CHAPTER 2
LITERATURE REVIEW
Reinforced concrete analysis is performed at a given section for either axial
force or bending moment or transverse shear loads. The axial force and bending
moment analysis usually idealises the stress-strain behavior of the concrete with a
rectangular stress block to simplify the calculations. More detailed, moment
curvature analysis may be performed with more complex stress-strain relationships.
The finite element analysis FEAPpv is designed for the use in an instructional
program to illustrate performance of different types of elements and modeling
methods and in a research and application environment which requires frequent
modifications to address new problem areas or analysis requirements. The subtopics below identify the definitions, history and explanation of stress, short column,
finite element analysis and isotropic material.
10
2.1
What is a Structure
A structure can be most broadly defined as an object whose purpose is to
carry a set of loads or forces from one place to another. In most cases the aim is to
transmit the applied loading from somewhere in space to the ground without
collapsing and without deforming excessively.
The identified common characteristics of structures are as listed and
explained below: 
structures are designed to carry loads

structures are usually either on the ground or on another structure with
reaction forces generated at the support points

the applied loads and reactions cause forces to be generated within the
members of the structure

the structural members must not collapse or deform excessively under
these forces
11
2.2
Basic Structural Principles
2.2.1
Forces and Displacements
Forces on a structure can arise from many sources, such as the structure’s
own weight, any objects placed on it, wind pressure and so forth. Force is a vector
quantity, that is, it has both magnitude and direction. The SI unit of force is Newton
(N), which is defined as the force required to impart an acceleration of one meter per
second per second to a mass of one kilogram (that is, 1 N = 1 kg m/s2).
An object placed on a structure will thus impart a vertical force equal to its
mass multiplied by the acceleration due to gravity. The forces on a body could also
give rise to moments, which tend to cause the body to rotate about an axis. The
moment of a force about an axis is simply equal to the magnitude of the force
multiplied by the perpendicular distance from the axis to the line of action of the
force.
The loads acting on a structure causes internal stresses, and so causes it to
deform. The deformations are usually expressed in terms of deflections (that is
straight line movements) and rotations about a point or axis. These deflections and
rotations are, of course, closely related to each other. In summary, the term force is
often used in practice to encompass both direct forces and moments, and likewise the
term displacement is often taken to include both translational movements or
deflections and rotations.
12
2.2.2
Axial Force and Bending Moment
Reinforced concrete analysis for axial force and bending moment is usually
performed by assuming a given strain value at the extreme compression fiber with a
linear strain distribution over the depth of the section. The stress distribution
typically assumes a rectangular stress block with a depth equal to some fraction of
the neutral axis depth and a magnitude equal to some fraction of the concrete
compressive strength as illustrated in Figure 2.1.
Figure 2.1: Stress Distribution
2.2.3
Sign Convention
Systems of forces, moments, displacements and rotations must be analysed
using a logical and consistent sign convention. It is extremely important that the
convention used is clearly stated and adhered to at all times, otherwise confusion and
errors are sure to occur.
13
The choice of a sign convention is far from straight forward. No system is
ideal for all circumstances; in particular, it proves impossible to prevent minus signs
from appearing in formulas and analysis.
When considering structures that can be idealised as two dimensional, it is
normally assumed that they lie within the x-y axis system. Systems of stresses and
strains in two dimensions are referred to conventional Cartesian (x, y) axes. Figure
2.2 shows the positive stresses acting on the faces of a small element that is a
positive force is one which acts in a positive direction on a positive face and viceversa. (Todd J. D., Williams M. S. (2000). “Structures: Theory and Analysis.”
London: Macmillan Press Ltd.)
Of the two subscripts, the first refers to the face of the element and the second
to the direction of force. Thus xx is stress acting in the x direction on a face whose
outward normal is also in the x direction, while yx is stress acting in the x direction
but on a face whose outward normal is in the y direction.
yy
Principle Directions
yx
xy

xx
xy
xx
yx
y
yy
x
Figure 2.2: Distribution of Internal Tractions
14
For computer application, (stiffness matrices, finite elements), the axis
systems used are in most respects the same as those described above. For a
computer, complete consistency is essential, and this requires changes to one or two
of the more intuitive convention used in hand methods.
First, the directionality of the external loads must be dealt with more
vigorously; if a load acts in a negative axis direction then is must be assigned a
negative value. Generally, for computer application, the sign of a displacement,
force or moment is governed solely by its direction in space and does not depend on
the face on which it acts.
2.2.4
Pinned Support
Structures generally rest on and transmit forces to either the ground or
another structure. The contact points via which the forces are transmitted are called
supports and the nature of supports plays a vital role in determining how the structure
carries the loads. Like loadings, structural supports come in wide variety of types
but are generally simplified to a few idealised cases for ease of analysis.
For the column foundation of this study, pinned supports have been
introduced at the foundation bottom. A pinned support is one which prevents the
structure from moving translationally in any direction at the support point, but
provides no resistance to rotation.
15
2.3
Stress
2.3.1
Definition of Stress
The effect that an internal force has on a member will depend on the
properties of the material from which the member is made, and on its dimensions;
obviously a very large member will be able to sustain greater forces than a smaller
one made from the same material.
This dependence on material size can be conveniently accounted for by
performing calculations in terms of stress rather than force. The SI unit of stress is
the Pascal (Pa). One Pascal is equal to one Newton per square meter.
In simple terms, stress can be defined as the force acting on a member or part
of a member divided by the area over which it acts. Stress is a useful quantity
because: 
It provides a measure how the internal forces is distributed through a
member

