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Butnariu, S., Analysis of mechanical structures using finite element method,
lecture notes, Ed. Universitatii Transilvania din Brasov, ISBN 978-606-190311-5 (CD), 2013.
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Silviu BUTNARIU
ANALYSIS OF MECHANICAL
STRUCTURES USING
FINITE ELEMENT METHOD
Lecture notes
CD inside
Editura Universităţii Transilvania din Braşov
2013
Silviu BUTNARIU
ANALYSIS OF MECHANICAL
STRUCTURES USING
FINITE ELEMENT METHOD
Lecture notes
Editura Universităţii Transilvania din Braşov
2013
1. INTRODUCTION
1.1. Preface
The finite element method is the most widely used numerical method for calculating
mechanical structures, however they may be complete in terms of geometry, applications
or materials.
The generality of the method, the simplicity of the basic concepts and the use of electronic
computers, explains the expansion and interest in this method.
Currently, this method, is aided with multiple applications extremely varied. As a result
of formulating concepts of maximum generalizations of this method, developed
algorithms and computer programs applied in a field of engineering were subsequently
transferred and adapted successfully without significant changes in other areas of applied
research such as fluid mechanics phenomena heat and mass transfer, electromagnetism,
mechanics, biomechanics, soil and rock mechanics, acoustics, technology and materials
processing, etc. .
Like any numerical method of calculation, this is an approximate method thus provides
approximate solutions that ensures sufficient accuracy for almost all engineering
calculations .
The interest for this method is reflected by the increase in the number of scientific
publications and conferences dedicated to this method. Important is the increasing the
number of finite element computer programs. The most recent software applications
dedicated for mechanical design have incorporated special software modules, which
allows numerical analysis with the finite element method or by other methods [36].
1.2. About PLM, CAD / CAE / CAM
In industry, Product Lifecycle Management (PLM) is the process of managing the entire
lifecycle of a product from its conception, through design and manufacture, to service and
disposal. PLM integrates people, data, processes and business systems and provides a
Analysis of Mechanical Structures using Finite Element Method
product information backbone for companies and their extended enterprise. Product
lifecycle management can be considered one of the four cornerstones of a manufacturing
corporation's information technology structure (Fig. 1). Within PLM there are five
primary areas:
x
x
x
x
x
Systems engineering (SE);
Product and portfolio (PPM);
Product design (CAx);
Manufacturing process management (MPM);
Product Data Management (PDM).
All companies need to manage communications and information with their customers
(CRM - customer relationship management), their suppliers and fulfillment (SCM-supply
chain), their resources within the enterprise (ERP-enterprise resource planning) and their
product planning and development [23].
Fig. 1 The Product Lifecycle Management (PLM) cycle
The following picture (Fig. 2) shows where CAD/CAM/CAE systems are used within the
overall product lifecycle, using the case of an automobile as an example [3].
Computer-aided design (CAD), also known as computer-aided design and drafting
(CADD) is the use of computer technology for the process of design and designdocumentation. Computer Aided Drafting describes the process of drafting with a
computer. Computer-aided design (CAD) is the use of computer systems to assist in the
6
Analysis of Mechanical Structures using Finite Element Method
Notice that the exact answer to this problem (can you prove this?) is
 x for 0  x  1
u exact  
2  x for 1  x  2
The displacement solution is:
How can you improve the approximation?
The stress within the bar:
184
Contents
1.
INTRODUCTION ...................................................................................................5
1.1.
PREFACE .......................................................................................................... 5
1.2.
ABOUT PLM, CAD / CAE / CAM .................................................................... 5
1.3.
ABOUT FEM AND FEA .................................................................................. 10
1.3.1.
A Brief History of the FEM and FEA.................................................... 10
1.3.2.
FEM in Engineering ............................................................................. 11
1.3.3.
Available Commercial FEM Software Packages ................................. 13
1.4.
2.
COURSE DESCRIPTION .................................................................................... 18
1.4.1.
Course Overview .................................................................................. 18
1.4.2.
Course Objectives ................................................................................. 19
1.4.3.
Recommended Background................................................................... 20
GENERAL PROBLEM DEFINITION ...............................................................21
2.1.
GENERAL CONSIDERATIONS ON THE METHOD ................................................ 21
2.2.
FEM TERMINOLOGY ...................................................................................... 26
2.3.
IDEALIZATION ............................................................................................... 28
2.3.1.
Models .................................................................................................. 28
2.3.2.
Mathematical Models ........................................................................... 28
2.3.3.
Implicit vs. Explicit Modeling ............................................................... 29
2.4.
