• Structural Engineering and Mechanics
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• v.27 no.5
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• pp.555-574
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• 2007

Static response of an elastic beam on a two-parameter tensionless foundation is investigated by assuming that the beam is symmetrically subjected to a uniformly distributed load and concentrated edge loads. Governing equations of the problem are obtained and solved by pointing out that a concentrated edge foundation reaction in addition to a continuous foundation reaction along the beam axis in the case of complete contact and a discontinuity in the foundation reactions in the case of partial contact come into being as a direct result of the two-parameter foundation model. The numerical solution of the complete contact problem is straightforward. However, it is shown that the problem displays a highly non-linear character when the beam lifts off from the foundation. Numerical treatment of the governing equations is accomplished by adopting an iterative process to establish the contact length. Results are presented in figures to demonstrate the linear and non-linear behavior of the beam-foundation system for various values of the parameters of the problem comparatively.

• Structural Engineering and Mechanics
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• v.42 no.1
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• pp.1-12
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• 2012

In this study, Pasternak foundation model, which is a two parameter foundation model, is used to analyze the behavior of laterally loaded beams embedded in semi-infinite media. Total potential energy variation of the system is written to formulate the problem that yielded the required field equations and the boundary conditions. Shear force discontinuities are exposed within the boundary conditions by variational method and are validated by photo elastic experiments. Exact solution of the deflection of the beam is obtained. Both foundation parameters are obtained by self calibration for this particular problem and loading type in this study. It is shown that, like the first parameter k, the second foundation parameter G also depends not only on the material type but also on the geometry and the loading type of the system. On the other hand, surface deflection of the semi infinite media under singular loading is obtained and another method is proposed to determine the foundation parameters using the solution of this problem.

• Structural Engineering and Mechanics
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• v.21 no.5
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• pp.519-537
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• 2005

A new generalized Bernoulli/Timoshenko beam-column element on a two-parameter elastic foundation is presented herein. This element is based on the exact solution of the differential equation which describes the deflection of the axially loaded beam resting on a two-parameter elastic foundation, and can take into account shear deformations, semi - rigid connections, and rigid offsets. The equations of equilibrium are formulated for the deformed configuration, so as to account for axial force effects. Apart from the stiffness matrix, load vectors for uniform load and non-uniform temperature variation are also formulated. The efficiency and usefulness of the new element in reinforced concrete or steel structures analysis is demonstrated by two examples.

• Structural Engineering and Mechanics
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• v.65 no.5
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• pp.505-511
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• 2018

Recently, a new foundation model called "Dynamic foundation model" was proposed for the dynamic analysis of structures on the foundation. This model includes a linear elastic spring, shear layer, viscous damping and the special effects of mass density parameter of foundation during vibration. However, the relationship of foundation property parameters with the experimental parameter of the influence of foundation mass also has not been established in previous research. Hence, the purpose of the paper presents a simple experimental model in order to establish relationships between foundation properties such as stiffness, depth of foundation and experimental parameter of the influence of foundation mass. The simple experimental model is described by a steel plate connected with solid rubber layer as a single degree of freedom system including an elastic spring connected with lumped mass. Based on natural circular frequencies of the experimental models determined from FFT analysis plots of the time history of acceleration data, the experimental parameter of the influence of foundation mass is obtained and the above relationships are also discussed.

• Structural Engineering and Mechanics
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• v.7 no.3
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• pp.259-276
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• 1999

An analytical solution for the shape functions of a beam segment supported on a generalized two-parameter elastic foundation is derived. The solution is general, and is not restricted to a particular range of magnitudes of the foundation parameters. The exact shape functions can be utilized to derive exact analytic expressions for the coefficients of the element stiffness matrix, work equivalent nodal forces for arbitrary transverse loads and coefficients of the consistent mass and geometrical stiffness matrices. As illustration, each distinct coefficient of the element stiffness matrix is compared with its conventional counterpart for a beam segment supported by no foundation at all for the entire range of foundation parameters.

• Structural Engineering and Mechanics
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• v.37 no.1
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• pp.61-77
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• 2011

Static and dynamic responses of a completely free elastic beam resting on a two-parameter tensionless Pasternak foundation are investigated by assuming that the beam is symmetrically subjected to a uniformly distributed load and concentrated load at its middle. Governing equations of the problem are obtained and solved by paying attention on the boundary conditions of the problem including the concentrated edge foundation reaction in the case of complete contact and lift-off condition of the beam ina two-parameter foundation. The nonlinear governing equation of the problem is evaluated numerically by adopting an iterative procedure. Numerical results are presented in figures to demonstrate the non-linear behavior of the beam-foundation system for various values of the parameters of the problem comparatively by considering the static and dynamic loading cases.

