Element Coordinate System Research Articles

Nondimensional modelling of 2D beam for slope discontinuity problem in ANCF

Various mechanical and structural systems possess slope discontinuity. Some bodies with slope discontinuity can be modeled using absolute nodal coordinate formulation, which can accurately express the rigid body mode when using the element and global shape functions. However, at the intersection of slope discontinuities, it is not easy to construct the equation of motion because the orientation changes. The problem of modeling at the intersection can be solved by defining the global slope vector in the body coordinate system instead of the element coordinate system. The analysis time in absolute nodal coordinate formulation is mostly greater than that in floating frame of reference formulation owing to the highly non-linear stiffness matrix. In this study, based on the work of Shabana and Mikkola, the analysis efficiency was demonstrated using the nondimensional model for the slope discontinuity problem in the absolute node coordinate system.

On the study of Rigid Body Inertia

In the classical finite element literature beams and plates are not considered as isoparametric elements since infinitesimal rotations are used as nodal coordinates. As a consequence, exact modeling of an arbitrary rigid body displacement cannot be obtained, and rigid body motion does not lead to zero strain. In order to circumvent this problem in flexible multibody simulations, an intermediate element coordinate system, which has an origin rigidly attached to the origin of the deformable body coordinate system and has axes which are parallel to the axes of the element coordinate system in the undeformed configuration was introduced. The correct equations of motion, however, can be obtained if the coordinates are defined in terms of global slopes. Using this new definition of the element coordinates, an absolute nodal coordinate formulation that leads to a constant mass matrix for the element can be developed. Using this formulation, in which no infinitesimal or finite rotations are used as nodal coordinates, beam and plate elements can be treated as isoparametric elements. G M T Detect language Afrikaans Albanian Arabic Armenian Azerbaijani Basque Belarusian Bengali Bosnian Bulgarian Catalan Cebuano Chichewa Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Esperanto Estonian Filipino Finnish French Galician Georgian German Greek Gujarati Haitian Creole Hausa Hebrew Hindi Hmong Hungarian Icelandic Igbo Indonesian Irish Italian Japanese Javanese Kannada Kazakh Khmer Korean Lao Latin Latvian Lithuanian Macedonian Malagasy Malay Malayalam Maltese Maori Marathi Mongolian Myanmar (Burmese) Nepali Norwegian Persian Polish Portuguese Punjabi Romanian Russian Serbian Sesotho Sinhala Slovak Slovenian Somali Spanish Sundanese Swahili Swedish Tajik Tamil Telugu Thai Turkish Ukrainian Urdu Uzbek Vietnamese Welsh Yiddish Yoruba Zulu Afrikaans Albanian Arabic Armenian Azerbaijani Basque Belarusian Bengali Bosnian Bulgarian Catalan Cebuano Chichewa Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Esperanto Estonian Filipino Finnish French Galician Georgian German Greek Gujarati Haitian Creole Hausa Hebrew Hindi Hmong Hungarian Icelandic Igbo Indonesian Irish Italian Japanese Javanese Kannada Kazakh Khmer Korean Lao Latin Latvian Lithuanian Macedonian Malagasy Malay Malayalam Maltese Maori Marathi Mongolian Myanmar (Burmese) Nepali Norwegian Persian Polish Portuguese Punjabi Romanian Russian Serbian Sesotho Sinhala Slovak Slovenian Somali Spanish Sundanese Swahili Swedish Tajik Tamil Telugu Thai Turkish Ukrainian Urdu Uzbek Vietnamese Welsh Yiddish Yoruba Zulu Text-to-speech function is limited to 200 characters Options : History : Feedback : Donate Close

An explicit algorithm for geometrically nonlinear transient analysis of spatial beams using a corotational total Lagrangian finite element formulation

An explicit method for nonlinear transient dynamic analysis of spatial beams with finite rotations using a corotational total Lagrangian finite element formulation is presented. The kinematics of the beam element is described in the current element coordinate system constructed in the current configuration of the beam element. The element deformation and inertia nodal forces are derived by the virtual work principle, the d'Alembert principle, and the consistent linearization of the geometrically nonlinear beam theory. A nodal rotation vector is used to represent the finite rotation of a base coordinate system rigidly attached to each node of the discretized structure. A numerical procedure of explicit method is proposed for the solution of the nonlinear equations of motion. The standard central difference method is applied to the incremental displacement vector and the incremental rotation vector, and the time derivatives of displacement vector and rotation vector. The nodal orientations are updated by the incremental nodal rotation vectors. The values of nodal rotation vectors are reset to zero in the current configuration.In order to assess the efficiency and the accuracy of the proposed method, numerical examples are studied and compared with the results obtained using the implicit method based on the Newmark method.

