Plate-foundation System Research Articles

Weak-form differential quadrature element analysis of plate on a tensionless and frictional foundation using a higher-order kinematics

Plate-foundation systems, as a structure common in engineering practices, have received extensive scientific research. A novel technical scheme is proposed to improve the computational efficiency in predicting the nonlinear responses of a plate on the foundation. The core work of this technical scheme lies in two aspects. The first aspect is to create a novel plate-foundation system, where a higher order kinematical assumption common in Carrera Unified Formulation is used to model the plate, and a plate-foundation interaction model taking into account the tensionless and frictional characteristics is created. The adjustable kinematical parameters of the higher-order kinematical assumption endow the plate with flexible accuracy. The second aspect is to derive a two-dimensional (2D) differential quadrature (DQ) formula for 2D interpolation based on the one-dimensional (1D) DQ formula, and thus a weak-form differential quadrature element method (WDQEM) for nonlinear quasi-static analysis of the plate-foundation system is proposed by using the principle of virtual work and the 2D interpolation derived. The proposed WDQEM supports flexible nodal configuration in the element, making it degree-of-freedom-flexible and suitable to carry out both p- and h-version strategies in the numerical simulation. Several numerical examples are given to (i) verify the proposed technical scheme and the corresponding computer codes; (ii) to investigate the numerical performance of the WDQEM through comparison between the classical plate model and the 3D solid model, and (iii) to study some physical behaviors related to the tensionless and frictional characteristics of the plate-foundation system. The example results show that (i) the WDQEM converges much faster than the conventional finite element method due to the higher order approximation; (ii) the higher order kinematics, due to its capability of adjusting the accuracy of kinematical assumptions, can effectively reduce the computational scale; (iii) the smoothness problem in the post process of stresses analysis is greatly alleviated, (iv) the tensionless and frictional features of the plate-foundation system should be considered when evaluating both the vertical and horizontal bearing capacities of the plate.

FE Analysis of FGM Plates on Arbitrarily Orthotropic Pasternak Foundations for Membrane Effects

In this study, the finite element analysis of sigmoid functionally graded material (S-FGM) plates resting on orthotropic Pasternak elastic foundations with different material angles is presented. For modelling, SAP2000 software package is used in which the required adjustments are made to obtain the expected behaviour of the plate-foundation system. The plate is modelled both using solid elements and layered shell elements by defining a number of solid elements and layers in the thickness direction having elastic properties equivalent to the properties of the S-FGM plate. The interaction between the plate and the foundation is treated by equalizing the vertical displacements of the plate and foundation nodal points. The orthotropic Pasternak foundation is modelled using plane strain elements with some adjustments to the elastic properties. The membrane effects of the simply supported S-FGM plate on Pasternak foundation are considered by assigning the edge boundaries of the system as pinned supports and these effects are excluded by converting all boundary nodes into roller supports except one of the corner nodes of the plate and the foundation due to the stability requirement. A number of verification examples are performed to demonstrate the convenience and robustness of the proposed model. This work can be easily extended to static and dynamic analyses of FGM plates with various geometries resting on arbitrarily orthotropic Pasternak elastic foundations for further studies.

Numerical solutions for magneto–electro–elastic laminated plates resting on Winkler foundation or elastic half-space

This article investigates the bending response performances of the magneto–electro–elastic (MEE) laminated plates resting on the Winkler foundation or the elastic half-space subjected to a transverse mechanical loading .By assuming that the foundation is not electrically and magnetically conductive, the scaled boundary finite element method (SBFEM) based on the three-dimensional (3D) theory of elasticity is applied for both the simulation of the MEE laminated plate and the elastic half-space. The SBFEM model considers the generalized displacement involving the elastic displacement, electric potential and magnetic potential as the nodal degree of freedom for the MEE laminated plates, and only the in-plane of the MEE laminated plate or the boundary of the elastic half-space needs to be discretized leading to reduce the spatial dimension by one. Furthermore, in the SBFEM, the governing equations can be solved by using an analytical approach in the radial direction of the scaled coordinate system, so that it is particularly suitable for the simulation of the elastic half-space. For the Winkler foundation–plate system, the global stiffness coupling governing equation that includes the interaction between the MEE laminated plate and the Winkler foundation is derived directly from the 3D elasticity equations of the MEE laminated plate by assuming that the foundation reactions are proportional to the transverse displacements of the plate structure. While for the MEE laminated plate-half-space system, the whole domain is divided into three sub-domains including the MEE laminated plate structure, the near and semi-infinite far foundation systems based on the sub-structure method, and then the stiffness matrix of each sub-domain can be determined by means of the SBFEM. As a result, the global stiffness equation of the plate-half-space system can be assembled according to the principle of the degree of freedom matching at the same nodes. The numerical results obtained for limiting cases by using the proposed method were compared with the published works and showed excellent agreements with the solutions based on the analytical and numerical approaches, so that the accuracy and applicability of the proposed formulations for the analysis of the interaction problems between the MEE laminated plate and the Winkler foundation or elastic half-space can be verified. Moreover, several numerical examples with various material properties, geometries, stacking sequences, aspect ratios, and supported boundary conditions were presented to show the effects of which on the responses of the plate–foundation system.

