Three-parameter Elastic Foundation Research Articles

Fundamental frequencies of cracked FGM beams with influence of porosity and Winkler/Pasternak/Kerr foundation support using a new quasi-3D HSDT

In this study, we have introduced, for the first time, a novel integral quasi-3D higher-order shear deformation theory (HSDT) employing a third-order shape function. This approach is employed to analyze the free vibration characteristics of a cracked porous functionally graded material (FGM) beam supported on a three-parameter elastic foundation (Winkler/Pasternak/Kerr). This new Quasi HSDT introduces a stretching effect that surpasses the capabilities of FSDT and other HDST. The employed shape function satisfies the conditions of shear stress nullity at both the higher and lower facets without the need for correction factors. The study incorporates a mathematical model representing Winkler/Pasternak/Kerr foundation types into the Hamiltonian to derive the equations of motion. The FGM beam studied in this paper is assumed to be composed of materials with a distribution that varies according to a power law along its height. Our results are compared with previous studies and we reinforce our findings with a parametric study assessing the impact of crack attributes on the natural frequencies of the FG plate. This study presents an advanced integral quasi-3D HSDT, applied for the first time, to analyze the behavior of FG beams resting on a three-parameter foundation.

Postbuckling responses of porous FGM spherical caps and circular plates including edge constraints and nonlinear three-parameter elastic foundations

An analytical investigation on nonlinear stability of porous functionally graded spherical caps and circular plates under uniform external pressure and elevated temperature, respectively, is presented in this article. Porosities are evenly and unevenly distributed within the structures and the effective properties of functionally graded material (FGM) are determined according to a modified rule of mixture. Governing equations are established within the framework of classical theory taking into account geometrically nonlinear terms, initial geometric imperfection, and interactive pressure from nonlinear three-parameter elastic foundation. Both temperature dependence of material properties and tangential elasticity of edge constraint are included. Approximate analytical solutions are assumed to satisfy clamped condition of boundary edge and Galerkin method is applied to derive load-deflection relations from which buckling loads and postbuckling paths are determined. Numerical results indicate that porosities have deteriorative and beneficial influences on load-carrying capacity of structures under external pressure and elevated temperature, respectively. Furthermore, the snap-through buckling response of pressurized FGM spherical cap can be alleviated and even eliminated by virtue of elastic foundations with suitable stiffness parameters.

Large deflection analysis of functionally graded saturated porous rectangular plates on nonlinear elastic foundation via GDQM

In the current study, large deflection analysis of a functionally graded saturated porous (FGSP) rectangular plate subjected to transverse loading which is located on a nonlinear three-parameter elastic foundation is provided. The constitutive law for the porous materials is written based on Biot's model which considers the effect of fluids within the pores. The mechanical properties of the plate are changed through its thickness according to different functions which are called porosity distributions. The shear deformation effects are taken into account, accordingly, the first-order shear deformation theory (FSDT) is used to describe the displacement components of the plate. Employing the Minimum total potential energy principle and calculus of variation, the governing equations, and associated boundary conditions are extracted. A generalized differential quadrature method (GDQM) is used to solve them for various boundary conditions. The results for the simpler state are validated with the previously published works and then the effects of different parameters on the deflection of the plate are investigated. It is seen increasing the porosity and Skempton coefficient, enhances and reduces the deflection of the structure, respectively. The results of this study may help to design and manufacture more reliable engineering structures that are exposed to loads.

Limit load analysis and imperfection sensitivity of heated or compressed FGM beams on nonlinear softening elastic foundation

Analysis of nonlinear response of thick FGM beams subjected to either thermal or mechanical load is performed. Beam is in contact with a three-parameter elastic foundation. Nonlinear component of the foundation may be of the hardening or softening type. Equilibrium equations of the clamped beam are obtained within the framework of von Karman assumptions and third-order shear deformation theory and solved using the two-step perturbation technique. It is shown that when the FGM beam is in contact with a sufficiently soft foundation, the post-buckling path of the heated/compressed FGM beam is unstable and therefore, the structure is imperfection sensitive. Communicated by Chandrika Prakash Vyasarayani

Nonlinear Static and Dynamic Thermal Buckling Analysis of Spiral Stiffened Functionally Graded Cylindrical Shells with Elastic Foundation

