Finite rotation analysis and consistent linearization using projectors

Nour-Omid, B.; Rankin, C.

doi:10.1016/0045-7825(91)90248-5

Cited by 247 publications

(165 citation statements)

References 22 publications

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“…This demonstration is in agreement with the more rigorous theoretical treatment by Simo & Vu-Quoc (1986) and Nour-Omid & Rankin (1991). The symmetric part of the correction stiffness matrix (16) coincides with the additional stiffness matrix (14), and hence the semi-tangential adjustment matrix, for When the modified Newton-Raphson iteration method is used, in which case the structure tangent stiffness matrix is determined at the start of each new increment only following the convergence of the last increment, the correction stiffness matrix (16) reduces to the additional stiffness matrix (14).…”

Section: Other Intricate Issues Associated With Spatial Rotationssupporting

confidence: 76%

Spatial Rotation Kinematics and Flexural–Torsional Buckling

Teh

2005

J. Eng. Mech.

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Abstract:This paper aims to clarify the intricacies of spatial rotation kinematics as applied to 3D stability analysis of metal framed structures with minimal mathematical abstraction. In particular, it discusses the ability of the kinematic relationships traditionally used for a spatial Euler-Bernoulli beam element, which are expressed in terms of transverse displacement derivatives, to detect the flexural-torsional instability of a cantilever and of an L-shaped frame. The distinction between transverse displacement derivatives and vectorial rotations is illustrated graphically. The paper also discusses the symmetry and asymmetry of tangent stiffness matrices derived for 3D beam elements, and the concepts of semi-tangential moments and semi-tangential rotations. Finally, the fact that the so-called vectorial rotations are independent mathematical variables are pointed out.

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Section: Other Intricate Issues Associated With Spatial Rotationssupporting

confidence: 76%

Spatial Rotation Kinematics and Flexural–Torsional Buckling

Teh

2005

J. Eng. Mech.

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“…The calculations are performed in the local element coordinate system after the finite rotation transformation 41 . We use a projector-consistent stress-stiffness matrix 42,43 to achieve a quadratic rate of convergence.…”

Section: Finite Element Modelmentioning

confidence: 99%

Finite-strain, finite-size mechanics of rigidly cross-linked biopolymer networks

et al. 2013

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The network geometries of rigidly cross-linked fibrin and collagen type I networks are imaged using confocal microscopy and characterized statistically. This statistical representation allows for the regeneration of large, three-dimensional biopolymer networks using an inverse method. Finite element analyses with beam networks are then used to investigate the large deformation, nonlinear elastic response of these artificial networks in isotropic stretching and simple shear. For simple shear, we investigate the differential bulk modulus, which displays three regimes: a linear elastic regime dominated by filament bending, a regime of strain-stiffening associated with a transition from filament bending to stretching, and a regime of weaker strain-stiffening at large deformations, governed by filament stretching convolved with the geometrical nonlinearity of the simple shear strain tensor. The differential bulk modulus exhibits a corresponding strain-stiffening, but reaches a distinct plateau at about 5 % strain under isotropic stretch conditions. The small-strain moduli, the bulk modulus in particular, show a significant size-dependence up to a network size of about 100 mesh sizes. The large-strain differential shear modulus and bulk modulus show very little size-dependence.

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“…Crisfield and Moita [17] presented a unified co-rotational framework for solids, shells and beams, which has strong links with the work done by Nour-Omid and Rankin [6] and inherits the merit of element-independence. It can be regarded as a simplified version since some terms are omitted in the derivation of the internal forces and the tangent stiffness.…”

Section: Introductionmentioning

confidence: 99%

“…The geometrical stiffness is independent of the local element and therefore any linear element can be extended to geometrically nonlinear analysis with this co-rotational algorithm. Rankin and Brogan [4] presented the EICR algorithm as a general framework, then authors (Rankin and Nour-Omid [5]; Nour-Omid and Rankin [6]) pointed out that the element-independent approach presented in reference (Rankin and Brogan [4]) cannot keep the internal force filed self-equilibrium and further proposed a projector operator to improve its performance. In addition, due to the non-additive nature of large rotations in a three-dimensional space, a pertaining transformation matrix was presented in reference (Nour-Omid and Rankin [6]).…”

