Unloaded Edges Research Articles

Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio

Abstract Mechanical metamaterials with negative Poisson’s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buckling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy’s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain–displacement relationship, temperature-dependent material properties, thermal effects, and the plate–substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and postbuckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.

Nonlinear thermomechanical vibration of initially stressed functionally graded plates with porosities

AbstractThe large‐amplitude vibration of initially stressed functionally graded material (FGM) rectangular plates with porosities is investigated in this paper. All edges of the plate are simply supported and uncompressed edges are tangentially restrained. Initial stresses are caused by in‐plane compressive load acting on freely movable edges or thermally induced forces at restrained edges. Pores present in FGM through even and uneven distributions. Material properties are assumed to be temperature‐dependent and, due to the presence of the pores, the effective properties of FGM are determined using a modified rule of mixture. Governing equations in terms of deflection and stress function are established based on the classical plate theory taking into account von Kármán nonlinearity and initial geometric imperfection. These equations are solved using approximate analytical solutions along with the Galerkin method to obtain a time differential equation containing quadratic and cubic nonlinear terms. Nonlinear frequencies are determined by means of numerical integration employing the fourth‐order Runge–Kutta scheme. Parametric studies are carried out to analyze the effects of material and geometry properties, porosity volume fraction and distribution, pre‐existent compressive and thermal loads, imperfection and degree of tangential constraints of unloaded edges on the natural frequencies and nonlinear to linear frequency ratio of FGM plates.

Buckling of shear-deformable unsymmetrically laminated plates

The couplings of unsymmetric laminated structures and the comparatively low transverse shear stiffnesses significantly influence their stability behaviour. The aim of this work is to improve the purely analytical stability analysis of unsymmetric laminated structures. This problem can be reduced to one single plate with the use of the discrete plate theory. In this study, laminates under uniaxial compressive load are considered that are simply supported at all edges and have rotational restraints on the unloaded edges (SRSR) or have a free edge (SFSR). Therefore, the present study provides explicit solutions for the buckling load in two ways. Firstly, the first explicit solution for the buckling load of unsymmetric laminates with the mentioned boundary conditions is developed, which takes the bending–extension couplings into account in a direct way. The method is an energy-based method in the form of the Rayleigh-quotient for classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and third order shear deformation theory (TSDT). Secondly, the reduced bending (RBS) method is investigated for the first time in the framework of TSDT. The comparison of unsymmetric cross-ply and angle-ply laminates shows a good agreement with finite element analysis (FEA). Due to the high computational efficiency of the presented methods, they are excellently suited for pre-design and optimization of composite structures, with simultaneous high usability in practical application.

A new model for calculating the elastic local buckling stress of steel plates in square CFST columns

Previous studies into the elastic local buckling response of the steel plates in concrete-filled steel tubular (CFST) sections have considered the effect of concrete infill on the plate boundary conditions and on the buckling mode shape (i.e. one-way buckling only), but have not considered the influence of hoop stresses. This is addressed in the current paper, where a new model is proposed in which the unloaded plate edges are assumed to be elastically restrained and account is taken of the influence of hoop stresses. Tensile hoop stresses are shown to delay the occurrence of local buckling in the steel plates of square and rectangular CFST sections, though their beneficial influence reduces with increasing rotational edge restraint. The model is verified against existing experimental results, revealing better predictions of elastic local buckling stresses compared to previous approaches. The influence of the key parameters—tensile hoop stresses, varying boundary conditions and differing width-to-thickness (b/t) ratios is subsequently explored using the developed model, where it is shown that the b/t ratio remains the dominant factor in defining the elastic local buckling stress of steel plates in CFST sections. Finally, an alternative equivalent thickness approach to allow for the influence of the hoop stresses is proposed.

