Orthogonal Point Collocation Method Research Articles

The postbuckling analysis of laminated circular plate with elliptic delamination

Based on the Von Karman plate theory, considering the effect of transverse shear deformation, and using the method of the dissociated three regions, the postbuckling governing equations for the axisymmetric laminated circular plates with elliptical delamination are derived. By using the orthogonal point collocation method, the governing equations, boundary conditions and continuity conditions are transformed into a group of nonlinear algebraically equation and the equations are solved with the alternative method. In the numerical examples, the effects of various elliptical in shape, delamination depth and different material properties on buckling and postbuckling of the laminated circular plates are discussed and the numerical results are compared with available data.

Axisymmetric buckling of laminated thick annular spherical cap

Axisymmetric buckling analysis is presented for moderately thick laminated shallow annular spherical cap under transverse load. Buckling under central ring load and uniformly distributed transverse load, applied statically or as a step function load is considered. The central circular opening is either free or plugged by a rigid central mass or reinforced by a rigid ring. Annular spherical caps have been analysed for clamped and simple supports with movable and immovable inplane edge conditions. The governing equations of the Marguerre-type, first order shear deformation shallow shell theory (FSDT), formulated in terms of transverse deflection w, the rotation ψ of the normal to the midsurface and the stress function Φ, are solved by the orthogonal point collocation method. Typical numerical results for static and dynamic buckling loads for FSDT are compared with the classical lamination theory and the dependence of the effect of the shear deformation on the thickness parameter for various boundary conditions is investigated.

Axisymmetric static and dynamic buckling of laminated thick truncated conical cap

Axisymmetric buckling analysis is presented for moderately thick laminated shallow, truncated conical caps under transverse load. Buckling under uniformly distributed loads and ring loads applied statically or as step function loads is considered. Marguerre-type, first-order shear deformation shallow shell theory is formulated in terms of transverse deflection w, the rotation ψ of the normal to the mid-surface and the stress function Φ. The governing equations are solved by the orthogonal point collocation method. Truncated conical caps with a circular opening, which is either free or plugged by a rigid central mass, have been analysed for clamped and simple supports with movable and immovable edge conditions. Typical numerical results are presented illustrating the effect of various parameters.

Axisymmetric buckling of laminated, moderately thick shallow conical cap

Axisymmetric buckling and postbuckling analysis is presented for a moderately thick, laminated shallow conical cap under static transverse load. Marguerre-type, first-order shear deformation shallow-shell theory is formulated in terms of transverse deflection w, the rotation ψ of the normal to the midsurface and the stress function Φ. The governing equations are solved by the orthogonal point-collocation method. Clamped conical caps and simple supports with movable and immovable edge conditions are considered. Typical numerical results are presented, illustrating the effect of various parameters. The dependence of the effect of the shear deformation on the thickness parameter, boundary conditions, ratio of Young's moduli and cap height is investigated.

Axisymmetric postbuckling of moderately thick laminated annular plate using FSDT

This work presents axisymmetric postbuckling response of polar orthotropic laminated moderately thick circular and annular plates subjected to uniformly distributed inplane radial compressive load at the outer edge. First-order shear deformation theory (FSDT) is used in terms of deflection w, rotation ψ of the normal to the midplane and the stress function Φ. These field variables are expanded in polynomials and orthogonal point collocation method is used to discretise the governing equations. The buckling load is obtained by solving a linear eigenvalue problem. Postbuckling loads for different deflections are obtained by solving nonlinear equations by Newton–Raphson method. Clamped and simply supported circular and annular plates with free hole or reinforced by a rigid ring are considered. The results are compared with the classical lamination theory to assess the dependence of the effect of shear deformation on boundary conditions, ratio of Young's moduli, lamination scheme, number of layers, the radius-to-thickness ratio and annular ratio. The response of unsymmetrically laminated plates under inplane radial load is also presented.

Geometrically nonlinear axisymmetric analysis of thick laminated annular plate using FSDT

This paper presents geometrically nonlinear, axisymmetric, static and transient, moderately large deflection response of a laminated, composite, moderately thick annular plate subjected to uniformly distributed ring loads. The layers are considered to be polar orthotropic. Rotational inertia is included but the inplane inertia is neglected. First-order shear deformation theory is used in terms of the deflection w, rotation ψ of the midplane and the stress function Φ. The governing equations are solved by the orthogonal point collocation method and Newmark-β scheme. The effect of the radius-to-thickness ratio of the plate on the deflection and stress response is studied for symmetrically and unsymmetrically laminated plates.

