Finite element method Among the up-to-date methods of stress state analysis, finite element method (abbreviated as FEM below, or often as FEA for analyses.

Slides:



Advertisements
Similar presentations
Finite element method Among the up-to-date methods of stress state analysis, the finite element method (abbreviated as FEM below, or often as FEA for analyses.
Advertisements

MANE 4240 & CIVL 4240 Introduction to Finite Elements
Modelling and Simulation for dent/deformation removal Henry Tan Tuesday, 24/2/09.
Finite Element Model Generation Model size Element class – Element type, Number of dimensions, Size – Plane stress & Plane strain – Higher order elements.
Introduction to Finite Element Method
MANE 4240 & CIVL 4240 Introduction to Finite Elements
By S Ziaei-Rad Mechanical Engineering Department, IUT.
NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS 1)Mechanics of Materials Approach (A) Complex Beam Theory (i) Straight Beam (ii) Curved Beam (iii)
APPLIED MECHANICS Lecture 10 Slovak University of Technology
Chapter 17 Design Analysis using Inventor Stress Analysis Module
Lecture 2 – Finite Element Method
Finite Element Primer for Engineers: Part 2
Copyright 2001, J.E. Akin. All rights reserved. CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: –Stress Analysis.
FEA Simulations Usually based on energy minimum or virtual work Component of interest is divided into small parts – 1D elements for beam or truss structures.
Unit 3: Solid mechanics An Introduction to Mechanical Engineering: Part Two Solid mechanics Learning summary By the end of this chapter you should have.
Finite Element Method in Geotechnical Engineering
MECh300H Introduction to Finite Element Methods
Introduction to Finite Element Analysis for Structure Design Dr. A. Sherif El-Gizawy.
ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 12: Isoparametric CST Area Coordinates Shape Functions Strain-Displacement Matrix Rayleigh-Ritz.
ECIV 720 A Advanced Structural Mechanics and Analysis
FE Modeling Strategy Decide on details from design Find smallest dimension of interest Pick element types – 1D Beams – 2D Plate or.
MCE 561 Computational Methods in Solid Mechanics
COMPUTER-AIDED DESIGN The functionality of SolidWorks Simulation depends on which software Simulation product is used. The functionality of different producs.
III Solution of pde’s using variational principles
MANE 4240 & CIVL 4240 Introduction to Finite Elements
EMA 405 Introduction. Syllabus Textbook: none Prerequisites: EMA 214; 303, 304, or 306; EMA 202 or 221 Room: 2261 Engineering Hall Time: TR 11-12:15 Course.
Chapter 5 Vibration Analysis
The Finite Element Method
Introduction to virtual engineering László Horváth Budapest Tech John von Neumann Faculty of Informatics Institute of Intelligent Engineering.
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the.
Finite element method Among up-to-date methods of mechanics and specifically stress analyses, finite element method (abbreviated as FEM below, or often.
ME 520 Fundamentals of Finite Element Analysis
School of Civil EngineeringSpring 2007 CE 595: Finite Elements in Elasticity Instructors: Amit Varma, Ph.D. Timothy M. Whalen, Ph.D.
Finite Element Method.
1 SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS Ahlem ALIA presented by Nicolas AQUELET Laboratoire de Mécanique.
Image courtesy of National Optical Astronomy Observatory, operated by the Association of Universities for Research in Astronomy, under cooperative agreement.
Chapter 6. Plane Stress / Plane Strain Problems
Chapter Five Vibration Analysis.
Applied mechanics of solids A.F. Bower.
Illustration of FE algorithm on the example of 1D problem Problem: Stress and displacement analysis of a one-dimensional bar, loaded only by its own weight,
Finite Element Analysis
HEAT TRANSFER FINITE ELEMENT FORMULATION
HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering.
MECH4450 Introduction to Finite Element Methods
CHAP 3 WEIGHTED RESIDUAL AND ENERGY METHOD FOR 1D PROBLEMS
CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: –Stress Analysis –Thermal Analysis –Structural Dynamics –Computational.
Variational formulation of the FEM Principle of Stationary Potential Energy: Among all admissible displacement functions u, the actual ones are those which.
III. Engineering Plasticity and FEM
1 CHAP 3 WEIGHTED RESIDUAL AND ENERGY METHOD FOR 1D PROBLEMS FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim.
ANSYS Basic Concepts for ANSYS Structural Analysis
Introduction to Finite Element Method
Finite Element Method Weak form Monday, 11/4/2002.
Finite Element Method in Geotechnical Engineering
Overview of Finite Element Methods
Overview of Loads ON and IN Structures / Machines
MANE 4240 & CIVL 4240 Introduction to Finite Elements
CAD and Finite Element Analysis
FEM : Finite Element Method 2017.
FEA Introduction.
Materials Science & Engineering University of Michigan
FEA convergence requirements.
FEA Simulations Boundary conditions are applied
Implementation of 2D stress-strain Finite Element Modeling on MATLAB
Analysis on the principle of finite elements (FEM/FEA)
Analytical Tools in ME Course Objectives
Chapter 2 Rudiment of Structural Analysis and FEM
8-1 Introduction a) Plane Stress y
Realistic multiphysics analysis
ANALYSIS OF BEAM BY USING FEM
Finite element analysis of the wrinkling of orthotropic membranes
Presentation transcript:

Finite element method Among the up-to-date methods of stress state analysis, finite element method (abbreviated as FEM below, or often as FEA for analyses as well) dominates clearly nowadays; it is used also in other fields of engineering analyses (heat transfer, streaming of liquids, electricity and magnetism fields, etc.). In mechanics, the FEM enables us to solve the following types of problems: stress-state analysis under static, cyclic or dynamic (including impact) loading, incl. various non-linear problems; natural as well as forced vibrations (eigenvalues of frequencies), with or without damping; contact problems (contact pressure distribution); stability problems (buckling of structures); stationary or non-stationary heat transfer and evaluation of temperature stresses (incl. residual stresses).

