A virtual work derivation of the Biot consolidation finite element formulation

1989 ◽  
Vol 6 (2) ◽  
pp. 158-162 ◽  
Author(s):  
D. Ding ◽  
D.J. Naylor
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Finite Element Formulation ◽  
Element Formulation

Two-Dimensional Geometrically Nonlinear Finite Element Formulation for Analysis of Straight Pipes

2020 ◽  
Author(s):  
Saher Attia ◽  
Magdi Mohareb ◽  
Michael Martens ◽  
Nader Yoosef Ghodsi ◽  
Yong Li ◽  
...  
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Strain Tensor ◽  
Structural Response ◽  
Small Strain ◽  
Nonlinear Finite Element ◽  
Finite Element Formulation ◽  
Single Element ◽  
Element Formulation ◽  
Geometrically Nonlinear

Abstract The paper presents a new and simple geometrically nonlinear finite element formulation to simulate the structural response of straight pipes under in-plane loading and/or internal pressure. The formulation employs the Green-Lagrange strain tensor to capture finite deformation-small strain effects. Additionally, the First Piola-Kirchhoff stress tensor and Saint Venant-Kirchhoff constitutive model are adopted within the principle of virtual work framework in conjunction with a total Lagrangian approach. The formulation is applied for a cantilever beam under three loading conditions. Results are in good agreement with shell models in ABAQUS. Although the solution is based on a single element, the formulation provides reasonable displacement and stress predictions.

A Finite Element Formulation for the Analysis of Horizontally-Curved Thin-Walled Beams

2015 ◽  
Author(s):  
Ashkan Afnani ◽  
Vida Niki ◽  
R. Emre Erkmen
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Curved Beams ◽  
Equilibrium Equations ◽  
Rotation Tensor ◽  
Initial Curvature ◽  
Horizontally Curved ◽  
Thin Walled

In this study, a finite element formulation is developed for the elastic analysis of thin-walled curved beams. Using a second-order rotation tensor, the strains of the deformed configuration are calculated in terms of the displacement values and the initial curvature. The principle of virtual work is then used to obtain the nonlinear equilibrium equations, based on which a finite element beam formulation is developed. The accuracy of the method is confirmed through comparisons with test results and shell-type finite element formulations and other curved beam formulations from the literature. It is also shown that the results of the developed formulation are very accurate for cases where initial curvature is very large.

Dynamics of Thick Viscoelastic Beams

1997 ◽  
Vol 119 (3) ◽  
pp. 273-278 ◽  
Author(s):  
A. R. Johnson ◽  
A. Tessler ◽  
M. Dambach
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Constitutive Law ◽  
Finite Element Formulation ◽  
Internal Variables ◽  
Element Formulation ◽  
Modal Interactions ◽  
Cantilevered Beam ◽  
Thick Beam

A viscoelastic higher-order thick beam finite element formulation is extended to include elastodynamic deformations. The material constitutive law is a special differential form of the Maxwell solid, which employs viscous strains as internal variables to determine the viscous stresses. The total time-dependent stress is the superposition of its elastic and viscous components. In the constitutive model, the elastic strains and the conjugate viscous strains are coupled through a system of first-order ordinary differential equations. The use of the internal strain variables allows for a convenient finite element formulation. The elastodynamic equations of motion are derived from the virtual work principle. Computational examples are carried out for a thick orthotropic cantilevered beam. Relaxation, creep, relaxation followed by free damped vibrations, and damping related modal interactions are discussed.

A Nonlinear Multi-Field Coupling Finite Element Method for Polarization Switching in Ferroelectrics

2008 ◽  
Author(s):  
Jie Wang ◽  
Marc Kamlah
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Three Dimensional ◽  
Polarization Switching ◽  
Ferroelectric Materials ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Polarization Distribution ◽  
Field Coupling ◽  
Simulated System

A three-dimensional nonlinear finite element formulation for ferroelectric materials is developed based on a principle of virtual work. The formulation includes the coupling of three physical fields, namely polarization field, electric field and strain field. The developed finite element formulation is employed to investigate the polarization distribution near a flaw in a ferroelectric single crystal under mechanical loadings. It is found that the polarization switching takes place near the flaw tip if the loadings exceed a critical value. In the simulation, we do not take any prior assumptions, i.e. without any switching criterion, on the polarization switching. The polarization switching is a result of the minimization of the total energy in the simulated system.

Control of Largely-Deformed Nonlinear Electro/Elastic Beam and Plate Systems: Finite Element Formulation and Analysis

2002 ◽  
Author(s):  
D. W. Wang ◽  
H. S. Tzou ◽  
H.-J. Lee
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Nonlinear Dynamic Analysis ◽  
Elastic Beam ◽  
Shell Element ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Control Analysis ◽  
Control Force ◽  
Piezoelectric Shell

Adaptive structures involving large imposed deformation often go beyond the boundary of linear theory and they should be treated as “nonlinear” structures. A generalized nonlinear finite element formulation for vibration sensing and control analysis of laminated electro/elastic nonlinear shell structures is derived based on the virtual work principle. A generic curved triangular piezoelectric shell element is proposed based on the layerwise constant shear angle theory. The dynamic system equations, equations of electric potential output and feedback control force defined in a matrix form are derived. The modified Newton-Raphson method is adopted for nonlinear dynamic analysis of large and complex piezoelectric/elastic/control structures. The developed piezoelectric shell element and finite element code are validated and then applied to control analysis of flexible electro-elastic (piezoelectric/elastic) structural systems. Vibration control of constant-curvature electro/elastic beam and plate systems is studied. Time-history responses of free and controlled nonlinear electro/elastic beam and plate systems are presented and nonlinear effects discussed.

scholarly journals A Few Good Reasons to Consider a Beam Finite Element Formulation on the Lie Group SE(3)

2013 ◽  
Author(s):  
Valentin Sonneville ◽  
Olivier Brüls
Keyword(s):  
Finite Element ◽  
Lie Group ◽  
Virtual Work ◽  
Interpolation Method ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Tangent Stiffness Matrix ◽  
Rigorous Framework ◽  
Internal Forces ◽  
Constant Deformation

Based on an original interpolation method we develop a beam finite element formulation on the Lie group SE(3) which relies on a mathematically rigorous framework and provides compact notations. We work out the beam kinematics in the SE(3) context, the beam deformation measure and obtain the expression of the internal forces using the virtual work principle. The proposed formulation exhibits important features from both the theoretical and numerical points of view. The approach leads to a natural coupling of position and rotation variables and thus differs from classical Timoshenko/Cosserat formulations. We highlight several important properties such as a constant deformation measure over the element, an invariant tangent stiffness matrix under of rigid motions or the absence of shear locking.

scholarly journals Finite Element Formulation of Forced Vibration Problem of a Prestretched Plate Resting on a Rigid Foundation

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
M. Eröz ◽  
A. Yildiz
Keyword(s):  
Finite Element ◽  
Forced Vibration ◽  
Virtual Work ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Virtual Work Principle ◽  
Loading Case ◽  
Linearized Theory

The three-dimensional linearized theory of elastodynamics mathematical formulation of the forced vibration of a prestretched plate resting on a rigid half-plane is given. The variational formulation of corresponding boundary-value problem is constructed. The first variational of the functional in the variational statement is equated to zero. In the framework of the virtual work principle, it is proved that appropriate equations and boundary conditions are derived. Using these conditions, finite element formulation of the prestretched plate is done. The numerical results obtained coincide with the ones given by Ufly and in 1963 for the static loading case.

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