Buckling analysis of shear‐deformable orthotropic laminated plates with rotational restraints

AbstractBuckling of shear‐deformable laminated plates is investigated in the present study. A closed‐form method to determine the buckling load is developed within the framework of Reddy's Third Order Shear Deformation Theory (TSDT). The considered orthotropic plates are under uniaxial compression and simply supported at all edges with rotational restraints at the unloaded edges. This allows the modelling of a number of structural situations, e.g. local buckling of beams or the skin fields of stiffened composite panels, for which only the rotational stiffness of the surrounding structure is required. The analysis approach is based on the Ritz method and employs simple shape functions with few variables for the deflection and rotations, which finally leads to an explicit approximate solution for the critical buckling load. The comparison to the Lévy‐type solution shows a satisfactory agreement. The new approach is very simple and highly computationally efficient, which provides a great advantage for example in optimisations.

Buckling of partially-compressed laminated composite plates

The web buckling is one failure mode of web crippling for thin-walled fiber-reinforced plastic (FRP) composite beams subjected to transverse concentrated or distributed compression load from both the top and bottom bearing plates, and the web between the top and bottom flanges can be modeled as a plate with the rotationally-restrained (RR) and partially-compressed loaded edges and the simply-supported (SS) unloaded edges. Explicit solution and universal analytical charts/formulas for buckling analysis of such RRSS composite plates partially compressed along the transverse direction are obtained. The unique Heaviside function and trigonometric series expansion are used to express the partial edge load with effective load length in the governing equilibrium equation. The explicit solution is established for the orthotropic plates subjected to concentrated point compression load in the middle; while the generic buckling analytical curves are developed for infinitely long laminated composite plates transversely compressed partially at the middle of edges as well as near the corners. The accuracy of the present method is validated with available simple analytical solution, numerical finite element method and experimental data. The effect of rotational restraint stiffness, plate aspect ratio, the location and length of partial edge compression load on buckling behavior of laminated composite plates is investigated in the parametric study. A practical guideline for laminated composite plates with the elastic rotational restraints along the loaded edges and partially compressed along the transverse direction is proposed, and it can be further applied in discrete plate analysis to predict the web buckling strength (one failure mode in web crippling) of FRP composite structural shapes under transverse loading at the locations of load bearing or supports.

Ultimate strength behavior of laser-welded web-core sandwich plates under in-plane compression

This paper numerically investigates the ultimate strength behavior of the laser-welded web-core sandwich plates under in-plane compression. Both geometric nonlinearity and material nonlinearity are taken into consideration. Besides the global deflection of the sandwich plate, the local deflections of the face plates are also introduced in the ultimate strength analyses to study the influence of the local buckling on the ultimate strength. The influence of the magnitudes of the initial deflections on the ultimate strength of the web-core sandwich plates is studied. Four boundary conditions are considered, and the result shows that the boundary conditions on the unloaded edges, i.e. the simply supported or clamped boundary conditions, have little effect on the ultimate strength of the sandwich plate. The ultimate strength of the sandwich plates shows a reduction when the laser weld rotation stiffness is accounted for, and transverse shear stiffness has a large effect on the ultimate strength. The contact between the web plates and face plates in the joints increases the ultimate strength of the sandwich plates, and the smaller root gap results in the higher ultimate strength.

Thermomechanical postbuckling of higher order shear deformable CNT-reinforced composite plates with elastically restrained unloaded edges

Buckling and postbuckling behavior of carbon nanotube (CNT) reinforced thick composite plates resting on elastic foundations and subjected to thermomechanical loads are investigated in this paper. The plates are subjected to uniform uniaxial compression in a thermal environment or the combined action of nondestabilizing preexisting uniaxial compression and uniform temperature rise. CNTs are reinforced into matrix through functionally graded distributions. The properties of constitutive materials are assumed to be temperature dependent and effective properties of CNT-reinforced composite are determined according to an extended rule of mixture. Governing equations are based on a higher order shear deformation theory taking von Kárman nonlinearity, initial geometrical imperfection, elasticity of tangential restraints of unloaded edges and plate-foundation interaction into consideration. Analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain nonlinear load-deflection relations. Numerical analyses are carried out to show the effects of CNT distribution patterns, preexisting loads, initial imperfection, degree of in-plane constraint, and elastic foundations on the nonlinear thermomechanical stability of CNT-reinforced composite plates.

