Finite Element Analysis in Mechanical Design Instructor's Notes and Tutorials

349 Pages·2010·7.67 MB·English

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Prepared by:
AAbbuull FFaazzaall MM.. AArriiff
Associate Professor, Mechanical Engineering
King Fahd University of Petroleum & Minerals.
Table of Contents

SECTION 1: INTRODUCTION TO FINITE ELEMENT ANALYSIS ................................... 7 
1.1 Finite Element Method .............................................................................................. 7 
1.2 Review Topics ............................................................................................................ 18 
1.3 General Steps of the Finite Element Method ................................................... 28 
1.4 Finite Element Types................................................................................................ 29 
1.5 Types of Element Formulation Methods ............................................................ 29 
1.6 Derivation of Spring Element Equations using Direct Method ................... 30 
1.7 Examples of Spring Systems .................................................................................. 34 
1.8 Bar Element Formulation using Direct Method ............................................... 38 
1.9 Bar Element Formulation using Potential Energy Approach ........................ 40 
1.10 Bar Element Formulation using Weighted Residual Approach ................. 44 
1.11 FEA: Modeling, Errors, and Accuracy. ............................................................ 49 
1.12 Responsibilty of Users. ......................................................................................... 50 
SECTION 2: STATIC ANALYSIS USING ANSYS ................................................................ 52 
2.1 Overview of Structural Analysis ........................................................................... 52 
2.2 Static Analysis Procedure ....................................................................................... 52 
SECTION 3: GEOMETRIC MODELING .............................................................................. 61 
3.1 Typical Steps Involved in Model Generation Within ANSYS ....................... 61 
3.2 Importing Solid Models Created in CAD systems ........................................... 61 
3.3 Solid Modeling ........................................................................................................... 63 
Tutorial 1: Solid Modeling using 2D Primitives ...................................................... 65 
Tutorial 2: Solid Modeling using 3D Primitives ...................................................... 73 
SECTION 4: STATIC ANALYSIS USING LINE ELEMENTS ............................................ 81 
Tutorial 3: Static Analysis using TRUSS Elements .................................................. 82 
Tutorial 4(a): Static Analysis using BEAM Elements ............................................. 95 
Tutorial 4(b): Static Analysis using BEAM Elements with Distributed Load . 99 
SECTION 5: STATIC ANALYSIS USING AREA ELEMENTS ......................................... 104 
Tutorial 5: Static Analysis using Area Elements- Plane Problem (Bracket) . 105 
Tutorial 6: Static Analysis using Area Elements- Plane Problem (Wrench) .. 122 
SECTION 6: STATIC ANALYSIS USING VOLUME ELEMENTS ................................... 127 
Tutorial 7: Static Analysis using Volume Elements-Component Design ......... 128 
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Tutorial 8: Static Analysis using Volume Elements-Assembly Design ............. 135 
SECTION 7: THERMAL-STRESS ANALYSIS ................................................................... 143 
Tutorial 9: Thermal Analysis of Mechanical Structure ......................................... 144 
Tutorial 10(a): Thermal-Stress Analysis - Sequential Coupled Field ............... 151 
Tutorial 10(b): Thermal-Stress Analysis – Direct Coupled Field ..................... 158 
SECTION 8: DYNAMIC ANALYSIS .................................................................................... 163 
Tutorial 11: Harmonic Analysis of Structure ......................................................... 164 
Tutorial 12: Modal Analysis of Structure ................................................................ 171 
SECTION 9: COMPOSITE MATERIALS ............................................................................ 177 
9.1 Composites - A Basic Introduction ................................................................... 177 
9.2. Modeling Composites using ANSYS ................................................................. 183 
Tutorial 13: Simply Supported Laminated Plate Under Pressure ..................... 193 
SECTION 10: PROBABILISTIC DESIGN ANALYSIS ...................................................... 212 
10.1 Probabilistic Design ............................................................................................. 212 
10.2 Probability Distributions .................................................................................... 217 
10.3 Choosing a Distribution for a Random Variable ......................................... 228 
10.4 Probabilistic Design Techniques ....................................................................... 232 
10.5 Postprocessing Probabilistic Analysis Results .............................................. 234 
Tutorial 14: Probabilistic Design Analysis of Circular Plate Bending. ........... 241 
11.0 APDL PROGRAMMING .............................................................................................. 266 
11.1 Introduction ......................................................................................................... 266 
11.2 Create The Analysis File ................................................................................... 266 
Tutorial 15: Stress Analysis of Bicycle Wrench. .................................................... 271 
Tutorial 16: Heat Loss from a Cylindrical Cooling Fin. ...................................... 278 
12.0 DESIGN OPTIMIZATION ........................................................................................... 287 
12.1 Introduction ......................................................................................................... 287 
12.2 Design Optimization Using Ansys ................................................................. 294 
Tutorial 17: Design Optimization Tutorial ............................................................. 321 
EExxeerrcciissee PPrroobblleemmss .............................................................................................................. 339 
RReeffeerreenncceess ............................................................................................................................... 345 
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3
PPRREEFFAACCEE

