Co-rotational Finite Element Formulation Research Articles

Corotational Finite Element Dynamic Analysis of Space Frames with Geometrically Nonlinear Behavior Based on Tait–Bryan Angles

The aim of this study is to compose a corotational finite element formulation for space frames with geometrically nonlinear behavior under dynamic loads. Using a moving frame through three successive rotations similar to Euler angles is one of the oldest techniques; however, there are still some gaps that require attention, mainly due to singularity. Hence, alternative techniques had been developed, sometimes elusive and computationally expensive. In this paper, we went back to the old technique and filled the gaps. Three-coordinate systems are used, i.e., the fixed global coordinate system, the fixed local coordinate system that is attached individually to every element, and the corotational local frame for each element that moves and rotates with the element. The deformation is always small relative to the corotational frame. The successive rotations between different coordinate systems are expressed using Tait–Bryan angles. Lagrange’s equation is used to derive the equation of motion, and the stiffness and mass matrices are obtained using the Euler–Bernoulli beam model. A MATLAB code is developed based on the Newton–Raphson method and the Newmark direct integration implicit method. In traditional techniques, singularity is attained when any rotation angle in the fixed local frame approaches π / 2 , and if any is greater than π / 2 , the techniques could fail to specify the location of the element. In this paper, each case is treated with a proper procedure, and special handling of trigonometric formulations prevents singularity and correctly specifies the location of elements in all situations. Different examples of beams and frames are analysed. While the method is not intricate, it is timesaving, is highly effective, provides more stable and robust analysis, and gives sufficiently accurate results. Compared to the parametrization of the finite rotations technique, the method has a significant reduction in the convergence rate because it avoids the storage of joint orientation matrices.

Plastic analysis of steel arches and framed structures with various cross sections

This paper presents a displacement-based numerical methodology following the Euler-Bernoulli theory to simulate the 2 nonlinear behavior of steel structures. It is worth emphasizing the adoption of co-rotational finite element formulations considering large displacements and rotations and an inelastic material behavior. The numerical procedures proposed considers plasticity concentrated at the finite elements nodes, and the simulation of the steel nonlinear behavior is approached via the Strain Compatibility Method (SCM), where the material constitutive relation is used explicitly. The SCM is also applied in determining the sections bearing capacity. Moreover, the present numerical approach is not limited to a specific structural member cross-sectional typology, with the residual stress models introduced explicitly in subareas of steel cross-sections generated by a 2D discretization. Finally, results consistent with the literature and with low processing time are presented.

Corotational Finite Element Formulation for Static Nonlinear Analyses with Enriched Beam Elements

The increasing reliance of aerospace structures on numerical analyses encourages the development of accurate, yet computationally efficient, models. Finite element (FE) beam models have, in particular, become widely used approximations during preliminary design stages and to investigate novel concepts, for example, aeroelastic tailoring. Over the last 50 years, developments in hp-FE methods based on elements of variable size (h) and polynomial degree (p) have helped reduce the computational cost of numerical analyses. Concurrently, many structures, including aircraft wings and wind turbine blades, have gradually increased in length, slenderness, and complexity. As a result, modern blades and wings regularly operate beyond the range of linear deformation, requiring nonlinear analyses for which hp-FE methods are often not readily applicable. The aim of this paper is, therefore, to derive a corotational FE formulation for enriched three-, four-, and five-noded beam elements, suitable for nonlinear hp-FE refinement. To this end, the mathematical formulation is derived to incorporate enriched elements within the corotational FE beam framework. The proposed formulation is then validated against multiple nonlinear benchmark problems and an experimental case study.

Second-order inelastic analysis of shallow and non-shallow steel arches

This work presents a second-order inelastic analysis of steel arches. The analysis of shallow and non-shallow arches with several cross sections and boundary and loads conditions are discussed. The computational platform used is the homemade CS-ASA, which performs advanced nonlinear static and dynamic analysis of structures. The nonlinear geometric effects are considered using a co-rotational finite element formulation; the material inelasticity is simulated by coupling the Refined Plastic Hinge Method (RPHM) with the Strain Compatibility Method (SCM), and the static nonlinear solution is based on an incremental-iterative strategy including continuation techniques. In the simulated nonlinear steel arch models, special attention is given to the equilibrium paths, the influence of rise-to-span ratio, support and loading conditions and full yield curves among other factors. The numerical results obtained show good agreement with those from literature and highlight that the arch rise-to-span ratio has great influence on the structure resistance and that the shallow arches can lose stability through the snap-through phenomenon.