Section: Introductionmentioning

confidence: 99%

“…Rankin and Brogan [4] presented the EICR algorithm as a general framework, then authors (Rankin and Nour-Omid [5]; Nour-Omid and Rankin [6]) pointed out that the element-independent approach presented in reference (Rankin and Brogan [4]) cannot keep the internal force filed self-equilibrium and further proposed a projector operator to improve its performance. In addition, due to the non-additive nature of large rotations in a three-dimensional space, a pertaining transformation matrix was presented in reference (Nour-Omid and Rankin [6]). The detailed derivation of the EICR method also can be found in references (Crisfield [7]; Felippa and Haugen [8]), in which Felippa and Haugen [8] introduced several related techniques, such as the mathematics of finite rotations, the fitting methods to satisfy invariance to nodal ordering and so on.…”

Section: Introductionmentioning

confidence: 99%

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A Co-Rotational Framework for Quadrilateral Shell Elements Based on the Pure Deformational Method

Tang

Liu

Chan

2018

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ABSTRACT:In this paper, a novel co-rotational framework for quadrilateral shell element allowing for the warping effect based on the pure deformational method is proposed for geometrically nonlinear analysis. This new co-rotational framework is essentially an element-independent algorithm which can be combined with any type of quadrilateral shell element. As the pure deformational method is adopted, the quantities of the shell element can be reduced, such that it needs less computer storage and therefore enhances the computational efficiency. In the proposed framework, the geometrical stiffness is derived with a clear physical meaning and regarded as the variations of the nodal internal forces due to the motions and deformations of shell element. Furthermore, the warping effect of quadrilateral shell element is considered, and for this reason, the structures undergoing significant warping can be efficiently solved by the proposed formulation. Finally, several benchmark examples are used to verify the validation and accuracy of the proposed method for geometrically nonlinear analysis.Keywords: element-independence, co-rotational method, flat quadrilateral shell element, pure deformation, geometrically nonlinear analysis, explicit tangent stiffness DOI: 10.18057/IJASC. 2018.14.1.6 INTRODUCTIONShell structures and steel frames made up of thin-wall members are commonly used in civil and structural engineering. Generally, it needs huge computer time to analyze a structure by finite shell elements compared with beam-column elements, especially in nonlinear analyses. Thus, the development of shell elements with high performance as well as the co-rotational framework with improved computational efficiency continuously attracts the attention of many researchers.There are three common methods based on the Lagrangian kinematic description for geometrically nonlinear analyses, i.e. total Lagrangian (TL), updated Lagrangian (UL) and co-rotational approaches. The last method is latest and attracts more attention than the others recently due to its simplicity and efficiency. The basic idea of the co-rotational method is that the total motions of an element can be divided into two parts, including the rigid body motions and the pure deformations.To decompose them, a local frame is attached to and co-rotated with the element. Then, the translations and rotations of the local frame are taken as the rigid body movements, while the deformations of the element produced in the local frame are regarded as the pure deformations. In reality, the classification of the Lagrangian kinematic descriptions for geometrically nonlinear analysis is not absolute. The co-rotational approach can be incorporated with either the total Lagrangian (TL) or the updated Lagrangian (UL) formulations. In the following, some research works related to the co-rotational description are discussed.

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Finite rotation analysis and consistent linearization using projectors

Cited by 247 publications

References 22 publications

Spatial Rotation Kinematics and Flexural–Torsional Buckling

Spatial Rotation Kinematics and Flexural–Torsional Buckling

Finite-strain, finite-size mechanics of rigidly cross-linked biopolymer networks

A Co-Rotational Framework for Quadrilateral Shell Elements Based on the Pure Deformational Method

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