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This paper presents a theoretical and experimental analysis of buckling load of the circular tube with flattened ends. The buckling tests were conducted on the steel tubes with different length and diameter, and the critical buckling force was determined from the measured relation between the lateral displacement and axial force. Analytical solutions for the critical buckling force of the circular tubes were derived in the series form, and a numerical procedure based on the finite difference method and quasi-Newton method was developed to determine the critical buckling load. The results show that both analytical and numerical solutions were in agreement with those measured from the experiment. Moreover, the effect of flattened part length on the value of the critical buckling force was investigated there. The paper provides a mathematical model of mentioned case and gives us some simplifications calculating the critical buckling force. [ABSTRACT FROM AUTHOR]

Beams (columns) subjected to axially distributed load (e.g., self-weight) are commonly treated using the classical Euler-Bernoulli beam theory, which ignores the transverse shear effect. Adopting the Engesser and Haringx shear theories, respectively, we study in this article the stability and initial post-buckling of sandwich (or laminated composite) beams under terminal force and axially distributed load. Nonlinear governing equations are derived from geometrical compatibility, equilibrium of forces, and moments. The critical buckling load, modal shapes of deformation, and shear force together with bending moment at buckling can be obtained by using the Galerkin's method in terms of trigonometric functions, and the initial post-buckled configuration of the beam is determined employing the shooting method. Predictions based on the Engesser theory agree with finite element simulation results, while the Haringx theory overestimates the buckling load. The effects of transverse shear and various different end constraint conditions on static buckling and initial post-buckling of sandwich beams are systematically explored. [ABSTRACT FROM AUTHOR]

In the present work, an innovative approach is developed and employed to investigate the buckling load of composite laminated plates using polar representation of the fourth-order flexural stiffness tensor. Based on the Rayleigh-Ritz method for calculating the buckling load of orthotropic uncoupled laminates, a new approach is proposed for calculating the critical buckling load as a function of fiber orientation and the number of layers. A new formula is also represented for measuring the possible numbers of buckling modes based on the length-to-width ratio of the plate. Moreover, the new methods make it easy to handle buckling problems, optimization and design of laminated plates subjected to in-plane loads. The efficiency and precision of this method are proved by numerical examples. The anisotropic parts of the buckling load are calculated, and their polar variations are investigated. [ABSTRACT FROM AUTHOR]

Buckling of a column made of functionally graded material is investigated. The functional grading is performed in the longitudinal direction. The following problem is addressed: determine the variation of the elastic modulus with the axial coordinate such that the buckling load exceeds the preselected value. In this study, only polynomial variation of the buckling mode is considered and only the case of cantilever column is treated. [ABSTRACT FROM AUTHOR]

The elastic buckling behavior of quadrilateral single-layered graphene sheets (SLGS) under bi-axial compression is studied employing nonlocal continuum mechanics. Small-scale effects are taken into consideration. The principle of virtual work is employed to derive the governing equations. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The buckling load of skew, rhombic, trapezoidal, and rectangular nanoplates considering various geometrical parameters are obtained. It is shown that nonlocal effects are very important in arbitrary quadrilateral graphene sheets and their inclusion results in smaller buckling loads. Also the effects of geometrical parameters such as aspect ratio, angle, and mode number on the buckling load decrease when scale coefficient increases, for all arbitrary quadrilateral SLGS. [ABSTRACT FROM AUTHOR]

In this study, the influence of nonuniformity of eccentricity of stringers on the general axial buckling load of stiffened laminated cylindrical shells with simply supported end conditions is investigated. The critical loads are calculated using Love's First-order Shear Deformation Theory and solved using the Rayleigh-Ritz procedure. The effects of the shell length-to-radius ratio, shell thickness-to-radius ratio, number of stringers, and stringers depth-to-width ratio on the buckling load of nonuniformly eccentric shells, are examined. The research demonstrates that an appropriate nonuniform distribution of eccentricity of stringers leads the buckling load to increase significantly. [ABSTRACT FROM AUTHOR]

