G.Venkateswara Rao

Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates

A finite element formulation for the large amplitude free oscillations of beams and orthotropic circular plates is presented in this paper. The present formulation does not need the knowledge of longitudinal/inplane forces developed due to large displacements and thus avoids the use of corresponding geometric stiffness matrices, which were used in earlier finite element formulations. The convergence of the results obtained by using the present formulation is very good. Comparison of the present results with the earlier work wherever possible confirms the reliability and effectiveness of the present finite element formulation.

Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates

Large amplitude flexural vibrations of slender beams, and thin circular and rectangular plates have been studied when a compatible longitudinal or inplane mode is coupled with the fundamental flexural mode. It is shown that the effect of longitudinal or inplane deformation and inertia is to reduce the non-linearity in the flexural frequency-amplitude relationship. Further, for slender beams and thin plates, the effect of longitudinal or inplane inertia is negligible.

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.

Re-investigation of large-amplitude free vibrations of beams using finite elements

The dynamic finite element matrix equations are reduced to a scalar second order non linear differential equation, solved by using the direct integration method.

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.

Intralaminar fracture toughness of a cross-ply laminate and its constituent sub-laminates

Abstract The intralaminar mode I fracture toughness values for a M55J/M18 carbon/epoxy cross-ply laminate [0°/90°]15 (alternate 0 and 90° layers) and its constituent sub-laminates, namely the [0°]30 and [90°]30 laminates are theoretically evaluated on the basis of a modified crack-closure integral (MCCI) method corresponding to the fracture loads obtained by testing C(T) specimens. A simple relationship between the intralaminar fracture toughness at the specimen level of a cross ply and its constituent sub-laminates is presented. Test data for fracture loads and crack-opening displacements at the opposite end (far end) to the crack tip of the three types of C(T) specimens are obtained. A comparison of fracture toughness values KIc and crack-opening displacements obtained from tests with theoretical predictions by finite-element analysis shows good agreement for the type of laminates considered. Comparison of the analytically determined KIc of the cross-ply C(T) specimen based on the new relationship using the respective values of KIc of its constituent C(T) specimens shows good agreement with test data.

A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation

A modified finite element formulation to study the post-buckling behaviour of elastic circular plates is presented in this paper. A discussion on the derivation of nonlinear stiffness matrix for post-buckling analysis is included and the present results are compared with continuum solutions.

A finite element formulation for large amplitude flexural vibrations of thin rectangular plates

Large amplitude flexural vibrations of rectangular plates are studied in this paper using a direct finite element formulation. The formulation is based on an appropriate linearisation of strain displacement relations and uses an iterative method of solution. Results are presented for rectangular plates with various boundary conditions using a conforming rectangular element. Whenever possible the present solutions are compared with those of earlier work. This comparison brings out the superiority of the proposed formulation over the earlier finite element formulation.

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.

Axisymmetric vibrations of circular plates including the effects of geometric non-linearity, shear deformation and rotary inertia

In this paper, axisymmetric vibrations of circular plates are studied, with inclusion of the effects of geometric non-linearity, shear deformation and rotatory inertia. The finite element method is employed to formulate the problem. Both simply supported and clamped plates are considered. The results indicate that the influence of geometric non-linearity is greater when the effects of shear deformation and rotatory inertia are included than when they are neglected.