Co-rotational Approach Research Articles

An improved solid‐shell element based on ANS and EAS concepts

SummaryThis paper presents an eight‐node nonlinear solid‐shell element for static problems. The main goal of this work is to develop a solid‐shell formulation with improved membrane response compared with the previous solid‐shell element (MOS2013), presented in . Assumed natural strain concept is implemented to account for the transverse shear and thickness strains to circumvent the curvature thickness and transverse shear locking problems. The enhanced assumed strain approach based on the Hu–Washizu variational principle with six enhanced assumed strain degrees of freedom is applied. Five extra degrees of freedom are applied on the in‐plane strains to improve the membrane response and one on the thickness strain to alleviate the volumetric and Poisson's thickness locking problems. The ensuing element performs well in both in‐plane and out‐of‐plane responses, besides the simplicity of implementation. The element formulation yields exact solutions for both the membrane and bending patch tests. The formulation is extended to the geometrically nonlinear regime using the corotational approach, explained in . Numerical results from benchmarks show the robustness of the formulation in geometrically linear and nonlinear problems. Copyright © 2016 John Wiley & Sons, Ltd.

Multi-objective optimization of a type of ellipse-parabola shaped superelastic flexure hinge

Abstract. Flexure hinges made of superelastic materials is a promising candidate to enhance the movability of compliant mechanisms. In this paper, we focus on the multi-objective optimization of a type of ellipse-parabola shaped superelastic flexure hinge. The objective is to determine a set of optimal geometric parameters that maximizes the motion range and the relative compliance of the flexure hinge and minimizes the relative rotation error during the deformation as well. Firstly, the paper presents a new type of ellipse-parabola shaped flexure hinge which is constructed by an ellipse arc and a parabola curve. Then, the static responses of superelastic flexure hinges are solved via non-prismatic beam elements derived by the co-rotational approach. Finite element analysis (FEA) and experiment tests are performed to verify the modeling method. Finally, a multi-objective optimization is performed and the Pareto frontier is found via the NSGA-II algorithm.

An efficient finite element formulation for nonlinear analysis of clustered tensegrity

Purpose– The purpose of this paper is to propose an efficient finite element formulation for nonlinear analysis of clustered tensegrity that consists of classical cables, clustered cables and bars.Design/methodology/approach– The derivation of the finite element formulation is based on the co-rotational approach, which decomposes a geometrically nonlinear deformation into a large rigid body motion and a small-strain deformation. A tangent stiffness matrix of a clustered cable is proposed and the Newton-Raphson scheme is employed to solve the nonlinear equation.Findings– The derived tangent stiffness matrix, including an additional stiffness terms that describes the slide effect of pulleys, can regress to the stiffness matrix of a classical cable, which is convenient for the implementation of finite element procedure. Two typical numerical examples show that the proposed formulation is accurate and requires less iteration than the force density method.Originality/value– The co-rotational formulation of a clustered cable is originally proposed, although some mature methods, such as the TL, Force Density and Dynamic Relaxation method, have been applied to nonlinear analysis of clustered tensegrity. The proposed co-rotational formulation proved efficient.

A 3D PVP co-rotational formulation for large-displacement and small-strain analysis of bi-modulus materials

An efficient computational method is developed for large-displacement and small-strain analysis of 3D bi-modulus materials which are often found in civil, composite and biological engineering. Based on the parametric variational principle (PVP), a unified constitutive equation of 3D bi-modulus materials is proposed to deal with the problem of numerical instability. The small-strain bi-modulus problem is transformed into a standard linear complementarity problem that can be solved easily by the classic Lemke׳s algorithm. By using the co-rotational approach, the local PVP formulation is combined with an existing co-rotational formulation and a new tangent stiffness matrix, including a parametric variable, is derived for geometrically nonlinear analysis. Traditional stress iteration is not required for calculation of the nodal internal force when the Newton–Raphson scheme or Arc-length method is employed to solve the material and geometric nonlinear problem. Convergence of the proposed algorithm is improved greatly in contrast to the traditionally iterative solution. Also, the proposed algorithm is used to simulate a unilateral contact behavior of staggered bio-composites and the effective elasticity modulus of composites is determined accurately.

Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

AbstractA new mixed tetrahedral element, particularly suited for the analysis of structures exhibiting nonlinear material and geometric behavior, is here presented. Its derivation is based on a Hu–Washizu type formulation, including also rotation and skew‐symmetric stress fields as independent variables, instrumental to equip the element with nodal rotational degrees of freedom. A Gauss‐point‐discontinuous interpolation is selected for the total strain field, in order to account for its possibly highly nonlinear spatial distribution due to inelastic strains. Accordingly, the resulting tetrahedron can properly describe inelastic effects occurring over a space scale smaller than the element size. An original and efficient iterative procedure is proposed to perform the element state determination. Geometric nonlinearities are treated by means of the corotational approach. A numerical simulation is presented to analyze the element performances. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Geometrically nonlinear analysis of hybrid beam–column with several encased steel profiles in partial interaction

This article presents a new co-rotational finite element for the large displacement analysis of hybrid steel–concrete beam/column with several encased steel profiles. The advantage of using the co-rotational approach is that the geometrical linear finite element formulation can be reused and automatically be transformed into geometrical nonlinear formulation. The exact stiffness matrix derived from the analytical solution of the governing equations for hybrid beam with longitudinal partial interaction is used for the local formulation. As a result, internal nodes used to avoid curvature locking encountered in low order polynomial finite elements are not required. Finally, five numerical applications are presented in order to illustrate the performance of the proposed formulation.

Nonlinear Beam-Column Element Under Consistent Deformation

A new nonlinear beam-column element capable of considering the shear deformation is proposed under the concept of consistent deformation. For the traditional displacement interpolation function, the beam-column element produces membrane locking under large deformation and shear locking when the element becomes slender. To eliminate the membrane and shear locking, force equilibrium equations are employed to derive the displacement function. Numerical examples herein show that membrane locking in the traditional nonlinear beam-column element could cause a considerable error. Comparison of the present improved formulae based on the Timoshenko beam theory with that based on the Euler–Bernoulli beam theory indicates that the present approach requires several additional parameters to consider shear deformation and it is suitable for computer analysis readily for incorporation into the frames analysis software using the co-rotational approach for large translations and rotations. The examples confirm that the proposed element has higher accuracy and numerical efficiency.

Dynamic Nonlinear Co-rotational Formulationfor Two-dimensional Continua

This paper presents an extension of BATTINI´s formulation to two-dimensional analysis of nonlinear dynamical problems. The main interest of the co-rotational approach is to separate rigid body motions from pure deformations at the local element level through co-rotated framework. Employing co-rotated framework to derive both internal and inertial terms is the objective of the formulation presented in this study. Numerical examples are presented to illustrate the ability of the proposed co-rotational formulation.

Geometrically nonlinear elasto-plastic analysis of clustered tensegrity based on the co-rotational approach

An efficient finite element formulation is developed for geometrically nonlinear elasto-plastic analyses of classical and clustered tensegrities based on the co-rotational approach. Large deformation of a clustered cable is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. The merit of the proposed method is that elasto-plastic behaviors of cables and bars can be described using engineering strain and stress is calculated directly by the small-deformation elasto-plastic theory, although geometrical nonlinearity must be taken into account. A tangent stiffness matrix, including an additional stiffness that describes the sliding effect of running pulleys is derived and nonlinear iteration is implemented by the Newton–Raphson scheme. Several numerical examples are employed to validate the proposed formulation. It can be found that the proposed formulation requires less iteration than the force density method.

A NURBS‐based finite element model applied to geometrically nonlinear elastodynamics using a corotational approach

SummaryA numerical model to deal with nonlinear elastodynamics involving large rotations within the framework of the finite element based on NURBS (Non‐Uniform Rational B‐Spline) basis is presented. A comprehensive kinematical description using a corotational approach and an orthogonal tensor given by the exact polar decomposition is adopted. The state equation is written in terms of corotational variables according to the hypoelastic theory, relating the Jaumann derivative of the Cauchy stress to the Eulerian strain rate.The generalized‐α method (Gα) method and Generalized Energy‐Momentum Method with an additional parameter (GEMM+ξ) are employed in order to obtain a stable and controllable dissipative time‐stepping scheme with algorithmic conservative properties for nonlinear dynamic analyses.The main contribution is to show that the energy–momentum conservation properties and numerical stability may be improved once a NURBS‐based FEM in the spatial discretization is used. Also it is shown that high continuity can postpone the numerical instability when GEMM+ξ with consistent mass is employed; likewise, increasing the continuity class yields a decrease in the numerical dissipation. A parametric study is carried out in order to show the stability and energy budget in terms of several properties such as continuity class, spectral radius and lumped as well as consistent mass matrices. Copyright © 2015 John Wiley & Sons, Ltd.

