness will increase track modulus (Stewart, 1985; Selig and Li, 

1994). This effect is caused by the load being spread over a larger area. The system

presented in this research 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 (Read, et al., 1994; Ebersohn, et al., 1993). Large variations in track 

modulus, such as those often found near bridges and crossings, have been shown to increase 

dynamic loading (Zarembski and Palese, 2003; Davis, et al., 2003). Increased dynamic 

loading reduces the life of the track components, resulting in shorter maintenance cycles 

(Davis, et al., 2003). 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 (Zarembski 

and Palese, 2003). Ride quality, as indicated by vertical acceleration, is also strongly 

dependent on track modulus. 

The economic constraints of 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 (Carr and Greif, 2000; Heelis, et al., 1999). These 

forces become significant as rail vehicles reach 50 km/hr (30 mph) (Kerr, 1976) and rail 

deflections increase with higher vehicle speeds up to a critical speed (Heelis, et al., 1999). It 

is suggested that track with a high and consistent modulus will allow for higher train speeds 

and therefore increased performance and revenue (Heelis, et al., 1999). 

3 
Previous localized field-testing has shown that it is possible to measure areas of low track 

modulus, variable track modulus, void deflection, variable total deflection, and inconsistent 

rail deflection (Sussmann, et al., 2001; Ebersohn and Selig, 1994). In the past such systems 

have been used to identify sections of track with poor performance. These measurements 

have been useful; however, they are expensive and are only being made over short distances 

(~tens of meters). The ability to make these measurements continuously over large sections 

of track is desirable (Ebersohn and Selig, 1994; Read, et al., 1994). 

This research presents the design, analysis, and testing of an on-board, non-contact track 

modulus measurement system. On-board measurements are difficult because no stable 

reference frame exists. The system presented here measures vertical rail displacement relative 

to the wheel/rail contact point, and these relative measurements are combined with an 

analytical model of the track structure and the measured vehicle loads (weight and dynamic 

forces) to estimate the track modulus. 

The sensor system consists of a digital vision system and two line lasers, as seen in Figure 1. 

The camera and line lasers are rigidly attached to a bracket mounted to the sideframe 

(structural member that connects axles) of a truck (two-axle assembly). These lasers are 

projected at an acute angle (~30 ̊). They cross and create curves (because of the curved 

profile of the rail head) across the surface of the rail. On softer track the rail will rise relative 

to the wheel/rail contact point, and the laser lines as observed by the camera will move closer 

together. Conversely, the distance between the lasers will increase on stiffer track. The 

minimum distance between these lines, d, can be related mathematically to the track modulus. 

Using line lasers allows the system to compensate for lateral movement of the rail relative to 

the camera and for changes in rail profile. 

Figure 1. Measurement System 

This research presents the mathematical models used to estimate track modulus from the laser 

measurements. The system is evaluated using computer simulations and field testing. Field 

testing was conducted at several locations on Class III, IV, and V tracks. Independent 

trackside measurements of rail deflection are used to confirm the on-board measurements. 

1.2 Background and Previous Work

The measurement of vertical track modulus is difficult. Previous measurement systems can 

be placed in two categories: trackside measurements and onboard measurements. With the 

trackside approach a section of track is closed to rail traffic, and a work crew uses specialized 

4 
equipment to make measurements at various discrete locations. Onboard measurements are 

made from a moving railcar and are more desirable because they can be made with less 

interruption of rail traffic and over longer distances. However, current onboard systems are 

still very labor intensive and move at only slow speeds and are therefore limited to short 

distances (e.g., hundreds of meters) and still interrupt traffic. 

1.2.1 Trackside Measurement of Track Modulus 

In all trackside methods rail deflection is measured before and after a static point load is 

applied. Differences lie in the number of deflection measurements made and how those 

measurements are used to estimate track modulus. Common trackside approaches include the 

Beam on Elastic Foundation and the Deflection Basin method. 

