A moving element method using timoshenko’s beam theory for dynamic analysis of train - Track systems

26 Ho Chi Minh City Open University Journal of Science–No.4(1) 2014 
 A MOVING ELEMENT METHOD USING TIMOSHENKO’S BEAM 
THEORY FOR DYNAMIC ANALYSIS OF TRAIN-TRACK SYSTEMS 
 Nguyen Minh Nhan1, Nguyen Thoi Trung1,2, Nguyen Van Thanh3, 
 Luong Van Hai3, Bui Xuan Thang22 
 1Division of Computational Mathematics and Engineering (CME), Institute for 
 Computational Science (INCOS), Ton Duc Thang University, Viet Nam 
 2Department of Mechanics, Faculty of Mathematics & Computer Science, VNU-HCM 
 University of Science, Viet Nam 
 3Faculty of Civil Engineering, VNU-HCM University of Technology, Viet Nam 
 (Received: 23/02/2014; Revised: 02/04/2014; Accepted: 16/06/2014) 
 ABSTRACT 
 The paper presents a dynamic analysis of train-track systems supported by viscoelastic foundations 
by combining Timoshenko’s beam theory and moving element method (MEM). In the proposed method, a 
three-node beam element is utilized to get a high order approximation for the deflection of Timoshenko 
beam. The reduced integral method is applied in order to avoid the shear-locking phenomenon when 
computing the shear strain energy of the rail beam. In addition, the behavior of train-track system with 
respect to time is deduced by using Newmark’s constant acceleration method. Numerical results show 
that the proposed method is free of shear locking and gives a good agreement with Koh et al.’s method 
using Euler-Bernoulli beam theory. 
 Keywords: train-track system, moving element method (MEM), Timoshenko beam theory, 
three node beam element, shear-locking phenomenon. 
2 Correspondence to: Bui Xuan Thang, Faculty of Mathematics and Computer Science, University of Science, Vietnam National 
University – HCMC, 227 Nguyen Van Cu, District 5, Hochiminh city, Vietnam. 
E-mail address: bxthang@hcmus.edu.vn ; bxthang071@yahoo.com.vn 
 Ho Chi Minh City Open University Journal of Science–No. 4(1) 2014 27 
 1. Introduction development of digital computers, many 
 Moving-load dymanic problems numerical methods such as finite element 
which are very common in engineering method (FEM), mesh-free method, finite 
have received a lot of interests from difference method have been proposed. 
researchers all around the world from quite These new methods have made the 
early. It was believed that in 1847, the researching of mechanics in general and of 
collapse of Stephanson’s Bridge in moving load problems in particular 
England motivated researchers to find out become more convenient and expandable. 
accurately the effects of moving loads to Along with this, all branches of transport 
structures [1]. Among a large number of have obtained many great advances with 
structures subjected to moving load, huge increase in speed and weight of 
transport engineering structures like vehicles. This fact has made the 
railways, bridges or pavements have researching of dynamic responses of 
gained much concern from beginning stage structures under moving vehicles become 
[2]. more and more important. Although 
 proving the strength in analyzing various 
 Up to now many models have been developedstructures to study in engineering,the behavior of finite the transport element structures under moving loads. In these models, structural components which are usually considered as beams, plates or shells are idealized by some structural theories such as Euler
Kirchhoff’s plate theory. Moving loads method (FEM) [8] has encountered many 
have also been split into many cases such obstacles when facing with moving load 
as moving constant or time depended problems. For example, a very large 
forces, moving masses and moving vehicle domain of structure subjected to high speed 
systems. Besides that, to reflect the reality, moving load requires a mesh with a lot of 
the effect of foundation on which the degrees of freedom (DOFs). Besides that, 
structures lie has been counted and divided keeping track of load positions related to 
into many cases like elastic and mesh nodes is unavoidable. These 
viscoelastic foundations. restrictions have increased much 
 Initially, analytic methods have been computational cost. Therefore, recently, to 
used widely to study the effect of moving solve this kind of problems, some new 
load to transport structures. Some of these approaches which commonly use moving 
methods which can be named here are the coordinates have appeared and improved 
method of using Green’s function and their efficiency. 
integral equations [3], method of expansion In 2000 and 2001, Chen and Huang 
of the eigen-functions [4], Galerkin’s [9, 10] studied infinite Timoshenko beam 
method [5] and the Fourier transform subjected to a moving load on viscoelastic 
method (FTM) [6, 7]. For example, in 1958 foundation. By using a moving coordinate 
and 1959, P. M. Mathews [6, 7] made an as Matthew [6, 7], the dynamic stiffness 
analysis of vibration of an infinite uniform matrix of the beam with the velocity 
beam on e ...  x
Then the strain and stress fields are easily deduced as 
  
  z  E  Ez
 x x x
 x x , (2) 
 w
 w 
 xz  xz GG  xz   
 x x 
where  is the shear coefficient which is usually given by 5/ 6 . 