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.

Prediction of dynamic stresses using flexible multibody system algorithms: Application to tracked vehicle upper structure

The objective of this investigation is to develop a general procedure for predicting the dynamic stresses when the nonlinear floating frame of reference formulation is used to model flexible components in multibody system applications. The procedure utilizes the concept of the floating frame of reference intermediate finite element coordinate system and the general continuum mechanics approach to develop accurate definitions of the floating frame of reference kinematic equations that enter into the formulations of the strains. These general equations are used to design a computational algorithm for the calculation of the dynamic stresses. In order to demonstrate the use of the procedure developed in this investigation for the stress prediction, a tracked vehicle system is used as the study model. Taking into consideration the actual endurance test, a model for a bulldozer upper structure is developed using computer aided engineering solution framework which includes flexible multibody system algorithms. In order to obtain the dynamic stresses, the flexible structure of the tracked vehicle is modeled using Mindlin shell finite elements. The equations of motion are developed using the finite element/floating frame of reference formulation that allows for effectively reducing the number of degrees of freedom by neglecting high-frequency modes of vibration using component mode synthesis techniques. The stresses are obtained using a procedure that takes into consideration the dynamic coupling between the rigid body motion and the elastic deformations. The use of the general continuum mechanics theory to formulate the strain and stress equations allows for applying the procedure developed in this paper to other element types. Kinematic relationships are used to formulate several components and elements of the undercarriage including the idler cushion system and the maximum reverse angle. Numerical results are presented in order to demonstrate the use of the general floating frame of reference procedure in the stress analysis of the tracked vehicle undercarriage system.

Formulation of the 2-Dimensional Beam Defined by the Absolute Nodal Coordinates with the Algebraic Constraints (The 1st Report, Formulation Based on the Canonical Theory)

This paper describes the development of the analytical procedure of the flexible beam in the absolute nodal coordinate formulation (ANCF) by using the canonical theory. The elastic force in the ANCF has the complicated expression when the strain energy is defined by the coordinates in the element coordinate system. In this paper, we introduce the artificial algebraic constraints into the analytical procedure and make the constrained system on purpose to simplify the derivation of the elastic force. Those constraints lead to the redundant degree of freedom and the primary constraints caused by the definition of momentum. In order to formulate this system, the Dirac's approach with the Poisson bracket is introduced. As a result, the equations of motion are obtained in the form including the expanded degrees of freedom. Moreover, the reduced form of the equations of motion shows that the elastic force of this system is expressed as the right-hand side of the evolutionary equation for the momentum in the canonical equation. The elastic force obtained by this approach is the general expression for the two dimensional problem of the flexible beam defined in the ANCF. The proposed method is applied to the two dimensional beam element with the assumption of the Bernoulli and Euler. Then, the derived elastic force is compared with the previous research. The equivalence of these formulation is shown by the linear strain model.

Formulation of the Multi-Body System Based on the Canonical Theory (Formulation of the System with Constraints for Coupling of Elements)

This paper describes the application of the canonical theory to a constrained multibody system in the absolute nodal coordinate formulation. In the canonical theory, the constraints make it difficult to formulate the equations of motion because of the primary constraints of the Hamiltonian system. In order to solve this problem, the Dirac's approach with Poisson's bracket is used to derive the canonical equations. The governing equation of multibody systems with constraints is usually expressed as the differential-algebraic equations. In contrast, the canonical theory for the constrained system proposed by Dirac leads to the ordinary differential equations without constraints. In this study, it is demonstrated that the linear and the nonlinear canonical equations which the elastic force is formulated in the element coordinate system. The natural frequency and the eigenmode of the beam are obtained by the linearized canonical equations. These results show that the natural frequency and the eigenmode satisfy the boundary conditions introduced by the constraints. Moreover, the numerical examples of the nonlinear problems with the canonical equation and the differential-algebraic equation are presented in order to verify the proposed formulation.