The Beam on an Elastic Foundation method uses a structural model, known as the Winkler 

Model, to represent the track system. The Winkler Model represents a point load applied to 

an infinite Bernoulli beam on an infinite elastic foundation. Trackside measurements of the 

deflection at the point where the load is applied are taken for a known load and modulus can 

then be calculated using (Selig and Li, 1994; Cai et al, 1994): 

3

4

0

3

1

1

1

4

1















 



 



 = w

P

E u (1)

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 (Zarembski and Choros, 1980). Its major 

limitation is that it provides only information for a single point. In reality, the modulus may 

be very different only a meter away. In addition, if multiple loads are present (as is the case 

with multi-axle railway vehicles used to apply the point load), small deflections must be 

assumed, and superposition is needed. In this case the Winkler Model cannot be simplified as 

in (1) and an iterative solution is required (Selig and Li, 1994; Cai et al., 1994). Slack in the 

rail can cause non-linearity in the load/deflection relationship. Therefore, a small load should 

be applied to determine the zero displacement position for the measurement (Ebersohn and 

Selig, 1994; Read et al., 1994). A heavy load is then applied and used as the loaded 

measurement. This further complicates the technique. 

The second trackside technique, the Deflection Basin method, uses the vertical equilibrium of 

the loaded rail to determine 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. Using a force balance this deflected area, or deflection basin, can be shown 

to be proportional to the integral of the rail deflection (Selig and Li, 1994; Cai et al., 1994): 

5 
 (2) ( ) ( ) ∫ ∫ ∞

−∞

∞

−∞ = = δ dx = uAδ P q x dx u x

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 (area between the original and deflected rail positions).

x is the longitudinal distance along the track. 

The multiple deflection measurements result here in longer traffic delays (Selig and Li, 1994); 

heavy and light loads are again used to eliminate slack. Some researchers question the basic 

assumptions of this method (Kerr and Shenton, 1985). 

All the above methods are time consuming and expensive, and all suffer from the major 

limitation that the measured modulus is only valid along a small length of track. 

1.2.2 Onboard Track Modulus Measurement 

Previous onboard track modulus measurement systems are similar to systems developed by 

the military and used in highway research (Carr, 1999). The systems use a long rigid truss 

that rides on two unloaded wheels. This truss creates a straight line, or cord, that is used as a 

reference for the measurement. A third wheel is then used to apply a load at midpoint of the 

cord (or truss), and the relative displacement between the loaded wheel and the unloaded truss 

is measured. The truss must be long enough, generally 30.48 m (100 ft), so that the two 

endpoints are not affected by the load at the center of the truss. This method again requires 

two measurements, one with a light load, made with a similar truss, and the heavy load, to 

distinguish between changes in geometry and changes in modulus. The output of this 

approach is a measurement of the relative displacement of the loaded wheel with respect to 

the unloaded wheel, and track modulus is then estimated. 

One vehicle, called the Track Loading Vehicle (TLV), uses this approach (Thompson and Li, 

2002). This vehicle is capable of measuring track modulus at speeds of 16.1 km/hr (10 mph). 

The TLV uses two cars, each with a center load bogie capable of applying loads from 4.45 kN 

to 267 kN (1 to 60 kips). A light load (13.3 kN or 3 kips) is applied by the first vehicle while 

a heavier load is applied by the second vehicle. A laser-based system on each vehicle 

measures the deflections of the rail caused by the center load bogies. The test procedure 

involves two passes over a section of track, first applying a 44.5 kN (10 kip) load and then a 

178 kN (40 kip) load (Thompson and Li, 2002). 

The TLV is operational, however, it still has limitations. First, tests are often performed at 

speeds below 16.1 km/hr (10 mph) so it is difficult to test long section of track (hundreds of 

miles). Second, significant expense in equipment and personnel are required for operation. 

For these reasons the TLV has not yet been widely implemented. 

This research presents a simple method to determine track modulus from a moving railcar. 

The sensor system uses a non-contact structured-lighting vision system. The proposed system

6 
is inexpensive and does not require significant support equipment. It has the potential to 

operate at higher speeds and could potentially be automated. This research presents the 

theory and design of the system. 

1.3 Models of Rail Deflection 

The following presents two mathematical planar models of rail deflection. The first is the 

widely accepted Winkler Model mentioned above, and the second is a new discrete support 

(crosstie) model used to estimate track modulus in this work. The models are compared, but 

the field results presented below show the discrete support model is needed when track 

modulus changes over short distances. Both models assume knowledge of the rail’s elastic 

modulus and cross-sectional geometry which are commonly available for all commercial rail. 

