Cengage Learning Finite Element Method Technical Guide

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A First Course in the Finite Element Method,
Fifth Edition
Daryl L. Logan
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CHAPTER
1
INTRODUCTION d
CHAPTER OBJECTIVES
.
.
.
.
.
.
.
.
To present an introduction to the finite element method.
To provide a brief history of the finite element method.
To introduce matrix notation.
To describe the role of the computer in the development of the finite element
method.
To present the general steps used in the finite element method.
To illustrate the various types of elements used in the finite element method.
To show typical applications of the finite element method.
To summarize some of the advantages of the finite element method.
Prologue
The finite element method is a numerical method for solving problems of engineering
and mathematical physics. Typical problem areas of interest in engineering and mathematical physics that are solvable by use of the finite element method include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
For problems involving complicated geometries, loadings, and material properties, it is generally not possible to obtain analytical mathematical solutions. Analytical
solutions are those given by a mathematical expression that yields the values of the
desired unknown quantities at any location in a body (here total structure or physical
system of interest) and are thus valid for an infinite number of locations in the body.
These analytical solutions generally require the solution of ordinary or partial differential equations, which, because of the complicated geometries, loadings, and material
properties, are not usually obtainable. Hence we need to rely on numerical methods,
such as the finite element method, for acceptable solutions. The finite element formulation of the problem results in a system of simultaneous algebraic equations for solution, rather than requiring the solution of differential equations. These numerical
methods yield approximate values of the unknowns at discrete numbers of points in
the continuum. Hence this process of modeling a body by dividing it into an equivalent system of smaller bodies or units (finite elements) interconnected at points common to two or more elements (nodal points or nodes) and/or boundary lines and/or
surfaces is called discretization. In the finite element method, instead of solving the
1
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2
d
1 Introduction
problem for the entire body in one operation, we formulate the equations for each
finite element and combine them to obtain the solution of the whole body.
Briefly, the solution for structural problems typically refers to determining the
displacements at each node and the stresses within each element making up the structure that is subjected to applied loads. In nonstructural problems, the nodal unknowns
may, for instance, be temperatures or fluid pressures due to thermal or fluid fluxes.
This chapter first presents a brief history of the development of the finite element
method. You will see from this historical account that the method has become a practical one for solving engineering problems only in the past 55 years (paralleling the
developments associated with the modern high-speed electronic digital computer).
This historical account is followed by an introduction to matrix notation; then we
describe the need for matrix methods (as made practical by the development of the
modern digital computer) in formulating the equations for solution. This section discusses both the role of the digital computer in solving the large systems of simultaneous algebraic equations associated with complex problems and the development of
numerous computer programs based on the finite element method. Next, a general
description of the steps involved in obtaining a solution to a problem is provided.
This description includes discussion of the types of elements available for a finite
element method solution. Various representative applications are then presented to
illustrate the capacity of the method to solve problems, such as those involving complicated geometries, several different materials, and irregular loadings. Chapter 1
also lists some of the advantages of the finite element method in solving problems of
engineering and mathematical physics. Finally, we present numerous features of computer programs based on the finite element method.
d
1.1 Brief History
d
This section presents a brief history of the finite element method as applied to both
structural and nonstructural areas of engineering and to mathematical physics. References cited here are intended to augment this short introduction to the historical
background.
The modern development of the finite element method began in the 1940s in the
field of structural engineering with the work by Hrennikoff [1] in 1941 and McHenry
[2] in 1943, who used a lattice of line (one-dimensional) elements (bars and beams)
for the solution of stresses in continuous solids. In a paper published in 1943 but not
widely recognized for many years, Courant [3] proposed setting up the solution of
stresses in a variational form. Then he introduced piecewise interpolation (or shape)
functions over triangular subregions making up the whole region as a method to
obtain approximate numerical solutions. In 1947 Levy [4] developed the flexibility or
force method, and in 1953 his work [5] suggested that another method (the stiffness
or displacement method) could be a promising alternative for use in analyzing statically redundant aircraft structures. However, his equations were cumbersome to
solve by hand, and thus the method became popular only with the advent of the
high-speed digital computer.
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1.1 Brief History
d
3
In 1954 Argyris and Kelsey [6, 7] developed matrix structural analysis methods
using energy principles. This development illustrated the important role that energy
principles would play in the finite element method.
The first treatment of two-dimensional elements was by Turner et al. [8] in 1956.
They derived stiffness matrices for truss elements, beam elements, and two-dimensional
triangular and rectangular elements in plane stress and outlined the procedure
commonly known as the direct stiffness method for obtaining the total structure stiffness matrix. Along with the development of the high-speed digital computer in the
early 1950s, the work of Turner et al. [8] prompted further development of finite element stiffness equations expressed in matrix notation. The phrase finite element was
introduced by Clough [9] in 1960 when both triangular and rectangular elements
were used for plane stress analysis.
A flat, rectangular-plate bending-element stiffness matrix was developed by
Melosh [10] in 1961. This was followed by development of the curved-shell bendingelement stiffness matrix for axisymmetric shells and pressure vessels by Grafton and
Strome [11] in 1963.
Extension of the finite element method to three-dimensional problems with the
development of a tetrahedral stiffness matrix was done by Martin [12] in 1961, by
Gallagher et al. [13] in 1962, and by Melosh [14] in 1963. Additional three-dimensional
elements were studied by Argyris [15] in 1964. The special case of axisymmetric solids
was considered by Clough and Rashid [16] and Wilson [17] in 1965.
Most of the finite element work up to the early 1960s dealt with small strains
and small displacements, elastic material behavior, and static loadings. However,
large deflection and thermal analysis were considered by Turner et al. [18] in 1960
and material nonlinearities by Gallagher et al. [13] in 1962, whereas buckling problems were initially treated by Gallagher and Padlog [19] in 1963. Zienkiewicz et al.
[20] extended the method to visco-elasticity problems in 1968.
In 1965 Archer [21] considered dynamic analysis in the development of the
consistent-mass matrix, which is applicable to analysis of distributed-mass systems
such as bars and beams in structural analysis.
With Melosh’s [14] realization in 1963 that the finite element method could be
set up in terms of a variational formulation, it began to be used to solve nonstructural
applications. Field problems, such as determination of the torsion of a shaft,
fluid flow, and heat conduction, were solved by Zienkiewicz and Cheung [22] in
1965, Martin [23] in 1968, and Wilson and Nickel [24] in 1966.
Further extension of the method was made possible by the adaptation of weighted
residual methods, first by Szabo and Lee [25] in 1969 to derive the previously known
elasticity equations used in structural analysis and then by Zienkiewicz and Parekh [26]
in 1970 for transient field problems. It was then recognized that when direct formulations and variational formulations are difficult or not possible to use, the method of
weighted residuals may at times be appropriate. For example, in 1977 Lyness et al. [27]
applied the method of weighted residuals to the determination of magnetic field.
In 1976, Belytschko [28, 29] considered problems associated with large-displacement nonlinear dynamic behavior, and improved numerical techniques for solving
the resulting systems of equations. For more on these topics, consult the texts by
Belytschko, Liu, Moran [58], and Crisfield [61, 62].
