Two-parameter elastic foundations are more accurate than a one-parameter (Winkler) foundation and are simpler than semi-infinite elastic continuum foundation models. Two kinds of finite elements are formulated in this paper to analyze beams on one-or two-parameter foundations. Models include Winkler, Filonenko-Borodich, Pasternak, generalized, and Vlasov foundations. One of the two kinds of elements is based on the exact displacement function. The other is based on a cubic displacement function. The stiffness matrix and consistent load vector is derived for each. Numerical tests show that the element based on the exact displacement function can give exact numerical results even if the number of elements is very small. The element based on a cubic function may require a fine mesh to give acceptable results.

Beam elements on two-parameter elastic foundations

Realizing the inadequacy of the classical Winkler model to represent an elastic continuum, Vlasov and Leont'ev developed a two‐parameter model for a beam on an elastic foundation. The Vlasov model required an estimation of a parameter, 7, that controls the decay of stresses in the continuum. Vallabhan and Das developed a unique iterative technique to determine a consistent value of the y parameter. Their model is called a “modified” or “refined Vlasov” model. Here, the finite different technique is employed to solve the fourth‐order differential equations. The results of sample problems obtained by using the modified Vlasov model are compared with more sophisticated finite element solutions.

Modified vlasov model for beams on elastic foundations

Even though the two‐parameter model developed by Vlasov for beams on elastic foundations represents the interaction between the beams and the foundation better than the Winkler model, it requires an estimation of a third parameter, γ, which represents the distribution of the displacements within the foundation. Using variational calculus, the value of γ is shown to be a function of some nondimensional parameters of the beam, the foundation, and the mode of loading. By using an iterative procedure, a consistent value of γ is calculated. An example of a beam carrying a uniformly distributed load is solved and the results are compared with a more rigorous finite element solution.

Parametric Study of Beams on Elastic Foundations

A two-parameter model has been proposed previously for predicting the response of laterally loaded single piles in homogenous soil. A disadvantage of the model is that at high Poisson's ratio, unreliable results may be obtained. In this paper, a new load transfer approach is developed to simulate the response of laterally loaded single piles embedded in a homogeneous medium, by introducing a rational stress field. The approach can overcome the inherent disadvantage of the two-parameter model, although developed in a similar way. Generalized solutions for a single pile and the surrounding soil under various pile-head and base conditions were established and presented in compact forms. With the solutions, a load transfer factor, correlating the displacements of the pile and the soil, was estimated and expressed as a simple equation. Expressions were developed for the modulus of subgrade reaction for a Winkler model as a unique function of the load transfer factor. Simple expressions were developed for estimating critical pile length, maximum bending moment, and the depth at which the maximum moment occurs. All the newly established solutions and/or expressions, using the load transfer factor, offer satisfactory predictions in comparison with the available, more rigorous numerical approaches. The current solutions are applicable to various boundary conditions, and any pile-soil relative stiffness.

Load transfer approach for laterally loaded piles

Theoretical and experimental investigation of local bending effects in sandwich plates

The Vlasov foundation model

By studying the response of a two layered foundation to surface loading, the Kerr foundation model is shown to be identical to a Vlasov foundation model. As a consequence of the analysis, expressions for the shear and spring constants of the Kerr foundation, are obtained.

On the Vlasov and Kerr foundation models

The extended Kantorovich method of variational calculus is used as the basis of a finite element method. This method has wide applicability in many fields of study; such as soil mechanics, vibration of membranes, fluid mechanics etc. Here the method is applied to the problem of magneto-hydrodynamic (MHD) duct flow, and excellent agreement is found between the obtained results and the analytical solution. Several of the disadvantages inherent in the usual form of the extended Kantorovich method are discussed, and means of overcoming them presented.

J. Xenophontos

Papers

The Vlasov foundation model

On the Vlasov and Kerr foundation models

An iterative finite element method using the Kantorovich technique