Postbuckling Equilibrium Paths Research Articles

Non-linear analysis of functionally graded fiber reinforced composite laminated plates, Part I: Theory and solutions

This paper deals with the large amplitude vibration, non-linear bending and postbuckling of fiber reinforced composite laminated plates resting on an elastic foundation in hygrothermal environments. Two kinds of fiber reinforced laminated plates, namely, uniformly distributed and functionally graded reinforcements, are considered. The material properties of fiber reinforced laminated plates are estimated through a micromechanical model and are assumed to be temperature-dependent and moisture-dependent. The motion equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction and the hygrothermal effect. A two-step perturbation technique is employed to determine the non-linear to linear frequency ratios of plate vibration, the load-deflection and load-bending moment curves of plate bending, and postbuckling equilibrium paths of laminated plates.

Buckling and Post-Buckling Behavior for Roof Strata in Longwall Mining

In accordance with the occurrence behavior of roof strata and the fracture characteristics of key stratum in shallow seam longwall mining, this paper studied the post-buckling behaviors of key roof stratum in the process of mining by using initial post-buckling theory, which derived a critical load and a breaking span of the main roof during the first weighting, determined the final subsidence for broken key stratum, and presented an application with the example of Daliuta 1203 face. The results indicate that the rock blocks a in are state of non-equilibrium after main roof breaking, the equilibrium path of main roof is unstable from breaking to final subsidence; thick unconsolidated layers above roof have effect on post-buckling behaviors of key stratum; the stability for bifurcation point equilibrium configuration and post-buckling equilibrium path of roof strata could be revealed and an effective method for determining displacement field of imperfection structure could be provided by using initial post -buckling theory.

Nonlinear vibration and postbuckling analysis of a single layer graphene sheet embedded in a polymer matrix

Nonlinear vibration and postbuckling behavior of a single layer graphene sheet (SLGS) embedded in a polymer matrix aroused by the nonlinear van der Waals (vdW) forces are investigated using the Kirchhoff plate theory. The interfacial vdW forces are described by a nonlinear function in terms of the graphene deflection. Through harmonic balance method, the nonlinear relation between deflection amplitudes and resonant frequencies of free vibrations of the SLGS and its postbuckling equilibrium path are derived. It is found that variation of resonant frequencies of an embedded SLGS is less dependent on the graphene aspect ratio and mode numbers as compared with a free-standing one. In-plane load effects upon the vibrational behavior of the SLGS and its postbuckling are also discussed. Simulation results have demonstrated the significance of considering the surrounding medium effect and its nonlinearity in the study of the vibration and buckling of the embedded graphene with applications in nanocomposties.

Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations

Abstract This paper investigates compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise are presented of a sandwich plate with carbon nanotube-reinforced composite (CNTRC) face sheets resting on an elastic foundation. The material properties of CNTRC face sheets are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations of the plate are based on a higher-order shear deformation plate theory that includes plate-foundation interaction. The thermal effects are also included and the material properties of both CNTRC face sheets and homogeneous core layer are assumed to be temperature-dependent. A two-step perturbation technique is employed to determine buckling loads (temperature) and postbuckling equilibrium paths. The numerical illustrations concern the compressive and thermal postbuckling behavior of perfect and imperfect, sandwich plates with functionally graded CNTRC face sheets resting on Pasternak elastic foundations under different thermal environmental conditions, from which results for the sandwich plate with uniformly distributed CNTRC face sheets are also obtained for comparison purposes. The results reveal that the foundation stiffness, the temperature changes, the nanotube volume fraction of face sheet, and the core-to-face sheet thickness ratio have significant effects on the compressive buckling load and postbuckling behavior of the sandwich plate, whereas this effect on the thermal postbuckling behavior is less pronounced for the same sandwich plate.

