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Abstract

This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the element independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.

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... Crisfield et al. [18] provided a comprehensive discussion of the consistency of corotational formulations for different elements. Felippa and Haugen [19] summarized existing EICR formulation and presented a unified theoretical framework for smallstrain corotational finite elements. The CRF exhibits efficient processing capabilities in dealing with dynamic and static problems, especially suitable for the nonlinear behavior of complex structures. ...
... The concept of finite rotations needs to be introduced, and Argyris [40] provided a detailed and comprehensive explanation of the intricacies of finite rotations. Many previous studies have been conducted based on this approach for the investigation of spatial rotations [17,19]. The special Euclidean group SE(3) is another way to describe spatial translations and rotations. ...
... To obtain the transformation relation between the element's global frame and the local frame, it is necessary to establish the projection relations between the displacement vector in the global frame and the deformation displacement vector in the corotational frame [19]. Combining Eq. (20), the variation expression for the elastic translation of nodes in the corotational frame can be represented as follows ...
Article
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This paper presents a new framework for corotational shell elements. The traditional corotational formulation uses local element frames, which greatly facilitates the calculation of elastic forces. However, all force vectors eventually need to be transformed into the global frame, resulting in the loss of invariance. The special Euclidean group SE(3) is introduced to describe the kinematics of the corotational shell element in the local frame. The equations of motion are established, in which the internal forces, inertial forces and tangent matrices are systematically derived in the SE(3) framework. The force vectors and their derivative matrices under the SE(3) description eliminate the effect of the rigid body motion, which is only related to the local deformation of the elements. Some examples are used to verify the validity and efficiency of the presented corotational shell element based on SE(3) to handle geometrically nonlinear problems. The results demonstrate that the SE(3) framework has higher computational efficiency with larger step size compared to the Lie group R³ × SO(3). According to the framework invariance brought by SE(3), a constant mass matrix during iterations is adopted to deal with the nonlinear problems with large rotation and small strain, which can significantly reduce the computational time. In summary, the results of the study show that the SE(3) framework has better characteristics and broader application prospects.
... Based on the mesoscopic Boltzmann equation, LBM features its high computational performance due to the linear local collision operation and the ease for coding and parallelization. As for numerical methods for simulating the solid structure, the corotational FEM [8,15] based on a beam or shell model appears to be an adequate candidate, as energy harvesting devices are usually integrated in slender solid structures with significantly large displacements, and the corotational FEM can efficiently simulate the large motion of this type of slender structure. Hence, it can be expected to provide an efficient numerical framework for simulating FSPEI problems by combining these two numerical methods mentioned above. ...
... In the present numerical framework, the corotational beam formulation [7,8,15,27] is adopted for the finite element method to be able to take into account large displacements and rotations of the beam. The main idea of the corotational formulation is to decompose the motion of each beam element into a rigid body part and a deformational part which is locally small in the corotating coordinate system. ...
... As stated in [15], the corotational formulation follows the assumption: 'displacements and rotations may be arbitrarily large, but deformations must be small.' Hence, the corotational configuration Ω e can be considered as constant in the following spacial integral with s ∈ [0, ℓ 0 e ], where ℓ 0 e denotes the initial length of the eth beam-element. ...
Article
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In the present work, we propose a numerical coupling scheme combining the corotational Finite Element Method (FEM) for solid beams, the regularized Lattice Boltzmann Method (LBM) for fluid flows and a PiezoElectric Model (PEM) for energy harvesting module. The proposed coupling method can be employed to numerically solve Fluid-Structure-PiezoElectric Interaction (FSPEI) multiphysics problems which are the fundamental phenomena in the considered energy harvesting applications. In the proposed coupling method, the FEM and LBM are strongly coupled through the implicit Immersed Boundary Method (IBM) ensuring exactly the no-slip condition. Comparing with volumetric FEMs, the adopted corotational beam formulation can not only allow for large structural displacements, but also help improve the computational efficiency, as the solid structure for energy harvesting devices is usually slender under large motion. The piezoelectric effects are incorporated into the corotational beam dynamics with the help of the virtual work principle. The FEM, LBM and PEM are coupled together in a strong way in the sense that the time integrations are carried out without staggered communications. The proposed coupling scheme is assessed by a series of validation test-cases with increasing complexity, in which a good agreement is obtained with references.
... This class still allows efficient simulations of threedimensional bodies with appropriate modal reduction. Finally, we encounter nonlinearly-elastic multibody systems as The term corotational was already used in the sixties/seventies by the continuum mechanics / finite element community as mentioned in [33]. The article [33] presents not only a unified framework but also a literature review of corotational finite element formulations in structural dynamics, where the corotational idea in finite element methods is attributed to [34,35] according to [36]. ...
... Finally, we encounter nonlinearly-elastic multibody systems as The term corotational was already used in the sixties/seventies by the continuum mechanics / finite element community as mentioned in [33]. The article [33] presents not only a unified framework but also a literature review of corotational finite element formulations in structural dynamics, where the corotational idea in finite element methods is attributed to [34,35] according to [36]. More recent information of the topic may be also found in [37]. ...
... The term is indeed less common in the flexible multibody dynamics community and often contrasted to classical flexible multibody dynamics formulations such as the FFRF [2]. However, the fundamental kinematic assumption of an additive decomposition of the total displacement field into an arbitrarily large rigid body motion and superimposed small deformations is the same [34,33]. Although, compared to corotational finite element formulations, where each finite element has one moving 2 frame per element, corotational flexible multibody dynamics formulations have one moving frame per body or, at most, per substructure, i.e., a subdivision of one body into multiple parts -hence, small deformations with respect to each substructure may end up representing "larger" deformations on a body level. ...
Article
Corotational formulations play an important role for flexible multibody dynamics systems, because they reflect the nature of many technical systems undergoing arbitrarily large rigid body motions but small deformations within each body. This paper defines flexible multibody dynamics and corotational formulations in this context. Furthermore, the «ingredients» and workflow of a flexible multibody dynamics simulation are briefly addressed for the reader less familiar with the topic. This part also points to major review papers and textbooks in the field, and embeds the unified formulations in the literature. The paper's main part presents state-of-the-art corotational flexible multibody dynamics formulations in a systematic and unified way. In this formulation part, the standard integral-based floating frame of reference formulation with modal reduction and with the conventionally employed lumped mass approximation is presented, and its drawbacks highlighted. Then, the so-called nodal-based, i.e., space-wise discretized, equations of motion are presented for several up-to-date nodal-displacement-based formulations within a unified framework. This approach clearly shows the equivalence of the presented formulations, and highlights the fact that the formulations differ only in the choice of degrees of freedom. Moreover, this contribution also intends to reduce the information and complexity within the scientific literature, since this unified framework allows the derivation of these formulations with significantly less effort.
... However, beam FE models based on corotational approaches [19][20][21][22] have been largely preferred. These allow the decoupling of the rigid body motions of the element, treated under the assumption of large displacement, from the deformation displacements, which allows the adoption of constitutive relationships formulated under the hypothesis of small deformations [6,[23][24][25]. ...
