Geometrically nonlinear formulation for the axi-symmetric transition finite elements

This paper presents a geometrically nonlinear formulation for the axi-symmetric transition finite elements using total lagrangian approach. The basic element is formulated using properties of the axi-symmetric solids and the axi-symmetric shells. A novel feature of the formulation presented here is that the restriction on the magnitude of the rotations for the shell nodes of the transition element is eliminated. This is accomplished by retaining true nonlinear functions of nodal rotations in the definition of the element displacement field. Such transition elements are essential for geometrically nonlinear applications requiring both axi-symmetric solids and the axi-symmetric shells. They ensure proper connection of the axi-symmetric solid portion of the structure to the shell like portion of the structure. It is shown that the selection of different stress and strain components at the integration points does not effect the overall linear response of the element. However, in the geometrically nonlinear formulation, it is necessary to select appropriate components of the stresses and the strains at the integration point for accurate and converging element behavior. Numerical examples are presented to demonstrate such characteristics of the transition elements.     

Geometrically non-linear formulation for two dimensional curved beam elements

This paper presents a geometrically non-linear formulation using a total lagrangian approach for the two dimensional curved beam elements. The beam element is derived using linear, paralinear and cubic-linear plane stress elements. The basic element geometry is constructed using the coordinates of the nodes on the element center line (η = 0) and the nodal point normals. The element displacement field is described using two translations of the node on the center line and a rotation about the axes normal to the plane containing the center line of the element. The existing beam element formulations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such a restriction. This is accomplished by retaining non-linear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting appropriate non-linear functions representing the effects of nodal rotations. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the behavior and the accuracy of the two dimensional beam elements for geometrically non-linear applications. In all cases comparisons made with theory and/or other published data show that the beam elements product accurate results and permit large load increments with good convergence characteristics.     

A unified formulation of small-strain corotational finite elements: I. Theory

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.     

Nonlinear finite element analysis of shells: Part I. three-dimensional shells

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.     

Static analysis of thin rotational shells

The purpose of the research presented here was to determine the elastic stresses induced in a thin shell of revolution. The method adopted was that of finite elements.A curved cone element was therefore developed which could predict the static response in thin loaded shells of revolution. The imposed loading can be axisymmetric or asymmetric. The element which is of isoparametric formulation is thought to have the salient advantages of having true nodal slope conformity, variable thickness, optional degrees of freedom within each element and ability to respond to lateral as well as inplane loads. Some of the numerical results using the present element have been included.The new formulation should be of special interest to engineers who have access to small computers since almost all the examples listed were solved on such a machine. It is hoped that, a moderate variety of engineering problems could now be solved more economically.     

Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation

In the finite element literature, beams and platesare not treated as isoparametric elements becausethe displacement gradients at the nodes are used todescribe infinitesimal rotations. Such alinearization of the slopes leads to incorrect rigidbody equations of motion when the beams and platesrotate as rigid bodies. A simple and efficient absolute nodal coordinate formulation, in which noinfinitesimal or finite rotations are used as nodalcoordinates, can be systematically developed andefficiently used in many large deformationapplications as well as in the analysis of curvedstructures. This formulation leads to a constantmass matrix, and as a consequence, an efficientprocedure can be used for solving for the nodalaccelerations. In the absolute nodal coordinateformulation, the displacement gradients at the nodesare determined in the undeformed referenceconfiguration using simple rigid body kinematics.The problems that arise from the linearization ofthe slopes and the use of the finite rotations asnodal coordinates are discussed in this paper.     

A symmetric finite element formulation for the inverse problem of aquifer transmissivity

This paper presents a symmetric isoparametric finite element formulation for the inverse problem of aquifer transmissivity calculation with known piezometric head. An important aspect of the present formulation is that the groundwater flow equation describing the aquifer behavior is transformed into a second-order differential equation by introducing an artificial variable φ. The two-dimensional, line and transition elements derived based on the weak formulation of this transformed equation possess symmetric matrices. In the formulation of the line elements φ and its derivative in η direction are retained as primary variables. This permits modelling of sudden changes in aquifer width. The transition elements provide a natural connecting link between the two-dimensional elements and the line elements. The line elements provide an efficient means of modelling aquifers with unidirectional flow. Numerical examples are given. A comparison of the results obtained here with the Galerkin finite element solution (nonsymmetric formulation) clearly demonstrates the superiority of the formulation presented here.     

Isoparametric line and transition finite elements with temperature and temperature gradients as primary variables for two dimensional heat conduction

This paper presents isoparametric line and transition finite element formulation for two dimensional heat conduction. The element properties are derived using weak formulation of the Fourier heat conduction equation and the element approximation where nodal temperatures and the nodal temperature gradients are retained as primary variables. The formulation permits linear temperature distribution through the element thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the elements. Furthermore, the elements can have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented to illustrate their applications, and a comparison is made with theoretical results.     

