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Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

The mechanical and thermal stability equations of asymmetric functionally graded discs subjected to the Pasternak foundation are developed by employing Hamilton’s energy principle based on the first-order shear theory. The material properties are temperature-dependent and vary according to power-law and exponentially in thickness and radial direction, respectively. Accordingly, the temperatu re is also varying in both directions. Using the well-developed differential quadrature method, stability equations are discretized along with the boundary conditions, leading to a complete algebraic linear equations system. The validation of results is performed to certify the results. Numerical and illustrative results are presented to study the effect of elastic foundation parameters, graded indexes, nodal lines, and boundary conditions on thermal and mechanical buckling. Also, the impact of compressive in-plane force on thermal buckling and thermal environment on mechanical buckling is presented.

This study examined the free vibration of a three-layered annular microplate, whose core and face sheets are composed of functionally graded saturated porous (FGSP) materials and functionally graded-graphene platelet-reinforced composites (FG-GPLRC), respectively. The microplate is supported by an elastic base, with the mechanical characteristics of all layers varying along the thickness direction. Employing the extended dynamic formulation of Hamilton’s principle, the equations of motion and boundary conditions are derived from the modified version of couple stress and first-order shear deformation theories, subsequently solved using the generalized differential quadrature method as an effective numerical approach. The study examines the impact of many parameters, including pore distributions, porosity coefficient, pore compressibility, dispersion patterns of graphene platelets, elastic foundation, small-scale parameter, and microplate aspect ratio. The results of this study may be beneficial for the construction of lightweight and sophisticated buildings.

The governing differential equations for the out-of-plane, free vibration of circular curved beams resting on elastic foundations are derived and solved numerically. The formulation takes into consideration the effects of rotary inertia and transverse shear deformation. The lowest three natural frequencies are calculated for beams with hinged–hinged, hinged-clamped, and clamped–clamped end constraints. The effects of various system parameters as well as rotary inertia and shear deformation on the natural frequencies are investigated.

In this study, a special class of closed-form solutions for inhomogeneous beam-columns on elastic foundations is investigated. Namely the following problem is considered: find the distribution of the material density and the flexural rigidity of an inhomogeneous beam resting on a variable elastic foundation so that the postulated trigonometric mode shape serves both as vibration and buckling modes. Specifically, for a simply-supported beam on elastic foundation, the harmonically varying vibration mode is postulated and the associated semi-inverse problem is solved that result in the distributions of flexural rigidity that together with a specific law of material density, an axial load distribution and a particular variability of elastic foundation characteristics satisfy the governing eigenvalue problem. The analytical expression for the natural frequencies of the corresponding homogeneous beam-column with a constant characteristic elastic foundation is obtained as a particular case. For comparison the obtained closed-form solution is contrasted with an approximate solution based on an appropriate polynomial shape, serving as trial function in an energy method.

Postbuckling, nonlinear bending, and nonlinear vibration analyses are presented for a simply supported Euler–Bernoulli beam resting on a two-parameter elastic foundation. The nonlinear model is introduced by using the exact expression of the curvature. Two kinds of end conditions, namely movable and immovable, are considered. The nonlinear equation of motion, including beam–foundation interaction, is derived separately for these two kinds of end conditions. The analysis uses a two-step perturbation technique to determine the postbuckling equilibrium paths of an axially loaded beam, the static large deflections of a bending beam subjected to a uniform transverse pressure, and the nonlinear frequencies of a beam with or without initial stresses. The numerical results confirm that the foundation stiffness has a significant effect on the nonlinear behavior of Euler–Bernoulli beams. The results also reveal that the end condition has a great effect on the nonlinear bending and nonlinear vibration behaviors of Euler–Bernoulli beams with or without elastic foundations.

In-plane bending loads occur in many thin-walled structures, including web core sandwich panels (foam-filled panels with interior webs) under transverse loading. The design of such structures is limited in part by local buckling of the thin webs and the subsequent impact on stiffness and strength. However, the core material can have a significant impact on web buckling strength and thus must be considered in design. This paper presents solutions for the buckling strength of simply supported plates under in-plane bending loads. The location of the neutral bending axis is allowed to vary and is characterized by a load parameter. A Pasternak model is used to account for the resistance of the foundation to compression and shear. Using the principle of minimum potential energy, buckling solutions are developed for infinitely long plates and representative foundation materials. The solutions match known results for two special cases: Uniform loading with variable foundation, and bending loads with no foundation. An order of magnitude increase in buckling strength is possible, depending on loading and foundation stiffness. The results suggest an important avenue for future development of lightweight structures, including sandwich panels and structures such as plate girders that are not typically associated with the use of foam filling.