Nonlinear oscillations and dynamic stability of an elastoplastic pyramidal truss

Pyramidal space trusses are currently used in many present-day engineering applications, either as main parts or as a constitutive element. The present work studies, using a corotational finite element formulation considering large displacements and rotations and an elastoplastic material behavior with isotropic strain hardening, the nonlinear dynamic behavior of a pyramidal space truss. A bilinear hysteretic model with kinematic hardening is used to represent the material nonlinearity. An appropriate strain measure in such processes is the logarithmic strain. However, the engineering strain and the quadratic strain measures have been usually used to perform the elastic stability analysis of such structures. Thus, in the present work results using both the quadratic and the logarithmic strain measures are compared. First, the static nonlinear behavior is investigated in order to understand the influence of nonlinearities and static preload on the potential energy landscape, which controls the dynamics of the truss. These structures exhibit a two-well potential function that leads to coexisting in-well and cross-well solution branches, resulting in several coexisting periodic, quasi-periodic and chaotic attractors. The influence of the constitutive law, strain measure, truss geometry and support flexibility on the static stability is then investigated. Based on these results, a detailed parametric analysis of the nonlinear truss response under vertical load is conducted using bifurcations diagrams of the Poincare map to investigate the effect of the constitutive law, strain measure, truss geometry and load control parameters on the bifurcation scenario, coexisting attractors and dynamic buckling loads. The effect of the strain hardening parameter is particularly emphasized, allowing a more comprehensive picture of the dynamical behavior exhibited by the bistable system. Properties of the response are further illustrated by samples of time and phase plane responses and the related Poincare section plots. In addition to known behaviors, a rich class of solutions and bifurcations, including jump phenomena, symmetry-breaking, period-doubling cascades, fold and chaos is detected.

Stability and load capacity of an elasto-plastic pyramidal truss

The present work investigates, using a corotational finite element formulation considering large displacements and rotations, geometric imperfections and an elasto-plastic material constitutive law, the nonlinear behavior of a pyramidal space truss under vertical and horizontal loads. Additionally novel analytical closed-form solutions for the nonlinear equilibrium paths, natural frequencies and critical parameters are derived for comparison purposes and investigation of the influence of geometric and material parameters on the truss stability. This geometry has an immediate practical interest since these structures are currently used in many present-day applications, either as main structural component or as a constitutive element. Four types of nonlinear instability phenomena are investigated theoretically and numerically: saddle-node (limit point) bifurcation, pitchfork bifurcation, individual Eulerian bars buckling and elasto-plastic buckling. Through a detailed parametric analysis the interaction between these buckling phenomena, using the FE tool, is investigated. In addition, the effect of elastic supports is considered. The influence of these phenomena on the load carrying capacity of the structure and its imperfection sensitivity is duly discussed in the present work. Finally, a detailed parametric analysis of a large roof composed of pyramidal truss units is conducted and the influence of the geometric parameters on its bistable behavior and load carrying capacity is investigated.

A Corotational Formulation Based on Hamilton’s Principle for Geometrically Nonlinear Thin and Thick Planar Beams and Frames

A corotational finite element formulation for two-dimensional beam elements with geometrically nonlinear behavior is presented. The formulation separates the rigid body motion from the pure deformation which is always small relative to the corotational element frame. The stiffness matrices and the mass matrices are evaluated using both Euler-Bernoulli and Timoshenko beam models to reveal the shear effect in thin and thick beams and frames. The nonlinear equilibrium equations are developed using Hamilton’s principle and are defined in the global coordinate system. A MATLAB code is developed for the numerical solution. In static analysis, the code employed an iterative method based on the full Newton-Raphson method without incremental loading, while, in dynamic analysis, the Newmark direct integration implicit method is also utilized. Several examples of flexible beams and frames with large displacements are presented. Not only is the method simple and time-saving, but it is also highly effective and highly accurate.