Free Vibration of Functionally Graded Material Beams in a Uniform Electric Field

Free vibration of thermal post-buckled functionally graded material (FGM) beams in a uniform electric field subjected to both temperature rise and voltage is studied. Voltage has a great influence on the deformation of the beam, and Thermo-electric post-buckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity and the material gradient parameters are plotted. It is found that the three lowest frequencies of the pre-buckled beam decrease with an increase in the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Post-Buckling Analysis of FGM Beam Subjected to Non-Conservative Forces and In-Plane Thermal Loading

Based on the Kirchhoff large deformation theory, the post-buckling behavior of right movable simply supported FGM beam subjected to non-conservative forces and in-plane thermal loading was analyzed in this paper. The temperature-dependent and spatially dependent material properties of the FGM beam were assumed to vary through the thickness. The nonlinear governing equations of FGM beam subjected to a uniform distributed tangential load along the central axis and in-plane thermal loading were derived. Then, a shooting method and Runge-kutta method are employed to numerically solve the resulting equations. The post-buckling equilibrium paths of the FGM beam with different parameters were plotted, and the effects of non-conservative force, temperature, gradient index of FGM on the post-buckling behavior of right movable simply supported FGM beams were analyzed.

On the mechanics of thin-walled angle column instability

This paper reports the results of a numerical investigation aimed at providing fresh insight on the mechanics underlying the buckling and (mostly) post-buckling behaviour of short-to-intermediate equal-leg thin-walled angle steel columns exhibiting fixed and pinned (but with the secondary warping prevented) end supports. Although most of the numerical results presented and discussed were obtained through Abaqus shell finite element analyses, the paper also includes some GBT-based critical stresses and buckling mode shapes, whose interpretation help clarifying the distinction between local and global buckling. The shell finite element results displayed consist of (i) elastic post-buckling equilibrium paths and (ii) curves and diagrams providing the evolution, along a given path, of the column deformed configurations and normal stress distributions.

Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation

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.

Postbuckling of functionally graded fiber reinforced composite laminated cylindrical shells, Part I: Theory and solutions

Buckling and postbuckling behavior are presented for fiber reinforced composite (FRC) laminated cylindrical shells subjected to axial compression or a uniform external pressure in thermal environments. Two kinds of fiber reinforced composite laminated shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The governing equations are based on a higher order shear deformation shell theory with von Kármán-type of kinematic non-linearity and including the extension-twist, extension-flexural and flexural-twist couplings. The thermal effects are also included, and the material properties of FRC laminated cylindrical shells are estimated through a micromechanical model and are assumed to be temperature dependent. The non-linear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths of FRC laminated cylindrical shells.

A NOVEL TECHNIQUE FOR NONLINEAR ANALYSIS OF BEAMS ON TWO-PARAMETER ELASTIC FOUNDATIONS

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.

New Results of Buckling and Postbuckling Behaviors for Fiber-Reinforced Composite Shear Deformable Anisotropic Laminated Cylindrical Shell Subjected to Torsion

New results of buckling and postbuckling analysis are presented for a shear deformable anisotropic laminated cylindrical shell of finite length subjected to torsion. The governing equations are based on a higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity and including the extension/twist, extension/flexural and flexural/twist couplings. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The postbuckling equilibrium path is unstable for a moderately thick laminated cylindrical shell under torsion and the shell structure is virtually imperfection-sensitive.

Exact Mathematical Model and its Numerical Solution of a Two-Layer Beam Subjected to Non-Uniform Temperature Rise

Based on accurately considering the axial extension, geometrically nonlinear governing equations for a two-layer beam subjected to thermal load were formulated. By using a shooting method, the strongly nonlinear ordinary differential equations with two-point boundary conditions were solved and numerical solution for thermal post-buckling and bending deformation of a two-layer beam with pinned-pinned ends and subjected to transversely non-uniform temperature rising were obtained. As an example, equilibrium paths and configuration for a beam laminated by brass and steel are presented and characteristic curves of the nonlinear deformation changing with the thermal load were plotted. Effects of the geometric and material parameters on the deformation of the beam were discussed and analyzed in detail. The theoretical analysis and numerical results show that the bending deformation and the stretching-bending coupling terms of beam subjected to uniform or non-uniform temperature rising can be produced because of the non-homogenous distribution of the material properties. The bending deformation resulted from transversely temperature rise is primary deformation when values of average temperature rise parameter is under critical temperature, however, the curves become the thermal post-buckling equilibrium paths with the increment of average temperature rise when values of average temperature rise parameter exceed critical temperature.