Considering importance and growing need of some topics, two new topics have been
included in the revised edition of Instrictor’s Note and Tutorials Manual for Finite
Element Analysis in Mechanical Design: Composite Materials and Probabilistic Design
Analysis. In addition, the complete manual has been reorganized and divided into ten
sections.

In Section 1, the basic concepts of finite element method has been introduced. It
contains topics related to review material, general steps of FEA, element types, element
formulation methods, formulation on 1-D line elements using various methods, sources
of error and, finally, the responsibilty of users.

More details on geometric modeling has been added in Section 3. It includes typical
steps involved in model generation within ANSYS, importing solid models created in
CAD systems and solid modeling using ANSYS preprocessor.

Lecture notes on composite materials and modeling of composite materials using finite
element method has been included in Section 9. It also includes a tutorial on the finite
element analysis of composite structure using ANSYS.

Section 10 consists of lecture notes on probabilistic design analysis using finite element
method and a tutorial.

The author gratefully acknowledges the support of the university management and
department chairman.

Abul-Fazal M. Arif
Associate Professor
Mechanical Engineering Department
August, 2005.

4
PPrreeffaaccee TToo FFiirrsstt EEddiittiioonn

The Finite Element Method (FEM) is a well-established technique for analyzing the structural behavior of
mechanical components and systems. In recent years, the use of finite element analysis as a design tool has grown
rapidly. Easy to use commercial software have become common tools in the hands of students as well as
practicing engineers. Unfortunately, many students who lack the proper training or understanding of the
underlying concepts have been using these tools.
Appreciating the importance of the topic and the need to train our graduates properly in finite element analysis
technique, the Department of Mechanical Engineering introduced a new course, titled “Finite Element Analysis in
Mechanical Design”, as a special topic during 1999-2000. Considering the interest shown by the students and the
success of the new course, it was changed to a new regular elective course as ME 489 during 2000-2001. Since
then this course was offered twice during the academic years 2001-2002 and 2002-2003 with an average of about
20 registered senior undergraduate and graduate students. It is also scheduled to be offered during semester 031
of academic year 2003-2004.

The objectives of this course includes:
 To teach students the basic concepts in the linear finite element method (FEM) as related to solving
engineering problems in solids and heat transfer.
 To provide students with a working knowledge of finite element analysis tools and their use in mechanical
design.

The topics covered in this course includes: Introduction to finite element; Finite Element Formulation;
Introduction to a general FE Software (ANSYS); Development of Beam, Frames and Grid Equations; 2-D
elasticity problems; Dynamic Analysis; and Heat Transfer Problems.

Although ME489 with a format of (3-0-3) does not have a separate lab, but it is embedded in the lecture as
computer session. During last offering of this course in 021 semester, about 10 computer sessions were conducted
so that the students get exposure to a general-purpose finite element analysis software, gain insight into
appropriate use of Finite Element Modeling, understand how to control modeling errors, benefit from hands-on
exercise at the computer workstation, and understand the safe use of the FEM in support of designing complex
load bearing components and structures. These sessions were very successful with the students and encouraged
them in using FEA to solve various design problems and their term project.