... For linear elastic material, Young's modulus = 70608 MPa and Poisson ratio = 0.3 are assumed. In addition to the Chicon's solution, a reference response is computed with standard force-based beam elements, that is, based on linear kinematic for the description of the intra-element deformations, but accounting for large displacements and rotations of the end nodes by means of the corotational formulation [6,23,24,47]. For this solution, an overkilling mesh is considered, made of 88 FEs, i.e., 40 FEs for the column and 48 FEs for the beam, with 8 FEs on the left-hand side of the load (on the length − ). ...
Article
In force-based beam finite elements, cross-section transverse displacements are often needed for post-processing purposes and for geometrically nonlinear structural analysis. This involves the complex integration of the cross-section strains along the beam axis, typically done by the Curvature and Shear Based Displacement Interpolation (CSBDI) technique. Although, the CSBDI is sufficiently accurate for standard applications, this may cause numerical issues when many quadrature cross-sections are placed along the element length. This work presents a novel technique for computing the transverse displacements of a 3D Timoshenko beam, based on a finite difference approximation of the bending and shear compatibility conditions, which avoids the issues of the CSBDI. The proposed technique is introduced in a force-based finite element formulation with moderately large deformations, endowed with a corotational approach, suitable for analyzing geometrically nonlinear framed structures. Detailed investigation of the accuracy and efficiency of the proposed technique is conducted comparing its performance with that of the CSBDI approach.
... CR formulations were developed for one, two-and threedimensional finite elements, such as beams [8,9], plates [10,11], shells [12], and bricks [13,14], while unified CR formulations for general elements are also available [13,15]. The CR approach has been exploited in various fields of structural analysis, including applications to masonry walls both Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ...
... For completeness, the Abaqus solution obtained with an overkill mesh of CPS4 elements incorporating a finite elasticity formulation is also reported. It can be observed that, at least in the present example, the incidence of non-infinitesimal strains emerges at the final stages of the loading history, as soon as the deformed cantilever tends to take an almost vertical configuration so that any further change of configuration should be ascribed to axial finite elastic deformations rather than to large displacements and rotations, a circumstance where Total or Updated Lagrangian finite elasticity formulations are more accurate [15]. The example confirms nevertheless that the presented CR-EVE approach leads to satisfactory results in terms of accuracy with respect to the homologous reference numerical solutions even when coarse meshes are employed. ...
Article
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An enhanced virtual element formulation for large displacement analyses is presented. Relying on the corotational approach, the nonlinear geometric effects are introduced by assuming nodal large displacements but small strains in the element. The element deformable behavior is analyzed with reference to the local system, corotating with the element during its motion. Then, the large displacement-induced nonlinearity is accounted for through the transformation matrices relating the local and global quantities. At the local level, the Virtual Element Method is adopted, proposing an enhanced procedure for strain interpolation within the element. The reliability of the proposed approach is explored through several benchmark tests by comparing the results with those evaluated by standard virtual elements, finite element formulations, and analytical solutions. The results prove that: (i) the corotational formulation can be efficiently used within the virtual element framework to account for geometric nonlinearity in the presence of large displacements and small strains; (ii) the adoption of enhanced polynomial approximation for the strain field in the virtual element avoids, in many cases, the need for ad-hoc stabilization procedures also in the nonlinear geometric framework.
... This method, first presented for SOFA in [17], relies on a representation based on three-dimensionnal Timoschenko beam theory [18] and a specific corotational formulation to account for large displacements [19,20]. It is implemented within the BeamAdapter 4 plugin of the SOFA framework. ...
... Although ambitious in terms of the realism targeted, the complexity of the model must remain limited. The approach followed is that of the use of three-dimensional beam theory [18] and corotational method [19]. The model was implemented using finite element method within a physics-based simulation platform, the SOFA framework. ...
Chapter
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Cable-Driven Parallel Robots (CDPRs) are a type of robot that is growing in popularity for different kinds of applications. However, the use of cables instead of rigid links makes the modelling of this robot a complex task, and therefore their trajectory planning and control are challenging. Assumptions such as inelastic, massless and non-sagging cables made when the CDPR is small are no longer valid when the robot becomes large. This paper presents a CDPR dynamic model taking into account cable elasticity and sagging, and its implementation within an open-source framework, named SOFA. Finally, the simulation results are compared to experiments conducted on a suspended CDPR.KeywordsCable-Driven Parallel RobotsDynamic ModelingFinite Element Method Cable ModelBeam Theory
... The core module of Project Chrono (Chrono::Engine) supports the modelling of non-linear Finite Element Analysis (FEA) to solve flexible multibody systems [29]. In this work, the flexible elements are solved with the co-rotational (CR) approach, whose theory can be seen for instance in Belytschko and Glaum [49], and Felippa and Haugen [50]. The CR concept is a Finite Element Method (FEM) that allows large displacements and rotations, but strains and deformations must be small when linear systems are considered. ...
... Finally, f in and K e are transformed from the local system into global coordinates following the approach presented in [50] and in [52]. The method can be applied to beam elements composed of two nodes and 6-DOFs such as the classical Euler-Bernoulli beams available in Project Chrono. ...
Article
This work proposes a two-way coupling between a Smoothed Particle Hydrodynamics (SPH) model-based named DualSPHysics and a Finite Element Analysis (FEA) method to solve fluid–structure interaction (FSI). Aiming at having a computationally efficient solution via spatial adjustable resolutions for the two phases, the SPH-FEA coupling herein presented implements the Euler–Bernoulli beam model, based on a simplified model that incorporates axial and flexural deformations, to introduce a solid solver in the DualSPHysics framework. This approach is particularly functional and very precise for slender beam elements undergoing large displacements, and large deformations can also be experienced by the structural elements due to the non-linear FEA implementation via a co-rotational formulation. In this two-way coupling, the structure is discretised in the SPH domain using boundary particles on which the forces exerted by fluid phases are computed. Such forces are passed over to the FEA structural solver that updates the beam shape and, finally, the particle positions are subsequently reshuffled to represent the deformed shape at each time step. The SPH-FEA coupling is validated against four reference cases, which prove the model to be as accurate as other approaches presented in literature.
... A correct and effective finite element model must be established for describing the geometrical nonlinear effect of the telescopic boom. At present, three approaches are often used for the finite element analysis of nonlinear solid and structural mechanics, namely total Lagrangian (TL; Pai et al., 2000;Nanakorn and Vu, 2006), updated Lagrangian (UL; Yang et al., 2007;Iu and Bradford, 2010), and co-rotational (CR) formulations (Crisfield and Moita, 1996;Felippa and Haugen, 2005;Li, 2007). Specifically, the CR formulation is suitable for describing the geometric nonlinearity of slender structures whose displacements and rotations may be arbitrarily large, while the local deformations are small. ...
... Therefore, this paper uses Cardan angles to establish the transformation matrix between local coordinate system and global coordinate system. Crisfield and Moita (1996) presented a unified formulation of the co-rotational approach for 3D elements with both translational and rotational degrees of freedom (Battini and Pacoste, 2002;Felippa and Haugen, 2005). The local coordinate system of the substructure is established at one side node of the substructure, and it is described by the global rotational angles (Cardan angles). ...
