Elastic Foundation Coefficients Research Articles

Nonlinear Static and Dynamic Buckling Analyses of Imperfect FGP Cylindrical Shells Resting on Nonlinear Elastic Foundation Under Axial Compression

In this paper, semi-analytical and analytical methods for the nonlinear static and dynamic buckling analyses of imperfect functionally graded porous (FGP) cylindrical shells subjected to axial compression are presented. The structure is embedded within a generalized nonlinear elastic foundation, treated as a two-parameter Winkler–Pasternak foundation augmented by a nonlinear cubic stiffness. The material property of the shell changes continuously through the thickness. Two types of FGP distributions, i.e. uniform porosity distribution (UPD) and nonuniform porosity distribution (NPD), are considered. By applying the Galerkin’s method to the von Kármán equations, the buckling of the shells was solved. The fourth-order Runge–Kutta method is utilized to obtain the responses of nonlinear dynamic buckling (NDB). The results obtained for some special cases are compared with those available elsewhere. The effects of various geometrical properties, material parameters and elastic foundation coefficients are investigated on the nonlinear static buckling (NSB) and dynamic buckling (DB) analyses of the shells. It was shown that various types of porosity, imperfection and the elastic foundation parameters have a strong effect on the buckling behaviors of the FGP cylindrical shells.

Free vibration of sandwich micro-beam with porous foam core,GPL layers and piezo-magneto-electric facesheets via NSGT

The aim of this research is to investigate free vibration of a novel five layer Timoshenko microbeam which consists of a transversely flexible porous core made of Al-foam, two graphen platelets (GPL) nanocomposite reinforced layers to enhance the mechanical behavior of the structure as well as two piezo-magneto-electric face sheets layers. This microbeam is subjected to a thermal load and resting on Pasternak\'s foundation. To accomplish the analysis, constitutive equations of each layer are derived by means of nonlocal strain gradient theory (NSGT) to capture size dependent effects. Then, the Hamilton\'s principle is employed to obtain the equations of motion for five layer Timoshenko microbeam. They are subsequently solved analytically by applying Navier\'s method so that discretized governing equations are determined in form of dynamic matrix giving the possibility to gain the natural frequencies of the Timoshenko microbeam. Eventually, after a validation study, the numerical results are presented to study and discuss the influences of various parameters such as nonlocal parameter, strain gradient parameter, aspect ratio, porosity, various volume fraction and distributions of graphene platelets, temperature change and elastic foundation coefficients on natural frequencies of the sandwich microbeam.

Sensitivity analysis of overburden water-resistant strata stability based on the mechanical properties of backfill body

Strata movement control during coal seam’s backfill mining is very much related to the mechanical properties of backfill body in goaf. A mechanical model of the structural water-resistant strata (SWRS) of stope overburden in backfill water-preserved mining (BWPM) was established on the basis of the key strata theory in ground control. This approach was conducted to investigate the mechanical properties of backfill body on the overburden water-resistant strata stability. The proposed model was based on the compressive mechanical properties of the filled gangue, which considered the linear and nonlinear elastic mechanical properties of backfill body. The stability mechanics criterion of the SWRS of stope overburden in BWPM on the nonlinear elastic foundation was deduced by introducing the tensile yield criterion. The maximum tensile stress (MTS) that acts on the SWRS of stope overburden decreased with the increase of the backfill body’s linear elastic foundation coefficient of k1 and the nonlinear elastic foundation coefficients of k2 and k3. However, the MTS rapidly decreased with the increase of k1 and slowly decreased with the increase of k2 and k3. The sensitivity analysis indicated that k1 exhibited the greatest effect on the MTS that acts on the SWRS, thereby accounting for 72.38%. In addition, k2 and k3 showed less effect on the MTS that acts on the SWRS, thereby only accounting for 19.56% and 8.06%, respectively. Results showed that the backfill body’s linear elastic foundation coefficient plays a very important role in controlling the stability of the SWRS of stope overburden in BWPM. Increasing k1 is more beneficial to the stability control of the water-resistant strata of stope overburden than increasing k2 and k3, especially the latter.