The ways in which a member responds to a certain amount of stress
are functions solely of the material properties, and are independent of
the member dimensions
16
2.3.2
Concept of Stress
The concept of stress originated from the study of strength and failure of
solids. The stress field is the distribution of internal "tractions" that balance a given
set of external tractions and body forces.
Figure 2.3: Internal Tractions
Initially, the external traction T that represents the force per unit area acting
at a given location on the body's surface is looked at. Traction T is a bound vector,
which means T cannot slide along its line of action or translate to another location
and keep the same meaning.
In other words, a traction vector cannot be fully described unless both the
force and the surface where the force acts on have been specified. Given both F and
s, the traction T can be defined as
17
The internal traction within a solid, or stress, can be defined in a similar
manner. Suppose a random slice is made across the solid shown in Figure 2.3,
leading to the free body diagram shown at right.
Surface tractions would appear on the exposed surface, similar in form to the
external tractions applied to the body's exterior surface. The stress at point P (see
Figure 2.4) can be defined using the same equation as was used for T.
Figure 2.4: Stress at Point P
Stress therefore can be interpreted as internal tractions that act on a defined
internal datum plane. One cannot measure the stress without first specifying the
datum plane.
18
2.3.3
The Stress Tensor (or Stress Matrix)
Surface tractions, or stresses acting on an internal datum plane, are typically
decomposed into three mutually orthogonal components. One component is normal
to the surface and represents direct stress. The other two components are tangential
to the surface and represent shear stresses.
What is the distinction between normal and tangential tractions, or
equivalently, direct and shear stresses? Direct stresses tend to change the volume of
the material (e.g. hydrostatic pressure) and are resisted by the body's bulk modulus
(which depends on the Young's modulus and Poisson ratio). Shear stresses tend to
deform the material without changing its volume, and are resisted by the body's shear
modulus.
2.3.4
Internal Stresses in Tension Members
This section considers the internal forces developed at a specific point within
the cross section of a member. The idea of fundamental importance introduced is the
concept of stress, or the force intensity per unit. Consider a simple tension member
as illustrated in Figure 2.5. It is evident that the internal tension present is not
actually concentrated at a specific point (as the arrows symbolizing internal force in
Figure 2.5 would seem to indicate) within the cross section of the member.
It is actually rather distributed over the entire cross section. The total internal
force necessary to equilibrate the external force acting on the member is in actuality
the resultant of all distributed forces, or stresses, acting at the cross section (Schodek
D. L. (1998). “Structures.” New Jersey: Prentice Hall, Inc.)
19
P
P
P
Tension stresses are
assumed uniformly
distributed when
member is axially
loaded.
X
X
P1
Stress = force/area
P
P
a) Loaded Member
b) Internal Forces
at Section X-X
P
c) Tension Stresses
at Section X-X
Figure 2.5: Tension Members
With respect to a simple element carrying a tension force, it is reasonable to
assume that if the external force acts along the axis of the member and at the centroid
or point of symmetry of the cross section, the stresses developed at the cross section
are of uniform intensity. Their resultant would have the same line of action as the
external force present.
When stresses are uniformly distributed, their magnitude is given by stress is
equivalent to force divided by area (P/A). Stresses of this type are normally called
axial or normal stresses. The stresses developed in a member loaded in direct
compression can be similarly described.
20
The assumption that stresses associated with axial loads are uniformly
distributed across a cross section is a reasonable one when the load is applied in a
purely axial way and the member involved is straight and of uniform cross section.
At unique points, such as where the external load is applied or at
discontinuities in the member, more complex stress patterns may occur. The
assumption, however, is generally quite good and perfectly acceptable as a working
hypothesis for preliminary design purposes.
2.3.5
Definition of Poisson’s Ratio
Poisson's ratio  is the ratio of transverse contraction strain to longitudinal
extension strain in the direction of stretching force. Tensile deformation is
considered positive and compressive deformation is considered negative. The
definition of Poisson's ratio contains a minus sign so that normal materials have a
positive ratio.
Poisson's ratio is related to elastic modulus K, the bulk modulus; G as the
shear modulus; and E, Young's modulus, by the following. The elastic modulus’ are
measures of stiffness. They are ratios of stress to strain. Stress is force per unit area,
with the direction of both the force and the area specified.
The theory of isotropic elasticity allows Poisson's ratios in the range from 1
to 1/2. Physically the reason is that for the material to be stable, the stiffness must be
positive; the bulk and shear stiffness are interrelated by formulas which incorporate
Poisson's ratio.
21
2.3.6
Definition of Elastic Modulus
Elastic Modulus is the ratio of stress to strain. For tensile or compressive
stress to strain, the elastic modulus is called Young’s Modulus, E. Young’s modulus
is really a measure of stiffness since the larger the value, the stiffer the material.
elastic limit
Stress
(Force/Area)
failure point
Strain (L/L0)
Figure 2.6: Stress Vs Strain (Elastic Limit)
The elastic limit is caused by the failure of the material to hold together. The
linear relationship shown in Figure 2.5 is called the Hooke’s Law. The slope of this
curve is Young’s Modulus (i.e. a tensile elastic modulus).
22
2.3.7
Young’s Modulus
2.3.7.1 Founder of Young’s Modulus – Thomas Young
Thomas Young from 1773 until 1829 was an English physicist, physician,
and Egyptologist. An authority on the mechanism of vision and on optics, he stated
(1807) a theory of color vision now known as the Young-Helmholtz theory, studied
the structure of the eye, and described the defect called astigmatism.
He is especially noted for reviving the wave theory of light as opposed to the
corpuscular theory, advancing as proof a demonstration based upon the principle of
interference of light, which he first formulated in 1801. Young's versatility is
evidenced by his contributions to the theory of and establishment of a coefficient of
elasticity, Young's modulus.
2.3.7.2 Definition of Young’s Modulus
For Thomas Young , Young’s modulus is a number representing the ratio of
stress to strain for a wire or bar of a given substance. According to Hooke's law the
strain is proportional to stress, and therefore the ratio of the two is a constant that is
commonly used to indicate the elasticity of the substance.
Young's modulus is the elastic modulus for tension, or tensile stress, and is
the force per unit cross section of the material divided by the fractional increase in
length resulting from the stretching of a standard rod or wire of the material.
23
Within the limits of elasticity, the ratio of the linear stress to the linear strain
is termed the modulus of elasticity or Young's Modulus and may be written Young's
Modulus, or E = (Stress/Strain). It is this property that determines how much a bar
will sag under its own weight or under a loading when used as a beam within its limit
of proportionality.
2.3.7.3 Elasticity
Elasticity is the ability of a body to resist a distorting influence or stress and
to return to its original size and shape when the stress is removed. All solids are
elastic for small enough deformations or strains, but if the stress exceeds a certain
amount known as the elastic limit, a permanent deformation is produced. Both the
