Analysis of a Beam Made of Physical Nonlinear Material on Nonlinear Elastic Foundation under a Moving Concentrated Load

2 e 1 2 [ q -K W(1 K W )]. = + the vibration equation of motion was derived using Hamilton principle and Euler Lagrange equation. The amplitude of vibration, circular frequency, bending moment, stress and deflection of the beam can be calculated by the presented solution. Considering the response of the beam, in the sense of its resonance, it was found that there is no critical velocity when the behavior of the beam and foundation material is assumed to be physically nonlinear and there are finite values for


INTRODUCTION
Presently there are many structures made from materials which are not subject to the Hook's law. Therefore there is a great tendency to study stress and strain at elements of structures made of physical nonlinear material under various statically and dynamical loads. In linear theory the property of material is not taken into consideration. However, all of relevant parameters are taken into consideration at the theory of nonlinearity. Thus, physical nonlinear theory for small deformations demonstrates an exact calculation method for the analysis of stress, strain and other internal forces at structural elements.
Finally the relationship between stress and strain in the case of physical nonlinear is presented by Hans Kaudrer [1] .
where, i, j = 1, 2, 3 Where K and G at small deformation are volume contraction and shear elastic module respectively. K (σ ) is average stress function and l(t 0 2 ) is shear stress function therefore: Researches have demonstrated that K(σ ) at physical nonlinear material at average relative deformation is close to the straight line K(σ ) = 1 and the two first sentences of shear stress function are enough.
Where is the physical nonlinear coefficient. The relationship between stress and strain at beams by simplification of Eq. 1 for two dimensional surfaces is: The purpose of this study was the analysis of a beam made of physical nonlinear material under moving load on nonlinear elastic foundation.

MATERIALS AND METHODS
It is assumed that the moving concentrated load along the simply supported prismatic beam laid on nonlinear elastic foundation shown in Fig. 1. The load P moves on the beam with a constant velocity V. The reaction of the nonlinear elastic foundation is: Where k 1 , k 2 are coefficients with respect to nonlinear elastic foundation which are determined experimentally, where W is the beam deflection.
The principle of Hamilton for this beam is as follow: The potential and kinetic energy of this system can be written as follows.
The research of the external moving load is given by [4] .
By substitution of expressions 7, 8, 10 in 6, we will have: Substituting, in expression 11 the principle of Hamilton Will be as follows: The deformation of the beam is found from the following expression: Where p(ξ) and q(τ) are coordinate and generalized function respectively. Substituting expression 13 in 12 we will have: Sink .Sink 0 l k 1 It is assumed that P(ξ) is known so the Hamilton integral will be as follows: Where: For Integral 15, Euler formula gives [2] : Where: By substituting expression 22 at 20 and comparing the coefficients of Sin nητ . We will get a lot of cubic nonlinear algebraic equations. There is no exact solution for these equations, there for, we emply an approximated method to solve this equation. For this purpose we take the three constraints Eq. 22 and in this case, we will have system of nonlinear equations. Solving that equation by method of Zeidel, we will have: When the load is out of the beam: differential equation of free vibration will be: Finally, by solving Eq. 27 we find the period of vibration: Where, Q-is amplitude of vibration And circular acceleration of vibration will be:

RESULTS AND DISCUSSION
Now the obtained analytic solution is applied to the following example. Figure 2 shows the cross section of a railway beam. The material of beam is considered steel which is made of a nonlinear material. The specification of the material are as follows:  Table 1. Based on Table 1 resonance curve is drawn (Fig. 3).
When k 2 = 0.0 that is nonlinear beam on elastic foundation (Winkler Theorem), in this case, the deformation, stress and bending moment at the middle span is determined and is showed at the Table 3.