Transverse Shear Effect on Flutter of Composite Panels

The effect of transverse shear deformation on the supersonic flutter of composite panels has been investigated using the finite element method. First‐order shear‐deformation laminated‐plate theory and quasi‐steady aerodynamic theory are employed for the analysis. The total displacement of the plate is expressed as the sum of the displacement due to bending and the displacement due to shear deformation. Thus, the aerodynamic pressure induced by the plate motion is also the sum of the pressure induced by bending deformation and the pressure induced by shear deformation. Numerical results show that the transverse shear deformation may have a significant effect on the flutter boundary if aerodynamic damping were small or neglected in the determination of flutter boundary.     

Plastic Deformations of Impulsively Loaded, Rigid-Plastic Beams

Rigid body dynamics is used to determine the deformation of a fixed-end, rigid-plastic beam subjected to uniformly distributed impulsive loading. The proposed solution methodology allows calculations of deformations at plastic hinges and can be used to establish rigid-plastic fracture criteria for rigid-plastic beams. Unlike previous solutions to this problem, rotary inertia and the shear deformations at the support are considered. The solution for beam deformations is described in three phases: shear, bending, and membrane. Each phase ends when the corresponding component of the strain rate vector vanishes. The initial shear phase is completed when the transverse shear velocity at the support vanishes. The beam then undergoes only rigid body rotation and axial stretching at plastic hinges in the bending phase. The bending phase ends when the angular velocity vanishes. In the membrane phase, the beam acts like a string until the transverse velocity vanishes. It has been found that beams subjected to low impulse velocity attain permanent deformation in the bending phase, while beams subjected to high impulse velocity reach permanent deformation in the membrane phase. The predictions of the beam deflections using the proposed methodology are within 15% of the experimental results.     

Mixed-Mode Fracture of Hybrid Material Bonded Interfaces under Four-Point Bending

A combined analytical and experimental approach is presented to characterize mixed-mode fracture of hybrid material bonded interfaces under four-point bending load, and closed-form solutions of compliance and energy release rate (ERR) of the mixed-mode fracture specimens are provided. The transverse shear deformations in each sublayer of bimaterial bonded beams are included by modeling the specimen as individual Timoshenko beams, and the effect of interface crack-tip deformation on the compliance and ERR are taken into account by applying the interface deformable bilayer beam theory (flexible-joint model). The higher accuracy of the present analytical solutions for both the compliance and ERR of mixed-mode fracture specimens is manifested by comparing them with the solutions predicted by the conventional beam theory (CBT) and finite-element analysis (FEA). As an application example, the fracture of wood–fiber-reinforced plastic (FRP) bonded interface is experimentally evaluated by using mixed-mode fracture specimens [i.e., four-point asymmetric end-notched flexure (4-AENF) and four-point mixed-mode bending (4-MMB)], and the corresponding values of critical ERRs are obtained. Comparisons of the compliance rate change and the resulting critical ERR based on the CBT, the present theoretical model, and FEA demonstrate that the crack-tip deformation plays an important role in accurately characterizing the mixed-mode fracture toughness of hybrid material bonded interfaces under four-point bending load.     

Finite Beam Element Considering Shear-Lag Effect in Box Girder

A finite-element method considering the interaction of the bending and shear-lag deformation of a box girder was established. Meanwhile a shear-lag-induced stiffness matrix was defined. The stiffness matrices considering the effect of the shear lag were deduced. At each node of the beam element, two shear-lag degrees of freedom were used as boundary conditions for the box girders. The proposed formulations were then applied to analyze the effects of the shear lag on the deflection, the internal forces, and the shear-lag coefficients in the simply supported cantilever and continuous box beams under uniformly distributed and concentrated loads. The numerical results obtained using the proposed procedure were in good agreement with those using the finite-shell-element method, the finite-stringer method, the analytical method based on the variational principle, and the model tests. The proposed method is reliable and more effective for the analysis of the shear lag in the actual box-girder structures.     

