Journal of Engineering Mechanics

An analytical study of local buckling of discrete laminated plates or panels of fiber-reinforced plastic structural shapes is presented. Two cases of composite plate analyses with two opposite edges simply supported and the other two opposite edges either both rotationally restrained or one rotationally restrained and the other free, are studied. Generic loading cases with combined linearly varying axial and in-plane shear loading are considered. A variational formulation of the Ritz method is used to establish the eigenvalue problem for the local buckling behavior, and explicit expressions for predictions of the plate buckling stress resultants, in terms of the rotationally restrained stiffness, the plate aspect ratio, and the ratios of applied stress resultants, are developed. Based on different boundary and loading conditions, simple and explicit local buckling solutions for several special cases are further reduced. Validity of the explicit solutions presented is demonstrated by a good agreement of comparisons between the present predictions and available solutions in the literature. Parametric studies are further conducted, and the influences of several parameters such as the rotationally restrained stiffness, biaxial loading stress ratio, material flexural orthotropy, and linearly varying axial loading stress gradient on the local buckling stress resultants are discussed.

An analytical pilot model for interactive buckling in sandwich struts with cores made from a functionally graded material based on total potential energy principles is presented. Using a Timoshenko beam approach, a system of nonlinear differential and integral equations is derived that predicts critical and secondary instabilities. These are validated against numerical simulations performed within the commercial finite-element package Abaqus. Good agreement is found, and this offers encouragement for more elaborate models to be devised that can account for face-core delamination—a feature where functionally graded materials are known to offer distinct advantages.

This paper presents an efficient stochastic buckling model for studying the structural reliability of layered composite columns with interlayer slip between the layers and random material and loading parameters. The model is based on the exact buckling model, response surface method, and Monte Carlo simulations. The probability of failure is investigated for a different number of random variables, sample points, and various degrees of response surfaces. The results show that the probability of failure is considerably affected by the type (deterministic or probabilistic) of the loading and its distribution.

The effect of fiber orientation is studied on the failure load of a laminated curved panel subjected to uniaxial compression for a combination of simply supported and clamped boundary conditions. The failure modes are specified as buckling and first-ply failure, with the failure load defined as the minimum of these two loads. The panel is taken as a shear-deformable symmetrically laminated angle-ply laminate, and the effect of fiber orientation on the failure load is investigated by considering several laminations consisting of 0, 90, and $\theta °$ ply angles and by determining the failure load for different aspect ratios and panel thicknesses. The best ply angles for each stacking sequence are determined to maximize the failure load and the stacking sequence giving the highest failure load. The numerical results are obtained using an eight-noded shell finite element that avoids the parasitic shear or shear locking problem. It is observed that the rotational restraints at the curved edges have a major effect on the failure load.

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.

This paper presents the formulation of a generalized beam theory (GBT) to analyze the vibration behavior of composite thin-walled prismatic members displaying straight axis, open-section, and arbitrary orthotropy. It accounts for the effects of (1) the cross section in-plane deformation, (2) geometric and material couplings, (3) primary and secondary warping, and (4) rotary inertia. First, the GBT equilibrium equations and boundary conditions are derived, and their terms are physically interpreted, i.e., related to the member mechanical properties. Then, a few remarks on the cross-section mechanical properties appearing in the linear stiffness and inertia terms are presented. Finally, to clarify the concepts involved in the proposed GBT formulation and illustrate its application and capabilities, an in-depth study concerning the local and global vibration behavior of lipped channel members with (1) simply supported end sections and wall cross-ply orthotropy and (2) fixed (clamped) end sections and wall nonaligned orthotropy are presented and discussed in detail.

This paper presents the incorporation of shear deformation effects into a generalized beam theory (GBT) formulation developed to analyze the first-order (linear) and buckling behavior of composite thin-walled members made of laminated plates displaying arbitrary orthotropy, often designated as anisotropic laminates. Unlike other existing beam theories, the proposed GBT formulation incorporates in a unified fashion (1) elastic coupling effects, (2) warping effects, (3) cross-section in-plane deformation, and (4) shear deformation. The main concepts and procedures involved in the currently available GBT are adapted and/or modified to account for the specific aspects associated with shear deformation. In particular, the GBT equilibrium equations and boundary conditions are derived, and their terms are physically interpreted. A lipped channel section is considered to illustrate the performance of a GBT cross-section analysis, namely, the operations required to determine the (additional) set of shear deformation modes. Finally, to clarify the concepts involved in the proposed GBT formulation and illustrate its application and capabilities, two numerical examples are presented and discussed in detail: the first concerns the first-order and buckling behaviors of a lipped channel column exhibiting nonaligned orthotropy; and the second assesses the influence of shear deformation on the buckling behavior of lipped channel columns with cross-ply orthotropy.

