Analysis of Buckling and Vibrations of Non-Prismatic Beam on Winkler-Pasternak Foundation with Finite Element and Riley-Ritz Methods

In the analysis of structures such as building foundations, railway rails, reservoirs, airport runways and docks, it is necessary to consider the effect of elastic foundation in modeling. For this reason, different theories such as Winkler, Pasternak, and Reisner have been introduced. On the other hand, a member with a variable cross-section has a higher load-bearing capacity than a prismatic member with a larger cross-section. Nowadays, engineers use members with variable cross-section in the design of structural members to minimize the consumption of materials and the weight of the structure and to increase the critical buckling load capacity. In this article, the buckling and free vibrations of non-prismatic Theory Euler-Bernoulli on the Winkler-Pasternak foundation are investigated by the finite element and Riley-Ritz methods. Non-prismatic cross section changes are investigated in three cases: linear changes in moment of inertia, cubic changes in moment of inertia and fourth order changes in moment of inertia. The foundation effect is also modeled in the equation according to the theory of Winkler and Pasternak. In the first step, the governing differential equation is derived using Hamilton's method. In the next step, the weak form of the differential equation is calculated and Hermitian interpolation functions are used as a transverse displacement function and weight function. In the last step, after extracting the material stiffness matrices, the geometric stiffness and the mass matrix, finally, the eigenvalues of the equation are checked. The results show that the simultaneous increase of the slope of the section and Winkler's spring constant and Pasternak's shear constant have a significant effect on increasing the effective length factor and reducing the load-bearing capacity in all the boundary conditions of different supports. Also, the simultaneous increase of the slope of the cross-section and the spring constant of Winlaker and the shear constant of Pasternak depending on the type of boundary conditions of the support causes an increase or decrease in the dimensionless natural frequency. For the practicality of the results in engineering calculations, aligned curves are used to present the results and display the graphs. The results of previous research are used for verification. There is an acceptable agreement between the present results and previous research.