1.3.1 The Winkler Model 

The Winkler Model describes the deflection of an infinitely long beam resting on an infinite 

continuous, uniform, elastic foundation in response to a single point load. In this model the 

deflection of the beam under an applied load is linearly proportional to the pressure between 

the base of the rail and the foundation. This model has been shown to be an effective method 

for determining track modulus (Raymond, 1985; Meyer, 2002), and derivations can be found 

in Kerr, 1976; Boresi and Schmidt, 2003. The vertical deflection of the rail, y, as a function 

of longitudinal distance along the rail x (referenced from the applied load) is given by: 

( ) e [ ( ) x ( x) u

P

y x x β β ] β β cos sin

2 = − + − ⋅ (3)

where: 

4

1

4 













 = EI z

u β (4)

where: 

P is the load on the track.

u is the track modulus.

E is the modulus of elasticity of the rail.

Iz is the moment of inertia of the rail. 

x 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) (Boresi and Schmidt, 2003). 

Figure 2 shows a plot of the rail deflection given by the Winkler Model over the length of a 

four-axle coal hopper. 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). Figure 

2 shows larger rail displacements with respect to the wheel/rail contact point occur with low 

track modulus. 

7 
Figure 2. Relative Rail Displacement under a Railcar 

Relative to the wheel/rail contact point (as in Figure 2) the rail displacement is most sensitive 

to track modulus far from the wheel contact point. This variation is used in the measurement 

system proposed in this research. The sensors that measure track displacement in this work 

are located approximately 1.25 m (48 in.) toward the center of the car from the third axle 

(indicated by a dashed line in Figure 2). 

1.3.2 Discrete Support Model 

The discrete support model describes the rail supported on a number of discrete springs with a 

single force applied. The discrete springs represent support at the crossties, and the single 

applied load represents one railcar wheel. At this time, tie support is modeled by linearly 

elastic springs. This model is more applicable at low speeds, but future work will include 

viscoelastic behavior. It is expected that measurements of track modulus may vary with train 

speed because of damping. However, the experimental results in this research suggest rate of 

deflection may not be significant at speeds below 48 km/hr (30 mph) (Section 1.6.4). 

The discrete tie model is shown to be useful in the experimental results (Section 1.6.3) 

because track modulus can vary from tie to tie. In Figure 3, the proposed model considers 

only finite lengths of rail and a finite number of ties. To reduce the model’s computational 

requirements (so it can be implemented in real time), 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), and the rail shape is continuous. 

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. The principles of 

stationary potential energy and Castigliano’s theorem on deflections are applied (Boresi and 

Schmidt, 2003). For these methods to be applicable, small displacements and linear elastic 

behavior is assumed. 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). 

8 
Figure 3. Discrete Model and Free Body Diagram 

The discrete support model is similar to the Winkler Model when the ties are uniformly 

spaced, have uniform stiffness, and the rail is long. Nine ties are used in this work, and 

experimental trackside measurements have shown this to be sufficient (Norman, 2004). 

The moment and shear force in the cantilevered sections of the model (Figure 3(A) and 3(C)) 

can now be calculated. Static equilibrium requires the moment and shear force for Section A 

to be: 

1 1 M M V x = A + A (5)

V1 =VA (6)

Likewise, for Section C: 

2 2 M M V x = C + C (7)

V2 =VC (8)

The forces in the springs can be determined with energy methods (Figure 3(B)). Section B is 

split into segments separated by the springs, and the segment’s internal moments are found (as 

in 9 and 11 above) to determine the beam’s strain energy from the internal moments, Figure 4. 

Energy from shear force is small and is neglected (Norman, 2004). 

9 
Figure 4. Middle Segment of Discrete Tie Model 

The equations for the internal moments in each segment can be written, (13)-(22), where 

moments M1 and M2, and the shear forces V1 and V2 are given by equations 9-12. The lengths 

of each of the segments (i.e., tie spacing) in the beam are given by L1 – L10 and the spring 

forces are denoted by F1 – F9 (Figure 4). 

