Resilient Modulus of Soft Soil beneath High Speed ...

Heelis et al

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Resilient modulus of soft soil beneath high speed rail lines 2

Michae l Heelis, Research Assistant , School of Civil Eng ineer ing, Nott ingha m Univers ity, NG7 2RD, -44-1159513907, Fax –44-115-9513898, [email protected]. Andrew Dawson, Senior Lecturer , School of Civil Engineering, Nottingham University, NG7 2RD, -44-1159513907, Fax –44-115-9513898, [email protected]. Andrew Collop, Lecturer, School of Civil Engineering, Nottingham University, NG7 2RD, -44-115-9513907 Fax –44-115-9513898, [email protected] David Chapman, Lecturer, School of Civil Engineering, Nottingham University, NG7 2RD, -44-115-9513907 Fax –44-115-9513898, [email protected] Victor Krylov, Professor of Acoustics and Vibrations, Department of Civil Engineering, Nottingham Trent University, Burton Street, NG9 1PZ. -44-115-9486450, Fax –44-115-9486450, [email protected]

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INTRODUCTION Due to the increasing congestion in the skies over the World’s major conurbations, passenger travel by airplane is increasingly seen as a non-environmentally friendly method of mass transit. The alternative of High speed railroads has been identified Worldwide as the form of transit that will see in the New Millenium for medium distance (< 1000 km) journeys. For example, there are the Channel Tunnel Rail Link in the United Kingdom, the T.G.V. in France and the I.C.E in Germany and the Shinkansen (‘Bullet’) trains of Japan. The building of the railroad structure itself is a significant cost to such a project and so recently there has been an increase in research in this area. Many of the design problems are caused by the tendency of modern railroad embankments to be flexible because the embankment height is limited by the two requirements that visibility and noise pollution should be reduced as much as possible. On ground with a low resilient modulus, track-soil bending waves may result in significant transient train-induced soil deflections while ride quality and maintenance costs also have a direct relationship to the magnitude of these deflections. Several analytical models can been used to predict the propagation of bending waves in the track/embankment system. A model of a Euler beam on a Winkler foundation is studied and used to demonstrate how the magnitude of the resulting displacements are dependent on train speed and track damping. The methods by which the model parameters may be calculated are discussed and their relative advantages and disadvantages are considered. From this study it is possible to form practical suggestions on methods by which the design of rail tracks can be adjusted to limit the maximum transient deflections as a train passes. MEASUREMENT The dynamic movements of the soils as trains pass over them are typically modelled by loads crossing a homogeneous elastic foundation. The use of Finite Element Modelling can incorporate the concept of layers of soil with different geotechnical properties. Either form of modelling requires the accurate input of geotechnical data. Selig and Waters (1) used the Geotrack model to calculate the strain in the sub-ballast of a railway embankment under static loading. Their investigation indicated that strains of 4x10-4 can be expected for typical modern trains. As the sub-ballast rests directly on the subgrade it would be expected that this would also be the expected maximum value of strain experienced by the subgrade. Selig and Waters (1) do not investigate low modulus soil conditions but their analysis gives us an indication of the order of magnitude for the expected strain levels. Ishihara (2) suggests that, for these medium levels of strain, visco-elastic models are appropriate and, therefore, not only are shear moduli and Poisson’s ratios required but also a measure of damping. Hunt (3) and