Free vibration of stiffened functionally graded circular cylindrical shell resting on Winkler–Pasternak foundation with different boundary conditions under thermal environment

In this paper, the analytical free-vibration analysis of a stiffened functionally graded circular cylindrical shell resting on a Winkler–Pasternak foundation is reported. Various boundary conditions and the thermal effect are considered. Material properties are assumed to be temperature-dependent and vary continuously across the shell’s thickness according to the power law distribution of the volume fraction of constituents. In order to derive the governing equations of the cylindrical shell structure, Hamilton’s principle together with the first-order shear deformation theory and the Lekhnitsky smeared stiffener technique is applied. The natural frequencies of the shell are determined by applying the Galerkin method together with the beam functions for the axial displacement fields. A good agreement between the present results and those available in the literature is observed. In numerical investigations, the influence of temperature field, material volume fraction index, elastic foundation coefficients, boundary conditions, and geometrical ratios on the fundamental natural frequencies of the shell is also given and discussed in detail.

Study on the Thin Plate Model with Elastic Foundation Boundary of Overlying Strata for Backfill Mining

In order to better study the movement principles of overlying strata during backfill mining, we established a thin plate model on an elastic foundation with elastic foundation boundary of the main roof. And by the finite difference method, the variation principles of the main roof’s principal moments and maximum subsidence ω0 with the elastic foundation coefficient k1 of the coal seam, the elastic foundation coefficient k2 of backfill body, the thickness h, Young’s modulus E, and Poisson’s ratio μ of main roof are calculated and studied. Using these calculations, we were able to determine that the main roof had three principal bending moment extreme points, including Mzz in backfill areas, Mc of the long side area, and Md of the short side area. The distance Lc of Mc advancing coal wall continuously increased with the increase in k2, while the principal moment of main roof’s middle area decreased with an increase in k2; when k2 became larger, the maximum principal moment in the midpoint of main roof transferred to the surrounding and the maximum principal moments was in four-corner area; Mc and Md decreased with an increase in k2, and Md was more sensitive to k2 than Mc; and Md decreased significantly with the increase in k2. Lc continuously decreased with the increase in k1, while Mc, Md and Mzz increased with the increase in k1 and the reduced amplitude of Mzz was the minimum. The effect of μ on principal bending moments and ω0 was very small; The growth rate of Mzz was the largest when E or h increased. Md, Mzz, and Lc remained unchanged when k1, k2, and Young’s modulus E of the main roof increased while the ratio value remained constant (k1/k2/E). Finally, the theoretical calculations were applied to the I26 backfill working face in the Xingdong mine to calculate the final subsidence amounts of the main roof. Field observations and theoretical calculations were about 48 mm, verifying the method’s applicability.

Nonlinear static and dynamic thermal buckling analysis of imperfect multilayer FG cylindrical shells with an FG porous core resting on nonlinear elastic foundation

In this article, a semi-analytical and analytical method for the nonlinear static and dynamic thermal buckling behavior of imperfect multilayer FG cylindrical shells with an FG porous core in the thermal environment is investigated. The structure is embedded within a generalized nonlinear elastic foundation which is composed of a two-parameter Winkler-Pasternak foundation augmented by a nonlinear cubic stiffness. Two types of multilayer FG cylindrical shells with an FG porosity core distribution consist of a uniform porosity distribution and a non-uniform porosity distribution core are considered. Using the Galerkin method with regard to the von Kármán equations, the discretized motion equation is obtained. The fourth-order Runge-Kutta method is utilized to obtain the nonlinear dynamic thermal buckling responses. The effects of various geometrical characteristics, material parameters, and elastic foundation coefficients are investigated on the nonlinear static thermal buckling and dynamic thermal buckling behavior of the multilayer FG system with an FG porous core. The results show that the various types of porosity core, imperfection and the elastic foundation parameters have strongly effect on the thermal buckling behaviors of the multilayer FG cylindrical shells with an FG porous core.

