Steady-State Stresses in a Half-Space Due to Moving Wheel-Type Loads with Finite Contact Patch

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1 Transaction A: Civil Engineering Vol. 7, No. 5, pp. 387{395 c Sharif University of Technology, October 2 Steady-State Strees in a Half-Space Due to Moving Wheel-Type Loads with Finite Contact Patch Abstract. M. Dehestani ;, A. Vafai and M. Mod In this paper, the steady-state strees in a homogeneous isotropic half-space under a moving wheel-type load with constant subsonic speed, prescribed on a nite patch on the boundary, are investigated. Navier's euations of motion in 2D case were modied via Stokes-Helmholtz resolution to a system of partial dierential euations. A double Fourier-Laplace transformation procedure was employed to solve the system of partial dierential euations in a new moving reference system, regarding the boundary conditions. The eects of force transmiion from the contact patch to the half-space have been considered in the boundary conditions. Utilizing a property of Laplace transformation leads to transformed steady-states strees for which inverse Fourier transformation yielded the steady-state strees. Considering two types of uniform and parabolic force transmiion mechanism and a comparison between the pertaining results demonstrated that the parabolic load transmiion induce lower strees than the uniform one. Results of the problem for various speeds of moving loads showed that the strees increase as the moving loads' speeds increase to an extremum speed known as CIS. After the CIS speed, strees' absolute values decrease for higher speeds. Eventually CIS values for homogeneous half-spaces with dierent material properties were obtained. Keywords: Moving load; Half-space; Wave propagation; Contact patch. INTRODUCTION The importance of the transportation engineering is intensifying due to the increase in all kinds of communications. New developments in technologies are employed to provide more facilities in transportation engineering. Hence structures which are modeled as half-spaces are subjected to larger moving loads with higher speeds. Thus understanding the exact dynamic behavior of structures subjected to moving loads, which enhance the design and construction of these structures, is of signicant interest. Dynamic responses of simple structures such as beams and plates, subjected to moving loads, have been investigated by many researchers. Recent investigations for simple structures under moving loads are devoted. Department of Civil Engineering, Sharif University of Technology, Tehran, P.O. Box , Iran. *. Corresponding author. dehestani@gmail.com Received 25 May 2; received in revised form 25 July 2; accepted 4 September 2 to special conditions and case studies. Andersen et al. [] reviewed numerical methods for analysis of structure and ground vibration from moving loads. They investigated the main theoretical aspects of analysis of moving loads in a local coordinate system, and addreed the steps in the nite-element and boundary element method formulations. They also studied, in their work, problems in describing material diipation in the moving reference system. In case studies of simple structures, identication of moving loads on bridges was investigated by Yu and Chan [2], who provided a review on recent progrees on identication of moving loads on bridges. Yu and Chan in their study introduced the theoretical background of four identication methods and performed numerical simulations, illustrative examples and comparative studies on the eects of dierent parameters. Xia et al. [3] investigated the dynamic interaction of a long suspension bridge under running trains. They found that the dynamic responses of the long suspension bridge under the running train are relatively