Article
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The instability load for the telescopic boom of an all-terrain crane is investigated in this paper. Combined with structural characteristics of the telescopic boom, each boom section is divided into several substructures, and the fixed-body coordinate system of each substructure is established based on the co-rotational method. A 3D Euler–Bernoulli eccentric beam element of the telescopic boom is derived. On the premise of considering the discretization of gravity and wind load, internal degrees of freedom of the substructure are condensed to the boundary nodes, forming a geometrical nonlinear super element. According to the nesting mode of the telescopic boom, a constraint way is established. The unstressed original length of the guy rope is calculated with a given preload so as to establish the equilibrium equations of the boom system with the external force of the guy rope and the corresponding tangent stiffness matrix. Regarding the above work, a new method for calculating the structural equilibrium path and instability load of telescopic boom structure is presented by solving the governing equations in a differential form. Finally, the method is validated by examples with different features.
... This is a Mindlin-Reissner model enriched by independent in-plane displacements of the soft layers. The same work implements a locking-free shell nite element with the geometrically nonlinear model recovered by the co-rotational approach [5]. The kinematics with independent shear deformations of the soft layers [3] results useful also for including thermal and viscous eects [6] and for modeling more general boundary conditions. ...
... The Finite Element Analysis (FEA) structural solver integrated in the core module of Project Chrono (Chrono::Engine) [18], supports the simulation of flexible structures. This library implements a non-linear FEA by using a co-rotational (CR) approach (see for instance [26]), which allows the deformable elements under this formulation to be capable of experiencing large deformations and rotations. The flexible elements presented in this work are solved as corotated 3D Euler-Bernoulli, whose formulation is based on the work presented in [27]. ...
Conference Paper
In the context of solving fluid-elastic structure interaction (FSI) problems involving ultra-thin elements, this paper presents a novel approach by using the extended two-way coupling between the DualSPHysics model and the Finite Element Analysis (FEA) structural solver integrated in Project Chrono. The flexible structure herein presented is based on the Euler-Bernoulli beam model, which incorporates axial and flexural deformations in 3D. The beam element is embedded into the Smoothed Particle Hydrodynamics (SPH) domain using an envelope subdomain that is discretized using dummy boundary particles. The presented dummy envelope serves as a decoupling interface for the geometrical properties of the structure, allowing for ultra-thin structures smaller than the initial inter-particle distance. Without the use of variable or adaptive particle resolutions , this setup extends the computational range of SPH as a tool to solve engineering problems with thin structural elements. To confirm the validity of this approach, the coupling is validated against two experimental setups consisting of flexible vegetation swaying under the action of oscillatory flow.
... In this contribution, a lumped mass matrix, in which a diagonal mass matrix (from the mass density ρ) is integrated over the volume of each element is employed. The stiffness matrix K is computed based on the corotational FEM, in which the rigid body motion from total finite element displacements is extracted (since it does not contribute to element deformations, see [34]). Note that the corotational FEM formulation makes it possible to handle large rotations for both, needle as well as tissue. ...
Preprint
We present an error-controlled mesh refinement procedure for needle insertion simulation and apply it to the simulation of electrode implantation for deep brain stimulation, including brain shift. Our approach enables to control the error in the computation of the displacement and stress fields around the needle tip and needle shaft by suitably refining the mesh, whilst maintaining a coarser mesh in other parts of the domain. We demonstrate through academic and practical examples that our approach increases the accuracy of the displacement and stress fields around the needle without increasing the computational expense. This enables real-time simulations. The proposed methodology has direct implications to increase the accuracy and control the computational expense of the simulation of percutaneous procedures such as biopsy, brachytherapy, regional anesthesia, or cryotherapy and can be essential to the development of robotic guidance.
... Using linear elasticity for modelling of soft tissues results in artifacts for large rotational deformation [35]. To overcome this issue, the stiffness matrix is computed based on the corotational formulation [36], in which the rigid motion can be extracted from the total finite element displacements. The element nodal internal force becomes ...
Preprint
This paper describes the use of the corotational cut Finite Element Method (FEM) for real-time surgical simulation. Users only need to provide a background mesh which is not necessarily conforming to the boundaries/interfaces of the simulated object. The details of the surface, which can be directly obtained from binary images, are taken into account by a multilevel embedding algorithm applied to elements of the background mesh that cut by the surface. Boundary conditions can be implicitly imposed on the surface using Lagrange multipliers. The implementation is verified by convergence studies with optimal rates. The algorithm is applied to various needle insertion simulations (e.g. for biopsy or brachytherapy) into brain and liver to verify the reliability of method, and numerical results show that the present method can make the discretisation independent from geometric description, and can avoid the complexity of mesh generation of complex geometries while retaining the accuracy of the standard FEM. Using the proposed approach is very suitable for real-time and patient specific simulations as it improves the simulation accuracy by taking into account automatically and properly the simulated geometry.
... In the various corotational frameworks, implementations of beam theory can be found e.g. in work of Refs. [5,11,17]. The generalized strain beam formulation used in the current work can be considered a corotational approach, though formulated in absolute coordinates and without rotation interpolation issues [7,21,33]. ...
Article
Full-text available
This paper presents the stiffness formulation of a beam element with the relevant third-order nonlinear geometric effects for relatively wide and thin rectangular beams, in particular when loaded in the plane and simultaneously deformed out of the plane. The element is initially straight in its undeformed configuration. The formulation is based on Timoshenko beam theory with nonuniform torsion and Wagner effects. The derivation is carried out by means of the Hellinger–Reissner variational principle with custom interpolation functions. The element is incorporated into the generalized strain beam formulation for multibody systems. Numerical simulations of precision flexure mechanisms show that the use of a single third-order element per flexible member can already yield adequate performance, at a significant reduction of the necessary degrees of freedom and the computation time, compared with using multiple second-order elements in the generalized strain beam formulation.
... A Mindlin-Reissner model enriched by independent in-plane displacements of the soft layers is proposed in Refs. [5,16], where the equal rotation of the stiff layers is directly a variable and the geometrically nonlinear model is recovered by the co-rotational strategy [17]. Besides shell models, the solid-shell approach [18][19][20][21] is based on a 3D continuum model with an assigned kinematic approximation in the thickness direction. ...
Article
Glass laminates consist of stiff glass plies permanently shear-coupled by polymeric interposed layers. When an external temperature rise occurs, the interlayers undergo a dramatic stiffness decay. As a consequence, not only the sectional warping typical of alternating stiff/soft composites is intensified, but also the overall behavior may evolve counter-intuitively. When slender elements prone to geometric nonlinearities are involved, even small thermal variations in intensity or distribution may act as uncertainty factors, strongly affecting the outcome. This paper proposes an efficient, robust, and accurate numerical framework to perform the sensitivity analysis to thermo-mechanical actions in glass plates. A large deformation isogeometric Kirchhoff-Love shell model enriched with through-the-thickness warping is employed, together with a generalized arc-length method involving a suitable temperature parameter as an additional unknown, namely the thermal amplifier or a spatial distribution coefficient. Numerical experiments are presented to highlight the effects that even small temperature variations produce on the equilibrium paths and the influence of the stiffness loss in the interlayer on the structural behavior and the accuracy of the models.