Towards real-time simulation of deformable structures by means of co-rotational finite element formulation

Whereas typical Finite Element (FE) computations are performed off-line, many virtual-reality (VR) applications put a demand for interactive simulations involving deformable objects. Interactive simulation implies real-time or nearly real-time computation and graphical representation of modeled deformable objects. The growing computational power of modern conventional hardware calls for FEM developments in this direction. Depending on specific VR applications, the developments need to account for different aspects of physical behavior, with geometrically nonlinear deformations emerging as one of most important and frequent. This paper proposes a simplified co-rotational FE formulation that considers the overall finite element motion as a superposition of rigid-body rotation and deformation described by a linear model with respect to the co-rotated reference frame. By neglecting the stress stiffening effect and the dependence of the element stiffness matrix on the deformational displacements, the formulation aims at meeting the objectives of highly efficient simulation, under certain conditions even real-time simulation, and with acceptable deviation in accuracy compared to the rigorous nonlinear FE formulation. Computations with both solid and shell elements are addressed. A set of examples is provided to illustrate and discuss the aspects of accuracy and achievable simulation speed.

A buckling and postbuckling analysis of axially loaded thin-walled beams with point-symmetric open section using corotational finite element formulation

The axially loaded thin-walled beams with point symmetric open section are studied using a corotational finite element formulation. The kinematics of the beam element is described in the current element coordinate system. The element nodal forces are derived using the virtual work principle, and consistent second order linearization of the fully geometrically non-linear beam theory. Different axial load systems with the same centric resultant axial force but different resultant bimoments are considered. Numerical examples studied show that the effects of bimoments on the buckling and postbuckling behavior of point symmetric open section beams are remarked.

Corotational Finite Element Formulation for Virtual-Reality Based Surgery Simulators

Surgical simulation provides a means for trainees to develop surgical competence that encompasses requisite knowledge, technical and cognitive skills and decision-making ability. Considering virtual-reality based surgery simulators, the key requirement is sufficiently accurate and numerically efficient computation of deformation behavior of soft tissues, which is highly nonlinear. The paper offers a simplified geometrically nonlinear corotational finite element formulation to meet the imposed requirements. The approach is used in combination with a rather simple type of finite element and an appropriate solver is chosen for fast computation of dynamical behavior. The finite element formulation is enriched with a coupled-mesh technique to enable modelling of complex geometries by relatively simple computational models. A few examples of models of internal organs are provided to discuss the aspects of the developed tools.

Finite element modeling of cable galloping vibrations—Part I: Formulation of mechanical and aerodynamic co-rotational elements

Cables are widely used lightweight and efficient structural members affected by various nonlinearities. This paper is devoted to the development of a finite element approach to the study of the nonlinear dynamic behavior of cable structures under wind loading. Firstly, the formulation of a co-rotational beam-cable element is presented to account for cable geometrical nonlinearities within the context of an Updated Lagrangian approach. The Euler–Bernoulli kinematics is adopted in the co-rotated frame assuming small or moderate displacements and strains. Secondly, the formulation of an aerodynamic element, meant to be superimposed to the mechanical beam element, is developed to fully account for the nonlinearity of the aerodynamic forces. The wind interaction forces are computed within the framework of the quasi-steady theory. Appropriate procedures for applying the aforementioned elements in static and dynamic analyses are presented and applied to the study of the static configuration and the galloping vibrations of a suspended cable under steady and turbulent wind conditions. The results point out the importance of the first anti-symmetric in-plane mode on the galloping response of the iconic selected structure, and reveal the role of the mechanical and aerodynamic model by comparison with a different discretization based on cable finite elements. A mechanical model, such as the proposed co-rotational finite element formulation, which includes the description of torsional rotations is required to properly model the aerodynamic loads in dynamic galloping analyses under steady wind. The mechanical and aerodynamic assumptions, instead, play a less important role in the case of turbulent wind conditions, leading to an eminently buffeting response entailing a strong effect of the swing of the structure.