In this article, an analytical solution for buckling of moderately thick functionally graded (FG) sectorial plates is presented. It is assumed that the material properties of the FG plate vary through the thickness of the plate as a power function. The stability equations are derived according to the Mindlin plate theory. By introducing four new functions, the stability equations are decoupled. The decoupled stability equations are solved analytically for both sector and annular sector plates with two simply supported radial edges. Satisfying the edges conditions along the circular edges of the plate, an eigenvalue problem for finding the critical buckling load is obtained. Solving the eigenvalue problem, the numerical results for the critical buckling load and mode shapes are obtained for both sector and annular sector plates. Finally, the effects of boundary conditions, volume fraction, inner to outer radius ratio (annularity) and plate thickness are studied. The results for critical buckling load of functionally graded sectorial plates are reported for the first time and can be used as benchmark. [ABSTRACT FROM AUTHOR]

Post-buckling behaviour of sandwich plates with functionally graded material (FGM) face sheets under uniform temperature rise loading is considered. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation, which acts in both compression and tension. The derivation of equations is based on the first-order shear deformation plate theory. Thermomechanical non-homogeneous properties of FGM layers vary smoothly by the distribution of power law across the thickness, and temperature dependency of material constituents is taken into account. Using the non-linear von-Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect sandwich plates with FGM face sheets are derived. The boundary conditions for the plate are assumed to be simply supported in all edges. The governing equations are reduced to two coupled equation in terms of stress function and lateral deflection. Employing the single mode approach combined with Galerkin technique, an approximate closed-form solution is presented to calculate the critical buckling temperature and post-buckling equilibrium path of the plate. Presented numerical examples contain the influences of power law index, sandwich plate geometry, geometrical imperfection, temperature dependency, and the elastic foundation coefficients. [ABSTRACT FROM AUTHOR]

higher-order global-local theory is proposed based on the double-superposition concept for free vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates subjected to thermomechanical loads. In contrast to all theories proposed so far for analysis of the viscoelastic plates, the continuity conditions of the transverse shear and normal stresses at the layer interfaces and the nonzero traction conditions at the top and bottom surfaces of the sandwich plates are satisfied. Another novelty is that these conditions may be satisfied for viscoelastic plates with temperature-dependent material properties and nonlinear behaviors subjected to thermomechanical loads. Furthermore, transverse flexibility is also taken into account. Some dynamic buckling/wrinkling analyses of the viscoelastic plates are performed in the present paper, for the first time. Comparisons made between results of the paper and results reported by well-known references confirm the accuracy and the efficiency of the proposed theory and the relevant solution algorithm. [ABSTRACT FROM AUTHOR]

Thermoelastic buckling behavior of thick rectangular plate made of functionally graded materials is investigated in this article. The material properties of the plate are assumed to vary continuously through the thickness of the plate according to a power-law distribution. Three types of thermal loading as uniform temperature raise, nonlinear and linear temperature distribution through the thickness of plate are considered. The coupled governing stability equations are derived based on the Reddy's higher-order shear deformation plate theory using the energy method. The resulted stability equations are decoupled and solved analytically for the functionally graded rectangular plates with two opposite edges simply supported subjected to different types of thermal loading. A comparison of the present results with those available in the literature is carried out to establish the accuracy of the presented analytical method. The influences of power of functionally graded material, plate thickness, aspect ratio, thermal loading conditions and boundary conditions on the critical buckling temperature of aluminum/alumina functionally graded rectangular plates are investigated and discussed in detail. The critical buckling temperatures of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future. [ABSTRACT FROM AUTHOR]

In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular thin plate are studied. [ABSTRACT FROM AUTHOR]

In this paper, an analytical procedure is presented to study the vibrational behavior of rectangular plates subjected to different types of non-uniformly distributed in-plane loads. The prebuckling equations, which contain two coupled partial differential equations, are solved analytically by considering the in-plane constrains. The potential and kinetic energies of the plate are calculated based on the first-order shear deformation theory, and the Ritz method is used to obtain the corresponding eigenvalue problem from Hamilton's principle. By parametric study, the effects of plate aspect ratio, thickness ratio and intensity of four types of in-plane load profiles, i.e., constant, parabolic, cosine and triangular on vibrational frequency and buckling load of the plate, are investigated. Comparison of the obtained results with the finite element solution shows the accuracy of the presented method for solving similar problems. [ABSTRACT FROM AUTHOR]