resistance to stress and the elastic limit depend on the composition of the solid.
Some different kinds of stresses are tension, compression, torsion, and
shearing (see strength of materials ). For each kind of stress and the corresponding
strain there is a modulus, i.e., the ratio of the stress to the strain; the ratio of tensile
stress to strain for a given material is called its Young's modulus . Hooke's law (for
Robert Hooke) states that, within the elastic limit, strain is proportional to stress.
24
2.3.7.4 Strength of Materials
Strength of materials is a measurement in engineering of the capacity of
metal, wood, concrete, and other materials to withstand stress and strain. Stress is
the internal force exerted by one part of an elastic body upon the adjoining part, and
strain is the deformation or change in dimension occasioned by stress.
When a body is subjected to pull, it is said to be under tension, or tensional
stress, and when it is being pushed, i.e., is supporting a weight, it is under
compression, or compressive stress. Shear, or shearing stress, results when a force
tends to make part of the body or one side of a plane slide past the other.
Torsion, or torsional stress, occurs when external forces tend to twist a body
around an axis. Materials are considered to be elastic in relation to an applied stress
if the strain disappears after the force is removed. The elastic limit is the maximum
stress a material can sustain and still return to its original form.
According to Hooke's law, the stress created in an elastic material is
proportional to strain, within the elastic limit. In calculating the dimensions of
materials required for specific application, the engineer uses working stresses that are
ultimate strengths, or elastic limits, divided by a quantity called factor of safety. In
laboratories materials are frequently “tested to destruction.”
They are deliberately overloaded with the particular force that acts against the
property or strength to be measured. Changes in form are measured to the millionth
of an inch. Static tests are conducted to determine a material's elastic limit, ductility,
hardness, reaction to temperature change, and other qualities.
25
Dynamic tests are those in which the material is exposed to a combination of
expected operating circumstances including impact (e.g., a shell against a steel tank),
vibration, cyclic stress, fluctuating loads, and fatigue. Polarized light, x-rays,
ultrasonic waves, and microscopic examination are some of the means of testing
materials.
2.3.7.5 Hooke’s Law
Hooke’s Law is the principle that the stress applied to stretch or compress a
body is proportional to the strain or to the change in length thus produced, so long as
the limit of elasticity of the body is not exceeded.
In physics, Hooke's law of elasticity states that if a force (F) is applied to an
elastic spring or prismatic rod (with length L and cross section A), its extension is
linearly proportional to its tensile stress σ and modulus of elasticity (E).
It is named after the 17th century physicist Robert Hooke. The law holds up
to a limit, called the elastic limit, or limit of elasticity, after which the metal will
enter a condition of yield and the spring will suffer plastic deformation up to the
plastic limit or limit of plasticity, after which it will eventually break if the force is
further increased.
26
Figure 2.7: Limit of Elasticity
2.3.7.6 Hooke’s Law for Isotropic Materials
Most metallic alloys and thermo set polymers are considered isotropic, where
by definition the material properties are independent of direction. Such materials
have only 2 independent variables (i.e. elastic constants) in their stiffness and
compliance matrices, as opposed to the 21 elastic constants in the general anisotropic
case.
The two elastic constants are usually expressed as the Young's modulus E and
the Poisson's ratio . However, the alternative elastic constants K (bulk modulus)
and/or G (shear modulus) can also be used. For isotropic materials, G and K can be
found from E and  by a set of equations, and vice-versa.
27
2.3.7.7 Definition of Isotropic Materials
Isotropic materials have uniform properties in all directions. The measured
properties of an isotropic material are independent of the axis of testing.
2.4
Short Columns
2.4.1
Definition of a Short Column
A column is a one-dimensional, vertical element which carries loads
primarily by axial compression, sometimes accompanied by significant bending
moments.
2.4.2
Members in Compression
Along with load-bearing walls, columns are the most common of all vertical
support elements. Strictly speaking, columns need not be only vertical. Rather, they
are rigid linear elements that can be inclined in any direction, but to which loads are
applied only at member ends. They are not normally subject to bending directly
induced by loads acting transverse to their axes.
28
Columns can be usually categorised in terms of their length. Short columns
tend to fail by crushing (a strength failure). Members in which the least crosssectional dimension present is appreciable relative to the length of the member are
short columns.
The load-carrying capacity of a short column is independent of the length of
the member and, when excessively loaded, the short column typically fails by
crushing. Consequently, its ultimate load-carrying capacity depends primarily on the
strength of the material used and its cross sectional area.
2.5
Finite Element Method
2.5.1
The role of Computers in Structural Analysis
A wide range of stiffness matrix computer programs is available for the
analysis of framed structures, while finite element software is widely used for the
analysis of continuum structures. The algorithms on which these programs are based
are derived from, and closely related to, the more traditional approaches, but by
writing them in matrix form they are rendered amenable to computational solution.
Therefore this study ventures into the usage of FEAPpv, as finite element software,
to obtain results for a short column analysis in stress distribution.
29
2.5.2
Historical Background of Computers in Structural Analysis
Since the dawn of history, structural engineering has been an essential part of
human endeavor. However, it was not until about the middle of the seventeenth
century those engineers began applying the knowledge of mechanics (mathematics
and science) in designing structures. Earlier engineering structures were designed by
trial and error and by using rule of thumb based on past experience.
The fact that some of the magnificent structures form earlier eras, such as
Egyptian pyramid (about 3000 B.C.), Greek temples (500 - 200 B.C), Roman
coliseums and aqueducts (200 B.C - A.D. 200), and the Gothic cathedrals (A.D. 1000
- 1500), still stand today is a testimonial to the ingenuity of their builders. Galileo
Galilei (1564 - 1642) is generally considered to be the originator of the theory of
structures. In his book entitled Two New Sciences, which was published in 1963,
Galileo analysed the failure of some simple structures, including cantilever beams.
Although Galileo’s predictions of the strength of beams were only
approximate, his work laid the foundation for future developments in the theory of
structures and ushered in a new era of structural engineering, in which the analytical
principles of mechanics and strength of materials would have major influence on the
design of structures. Following Galileo’s pioneering work, the knowledge of
structural mechanics advanced at a rapid pace in the second half of the seventeenth
century and into the eighteenth century.
Among the notable investigators of that period were Robert Hooke (16351703), who developed the law of linear relationship between the force and
deformation of the materials (Hooke’s law); Sir Isaac Newton (1667 - 1748), who
formulated the laws of motion and developed calculus; John Bernoulli (1667 -1748),
who formulated the principles of virtual work.
30
Leonhard Euler (1707 - 1783), who developed the theory of bucking of
columns; and C.A. de Coulomb (1736 -1806), who presented the analysis of bending