Bending, Buckling, and Vibration of Micro/Nanobeams by Hybrid Nonlocal Beam Model

The hybrid nonlocal Euler-Bernoulli beam model is applied for the bending, buckling, and vibration analyzes of micro/nanobeams. In the hybrid nonlocal model, the strain energy functional combines the local and nonlocal curvatures so as to ensure the presence of small length-scale parameters in the deflection expressions. Unlike Eringen’s nonlocal beam model that has only one small length-scale parameter, the hybrid nonlocal model has two independent small length-scale parameters, thereby allowing for a more flexible and accurate modeling of micro/nanobeamlike structures. The equations of motion of the hybrid nonlocal beam and the boundary conditions are derived using the principle of virtual work. These beam equations are solved analytically for the bending, buckling, and vibration responses. It will be shown herein that the hybrid nonlocal beam theory could overcome the paradoxes produced by Eringen’s nonlocal beam theory such as vanishing of the small length-scale effect in the deflection expression or the surprisingly stiffening effect against deflection for some classes of beam bending problems.     

Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams under Electrothermomechanical Loads

A new efficient electromechanically coupled geometrically nonlinear (of von Karman type) zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. The deflection is approximated as piecewise quadratic to explicitly account for the transverse normal strain due to thermal and electric fields. The longitudinal displacement is approximated as a combination of third order global variation and a layerwise linear variation. The shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are used to formulate the theory in terms of three primary displacement variables. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact two-dimensional (2D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 2D solution which neglects the prebuckling transverse strain effect.     

Spatial Stability of Shear Deformable Nonsymmetric Thin-Walled Curved Beams: A Centroid-Shear Center Formulation

An improved shear deformable curved beam theory to overcome the drawback of currently available beam theories is newly proposed for the spatially coupled stability analysis of thin-walled curved beams with nonsymmetric cross sections. For this, the displacement field is introduced considering the second order terms of semitangential rotations. Next the elastic strain energy is newly derived by using transformation equations of displacement parameters and stress resultants and considering shear deformation effects due to shear forces and restrained warping torsion. Then the potential energy due to initial stress resultants is consistently derived with accurate calculation of the Wagner effect. Finally, equilibrium equations and force–deformation relations are obtained using a stationary condition of total potential energy. The closed-form solutions for in-plane and out-of-plane buckling of curved beams subjected to uniform compression and pure bending are newly derived. Additionally, finite-element procedures are developed by using curved beam elements with arbitrary thin-walled sections. In order to illustrate the accuracy and the practical usefulness of this study, closed-form and numerical solutions for spatial buckling are compared with results by available references and ABAQUS’ shell elements.     

Bending Solutions of Axisymmetric Levinson Plates in Terms of Corresponding Kirchhoff Solutions

This paper presents exact axisymmetric bending solutions of circular and annular plates based on the higher-order plate theory of Levinson. The solutions are displayed in terms of the corresponding Kirchhoff (or classical thin) plate solutions. These Kirchhoff-Levinson bending relationships are derived using the mathematical similarity of the governing equations of the two plate theories and the basis of load equivalence. The relationships allow one to readily deduce the more accurate Levinson plate solutions that account for the effect of transverse shear deformation, without having to solve the more complicated Levinson plate equations.     

On the Comparison of Timoshenko and Shear Models in Beam Dynamics

The classical Timoshenko beam model and the shear beam model are often used to model shear building behavior both for stability or dynamic analysis. This technical note questions the theoretical relationship between both models for large values of bending to shear stiffness parameter. The simply supported beam is analytically studied for both models. Asymptotic solutions are obtained for large values of bending to shear stiffness parameter. In the general case, it is proven that the shear beam model cannot be deduced from the Timoshenko model, by considering large values of bending to shear stiffness parameter. This is only achieved for specific geometrical parameter in the present example. As a conclusion, the capability of the shear model to approximate Timoshenko model for large values of bending to shear stiffness parameter is firmly dependent on the material and geometrical characteristics of the beam section and on the boundary conditions.     

Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

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.     

