Elastic Foundation Beam Analysis on Rigid Pavement and Piles

. In this study, the rigid pavement structure and pile foundation were modeled as the foundation of the Winkler elastic beam. for pavement with distribution loads and traffic loads, and variations in support. While for pile foundations with lateral loads. Solution of analytical differential equations with computer applications (CAS) Maple. The aim of this research are : (a). To know whether completion of algebraic computer application can be obtained by analytically method or numerically method, (b). To check whether analytical model is possible to be applied to various loads at any point in the beam or not, (c). Where of the strength or the displacement boundary to be located in the beam during load application, (d). To verify an approach possibility implementation of a simple computer algebra system, (e). To know whether achieved solutions which possible to solve ordinary differential equation (ODE) can be implemented in any mathematical system.


Introduction
The concept of beams on elastic foundations was first developed by E. Winkler in 1867. Since then it has been applied by Terzaghi and many others to geotechnical problems in which ground or subgrade is supporting media. Analysis of bending of beams on an elastic foundation is developed on the assumption that the reaction forces of the foundation are proportional at every point to the deflection of the beam at that point. This assumption was introduced by E. Winkler, formed the basis of H. Zimmermann's classic work on railroad track, published in 1888 [1,2,3]. Application example of the concept of elastic foundation consists of various structures including foundation beams and footings, piles, retaining walls, piles and many others [2,3,4,5,6]

Differential Equations of Elastic Foundation
By reviewing the foundation beams, along the span is supported by the ground and receives vertical loads. The reaction that occurs according to the basic assumption, the intensity of p at each point is proportional to the deflection of the beam (y) at that point Where ( ) h y x is the solution of homogeneous differential equation The integration constants C 1 , C 2 , C 3 , C 4 , appearing in equation are determined the following boundary conditions (BCs) [4,6,7,8]

Pile With Lateral Loads
The structural analysis of vertical piles with lateral loads gives almost the same boundary value problem (BVP) as a beam on an elastic foundation. The differences are in the loading which is concentrated at the pile's top and in the elastic foundation model. The pile foundation model includes variable spring stiffness while the beam model has constant stiffness of the bed's springs [3,4,5,9,10,11].

Fig. 2. Pile foundations with lateral load
In accordance with Equation (6), with load distributed (q) along the beam = 0, and the modulus of the foundation is a function of depth k=f(z), the following equation is obtained The loads q(x) can be presented by singularity functions as [4] ( ) (16) where a is the distance from the force P 1 to the left end of the beam; b is the distance from the beam's left end to the beginning of the force P 2 ; c is the distance from the beam's left end to the force P 3 . For δ(x − a) notation is the Dirac delta function. Equation (15, 16) is solved numerically with the CAS Maple application, with code : Sol:=evalf(dsolve({de,ibc},y(x))): The calculation results can be seen in Table 1 and Figure 4

Pile Foundations with Lateral Load
From Bowles [5], the pile is a steel section HP 360×174, b=378 mm, Moment Inertia Ix = 0.5080 Based on the results obtained analytically by solving differential equations, and comparing the results of analysis from Bowles [5] using the finite element method (FEM), for maximum deflection there is a difference of 0.28%, maximum moment of -0.16% and soil pressure (z = 0) difference of 0.30%.

Conclusion
Analysis of elastic foundation with the Winkler model can be used in general for the solution of rigid pavement structures, and soil-pile interaction approaches.
-Completion of algebraic computer applications can be obtained analytically or numerically.
-Analytical model can be applied to various loads at any point in the beam, -Application of conditions to the strength or displacement boundary at the end of the beam.
-Implementation of an approach to a simple computer algebra system -Solutions presented can be implemented in any mathematical system that can solve Ordinary Differential Equations (ODE) -The analytical solutions obtained can be used to analyze the effect of different parameters on structural behavior.