of elastic beams. In 1826 L.M.Navier (1785 - 1836) published a paper on elastic
behavior of structures, which is considered to be the first textbook on the modern
theory of strength of materials. The developments of structural mechanics continued
at a tremendous pace through the rest of the nineteenth century and in to the
twentieth century, when most of the classical methods of analysis of structures
described in this text were developed.
The important contributors of this period included B.P Clapeyron (1799 1864), who formulated the three moment equation for the analysis of continuous
beams; J.C.Maxwell (1831 - 1879), who presented the method of consistent
deformation and the law of reciprocal deflections; Otto Mohr (1835-1918), who
developed the conjugate-beam method for calculation of deflections and Mohr’s
circle of stress and strain and Alberto Castigliano (1847 - 1884), who formulated the
theorem of least work.
The others are C.E.Greene (1842 - 1903), who developed the moment - area
method; H.Muller-Breslau (1851 - 1925), who presented a principle for constructing
influence lines; G.A.Maney (1888 - 1947), who developed the slope-deflection
method, which is considered to be the precursor to be the matrix stiffness method;
and Hardy cross (1885 - 1959), who developed the moment-distribution method in
1992.
The moment-distribution method provided engineers with a simple iterative
procedure for analyzing highly statically indeterminate structures. This method,
which was widely used by structural engineers during the period from 1930-197,
contributed significantly to their understanding of the behavior of statically
indeterminate frames. Many structures designed during the period, such as high-rise
buildings, would not have been possible without the availability of the moment distribution method.
31
The availability of computers in the 1950s revolutionized structural analysis.
Because the computer can solve large systems of simultaneous equations, analyses
that took days and sometimes weeks in the computer era could now be preformed in
seconds. The development of the current computer-oriented methods of structural
analysis can be attributed to, among others, J.H.Argyris, R.W.Clough, S.Kelsey,
R.K.Livesley, H.C.Martin, M.T.Turner, E.L.Wilson, and O.C.Zienkiewicz.
2.5.3
What is Finite Element Analysis
FEA consists of a computer model of a material or design that is stressed and
analysed for specific results. It is used in new product design, and existing product
refinement. In case of structural failure, FEA may be used to help determine the
design modifications to meet the new condition. There are generally two types of
analysis that are used in industry: 2-D modeling, and 3-D modeling.
While 2-D modeling conserves simplicity and allows the analysis to be run
on a relatively normal computer, it tends to yield less accurate results. 3-D
modeling, however, produces more accurate results while sacrificing the ability to
run on all but the fastest computers effectively.
Within each of these modeling schemes, the programmer can insert numerous
algorithms (functions), which may make the system behave linearly or non-linearly.
Linear systems are far less complex and generally do not take into account plastic
deformation. Non-linear systems do account for plastic deformation, and many also
are capable of testing a material all the way to fracture.
32
2.5.4
History of Finite Element Analysis
In 1851, to derive the differential equation of the surface of minimum area
bounded by a given closed curve in space, Schellbach discretized a surface into right
triangles and wrote a finite difference expression for the total discretized area. He
proposed no other application or generalization of the idea.
FEA is now regarded as a way to avoid differential equations by replacing
them with an approximating set of algebraic equations. Starting in 1906, researchers
noted that a framework having many bars in a regular pattern behaves much like an
isotropic elastic body. Application to problems of plane elasticity and plate bending
was reported in 1941.
This work exploits well known methods for analysis of framed structures but
cannot be applied to bodies of arbitrary shape. Also, rather than discretization of a
continuum into smaller pieces, structural members of a different type are substituted.
The framework method may be regarded as a precursor to FEA rather than an early
form of it.
The FE method as we know it today seems to have originated with Courant in
his 1943 paper, which is the written version of a 1941 lecture to the American
Mathematical Society. Courant notes that the method suggests a wide generalisation
which provides great flexibility and seems to have great practical value. Practical
application did not appear until aeronautical engineers developed the method,
apparently without knowing of Courant’s work.
33
The name finite element was coined by Clough in 1960. Many new elements
for stress analysis were soon developed, largely by intuition and physical argument.
In 1963, FEA acquired respectability in academia when it was recognized as a form
of the Rayleigh-Ritz method, a classical approximation technique. Thus FEA was
seen not just as a special trick for stress analysis but as a widely applicable method
having a sound mathematical basis.
Papers about heat conduction and seepage flow using FEA appeared in 1965.
General-purpose computer program for FEA emerged in the 1960s and early 1970s.
Since the late 1970s, computer graphics of increasing power have been attached to
FE software, making FEA attractive enough to be used in actual design. Previously,
FEA was so tedious that it was used mainly to verify a design already completed or
to study a structure that had failed.
Computational demands of practical FEA are so extensive that computer
implementation is mandatory. Analyses that involve more than 100,000 degrees of
freedom are not uncommon. It is no accident that developments in FEA. The first
textbook about FEA appeared in1967.
By 1995, Mackerle estimated that about 3800 papers about FEA were being
published annually, and that the cumulative total of FEA publication amounted to
some 380 books, 400 conference proceedings, and 56,000 papers (excluding papers
on fluid mechanics). Mackerle also counted 310 general-purpose FE computer
programs.
34
2.5.5
The Basic Concept of the Finite Element Method
The most distinctive feature of the finite element method that separates form
the others is the division of a given domain into a set of simple sub domains, called
finite elements. Any geometric shape that allows computation of the solution or its
approximation, or provides necessary relations among the values of the solution at
selected points, called nodes, of the sub domain, qualifies as a finite element.
Other features of the method include seeking continuous, often polynomial,
approximations of the solution over each element in terms of nodal values, and
assembly of element equations by imposing the inter-element continuity of the
solution and balance of inter-element forces.
2.5.6
How does Finite Element Analysis Work
Finite element analysis (FEA) uses a complex system of points called nodes,
which make a grid called a mesh. This mesh is programmed to contain the material
and structural properties, which define how the structure will react to certain loading
conditions. Nodes are assigned at a certain density throughout the material
depending on the anticipated stress levels of a particular area.
Regions, which will receive large amounts of stress usually, have a higher
node density than those, which experience little or no stress. Points of interest may
consist of fracture point of previously tested material, fillets, corners, complex detail,
and high stress areas.
35
The mesh acts like a spider web in that from each node, there extends a mesh
element to each of the adjacent nodes. This web of vectors is what carries the
material properties to the object, creating many elements.
A wide range of objective functions (variables within the system) is available
for minimisation or maximization: 
Mass, volume, temperature

