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This paper investigates the parametric instability of functionally graded graphene reinforced nanocomposite plates that undergo a periodic uniaxial in-plane force and a uniform temperature rise. The plate is composed of multiple graphene platelet reinforced composite (GPLRC) layers in which graphene platelets (GPLs) are uniformly distributed in each individual layer with GPL concentration varying layer-wise across the plate thickness. The modified Halpin–Tsai model that takes into account the GPL geometry effect is employed to calculate the Young's modulus of the GPLRC. Based on the first-order shear deformation theory, the governing equations are deduced and then are solved by using the differential quadrature approach integrated with the Bolotin's method. A parametric study is undertaken to show the influences of GPL distribution pattern, concentration and geometry, temperature change, static in-plane force, plate geometry and boundary condition on the parametric instability of functionally graded multilayer GPLRC plates. It is found that the addition of a small amount of GPL reinforcements considerably increases the critical buckling load and natural frequencies but reduces the size of unstable region. The reinforcing effect is the best when the surface layers of the plate are GPL-rich. [ABSTRACT FROM AUTHOR]

On the basis of the nonlocal strain gradient theory, a size-dependent Euler–Bernoulli beam model is formulated and devoted to investigating the scaling effect on the post-buckling behaviors of functionally graded (FG) nanobeams with the von Kármán geometric nonlinearity. The developed beam model can incorporate the scaling effect of both nonlocal long-range force and microstructure-dependent strain mechanism. To simplify the redundancy of the governing equation and derive the closed-form solutions, a physical neutral surface is applied for removing the bending-stretching coupling due to geometric nonlinearity and the coupling rigidity between the extensional and bending rigidities of the though-thickness FG material. The closed-form solutions for the post-buckled configuration and the critical buckling force (CBF) are deduced in the case of hinged-hinged boundary conditions. The effects of scaling parameters and material property variation on the post-buckled configuration and the CBF are investigated in detail. It is found that the stiffness-hardening or stiffness-softening effect is dependent of the values of scaling parameters. [ABSTRACT FROM AUTHOR]

Buckling of the rectangular thin plates is a class of problems of fundamental importance in mechanical engineering. Although various theoretical and numerical approaches have been developed, benchmark analytic solutions are still rare due to the mathematical difficulty in solving the complex boundary value problems of the governing high-order partial differential equation. Actually, most available solutions can be categorized as either “accurate” for the plates with two opposite edges simply supported or “approximate” for those without two opposite edges simply supported. In this paper, we present the first work on the symplectic superposition method-based analytic buckling solutions of the rectangular thin plates. A Hamiltonian system-based variational principle via the Lagrangian multiplier method is proposed to formulate the thin plate buckling in the symplectic space. Then the governing equation is analytically solved for some fundamental subproblems which are superposed to yield the final solutions of the original problems. For each problem, a set of equations are produced with respect to the expansion coefficients of the quantities imposed on the plate edges. The existence of the nontrivial solutions of the equations sets the requirement that the determinant of the coefficient matrix be zero, which leads to a transcendental equation with respect to the buckling loads. The buckling mode shapes are obtained by substituting the nontrivial coefficient solutions into the mode shape solutions of the subproblems, followed by superposition. Four types of buckling problems are studied for the plates with combinations of clamped and simply supported edges, without two opposite edges simply supported. The developed method as well as the accurate analytic results is well validated by the finite element method and very few analytic results from the literature. [ABSTRACT FROM AUTHOR]

In this paper, recently developed inverse trigonometric zigzag theory by the authors is extended for the stability analysis of laminated composite and sandwich plates. The model satisfies the inter-laminar continuity at layer interfaces as well as the traction free boundary conditions at top and bottom surfaces of the plate. This theory assumes the non-linear distribution of transverse shear stresses. Green–Lagrange non-linear strain–displacement relations are used to represent geometric nonlinearity in buckling analysis. An efficient C 0 finite element is employed for discretization of the laminate. Numerical results on buckling load parameters are evaluated for laminated composite and sandwich plates covering with different features such as aspect ratios, span thickness ratios, modular ratios, loading and boundary conditions. To ensure the potentiality of the present theory, the evaluated results are validated with the three dimensional elasticity results and the existing results based on different deformation theories as well. Higher modes with corresponding mode shapes are also presented for laminated composite plates. The exemplary results with minimum percentage error with respect to exact results ascertain the accuracy and efficiency of the present theory. [ABSTRACT FROM AUTHOR]