There are many good textbooks already in existence that cover the theory of finite element methods. Similarly,
there is detailed users manuals are available for commercial software. But, these are useful for advanced students
and users. Therefore, there was a need to develop a computer session manual in line with the flow of the course
and utilizing the software platform available in the department. This manual will help both the students and the
instructors. Students will be able to acquire the required level of understanding and skill in modeling, analysis,
validation and report generation for various design problems. Whereas the instructors will be able to save time,
currently spent in computer sessions, to cover more topics such as structural dynamics and design optimization.

This manual could also be very helpful for the students of senior design project (ME 411 and ME 412), ME485
(Mechanical System Design), and ME590 (FEA for Large Deformation Problems). In addition, it could be used
for computer sessions of short courses on stress analysis techniques and Finite Element Analysis offered by the
Mechanical Engineering department.

After giving a brief introduction to the finite element analysis and modeling, various guided-tutorials have been
included in this manual. Several new tutorials have been developed and others adapted from different sources
including ANSYS manuals, ANSYS workshops and INTERNET resources. Tutorials have been arranged
according to the flow of the course and covers all ME489 related topics, such as solid modeling using 2D and 3D
primitives available in ANSYS, static structural analysis (truss, beam, 2D and 3D structures), dynamic analysis
(harmonic and modal analysis), and thermal analysis.

The author gratefully acknowledges the support and encouragement provided by Dr. Faleh A. Al-Sulaiman, the
Chairman of the Mechanical Engineering Department, in writing this manual and for recommending the request
for one-month summer support. I am thankful to Mr. Zahid Qamar and Mr. Munir Qureshi for their effort in some
of the tutorials included here.

Dr. Abul Fazal M. Arif,
Assistant Professor
5
Mechanical Engineering Department
August, 2003.
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SECTION 1: INTRODUCTION TO FINITE ELEMENT
ANALYSIS

1.1 Finite Element Method

The field of Mechanics can be subdivided into three major areas: Theoretical, Applied, and
Computational. Theoretical Mechanics deals with fundamental laws and principles of
mechanics studied for their intrinsic scientific value. Applied Mechanics transfers this
theoretical knowledge to scientific and engineering applications, especially through the
construction of mathematical models of physical phenomena. Computational Mechanics
solves specific problems by simulation through numerical methods implemented on digital
computers.

One of the most important advances in applied mathematics in the 20th century has been the
development of the Finite Element Method as a general mathematical tool for obtaining
approximate solutions to boundary-value problems. The theory of finite elements draws on
almost every branch of mathematics and can be considered as one of the richest and most
diverse bodies of the current mathematical knowledge.

1.1.1 Mathematical Modeling of Physical Systems

In general, engineering problems are mathematical models of physical situations. Two main
goals of engineering analysis are to be able to identify the basic physical principle(s) and
fundamental laws that govern the behavior of a system or a control volume and to translate
those principles into a mathematical model involving an equation or equations that can be
solved accurately to predict qualitative and quantitative behavior of the system. The resulting
mathematical model is frequently a single differential equation or a set of differential equations
with a set of corresponding boundary and initial conditions whose solution should be
consistent with and accurately represent the physics of the system. These governing equations
represent balance of mass, force, or energy. When possible, the exact solution of these
equations renders detailed behavior of a system under a given set of conditions.

In situations where the system is relatively simple, it may be possible to analyze the problem
by using some of the classical methods learned in elementary courses in ordinary and partial
differential equations. Far more frequently, however, there are many practical engineering
problems for which we can not obtain exact solutions. This inability to obtain an exact solution
may be attributed to either the complex nature of governing differential equations or the
difficulties that arise from dealing with the boundary and initial conditions. To deal with such
problems, we resort to numerical approximations. In contrast to analytical solutions, which
show the exact behavior of a system at any point within the system, numerical solutions
approximate exact solutions only at discrete points, called nodes.

Due to the complexity of physical systems, some approximation must be made in the process
of turning physical reality into a mathematical model. It is important to decide at what points in
the modeling process these approximations are made. This, in turn, determines what type of
analytical or computational scheme is required in the solution process. Let us consider a
diagram of the two common branches of the general modeling solution process as shown in
Figure 1.
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Figure 1

For many real world problems the second approach is in fact the only possibility. For instance
suppose that the aim is to find the thermo-mechanical stresses in an air-cooled turbine blade
depicted in Figure 2.

