The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
M. A. Bradford - One of the best experts on this subject based on the ideXlab platform.
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Effects of Approximations on the Lateral-Torsional Buckling and Postbuckling Analysis
2012Co-Authors: M. A. BradfordAbstract:Lateral-torsional buckling and postbuckling of beams can be analysed using finite element methods. In formulating a finite beam element, a Rotation Matrix is used to derive nonlinear strain-displacement relationships. Because of couplings between displacements and twist Rotations, components of the Rotation Matrix are lengthy and complicated. To facilitate the formulation, approximations are usually made to simplify the Rotation Matrix. A simplified small Rotation Matrix is often used in the lateral-torsional buckling analysis and a simplified second order Rotation Matrix is used for the lateraltorsional postbuckling analysis. However, the small Rotation and second order Rotation matrices do not describe Rotations accurately and introduce some approximations to the coupling between displacements and Rotations. This paper investigates the effects of the approximations on the lateral-torsional buckling and postbuckling analysis of beams. It is shown that a analytical model based on the small Rotation Matrix predicts incorrect buckling loads. A finite element model based on the second order Rotation Matrix may lead to poor predictions of the postbuckling behaviour.
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effects of approximations in analyses of beams of open thin walled cross section part i flexural torsional stability
International Journal for Numerical Methods in Engineering, 2001Co-Authors: M. A. BradfordAbstract:In formulating a finite element model for the flexural–torsional stability and 3-D non-linear analyses of thin-walled beams, a Rotation Matrix is usually used to obtain the non-linear strain–displacement relationships. Because of the coupling between displacements, twist Rotations and their derivatives, the components of the Rotation Matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the Rotation Matrix. A simplified small Rotation Matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin-walled beams of open cross-section. However, the approximations in the small Rotation Matrix may lead to the loss of some significant terms in the stability stiffness Matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin-walled beams, while a companion paper investigates the effects of approximations in the 3-D non-linear analysis. It is found that a finite element line model based on a small Rotation Matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin-walled beams, modification of the model is needed, or a finite element model based on a second-order Rotation Matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.
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Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability
International Journal for Numerical Methods in Engineering, 2001Co-Authors: M. A. BradfordAbstract:In formulating a finite element model for the flexural–torsional stability and 3-D non-linear analyses of thin-walled beams, a Rotation Matrix is usually used to obtain the non-linear strain–displacement relationships. Because of the coupling between displacements, twist Rotations and their derivatives, the components of the Rotation Matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the Rotation Matrix. A simplified small Rotation Matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin-walled beams of open cross-section. However, the approximations in the small Rotation Matrix may lead to the loss of some significant terms in the stability stiffness Matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin-walled beams, while a companion paper investigates the effects of approximations in the 3-D non-linear analysis. It is found that a finite element line model based on a small Rotation Matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin-walled beams, modification of the model is needed, or a finite element model based on a second-order Rotation Matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.
Andrew Staniforth - One of the best experts on this subject based on the ideXlab platform.
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Treatment of vector equations in deep‐atmosphere, semi‐Lagrangian models. II: Kinematic equation
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Nicholas W. Wood, A A White, Andrew StaniforthAbstract:Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain Rotation Matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This Rotation Matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the Rotation Matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the Rotation Matrix in the spherical polar case involves three matrices, one of which represents Rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total Rotation Matrix results when the great circle Rotation Matrix is replaced by the identity Matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere Rotation Matrix agrees with that used by ECMWF and Meteo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
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treatment of vector equations in deep atmosphere semi lagrangian models ii kinematic equation
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Nigel Wood, A A White, Andrew StaniforthAbstract:Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain Rotation Matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This Rotation Matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the Rotation Matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the Rotation Matrix in the spherical polar case involves three matrices, one of which represents Rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total Rotation Matrix results when the great circle Rotation Matrix is replaced by the identity Matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere Rotation Matrix agrees with that used by ECMWF and Meteo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
A A White - One of the best experts on this subject based on the ideXlab platform.
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Treatment of vector equations in deep‐atmosphere, semi‐Lagrangian models. II: Kinematic equation
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Nicholas W. Wood, A A White, Andrew StaniforthAbstract:Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain Rotation Matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This Rotation Matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the Rotation Matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the Rotation Matrix in the spherical polar case involves three matrices, one of which represents Rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total Rotation Matrix results when the great circle Rotation Matrix is replaced by the identity Matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere Rotation Matrix agrees with that used by ECMWF and Meteo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
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treatment of vector equations in deep atmosphere semi lagrangian models ii kinematic equation
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Nigel Wood, A A White, Andrew StaniforthAbstract:Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain Rotation Matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This Rotation Matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the Rotation Matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the Rotation Matrix in the spherical polar case involves three matrices, one of which represents Rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total Rotation Matrix results when the great circle Rotation Matrix is replaced by the identity Matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere Rotation Matrix agrees with that used by ECMWF and Meteo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
Nigel Wood - One of the best experts on this subject based on the ideXlab platform.
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treatment of vector equations in deep atmosphere semi lagrangian models ii kinematic equation
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Nigel Wood, A A White, Andrew StaniforthAbstract:Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain Rotation Matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This Rotation Matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the Rotation Matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the Rotation Matrix in the spherical polar case involves three matrices, one of which represents Rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total Rotation Matrix results when the great circle Rotation Matrix is replaced by the identity Matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere Rotation Matrix agrees with that used by ECMWF and Meteo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
Nicholas W. Wood - One of the best experts on this subject based on the ideXlab platform.
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Treatment of vector equations in deep‐atmosphere, semi‐Lagrangian models. II: Kinematic equation
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Nicholas W. Wood, A A White, Andrew StaniforthAbstract:Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain Rotation Matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This Rotation Matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the Rotation Matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the Rotation Matrix in the spherical polar case involves three matrices, one of which represents Rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total Rotation Matrix results when the great circle Rotation Matrix is replaced by the identity Matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere Rotation Matrix agrees with that used by ECMWF and Meteo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