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4
d
1 Introduction
A relatively new field of application of the finite element method is that of bioengineering [30, 31]. This field is still troubled by such difficulties as nonlinear materials,
geometric nonlinearities, and other complexities still being discovered.
From the early 1950s to the present, enormous advances have been made in the
application of the finite element method to solve complicated engineering problems.
Engineers, applied mathematicians, and other scientists will undoubtedly continue to
develop new applications. For an extensive bibliography on the finite element method,
consult the work of Kardestuncer [32], Clough [33], or Noor [57].
d
d
1.2 Introduction to Matrix Notation
Matrix methods are a necessary tool used in the finite element method for purposes of
simplifying the formulation of the element stiffness equations, for purposes of longhand solutions of various problems, and, most important, for use in programming
the methods for high-speed electronic digital computers. Hence matrix notation represents a simple and easy-to-use notation for writing and solving sets of simultaneous
algebraic equations.
Appendix A discusses the significant matrix concepts used throughout the text.
We will present here only a brief summary of the notation used in this text.
A matrix is a rectangular array of quantities arranged in rows and columns that is
often used as an aid in expressing and solving a system of algebraic equations. As examples
of matrices that will be described in subsequent chapters, the force components ðF1x ;
F1y ; F1z ; F2x ; F2y ; F2z ; . . . ; Fnx ; Fny ; Fnz Þ acting at the various nodes or points ð1; 2; . . . ; nÞ
on a structure and the corresponding set of nodal displacements ðu1 ; v1 ; w1 ;
u2 ; v2 ; w2 ; . . . ; un ; vn ; wn Þ can both be expressed as matrices:
9
8
F1x >
>
>
>
>
>
>
>
>
F1y >
>
>
>
>
>
>
>
>
>
>
F
1z
>
>
>
>
>
>
>
>
>
F
2x >
>
>
>
>
>
=
< F2y >
fF g ¼
F2z >
>
>
>
>
.. >
>
>
>
>
>
. >
>
>
>
>
>
>
>
>
>
>
F
nx
>
>
>
>
>
>
>F >
>
ny >
>
>
>
>
;
:
Fnz
8 9
u1 >
>
>
>
>
>
>
>
>
v1 >
>
>
>
>
>
>
>
>
>
>
w
1
>
>
>
>
> >
>
>
>
u
2 >
>
>
>
>
>
=
< v2 >
fdg ¼
w2 >
>
>
>
>
.. >
>
>
>
>
>
. >
>
>
>
>
>
>
>
>
>
>
u
n
>
>
>
>
>
>
>v >
>
n >
>
>
>
;
: >
wn
ð1:2:1Þ
The subscripts to the right of F identify the node and the direction of force, respectively. For instance, F1x denotes the force at node 1 applied in the x direction. The x,
y, and z displacements at a node are denoted by u, v, and w, respectively. The subscript
next to u, v, and w denotes the node. For instance, u1 , v1 , and w1 denote the displacement components in the x, y, and z directions, respectively, at node 1. The matrices in
Eqs. (1.2.1) are called column matrices and have a size of n1. The brace notation {}
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1.2 Introduction to Matrix Notation
d
5
will be used throughout the text to denote a column matrix. The whole set of force or
displacement values in the column matrix is simply represented by {F } or {d}.
The more general case of a known rectangular matrix will be indicated by use of
the bracket notation ½ . For instance, the element and global structure stiffness matrices ½k and ½K, respectively, developed throughout the text for various element types
(such as those in Figure 1–1 on page 10), are represented by square matrices given as
2
3
k11 k12 . . . k1n
6
7
6 k21 k22 . . . k2n 7
ð1:2:2Þ
½k ¼ 6
.. 7
..
6 ..
7
4 .
. 5
.
kn1
kn2
K11
6
6 K21
½K ¼ 6
6 ..
4 .
K12
K22
..
.
Kn1
Kn2
2
and
...
knn
3
. . . K1n
7
. . . K2n 7
.. 7
7
. 5
. . . Knn
ð1:2:3Þ
where, in structural theory, the elements kij and Kij are often referred to as stiffness
influence coefficients.
You will learn that the global nodal forces {F } and the global nodal displacements {d} are related through use of the global stiffness matrix [K ] by
fF g ¼ ½Kfdg
ð1:2:4Þ
Equation (1.2.4) is called the global stiffness equation and represents a set of simultaneous equations. It is the basic equation formulated in the stiffness or displacement
method of analysis.
To obtain a clearer understanding of elements Kij in Eq. (1.2.3), we use Eq.
(1.2.1) and write out the expanded form of Eq. (1.2.4) as
9 2
8
>
F1x >
>
>
> 6 K11
>
>
= 6K
< F1y >
21
¼6
.
.
6
.
> 4 .
> .. >
>
>
>
>
>
;
:
Kn1
Fnz
K12
K22
Kn2
38 9
. . . K1n >
>
> u1 >
>
> >
7>
. . . K2n 7< v1 =
7 .
7> . >
5>
. >
>
>
>
;
: >
. . . Knn
wn
ð1:2:5Þ
Now assume a structure to be forced into a displaced configuration defined by
u1 ¼ 1; v1 ¼ w1 ¼ wn ¼ 0. Then from Eq. (1.2.5), we have
F1x ¼ K11
F1y ¼ K21 ; . . . ; Fnz ¼ Kn1
ð1:2:6Þ
Equations (1.2.6) contain all elements in the first column of [K]. In addition, they
show that these elements, K11 ; K21 ; . . . ; Kn1 , are the values of the full set of nodal
forces required to maintain the imposed displacement state. In a similar manner, the
second column in [K] represents the values of forces required to maintain the
displaced state v1 ¼ 1 and all other nodal displacement components equal to zero.
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6
d
1 Introduction
We should now have a better understanding of the meaning of stiffness influence
coefficients.
Subsequent chapters will discuss the element stiffness matrices [k] for various
element types, such as bars, beams, plane stress and three-dimensional stress. They
will also cover the procedure for obtaining the global stiffness matrices [K] for various
structures and for solving Eq. (1.2.4) for the unknown displacements in matrix {d}.
Using matrix concepts and operations will become routine with practice; they
will be valuable tools for solving small problems longhand. And matrix methods are
crucial to the use of the digital computers necessary for solving complicated problems
with their associated large number of simultaneous equations.
d
1.3 Role of the Computer
d
As we have said, until the early 1950s, matrix methods and the associated finite element method were not readily adaptable for solving complicated problems. Even
though the finite element method was being used to describe complicated structures,
the resulting large number of algebraic equations associated with the finite element
method of structural analysis made the method extremely difficult and impractical to
use. However, with the advent of the computer, the solution of thousands of equations
in a matter of minutes became possible.