Free vibration of FG-GPLRC spherical shell on two parameter elastic foundation

In the present research, the free vibration analysis of functionally graded (FG) nanocomposite deep spherical shells reinforced by graphene platelets (GPLs) on elastic foundation is performed. The elastic foundation is assumed to be Winkler-Past ernak-type. It is also assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the nanocomposite shell. Volume fraction of the graphene platelets as nanofillers may be different in the layers. The modified HalpinTsai model is used to approximate the effective mechanical properties of the multilayer nanocomposite. With the aid of the first order shear deformation shell theory and implementing Hamilton\'s principle, motion equations are derived. Afterwards, the generalized differential quadrature method (GDQM) is utilized to study the free vibration characteristics of FG-GPLRC spherical shell. To assess the validity and accuracy of the presented method, the results are compared with the available researches. Finally, the natural frequencies and corresponding mode shapes are provided for different boundary conditions, GPLs volume fraction, types of functionally graded, elastic foundation coefficients, opening angles of shell, and thickness-to-radius ratio.

A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin\'s approach

In this work, the buckling analysis of material sandwich plates based on a two-parameter elastic foundation under various boundary conditions is investigated on the basis of a new theory of refined trigonometric shear deformation. This theory includes indeterminate integral variables and contains only four unknowns in which any shear correction factor not used, with even less than the conventional theory of first shear strain (FSDT). Applying the principle of virtual displacements, the governing equations and boundary conditions are obtained. To solve the buckling problem for different boundary conditions, Galerkin\'s approach is utilized for symmetric EGM sandwich plates with six different boundary conditions. A detailed numerical study is carried out to examine the influence of plate aspect ratio, elastic foundation coefficients, ratio, side-to-thickness ratio and boundary conditions on the buckling response of FGM sandwich plates. A good agreement between the results obtained and the available solutions of existing shear deformation theories that have a greater number of unknowns proves to demonstrate the precision of the proposed theory.

Combination resonances of imperfect SSFG cylindrical shells rested on viscoelastic foundations

The present paper investigates the combination resonance behavior of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal and external functionally graded stiffeners under two-term large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. With regard to classical plate theory of shells, von-Karman equation and Hook law, the relations of stress-strain are derived for shell and stiffeners. The spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. According to the Galerkin method, the discretized motion equation is obtained. The combination resonance is obtained by using the multiple scales method. Finally, the influences of the stiffeners angles, foundation type, the nonlinear elastic foundation coefficients, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

Nonlinear thermal stability and snap-through buckling of temperature-dependent geometrically imperfect graded nanobeams on nonlinear elastic foundation

In this research, thermal postbuckling and nonlinear thermal bending of size-dependent functionally graded (FG) perfect/imperfect nanobeams in-contact with a nonlinear elastic foundation and exposed to thermal loading are first scrutinized. The second objective of this research is to investigate snap-through instability in a thermally postbuckled FG nanobeam due to uniform lateral load. The geometrical imperfection of the nanobeam is taken into account. Thermo-mechanical material properties are temperature-dependent and are assumed to vary continuously throughout the nanobeam thickness on the basis of the power-law model. The nonlinear equilibrium equations are derived according to the Euler-Bernoulli beam theory in conjunction with the nonlinear von-Karman assumptions based on the nonlocal elasticity theory. Using Chebyshev polynomial of the first kind, Ritz method is utilized into the principle of virtual displacement to form nonlocal governing equations. Three different methodologies, including direct displacement control approach, Newton-Raphson iterative method, and cylindrical arch-length scheme are utilized to investigate the nonlinear thermal stability curves and snap-through phenomenon through limit points of a thermally postbuckled FG nanobeam. The primary purpose of this research is to explore the influences of the nonlocal parameter, nonlinear elastic foundation coefficients, power-law index, imperfection amplitude as well as the different types of boundary conditions on the nonlinear thermal stability and snap-through phenomenon of the FG nanobeam.

