Free vibration of axially loaded rectangular composite beams using refined shear deformation theory

Free vibration of axially loaded rectangular composite beams with arbitrary lay-ups using refined shear deformation theory is presented. It accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply axial-flexural coupled vibration. A displacement-based one-dimensional finite element model is developed to solve the problem. Numerical results are obtained for rectangular composite beams to investigate effects of fiber orientation and modulus ratio on the natural frequencies, critical buckling loads and load–frequency curves as well as corresponding mode shapes.     

Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects

This article presents the buckling analysis of isotropic nanoplates using the two variable refined plate theory and nonlocal small scale effects. The two variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Nonlocal governing equations of motion for the nanoplate are derived from the principle of virtual displacements. The closed-form solution for buckling load of a simply supported rectangular nanoplate subjected to in-plane loading has been obtained by using the Navier’s method. Numerical results obtained by the present theory are compared with available exact solutions in the literature. The effect of nonlocal scaling parameter, mode numbers and aspect ratios of the nanoplates on buckling load are investigated and discussed in detail in the present work. It can be concluded that the present theory, which does not require shear correction factor, is not only simple but also comparable to the first-order and higher order shear deformable theory.     

Locking-free finite elements for shear deformable orthotropic thin-walled beams

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are presented. The assumed kinematical model is based on Timoshenko–Reissner theory so as to take shear strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms arise between bending in the two principal planes and between bending and torsion. The adopted model holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared with the available numerical examples. In particular, some shape functions are obtained from ‘modified’ Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections are performed to verify convergence rate and accuracy of the proposed formulations, especially in the presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models are compared with results given by plate-shell models. Copyright © 2007 John Wiley & Sons, Ltd.     

An accurate two-node finite element for shear deformable curved beams

An accurate two-node (three degrees of freedom per node) finite element is developed for curved shear deformable beams. The element formulation is based on shape functions that satisfy the homogeneous form of the partial differential equations of motion which renders it free of shear and membrane locking. The element is demonstrated to converge to the results obtained from a shear deformable straight beam when the beam becomes shallower. Numerical examples were performed to demonstrate the accuracy and efficiency with respect to previously published formulations. © 1998 John Wiley & Sons, Ltd.     

Super convergent shear deformable finite elements for stability analysis of composite beams

The super convergent finite beam elements are newly presented for the spatially coupled stability analysis of composite beams. For this, the theoretical model applicable to the thin-walled laminated composite I-beams subjected to the axial force is developed. The present element includes the transverse shear and the warping induced shear deformation by using the first-order shear deformation beam theory. The stability equations and force–displacement relationships are derived from the principle of minimum total potential energy. The explicit expressions for the seven displacement parameters are then presented by applying the power series expansions of displacement components to simultaneous ordinary differential equations. Finally, the element stiffness matrix is determined using the force–displacement relationships. In order to demonstrate the accuracy and the superiority of the beam element developed by this study, the numerical solutions are presented and compared with the results obtained from other researchers, the isoparametric beam elements based on the Lagrangian interpolation polynomial, and the detailed three-dimensional analysis results using the shell elements of ABAQUS. The effects of shear deformation, boundary condition, fiber angle change, and modulus ratios on buckling loads are investigated in the analysis.     

A C1 finite element including transverse shear and torsion warping for rectangular sandwich beams

A new three‐noded C¹ beam finite element is derived for the analysis of sandwich beams. The formulation includes transverse shear and warping due to torsion. It also accounts for the interlaminar continuity conditions at the interfaces between the layers, and the boundary conditions at the upper and lower surfaces of the beam. The transverse shear deformation is represented by a cosine function of a higher order. This allows us to avoid using shear correction factors. A warping function obtained from a three‐dimensional elasticity solution is used in the present model. Since the field consistency approach is accounted for interpolating the transverse strain and torsional strain, an exact integration scheme is employed in evaluating the strain energy terms. Performance of the element is tested by comparing the present results with exact three‐dimensional solu‐tions available for laminates under bending, and the elasticity three‐dimensional solution deduced from the de Saint‐Venant solution including both torsion with warping and bending. In addition, three‐dimensional solid finite elements using 27 noded‐brick elements have been used to bring out a reference solution not available for sandwich structures having high shear modular ratio between skins and core. A detailed parametric study is carried out to show the effects of various parameters such as length‐to‐thickness ratio, shear modular ratio, boundary conditions, free (de Saint‐Venant) and constrained torsion. Copyright © 1999 John Wiley & Sons, Ltd.     

