Vibration and buckling of composite beams using refined shear deformation theory

Vibration and buckling analysis of composite beams with arbitrary lay-ups using refined shear deformation theory is presented. The theory accounts for the parabolical variation of shear strains through the depth of beam. Three governing equations of motion are derived from Hamilton's principle. The resulting coupling is referred to as triply coupled vibration and buckling. A two-noded C1 beam element with five degree-of-freedom per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are obtained for composite beams to investigate effects of fibre orientation and modulus ratio on the natural frequencies, critical buckling loads and corresponding mode shapes

Static behavior of composite beams using various refined shear deformation theories

Static behavior of composite beams with arbitrary lay-ups using various refined shear deformation theories is presented. The developed theories, which do not require shear correction factor, account for parabolical variation of shear strains and consequently shear stresses through the depth of the beam. In addition, they have strong similarity with Euler–Bernoulli beam theory in some aspects such as governing equations, boundary conditions, and stress resultant expressions. A two-noded C1 finite element with six degree-of-freedom per node which accounts for shear deformation effects and all coupling coming from the material anisotropy is developed to solve the problem. Numerical results are performed for symmetric and anti-symmetric cross-ply composite beams under the uniformly distributed load and concentrated load. The effects of fiber angle and lay-ups on the shear deformation parameter and extension-bending-shear-torsion response are investigated

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

A nonlocal sinusoidal plate model for micro/nanoscale plates

A nonlocal sinusoidal plate model for micro/nanoscale plates is developed based on Eringen’s nonlocal elasticity theory and sinusoidal shear deformation plate theory. The small scale effect is considered in the former theory while the transverse shear deformation effect is included in the latter theory. The proposed model accounts for sinusoidal variations of transverse shear strains through the thickness of the plate, and satisfies the stress-free boundary conditions on the plate surfaces, thus a shear correction factor is not required. Equations of motion and boundary conditions are derived from Hamilton’s principle. Analytical solutions for bending, buckling, and vibration of simply supported plates are presented, and the obtained results are compared with the existing solutions. The effects of small scale and shear deformation on the responses of the micro/nanoscale plates are investigated

Post-buckling of functionally graded microplates under mechanical and thermal loads using isogeometric analysis

The present study uses the isogeometric analysis (IGA) to investigate the post-buckling behavior of functionally graded (FG) microplates subjected to mechanical and thermal loads. The modified a strain gradient theory with three length scale parameters is used to capture the size effect. The Reddy third-order shear deformation plate theory with the von Kámán nonlinearity (i.e., small strains and moderate rotations) is employed to describe the kinematics of the microplates. Material variations in the thickness direction of the plate are described using a rule of mixtures. In addition, material properties are assumed to be either temperature-dependent or temperature-independent. The governing equations are derived using the principle of virtual work, which are then discretized using the IGA approach, whereby a C2-continuity requirement is fulfilled naturally and efficiently. To trace the post-buckling paths, Newton’s iterative technique is utilized. Various parametric studies are conducted to examine the influences of material variations, size effects, thickness ratios, and boundary conditions on the post-buckling behavior of microplates

Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis

This study aims to investigate the postbuckling response of functionally graded (FG) nanoplates by using the nonlocal elasticity theory of Eringen to capture the size effect. In addition, Reddy’s third-order shear deformation theory is adopted to describe the kinematic relations, while von Kaman’s assumptions are used to account for the geometrical nonlinearity. In order to calculate the effective material properties, the Mori-Tanaka scheme is adopted. Governing equations are derived based on the principle of virtual work. Isogeometric analysis (IGA) is employed as a discretization tool, which is able to satisfy the C1-continuity demand efficiently. The Newton-Raphson iterative technique with imperfection is employed to trace the postbuckling paths. Various numerical studies are carried out to examine the influences of gradient index, nonlocal effect, ratio of compressive loads, boundary condition, thickness ratio and aspect ratio on the postbuckling behaviour of FG nanoplates

Size-dependent behaviour of functionally graded sandwich microbeams based on the modified couple stress theory

Abstract Static bending, buckling and free vibration behaviours of size-dependent functionally graded (FG) sandwich microbeams are examined in this paper based on the modified couple stress theory and Timoshenko beam theory. To avoid the use of a shear correction factor, equilibrium equations were used to compute the transverse shear force and shear stress. Two types of sandwich beams were considered: (1) homogeneous core and FG skins and (2) FG core and homogeneous skins. Numerical results were presented to illustrate the small scale effects on the behaviours of FG sandwich beams. The results reveals that the inclusion of the size effects results in an increase in the beam stiffness, and consequently, leads to a reduction of deflections and stresses and an increase in natural frequencies and critical buckling loads. Such effects are more pronounced when the beam depth was small, but they become negligible with the increase of the beam depth

A quasi-3D hyperbolic shear deformation theory for functionally graded plates

A quasi-3D hyperbolic shear deformation theory for functionally graded plates is developed. The theory accounts for both shear deformation and thickness-stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate without requiring any shear correction factor. The benefit of the present theory is that it contains a smaller number of unknowns and governing equations than the existing quasi-3D theories, but its solutions compare well with 3D and quasi-3D solutions. Equations of motion are derived from the Hamilton principle. Analytical solutions for bending and free vibration problems are obtained for simply supported plates. Numerical examples are presented to verify the accuracy of the present theory

Free vibration of axially loaded composite beams using a four-unknown shear and normal deformation theory

This paper presents free vibration of composite beams under axial load using a four-unknown shear and normal deformation theory. The constitutive equation is reduced from the 3D stress-strain relations of orthotropic lamina. The governing differential equations of motion are derived using the Hamilton’s principle. A two-node C1 beam element is developed by using a mixed interpolation with linear and Hermite-cubic polynomials for unknown variables. Numerical results are computed and compared with those available in the literature and commercial finite element software (ANSYS and ABAQUS). The comparison study illustrates the effects of normal strain, lay-ups and Poisson’s ratio on the natural frequencies and load-frequency curves of composite beams