Orthogonal Point Collocation Method Research Articles

Axisymmetric Postbuckling of Orthotropic Thin Tapered Annular Plates

This paper explores the geometrically nonlinear postbuckling response of cylin drically orthotropic thin tapered annular plates subjected to inplane radial compressive load at the outer edge. Clamped and simply supported plates with a free hole and a plugged hole have been analysed for the case of linear variation of thickness. The orthogonal point collocation method with zeros of a Legendre polynomial as colloca tion points has been employed to solve nonlinear governing differential equations ex pressed in terms of transverse displacement w and stress function ψ. Results have been presented for three values of annular ratio, five values of taper ratio and four values of orthotropic parameter and their effect on the buckling and postbuckling loads has been investigated.

Nonlinear axisymmetric vibration of orthotropic thin annular plates with a rigid central mass

Large amplitude axisymmetric vibrations of uniform and tapered orthotropic annular plates with a rigid central mass are studied. A time-mode solution of the von Kármán-type-governing equations is obtained using the Kantorovich averaging method. The orthogonal point collocation method is used for spatial discretization. The effect of various parameters is studied.

Nonlinear static and dynamic analysis of circular plates and shallow spherical shells using the collocation method

AbstractThe present work investigates the efficacy and applicability of interior global orthogonal point collocation method to the axisymmetric nonlinear analysis of elastic circular plates and shallow spherical shells subjected to uniformly distributed transverse load. Spacewise discretisation has been carried out using a polynomial expansion with the zeros of a Chebyshev polynomial as collocation points. Timewise integration has been carried out with Newmark k‐β scheme corresponding to average acceleration method. The static response and snap‐through buckling results, as well as, the dynamic response and dynamic buckling results under a uniformly distributed step load have been obtained and found to agree closely with the available results.

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

This study deals with the geometrically nonlinear axisymmetric static and transient analysis of cylindrically orthotropic elastic thin tapered circular plates subjected to uniformly distributed and discrete central loads. Differential equations in terms of transverse displacement w and stress function ψ have been employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in space domain and Newmark-β scheme in time domain have been used. Step function dynamic loads are considered. Static and dynamic results have been presented for isotropic and orthotropic immovable clamped and simply supported plates with linearly varying thickness for three values of taper ratios and the effect of varying thickness has been investigated. A simple approximate method is used to predict the maximum dynamic response to step load from the results for static loads and is found to yield sufficiently accurate results.

Nonlinear response of orthotropic thick shallow conical springs

This study deals with the geometrically nonlinear axisymmetric deflection characteristics of orthotropic thick shallow truncated conical springs. Marguerre type governing equations including the effect of shear deformation have been formulated in terms of normal displacement ω, shear rotation φ and stress function ψ. Polynmial approximations are used for these field variables and the discretised equations are obtained using the orthogonal point collocation method. The influence of thickness-to-radius ratio, orthotropic parameter and edge conditions has been investigated. Significant influence of shear deformation on the response is found for moderately thick high-modulus composite clamped caps. Its effect is less pronounced for simply-supported caps.

Nonlinear axisymmetric static and transient analysis of orthotropic thick annular plates

This paper deals with the geometrically nonlinear axisymmetric static and transient response of moderately thick cylindrically orthotropic circular plates subjected to uniformly distributed and ring loads. Immovable clamped and simply supported annular plates with and without a rigid plug subjected to static and step function loads have been considered. Shear deformation and rotary inertia have been included. Orthogonal point collocation method and Newmark-β scheme have been employed to solve the differential equations expressed in terms of transverse displacement ω, shear rotation φ and stress function ψ. The effect of transverse shear has been investigated for isotropic and orthotropic plates. A simple approximate method has also been used to predict the maximum dynamic response to step loads from the results for static loads.

Non-linear axisymmetric transient analysis of orthotropic thin annular plates with a rigid central mass

The geometrically non-linear, axisymmetric transient elastic response is determined of cylindrically orthotropic thin annular plates with a rigid central mass subjected to a uniformly distributed load on the plate as well as a central load on the rigid mass. The dynamic analogue of the von Kármán equations in terms of the normal displacement w and the stress function Ψ are employed. The displacement w and stress function Ψ are expanded in finite power series and the orthogonal point collocation method in the space domain and the Newmark β scheme in the time domain are used. The response of isotropic and orthotropic, clamped as well as simply supported, annular plates with a rigid central mass, subjected to step function and sinusoidal pulse loads, has been calculated for two values of the annular ratio. The influence of the mass ratio and the magnitude of the step load on the deflection response has been determined. The effect of mass ratio, amplitude and duration of sinusoidal pulse on the deflection response has also been studied.

