Geometrically Exact Analysis of Spatial Frames

1993 ◽  
Vol 46 (11S) ◽  
pp. S118-S128 ◽  
Author(s):  
Paulo M. Pimenta ◽  
Takashi Yojo
Keyword(s):  
Finite Element ◽  
Weak Form ◽  
Small Strain ◽  
Exact Expression ◽  
Galerkin Projection ◽  
Equilibrium Equations ◽  
Rodrigues Formula ◽  
Linear Elastic ◽  
3D Space ◽  
Shear Distortion

A fully nonlinear, geometrically exact, finite strain rod model is derived from basic kinematical assumptions. The model incorporates shear distortion in bending and can take account of torsion warping. Rotation in 3D space is handled with the aid of the Euler-Rodrigues formula. The accomplished parametrization is simple and does not require update algorithms based on quaternions parameters. Weak and strong forms of the equilibrium equations are derived in terms of cross section strains and stresses, which are objective and suitable for constitutive description. As an example, an invariant linear elastic constitutive equation based on the small strain theory is presented. The attained formulation is very convenient for numerical procedures employing Galerkin projection like the finite element method and can be readily implemented in a finite element code. A mixed formulation of Hu-Washizu type is also derived, allowing for independent interpolation of the displacement, strain and stress fields within a finite element. An exact expression for the Fre´chet derivative of the weak form of equilibrium is obtained in closed form, which is always symmetric for conservative loading, even far from an equilibrium state and is very helpful for numerical procedures like the Newton method as well as for stability and bifurcation analysis. Several numerical examples illustrate the usefulness of the formulation in the lateral stability analysis of spatial frames. These examples were computed with the code FENOMENA, which is under development at the Computational Mechanics Laboratory of the Escola Polite´cnica.

scholarly journals BEAM ELEMENT UNDER FINITE ROTATIONS

2021 ◽  
Vol 30 ◽  
pp. 87-92
Author(s):  
Emma La Malfa Ribolla ◽  
Milan Jirásek ◽  
Martin Horák
Keyword(s):  
Cross Sections ◽  
Shooting Method ◽  
Small Strain ◽  
Beam Model ◽  
Equilibrium Equations ◽  
Finite Rotations ◽  
Nonlinear Beam ◽  
Linear Elastic ◽  
Beam Elements ◽  
Large Rotations

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

Generalized Topology Optimization of Linear Elastic Structures Subjected to Thermal Loads

1993 ◽  
Author(s):  
Helder C. Rodrigues ◽  
Paulo A. Fernandes
Keyword(s):  
Finite Element ◽  
Topology Optimization ◽  
Computational Model ◽  
Asymptotic Methods ◽  
Generalized Topology ◽  
Thermal Loads ◽  
Equilibrium Equations ◽  
Topology Optimization Problem ◽  
Finite Element Approximations ◽  
Linear Elastic

Abstract This paper presents the development of a computational model for the generalized topology optimization problem, using a material distribution approach, of 2-D linear elastic solids subjected to thermal loads, with compliance objective function and an isoperimetric constraint on volume. The model relies on homogenization asymptotic methods to characterize the influence of the material periodic microstructure and a finite element displacement formulation is used to approximate the homogenized equilibrium equations obtained. The computational model developed is tested in several examples considering different finite element approximations and the influence of the design variables (material density and orientation) is analyzed.

scholarly journals Discrete Averaging Relations for Micro to Macro Transition

2016 ◽  
Vol 83 (8) ◽  
Author(s):  
Chenchen Liu ◽  
Celia Reina
Keyword(s):  
Finite Element ◽  
Large Deformations ◽  
Fe Simulation ◽  
Weak Form ◽  
Volume Element ◽  
Shape Functions ◽  
Additional Error ◽  
Equilibrium Equations ◽  
Solid Foundation ◽  
Multiscale Finite Element

The well-known Hill's averaging theorems for stresses and strains as well as the so-called Hill–Mandel condition are essential ingredients for the coupling and the consistency between the micro- and macroscales in multiscale finite-element procedures (FE2). We show in this paper that these averaging relations hold exactly under standard finite-element (FE) discretizations, even if the stress field is discontinuous across elements and the standard proofs based on the divergence theorem are no longer suitable. The discrete averaging results are derived for the three classical types of boundary conditions (BC) (affine displacement, periodic, and uniform traction BC) using the properties of the shape functions and the weak form of the microscopic equilibrium equations without further kinematic constraints. The analytical proofs are further verified numerically through a simple FE simulation of an irregular representative volume element (RVE) undergoing large deformations. Furthermore, the proofs are extended to include the effects of body forces and inertia, and the results are consistent with those in the smooth continuum setting. This work provides a solid foundation to apply Hill's averaging relations in multiscale FE methods without introducing an additional error in the scale transition due to the discretization.

scholarly journals Finite Element Modelling and Mechanical Characterization of Graphyne

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Ricardo Couto ◽  
Nuno Silvestre
Keyword(s):  
Finite Element ◽  
Elastic Properties ◽  
Density Functional ◽  
Computational Models ◽  
Mechanical Characterization ◽  
Computational Simulation ◽  
Md Simulations ◽  
Small Strain ◽  
Fe Model ◽  
Linear Elastic