2 388 M. Dehestani, A. Vafai and M. Mod small, and the eects of bridge motion on the runability of the railway vehicles are insignicant. Mehri et al. [4] presented an exact and direct modeling techniue based on the dynamic Green function for modeling beam structures with various boundary conditions, subjected to a constant load moving at constant speed. Inuences of the inertial loads on simple structures were investigated in many papers. Moving loads with inertial eects are known as moving maes. In this eld, Akin and Mod [5] introduced an analyticalnumerical method to solve the governing partial dierential euation of the beam under a moving ma. They ignored the eects of the speed of the moving load. Dehestani et al. [6] investigated the strees in thinwalled nite beams with various boundary conditions subjected to a moving ma under a pulsating force. They have used separation of variables method to solve the partial dierential euations of beams wherein the eects of the speeds of moving maes were considered. Accomplished works on the semi-innite regions under moving loads are not numerous compared with works on the inuences of moving loads on simple structures such as beams and plates. In this eld, Celebi and Schmid [7] studied ground vibrations induced by moving loads. They presented two numerical freuency domain methods in order to analyze the propagation of surface vibrations in the free eld adjacent to railway lines induced by specic moving loads acting on the surface of a layered and homogeneous half-space with dierent material properties. Bierera and Bode [8] presented a semi-analytical model in time domain, for moving loads with subsonic speeds. They obtained the vertical displacements induced by moving loads with constant and time-varying amplitudes at a xed observation point at the surface of the 3D half-space in time domain. The authors also introduced a semi-analytical, discrete model based on Green's functions for a suddenly applied, stationary surface point load with Heaviside time dependency to solve the non-axisymmetric, initial boundary value problem. They compared results for the transient and the steady-state ground motions to the analytical solutions. Galvin and Dominguez [9] employed a 3D time domain boundary element formulation for elastic solids to evaluate the soil motion due to high-speed moving loads and in particular, to high-speed trains. Their results demonstrated that the boundary element approach can be used for actual analyses of high-speed train-induced vibrations. In this paper, steady-state strees in a homogeneous, isotropic half-space under a moving load with constant subsonic speeds are investigated. Transmiion of the load to the half-space is accomplished through a nite contact patch which is more realistic and was ignored in previous studies. MATHEMATICAL MODEL Description of the Model Consider a homogeneous, isotropic, elastic half-space y with no body forces, as shown in Figure, subjected to a wheel-type load with a nite rectangular, 2a, patch. Half-space is under the action of a load P (x; t) which is prescribed through the patch on the surface boundary and moves at right angle to the positive direction of axis x from innity to innity at uniform speed V. In Figure, represents the density of the half-space material and, and are Lame's elastic constants for plane stre condition in which: = = E 2 ; () E 2( + ) ; (2) where E and are elastic modulus and Poion's ratio, respectively. Governing Euations Navier's Euation of motion for a homogeneous, isotropic half-space with no body forces can be expreed as: u = r 2 u + ( + )rr:u; (3) where u stands for the displacement vector component. Applying the Stokes-Helmholtz resolution to the displacement eld yields: u = grad + curl ; (4) Figure. Moving wheel-type load on a homogeneous, elastic, isotropic half-space.

3 Steady-State Strees in a Half-Space 389 where and are scalar and vector-valued functions, respectively, for which on account of the denitene, we should have: i;i = : (5) Substituting Euation 4 into Euation 3 leads to the fact that Navier's euation is satised, if the functions and are solutions of the wave-type euations: r 2 = c 2 ; 2 k r 2 k = c 2 ; (7) where c = +2 and c 2 = are dilatational disturbance propagation velocity and shear waves propagation velocity, respectively. Boundary Conditions Surface of the half-space is aumed to be subjected to wheel-type loads. In order to investigate the force transmiion mechanism in contact patch between load and the boundary, two types of inuence mechanism are considered. For the rst case, aume that the load Q(t) is applied uniformly on the patch for which the traction on the surface takes the form: P (x; t)= Q(t) fh(x V t+a) H(x V t a)g; (8) 4ab where H denotes the Heaviside function. It is much more accurate to consider the normal force (preure) distribution on the contact patch to be parabolic []. Thus in the second case, the load Q(t) is applied through a parabolic function to the patch. Therefore after some manipulations the traction on the surface would be obtained as: P 2 (x; t) = 3Q(t) x V t 2! 8ab a fh(x V t + a) H(x V t a)g: (9) Hence the problem is to obtain the distribution of the strees in D elastic half-space medium with the boundary conditions for y = to be: 22 = P i (x; t); () 2 = : () The other boundary conditions can be obtained from the fact that innitely far into the bulk of the medium the strees must be vanished. Thus the functions and must be obtained in such a way that all strees approach to zero as y tends to innity. ANALYTICAL SOLUTION Regardingthe special type of the inuence for the moving loads on structures, it is convenient to implement a moving reference system instead of a xed reference system as: 8 >< >: = x y = y = t V t (2) Thus the governing Euations 6 and 7 take the forms: ( kv 2 2 k 2 @y2 =; (3) ( k2v 2 2 k2 2 @y2 =; (4) where k 2 = c 2 and k2 = c2 2. In most of the applicable cases, speeds of the surface moving loads are not higher than c 2 of that half-space. Hence in this study, it is aumed that the speeds are in subsonic range in which k2v 2 2 <. In order to solve the system of Euations 3 and 4 as the governing euations (regarding the boundary conditions) a concurrent Fourier-Laplace integral transformation of position and time is considered as: Z Z ^(; y; s)= (; y; )e i e s dd!(; y; ) = 4 2 i Z ^ (; y; s)= Z = 4 2 i Z s s Z+i i ^(; y; s)e s e i dsd; (5) (; y; )e i e s dd! (; y; ) Z s s Z+i i ^ (; y; s)e s e i dsd: (6) Employing the concurrent Fourier-Laplace integral transformation to Euations 3 and 4, regarding the boundary conditions, would give rise to: ^(; y; s)=a(; s) exp (k(s 2 + iv ) ) 2 y ; (7) ^ (; y; s)=b(; s) exp (k2(s+iv 2 ) ) 2 y ; (8) where functions A(; s) and B(; s) should be obtained using boundary conditions in Euations and.

