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An efficient modelling technique based on one dimensional (1D) beam finite element analysis for buckling of thin-walled laminated composite beams having open/closed sections is proposed. The formulation derived has sufficient generality for accommodating arbitrary stacking sequences of the individual beam section walls, and includes all possible couplings between axial, shear, bending and torsional modes of deformation. The effects of transverse shear deformation of the section walls and out-of-plane warping of the beam section are considered where provision exists to restrain or allow warping deformation. The incorporation of shear deformation leads to a problem in the finite element implementation of the proposed beam kinematics, but this is successfully addressed adopting a novel modelling concept. Numerical results obtained for the sample cases of open sections I beams and closed section box beams are presented. The numerical results are benchmarked/compared to data available in open literature, and it is shown that the proposed model performs very well. Finally, a study of the effect of axial and end moment loading, acting alone or in combination, on the buckling response of thin-walled composite beams is presented. [ABSTRACT FROM AUTHOR]

In this paper, single variable beam theories taking into account effect of transverse shear deformation are developed and applied for the bending, buckling and free vibration analysis of thick isotropic beams. The most important feature of the present beam theories is that unlike any other higher order theory, the proposed class of theories contains only one unknown variable and does not require shear correction factor. The displacement field of the present theories is built upon the classical beam theory. The theories account for parabolic distribution of transverse shear stress using constitutive relations, satisfying the traction free conditions at top and bottom surfaces of the beam. Governing differential equation and boundary conditions of these theories are obtained using the principle of virtual work. Results obtained for the displacements, stresses, fundamental frequencies and critical buckling loads of simply supported isotropic solid beams are compared with those obtained by other theories to validate the accuracy of the present theories.

In this paper, a discrete model of the planar Cosserat rod is presented. Based on the calculus of variations, the equilibrium equations of the model are derived for potential forces and hyperelastic material. Buckling of the structure under axial loading is thoroughly studied assuming linear elasticity. Dimensionless stiffness parameters are introduced, and analytical solutions are given for the critical loads and the corresponding buckled shapes of the model. Classification of the axially loaded structure is accomplished based on the number, sign, and physical admissibility of its buckling loads. It is revealed that the model can possess several buckling modes under tension. [ABSTRACT FROM AUTHOR]

The major novelty of the paper in the study, post-buckling of simply supported FGM beams using various theory, classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), parabolic shear deformation beam theory (PSDBT) and exponential shear deformation beam theory (ESDBT). Governing equations of FGM beam for post-buckling problem were found by applying Hamilton principle and Navier type solution method was used to solve post-buckling problem. It is assumed that elasticity modulus is changing in the thickness direction and all other material properties are taken to be constant. Variation of elasticity modulus in the thickness direction, are described by a simple power law distribution in terms of the volume fractions of constituents. The shear effect is shown to have a significant contribution to both the buckling and post-buckling behaviors. Results of this analysis show that classical and first-order theories underestimate the amplitude of buckling while all higher order theories, considered in this study, yield very close results for the static post-buckling response. [ABSTRACT FROM AUTHOR]

In the present study, a simple trigonometric shear deformation theory is applied for the bending, buckling and free vibration of crossply laminated composite plates. The theory involves four unknown variables which are five in first order shear deformation theory or any other higher order theories. The in-plane displacement field uses sinusoidal function in terms of thickness co-ordinate to include the shear deformation effect. The transverse displacement includes bending and shear components. The present theory satisfies the zero shear stress conditions at top and bottom surfaces of plates without using shear correction factor. Equations of motion associated with the present theory are obtained using the dynamic version of virtual work principle. A closed form solution is obtained using double trigonometric series suggested by Navier. The displacements, stresses, critical buckling loads and natural frequencies obtained using present theory are compared with previously published results and found to agree well with those. [ABSTRACT FROM AUTHOR]

Abstract: In this study, the vibration and buckling of functionally graded (FG) orthotropic cylindrical shells under external pressures is investigated using the shear deformation shell theory (SDST). The basic equations of shear deformable FG orthotropic cylindrical shells are derived using Donnell shell theory and solved using the Galerkin method. Parametric studies are made to investigate effects of shear deformation, orthotropy, compositional profiles and shell characteristics on the dimensionless frequency parameter and critical external pressures. Some comparisons among various theories have been performed in order to show the differences between the parabolic shear deformation theory (PSDT) and several higher-order shear deformation theories (HSDTs). [Copyright &y& Elsevier]

