ON THE RELATIONSHIP BETWEEN LOAD AND DEFLECTION IN RAILROAD TRACK STRUCTURE

Transcription

1 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 Mahmood Fateh, Gary Carr Federal Railroad Administration Office of Research and Development 1200 Ne Jersey Avenue SE Washington, DC ABSTRACT Track Modulus, defined as ratio beteen the rail deflection and the vertical contact pressure beteen the rail base and track foundation, is an important parameter in determining track quality and safety. The Winkler model is a idely used mathematical epression 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 beteen the rail base and track foundation increases linearly ith vertical deflection. Hoever, it is idely 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 ne model of track deflection here the elastic foundation beneath the rail has a cubic polynomial relationship beteen applied pressure and vertical deflection. This ne cubic model is compared to other models of railroad track structure, including the Winkler, Pasternak, and Discrete Support models, as ell as ith eperimental data. It is shon that the cubic model is a better representation of real track structure. INTRODUCTION Background The relationship beteen 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, hich 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

2 Lu et al. 2 resulting vertical deflection. Track stiffness relates load to deflection hile track modulus relates a distributed load to deflection. Railay 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 heels and is supported on a tie pad and held in place ith fasteners to crossties. The crossties rest on a layer of rock ballast and subballast used to provide drainage. The soil belo 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 ith increasing subgrade resilient modulus, and decreases ith increasing subgrade layer thickness 2. Ballast layer thickness and fastener stiffness are the net most important factors 2, 4. Increasing the thickness of the ballast layer and or increasing fastener stiffness ill 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 lo track modulus and large variations in track modulus are undesirable. Lo track modulus has been shon 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 shon 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 shon 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 shon that at high speeds there ill 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 ith higher vehicle speeds up to a critical speed 11. It is suggested that track ith a high and consistent modulus ill allo for higher train speeds and therefore increased performance and revenue 11.

3 Lu et al. 3 An improved mathematical understanding of the relationship beteen loads and deflections ill 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 shos a free load and deflection diagram of the rail under a one-heel load Figure 1, top. Here, the rail is considered as a continuously supported beam here represents the distance along the beam and represents the vertical beam deflection. The approimation 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, here + P p d. The supporting structure supports the bottom of the rail ith a reaction distributed force, q. = 0 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 and p causes curvature in the beam as given by the folloing differential equation: 4 d EI = q p, or 4 d 1 4 d EI + p = q 4 d The solution to the differential equation is dependent upon the boundary conditions of the beam as ell as the loading conditions. A free body diagram that shos sections of the beam is shon 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, on each half of the infinite beam, or: P p d = 0 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.

4 Lu et al. 4 d d = 0 = 0 2b The above differential equation and boundary conditions can no be set up and solved in different ays to represent various track behavior. Four such solutions are defined in Section 3. FIELD MEASUREMENTS OF TRACK MODULUS Figure 3 shos the eperimental results of the track responses under various applied loads. Rail deflection as measured at given locations using linear variable differential transformers LVDTs as a short, slo moving train of knon eight passed. The ales of the train eighed N lbf, N lbf, and N 6890 lbf. The LVDTs ere 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 shon in Figure 3. Here the LVDTs ere placed at 1m 3ft increments along the track =1m, 2m, 3m, 4m. These measurements, along ith many others dating back to the Talbot Report 13 clearly indicate that the vertical rail deflections are not linearly proportional to the heel 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 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 ith each method assuming a different definition of track modulus that approimate the non-linear behavior of real track. Consider the definitions of track modulus represented in Figure 4 and described in the folloing sections. Beam On an Elastic Foundation BOEF Method The most straightforard method to estimate track modulus at a given track location is to simply measure the vertical deflection at the point 0= o of an applied knon 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 beteen rail supporting load p and deflection is linear and elastic i.e. p=u as in 2, 1. These assumptions lead to the Winkler model as described in Section 3.1. The resulting track modulus is given by:

5 Lu et al u = 4 EI P here: 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 0 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. Hoever, as shon in Figure 3, it is clear that this linear approimation 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 loads 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 loads, hich makes it more time consuming 2. Using a force balance this deflected area, or deflection basin, can be shon to be proportional to the integral of the rail deflection 2, 1: P d = uδ = q d = uaδ 4 here: P is the load on the track q is the vertical supporting force per unit length u is the track modulus δ is the vertical rail deflection A δ is the deflection basin area