4 39 M. Dehestani, A. Vafai and M. Mod Euation 4 and elastic constitutive relation between strains and strees in elastodynamic theory yield the transformed strees in the 2D case as: ^ (; y; s) = 2 (:5k 2 2 k 2 )(s + iv ) 2 2 ^ + 2i k 2 2(s + iv ) ^ ; (9) ^ 2 (; y; s) = 2i k 2 (s + iv ) ^ + 2 :5k 2 2(s + iv ) ^ ; (2) ^ 22 (; y; s) = 2 :5k 2 2(s + iv ) ^ 2i k 2 2(s + iv ) ^ : (2) Transformed types of boundary conditions in Euations and and transformed strees given by Euations 2 and 2 for y = establish a system of euation from which functions A(; s) and B(; s) can be obtained as: A(; s) = :5k 2 2(s + iv ) F (; s) B(; s) = i k2 (s + iv ) F (; s) where: F (; s) = :5k 2 2(s + iv ) P (; s); (22) P (; s); (23) 2 k 2 2(s+iV ) k 2 (s+iv ) : (24) Transformed Boundary Strees Concurrent Fourier-Laplace integral transforms for the boundary strees given by Euations 8 and 9 can be obtained by the following euation: Z Z ^P (; s) = P (; )ei e s dd; (25) as: ^P (; s) = ^Q(s) ^P 2 (; s) = 3 ^Q(s) 8b sin(a) ; (26) a sin(a) a 3 3 cos(a) 2 ; (27) where: Z ^Q(s) = LfQ()g = Q()e s d: (28) Because of the fact that for the case of a moving object, the length of the contact patch 2a is very small, the evaluation of the inverse formidable integrals whose integrands are given in Euations 9, 2 and 2 may not be worth the eort. Thus it is convenient to obtain the asymptotic expansions of Euations 26 and 27 as a tends to zero: ^P (; s) = ^Q(s) p (); (29) ^P 2 (; s) = ^Q(s) p 2(); (3) where: p () = p 2 () = a O(6 ); (3) a O(6 ): (32) Obtaining the Steady-State Strees The steady-state strees are known as the strees after suciently large time duration which takes a constant value with respect to the time. In order to obtain the steady-state strees, the strees should be evaluated for time tending to innity. To this aim, consider the Laplace transformation for the rst derivative of the strees Z (; y; L ij (; y; ) = e = slf ij (; y; )g ij (; y; ): (33) Use of Euation 33 would lead to: ij (; y) = lim! ij(; y; ) = lim(s^ ij (; y; s)) s! Z = 2 lim s^ ij (; y; s)e i d: (34) s! Performing the limitations in the expanded form of the strees yield the steady-state strees as: Z (; y) = 4b C e jjy +D e jj2y p i ()e i d; (35) Z 2(; y)(; y) = i 4b jj C 2 e jjy +D 2 e jj2y p i ()e i d; (36)