A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory

In the present research, free vibration study of functionally graded (FG) nanobeams with graded nonlocality in thermal environments is performed according to the third-order shear deformation beam theory. The present nanobeam is subjected to uniform and nonlinear temperature distributions. Thermo-elastic coefficients and nonlocal parameter of the FG nanobeam are graded in the thickness direction according to power-law form. The scale coefficient is taken into consideration implementing nonlocal elasticity of Eringen. The governing equations are derived through Hamilton's principle and are solved analytically. The frequency response is compared with those of nonlocal Euler–Bernoulli and Timoshenko beam models, and it is revealed that the proposed modeling can accurately predict the vibration frequencies of the FG nanobeams. The obtained results are presented for the thermo-mechanical vibrations of the FG nanobeams to investigate the effects of material graduation, nonlocal parameter, mode number, slenderness ratio, and thermal loading in detail. The present study is associated to aerospace, mechanical, and nuclear engineering structures that are under thermal loads.     

Torsional buckling and postbuckling of double-walled carbon nanotubes by nonlocal shear deformable shell model

This paper presents an investigation on the buckling and postbuckling of double-walled carbon nanotubes (CNTs) subjected to torsion in thermal environments. The double-walled carbon nanotube is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects and van der Waals interaction forces. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent and are obtained from molecular dynamics (MD) simulations. The small scale parameter _e0a$e_{0}a$ is estimated by matching the buckling torque of CNTs observed from the MD simulation results with the numerical results obtained from the nonlocal shear deformable shell model. The results show that buckling torque and postbuckling behavior of CNTs are very sensitive to the small scale parameter _e0a$e_{0}a$. The results reveal that the size-dependent and temperature-dependent material properties have a significant effect on the torsional buckling and postbuckling behavior of both single-walled and double-walled CNTs.     

A new shear and normal deformation theory for isotropic,transversely isotropic,laminated composite and sandwich plates

In the present study, a sinusoidal shear and normal deformation theory taking into account effects of transverse shear as well as transverse normal is used to develop the analytical solution for the bidirectional bending analysis of isotropic, transversely isotropic, laminated composite and sandwich rectangular plates. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and traction free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. The displacement field uses sinusoidal function in terms of thickness coordinate to include the effect of transverse shear and the cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal. The kinematics of the present theory is much richer than those of the other higher order shear deformation theories, because if the trigonometric term is expanded in power series, the kinematics of higher order theories are implicitly taken into account to good deal of extent. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. The Navier solution for simply supported laminated composite plates has been developed. Results obtained for displacements and stresses of simply supported rectangular plates are compared with those of other refined theories and exact elasticity solution wherever applicable.     

A simple hybrid stress element for shear deformable plates

A new quadrilateral 4‐node element for shear deformable plates is developed based on the hybrid stress formulation. The element is designed to be simple, stable, free of locking and to pass all the patch tests. To this purpose, special attention is devoted to select displacement and stress approximations. The standard displacement interpolation is enhanced by linking the transverse displacement to the nodal rotations and an appropriate stress approximation is rationally derived. In particular, the assumed stress approximation is equilibrated within each element, co‐ordinate invariant and ruled by the minimum number of parameters. Excellent element performance is demonstrated by a wide experimental evaluation. Copyright © 2005 John Wiley & Sons, Ltd.     