A 4-Node Co-Rotational Quadrilateral Composite Shell Element

A 4-node co-rotational quadrilateral composite shell element is presented. The local coordinate system of the element is a co-rotational framework defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product. Thus, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. Compared with other existing co-rotational finite-element formulations, the present element has two features: (i) The two smallest components of the mid-surface normal vector at each node are defined as the rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure; (ii) both element tangent stiffness matrices in the local and global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of strain energy with respect to the local nodal variables and, through chain differentiation with respect to the global nodal variables. In the modeling of composite structures, the first-order shear deformable laminated plate theory is adopted in the local element formulation, where both the thickness deformation and the normal stress in the direction of the shell thickness are ignored, and an assumed strain method is employed to alleviate the membrane and shear locking phenomena. Several examples involving composite plates and shells with large displacements and large rotations are presented to testify to the reliability and convergence of the present formulation.

AN EFFICIENT CO-ROTATIONAL FEM FORMULATION USING A PROJECTOR MATRIX

Co-rotational finite element (FE) formulations can be seen as a very efficient approach to resolving geometrically nonlinear problems in the field of structural mechanics. A number of co-rotational FE formulations have been well documented for shell and beam structures in the available literature. The purpose of this paper is to present a co-rotational FEM formulation for fast and highly efficient computation of large three-dimensional elastic deformations. On the one hand, the approach aims at a simple way of separating the element rigid-body rotation and the elastic deformational part by means of the polar decomposition of deformation gradient. On the other hand, a consistent linearization is introduced to derive the internal force vector and the tangent stiffness matrix based on the total Lagrangian formulation. It results in a non-linear projector matrix. In this way, it ensures the force equilibrium of each element and enables a relatively straightforward upgrade of the finite elements for linear analysis to the finite elements for geometrically non-linear analysis. In this work, a simple 4-node tetrahedral element is used. To demonstrate the efficiency and accuracy of the proposed formulation, nonlinear results from ABAQUS are used as a reference.

Co-rotational finite element formulation used in the Koiter–Newton method for nonlinear buckling analyses

The Koiter–Newton approach is a novel reduced order modeling technique for buckling analysis of geometrically nonlinear structures. The load carrying capability of the structure is achieved by tracing the entire equilibrium path in a stepwise manner. At each step a reduced order model generated from Koiter׳s asymptotic expansion provides a nonlinear prediction for the full model, corrected by a few Newton steps. The construction of the reduced order model requires derivatives of the strain energy with respect to the degrees of freedom up to the fourth order, which is two orders more than traditionally needed for a Newton based nonlinear finite element technique. In this paper we adopt the co-rotational formulation to facilitate these complex differentiations. We extend existing co-rotational beam and shell element formulations to make them applicable for the high order derivatives of the strain energy. The geometrical nonlinearities are taken into account using derivatives of the local co-rotational frame with respect to global degrees of freedom. This is done outside the standard element routines and is thus independent of the element type. We utilize three configurations and the nonlinear rotation matrix to describe finite rotations of the shell accurately, and profit from the automatic differentiation technique to optimize the programming of high order derivatives. The performance of the proposed approach using the co-rotational formulation is demonstrated using benchmark examples of isotropic and laminated composite structures.

A Corotational Formulation for Large Displacement Analysis of Functionally Graded Sandwich Beam and Frame Structures

A corotational finite element formulation for large displacement analysis of planar functionally graded sandwich (FGSW) beam and frame structures is presented. The beams and frames are assumed to be formed from a metallic soft core and two symmetric functionally graded skin layers. The Euler-Bernoulli beam theory and von Kármán nonlinear strain-displacement relationship are adopted for the local strain. Exact solution of nonlinear equilibrium equations for a beam segment is employed to interpolate the displacement field for avoiding the membrane locking. An incremental-iterative procedure is used in combination with the arc-length control method to compute the equilibrium paths. Numerical examples show that the proposed formulation is capable of evaluating accurately the large displacement response with just several elements. A parametric study is carried out to highlight the effect of the material distribution, the core thickness to height ratio on the large displacement behaviour of the FGSW beam, and frame structures.

