International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)645
Dynamic Analysis of Railway track
Sunita Kumari
1, Pragyan P Sahoo
2, Dr. V A Swant
31M.Tech Student, 2Research Scholar, 3Assistant Professor, Address, Deptt. Of Civil Engg. IIT Roorkee-247667,India
Abstract— In the present analysis, an infinite Euler– Bernoulli beam of constant cross-section resting on an elastic foundation is considered. The beam and foundation are assumed to be homogeneous and isotropic. The foundation is modeled using one parameter without damping. The beam is subjected to a constant point load moving with a constant speed along the beam. An effort has been made to find the solution of the governing differential equation analytically. The equations governing different responses such as beam deflection, bending moment have been obtained in closed form for undamped case. The maximum displacement occurs behind the moving load. The configurations of the steady state solutions may change as damping varies.At subcritical speed the absolute value of the deflection of the beam increases with increase of the load velocity. Peak maximum deflection appears when the load travels at the critical speed
Keywords— CVR, Euler–Bernoulli beam, Winkler beam I. INTRODUCTION
Recent findings indicate one of major causes of damages, which are attributed to the resonant behaviors, in a railway track. Basically, when a railway track is excited to generalize dynamic loading, the railway track deforms and then vibrates for certain duration. Dynamic responses of the railway track and its components are the key to evaluate the structural capacity of railway track and its components. The track vibrations can cause the crack damage in railway sleepers or fasteners, or even the breakage of ballast support. When the foundation is modeled as an elastic foundation, a critical velocity is found for the existing for moving load. Waves excited by a moving load with supercritical velocity propagate in a different way as they do when the load velocity is subcritical. Because high speed vehicles (e.g., trains and automobiles) are getting extensively adopted as the surface transportation carriers, hence considerable attention should be paid to the response of the transportation structure. The main objective of this study is to discuss the vibration and stability of a Bernoulli–Euler beam resting on a Winkler-type elastic foundation subjected to a static axial force under a moving load. A distributed load with a constant advance velocity was considered instead of a point load because moving loads in practice have normally a finite area over which they are distributed and the point load represents only an extreme case.
Damping of linear hysteretic nature was considered in this study, it is assumed that the foundation displacement at a point is dependent only on the force acting on the point by conducting parametric studies.
II. LITERATURE REVIEW
The approach of beams on elastic foundation is used by many researchers for the analysis of pavement. The problem concerned with the dynamic behavior of an Euler-Bernoulli beam. Excited by a moving load has received considerable attention in civil engineering in the recent years.
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)646
The governing equations in the form of deflection, bending moment and shear force were obtained in closed form for the undamped case. For the case where damping is also present, numerical results were obtained for both underdamping and overdamping cases. It is being reported that in the ideal undamped case, the steady state will never be attained at supercritical speed of the load. Also the effect of difference in the modeling of the foundation was found to be insignificant.
Kien (2008) adopted the FEM to investigate the dynamic response of prestressed Timoshenko beam resting on two-parameter elastic foundation subjected to a moving concentrated harmonic load. It was reported that the critical velocity at which the dynamic deflection of the beam reaches a peak value, is governed by the foundation stiffness. Winkler foundation subject to a platoon of moving loads with uniform line distributions was presented by Sun and Luo (2008). Numerical methods based on the fast Fourier transform were presented for efficient computation of dynamic response of the beam.
In the present analysis, which is the extension of Mallik et al. (2006), an infinite Euler–Bernoulli beam of constant cross-section resting on an elastic foundation is considered. The beam and foundation are assumed to be homogeneous and isotropic. The foundation is modeled using both one parameter without damping. The beam is subjected to a constant point load moving with a constant speed along the beam. An effort has been made to find the solution of the governing differential equation analytically. The equations governing different responses such as beam deflection, bending moment have been obtained in closed form for the undamped case.
III. MODELING OF BEAMS ON ELASTIC FOUNDATION
SUBJECTED TO MOVING LOADS
The differential equation of motion for an Euler– Bernoulli beam, resting on a two-parameter foundation and subjected to a moving load is given by
4 2 2
1
4 2 2 , .1
w w w dw
EI k kw c P x t Eq
dt
x dx t
Where w(x,t) is the transverse deflection of the beam(m), E is the Young’s modulus of the beam material(N/m2), I is the second moment area of the beam cross section about its neutral axis(m4), k is the spring constant (first parameter)per unit beam length(N/m2), k1 is the shear
parameter (second parameter) of the soil (N), ρ is the mass per unit length of the beam (kg/m), c is the coefficient of viscous damping per unit length of the beam (Ns/m2), P(x, t) is the applied moving load per unit length (N/m), x is the space coordinate measured along the length of the beam (m), t is the time (s).