6 Lu et al. 6 area beteen the original and deflected rail positions 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 etremely time consuming and only reveal the track modulus at a given point. As shon in Figure 3, these measurements could change dramatically for a point just centimeters aay. Heavy-Light Load Method Many have represented the load/defection curve as pieceise linear ith a lo stiffness at lo 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 lo stiffness is eperienced until the tie contacts the ballast resulting in a higher stiffness. This leads to a measurement of track stiffness using to loads, Figure 4, that are ideally both in the high stiffness range e.g. slack is removed 16, 6, 17. here: P2 P1 k = k is the track stiffness P 1 and P 2 are the applied loads 1 and 2 are the corresponding deflections Again, a linear assumption is used to then transform the stiffness measurements of the to loads to track modulus substitute P k = o into Equation 3. The clear difficulty ith this measurement is that the real load/deflection relationship is not pieceise linear and the resulting stiffness varies ith the selection of the to loads, P 1 and P 2. 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

7 Lu et al. 7 defined as the nominal ale load for a given freight line e.g. 160kN or 286,000/8=36kips. This can be epressed mathematically as the derivative of the pressure deflection curve evaluated at the characteristic load P*: u * p = * P 6 here: 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 ith the linear assumptions as described above or ith 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=u. The BOEF differential equation then becomes: 4 d EI + u = q 4 d 7 This model has been shon 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,, as a function of longitudinal distance along the rail referenced from the applied load is given by: Pβ 2u β = e [ cos β + sin β ] 8 here:

8 Lu et al. 8 4 u β = EI 1 4 here: 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 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-ale coal hopper is shon in Figure 5. The deflection is shon relative to the heel/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 ith an elastic modulus of GPa 30,000,000 psi and an area moment of inertia of 2704 cm in 4. The limitations of the Winkler model are clear given the idely accepted non-linearity of track structure. Hoever, this model is often used because it does provide a clear closed form solution to the relationship beteen load and deflection in track structure. Discrete Support Model A second model assumes a similar linear relationship beteen 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 hen 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 heel 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

9 Lu et al. 9 computational requirements, the rail is assumed to etend beyond the ties and is fied at a large distance from the last tie. This ensures the boundary conditions are ell defined the rail is flat, far aay, or =0 and =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 unknons. The moment and shear force in the cantilevered sections of the model Figure 6A and C can no be calculated. Static equilibrium requires the moment, for Section A, to be: M =, and M = 2 M + C VC 2 1 M + A VA 1 9 Similar equations can be ritten for the sections of beam beteen each of the discrete supports. This leads to N+4 equations here N is the number of discrete supports used in the model. No, the total system energy can be ritten 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 M 2EI i = d + F 2k 2 i i 10 Where M i 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, F i is the force in each support spring and k i is the stiffness of each spring. Castigliano s theorem can no be used to create equations needed to solve for the unknon spring forces and boundary conditions moment and shear forces: U F i U = M A U = M B U U = 0, and = = 0 V V A B 11 With these relationships, a set of N+4 equations and N+4 unknons can be developed by substituting the moment epressions into Equation 12. These epressions can be ritten in matri form: 12

10 Lu et al. 10 MF = P here: P is the load vector M is a N+4 N+4 matri of the eternal 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. Hoever, the discrete model has the additional ability to represent non-uniform track. Figure 7B compares the deflections from the to models for uniform modulus and a single applied load. The continuous line represents the Winkler model and the boes indicate the tie locations in the discrete model. The track modulus used in the Winkler model as 20.7 MPa 3000 lbf/in./in. and the corresponding tie stiffness as 10.5 MN/m 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 lbf as applied over the center tie. The deflection predicted by both models is very similar ith a maimum variation of 6.47%. The clear advantage of the discrete support model is that it can represent non-uniform track. In Figure 7C the stiffness of the 3rd tie from the left end has been decreased by 50% to 5.25 MN/m or lbf/in.. In Figure 7D, the stiffness of the 3rd tie has been increased by 100% to 21.0 MN/m or lbf/in.. The track deflection ith a single soft tie Figure 7C is no longer symmetric about the loading point. The rail is deflected more on the left side of the load here the soft tie is located. The maimum deflection of the rail as also slightly increased by approimately mm in.. Figure 7D shos the rail deflection here the stiffness of the 3rd tie has been doubled to 21.0 MN/m lbf/in.. The discrete model shos that the deflection near the stiff tie and the maimum deflection have both decreased by approimately mm in.. The results from these eamples sho that 1 the to models give similar results for similar inputs, and 2 the deflection curve can be affected by a single tie.