... (5) Kinematics can be either corotational or linear. The corotational kinematics is based on the work of Felippa et al., i.e., the EICR (Felippa, 2000;Felippa & Haugen, 2005) (Element Independent Corotational formulation). Quaternions are used to handle finite rotations. ...
Article
Full-text available
To simulate the realistic nonlinear behavior of reinforced concrete shear walls, which are an efficient structural system for resisting lateral forces, it is crucial to choose the appropriate numerical models and constitutive materials. Because of this, the current research, through two research portions, examines the seismic response of RC shear walls using a combined model of a multi-layer shell element and fiber beam element. This combined model of RC shear walls is guaranteed to be used in sequential and parallel analyses by comparing the results of using several model types of concrete’s nonlinear materials and shell elements. First, this combined model is used to calibrate the hysteretic curve resulting from the experiment of an RC shear wall subjected to a cyclic load. Second, the seismic response of RC shear walls in a building, with critical dimensions and numbers of shear walls, designed according to the Syrian Arab Code and subjected to a strong earthquake is assessed. The analysis’s findings demonstrate that the combined model of the RC shear wall and the experimental data from the hysteretic curve calibration agree well. It also notes that the performance level of the studied building, for all material types, reached collapse and collapse prevention levels according to damage and story drift ratio approaches, respectively, indicating that the shear walls’ design using the equivalent static method is unsafe for this studied building. This emphasizes the necessity of using sophisticated nonlinear models of materials and elements along with performance-based seismic design to develop the shear walls’ actual behavior.
... A corotational approach [27][28][29][30] is employed to use the linear version of MISS-4C presented above to conduct geometrically nonlinear static analyses [12,31]. First, the FE mixed energy Eq. ...
Article
Full-text available
Variable stiffness (VS) composite laminates provide larger freedom to design thin- walled structures than constant stifness (CS) composite laminates. They showed to allow the redistributing of stresses, improving buckling and post-buckling performance and, therefore, reducing material weight and costs. This work extends a recently developed mixed shell element, MISS-4C, to the postbuckling analysis of VS composite laminate structures. MISS-4C has a linear elastic closed-form solution for the stress interpolation of symmetric composite materials. Its stress field interpolation is obtained by the minimum number of parameters, making it an isostatic element. Moreover, its kinematic is only assumed along its contour, leading to an efficient evaluation of all operators obtained through analytical integration along the element contour. MISS-4C uses a corotational approach within a fast multi-modal Koiter algorithm to efficiently obtain the initial post-buckling response of VS composite laminate structures. First, the element performance is investigated by analysing a carbon fiber VS composite laminate plate subjected to compressive stresses. Numerical results obtained with MISS-4C are compared with those obtained with the MISS-4 element, showing good accuracy and a high convergence rate. Subsequently, the structural response of a glass bridge VS composite laminate girder of a short length bridge is optimised through a multi-objective optimisation that exploits the robustness of the MISS-4C element and the efficiency of the multi-modal Koiter algorithm.
... 106-112). The determination of the structure's nonlinear response with 3DFloat is based on a corotated FEM approach [41]. In this approach, the element equations are formulated at each time step in a coordinate system attached to a reference configuration, which represents a deformed state of the previous time step. ...
Article
Full-text available
As the industry transitions toward Floating Offshore Wind Turbines (FOWT) in greater depths, conventional chain mooring lines become impractical, prompting the adoption of synthetic fiber ropes. Despite their advantages, these mooring lines present challenges in inspection due to their exterior jacket, which prevents visual assessment. The current study focuses on vibration-based Structural Health Monitoring (SHM) in FOWT synthetic mooring lines under uncertainty arising from varying Environmental and Operational Conditions (EOCs). Six damage detection methods are assessed, utilizing either multiple models or a single functional model. The methods are based on Vector Autoregressive (VAR) or Transmittance Function Autoregressive with exogenous input (TF-ARX) models. All methods are evaluated through a Monte Carlo study involving 1100 simulations, utilizing acceleration signals generated from a finite element model of the OO-Star Wind Floater Semi 10 MW wind turbine. With signals from only two measuring positions, the methods demonstrate excellent results, detecting the stiffness reduction of a mooring line at levels 10% through 50%. The methods are also tested for healthy cases, with those utilizing TF-ARX models achieving zero false alarms, even for EOCs not encountered in the training data.
... This is a Mindlin-Reissner model enriched by independent in-plane displacements of the soft layers. The same work implements a locking-free shell finite element with the geometrically nonlinear model recovered by the co-rotational approach [31]. The kinematics with independent shear deformations of the soft layers [22] results useful also for including thermal and viscous effects [32] and for modeling more general boundary conditions. ...
Article
This paper presents a large deformation Kirchhoff-Love shell model hierarchically enhanced with through-the-thickness warping functions, arbitrarily chosen by the user. Two unknowns are introduced for each of them, representing its amplitudes in two directions tangent to the shell surface. NURBS are used to approximate reference surface displacement and warping amplitudes in the weak form. The transverse shear strains are linear functions of the warping parameters only and naturally free from locking. A patch-wise reduced integration avoids membrane locking and improves efficiency. Particular attention is paid to the modeling of composites made up of multiple stiff layers coupled with soft interlayers. The alternating layup with high stiffness ratios induces a significant sectional warping with transverse shear strains concentrated in the soft layers. Two warping models are investigated: (WI) all stiff layers maintain the same director orthogonal to the deformed surface with independent transverse shear deformations of the soft layers; (WZ) a single zigzag function linking these deformations. The numerical tests confirm the great accuracy of the hierarchic shell model in reproducing the solid solution with a small number of discrete parameters, provided that the correct warping model is chosen. WI is reliable for all alternating layups. WZ reduces the unknowns to five per surface point, regardless of the number of layers, and is accurate for uniform soft layers.
... In the formulation of small-strain corotational 2D shell elements, Felippa and Haugen (2005) indicated that the geometric stiffness matrix is non-symmetric at the element level. However, similar to Crisfield's assertion, it was also stated that the assembled global geometric stiffness becomes symmetric as global equilibrium is approached if there are no applied nodal moments and the displacement boundary conditions are conserving. ...
... The corotational formulation(CR) is one of the three geometrically nonlinear formulations, the other two are the total Lagrangian(TL) and updated Lagrange (UL) [25]. The main difference between these three methods is the choice of reference configuration. ...
... In the formulation of small-strain corotational 2D shell elements, Felippa and Haugen (2005) indicated that the geometric stiffness matrix is non-symmetric at the element level. However, similar to Crisfield's assertion, it was also stated that the assembled global geometric stiffness becomes symmetric as global equilibrium is approached if there are no applied nodal moments and the displacement boundary conditions are conserving. ...