The transverse vibrations of cracked beams with rectangular cross sections resting on Pasternak and generalized elastic foundations are considered. Both the Euler–Bernoulli (EB) and Timoshenko beam (TB) theories are used. The open edge crack is represented as a rotational spring whose compliance is obtained by the fracture mechanics. By applying the compatibility conditions between the beam segments at the crack location and the boundary conditions, the characteristic equations are derived, from which the nondimensional natural frequencies are solved as the roots. Sample numerical results showing the effects of crack depth, crack location, foundation type and foundation parameters on the natural frequencies of the beam are presented. It is observed that the existence of crack reduces the natural frequencies, whereas the elasticity of the foundation increases the stiffness of the system and thus the natural frequencies. It is also observed that the type of elastic foundation has a significant effect on the natural frequencies of the cracked beam.

The free vibration of functionally graded (FG) beams with various boundary conditions resting on a two-parameter elastic foundation in the thermal environment is studied using the third-order shear deformation beam theory. The material properties are temperature-dependent and vary continuously through the thickness direction of the beam, based on a power-law distribution in terms of the volume fraction of the material constituents. In order to discretize the governing equations, the differential quadrature method (DQM) in conjunction with the Hamilton’s principle is adopted. The convergence of the method is demonstrated. In order to validate the results, comparisons are made with solutions available for the isotropic and FG beams. Through a comprehensive parametric study, the effect of various parameters involved on the FG beam was studied. It is concluded that the uniform temperature rise has more significant effect on the frequency parameters than the nonuniform case.

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

This technical note presents a static–dynamic relationship for the flexural free vibration analysis of beams in tension with some specific boundary conditions. It is shown to be possible that a free vibration system can be solved via a static analysis approach to determine the natural frequencies of the beam with tension forces. The key idea of this study is to substitute the real natural frequency parameters with zero or negative elastic foundation stiffness, thereby allowing one to obtain the natural frequencies by analyzing the case with negative foundation elastic constant. This static approach for vibration problems can be extended for more complicated engineering structural systems.

This paper investigates the nonlinear instability of eccentrically stiffened functionally graded (ES-FG) sandwich truncated conical shells subjected to the axial compressive load. The core of the FG sandwich truncated conical shells, assumed to be thin, is made of pure metal or ceramic materials and the two skin layers are made of a FG material. The shell reinforced by orthogonal stiffeners (stringers) is also made of FG materials. The change of spacing between the stringers in the meridional direction is considered. The governing equations are derived using the Donnell shell theory with von Karman geometrical nonlinearity along with the smeared technique for stiffeners. The resulting coupled set of three nonlinear partial differential equations with variable coefficients in terms of displacement components are solved by the Galerkin’s method. The closed-form expressions for determining the critical buckling load and for analyzing the postbucking load–deflection curves are obtained. The accuracy of present formulation is verified by comparing the results obtained with available ones in the literature. The effects of various parameters such stiffeners, foundations, material properties, geometric dimensions on the stability of the shells are studied in detail.

In this paper, the sound radiation behaviors of the functionally graded porous (FGP) plate with arbitrary boundary conditions and resting on elastic foundation are studied by means of the modified Fourier series method. It is assumed that a total of three types of porosity distributions are considered in the present study. The material parameters are determined according to the porosity coefficient used to denote the size of pores in the plate. The governing equations of the FGP plate are derived by utilizing the Hamilton’s principle on the basis of the first-order deformation theory (FSDT). Each displacement component of the FGP plate is expanded as the Fourier cosine series combined with auxiliary polynomial functions introduced to enhance the convergence rate of the series expansions. The acoustic response of the FGP plate due to a concentrated harmonic load is calculated by evaluating the Rayleigh integral. Good agreements are attained by comparing the present results with those in available literatures, which show the accuracy and versatility of the developed method in this paper. Finally, the influences of the porosity distribution type, porosity coefficient, boundary condition and elastic foundation on the sound radiation of the FGP plate are analyzed in detail.

Nonlinear buckling analysis for honeycomb auxetic-core sandwich toroidal shell segments with CNT-reinforced face sheets surrounded by elastic foundations under the radial pressure is presented in this study. The basic equation system of shells is established based on the von Kármán–Donnell nonlinear shell theory, combined with Stein and McElman approximation. Meanwhile, the foundation-shell elastic interaction is simulated by the foundation model based on the Pasternak assumption. The Galerkin procedure is utilized to achieve the pre-buckling and post-buckling responses for the shell, from which the radially critical buckling load is determined. Numerical analysis shows the various influences of auxetic-core layer, CNT-reinforced face sheets, and elastic foundation on the pre-buckling and postbuckling behavior of sandwich shells with CNT reinforced face sheets.