11 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, knon as the Pasternak foundation, vertical displacement of one section of the elastic foundation ill result in displacement of neighboring sections of the elastic foundation e.g. a mattress here the springs are tied together. This distinction is most prevalent hen the beam has lo bending stiffness i.e. lo EI. Here, the supporting distributed load, p is given by: 2 d p = G 2 d p + u p 13 here u p is a track modulus and G p is a shear modulus. Substituting into Equation 1 gives the folloing governing differential equation: 4 d EI 4 d 2 d G p + u p = q d 2 14 The solution from Kerr**** for a single applied load P acting at =0, is P = 2u p 2 β α e ακ [ κ cos κ + α sin κ ], < < 15 here: u p β 2 = ; α, κ = ± β 4EI 2 ± G p 4EI 16 The resulting relationship beteen applied load, P, and deflection, o, is still linear as in the Winkler Model, Figure 8. Here G p =60GPa8,702,400psi, I=3663cm 4 88in 4, E=206.8GPa 30,000,000psi are used as parameters. Hoever, the effective stiffness of the Pasternak model is higher because more of the elastic foundation is involved in producing reaction supporting pressure. The difference beteen the Pasternak model and the Winkler model are more evident hen either the beam is not stiff lo EI or the shear modulus is high. Figure 9 shos the correlation beteen the deflections of the to models, for an identical beam under identical loads, ith to 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.

12 Lu et al. 12 Nonlinear Cubic Model The limitation ith 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 shoed that the vertical rail deflections ere not linearly proportional to the heel loads. An etensive eperimental study conducted by Zarembski and Choros 14 also clearly documented this nonlinear response. Here, a ne model is proposed that represents the relationship beteen vertical rail deflection and the rail support distributed load as a cubic polynomial. To define this relationship the eperimental results of Zarembski and Choros 14 are plotted in Figure 10 along ith a cubic polynomial curve fit. The polynomial fit is ecellent R 2 = Using a cubic polynomial has several advantages. First, it clearly captures the behavior of real track Figure 10 in that it provides for lo stiffness at lo loads and higher stiffness at higher loads. Also, negative displacement of the track track lift does not result in significant donard 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 don. Here, the supporting distributed load p has a cubic relationship beteen p and : p = u + u Note, that symmetry about the applied load requires the second order term to vanish. Substituting into the BOEF model gives the folloing differential equation. 4 d 3 EI + u1+ u3 = q 4 d 18 Equation 19 is a nonlinear differential equation and a closed form analytical solution is not straightforard. One analytical approimation based on the Cunningham s method can be found in McVey 23. Hoever, a numerical solution for this Boundary Value Problem BVP can be obtained. The BVP can be ritten in state space notation as: = func, 19

13 Lu et al. 13, func = = Given equation 19 the BVP becomes: + = u u EI As the name implies, the fourth order BVP described above requires the value of four boundary conditions, here: o = = = = = = = = No, since the BVP can have more than one correct solution, an initial guess for the last boundary condition that ill cause the solution to converge to the epected solution. In this ork, the initial guess is provided by the Winkler model evaluated at =0 and u=u 3 given by: here: 2 0 = = = EI u u P o β β 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 d u u The unique solution that satisfies all these constraints ill give the rail deflection. Any number of numerical techniques can be used to solve this ell posed BVP. In this ork the bvp4c function in Matlab 24 as used

14 Lu et al. 14 As the cubic model closely represents the deflection test data for the hole range of heel 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 ill be spread over a ider span than it is for the linear model as shon in Figure 11. Meanhile, the deflection at the contact point for the cubic model is slightly larger than the one for the Winkler model hen 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: p = 3 u + u * 1 3 u = = u1 + * P * P 3u 3 2 * P 25 This definition of track modulus is compared to the Winkler model for a given measurement of load of N 34kips and displacement of 0.254cm 0.1 in Figure 12. In this Figure the load deflection curve is plotted from the eperimental data of Zarembski and Choros 14 shon in Figure 10. It is clear that for single data points at higher loads the Winkler model ill alays underestimate the actual track modulus Figure 12. The Winkler model ill also poorly represent changes in deflection ith respect to changes in load at these higher values. It is also clear from these data that any to choices of loads as in the Heavy-Light load definition of track modulus ill give a different value of track modulus. CONCLUSIONS Due to the idely 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 approimation to the nonlinear behavior, but there are still some discrepancies beteen the pieceise linear approimation and the real continuous nonlinear track behavior. The cubic model clearly captures the behavior of real track in that it provides for lo stiffness at lo loads and higher stiffness at higher loads. It represents the real track structure under the hole range of heel loads.