The first modern-day commercial computer appears to have been the Univac,
IBM 701 which was developed in the 1950s. This computer was built based on
vacuum-tube technology. Along with the UNIVAC came the punch-card technology
whereby programs and data were created on punch cards. In the 1960s, transistorbased technology replaced the vacuum-tube technology due to the transistor’s reduced
cost, weight, and power consumption and its higher reliability. From 1969 to the late
1970s, integrated circuit-based technology was being developed, which greatly
enhanced the processing speed of computers, thus making it possible to solve
larger finite element problems with increased degrees of freedom. From the late
1970s into the 1980s, large-scale integration as well as workstations that introduced a
windows-type graphical interface appeared along with the computer mouse. The first
computer mouse received a patent on November 17, 1970. Personal computers had
now become mass-market desktop computers. These developments came during the
age of networked computing, which brought the Internet and the World Wide Web.
In the 1990s the Windows operating system was released, making IBM and IBMcompatible PCs more user friendly by integrating a graphical user interface into the
software.
The development of the computer resulted in the writing of computational programs. Numerous special-purpose and general-purpose programs have been written
to handle various complicated structural (and nonstructural) problems. Programs
such as [46–56] illustrate the elegance of the finite element method and reinforce
understanding of it.
In fact, finite element computer programs now can be solved on single-processor
machines, such as a single desktop or laptop personal computer (PC) or on a cluster of
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1.4 General Steps of the Finite Element Method
d
7
computer nodes. The powerful memories of the PC and the advances in solver programs have made it possible to solve problems with over a million unknowns.
To use the computer, the analyst, having defined the finite element model, inputs
the information into the computer. This information may include the position of the
element nodal coordinates, the manner in which elements are connected, the material
properties of the elements, the applied loads, boundary conditions, or constraints,
and the kind of analysis to be performed. The computer then uses this information
to generate and solve the equations necessary to carry out the analysis.
d
1.4 General Steps of the Finite Element Method
d
This section presents the general steps included in a finite element method formulation
and solution to an engineering problem. We will use these steps as our guide in developing solutions for structural and nonstructural problems in subsequent chapters.
For simplicity’s sake, for the presentation of the steps to follow, we will consider
only the structural problem. The nonstructural heat-transfer, fluid mechanics and electrostatics problems and their analogies to the structural problem are considered in
Chapters 13 and 14.
Typically, for the structural stress-analysis problem, the engineer seeks to determine displacements and stresses throughout the structure, which is in equilibrium
and is subjected to applied loads. For many structures, it is difficult to determine the
distribution of deformation using conventional methods, and thus the finite element
method is necessarily used.
There are two general direct approaches traditionally associated with the finite
element method as applied to structural mechanics problems. One approach, called
the force, or flexibility, method, uses internal forces as the unknowns of the problem.
To obtain the governing equations, first the equilibrium equations are used. Then necessary additional equations are found by introducing compatibility equations. The
result is a set of algebraic equations for determining the redundant or unknown
forces.
The second approach, called the displacement, or stiffness, method, assumes the
displacements of the nodes as the unknowns of the problem. For instance, compatibility conditions requiring that elements connected at a common node, along a common
edge, or on a common surface before loading remain connected at that node, edge, or
surface after deformation takes place are initially satisfied. Then the governing equations are expressed in terms of nodal displacements using the equations of equilibrium
and an applicable law relating forces to displacements.
These two direct approaches result in different unknowns (forces or displacements) in the analysis and different matrices associated with their formulations (flexibilities or stiffnesses). It has been shown [34] that, for computational purposes, the displacement (or stiffness) method is more desirable because its formulation is simpler for
most structural analysis problems. Furthermore, a vast majority of general-purpose
finite element programs have incorporated the displacement formulation for solving
structural problems. Consequently, only the displacement method will be used
throughout this text.
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8
d
1 Introduction
Another general method that can be used to develop the governing equations for
both structural and nonstructural problems is the variational method. The variational
method includes a number of principles. One of these principles, used extensively
throughout this text because it is relatively easy to comprehend and is often introduced
in basic mechanics courses, is the theorem of minimum potential energy that applies to
materials behaving in a linear-elastic manner. This theorem is explained and used in
various sections of the text, such as Section 2.6 for the spring element, Section 3.10
for the bar element, Section 4.7 for the beam element, Section 6.2 for the constantstrain triangle plane stress and plane strain element, Section 9.1 for the axisymmetric
element, Section 11.2 for the three-dimensional solid tetrahedral element, and Section
12.2 for the plate bending element. A functional analogous to that used in the theorem
of minimum potential energy is then employed to develop the finite element equations
for the nonstructural problem of heat transfer presented in Chapter 13.
Another variational principle often used to derive the governing equations is the
principle of virtual work. This principle applies more generally to materials that behave
in a linear-elastic fashion, as well as those that behave in a nonlinear fashion. The principle of virtual work is described in Appendix E for those choosing to use it for developing the general governing finite element equations that can be applied specifically to
bars, beams, and two- and three-dimensional solids in either static or dynamic systems.
The finite element method involves modeling the structure using small interconnected elements called finite elements. A displacement function is associated with each
finite element. Every interconnected element is linked, directly or indirectly, to every
other element through common (or shared) interfaces, including nodes and/or boundary lines and/or surfaces. By using known stress/strain properties for the material
making up the structure, one can determine the behavior of a given node in terms of
the properties of every other element in the structure. The total set of equations
describing the behavior of each node results in a series of algebraic equations best
expressed in matrix notation.
We now present the steps, along with explanations necessary at this time, used in
the finite element method formulation and solution of a structural problem. The purpose of setting forth these general steps now is to expose you to the procedure generally followed in a finite element formulation of a problem. You will easily understand
these steps when we illustrate them specifically for springs, bars, trusses, beams, plane
frames, plane stress, axisymmetric stress, three-dimensional stress, plate bending, heat
transfer, fluid flow and electrostatics in subsequent chapters. We suggest that you
review this section periodically as we develop the specific element equations.
Keep in mind that the analyst must make decisions regarding dividing the structure or continuum into finite elements and selecting the element type or types to be
used in the analysis (step 1), the kinds of loads to be applied, and the types of boundary conditions or supports to be applied. The other steps, 2 through 7, are carried out
automatically by a computer program.
Step 1 Discretize and Select the Element Types
Step 1 involves dividing the body into an equivalent system of finite elements with
associated nodes and choosing the most appropriate element type to model most
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1.4 General Steps of the Finite Element Method
d
9
closely the actual physical behavior. The total number of elements used and their variation in size and type within a given body are primarily matters of engineering judgment. The elements must be made small enough to give usable results and yet large
enough to reduce computational effort. Small elements (and possibly higher-order elements) are generally desirable where the results are changing rapidly, such as where
changes in geometry occur; large elements can be used where results are relatively constant. We will have more to say about discretization guidelines in later chapters, particularly in Chapter 7, where the concept becomes quite significant. The discretized
body or mesh is often created with mesh-generation programs or preprocessor programs available to the user.
The choice of elements used in a finite element analysis depends on the physical
makeup of the body under actual loading conditions and on how close to the actual
behavior the analyst wants the results to be. Judgment concerning the appropriateness
of one-, two-, or three-dimensional idealizations is necessary. Moreover, the choice
of the most appropriate element for a particular problem is one of the major tasks
that must be carried out by the designer/analyst. Elements that are commonly
employed in practice—most of which are considered in this text—are shown in
Figure 1–1.