Based on Eringen’s nonlocal elasticity theory, a new nonlinear, nonlocal constitutive relation in polar coordinates is presented for axisymmetric bending of annular graphene-like nanoplate. Instead of the common strain gradient theory, an iterative procedure is developed to solve the coupled nonlinear constitutive relations and to express the nonlocal stresses asymptotically in stress gradients with increasing orders. Subsequently, the nonlocal strain energy is formulated similarly to that of gradient elasticity, and the potential energy of external forces is obtained. Because analytical solutions are not available to date, a computational approach is developed to compute the minimum total energy and to establish the bending equilibrium condition of the nonlocal nanoplate. In a mathematically asymptotic manner, the nonlocal bending deflection function is approximated by finite polynomials that satisfy the admissible geometric boundary conditions. A numerical algorithm based on a minimum energy approach is subsequently developed to solve the coefficients of the nonlocal deflection function. To demonstrate the accuracy and computational efficiency, four practical examples with different boundary conditions and external loadings are presented. After verifying numerical convergence with increasing orders of polynomials, the numerical solutions show that the dimensionless maximum deflection decreases with increasing nonlocal effect. The analytical and numerical solutions presented here will assist in behavioral analyses for graphene and graphene-like structures and their performances.

This paper deals with the lateral-torsional buckling of elastic microstructured beams. A linear gradient elasticity approach is presented. The elastic constitutive law is linear for the generalized beam model, expressed with first- and second-order moment variables in a sixth-order dimensional stress-strain space. The boundary conditions (including the higher-order boundary conditions) are derived from application of the variational principle applied to second-grade elastic models. The effect of prebuckling deformation is taken into consideration based on the Kirchhoff-Clebsch theory. Some analytical solutions are presented for a hinged-hinged strip beam. The lateral-torsional buckling moment is sensitive to small length terms inherent in the microstructured constitutive law. It is shown that the gradient parameter tends to increase the critical lateral-torsional buckling moment, a conclusion different from the one obtained with an Eringen-based nonlocal model. This tendency is consistent with that observed for the in-plane stability analysis (based on a second-grade model), for the lateral buckling of a hinged-hinged axially loaded column. The effect of higher-order boundary conditions is also studied with the vanishing of the couple stress components. It is numerically shown, through the resolution of a sixth-order differential equation, that the lateral-torsional buckling moment is not significantly affected by the choice of higher-boundary conditions for sufficiently small ratio between the characteristic length and the total length of the structure. We finally notice that the introduction of the warping effect can be understood as a special case of gradient elasticity theory.

This paper is devoted to the analytical and numerical modeling of the lateral-torsional stability of horizontally layered composite beams. Composite beams are classified as horizontally layered beams with interlayer slip or sandwich beams with a weak shear core. The governing differential equations of the out-of-plane behavior of horizontally layered composite beams are supported by variational arguments. In the theoretical analysis, a distinction is made between the influence of the shear connection at the interface with respect to the in-plane or transversal deformations and to the out-of-plane or lateral deformations, respectively. Some engineering results are presented for a partially composite beam under pure bending moment. In the case of noncomposite in-plane action (orthotropic connection), a simple closed-form solution is derived for the lateral-torsional buckling moment, and it is shown that the exact dimensionless buckling moment depends only on two structural parameters for beams composed of two identical subelements. The results are analogous to those obtained for the in-plane buckling of partially composite or sandwich-type beams, where the buckling moment increases with the stiffness of the shear connection. Prandtl’s valid solution for lateral-torsional buckling of ordinary beams is also found for composite beams in the case of noncomposite action in both the transversal and lateral directions. A generalization of Prandtl’s valid solution for composite beams with partial composite action in the lateral direction and noncomposite action in the transversal direction is derived. It is shown that the lateral-torsional buckling formulas are strongly affected by the kinematics of the connected shear layer. Also, the lateral-torsional buckling of partially composite beams with both in-plane and out-of-plane slip behavior is analyzed using the Rayleigh-Ritz method. This mathematical problem leads to a system of differential equations with nonuniform coefficients. An approximated solution is derived for the isotropic connection with isotropic noncomposite actions, whereas an exact solution is presented for the orthotropic connection with noncomposite in-plane action. Finally, the Rayleigh-Ritz approach is compared with some numerical results associated with the exact resolution of the differential equations with nonuniform coefficients. The Rayleigh-Ritz approach appears to be efficient to capture the main phenomena, including the nonmonotonic dependence of the buckling load to the connection parameter.

Wood I-joists are efficient and lightweight structural members that are well suited for long-span floor and rafter applications. However, because of the thin-wall nature of the web and relatively low stiffness, local web buckling of the joist is an issue that must be considered. To increase the buckling capacity of these joists, investigators at West Virginia Univ. and Univ. of Idaho developed web panels with a sinusoidal geometry, instead of the flat web panels commonly used for prefabricated I-joists. This study presents analytical modeling, experimental testing, and finite-element (FE) analysis of the local buckling capacities of these sinusoidal web panels under compression. The flanges and web are not rigidly connected; thus, this problem can be described as the instability of a sinusoidal shell with two rotationally restrained loaded edges under compression. Using an energy method, the critical compression buckling stresses were obtained in terms of the elastic rotational restraint stiffness. The analytical solution was verified by FE analysis. Compression tests were performed for joists with both flat and sinusoidal webs, each at two different heights, to evaluate the elastic rotational restraining effect and illustrate the increase of the buckling capacities for sinusoidal webs compared with flat webs. Finally, a parametric study was conducted to study the core aspect ratio effect on the buckling load. The method described in this paper can be further extended to determine local buckling capacities of other structural shapes with elastic restraints along the loaded edges.

