# Dynamic response of rigid pavement under moving traffic load with variable velocity/Standziuju dangu dinamine reakcija veikiant nepastoviu greiciu judancio transporto apkrovai/Ar mainigu atrumu braucosas kustigas satiksmes slodzes izraisita stingas segas dinamiska reakcija/ Jaiga katte dunaamiline reaktsioon muutuva kiirusega liikuva liikluskoormuse all.

1. Introduction

2. Governing equations and solution analysis

Based on the classical small deflection theory of plate, the governing equation for dynamic lateral deflection w(x, y, t) in an x-y Cartesian coordinate system is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where D--the flexural rigidity of the plate defined by D = [Eh.sup.3]/12(1 - [[mu].sup.2], MPa.[m.sup.3]; [rho]--density, MN/[m.sup.2]; h--thickness, m; E--modulus, MPa; [mu],--Poisson's ratio; C--damping coefficient of the plate respectively, MN/[m.sup.3]; K--the reaction modulus of the foundation, MN/[m.sup.3]; F(x, y, t) the external dynamic load acting on the plate surface, N; t--denotes time, s.

The plate is assumed to extend to infinity in the horizontal plane. The load pressure within the contact area is assumed to be uniformly distributed in a rectangular area. In this paper, the loads moving along the x direction are considered, which can be represented in the form

F(x,y,t) = [F.sub.0]{U[x + b - X(t)]U[y + c] - U[x - b - X(t)]U[y - c]}, (2)

where [F.sub.0]--the amplitude function of the load, N; U--the unit step function; b and c--the half lengths of the rectangle sides of the load distribution, respectively, m; X(t) denotes a function describing the motion of the force at time t defined as

X(t) = [x.sub.0] + [V.sub.t] + 1/2 [at.sup.2], (3)

where [x.sub.0]--the load position, m; V--the initial speed, m/s; a--the constant acceleration, m/[s.sup.2]. This function describes a uniform decelerating or accelerating motion. The uniform velocity type of motion is given by

X(t) = [x.sub.0] + Vt. (4)

Eqs (1)-(3) are total formulations for the analytical model of the plate on elastic foundation, subjected to moving concentrated loads.

In order to solve the problem described above a double dimension Fourier transform is adopted as defined by following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

The inverse Fourier transform is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Using the Fourier transform defined as Eq (5), the Eq (1) can be presented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Fourier transforming Eq (3) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Substituting Eq (8) into Eq (7) and using Duhamel Integration and the inverse Fourier transform, the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Using Mathematica software and integrating for t in Eq (9) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] dt is the error function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The transverse stresses are the longitudinal stresses at the bottom of the plate that can be obtained respectively from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The stresses considered in this study are tensile stresses in the longitudinal direction at the bottom of the plate because the transverse stresses are smaller than the longitudinal stresses.

3. Dynamic response to moving accelerated/decelerated load

An infinite plate on the Winkler foundation subjected to a moving traffic load with variable velocity (accelerating or decelerating) is considered. The concerned data used in all the numerical examples are as follows: E = 3.45 x [10.sup.4] MPa, [mu] = 0.15, K = 1.36 MN/[m.sup.3], C = 2.0 x [10.sup.5] MN/[m.sup.3], b = 0.35 m, h = 0.30 m, c = 0.25 m, [F.sub.0] = 1.0 x [10.sup.5] N, t = 1.0 s, [x.sub.0] = 0.0. A range of values of acceleration and velocity are examined in these examples from which the effect of acceleration and velocity on the dynamic response of the plate are well investigated.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Figs 1 and 2 show the max deflections of the plate with different acceleration, deceleration and velocity at time of 1 s and at load position of 0.5 m. Fig. 3 shows stresses of the plate with different acceleration and velocity at time of 1 s and at load position of 0.5 m. It can be seen that the max deflections and stresses of the plate change with the load varying acceleration, deceleration and velocity and have different max values. This feature can also be seen in Figs 4-9 which illustrate the dynamic deflections and stresses of the plate under the different varying load acceleration, deceleration and velocity when one of them is fixed.

From Fig. 4, it can be seen that when the load acceleration is fixed, the deflection of the plate increases with the load varying velocity and there are peak values. At the acceleration of 0.2 m/[s.sup.2] and the velocity of 100 km/h, the displacement reaches the max value. When the acceleration is given 2 m/[s.sup.2] the displacement will have a peak value at the velocity of 130 km/h.

Fig. 5 shows the results of displacement with different velocities when the deceleration is given. It can be seen that these are similar to the results shown in Fig. 4; the deflection of the plate increases with the load varying velocity and there are peak values. But the increasing speed is larger than that when the load moves with acceleration. Also, at the deceleration of -0.2 m/[s.sup.2] and velocity of 100 km/h the displacement has max value. But when deceleration is fixed at -0.2 m/[s.sup.2], the displacement has a peak value at the velocity of 190 km/h.

Figs 6 and 7 illustrate the results of displacement with different acceleration and deceleration when the load moving velocity is fixed. It can be seen that the displacement reduces with the acceleration and deceleration increasing when the load moving velocity is fixed.

