Mixed-displacement finite-element analysis with particular application using plane-stress triangles

1972 ◽  
Vol 7 (4) ◽  
pp. 243-252 ◽  
Author(s):  
G M McNeice ◽  
S F Hunnisett
Keyword(s):  
Finite Elements ◽  
Plane Stress ◽  
Strain Energy ◽  
Degrees Of Freedom ◽  
High Strain ◽  
Energy Content ◽  
Three Dimensional ◽  
Triangular Grid ◽  
Element Analysis ◽  
Displacement Approach

Finite elements which have different displacement functions for each boundary are termed ‘mixed-displacement’ elements. The mixed-displacement approach to finite elements is applied to plane-stress problems in which mixed-strain triangles are formed from the linear-strain triangle. It is shown that by using these triangles in conjunction with existing linear- and constant-strain triangles, higher strain-energy content is produced with fewer active degrees of freedom than are otherwise necessary. A basic triangular grid is first assumed and any number of additional midside nodes can be inserted in regions of high strain gradients. Reference to three-dimensional isoparametric elements is given.

Analysis of Shape Memory Alloy Components Using Beam, Shell, and Continuum Finite Elements

2010 ◽  
Author(s):  
Darren Hartl ◽  
Tyler Zimmerman ◽  
Matthew Dilligan ◽  
James Mabe ◽  
Frederick Calkins
Keyword(s):  
Finite Elements ◽  
Shape Memory Alloy ◽  
Shape Memory ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Constitutive Models ◽  
Analysis Tool ◽  
Three Dimensional Model ◽  
Restoring Force ◽  
Element Analysis

This work discusses the increased capabilities of a three-dimensional analysis tool for shape memory alloy engineering components. As the number and complexity of proposed SMA applications increases, engineers and designers must seek out or develop more capable predictive methods. Three-dimensional models implemented in a continuum finite element analysis (FEA) framework can be applied to most SMA component geometries. However, such methods may require fine meshes in 3-D space, resulting in many degrees of freedom and potentially long analysis times. On the other hand, constitutive models implemented in one dimension can be simple and fast, but are restricted to a limited class of problems for which such reductions are appropriate (e.g., rods and beams). More recently, engineers have begun investigating more complex SMA bending components for which 2-D shell elements might provide a computationally efficient FEA discretization. Here we consider a single modeling tool (a material subroutine) that combines 1-D, 2-D, and 3-D implementations for use in a general FEA framework. As an example analysis case, we consider an SMA bending element that has been adhesively bonded to a carbon fiber-reinforced polymer (CFRP) laminate and is subjected to thermally-induced actuation. The active SMA and passive composite components are bonded in a pre-stressed configuration such that the elastic laminate provides a variable restoring force to the SMA during transformation, resulting in repeatable actuation cycles. This two-part bonded configuration is analyzed using different types of finite elements (1-D beam, 2-D shell, and full 3-D continuum elements). The constitutive behavior of the shape memory alloy is defined using an established three-dimensional model based on continuum thermodynamics and motivated by the methods of classical plasticity. A user material subroutine (UMAT) in an Abaqus Unified FEA framework is used to implement the model. The methodology for capturing 1-D, 2-D, and 3-D thermomechanical response in a single such UMAT is described. The run times of the various analyses are compared, and the relative accuracies of the results are discussed.

An Exact Three-Dimensional Beam Element With Nonuniform Cross Section

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
M. Jafari ◽  
M. J. Mahjoob
Keyword(s):  
Cross Section ◽  
Stiffness Matrix ◽  
Degrees Of Freedom ◽  
Direct Method ◽  
Three Dimensional ◽  
Curved Beams ◽  
Element Analysis ◽  
Shear Loads ◽  
Line Passing ◽  
Exact Stiffness Matrix

In this paper, the exact stiffness matrix of curved beams with nonuniform cross section is derived using direct method. The considered element has two nodes and 12 degrees of freedom, with three forces and three moments applied at each node. The noncoincidence effect of shear center and center of area is also considered in this element. The deformations of the beam are due to bending, torsion, tensile, and shear loads. The line passing through center of area is a general three-dimensional curve and the cross section properties may change arbitrarily along it. The method is extended to deal with distributed loads on the curved beams. The stiffness matrix of some selected types of beams is determined by this method. The results are compared (where possible) with previously published results, simple beam finite element analysis and analytic solution. It is shown that the determined stiffness matrix is exact and that any type of beam can be analyzed by this method.