Nonlinear static and dynamic analysis for laminated, annular, spherical caps of moderate thickness

Based on Timoshenko-Mindlin kinematic hypothesis, the shallow shell theory is extended to include the transverse shear deformation for the nonlinear axisymmetric dynamic analysis of the symmetric cross-ply shallow spherical shell. Using the orthogonal point collocation method and the Newmark scheme, an iterative solution is formulated. The numerical results for the nonlinear static and dynamic responses and dynamic buckling of these shallow spherical shells with circular holes under uniformly distributed static or dynamic normal impact loads are presented and compared with available data.

Nonlinear dynamic response and dynamic buckling of shallow spherical shells with circular hole

In this paper, the nonlinear equations of motion for shallow spherical shells with axisymmetric deformation including transverse shear are derived. The nonlinear static and dynamic response and dynamic buckling of shallow spherical shells with circular hole on elastically restrained edge are investigated. By using the orthogonal point collocation method for space and Newmark-β scheme for time, the displacement functions are separated and the nonlinear differential equations are replaced by linear algebraic equations to seek solutions. The numerical results are presented for different cases and compared with available data.

Non-linear vibration of orthotropic thin rectangular plates on elastic foundations

This study deals with the large amplitude free vibrations of rectilinearly orthotropic thin rectangular plates resting on elastic foundations. The von Kármán-type of governing equations are employed. The displacements are assumed to be harmonic in time and are expanded space-wise in terms of polynomial basis functions which satisfy the boundary conditions exactly. The time co-ordinate is eliminated by the Kantorovich averaging method. The orthogonal point collocation method is used to obtain the discretised equations. The non-linear eigenvalue problem is solved by an iterative method. Numerical results are presented for the linear frequency of the fundamental mode and for the ratio of the non-linear frequency to the linear frequency. Plates with opposite edges clamped or simply supported with immovable in-plane conditions and resting on Winkler, Pasternak and non-linear Winkler foundations are considered. The effects of foundation parameters, material parameters, edge conditions and the aspect ratio on the non-linear vibration behaviour have been investigated.

Nonlinear static analysis of rectangular plates on elastic foundations by the orthogonal point collocation method

The conventional interior point collocation method is greatly improved if the collocation points are located at the zeros of orthogonal polynomials. In this paper, the simplicity and good accuracy of this orthogonal point collocation method is demonstrated for the solution of some geometrically nonlinear problems of moderately large deflections of rectangular plates subjected to static loads. Von Kármán-type governing equations are employed. Clamped and simply supported plates with immovable inplane conditions at the edges are considered. Plates which are simply supported at two opposite edges and clamped at the other two edges are also analyzed. Winkler and Pasternak models of elastic foundation are used. The present results agree quite well with the available ones. New results are presented for orthotropic plates resting on Winkler and Pasternak foundations.

Nonlinear forced vibration of orthotropic thin rectangular plates

This study deals with the large amplitude forced vibrations of specially orthotropic thin rectangular plates. Geometric nonlinearity is included in accordance with von Kármán type large deflection plate theory. The transverse displacement is assumed to be harmonic at the same frequency as that of the uniformly distributed harmonic force. The displacements are expanded space-wise in terms of polynomial basis functions which satisfy the boundary conditions. Time is eliminated by averaging over the time-period of the harmonic force. The orthogonal point collocation method is used to obtain the discretized equations. These nonlinear equations are solved by an iterative procedure. Plates with all edges clamped or simply-supported and immovable inplane are considered. The relations of amplitude and frequency ratio for rectangular plates made of various orthotropic materials are presented.

Large deflection axisymmetric analysis of orthotropic annular plates on elastic foundations

Large deflection axisymmetric response analyses of cylindrically orthotropic thin annular plates, resting on annular elastic foundations and subjected to uniformly distributed loads are presented. Static and step function loads applied to clamped and simply supported annular plates are considered. The natural boundary conditions employed at the hole are consistent with those obtained from Hamilton's principle. Von Kármán type governing equations are solved using the orthogonal point collocation method in conjunction with the Newmark-β scheme. The maximum response to step loads obtained from static analysis is shown to agree very well with that obtained from a transient analysis. Thus one static analysis is sufficient to obtain the maximum response to a whole set of step loads, instead of separate transient solutions for each load. New results are presented to study the effects of the foundation parameters on the plate response.

Nonlinear transient response of orthotropic thin rectangular plates on elastic foundations

This paper presents geometrically nonlinear transient analysis of rectilinearly orthotropic thin rectangular plates resting on Winkler and Pasternak foundations for uniformly distributed step function and sinusoidal pulse loadings. The orthogonal point collocation method in the space domain and Newmark-β scheme in the time domain are employed. Immovable clamped and simply-supported plates are analysed. An approximate method is used to predict the maximum dynamic response to step loads from the results for static loads and is found to yield sufficiently accurate results.