... This is a Mindlin-Reissner model enriched by independent in-plane displacements of the soft layers, for which the equal rotation of the stiff layers is a direct variable. The same work provides a locking-free shell finite element and the geometrically nonlinear model is recovered by the co-rotational strategy, suitable for small strain problems (Felippa and Haugen, 2005). Further developments are reported in Liang et al. (2016), in particular for what concerns the modeling of creep in the viscoelastic interlayers. ...
Article
Solid-shell models are developed for the geometrically nonlinear analysis of multi-layered composite structures made of alternating layers with large difference in material properties. Exemplificative applications are presented for laminated glass, in which a number of stiff plies of glass are permanently shear-coupled by soft interlayers. The sectional warping due to significant transverse shear strains in the soft layers makes theories of laminated plates based on the plane-section hypothesis unreliable. The proposed approach is based on a geometrically exact solid-shell finite element model with one element per layer in the thickness direction, as alternative to solid discretization. The element approximation is based on the displacement nodal values at the top and bottom surfaces of the layers, with a natural C0 continuity. An alternative solid-shell model with fewer parameters is derived imposing the equal finite rotation of the stiff layers at each surface point by a local rotation-free re-parametrization of the nodal displacements and enforcing the plane stress condition. The approach permits an easy coupling with a fully solid discretization, e.g. to model connections, and is based on a simple strain measure quadratic in the displacement unknowns and suitable for finite strains. Extensive numerical examples for laminated glass plates and curved shells susceptible to large deflections and buckling are provided, comparing the results with those from a fully solid approach.
... The CR method [30][31][32][33][34][35][36] can be taken as a miscellany of above formulations, where the total change of configuration is shared into a rigid body aliquot and a purely deformation one. The rigid body contribution does not deform the element and is associated to a global coordinate system. ...
Chapter
This paper presents a 2D curvilinear beam finite element model focusing the interest on its use for non-linear analysis caused by very large displacements, addressed with the Update Lagrangian strategy. The method allows using very long curvilinear beams even when high geometric nonlinearities occur. This is due inasmuch the proposed formulation does not require any pre-set shape function that would inevitably force to use a huge number of elements to achieve reliability. The lack of shape-functions is overcome using the integration of the compatibility equations, that provide the whole internal displacement field from the only knowledge of the element nodal degree of freedom. Section-slices subdivision allows to sum, not to assemble, the flexibility contribute of each slice and consequently to build up the end-to-end tangent stiffness matrix of a generic curvilinear beam element. Moreover, the flexibility feature of every slice can be deduced analytically once and for all. To validate the proposed element some comparisons are carried out with analytical and numerical solutions obtained with Runge–Kutta integration method or cubic isoparametric finite elements.KeywordsLarge displacement analysisFinite element methodCurvilinear beam element
... Reference [29] proposed a different formulation of the CR method, of which a review can be found in [30]. Briefly, the beam was discretized into a number of shorter elements that undergo a relative deflection smaller than the beam one, allowing the application of the standard Euler-Bernoulli theory to estimate the deformation of these shorter elements. ...
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There is a growing application of passive exoskeletons in the industrial sector with the purpose to reduce the incidence of work-related musculoskeletal disorders (MSDs). Nowadays, while many passive shoulder exoskeletons have been developed to support overhead tasks, they present limitations in supporting tasks such as load lifting and carrying. Further developments are therefore needed to have a wider application of these devices in the industrial sector. This paper presents a modelling procedure of a passive non-rigid exoskeleton for shoulder support that can be used to evaluate the device in its development phase. The modelling began with the definition of the equations to describe the exoskeleton kinematics and dynamics to obtain the support force profile provided by the device over the shoulder flexion angle. A musculoskeletal simulation software was then used to evaluate the effect of the device on the human body. The computed support force profile is in agreement with the purpose of the device, with the maximal support force obtained for a shoulder flexion angle of 85–90°. The maximum support force value had the same magnitude as the one reported by the device user manual (3.5 kg). In particular, for a determined exoskeleton configuration, the maximum support force value computed was 34.3 N, equal to the reported by the manufacturer. The subsequent musculoskeletal simulation showed the ability of the device to reduce the muscular activation of agonist muscles such as the anterior deltoid (−36.01%) compared to the case when the exoskeleton is not used. The musculoskeletal results showed a positive effect of the device on the joint reaction forces at the glenohumeral joint with a reduction up to 41.91%. Overall the methodology and the mathematical model proposed can be used to further develop these devices, making them suitable for a wider range of tasks.
... The method that applies this approach is the corotational formulation, see, e.g., [24][25][26]. Because the elastic deformation can be described linearly to this frame, reduced-order models can be used in the corotational formulation, see, e.g., [11,27,28]. ...
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Design and optimization of flexure mechanisms and real time high bandwidth control of flexure based mechanisms require efficient but accurate models. The flexures can be modeled using sophisticated beam elements that are implemented in the generalized strain formulation. However, complex shaped frame parts of the flexure mechanisms could not be modeled in this formulation. The generalized strain formulation for flexible multibody analysis defines the configuration of elements using a combination of absolute nodal coordinates and deformation modes. This paper defines a multinode superelement in this formulation, i.e., an element having its properties derived from a reduced linear finite element model. This is accomplished by defining a local element frame with the coordinates depending on the absolute nodal coordinates. The linear elastic deformation is defined with respect to this frame, where rotational displacements are defined using the off-diagonal terms of local rotation matrices. The element frame can be defined in multiple ways; the most accurate results are obtained if the resulting elastic rotations are as small as possible. The inertia is defined in two different ways: the so-called “full approach” gives more accurate results than the so-called “corotational approach” but requires a special term that is not available from standard finite element models. Simulations show that (flexure based) mechanisms can be modeled accurately using smart combinations of superelements and beam elements.
... The details of the CR formulation are provided in Refs. [63,64]. Moreover, the general procedure of the finite element method derives a solution by defining the global stiffness matrix K G and global load vector f G to describe the motion of the discretized structure after elemental formulation. ...
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To develop a predictive digital twin for future structural design or maintenance, a real-time solution for structural analysis is essential. However, a large-scale nonlinear structural analysis still requires recursive procedures that incur high computational costs. In this study, we propose a neural-network-based model order reduction method for a given parameter space. It is realized by combining an autoencoder with a deep neural network to efficiently address high-dimensional data. The key aspects of the proposed approach include the integration of projection-based model reduction for data mining and multistep model reduction. Moreover, the combination of two network architectures, which can learn a direct relationship between the parameter and the nonlinear displacement field, was considered. Transfer learning over the time span of interest was performed to broaden the time history prediction of nonlinear structural dynamics. The proposed approach was compared with the full-order model by considering numerical examples of nonlinear structural dynamics to demonstrate its efficiency and accuracy. As a result, the real-time prediction of nonlinear structural dynamics was achieved. Moreover, the proposed approach showed excellent computational efficiency in parameterized nonlinear structural analyses.