15 Lu et al. 15 Combined ith the proposed definition of track modulus at characteristic load, the cubic model can sensitively demonstrate the changes in deflection ith respect to the changes in load at higher values, hich the linear Winkler model ill poorly represent. ACKNOWLEDGEMENTS This ork is supported under a grant from the Federal Railroad Administration. The authors ould specifically like to thank Mahmood Fateh and Gary Carr ith FRA and William GeMeiner of the UPRR. We ould also like to thank BNSF and UPRR for track access and logistical support. REFERENCES [1] Cai, Z., Raymond, G. P., and Bathurst, R. J., 1994, Estimate of Static Track Modulus Using Elastic Foundation Models, Transportation Research Record 1470, pp [2] Selig, E. T., and Li, D., 1994, Track Modulus: Its meaning and Factors Influencing It, Transportation Research Record 1470, pp [3] Li, Dingqing, and Selig, Ernest T., 1994, Resilient Modulus for Fine-Grained Subgrade Soils, Journal of Geotechnical Engineering, Vol. 120, No. 6, pp [4] Li, Dingqing, and Selig, Ernest T., 1998, Method for Railroad Track Foundation Design I: Development, Journal of Geotechnical And Geoenvironmental Engineering, Vol. 68, No. 7-8, pp [5] Steart, Henry E., 1985, Measurement and Prediction of Vertical Track Modulus, Transportation Research Record 1022, pp [6] Read, D., Chrismer, S., Ebersohn, W., and Selig, E., 1994, Track Modulus Measurements at the Pueblo Soft Subgrade Site, Transportation Research Record 1470, pp [7] Ebersohn, W., Trevizo, M. C., and Selig, E. T., 1993, Effect of Lo Track Modulus on Track Performance, International Heavy Haul Association, Proc. Of Fifth International Heavy Haul Conference pp [8] Zarembski, A. M., and Palese, J., August 2003, Transitions eliminate impact at crossings, Railay Track & Structures, pp [9] Davis, D. D., Otter, D., Li, D., and Singh, S., December 2003, Bridge approach performance in revenue service, Railay Track & Structures, pp

16 Lu et al. 16 [10] Carr, Gary A. and Greif, Robert, Vertical Dynamic Response of Railroad Track Induced by High Speed Trains, Proc. of the ASME/IEEE Joint, pp [11] Heelis, M. E., Collop, A. C., Chapman, D. N. and Krylov, V., 1999, Predicting and measuring vertical track displacements on soft subgrades, Railay Engineering. [12] Kerr, Arnold D. On the Stress Analysis of Rails and Ties, Proceedings American Railay Engineering Association, 1976, Vol. 78, pp [13] ASCE-AREA Special Committee on Stresses in Railroad Track, Bulletin of the AREA, First Progress Report Vol 19,1918, Second Progress Report Vol 21,1920. [14] Zarembski, Allan M. and Choros, John. On the measurement and calculation of vertical track modulus, Proceedings American Railay Engineering Association, 1980, Vol. 81, pp [15] Kerr, A. D. and Shenton, H. W.,1986, Railroad Track Analyses and Determination of Parameteres, Journal of Engineering Mechanics, Vol. 112, No.11, pp [16] Ebersohn, W., and Selig, E. T., 1994, Track Modulus on a Heavy Haul Line, Transportation Research Record 1470, pp [17] Kerr, A.D., 2003, Fundamentals of Railay Track Engineering, Simmons-Boardman Books, Inc., Omaha, pp [18] Arnold, R., Lu, S., et al, 2006, Measurement of Vertical Track Modulus from a Moving Railcar, AREMA Conference Proceedings [19] Raymond, G. P., 1985, Analysis of Track Support and Determination of Track Modulus, Transportation Research Record 1022, pp [20] Meyer, Marcus B. 2002, Measurement of Railroad Track Modulus on a Fast Moving Railcar, University of Nebraska Lincoln. May [21] Boresi, Arthur P. and Schmidt, Richard J Advanced Mechanics of Materials 6 th Edition. John Wiley & Sons, Ne York, NY: Chap. 5,10. [22] Norman, Christopher D. 2004, Measurment of Track Modulus from a Moving Railcar, Masters Thesis, University of Nebraska Lincoln. August 2004.

17 Lu et al. 17 [23] McVey, Brian. 2006, A Nonlinear Approach to Measurementt of Vertical Track Deflection from a Moving Railcar, Masters Thesis, University of Nebraska Lincoln. May [24] Kierzenka J. and Shampine L. F., 2001, A BVP Solver based on Residual Control and the MATLAB PSE, ACM TOMS, Vol. 27, No. 3, pp

18 Lu et al. 18 Figure 1: Free Body Diagram of the Rail

19 Figure 2: Boundary Condition of the Rail

20 Figure 3: Deflection of Track Under Three Loads

21 Figure 4: Various Representations of Track Modulus

22 Figure 5: Relative Rail Displacement Under a Railcar

23 Figure 6: Discrete Model and Free Body Diagram

24 Figure 7: Comparison of Winkler and Discrete Models

25 Figure 8: Stiffness of Winkler and Pasternak Models

26 Figure 9: Comparison of Winkler and Pasternak Models

27 Figure 10: Eperimental Data and Curve Fitting

28 Figure 11: Comparison of Cubic and Winkler Models

29 Figure 12: Modulus Calculations in Winkler and Cubic Model

30 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: Eperimental Data and Curve Fitting Figure 11: Comparison of Cubic and Winkler Models Figure 12: Modulus Calculations in Winkler and Cubic Model