1 1

1

for 0 x L

M M

≤ ≤

= (9)

[ ( ) ( )]

1 1 1 2

1 1 1 1 1 1 1

for L x L L

M M V x L F x L

≤ ≤ +

= − + − + − (10)

[ ] ( ) ( ) ( )

1 2 1 1 2 3

1 1 1 1 1 1 1 2 1 1 2

for L L x L L L

M M V x L F x L F x L L

+ ≤ ≤ + +

= − + − + − + − − (11)

( ) ( ) ( )

( )

1 2 3 1 1 2 3 4

3 1 1 2 3

1 1 1 1 1 1 1 2 1 1 2

for L L L x L L L L

F x L L L

M V x L F x L F x L L

M

+ + ≤ ≤ + + +





 





+ − − −

+ − + − + − − = − (12)

( ) ( ) ( )

( ) (

1 2 3 4 1 1 2 3 4 5

3 1 1 2 3 4 1 1 2 3 4

1 1 1 1 1 1 1 2 1 1 2

for L L L L x L L L L L

F x L L L F x L L L L

M V x L F x L F x L L

M

+ + + ≤ ≤ + + + +





 





+ − − − + − − − −

+ − + − + − − = − ) (13)

( ) ( ) ( )

( ) (

10 9 8 7 2 10 9 8 7 6

7 2 10 9 8 6 2 10 9 8 7

2 2 2 10 9 2 10 8 2 10 9

for L L L L x L L L L L

F x L L L F x L L L L

M V x L F x L F x L L

M

+ + + ≤ ≤ + + + +





 





+ − − − + − − − −

+ − + − + − − = ) (14)

( ) ( ) ( )

( )

10 9 8 2 10 9 8 7

7 2 10 9 8

2 2 2 10 9 2 10 8 2 10 9

for L L L x L L L L

F x L L L

M V x L F x L F x L L

M

+ + ≤ ≤ + + +





 





+ − − −

+ − + − + − − = (15)

10
[ ] ( ) ( ) ( )

10 9 2 10 9 8

2 2 2 10 9 2 10 8 2 10 9

for L L x L L L

M M V x L F x L F x L L

+ ≤ ≤ + +

= + − + − + − − (16)

[ ( ) ( )]

10 2 10 9

2 2 2 10 9 2 10

for L x L L

M M V x L F x L

≤ ≤ +

= + − + − (17)

2 10

2

for 0 x L

M M

≤ ≤

= − (18)

In the above equations, the shear forces, moments, and spring forces are all unknown; 

however, one spring force can be determined by a vertical force balance: 

F1 = P −V1 −V2 − F2 − F3 − F4 − F5 − F6 − F7 − F8 − F9 (19)

where P is a known wheel load (e.g., 157 kN or 35 kips). The strain energy can now be 

written where ki is the stiffness of spring i: 

9

2

9

8

2

8

7

2

7

6

2

6

5

2

5

4

2

4

3

2

3

2

2

2

1

2

1

2

2

2

2

2

2

2

2

0

2

2

1

2

1

2

1

2

1

2

0

1

2

2 2 2 2 2 2

2 2 2 2

2 2 2

2 2 2

2 2 2

10 9 8 7 6

10 9 8 7

10 9 8 7

10 9 8

10 9 8

10 9

10 9

10

1 2 3 4 5 10

1 2 3 4

1 2 3 4

1 2 3

1 2 3

1 2

1 2

1

1

k

F

k

F

k

F

k

F

k

F

k

F

k

F

k

F

k

F dx

EI

M

dx

EI

M dx

EI

M dx

EI

M

dx

EI

M dx

EI

M dx

EI

M

dx

EI

M dx

EI

M dx

EI

M U

L L L L L

L L L L

L L L L

L L L

L L L

L L

L L

L

L L L L L L

L L L L

L L L L

L L L

L L L

L L

L L

L

L

+ + + + + +

+ + +

+ + +

+ + +

= + + +

∫

∫ ∫ ∫

∫ ∫ ∫

∫ ∫ ∫

+ + + +

+ + +

+ + +

+ +

+ +

+

+

+ + + +

+ + +

+ + +

+ +

+ +

+

+

(20)

Castigliano’s theorem is used to create the number of equations needed to solve for the 

unknown spring forces and boundary conditions (moment and shear force). In this case, 12 

unknown variables exist (8 spring forces, 2 reaction moments, and 2 reaction forces). From

Castigliano’s theorem:

0, and 0

0,

2 3 4 5 6 7 8 9

= ∂

∂ = ∂∂ = ∂∂ = ∂∂

= ∂

∂ = ∂∂ = ∂∂ = ∂∂ = ∂∂ = ∂∂ = ∂

∂ = ∂∂

A B A VB

U

V

U

M

U

M

U

F

U

F

U

F

U

F

U

F

U

F

U

F

U

F

U

(21)

With these relationships a set of 12 equations and 12 unknowns is developed by substituting 

the moment expressions (9-23) into (24). These expressions can be written in matrix form: 

MF = P (22)

11
where: 

P is the load vector.