Investigation on steady state deformation and free vibration of a rotating inclined Euler beam

The steady state deformation and infinitesimal free vibration around the steady state deformation of a rotating inclined Euler beam at constant angular velocity are investigated by the corotational finite element method combined with floating frame method. The element nodal forces are derived using the consistent second order linearization of the nonlinear beam theory, the d'Alembert principle and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The equations of motion of the system are defined in terms of an inertia global coordinate system, which is coincident with a rotating global coordinate system rigidly tied to the rotating hub. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady state deformation and natural frequency of the rotating inclined beam.

A Corotational Finite Element Method Combined with Floating Frame Method for Large Steady‐State Deformation and Free Vibration Analysis of a Rotating‐Inclined Beam

A corotational finite element method combined with floating frame method and a numerical procedure is proposed to investigate large steady‐state deformation and infinitesimal‐free vibrationaround the steady‐state deformation of a rotating‐inclined Euler beam at constant angular velocity. The element nodal forces are derived using the consistent second‐order linearization of the nonlinear beam theory, the d′Alembert principle, and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The governing equations for linear vibration are obtained by the first‐order Taylor series expansion of the equation of motion at the position of steady‐state deformation. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady‐state deformation and natural frequency of the rotating beam with different inclined angle, angular velocities, radius of the hub, and slenderness ratios.

Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.

Investigation of the design of a metal-lined fully wrapped composite vessel under high internal pressure

In this study, a Type III composite pressure vessel (ISO 11439:2000) loaded with high internal pressure is investigated in terms of the effect of the orientation of the element coordinate system while simulating the continuous variation of the fibre angle, the effect of symmetric and non-symmetric composite wall stacking sequences, and lastly, a stacking sequence evaluation for reducing the cylindrical section-end cap transition region stress concentration. The research was performed using an Ansys® model with 2.9 l volume, 6061 T6 aluminium liner/Kevlar® 49-Epoxy vessel material, and a service internal pressure loading of 22 MPa. The results show that symmetric stacking sequences give higher burst pressures by up to 15%. Stacking sequence evaluations provided a further 7% pressure-carrying capacity as well as reduced stress concentration in the transition region. Finally, the Type III vessel under consideration provides a 45% lighter construction as compared with an all metal (Type I) vessel.

Gradient Deficient Curved Beam Element Using the Absolute Nodal Coordinate Formulation

In this investigation, a gradient deficient beam element of the absolute nodal coordinate formulation is generalized to a curved beam for the analysis of multibody systems, and the performance of the proposed element is discussed by comparing with the fully parametrized curved beam element and the classical large displacement beam element with incremental solution procedures. Strain components are defined with respect to the initially curved configuration and described by the arc-length coordinate. The Green strain is used for the longitudinal stretch, while the material measure of curvature is used for bending. It is shown that strains of the curved beam can be expressed with respect to those defined in the element coordinate system using the gradient transformation, and the effect of strains at the initially curved configuration is eliminated using one-dimensional Almansi strain. This property can be effectively used with a nonincremental solution procedure employed for the absolute nodal coordinate formulation. Several numerical examples are presented in order to demonstrate the performance of the gradient deficient curved beam element developed in this investigation. It is shown that the use of the proposed element leads to better element convergence as compared with the fully parametrized element and the classical large displacement beam element with incremental solution procedures.

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.

Inherent Strain Method and Thermal Elastic-Plastic Analysis of Welding Deformation of a Thin-Wall Beam

AbstractThe transient thermal process of a thin-wall beam with CO2Gas Metal Arc Welding (GMAW) is analyzed by Finite Element Analysis Method (FEA). The thermal input is simplified as transient section body heat sources and loaded as its actual sequence in the analysis. The transient temperature field obtained can represent the basic characteristics of the real welding process and can be used as the foundation of thermal elastic-plastic analysis. Based on the temperature field, thermal elastic-plastic FEA is performed on the thin-wall beam. The distribution and change of the welding deformation, stress and strain are obtained and compared with the experiment results. Also an improvement can be presented on the inherent strain method. Using the inherent strain method, the welding deformation of the thin-wall beam is calculated. The temperature loading method is developed to load the variable inherent strain value expediently. The loading of inherent strain value on spatial welding line that is unparallel to the global coordinate axis is achieved with the application of element coordinate system. Comparison with the experiment results shows that both the thermal-elastic-plastic analysis and inherent strain analysis method can be used to predict the welding deformation effectively, the results calculated by both the thermal-elastic-plastic analysis and inherent strain analysis are close to the test measure results.

Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements

In this paper, the treatment of the slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is discussed. The paper explains the fundamental problems associated with developing a constant transformation that accounts for the slope discontinuities in the case of gradient deficient ANCF finite elements. A procedure that allows for the treatment of slope discontinuities in the case of gradient deficient finite elements which do not employ full parameterization is proposed for the special case of commutative rotations. The use of the proposed procedure leads to a constant orthogonal element transformation that describes the element initial configuration. As a consequence, one obtains in the case of large deformation and commutative rotations, a constant mass matrix for the structures. In order to achieve this goal, the concept of the intermediate finite element coordinate system is invoked. The intermediate finite element coordinate system used in this investigation serves to define the element reference configuration, follows the rotation of the structure, and maintains a fixed orientation relative to the structure coordinate system. Since planar rotations are always commutative, the procedure proposed in this investigation is applicable to all planar gradient deficient ANCF finite elements.

Co-rotational finite element formulation for thin-walled beams with generic open section

A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with generic open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The element nodal forces are derived using the virtual work principle. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. The way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of the terms in the element nodal force and tangent stiffness matrix, which will converge to zero with the decrease of element size, on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples.

A Linear Beam Finite Element Based on the Absolute Nodal Coordinate Formulation

Abstract In this paper, a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation is proposed. The nonlinear elastic forces of the beam element are obtained using a continuum mechanics approach, without employing a local element coordinate system. In this study, linear polynomials are used to interpolate both the transverse and longitudinal components of the displacement. This is different from other absolute nodal-coordinate-based beam elements where cubic polynomials are used in the longitudinal direction. The use of linear interpolation polynomials leads to the phenomenon known as shear locking. This defect is avoided through the adoption of selective integration within the numerical integration method. The proposed element is verified using several numerical examples. The results of the proposed element are compared to analytical solutions and the results for an existing shear deformable beam element. It is shown that by using the proposed element, accurate linear and nonlinear static deformations, as well as realistic dynamic behavior including the capturing of the centrifugal stiffening effect, can be achieved with a smaller computational effort than by using existing shear deformable two-dimensional beam elements.

A Non-Incremental Nonlinear Finite Element Solution for Cable Problems

In this investigation, a nonlinear finite element method for the large deformation and rotation of cable problems is presented. This method is based on finite element absolute nodal coordinate formulation that guarantees the continuity of all displacement gradients and leads to a constant mass matrix. The classical cable theory is first reviewed and the assumptions used in this linear theory are defined in order to demonstrate the basic differences between the linear theory and the nonlinear finite element formulation proposed in this paper for cable applications. The elastic cable forces in the absolute nodal coordinate formulation are obtained using a general continuum mechanics approach that accounts for the effect of all geometric nonlinearities. It is shown in this investigation that the use of the general continuum mechanics approach leads to a simpler and more efficient formulation as compared to the use of the assumptions of the linear theory that employs a local finite element coordinate system. The results obtained using the absolute nodal coordinate formulation show a good agreement with the results obtained using the classical cable theory when linear cable problems are considered. In particular it is shown that the use of perturbation methods to linearize the finite element equations of motion leads to modal characteristics results that are in a good agreement with the linear theory. The results of this investigation obtained using explicit numerical integration also show the potential of the proposed finite element formulation in the nonlinear analysis of cables that experience large rotations and deformations. The generalization of the procedure presented in this paper to three-dimensional cable problems is demonstrated and the computer implementation in multibody algorithms is discussed.

Use of the Finite Element Absolute Nodal Coordinate Formulation in Modeling Slope Discontinuity

A large rigid body rotation of a finite element can be described by rotating the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities. The analysis presented in this paper also demonstrates that objectivity is not an issue when the absolute nodal coordinate formulation is used due to the fact that this formulation automatically accounts for the proper coordinate transformations.

A co-rotational formulation for thin-walled beams with monosymmetric open section

A consistent co-rotational total Lagrangian finite element formulation and numerical procedure for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with monosymmetric open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system, which is constructed at the current configuration of the beam element. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order term of the twist rate of the beam axis is considered in element nodal forces. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.