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:
The dynamic response of a Timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjected to motion of a traveling mass moving with a constant velocity is studied. Primarily, the beam’s nonlinear governing coupled PDEs of motion for the lateral and longitudinal displacements as well as the beam’s cross-sectional rotation are derived using Hamilton’s principle. On deriving these nonlinear coupled PDEs the stretching effect of the beam’s neutral axis due to the beam’s fixed end conditions in conjunction with the von-Karman strain-displacement relations is considered. To obtain the dynamic responses of the beam under the act of a moving mass, derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic responses are obtained using mode summation technique. Furthermore, after verification of our results with other sources in the literature a parametric study on the dynamic response of the beam is conducted by changing the velocity of the moving mass, damping coefficient, and stiffnesses of the foundation including linear and cubic nonlinear parts, respectively. It is observed that the inclusion of geometrical and foundation stiffness nonlinearities into the system in presence of the foundation damping will produce significant effect in the beam’s dynamic response. 1. Introduction The topic of vibration study of structural elements such as strings, beams, plates, and shells under the act of a moving mass is of great interest and importance in the field of structural dynamics. It should be noted that the review of numerous reported studies related to the dynamic behavior of mechanical/structural systems discloses that almost linear behavior of such systems is considered. Indeed, in reality such systems inherently and naturally have nonlinear behavior, for example, due to the geometrical nonlinearity or when they are subjected to external loadings comparatively large enough. As we will see later on in the modeling of the problem, the stretching of the beam’s neutral axis due to fixed ends condition adds another nonlinearity to the dynamical behavior of the system. In addition, there are some other external distinct mechanical elements having nonlinear behavior attached to such structures like shock energy absorbers or viscoelastic foundations which will add further other nonlinearities in the model analysis. From mechanical point of view, any beam structure can be modeled as a thin or thick beam for which different theories usually can be implemented. In extending the issue of the

Abstract:
An efficient analytical method for vibration analysis of a Euler-Bernoulli beam on elastic foundation with elastically restrained ends has been reported. A Fourier sine series with Stoke’s transformation is used to obtain the vibration response. The general frequency determinant is developed on the basis of the analytical solution of the governing differential equation for all potential solution cases with rigid or restrained boundary conditions. Numerical analyses are performed to investigate the effects of various parameters, such as the springs at the boundaries to examine how the elastic foundation parameters affect the vibration frequencies. 1. Introduction Beams resting on elastic foundations have wide application in engineering practice. The vibration analysis of beams is investigated using various elastic foundation models, such as, Vlasov, Pasternak, and Winkler models. A number of studies have been performed to predict the dynamic response of beams on elastic foundations with different boundary conditions. Numerous works have been performed to explore the static deflection and vibration response of the beams resting on various elastic foundations. Chun [1] has investigated free vibration of hinged beam. Maurizi et al. [2] have considered the vibration frequencies for a beam with different boundary conditions. Vibration of beams on partial elastic foundations has been studied by Doyle and Pavlovic [3]. Laura et al. [4] have investigated beams which carry concentrated masses subject to an axial force. Abbas [5] has investigated vibration of Timoshenko beams with elastically restrained ends. Free vibration and stability behavior of uniform beams and columns with nonlinear elastic end rotational restraints has been considered by Rao and Naidu [6]. Free vibration behaviour of an Euler-Bernoulli beam resting on a variable Winkler foundation has been considered by Kacar et al. [7]. Civalek [8] has implemented differential quadrature and harmonic differential quadrature methods for buckling analysis of thin isotropic plates and elastic columns. H. K. Kim and M. S. Kim [9] have considered vibration of beams with generally restrained boundary conditions. A number of studies have been reported investigating the free vibration of beams on elastic foundation [10–25]. Although vibration analysis of beams on elastic foundation is a widely studied topic, there are only few papers that exist in the literature pertaining to the analysis of beams with elastically restrained ends. In this study, an efficient method is introduced for the analysis of the free

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 problem of vibrations of fluid-conveying pipes resting on a two-parameter foundation model such as the Pasternak-Winkler model is studied in this paper. Fluid-conveying pipes with ends that are pinned-pinned, clampedpinned and clamped-clamped are considered for study. The frequency expression is derived using Fourier series for the pinned-pinned case. Galerkin’s technique is used in obtaining the frequency expressions for the clamped-pinned and clamped-clamped boundary conditions. The effects of the transverse and shear parameters related to the Pasternak- Winkler model and the fluid flow velocity parameter on the frequencies of vibration are studied based on the numerical results obtained for various pipe end conditions. From the results obtained, it is observed that the instability caused by the fluid flow velocity is effectively countered by the foundation and the fluid conveying pipe is stabilized by an appropriate choice of the stiffness parameters of the Pasternak-Winkler foundation. A detailed study is made on the influence of Pasternak-Winkler foundation on the frequencies of vibration of fluid conveying pipes and interesting conclusions are drawn from the numerical results presented for pipes with different boundary conditions.