Bending Solutions of Sectorial Mindlin Plates from Kirchhoff Plates

This study presents exact relationships between the bending solutions of sectorial plates based on the Kirchhoff (or classical thin) plate theory and the Mindlin plate theory. While the former plate theory neglects the effect of transverse shear deformation, the latter theory allows for this effect, which becomes significant when dealing with thick plates and sandwich plates. The considered sectorial plates have simply supported radial edges, while the circular curved edge may be either simply supported, or clamped or free. The availability of such relationships allow easy conversion of the existing Kirchhoff sectorial plate solutions into the corresponding Mindlin solutions, thus bypassing the need to solve the more complicated bending equations of the Mindlin plates. The use of the relationships is illustrated using some sectorial plate examples, and sample solutions obtained were checked with previous researchers' results and those computed from the software ABAQUS.     

In-Plane Free Vibration of Symmetric Cross-Ply Laminated Circular Bars

A parametric study is performed to investigate influences of the opening angles, the slenderness ratios, the material types, the boundary conditions, and the thickness-to-width ratios of the cross section on the in-plane natural frequencies of symmetric cross-ply laminated circular composite beams. Governing equations are obtained based on the classical beam theory. The transfer matrix method is successfully applied to calculate exact natural frequencies with the help of an effective numerical algorithm, which was previously used for isotropic materials. The effects of the shear deformation, the axial deformation, and the rotary inertia are included in the formulation based on the first-order shear deformation theory. The physical system is considered as a continuous system. To verify the present theory, two examples are worked out for straight beams. A quite good agreement is observed with the reported results.     

Fracture Behavior of Multidirectional DCB Specimen: Higher-Order Beam Theories

Mathematical models, for the stress analysis of symmetric multidirectional double cantilever beam (DCB) specimen using classical beam theory, first and higher-order shear deformation beam theories, have been developed to determine the Mode I strain energy release rate (SERR) for symmetric multidirectional composites. The SERR has been calculated using the compliance approach. In the present study, both variationally and nonvariationally derived matching conditions have been applied at the crack tip of DCB specimen. For the unidirectional and cross-ply composite DCB specimens, beam models under both plane stress and plane strain conditions in the width direction are applicable with good performance where as for the multidirectional composite DCB specimen, only the beam model under plane strain condition in the width direction appears to be applicable with moderate performance. Among the shear deformation beam theories considered, the performance of higher-order shear deformation beam theory, having quadratic variation for transverse displacement over the thickness, is superior in determining the SERR for multidirectional DCB specimen.     

Analysis of Thick Circular Plates Undergoing Large Deflections

This investigation considers the effect of transverse shear deformation on bending of the axisymmetrically loaded isotropic and orthotropic circular and annular plates undergoing large deflection. The analysis treats the nonlinear terms of lateral displacement as fictitious loads acting on the plate. The solution of a von Kármán‐type plate is, therefore, reduced to a plane problem in elasticity and a linear plate‐bending problem. Results are presented for simply supported and clamped plates and are in good agreement with the available solutions. For plates considered in this study, the influence of shear deformation on lateral displacement becomes more significant as the orthotropic parameter increases. The linear and nonlinear solutions for orthotropic plates deviate at a low value of the maximum deflection‐to‐thickness ratio (w/h). Consequently, the extent of w/h within which the small‐deflection theory is applicable to orthotropic plates is much lower than the value of about 0.4 typically used for isotropic plates, and it depends, in general, on the degree of orthotropy. The technique employed in this study is well suited for the analysis of nonlinear plate problems.     

Buckling of Delaminated Composite Beams with Shear Deformation Effect

A general 1D model of composite delaminated beams with shear deformation effect is derived for buckling behavior. The constitutive models of composite laminated beams are derived from the classical 2D laminate theory. The present cylindrical bending models can be used—with much greater accuracy than their well-known plane-strain and plane-stress counterparts—as upper and lower bounds toward one of which the behavior tends, depending on the width-to-length ratio. The analysis is based on the first-order Timoshenko-Mindlin kinematic approach. The differential equations are solved with the aid of a specially developed, very efficient interlaced finite-difference scheme eliminating the “shear locking” phenomenon. A parametric study of the shear deformation effect associated with various constitutive models is carried out for angle-ply delaminated laminate. It was found that the most significant difference between the models is associated with the mix of local and global modes.     

