FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS"

Transcription

1 FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC.

2 CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT METHOD: THE BIG PICTURE Discretization and Element Equations / Plane Trass Element / Triangular Element for Two-Dimensional Heat Flow / General Remarks on Finite Element Discretization / Triangular Element for Two-Dimensional Stress Analysis / Assembly of Element Equations / Boundary Conditions and Nodal Solution / Essential Boundary Conditions by Rearranging Equations / Essential Boundary Conditions by Modifying Equations / Approximate Treatment of Essential Boundary Conditions / Computation of Reactions to Verify Overall Equilibrium / Element Solutions and Model Validity / Plane Truss Element / Triangular Element for Two-Dimensional Heat Flow / Triangular Element for Two-Dimensional Stress Analysis / Solution of Linear Equations / Solution Using Choleski Decomposition / Conjugate Gradient Method / 62

3 Vi CONTENTS 1.6 Multipoint Constraints / Solution Using Lagrange Multipliers / Solution Using Penalty Function / Units / 83 MATHEMATICAL FOUNDATION OF THE FINITE ELEMENT METHOD Axial Deformation of Bars / Differential Equation for Axial Deformations / Exact Solutions of Some Axial Deformation Problems / Axial Deformation of Bars Using Galerkin Method / Weak Form for Axial Deformations / Uniform Bar Subjected to Linearly Varying Axial Load / Tapered Bar Subjected to Linearly Varying Axial Load / One-Dimensional BVP Using Galerkin Method / Overall Solution Procedure Using Galerkin Method / Higher Order Boundary Value Problems / Rayleigh-Ritz Method / Potential Energy for Axial Deformation of Bars / Overall Solution Procedure Using the Rayleigh-Ritz Method / Uniform Bar Subjected to Linearly Varying Axial Load / Tapered Bar Subjected to Linearly Varying Axial Load / Comments on Galerkin and Rayleigh-Ritz Methods / Admissible Assumed Solution / Solution Convergence the Completeness Requirement / Galerkin versus Rayleigh-Ritz / Finite Element Form of Assumed Solutions / Linear Interpolation Functions for Second-Order Problems / Lagrange Interpolation / Galerkin Weighting Functions in Finite Element Form / Hermite Interpolation for Fourth-Order Problems / Finite Element Solution of Axial Deformation Problems / Two-Node Uniform Bar Element for Axial Deformations / Numerical Examples / 155 ONE-DIMENSIONAL BOUNDARY VALUE PROBLEM ,-Selected Applications of ID BVP / Steady-State Heat Conduction / Heat How through Thin Fins / 175

4 Vii Viscous Fluid Flow between Parallel Plates Lubrication Problem / Slider Bearing / Axial Deformation of Bars / Elastic Buckling of Long Slender Bars / Finite Element Formulation for Second-Order ID BVP / Complete Solution Procedure / Steady-State Heat Conduction / Steady-State Heat Conduction and Convection / Viscous Fluid Flow Between Parallel Plates / Elastic Buckling of Bars / Solution of Second-Order ID BVP / A Closer Look at the Interelement Derivative Terms / TRUSSES, BEAMS, AND FRAMES Plane Trasses / Space Trasses / Temperature Changes and Initial Strains in Trasses / Spring Elements / Transverse Deformation of Beams / Differential Equation for Beam Bending / Boundary Conditions for Beams / Shear Stresses in Beams / Potential Energy for Beam Bending / Transverse Deformation of a Uniform Beam / Transverse Deformation of a Tapered Beam Fixed at Both Ends / Two-Node Beam Element / Cubic Assumed Solution / Element Equations Using Rayleigh-Ritz Method / Uniform Beams Subjected to Distributed Loads / Plane Frames / Space Frames / Element Equations in Local Coordinate System / Local-to-Global Transformation / Element Solution / Frames in Multistory Buildings / 293