Figure 2

The complex three-dimensional geometry of the blade along with the combined thermal and
mechanical loadings makes the analysis of the blade a formidable task. Nevertheless, many
powerful commercial finite element packages are available that can be implemented to perform
this task with relative ease.

Figure 3
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1.1.2 Basic Concept of Numerical Methods

The basic concept of these methods is based on the idea of building a complicated object with
simple blocks, or, dividing a complicated object into small and manageable pieces. Application
of this simple idea can be found everywhere in everyday life as well as in engineering.
Examples include Lego (kids’ play), buildings, and approximation of the area of a circle:

“ElementSi”


S  1 R2 sin
Area of one triangle:
i 2 i
Area of the circle:

N
S  S  1 R2N sin(2) R2 as N  
N i 2 N
i1

where N = total number of triangles ( elements). The first step of any numerical procedure is
discretization. This process divides the medium of interest into a number of small subregions
and nodes.

There are two common classes of numerical methods: (1) Finite Difference Methods and (2)
Finite Element Methods (FEM). With finite difference methods, the differential equation is
written for each node, and the derivatives are replaced by difference equations. This approach
results in a set of simultaneous linear equations. Although finite difference methods are easy to
understand and employ in simple problems, they become difficult to apply to problems with
complex geometries or complex boundary conditions. This situation is also true for problems
with nonisotropic properties. In constrast, the finite element method uses integral formulations
rather than difference equations to create a system of algebraic equations. Moreover, an
approximate continous function is assumed to represent the solution for each element. The
complete solution is then generated by connecting or assembling the individual solutions,
allowing for continuity at the interelemental boundaries.

Thus, Finite Element Method (FEM) is a numerical analysis technique for obtaining
approximate solutions to a wide variety of engineering problems.

1.1.3 A Brief History of FEM:

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the
Ritz method of numerical analysis and minimization of variational calculus to obtain
approximate solutions to vibration systems. Shortly thereafter, a paper published in 1956 by M.
J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp established a broader definition of
numerical analysis. The paper centered on the "stiffness and deflection of complex structures".
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By the early 70's, FEA was limited to expensive mainframe computers generally owned by the
aeronautics, automotive, defense, and nuclear industries. Since the rapid decline in the cost of
computers and the phenomenal increase in computing power, FEA has been developed to an
incredible precision. Present day super computers are now able to produce accurate results for
all kinds of parameters.
1943 ----- Courant (Variational methods)
1956 ----- Turner, Clough, Martin and Topp (Stiffness)
1960 ----- Clough (“Finite Element”, plane problems)
1970s ----- Applications on mainframe computers
1980s ----- Microcomputers, pre- and postprocessors
1990s ----- Analysis of large structural systems.

1.1.4 The FEM Analysis Process

A model-based simulation process using FEM consists of a sequence of steps. This sequence
takes two basic configurations depending on the environment in which FEM is used. These are
referred to as the Mathematical FEM and the Physical FEM.

The Mathematical FEM
As depicted in Figure 4, the centerpiece in the process steps of the Mathematical FEM is the
mathematical mode, which is often an ordinary or partial differential equation in space and
time. Using the methods provided by the Variational Calculus, a discrete finite element model
is generated from of the mathematical model. The resulting FEM equations are processed by
an equation solver, which provides a discrete solution. In this process we may also think of an
ideal physical system, which may be regarded as a realization of the mathematical model. For
example, if the mathematical model is the Poisson’s equation, realizations may be a heat
conduction problem. In Mathematical FEM this step is unnecessary and indeed FEM
discretizations may be constructed without any reference to physics.

The concept of error arises when the discrete solution is substituted in the mathematical and
discrete models. This replacement is generically called verification. The solution error is the
amount by which the discrete solution fails to satisfy the discrete equations. This error is
relatively unimportant when using computers. More relevant is the discretization error, which
is the amount by which the discrete solution fails to satisfy the mathematical model.

Figure 4

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