In this paper, an examination on the backbone curves of nanocomposite beams reinforced with graphene platelet (GPL) on elastic foundation exposed to a temperature increment is accomplished. By means of the Hamilton principle and in the framework of a third-order shear deformation beam theory, i.e. Reddy’s beam theory (RBT), along with the von Karman nonlinear strains, the nonlinear motion equations are disclosed. The Halpin–Tsai micromechanical model is exploited in order to release the effective modulus of elasticity of the nanocomposite beam while the thermal expansion coefficient, Poisson’s ratio and the mass density are estimated resorting to the rule of mixtures. Elastic foundation, and uniform temperature (UD) rising impacts are incorporated during the mechanical modeling of the nanocomposite beam. The Ritz scheme together with an iterative procedure reveals the nonlinear natural frequency associated to an assumed deflection in order to sketch the corresponding backbone curve. The outcomes are validated in comparison with the available results. Some case studies are established for the sake of clarifying the impressions of the distribution pattern of the GPL, the beam length to its thickness ratio, the weight fraction of the GPL, the elastic foundation, the boundary condition type, and the temperature changes on the first backbone curve of the nanocomposite beam. It is elucidated that the increment of the weight fraction of the GPL with X/O distribution pattern decreases/increases the hardening behavior of clamped–clamped (C-C) and simply-supported beam, while the softening behavior of a clamped-free (C-F) nanocomposite beam is independent of the division pattern, and the weight fraction of the GPL. Moreover, the temperature increment unlike the elastic foundation develops the hardening behavior of the backbone curves of simply supported and C-C nanocomposite beams. Although the backbone curve associated to a C-F nanocomposite beam is invariant with respect to the temperature, the elastic foundation develops the softening trend of C-F nanocomposite beams.

Excessive vibration has always been a serious problem for conical shell structures, while the application of the graphene-based free-constrained layer (GFCL) based on carbon fiber-reinforced composite (CFRC) structure is a novel way to improve structural performance. An analytical model for vibration and dynamic characteristics of the GFCL-CFRC conical shell resting on the Winkler–Pasternak elastic foundation with arbitrary boundary conditions is constructed, and four types of GFCL porosity distribution and GFCL dispersion pattern are considered in this model. The multi-segment technique and virtual spring technique are utilized to simulate arbitrary boundary conditions. Then, the first-order shear deformation theory (FSDT) and Hamilton’s principle are employed to obtain the motion equation of the GFCL-CFRC conical shell, and the motion equation of the GFCL-CFRC conical shell is solved by the Ritz method. In conclusion, the dispersion mode of GFCL, thickness ratio of GFCL, and fiber angle have influence on the dynamic performance. With a reasonable design, the dynamic performance of the GFCL-CFRC conical shell can be further improved.

The nonlinear buckling response of laminated composite cylindrical shells reinforced with graphene nanoplatelets (GNPs) is studied in this paper. The functionally graded (FG) shell reinforced by GNPs is analytically studied under external pressure and uniform temperature rise loadings. It is also assumed that the GNP-reinforced laminated composite shell is in contact with an elastic foundation. Various types of profiles are employed for the GNP distribution patterns in the shell thickness including 10 nanocomposite layers. The nonlinear strain-displacement relations of the shallow cylindrical panel are established utilizing the third-order shear deformation shell theory. Governing equilibrium equations of the laminated GNP-reinforced composite shell are formulated employing the principle of virtual displacement. The coupled system of nonlinear differential equations is solved analytically for the hinged–hinged and fixed–fixed boundaries of the shell using a perturbation-based technique. Correctness of presented formulations and obtained solutions is proved by comparisons with results from previous studies for an isotropic cylindrical shell. Novel numerical results reveal that the material properties, geometrical characteristics and load parameters significantly affect on the buckling behavior of laminated composite cylindrical shells.

Vibration behaviors of elastically supported functionally graded (FG) sandwich beams resting on elastic foundations under moving loads are investigated. The transformed-section method is first applied to establish the bending vibration equations of FG sandwich beams, then the Chebyshev collocation method is used to study free and forced vibrations. Two types of sandwich beams with FG faces-isotropic core and isotropic faces-FG core are considered. The material properties of FG materials are assumed to vary across the beam thickness according to a simple power function. Regarding the free vibration analysis, bending vibration frequencies are calculated numerically by forming a matrix eigenvalue equation. As for the forced vibration analysis, the backward differentiation formula method is employed to solve the time-dependent ordinary differential equations to obtain the dynamic deformations of the beam. To ensure the accuracy of the proposed model, some calculated results are compared with those in the published literature. Parametric studies are then performed to demonstrate the effects of material gradient indexes, stack types, layer thickness ratios, slenderness ratios, excitation frequency and speed of moving loads, and foundation and support stiffness parameters on the dynamic characteristics of FG sandwich beams.

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.

This paper presents a novel shear deformation theory for analyzing porous microbeams’ bending, buckling, and free vibration resting on a foundation. The proposed shear function incorporating three kinetic variables satisfies zero-traction boundary conditions on the top and bottom surfaces of the beams and does not require a shear correction factor. The modified couple stress theory accounts for the size-dependent effects, and the governing equations are derived from Lagrange’s equation using the proposed shear function. Legendre–Ritz functions are developed to analyze the porous microbeams’ buckling, free vibration, and bending behaviors. The effects of material length scale parameter, porosity, span-to-height ratio, boundary condition, and foundation parameter on the mechanical responses of beams are investigated. Numerical results demonstrate the accuracy and efficiency of the proposed theory and can serve as benchmarks for future analysis of porous microbeams on elastic foundations.