Graphyne is an allotrope of carbon with excellent mechanical, electrical, and optical properties. The scientific community has been increasingly interested in its characterization and computational simulation, using molecular dynamics (MD) simulations and density functional theory (DFT). The present work presents, for the first time (to the authors’ knowledge), a finite element (FE) model to evaluate the elastic properties of graphyne. After presenting a brief literature review on the latest developments of graphyne and its mechanical characterization through computational methods, the FE model of graphyne sheet is presented in detail and the calculation of its elastic properties described. The linear elastic properties (Young’s modulus, Poisson’s ratio, bulk modulus, and shear modulus) obtained from the proposed FE models are in general agreement with those previously obtained by other authors using more complex computational models (MD and DFT). The influence of van der Waals (vdW) interatomic forces on the linear elastic properties of planar graphyne is negligible and can be disregarded if small strain hypothesis is adopted. The FE models also show that graphyne exhibits marginal orthotropic behavior, that is, “quasi-isotropic” behavior, a fact that agrees with the conclusions reported by other researchers.

An Eight-Node Hexahedral Finite Element with Rotational DOFs for Elastoplastic Applications

2019 ◽  
Vol 11 (1) ◽  
pp. 54-66
Author(s):  
Ayoub Ayadi ◽  
Kamel Meftah ◽  
Lakhdar Sedira ◽  
Hossam Djahara
Keyword(s):  
Mathematical Model ◽  
Finite Element ◽  
Plastic Zone ◽  
Displacement Vector ◽  
Small Strain ◽  
Benchmark Problems ◽  
Plasticity Model ◽  
Vector Approximation ◽  
Calculation Code

Abstract In this paper, the earlier formulation of the eight-node hexahedral SFR8 element is extended in order to analyze material nonlinearities. This element stems from the so-called Space Fiber Rotation (SFR) concept which considers virtual rotations of a nodal fiber within the element that enhances the displacement vector approximation. The resulting mathematical model of the proposed SFR8 element and the classical associative plasticity model are implemented into a Fortran calculation code to account for small strain elastoplastic problems. The performance of this element is assessed by means of a set of nonlinear benchmark problems in which the development of the plastic zone has been investigated. The accuracy of the obtained results is principally evaluated with some reference solutions.

scholarly journals Fatigue Crack Growth Analysis with Extended Finite Element for 3D Linear Elastic Material

Metals ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 397
Author(s):  
Yahya Ali Fageehi
Keyword(s):  
Finite Element ◽  
Fatigue Crack ◽  
Fatigue Life ◽  
Crack Propagation ◽  
Crack Growth ◽  
Mixed Mode ◽  
Propagation Path ◽  
Loading Conditions ◽  
Extended Finite Element ◽  
Linear Elastic

This paper presents computational modeling of a crack growth path under mixed-mode loadings in linear elastic materials and investigates the influence of a hole on both fatigue crack propagation and fatigue life when subjected to constant amplitude loading conditions. Though the crack propagation is inevitable, the simulation specified the crack propagation path such that the critical structure domain was not exceeded. ANSYS Mechanical APDL 19.2 was introduced with the aid of a new feature in ANSYS: Smart Crack growth technology. It predicts the propagation direction and subsequent fatigue life for structural components using the extended finite element method (XFEM). The Paris law model was used to evaluate the mixed-mode fatigue life for both a modified four-point bending beam and a cracked plate with three holes under the linear elastic fracture mechanics (LEFM) assumption. Precise estimates of the stress intensity factors (SIFs), the trajectory of crack growth, and the fatigue life by an incremental crack propagation analysis were recorded. The findings of this analysis are confirmed in published works in terms of crack propagation trajectories under mixed-mode loading conditions.

Topology Synthesis of Compliant Mechanisms for Nonlinear Force-Deflection and Curved Path Specifications

1999 ◽  
Vol 123 (1) ◽  
pp. 33-42 ◽  
Author(s):  
A. Saxena ◽  
G. K. Ananthasuresh
Keyword(s):  
Finite Element ◽  
Linear Models ◽  
Compliant Mechanisms ◽  
Synthesis Method ◽  
Design Sensitivity ◽  
Finite Element Models ◽  
Geometrically Nonlinear ◽  
Linear Elastic ◽  
Nonlinear Design ◽  
Nonlinear Force

Optimal design methods that use continuum mechanics models are capable of generating suitable topology, shape, and dimensions of compliant mechanisms for desired specifications. Synthesis procedures that use linear elastic finite element models are not quantitatively accurate for large displacement situations. Also, design specifications involving nonlinear force-deflection characteristics and generation of a curved path for the output port cannot be realized with linear models. In this paper, the synthesis of compliant mechanisms is performed using geometrically nonlinear finite element models that appropriately account for large displacements. Frame elements are chosen because of ease of implementation of the general approach and their ability to capture bending deformations. A method for nonlinear design sensitivity analysis is described. Examples are included to illustrate the usefulness of the synthesis method.

The Equilibrium Cell-Based Smooth Finite Element Method for Shakedown Analysis of Structures

2019 ◽  
Vol 16 (05) ◽  
pp. 1840013 ◽  
Author(s):  
P. L. H. Ho ◽  
C. V. Le ◽  
T. Q. Chu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimization Problem ◽  
Conic Programming ◽  
Equilibrium Equations ◽  
Shakedown Analysis ◽  
Numerical Examples ◽  
Elastic Stresses ◽  
Gauss Points ◽  
Element Method

This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretized mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.

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