Finite element displacement method for large amplitude free flexural vibrations of beams and plates

A finite element method to determine the nonlinear frequency of beams and plates for large amplitude free vibrations is presented. The equation of motion is characterized by the basic stiffness, mass, geometrical stiffness and the associated inplane force matrices. The procedures for solving the system equations of motion are discussed, and the explicit formulations of the geometrical stiffness and inplane force matrices of a rectangular plate element are given. Examples of large amplitude free oscillations for rectangular plates and beams with various boundary conditions are given. Characteristics of convergence are investigated. In all the cases where comparisons with previous investigations are made, good agreement has been obtained. It indicates that the present method will give results entirely adequate for engineering purposes.     

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.     

On various formulations of large amplitude free vibrations of beams

Various finite element formulations of large amplitude free vibrations of beams with immovably supported ends are discussed in this paper. Analytical formulation based on the Rayleigh-Ritz method is also presented. Numerical results of the analytical approach are seen to be in good agreement with some of these finite element formulations. Mixed finite element formulations based on two methods are derived to study the large amplitude free vibrations of beams. The mixed finite element methods also show good agreement with the analytical and the above finite element formulations. Various points of view raised from time to time on the applicability of these formulations can now be clarified through these formulations and the numerical results. The weakness of the so-called improved Ritz-type finite element model in predicting the nonlinear frequency ratio is highlighted through various results of the above formulations. As a typical example, a hinged-hinged beam on immovable ends is considered for all the above formulations and the nonlinear frequencies show a good agreement amongst themselves at all amplitude levels.     

Large amplitude vibrations of circular plates with mixed boundary conditions

In this paper, a method of analysing large amplitude vibrations of circular plates with mixed boundary conditions is explained and is illustrated with an example where part of the boundary is clamped and the remaining simply-supported     

Finite element bending analysis of nonuniform circular plates

The finite element method is applied to the small deflection bending analysis of nonuniform thin axisymmetric circular plates made of linear elastic material. Elements with annular and circular geometry with only 4 degrees of freedom are used in the analysis of both symetrically and nonsymmetrically loaded plates. Non-symmetric loads are expanded in Fourier series and elements restricted to deform with specified number of nodal diameters are used for each component of loading. The method is checked with several numerical examples. Although applicable to only axisymmetric plates, the method gives better results compared to other finite element methods besides offering savings in computer storage and time.     

Large amplitude free flexural vibrations of thin plates of arbitrary shape

A finite element formulation is developed for analyzing large amplitude free flexural vibrations of elastic plates of arbitrary shape. Stress distributions in the plates, deflection shape and nonlinear frequencies are determined from the analysis. Linearized stiffness equations of motion governing large amplitude oscillations of plates, quasi-linear geometrical stiffness matrix, solution procedures, and convergence characteristics are presented. The linearized geometrical stiffness matrix for an eighteen degrees-of-freedom conforming triangular plate element is evaluated by using a seven-point numerical integration. Nonlinear frequencies for square, rectangular, circular, rhombic, and isosceles triangular plates, with edges simply supported or clamped, are obtained and compared with available approximate continuum solutions. It demonstrates that the present formulation gives results entirely adequate for many engineering purposes.     

Finite element free vibration of eccentrically stiffened plates

An isoparametric stiffened plate element is introduced for the free vibration analysis of eccentrically stiffened plates. The element has the ability to accommodate irregular boundaries. Moreover, the formulation considers shear deformation, hence, the formulation is applicable to both thick and thin plates. In the present formulation, the stiffeners can be placed anywhere within the plate element and they need not necessarily follow the nodal lines. In addition, the effects of lumped and consistent mass matrices on natural frequencies of stiffened plates are investigated. The effects of several parameters of the stiffener—eccentricity, shape, torsional stiffness etc.—on the natural frequencies of the stiffened plates are studied.     

Post-buckling of cylindrically orthotropic linearly tapered circular plates by finite element method

The post-buckling behavior of tapered circular plates with cylindrically orthotropic material properties is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and orthotropy parameter. Both simply supported and clamped boundary conditions are considered.     

Finite element iterative method for optimal elastic design of circular plates

An iterative procedure is presented for the optimal elastic design of circular or annular plates under axisymmetric loads. The deflection or slope at a given radius is prescribed, and technological constraints are taken into account. Some examples are discussed.     

Nonlinear flexural vibrations of laminated orthotropic plates

Large amplitude free flexural vibrations of laminated orthotropic plates are studied using C⁰ shear flexible QUAD-8 plate element. The nonlinear governing equations are solved using the direct iteration technique. Numerical results are obtained for isotropic, orthotropic and cross-ply laminated plates with simply-supported boundary conditions on immovable edges. It is observed that hardening behaviour is increased for thick plates and orthotropic plates.     