Three-Dimensional Solutions for Rotor Blades Using High-Order Geometrical Nonlinear Beam Finite Elements

2019 ◽  
Vol 64 (3) ◽  
pp. 1-10
Author(s):  
Matteo Filippi ◽  
Alfonso Pagani ◽  
Erasmo Carrera
Keyword(s):  
Finite Elements ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Shear Deformation Theory ◽  
High Order ◽  
Rotor Blades ◽  
Constitutive Law ◽  
Displacement Fields ◽  
Cross Sectional ◽  
The Cross

This paper proposes a geometrically nonlinear three-dimensional formalism for the static and dynamic study of rotor blades. The structures are modeled using high-order beam finite elements whose kinematics are input parameters of the analysis. The displacement fields are written using two-dimensional Taylor- and Lagrange-like expansions of the cross-sectional coordinates. As far as the Taylor-like polynomials are concerned, the linear case is similar to the first-order shear deformation theory, whereas the higher-order expansions include additional contributions that describe the warping of the cross section. The Lagrange-type kinematics instead utilizes the displacements of certain physical points as degrees of freedom. The inherent three-dimensional nature of the Carrera unified formulation enables one to include all Green–Lagrange strain components as well as all coupling effects due to the geometrical features and the three-dimensional constitutive law. A number of test cases are considered to compare the current solutions with experimental and theoretical results reported in terms of large deflections/rotations and frequencies related to small amplitude vibrations.

The Application of Harmonic Finite Elements in the Seismic Response Spectrum Analysis of a Skirt Supported Vessel

2010 ◽  
Author(s):  
Bikramjit Singh Antaal ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Seismic Response ◽  
Spectrum Analysis ◽  
Degrees Of Freedom ◽  
Response Spectrum ◽  
Response Spectrum Analysis ◽  
Element Analysis ◽  
Structure Interaction ◽  
Input Response

This paper describes the finite element considerations employed in a seismic response spectrum analysis of a skirt supported, liquid containing pressure vessel. Like many axisymmetric cylindrical vessels, the gross seismic response to an input response spectrum can be categorized by a simplified lump mass model that includes both the mass of the vessel proper in combination with the associated mass of multiple fluid levels. This simplified response may be utilized to determine the initial sizing of the supporting configuration, such as a skirt, but lacks the ability to properly address the fluid-structure interaction that creates sloshing loads on the vessel walls. The most obvious method to address the fluid-structure interaction when considering the finite element method is to build a three-dimensional model of the vessel proper, including, but not limited to the shell courses, the top and bottom heads (for a vertical vessel), and the support skirt. The inclusion of the fluid effects may now be incorporated with a “contained fluid” finite element, however, for vessels of any significant volume, the number of finite elements can easily exceed 100,000 and the number of degrees of freedom can sore from as few as 300,000 to as many as 500,000 or more. While these types of finite element analysis problems can be solved with today’s computer hardware and software, it is not desirable in any analysis to have that volume of information that has to be reviewed and approved in a highly regulated nuclear QA environment (if at all possible). With these items in mind, the methodology described in this paper seeks to minimize the number of degrees of freedom associated with a response spectrum analysis of a liquid filled, skirt supported vertical pressure vessel. The input response spectra are almost always provided in Cartesian coordinates, while many, if not most liquid containing pressure vessels are almost always axisymmetric in geometry without having benefit of being subjected to an axisymmetric load (acceleration in this case) due to the specified seismic event. The use of harmonic finite elements for both the vessel structure and the contained fluid medium permit the efficiencies associated with an axisymmetric geometry to be leveraged when the seismic response spectrum is formulated in terms of a Fourier series and combined to regain the effects of the two orthogonal, horizontally applied accelerations as a function of frequency. The end result as discussed and shown in this paper is a finite element model that permits a dense mesh of both the fluid and the structure, while economizing on the number of simultaneous equations required to be solved by the chosen finite element analysis.