Experimental and numerical study on ultimate strength of stiffened columns under axial compression

The ultimate strength of six stiffened columns under axial compressive loading are studied by experimental and numerical methods. The web local buckling mode observed in specimens is analyzed based on the measured strain and displacement during the test. The effects of measured stress-strain curve and elastic/perfectly plastic curve of material to the ultimate strength of specimens are investigated by numerical calculations. The simply supported boundaries at the loaded edges and free boundaries at unloaded edges were produced based on the horizontal test fixture. The initial geometrical imperfections were measured and tensile tests with different thicknesses were conducted to get the true stress-strain curve. The strain gauges and displacement meters were installed to analyze the failure modes and it indicated that the local web buckling occurred first and dominated the collapse. The ultimate strength calculated by numerical analysis reaches a good agreement with experimental results. The ultimate strength calculated with the elastic/perfectly plastic curve is approximately 10% larger than that with the true stress-strain curve. It indicates that the elastic/perfectly plastic material constitutive model will not always overestimate the ultimate strength as the non-linear property between the proportional limit stress and 0.2% proof stress of material would reduce the ultimate strength.

Criterion of proppant pack mobilization by filtrating fluids: Theory and experiments

The goal of hydraulic fracturing is placing a granular material, proppant, in the created fracture to sustain its conductivity and attain economical production rates. After fracture closure on the placed proppant and subsequent pressure drawdown in a well, the consolidated proppant can be dragged by the producing fluids and undesirably be produced back to the well. Rough estimations or accurate simulations of this phenomenon require a valid quantitative criterion of proppant mobilization in a fracture.In this work, we analytically derive such criterion from the fundamental principles. The criterion is expressed in terms of the minimum fluid filtration velocity that destabilizes the proppant. It is sensitive to such important factors as cohesion, thickness and permeability of a proppant pack, mean grain size, roughness and softness of fracture walls, and filtrating fluid viscosity. The developed theory considers the existing unloaded edge of a proppant pack, its granular structure, and related finite cohesion of particles holding each other in a pack. These key properties are shown to play a dominant role for mobilization of a proppant in hydraulic fractures.On the other hand, we conduct an extensive series of laboratory experiments mimicking subterranean fracture closure on proppant, fluid filtration through it, and registering proppant flowback. In the experiments, we try a wide range of commercial proppants, less- and more-viscous fluids, and stiff and soft fracture walls. We associated proppant flowback onset with the change of proppant pack height. We also show how proppant bridging depends on the particle embedment into soft walls and illustrate various geometries of proppant washing out, giving clues to various scenarios of proppant flowback evolution.Finally, we observe a good correspondence of our theory and experimental observations. Theoretical hypotheses of unstressed edge and finite cohesion of a granular solids serve as the only means to explain the observed range of filtration velocity magnitudes for proppant mobilization and their independence from the value of confining stress. The reciprocity between the cohesion and critical velocity revealed by the theory is confirmed by experiments. It allows measuring only one of them to estimate another.

Analysis of inelastic buckling of rectangular plates with a free edge using polynomial deflection functions

AbstractThe inelastic buckling behaviour of different rectangular thin isotropic plates having a free edge is studied. Various combinations of boundary conditions are subject to in-plane uniaxial compression and each rectangular plate is bounded by an unloaded free edge. The characteristic deflection function of each plate is formulated using a polynomial function in form of Taylor–Maclaurin series. A deformation plasticity approach is adopted and the buckling load equation is modified using a work principle technique. Buckling coefficients of the plates are calculated for various aspect ratios and moduli ratios. Findings obtained from the investigation are found to reasonably agree with data published in the literature.

Local buckling of rectangular concrete-filled steel tubular columns with binding bars under eccentric compression