5 Viii CONTENTS TWO-DIMENSIONAL ELEMENTS Selected Applications of the 2D BVP / Two-Dimensional Potential Flow / Steady-State Heat Flow / Bars Subjected to Torsion / Waveguides in Electromagnetics / Integration by Parts in Higher Dimensions / Finite Element Equations Using the Galerkin Method / Rectangular Finite Elements / Four-Node Rectangular Element / Eight-Node Rectangular Element / Lagrange Interpolation for Rectangular Elements / Triangular Finite Elements / Three-Node Triangular Element / Higher Order Triangular Elements / 371 MAPPED ELEMENTS Integration Using Change of Variables / One-Dimensional Integrals / Two-Dimensional Area Integrals / Three-Dimensional Volume Integrals / Mapping Quadrilaterals Using Interpolation Functions / Mapping Lines / Mapping Quadrilateral Areas / Mapped Mesh Generation / Numerical Integration Using Gauss Quadrature / Gauss Quadrature for One-Dimensional Integrals / Gauss Quadrature for Area Integrals / Gauss Quadrature for Volume Integrals / Finite Element Computations Involving Mapped Elements / Assumed Solution / Derivatives of the Assumed Solution / Evaluation of Area Integrals / Evaluation of Boundary Integrals / Complete Mathematica and MATLAB Solutions of 2D BVP Involving Mapped Elements / Triangular Elements by Collapsing Quadrilaterals / Infinite Elements / One-Dimensional BVP / Two-Dimensional BVP / 458

6 7 ANALYSIS OF ELASTIC SOLIDS Fundamental Concepts in Elasticity / Stresses / Stress Failure Criteria / Strains / Constitutive Equations / Temperature Effects and Initial Strains / Governing Differential Equations / Stress Equilibrium Equations / Governing Differential Equations in Terms of Displacements / General Form of Finite Element Equations / Potential Energy Functional / Weak Form / Finite Element Equations / Finite Element Equations in the Presence of Initial Strains / Plane Stress and Plane Strain / Plane Stress Problem / Plane Strain Problem / Finite Element Equations / Three-Node Triangular Element / Mapped Quadrilateral Elements / Planar Finite Element Models / Pressure Vessels / Rotating Disks and Flywheels / Residual Stresses Due to Welding / Crack Tip Singularity / TRANSIENT PROBLEMS Transient Field Problems / Finite Element Equations / 5^ Triangular Element / Transient Heat Flow / Elastic Solids Subjected to Dynamic Loads / Finite Element Equations / Mass Matrices for Common Structural Elements / Free-Vibration Analysis / Transient Response Examples / 573

7 9 p-formulation p-formulation for Second-Order ID BVP / Assumed Solution Using Legendre Polynomials / Element Equations / Numerical Examples / ^-Formulation for Second-Order 2D BVP / p-mode Assumed Solution / Finite Element Equations / Assembly of Element Equations / Incorporating Essential Boundary Conditions / Applications / 624 A USE OF COMMERCIAL FEA SOFTWARE 641 A.I ANSYS Applications / 642 A. 1.1 General Steps / 643 A. 1.2 Trass Analysis / 648 A. 1.3 Steady-State Heat Flow / 651 A. 1.4 Plane Stress Analysis / 655 A.2 Optimizing Design Using ANSYS / 659 A.2.1 General Steps / 659 A.2.2 Heat Flow Example / 660 A.3 ABAQUS Applications / 663 A.3.1 Execution Procedure / 663 A.3.2 Trass Analysis / 665 A.3.3 Steady-State Heat Flow / 666 A.3.4 Plane Stress Analysis / 671 B VARIATIONAL FORM FOR BOUNDARY VALUE PROBLEMS 676 B.I Basic Concept of Variation of a Function / 676 B.2 Derivation of Equivalent Variational Form / 679 B.3 Boundary Value Problem Corresponding to a Given Functional / 683 BIBLIOGRAPHY 687 INDEX 695

CAD and Finite Element Analysis

CAD and Finite Element Analysis CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: Stress Analysis Thermal Analysis Structural Dynamics Computational Fluid Dynamics (CFD) Electromagnetics Analysis...

More information

Finite Element Formulation for Plates - Handout 3 -

Finite Element Formulation for Plates - Handout 3 - Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the

More information

1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples.

1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples. M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper - I MCEGS 106-2 FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL

More information

Finite Element Formulation for Beams - Handout 2 -

Finite Element Formulation for Beams - Handout 2 - Finite Element Formulation for Beams - Handout 2 - Dr Fehmi Cirak (fc286@) Completed Version Review of Euler-Bernoulli Beam Physical beam model midline Beam domain in three-dimensions Midline, also called

More information

Finite Element Methods (in Solid and Structural Mechanics)

Finite Element Methods (in Solid and Structural Mechanics) CEE570 / CSE 551 Class #1 Finite Element Methods (in Solid and Structural Mechanics) Spring 2014 Prof. Glaucio H. Paulino Donald Biggar Willett Professor of Engineering Department of Civil and Environmental

