Effect Of Elastic Foundation Research Articles

Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of embedded restrained nanobeams enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small-scale effects on the dynamical response, Eringen's differential form of nonlocal elasticity is used which employs a scale (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories, including the effect of nonlocality and employing the Fourier sine series together with Stokes' transformation, systems of linear equations are obtained and solved using the coefficient matrices. The combined effects of elastic boundary conditions, elastic foundation, dispersion patterns and volume fractions of carbon nanotubes, and nonlocal parameter are examined by solving eigenvalue problems constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube-reinforced nanobeams under different rigid or restrained boundary conditions, including Winkler-Pasternak type elastic foundation. A comprehensive parametric study is performed, focusing on various effects for the free vibrational response of the composite nanobeam reinforced with carbon nanotubes. It is concluded that adding a small amount of carbon nanotube material can reinforce the stiffness of the composite nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium, and boundary conditions.

Nonlinear Buckling and Postbuckling Response of Porous FGM Shallow Spherical Caps and Circular Plates with Nonlinear Elastic Foundation Effects Using the Ritz Energy Method

Nonlinear Buckling and Postbuckling Response of Porous FGM Shallow Spherical Caps and Circular Plates with Nonlinear Elastic Foundation Effects Using the Ritz Energy Method

The Nonlinear Bending of Sector Nanoplate via Higher-Order Shear Deformation Theory and Nonlocal Strain Gradient Theory

In this context, the nonlinear bending investigation of a sector nanoplate on the elastic foundation is carried out with the aid of the nonlocal strain gradient theory. The governing relations of the graphene plate are derived based on the higher-order shear deformation theory (HSDT) and considering von Karman nonlinear strains. Contrary to the first shear deformation theory (FSDT), HSDT offers an acceptable distribution for shear stress along the thickness and removes the defects of FSDT by presenting acceptable precision without a shear correction parameter. Since the governing equations are two-dimensional and partial differential, the extended Kantorovich method (EKM) and differential quadrature (DQM) have been used to solve the equations. Furthermore, the numeric outcomes were compared with a reference, which shows good harmony between them. Eventually, the effects of small-scale parameters, load, boundary conditions, geometric dimensions, and elastic foundations are studied on maximum nondimensional deflection. It can be concluded that small-scale parameters influence the deflection of the sector nanoplate significantly.

Elastic foundation effects on dynamic characteristics of agglomerated hybrid laminated truncated conical nanocomposite panels and shells

Elastic foundation effects on dynamic characteristics of agglomerated hybrid laminated truncated conical nanocomposite panels and shells

Effects of multiple edge cracks, shear force, elastic foundation, and boundary conditions on bucking of small-scale pillars

The buckling instability of micro- and nanopillars can be an issue when designing intelligent miniaturized devices and characterizing composite materials reinforced with small-scale beam-like particles. Analytical modeling of the buckling of miniaturized pillars is especially important due to the difficulties in conducting experiments. Here, a well-posed stress-driven nonlocal model is developed, which allows the calculation of the critical loads and buckling configurations of the miniaturized pillars on an elastic foundation and with arbitrary numbers of edge cracks. The discontinuities in bending slopes and deflection at the damaged cross-sections due to the edge cracks are captured through the incorporation of both rotational and translational springs. A comprehensive analysis is conducted to investigate the instability of pillars containing a range of one to four cracks. This analysis reveals interesting effects regarding the influence of crack location, nonlocality, and elastic foundation on the initial and subsequent critical loads and associated buckling configurations. The main findings are: (i) the shielding and amplification effects related to a system of cracks become more significant as the dimensions of pillars reduce, (ii) the influence of the shear force at the damaged cross-section related to the translational spring must not be neglected when dealing with higher modes of buckling and long cracks, (iii) an elastic foundation decreases the effects of the cracks and size dependency on the buckling loads, and (iv) the effects of the edge cracks on the critical loads and buckling configurations of the miniaturized pillars are highly dependent on the boundary conditions.