5 Steady-State Strees in a Half-Space 39 22(; y) = Z C 22 e jj y 4b +D 22 e jj 2y p i ()e i d; (37) p where i = i 2, i = k i V and p i () should be substituted from Euations 3 or 32, depending on the case. stands for the steady-state form of the load which is eual to: = lim Q() = lim! (s ^Q(s)): (38) s! Constant parameters C ij and D ij can be expreed as: C = (2 :5 2 2 )( :5 2 2) ( :5 2 2 )2 2 ; (39) C 2 = C 22 = D = D 2 = D 22 = ( :5 2 2) ( :5 2 2 )2 2 ; (4) ( :5 2 2) 2 ( :5 2 2 )2 2 ; (4) 2 ( :5 2 2 )2 2 ; (42) ( :5 2 2) ( :5 2 2 )2 2 ; (43) 2 ( :5 2 2 )2 2 : (44) The integrations in Euations 35, 36 and 37 can be carried out, concerning the identities: Z Z 2m cos()e iy d = I (m) i (; y) cos(m); (45) 2m sin()e iy d = J (m) i (; y) cos(m); (46) where m = ; ; 2; and: I (m) i (; y) i 2m i 2y2 + 2 ; (47) J (m) i (; 2m i 2y2 + 2 : (48) Hence nal expreions for the steady state strees in the half-space for the two cases introduced in the previous sections can be obtained as: Case : (; y) = C I () +D I () 2 2(; y)(; y)= 22(; y) = Case 2: (; y) = 2(; y)(; y) = +D 2 J () 2 6 I() + a4 2 I(2) 6 I() 2 + a4 2 I(2) 2 C 2 J () C 22 I () +D 22 I () 2 C 4 I() +D 4 I() 2 +D 2 4 J() 2 22(; y) = C 2 4 J() +D 22 4 I() 2 6 J() 2 + a4 2 J(2) 2 ; (49) 6 J() + a4 2 J(2) ; (5) 6 I() + a4 2 I(2) 6 I() 2 + a4 2 I(2) 2 : (5) 4 I() + a4 2 I(2) 4 I() 2 + a4 2 I(2) 2 4 J() + a4 2 J(2) 4 J() 2 + a4 2 J(2) 2 C 22 4 I() NUMERICAL EXAMPLE ; (52) ; (53) 4 I() + a4 2 I(2) 4 I() 2 + a4 2 I(2) 2 : (54) Consider an isotropic, homogeneous semi-innite region, shown in Figure, with = 5 MPa, = :33 and = 2 kg/m 3, subjected to a moving load of 2 KN with constant speeds and prescribed on a patch with 2a = :8 m and = :2 m. Steady-state strees with respect to the position of the moving load for constant depth of y = :5 m and constant speed of V = :3 k2, in Cases and 2, have been evaluated and are shown in Figures 2 and 3, respectively. A comparison between the values of the strees in two cases demonstrates that the half space experiences higher strees when the load

6 392 M. Dehestani, A. Vafai and M. Mod Figure 2. Variations of strees at y = :5 m with respect to the position of a moving load with V = :3k2 in Case. is prescribed uniformly on the nite patch, compared with the case in which the load is prescribed in the parabolic form. This shows the importance of the circumstance in which the load is prescribing on the half-space boundary. Figures 4 and 5 show the variations of strees with respect to various depths in the half-space for points under the moving load, =, and constant speed of V = :3 k2. Results demonstrated that the Figure 3. Variations of strees at y = :5 m with respect to the position of a moving load with V = :3k2 in Case 2. strees decay uickly with increase in the depth. It should be noted that for =, the shear strees in two cases should be vanished due to the symmetry of the problem denition. Figures 6 and 7 show the variations of the strees for the point with coordinates (; y) = (; :5) subjected to moving loads with various subsonic speeds in two cases. Results demonstrate that the strees increase as the moving loads' speeds increase to an

7 Steady-State Strees in a Half-Space 393 Figure 4. Variations of the strees for = at various depths in the half space subjected to a moving load with V = :3k2 in Case. Figure 6. Variations of the strees for the point with coordinates (; y) = (; :5) subjected to moving loads with various subsonic speeds in Case. Figure 5. Variations of the strees for = at various depths in the half space subjected to a moving load with V = :3k2 in Case 2. Figure 7. Variations of the strees for the point with coordinates (; y) = (; :5) subjected to moving loads with various subsonic speeds in Case 2.