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Fryba (4) have already shown how the level of system damping is a major factor in the magnitudes of the transient vibrations of trains. Laboratory Measurement To determine the correct stress-strain response (incorporating stiffness, visco-elastic and damping effects) is not a straightforward task. Initially, repeated load triaxial testing was considered. However, the triaxial test is not appropriate for two main reasons. Not only are the strain levels that must be applied to the sample significantly higher than are to be expected in the in-situ material, but also it is difficult to measure damping with a cyclic triaxial test. The piece of equipment more suited for these typical ballast strain levels, and for the calculation of damping, is the resonant column apparatus (2). Peat samples from the Rainham Marsh site on the proposed Channel Tunnel Rail Link (U.K.) route have been tested using a resonant column machine at the Instituto Superior Tecnico, Lisbon, Portugal. The standard geotechnical tests were also performed on the peat samples. The peat was found to have moisture content of 450-500 % from standard moisture content tests (at 1050C). The dried specimens were further dried in an oven at 4500C and the ash content was found to be 25% indicating that the solid material was some 75% organic (5). The specific density of the peat was calculated to be 1.86. Samples of peat were tested and found to have a unit weight of 9.82 kN/m3. In the resonant column test, a solid cylindrical column of soil specimen is fixed in place in a triaxial cell and set in motion in a torsional mode of vibration. The frequency of the drive system is changed until the first torsional resonance of the soil specimen is encountered. With this resonant frequency identified, and the sample geometry and conditions of end-restraint known, it is possible to back-calculate the velocity of shear wave propagation through the soil specimen. The wave velocity can be used to calculate the shear stiffness of the soil specimen once the density of the specimen has been calculated. This procedure is repeated several times for different amplitudes of driving force and, hence, imposed shear-strain levels, and also for different confining pressures. The process is specified in ASTM D 4015 (6). The results for a fully saturated sample (the samples were from a depth of approximately 5m) and for an overburden pressure of 50 kPa are shown in Figures 1 and 2. Note the shear stiffness is apparently increasing with time even after a consolidation time of approximately 8000 minutes. This is typical for peat (7,8) and creep may continue for as long as the sample is loaded, this would preclude the building of an embankment directly on the peat layers and some form of soil stabilization would be required.

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In-situ Measurement A Continuous Surface Wave System (CSWS) supplied by GDS Ltd, Surrey U.K. has also been used on a railway line to provide estimates of the resilient modulus of the soil. Two tests were performed on a 6m embankment overlying a peaty clay subgrade. The first test was performed with geophones on the ties of a railway line whereas the second test was performed on the embankment adjacent to the first test but off the railway line (on the line of track no longer in use, Figure 3). Therefore, the difference between the two tests was due to the presence of ties and the track in the first test. The results are plotted in Figure 4. It can be seen that for the off-track data the upper 2 - 2.5 m of the soil stiffness profile is approximately constant with depth indicating the response of the ballast. There is then a section down to a depth of approximately 7 –8 m with a uniform increase, indicating the stiffness of the imported clay that makes up the embankment. After which there is a drop in the soil stiffness at a depth of approximately 9 –10 m and then another section with an increase in soil stiffness with depth. This indicates the peaty clay subgrade that is less stiff than the clay of the embankment itself. Note that there appears to be an increase in the shear soil stiffness reported for shallow depths when the test was performed between the tracks and ties. This is due to the stiffness of the track, which is, of course, much stiffer than the ballast or subgrade. The reduction in stiffness (for the test between the track and ties) between 5 and 9 meters implies that the wave velocity is less for a particular wavelength of vibration (between 15 and 25 m) (Figure 5). The wave velocity is no longer associated with shear waves in the ground (or Rayleigh waves) but with the bending waves in the track/embankment system. PREDICTION The simplest analytical approach commonly used to model ground response to a travelling train is a two dimensional model of a point load moving on an infinite beam on an elastic foundation comprising of a series of discrete springs. This model is linear elastic and, therefore, superposition can be used to provide the solution for loads applied by a multi-axle train. Because the springs are discrete there is no shear coupling in the ‘soil’ model along the axis of the beam. It is recognized that this model may not cover the complexities of non-linear and time-varying organic soils, such as illustrated in Figure 1. However, as a first approach it is useful it does provide a model that can be solved relatively simply in closed form (4). A Kerr (9) or a similar model could model shear interaction. However, Kneifati (10) showed that the Winkler model is sufficient to predict deflections away from any end effects of the beam. The railway track under consideration is modeled as an infinite beam and, therefore at least initially the Winkler model is an appropriate initial model. The defining differential equation of this problem is,

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EI

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d 4 v( x, t ) d 2 v( x, t ) + + kv ( x , t ) = Pδ ( x − ct ) µ dx 4 dx 2

(1)

where EI is the flexural rigidity of the beam resting on the foundation, v is the vertical displacement, x is the distance along the beam, t is time, k is the stiffness coefficient of the Winkler foundation, µ is the mass per unit length of the beam, P is the load and δ(x - ct) is the Dirac function of a point unit load moving with velocity c in m/s. It has to be decided whether EI represents the bending stiffness of the rail only, or the rail and ballast, or even, the rail, ballast and embankment. The literature indicates different views on this, with Hunt (3) using the entire embankment and Fortin (11) using only the rail and ballast. Both authors claim good correlation between their models and measured data. In order to produce finite oscillations a damping factor is introduced to form a visco-elastic soil model. This is suitable for the strain levels expected (2) and has damping which is proportional to the vertical velocity and, hence, Equation (1) becomes (4):