... Many other contributions have been then proposed, especially for the computational efficiency of the method, e.g., [14,15,16]. Felippa and Haugen provided a comprehensive overview on derivation aspects of corotational finite elements in Ref. [17]. A central issue in developing nonlinear 3D beam formulations concerns the treatment of the rotational field because of its noncommutative nature. ...
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This article investigates the dynamic nonlinear response of three-dimensional structures using variable-kinematics finite beam elements obtained with the Carrera Unified Formulation. The formalism enables one to consider the three-dimensional form of displacement–strain relations and constitutive law. The deformation mechanisms and the associated couplings are described consistently with the selected kinematic model. The Hilbert–Hughes–Taylor method and the iterative Newton–Raphson scheme are adopted to solve the motion equations derived in a total Lagrangian scenario. Various models have been obtained by using Taylor- and Lagrange-like expansions. The capabilities of the beam elements are assessed considering isotropic, homogeneous structures with compact and thin-walled sections.
... We note that the corotational formulation is a mechanically consistent approach for expressing geometrically nonlinear behaviors, and thus the tangential stiffness matrix derived by the corotational formulation should be equivalent with other formulations, such as the total and updated Lagrange approaches in global coordinates (Felippa and Haugen 2005). Consequently, the corotational formulation does not necessarily provide a positive tangential stiffness, and thus cannot completely avoid mesh distortion in low-stiffness regions, as indicated in Sect. ...
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Topology optimization for three-dimensional (3D) thin plate structure is an attractive methodology for versatile industrial and biomedical applications. For this perspective, the topology optimization requires an appropriate treatment of the 3D geometric nonlinearity of thin structures that avoids numerical instabilities, which is a well-known challenge in topology optimization. This paper develops a density-based topology optimization for thin plates and considers geometric nonlinearity using a 3D corotational triangle element formulation. The corotational formulation is an approach for expressing a finite deformation by dividing small strains and finite rotations into local element coordinates. Thus, the mechanical behavior in local coordinates can be assumed to be linearly elastic behavior that follows the small strain theorem. This technique is expected to be effective and stable for topology optimization with geometric nonlinearity. Complementary work minimization with volume constraints was applied for density-based topology optimization of the plate structure by a solid isotropic material with penalization method. Numerical examples of two benchmarks demonstrated consistencies with existing related works. We conducted topology optimization of an ankle-foot orthosis (AFO) as a biomedical application and showed the capabilities of the proposed methodology and the minimum increases of the complementary work with an optimum design with a volume reduction ratio. These achievements highlight the capabilities of the developed topology optimization as an efficient framework and feasibilities for a new orthosis design.
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When the free standing riser (FSR) is in service in the ocean, its mechanical properties are affected by various factors, including complex ocean current forces, buoyancy of the buoyancy can, and torque caused by the deflection of the upper floating body. These loads have a great influence on the deformation and internal force of the FSR. The static performance of FSR is investigated in this research under various working conditions. The finite element model of FSR is established based on the co-rotational method. The arc length approach is used to solve the model. The load is exerted in increments. The current load on the riser changes with the configuration of the riser. The accuracy of the numerical method is verified by Abaqus software. The calculation time is also compared. Then, the effects of uniform current, actual current and floating body yaw motion on FSR are studied by parameter analysis. Additionally, the influence of the FSR on the ocean current after the failure of part of the buoyancy can chamber is analyzed. The results show that the numerical model based on the co-rotational method can effectively simulate the large rotation and torsion behavior of FSR. This method has high computational efficiency and precision, and this method can quickly improve the efficiency of numerical calculation of static analysis of deep-water riser. The proposed technology may serve as an alternative to the existing proprietary commercial software, which uses a complex graphical user interface.
Article
The geometric nonlinearity is an important issue for the structural systems which have a tendency of showing the large displacements. Although the axial forces have a big effect in the geometric nonlinearity, the contribution of shear forces has to also be considered. For this purpose, a Timoshenko-based beam element, which is employed to construct the structural system, is utilized to investigate the shear effect on the geometrically nonlinear structural behavior. While MATLAB scripts are included for displaying how to form the corresponding tangent stiffness matrices, the shear effect is evaluated using both the benchmark application tests and OpenSees program. Therefore, the other main contribution of this study is to give the additional suggestions for the element formulation which is developed for Timoshenko beam in OpenSees program. Consequently, it is shown that the shear effect on the structural behavior has an importance for the structural instability.
Article
DSR (Deep steep riser) is a new riser structure that reduces the ultra-high-tension load caused by the riser self-weight. In this paper, the mechanical behavior of DSR under different buoyancy module configurations and different ocean currents is studied. The finite element model of DSR is established based on co-rotational coordinate method. The model is solved by arc length method. The accuracy of the numerical method is verified by Abaqus software. Then, the effects of buoyancy module length and buoyancy factor on DSR are analyzed. Finally, the influence of different current incidence angles and velocities on DSR is evaluated. The results show that the DSR model based on the co-rotational coordinate method can effectively simulate the nonlinear behavior of large deformation of DSR. The method is simple, flexible and computationally efficient. This method can quickly improve the efficiency of numerical calculation in static analysis of deepwater riser. And DSR is feasible under certain conditions.
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The present work describes a co-rotating shear flexible beam element without shear locking and integrating Euler-Bernoulli's and Timoshenko's beam theories. The co-rotational kinematics is based on the separation of the motion in deformational and rigid body components. The deformation of the beam element is composed by three natural modes of deformation: the extension mode, the symmetric bending mode, and the anti-symmetric bending mode. The respective generalized stresses from these natural modes are self-balanced, allowing the achievement of a consistent tangent stiffness matrix. In this paper, it is detailed and deduced all the algebraic steps for the deduction of the elastic stiffness matrix, the geometric stiffness matrix, and the co-rotation stiffness matrix. Some examples are presented and the numerical results demonstrate that the beam element here presented is able to handle large rotations.
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The corotational (CR) method is effective for extending linear elements to geometrically nonlinear analysis, but faces challenges with penalty elements based on modified couple stress theory due to C ¹ continuity requirements. This paper presents an alternative CR formulation for geometrically nonlinear analysis of couple stress elasticity, approximating rigid body rotation incrementally. Then, a 4-node, 12-DOF plane element US-Q4[Formula: see text]-CS for size-dependent materials is extended for geometrically nonlinear analysis. Numerical examples demonstrate the new nonlinear element’s excellent performance in simulating size effects for the case of geometric nonlinearity.
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This paper presents a geometrically nonlinear three-dimensional force-based beam finite element that efficiently accounts for the effects of rigid joint offsets. The model imposes the element equilibrium in the local reference system referring to the element deformed configuration, considering the von Kármán nonlinear terms, and exploits a corotational formulation to account for rigid large displacements and rotations of the beam. https://authors.elsevier.com/a/1iAan_12dr6Y7h
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Shape sensing, which is the real-time monitoring of deformed shapes using discrete surface strain, is a fundamental approach to ensure structural safety, reliability, and affordability. Large deformation shape sensing is obviously more important because large deformations can result in structural damage and failure. Nevertheless, there are few effective methods for the shape sensing of large deformations. Based on Timoshenko beam theory, this paper establishes a new method, called analogy stiffness upgrading (ASU), to reconstruct nonlinear deformation. In this method, the inverse finite element method (iFEM) is used to predict the initial displacement field and compute the analogy stiffness matrix. Then, the analogy stiffness matrix is upgraded by using coordinate transformation from a co-rotational procedure. Through iterative computation, the real displacement field is finally obtained when the rotation angle calculated from the input strain data is the same as the integral result from the section strain data. Numerical examples and model tests are carried out to verify the ASU method. It is evident from the results that the ASU method can predict largely deformed shapes of beam structures with superior precision.