In the present study, the buckling behaviour of laminated composite skew plates with different boundary conditions subjected to linearly varying in-plane loads are presented. The skew plate is modelled based on higher order shear deformation theory, which accurately predicts the buckling behaviour for the thick plate. The in-plane stress distribution within the skew plate due to linearly varying in-plane load is equal to the applied in-plane edge load in the pre-buckling range. Using these in-plane stress distributions, the total potential energy functional is formulated. Total potential energy is a function of the total strain energy of skew plate and potential energy due to in-plane stress distributions. The total strain energy of skew plate contains membrane energy, bending energy, additional bending energy due to additional change in curvature and shear energy due to shear deformation, respectively. The total potential energy functionals mapped from physical domain to computational domain over which a set of orthonormal polynomials satisfying the essential boundary conditions is generated by Gram–Schmidt orthogonalization process. Using a Rayleigh-Ritz method in conjunction with Boundary Characteristics Orthonormal Polynomials, the total potential energy functional is converted into sets of algebraic equations. Finally, these algebraic equations are rearranged as a linear eigenvalue problem, which is solved to obtain the critical buckling loads. The numerical results are presented for different skew angles, boundary conditions, length to thickness ratios, aspect ratios and in-plane loadings. It is observed that the critical buckling load increase with the increase of skew angle as well as change in the mode shape at a lower aspect ratio with the increase of skew angle. [ABSTRACT FROM AUTHOR]

A new size-dependent beam model is introduced on the basis of hyperbolic shear deformation beam and modified strain gradient theory. The governing differential equations and corresponding boundary conditions are obtained with the aid of minimum total potential energy principle. The static bending and buckling behaviors of simply supported microbeams embedded in an elastic medium are investigated. The interactions between the microbeam and elastic medium are simulated by Winkler foundation model. Navier solution procedure is employed to obtain analytical solutions for deflections under sinusoidal load and critical buckling loads. The effects of material length scale parameter, length-to-thickness ratio, shear correction factors and Winkler modulus on the bending and buckling responses of microbeams are discussed in detail. The results are comparatively presented with the results of other beam theories. It is observed that the new results predicted by the present model and the results evaluated by sinusoidal shear deformation beam model are in good agreement. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new microstructure-dependent sinusoidal beam model for buckling of microbeams is presented using modified strain gradient theory. This microbeam model can take into consideration microstructural and shear deformation effects. The equilibrium equations and corresponding boundary conditions in buckling are derived with the minimum total potential energy principle. Buckling problem of a simply supported microbeam subjected to an axial compressive force is analytically solved by Navier solution procedure. Influences of thickness-to-length scale parameter and slenderness ratios on buckling behavior are discussed in detail. It is observed that the size dependency becomes more important when the thickness of the microbeam is closer to material length scale parameter. Also, it can be said that the effects of shear deformation are more considerable for short and thick beams with lower slenderness ratios. [Copyright &y& Elsevier]

Abstract: In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin''s approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented. [Copyright &y& Elsevier]

Abstract: This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge. [Copyright &y& Elsevier]

Abstract: In this paper, lower and upper bounds of the buckling load of a composite beam with two non-overlapping delaminations are obtained by developing analytical models. The characteristic equation governing the delamination buckling is derived by using Euler–Bernoulli beam theory, performing proper linearization and by imposing the appropriate continuity and boundary conditions. The effects of the differential stretching and the bending–extension coupling are considered. The accuracy of the approach is verified by comparing results with previously published data and a separately carried out finite element analysis. The effects of the dimensions of the delaminations, their thicknesswise and spanwise locations on the lower and upper bounds of the buckling load are investigated in detail. The longer of the two non-overlapping delaminations dominates the buckling behavior. Composite beams with anti-symmetric non-overlapping delaminations withstand lower buckling loads than the composite beams with symmetric non-overlapping delaminations. The lower and upper bounds of the buckling load will be useful to gauge the working range of bridging and give guidelines for practical applications. [Copyright &y& Elsevier]

This paper examines the buckling and collapse of cylindrical shells under axial load with one end radially and tangentially fixed, with varying axial fixity, and the other end free. The bifurcation loads are found for elastic cylinders, while collapse loads are found for both elastic and elastic-perfectly plastic cylinders. The varying axial restraint is applied in the form of linear springs. The eigenvalue buckling loads are calculated with conditions matching those of a classical analysis. Bifurcation loads are shown to be a function of the axial restraint; as the axial restraint is increased, the bifurcation load increases dramatically, until it reaches that of a semi-infinite, open ended cylinder. A non-dimensional form of the axial spring stiffness is proposed, and shown to be applicable across a range of geometries.The collapse load and imperfection sensitivity of cylinders with the boundary conditions examined here is also found to be a function of the axial restraint. Cylinders with low axial restraint are shown to be imperfection insensitive, with collapse loads above, or close to, the bifurcation load. As the amount of axial restraint increases, the collapse behaviour displays a degree of imperfection sensitivity associated with more usual boundary conditions. [Copyright &y& Elsevier]