More information

CAE -Finite Element Method

CAE -Finite Element Method 16.810 Engineering Design and Rapid Prototyping Lecture 3b CAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 16, 2007 Numerical Methods Finite Element Method Boundary Element Method

More information

820446 - ACMSM - Computer Applications in Solids Mechanics

820446 - ACMSM - Computer Applications in Solids Mechanics Coordinating unit: 820 - EUETIB - Barcelona College of Industrial Engineering Teaching unit: 737 - RMEE - Department of Strength of Materials and Structural Engineering Academic year: Degree: 2015 BACHELOR'S

More information

Finite Element Method for Engineers. From Theory to Practice

Finite Element Method for Engineers. From Theory to Practice Finite Element Method for Engineers From Theory to Practice Finite Element Method for Engineers From Theory to Practice C.V. Girija Vallabhan Mehmet Zülfü Aş1k Narosa Publishing House New Delhi Chennai

More information

Programming the Finite Element Method

Programming the Finite Element Method Programming the Finite Element Method FOURTH EDITION I. M. Smith University of Manchester, UK D. V. Griffiths Colorado School of Mines, USA John Wiley & Sons, Ltd Contents Preface Acknowledgement xv xvii

More information

Types of Elements

Types of Elements chapter : Modeling and Simulation 439 142 20 600 Then from the first equation, P 1 = 140(0.0714) = 9.996 kn. 280 = MPa =, psi The structure pushes on the wall with a force of 9.996 kn. (Note: we could

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 01 Welcome to the series of lectures, on finite element analysis. Before I start,

More information

List of Problems Solved Introduction p. 1 Concept p. 1 Nodes p. 3 Elements p. 4 Direct Approach p. 5 Linear Spring p. 5 Heat Flow p.

List of Problems Solved Introduction p. 1 Concept p. 1 Nodes p. 3 Elements p. 4 Direct Approach p. 5 Linear Spring p. 5 Heat Flow p. Preface p. v List of Problems Solved p. xiii Introduction p. 1 Concept p. 1 Nodes p. 3 Elements p. 4 Direct Approach p. 5 Linear Spring p. 5 Heat Flow p. 6 Assembly of the Global System of Equations p.

More information

CAE -Finite Element Method

CAE -Finite Element Method 16.810 Engineering Design and Rapid Prototyping CAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 11, 2005 Plan for Today Hand Calculations Aero Æ Structures FEM Lecture (ca. 45 min)

More information

INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS

INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS INTEGRAL METHODS IN LOW-FREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,

More information

EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss

EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000

More information

Finite Element Method

Finite Element Method 16.810 (16.682) Engineering Design and Rapid Prototyping Finite Element Method Instructor(s) Prof. Olivier de Weck deweck@mit.edu Dr. Il Yong Kim kiy@mit.edu January 12, 2004 Plan for Today FEM Lecture

More information

Finite Element Method (ENGC 6321) Syllabus. Second Semester 2013-2014

Finite Element Method (ENGC 6321) Syllabus. Second Semester 2013-2014 Finite Element Method Finite Element Method (ENGC 6321) Syllabus Second Semester 2013-2014 Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered

More information

AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS

AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS By: Eduardo DeSantiago, PhD, PE, SE Table of Contents SECTION I INTRODUCTION... 2 SECTION II 1-D EXAMPLE... 2 SECTION III DISCUSSION...

More information

Stress Recovery 28 1

Stress Recovery 28 1 . 8 Stress Recovery 8 Chapter 8: STRESS RECOVERY 8 TABLE OF CONTENTS Page 8.. Introduction 8 8.. Calculation of Element Strains and Stresses 8 8.. Direct Stress Evaluation at Nodes 8 8.. Extrapolation

More information

Applied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama

Applied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama Applied Finite Element Analysis M. E. Barkey Aerospace Engineering and Mechanics The University of Alabama M. E. Barkey Applied Finite Element Analysis 1 Course Objectives To introduce the graduate students

More information

Element Selection Criteria

Element Selection Criteria Element Selection Criteria Element Selection Criteria Overview Elements in ABAQUS Structural Elements (Shells and Beams) vs. Continuum Elements Modeling Bending Using Continuum Elements Stress Concentrations

More information

CIEG 605 Intermediate Topics in Finite Element Analyses

CIEG 605 Intermediate Topics in Finite Element Analyses CIEG 605 Intermediate Topics in Finite Element Analyses 2016 Fall Semester General Description from Course Catalog Topics include finite element analyses of linear steady-state scalar field problems, linear