The primary line elements [Figure 1–1(a)] consist of bar (or truss) and beam elements. They have a cross-sectional area but are usually represented by line segments.
In general, the cross-sectional area within the element can vary, but throughout this
text it will be considered to be constant. These elements are often used to model trusses
and frame structures (see Figure 1–2 on page 16, for instance). The simplest line element (called a linear element) has two nodes, one at each end, although higher-order
elements having three nodes [Figure 1–1(a)] or more (called quadratic, cubic, etc., elements) also exist. Chapter 10 includes discussion of higher-order line elements. The line
elements are the simplest of elements to consider and will be discussed in Chapters 2
through 5 to illustrate many of the basic concepts of the finite element method.
The basic two-dimensional (or plane) elements [Figure 1–1(b)] are loaded by
forces in their own plane (plane stress or plane strain conditions). They are triangular
or quadrilateral elements. The simplest two-dimensional elements have corner nodes
only (linear elements) with straight sides or boundaries (Chapter 6), although there
are also higher-order elements, typically with midside nodes [Figure 1–1(b)] (called
quadratic elements) and curved sides (Chapters 8 and 10). The elements can have
variable thicknesses throughout or be constant. They are often used to model a wide
range of engineering problems (see Figures 1–3 and 1–4 on pages 17 and 18).
The most common three-dimensional elements [Figure 1–1(c)] are tetrahedral
and hexahedral (or brick) elements; they are used when it becomes necessary to perform a three-dimensional stress analysis. The basic three-dimensional elements
(Chapter 11) have corner nodes only and straight sides, whereas higher-order elements
with midedge nodes (and possible midface nodes) have curved surfaces for their sides
[Figure 1–1(c)].
The axisymmetric element [Figure 1–1(d)] is developed by rotating a triangle or
quadrilateral about a fixed axis located in the plane of the element through 360 . This
element (described in Chapter 9) can be used when the geometry and loading of the
problem are axisymmetric.
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10
d
1 Introduction
y
y
1
2
x
2
1
3
x
(a) Simple two-noded line element (typically used to represent a bar or beam element) and the
higher-order line element
3
y
3
4
1
2
1
2
x
Triangulars
Quadrilaterals
(b) Simple two-dimensional elements with corner nodes (typically used to represent plane stress/
strain) and higher-order two-dimensional elements with intermediate nodes along the sides
y
1
7
8
x
z
2
4
4
5
3
1
6
3
2
Regular hexahedral
Tetrahedrals
Irregular hexahedral
(c) Simple three-dimensional elements (typically used to represent three-dimensional stress state)
and higher-order three-dimensional elements with intermediate nodes along edges
z
4
3
1
1
2
3
2
Quadrilateral ring
q
Triangular ring
r
(d) Simple axisymmetric triangular and quadrilateral elements used for axisymmetric problems
Figure 1–1 Various types of simple lowest-order finite elements with corner
nodes only and higher-order elements with intermediate nodes
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1.4 General Steps of the Finite Element Method
d
11
Step 2 Select a Displacement Function
Step 2 involves choosing a displacement function within each element. The function is
defined within the element using the nodal values of the element. Linear, quadratic,
and cubic polynomials are frequently used functions because they are simple to work
with in finite element formulation. However, trigonometric series can also be used.
For a two-dimensional element, the displacement function is a function of the coordinates in its plane (say, the x- y plane). The functions are expressed in terms of the
nodal unknowns (in the two-dimensional problem, in terms of an x and a y component). The same general displacement function can be used repeatedly for each
element. Hence the finite element method is one in which a continuous quantity,
such as the displacement throughout the body, is approximated by a discrete model
composed of a set of piecewise-continuous functions defined within each finite domain
or finite element.
Step 3 Define the Strain= Displacement and Stress=Strain
Relationships
Strain/displacement and stress/strain relationships are necessary for deriving the equations for each finite element. In the case of one-dimensional deformation, say, in the
x direction, we have strain ex related to displacement u by
du
ð1:4:1Þ
dx
for small strains. In addition, the stresses must be related to the strains through the
stress/strain law—generally called the constitutive law. The ability to define the material behavior accurately is most important in obtaining acceptable results. The simplest
of stress/strain laws, Hooke’s law, which is often used in stress analysis, is given by
ex ¼
sx ¼ Eex
ð1:4:2Þ
where sx ¼ stress in the x direction and E ¼ modulus of elasticity.
Step 4 Derive the Element Stiffness Matrix and Equations
Initially, the development of element stiffness matrices and element equations was
based on the concept of stiffness influence coefficients, which presupposes a background in structural analysis. We now present alternative methods used in this text
that do not require this special background.
Direct Equilibrium or Stiffness Method
According to this method, the stiffness matrix and element equations relating nodal
forces to nodal displacements are obtained using force equilibrium conditions for a
basic element, along with force/deformation relationships. Because this method is
most easily adaptable to line or one-dimensional elements, Chapters 2, 3, and 4 illustrate this method for spring, bar, and beam elements, respectively.
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12
d
1 Introduction
Work or Energy Methods
To develop the stiffness matrix and equations for two- and three-dimensional elements, it
is much easier to apply a work or energy method [35]. The principle of virtual work (using
virtual displacements), the principle of minimum potential energy, and Castigliano’s theorem are methods frequently used for the purpose of derivation of element equations.
The principle of virtual work outlined in Appendix E is applicable for any material behavior, whereas the principle of minimum potential energy and Castigliano’s
theorem are applicable only to elastic materials. Furthermore, the principle of virtual
work can be used even when a potential function does not exist. However, all three
principles yield identical element equations for linear-elastic materials; thus which
method to use for this kind of material in structural analysis is largely a matter of convenience and personal preference. We will present the principle of minimum potential
energy—probably the best known of the three energy methods mentioned here—in
detail in Chapters 2 and 3, where it will be used to derive the spring and bar element
equations. We will further generalize the principle and apply it to the beam element
in Chapter 4 and to the plane stress/strain element in Chapter 6. Thereafter, the principle is routinely referred to as the basis for deriving all other stress-analysis stiffness
matrices and element equations given in Chapters 8, 9, 11, and 12.
For the purpose of extending the finite element method outside the structural stress
analysis field, a functional1 (a function of another function or a function that takes functions as its argument) analogous to the one to be used with the principle of minimum
potential energy is quite useful in deriving the element stiffness matrix and equations
(see Chapters 13 and 14 on heat transfer and fluid flow, respectively). For instance, letting p denote the functional and f ðx; yÞ denote a function f of two variables x and y,
we then have p ¼ pð f ðx; yÞÞ, where p is a function of the function f . A more general
form of a functional depending on two independent variables uðx; yÞ and vðx; yÞ,
where independent variables are x and y in Cartesian coordinates, is given by
ðð
ð1:4:3Þ
p¼
F ðx; y; u; v; u;x ; u;y ; v;x ; v;y ; u;xx ; . . . v;yy Þdx dy
where the comma preceding the subscripts x and y denotes differentiation with respect
qu
, etc.
to x or y, i.e., u;x ¼ qx
Methods of Weighted Residuals
The methods of weighted residuals are useful for developing the element equations;
particularly popular is Galerkin’s method. These methods yield the same results as
the energy methods wherever the energy methods are applicable. They are especially
useful when a functional such as potential energy is not readily available. The
weighted residual methods allow the finite element method to be applied directly to
any differential equation.