Observations on Higher‐Order Beam Theory

A parabolic shear‐deformation beam theory assuming a higher‐order variation for axial displacement has been recently presented. In this theory, the axial displacement variation can be selected so that it results in a suitable admissible transverse shear‐strain variation across the depth of the beam. This paper examines several transverse shear‐strain variations that can go with the aforementioned higher‐order theory. Apart from the usual simple parabolic variation, six other shear‐strain variations are considered: the sinusoidal variation, cubic, quartic, quintic, and sixth‐order polynomials. All these variations for transverse shear‐strain satisfy the requirement that the shear strain be zero at the extreme fibers (z? = ?±h/2) and nonzero elsewhere along the depth of the beam. Comparison of the results from this paper with results from others show that the simple parabolic distribution for transverse shear strain gives most accurate results. Also, Timoshenko's theory (with a shear factor of five‐sixths) and the current formulation which uses the parabolic shear‐strain distribution, give identical values for deflections.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

Measurement of the Timoshenko Shear Stiffness. II: Effect of Transverse Compressibility

This is the second of two papers devoted to the issue of measuring the Timoshenko shear stiffness of thin-walled composite beams. In the first paper, the effect of warping on the effective Timoshenko shear stiffness, as measured through bending tests, was studied. The bending test was simulated using finite-element analysis, and the results indicated that the warping effect was minimal. On the other hand, the evidence suggests that transverse flexibility may have a significant influence on the effective Timoshenko shear stiffness, decreasing the effective shear stiffness at shorter test spans. The purpose of the present study is to further investigate this effect and to explore the use of a sandwich theory to predict the measurement error. A higher-order sandwich theory, which captures the transverse strain at concentrated loads and supports, is applied to a commercially available thin-walled composite beam. The results indicate that the sandwich model does capture the decrease in the effective shear stiffness at short spans, and the dependence of the shear stiffness on span-to-depth ratio is similar to that calculated in the first paper, using the finite-element method.     

High-Accuracy Analysis of Beams of Bimodulus Materials

Recently, several authors have treated the problems of bending of beams of bimodulus materials. The present paper, applies Levinson beam theory, which includes shear deformation and warping of the cross section, to bending analysis of thick rectangular beams with bimodulus materials. Many numerical results are obtained by use of the transfer matrix approach and compared with the methods of Bernoulli-Euler beam theory, Timoshenko beam theory, and Levinson beam theory with various boundaries. Also, the neutral-surface location and displacements for beams of bimodulus materials are calculated.     

Transverse Shear Stiffness of Composite Honeycomb Core with General Configuration

Based on the zeroth-order approximation of a two-scale asymptotic expansion, equivalent elastic shear coefficients of periodic structures can be evaluated via the solution of a local function τklij(y), and the homogenization process reduces to solving the local function τklij(y) by invoking local periodic boundary conditions. Then, effective transverse shear stiffness properties can be analytically predicted by reducing a local problem of a given unit cell into a 2D problem. In this paper, an analytical approach with a two-scale asymptotic homogenization technique is developed for evaluation of effective transverse shear stiffness of thin-walled honeycomb core structures with general configurations, and the governing 3D partial differential equations are solved with the assumptions of free warping constraints and constant variables through the core wall thickness. The explicit formulas for the effective transverse shear stiffness are presented for a general configuration of honeycomb core. A detailed study is given for three typical honeycomb cores consisting of sinusoidal, tubular, and hexagonal configurations, and their solutions are validated with existing equations and numerical analyses. The developed approach with certain modifications can be extended to other sandwich structures, and a summary of explicit solutions for the transverse shear stiffness of common honeycomb core configurations is provided. The lower bound solution provided in this study is a reliable approximation for engineering design and can be efficiently used for quick evaluation and optimization of general core configurations. The upper bound formula, based on the assumption of uniform shear deformation, is also given for comparison. Further, it is expected that with appropriate construction in the displacement field, the more accurate transverse stiffness can be analytically attained by taking into account the effect due to the face-sheet constraints.