More information

Pro/ENGINEER Mechanica Simulation using Pro/ENGINEER Wildfire 5.0

Pro/ENGINEER Mechanica Simulation using Pro/ENGINEER Wildfire 5.0 Pro/ENGINEER Mechanica Simulation using Pro/ENGINEER Wildfire 5.0 Overview Course Code TRN-2235-S Course Length 40 Hours This course is designed for new users who want to test, validate, and optimize product

More information

An Overview of the Finite Element Analysis

An Overview of the Finite Element Analysis CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry

More information

Graduate Courses in Mechanical Engineering

Graduate Courses in Mechanical Engineering Graduate Courses in Mechanical Engineering MEEG 501 ADVANCED MECHANICAL ENGINEERING ANALYSIS An advanced, unified approach to the solution of mechanical engineering problems, with emphasis on the formulation

More information

LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite

More information

Course in. Nonlinear FEM

Course in. Nonlinear FEM Course in Introduction Outline Lecture 1 Introduction Lecture 2 Geometric nonlinearity Lecture 3 Material nonlinearity Lecture 4 Material nonlinearity continued Lecture 5 Geometric nonlinearity revisited

More information

The elements used in commercial codes can be classified in two basic categories:

The elements used in commercial codes can be classified in two basic categories: CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for

More information

Frequently Asked Questions about the Finite Element Method

Frequently Asked Questions about the Finite Element Method Frequently Asked Questions about the Finite Element Method 1. What is the finite element method (FEM)? The FEM is a novel numerical method used to solve ordinary and partial differential equations. The

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko

BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko Uemuet Goerguelue Two mathematical models, namely the shear-deformable (Timoshenko) model and the shearindeformable (Euler-Bernoulli)

More information

Solving Direct and Inverse Heat Conduction Problems

Solving Direct and Inverse Heat Conduction Problems Jan Taler Piotr Duda Solving Direct and Inverse Heat Conduction Problems 4y Springer Part I Heat Conduction Fundamentals 1 1 Fourier Law 3 Literature 6 2 Mass and Energy Balance Equations 7 2.1 Mass Balance

More information

ANALYSIS OF STRUCTURAL MEMBER SYSTEMS JEROME J. CONNOR NEW YORK : ':,:':,;:::::,,:

ANALYSIS OF STRUCTURAL MEMBER SYSTEMS JEROME J. CONNOR NEW YORK : ':,:':,;:::::,,: ANALYSIS OF JEROME J. CONNOR, Sc.D., Massachusetts Institute of Technology, is Professor of Civil Engineering at Massachusetts Institute of Technology. He has been active in STRUCTURAL MEMBER teaching

More information

TWO-DIMENSIONAL FINITE ELEMENT ANALYSIS OF FORCED CONVECTION FLOW AND HEAT TRANSFER IN A LAMINAR CHANNEL FLOW

TWO-DIMENSIONAL FINITE ELEMENT ANALYSIS OF FORCED CONVECTION FLOW AND HEAT TRANSFER IN A LAMINAR CHANNEL FLOW TWO-DIMENSIONAL FINITE ELEMENT ANALYSIS OF FORCED CONVECTION FLOW AND HEAT TRANSFER IN A LAMINAR CHANNEL FLOW Rajesh Khatri 1, 1 M.Tech Scholar, Department of Mechanical Engineering, S.A.T.I., vidisha

More information

Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31)

Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Outline -1-! This part of the module consists of seven lectures and will focus

More information

Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16)

Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16) Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16) The Stiffness method provides a very systematic way of analyzing determinate and indeterminate structures. Recall Force

More information

The Basics of FEA Procedure

The Basics of FEA Procedure CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring

More information

Feature Commercial codes In-house codes

Feature Commercial codes In-house codes A simple finite element solver for thermo-mechanical problems Keywords: Scilab, Open source software, thermo-elasticity Introduction In this paper we would like to show how it is possible to develop a

More information

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

MASTER OF SCIENCE IN MECHANICAL ENGINEERING MASTER OF SCIENCE IN MECHANICAL ENGINEERING Introduction There are over 22 schools in Mindanao that offer Bachelor of Science in Mechanical Engineering and majority of their faculty members do not have

More information

COMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS

COMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, March-April 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2

More information

3 Concepts of Stress Analysis

3 Concepts of Stress Analysis 3 Concepts of Stress Analysis 3.1 Introduction Here the concepts of stress analysis will be stated in a finite element context. That means that the primary unknown will be the (generalized) displacements.