M is a 12 x 12 matrix of coefficients of the external forces.

F is a column vector of the external forces F2 – F9, MA, VA, MB, and VB . 

The solution to this matrix equation gives the forces in each of the springs. The spring 

deflections are: 

i

i

i k

F d = (23)

where: 

di is the deflection of spring i. 

Fi is the force in spring i. 

ki is the stiffness of spring i.

1.3.3 Comparison of Winkler and Discrete Models 

The two models are now compared. Experimental results have shown the Winkler Model is a 

good representation of track deflection (Zarembski and Choros, 1980; Norman, 2004). 

Therefore, the discrete model should give results similar to the Winkler Model for similar 

inputs. The discrete model will, however, have the additional ability to represent non-uniform

track. 

12
Figure 5. Comparison of Winkler and Discrete Models 

13
Figure 5(B) compares the deflections from the two models for uniform modulus. 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) (representing a single vehicle wheel) was 

applied over the center tie. The deflection predicted by both models is very similar with a 

maximum error of 6.47 percent (assuming the accepted Winkler Model is correct). 

Real track has non-uniform modulus, and these differences can be represented by the discrete 

model. In Figure 5(C), the stiffness of the third tie from the left end has been decreased by 50 

percent (to 5.25 MN/m or 30000 lbf/in.). In Figure 5(D), the stiffness of the third tie has been 

increased by 100 percent (to 21.0 MN/m or 120000 lbf/in.). In both cases the Winkler Model 

is shown with a uniform modulus. 

The track deflection with a single soft tie (Figure 5(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 5(D) shows the rail deflection where the stiffness of the 

third 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 the two 

models give similar results for similar inputs and the deflection curve can be affected by a 

single tie. 

1.4 Modeling and Simulation 

1.4.1 Dynamic Railcar Model 

A planar railcar model was used for each rail, and this model is used to predict the 

performance of the sensor system presented in this research. This prediction is then compared 

to the results of the experimental field measurements presented in Section 6. The model 

allows for horizontal (x) and vertical (y) motion, as well as planar rotation of the car body and 

both trucks. In Figure 6, the car body and both trucks each have an associated mass, and the 

car body is separated from each truck by a spring/damper suspension system. The model 

represents dynamic interaction between the railcar and the track model presented in the 

previous section through the wheel contact loads (P1 - P4). Changes in track modulus or 

changes in track geometry will cause changes in dynamic loading at each wheel. 

The equations of motion can now be written for the car body: 

A B car car car F + F − M g = M &

y& (28)

A B car car F L F L I θ

&& − = (29)

and for each truck (assuming small angles): 

14
A A A A P + P − F − M g = M &

y& 1 2 (30)

A A Pl P l I θ

&& 1 − 2 = (31)

B B B B P + P − F − M g = M &

y& 3 4 (32)

B B P l P l I θ

&& 3 − 4 = (33)

In the above equations, L is the distance between the car body center and truck center. The 

distance between each wheel and the truck center is l. 

Expressions for the forces FA and FB can be determined from the free body diagrams for the 

suspension system (spring and dampers): 

( ) ( ) A A A car car A A car L car F = k y − y − Lsinθ +b y& − y& − sinθ (34)

( ) ( ) B B B car car B B car L car F = k y − y + Lsinθ + b y& − y& + sinθ (35)

Figure 6. Railcar Free Body Diagrams 

The horizontal motion is considered an input derived from the train velocity, and it is assumed 

that each wheel maintains rail contact. The angle of each truck, therefore, is related to the 

displacement of its wheels. For Truck A, the relationship is: 

15
2

1 2 y y y A

+ = (36)

l

y y A 2

sin 1 − 2 θ = (37)

where each yi corresponds to the displacement at the wheel contact points (e.g., y1

corresponds to the displacements at load P1) and l is the length of the truck. The deflection of 

the wheels can then be shown to be: 

A A y1 = y + lθ (38)

A A y2 = y − lθ (39)

Wheel displacements are now expressed in terms of truck displacements and angles.