Effect of a Four-Parameter Kerr-Type Elastic Foundation on the Large-Amplitude Free Vibration of the Nanocomposite Beams: A Dual-FG Modeling

This study examines the free vibration of a unique nanocomposite beam, characterized by large-amplitude motions. The composition of the material comprises functionally graded (FG) polymers fortified with graphene platelet (GPL) nanofillers, forming a nanocomposite structure with a dual FG designation. The arrangement of graphene platelet nanofillers within the dual-phase matrix follows one of five linear functionally graded models. To ascertain the behavior of the nanocomposite, the homogenization process employs the Voigt rule of mixtures for the power law-based FG matrix, while the Halpin–Tsai homogenization procedure is employed for the functionally graded graphene platelet (FG-GPL) reinforced nanocomposite. The beam is supported by a novel four-parameter nonlinear Kerr elastic foundation, located at the beam’s lower surface, which governs the linear and nonlinear frequencies. Employing the Karama higher-order shear deformation theory, the beam’s displacements are estimated, and the strains are described by the Von-Kármán geometrically nonlinear relations. Solving the problem necessitates the utilization of the generalized differential quadrature (GDQ) method, coupled with a bilateral iterative displacement control (BIDC) technique. Notably, the equations are rendered independent of the time variable implementing the weighted residual Galerkin procedure. The obtained frequencies of the GPL-reinforced beam with an isotropic homogeneous matrix are compared against those documented in existing literature, ensuring validation. Subsequently, parametric results are presented to assess the impact of the novel nanocomposite features on the linear and nonlinear frequencies of beams.

Geometrically Nonlinear Buckling of FG-CNTRC Plates Stiffened by FG-CNTRC Stiffeners Subjected to Combined Loads Using Nonlinear Reddy’s HSDT

The nonlinear buckling and postbuckling of functionally graded carbon nanotube reinforcement composite (FG-CNTRC) plates resting on the nonlinear effect of elastic foundation under combined load including external pressure and axial compression loads with uniformly distributed temperature rises are fully investigated in this paper. The FG-CNTRC plates are reinforced by FG-CNTRC stiffeners and the plate-foundation interaction is modeled using a nonlinear model. The higher-order shear deformation plate theory with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Additionally, by using the anisotropic higher-order shear deformation beam theory, a new smeared stiffener technique is successfully developed for FG-CNTRC stiffeners. Galerkin’s method is used to achieve the equilibrium equation system in the nonlinear algebraic forms, and the critical load and postbuckling load–deflection curve expression are explicitly determined. The numerical investigations discuss the influences of FG-CNTRC stiffeners, hardening/softening nonlinear elastic foundation, uniformly distributed temperature rises, geometrical and material features on nonlinear stability of stiffened FG-CNTRC plates.

Hygrothermally-Induced Vibration Analysis of Porous FGM Rectangular Mindlin Plates Resting on Elastic Foundation

The nonlinear vibration response of rectangular plates made of functionally graded porous materials (FGPs) induced by hygrothermal loading is investigated in this article using a numerical approach. The effect of elastic foundation on the vibrations is taken into account according to the Winkler–Pasternak model. Hygroscopic stresses produced due to nonlinear rise in moisture concentration are also considered. The temperature-dependent material properties of plate are computed based on the modified Voigt’s rule of mixture and Touloukian experiments for even and uneven distribution patterns of porosity. Within the framework of the first-order shear deformation plate theory and von-Kármán nonlinearity, Hamilton’s principle is utilized in order to derive the equations of motions. To achieve the temporal evolution of maximum lateral deflection of hygrothermally-induced plates, the generalized differential quadrature (GDQ) and Newmark integration methods are employed. Selected numerical results are presented to study the influences of temperature distribution, porosity volume fraction, moisture concentration, geometrical parameters, elastic foundation parameters and FG index on the geometrically nonlinear vibrations of FG porous plates with various boundary conditions.

A new analytical approach to the nonlinear buckling and postbuckling behavior of functionally graded graphene reinforced composite laminated cylindrical, parabolic, and half‐sinusoid shallow imperfect panels

AbstractThis paper presents and analyzes the nonlinear buckling responses of three types of shallow imperfect panels (cylindrical panel, parabolic panel, and half‐sinusoid panel) made from functionally graded graphene reinforced composite (FG‐GRC) on nonlinear elastic foundations subjected to axial compressive load in thermal environments. The nonlinear governing equations are formulated on Reddy's higher‐order shear deformation shell theory (HSDST) and take into account von Karman‐type nonlinearity. A new approximation technique to determine the stress function in average sense is developed and Galerkin's method is used to obtain the algebraically nonlinear equation system. Then, the simple calculation process can be used to solve the obtained equation systems, and the formulations to calculate the critical buckling loads and postbuckling load‐deflection curves are expressed in explicit form. The results can be flexibly applied to FG‐GRC panels with different curvatures in engineering designs. The effects of panel types, geometrical parameters, temperature increase, initial imperfection, and nonlinear elastic foundations on the critical buckling loads and postbuckling curves of cylindrical, parabolic, and half‐sinusoid FG‐GRC panels are discussed in numerical results. Numerical results also show a small disadvantage in the load‐carrying capacity of cylindrical panels compared to parabolic panels and half‐sinusoid shallow panels.Highlights Postbuckling of cylindrical, parabolic, and half‐sinusoid panels are studied. The panels are made of functionally graded graphene reinforced composite. Reddy's higher‐order shear deformation shell theory is applied. A new approximation technique to determine the stress function is developed. Critical buckling loads and postbuckling expressions are explicitly obtained.