More information

Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) Introduction to the Finite Element Method (FEM) Lecture 1 The Direct Stiffness Method and the Global StiffnessMatrix Dr. J. Dean 1 Introduction The finite element method (FEM) is a numerical technique

More information

CIVL 7/8117 Chapter 3a - Development of Truss Equations 1/80

CIVL 7/8117 Chapter 3a - Development of Truss Equations 1/80 CIV 7/87 Chapter 3a - Development of Truss Equations /8 Development of Truss Equations Having set forth the foundation on which the direct stiffness method is based, we will now derive the stiffness matri

More information

THERMAL CRACKING RESPONE OF REINFORCED CONCRETE BEAM TO GRADIENT TEMPERATURE

THERMAL CRACKING RESPONE OF REINFORCED CONCRETE BEAM TO GRADIENT TEMPERATURE THERMAL CRACKING RESPONE OF REINFORCED CONCRETE BEAM TO GRADIENT TEMPERATURE L. DAHMANI, M.KOUANE Abstract In this paper are illustrated the principal aspects connected with the numerical evaluation of

More information

Numerical Methods for Engineers

Numerical Methods for Engineers Steven C. Chapra Berger Chair in Computing and Engineering Tufts University RaymondP. Canale Professor Emeritus of Civil Engineering University of Michigan Numerical Methods for Engineers With Software

More information

Shell Elements in ABAQUS/Explicit

Shell Elements in ABAQUS/Explicit ABAQUS/Explicit: Advanced Topics Appendix 2 Shell Elements in ABAQUS/Explicit ABAQUS/Explicit: Advanced Topics A2.2 Overview ABAQUS/Explicit: Advanced Topics ABAQUS/Explicit: Advanced Topics A2.4 Triangular

More information

Asymptotic Analysis of Fields in Multi-Structures

Asymptotic Analysis of Fields in Multi-Structures Asymptotic Analysis of Fields in Multi-Structures VLADIMIR KOZLOV Department of Mathematics, Linkoeping University, Sweden VLADIMIR MAZ'YA Department of Mathematics, Linkoeping University, Sweden ALEXANDER

More information

Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction

Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.

More information

Finite Elements for 2 D Problems

Finite Elements for 2 D Problems Finite Elements for 2 D Problems General Formula for the Stiffness Matrix Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows,

More information

FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633

FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633 FINITE ELEMENT : MATRIX FORMULATION Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 76 FINITE ELEMENT : MATRIX FORMULATION Discrete vs continuous Element type Polynomial approximation

More information

ESSENTIAL COMPUTATIONAL FLUID DYNAMICS

ESSENTIAL COMPUTATIONAL FLUID DYNAMICS ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov WILEY JOHN WILEY & SONS, INC. CONTENTS PREFACE xv 1 What Is CFD? 1 1.1. Introduction / 1 1.2. Brief History of CFD / 4 1.3. Outline of the Book / 6 References

More information

CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS

CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS 1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the

More information

Tower Cross Arm Numerical Analysis

Tower Cross Arm Numerical Analysis Chapter 7 Tower Cross Arm Numerical Analysis In this section the structural analysis of the test tower cross arm is done in Prokon and compared to a full finite element analysis using Ansys. This is done

More information

A quadrilateral 2-D finite element based on mixed interpolation of tensorial

A quadrilateral 2-D finite element based on mixed interpolation of tensorial A quadrilateral 2-D finite element based on mixed interpolation of tensorial components Eduardo N. Dvorkin and Sara I. Vassolo* Instituto de Materiales y Estructuras, Facultad de Ingenieria, Universidad

More information

Module 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction

Module 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction Module 1 : Conduction Lecture 5 : 1D conduction example problems. 2D conduction Objectives In this class: An example of optimization for insulation thickness is solved. The 1D conduction is considered

More information

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, March-April 2016, pp. 45-66, Article ID: IJCIET_07_02_004 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2

More information

Numerical Analysis An Introduction

Numerical Analysis An Introduction Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs

More information

Reliable FE-Modeling with ANSYS

Reliable FE-Modeling with ANSYS Reliable FE-Modeling with ANSYS Thomas Nelson, Erke Wang CADFEM GmbH, Munich, Germany Abstract ANSYS is one of the leading commercial finite element programs in the world and can be applied to a large