In this paper, an analytical method is used to study the nonlinear static and dynamic thermal buckling analysis of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells. The SSFG cylindrical shell is surrounded by a linear and nonlinear elastic foundation. The proposed linear model is based on the two-parameter elastic foundation (Winkler and Pasternak). A three-parameter elastic foundation with hardening/softening cubic nonlinearity is used for nonlinear model. The material properties are temperature dependent and assumed to be continuously graded in the thickness direction. Also, for thermal buckling analysis, the uniform and linear temperature distribution in thickness direction is considered. The SSFG cylindrical shells are considered with various angles for spiral stiffeners. The strain–displacement relations are obtained based on the von Kármán nonlinear equations and the classical plate theory of shells. The smeared stiffener technique and the Galerkin method are applied to solve the nonlinear problem. In order to find the nonlinear dynamic thermal buckling responses, the fourth-order Runge–Kutta method is used. To validate the results, comparisons are made with those available in literature and good agreements are shown. The effects of various geometrical and material parameters are investigated on the nonlinear static and dynamic thermal buckling response of SSFG cylindrical shells.

Buckling and Postbuckling of Concentrically Stiffened Piezo-Composite Plates on Elastic Foundations

This research presents the modeling and analysis for the buckling and postbuckling behavior of sandwich plates under thermal and mechanical loads. The lay-up configurations of plates are laminated composite with concentric stiffener and surface mounted piezoelectric actuators. The plates are in contact with a three-parameter elastic foundation including softening and/or hardening nonlinearity. Several types of grid shapes of stiffeners are studied such as ortho grid, angle grid, iso grid, and orthotropic grid. The equations of structures are formulated based on the classical lamination theory incorporating nonlinear von-Karman relationships. The stress function and Galerkin procedure are applied to derive explicit formulations of the equilibrium paths. New results are introduced to give the influences of voltage through the thickness of piezoelectric actuators, different stiffeners, and nonlinear elastic foundations.

Geometrically nonlinear analysis of shear deformable FGM shallow pinned arches on nonlinear elastic foundation under mechanical and thermal loads

The present investigation deals with the large amplitude response of shallow thick circular arches subjected to thermal, mechanical, or thermomechanical loads. The structure is resting on a three-parameter elastic foundation containing the Winkler springs, the shear Pasternak layer, and hardening nonlinear springs. The arch is made of a through-the-thickness functionally graded material. Except for Poisson’s ratio, the other properties of the arch are assumed to be temperature and position dependent. Each property is estimated according to the rule of mixtures in terms of the volume fraction of the constituents. The case of uniform temperature rise and uniform radial lateral load is considered. The governing equations of the arch are obtained with the aid of the third-order shear deformation arch theory and the von Karman type of geometrical nonlinearity. The developed coupled and nonlinear equilibrium equations are solved using the two-step perturbation technique for the case of simply supported arches. Closed-form expressions are developed for nonlinear bending equilibrium path of shallow arches under thermal and mechanical loads. Results are provided for different power law indices, geometrical parameters, and the foundation stiffness.

Nonlinear dynamic analysis of spiral stiffened functionally graded cylindrical shells with damping and nonlinear elastic foundation under axial compression

The semi-analytical method to study the nonlinear dynamic behavior of simply supported spiral stiffened functionally graded (FG) cylindrical shells subjected to an axial compression is presented. The FG shell is surrounded by damping and linear/nonlinear elastic foundation. The proposed linear model is based on the two-parameter elastic foundation (Winkler and Pasternak). A three-parameter elastic foundation with hardening/softening cubic nonlinearity is used for nonlinear model. The material properties of the shell and stiffeners are assumed to be FG. Based on the classical plate theory of shells and von Karman nonlinear equations, smeared stiffeners technique and Galerkin method, this paper solves the nonlinear vibration problem. The fourth order Runge-Kutta method is used to find the nonlinear dynamic responses. Results are given to consider effects of spiral stiffeners with various angles, elastic foundation and damping coefficients on the nonlinear dynamic response of spiral stiffened simply supported FG cylindrical shells.