EI

d 4 v( x, t ) d 2 v( x, t ) dv ( x , t ) + µ + 2 µω b + kv ( x , t ) = Pδ ( x − ct ) 4 2 dx dx dx

(2)

where ωb is defined as the circular frequency of damping (4). 2µωb is often defined as C (not to be confused with c, the velocity) the damping coefficient in Ns/m/m. The solution to the above equations is highly dependent on the velocity, c and the damping of the system. These are characterised by two non-dimensional parameters, α, and β, defined by

 µ2 α = c   4 kEI

  

1

(3)

4

(4)

 1   β = C   4kµ 

Note that β in this equation is identical to the damping coefficient that was calculated in the resonant column test. The solution provides the displacement, v(x), at any point at distance, x, from the applied load which is expressed in terms of another non-dimensional parameter, ϖ(x), that is defined by,

(

2 2 4 k 3 EI ϖ (x ) = v (x ) P

)

(5)

The solution for the static problem gives a displacement directly under the applied load equal to ϖ = 1. It should be noted that the displacement is not directly proportional to the coefficient of subgrade reaction, 1/k,

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but instead varies with k-3/4. The displacement is proportional to the applied load as expected. In Figure 6 the non-dimensional displacement directly under the applied load is plotted for a variety of levels of damping and for different non-dimensional speeds. The effect of damping is, evidently, very important for the resulting displacements. The displacement under the load relates directly to the ride quality of the train as a whole, however this is not always the maximum displacement that the track experiences. The maximum displacement, however, does not necessarily occur directly under the applied load and, therefore, this parameter is plotted in Figure 7. This quantity relates to the movement in the ballast and subgrade and therefore the amount of maintenance the section of the track will require. Note that there is not such a large reduction in the magnitude of the maximum displacement after the critical speed has been exceeded. In Figure 8 the predicted response of the track is plotted for a train moving right to left. A constant resilient modulus is assumed and, therefore, the principle of superposition may be used and the response of a real train can be simulated by the addition of several point loads with offsets that represent the axle spacing. Care should be taken to ensure that the wavelengths of the vibrations are not such that the vibrations from neighboring axles constructively interfere. MODELLING The main problem with the concept of the coefficient of subgrade reaction (the Winkler model) is how to relate the coefficient of subgrade reaction, k, to the measurable resilient modulus, Es, and Poissons Ratio, νs, for the of soils. For a particular embankment on the East Coast Mainline in the United Kingdom studied by Hunt (3) the values of k using the various methods available are compared. For the subgrade, Es = 10 MN/m2, and νs = 0.3 (this is approximately equal to Gs = 3.5 MN/m2 somewhat more compliant than the soil tested from the Rainham Marsh site, where Gs rises to > 10 MN/m2 after 8000 minutes, Figure 1). Following Hunt (3), the beam is assumed to be part of the embankment that supports the track (Figure 9). The flexural stiffness of the beam (EI) is calculated to be 152.5 MNm2 and the width of the beam resting on the subgrade (the Winkler foundation) is 5.6 m. Five methods (Table 1) are now used to estimate the appropriate value of k. In the case of the Vlaslov model (12) an assumption that the strain reduces exponentially with depth (hence µB = 1.5) is required. Finite element analyses using plane strain elements in two orthogonal directions. The first takes a cross-section through the track/ballast and subgrade (similar to Figure 9). The second Finite Element Model is at right angles to this along the track (in the direction of train movement). The values of coefficient of subgrade reaction were calculated by dividing the applied load by the magnitude of the deflection