Strain energy, stress strain

Force, displacement, velocity, acceleration

Synthetic (User defined)
There are multiple loading conditions, which may be applied to a system. Each
FEA program may come with an element library, or one is constructed over time.
Some sample elements are: 
Rod elements

Beam elements

Plate/Shell/Composite elements

Shear panel

Solid elements

Spring elements
36

Mass elements

Rigid elements

Viscous damping elements
Many FEA programs also are equipped with the capability to use multiple
materials within the structure such as: 
Isotropic, identical throughout

Orthotropic, identical at 90 degrees

General anisotropic, different throughout
2.5.7
Use of General Purpose of Finite Element Analysis
The use of general purpose of FEA software has been generally summarized and
it involves the following steps: 
Preprocessing
-
Input data describes geometry, material properties, loads, and
boundary conditions Software can automatically prepare much of the
FE mesh, but must be given direction as to the type of element and the
mesh density desired. The analyst must choose one or more element
formulations that suit the mathematical model, and state how large or
how small element should be in selected portions of the FE model.
All data should be reviewed for correctness before proceeding.
37

Numerical analysis
-
Software automatically generates matrices that describe the behavior
of each element, combines these matrices into a large matrix equation
that represents the FE structure. Solves this equation to determine
values of field quantities at nodes. Substantial additional calculations
are performed if behavior is nonlinear or time dependent.