 If we set two operators of bending and shear strain and the displacement field of 
Timoshenko beam which contains only the deflection w and rotation  as 
   w 
 L 0,, L 1 u 
 1 2 , (3) 
 x x  
then the energy strain of Timoshenko beam which is a sum of bending and shear 
component is written briefly as 
30 Ho Chi Minh City Open University Journal of Science–No.4(1) 2014 
 12 1 2
 U EI L u dx  GA L u dx , (4) 
 2 LL1 2 2
where L is the length of the beam segment any compatible and small virtual 
and A is the area of beam cross section. displacements imposed on the body, the 
 total internal virtual work done W must be 
 2.1. Energy method for dynamic I
analysis of train-track system equal to the total external virtual work 
 done W . 
 In this section, we use the principle of E
virtual work to formulate the weak form of The internal virtual work of rail beam 
Timoshenko beam on viscoelastic can be expressed as 
foundation. This principle states that for 
 W  U EI L  u TT L u dx kGA L  u L u dx , (5) 
 I 1 1 L 2 2
whereas the external virtual work done W E force of beam, the elastic and damping 
consists of the works done by the dynamic forces of the foundation. 
force at the wheel contact point, the inertial 
 WWWWW 1 2 3 4 . (6) 
 EEEEE
Particularly, 
 W1 F t Dirac x Vt  wdx W3  w kwdx
 E L E L
 , (7) 
 W2  w mwdx W4  w cwdx
 E L E L
in which F t given in Eq. (21) is the that the contact point is at x 0 when time 
dynamic force at the wheel contact point t 0 . To avoid keeping track of the load 
and  Dirac denotes the Dirac-delta function. position, a moving coordinate system is 
 defined as follow 
 Note that x-coordinate is a fix 
coordinate in the longitudinal direction of 
the beam with the origin is chosen such 
 r x Vt . (8) 
 Applying the chain rule we can easily the fix coordinate x and the derivatives 
transform the derivatives with respect to with respect to time t as follows 
     
 V
 x r t t  r . (9) 
 2  2  2  2  2  2
 2VV 2
 2 2 2 2 2
 x r t t r  t r
 At the point of time t 0 , in the fixed arbitrary point of time t this position is 
co-ordinate, the position of the beam vt L/,/2 vt L 2 . By using the moving 
segment is  LL/,/2 2 and in general, at coordinate system above the interesting 
 Ho Chi Minh City Open University Journal of Science–No. 4(1) 2014 31 
domain of beam is always  LL/,/2 2 and In the moving coordinate we have the 
the position at which the moving force internal and external virtual works are 
affect the beam is always r 0 . So there’s rewritten as follows. 
no need to keep track of moving load. 
 T T
 WI EI L1 u L 1 u dr kGA L 2  u L 2 u dr (10) 
 L L
where 
   w 
 L1 0 ,, L 2 1 u (11) 
 r  r  
are the two operators of bending and shear Similarly, the total external virtual work in 
strain and the displacement field of this moving coordinate is given by
Timoshenko beam in moving coordinate. 
 W F t Dirac r  wdr 
 E L
 w 2 w w . (12)
  w mw 2mV cw mV2 cV kw dr
 L 2 
 r r r 
 2.2. Foundation of MEM discretize the displacement field (Figure 1). 
 As in equation (12) we can see that Consider a typical three node element 
there is a second order derivative with whose length is le . The shape functions for 
respect to r so we use three node elements to this element is given by 
 l2 3 l r 2 r 2 4r l r r l 2 r 
 N e e ,,NN e e . (13) 
 1 l 2 2l 2 3 l2
 e e e
Then the displacement field is approximated as follows 
 u  N qe 
 , (14) 
 w N q
 w e 
 T 
in which qe  w1 1 w 2  2 w 3  3 is a vector of six DOFs of a beam element and 
the matrices of shape functions are 
 NNN0 0 0 
 NN 1 2 3 NNN0 0 0 . (15) 
   0NNN 0 0 w  1 2 3 
 1 2 3 
Substitute Eq. (14) into Eq. (10) and (12), we have the approximated internal and external 
virtual works are given by 
 Ne l l
 T eTT e 
 WI q e EI L1 N L 1 N dr  GA L 2 N L 2 N dr q e  , (16). 