A Galerkin formulation for shear deformable plate bending dynamics

A Galerkin boundary element formulation for shear deformable plate bending dynamics is developed. The formulation makes use of the static fundamental solutions for the weighted residual integral equations. The domain integrals carrying the inertia terms and generic static loads are considered as body forces and approximated with boundary values using the dual reciprocity method. The load is modelled as a series of impact loads of time varying intensity and moving in space in a predetermined path. The formulation was implemented and tested solving a benchmark problem. The results are compared with finite element solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Bending and free vibration analysis of shear deformable laminated composite beams by finite element method

In the present investigation a higher-order shear deformation theory and the conventional first-order theory are used to develop a finite element method to analyse accurately the bending and free vibration behaviour of laminated composite beams, using nine-noded isoparametric elements. The higher-order theory assumes all the displacement components, u, v and w, which contain variation up to a cubic power of z. The effects of various parameters such as fibre orientation, stacking sequence, span-to-thickness ratio and support condition on the non-dimensionalised deflections, stresses and fundamental frequencies are investigated. Cases where only the higher-order theory is likely to yield accurate results are highlighted.     

Dynamic analysis of thin-walled and variable open section beams with shear flexibility

The equations of motion of thin-walled beams with open cross-section, considering the effects of shear flexibility and rotatory inertia in the stress resultants, as well as variable cross-sectional properties are presented using a state variables approach. These equations are based on Vlasov's theory of thin-walled beams, which is modified to include the effects indicated above. The resulting equations are used in the determination of the natural frequencies of an open cross-section prestressed concrete beam. Comparisons with other formulations are also presented employing the waves dispersion equation.     

A new boundary element formulation for shear deformable shells analysis

A new domain‐boundary element formulation to solve bending problems of shear deformable shallow shells having quadratic mid‐surface is presented. By regrouping all the terms containing shells curvature and external loads together in equilibrium equation, the formulation can be formed by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. The boundary is discretized into quadratic isoparametric element and the domain is discretized using constant cells. Several examples are presented, and the results shows a good agreement with the finite element method. Copyright © 1999 John Wiley & Sons, Ltd.     

A nonlocal continuum theory for diatomic elastic solids

In this study, taking the effects of long range intermolecular forces into account, a mechanical model for diatomic solids is presented. The model we use is based upon the assumption that a diatomic solid may be considered to consist of two simple elastically interacting media which are initially overlapping but may have relative motions at a later time $\text{t}$. Based on this assumption, the kinematics and balance laws, i.e. conservation of mass, balance of momenta, conservation of energy and equation of entropy production are presented. The nonlinear and linear constitutive equations and the discussion of some special cases are introduced in Sections 3 and 4, respectively. To illustrate the theory, in the final part, the propagation of time harmonic waves in such a medium is studied, and by comparing present result with those of lattice dynamics, the explicit form of influence functions is obtained.     

The shear coefficient for quartz crystal of rectangular crosssection in Timoshenko's beam theory

In this paper, we derive the shear coefficient K for quartz crystal of rectangular cross section in Timoshenko's beam theory of three-dimensional (3-D) elasticity. The theoretical result gives K as a function of the transformed elastic stiffness constants C₂₂ ^', cˆ₄₄, C₆₆'. For an isotropic material it agrees completely with that derived by G.R. Cowper (1966). In addition, a value of K and its temperature coefficient versus cut angle is calculated so that K has a value of 0.796-0.867 and its temperature coefficient of the first order varies from -15.3×10 ^-6/°C to +13.2×10^-6/°C, when a Z-plate of quartz crystal is rotated with a cut angle &thetas; of 0-180° about the x axis. The values of K are then compared with those for isotropic materials obtained by other authors     

Combined micro- and nano-scale surface textures for enhanced near-infrared light harvesting in silicon photovoltaics

As silicon photovoltaics evolve towards thin-wafer technologies, efficient optical absorption for the near-infrared wavelengths has become particularly challenging. In this work, we present a solution that employs combined micro- and nano-scale surface textures to increase light harvesting in the near-infrared for crystalline silicon photovoltaics, and discuss the associated antireflection and scattering mechanisms. The surface textures are achieved by uniformly depositing a layer of indium-tin-oxide nanowhiskers on micro-grooved silicon substrates using electron-beam evaporation. The nanowhiskers facilitate optical transmission in the near-infrared by functioning as impedance matching layers with effective refractive indices gradually varying from 1 to 1.3. Materials with such unique refractive index characteristics are not readily available in nature. As a result, the solar cell with combined textures achieves over 90% external quantum efficiencies for a broad wavelength range of 460-980 nm, which is crucial to the development of advanced thin-substrate silicon solar cells.