Post processing
-
The FEA solution and quantities derived from it are listed or
graphically displayed. This step is also automatic, except that the
analyst must tell the software what lists or displays to prepare. In
stress analysis, typical displays include the deformed shape, with
deformations exaggerated and probably animated, and stresses of
various types on various planes.
2.5.8
Benefits from Using the Finite Element Method
The most obvious benefit is that the FEM can provide solutions to many
complicate problems that would be intractable by other techniques. This has been
most apparent in the fields of mechanical and structural engineering; in fact, it is
generally recognized that over the last quarter century the FEM has produced a
revolution in these fields. It now appears that similar revolutions are on the horizon
as the FEM continues to spread to other disciplines.
The FEM is a very modular technique. The element equation can be used
repeatedly, not only for all the elements in a particular mesh but also for all other
problems and in other programs. New types of elements (and hence new sets of
elements equations) can be added to programs as the need arises, gradually building
up an element library, that can be moved form program to program.
38
A user can select from element library to create a mesh of different element
types, much like a child using different blocks to build a structure. This has a
definite impact on human resources, because when a person develops an FE
computer program, it can be used to solve not just one specific problem but a whole
class of problems that differ substantially in geometry, boundary conditions, and
other properties. This is a capital investment for the future.
All FE applications are basically the same type of mathematical techniques;
most of which can be learned at the undergraduate college level and some of which
are introduced at high school level. Thus considerable proficiency can be gained by
the average user form a few years of formal schooling plus some practical
experience, and the repeated exposure to the same techniques during subsequent
years maintains a sharpness of skill that yields efficient use of ones time. In short, a
few techniques will solve a broad spectrum of problems.
2.6
Types of Engineering Analysis
Structural analysis consists of linear and non-linear models. Linear models
use simple parameters and assume that the material is not plastically deformed. Nonlinear models consist of stressing the material past its elastic capabilities. The
stresses in the material then vary with the amount of deformation.
Vibrational analysis is used to test a material against random vibrations,
shock, and impact. Each of these incidences may act on the natural vibrational
frequency of the material, which, in turn, may cause resonance and subsequent
failure.
39
Fatigue analysis helps designers to predict the life of a material or structure
by showing the effects of cyclic loading on the specimen. Such analysis can show
the areas where crack propagation is most likely to occur. Failure due to fatigue may
also show the damage tolerance of the material.
Heat Transfer analysis models the conductivity or thermal fluid dynamics of
the material or structure. This may consist of a steady state or transient transfer.
Steady-state transfer refers to constant thermo properties in the material that yield
linear heat diffusion.
2.7
Results of Finite Element Analysis
Finite element analysis has become a solution to the task of predicting failure
due to unknown stresses by showing problem areas in a material and allowing
designers to see all of the theoretical stresses within. This method of product design
and testing is far superior to the manufacturing costs, which would accrue if each
sample was actually built and tested.
2.8
Why Use FEAPpv
FEAPpv is a code designed for research and educational use and is able to
solve a wide variety of problem in linear and non-linear solid continuum mechanics.
The current FEAP system contains a general element library.
40
Elements are available to model one, two or three-dimensional problems in
linear and/or non-linear structural and solid mechanics and for linear heat conduction
problems. Each solid element accesses a material model library.
Material models are provided for elasticity, viscoelasticity, plasticity, and
heat transfer constitutive equations. Elements also provide capability to generate
mass and geometric stiffness matrices for structural problems and to compute output
quantities associated for each element (e.g., stress, strain), including capability of
projecting these quantities to nodes to permit graphical outputs of result contours.
Users also may add an element to the system by writing and linking a single
module to the FEAP system. Details on specific requirements to add an element as
well as other optional features available are included in the FEAP Programmers
Manual.
The system libraries that are available in FEAPpv are: 
finite element library

(thermo-mechanics, small and large deformation)

solution strategy library non-linear problems

with simple mesh generator

with graphic post-processor
41
Using FEAPpv it is possible to introduce: 
elements defined by the user for the solution of specific problems modeled by
partial differential equations, PDE (solid mechanics, fluid mechanics, thermomechanics etc.)

new solution strategies

new mesh generators
2.9
Founder of FEAP/FEAPpv
2.9.1
Biographical Sketch of Professor Robert L. Taylor
Robert L. Taylor received his graduate training in the Department of Civil
Engineering at the University of California, Berkeley (MS 1958 and PhD 1963). The
areas of research for Professor Robert L. Taylor are mechanics of solids,
computational mechanics, finite element methods, finite element software.
2.9.2
Biographical Sketch of Emeritus Professor O.C. Zienkiewicz
Dr. O. C. Zienkiewicz, CBE, FRS, FEng. is Professor Emeritus and Director
of the Institute for Numerical Methods in Engineering at the University of Wales, in
Swansea. He was born in Caterham, England in 1921. His contributions range over
42
the wide field of mechanics and engineering. Since his first paper in 1947 dealing
with numerical approximation to the stress analysis of dams (using relaxation and
finite differences) he has published nearly 600 papers and written or edited 25 books.
He is one of the early pioneers of the finite element method and the first to
realise its general potential for solution of problems outside the area of solid
mechanics. His books on the Finite Element Method were the first to present the
subject and remained unchallenged till 1972 when many others entered the area. The
"The Finite Element Method IV, McGraw Hill 1989 - 1991 (2 Vols, pp 1400) and the
current Finite Element Method V, Butterworth and Heinemann 2000 (3 Vols, pp
1482) were both written together with Professor R. L. Taylor of Berkeley, California
and remain today the standard reference text.
2.10
Difference of FEAP and FEAPpv
Generally, FEAPpv is not extended or modified on a regular basis.
Corrections to errors found by users will be incorporated into the program. On the
other hand, FEAP is continuously developed and extended on a regular basis. New
users are encouraged to try the free version, FEAPpv, to learn about the structure and
general format of use. Once this is known, the migration to FEAP is straight forward
since both programs share the same input format (which can consist of constants,
variables, paramaters, and expressions as described in the user manuals).
FEAP and FEAPpv have the same basic input style; however, the number
of input commands is different. In particular, FEAPpv does not have the ability to
solve problems which contain rigid body parts in the finite deformation solution
option. The node and element options are also not available in FEAPpv. Finally,
only 5 user mesh functions are provided in FEAPpv whereas 10 exist in FEAP.
43
The loop feature in mesh input for FEAP permits generation of problems with
sub-structure forms or repetitive patterns. For example see the wheel problem in the
FEAP User Manual. The user manuipulation options available in each program are
similar except for the following which are included in FEAP but not in FEAPpv:

Master/Slave options in small deformation which permit, for example, rigid
floor diaphragms or beams in in-plane or axial deformations.

Multi-body dynamics with joint connections in large displacement analysis.
This feature currently works only with the energy-momentum conserving
elements and time integration methods.

Profile optimization for use with variable-band linear equation solvers.

Real and complex arithmetic solution options (N.B. No elements are included
with complex module, etc.).

Two and three-dimensional contact interaction capability.

Partitioned solution option.