8 394 M. Dehestani, A. Vafai and M. Mod An analytical approach has been used to obtain the steady-state strees in a homogeneous, isotropic halfspace subjected to a moving load with a nite contact patch. To this aim, a concurrent double integral transformation was employed to solve the system of wave-type euations pertaining to the Navier's euation of motion. A comparison between two types of the load transmiion in the contact patch demonstrated that a parabolic load transmiion induce lower strees compared with its corresponding uniform load transmiion. Half-space experiences higher strees for higher speeds of the moving loads until reaching to an extremum speed known as CIS. After the critical inuential speed, the absolute values of the strees decrease for higher speeds. Obtaining the CIS values for dierent half-space materials showed that the CIS values increase as the ratio between the dilatational disturbance propagation velocity and propagation velocity of shear waves, c =c 2, increases. REFERENCES Figure 8. CIS values for various k 2 =k. extremum speed. As shown in Figures 6 and 7, after the extremum speed, strees' absolute values decrease for higher speeds. The extremum speed is euivalent to the Critical Inuential Speed (CIS) which is introduced in [2]. CIS which should be le than k2 can be obtained from the solutions of the following euation: :5k 2 2(CIS) 2 2 ( k 2 (CIS)2 ) ( k 2 2 (CIS)2 ) = : (55) Solution of Euation 55 for various half-space materials with dierent k and k 2 values are shown in Figure 8. This gure demonstrates that with increase in the Poion's ratio, the critical inuential speeds approach to a nite limit. CONCLUDING REMARKS. Andersen, L., Nielsen, S.R.K. and Krenk, S. \Numerical methods for analysis of structure and ground vibration from moving loads", Computers and Structures, 85, pp (27) 2. Yu, L. and Chan, T.H.T. \Recent research on identication of moving loads on bridges", Journal of Sound and Vibration, 35, pp. 3-2 (27). 3. Xia, H., Xu, Y.L., Chan, T.H.T. and Zakeri, J.A. \Dynamic responses of railway suspension bridges under moving trains", Scientia Iranica, 4(5), pp (27) 4. Mehri, B., Davar, A. and Rahmani, O. \Dynamic green function solution of beams under a moving load with dierent boundary conditions", Scientia Iranica, Transaction B: Mechanical Engineering, 6(3), pp (29). 5. Akin, J.E. and Mod, M. \Numerical solution for response of beams with moving ma", ASCE Journal of Structural Engineering, 5(), pp. 2-3 (989). 6. Dehestani, M., Vafai, A. and Mod, M. \Strees in thin-walled beams subjected to a traversing ma under a pulsating force", Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, (2), DOI:.243/954462JMES Celebi, E. and Schmid, G. \Investigation of ground vibrations induced by moving loads", Engineering Structures, 27, pp (25) 8. Bierer, T. and Bode, C. \A semi-analytical model in time domain for moving loads", Soil Dynamics and Earthuake Engineering, 27, pp (27) 9. Galvin, P. and Dominguez, J. \Analysis of ground motion due to moving surface loads induced by highspeed trains", Engineering Analysis with Boundary Elements, 3, pp (27).. Rajamani, R., Mechanical Engineering Series, Vehicle Dynamics and Control, st Ed., Springer (25). Zheng, D.Y., Au, F.T.K. and Cheung, Y.K. \Vibration of vehicle on compreed rail on viscoelastic foundation", Journal of Engineering Mechanics, 26(), pp (2). 2. Dehestani, M., Mod, M. and Vafai, A. \Investigation of critical inuential speed for moving ma problems on beams", Applied Mathematical Modelling, 33, pp (29). BIOGRAPHIES Mehdi Dehestani was born in 982. He received his B.S., M.S. and Ph.D. degrees in Structural and Earthuake Engineering from the Department of Civil

9 Steady-State Strees in a Half-Space 395 Engineering, Sharif University of Technology, Tehran, Iran, since 2 to 2. He is currently an aistant profeor of Structural and Earthuake Engineering in Department of Civil Engineering at Babol University of Technology, Babol, Iran. His research interest is in the eld of Theoretical and Applied Mechanics. Abolhaan Vafai, Ph.D., is a Profeor of Civil Engineering at Sharif University of Technology. He has authored/co-authored numerous papers in dierent elds of Engineering: Applied Mechanics, Biomechanics, Structural Engineering (steel, concrete, timber and oshore structures). He has also been active in the area of higher education and has delivered lectures and published papers on challenges of higher education, the future of science and technology and human resources development. Maood Mod, is a profeor of Structural and Earthuake Engineeringat Sharif University of Technology, Iran. He has taught and instructed Basic and Advance Engineering courses in the eld of Structural Mechanics and Earthuake Engineering. His research interest is in the area of Engineering Mechanics and Structural Dynamics, Application and Implementation of Finite Element Techniue in Static and Dynamic Problems with emphasis on Theory and Design.