A combined boundary-finite element procedure for dynamic analysis of plates with fluid and foundation interaction considering free surface effect

This study combines a mixed finite element formulation and a boundary element procedure for the free vibration analysis of Mindlin plates resting on Pasternak foundation and interacting with a quiescent fluid with a free surface on the other side. The plate-foundation system is represented by a two field mixed finite element formulation, based on the Hellinger-Reissner variational principle, and the fluid-structure interaction is incorporated into the analysis through a boundary element solution. The proposed formulation provides the added mass matrix in terms of the plate deflection, which is appended into the plate equation of motion. The method is applied to the free vibration problem of circular and elliptical bottom plates of rigid fluid storage tanks supported by elastic foundation. Effects of system parameters, such as thickness to width and fluid depth ratios, foundation parameters and plate ellipticity, are extensively studied.

Dynamic analysis of Mindlin plates on viscoelastic foundations under a moving vehicle by CS-MIN3 based on C0-type higher-order shear deformation theory

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed to improve the performance of the existing three-node Mindlin plate element (MIN3) for static and dynamic analyses of Mindlin plates. In this paper, the CS-MIN3 is extended to the C0-type higher-order shear deformation plate theory (C0-HSDT) and incorporated with damping-spring systems for dynamic analyses of Mindlin plates on the viscoelastic foundation subjected to a moving vehicle. The plate-foundation system is modeled as a discretization of triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. A two-step process for transforming the weight of a four-wheel vehicle into loads at nodes of elements is presented. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results. A parametric examination is also conducted to determine the effects of various parameters on the dynamic response of the plates on the viscoelastic foundation subjected to the moving vehicle.

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise deformation theory for dynamic response of composite plates resting on viscoelastic foundation

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamics analyses of Mindlin plates. In this paper, the CS-FEM-DSG3 is extended to the layerwise deformation theory for dynamic response of sandwich and laminated composite plates resting on viscoelastic foundation subjected to a moving mass. The plate-foundation system is modeled as a discretization of triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results.

A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation

A cell-based smoothed finite element method using three-node Mindlin plate element (CS-FEM-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS-FEM-MIN3 is extended and incorporated with damping-spring systems for dynamic responses of sandwich and laminated composite plates resting on viscoelastic foundation subjected to a moving mass. The plate-foundation system is modeled as a discretization of three-node triangular plate elements supported by discrete springs and dashpots at the nodal points representing the viscoelastic foundation. The position of the moving mass with specified velocity on triangular elements at any time is defined, and then the moving mass is transformed into loads at nodes of elements. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of others available numerical results. A parametric examination is also conducted to determine the effects of various parameters on the dynamic response of the plates on the viscoelastic foundation subjected to a moving mass.

A parametric study for thick plates resting on elastic foundation with variable soil depth

The soil depth is generally considered to be constant for the analysis of plates resting on elastic foundation in the literature. However, it is most reasonable to have a variable subsoil depth as the plate dimensions get larger. In present study, linearly varying subsoil depth is considered as well as constant, linear and quadratic variation of modulus of elasticity with subsoil depth. Also, a parametric study is performed to demonstrate the behavior of thick plates on elastic foundations with variable soil depth. Modified Vlasov Model is used for the analysis of the plate foundation system, and 8-noded Mindlin plate element incorporating shear strain throughout plate thickness is used for the finite element model. Numerical examples are obtained from the literature to compare results and to show the influence of variable soil stratum depth on the behavior of plates. Displacements, bending moments, and shear forces are presented in tabular and graphical formats. As far as results are compared, it can be concluded that variable soil depth significantly affects the variation of the displacements and therefore the internal forces of the plate while keeping it constant ends up with unrealistic results.