Fig. 8 presents the results of the stresses of the plate with different acceleration when the load varying velocity is fixed. It can be seen that the stresses of the plate reduce with the increasing acceleration and deceleration when the load varying velocity is fixed. At small values of velocity (less than 50 km/h) the reduction is slow. At large values of velocity the reduction is rapid.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Fig. 9 is the results of the stresses of the plate with different velocity when the load varying acceleration is fixed. It can be seen that the stresses of the plate increase with the acceleration increasing when the load varying velocity is fixed. At the acceleration of 0.2 m/[s.sup.2] and the velocity of 140 km/h the stresses reach the max value. When the acceleration is given 2 m/[s.sup.2] the plate stresses reach peak values at the velocity of 170 km/h.

4. Conclusions

In this paper, the dynamic response of an infinite plate on an elastic foundation subjected to a moving traffic load with variable velocity (accelerating or decelerating) is investigated via a triple Fourier transform. The effects of the load varying velocity, acceleration and deceleration are also discussed. The numerical results show that the max deflections and stresses of the plate change with the load varying acceleration, deceleration and velocity. Both the deflections and the stresses reach different max values. The dynamic displacements and stresses increase with the load varying velocity and decrease with the load varying acceleration and deceleration. These phenomena suggest that the design of rigid pavements should carefully consider the effect of the dynamic load caused by variable moving vehicles.

doi: 10.3846/bjrbe.2012.07

Received 19 May 2010; accepted 24 February 2011

References

Auersch, L. 2008. The Effect of Critically Moving Loads on the Vibrations of Soft Soils and Isolated Railway Tracks, Journal of Sound and Vibration 310(3): 587-607. http://dx.doi.org/10.1016/j.jsv.2007.10.013

Degrande, G.; Schillemans, L. 2001. Free Field Vibrations during the Passage of a Thalys High-Speed Train at Variable Speed, Journal of Sound and Vibration 247(1): 131-144. http://dx.doi.org/10.1006/jsvi.2001.3718

Gbadeyan, J. A.; Oni, S. T. 1992. Dynamic Response to Moving Concentrated Masses of Elastic Plates on a Non-Winkler Elastic-Foundation, Journal of Sound and Vibration 154(2): 343-358. http://dx.doi.org/10.1016/0022-460X(91)90585-8

Huang, M. H.; Thambiratnam, D. P. 2002. Dynamic Response of Plates on Elastic Foundation to Moving Loads, Journal of Engineering Mechanics-ASCE 128(9): 1016-1022. http://dx.doi.org/ 10.1061/(ASCE)0733-9399(2002)128:9(1016)

Hussein, M. F. M.; Hunt, H. E. M. 2006. Modelling of Floating slab Tracks with Continuous Slabs under Oscillating Moving Loads, Journal of Sound and Vibration 297(1-2) 37-54. http://dx.doi.org/10.1016/j.jsv.2006.03.026

Kim, S. M.; Roesset, J. M. 1998. Moving Loads on a Plate on Elastic Foundation, Journal of Engineering Mechanics 124(9): 1010-1017. http://dx.doi.org/ 10.1061/(ASCE)07339399(1998)124:9(1010)

Lefeuve-Mesgouez, G.; Le Houedec, D.; Peplow, A. T. 2000. Ground Vibration in the Vicinity of a High-Speed Moving Harmonic Strip Load, Journal of Sound and Vibration 231(5): 1289-1309. http://dx.doi.org/10.1006/jsvi.1999.2731

Lin, Y. H.; Trethewey, M. W. 1990W. Finite Element Analysis of Elastic Beams Subjected to Moving Dynamic Loads, Journal of Sound and Vibration 136(2): 323-342. http://dx.doi.org/10.1016/0022-460X(90)90860-3

Sun, L. 2006. Analytical Dynamic Displacement Response of Rigid Pavements to Moving Concentrated and Line Loads, International Journal of Solids and Structures 43(14-15): 4370-4383. http://dx.doi.org/10.1016/j.ijsolstr.2005.06.105

Suzuki, S. I. 1977. Dynamic Behavior of a Finite Beam Subjected to Travelling Loads with Acceleration, Journal of Sound and Vibration 55(1): 65-70. http://dx.doi.org/ 10.1016/0022460X(77)90583-1

Vallabhan, C. V. G.; Straughan, W. T.; Das, Y. C. 1991. Refined Model for Analysis of Plates on Elastic Foundations, Journal of Engineering Mechanics 117(12): 2830-2844. http://dx.doi. org/10.1061/(ASCE)0733-9399(1991)117:12(2830)

Yang Zhong (1) ([mail]), Yuanyuan Gao (2), Mingliang Li (3)

School of Civil and Hydraulic Engineering, Dalian University of Technology, No.4 Linggong Street, 116024 Dalian, China

E-mails: (1) zhongyang58@163.net; (2) gaoyuanyuan2286@163.com; (3) lis.1221@163.com
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