Abstract:
In the present study, an investigation is carried out to determine
the effect of soil–rock and rock–rock foundation systems on
dynamic response of block foundations under vertical mode of
vibration. The half-space theory is used for the analysis of
foundation resting on homogeneous soil and rocks. The finite
element program having transmitting boundaries is considered
for layered system considering soil–rock and rock–rock
combinations. The analysis is carried out in details for soil–
rock and weathered rock–rock systems and the different
equations are presented for above combinations. The effect of
top layer thicknesses, shear wave velocity and eccentric
moments are also simulated. The rock–rock systems considered
are sandstone, shale and limestone underlain by basalt rock. It
is interpreted that as the shear wave velocity ratio increase the
natural frequency increases and the peak displacement
amplitude decreases.

Abstract:
This paper presents formulations for a Timoshenko beam subjected to an accelerating mass using spectral element method in time domain (TSEM). Vertical displacement and bending rotation of the beam were interpolated by Lagrange polynomials supported on the Gauss-Lobatto-Legendre (GLL) points. By using GLL integration rule, the mass matrix was diagonal and the dynamic responses can be obtained efficiently and accurately. The results were compared with those obtained in the literature to verify the correctness. The variation of the vibration frequencies of the Timoshenko and moving mass system was researched. The effects of inertial force, centrifugal force, Coriolis force, and tangential force on a Timoshenko beam subjected to an accelerating mass were investigated. 1. Introduction Dynamic response of structures subjected to a moving force or moving mass is an important issue in engineering problems. For example, the trains have experienced great advances characterized by increasingly higher speeds and weights of vehicles. As a result, the dynamic response, as well as stresses, can be significantly higher than that before or static loads. The problem arose from the observations is a structure subjected to moving masses. Many researchers studied these problems and many studies are presented in the literatures. For examples, references [1–3] assuming the moving load to be a moving force have given some analytical solutions. Abu-Hilal [4] studied the dynamic response of a double Euler-Bernoulli beam due to a moving constant load. Fryba [5] extensively analyzed the solution of moving loads on structures. Rao [6] gave a detailed analysis of the vibration of a beam excited by a moving oscillator using a perturbation method. Zibdeh and Juma [7] presented the dynamic response of a rotating beam subjected to a random moving load using analytical and numerical methods. Akin and Mofid [8] investigated the dynamic behavior of Bernoulli-Euler beams carrying a moving mass with different boundary conditions using analytical-numerical method, and achievements by other researchers are presented in literatures [9–14]. The studies mentioned are based on Bernoulli-Euler beam, while the moment of inertia and shear deformation should be taken into account when ratio of the height to span is large. References [15–20] studied the dynamic response of Timoshenko beams subject to moving force. Ross [21] studied the problem of a viscoelastic Timoshenko beam subjected to a step-loading using the Laplace transform method. Katz et al. [22] solved the dynamic response of a rotating

Abstract:
In this study a discrete semi analytical method based on a combination of the Finite Elements, Thin Layers and Integral Equations Methods is presented to evaluate the vertical static and dynamic linear response of a pile foundation subjected to a vertical load. This study, which takes into account the soil-pile interaction effects, is able to treat the case of an arbitrary shaped pile. However, the presentation will be limited to the case of a massless square pile with linear elastic behaviour. The pile is embedded in a viscoelastic soil-layer overlaying a bedrock and subjected to static or/and harmonic vertical load, applied at the head of the pile. The aim of this study is to characterise the response of a pile through the force-displacement relationship. Numerical results are presented to illustrate the accuracy of this study.

Abstract:
the response of simply supported rectangular plates carrying moving masses and resting on variable winkler elastic foundations is investigated in this work. the governing equation of the problem is a fourth order partial differential equation. in order to solve this problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. for the solutions of these equations, a modification of the struble's technique and method of integral transformations are employed. numerical results in plotted curves are then presented. the results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor r0 increases. furthermore, for fixed value of r0, the displacements of the simply supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus f0 increases. the results further show that, for fixed r0 and f0, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. therefore, the moving force solution is not a safe approximation to the moving mass problem. hence, safety is not guaranteed for a design based on the moving force solution. also, the analyses show that the response amplitudes of both moving force and moving mass problems decrease both with increasing foundation modulus and with increasing rotatory inertia correction factor. the results again show that the critical speed for the moving mass problem is reached prior to that of the moving force for the simply supported rectangular plates on variable winkler elastic foundation.

Abstract:
Although the Winkler’s model is a poor representation of the many practical subgrade or subbase materials, it is widely used in soil-structure problems for almost one and a half century. The foundations represented by Winkler model can not sustain shear stresses, and hence discontinuity of adjacent spring displacements can occur. This is the prime shortcoming of this foundation model which in practical applications may result in significant inaccuracies in the evaluated structural response. In order to overcome these problem many researchers have been proposed various mechanical foundation models considering interaction with the surroundings. Among them we shall mention 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 Filonenko-Borodich, Pasternak, generalized, and Vlasov foundations. Mathematically, the equations to describe the reaction of the two-parameter foundations arc equilibrium ones, and the only difference is the definition of the parameters. For the convenience of discussion, the Pasternak foundation is adopted in present paper. In order to analyse the bending behavior of a Euler-Bernoulli beam resting on two-parameter elastic foundation a (displacement) Finite Element (FE) formulation, based on the cubic displacement function of the governing differential equation, is introduced. The resulting effects of shear stiffness of the Pasternak model on the mechanical quantities are discussed in comparison with those of the Winkler’s model. Some numerical case studies illustrate the accuracy of the formulation and the importance of the soil shearing effect in the vertical direction, associated with continuous elastic foundation.