Abstract:
This paper deals with linear elastic behavior of deep beams resting on linear and nonlinear Winkler type elastic foundations with both compress ional and tangential resistances. The basic or governing equations of beams on nonlinear elastic Winkler foundation are solved by finite difference method. The finite element method in Cartesian coordinates is formulated using two dimensional plane stress isoparametric finite elements to model the deep beam and elastic springs to model the foundation. Two computer programs coded in Fortran_77 for the analysis of beams on nonlinear elastic foundations are developed. Comparisons between the two methods and other studies are performed to check the accuracy of the solutions. Good agreement was found between the solutions with percentage difference of 3%. Several important parameters are incorporated in the analysis, namely, the vertical subgrade reaction, horizontal subgrade reaction and beam depth to trace their effects on deflections, bending moments and shear forces.

Abstract:
Critical velocities are investigated for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. The determination of critical velocities ultimately comes down to discrimination of the existence of multiple real roots of an algebraic equation with real coefficients of the 4th degree, which can be solved by employing Descartes sign method and complete discrimination system for polynomials. Numerical calculations for the European high-speed rail show that there are at most four critical velocities for an infinite Timoshenko beam, which is very different from those gained by others. Furthermore, the shear wave velocity must be the critical velocity, but the longitudinal wave velocity is not possible under certain conditions. Further numerical simulations indicate that all critical velocities are limited to be less than the longitudinal wave velocity no matter how large the foundation stiffness is or how high the loading frequency is. Additionally, our study suggests that the maximum value of one group velocity of waves in Timoshenko beam should be one “dangerous” velocity for the moving load in launching process, which has never been referred to in previous work. 1. Introduction Moving-load problems have received a great deal of attention worldwide in the past several decades. The earliest moving-load problems are about railway bridges excited by traveling trains. Then the application areas gradually have been extended to high-speed commuter trains, missile sled test tracks, high-speed projectile launchers, and so on. “Critical velocity” is a phenomenon that has been found in beams subjected to moving loads. A resonant wave in a beam can be induced when a load moves at the critical velocity, which results in an unbounded increase of the displacements, rotation, and bending moments of the beam for an undamped case. In reality, damping always exists, but even so, a very large deflection of the beam may occur if the moving load approaches the critical velocity [1–3]. Researches show that the “critical velocity” phenomenon may shorten life of launchers due to high stress, fatigue, premature wear, and gouging of rails [4]. Therefore, it is very necessary to determine the critical velocity for a beam subjected to a moving load. Timoshenko [3] firstly derived an expression for the critical velocity of a concentrated load moving along the Bernoulli-Euler beam resting on a continuous elastic foundation. It was proposed that the critical velocity is analogous to an additional longitudinal compressive force

Abstract:
The buckling of axially compressed beams resting on elastic foundation is considered, accounting for discontinuous (unbonded) contact between beam and subgrade which is the case of real structural response. Using Galerkin’s method a two-region contact/non-contact configuration was revealed as the only possible post-buckling deformation for both pinned – pinned and fixed – fixed boundary conditions, a fact strongly contradicting relevant results where continuous contact was assumed.

Abstract:
An improved model for bending of thin viscoe-lastic plate resting on Winkler foundation is presented. The thin plate is linear viscoelastic and subjected to normal distributed loading, the effect of normal stress along the plate thickness on the deflection and internal forces is taken into account. The basic equations for internal forces and stress distribution are derived based on the general viscoelastic theory under small deformation condition. The reduced equations for elastic case are given as well. It is shown that the proposed model reveals a larger flex-ural rigidity compared to that in classic models, in which the normal stress along the plate thickness is neglected.

Abstract:
In this paper, we employ the theory of matrices and continued fractions for the solution of the bending problem of continuous beams on elastic foundation with unyielding supports. End moments are obtained in explicit expressions. Accurate numerical results may be calculated from these expressions directly without salving simultaneous equations.