Nonlinear Axisymmetric Static and Transient Response of Orthotropic Annular Shallow Spherical Caps

ABSTRACT This investigation deals with the nonlinear axisymmetric static and transient response of orthotropic annular shallow spherical caps with a free edge hole and with a rigid central mass, subjected to uniformly distributed load and a central load. The dynamic analogue of Marguerre equations, in terms of normal displacement w and stress function Ψ, are employed. An orthogonal point collocation method is used for spatial discretization and a Newmark-β scheme is used for time integration. Static load, step function load, and sinusoidal pulse load are considered. The influence of annular ratio on the response is studied for isotropic (β = 1) and orthotropic (β = 3) clamped caps, with a rise to thickness ratio of 1.5.

Circular plates on Pasternak elastic foundations

AbstractThis study deals with the geometrically nonlinear axisymmetric static and transient response of cylindrically orthotropic thin circular plates resting on Pasternak elastic foundations subjected to uniformly distributed loads. Clamped and simply‐supported plates with radially movable and immovable edges have been considered. The von‐Kármán type nonlinear governing equations have been solved using orthogonal point collocation method in the space domain and Newmark‐β scheme in the time domain. The maximum transient response to step loads has also been predicted from static response. The influence of the foundation parameters on the response has been investigated.

Nonlinear Analysis of Thick Circular Plates

This paper is concerned with the geometrically nonlinear axisymmetric static and transient analysis of moderately thick cylindrically orthotropic circular plates subjected to uniformly distributed and discrete central loads. Shear deformation and rotary inertia are included in the analysis. Differential equations in terms of the transverse displacement, w, the rotation of the normal to the middle surface, ϕ, and the stress function, ψ, are employed. These three field variables are expanded in finite power series and the discretized equations are obtained by using the orthogonal point collocation method in the space domain and the Newmark‐β scheme in the time domain. Results are presented for immovable clamped and simply supported plates for static and step function loads. The effect of transverse shear is investigated for isotropic, transversely isotropic and orthotropic plates.

Nonlinear axisymmetric static and transient analysis of orthotropic thin tapered annular plates

This paper considers the geometrically nonlinear axisymmetric static and transient responses of cylindrically orthotropic thin tapered annular plates subjected to uniformly distributed and ring loads. Immovable clamped and simply supported annular tapered plates with and without a rigid plug subjected to static and step function loads have been analysed. Orthogonal point collocation method and Newmark-β scheme have been employed to solve nonlinear governing differential equations expressed in terms of transverse displacement ω and stress function Ψ. Results have been presented for isotropic and orthotropic annular tapered plates with linear variation of thickness for two annular ratios and three taper ratios, and the effect of varying thickness has been investigated. A simple approximate method has been used to predict the maximum dynamic response to step loads from the results for static loads. This method gives sufficiently accurate results for engineering applications.

Nonlinear axisymmetric vibration of orthotropic thin circular plates with elastically restrained edges

This paper deals with the large-amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates of varying thicknesses with edge elastically restrained against rotation and in-plane displacement. Geometric nonlinearity due to moderately large deflections is included. Linear, parabolic and cubic variations of thickness are considered. Harmonic vibrations are assumed and time is eliminated from the von Kármán-type governing equations by the Kantorovich averaging method. The orthogonal point collocation method is used for spatial discretisation. Results are presented for the linear frequency of first axisymmetric mode and for the amplitude-period response. The effect of taper ratio, orthotropic parameter and rotational and in-plane stiffness of the support on the nonlinear vibration behaviour is investigated.

Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.

Axisymmetric static and dynamic buckling of orthotropic shallow conical caps

This study deals with the axisymmetric static and dynamic buckling of elastic polar orthotropic thin shallow conical caps subjected to uniformly distributed loads. Static and step function conservative loadings on conical caps with clamped immovable, simply supported immovable, and simply supported movable edges have been considered. The governing equations are formulated in terms of normal displacement w and stress function i//. Orthogonal point collocation method is used for spatial discretization and Newmark-/? scheme is used for time marching. The present results for the isotropic clamped immovable conical caps are in good agreement with the available results. The influence of orthotropic parameter /? and the edge conditions on the static and dynamic buckling loads has been investigated. Dynamic buckling loads obtained from static analysis have been found to agree well with the dynamic buckling loads based on transient response.