Since the trucks are far apart and the rotation of the car body is small, it is assumed no 

interaction exists between the wheel loads at Truck A and Truck B (however, interaction can 

occur from the left to the right rail for a given truck). Therefore, if the Winkler Model 

(Equation 7) is used and the deflections are superposed, the deflection at wheels 1 and 2 of 

Truck A are: 

2

2

2 2

1

1 1

1 2 2

α β β

u

P

u

P

y = + (40)

1

1

1 1

2

2 2

2 2 2

α β β

u

P

u

P

y = + (41)

where: e ( x x) x

1 1 1 cos sin 1 α β β β = + − (42)

e ( x x) x

2 2 2 cos sin 2 α β β β = + − (43)

Equations 40 and 41 determine the deflections at each of the applied loads, P1 and P2. These 

equations are the result of using the Winkler Model (Equation 7) and superposition of both 

loads. 

Finally, the wheel contact forces (P1...P4) are determined assuming a quasistatic interaction 

with the rail: 

1 2

1 2 2

1

1

1 1

2

α α

α

β +

− = u y y P (44)

1 2

2 1 1

2

2

2 1

2

α α

α

β +

− = u y y P (45)

16
1.4.2 Sensor Model 

The track and car models above are used with a model of the sensor system to emulate the 

measurement system. The sensor model is a kinematic relationship between the sensors and 

rail. The sensor system has two line lasers and a camera. The line lasers intersect the rail 

surface at an acute angle to create curves across the surface of the rail (Figure 1). The camera 

then observes the distance between the two laser lines, d. A computer uses a mathematical 

model to calculate the track modulus from this reading. 

Figure 7 shows the kinematics of the sensor system. It is assumed that the sensor system is 

rigid with respect to the wheel contact point (H is constant). This is a reasonable assumption 

as the instrument beam, truck, and wheels are all massive, nearly rigid elements, and these 

elements do not include the suspension of the railcar. Rotation of the side frame (A) could 

cause this distance (H) to change, but this rotation has been experimentally shown to be 

insignificant (Norman, 2004). This value (H) is needed for calibration of the system, and this 

can be accomplished by taking measurements over a location of known modulus (Norman, 

2004). 

The sensor system measures the distance between the camera image plane and the rail surface, 

h. The displacement of the rail surface with respect to the wheel/rail contact plane, yr

( yr = H − h ), can then be found. The displacement, yr, can be related to the absolute rail 

deflection (with respect to the undeflected/unloaded rail), ycamera, and the absolute deflection 

of the wheel/rail contact point, ywheel ( r camera wheel y = y − y ). 

Figure 7. Sensor System Model 

Finally, height of the sensor system above the rail surface, h, can be geometrically related to 

the sensor reading (d) giving the distance between the laser curves, Figure 8. 

17
Figure 8. Sensor System Geometry 

The height of the sensor system above the rail (h) can be related to the sensor reading (d) with 

the following geometry: 

(L1 + l

1 )tanθ1 = h (49)

(L2 + l2 )tanθ2 = h (50)

1 2 d = l +l (51)

where: 

L1 and L2 are the horizontal positions of the lasers from the camera.

θ1 and θ2 are the angles between the lasers and the horizontal.

l1 and l2 are the distances from the camera’s centerline and the rail/laser intersection. 

Substituting l1 and l2 from (49) and (50) into (51) gives (d): 

( ) 1 2

1 2 tan tan

L L h h d = + − +

θ θ (52)

Combining this information with the track model (e.g., Winkler Model (Equation 7)), the 

sensor reading (d) can be related to the track modulus (u). The relationship is nonlinear 

because of the nonlinear track model. Figure 9 shows curves for the system for three different 

distances between the camera image plane and the wheel/rail contact point (H). Changing this 

distance shifts the curve and changes the system’s sensitivity. Therefore, the sensor system

should be mounted at a height (H) that is appropriate for the type of track to be measured. 

This parameter (H) must be found to calibrate the system using trackside measurements as 

described in the following section. 

18
Figure 9. Track Modulus (u) versus Sensor Measurement (d) 

1.5 Trackside Measurements 

The sensor measurements are first calibrated using independent trackside measurements. 

Then, other independent trackside measurements are then made to verify the system’s 

performance. True track modulus was determined by making trackside deflection 

measurements. In Figure 10, rail deflection was measured at a given location using linear 

variable differential transformers (LVDTs) as a slow moving train passed. The LVDTs were 

mounted to steel rods driven (about 1 m (3’)) into the subgrade to provide a stable reference. 

