Modelling of moving load effect on concrete pavements

Abstract. Concrete roads may have different layout arrangement. The structures made of isolated prefabricated slabs are used for the construction of temporary roadways. The paper deals with numerical simulation of moving load effect on isolated slabs on elastic foundation. The computing model of a lorry and computing model of a slab are introduced. The deflections at the middle of the slab and tire forces of vehicle are modeled under various conditions. The influence of speed of vehicle motion and influence of initial conditions are evaluated. The results are presented in time domain in graphical and numerical manner.


Introduction
Concrete slabs are widely used in road building.Isolated plates are usually used for the temporary pavement structures.Knowledge of stress and strain is required for correct design of the slabs.Numerical methods enable to simulate moving load effect in dynamic regime with satisfactory precision.The computing models of vehicles can be created on various levels.For the purpose of this task the plane computing model is adopted.The pavement is modelled as the thin slab on elastic foundation.The assumption about deflection plate of the slab is introduced to reduce the partial differential equation of motion on ordinary differential equation.The problem is solved by numerical way.The results depend on the speed of the vehicle and on its initial conditions.The deflections in the middle of the slab and tire forces are calculated.The results are presented in numerical and graphical manner.

Computing model of vehicle
A plane computing model of the Tatra 815 lorry is used for the solution of this problem [1].It is the multibody computing model with 8 degrees of freedom, Fig. 1.The model has three mass bodies with five degrees of freedom and three massless degrees of freedom.Functions ri(t), (i = 1 -5) describe the vibration of mass bodies.The model can simulate the heave and pitch effects of mass bodies.The model is excited kinematically by running along the road.Contact forces Fj(t), (j = 3, 4, 5) correspond to the massless degrees of freedom.The equations of motion for the calculation of functions ri(t) have the form of ordinary differential equations.
The terms for calculation of tire forces are Values di(t), (i = 1 -4) represent the deformation of connected members of the model in time t.The derivations with respect to time are denoted by the dot above the symbol of dependent variable.Gj (j = 3, 4, 5) represents the gravity forces acting in the contact points.

Computing model of slab
The slab computing model is created in the sense of the Kirchhoff theory of thin slabs on elastic foundation [2].The equation of motion describing the slab vibration has the form It is a partial differential equation, which will be solved using the Fourier method.The assumption about the shape of deflection plate of slab w0(x,y) due to the load is adopted and wanted function w(x,y,t) describing the shape of deflection plate of the slab in time t is expressed as where (5) In case of discrete moving load contact force Fj(t) must be transformed on continuous load p(x,y,t).It may be done by the advance proposed by Dirac [2] , The convergence of the series in equation ( 10) is possible without major detriment to accuracy, taking into account only the first member of the series.Then the expression (10) is simplified   Synchronous vibration of vehicleslab in our case is described by a system of the 2 nd order ordinary differential equations.Equations ( 1) describe the vibration of vehicle and equation ( 12) describes the slab vibration.The computer program in MATLAB was created for the numerical solution of equations of motion and for displaying of the obtained results.The 4 th order Runge-Kutta step-by-step integration method was used for the numerical solution [4].

Results of numerical solution
Vehicle always enters the slab vibrant.The results of solution are influenced by vehicle initial conditions at the moment of vehicle entry on the slab.Two variants were considered during numerical solution, Fig. 3.

Conclusion
The movement of vehicles along isolated concrete slabs on the elastic foundation is a real engineering task and can be solved in a numerical way.The slab dynamic deflections, internal forces and the vehicle tire forces can be observed.The task was analyzed under two variants of the vehicle initial conditions.The vehicle enters the slab with aggravated springs (Variant a) and the vehicle enters the slab with lightened springs (Variant b).
The absolute maxima of the middle slab vertical dynamic deflections in the interval of speed of vehicle motion 5 -130 km/h are practically the same for both variants (Variant a, wmax = 0.07562 mm, V = 120 km/h, Variant b, wmax = 0.07728 mm, V = 120 km/h).The absolute minima of the middle slab vertical dynamic deflections in the interval of speed of vehicle motion 5 -130 km/h are different for individual variants (Variant a, wmin = 0.07261 mm, V = 100 km/h, Variant b, wmin = 0.07349 mm, V = 5 km/h).

1
b, f describe the stiffness, damping and friction properties of connecting members, m, Iy represent the mass and mass moment of inertia of mass bodies.

Fig. 3 .Fig. 4 .Fig. 5 .
Fig. 3. Initial conditions of vehicle.Numerical solution was realized in dependence on the speed of the vehicle motion in the interval of speeds 5 -130 km/h with the step of 5 km/h.For every speed of the vehicle motion the time course of the middle slab vertical displacements and the time course of contact forces were calculated.Maxima of middle slab vertical displacements and dispersion of contact forces are plotted versus speed of the vehicle motion.Demonstration of the obtained results for Variant a is presented in Figs. 4, 5, 6, 7 and for Variant b in Figs. 8, 9, 10, 11.