Nonlinear Thermal/Mechanical Buckling of Orthotropic Annular/Circular Nanoplate with the Nonlocal Strain Gradient Model.

This article presents the nonlinear investigation of the thermal and mechanical buckling of orthotropic annular/circular single-layer/bilayer nanoplate with the Pasternak and Winkler elastic foundations based on the nonlocal strain gradient theory. The stability equations of the graphene plate are derived using higher-order shear deformation theory (HSDT) and first-order shear deformation theory (FSDT) considering nonlinear von Karman strains. Furthermore, this paper analyses the nonlinear thermal and mechanical buckling of the orthotropic bilayer annular/circular nanoplate. HSDT provides an appropriate distribution for shear stress in the thickness direction, removes the limitation of the FSDT, and provides proper precision without using a shear correction coefficient. To solve the stability equations, the differential quadratic method (DQM) is employed. Additionally, for validation, the results are checked with available papers. The effects of strain gradient coefficient, nonlocal parameter, boundary conditions, elastic foundations, and geometric dimensions are studied on the results of the nondimensional buckling loads. Finally, an equation is proposed in which the thermal buckling results can be obtained from mechanical results (or vice versa).

Effect of elastic foundation and porosity on buckling response of linearly varying functionally graded material plate

This work presents the buckling response of porous plate made of functionally graded materials (FGMs). A tapered FGM plate with uniaxial and biaxial loading under numerous boundary conditions was investigated. Simple power law and sigmoidal law are applied to compute the tailored material properties. Further, the effect of Pasternak foundation on the buckling response for a plate is computed, considering the five types of porosity pattern. The first order-shear deformation theory (FSDT) is used to describe the kinematic field of the plate. The variational principle is used to derive the equation of motions. Subsequently, the solution for buckling analysis is found by applying the Galerkin’s-Vlasov method. It has been seen that the effect of elastic foundation largely influenced the buckling response of FGM plate, especially Pasternak foundation has significant effect when comparing with Winkler’s foundation. Similarly, buckling load, subjected to uniaxial or biaxial compression of plate, decreases greatly for porous material. Numerical examples for computing buckling response are compared for accuracy and verification purpose of the present formulation, with earlier published results. Considering the effect of thickness variation, span ratio, compressive load conditions, foundation stiffness and porosity on buckling analysis are estimated and established as a benchmark result in tabulated form. Moreover, parametric studies have been done, by keeping in view, to explore the utility of FGM tapered plate and the responses represented in the graphically form.

Dynamic characteristic analysis of beam structures with nonlinear elastic foundations and boundaries

The research of structures with elastic foundations has attracted extensive attention and many dynamic modeling and analysis methods for beam structures with elastic foundations have been proposed in current mainstream research. However, the low-dimension and high-accuracy models are more in line with the requirements of engineering practice. Hence, this paper aims to establish a novel nonlinear dynamic model of beam structures with nonlinear elastic foundations and boundaries. The global modes are first obtained by using the global mode method, and the forced vibration response analysis of the beam structures is subsequently carried out by developing a corresponding solving procedure. Thereafter, case studies are conducted on a cantilever beam with nonlinear elastic foundations and boundaries to validate the established nonlinear dynamic model. The consistency of the results obtained by the established model and finite element analysis demonstrates the accuracy of the established model. Finally, the effects of elastic foundations, boundaries, external excitation amplitude and nonlinear coupling terms on the dynamic characteristics of the cantilever beam structure are analyzed through the comparative case studies.