More information

FINITE ELEMENT METHOD (FEM): AN OVERVIEW. Dr A Chawla

FINITE ELEMENT METHOD (FEM): AN OVERVIEW. Dr A Chawla FINITE ELEMENT METHOD (FEM): AN OVERVIEW Dr A Chawla ANALYTICAL / MATHEMATICAL SOLUTIONS RESULTS AT INFINITE LOCATIONS CONTINUOUS SOLUTIONS FOR SIMPLIFIED SITUATIONS ONLY EXACT SOLUTION NUMERICAL (FEM)

More information

Vehicle-Bridge Interaction Dynamics

Vehicle-Bridge Interaction Dynamics Vehicle-Bridge Interaction Dynamics With Applications to High-Speed Railways Y. B. Yang National Taiwan University, Taiwan J. D. Yau Tamkang University, Taiwan Y. S. Wu Sinotech Engineering Consultants,

More information

BUCKLING OF BARS, PLATES, AND SHELLS. Virginia Polytechnic Institute and State University Biacksburg, Virginia 24061-0219

BUCKLING OF BARS, PLATES, AND SHELLS. Virginia Polytechnic Institute and State University Biacksburg, Virginia 24061-0219 BUCKLING OF BARS, PLATES, AND SHELLS ROBERT M. JONES Science and Mechanics Professor Emeritus of Engineering Virginia Polytechnic Institute and State University Biacksburg, Virginia 24061-0219 Bull Ridge

More information

ANALYTICAL AND EXPERIMENTAL EVALUATION OF SPRING BACK EFFECTS IN A TYPICAL COLD ROLLED SHEET

ANALYTICAL AND EXPERIMENTAL EVALUATION OF SPRING BACK EFFECTS IN A TYPICAL COLD ROLLED SHEET International Journal of Mechanical Engineering and Technology (IJMET) Volume 7, Issue 1, Jan-Feb 2016, pp. 119-130, Article ID: IJMET_07_01_013 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=7&itype=1

More information

CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION

CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION CHAP FINITE EEMENT ANAYSIS OF BEAMS AND FRAMES INTRODUCTION We learned Direct Stiffness Method in Chapter imited to simple elements such as D bars we will learn Energ Method to build beam finite element

More information

Finite Element Analysis

Finite Element Analysis A Course Material on By Mr. L.VINOTH ME ASSISTANT PROFESSOR DEPARTMENT OF MECHANICAL ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM 638 56 QUALITY CERTIFICATE This is to certify that the e-course

More information

Deflections. Question: What are Structural Deflections?

Deflections. Question: What are Structural Deflections? Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the

More information

Chapter 4. Shape Functions

Chapter 4. Shape Functions Chapter 4 Shape Functions In the finite element method, continuous models are approximated using information at a finite number of discrete locations. Dividing the structure into discrete elements is called

More information

INTRODUCTION TO THE FINITE ELEMENT METHOD

INTRODUCTION TO THE FINITE ELEMENT METHOD INTRODUCTION TO THE FINITE ELEMENT METHOD G. P. Nikishkov 2004 Lecture Notes. University of Aizu, Aizu-Wakamatsu 965-8580, Japan niki@u-aizu.ac.jp 2 Updated 2004-01-19 Contents 1 Introduction 5 1.1 What

More information

INJECTION MOLDING COOLING TIME REDUCTION AND THERMAL STRESS ANALYSIS

INJECTION MOLDING COOLING TIME REDUCTION AND THERMAL STRESS ANALYSIS INJECTION MOLDING COOLING TIME REDUCTION AND THERMAL STRESS ANALYSIS Tom Kimerling University of Massachusetts, Amherst MIE 605 Finite Element Analysis Spring 2002 ABSTRACT A FEA transient thermal structural

More information

Volumetric locking in finite elements

Volumetric locking in finite elements Volumetric locking in finite elements on the relation between constraint ratio and locking behavior Bachelor Final Project Bart Vossen MT 08.31 July 31, 2008 Supervisors: H. R. Javani Joni, MSc dr. ir.

More information

Study on Pressure Distribution and Load Capacity of a Journal Bearing Using Finite Element Method and Analytical Method

Study on Pressure Distribution and Load Capacity of a Journal Bearing Using Finite Element Method and Analytical Method International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:1 No:5 1 Study on Pressure Distribution and Load Capacity of a Journal Bearing Using Finite Element Method and Method D. M.