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under the applied load. It was found that the results for all the methods gave results of approximately the same order of magnitude (Table 1). The Biot (13) and Vlaslov (12) models give similar results for a large range of subgrade Resilient Modulus (with a Poisson ratio kept constant at 0.3). However, the values for the Vesic (14) model give significantly higher values of coefficient of subgrade reaction (Figure 10). If the Resilient Modulus of the subgrade is kept constant at 10 MN/m2 then the Vesic and Biot models both exhibit small and approximately linear variations with Poisson ratio (Figure 11). Note that if a Poisson ratio of 0.5 is used, then the Vlaslov model will give an infinite coefficient of subgrade reaction (Table 1). The approach suggested by Fortin (11) uses a different definition of the beam for the bending wave problem. In this case the beam comprises the rail only. The flexural stiffness (EI) of typical track can be calculated to be approximately 10 MNm2 (1). This will decrease the values calculated by Vesic, for example, by approximate 25%. However, the embankment is then taken as part of the subgrade and it would be expected that this would increase the effective stiffness of the foundation. The mass of the (‘rail’) beam also includes the mass of the ties and the ballast. Ongoing studies are being conducted into which model (Hunt or Fortin) is more appropriate or if a different model of the beam should be used. The problem with the Hunt model is that Hunt measured negligible displacements at the bottom of the embankment due to the passage of trains. This implies that it is not behaving like a true beam. The model ignores the stiffness of the track, which, as previously mentioned, does have a stiffness of a magnitude which is of the same order as that calculated for the embankment ‘beam’. It also ignores the mass of the beam and ballast. Fortin’s model ignores the embankment mass. The effect of the stiffness of the embankment could be incorporated into a layered soil model in order to calculate the coefficient of subgrade reaction. IMPLICATIONS In the modelling section it has been shown that the soil structure models can give values of the subgrade reaction that vary between 10.10 to 16.54 MN/m2 for the embankment previously considered. In order to make a comparison into how variations in the particular parameters effect the overall displacement, a particular solution will be used and then the parametric variations of ±10% will be considered (Table 2). The standard solution will be for an axle of weight 20 kN, travelling at 100 mph (45 m/s), on an embankment with flexural stiffness EI = 152.5x106 N/mm2, mass per unit length of µ = 20124 kg/m, and damping C = 57623.4 Ns/m/m (taken so that β = 0.05). The coefficient of subgrade reaction is, k = 16.5 kN/m/m. This is the same model as in the analysis

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performed by Hunt (3) and has a critical velocity of 70.60 m/s (this is the speed that bending waves travel along the system). Table 2 indicates that if k is increased by 10 % the critical velocity is increased by 2.4% (to 72.29 m/s) and the damping is decreased by 4.7% (to 0.048). However, the displacement parameter is not a linear function of these parameters, and therefore the simple relations in Table 2 (which apply for all values of parameters) cannot be formed for the displacement. Instead it is necessary to choose a particular speed and look at the variations about that speed (the speed chosen here will be that of a typical train (100 mph)). If the embankment height, h, is increased (say by +10%) then two of the defining parameters will also change. The mass per unit length of the embankment (assuming constant material density) will also increase by 10% and the second moment of area of the beam, I, will increase by 33.1 %. The effect of this on the displacement has also been examined. In Table 3 it is shown that a 10% increase in the embankment height will reduce the deflection underneath the load by 8.3%. DESIGN EXAMPLE It is desired to design a track for trains with a maximum axle load of 20kN. Trains speeds are not to be limited by the dynamic magnification effect. If the damping of the rail-subgrade system (β) is assumed constant at 5% then the maximum non-dimensional deflection for any speed is 2.6 times the static deflection value. It is possible to calculate the minimum permissible coefficient of subgrade reaction required to prevent deflections exceeding a certain value. In Figure 12 this lower limit has been plotted for a range of beam flexural stiffnesses. The coefficient of subgrade reaction can then be related to the Young’s Modulus and Poisson ratio of the subgrade by any of the three analytical methods or by using a Finite Element Model. For example, point A on Figure 12 requires a minimum coefficient of subgrade reaction of 18 MN/m/m, which Figure 10 shows that the matching Resilient Modulus values (with a Poisson Ratio of 0.35) are between 12 and 20 MN/m2, depending on which model used. If a particular site did not provide the resilient modulus then soil improvement techniques would be required.

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CONCLUSIONS The phenomenon of dynamic magnification of vertical displacements of a railway line crossing soft soil has been identified and analyzed. The model used can be defined using two non-dimensional parameters. The first relates to the relative speed of the train compared to the stiffness of the track foundation. The second nondimensional parameter relates to the damping of the overall system. The foundation model used is a Winkler model and, therefore, a coefficient of subgrade reaction is required. It is possible to relate the resilient soil modulus to the coefficient of subgrade reaction and to provide design criteria for trains crossing soils with a low resilient modulus.