The analytical solution of a buckling force of a pretwisted delaminated composite column with a proper consideration of the extensional and bending stiffness coupling and transverse shear effect is presented. The system of homogenous linearized differential equations with nonconstant coefficients obtained for this problem is solved with the help of the mathematical theory of analytic differential systems. The parametric studies are presented considering the effect of slenderness, the length of delamination, the asymmetrical position of delamination, and the transverse shear on the critical buckling force. [ABSTRACT FROM AUTHOR]

This paper presents a study on the thermal postbuckling response of a shear deformable functionally graded cylindrical shell of finite length embedded in a large outer elastic medium. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely the Voigt model and Mori-Tanaka model, are considered. The governing equations are based on a higher-order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature dependent. The governing equations are solved by a singular perturbation technique. The numerical results show that in some cases the FGM cylindrical shell with intermediate volume fraction index does not have intermediate buckling temperature and thermal postbuckling strength. The results reveal that Voigt model and Mori-Tanaka model have the same accuracy for predicting the thermal buckling and postbuckling behavior of FGM shells. The results confirm that for the case of heat conduction, the postbuckling equilibrium path for geometrically perfect FGM cylindrical shells with simply supported boundary conditions is no longer of the bifurcation type. [ABSTRACT FROM AUTHOR]

Highlights: [•] Shear and torsional rigidities of trusses are theoretically derived. [•] Out-of-plane buckling loads of pin-ended circular truss arches are derived. [•] Effect of shear deformation on buckling loads of truss arches is included. [Copyright &y& Elsevier]

Meshless collocations utilizing Gaussian and Multiquadric radial basis functions for the stability analysis of orthotropic and cross ply laminated composite plates subjected to thermal and mechanical loading are presented. The governing differential equations of plate are based on higher order shear deformation theory considering two different transverse shear stress functions. The plate governing differential equations are discretized using radial basis functions to cast a set of simultaneous equations. The convergence of both radial basis functions is studied for different values of shape parameters. Several numerical examples are undertaken to demonstrate the accuracy of present method and the effects of orthotropy ratio of the material, span to thickness ratio of the plate, and fiber orientation on critical load/temperature are also presented. [ABSTRACT FROM AUTHOR]

Abstract: Buckling analysis of functionally graded micro beams based on modified couple stress theory is presented. Three different beam theories, i.e. classical, first and third order shear deformation beam theories, are considered to study the effect of shear deformations. To present a profound insight on the effect of boundary conditions, beams with hinged-hinged, clamped–clamped and clamped–hinged ends are studied. Governing equations and boundary conditions are derived using principle of minimum potential energy. Afterwards, generalized differential quadrature (GDQ) method is applied to solve the obtained differential equations. Some numerical results are presented to study the effects of material length scale parameter, beam thickness, Poisson ratio and power index of material distribution on size dependent buckling load. It is observed that buckling loads predicted by modified couple stress theory deviates significantly from classical ones, especially for thin beams. It is shown that size dependency of FG micro beams differs from isotropic homogeneous micro beams as it is a function of power index of material distribution. In addition, the general trend of buckling load with respect to Poisson ratio predicted by the present model differs from classical one. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to propose a method for the evaluation of the elastic critical buckling load of columns in frames consisting of members susceptible to non-negligible shear deformations, such as built-up members in steel frames, based on Engesser''s approach. To that effect, a stability matrix is proposed and three general stability equations are derived for the cases of unbraced, partially braced and braced frames. Indicative graphic interpretation of the solutions for the stability equations of the braced and unbraced cases is shown. Slope-deflection equations for shear-weak members with semi-rigid connections are also derived and used for the presentation of a complete set of rotational stiffness coefficients, which are then used for the replacement of members converging at the bottom and top ends of the column in question by equivalent springs. All possible rotational and translational boundary conditions at the far end of these members, as well as the eventual presence of axial force, are considered. Five examples are presented, dealing with braced, unbraced and partially braced frames, with rigid and semi-rigid beam to column connections, loaded with concentrated or uniformly distributed loads, in a symmetrical or non-symmetrical pattern. In all cases the proposed approach is in excellent agreement with finite element results. [Copyright &y& Elsevier]