More information

Distance Learning Program

Distance Learning Program Distance Learning Program Leading To Master of Engineering or Master of Science In Mechanical Engineering Typical Course Presentation Format Program Description Clarkson University currently offers a Distance

More information

Back to Elements - Tetrahedra vs. Hexahedra

Back to Elements - Tetrahedra vs. Hexahedra Back to Elements - Tetrahedra vs. Hexahedra Erke Wang, Thomas Nelson, Rainer Rauch CAD-FEM GmbH, Munich, Germany Abstract This paper presents some analytical results and some test results for different

More information

Finite Element Simulation of Simple Bending Problem and Code Development in C++

Finite Element Simulation of Simple Bending Problem and Code Development in C++ EUROPEAN ACADEMIC RESEARCH, VOL. I, ISSUE 6/ SEPEMBER 013 ISSN 86-48, www.euacademic.org IMPACT FACTOR: 0.485 (GIF) Finite Element Simulation of Simple Bending Problem and Code Development in C++ ABDUL

More information

FREE VIBRATION OF SPHERICAL SHELL USING FINITE ELEMENT APPROACH

FREE VIBRATION OF SPHERICAL SHELL USING FINITE ELEMENT APPROACH Communication Science & technology N 4. January 04 COST FREE VIBRATION OF SPHERICAL SHELL USING FINITE ELEMENT APPROACH M. BOUAZZA,, A. LAIREDJ 3, D. OUINAS 4, and A. HAMOUINE Department of Civil Engineering,

More information

Introduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1

Introduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1 Introduction to Solid Modeling Using SolidWorks 2012 SolidWorks Simulation Tutorial Page 1 In this tutorial, we will use the SolidWorks Simulation finite element analysis (FEA) program to analyze the response

More information

PREDICTION OF MACHINE TOOL SPINDLE S DYNAMICS BASED ON A THERMO-MECHANICAL MODEL

PREDICTION OF MACHINE TOOL SPINDLE S DYNAMICS BASED ON A THERMO-MECHANICAL MODEL PREDICTION OF MACHINE TOOL SPINDLE S DYNAMICS BASED ON A THERMO-MECHANICAL MODEL P. Kolar, T. Holkup Research Center for Manufacturing Technology, Faculty of Mechanical Engineering, CTU in Prague, Czech

More information

MAE4700/5700: Finite Element Analysis for Mechanical and Aerospace Design

MAE4700/5700: Finite Element Analysis for Mechanical and Aerospace Design MAE4700/5700: Finite Element Analysis for Mechanical and Aerospace Design Cornell University, Fall 2009 Lectures: PHL 403, TTh, 10:10-11:25 am Recitations (ANSYS instruction) Rhodes 471, Fr 1:25-2:15 pm

More information

New approaches in Eurocode 3 efficient global structural design

New approaches in Eurocode 3 efficient global structural design New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beam-column FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural

More information

Integration of a fin experiment into the undergraduate heat transfer laboratory

Integration of a fin experiment into the undergraduate heat transfer laboratory Integration of a fin experiment into the undergraduate heat transfer laboratory H. I. Abu-Mulaweh Mechanical Engineering Department, Purdue University at Fort Wayne, Fort Wayne, IN 46805, USA E-mail: mulaweh@engr.ipfw.edu

More information

Contents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern

Contents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern Preface page xv 1 Vector algebra 1 1.1 Preliminaries 1 1.2 Coordinate System invariance 4 1.3 Vector multiplication 9 1.4 Useful products of vectors 12 1.5 Linear vector Spaces 13 1.6 Focus: periodic media

More information

The finite-element method

The finite-element method The finite-element method Version April 19, 21 [1, 2] 1 One-dimensional problems Let us apply the finite element method to the one-dimensional fin equation d 2 T dx 2 T = (1) with the boundary conditions

More information

Numerical Analysis Introduction. Student Audience. Prerequisites. Technology.

Numerical Analysis Introduction. Student Audience. Prerequisites. Technology. Numerical Analysis Douglas Faires, Youngstown State University, (Chair, 2012-2013) Elizabeth Yanik, Emporia State University, (Chair, 2013-2015) Graeme Fairweather, Executive Editor, Mathematical Reviews,

More information

(laboratory) (fall semester)

(laboratory) (fall semester) Civil Engineering Courses-1 CIV 211/Surveying Prerequisite: MAT 127 An introduction to the theory and applications of modern surveying processes. Students use optical and digital land surveying instruments

More information

Chapter 2: Load, Stress and Strain

Chapter 2: Load, Stress and Strain Chapter 2: Load, Stress and Strain The careful text- books measure (Let all who build beware!) The load, the shock, the pressure Material can bear. So when the buckled girder Lets down the grinding span,