Finite element free vibration analysis of eccentrically stiffened plates

A new finite element model is proposed for free vibration analysis of eccentrically stiffened plates. The formulation allows the placement of any number of arbitrarily oriented stiffeners within a plate element without disturbing their individual properties. A plate-bending element consistent with the Reissner-Mindlin thick plate theory is employed to model the behaviour of the plating. A stiffener element, consistent with the plate element, is introduced to model the contributions of the stiffeners. The applied plate-bending and stiffener elements are based on mixed interpolation of tensorial components (MITC), to avoid spurious shear locking and to guarantee good convergence behaviour. Several numerical examples using both uniform and distorted meshes are given to demonstrate the excellent predictive capability of this approach.     

Post-buckling behaviour of elastic circular plates using a simple finite element formulation

The post-buckling behaviour of elastic circular plates is studied in this paper using a simple finite element formulation. The final linearized eigenvalue problem is solved by using three numerical methods and all these methods are found to yield accurate results for moderately large deflections. Comparison of the present results with the results existing in literature shows the validity of this formulation.     

Finite element analysis of eccentrically stiffened plates in free vibration

A compound finite element model is developed to investigate eccentrically stiffened plates in free vibration. The plate elements and beam elements are treated as integral parts of a compound section, and not as independent bending components. The derivation is based on the assumptions of small deflection theory. In the orthogonally stiffened directions of the compound section, the neutral surfaces may not coincide. They lie between the middle surface of the plate and the centroidal axes of the stiffeners. The results of this study are compared with existing ones and with those of the orthotropic plate approximation. Modifications to the existing equivalent orthotropic rigidities are proposed.     

Trefftz-unsymmetric finite element for bending analysis of orthotropic plates

This work develops a new four-node quadrilateral displacement-based Trefftz-type plate element for bending analysis of orthotropic plates within the framework of the unsymmetric finite element method (FEM). In the present formulation, the modified isoparametric interpolations are employed to formulate the element’s test functions in which the deflection is effectively enriched by the nodal rotation degrees of freedom (DOFs). Meanwhile, the element’s trial functions are determined based on the Trefftz functions that can a prior satisfy the governing equations of orthotropic Mindlin–Reissner plates. Numerical benchmark tests reveal that the new unsymmetric plate element is free of shear locking problem and can produce satisfactory results for both the displacement and stress resultant. In particular, it exhibits quite good tolerances to the gross mesh distortion.

     

Nonlinear free bending vibrations of plates

A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

A finite element formulation for multilayered and thick plates

A bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates. The underlying theory used allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates. To illustrate the versatility of the multilayered element, three examples are presented and the results are compared with available exact solutions.     

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.     

Influence of the order of polynomial on the convergence in ritz finite element formulation to nonlinear vibrations of beams

The influence of the order of inplane polynomial on the convergence of solution, when a Ritz finite element formulation is used to study nonlinear vibrations of beams, is investigated here. Three types of polynomial distributions for the inplane displacement “u” are considered while the polynomial distribution for transverse displacement “w” is retained as cubic always. A hinged-hinged beam on immovable ends with different discretization is chosen as an example for the convergence study on the nonlinear hardening parameter. From the results obtained, it has been concluded that for a chosen cubic polynomial distribution for transverse displacement, a cubic polynomial distribution for the inplane displacement will be a compatible mode shape satisfying the physical aspects of the convergence and nature of bound for the nonlinear hardening parameter.     

The large amplitude vibration of hinged beams

The large amplitude free vibrations of a simply-supported beam with ends kept a constant distance apart is studied using the actual nonlinear equilibrium equations (i.e. specification of loads in terms of the deformed coordinates of the beam) and the exact nonlinear expression for curvature in addition to the nonlinearity arising from the axial force. A variable separable assumption, together with certain assumptions as to the behaviour of the time function defines an eigenvalue characteristic of the vibration. A numerically exact successive integration and iterative technique establishes the dependence of this quantity on the amplitude of vibrations. The hardening effect of nonlinearity is then interpreted in terms of the variation of this quantity with the amplitude of vibration. This new criteria to define nonlinearity, is compared with several existing in the literature. The present analysis allows the separation of the effects of stretching and large deflection equations on the nonlinear behaviour and the conclusion can be made, based on numerical evidence, that the predominant nonlinearity is due to stretching. The axial force at any station in the beam and the bending stress can also be computed in a numerically exact sense, at the point of maximum amplitude.