Conference Paper
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Summary: In this paper, the challenges arising in software implementation of large displacements are presented. Nonlinear analysis, including analysis of large displacement, have been implemented in a number of commercial software. However, the nonlinear analysis itself is time consuming, which initiates the need for less time-consuming analyses. In order to achieve that, the corotational formulation was used, which implies the separation of the element displacement into two parts: the displacement of the rigid body and the element deformation. This paper also presents the implementation algorithm, and some representative results are provided.
Thesis
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The present work deals with the development of an analysis-model applicable to large scale 3D beam structures of reinforced and prestressed concrete. The model is based on the finite element method and allows for large displacements through the Corotated Lagrangian description of motion and a variety of material nonlinearities in the short-time as well as the long-time regime. The loading may be both unidirectional and corotational. By relating all changes in loads, prescribed displacements, temperature, time and static system to a common history parameter, the response of a structure may be traced from the very start of construction to its completion, throughout the service life and finally into the ultimate load range. The key ingredient of the analysis-model is the new 3D shear-beam element formulation. It can handle the response of reinforced and prestressed concrete in each one and combinations of the axial, bending, shear and torsion modes. http://urn.nb.no/URN:NBN:no-37298
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A finite element,template is a parametrized,algebraic form,that reduces to specific finite elements by setting numerical,values to the free parameters. Following an outline of high performance elements, templates for Kirchhoff Plate-Bending Triangles (KPT) with 3 nodes and 9 degrees of freedom are studied. A 37-parameter template is constructed using the Assumed,Natural Deviatoric Strain (ANDES) approach. Specialization of this template includes well known,elements such as DKT and HCT. The question addressed here is: can these parameters,be selected to produce,high performance elements? The study is carried out by staged application of constraints on the free parameters. The first stage produces,element,families satisfying invariance and aspect ratio insensitivity conditions. Application of energy balance constraints produces specific elements. The performance,of such elements in a preliminary set of benchmark,tests is reported.
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This study is concerned with a finite element method for the analysis of large deformation problems. The method employs local Cartesian coordinate systems, attached to each element of the structure, which translate and rotate with the elements as the deformations proceed. The primary purpose of the present work was to develop a unified theory, based on the concept of a co-rotational coordinate system, for the analysis of geometrically and materially nonlinear structures. Nonlinear transformations that relate local and global variables are derived, and the global element tangent stiffness matrix is established. The relations between stress and deformation variables in the co-rotating and in the global system are shown.
Book
This paper discusses objective and efficient ways of dealing with large rotation problems in nonlinear finite element analysis. The methods described include a corotational, 'ghost reference' description, updating of strain and stress components, total and incremental virtual work expressions, definition of finite rotations at nodes and incremental updates of these rotations. The treatment of bending type elements with local, rotational freedom is described in further detail. The paper includes examples of large displacements for plate and shell type problems.
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1. Purpose of the non-linear theories. Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.
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The theoretical basis and implementation of computer methods for the transient analysis of solids and structures are described. In addition, certain classes of fluid-structure problems, in which the response of the structure is of primary interest and the behavior of the fluid can be simplified extensively, are considered. The latter include situations such as acoustic models of the fluid or situations where the flow velocity and effects of viscosity are small.
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This book embodies an approach to non-linear elasticity which marks a fundamental departure from classical and current trends. The basic theory was first published between the years 1934 and 1940 in seven papers listed at the end of this Preface. In addition to a systematic treatment of the general theory and extensions to viscoelasticity, the book includes comprehensive new developments and applications, many of which are presented here for the first time. The work is characterized by the use of cartesian concepts and of elementary mathematical methods that do not require a knowledge of the tensor calculus or other more specialized techniques. The explicit introduction of a local rotation field in the three-dimensional equations leads to a theory which separates the physics from the geometry and is equally valid for elastic and non-elastic materials, using either rectangular or curvilinear coordinates. As this book demonstrates, the scope of problems solved by these new methods goes far beyond the results which it has been possible to obtain by the more elaborate and less general traditional approach. New insights, leading to many discoveries and a unified outlook have been brought into such widely diversified areas as rubber elasticity, internal gravity waves in a fluid and tectonic folding in geodynamics. The theory provides rigorous and completely general equations governing the dynamics and stability of solids and fluids under initial stress in the context of small perturbations. It does not require that the medium be elastic or isotropic but is applicable to anisotropic, viscoelastic, or plastic media. No assumptions are introduced regarding the physical process by which the initial stress has been generated. The treatment of viscoelasticity, which constitutes a substantial portion of the book, incorporates some of the results established in my previous work on non-equilibrium thermodynamics. Non-linear theories of deformation and applications to problems of finite strain are obtained by extension ofthe concept of incremental deformation in a medium under initial stress. In contrast to the presentation in the papers listed at the end of this Preface, the concepts and methods are developed primarily in the context of the linearized mechanics of continuous media under initial stress as an independent theory.
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A general nonlinear theory for thin shells of arbitrary midsurface geometry is formulated in terms of a finite rotation vector and a stress-function vector. Compatibility equations, equilibrium equations, and boundary conditions are derived which are valid for shells undergoing arbitrarily large rotations and strains. For problems admitting a potential energy functional, a variational principle is formulated. The simplifications implied by small extensional strains are discussed. The theory contains, as special cases, Reissner’s equations for the axisymmetric deformation of shells of revolution, and the Sanders-Koiter linear shell theory.
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A new approach for enforcing invariance to rigid body motion is introduced. This invariance property is derived in terms of a projection matrix, P, which depends on the rigid body modes and acts on nodal forces, f, independent of the element formulation. The invariance property takes the form . Moreover, the same transformation enforces equilibrium of the element internal force vector when it is not initially satisfied. An equivalent transformation using P is also derived for the element stiffness matrix.This approach has been applied to a number of established elements, and is used to solve two text examples that expose deficiencies of elements. Numerical results indicate marked improvement in the performance of the elements tested.
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In a previous paper [Comput. Methods Appl. Mech. Engrg. 191 (2002) 1755], the authors have presented a 3D co-rotational elastic beam element including warping effects. This formulation is now further developed in order to incorporate elasto-plastic deformations. The element possesses seven degrees of freedom at each node and can be used to model beams with arbitrary cross-sections. Thus, within the present approach, the centroid and shear center of the cross-section are not necessarily coincident. The main purpose of this element is to model elasto-plastic instability problems. In this context, two methods of branch-switching are tested and discussed. In the first one, the bifurcation point is isolated by successive bisections and the branch-switching is operated by using the eigenvector associated to the negative eigenvalue. In the second one, introduced by Petryk, an energy approach is used to select automatically the stable post-bifurcation path. Six examples, including large displacement and stability problems, are used in order to assess the performances of the element.