Non-linear axisymmetric transient analysis of an orthotropic thin circular plate with an elastically restrained edge

Results are presented for the geometrically non-linear axisymmetric transient elastic stress and deflection responses of a cylindrically orthotropic thin circular plate with an elastically restrained edge, including both rotational and in-plane displacements. In the analysis the dynamic analogue of the von Kárman governing differential equations in terms of the normal displacement w and the stress function ψ are employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in the space domain and the Newmark-β scheme in the time domain are used. Four types of uniformly distributed transient loadings have been considered: step function, sinusoidal and N-shaped pulses, and exponentially decaying loads. The influence of the orthotropic parameter β and the elastic rotational and in-plane edge restraint parameters ( K b , K i ) on the large amplitude response has been investigated. The effect of a prescribed in-plane displacement on the non-linear transient response has also been studied.

Axisymmetric static and dynamic buckling of orthotropic shallow spherical caps with flexible supports

This investigation deals with the static and dynamic axisymmetric buckling of elastic orthotropic thin shallow spherical shells with elastically restrained edge for inplane and rotational displacements. Governing equations in terms of normal displacementw and stress function ψ have been employed. Orthogonal point collocation method is used for spatial discretisation and Newmark-β scheme is used for time-marching. The uniformly distributed static and step function conservative loadings normal to the underformed surface are considered. The present results are in good agreement with the available results. The influence of orthotropicity parameter β and the support stiffness parameters on the static and dynamic buckling loads has been investigated.

Axisymmetric postbuckling of orthotropic thin tapered circular plates

This study deals with the geometrically non-linear postbuckling response of fibre-reinforced cylindrically orthotropic thin tapered circular plates subjected to an in-plane radial compressive load. Clamped and simply-supported plates with linear variation of thickness have been analysed. Von Kármán equations for orthotropic plates of varying thickness in terms of transverse displacement ω and stress function Ψ have been employed. Monomials are used as basis functions for ω and Ψ. An orthogonal point collocation method with zeros of a Legendre polynomial as collocation points has been used to solve the governing differential equations. The effect of orthotropic parameter and taper ratio has been investigated for clamped and simply-supported plates.

Nonlinear axisymmetric transient analysis of orthotropic thin annular plates

The investigation deals with the geometrically nonlinear, axisymmetric, transient elastic response for stresses and deflection of cylindrically orthotropic thin annular plates subjected to uniformly distributed and ring loads. The dynamic analogues of von Kármán equations in terms of normal displacement w and stress function ϕ have been employed. The displacement w and stress function ϕ are expanded in finite power series. The orthogonal point collocation method in the space domain and the Newmark-β scheme in the time domain have been used. Three types of uniformly distributed dynamic loadings, namely, step function, sinusoidal pulse and exponentially decaying load, and a case of step function ring load at the free edge have been considered. Detailed results are presented for a clamped and simply supported annular plate with a free hole, and an annulus supported at the inner boundary and free at the outer edge, for different values of orthotropic parameter and annular ratio. The effect of elastic rotational and inplane edge restraints has also been studied.

Axisymmetric static and dynamic buckling of orthotropic shallow spherical cap with circular hole

Nonlinear axisymmetric static and dynamic buckling of clamped isotropic and cylindrically orthotropic elastic cap with central circular hole have been investigated. The governing equations are expressed in terms of normal displacement ω and stress function ψ. The orthogonal point collocation method has been used in the space domain and Newmark-β scheme in the time domain. The cases of shallow cap with free hole and with a hole plugged by rigid central mass are considered. Analysis has been carried out for uniformly distributed load and a ring load at the hole. Dynamic load is taken as a step function load. Detailed new results for static and dynamic buckling loads have been presented for the isotropic and orthotropic cases.

Non-linear axisymmetric static analysis of orthotropic thin annular plates

This investigation deals with the geometrically non-linear axisymmetric static analysis of elastic orthotropic thin annular plates subjected to uniformly distributed and ring loads. Non-linear differential equations in terms of the transverse displacement w and stress function ψ have been employed. Both w and ψ are expanded in finite power series and the orthogonal point collocation method has been used to obtain discretised algebraic equations from the governing differential equations. Results have been presented for annular plates with and without a rigid plug in the hole; simply supported, clamped and free outer edge conditions; three orthotropic parameters β; two annular ratios and two loading conditions. Results have also been presented illustrating the effect of an elastic rotational constraint at the edge and the effect of a prescribed inplane displacement of the edge.

Nonlinear Static and Transient Analysis of Orthotropic Thin Circular Plates with Elastically Restrained Edge Under Central Load

This study deals with the geometrically nonlinear axisymmetric static and transient analysis of cylindrically orthotropic thin circular plate with elastically restrained edge for rotation and inplane displacements, under a concentrated load at the center. Analogue of Von-Karman differential equations in terms of normal displacement w and stress function ψ have been employed. The displacement w and stress function ψ are expanded in finite power series. Orthogonal point collocation method in space do main and Newmark-β scheme in time domain have been used. New results are presented for orthotropic plate for edge conditions spanning the range of elastic edge restraints from the movable simply supported edge to the immovable clamped edge. The influence of orthotropic parameter and the elastic rotation and inplane edge restraint parameters on the response has been investigated.