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ACKNOWLEDGEMENTS The authors would like to acknowledge the assistance of Rail Link Engineering for providing suitable samples from the Rainham Marsh site in U.K. and, also, the Instituto Superior Technico, Lisbon, for providing the resonant column facility. The Engineering and Physical Science Research Council are also thanked for the funding of this research.

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REFERENCES 1.

Selig E.T. and Waters J.M., Track geotechnology and substructure management, Thomas Telford Publications, London, 1994.

2.

Ishihara, Soil behaviour in earthquake geotechnics, Oxford University Press, Oxford, 1996.

3.

Hunt G.A., Analysis of requirements for railway construction on soft ground British Railway Report No. LR TM 031, British Railways Research Derby, 1994.

4.

Fryba, L., , Noordhof Intrernational Publications, Groningen, Netherlands, 1972.

5.

ASTM D 2974, Standard Test Method for Moisture, Ash and Organic Matter of Peat and Other Organic Soils, ASTM, 1995.

6.

ASTM D 4015, Standard Test Methods for Modulus and Damping of Soils by the Resonant-Column Method, ASTM 1992.

7.

Berry and Vickers, Consolidation of Fibrous Peat, J. Geotech. Eng. Div. ASCE, Vol 101, GT 8, August 1975, pp. 741-753

8.

Boulanger R.W., Arulnathan R., Harder L.F. Jr, Torres R.A. and Driller M.W., Dynamic Properties of Sherman Island Peat, J. Geotech. and Geoenvironmental Engng, ASCE, Vol. 124, 1, January 1998, pp. 1220.

9.

Kerr A.D., Viscoelastic Winkler foundation with Shear Interactions, J. Engineering Mechancics, ASCE, 87, EM3, 1961, pp. 13-20.

10. Kneifati M.C., Analysis of plates in a Kerr foundation, J. Engineering Mechancics, ASCE, Vol 11, 11, 1985, pp.1325-1342. 11. Fortin J.P., Dynamic track deformation, French Railway Review, Vol 1, 1, 1983 pp.3-12. 12. Vlaslov V.L. and Leontiev N.H., Beams plates and shells on elastic foundations, Fizmatgiz, Moscow, 1956. 13. Biot A.M., Bending of an infinite beam on an elastic foundation, J.Applied Mechanics, ASME, Vol 4., 1937, A1-A7. 14. Vesic A.S., Beams on elastic subgrade and the Winkler hypotheisis, Proceedings 5th International Conf Soil Mech. Found. Engng, Paris, Vol. 1, 1963, pp. 845-850.

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LIST OF TABLES TABLE 1. Estimation of subgrade stiffness TABLE 2. Critical velocity and damping sensitivity to ±10% changes in defining parameters. TABLE 3. Displacement against changes in defining parameters

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TABLE 1. Estimation of subgrade stiffness Study

Equation

Coefficient

of

stiffness, k (MN/m2) Vesic

Biot

0.65 E s k= 1 − ν s2

12

Es B 4 EI 0.108

10.10

16.54

k=

0.95 Es  Es B 4    1 − ν s2  1 − ν s2 EI 

Vlaslov

k=

Es (1 − ν s ) µB (1 + ν s )(1 − 2ν s ) 2

Plane strain along track

Finite Element Analysis

14.61

Plane strain across track

Finite Element Analysis

10.45

(

)

10.10

soil

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TABLE 2. Critical velocity and damping sensitivity to ±10% changes in defining parameters. % change in critical velocity, vcr % change in

% change in damping factor, β

k

EI

µ

C

k

µ

+2.4%

+2.4%

-4.7%

+10%

-4.7%

-4.7%

-2.6%

-2.6%

+5.4%

-10%

+5.4%

+5.4%

parameter +10% -10%

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TABLE 3. Displacement against changes in defining parameters % change

+10%

% change in displacement c

C

µ

EI

P

h

+7.67%

-0.1%

+3.4%

-3.8%

+10%

-8.3%

-5.7%

+0.1%

-3.2%

+4.5%

-10%

+10.2%

-10%

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LIST OF FIGURES

Figure 1. Shear stiffness plotted against consolidation time. Figure 2. Properties of peat with respect to applied levels of strain. Figure 3. The location of the CSWS Testing. Figure 4. CSWS results from a railway embankment. Figure 5. Wave velocity plotted against wavelength for the CSWS Test. Figure 6. Non-dimensional plot of vertical displacement under a single axle. Figure 7. Non-dimensional plot of maximum vertical displacement. Figure 8. Deflection response of track to a single axle travelling at four constant speeds. Figure 9. Cross-section of the embankment Figure 10. Variation of coefficient of subgrade reaction with Resilient Modulus of soil for vs = 0.3. Figure 11. Variation of coefficient of subgrade reaction with Poisson ratio. Figure 12. Design criteria for Winkler model

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2

Shear Stiffness, Gs (MN/m )

12

10

8

6

4 0.1

1

10

100

1000

Time (min)

Figure 1. Shear stiffness plotted against consolidation time.