  0 0 
 e 1 
32 Ho Chi Minh City Open University Journal of Science–No.4(1) 2014 
 l
 qT e F t  Dirac r N T dr 
 e 0 w 
 Ne l
 W  qT e m NT N q 2 mV NT N c NT N q  dr . (17) 
 E e 0 wwe  wwrwwe,  
 e 1 l
  qT e mV2 NT N cV NT N k NT N q  dr 
 e 0 wwrr, wwr, wwe 
Finally, the weak form of the problem is deduced from the principle of virtual work, 
WWIE . Therefore 
 Ne Ne l
 qT F  qT e M q C q  K q  dr , (18) 
 e e  e 0 e e e e  e e 
 e 1 e 1
where 
 l
 F e F t  Dirac r N T dr
 e 0 w
 l
 MNN e mT dr
 e 0 w w 
 l
 C e 2mV NNT cNNT . (19). 
 e 0 w w, r w w 
 lTT le
 KLNLNLNLN EIe dr  GA dr 
 e 0 1 1 0 2 2 
 l
 e 2 T T T 
 mvNNNNNNw w,, rr cVw w r kw w dr
 0 
 For simplicity, the Gaussian quadrature governing equations of the rail beam. 
rule is used to compute these matrices 2.3. The coupled equations of 
above. In addition, reduced integration motion for the train-track model 
technique is applied when computing the 
 Besides the effect of moving gravity 
shear component of the stiffness matrix K e 
in order to overcome the shear-locking load, train-track vibration is also caused by 
phenomenon. Particularly, instead of using the roughness of the rail. Therefore it is 
three Gaussian points as normal (because necessary to include the rail corrugation in 
second order shape functions are used), we the formulation. By using Newton’s second 
only use two points here. Because the law, we can easy find the force equilibrium 
 equations for each mass in train-track 
virtual displacement vector q is 
 e model as 
arbitrary, it can be eliminated to give the 
 m3 u 3 k 3 u 2 u 3 c 3 u  2 u  3 m 3 g
 mukuu22221221 cuu   kuu 323323 cuu   mg 2 (20) 
 mukuw cuw   kuu cuu   mgkycy 
 11110110212212 1  1c 1 c 
 Fc
in which yc is the rail corrugation gravity. From the third equation in Eq. (20) 
expressed as a function of time, w w we have the wheel contact force is given 
 0 r 0 by 
is the rail deflection, both at wheel contact 
point, and g is the acceleration due to 
 F t c u w  y  k u w y . (21) 
 1 1 0c 1 1 0 c 
Eq. (20) then is coupled with the rail beam to create equations of motion for train-track 
 Ho Chi Minh City Open University Journal of Science–No. 4(1) 2014 33 
model 
 Mz Cz  Kz P. (22) 
 Here z is the displacement vector first paper of MEM [11]. Particularly, 
which consists all DOFs of beam and three parameters for track and vehicle are given 
DOFs of train system. M , C, K are as in 
structural matrices which obtained by with an addition is that the Poisson’s 
assembling the corresponding matrices of ratio of rail beam is  0. 3. Here we will 
the beam model and the train model. illustrate two cases of moving load, 
Finally, Newmark’s constant acceleration constant force at constant velocity for the 
method is used to solve the above dynamic first case and 3-DOF vehicle at constant 
equation. velocity for another. Both MEM-T and 
 3. Numerical examples MEM-E methods are implemented. To put 
 In this section, to verify the efficiency all of these methods in a same condition, 
of the proposed method, we will use some we only use a regular mesh of 100 beam 
problems proposed by Koh et al. in their elements as being used by Koh et al. [11]. 
 Table 1. Parameters of train-track system 
 Track parameters 
 L 50 m m 60.0 kg/m 
 E 2.00 x 1011 Pa k 1.00 x 107 N/m2 
 I 3.06 x 10-5 m4 c 4900 Ns/m2 
 Vehicle parameters 
 m1 350 kg m2 250 kg m3 3500 kg 
 9 6 5
 k1 8.00 x 10 N/m k2 1.26 x 10 N/m k3 1.41 x 10 N/m 
 5 3 3
 c1 6.70 x 10 Ns/m c2 7.10 x 10 Ns/m c3 8.87 x 10 Ns/m 
 Besides that, in the case of moving assumed to be smooth and the damping of 
force, the shear-locking phenomenon is foundation is removed. We only consider 
illustrated by comparing results from the effect of the total gravity load of three 
different methods when the thickness of mass moving at a constant velocity of 20 
rail beam changed. It’s assumed that h is m/s. Because the excitation is a constant 
the thickness of rail beam corresponding to force, the solution is time invariant at the 
the original data of Koh et al. and the area steady state. All time derivatives in Eq. 
of beam cross section and the second (22) are vanished. As a results, there’s no 
moment of area is given by A h and need to using Newmark’s method. Besides 
 solution from MEM-E method proposed by 
I Ah 2 /12 , respectively. 
 Koh at al. [11], another one called quasi-
 3.1. Constant force at constant static solution from Ref [15] is used as a 
velocity comparator. This solution is given 
 In this example, the rail beam is explicitly by 
34 Ho Chi Minh City Open University Journal of Science–No.4(1) 2014 
 w r Ae r cos r sin r (23) 
 1/ 4
in which A F/ 2 k with F is the constant moving load and k/ 4 EI . 