If loads are moving in the positive x direction with a constant advance velocity v, a moving coordinate ξ can be defined by (x-vt). Then, the governing differential equation in a moving Cartesian coordinate system can be expressed as
4 2 2
1
4 2 2 .2
w w w dw
EI k kw c P x vt Eq
x dx t dt
Where P is the concentrated load moves with a constant velocity V, P(x, t) = Pδ(x-vt) where δ is the Dirac’s delta function and x is measured from the location of the load at x = 0.
Using the two parameter model, the values of k and k1
are based on the constrained deformation of an elastic layer given by Vlazov and Leotiev[9]. For a single layer of thickness H with a linear variation of normal stresses, k and k1 per unit width are given by
; 1
1 1 2 6 1
s s
s s s
E E H
k k
H
Divide both sides of Eq. (2) by EI to get
4 2 2
1
4 2 2 .3
k
w w k w c w P
w x vt Eq
EI EI EI EI t EI
dx x t
IV. INFINITE BEAM ON AONE PARAMETER FOUNDATION
MODEL
Now considering one parameter model and without damping the above Eq. reduces to
4 2
4 2
w k w P
w x vt
EI EI EI
dx t
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)647
Now defining,
2 1 1 ; ; ; 2 2 k k c
a b c d
EI EI EI EI
For an infinite beam in the steady state, it is found that the response w becomes a function of (x-vt), rather than of (x, t). The issue of the stability of the moving mass on beam becomes easier to solve if the reference frame is related to the moving mass. Thus, the Dirac delta function that locates the position of the contact point between the moving mass and the beam becomes a space-dependent function only.
x
vt
And hence can be written as follows using the above defined parameters
4 2 2 4 2 4 2 2 2 4 2 22 0 .4
w w P
b w a x vt
EI
dx t
d w d w
b w av Eq
d d
Let
Us assume that the solution of the above equation as
m
w
e
This after substituting in Eq. (5) yields
4 2 2 2
2
0
m
av m
b
Eq.5Undamped case (velocity less than critical velocity) We define critical velocity of the moving load
;
cr cr
v
b a v
v
The roots of the equation as follows:
2 2 2 2
m
av
i
b
av
2
;
cr
With
av
b
v
b a
or
v
v
Thus four roots of the Eq. (6) will be in the form of complex conjugates and can be written as:
1 2
3 4
;
;
m
i
m
i
m
i
m
i
2 2 ; 2 2b av b av
Where
Assuming the solution in the form of sin and cos (trigonometric function) with exponential terms.
1 1
sin
2cos
0
w
e
C
C
for
Ensuring,w1vanishes as
2 3
sin
4cos
0
w
e
D
D
for
Ensuring,w1vanishes as
The boundary conditions are as follows:
1 2 1 2
1 2 1 2
0
0 ;
0
0
0
0 ;
0
0
w
w
w
w
P
w
w
w
w
EI
Appling boundary condition, the final solution is obtained as
1
2
sin
cos
0
4
cos
sin
0
4
Pe
w
for
EIb
Pe
w
for
EIb
V. PARAMETRIC STUDIES
The parametric study is carried out in order to get the response of moving loads in undamed case. Default values of parameters used for numerical computation. These parameters reflect typical structural and material properties of pavement. Using the parametric value deflection has been calculated using the above formulation. The variation in deflection and bending moment with new co-ordinate system ξ are calculated.
Table. 1 Soil and Beam Parameters
Parameters Assumed values ρ(kg/m) 25 EI(Nm2) 1.75×106 k (N/m2) 40.78×105
k1(N) 666875
P(N/m) 93.36×103 Es (N/m2) 3.73×106
International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 11, November 2012)648
VI. RESULTS AND DISCUSSION
Deflection without Damping:
Based on the above analysis, numerical computation has performed to examine the dynamic displacement response of the beam to a moving load. Fig.1 presents the displacement variation of beam with parameter
x
v t
where x varies from 0 to 50m, while time from t=0-1 second. The effect of different velocity ratio (such as 0.25, 0.5, 0.75 and 0.99) is illustrated in Figs. 1 to 4. It is observed that the defection at x=0 with time t=0 sec deflection is at peak and is found maximum deflection occurs beneath the load. With the absence of the damping the dynamic response is found to be symmetric for case x=0, i.e. the dynamic response of the moving load propagates along both the direction in the same way for
0
and
0
.Fig.1 Deflection Vs Distance with velocity ratio= 0.25 without damping
Fig.2 Deflection VsDistance with velocity ratio= 0.5 without damping
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Fig.4 Deflection Vs Distance with velocity ratio= 0.99 without damping
Fig. 1-2shows that with increase of distance i.e. at x=10m the deflection is almost same in both cases where as in Fig.3 it is more than the earlier observation. So it may be concluded that with increase of velocity ratio maximum deflection increases. This is true for subcritical speed whereas in Fig.4 the critical velocity ratio 0.99 close to critical velocity hence the deflection shoots high.