Based on the energy method, this article presents a theoretical study on the elastic local buckling of steel plates in rectangular concrete-filled steel tubular columns with binding bars subjected to eccentric compression. The formulas for elastic local buckling strength of the steel plates in eccentrically loaded rectangular concrete-filled steel tubular columns with binding bars are derived, assuming that the loaded edges are clamped and the unloaded edges of the steel plate are elastically restrained against rotation. Then, the experimental results are compared with these formulas, which exhibits good agreement. Subsequently, the formulas are used to study the elastic local buckling behavior of steel plates in rectangular concrete-filled steel tubular columns with binding bars under eccentric compression. It is found that the local buckling stress of steel plates in eccentrically loaded rectangular concrete-filled steel tubular columns with binding bars is significantly influenced by the stress gradient coefficient, width-to-thickness ratio, and the longitudinal spacing of binding bars. With the decrease of width–thickness ratios or the longitudinal spacing of binding bars or with the increase of the stress gradient coefficient, the local buckling stress increases. Furthermore, the influence of the longitudinal spacing of binding bar is more significant than the stress gradient coefficients. Finally, appropriate limitation for depth-to-thickness ratios ( D/ t), width-to-thickness ratios ( B/ t), and binding bar longitudinal spacing at various stress gradient coefficients ( α0) corresponding to different cross-sectional aspect ratios ( D/ B) are suggested for the design of rectangular concrete-filled steel tubular columns with binding bars under eccentric compression.

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.

Elastic-plastic local buckling model of rectangular CFT columns under axial compression

Assuming that the loaded edges of steel plates are clamped, while the unloaded edges are restrained against rotation, the elastic-plastic local buckling model of rectangular CFT columns was proposed. The formula for calculating elastic-plastic local buckling strength of steel plates in rectangular CFT columns were proposed using the deformation theory of plasticity. The calculated results by formula were compared with the experimental results, showing good agreement between the calculated results and the experimental results. Based on the proposed elastic-plastic local buckling model of rectangular CFT columns, the effects of width-to-wall thickness ratios on the elastic-plastic local buckling critical stress of the steel tube in the cases of different cross-sectional aspect ratios were investigated. Finally, the reasonable limitations for width-to-wall thickness ratios and the reasonable thickness ratios of the steel plates in the cases of different cross-sectional aspect ratios are suggested for engineering design, which can ensure that the elastic-plastic local buckling of the steel tube can not occur before the steel yield.

Nonlinear stability of CNT-reinforced composite cylindrical panels with elastically restrained straight edges under combined thermomechanical loading conditions

This article investigates the nonlinear stability of composite cylindrical panels (CPs) reinforced by carbon nanotubes (CNTs), resting on elastic foundations and subjected to combined thermomechanical loading conditions. CNTs are embedded into matrix phase through uniform distribution or functionally graded distribution. Material properties of constituents are assumed to be temperature dependent and effective elastic moduli of carbon nanotube–reinforced composite are estimated by the extended rule of mixture. Nonlinear governing equations of geometrically imperfect panels are based on first-order shear deformation theory accounting for elastic foundations and tangential constraint of straight edges. Analytical solutions are assumed to satisfy simply supported boundary conditions and closed-form expressions relating load and deflection are derived through Galerkin method. Numerical examples show the effects of preexisting nondestabilizing loads, distribution patterns, panel curvature, in-plane condition of unloaded edges, thermal environments, initial imperfection, and elastic foundations on the nonlinear stability of nanocomposite CPs under combined loading conditions.

Ultimate strength assessment of plated steel structures with random pitting corrosion damage

Pitting corrosion poses a threat to plated steel structures serving in aggressive corrosion environments. This paper involves numerical studies on the structural behaviour and ultimate strength reduction of plated steel structures due to random pitting damage. Stochastic simulations were used to model the random nature of the pitting corrosion varying pitting shape, depth and distribution. A series of nonlinear analyses were performed on unstiffened plates and stiffened panels to understand the mechanisms of structural collapse due to random pitting damage. Empirical formulae were derived respectively for the prediction of ultimate strength reductions of unstiffened plates and stiffened panels in terms of regression analysis from the numerical results. Random pitting corrosion induces a variation and reduction in ultimate strength, and can lead to a transition in failure mode. The collapse of pitted structures under uniaxial compression has a feature that the onset of plasticity initiates in the areas close to the unloaded edge of the structure, and propagates into a continuous plasticity region linking the pits with highly concentrated stress. The pitted area with intensively stress-concentrated pits undergoes a locally amplified deformation that determines the failure mode, leading to structural failure.