Material and geometric nonlinear analysis of reinforced concrete frames

The analysis of reinforced concrete structures until failure requires the consideration of geometric and material nonlinearities. However, nonlinear analysis is much more complex and costly than linear analysis. In order to obtain a computationally efficient approach to nonlinear analysis of reinforced concrete structures, this work presents the formulation of a nonlinear plane frame element. Geometric nonlinearity is considered using the co-rotational approach and material nonlinearity is included using appropriate constitutive relations for concrete and steel. The integration of stress resultants and tangent constitutive matrix is carried out by the automatic subdivision of the cross-section and the application of the Gauss quadrature in each subdivision. The formulation and computational implementation are validated using experimental results available in the literature. Excellent results were obtained.

NURBS-based three-dimensional analysis of geometrically nonlinear elastic structures

A comprehensive kinematical description with a corotational approach, which is based on exact polar decomposition for solid elements using isogeometric analysis is the main objective of this work. A numerical model using NURBS-based FEM is developed to deal with nonlinear problems in solid mechanics involving large rotations. From the theoretical viewpoint, the rate type form of state equation is formulated within the framework of isotropic infinitesimal elasticity. It is shown in particular that the state equation relates the Jaumann derivative of the Cauchy stress to the Eulerian strain rate. Illustrative examples are analyzed to verify the efficiency in solving a wide variety of strong nonlinear problems. • NURBS-based analysis is employed to deal with nonlinear problems in solid mechanics. • Kinematical description with a corotational-based approach is proposed. • The rate form of state equation is formulated within framework of large rotations. • Several examples of shells structures are analyzed numerically.

Large displacement behaviour of tapered cantilever Euler–Bernoulli beams made of functionally graded material

The large displacement behaviour of tapered cantilever Euler–Bernoulli beams made of functionally graded material subjected to end forces is studied by the finite element method. The effective Young’s modulus of the beams is assumed to be graded in the thickness direction by a power-law distribution. Based on the co-rotational approach, a finite element formulation is derived and employed in the study. An incremental/iterative procedure in combination with the arc-length control method is used in computing the large displacement response of the beams. The numerical results show that the derived formulation is capable to give accurate results by using just several elements. A parametric study is given to highlight the influence of the material inhomogeneity, taper ratio and taper type on the large displacement behaviour of the beams. The large displacement behaviour of beams composed of different constituent materials is also investigated and highlighted.

Force-based FE for large displacement inelastic analysis of two-layer Timoshenko beams with interlayer slips

This paper presents a novel finite element model for the fully material and geometrical nonlinear analysis of shear-deformable two-layer composite planar beam/column members with interlayer slips. We adopt the co-rotational approach where the motion of the element is decomposed into two parts: a rigid body motion which defines a local co-ordinate system and a small deformational motion of the element relative to this local co-ordinate system. The main advantage of this approach is that the transformation matrices relating local and global quantities are independent from the choice of the geometrical linear local element. The effect of transverse shear deformation of the layers is taken into account by assuming that each layer behaves as a Timoshenko beam element. The layers are assumed to be continuously connected and partial interaction is considered by adopting a continuous relationship between the interface shear flow and the corresponding slip. In order to avoid curvature and the shear locking phenomena, the local linear element is derived from the force-based formulation. The present model provides an efficient tool for the elastoplastic buckling analysis of two-layer shear deformable beam/column with arbitrary support and loading conditions. Finally, two numerical applications are presented in order to assess the performance of the proposed formulation.

Interactive real-time physics: An intuitive approach to form-finding and structural analysis for design and education

Real-time physics simulation has been extensively used in computer games, but its potential has yet to be fully realized in design and education. We present an interactive 3D physics engine with a wide variety of applications.In common with traditional FEM, the use of a local element stiffness matrix is retained. However, unlike typical non-linear FEM routines elements forces, moments and inertia are appropriately lumped at nodes following the dynamic relaxation method. A semi-implicit time integration scheme updates linear and angular momentum, and subsequently the local coordinate frames of the nodes. A co-rotational approach is used to compute the resultant field of displacements in global coordinates including the effect of large deformations. The results obtained compare well against established commercial software.We demonstrate that the method presented allows the making of interactive structural models that can be used in teaching to develop an intuitive understanding of structural behaviour. We also show that the same interactive physics framework allows real-time optimization that can be used for geometric and structural design applications.