10000

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damping

1

6

0.8

5 4

0.6

3 0.4

2

0.2 0 1.00E-06

Damping ratio, %

G/Gmax

G/Gmax

1

1.00E-05

1.00E-04 Strain

Figure 2. Properties of peat with respect to applied levels of strain.

0 1.00E-03

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Test 1

Test 2

6 m imported clay fill

Peaty Clay

Figure 3. The location of the CSWS Testing.

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Off track (on embankment) 0

100

200

Between tracks and ties 300

400

500

0.00

-5.00

Depth (m)

-10.00

-15.00

-20.00

-25.00

-30.00 Shear Modulus, Gmax (MPa)

Figure 3. CSWS results from a railway embankment.

600

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Off track (on embankment)

Between track and ties

700

Wave Velocity (m/s)

600 500 400 300 200 100 0 0.00

50.00

100.00

Wavelength (m)

Figure 5. Wave velocity plotted against wavelength for the CSWS Test.

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3

Displacement/static displacement ϖ

β = 0.05 2.5 2

β = 0.1

1.5

β = 0.2

1

β = 0.5

0.5 0 0

0.5

1

Speed/critical speed

1.5

α

Figure 6. Non-dimensional plot of vertical displacement under a single axle.

2

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4

β = 0.05

Displacement/static displacement ϖ

3.5 3

β = 0.1

2.5 2

β = 0.2

1.5 1

β = 0.5

0.5 0 0

0.5

1

Speed/critical speed

1.5

α

Figure 7. Non-dimensional plot of maximum vertical displacement.

2

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-8

5

-8

x 10

5

4

4

3

3

2

2 Response / m/N

Re sp 1 on se 0 / m/ -1 N

1 0 -1

-2

-2

-3

-3

-4

-4

-5 -200

-150

-100

-50

0 50 Distance / m

100

150

x 10

-5 -200

200

-150

-100

α = 0.5 (slow train) 5

4

4

3

3

2

2 Response / m/N

Response / m/N

100

150

200

150

200

-8

x 10

1 0 -1

-1 -2 -3

-4

-4

-100

-50

0 50 Distance / m

α = 1 (critical speed)

100

150

200

Direction of travel

0

-3

-150

x 10

1

-2

-5 -200

0 50 Distance / m

α = 0.9

-8

5

-50

-5 -200

-150

-100

-50

0 50 Distance / m

100

α = 1.4 (super critical train)

Graphs are for damping = 0.05, with EI = 152.5x106 Nm2, µ = 20124 kg/m, k = 16.5x106 N/m/m and, hence, c= 57623.4 Ns/m, speeds are proportion of critical speed of system = (4 EI k/µ2)1/4 = 70.7 m/s.

Figure 8. Deflection response of track to a single axle travelling at four constant speeds.

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3m

1 2

Beam

5.6 m

Figure 9. Cross-section of the embankment

2.6 m

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Coeficient of Subgrade Reaction 2 MN/m

Vesic

Biot

Vlaslov

180 160 140 120 100 80 60 40 20 0 0

20

40

60

Resilient Modulus of Soil MN/m

80

100

2

Figure 10. Variation of Coefficient of subgrade reaction with Resilient Modulus of soil for vs = 0.3.

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Coficient of Subgrade Reaction 2 MN/m

Vesic

Biot

Vlaslov

30 25 20 15 10 5 0 0.2

0.25

0.3

0.35

Poisson Ratio of Soil MN/m

0.4

0.45

2

Figure 11. Variation of coefficient of subgrade reaction with Poisson ratio.

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Minimum Coefficient of subgrade reaction (MN/m/m)

100 1 mm deflection A

10

5 mm deflection 1 10 mm deflection

0.1 0

50

100

150

200 2

Flexural Stiffness of the beam (MNm )

Figure 12. Design criteria for Winkler model