Thermomechanical nonlinear analysis of axially compressed carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations with tangentially restrained edges

ABSTRACTBuckling, postbuckling, and nonlinear responses of composite cylindrical panels reinforced by single-walled carbon nanotubes (CNTs), supported by an elastic foundation, exposed to elevated temperature and axially compressed by uniform load are investigated in this article. Distribution of CNTs is uniform or graded in the thickness direction and the effective properties of CNT-reinforced composite are assumed to be temperature dependent, and are estimated by extended rule of mixture through a micromechanical model. Governing equations are established based on thin shell theory taking von Kármán–Donnell nonlinearity, initial geometrical imperfection, Pasternak-type elastic foundation and tangential elastic constraints of boundary edges into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to derive explicit expressions of load–deflection relation from which critical buckling loads can be obtained. Unlike works in the literature, the present study accounts for elasticity of tangential restraint of two unloaded straight edges in model of cylindrical panel. The study also gives conditions for which bifurcation type buckling response can occur and novel findings in numerical examples.

Contact buckling behaviour of corrugated plates subjected to linearly varying in-plane loads

An analytical method is developed for analysing the contact buckling response of infinitely long, thin corrugated plates and flat plates restrained by a Winkler tensionless foundation and subjected to linearly varying in-plane loadings, where the corrugated plates are modelled as orthotropic plates and the flat plates are modelled as isotropic plates. The critical step in the presented method is the explicit expression for the lateral buckling mode function, which is derived through using the energy method. Simply supported and clamped edges conditions on the unloaded edges are considered in this study. The acquired lateral deflection function is applied to the governing buckling equations to eliminate the lateral variable. Considering the boundary conditions and continuity conditions at the border line between the contact and non-contact zones, the buckling coefficients and the corresponding buckling modes are found. The analytical solution to the buckling coefficients is also expressed through a fitted approximate formula in terms of foundation stiffness, which is verified through previous studies and finite element (FE) method.

Domain decomposition technique to simulate crack in nonlinear analysis of initially imperfect laminates

In this research, an effective computational technique is carried out for nonlinear and post-buckling analyses of cracked imperfect composite plates. The laminated plates are assumed to be moderately thick so that the analysis can be carried out based on the first-order shear deformation theory. Geometric non-linearity is introduced in the way of von-Karman assumptions for the strain-displacement equations. The Ritz technique is applied using Legendre polynomials for the primary variable approximations. The crack is modeled by partitioning the entire domain of the plates into several sub-plates and therefore the plate decomposition technique is implemented in this research. The penalty technique is used for imposing the interface continuity between the sub-plates. Different out-of-plane essential boundary conditions such as clamp, simply support or free conditions will be assumed in this research by defining the relevant displacement functions. For in-plane boundary conditions, lateral expansions of the unloaded edges are completely free while the loaded edges are assumed to move straight but restricted to move laterally. With the formulation presented here, the plates can be subjected to biaxial compressive loads, therefore a sensitivity analysis is performed with respect to the applied load direction, along the parallel or perpendicular to the crack axis. The integrals of potential energy are numerically computed using Gauss-Lobatto quadrature formulas to get adequate accuracy. Then, the obtained non-linear system of equations is solved by the Newton-Raphson method. Finally, the results are presented to show the influence of crack length, various locations of crack, load direction, boundary conditions and different values of initial imperfection on nonlinear and post-buckling behavior of laminates.

On the buckling of an isotropic rectangular plate uniformly compressed on two simply supported edges and with two free unloaded edges

This expository brief note elaborates on the solution to the buckling problem of a uniformly compressed rectangular plate for which the two opposite loaded edges and the unloaded edges are considered simply supported and free, respectively. While Reissner originally discussed briefly this problem in 1909, it was not until 1954 when Ishlinskii presented its rigorous mathematical solution in the form of a transcendental equation, the solution of which requires the use of an iterative numerical technique. Despite remaining almost unreferenced and unnoticed in the literature, the works of Reissner and Ishlinskii in connection with this specific stability problem are educationally instructive and practically relevant. Ishlinskii's solution forms the basis upon which a simple approximate closed-form solution, presented in this note, yields results that do not differ significantly from those computed iteratively using the transcendental equation. Results presented in this brief note can be used as benchmark solutions for validating structural engineering software packages.