The linearized buckling response of uniform steel beam-columns resting on a Pasternak foundation type is dealt with in this work. The corresponding two-point boundary value problem depends on the two parameters associated with the foundation model, that is, shear layer constant and Winkler spring coefficient, as well as on the axial loading. The fourth-order differential equation of buckling and the consequent characteristic one yield exact values of the buckling loads, which may lead to shape functions, being either one sinusoidal waveform or the sum of two different waveforms. The former case is associated with a single-mode response, while the latter with mode coupling. The conditions for which each of these cases characterizes the beam-column behavior are fully assessed and the dependence on the combination of the aforementioned parameters on the response is discussed in detail. It is found that regardless of the order of coupling, the corresponding mode is not related to rational values of buckling loads and hence this unfavorable phenomenon is excluded in structural design. [ABSTRACT FROM AUTHOR]

Investigation of the behaviour of plate girders (welded I girders) under the action of localized (patch) loading on a flange in the plane of the web is presented. Girders without vertical (transversal) stiffeners in the zone of load introduction, but with horizontal (longitudinal) stiffeners, are considered. An approximate solution for obtaining ultimate load is developed using failure (collapse) mechanism and following the upper bound theorem of plastic collapse. Procedure is based on the solution introduced by Roberts and Rocky and further elaborated by Roberts for the determination of the ultimate patch load for girders without longitudinal stiffeners. Procedure presented in this paper is adapted for the girders with longitudinal stiffeners. Proposed collapse mechanism is based on the experimental evidence. When the longitudinal stiffeners have enough rigidity, deformation at collapse, buckling of the web, is limited to the part of the web between loaded flange and longitudinal stiffener. Values of the ultimate loads obtained through developed procedure as well as the values from other, earlier authors' proposal that was accepted and applied in the British standard BS 5400 are compared with the values from authors' experiments and available database with the test results from other investigators. [ABSTRACT FROM AUTHOR]

Thermomechanical instability of shallow spherical shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators is studied in this paper. The governing equations are based on the classical shell theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the property of the FGMs varies continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials. The constituent materials of the functionally graded shell are assumed to be mixture of ceramic and metal. The analytical solutions are obtained for uniform external pressure, thermal loading, and constant applied actuator voltage. [ABSTRACT FROM AUTHOR]

Based on beam-shaped-function, the analytical solution for composite plates with arbitrary embedded delaminations is presented. The deflection function of the delaminated plate is composed by those of beams with the corresponding loading and support conditions, which can be easily and accurately derived from the beam analysis, and the deflection amplitude is derived by the minimum potential energy principle. The closed form solutions of displacements, stresses, and energy release rate of a composite plate containing an arbitrarily embedded rectangular delamination are obtained and compared with the three-dimensional finite element results to validate the accuracy of this present method. Furthermore, the influences of delamination depth, length, central position, and modulus mismatch ratio ( E/ E) of the upper and lower sublaminate on the energy release rate are discussed. [ABSTRACT FROM AUTHOR]

In this paper, a new functionally graded material (FGM) porous sandwich plate analysis model considering pores with only four variables is proposed. It is applied to the nonlinear bending of two different types of FGM porous sandwich plates under mechanical loading, and the bending of two different types of FGM porous sandwich plates is compared. The governing equations are obtained through the principle of virtual work. The simple supported FGM porous sandwich plates are solved using the Navier method. The obtained numerical results are compared with the results in the literature to verify the accuracy of the theory in this paper. Finally, the effects of volume fraction index, porosity, layer thickness ratio, side-thickness ratio and symmetry on the mechanical bending properties of FGM porous sandwich plates are studied in detail. By comparing the bending characteristics of two types of FGM porous sandwich plates under different influencing factors, it is found that the bending characteristics of different FGM porous sandwich plate types are also very different. This has great guiding significance for the design of FGM porous sandwich plates. [ABSTRACT FROM AUTHOR]

A new Bernoulli-Euler beam model is developed based on modified gradient elasticity theory. The governing equation and boundary conditions, which contain two internal length scales (i.e., lx and lz), are derived by the variational principle. The new model can be simplified to the classical beam theory when the two internal length scales vanish. The numerical examples of cantilever beams subjected to two typical loadings are presented. Results show that the size effect can be captured by the new model, and the deflection decreases with the internal length scales increasing. The influence of lz (the internal length scale along the beam thickness direction) on deflection is much greater than that of lx (the internal length scale along the beam length direction), and the increment of stiffness is mainly controlled by lz. The new beam model is convenient for engineering applications and designs. [ABSTRACT FROM AUTHOR]