Figure 11 shows a comparison of a typical trackside measurement and a curve fit using the 

Winkler Model. Again, the rail parameters (I, E) and applied load (car weight) are known; 

therefore, a track modulus (u) is chosen to fit the trackside measurements. At this example 

location the track modulus was determined to be 19.3 MPa (2800 lbf/in./in.). The trackside 

data (solid) match the model (dotted) very well with the exception of some high frequency 

noise caused by vibration. 

19
Figure 10. Trackside LVDT Measurements 

Figure 11. Comparison of Model and Trackside Data 

1.6 System Testing 

The system has been tested at various locations and for various speeds on Class III and IV 

tracks. The test setup will be briefly described and testing will be shown for three sections of 

tangent track. The first two test sites, labeled Site A and Site B, are at slow speeds (~8 km/hr 

(~5 mph)). Trackside measurements were used to verify the sensor system at these locations, 

and simulation results are also presented. Finally, at the third site, Site C, the sensor system is 

tested at various train speeds. Other system testing, not presented in this research, covers over 

125 km (70 mi) of Class V track and 42 km (25 mi) of Class IV track (McVey, 2005). The 

test sites presented in this research were chosen to demonstrate the accuracy of the 

measurements, show the importance and capabilities of the discrete tie model, and show the 

consistency of the measurements at different train speeds. 

20
1.6.1 Field Test Arrangement 

The sensor system consists of two cars. The first is a heavy vehicle with the sensors attached 

to the side frame of the truck. The second is a refurbished caboose used to carry human 

operators and support equipment, such as computers and a power source. The spring 

displacement on the heavy car’s suspension is measured to estimate the load (dynamic and 

static) at the wheels near the sensor system. 

Figure 12 shows typical sensor images. The video images show the line lasers intersecting 

the top of rail. An odometer is also displayed, showing the location of the measurement along 

the track (corresponding GPS measurements are recorded but not shown here). The video 

images are recorded and post-processed. The lasers are first isolated using color filtering and 

threshold algorithms. The curve of each laser is then fitted with a second order polynomial. 

Finally, the minimum distance between these two polynomials is found analytically. Curve 

fitting a polynomial makes the system robust to small variations in rail head profile (Norman, 

2004). 

Figure 12. Typical Test Images 

1.6.2 Test Site A–System Verification 

The first test presented is for a 10m (33’) section of tangent (straight) track. The sensor 

readings are compared to trackside measurements and the dynamic simulation. 

21
Figure 13. Site A: Trackside, Simulation, and Field 

Figure 13 shows the results where the black dots represent trackside measurements of 

modulus. Linear interpolation was used to estimate the track modulus between these five 

discrete measurements (straight solid line). This estimate of track modulus was used as the 

input to the track model used in the simulation. The sensor system’s performance was then 

simulated using the train car dynamics (Equations 44 and 45) and sensor mathematics 

(Equation 52) described above. The simulated modulus measurements are shown as a dashed 

line. The final curve (solid) represents the real measurements made by the sensor system. 

The simulation (dashed) gave very similar results to the trackside measurements. It is 

expected that the simulation results are similar to the trackside measurements because the 

trackside measurements were used as the input. However, small discrepancies occur because 

of dynamic effects and angular motion of the side frame. 

The measurements made by the sensor system also match the trackside measurements very 

well. The maximum discrepancy between the trackside modulus measurement and the on 

board sensor measurements occur at 22.3 m with a maximum error of ~5 MPa (709 

lbf/in./in.). This error is well within an acceptable tolerance for the sensor system. 

22
Figure 14. Track Geometry at Test Site A 

Figure 14 shows the track geometry measurements for Test Site A. The absolute height of the 

rail was measured for both rails using a surveyor’s transit. The figure also shows cross-level 

measurements (height of one rail with respect to the other). Such variations in track geometry 

have been a limitation to previous systems (Carr, 1999). These results show this sensor 

system, based on relative measurements, is robust to such variations. Extreme variations in 

track geometry will cause errors in modulus measurement if the errors occur over short 

distances (~1m). Such dramatic geometry changes are, however, not acceptable for 

commercial rail and are detectable with other commonly used track geometry systems 

(Norman, 2004). 

1.6.3 Test Site B–Usefulness of The Discrete Model 

Test Site B is a section of tangent Class III track. In Figure 15, the sensor measurements 

(dark solid line) are compared with track side measurements (boxes) and simulation. This test 

section is interesting because the real sensor measurements matched the trackside 

measurements well at each of these discrete locations (12.7, 14.22, 16.26, and 17.78 m 

referenced from the nearest milepost). The agreement between the sensor readings and the 

trackside measurements validates the sensor system. 