1 Another definition of a functional is as follows: A functional is an integral expression that implicitly contains
differential equations that describe the problem. A typical functional is of the form I ðuÞ ¼
Ð
F ðx; u; u 0 Þ dx where uðxÞ; x, and F are real so that I ðuÞ is also a real number. Here u 0 ¼ qu =qx.
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1.4 General Steps of the Finite Element Method
d
13
Galerkin’s method, along with the collocation, the least squares, and the subdomain weighted residual methods are introduced in Chapter 3. To illustrate each
method, they will all be used to solve a one-dimensional bar problem for which a
known exact solution exists for comparison. As the more easily adapted residual
method, Galerkin’s method will also be used to derive the bar element equations in
Chapter 3 and the beam element equations in Chapter 4 and to solve the combined
heat-conduction/convection/mass transport problem in Chapter 13. For more information on the use of the methods of weighted residuals, see Reference [36]; for additional applications to the finite element method, consult References [37] and [38].
Using any of the methods just outlined will produce the equations to describe
the behavior of an element. These equations are written conveniently in matrix
form as
8 9 2
f1 >
>
k11
>
>
>
>
>
>
6
>
>
k21
>
= 6
< f2 >
6
f3 ¼ 6
6 k31
>
.. >
>
>
6 ..
>
>
>
. >
> 4 .
>
>
;
: >
kn1
fn
k12
k22
k32
k13
k23
k33
38 9
. . . k1n >
>
> d1 >
>
> >
7>
. . . k2n 7>
d2 >
>
>
=
<
7
. . . k3n 7
d
3
7
.. 7>
>
> .. >
>
>
. 5>
. >
>
>
>
;
: >
. . . knn
dn
ð1:4:4Þ
or in compact matrix form as
f f g ¼ ½kfdg
ð1:4:5Þ
where f f g is the vector of element nodal forces, ½k is the element stiffness matrix
(normally square and symmetric), and fdg is the vector of unknown element nodal
degrees of freedom or generalized displacements, n. Here generalized displacements
may include such quantities as actual displacements, slopes, or even curvatures. The
matrices in Eq. (1.4.5) will be developed and described in detail in subsequent chapters
for specific element types, such as those in Figure 1–1.
Step 5 Assemble the Element Equations to Obtain the Global
or Total Equations and Introduce Boundary Conditions
In this step the individual element nodal equilibrium equations generated in step 4 are
assembled into the global nodal equilibrium equations. Section 2.3 illustrates this concept for a two-spring assemblage. Another more direct method of superposition
(called the direct stiffness method ), whose basis is nodal force equilibrium, can be
used to obtain the global equations for the whole structure. This direct method is illustrated in Section 2.4 for a spring assemblage. Implicit in the direct stiffness method is
the concept of continuity, or compatibility, which requires that the structure remain
together and that no tears occur anywhere within the structure.
The final assembled or global equation written in matrix form is
fF g ¼ ½Kfdg
ð1:4:6Þ
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14
d
1 Introduction
where fF g is the vector of global nodal forces, ½K is the structure global or total stiffness matrix, (for most problems, the global stiffness matrix is square and symmetric)
and fdg is now the vector of known and unknown structure nodal degrees of freedom
or generalized displacements. It can be shown that at this stage, the global stiffness
matrix ½K is a singular matrix because its determinant is equal to zero. To remove
this singularity problem, we must invoke certain boundary conditions (or constraints
or supports) so that the structure remains in place instead of moving as a rigid body.
Further details and methods of invoking boundary conditions are given in subsequent chapters. At this time it is sufficient to note that invoking boundary or support conditions results in a modification of the global Eq. (1.4.6). We also emphasize that the applied known loads have been accounted for in the global force
matrix fF g.
Step 6 Solve for the Unknown Degrees of Freedom
(or Generalized Displacements)
Equation (1.4.6), modified to account for the boundary conditions, is a set of simultaneous algebraic equations that can be written in expanded matrix form as
8 9 2
>
K11
F1 >
>
>
>
> 6
> >
<
F2 = 6 K21
¼6
.
6 .
>
> 4 ..
> .. >
>
>
>
>
: ;
Fn
Kn1
K12
K22
Kn2
38 9
. . . K1n >
>
> d1 >
>
> >
7>
. . . K2n 7< d2 =
7
.. 7> .. >
>
> . >
. 5>
>
;
: >
dn
. . . Knn
ð1:4:7Þ
where now n is the structure total number of unknown nodal degrees of freedom.
These equations can be solved for the ds by using an elimination method (such as
Gauss’s method) or an iterative method (such as the Gauss–Seidel method). These
two methods are discussed in Appendix B. The ds are called the primary unknowns,
because they are the first quantities determined using the stiffness (or displacement)
finite element method.
Step 7 Solve for the Element Strains and Stresses
For the structural stress-analysis problem, important secondary quantities of strain
and stress (or moment and shear force) can be obtained because they can be directly
expressed in terms of the displacements determined in step 6. Typical relationships
between strain and displacement and between stress and strain—such as Eqs. (1.4.1)
and (1.4.2) for one-dimensional stress given in step 3—can be used.
Step 8 Interpret the Results
The final goal is to interpret and analyze the results for use in the design/analysis process. Determination of locations in the structure where large deformations and large
stresses occur is generally important in making design/analysis decisions. Postprocessor computer programs help the user to interpret the results by displaying them in
graphical form.
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1.5 Applications of the Finite Element Method
d
1.5 Applications of the Finite Element Method
d
15
d
The finite element method can be used to analyze both structural and nonstructural
problems. Typical structural areas include
1. Stress analysis, including truss and frame analysis (such as pedestrian
walk bridges, high rise building frames, and windmill towers), and
stress concentration problems, typically associated with holes, fillets,
or other changes in geometry in a body (such as automotive parts,
pressures vessels, medical devices, aircraft, and sports equipment)
2. Buckling, such as in columns, frames, and vessels
3. Vibration analysis, such as in vibratory equipment
4. Impact problems, including crash analysis of vehicles, projectile
impact, and bodies falling and impacting objects
Nonstructural problems include
1. Heat transfer, such as in electronic devices emitting heat as in a personal
computer microprocessor chip, engines, and cooling fins in radiators
2. Fluid flow, including seepage through porous media (such as water
seeping through earthen dams), cooling ponds, and in air ventilation
systems as used in sports arenas, etc., air flow around racing cars,
yachting boats, and surfboards, etc.
3. Distribution of electric or magnetic potential, such as in antennas and
transistors
Finally, some biomechanical engineering problems (which may include stress
analysis) typically include analyses of human spine, skull, hip joints, jaw/gum tooth
implants, heart, and eye.