An enhanced efficient zigzag theory is presented for the static response in elastic composite plates under mechanical loading. The number of variables is six, which is one more than the conventional zigzag theory. Transverse shear stresses have been obtained through the use of constitutive equations in both symmetric and antisymmetric laminates under uniformly and sinusoidally applied mechanical load. The theory has a good representation of all the three displacement components. This is obtained by using individual descriptions for each layer as is observed in three-dimensional elasticity solution. Interlaminar continuity conditions on all displacement components, all transverse stresses and on the gradient of transverse normal stress as well as transverse shear-free conditions on the top and the bottom surfaces have been utilized to make the primary variables independent of number of layers in the laminate. Equilibrium equations and boundary conditions are derived from variational principle. Navier solution is obtained for simply supported square and rectangular plates. The accuracy of the present theory is assessed by comparison with three-dimensional (3D) elasticity solution. It is found that refinement of the transverse displacement alone is not sufficient to make the new theory capable of providing good accuracy in calculation of transverse stresses from constitutive equations, though some improvement is obtained in case of symmetric laminates. [ABSTRACT FROM AUTHOR]

In this work we consider an optimal design of corrugated cylindrical shells. The functionally graded (FG) corrugation with amplitude following a given shape is defined. The problem is solved as a linear one assuming the axially symmetric loading. The method of estimation of the stress-strain state and, consequently, the shell optimal design based on homogenization approach is proposed. The illustrative examples of the optimal FG corrugation shell subjected to hydrostatic load have demonstrated high efficiency of the employed method. [ABSTRACT FROM AUTHOR]

Clarifying the geometry of the fractional tangent space of a curve, fractional differential geometry of curves has already been revisited, Lazopoulos and Lazopoulos (), defining also the curvature vector. Fractional bending of a beam is introduced, applying Euler-Bernoulli bending principle. The proposed theory is implemented to the bending deformation of a cantilever beam under continuously distributed loading. [ABSTRACT FROM AUTHOR]

Considering thermal loads, dynamics and stability of an outer cylindrical shell conveying swirling fluid in the annulus between the inner shell-type body and the outer shell are investigated by the travelling wave approach. Shell motions are considered based on Donnell-type shell theory. The fluid forces are described by means of the potential flow theory, and thermal loads are determined by the thermo-elastic theory. The numerical analyses are conducted by a zero-level contour method. The study shows the effects of annular gaps and boundary conditions on stability of shells. The influences of angular flow on the critical axial velocity and axial flow on the critical annular velocity are discussed. Moreover, the thermal loads decrease the critical flow velocity markedly and the critical temperature rise is found. [ABSTRACT FROM AUTHOR]

The plate resting on the elastic foundation is an issue of mutual action between two media. The elastic foundation has important impact on static response of composite plates, while accurate prediction on transverse shear stresses of laminated composite plates resting on elastic foundation and subjected to hygro-thermo-mechanical effects is scare in literature. To fill this gap, this paper proposes a higher-order global-local model (HGLM) satisfying the continuity conditions of transverse shear stresses at interfaces. Based on the model HGLM, analytical solution for the governing differential equations of the simply supported composite plate subjected to mechanical loading, temperature and moisture concentrations is developed by employing Navier's technique followed by the procedure of the separation of variables. Comparing the present results with the available elasticity solutions, it is found that the proposed model HGLM can produce promising transverse shear stresses directly from constitutive equations. Based on the model HGLM, the elastic foundation parameters, stacking sequence and thermal expansion coefficients did have significant effects on the hygro-thermo-elastic response of composite plates. However, the other models lose capacity to simulate the effects of the elastic foundation parameters on displacements and stresses for hygrothermal expansion of composite plates. [ABSTRACT FROM AUTHOR]