Geometrical Nonlinear Analysis of Tensegrity Based on a Co-Rotational Method

Tensegrities are structures whose integrity is based on a balance between tension and compression. A numerical procedure is presented for the geometrical nonlinear analysis of tensegrity structures. This approach is based on a co-rotational method where the major component of geometrical non-linearity is treated by a co-rotational filter. This is achieved by separating rigid body motions from deformational displacements. The outcomes evince that the efficiency of the co-rotational approach is considerably greater than those using total Lagrangian and updated Lagrangian formulations for space rod elements, which have more rigid body movement modes than deformational modes. Numerical examples illustrate that the displacements of tensegrity systems depend on the applied force density coefficient and external loading values. Furthermore, in the analysis of tensegrity structures, constraints such as the yield strength of all elements and zero stiffness of string elements becoming slack at any equilibrium configuration must be allowed for.

An efficient multiscale method for 2D large displacement – Small strain analysis of heterogeneous materials

A multiscale computational method is developed for 2D large displacement – small strain analysis of heterogeneous structures. The co-rotational approach is employed to obtain the equivalent tangent stiffness matrix of the coarse element in the global coordinate system. The equivalent tangent stiffness matrix of the whole heterogeneous structure with the coarse-scale mesh can be assembled naturally. The equilibrium iterations for each load step can be performed directly on the macroscopic scale and thus a large number of computational resources will be saved. Moreover, a new local displacement correction technique is proposed to calculate the microscopic stress and strain results. Two numerical examples are carried out to verify the validity and efficiency of the proposed multiscale method. It can be found that the proposed method not only ensure the accuracy but also save the computational resources greatly by comparison with the traditional finite element method (FEM).

Large deflections of tapered functionally graded beams subjected to end forces

The large deflections of tapered functionally graded beams subjected to end forces are studied by using the finite element method. The material properties of the beams are assumed to vary through the thickness direction according to a power law distribution. A first order shear deformable beam element employed the exact polynomials to interpolate the transverse displacement and rotation, is formulated in the context of the co-rotational approach. The large deflection response of the beams is computed by using the arc-length control algorithm in combination with the Newton–Raphson iterative method. The numerical results show that the formulated element is capable to assess accurately the response of the beams by using just several elements. A parametric study is given to examine the influence of the material non-homogeneity, taper ratio as well as the aspect ratio on the large deflection behaviour of the beams.

A solid‐shell corotational element based on ANDES, ANS and EAS for geometrically nonlinear structural analysis

SUMMARYThis paper describes an eight‐node, assumed strain, solid‐shell, corotational element for geometrically nonlinear structural analysis. The locally linear kinematics of the element is separated into in‐plane (which is further decoupled into membrane and bending), thickness and transverse shear components. This separation allows using any type of membrane quadrilateral formulation for the in‐plane response. Assumed strain fields for the three components are constructed using different approaches. The Assumed Natural Deviatoric Strain approach is used for the in‐plane response, whereas the Assumed Natural Strain approach is used for the thickness and transverse shear components. A strain enhancement based on Enhanced Assumed Strain concepts is also used for the thickness component. The resulting element passes well‐known shell element patch tests and exhibits good performance in a number of challenging benchmark tests. The formulation is extended to the geometric nonlinear regime using an element‐independent corotational approach. Some key properties of the corotational kinematic description are discussed. The element is tested in several well‐known shell benchmarks and compared with other thin‐shell and solid‐shell elements available in the literature, as well as with commercial nonlinear FEM codes. Copyright © 2013 John Wiley & Sons, Ltd.

An efficient algorithm for mechanical analysis of bimodular truss and tensegrity structures

A new stable algorithm is developed for the geometric nonlinear analysis of bimodular truss and tensegrity structures based on the parametric variational principle (PVP) and the co-rotational approach. The bilinear constitutive model is proposed for modeling tensegrity structures which consist of prestressed struts and cables. The PVP for bimodular truss is first introduced for small deformation analysis, and then the geometric nonlinear formulation is derived by combining the PVP linear formulation with the co-rotational approach. The Newton–Raphson (NR) scheme combined with Lemke’s algorithm is employed to solve the resulting nonlinear complementarity equations. Numerical examples are given to demonstrate the validity and efficiency of the proposed PVP method for mechanical analysis of bimodular truss and tensegrity structures. Convergence behaviors of different computational schemes for geometric linear and nonlinear analyses are also discussed in detail, which shows the good numerical stability and efficiency of the proposed method.