Postbuckling Behavior of 3D Braided Rectangular Plates Subjected to Biaxial Compression

AbstractPostbuckling behavior of three-dimensional (3D) braided rectangular plates subjected to biaxial compression is presented. The 3D braided composite may be treated as a cell system, and the geometry of each cell is deeply dependent on its position in the cross section of the plate. Based on Reddy’s higher-order shear deformation plate theory and general von Karman–type equations that include the initial geometric imperfection of the plate, a perturbation technique is employed to determine postbuckling equilibrium paths of 3D braided rectangular plates. The results reveal that the geometric and physical properties have a significant effect on the postbuckling behavior of braided composite plates.

Nonlinear buckling behavior of 3D-braided composite cylindrical shells subjected to internal pressure loads in thermal environments

The nonlinear buckling behavior of a 3D-braided composite cylindrical shell of finite length subjected to internal pressure in thermal environments is considered. According to a new micromacromechanical model, a 3D-braided composite may be treated as a cell system where the geometry of each cell strongly depends on its position in the cross section of the cylindrical shell. The material properties of the epoxy matrix are expressed as linear functions of temperature. The governing equations are based on Reddy’s higher-order shear deformation theory of shells with a von Karman–Donnell-type kinematic nonlinearity and include thermal effects. The singular perturbation technique is employed to determine the buckling pressure and the postbuckling equilibrium paths of the shell.

Lateral–torsional buckling analysis of thin-walled beams including shear and pre-buckling deformation effects

In this paper, lateral–torsional buckling behavior of open-section thin-walled beams is investigated based on a geometrically nonlinear formulation, which considers the effects of shear deformations. A finite element numerical solution along with an incremental-iterative solution procedure is adopted to trace the pre-buckling as well as the post-buckling equilibrium paths. Formulation is applicable to a general type of open-section and load position effects are also included. Numerical results are validated through comparisons with experimental results and those based on other formulations presented in the literature. Comparisons have also been made between the results based on fully nonlinear analysis and linearized buckling analysis in order to illustrate the effects of pre-buckling deformations as well as the shear deformations on the buckling load predictions. Examples illustrate the influence of beam slenderness and moment gradient on the effects of pre-buckling deformations in predicting bucking loads.

Thermal Postbuckling Behavior of 3D Braided Rectangular Plates

Thermal postbuckling characteristics of 3D braided rectangular plates are investigated. A 3D braided composite may be treated as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the plate. Based on Reddy's higher-order shear deformation plate theory, a perturbation technique is employed to determine buckling temperatures and postbuckling equilibrium paths of simply supported 3D braided rectangular plates. The results reveal that the temperature-dependent properties, geometric parameter, fiber volume fraction and braiding angle have a significant effect on thermal postbuckling behavior of braided composite plates.

Postbuckling Behavior of Shear-Deformable Anisotropic Laminated Cylindrical Panels under External Lateral Pressure

A postbuckling analysis for a shear-deformable anisotropic laminated cylindrical panel of finite length is presented, subjected to external lateral pressure based on a boundary-layer theory of shell buckling. The material properties of each layer of the panel are assumed to be linearly elastic. The governing equations are based on a higher-order shear-deformation shell theory with a von Karman–Donnell type of kinematic nonlinearity and including extension/twist, extension/flexural, and flexural/twist couplings. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect moderately thick, anisotropic laminated cylindrical panels under different values of panel parameters and stacking sequence. The new finding reveals that a circumferential stress exists along with an associate shear stress and twisting when the moderately thick anisotropic laminated cylindrical panel is subjected to external lateral pressure.

Postbuckling of shear deformable stiffened anisotropic laminated cylindrical shell under axial compression

A postbuckling analysis is presented for a shear deformable anisotropic laminated cylindrical shell with stiffener of finite length subjected to axial compression. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity and including the extension/twist, extension/flexural and flexural/twist couplings. The ‘smeared stiffener’ approach is adopted for the beam stiffeners. This arrangement allows the beam stiffeners to be assembled directly into the global stiffness matrix. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, grid, axial, ring stiffened, and unstiffened shells. The results confirm that there exists a compressive stress along with an associate shear stress and twisting when the anisotropic shell is subjected to axial compression. The postbuckling equilibrium path is unstable for the moderately thick cylindrical shell under axial compression and the stiffened shell structure is imperfection-sensitive.

Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part II: Pressure-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.