Functional The fundamentals of FEM are quite different from the analytical methods of stress-strain analysis. While analytical methods of stress-strain analysis are based on the differential and integral calculus, FEM is based on the variation calculus which is generally not so well known; it seeks for a minimum of some functional. Explanation of the concept – analogy with functions: Function - is a mapping between two sets of numbers. It is a mathematical term for a rule which enables us to assign unambiguously some numerical value (from the image of mapping) to an initial numerical value (from the domain of mapping). Functional - is a mapping from a set of functions to a set of numbers. It is a rule which enables us to assign unambiguously some numerical value to a function (on the domain of the function or on its part). Definite integral is example of a functional.

Principle of minimum of the quadratic functional Among all allowable displacements (i.e. those which meet boundary conditions of the problem, its geometric and physical equations), only those displacements can come into existence between two close loading states (displacement change by a variation δu) which minimize the quadratic functional ΠL. This functional (called Lagrange potential) represents the total potential energy of the body, and the corresponding displacements, stresses and strains minimizing its value represent the seeked elasticity functions. This principle is called Lagrange variation principle. Lagrange potential ΠL can be written as follows: ΠL = W – P where W - total strain energy of the body P – total potential energy of external loads

Basic notions of FEM Finite element – a subregion of the solved body with a simple geometry. Node – a point in which the numerical values of the unknown deformation parameters are calculated. Base function – a function describing the distribution of deformation parameters (displacements) inside the element (between the nodes). Shape function – a function describing the distribution of strains inside the element, it represents a derivative of the base function. Discretization – transformation of a continuous problem to a solution of a finite number of non-continuous (discrete) numerical values. Mesh density – it influences the time consumption and accuracy of the solution. Matrixes (they are created by summarization of contributions of the individual elements) of displacements of stiffness of base functions Convergence – a property of the method, the solution tends to the reality (continuous solution) when the mesh (discretization) density increases. Percentual energy error gives an assessment of the total inaccuracy of the solution, it represents the difference between the calculated stress values and their values smoothed by postprocessing tools used for their graphical representation, transformed into difference in strain energies. Isoparametric element – element with the same order of the polynoms used in description of both geometry and base functions.

FEA of the stress concentration in a notch (tensed bar with shoulder)

Stress distribution in the dangerous cross section of the notch

Overview of basic types of finite elements They can be distinguished from the viewpoint of assumptions the element is based on (bar assumptions, axisymmetry, Kirchhoff plates, membrane shells, etc.), or for what a family of problems the element is proposed. 2D elements (plane stress, plain strain, axisymmetry) 3D elements (volume elements - bricks) Bar-like elements (either for tension-compression only, or for flexion or torsion as well) Shell elements Special elements (with special constitutive relations, contact or crack elements, etc.)

Types of elements – three-dimensional Deformation parameters Type of element Representation Tetrahedron Pentahedron Hexahedron with 8 nodes Hexahedron with 20 nodes

Types of elements – one-dimensional Type of element Representation Deformation parameters Link – (strut,bar) 1D element (tension only) in 2D in 3D Beam element (2D flexion only) in 2D Frame element (flexion + tension) in 3D – with torsion

Types of elements – two-dimensional Deformation parameters Type of element Representation Membrane shell elements Linear triangular shell element Quadratic triangular shell element Linear quadrilateral shell element Quadratic isoparametric quadrilateral shell element Plate element General (momentum) shell elements linear quadratic isoparametric

Basic types of constitutive relations soluble in FEA linear elastic anisotropic ─ elastic parameters are direction dependent (monocrystals, wood, fibre composites or multilayer materials), elastic-plastic (steel above the yield stress) with different types of behaviour above the yield stress (perfect elastic-plastic materials, various types of hardening), non-linear elastic ─ deformations are reversible, but non-linearly related to the stress (rubber and other elastomers, soft biological tissues), hyperelastic (showing elastic strain on the order of 101 – 102percents, non-linear always as well), Viscoelastic ─ the deformation is recoverable but time-dependent, i.e. not instantaneous but with some transition time when the material is out of equilibrium; the material shows creep, stress relaxation and hysteresis (plastics, pure alluminium, steel above 400°C) , Viscoplastic ─ plastic (permanent) deformation is time-dependent, etc.

Examples of non-linear problems: a FE mesh in a plastic guard of a pressure bottle valve

Distribution of Mises stresses in a plastic safety guard after impact, grey colour shows regions where yield stress Re=54MPa was exceeded.)

Assembly with a steel shock absorber after impact test (total assembly with bottle and plastic guard in the video)

Steel shock absorber original welded design under simulated impact test

Steel shock absorber new design with controlled plastic deformation (details in pdf file)

Comparison of different solutions of steel shock absorbers The area below the curve corresponds to the absorbed deformation energy.