Two-Dimensional Electro-Elastic Analysis of FG-CNTRC Cylindrical Laminated Pressure Vessels With Piezoelectric Layers Based on Third-Order Shear Deformation Theory

Abstract The purpose of this paper is to show the electro-elastic static behavior of cylindrical sandwich pressure vessels integrated with piezoelectric layers. The core is made of functionally graded carbon nanotube-reinforced composite (FG-CNTRC). The cylinder is embedded between two piezoelectric layers made of PZT-4. The effective material properties of reinforced core with carbon nanotubes (CNTs) are calculated based on rule of mixture. The constitutive relations are developed in cylindrical coordinate system based on a higher-order shear deformation theory for both core and piezoelectric layers. The employed higher-order theory is based on third-order variation of deformations along the thickness direction to improve the accuracy of numerical results. The method of eigenvalue–eigenvector is used for solution of system of governing equations along the longitudinal direction. The numerical results are provided along the longitudinal and radial directions in terms of significant parameters such as various patterns of CNTs, various volume fractions of CNTs, various elastic foundation coefficients, and various applied electrical potentials.

Buckling and vibration analysis of FG-CNT-reinforced composite rectangular thick nanoplates resting on Kerr foundation based on nonlocal strain gradient theory

This paper investigates the buckling and free vibration analysis of functionally graded carbon nanotube-reinforced composite thick rectangular nanoplates resting on a Kerr foundation under different boundary conditions. Quasi-three-dimensional hyperbolic shear deformation theory is employed to study the effects of transverse shear deformation and thickness stretching. To capture the small-size effects of nanoscale dimensions, the nonlocal strain gradient theory is used, which includes nonlocal parameters and length scale of the material. In this study, rectangular nanocomposite plates are reinforced by carbon nanotubes which are assumed to be graded through the thickness direction with four types of distributions, namely, uniformly, FG-O, FG-V, and FG-X. The governing equations and boundary conditions are extracted within Hamilton’s principle. They are discretized and numerically solved by utilizing a generalized differential quadrature method. The critical buckling loads and natural frequencies are determined by solving the eigenvalue problem. The accuracy of present results is validated with those available in the literature. Also, the effect of various factors, such as aspect ratio, length-to-thickness ratio, in-plane loading factor, length scale parameter, nonlocal parameter, volume fraction and dispersion profile of carbon nanotubes, elastic foundation coefficients, and different boundary conditions, on the buckling behavior and free vibration of nanoplates is investigated.

Thermoelastic behavior of sandwich plates with porous polymeric core and CNT clusters/polymer nanocomposite layers

In this paper, sandwich plates with one porous polymeric core and two carbon nanotube (CNT)/polymer nanocomposite outer layers were proposed as ultra-lightweight engineering structures. The distribution of pores and CNTs were assumed to be functionally graded (FG) along the thickness of core and nanocomposite layers, respectively. Moreover, the important effect of the formation of CNT clusters in nanocomposite layers were considered. Specifically, thermoelastic static responses of proposed porous nanocomposite sandwich plates (PNSPs) subjected to thermal gradient and mechanical loads were presented in terms of deflection and stress distributions. Governing thermoelastic equations derived from total energy function were facilitated using a mesh-free method in combination with two plate theories. Furthermore, Eshelby-Mori-Tanaka’s approach was utilized to estimate the temperature-dependent material properties of the resulted CNT clusters/polymer nanocomposite. The effects of CNTs and porosity characteristics, thermal and mechanical loadings, geometrical dimensions, elastic foundation coefficients and boundary conditions on the bending and stress distributions of the proposed PNSPs were investigated. The results showed that sandwich plates with FG porous cores had less deflection compared with those containing UD cores. Moreover, due to the formation of CNT clusters, adding more than 5% volume fraction of CNTs had an insignificant effect on the deflection of PNSPs.