We now present some typical applications of the finite element method. These
applications will illustrate the variety, size, and complexity of problems that can be
solved using the method and the typical discretization process and kinds of elements used.
Figure 1–2 illustrates a control tower for a railroad. The tower is a threedimensional frame comprising a series of beam-type elements. The 48 elements are
labeled by the circled numbers, whereas the 28 nodes are indicated by the uncircled
numbers. Each node has three rotation and three displacement components associated
with it. The rotations (ys) and displacements (ds) are called the degrees of freedom.
Because of the loading conditions to which the tower structure is subjected, we have
used a three-dimensional model.
The finite element method used for this frame enables the designer/analyst
quickly to obtain displacements and stresses in the tower for typical load cases, as
required by design codes. Before the development of the finite element method and
the computer, even this relatively simple problem took many hours to solve.
The next illustration of the application of the finite element method to problem solving is the determination of displacements and stresses in an underground
box culvert subjected to ground shock loading from a bomb explosion. Figure 1–3
shows the discretized model, which included a total of 369 nodes, 40 onedimensional bar or truss elements used to model the steel reinforcement in the box
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16
d
1 Introduction
v1
u1
w1
Figure 1–2 Discretized railroad control tower (28 nodes, 48 beam elements) with
typical degrees of freedom shown at node 1, for example (By Daryl L. Logan)
culvert, and 333 plane strain two-dimensional triangular and rectangular elements
used to model the surrounding soil and concrete box culvert. With an assumption
of symmetry, only half of the box culvert need be analyzed. This problem requires
the solution of nearly 700 unknown nodal displacements. It illustrates that different
kinds of elements (here bar and plane strain) can often be used in one finite
element model.
Another problem, that of the hydraulic cylinder rod end shown in Figure 1–4,
was modeled by 120 nodes and 297 plane strain triangular elements. Symmetry was
also applied to the whole rod end so that only half of the rod end had to be analyzed,
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1.5 Applications of the Finite Element Method
d
17
Figure 1–3 Discretized model of an underground box culvert (369 nodes, 40 bar
elements, and 333 plane strain elements) [39]
as shown. The purpose of this analysis was to locate areas of high stress concentration
in the rod end.
Figure 1–5 shows a chimney stack section that is four form heights high (or a
total of 32 ft high). In this illustration, 584 beam elements were used to model the vertical and horizontal stiffeners making up the formwork, and 252 flat-plate elements
were used to model the inner wooden form and the concrete shell. Because of the
irregular loading pattern on the structure, a three-dimensional model was necessary.
Displacements and stresses in the concrete were of prime concern in this problem.
Figure 1–6 shows the finite element discretized model of a proposed steel
die used in a plastic film-making process. The irregular geometry and associated
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18
d
1 Introduction
Figure 1–4 Two-dimensional analysis of a hydraulic cylinder rod end (120 nodes,
297 plane strain triangular elements)
Figure 1–5 Finite element model of a chimney stack section (end view rotated 45 )
(584 beam and 252 flat-plate elements) (By Daryl L. Logan)
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1.5 Applications of the Finite Element Method
d
19
z
y
(a)
(b)
Figure 1–6 (a) Model of a high-strength steel die (240 axisymmetric elements) used in
the plastic film industry (By Daryl L. Logan) and (b) the three-dimensional visual of the
die as the elements in the plane are rotated through 360 around the z-axis of symmetry
(See the full-color insert for a color version of this figure.)
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20
d
1 Introduction
Figure 1–7 Three-dimensional solid element model of a swing casting for a
backhoe frame
potential stress concentrations necessitated use of the finite element method to obtain
a reasonable solution. Here 240 axisymmetric elements were used to model the threedimensional die.
Figure 1–7 illustrates the use of a three-dimensional solid element to model a
swing casting for a backhoe frame. The three-dimensional hexahedral elements are
necessary to model the irregularly shaped three-dimensional casting. Two-dimensional
models certainly would not yield accurate engineering solutions to this problem.
Figure 1–8 illustrates a two-dimensional heat-transfer model used to determine
the temperature distribution in earth subjected to a heat source—a buried pipeline
transporting a hot gas.
Figure 1–9 shows a three-dimensional model of human pelvis which can be
used to study stresses in the bone and the cement layer between the bone and the
implant.
More recently, mechanical event simulation (MES), including nonlinear behavior and contact, such as in roll forming processes, has been studied using finite element
analysis [46], as shown in Figure 1–11 and wind mill generator stress analysis under
various loading conditions, including wind, ice, and earthquake while the blades are
rotating has been performed [46], as shown in Figure 1–12.
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1.5 Applications of the Finite Element Method
d
21
Figure 1–8 Finite element model for a two-dimensional temperature distribution in
the earth
Figure 1–9 Finite element model of a
human pelvis (Studio MacBeth=Science
Photo Library)
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The Loader Coupler
z
The Bucket
x
y
Figure 1–10 Finite element model of a 710G bucket with 169,595 elements and 185,026 nodes used (including 78,566 thin-shell
linear quadrilateral elements for the bucket and coupler, 83,104 solid linear brick elements to model the bosses, and 212 beam
elements to model lift arms, lift arm cylinders, and guide links) (Courtesy of Yousif Omer, Structural Design Engineer, Construction
and Forestry Division, John Deere Dubuque Works)
Linear Beams, The Lift Arm Cylinders
Linear Beams, The Loader Power Link
Parabolic Beam, The Loader Guide Link
Taper Beams, The Loader Lift Arm
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22
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1.6 Advantages of the Finite Element Method
d
23
Figure 1–11 Finite element model of contour roll forming or cold roll forming process (Courtesy
of Valmont West Coast Engineering) (See the full-color insert for a color version of this figure.)
Finally, the field of computational fluid dynamics (CFD) using finite element analysis
has recently been used to design ventilation systems, such as in large sports arenas,
and to study air flow around race cars and around golf balls when suddenly struck
by a golf club [63].
These illustrations suggest the kinds of problems that can be solved by the finite
element method. Additional guidelines concerning modeling techniques will be provided in Chapter 7.
d
1.6 Advantages of the Finite Element Method
d
As previously indicated, the finite element method has been applied to numerous
problems, both structural and nonstructural. This method has a number of advantages that have made it very popular. They include the ability to
1. Model irregularly shaped bodies quite easily
2. Handle general load conditions without difficulty
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24
d
1 Introduction
Figure 1–12 Finite element model showing the von Mises stress plot of a wind mill
tower at a critical time step using a nonlinear finite element simulation (Courtesy of
Valmont West Coast Engineering)
3. Model bodies composed of several different materials because the element
equations are evaluated individually
4. Handle unlimited numbers and kinds of boundary conditions
5. Vary the size of the elements to make it possible to use small elements
where necessary
6. Alter the finite element model relatively easily and cheaply
7. Include dynamic effects
8. Handle nonlinear behavior existing with large deformations and nonlinear
materials
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1.7 Computer Programs for the Finite Element Method
d
25
The finite element method of structural analysis enables the designer to detect
stress, vibration, and thermal problems during the design process and to evaluate design
changes before the construction of a possible prototype. Thus confidence in the acceptability of the prototype is enhanced. Moreover, if used properly, the method can
reduce the number of prototypes that need to be built.