Multi-objective optimization of snap-through instability of helicoidal composite imperfect beams using Bernstein polynomials method

This paper deals with the multi-objective optimization problems of snap-through instability of geometrically imperfect simply supported helicoidal composite beam with nonlinear orientation schemes. The objective function is to maximize the load required to initiate snap-through and minimize the instability region. It is assumed that the curved bioinspired beam is embedded on nonlinear elastic foundation and is subjected to different types of lateral loads. The presented formulation is based on Euler-Bernoulli theory, nonlinear higher order strain, and imperfect curvature. Bernstein polynomials (BPs) method is employed to solve the nonlinear nonhomogeneous fourth order integro-differential governing equation. Developed model can considered the effect of imperfection, lamination scheme, and elastic foundation on the nonlinear deflection and snap-through behaviors of helicoidal composite structures. Analytical closed form formula of the nonlinear load-deflection relationship is developed. The analytical relation is validated by comparing it with exact solution and numerical results available in literature. Multi-objective particle swarm optimization (MOPSO) algorithm is employed to determine Pareto-optimal set for limiting load and region of instability. In these analyses, optimal lamination parameter, amplitude of imperfection and elastic foundation constants are found for helicoidal composite beam, and the results are presented as 2D Pareto-optimal design points.

Determination of the Optimum Values of the Natural Frequency of the Nanocomposite Multilayer Organic Solar Cell by using Basic Differential Evolution Optimization Algorithms

By the analytical approach, the equation of natural frequency of nanocomposite multilayer organic solar cell (NMOSC) is analyzed in this work. The NMOSC is composed of 6 layers: Al, LiF, P3HT: PCBM, PEDOT: PSS, ITO and Glass. By using the classical plate theory, the Hooke’s Law, the geometrical compatibility equation and the nonlinear equilibrium equations are proposed. These equations combined with the Galerkin method in order to investigate the effect of elastic foundations, geometrical parameters and initial imperfection on the natural frequency. Besides, in order to determine the maximum natural frequency of NMOSC, basic differential evolution optimization algorithm (DE) was used with five variables: elastic foundations include Winkeler foudation and Pasternak foudation, temperature, width and length. In the numerical results, the influence of length to width ratio, imperfection, elastic foundations and temperature increment were evaluated in detail. The optimal results from DE algorithm are shown and compared.

Development of analytical and FEM solutions for static and dynamic analysis of smart piezoelectric laminated composite plates on elastic foundation

This paper proposes new analytical and finite element solutions for studying the effects of elastic foundations on the uncontrolled and controlled static and vibration responses of smart multi-layered laminated composite plates with integrated piezoelectric layers, acting as actuators and sensors. A non-polynomial higher-order plate theory with zigzag kinematics involving a trigonometric function and a local segmented zigzag function is adopted for the first time for modeling the deformation of a smart piezoelectric laminated composite plate supported on an elastic foundation. This model has only five independent primary variables like that of the first-order shear deformation theory, yet it considers the realistic parabolic behavior of the transverse shear stresses across the thickness of the laminated composites plates, and also maintains the continuity conditions of transverse shear stresses at the interfaces of the laminated plates. A two-parameter foundation model, namely Pasternak’s foundation, is used to model the deformation and shear interactions of the elastic foundation. The governing set of equations is derived by implementing Hamilton’s principle and variational calculus. Two different solution methods, namely, a generalized closed-form analytical solution of Navier-type, and a C0 isoparametric finite element (FE) formulation, are developed for solving the governing set of equations. The solutions in the time domain are obtained with Newmark’s average acceleration method. Comprehensive parametric studies are presented to investigate the influence of elastic foundation parameters, piezoelectric layers, loading, and boundary conditions on the static and dynamic responses of the smart composite plates with piezoelectric layers. The effects of the elastic foundations on the vibration control of the smart composite plates are also presented by coupling the piezoelectric actuator and sensor with a feedback controller. Several benchmark results are presented to show the influence of the various material and geometrical parameters on the controlled and uncontrolled responses of the smart plates, and also the significant effect of the elastic foundations on the static and dynamic responses of the smart structures. The results obtained are in very good agreement with the available literature, and it can be concluded that the proposed analytical solution and FE formulation can be efficiently used to model the static and dynamic electro-elastic behavior of smart laminated plates supported on elastic foundations.

Higher order computational model considering the effects of transverse normal strain and 2-parameter elastic foundation for the bending of laminated panels

In this study, the authors analyze laminated composite panels supported on an elastic foundation considering the effects of transverse normal strain. A 2-parameter, i.e., Winkler and Pasternak foundation model is assumed to represent the interaction between the panels and the foundation. The theory presented here takes into account the effects of transverse shear and normal strains. The theory plots realistic distributions of the transverse shear stress through the plate thickness and satisfies the shear-free conditions at the extreme surfaces of the panel. The differential equations of the present model are obtained from the principle of virtual work. The laminated composite panel resting on the elastic foundation is analyzed for simply supported boundary conditions. For the verification purpose, the presented problems are also solved using the Reddy's model, Mindlin's model, and the classical model. Good agreement is observed between the numerical results obtained using the present model and the other models.