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A new corotational procedure is developed which enables existing finite element formulations to be used in problems that contain arbitrarily large rotations. Through the use of a nonsingular large rotation vector, the contribution of the rigid body motion of the element to the total displacement field is removed before element computations are performed, with the result that almost any element can be easily upgraded to handle large rotations. This paper contains a derivation of the theory, an outline of the implementation into the STAGS code, and a demonstration of performance for problems involving large rotations and moderate strains.
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It is shown in this paper that Euler was first to derive the finite rotation formula which is often erroneously attributed to Rodrigues, while Rodrigues was responsible for the derivation of the composition formulae for successive finite rotations and the so-called Euler parameters of finite rotation. Therefore, based upon historical facts, the following nomenclature is suggested: Euler's finite rotation formula, Rodrigues' composition formulae of finite rotations, and Euler-Rodrigues parameters. The text of the paper contains modern symbols and formula forms, while the Appendices contain brief summaries from relevant historical sources with minor alterations in symbols at the most.
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The concept of finite elements for the analysis of shells is developed here with several important advances. Firstly, the Kirchhoff theory of shells is refined to include a transverse shear deformation. The refined theory admits simpler approximating functions while preserving continuity at the inter-sections of elements. Secondly, the motion of the element is decomposed into a rigid body motion followed by a deformation. The decomposition serves to extend existing formulations for linearly elastic elements to problems involving finite rotations and buckling. Thirdly, the Lagrange equations are introduced to derive the equations of the discrete system. The method yields the consistent inertial terms for any manner of motion, oscillatory or transient. Finally, the simplest approximating polynomials are introduced in the context of the shear-deformation theory. Further simplification is achieved by the introduction of constraints analogous to the Kirchhoff hypothesis of the continuum theory. The constraints provide a rational basis for neglecting the contribution of transverse shear in the strain energy. The resulting approximation converges rapidly to the Kirchhoff theory for examples cited.
Article
Due to the very non-linear behaviour of thin shells under collapse, numerical simulations are subject to challenges. Shell finite elements are attractive in these simulations. Rotational degrees of freedom do, however, complicate the solution. In the present study a co-rotated formulation is employed. The deformation of the shell is decomposed in to a contribution from large rigid body rotation and a strain producing term. A triangular assumed strain shell finite element is used. Hence, a high performance elastic element is combined with the co-rotated formulation. In the co-rotated co-ordinate system the plasticity is accounted for by a simplifyed Ilyushin stress resultant yield surface. The stress update is determined from the backward Euler difference, and a consistent geometrical and material tangent stiffness is derived. Comparison with other published analysis results show that the present formulation gives acceptable accuracy. Copyright © 1999 John Wiley & Sons, Ltd.
Article
A curved C**0 shell element is presented, which corrects several deficiencies in existing quadratic shell elements. The improvements realized in the present element include rank sufficiency without transverse shear locking, consistent membrane strain interpolation that admits inextensional bending without reduced integration, and adequate representation of curvature effects to capture the important membrane-bending coupling. The element can be constructed either by a nine-point integration rule or by a four-point integration rule with the proper rank compensating terms. Numerical experiments with the present element on several benchmark problems indicate that the element yields accurate and reliable solutions without any ostensible deficiency. The element is recommended for production analysis of shell structures.
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Article
This paper describes a co-rotational formulation for three-dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted ‘strain measures’. The latter relate to standard beam theory but are embedded in a continuously rotating frame. A set of numerical examples show that the element provides an excellent numerical performance.
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Today matter is universally regarded as composed of molecules. Though molecules cannot be discerned by human senses, they may be defined precisely as the smallest portions of a material to exhibit certain of its distinguishing properties, and much of the behavior of individual molecules is predicted satisfactorily by known physical laws. Molecules in their turn are regarded as composed of atoms; these, of nuclei and electrons; and nuclei themselves as composed of certain elementary particles. The behavior of the elementary particles has been reduced, so far, but to a partial subservience to theory. Whether these elementary particles await analysis into still smaller corpuscles remains for the future.
Book
This book is written for the practicing engineer. It is an attempt to bring together various strands of work on nonlinear finite elements. The developments in the book are related to computer applications; there are a number of Fortran listings, and many flow charts, for solving parts of nonlinear finite element problems. (Floppy disks with the Fortran source and data files are available from the publisher). This book takes an engineering rather than a mathematical approach to nonlinear finite elements. The first three chapters deal with truss elements. The author introduces basic concepts of nonlinear finite element analysis for simple truss systems with one degree of freedom. The solution schemes considered include an incremental (Euler), an iterative (Newton-Raphson), and a combined incremental and iteration approach (full or modified Newton-Raphson or the initial stress method). In chapter 2, the author introduces the shallow truss theory of chapter 1 to derive the finite element equations for a shallow truss slement with four degrees of freedom. A set of Fortran subroutines is given to solve simple bar-spring problems; some flowcharts are also provided. This chapter also contains data and solutions from a number of bar-spring problems.
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The line-spring finite element is a versatile numerical tool for performing engineering fracture mechanics analysis of surface cracked shells. The accuracy of the line-spring finite element solution for deep/medium-sized cracks has shown to be higher than the accuracy for the shallow-sized ones. Proper treatment of shallow cracks is important because they are the ones most frequently encountered in engineering practice. Accurate yield surfaces of plane-strain single edge-cracked specimens having shallow, as well as deep, cracks are developed here in order to improve the overall performance of the line-spring element. The yield surface is represented by equations that automatically satisfy the convexity requirement and that fit the result of limit load analyses. The present study addresses, furthermore, the performance of the backward Euler return algorithm for one of the accurate yield surface formulated here, by means of iso-error maps.
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The motion of a flexible body undergoing arbitrarily large rotations with respect to an inertial frame is split into a mean rigid body motion, defining a dynamical reference frame, and a relative motion taking into account the deformations.The mean motion is usually taken to satisfy the Tisserand conditions of zero relative momentum and angular momentum, a choice that, as shown in the paper, corresponds to a minimum value of the relative kinetic energy. The condition of zero angular momentum is however non linear and introduces discretization difficulties that can be overcome by another choice.The choice proposed in the paper minimizes the mean square of relative displacements. It preserves the zero momentum condition but linearizes the angular momentum condition in such a way that the relative displacements are representable exactly by an expansion in natural elastic vibration modes.Hamilton's principle is used to derive all the equations, including the mean motion ones, by using the concept of quasi-coordinates. Gravitational potential and thrust vectors, as locally oriented by the body motion and deformation, are accounted for. The equations are not limited to small distortion of the body, but to small strains.
Book
The book offers a unified view of problem solving and concentrates on the techniques for implementing computer programs.
Article
An improved algorithm is presented for integrating rate constitutive equations in large-deformation analysis. The algorithm is shown to be ‘objective’ with respect to large rotation increments.