This paper addresses the finite strip formulations for the stability analysis of viscoelastic composite plates with variable thickness in the transverse direction, which are subjected to in-plane forces. While the finite strip method is fairly well-known in the buckling analysis, hitherto its direct application to the buckling of viscoelastic composite plates with variable thickness has not been investigated. The equations governing the stiffness and the geometry matrices of the composite plate are solved in the time domain using both the higher-order shear deformation theory and the method of effective moduli. These matrices are then assembled so that the global stiffness and geometry matrices of a moderately thick rectangular plate are formed which lead to an eigenvalue problem that is solved to determine the magnitude of critical buckling load for the viscoelastic plate. The accuracy of the proposed model is verified against the results which have been reported elsewhere whilst a comprehensive parametric study is presented to show the effects of viscoelasticity parameters, boundary conditions as well as combined bending and compression loads on the critical buckling load of thin and moderately thick viscoelastic composite plates. [ABSTRACT FROM AUTHOR]

Abstract: The onset of buckling in square laminated multi-layered composite plates, subject to unidirectional in-plane loads, is investigated within the framework of a generalized higher-order shear deformation theory suitable to capture significant transverse shear and thickness-wise deformation effects. The displacement field is expanded in a Taylor series of the thickness coordinate with arbitrary polynomial degree; in turn, the series coefficients, expressed as a superposition of admissible functions, are determined according to the Rayleigh–Ritz method. Truly higher-order polynomial terms, along with a sufficient number of in-plane admissible functions, are shown to be necessary for convergence towards the fundamental buckling load multiplier. As a by-product, reduced-order models are identified for various plate geometries and lamination schemes. The sensitivity of the lowest buckling load with respect to the nondimensional parameters (the thickness ratio, the ratio between the elastic moduli, the ply angle) is investigated. In particular, the attention is focused on the cross-over phenomenon between the lowest two buckling eigenvalues in multi-layered composite square plates with different lamination schemes. The presented results shed light onto the buckling behavior of thick shear-deformable multi-layered plates. [Copyright &y& Elsevier]

Abstract: In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler–Bernoulli, Timoshenko, Reddy, Levinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes. [Copyright &y& Elsevier]

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory. [ABSTRACT FROM AUTHOR]

Abstract: In the present study, a new higher order shear deformable laminated composite plate theory is proposed. It is constructed from 3-D elasticity bending solutions by using an inverse method. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. It was observed that this theory gives most accurate results with respect to 3-D elasticity solutions for bending and stress analysis when compared with existing five degree of freedom shear deformation theories [Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51:745–52; Touratier M. An efficient standard plate theory. Int J Eng Sci 1991;29(8):901–16; Karama M, Afaq KS, Mistou S. Mechanical behaviour of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 2003;40:1525–46]. All shear deformation theories predict the vibration and buckling results with reasonable accuracy, generally within %2 for investigated problems. Previous exponential shear deformation theory of Karama et al. (2003) can be found as a special case. [Copyright &y& Elsevier]

Two new hyperbolic displacement models, HPSDT1 and HPSDT2, are used for the buckling and free vibration analyses of simply supported orthotropic laminated composite plates. The models contain hyperbolic expressions to account for the parabolic distributions of transverse shear stresses and to satisfy the zero shear-stress conditions at the top and bottom surfaces of the plates. The equation of motion for thick laminated rectangular plates subjected to in-plane loads is deduced through the use of Hamilton’s principle. Closed-form solutions are obtained by using the Navier technique, and then the buckling loads and the fundamental frequencies are found by solving eigenvalue problems. The accuracy of the models presented is demonstrated by comparing the results obtained with solutions of other higher-order models given in the literature. It is found that the theories proposed can predict the fundamental frequencies and buckling loads of cross-ply laminated composite plates rather accurately. [ABSTRACT FROM AUTHOR]

Abstract: The exact analytical solution of buckling in delaminated columns is presented. In order to investigate analytically the influence of axial and shear strains on buckling loads the geometrically exact beam theory is employed with no simplification of the governing equations. The critical forces are then obtained by the linearized stability theory. In the paper, we limit the studies to linear elastic columns with a single delamination, but with arbitrary longitudinal and vertical asymmetry of delamination and arbitrary boundary conditions. The studies of quantitative and qualitative influence of transverse shear are shown in detail and extensive results for buckling loads with respect to delamination length, thickness and longitudinal position are presented. [Copyright &y& Elsevier]