• Structural Engineering and Mechanics
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• v.90 no.6
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• pp.551-568
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• 2024

The present study deals with static and dynamic behaviors including forced vibrations of an elastic rectangular nano plate on the two-parameter foundation. Firstly, the rectangular plate is assumed to be subjected to uniformly distributed and eccentrically applied concentrated loads. The governing equations of the problem are derived by considering the dynamic response of the plate, employing a series of the Chebyshev polynomials for the displacement function and applying the Galerkin method. Then, effects of the non-essential boundary conditions of the plate, i.e., the boundary conditions related to the shearing forces, the bending moments and the corner forces, are included in the governing equation of motion to compensate for the non-satisfied boundary conditions and increase the accuracy of the Galerkin method. The approximate numerical solution is accomplished using an iterative process due to the non-linearity of the unilateral property of the two-parameter foundation. The plate under static concentrated load is investigated in detail numerically by considering a wide range of parameters of the plate and the foundation stiffnesses. Numerical treatment of the problem in the time domain is carried out by assuming a stepwise variation of the concentrated load and the linear acceleration procedure is employed in the solution of the system of governing differential equations derived from the equation of motion. Time variations of the contact region and those of the displacements of the plate are presented in the figures for various numbers of the two-parameter of the foundation, as well as the classical and nano parameters of the plate particularly focusing on the non-linearity of the problem due to the plate lift-off from the unilateral foundation. The effects of classical and nonlocal parameters and loading are investigated in detail. Definition of the separation between the plate and the two-parameter foundation is presented and applied to the given problem. The effect of the lift-off on the static and dynamic behavior of the rectangular plate is studied in detail by considering various loading conditions. The numerical study shows that the effect of nonlocal parameters on the behavior of the plate becomes significant, when nonlinearity becomes more profound, due to the lift-off of the plate. It is seen that the size effects are significant in static and dynamic analysis of nano-scaled rectangular plates and need to be included in the mechanical analyses. Furthermore, the corner displacement of the plate is affected more significantly from the lift-off, whereas it is less marked in the time variation of the middle displacement of the plate. Several numerical examples are presented to examine the sensibility of various parameters associated with nonlocal parameters of the plate and foundation. Both stiffening and softening nonlocal parameters behavior of the plate are identified in the numerical solutions which show that increasing the foundation stiffness decreases the extent of the contact region, whereas the stiffness of the shear layer increases the contact region and reduces the foundation settlement considerably.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.14 no.3
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• pp.401-411
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• 2001

This paper deals with the vibration analysis of the rectangular plates which are subjected to uniform in-plane stresses and supported on In-homogeneous Pasternak foundation. Two parametric foundation which Winkler foundation parameter and shear foundation parameter considered, is called by the Pasternak foundation. The values of Winkler foundation parameter of central and border zone of plate are chosen as k₁and k₂respectively, and the value of shear foundation is chosen as constant about all zone of plate. After composing global flexural stiffeness matrix, geometrical stiffeness matrix, mass matrix, and the stiffeness matrix of the Pasternak foundation, eigenvalue problems which are composed of these matrices are solved. The result shows that the shear foundation parameter must not be ignore when considering the stiffeness of foundation.

• Proceedings of the KSME Conference
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• 2000.04a
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• pp.596-601
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• 2000

The paper presents free vibration and stability analyses of a non-uniform Timoshenko beam resting on a two-parameter elastic soil. The soil parameters can vary along the spat and is assumed to be two-parameter model including the effects of both transverse shear deformation and elastic foundation Governing equations related to the vibration and the stability of the beam are derived from Hamilton's principle, and the resulting eigen-value problems can be solved to give natural frequencies and critical force by finite element method. Numerical results for both vibration and stability of beams under an axial force are presented and compared with other available solutions. Finally, vibration frequencies, mode shapes and critical forces are investigated for various thickness ratios, shear foundation parameter, Winkler foundation parameter and boundary conditions of tapered Timoshenko beams.

• Structural Engineering and Mechanics
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• v.15 no.2
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• pp.249-267
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• 2003

In this study, circular plates subjected to general type of loads and supported on a two-parameter elastic foundation are analysed. The stiffness, elastic bedding and soil shear effect matrices of a fully compatible ring sector plate element, developed by Saygun (1974), are obtained numerically assuming variable thickness of the element. Ring sector soil finite element is also defined to determine the deflection of the soil surface outside the domain of the plate in order to establish the interaction between the plate and the soil. According to Vallabhan and Das (1991) the elastic bedding (C) and shear parameters ($C_T$) of the foundation are expressed depending on the elastic constants ($E_s$, $V_s$) and the thickness of compressible soil layer ($H_s$) and they are calculated with a suitable iterative procedure. Using ring sector elements presented in this paper, permits the generalization of the loading and the boundary conditions of the soil outside the plate.