Optimization of cross-section of actively bent grid shells with strength and geometric compatibility constraints

The use of bending as self forming process allows the realization of shape-resistant systems, such as grid shell structures. Here, a numerical method for optimization of the cross-section of actively bent structures is introduced. For a given load distribution, the optimization objective consists of normalizing the bending stresses to a given value on the entire structure. In addition, strength and geometric compatibility constraints are taken into account. The method is demonstrated by numerical examples. Further, in order to handle the large displacements involved, a co-rotational Finite Element formulation is adopted and modified to take into account the changes in stiffness that occur in the forming process of active bending systems. The modified co-rotational formulation is solved for static equilibrium using a Dynamic Relaxation scheme, and is tested against the analytical solutions of some preliminary test cases, as well as experimental results, and shown to be ‘accurate’.

A 6-node co-rotational triangular elasto-plastic shell element

A 6-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element is defined by the vectors directing from one vertex to the other two vertices and their cross product. Based on such a co-rotational framework, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, resulting in the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite element formulations, both the element tangent stiffness matrices in the local and in the global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of the strain energy with respect to the local nodal variables and, through chain differentiation, with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell---Huber---Hencky---von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is employed. To overcome locking problems, the assumed linear membrane strains and shear strains are obtained by using the line integration method proposed by MacNeal, and the assumed higher-order membrane strains are obtained by enforcing the stationarity of the mixed displacement-strain canonical functional, these assumed strains are then employed to replace the corresponding conforming strains. The reliability and convergence of the present 6-node triangular shell element formulation are verified through two elastic plate patch tests as well as two elastic and five elasto-plastic plate/shell problems undergoing large displacements and large rotations.

A numerical framework for simulating fluid-structure interaction phenomena

In this paper, a numerical tool able to solve fluid-structure interaction problems is proposed. The lattice Boltzmann method is used to compute fluid dynamics, while the corotational finite element formulation together with the Time Discontinuous Galerkin method are adopted to predict structure dynamics. The Immersed Boundary method is used to account for the presence of an immersed solid in the lattice fluid background and to handle fluid-structure interface conditions, while a Volume-of-Fluid-based method is adopted to take trace of the evolution of the free surface. These ingredients are combined through a partitioned staggered explicit strategy, according to an efficient and accurate algorithm recently developed by the authors. The effectiveness of the proposed methodology is tested against two different cases. The former investigates the dam break phenomenon, involving the modeling of the free surface. The latter involves the vibration regime experienced by two highly deformable flapping flags obstructing a flow. A wide numerical campaign is carried out by computing the error in terms of interface energy artificially introduced at the fluid-solid interface. Moreover, the structure behavior is dissected by simulating scenarios characterized by different values of the Reynolds number. Present findings are compared to literature results, showing a very close agreement.

A partitioned approach for two-dimensional fluid–structure interaction problems by a coupled lattice Boltzmann-finite element method with immersed boundary

This paper focuses on a computational approach for nearly incompressible viscous fluids interacting with slender elastic structures. The lattice Boltzmann method is used to predict fluid dynamics and the corotational finite element formulation together with the time Discontinuous Galerkin method to predict structure dynamics. The coupling strategy is explicit and the fluid–structure interface conditions are handled by the Immersed Boundary method. A wide numerical testing proves the effectiveness of the proposed approach and its advantages with respect to similar strategies implementing the bounce-back scheme to enforce interface conditions.

A four‐node corotational quadrilateral elastoplastic shell element using vectorial rotational variables

SUMMARYA four‐node corotational quadrilateral elastoplastic shell element is presented. The local coordinate system of the element is defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product. This local coordinate system rotates rigidly with the element but does not deform with the element. As a result, the element rigid‐body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing corotational finite‐element formulations, the resulting element tangent stiffness matrix is symmetric owing to the commutativity of the local nodal variables in calculating the second derivative of strains with respect to these variables. For elastoplastic analyses, the Maxwell–Huber–Hencky–von Mises yield criterion is employed, together with the backward‐Euler return‐mapping method, for the evaluation of the elastoplastic stress state; the consistent tangent modulus matrix is derived. To eliminate locking problems, we use the assumed strain method. Several elastic patch tests and elastoplastic plate/shell problems undergoing large deformation are solved to demonstrate the computational efficiency and accuracy of the proposed formulation. Copyright © 2013 John Wiley & Sons, Ltd.