This paper presents an exact analytical solution for the stress distributions within an elastic hollow sphere subjected to diametrical point loads. The solution is suitable for both thin and thick hollow spheres. New variables are introduced in order to uncouple the system of governing equations so that explicit differential equations are obtained for displacement components and stress components. Moreover, Fourier-Legendre expansion technique is employed in order to determine the unknown coefficients in the analytical solutions for hollow spheres. The present solution can be considered as an extension of the classical solution by Hiramatsu and Oka (Int J Rock Mech Min Sci 3:89-99, ) for solid spheres under the point loads, which provided the theoretical basis for the point load strength test. Unlike in solid spheres, the stress concentrations within the hollow spheres under the point loads are usually developed at the joint point of the inner surface and the loading axis, and the thinner the hollow sphere, the larger the tensile stress concentrations developed at the inner surface. This numerical result indicates that the failure of the hollow spheres usually starts at the inner surface, and the normalized tensile stress at the inner surface increases with the increase in Poisson's ratio and internal pressure, but decreases with the increase in the size of the loading area. Moreover, significant shear stress zone is usually developed in the areas immediately inside the outer surface, and the maximum shear stress is often developed at the point immediately inside the outer surface jointing the edge of the loading area and the center of the hollow sphere. The present solution can be used to analyze the failure mechanism of bulk foams made up of hollow spheres in engineering. [ABSTRACT FROM AUTHOR]

Small-scale effects in nanobeams are effectively described by the E ringen model of nonlocal elasticity. The nonlocal elastostatic problem of B ernoulli-E uler nanobeams is here formulated in variational terms by recognizing that the nonlocality effect is equivalent to a bending curvature distortion prescribed on a corresponding local nanobeam, subjected to the same kinematic boundary constraints and applied loads. The conditions to be imposed for the kinematic integrability of the bending curvature field are also provided to evaluate the bending moment solution in statically indeterminate nonlocal nanobeams. Since the curvature distortion describing the nonlocality effect is kinematically integrable in statically determinate structures, bending moments do not exhibit small-scale effects in non-redundant nanobeams. The equivalence method illustrated in the present paper is resorted to for solving the nonlocal elastostatic problem of nanobeams under constant transversal load distributions. [ABSTRACT FROM AUTHOR]

Progressive deformation of honeycomb structures subjected to in-plane loading was approximately analyzed by using the collapse modes of hexagonal unit cells. The collapse modes were categorized as freely compressive, restricted compressive, and shear. Moreover, there were five characteristic deformation patterns, namely deformation bands. Average stresses of the collapsing honeycomb models were evaluated in terms of the plastic collapse stress per hinge and total number of hinges of progressively arising deformation bands. The displacements of the models were obtained by multiplying the displacement per cell with the number of collapsed cells. The present method was used to analyze progressive deformation of typical honeycomb structures. The validity of the stress-displacement relations derived for some structures was confirmed by comparing them with finite element method (FEM) results. Our method is much simpler than FEM but just as effective. [ABSTRACT FROM AUTHOR]

In this paper, the stresses developed in slightly compressible, finitely deformed half-space due to normal moving load has been discussed. It has been observed that this type of media have much effect on the existing normal and shearing stresses due to the load. The numerical calculations have been done for developed stresses at different depths. It has been observed that the stresses increase significantly in case of slightly compressible, deformed media. [ABSTRACT FROM AUTHOR]

Abstract  In this paper a general solution for the analysis of shear deformable stiffened plates subjected to arbitrary loading is presented. According to the proposed model, the arbitrarily placed parallel stiffening beams of arbitrary doubly symmetric cross section are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. Their unknown distribution is established by applying continuity conditions in all directions at the interfaces. The utilization of two interface lines for each beam enables the nonuniform distribution of the interface transverse shear forces and the nonuniform torsional response of the beams to be taken into account. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second-order effects. The analysis of the plate is based on Reissner’s theory, which may be considered as the standard thick plate theory with which all others are compared, while the analysis of the beams is performed employing the linearized second order theory taking into account shear deformation effect. Six boundary value problems are formulated and solved using the analog equation method (AEM), a BEM based method. The solution of the aforementioned plate and beam problems, which are nonlinearly coupled, is achieved using iterative numerical methods. The adopted model permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The effectiveness, the range of applications of the proposed method and the influence of shear deformation effect are illustrated by working out numerical examples with great practical interest. [ABSTRACT FROM AUTHOR]