The buckling response of symmetric laminates that possess strong flexural-twist coupling is studied using different methodologies. Such plates are difficult to analyze because of localized gradients in the mode shape. Initially, the energy method (Rayleigh-Ritz) using Legendre polynomials is employed, and the difficulty of achieving reliable solutions for some extreme cases is discussed. To overcome the convergence problems, the concept of Lagrangian multiplier is introduced into the Rayleigh-Ritz formulation. The Lagrangian multiplier approach is able to provide the upper and lower bounds of critical buckling load results. In addition, mixed variational principles are used to gain a better understanding of the mechanics behind the strong flexural-twist anisotropy effect on buckling solutions. Specifically, the Hellinger-Reissner variational principle is used to study the effect of flexural-twist coupling on buckling and also to explore the potential for developing closed-form solutions for these problems. Finally, solutions using the differential quadrature method are obtained. Numerical results of buckling coefficients for highly anisotropic plates with different boundary conditions are studied using the proposed approaches and compared with finite-element results. The advantages of both the Lagrangian multiplier theory and the variational principle in evaluating buckling loads are discussed. In addition, a new simple closed-form solution is shown for the case of a flexurally anisotropic plate with three sides simply supported and one long edge free.

This paper presents analytical approximate solutions for the initial postbuckling deformation of sandwich beams including transverse shear. The approximate procedure is based on the nonlinear beam equation (with transverse shear included) by coupling the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials; the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation. Analytical approximations to the resulting boundary condition problem are established by combining the methods of Newton linearization and harmonic balance. Unlike the classical method of harmonic balance, the linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of nonlinear algebraic equations, thereby establishing analytical approximate solutions. Illustrative examples are presented for a few typical sandwich construction configurations, and it is shown that the proposed approximate solutions are in excellent agreement with the numerical solution obtained via the shooting method for both small and large angles of rotation.

In this paper, the buckling behavior of generic higher-order shear beam models is investigated in a unified framework. This paper shows that most higher-order shear beam models developed in the literature (polynomial, sinusoidal, exponential shear strain distribution assumptions over the cross section) can be classified in a common gradient elasticity Timoshenko theory, whatever the shear strain distribution assumptions over the cross section. The governing equations of the bending/buckling problem are obtained from a variational approach, leading to a generic sixth-order differential equation. Buckling solutions are presented for usual archetypal boundary conditions such as pinned-pinned, clamped-free, clamped-hinge, and clamped-clamped boundary conditions. The results are then extended to general boundary conditions based on generalized linear elastic connection law including vertical and rotational stiffness boundary conditions. Engineering analytical solutions are derived in a dimensionless format. The model valid for macrostructures is generalized for micro- or nanostructures using the nonlocal integral Eringen’s model. The nonlocal framework is also developed in a variational consistent framework. Buckling solutions are finally presented for the nonlocal higher-order beam/colum models.

The analytical strip method (ASM) is extended in this paper to the buckling analysis of antisymmetric laminated composite plates with bending-extension coupling. A system of three equations of equilibrium governing the buckling response of antisymmetric laminated composite plates is reduced to a single eighth-order partial differential equation, expressed in terms of a displacement function. This equation is then solved analytically to generate the critical value of the in-plane forces ($N_x$, $N_y$) for antisymmetric cross-ply and angle-ply laminated and stiffened composite plates. The effects of various parameters, including plate aspect ratio, plate orthotropicity ratio, beam rigidity, and plate bending-extension coupling on the critical values of $N_x$ and $N_y$ on antisymmetric laminated composite plates are discussed. The results derived using the ASM are presented in tabular and graphical forms and compare very well with ones derived using finite-element analysis.

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.

A slender system with different types of load (conservative and nonconservative) is considered in this paper. The conservative load is the specific load: a generalized load with the force directed toward the positive pole. The nonconservative load (Beck’s load) was generated by a reaction engine. A set of two bolts (characteristic for a specific load), with mutual rotation limited by a rotational spring with a linear characteristic, was taken into account in the system. The boundary problem of the considered systems was formulated on the basis of Hamilton’s principle. The critical loads, both divergence and flutter, the regions of divergence and flutter instability, and the characteristic curves in the plane: load – natural frequency were determined on the basis of the kinetic criterion of stability. Numerical computations were carried out for different values of the parameters of the considered system, such as the rigidity of a rotational spring, parameters connected to the length of the bolts, and the inertia parameters of the translational and rotational mass placed at the loaded end of a column.