Large amplitude free vibration analysis of shear deformable FGM shallow arches on nonlinear elastic foundation

Current investigation deals with the small and large amplitude free vibrations of a curved beam (arch) resting on a nonlinear elastic foundation. The arch is made of a functionally graded material, where the properties are graded across the thickness. Uniform temperature elevation in the arch is also considered and material properties are assumed to be temperature dependent. The governing motion equations of the arch are established using a higher order arch theory which satisfies the traction free boundary conditions and the von Kármán type of non-linearity. The governing equations of the arch are solved for the case of an immovable pinned arch using the two step perturbation technique. Closed form expressions are given to estimate the nonlinear frequencies of the arch as a function of the mid-span deflection. Numerical results of this study are validated for the case of flat FGM beams on elastic foundation. Afterwards, novel numerical results are given to explore the influences of power law index, elastic foundation coefficients, length to thickness ratio, length to radius ratio, and the temperature effects.

Semi-analytical vibration analysis of FGM cylindrical shells surrounded by elastic foundations in a thermal environment

The natural frequency and transient response of FGM cylindrical shells under arbitrary boundary conditions are performed in present work. A novel semi-analytical method, which integrates Durbin’s inverse Laplace transform and the differential quadrature method, is developed to analyze the dynamical behavior of cylindrical shells. Durbin’s numerical inversion method is selected to gain time domain solutions. The trigonometric series expansion is used in the circumferential direction whereas the use of differential quadrature method provides numerical solutions in terms of axial direction. Comparisons show that the calculated natural frequencies are in good agreement with results in the literature. Convergence study illustrates that the developed method is rapidly convergent with the increase of sampling points, and the calculated transient response of the cylindrical shell is validated by comparing with Navier’s solution. The influences of boundary conditions, material graded indexes, temperature changes, elastic foundation coefficients and geometric parameters on transient response are analyzed. Numerical results indicate that the peak displacement of cylindrical shells increases with the increase of temperature changes and length-radius ratios or the decrease of elastic foundation coefficients and thickness-radius ratios.

Critical buckling temperature and force in porous sandwich plates with CNT-reinforced nanocomposite layers

This paper presents the thermal and mechanical buckling behaviors of lightweight polymeric nanocomposite sandwich plates containing uniformly dispersed (UD) pores resting on two-parameter elastic foundations. The outer layers of the proposed sandwich plates were assumed to be made of functionally graded (FG) carbon nanotube (CNT)-reinforced polymeric nanocomposite. For nanocomposite layers, the considerable effect of cluster formation of randomly-oriented CNTs was considered and material properties were evaluated through Eshelby-Mori-Tanaka (EMT)'s approach with the definition of cluster state. Furthermore, the first and third order shear deformation theories (FSDT and TSDT) of plates were employed to define the total energy function of sandwich plates. The governed buckling equations were facilitated using a mesh-free method. The effects of porosity, nanofiller characterizations, elastic foundation coefficients, sandwich plate dimensions and boundary conditions on buckling behavior were explored. The obtained results showed that porosity considerably improved thermal buckling behavior; however, it reduced the critical mechanical loads of sandwich plates. Moreover, adding CNTs into outer layers simultaneously improved the mechanical and thermal buckling responses of sandwich plates.

Functionally Graded Sandwich Circular Plate of Non-Uniform Varying Thickness with Homogenous Core Resting on Elastic Foundation: Investigation on Bending via Differential Quadrature Method

This paper illustrates the effectiveness of two functionally graded (FG) layers on a homogenous circular plate for bending analysis. The classical plate theory (CPT) serves as the basis of the analysis. Differential quadrature method (DQM) as semi-analytical method is employed to solve the governing equations. The material properties is varied to obey power-law in terms of the plate thickness direction. The plate is subjected to uniform transverse loading and resting on Winkler elastic foundation. In this study, the effect of the different profile of the plate thickness, elastic foundation coefficient, the volume fraction FG index, and effect of the boundary conditions, namely, simply supported and clamped edge on static response are demonstrated. The results are compared with finite element method and published literature that observed to be in accordance with each other.