THE THEORETICAL QUASI-GENERAL MODEL FOR FINITE ELEMENT ANALYSIS OF
A MECHANICAL SUBASSEMBLY ELEMENT .................................................................... 31
2.5. TYPES OF SOLVABLE PROBLEMS USING FINITE ELEMENTS ANALYSIS .................... 34
3.
2.6.
THE MODEL FOR ANALYSIS ............................................................................ 36
2.7.
SAMPLES OF ANALYSIS MODELS ..................................................................... 42
2.8.
THE GENERAL PROCEDURE OF FEA ................................................................ 43
PRE-PROCESSING PHASE ...............................................................................45
3.1.
GEOMETRY MODELING ................................................................................... 47
3.1.1.
General considerations about geometry modeling .............................. 47
3.1.2.
Basic concepts about 3D modeling ....................................................... 51
3.1.3.
Samples of the various 3D modeling software ...................................... 56
Analysis of Mechanical Structures using Finite Element Method
3.2.
3.2.1.
Modeling the mechanical behavior of materials................................... 70
3.2.2.
Modeling thermal behavior of materials .............................................. 73
3.3.
Discretization ....................................................................................... 74
3.3.2.
The Finite Elements .............................................................................. 75
3.3.3.
The meshing procedure......................................................................... 76
3.3.4.
Types of Finite Elements ...................................................................... 78
3.3.5.
Classification of Mechanical Elements ................................................ 83
3.3.6.
Assembly .............................................................................................. 86
3.3.7.
Modeling methods with elements ......................................................... 86
3.3.8.
Choosing finite elements and meshing parameters............................... 88
3.3.9.
Meshing example (1D, 2D, 3D) ............................................................ 91
BOUNDARY CONDITIONS. INTRODUCING SUPPORTS...................................... 100
3.4.1.
Boundary Conditions ......................................................................... 100
3.4.2.
Introducing suports ............................................................................. 101
3.4.3.
Restraints in CATIA ............................................................................ 103
3.4.4.
Restraints in ANSYS ............................................................................ 105
3.5.
2
MESHING (DISCRETIZATION) – TYPES OF FINITE ELEMENTS 3D, 2D, 1D ......... 74
3.3.1.
3.4.
4.
ASSIGN MATERIAL PROPERTIES ...................................................................... 70
LOAD MODELING .......................................................................................... 110
3.5.1.
Loads in CATIA .................................................................................. 111
3.5.2.
Loads in ANSYS ................................................................................. 113
SOLVING PHASE ..............................................................................................117
4.1.
GENERAL CONSIDERATIONS ABOUT SOLVING PHASE .................................... 118
4.2.
MODERN DESIGN PROBLEM .......................................................................... 120
4.3.
SOLVING METHODS ...................................................................................... 121
4.3.1.
Numerical Methods ............................................................................ 121
4.3.2.
Fundamental concepts ....................................................................... 122
4.3.3.
FEM formulation for a linear differential equation............................ 125
Analysis of Mechanical Structures using Finite Element Method
4.3.4.
4.4.
Linear Static Analysis ......................................................................... 131
4.4.2.
Vibration Analysis ............................................................................. 136
4.4.3.
Buckling Analysis ............................................................................... 140
4.4.4.
Thermal Analysis ............................................................................... 144
EXAMPLES FOR SOLVING COMMANDS .......................................................... 151
4.5.1.
Solving the Model in CATIA ............................................................... 151
4.5.2.
Solving the Model in ANSYS ............................................................... 152
POST-PROCESSING PHASE ...........................................................................153
5.1.
ABOUT PRE- AND POST-PROCESSOR .............................................................. 153
5.2.
GENERAL CONSIDERATIONS ABOUT THE POST-PROCESSING PHASE .............. 155
5.3.
RESULTS....................................................................................................... 157
5.3.1.
Displacements .................................................................................... 157
5.3.2.
Stresses .............................................................................................. 158
5.3.3.
Strains ................................................................................................ 160
5.4.
6.
TYPES OF ANALYSIS ..................................................................................... 130
4.4.1.
4.5.
5.
From Strong Form to Weak form (1D) .............................................. 126
SAMPLES OF POST-PROCESSING .................................................................... 161
5.4.1.
Post-processing in CATIA ................................................................. 161
5.4.2.
Post-processing in ANSYS ................................................................. 164
REFERENCES ....................................................................................................169
ANNEXES ....................................................................................................................173
A.1. CONVERGENCE OF DISPLACEMENT-BASED FEM ............................................... 173
A.2. VALUES CONVERGENCES (CATIA)................................................................... 175
A.3. PRINCIPLES OF MINIMUM POTENTIAL ENERGY AND RAYLEIGH-RITZ ................. 177
A.3.1. Potential energy of a system ..................................................................... 177
A.3.2. Rayleigh-Ritz Principle .............................................................................. 182
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