scholarly journals Nonlinear Mechanics of Beams With Partial Piezoelectric Layers

2019 ◽  
Vol 86 (10) ◽  
Author(s):  
Hamed Farokhi ◽  
Mergen H. Ghayesh
Keyword(s):  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Coupled Model ◽  
Beam Theory ◽  
Internal Resonances ◽  
Element Analysis ◽  
Nonlinear Dynamical ◽  
Dimensional Finite Element ◽  
Resonant Region ◽  
Three Dimensional Finite Element

Abstract This paper investigates the nonlinear static response as well as nonlinear forced dynamics of a clamped–clamped beam actuated by piezoelectric patches partially covering the beam from both sides. This study is the first to develop a high-dimensional nonlinear model for such a piezoelectric-beam configuration. The nonlinear dynamical resonance characteristics of the electromechanical system are examined under simultaneous DC and AC piezoelectric actuations, while highlighting the effects of modal energy transfer and internal resonances. A multiphysics coupled model of the beam-piezoelectric system is proposed based on the nonlinear beam theory of Bernoulli–Euler and the piezoelectric constitutive equations. The discretized model of the system is obtained with the help of the Galerkin weighted residual technique while retaining 32 degrees-of-freedom. Three-dimensional finite element analysis is conducted as well in the static regime to validate the developed model and numerical simulation. It is shown that the response of the system in the nonlinear resonant region is strongly affected by a three-to-one internal resonance.

Mixed Finite Element Analysis of Elastomeric Butt-Joints

2005 ◽  
Vol 129 (1) ◽  
pp. 11-18 ◽  
Author(s):  
P. A. Kakavas ◽  
G. I. Giannopoulos ◽  
N. K. Anifantis
Keyword(s):  
Finite Element ◽  
Strain Energy ◽  
Poisson Ratio ◽  
Mixed Finite Element ◽  
Three Dimensional ◽  
Butt Joint ◽  
Element Formulation ◽  
Butt Joints ◽  
Element Analysis ◽  
Strain Energy Density Function

This paper presents a mixed finite element formulation approximating large deformations observed in the analysis of elastomeric butt-joints. The rubber has been considered as nearly incompressible continuum obeying the Mooney/Rivlin (M/R) strain energy density function. The parameters of the model were determined by fitting the available from the literature uniaxial tension experimental data with the constitutive equation derived from the M/R model. The optimum value of the Poisson ratio is adjusted by comparing the experimentally observed diametral contraction of the model with that numerically obtained using the finite element method. The solution of the problem has been obtained utilizing the mixed finite element procedure on the basis of displacement/pressure mixed interpolation and enhanced strain energy mixed formulation. For comparison purposes, an axisymmetric with two-parameter M/R model and a three-dimensional (3D) with nine-parameters M/R model of the butt-joint are formulated and numerical results are illustrated concerning axisymmetric or general loading. For small strains the stress and/or strain distribution in the 2D axisymmetric butt-joint problem was compared with derived analytical solutions. Stress distributions along critical paths are evaluated and discussed.

Dynamic Finite Element Analysis of a Highly Parallel Robotic Surface

2011 ◽  
Author(s):  
Chris Salisbury
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Angular Displacement ◽  
Three Dimensional ◽  
Element Analysis ◽  
Revolute Joints ◽  
Stiffness Matrices ◽  
Joint Forces ◽  
Ricatti Equation ◽  
Optimal Dynamic

A novel three-dimensional robotic surface is devised using triangular modules connected by revolute joints that mimic the constraints of a spherical joint at each triangle intersection. The finite element method (FEM) is applied to the dynamic loading of this device using three dimensional (6 degrees of freedom) beam elements to not only calculate the cartesian displacement and force, but also the angular displacement and torque at each joint. In this way, the traditional methods of finding joint forces and torques are completely bypassed. An effiecient algorithm is developed to linearly combine local mass and stiffness matrices into a full structural stiffness matrix for the easy application of loads. An analysis of optimal dynamic joint forces is carried out in Simulink® with the use of an algebraic Ricatti equation.