Nonlinear buckling and post-buckling of imperfect FG porous sandwich cylindrical panels subjected to axial loading on elastic foundation

This paper deals with the nonlinear buckling and post-buckling of sandwich cylindrical panels with non-uniform porous core and functionally graded face sheets. The imperfect sandwich cylindrical panels are subjected to axial loading on elastic foundation. Based on the Donnell shell theory, with von Kármán geometrical nonlinearity, the governing equations are derived. The effects of elastic foundation, various panel geometrical characteristics, porosity parameters, and the thickness of the porous core are investigated. The effects of foundation parameters, porosity parameters, the thickness of the porous core, and material parameters are investigated.

Nonlinear thermo-mechanical buckling and postbuckling of sandwich FG-GPLRC spherical caps and circular plates with porous core by using higher-order shear deformation theory

The main aim of this research is to establish the algorithm for nonlinear thermo-mechanical buckling of sandwich functionally graded graphene platelet reinforced composite (FG-GPLRC) shallow spherical caps and circular plates with porous core under external pressure and/or uniformly distributed thermal load according to the higher-order shear deformation theory considering the von Karman nonlinearities. Sandwich spherical caps and circular plates are made by the porous core and two FG-GPLRC coatings and are assumed to be rested on an elastic foundation modeled by the Pasternak model. The equilibrium equation system in the form of nonlinear algebra can be approximately obtained using the Ritz energy method. The critical buckling loads and postbuckling curves can be explicitly determined. The effects of material parameters, geometrical parameters, porous core, and elastic foundation on thermo-mechanical buckling of sandwich spherical caps and circular plates with porous core and FG-GPLRC coatings are investigated and discussed in detail in the numerical investigation section.

Free Vibration Characteristics of Bidirectional Graded Porous Plates with Elastic Foundations Using 2D-DQM

This manuscript develops for the first time a mathematical formulation of the dynamical behavior of bi-directional functionally graded porous plates (BDFGPP) resting on a Winkler–Pasternak foundation using unified higher-order plate theories (UHOPT). The kinematic displacement fields are exploited to fulfill the null shear strain/stress at the bottom and top surfaces of the plate without needing a shear factor correction. The bi-directional gradation of materials is proposed in the axial (x-axis) and transverse (z-axis) directions according to the power-law distribution function. The cosine function is employed to define the distribution of porosity through the transverse z-direction. Equations of motion in terms of displacements and associated boundary conditions are derived in detail using Hamilton’s principle. The two-dimensional differential integral quadrature method (2D-DIQM) is employed to transform partial differential equations of motion into a system of algebraic equations. Parametric analysis is performed to illustrate the effect of kinematic shear relations, gradation indices, porosity type, elastic foundations, geometrical dimensions, and boundary conditions (BCs) on natural frequencies and mode shapes of BDFGPP. The effect of the porosity coefficient on the natural frequency is dependent on the porosity type. The natural frequency is dependent on the coupling of gradation indices, boundary conditions, and shear distribution functions. The proposed model can be used in designing BDFGPP used in nuclear, marine, aerospace, and civil structures based on their topology and natural frequency constraints.

Torsional buckling and postbuckling behavior of stiffened FG-GRCL toroidal shell segments surrounded by elastic foundation

In this study, a new algorithm for the nonlinear buckling behavior of the stiffened functionally graded graphene-reinforced composite laminated (FG-GRCL) toroidal shell segments in the thermal environment surrounded by the elastic foundation and subjected to torsional load is introduced. The FG-GRCL shell segment is stiffened by the FG-GRCL ring and/or stringer stiffener system. Based on the Donnell shell theory with the Stein and McElman approximation, considering the von Karman geometrical nonlinearity and the anisotropic smeared stiffener technique for FG-GRCL stiffeners, the formulations to investigate the critical buckling load and postbuckling torsional load-deflection curves are determined. A new algorithm is fully developed by adding the effect of circumferential stress into the stress function. The significant effects of temperature, stiffener system, geometrical parameters, and elastic foundation are considered and studied in detail.