Setting of order of ODE for each degree of freedom (e.g., in coupled transient
thermo-mechanical solutions heat equation is a first order ODE and the
mechanical part is a second order ODE or static.)
44
2.11
Commencing FEAPpv
2.11.1 Problem Definition of FEAPpv
To perform an analysis using the finite element method the first step is to
subdivide the region of interest into elements and nodes. In this process the analyst
must make a choice on:

the type of elements to use

where to place nodes

how to apply the loading and boundary restraints

the appropriate material model and values of its parameters in each element

any other aspects relating to the particular problem
The specification of the node and element data defines what we will
subsequently refer to as the finite element mesh or, for short, the mesh of the
problem. In order to complete a problem specification it is necessary also to specify
additional data, e.g., boundary conditions, loads, etc.
Once the analyst has defined a model of the problem to be solved it is
necessary to define the nodal and element data in a form which may be interpreted by
FEAPpv. The steps to define a mesh for FEAPpv would be further explained.
45
The second phase of a FEA is to specify the solution algorithm for the
problem. This may range from a simple linear static (steady state) analysis for one
loading condition to a more complicated transient non-linear analysis subjected to a
variety of loading conditions. FEAPpv permits the user to specify the solution
algorithm utilizing a solution command language. The execution of FEAPpv is
initiated by using the command FEAPpv.
2.11.2 Input Records Specification
Data input specifications in FEAPpv consist of records which may contain
from 1 to 255 characters of information in free format form. Each record can contain
up to 16 alphanumeric data items.
The maximum field width for any single data item is 15 characters (14
characters of data and 1 character for separating fields). Specific types of data items
are discussed below. Sets of records, called data sets, start with a text command
which controls input of one or more data items.
Data sets may be grouped into a single file (called the input data file) or may
be separated into several files and joined together using the include command
described below. Sets of records may also be designated as a save set and later read
again for reuse.
Each input record may be in the form of text and/or numerical constants,
parameters, or expressions. Text fields all start with the letters a through z (either
upper or lower case may be used; however, internally FEAPpv converts upper case
letters to lower case).
46
The remaining characters may be either letters or numbers. Constants are
conventional forms for specifying input data and may be integer or real quantities as
needed. Parameters consist of one or two characters to which values are assigned.
The first character of a parameter must be a letter (a to z); the second may be
a letter (a to z) or numeral (0 to 9). Expressions are combinations of constants,
parameters, and/or functions which can be evaluated as the required data input item.
2.11.3 Start of Problem and Control Infomation
The first part of an input data file contains the control data which consists of two
records: 
A start/title record which must have as the first four non-blank characters
FEAPpv (either upper or lower case letters may be used with the remainder
used as a problem title).

The second record contains problem size information consisting of:
o
NUMNP - Number of nodal points
o
NUMEL - Number of elements
o
NUMMAT - Number of material property sets
o
NDM - Space dimension of mesh
o
NDF - Maximum number of unknowns per node
o
NEN - Maximum number of nodes per element.
47
For standard input options FEAPpv can automatically determine the number
of nodes, elements, and material sets. Thus, on the control record the values of
NUMNP, NUMEL, and NUMMAT may be omitted (i.e., specified as zero). When
using automatic numbering it is generally advisable to use mesh input options, which
avoid direct specification of a node or element number.
Specification of nodal loads (forces), nodal displacements (displacements),
and boundary condition restraint codes have options which begin with E and C for
edge and coordinate related options, respectively. It is recommended these be used
whenever possible. After this, the material type and properties are defined.
2.11.4 The Coordinate Command
The coordinates of nodes may be specified using the COORdinate command.
For example, the commands to generate polar coordinates for an eleven node mesh
of a circular beam with radius 5 are given by: -
COORdinates
1 1 5.0 90.0
11 0 5.0 0.0
! Termination record
After the COORdinate command individual records defining each nodal point
and its coordinates are specified as: -
N, NG, X_N, Y_N, Z_N
48
Where;
N
Number of nodal point.
NG
Generation increment to next node.
X-N
value of x1 coordinate.
Y-N
value of x2 coordinate.
Z-N
value of x3 coordinate.
It is only necessary to specify the components corresponding to the spatial
dimension of the mesh. Thus for 2-dimensional meshes only X-N and Y-N need be
given.
2.11.5 The Element Command
The ELEMent command may be used to input the list of nodes connected to
an individual element. For elements where the maximum number of nodes is less or
equal to 13 (i.e. the NEN parameter on the control record), the records following the
command are given as: -
N, NG, MA, (ND_i, i=1,NEN)
Where;
N
Number of element.
NG
Generation increment for node numbers.
MA
Material identifier associated with element.
ND-i i-Node number defining element.
49
The second option available to specify the nodal quantities is based on
coordinates and is used to apply a common value to all nodes located at some
constant coordinate location called the edge value. The options EBOUndary,
EFORce, EDISplacement, EANGle are used for this purpose.
For example, if it is required to impose a zero displacement for the first
degree of freedom of all nodes located at y = 0:5. The edge boundary conditions
may set using;
EBOUndary
2 0.5 1 0
! Termination record
In the above the 2 indicates the second coordinate direction (i.e., x2 or y for
Cartesian coordinates) and 0:5 is the value of the x2 or y coordinate to be used to find
the nodes. The last two fields are the boundary condition pattern to apply to all the
nodes located.
That is, above we are indicating the first degree-of-freedom is to have
specified displacements and the second is to have specified forces. FEAPpv locates
all nodes which are within a small tolerance of the specified coordinate after the
mesh input is completed.
50
2.11.6 EBoundary
The boundary restraint conditions may be set along any set of nodes which
has a constant value of the i-coordinate direction (e.g., 1-direction (or x), 2-direction
(or y), etc.). The data to be supplied during the definition of the mesh consists of: 
i-coor { Direction of coordinate (i.e., 1 = x, 2 = y, etc.)

xi-value { Value of i-direction coordinate to be used during search (a
tolerance of 1/1000 of mesh size is used during search, any coordinate within
the gap is assumed to have the specified value).