Buckling and free vibration analysis of simply supported symmetric and antisymmetric cross-ply thick composite plates on elastic foundation are examined by a new hyperbolic displacement model in this paper. In this new model, inplane displacements vary as a hyperbolic function across the plate thickness, so account for parabolic distributions of transverse shear stresses and satisfy zero shear stress conditions at the top and bottom surfaces of the plate. In the analysis, the foundation is modeled as two parameter Pasternak type foundation, and Winkler type if the second foundation parameter is zero. The equation of motion for thick laminated rectangular plates resting on elastic foundation and subjected to inplane loads is obtained through Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then buckling loads and fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration and buckling of cross-ply laminated plates on elastic foundation are presented, and compared with the ones available in the literature. [ABSTRACT FROM AUTHOR]

The stability and second-order analyses of three-dimensional (3D) multicolumn systems including the effects of shear deformations along the span of each column are presented in a condensed manner. This formulation is an extension to an algorithm presented recently by the writer in 2002 and 2003 by which the critical load of each column, the total critical load, and the second-order response of a 3D multicolumn system with semirigid connections can be determined directly. The proposed solution includes not only the combined effects of flexural deformations and shear distortions along the columns in their two principal transverse axes, but also the effect of the shear forces along each member induced by the applied end axial force as the columns deform and deflect (as suggested by Haringx in 1947 and explained by Timoshenko and Gere in 1961) in their two principal transverse axes. The extended characteristic transcendental equations (corresponding to multicolumn systems with sidesway and twist uninhibited, partially inhibited, and totally inhibited) that are derived and discussed in this publication find great applications in the stability and second-order analyses of 3D multicolumn systems made of materials with relatively low shear stiffness such as orthotropic composite materials (fiber reinforced plastic) and multilayer elastomeric bearings used for seismic isolation of buildings. The phenomenon of buckling under axial tension in members with relatively low shear stiffness (observed by Kelly in 2003 in multilayer elastomeric bearings, and recently discussed by the writer in 2005) is captured by the proposed method. Tension buckling must not be ignored in the stability analysis of multicolumn systems made of columns in which the shear stiffness GAs is of the same order of magnitude as π2EI/h2. [ABSTRACT FROM AUTHOR]

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration. [ABSTRACT FROM AUTHOR]

This paper reports a detailed investigation concerning the local-plate and distortional elastic postbuckling behaviors of cold-formed steel lipped channel columns with web and flange intermediate stiffeners (the corresponding unstiffened lipped channel column postbuckling behaviors are often used as reference). This investigation relies on results obtained through geometrically nonlinear analyses based on a recently developed and numerically implemented generalized beam theory (GBT) formulation that incorporates (1) conventional (no shear deformation), (2) shear (nonlinear warping), and (3) transverse extension deformation modes. The numerical results shown provide the evolution, along a given local-plate or distortional postbuckling equilibrium path, of the column (1) deformed configuration and (2) relevant displacement profiles and/or stress distributions—mostly for validation purposes, some of them are compared with values yielded by shell finite element analyses performed by means of the code ABAQUS. In order to assess the influence of the member end support conditions, the paper also includes a comparison between the distortional postbuckling behaviors of columns with pinned / free-to-warp and fixed / warping-prevented end sections. Taking full advantage of the GBT unique modal features, all the above results are discussed in great detail and it becomes possible to unveil, explain, and/or shed some new light on several interesting and scarcely known behavioral aspects. In particular, one is able to provide illuminating and structurally (mechanically) meaningful explanations for the qualitative differences exhibited by the local-plate and distortional postbuckling behaviors of plain and stiffened (web and flanges) lipped channel columns. [ABSTRACT FROM AUTHOR]

Stability equations that evaluate the elastic critical load of columns in any type of construction with sidesway uninhibited, partially inhibited, and totally inhibited including the effects of bending and shear deformations are derived in a classical manner. The “modified” shear equation proposed by Timoshenko and Gere is utilized in the derived equations which can be applied to the stability of frames (“unbraced,” “partially braced,” and “totally braced”) with rigid, semirigid, and simple connections. The complete column classification and the corresponding three stability equations overcome the limitations of current methods. Simple criteria are presented that define the concept of minimum lateral bracing required by columns and plane frames to achieve nonsway buckling mode. Four examples are presented that demonstrate the effectiveness and accuracy of the proposed stability equations and the importance of shear deformations in columns with relatively low shear stiffness AsG such as in built-up metal columns or columns made of laminated composites (fiber-reinforced polymers). [ABSTRACT FROM AUTHOR]

