Concrete slab on elastic foundation under one axle load

The paper is dedicated to the dynamical analysis of an isolated concrete slab on an elastic foundation to moving load effect of one axle load. The computational models of one axle load and a slab are mathematically described by the system of ordinary differential equations. The equations of motion are solved numerically in the environment of program system MATLAB. The plate response and dynamic load are monitored at a certain speed and certain initial conditions, depending on the properties of the elastic subgrade.


Introduction
Reinforced concrete roadways are frequently used today [1]. Temporary road structures are often constructed as isolated prefabricated slabs. Smaller slabs are loaded with only one vehicle axle at a given time. The time courses of dynamic load and time courses of stress and strains at various conditions are needed for the proper design of the structure. The response of the slab to such a moving load is dependent on many factors -the speed of the vehicle, the initial conditions, the properties of the ground, the dimensions of the slab, the properties of the material, and so on. The computational model of one axle load and a computational model of a slab on an elastic foundation are created. Equations of motion are derived in the form of ordinary differential equations and they are solved numerically in the environment of MATLAB [2]. The influence of subgrade stiffness on slab vertical deflection and on dynamic load from the movement of one vehicle axle model is monitored. The results of the calculations are shown in the form of graphs supplemented with numerical data. vertical direction. The contact force F(t) corresponds to the mass-less degree of freedom. The model was already used in [3]. The equations of motion for unknown functions ri(t) have the meaning of ordinary differential equations The expression for contact force is as follow

Computational model of a slab
The computational model of thin slabs on an elastic foundation was adopted [4]. The model was already used in [3]. The equation of motion of the slab oscillation has the form This partial differential equation is solved by Fourier method. The assumption about the shape of bending plate of the slab w0(x,y) due to the load is adopted and the wanted function w(x,y,t) describing the shape of bending plate of the slab in time t is expressed as The function w0(x,y) is the known function and it dependents on the coordinates x, y only. The q(t) function has the meaning of generalized Lagrange coordinate. It represents the proportionality coefficient between dynamic and static deflection in the time t.  (4) and (5) in equation (3) and its further modification. In case of observing deflection in the middle of the slab (x = lx/2 a y = ly/2) the resulting shape of (3) is In (6)

Numerical solution and obtained results
The synchronous vibration of the system vehicleslab is mathematically described by the system of three 2 nd order ordinary differential equations. Two equations (1) describe vibration of vehicle and one equation (6) describes vibration of the slab. All numerical calculations and displaying of obtained results were realised in MATLAB [2]. The 4 th order Runge -Kutta step-by-step integration method was used for numerical solution.
Initial conditions are also part of the solution. It is assumed that the slab is at the beginning of the solution motionless. Initial conditions on the vehicle are considered in 2 variants.  Fig. 3, 5, 6 and for Variant 2 in Fig. 4, 7, 8.

Conclusion
Modulus of foundation compressibility greatly affects plate deflections. The initial conditions on the vehicle practically do not affect this value. In Variant 1, the deflection varies within the range of 0.126 ÷ 0.021 mm and for Variant 2 within the range of 0.134 ÷ 0.022 mm. Changing the modulus of foundation compressibility practically does not affect the contact forces. From an engineering point of view, the initial conditions on the vehicle do not affect the dispersion of contact forces. We are still moving between 6.978 ÷ 6.931 kN. The only change is that in Variant 1, we move at loading levels of approximately 64 ÷ 71 kN and in Variant 2 at loading levels of approximately 61 ÷ 68 kN. This work was supported by the Grant National Agency VEGA of the Slovak Republic, project number 1/0006/20.