Gajbir Singh

A two-noded locking–free shear flexible curved beam element

A new two-noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The element with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representation for radial displacement (w), the displacement field for tangential displacement (u) and section rotation (θ) are determined by employing force-moment and moment-shear equilibrium equations. This results in polynomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quartic polynomial representation for both u and θ for curved element configurations, which reduces to linear and quadratic polynomials for u and θ, respectively, for straight element configuration. These coupled polynomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C0 element is validated for beam having straight/curved geometries over a wide range of slenderness ratios. The results indicates that performance of the element is much superior to other elements of the same class. Copyright

Large-amplitude free vibrations of beams : a discussion on various formulations and assumptions

Various analytical formulations of the problem of large-amplitude free vibrations of simply supported beams with immovable ends based on the Rayleigh-Ritz method with one-term approximations for axial and transverse displacements are presented in this paper. Many controversial points, raised by different investigators at different times, such as whether axial displacements should be included in the strain energy or not, and whether or not the strain-displacement relationship can be linearized, are discussed herein. It is observed that the formulation wherein the axial displacement is neglected and, further, the quadratic term in the strain displacement relation is linearized leads to an equation of motion, which when solved, based on the simple harmonic oscillations assumption, yields exactly the same non-linear frequency as do the perturbation method, the Ritz-Galerkin method and the elliptical integral solution with axial displacement included, no linearization of non-linear terms and without the harmonic oscillations assumption.

Non-linear vibrations of simply supported rectangular cross-ply plates

A method of direct numerical integration of the frequency-ratio expression is proposed to study the non-linear free vibration behaviour of rectangular cross-ply laminates. The proposed method, even with single-term approximations for the admissible functions, yields results that agree very well with the existing perturbation solutions. Non-linear behaviour of the cross-ply laminates is also studied with the harmonic oscillations assumption, by using the conservation of energy and the modal equation. The results are found to be lower and upper bounds to those obtained from the direct numerical integration method. It is also observed that the arithmetic mean of the two solutions with the harmonic oscillations assumption matches very well with that of the direct numerical integration method. Non-linear vibration characteristics are obtained for several configurations of cross-ply plates. Results for orthotropic and isotropic plates are also obtained as special cases.

Free vibration of arches using a curved beam element based on a coupled polynomial displacement field

Abstract The performance of a curved beam finite element with coupled polynomial distributions for normal displacement ( w ) and tangential displacement ( u ) is investigated for in-plane flexural vibration of arches. A quartic polynomial distribution for u is derived from an assumed cubic polynomial field for w using force–moment equilibrium equations. The coupling of these displacement fields makes it possible to express the strain field in terms of only six generalized degrees of freedom leading to a simple two-node element with three degrees of freedom per node. Numerical performance of the element is compared with that of the other curved beam elements based on independently assumed field polynomials. The formulation is shown to be free from any spurious constraints in the limit of inextensional flexural vibration modes and hence does not exhibit membrane locking. The resulting well-conditioned stiffness matrix with consistent mass matrix shows excellent convergence of natural frequencies even for very thin deep arches and higher vibrational modes. The accuracy of the element for extensional flexural motion is also demonstrated.

Buckling and post-buckling analysis of moderately thick laminated rectangular plates

Abstract The first-order shear deformation theory formulated by Mindlin, associated with von Karmans non-linear strain-displacement relationships is employed to investigate the buckling and post-buckling of moderately thick laminated plates. An eight-node isoparametric plate finite element with 5 d.f, per node is developed for this purpose. The plate is assumed to be subjected to uni or biaxial compression and the plate edges are allowed to move in the load direction. The assembled finite element equations are solved along with constraint equations enforcing the same displacement (in the load direction) at all the nodes on an edge. The effects of boundary conditions, aspect ratio, side to thickness ratio and lay-up sequence on the buckling and post-buckling behaviour are studied in detail. Some changes in the mode shape coupled with a drop in the load carrying capacity of the plate are also reported herein.

Buckling of moderately thick rectangular composite plates subjected to partial edge compression

Abstract The objective of the present paper is to investigate the influence of partial edge compression on the critical loads of moderately thick laminated plates. Towards this, an eight node isoparametric plate element is developed. The element has five degrees of freedom per node. The computer code developed accepts two sets of boundary conditions, one for pre-buckling stress analysis and the second for stability analysis. This flexibility is proposed to exploit the mid-line symmetry conditions. Two types of partial edge compression, viz., (I) uniform partial edge compression near the corners and (II) uniform partial edge compression at the middle of edges are considered. The effect of percentage of loaded edge length on the critical load of thin and thick composite plates with simply supported and clamped edge conditions is studied in detail.

Thermal Postbuckling Behavior of Laminated Composite Plates

Thermal buckling and postbuckling behavior of shear deformable laminated composite plates is investigated by employing a four-node rectangular C(sup 1) continuous finite element. The investigation reveals that the postbuckling path may not remain stable throughout. It is shown that secondary instabilities coupled with changes in the spatial deformation do take place from the postbuckling path. 15 refs.

Analysis of the Nonlinear Vibrations of Unsymmetrically Laminated Composite Beams

Large-amplitude free vibrations of unsymmetrically laminated beams using von Karman large deflection theory are investigated herein. One-dimensional Finite elements based on classical lamination theory, first-order shear-deformation theory, and higher-order shear-deformation theory having 8, 10, and 12 degrees of freedom per node, respectively, are developed.

Large amplitude free vibration of simply supported antisymmetric cross-ply plates

The presence of bending-extension coupling in antisymmetric cross-ply plates results in the cubic nonlinear term in addition to the fourth-power term in the energy balance equation. A direct numerical integration method has been proposed to study the large amplitude free vibrations of such plates

Thermal buckling of cross-ply composite laminates

Abstract Thermal buckling of antisymmetric cross-ply composite laminates is investigated in this paper. A one-dimensional finite element having two nodes and six degrees of freedom, namely axial displacement, transverse displacement, rotation of the normal to the beam axis and their derivatives with respect to the beam coordinate axis at each node, is developed based on the first-order shear deformation theory. The element elastic stiffness matrix and the geometric stiffness matrix (based on thermal stresses) each of order 12 x 12 are derived to compute the bifurcation thermal load (critical temperature). Correspondence of thermal critical load parameter and mechanical critical load parameter in the case of isotropic and composite laminates is discussed. Results are presented for various boundary conditions, lay-up sequences and slenderness ratios.