On the geometrical stiffness of a beam in space—a consistent V.W. approach

The application of the standard virtual work expressions to the large displacement-small strain domain merely requires the replacement of the standard linear strain-displacement relations by the quadratic ones. In fact, the geometrical stiffness matrix of an arbitrary finite element can be derived immediately from the virtual work of the second order terms in the strains.A correct geometrical stiffness is, however, only obtained if the prerequisites of the energy theorems are strictly observed. This proves to be a rather difficult task as soon as the elements contain rotational freedoms. In this case, the solution demands a comprehensive understanding of the true nature of the strains, stresses and nodal displacements, rotations as well as forces, moments.After an extensive study of the relevant entities the above principle is successfully applied to the derivation of the geometrical stiffness of a beam element in space. The consistency of the present approach is demonstrated by the full agreement with the prior results of the authors based on the natural mode technique [2,3].Some numerical examples demonstrate the practical importance of the present development for the geometrically nonlinear analysis of three-dimensional frame structures.     

Transition finite elements with temperature and temperature gradients as primary variables for axisymmetric heat conduction

This paper presents finite element formulation for a special class of elements referred to as “transition finite elements” for axisymmetric heat conduction. The transition elements are necessary in applications requiring the use of both axisymmetric solid elements and axisymmetric shell elements. The elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodal temperatures as well as nodal temperature gradients are retained as primary variables. The weak formulation of the Fourier heat conduction equation is constructed in the cylindrical co-ordinate system (r, z). The element geometry is defined in terms of the co-ordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The element temperature field is approximated in terms of element approximation functions, nodal temperatures and the nodal temperature gradients. The properties of the transition elements are then derived using the weak formulation and the element temperature approximation. The formulation presented here permits linear temperature distribution through the element thickness. Convective boundaries as well as distributed heat flux is permitted on all four faces of the element. Furthermore, the element formulation also permits distributed heat flux and orthotropic material behaviour. Numerical examples are presented, first to illustrate the accuracy of the formulation and second to demonstrate its usefulness in practical applications. Numerical results are also compared with the theoretical solutions.     

A new locking-free formulation for planar,shear deformable,linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.     

p-Version axisymmetric shell element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order p_ξ and p_η. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to p_ξ + 1 and p_η+ 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C⁰ continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element.

The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete axisymmetric state of stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells. The formulation presented here removes virtually all of the drawbacks present in the existing GNL axisymmetric shell finite element formulations and has many additional benefits. First, the currently available GNL axisymmetric shell finite element formulations are based on fixed interpolation order and thus are not hierarchical and have no mechanism for p-level change. Secondly, the element displacement approximations in the existing formulations are either based on linearized (with respect to nodal rotation) displacement fields in which case a true Lagrangian formulation is not possible and the load step size is severely limited or are based on nonlinear nodal rotation functions approach in which case though the kinematics of deformation is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. Such limitations and difficulties do not exist in the present formulation. The hierarchical displacement approximation used here does not involve traditional nodal rotations that have been used in the existing shell element formulations, thus the difficulties associated with their use are not present in this formulation.

Incremental equations of equilibrium are derived and solved using the standard Newton method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the preset formulation. The results obtained from the present formulation are compared with those available in the literature.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Hybrid-stress isoparametric elements for moderately thick and thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of moderately thick and thin multilayer laminated composite plates is presented. The element displacement behavior is characterized by laminate reference-surface inplane and transverse displacements and laminate nonnormal crosssection rotations; as a result, the number of degrees of freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. By isolating and analytically integrating all through-thickness contributions to the element matrices, the computation time for the element stiffness generation becomes nearly independent of the number of layers, and thus a computationally efficient element is produced. The formulation is used to develop an 8-node isoparametric multilayer plate element which is naturally invariant, of correct rank, and nonlocking in the thin-plate limit. Results for selected example problems show the range of applicability and convergence behavior of the element.     

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

This paper concerns the analytical formulation and finite element modelling of arbitrary active constrained layer damping (ACLD) treatments applied to beams. A partial layerwise theory is utilized to define the displacement field of beams with an arbitrary number of elastic, viscoelastic and piezoelectric layers attached to both surfaces, and a fully coupled electro-mechanical theory is considered for modelling the behavior of the piezoelectric layers. The damping of the viscoelastic layers is modelled by the complex modulus approach. The weak forms of the analytical formulation, governing the motion and electric charge equilibrium, are presented. Based on the weak forms, a one-dimensional finite element (FE) model is developed, with the nodal mechanical degrees of freedom being the axial displacement, transverse displacement and the rotation of the mid-plane of the host beam and the rotations of the individual layers, and the electrical elemental degrees of freedom being the electrical potential difference of each piezoelectric layer. Frequency response functions were measured experimentally and evaluated numerically for a freely suspended aluminium beam with an ACLD patch. In order to validate the FE model the results are presented and discussed.     

Finite element free vibration analysis of conoidal shells

Doubly curved conoidal shells are increasingly used for various industrial structures. Conoidal shells are aesthetically appealing and, being ruled surfaces, provide ease of casting. The variation of curvature is the difficulty enountered in the analysis of these shells. The finite element method is used here for the analysis of generalized doubly curved shells and is applied to truncated and full conoids of different boundary conditions, aspect ratios and degrees of truncation. An eight-noded isoparametric finite element with five degrees of freedom per node, including three translations and two rotations, is utilized. The accuracy is checked by comparing the results obtained by the present analysis with those existing in the literature. Results are presented for different conoidal shells and a set of conclusions are arrived at based on a parametric study.     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.