Even though the finite element method was initially used for structural analysis,
it has since been adapted to many other disciplines in engineering and mathematical
physics, such as fluid flow, heat transfer, electromagnetic potentials, soil mechanics,
and acoustics [22–24, 27, 42–44].
d
1.7 Computer Programs for the Finite
Element Method
d
There are two general computer methods of approach to the solution of problems by
the finite element method. One is to use large commercial programs, many of which
have been configured to run on personal computers (PCs); these general-purpose programs are designed to solve many types of problems. The other is to develop many
small, special-purpose programs to solve specific problems. In this section, we will discuss
the advantages and disadvantages of both methods. We will then list some of the
available general-purpose programs and discuss some of their standard capabilities.
Some advantages of general-purpose programs:
1. The input is well organized and is developed with user ease in mind.
Users do not need special knowledge of computer software or
hardware. Preprocessors are readily available to help create the finite
element model.
2. The programs are large systems that often can solve many types of
problems of large or small size with the same input format.
3. Many of the programs can be expanded by adding new modules for
new kinds of problems or new technology. Thus they may be kept
current with a minimum of effort.
4. With the increased storage capacity and computational efficiency of
PCs, many general-purpose programs can now be run on PCs.
5. Many of the commercially available programs have become very
attractive in price and can solve a wide range of problems [45–56].
Some disadvantages of general-purpose programs:
1. The initial cost of developing general-purpose programs is high.
2. General-purpose programs are less efficient than special-purpose
programs because the computer must make many checks for each
problem, some of which would not be necessary if a special-purpose
program were used.
3. Many of the programs are proprietary. Hence the user has little access
to the logic of the program. If a revision must be made, it often has to
be done by the developers.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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26
d
1 Introduction
Some advantages of special-purpose programs:
1.
2.
3.
4.
The programs are usually relatively short, with low development costs.
Small computers are able to run the programs.
Additions can be made to the program quickly and at a low cost.
The programs are efficient in solving the problems they were designed
to solve.
The major disadvantage of special-purpose programs is their inability to solve
different classes of problems. Thus one must have as many programs as there are different classes of problems to be solved. A list of special-purpose, public-domain finiteelement programs is given in the website [60].
There are numerous vendors supporting finite element programs, and the interested user should carefully consult the vendor before purchasing any software. However, to give you an idea about the various commercial personal computer programs
now available for solving problems by the finite element method, we present a partial
list of existing programs.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Algor [46]
Abaqus [47]
ANSYS [48]
COSMOS/M [49]
GT-STRUDL [50]
LS-DYNA [59]
MARC [51]
MSC/NASTRAN [52]
NISA [53]
Pro/MECHANICA [54]
SAP2000 [55]
STARDYNE [56]
Standard capabilities of many of the listed programs are provided in
the preceding references and in Reference [45]. These capabilities include
information on
1. Element types available, such as beam, plane stress, and threedimensional solid
2. Type of analysis available, such as static and dynamic
3. Material behavior, such as linear-elastic and nonlinear
4. Load types, such as concentrated, distributed, thermal, and displacement (settlement)
5. Data generation, such as automatic generation of nodes, elements, and
restraints (most programs have preprocessors to generate the mesh for
the model)
6. Plotting, such as original and deformed geometry and stress and
temperature contours (most programs have postprocessors to aid in
interpreting results in graphical form)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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References
d
27
7. Displacement behavior, such as small and large displacement and buckling
8. Selective output, such as at selected nodes, elements, and maximum or
minimum values
All programs include at least the bar, beam, plane stress, plate-bending, and threedimensional solid elements, and most now include heat-transfer analysis capabilities.
Complete capabilities of the programs and their cost are best obtained through
program reference manuals and websites, such as References [46–56, 59].
d
References
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Nov., Dec. 1954 and Feb., Mar., Apr., May 1955.
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London, 1960 (collection of papers published in Aircraft Engineering in 1954 and 1955).
[8] Turner, M. J., Clough, R. W., Martin, H. C., and Topp, L. J., ‘‘Stiffness and Deflection
Analysis of Complex Structures,’’ Journal of Aeronautical Sciences, Vol. 23, No. 9,
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American Society of Civil Engineers, 2nd Conference on Electronic Computation, Pittsburgh, PA, pp. 345–378, Sept. 1960.
[10] Melosh, R. J., ‘‘A Stiffness Matrix for the Analysis of Thin Plates in Bending,’’ Journal of
the Aerospace Sciences, Vol. 28, No. 1, pp. 34–42, Jan. 1961.
[11] Grafton, P. E., and Strome, D. R., ‘‘Analysis of Axisymmetric Shells by the Direct Stiffness Method,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 1,
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[12] Martin, H. C., ‘‘Plane Elasticity Problems and the Direct Stiffness Method,’’ The Trend in
Engineering, Vol. 13, pp. 5–19, Jan. 1961.
[13] Gallagher, R. H., Padlog, J., and Bijlaard, P. P., ‘‘Stress Analysis of Heated Complex
Shapes,’’ Journal of the American Rocket Society, Vol. 32, pp. 700–707, May 1962.
[14] Melosh, R. J., ‘‘Structural Analysis of Solids,’’ Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, pp. 205–223, Aug. 1963.
[15] Argyris, J. H., ‘‘Recent Advances in Matrix Methods of Structural Analysis,’’ Progress in
Aeronautical Science, Vol. 4, Pergamon Press, New York, 1964.
[16] Clough, R. W., and Rashid, Y., ‘‘Finite Element Analysis of Axisymmetric Solids,’’ Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil
Engineers, Vol. 91, pp. 71–85, Feb. 1965.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Licensed to: CengageBrain User
28
d
1 Introduction
[17] Wilson, E. L., ‘‘Structural Analysis of Axisymmetric Solids,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 3, No. 12, pp. 2269–2274, Dec. 1965.
[18] Turner, M. J., Dill, E. H., Martin, H. C., and Melosh, R. J., ‘‘Large Deflections of Structures Subjected to Heating and External Loads,’’ Journal of Aeronautical Sciences, Vol. 27,
No. 2, pp. 97–107, Feb. 1960.
[19] Gallagher, R. H., and Padlog, J., ‘‘Discrete Element Approach to Structural Stability
Analysis,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 1, No. 6,
pp. 1437–1439, 1963.
[20] Zienkiewicz, O. C., Watson, M., and King, I. P., ‘‘A Numerical Method of Visco-Elastic
Stress Analysis,’’ International Journal of Mechanical Sciences, Vol. 10, pp. 807–827, 1968.
[21] Archer, J. S., ‘‘Consistent Matrix Formulations for Structural Analysis Using FiniteElement Techniques,’’ Journal of the American Institute of Aeronautics and Astronautics,
Vol. 3, No. 10, pp. 1910–1918, 1965.