If it is assumed that the track modulus varies linearly between these discrete measurements, 

however, large disagreement exists between the sensor readings and this linearly interpolated 

estimate of modulus. The discrete tie model, where each tie can take on a different stiffness 

value, is needed to estimate track modulus at these intermediate locations. Using this model 

allows variable tie stiffness to be included in the simulation. Because of the agreement 

between the sensor readings and the independent discrete trackside measurements it is

believed that the variation between the trackside measurements, is real (softer and/or stiffer 

23
ties) and that the discrete tie model is needed to produce a good estimate of modulus between 

these locations. 

The results of the simulation using the discrete tie model and placing the four stiffer ties and 

two softer ties the simulation results (dashed) matched the sensor system measurements 

(solid). Allowing individual ties to have different stiffness values led to a simulated estimate 

that matched both the sensor readings and the trackside measurements. 

Figure 15. Mile Post 31 East: Trackside Data, Discrete Model Simulation, and 

Sensor Measurements 

Measurements of track geometry were made at Test Site B (similar to Figure 14), and no 

correlation existed between changes in geometry and changes in track modulus (Norman, 

2005). This again suggests that the measurement system can accurately estimate track 

modulus in the presence of normal variations in track geometry. 

1.6.4 Test Site C–Effects of Speed 

In Figure 16, Test Site C is a 60 m (200’) section of Class IV tangent track where several 

measurements were repeated at different train speeds 8, 16, 32, 48 and 65 km/hr (5, 10, 20 

,30, 50 mph). It can be seen that modulus measurements are consistent at each of these train 

speeds with an average standard deviation between the measurements of only 345 lbf/in./in. 

At higher train speeds where abrupt variations (spikes) exist in track modulus, an 

overestimate of these variations occurs. This suggests slight error in measurements resulting 

from dynamic loads. 

24
Figure 16. Effects of Speed–Test Site C 

Figure 17 shows the averages of several measurements for both rails at Test Site C. These 

results show good agreement between the rails with two notable exceptions where the right 

rail is stiffer than the left and where the left is stiffer than the right. These can be generally 

attributed to neighboring crossties that are not parallel. This causes two ties to be very close 

on one rail and have wider spacing on the other. 

Figure 17. Variation between Rails–Test Site C 

25
2 SUMMARY AND CONCLUSIONS 

The modulus of railroad track plays a major role in its ability safely support rail vehicles. 

This research presented the design, modeling, and testing of a sensor system capable of 

measuring track modulus over long distances from a moving railcar. 

The report presents the Winkler Model and a derivation of a discrete tie model as methods to 

relate track deflection to track modulus. These models are used to calculate track modulus 

from the measurements made by the sensor system. The sensor system was modeled to show 

how the measurements are related to track modulus. In addition, a dynamic model of a railcar 

is presented, and this model is used to simulate sensor output. 

The proposed sensor system was tested at several locations on various types of track. 

Independent trackside measurements of track modulus and simulations results were used to 

verify the accuracy of the sensor system. The testing showed that the system was able to 

measure the correct trend in track modulus (Test Sites A and B). They also suggest the 

system’s ability to correctly resolve modulus changes from tie to tie (Site B). Finally, test 

results showed the system’s ability to consistently measure track modulus at train speeds up to 

65 km/hr (40 mph) and to identify modulus variations between both rails (Site C). 

26
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27
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Car,” Research Results, RR01-01. 

http://www.fra.dot.gov/downloads/Research/rr01_01.pdf

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Measurements 5th Edition, Addison-Wesley Publishing Company, Inc. Reading, MA: 

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28
29 

ABBREVIATIONS & ACRONYMS 

BNSF Burlington Northern Santa Fe 

FRA Federal Railroad Administration 

ft feet 

GPa gigapascal 

GPS Global Positioning System 

hr hour 

in. inch 

km kilometer 

kN kilonewton 

lbf pounds-force 

LVDT linear variable differential transformers 

m meter 

mi mile 

mm millimeter 

MN meganewton 

MPa megapascal 

mph miles per hour 

OPPD Omaha Public Power District 

psi pounds per square inch 

TLV track loading vehicle 

UP Union Pacific Railroad 

UPRR Union Pacific Railroad 

VTM vertical track modulus