ibc(1) { Restraint conditions for all nodes with value of

ibc(2) search.(0 = boundary code remains as previously set ... > 0 denotes a
fixed dof, < 0 resets previously

ibc(ndf) defined boundary codes to 0.)
The EBOU command may be used with two options. Using the EBOU, SET
option replaces previously defined conditions at any node by the pattern specified.
Using the EBOU, ADD option accumulates the specified boundary conditions with
previously defined restraints.
The default mode is ADD. Boundary restraint conditions may also be specified
using the BOUN and CBOU commands. The data is order dependent with data
defined by DISP processed first, EDIS processed second and the CDIS data
processed last. The value defined last is used for any analysis.
Example: EBOUndary
51
All the nodes located on the x3 = z = 0 plane are to have restraints on the 3rd
and 6th degrees of freedom. This may be specified using the command set: -
EBOUndaray
3 0.0 0 0 1 0 0 1
Where non-zero values indicate a restrained degree of freedom and a zero an
unrestrained degree of freedom. Non-zero displacements may be specified for
restrained dof's and non-zero forces for unrestrained dof's.
2.11.7 The Element Library
FEAPpv contains a library of standard elements and material models which
may be employed to solve a wide range of problems in solid and structural
mechanics, and heat transfer analysis. In addition, users may program and add new
elements to the program. The type of element to be employed in an analysis is
specified as part of the MATErial data sets. The first record of each material data set
also contains the material property number.
Each material property number is an integer ranging from one (1) to the
maximum number of material sets specified on the control data record (which
immediately follows the FEAPpv start/title record); however, as noted earlier, the
maximum number on material sets on the control data may be specified as zero and
FEAPpv will automatically compute the maximum number of material sets from the
input data.
52
The second record of the material set data defines the type of element to be
used. For this study the type of element used is: 
SOLId - A solid element is used to solve continuum problems with either
small or large deformations. Options exist to use finite elements based on
displacement or mixed formulations. Small deformation elements contain a
library of models for elastic, viscoelastic, or elastoplastic constitutive
equations. Finite deformation elements contain only elastic material models.
For two dimensional problems each element is a quadrilateral with between 4
and 9-nodes. The two dimensional displacement formulations also permit use
of 3 or 6-node triangular elements. The degrees of freedom on each node are
displacements, in the coordinate directions. The degrees of freedom are
ordered as: 2-D Plane problems, ux; uy, coordinates are x; y; 2-D
Axisymmetric problems, ur; uz, coordinates are r; z.
2.11.8 Material Models – Isotropic Linear Elastic Models
The isotropic models require less data since now only two independent elastic
parameters are needed to define ^C. These are taken as Young's modulus, E, and
Poisson's ratio,  and for an isotropic material the elastic compliance array is: -
53
2.11.9 Force
The FORCe command is used to specify the values for nodal boundary
forces. For each node to be specified a record is entered with the following
information: 
node { Number of node to be specified

ngen { Increment to next node, if generation

is used, otherwise 0.

f (1,node) { Value of force for 1-dof

f (2,node) { Value of force for 2-dof etc., to ndf directions
When generation is performed, the node number sequence for node1-node2
sequence shown at top will be: -
node1, node1+ngen1, node1+2_ngen1, .... , node2
The values for each force will be a linear interpolation between the node1 and
node2 values for each degree-of-freedom.
While it is possible to specify both the force and the displacement applied to
a node, only one can be active during a solution step. The determination of the
active value is determined from the boundary restraint condition value.
54
If the boundary restraint value is zero and you use one of the force-commands
a force value is imposed, whereas, if the boundary restraint value is non-zero and you
use one of the displacement-commands a displacement value is imposed. (See
BOUNdary, CBOUndary, or EBOUndary pages for setting boundary conditions).
It is possible to change the type of boundary restraint during execution by
resetting the boundary restraint value. Force conditions may also be specified using
the EFORce and CFORce commands. The data is order dependent with data defined
by FORCe processed first, EFORce processed second and the CFORce data
processed last. The value defined last is used for any analysis.
2.11.10 Problem Solving
Each problem is solved by using a set of the command language statements
which together form the algorithm defining the particular solution method employed.
The commands to solve a linear static problem are: 
BATCh !initiate batch execution

TANG !form tangent matrix

FORM !form residual

SOLVe !solve equations

DISPlacement,ALL !output all displacements

STREss,ALL !output all element stresses
55

REACtion,ALL !output all nodal reactions.

END !end of batch program
The command sequence TANG, FORM and SOLVe is the basic solution step in
FEAPpv and for simplicity (and efficiency) may be replaced by the single command
TANG,, 1.
This single statement is more efficient in numerical operations since it involves
only a single process to compute all the finite element tangent and residual arrays,
whereas the three statement form requires one for TANG and a second for FORM.
Thus, BATCh !initiate batch execution, TANG,,1 !form and solve,
DISPlacement,ALL !output all displacements, STREss,ALL !output all element
stresses, REACtion,ALL !output all nodal reactions, END !end of batch program is
the preferred solution form. Some problems have tangent matrices which are
unsymmetric.
For these situations the TANGent command should be replaced by the
UTANgent command. The commands PRINt and NOPRint may be used to control
or prevent information appearing on the screen – information always goes to the
output file. Printing to the screen is the default mode. Additional commands may be
added to the program given above.
For example, inserting the following command after the solution step (i.e., the
TANG,,1 command) will produce a screen plot of the mesh:
PLOT, MESH! plot mesh
56
2.11.11 End
The last mesh command must be END. This terminates the mesh input and
returns to the control program, which may then perform additional tasks on the data
or STOP execution. Immediately following the END mesh command any additional
data required to manipulate the mesh (e.g., TIE, LINK, ELINk, PARTition ORDEr,
RIGId and JOINt should be given prior to initiation of a problem solution using
BATCh and/or INTEractive.