Dynamic and stability analysis of functionally graded material sandwich plates in hygro-thermal environment using a simple higher shear deformation theory

A new simple “four-variable shear deformation” plate model is proposed in this work to demonstrate the hygro-thermal environment effects on dynamic and buckling of functionally graded material “sandwich plates” supported by “Winkler–Pasternak” elastic foundations. Equations of motion are obtained from Hamilton’s principle containing the hygro-thermal influences. This model uses only “four variables” and considers trigonometric variation of “transverse shear stress.” In addition, these stresses become zero at the upper and lower surfaces of the “sandwich plate.” The novelty of this formulation is the addition of the integral term in the field of displacement, which leads to a reduction of the number of unknowns and basic equations. Various kinds of functionally graded material “sandwich plates” are examined in this study. To verify the validity of the model, the calculated results are compared with the existing results. Comparison investigations reveal that the computed results of the proposed formulation are in excellent agreement with those of other “higher shear deformation theories.” Parametric investigation is established to demonstrate the effects of temperature rise, moisture condition, elastic foundation coefficients, and power law index on the vibration and buckling of functionally graded material “sandwich plates.”

Mindlin’s strain gradient theory for vibration analysis of FG-CNT-reinforced composite nanoplates resting on Kerr foundation in thermal environment

This article examines the application of simplified Mindlin’s strain gradient theory to free vibration analysis of functionally graded carbon nanotube–reinforced composite (FG-CNTRC) thick rectangular nanoplates resting on Kerr elastic foundation in thermal environment. The theory contains only one length scale parameter corresponding to strain gradient effects. Also, a quasi-3D hyperbolic theory considering transverse shear deformation and thickness stretching effects is employed to present the formulation. In this study, properties of the carbon nanotubes (CNTs) and the polymeric matrix are assumed to be temperature dependent. Distribution of CNTs across the thickness of the nanoplate is considered to be uniform (UD) or functionally graded (FG-X, FG-V, and FG-O). According to Hamilton’s principle and the generalized differential quadrature method, the governing equations and associated boundary conditions are obtained and discretized, respectively. The natural frequencies of FG-CNTRC nanoplates are determined by solving eigenvalue problem. The numerical results of the present formulation are compared with those available in the literature to explain the accuracy of the suggested theories. Then, parametric studies are presented to examine the effects of elastic foundation coefficients, size parameter, temperature change, volume fraction and dispersion profile of CNTs, aspect ratio, length-to-thickness ratio, and different boundary conditions on vibration behavior of FG-CNTRC nanoplates. The results confirmed that size parameter and changes in temperature play an important role in determining natural frequencies. In addition, the shear layer parameter of Kerr foundation has more influence with respect to the coefficients of the upper and lower layers.

Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation

In this paper, an analytical method is used to study the nonlinear primary resonance of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal stiffeners. The SSFG cylindrical shell is surrounded by linear and nonlinear elastic foundation and the effect of structural damping on the system response is also considered. The material properties of the shell and stiffeners are assumed to be continuously graded in the thickness direction. Three-parameter nonlinear elastic foundation model is consists of two-parameter linear elastic foundation (Winkler and Pasternak) and one hardening/softening cubic nonlinearity parameter. Based on the von Karman nonlinear equations and the classical plate theory of shells, the strain–displacement relations are derived. The smeared stiffener technique is used to the model of the internal stiffeners. Using the Galerkin method, the partial differential equations of motion are discretized. The nonlinear primary resonance is analyzed by means of the multiple scales method. The effects of various geometrical characteristics, material parameters and elastic foundation coefficients are investigated on the nonlinear primary resonance.