Abstract:
The traditional way to overcome the shortcomings of the Winkler foundation model is to incorporate spring coupling by assemblages of mechanical elements such as springs, flexural elements (beams in one-dimension, 1-D, plates in 2-D), shear-only layers and deformed, pretensioned membranes. This is the class of two-parameter foundations ? named like this because they have the second parameter which introduces interactions between adjacent springs, in addition to the first parameter from the ordinary Winkler?s model. This class of models includes Wieghardt, Filonenko-Borodich, Hetényi and Pasternak foundations. Mathematically, the equations to describe the reaction of the two-parameter foundations are equilibrium, and the only difference is the definition of the parameters. In order to analyse the bending behavior of a Euler-Bernoulli beam resting on linear variable two-parameter elastic foundation a (displacement) Finite Element (FE) formulation, based on the cubic displacement function of the governing differential equation, is introduced.

Abstract:
Purpose: The main issue of this paper is to present results of finite element analysis of beams elements onunilateral elastic foundation received with a use of special finite elements of zero thickness designated forfoundation modelling.Design/methodology/approach: Computer strength analysis with a use of Finite Element Method (FEM)was carried out.Findings: The paper presents possibilities of special finite elements of zero thickness which enable takinginto consideration unilateral contact in construction-foundation interaction as well as an impact of surroundingconstruction environment to its behaviour.Research limitations/implications: Further researches should concentrate on taking into consideration amulti-layer aspects as well as elasto-plasticity of foundation.Practical implications: Modern engineering construction on elastic foundation analyze require not onlystandard analysis on Winkler (one parameter) foundation but also calculation of construction on two-parameterfoundation which will take into consideration a possibility of loosing contact between construction and foundation(unilateral contact).Originality/value: The paper can be useful for person who performs strength analysis of beams on elasticfoundation with a use of finite element method.

Abstract:
Nonlinear beam resting on linear elastic foundation and subjected to harmonic excitation is investigated. The beam is simply supported at both ends. Both linear and nonlinear analyses are carried out. Hamilton’s principle is utilized in deriving the governing equations. Well known forced duffing oscillator equation is obtained. The equation is analyzed numerically using Runk-Kutta technique. Three main parameters are investigated: the damping coefficient, the natural frequency, and the coefficient of the nonlinearity. Stability regions for first mode analyses are unveiled. Comparison between the linear and the nonlinear model is presented. It is shown that first mode shape the natural frequency could be approximated as square root of the sum of squares of both natural frequency of the beam and the foundation. The stretching potential energy is proved to be responsible for generating the cubic nonlinearity in the system.

Abstract:
The classical flexure problem of non-linear incompressible elasticity is revisited for elastic materials whose mechanical response is different in tension and compression---the so-called bimodular materials. The flexure problem is chosen to investigate this response since the two regions, one of tension and one of compression, can be identified easily using simple intuition. Two distinct problems are considered: the first is where the stress is assumed continuous across the boundary of the two regions, which assumption has a sound physical basis. The second problem considered is more speculative: it is where discontinuities of stress are allowed. It is shown that such discontinuities are necessarily small for many applications, but might nonetheless provide an explanation for the damage incurred by repeated flexure. Some experimental evidence of the possibility of bimodularity in elastomers is also presented.

Abstract:
The key aspect in the design of flexible structural elements in contact with bearing soils is the way in which soil reaction, referred to qualitatively as soil’s reactive pressure (p), is assumed or accounted for in analysis. A magnitude and distribution of p might be preliminary assumed, or some mathematical relationships could be incorporated into the analysis itself, so that p is calculated as part of the analysis. In order to eliminate the bearing soil reaction as a variable in the problem solution, the simplified continuum approach is presented. This idealization provides much more information on the stress and deformation within soil mass compared to ordinary Winkler model, and it has the important advantage of the elimination of the necessity to determine the values of the foundation parameters, arbitrarily, because these values can be computed from the material properties (deformation modulus, Es, Poisson ratio, νs and depth of influence zone, H, along the beam) for the soil. A numerical investigatio proach is also presented.