Hence it may be concluded that most of the deflection occurs under the load as the energy of the deformation in the beam foundation system propagates with a speed lower than that of load. At a distance away from the load the deflection gradually diminishes as absorption of radiation energy increases with distance.
Bending Moment without Damping
It is clearly seen that the maximum negative bending moment occurs at a point shifted ahead of the load and the amount of shifting decreases with increasing speed. The maximum positive bending moment however occurs near the load as long as v<vcr. This can be clearly observed from Figs.11 to14 i.e. with increase of load velocity ratio maximum positive bending moment increases at x=0 at t=0 sec. Here it is assumed the upward bending moment positive while downward bending moment is negative. Negative bending moment in the profile signifies that there is a downward settlement of the pavement i.e. net moving load pressure becomes more than the safe bearing capacity of soil.
With the distance away from the load the response gradually dies down. Here also from the figure it may be inferred that maximum shear force acting at point where the profile of the bending moment changes. For subcritical velocity, the bending moment distribution is almost symmetrical about the load.
[image:5.612.337.529.257.435.2]Again considering, the increasing effect of the load velocity ratio, negative bending moment increases rather than the positive bending moment. This can be justified from Fig. 5 -6.
Fig.5 B.M Vs Distance with velocity ratio=.25 without damping
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Fig.7 B.MVs Distance with velocity ratio= 0.75 without damping
Fig.8 B.M Vs Distance with velocity ratio= 0.99 without damping
VII. CONCLUSION
A Bernoulli–Euler beam resting on an elastic foundation subject to moving dynamic loads is studied in this paper. The steady-state deflection and bending moment has been studied for undamped case.
The deflection response of the beam with elastic foundation is symmetric with respect to time t=0 or space x=0, while it is asymmetric when damping is considered. The maximum displacement occurs behind the moving load. The configurations of the steady state solutions may change as damping varies.
At subcritical speed the absolute value of the deflection of the beam increases with increase of the load velocity. Peak maximum deflection appears when the load travels at the critical speed.
REFERENCES
[1] Cifuentes, A.O., (1989). Dynamic response of a beam excited by a moving mass. FiniteElements in Analysis and Design. 237-246. [2] Harr, M. E., Davidson, J. L. and Ramaswamy, S. V. (1969). Euler
Beams on a two parameter foundation model. ASCE .J. Soil. Mechanics and Foundation. 95, 933-948
[3] Fryba, L. (1972). Vibration of solids and structures under moving loads. Noordhoff International Publishing, Groningen, The Netherlands
[4] J.T. Kenny Jr.(1954), Steady state vibrations of beam on elastic foundation for moving load, Journal of Applied Mechanics, Transactions ASME 21 , 359–364..
[5] Kien, N. D. (2008). Dynamic response of prestressedtimoshenko beams resting on two-parameter foundation to moving harmonic load. TechnischeMechanik. 28(3-4),237–258.
[6] Mallik, A. K., Chandra, S., & Singh, A. B. (2006). Steady-state response of an elastically supported infinite beam to a moving load. Journal of Sound and Vibration, 291(3-5), 1148–1169.
[7] Sun, L., Feiquan, Luo (2008). Steady-State Dynamic Response of a Bernoulli–Euler Beam on a Viscoelastic Foundation Subject to a Platoon of Moving Dynamic Loads. J. Appl. Mech. ASME. 130, 051002-3-19.
[8] Zhaohua, F. and Cook, R. D. (1983). Beam elements on two-parameter elastic foundations. J. Eng. Mech .Div., ASCE. 109, 1390-1402. F. Richards, S. E. Davidson and S.A Long, Dual band reactively loaded microstrip patch antenna, IEE Trans. Antenna Propa., 33 (1985) 556-561.