[22] Zienkiewicz, O. C., and Cheung, Y. K., ‘‘Finite Elements in the Solution of Field Problems,’’ The Engineer, pp. 507–510, Sept. 24, 1965.
[23] Martin, H. C., ‘‘Finite Element Analysis of Fluid Flows,’’ Proceedings of the Second Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, Ohio,
pp. 517–535, Oct. 1968. (AFFDL-TR-68-150, Dec. 1969; AD-703-685, N.T.I.S.)
[24] Wilson, E. L., and Nickel, R. E., ‘‘Application of the Finite Element Method to Heat
Conduction Analysis,’’ Nuclear Engineering and Design, Vol. 4, pp. 276–286, 1966.
[25] Szabo, B. A., and Lee, G. C., ‘‘Derivation of Stiffness Matrices for Problems in Plane
Elasticity by Galerkin’s Method,’’ International Journal of Numerical Methods in Engineering, Vol. 1, pp. 301–310, 1969.
[26] Zienkiewicz, O. C., and Parekh, C. J., ‘‘Transient Field Problems: Two-Dimensional and
Three-Dimensional Analysis by Isoparametric Finite Elements,’’ International Journal of
Numerical Methods in Engineering, Vol. 2, No. 1, pp. 61–71, 1970.
[27] Lyness, J. F., Owen, D. R. J., and Zienkiewicz, O. C., ‘‘Three-Dimensional Magnetic Field
Determination Using a Scalar Potential. A Finite Element Solution,’’ Transactions on
Magnetics, Institute of Electrical and Electronics Engineers, pp. 1649–1656, 1977.
[28] Belytschko, T., ‘‘A Survey of Numerical Methods and Computer Programs for Dynamic
Structural Analysis,’’ Nuclear Engineering and Design, Vol. 37, No. 1, pp. 23–34, 1976.
[29] Belytschko, T., ‘‘Efficient Large-Scale Nonlinear Transient Analysis by Finite Elements,’’
International Journal of Numerical Methods in Engineering, Vol. 10, No. 3, pp. 579–596, 1976.
[30] Huiskies, R., and Chao, E. Y. S., ‘‘A Survey of Finite Element Analysis in Orthopedic Biomechanics: The First Decade,’’ Journal of Biomechanics, Vol. 16, No. 6, pp. 385–409, 1983.
[31] Journal of Biomechanical Engineering, Transactions of the American Society of Mechanical
Engineers, (published quarterly) (1st issue published 1977).
[32] Kardestuncer, H., ed., Finite Element Handbook, McGraw-Hill, New York, 1987.
[33] Clough, R. W., ‘‘The Finite Element Method After Twenty-Five Years: A Personal View,’’
Computers and Structures, Vol. 12, No. 4, pp. 361–370, 1980.
[34] Kardestuncer, H., Elementary Matrix Analysis of Structures, McGraw-Hill, New York, 1974.
[35] Oden, J. T., and Ripperger, E. A., Mechanics of Elastic Structures, 2nd ed., McGraw-Hill,
New York, 1981.
[36] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Academic
Press, New York, 1972.
[37] Zienkiewicz, O. C., The Finite Element Method, 3rd ed., McGraw-Hill, London, 1977.
[38] Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J., Concepts and Applications of
Finite Element Analysis, 4th ed., Wiley, New York, 2002.
[39] Koswara, H., A Finite Element Analysis of Underground Shelter Subjected to Ground Shock
Load, M.S. Thesis, Rose-Hulman Institute of Technology, 1983.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Licensed to: CengageBrain User
Problems
d
29
[40] Greer, R. D., ‘‘The Analysis of a Film Tower Die Utilizing the ANSYS Finite Element Package,’’ M.S. Thesis, Rose-Hulman Institute of Technology, Terre Haute, Indiana, May 1989.
[41] Koeneman, J. B., Hansen, T. M., and Beres, K., ‘‘The Effect of Hip Stem Elastic Modulus
and Cement/Stem Bond on Cement Stresses,’’ 36th Annual Meeting, Orthopaedic Research
Society, Feb. 5–8, 1990, New Orleans, Louisiana.
[42] Girijavallabham, C. V., and Reese, L. C., ‘‘Finite-Element Method for Problems in Soil
Mechanics,’’ Journal of the Structural Division, American Society of Civil Engineers, No.
Sm2, pp. 473–497, Mar. 1968.
[43] Young, C., and Crocker, M., ‘‘Transmission Loss by Finite-Element Method,’’ Journal of
the Acoustical Society of America, Vol. 57, No. 1, pp. 144–148, Jan. 1975.
[44] Silvester, P. P., and Ferrari, R. L., Finite Elements for Electrical Engineers, Cambridge
University Press, Cambridge, England, 1983.
[45] Falk, H., and Beardsley, C. W., ‘‘Finite Element Analysis Packages for Personal Computers,’’ Mechanical Engineering, pp. 54–71, Jan. 1985.
[46] Algor Interactive Systems, 150 Beta Drive, Pittsburgh, PA 15238.
[47] Web site http://www.abaqus.com.
[48] Swanson, J. A., ANSYS-Engineering Analysis Systems User’s Manual, Swanson Analysis
Systems, Inc., Johnson Rd., P.O. Box 65, Houston, PA 15342.
[49] COSMOS/M, Structural Research & Analysis Corp., 12121 Wilshire Blvd., Los Angeles,
CA 90025.
[50] web site http://ce6000.cegatech.edu.
[51] web site http://www.mscsoftware.com.
[52] MSC/NASTRAN, MacNeal-Schwendler Corp., 600 Suffolk St., Lowell, MA, 01854.
[53] web site http://emrc.com.
[54] Toogood, Roger, Pro/MECHANICA Structure Tutorial, SDC Publications, 2001.
[55] Computers & Structures, Inc., 1995 University Ave., Berkeley, CA 94704.
[56] STARDYNE, Research Engineers, Inc., 22700 Savi Ranch Pkwy, Yorba Linda, CA 92687.
[57] Noor, A. K., ‘‘Bibliography of Books and Monographs on Finite Element Technology,’’
Applied Mechanics Reviews, Vol. 44, No. 6, pp. 307–317, June 1991.
[58] Belytschko, T., Liu W. K., and Moran, B., Nonlinear Finite Elements For Continua and
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[60] web site http://homepage.usak.ca/~ijm451/finite/fe resources.
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[63] ANSYS Advantage, Vol. 1, Issue 1, 2007.
d
Problems
1.1 Define the term finite element.
1.2 What does discretization mean in the finite element method?
1.3 In what year did the modern development of the finite element method begin?
1.4 In what year was the direct stiffness method introduced?
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
30
d
1 Introduction
1.5 Define the term matrix.
1.6 What role did the computer play in the use of the finite element method?
1.7 List and briefly describe the general steps of the finite element method.
1.8 What is the displacement method?
1.9 List four common types of finite elements.
1.10
Name three commonly used methods for deriving the element stiffness matrix and
element equations. Briefly describe each method.
1.11
To what does the term degrees of freedom refer?
1.12
List five typical areas of engineering where the finite element method is applied.
1.13
List five advantages of the finite element method.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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