Optimum Design of Braced Steel Space Frames including Soil-Structure Interaction via Teaching-Learning-Based Optimization and Harmony Search Algorithms

Optimum design of braced steel space frames including soil-structure interaction is studied by using harmony search (HS) and teaching-learning-based optimization (TLBO) algorithms. A three-parameter elastic foundation model is used to incorporate the soil-structure interaction effect. A 10-storey braced steel space frame example taken from literature is investigated according to four different bracing types for the cases with/without soil-structure interaction. X, V, Z, and eccentric V-shaped bracing types are considered in the study. Optimum solutions of examples are carried out by a computer program coded in MATLAB interacting with SAP2000-OAPI for two-way data exchange. The stress constraints according to AISC-ASD (American Institute of Steel Construction-Allowable Stress Design), maximum lateral displacement constraints, interstorey drift constraints, and beam-to-column connection constraints are taken into consideration in the optimum design process. The parameters of the foundation model are calculated depending on soil surface displacements by using an iterative approach. The results obtained in the study show that bracing types and soil-structure interaction play very important roles in the optimum design of steel space frames. Finally, the techniques used in the optimum design seem to be quite suitable for practical applications.

Thermoelastic response of a microbeam embedded in visco-Pasternak’s medium based on GN-III model

ABSTRACTThis article investigates the effect of visco-Pasternak’s foundation on the field quantities of thermoelastic microbeams. Classical Euler–Bernoulli’s beam theory is used for description of deformation components of the microbeam based on a symmetric distribution. A generalized thermoelasticity theory is used to get the coupling equations of a simply supported microbeam resting on a three-parameter elastic foundation and subjected to a heat source. The deflection, axial displacement, temperature, bending moment, and axial stress of the microbeam are evaluated in terms of different parameters and are plotted for sensing the effect of all used parameters. Some tabulated results that investigate the visco-Pasternak’s parameters are presented for future comparisons. The effect of time parameter is also illustrated.

Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field

Abstract In this paper a nonlinear investigation is presented for impact response of functionally graded material plates which are resting on a nonlinear three-parameter elastic foundation with simply supported end condition. In foundation, the influence of linear Winkler and Pasternak parameter is taken into account. The governing equations of the plate are derived based on a Reddy's higher order shear deformation theory with the von Karman-type of kinematic nonlinearity. The effect of thermal field is considered and material properties of the plates are assumed to be temperature dependent, as well and vary along the thickness according to Voigt model. In addition, by using a modified Hertz's contact law, the influence of material properties of the substrate layers on impact response is considered. The equations of motion are solved via an analytical procedure. The effects of deformation theories as well as thermal field, foundation parameters and impactor's parameters are discussed in results and discussion section.

Optimum design of steel space frames including soil-structure interaction

The effect of soil-structure interaction on the optimum design of steel space frames is investigated using metaheuristic algorithms. Three-parameter elastic foundation model is used to incorporate ...

Nonlinear thermal dynamic buckling of FGM beams

Abstract Based on the nonlinear dynamic analysis, dynamic buckling and imperfection sensitivity of the FGM Timoshenko beam subjected to sudden uniform temperature rise are studied. Initial geometric imperfection of the beam is also taken into account. It is assumed that during deformation the beam is resting over a conventional three-parameter elastic foundation with softening/hardening cubic non-linearity. The analysis is performed with considering temperature dependency assumption of each thermo-mechanical property of the FGM beam. The governing nonlinear dynamic equations are derived based on the generalized Hamilton principle. In the spatial approximation of the problem, a set of ordinary differential equations in time is obtained by the conventional multi-term Ritz method. These equations are converted into a set of algebraic equations by utilizing the Newmark family of time approximation scheme. The obtained non-linear algebraic equations are solved via the well known Newton–Raphson iterative scheme. The Budiansky–Roth criterion is used to detect the unbounded motion type of dynamic buckling. Results reveal that for beams with stable post-buckling equilibrium path, no dynamic buckling occurs according to the Budiansky–Roth criterion. However, dynamic buckling may occur for the FGM beams resting on sufficiently stiff softening elastic foundation due to their unstable post-buckling equilibrium paths.

Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko׳s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.

Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation

In this paper, static and dynamic buckling of an FGM beam subjected to uniform temperature rise loading and uniform compression are studied. Material properties of the beam are assumed to be graded across the thickness. Each thermo-mechanical property of the beam is assumed to be temperature dependent. Initial imperfection of the beam is also taken into account. The beam is resting over a three-parameter elastic foundation with hardening/softening cubic nonlinearity which acts in tension as well as in compression. Nonlinear governing equations are obtained based on the static version of virtual displacements and are solved via the multi-term Galerkin method. Dynamic buckling load levels are estimated based on the well-known Hoff-Simitses criterion. Results reveal that for sufficiently stiff softening elastic foundation, post-buckling equilibrium path becomes unstable. Furthermore, when the thermal post-buckling equilibrium path is stable, no dynamic buckling occurs according to this criterion.

Dynamics of Piezoelectric Smart Beams Using a Three-Parameter Elastic Foundation Model

Piezoelectric patches are usually externally bonded to host structures as sensors and/or actuators. The performance and integrity of this type of smart structures are determined by the interface stresses within the adhesive layer. In this study, the dynamic behavior of a piezoelectric smart beam is analyzed using a novel three-parameter elastic foundation model. Closed-form solutions of the deformations, interfacial stresses, strain transfer, and sensing charge output are obtained. The new interfacial stress solutions overcome the inherent shortcomings in the existing two-parameter elastic foundation model by not only satisfying the zero shear stress boundary condition at the free edge, but also correctly predicting two different peeling stress distributions along the piezoelectric patch/adhesive interface and the host beam/adhesive interface. When the thick adhesive layer is used, the present three-parameter elastic foundation model exhibits high accuracy; while the existing two-parameter elastic foundation model significantly overestimates the response of the smart beam.

Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM

The effects of three-parameter elastic foundations and thermo-mechanical loading on axisymmetric large deflection response of a simply supported annular FGM plate are investigated. An annular FGM plate, resting on a three-parameter elastic foundation under a transverse uniform loading and a transverse non-uniform temperature, is considered. The mechanical and thermal properties of the FGM plate are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The mathematical modeling of the plate and the resulting nonlinear governing equations of equilibrium are derived based on the first-order shear deformation theory (FSDT) in conjunction with nonlinear von Karman assumptions. A polynomial-based differential quadrature method is used as a simple but powerful numerical technique to discretize the nonlinear governing equations and to implement the boundary conditions. Finally, the effects of certain parameters, such as nonlinear foundations stiffness, volume fraction index, and temperature, on the axisymmetric large deflection response of the FGM plate are obtained and discussed in detail.

A three-parameter elastic foundation model for interface stresses in curved beams externally strengthened by a thin FRP plate

Externally bonding of fiber reinforced polymer (FRP) plates or sheets has become a popular method for strengthening reinforced concrete structures. Stresses along the FRP–concrete interface are of great importance to the effectiveness of this type of strengthening because high stress concentration along the FRP–concrete interface can lead to the FRP debonding from the concrete beam. In this study, we develop an analytical solution of interface stresses in a curved structural beam bonded with a thin plate. A novel three-parameter elastic foundation model is used to describe the behavior of the adhesive layer. This adhesive layer model is an extension of the two-parameter elastic foundation commonly used in existing studies. It assumes that the shear stress in the adhesive layer is constant through the thickness, and the interface normal stresses along two concrete/adhesive and adhesive/FRP interfaces are different. Closed-form solutions are obtained for these two interfacial normal stresses, shear stress within the adhesive layer, and beam forces. The validation of these solutions is confirmed by finite element analysis.

Vibration of Timoshenko beams on three-parameter elastic foundation

In the present paper, the process of the formulation of the equations of dynamic equilibrium and of the respective equations of natural vibration of Timoshenko beams on three-parameter elastic foundation within the framework of the second order theory is presented. The corresponding mass matrices are also formulated. In order to define the values of the parameters of the three-parameter model, an analytical method is applied. With the aid of two numerical examples, it is proved that the three-parameter model is capable of yielding natural period values that sufficiently approximate those derived by the solution using 2D finite element soil models.

Exact matrices for beams on three-parameter elastic foundation

In the present paper, the process of the formulation of the stiffness and transfer matrices and of the load vectors of Timoshenko beams on Kerr-type three-parameter elastic foundation is presented. These matrices are developed on the basis of the exact solution of the governing differential equations for all potential solution cases. The reliability and precision of the modeling of beams resting on elastic foundation through the use of these matrices is assessed by means of two numerical examples.