 Table 2. Deflection of Beam at Contact Point (mm) 
 Deflection of Beam at Contact Point (mm) 
 Quasi-static (MEM-E) 
 [Error! [Error! 
 Thickness MEM-T MEM-T (Full 
 Reference Reference 
 (Reduced Int) Int) 
 source not source not 
 found.] found.] 
 8h -0.3380 -0.3381 -0.3393 -0.3408 
 4h -0.5684 -0.5685 -0.5699 -0.5731 
 2h -0.9560 -0.9562 -0.9580 -0.9679 
 h -1.6078 -1.6088 -1.6110 -1.6455 
 h/2 -2.7039 -2.7075 -2.7111 -2.7845 
 h/4 -4.5474 -4.5458 -4.5663 -4.4142 
 h/8 -7.6478 -7.4411 -7.6797 -6.0722 
 Table 3. Error Percent to Quasi-static solution 
 Error Percent to Quasi-static solution (%) 
 Koh et al. [Error! 
 Thickness Reference source MEM-T (Reduced 
 MEM-T (Full Int) 
 not found.] (MEM- Int) 
 E) 
 8h 0.0314 0.3923 0.8401 
 4h 0.0096 0.2650 0.8250 
 2h 0.0258 0.2069 1.2472 
 h 0.0662 0.1988 2.3461 
 h/2 0.1335 0.2672 2.9792 
 h/4 -0.0353 0.4155 -2.9302 
 h/8 -2.7035 0.4170 -20.6023 
 The deflections of beam at contact the quasi-static solution. It’s obvious that 
point corresponding to different cases of MEM-T using reduced integration gives 
beam thickness are computed and good agreement with MEM-E and quasi-
presented in Table 2. Table 3 shows the static solution. In addition, MEM-T using 
error percent of other methods compared to reduced integration is also free of shear-
 Ho Chi Minh City Open University Journal of Science–No.4 (1) 2014 35 
locking when the rail beam is thinner the beam is thinner or thicker, the proposed 
whereas MEM-T using full integration is method gives excellent agreements with 
not. Figure 2, Figure 3 represents the rail both MEM-E method and quasi-static 
displacement profiles computed by the solution. 
mentioned methods. Clearly, no matter if 
 Figure 2. Rail displacement profile corresponding to some cases of beam thickness 
 Figure 3. Rail displacement profile corresponding to some cases of beam thickness 
 3.2. 3-DOF vehicle at constant We now consider a moving vehicle load 
velocity 
36 Ho Chi Minh City Open University Journal of Science–No.4(1) 2014 
instead of a pure force. In this case, the rail is given by a periodic function, 
corrugation is counted as an excitation and 
 y x ysin2 x /  y sin 2 Vt /  (24) 
 c c0 c c 0 c 
where the amplitude and wavelength are The displacement of beam at wheel contact 
given by yc 0 0. 5 mm, c 0. 5 m, point and displacements of three masses 
respectively. The vehicle is modeled as a which represent three components of the 
spring-mass-damper moving on the rail train is taken to study. 
beam at a constant velocity, V 20 m/s. 
 Figure 4. Displacements of train masses and displacement 
 of beam at the contact point 
 The dynamic equations of MEM is c / V = 0.025s. In these results, the static 
solved by using Newmark’s constant responses due to self-weight of the three 
acceleration with a time step of 0. 0001s and masses are excluded. We can see an 
at-rest initial conditions. As in previous excellent agreement between MEM-T using 
example, MEM-E and MEM-T with reduced integration and MEM-E proposed 
reduced or full integration used are by Koh et al. 
implemented. Figure 4 shows the dynamic 
 4. Conclusion 
responses of the rail displacement at the 
contact point and the displacements of three In this paper, the moving element 
masses in a typical corrugation cycle, T= method (MEM) is extended to analyze the 
 dynamic behaviors of train-track system 
 Ho Chi Minh City Open University Journal of Science–No. 4(1) 2014 37 
which is modeled by a Timoshenko beam theory. The obtained results are very 
on viscoelastic foundation subjected to a promising to extend to analyze the 
moving spring-mass-damper system. The dynamic behavior of beam structures made 
proposed method is hence called as the by composite and FGM. 
MEM-T. The coupled train-track Acknowledgements 
governing equations are then established 
and solved by Newmark’s constant This work was supported by Vietnam 
acceleration. In the MEM-T, the reduced National Foundation for Science & 
integration is used to avoid the shear Technology Development (NAFOSTED), 
locking. The numerical examples show that Ministry of Science & Technology, under 
the results by the MEM-T agree well with the basic research program (Project No.: 
those by Koh et al. [11] using Euler beam 107.02-2012.05).
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