Analytical solution to the elastic bending of long and rectangular thin plate resting on rubber foundation

In this study, the elastic bending of long and rectangular plate resting on unilateral rubber foundation under a concentrated load has been investigated. A new continuum model has been developed to study the rubber foundation reaction. A simplified one-parameter model could thus be obtained for the foundation reaction permitting the parameter to be directly determined without resorting to trial estimation. The governing differential equations have been derived for the plate–foundation system. A numerical–analytical method has been proposed for the exact and closed-form solution of the equations. Several examples have been presented in order to draw a comparison between the results of the proposed solution and the conventional numerical solution. The results have also been compared with the Winkler's mechanical model. Finally, the effect of various parameters on the behaviour of the plate–foundation system has been studied. The results indicate that the length of contact region between the plate and the foundation is influenced by the mechanical and geometrical parameters of the plate and also on the foundation modulus but does not depend on the loading.

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate–foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.

Three-dimensional free vibration of thick circular plates on Pasternak foundation

This paper describes a study of the three-dimensional vibration characteristics of thick circular plates resting on elastic foundation. The analysis is based on the three-dimensional small-strain, linear and exact elasticity theory. The foundation is described by the Pasternak model. The Ritz method is used to derive the frequency equation of the plate-foundation system by augmenting the strain energy of the plate with the elastic potential energy of the foundation. A set of Chebyshev polynomials multiplied by a boundary function is adopted as the admissible functions of the displacement components in each direction. For plates with free edges, the effect of foundation medium beyond the edge of the plate has been considered by introducing the generalized shearing force concept in the analysis. The convergence and comparison studies demonstrate the correctness and accuracy of the present method. It is shown that the present method has rapid convergent rate, stable numerical operation and very high accuracy. Parametric investigations on the dynamic behavior of thick circular plates resting on elastic foundation have been carried out with respect to various thickness–radius ratios, foundation stiffness parameters and boundary conditions. Results known for the first time have been reported and discussed in detail. Some significant conclusions have been drawn.

FORMULATIONS OF THE DEFORMABLE FOUNDATION CONTACT PROBLEMS

Up to now in many works a construction on elastic foundation was modelled by the Winkler's hypothesis. In this paper the flexible plate-foundation system is investigated as an interaction of separate elastic bodies, which lie on the contact surface. Such a problem formulation allowed to make more exact the stress and strain distribution in the construction and the contact surface, to estimate the dead weight of the foundation. The presented numerical examples are realised using computer programme COSMOS/M. The obtained results show evidently the expediency of the new method application in practice.

FORMULATIONS OF THE DEFORMABLE FOUNDATION CONTACT PROBLEMS

Abstract Up to now in many works a construction on elastic foundation was modelled by the Winkler's hypothesis. In this paper the flexible plate-foundation system is investigated as an interaction of separate elastic bodies, which lie on the contact surface. Such a problem formulation allowed to make more exact the stress and strain distribution in the construction and the contact surface, to estimate the dead weight of the foundation. The presented numerical examples are realised using computer programme COSMOS/M. The obtained results show evidently the expediency of the new method application in practice.

Nonaxisymmetric Unbonded Contact of Plates on Tensionless Winkler Foundations*

ABSTRACT The unbonded contact problem for an annular plate resting on a tensionless Winkler elastic foundation is investigated. The problem is solved by minimizing the total potential energy of the plate-foundation system. Two alternative functions that include either Bessel or Kelvin functions are used to define the plate displacement. By satisfying a set of four boundary conditions and using the Ritz method, the complete displacement function for the plate is obtained iteratively. To improve the accuracy and convergence of the iterative approach, the solution requires a complete revision of the assumed displacement functions after every iteration. It is proven that the selected displacement functions and the suggested procedure make the approach highly accurate. Comparisons with results obtained by other approaches, in addition to other examples, are presented.