Experimental determination of the flexural and torsional properties of pultruded FRP profiles, based on full section and coupon tests, is described. Results for a range of I-profiles indicates that transverse shear moduli, determined from full section three point bending tests, are influenced significantly by localised deformation at the supports. Closed form solutions for the influence of shear deformation on global flexural, torsional and lateral buckling of pultruded FRP profiles have been developed. Parametric studies for a range of I-profiles indicate that shear deformation can reduce flexural and torsional buckling loads by up to 10%. For wide flange I-profiles pre-buckling displacements can increase lateral buckling moments by more than 20% while the influence of shear deformation is relatively minor. [ABSTRACT FROM AUTHOR]

The mechanical behavior of hybrid beams made of an isotropic core (syntactic foam) and a fiber reinforced plastic skin is investigated theoretically and experimentally. The beams are subjected to three point bending tests. The analytical approach developed is compared to the experimental evidence reported in this study and a very good agreement is found. [ABSTRACT FROM AUTHOR]

Fiber-Reinforced Polymer (FRP) composites are considered to be a potentially promising material for use as pile foundations in harsh environments such as industrial or matine conditions. As with piles made of other materials, FRP piles may buckle under extreme loading situations during their installation by driving orjacking into soils or when they are subjected to permanent superstructure loads. Since FRP composites generally have anisotropic properties, relatively low moduli, and relatively high elastic to shear modulus ratios compared to steel, for example, the shear deformation effect plays a more important role in dictating the buckling loads of FRP piles. In this paper, the Timoshenko beam theory is adopted to derive solutions which include the shear deformation effect for buckling loads of vertically loaded transversely isotropic FRP piles with five different boundary conditions. Parametric studies show that the buckling loads for the cases where the shear defonnation effect is considered are always lower than when the shear deformation effect is ignored. [ABSTRACT FROM PUBLISHER]

• The buckling of laminated shallow arches including the shear deformation is studied. • Analytical solutions are obtained for both fixed and pinned laminated arches. • A specific parameter defining a switch between buckling and pre-buckling is proposed. • The effect of transverse shear strain on the critical buckling load cannot be ignored. This paper is concerned with the nonlinear in-plane buckling of shear deformable laminated composite shallow arches under a uniform radial loading. The virtual work method is used to establish both governing differential equations and buckling equilibrium equations based on the first order shear deformation theory to include the effect of shear deformation for which analytical solutions for both limit point buckling and bifurcation point buckling are derived. A specific parameter that defines the switch between buckling and pre-buckling, limit point buckling and bifurcation buckling is proposed and defined. The effect of shear deformation on the buckling load is discussed in detail. It is observed from typical equilibrium paths of the arch that the shear deformation decreases the critical buckling load of laminated arches and this effect becomes more important and cannot be neglected for fixed arches with slenderness ratio S / r x < 150 and pinned arches with S / r x < 100. Direct comparisons with finite element results demonstrate that the proposed analytical solutions can provide a good prediction for the nonlinear buckling of shallow laminated arches under a uniform radial loading. [ABSTRACT FROM AUTHOR]

A new theory for the buckling and nonlinear analysis of circular arches with shear deformation is developed, the theory is based on the variational principle in which the nonlinear strain energy of all stress components are included. For arches subjected to a uniform constant-directed radial load, internal forces and deformations including shear deformation are determined, then the buckling of arches only considering uniform axial forces and incorporating the effect of pre-buckling deformations and all internal forces are respectively carried out, with the results of the latter are fully validated by FE predictions, thus the proposed theory is verified. The buckling of arches with shear deformation is then viewed as an interactive buckling between flexural and shear buckling, Simplified formulas for the critical loads and the critical axial forces of arches for 4 types of loads are presented, the comparison shows excellent agreement between the proposed formulas and FE analysis. • A buckling theory of pin-ended circular arches including shear deformation is developed. • The variety of internal forces including shear deformation is determined. • Analytical solutions of the critical load considering the effect of shear deformation is derived. • The buckling of arches is viewed as an interactive buckling between flexural and shear buckling. [ABSTRACT FROM AUTHOR]