A new finite element formulation for vibration analysis of thick plates

A new procedure for determining properties of thick plate finite elements, based on the modified Mindlin theory for moderately thick plate, is presented. Bending deflection is used as a potential function for the definition of total (bending and shear) deflection and angles of cross-section rotations. As a result of the introduced interdependence among displacements, the shear locking problem, present and solved in known finite element formulations, is avoided. Natural vibration analysis of rectangular plate, utilizing the proposed four-node quadrilateral finite element, shows higher accuracy than the sophisticated finite elements incorporated in some commercial software. In addition, the relation between thick and thin finite element properties is established, and compared with those in relevant literature

Kriging-Based Timoshenko Beam Elements with the Discrete Shear Gap Technique

Kriging-based finite element method (K-FEM) is an enhancement of the FEM through the use of Kriging interpolation in place of the conventional polynomial interpolation. In this paper, the K-FEM is developed for static, free vibration, and buckling analyses of Timoshenko beams. The discrete shear gap technique is employed to eliminate shear locking. The numerical tests show that a Kriging-Based beam element with cubic basis and three element-layer domain of influencing nodes is free from shear locking. Exceptionally accurate displacements, bending moments, natural frequencies, and buckling loads and reasonably accurate shear force can be achieved using a relatively course mesh

Study of the Discrete Shear Gap Technique in Timoshenko Beam Elements

A major difficulty in formulating a finite element for shear-deformable beams, plates, and shells is the shear locking phenomenon. A recently proposed general technique to overcome this difficulty is the discrete shear gap (DSG) technique. In this study, the DSG technique was applied to the linear, quadratic, and cubic Timoshenko beam elements. With this technique, the displacement-based shear strain field was replaced with a substitute shear strain field obtained from the derivative of the interpolated shear gap. A series of numerical tests were conducted to assess the elements performance. The results showed that the DSG technique works perfectly to eliminate the shear locking. The resulting deflection, rotation, bending moment, and shear force distributions were very accurate and converged optimally to the corresponding analytical solutions. Thus the beam elements with the DSG technique are better alternatives than those with the classical selective-reduced integration

Finite element analysis of plate and beam models

We consider linear mathematical models for elastic plates and beams. To be specific, we consider the Euler-Bernoulli, Rayleigh and Timoshenko theories for beams and the Kirchhoff and Reissner-Mindlin theories for plates. The theories mentioned above refer to the partial differential equations that model a beam or plate. The contact with other objects also need to be modelled. The equations that result are referred to as “interface conditions". We consider three problems concerning interface conditions for plates and beams: A vertical slender structure on a resilient seating, the built in end of a beam and a plate-beam system. The vertical structure may be modelled as a vertically mounted beam. How- ever, the dynamics of the seating must be included in the model and this increases the complexity of a finite element analysis considerably. We show that the interface conditions and additional equations can be accommodated in the variational form and that the finite element method yields excellent results. Although the Timoshenko model is considered to be better than the Euler- Bernoulli model, some authors do not agree that it is an improvement for the case of a cantilever beam. In a modal analysis of a two-dimensional beam model, we show that the Timoshenko model is not only better, but it provides good results when the beam is so short that one is reluctant to use beam theory at all. In applications, structures consisting of linked systems of beams and plates are encountered. We consider a rectangular plate connected to two beams. Combining the Reissner-Mindlin plate model and the Timoshenko beam model can be seen as a first step towards a better model while still avoiding the complexity of a fully three-dimensional model. However, the modelling of the plate-beam system is more complex than in the case of the classical theory and the mathematical analysis and numerical analysis present additional difficulties. A weak variational form is derived for all the model problems. This is necessary to apply general existence and uniqueness results. It is also necessary to apply general convergence results and derive error bounds. The setting for the weak variational forms are product spaces. This is due to the complex nature of the model problems.Thesis (PhD (Mathematics and Applied Mathematics))--University of Pretoria, 2006.Mathematics and Applied Mathematicsunrestricte

Shooting method for free vibration of FGM Reissner-Mindlin circular plates resting on elastic foundation in thermal environments

This paper presents a free vibration analysis of functionally graded Reissner-Mindlin circular plates with various supported boundaries in thermal environments. A FGM consisting of metal and ceramic was considered in the study. Based on the geometric equation, physical equation and equilibrium equation of thick plate, taking into account the transverse shearing deformation, the free vibration equation of the axisymmetric FGM moderately thick circular plates was derived in terms of the middle surface angles of rotation and lateral displacement. The material properties of the plate were assumed to vary continuously in the thickness direction according a power law. By using shooting method to solve the coupled ordinary differential equations with different boundary conditions, the natural frequencies of FGM thick circular plates were obtained numerically. The effects of material gradient property, thickness ratio and boundary conditions on the natural frequencies were discussed in detail

Much ado about shear correction factors in Timoshenko beam theory

AbstractMany shear correction factors have appeared since the inception of Timoshenko beam theory in 1921. While rational bases for them have been offered, there continues to be some reluctance to their full acceptance because the explanations are not totally convincing and their efficacies have not been comprehensively evaluated over a range of application. Herein, three-dimensional static and dynamic information and results for a beam of general (both symmetric and non-symmetric) cross-section are brought to bear on these issues. Only homogeneous, isotropic beams are considered. Semi-analytical finite element (SAFE) computer codes provide static and dynamic response data for our purposes. Greater clarification of issues relating to the bases for shear correction factors can be seen. Also, comparisons of numerical results with Timoshenko beam data will show the effectiveness of these factors beyond the range of application of elementary (Bernoulli–Euler) theory.An issue concerning principal shear axes arose in the definition of shear correction factors for non-symmetric cross-sections. In this method, expressions for the shear energies of two transverse forces applied on the cross-section by beam and three-dimensional elasticity theories are equated to determine the shear correction factors. This led to the necessity for principal shear axes. We will argue against this concept and show that when two forces are applied simultaneously to a cross-section, it leads to an inconsistency. Only one force should be used at a time, and two sets of calculations are needed to establish the shear correction factors for a non-symmetrical cross-section