More information

Tutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0

Tutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0 Tutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0 1 Problem Description Design a gantry crane meeting the geometry presented in Figure 1 on page #325 of the course textbook

More information

NUMERICAL METHODS TOPICS FOR RESEARCH PAPERS

NUMERICAL METHODS TOPICS FOR RESEARCH PAPERS Faculty of Civil Engineering Belgrade Master Study COMPUTATIONAL ENGINEERING Fall semester 2004/2005 NUMERICAL METHODS TOPICS FOR RESEARCH PAPERS 1. NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS - Matrices

More information

Indeterminate Analysis Force Method 1

Indeterminate Analysis Force Method 1 Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to

More information

Elementary Differential Equations

Elementary Differential Equations Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College

More information

Chapter 4: Summary and Conclusions

Chapter 4: Summary and Conclusions Chapter 4: Summary and Conclusions 4.1 Summary Three different models are presented and analyzed in this research for the purpose of studying the potential of using post-buckled or pre-bent elastic struts

More information

Unit M4.3 Statics of Beams

Unit M4.3 Statics of Beams Unit M4.3 Statics of Beams Readings: CD 3.2-3.6 (CD 3.8 -- etension to 3-D) 16.003/004 -- Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute of Technology EARNING OBJECTIVES

More information

Structural Integrity Analysis

Structural Integrity Analysis Structural Integrity Analysis 1. STRESS CONCENTRATION Igor Kokcharov 1.1 STRESSES AND CONCENTRATORS 1.1.1 Stress An applied external force F causes inner forces in the carrying structure. Inner forces

More information

Introduction to the Finite Element Method (FEM)

Introduction to the Finite Element Method (FEM) Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the one-dimensional

More information

Interaction between plate and column buckling

Interaction between plate and column buckling Delft, University of Technology Engineering office of Public works Rotterdam Interaction between plate and column buckling Master Thesis Name: Alex van Ham Student number: 1306138 Email: vanham.alex@gmail.com

More information

ENGI 8673 Subsea Pipeline Engineering Faculty of Engineering and Applied Science

ENGI 8673 Subsea Pipeline Engineering Faculty of Engineering and Applied Science GUIDANCE NOTE LECTURE 12 THERMAL EXPANSION ANALYSIS OVERVIEW The calculation procedure for determining the longitudinal pipeline response can be formulated on the basis of strain. The longitudinal strain

More information

Opto-Mechanical I/F for ANSYS

Opto-Mechanical I/F for ANSYS Abstract Opto-Mechanical I/F for ANSYS Victor Genberg, Keith Doyle, Gregory Michels Sigmadyne, Inc., 803 West Ave, Rochester, NY 14611 genberg@sigmadyne.com Thermal and structural output from ANSYS is

More information

Simulation of Residual Stresses in an Induction Hardened Roll

Simulation of Residual Stresses in an Induction Hardened Roll 2.6.4 Simulation of Residual Stresses in an Induction Hardened Roll Ludwig Hellenthal, Clemens Groth Walzen Irle GmbH, Netphen-Deuz, Germany CADFEM GmbH, Burgdorf/Hannover, Germany Summary A heat treatment

More information

Development of Membrane, Plate and Flat Shell Elements in Java

Development of Membrane, Plate and Flat Shell Elements in Java Development of Membrane, Plate and Flat Shell Elements in Java by Kaushalkumar Kansara Thesis submitted to the Faculty of the Virginia Polytechnic Institute & State University In partial fulfillment of

More information

Strip Crown Prediction: Developing a Refined Dynamic Roll-Stack Model for the Hot Rolling Process

Strip Crown Prediction: Developing a Refined Dynamic Roll-Stack Model for the Hot Rolling Process Strip Crown Prediction: Developing a Refined Dynamic Roll-Stack Model for the Hot Rolling Process By Derek E. Slaughter Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University

More information

Dynamic Load and Stress Analysis of a Crankshaft

Dynamic Load and Stress Analysis of a Crankshaft 27-1-28 Dynamic Load and Stress Analysis of a Crankshaft Farzin H. Montazersadgh and Ali Fatemi The University of Toledo Copyright 27 SAE International ABSTRACT In this study a dynamic simulation was conducted

More information

4.3 Draft 5/27/04 2004 J.E. Akin 1

4.3 Draft 5/27/04 2004 J.E. Akin 1 4. Draft 5/7/04 004 J.E. Akin Chapter INTRODUCTION. Finite Element Methods The goal of this text is to introduce finite element methods from a rather broad perspective. We will consider the basic theory

More information