Three-Dimensional Creep Analysis of Solder Joints in Surface Mount Devices

1995 ◽  
Vol 117 (1) ◽  
pp. 20-25 ◽  
Author(s):  
Pardeep K. Bhatti ◽  
Klaus Gschwend ◽  
Abel Y. Kwang ◽  
Ahmer R. Syed
Keyword(s):  
High Performance ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Nonlinear Material ◽  
Computational Time ◽  
Nonlinear Creep ◽  
Element Analysis ◽  
Surface Mount ◽  
Creep Analysis ◽  
Highly Nonlinear

Three-dimensional finite element analysis has been applied for determining time-dependent solder joint response of leaded surface mount components under thermal cycling. Two main challenges are the geometric complexity in mesh development and computationally intensive analysis because of the highly nonlinear material properties. Advanced techniques have been applied, including multi-point constraints for mesh transition, which reduces the number of degrees of freedom in the model, and substructuring, which effectively reduces computational time in the iterative analysis. The result is a generic approach for nonlinear creep analysis using commercial FEA software on a high performance workstation. Illustrations are provided for J and gullwing leaded packages.

A novel direct resolution method for coupled systems in finite element analysis

2020 ◽  
Vol 39 (5) ◽  
pp. 1271-1280
Author(s):  
Youcef Boutora ◽  
Noureddine Takorabet
Keyword(s):  
Finite Elements ◽  
Linear Systems ◽  
Symmetric Matrix ◽  
Direct Method ◽  
Three Dimensional ◽  
Coupled Systems ◽  
Resolution Method ◽  
Element Analysis ◽  
Content Type ◽  
Magnetostatic Problem

Purpose This paper aims to propose a novel direct method for indefinite algebraic linear systems. It is well adapted for sparse linear systems, such as those of two-dimensional (2-D) finite elements problems, especially for coupled systems. Design/methodology/approach The proposed method is developed on an example of an indefinite symmetric matrix. The algorithm of the method is given next, and a comparison between the numbers of operations required by the method and the Cholesky method is also given. Finally, an application on a magnetostatic problem for classical methods (Gauss and Cholesky) shows the relative efficiency of the proposed method. Findings The proposed method can be used advantageously for 2-D finite elements in stepping methods without using a block decomposition of matrices. Research limitations/implications This method is advantageous for direct linear solving for 2-D problems, but it is not recommended at this time for three-dimensional problems. Originality/value The proposed method is the first direct solver for algebraic linear systems proposed since more than a half century. It is not limited for symmetric positive systems such as many of direct and iterative methods.

Finite Elements Developed in Cylindrical Coordinates for Three-Dimensional Tire Analysis

1997 ◽  
Vol 25 (1) ◽  
pp. 2-28 ◽  
Author(s):  
K. T. Danielson ◽  
A. K. Noor
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Three Dimensional ◽  
Cylindrical Coordinates ◽  
Element Analysis ◽  
Cartesian Coordinates ◽  
Automobile Tire ◽  
Commercial Code ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

Abstract Finite elements developed in cylindrical coordinates are presented for three-dimensional analysis of tires. In contrast to elements formulated in Cartesian coordinates, these elements allow the exact representation of circular shapes. The exact modeling of circular geometries can provide better finite element predictions and reduce the number of elements needed around the tire circumference. Numerical results are presented for the application of this formulation to the analysis of a radial automobile tire subjected to rim mounting, nonconservative inflation pressure, and rigid pavement contact. The predictions of the foregoing finite elements are compared to experimental data and to predictions of a commercial code using finite elements developed in Cartesian coordinates. The comparisons demonstrate the accuracy and the advantages of the cylindrical coordinate formulation for the three-dimensional finite element analysis of tires.

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