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**Note**Institute:
**
JNTUK KAKINADA
**Course:
**
B.Tech
**Specialization:
**Civil Engineering**Offline Downloads:
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1
Lecture 1: Introduction
electromagnetic and structural working environments. FEM allows entire designs to be constructed,
refined and optimized before the design is manufactured. This powerful design tool has significantly
improved both the standard of engineering designs and the methodology of the design process in
many industrial applications. The use of FEM has significantly decreased the time to take products
from concept to the production line. One must take the advantage of the advent of faster generation
of personal computers for the analysis and design of engineering product with precision level of
accuracy.
1.1.2 Background of Finite Element Analysis
The finite element analysis can be traced back to the work by Alexander Hrennikoff (1941) and
Richard Courant (1942). Hrenikoff introduced the framework method, in which a plane elastic
medium was represented as collections of bars and beams. These pioneers share one essential
characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually
called elements.
•
In 1950s, solution of large number of simultaneous equations became possible because of the
digital computer.
•
In 1960, Ray W. Clough first published a paper using term “Finite Element Method”.
•
In 1965, First conference on “finite elements” was held.
•
In 1967, the first book on the “Finite Element Method” was published by Zienkiewicz and
Chung.
•
In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering
problems.
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1.1.1 Introduction
The Finite Element Method (FEM) is a numerical technique to find approximate solutions of partial
differential equations. It was originated from the need of solving complex elasticity and structural
analysis problems in Civil, Mechanical and Aerospace engineering. In a structural simulation, FEM
helps in producing stiffness and strength visualizations. It also helps to minimize material weight
and its cost of the structures. FEM allows for detailed visualization and indicates the distribution of
stresses and strains inside the body of a structure. Many of FE software are powerful yet complex
tool meant for professional engineers with the training and education necessary to properly interpret
the results.
Several modern FEM packages include specific components such as fluid, thermal,

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•
•
•
In the 1970s, most commercial FEM software packages (ABAQUS, NASTRAN, ANSYS,
etc.) originated. Interactive FE programs on supercomputer lead to rapid growth of CAD
systems.
In the 1980s, algorithm on electromagnetic applications, fluid flow and thermal analysis were
developed with the use of FE program.
Engineers can evaluate ways to control the vibrations and extend the use of flexible,
deployable structures in space using FE and other methods in the 1990s. Trends to solve fully
coupled solution of fluid flows with structural interactions, bio-mechanics related problems
with a higher level of accuracy were observed in this decade.
1.1.3 Numerical Methods
The formulation for structural analysis is generally based on the three fundamental relations:
equilibrium, constitutive and compatibility. There are two major approaches to the analysis:
Analytical and Numerical. Analytical approach which leads to closed-form solutions is effective in
case of simple geometry, boundary conditions, loadings and material properties. However, in reality,
such simple cases may not arise. As a result, various numerical methods are evolved for solving such
problems which are complex in nature. For numerical approach, the solutions will be approximate
when any of these relations are only approximately satisfied. The numerical method depends heavily
on the processing power of computers and is more applicable to structures of arbitrary size and
complexity. It is common practice to use approximate solutions of differential equations as the basis
for structural analysis. This is usually done using numerical approximation techniques. Few
numerical methods which are commonly used to solve solid and fluid mechanics problems are given
below.
•
Finite Difference Method
•
Finite Volume Method
•
Finite Element Method
•
Boundary Element Method
•
Meshless Method
The application of finite difference method for engineering problems involves replacing the
governing differential equations and the boundary condition by suitable algebraic equations. For
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With the development of finite element method, together with tremendous increases in computing
power and convenience, today it is possible to understand structural behavior with levels of
accuracy. This was in fact the beyond of imagination before the computer age.

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example in the analysis of beam bending problem the differential equation is reduced to be solution
of algebraic equations written at every nodal point within the beam member. For example, the beam
equation can be expressed as:
d 4w
q
(1.1.1)
=
4
EI
dx
To explain the concept of finite difference method let us consider a displacement function variable
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namely w = f ( x)
Fig. 1.1.1 Displacement Function
Now, w f (x x) f (x)
dwΔw
= Lt
dxΔx Δx0
Thus,
So,
f(x + Δx) - f(x) 1
= Lt
ΔxΔx
0
h
= wi+1 - wi
1
d 2w d 1
1
= wi+1 - wi = 2 wi+2 - wi+1 - wi+1 + wi = 2 wi+2 - 2 wi+1 + wi
2
dx
dx h
h
h
d 3w 1
= wi+3 - wi+2 - 2wi+2 + 2wi+1 + wi+1 - wi
dx 3 h 3
1
= 3 wi+3 - 3wi+2 + 3wi+1 - wi
h
(1.1.2)
(1.1.3)
(1.1.4)

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d 4w 1
= wi+4 - wi+3 - 3wi+3 + 3wi+2 + 3wi+2 - 3wi+1 - wi+1 + wi
dx 4 h 4
1
= 4 wi+4 - 4wi+3 +6wi+2 - 4wi+1 + wi
(1.1.5)
h
1
= 4 wi+2 - 4wi+1 +6wi - 4wi-1 + wi-2
h
Thus, eq. (1.1.1) can be expressed with the help of eq. (1.1.5) and can be written in finite difference
form as:
( wi − 2 − 4 wi −1 + 6 wi − 4 wi +1 + wi + 2 ) =
q 4
h
EI
(1.1.6)
Thus, the displacement at node i of the beam member corresponds to uniformly distributed load can
be obtained from eq. (1.1.6) with the help of boundary conditions. It may be interesting to note that,
the concept of node is used in the finite difference method. Basically, this method has an array of
grid points and is a point wise approximation, whereas, finite element method has an array of small
interconnecting sub-regions and is a piece wise approximation.
Each method has noteworthy advantages as well as limitations. However it is possible to
solve various problems by finite element method, even with highly complex geometry and loading
conditions, with the restriction that there is always some numerical errors. Therefore, effective and
reliable use of this method requires a solid understanding of its limitations.
1.1.4 Concepts of Elements and Nodes
Any continuum/domain can be divided into a number of pieces with very small dimensions. These
small pieces of finite dimension are called ‘Finite Elements’ (Fig. 1.1.3). A field quantity in each
element is allowed to have a simple spatial variation which can be described by polynomial terms.
Thus the original domain is considered as an assemblage of number of such small elements. These
elements are connected through number of joints which are called ‘Nodes’. While discretizing the
structural system, it is assumed that the elements are attached to the adjacent elements only at the
nodal points. Each element contains the material and geometrical properties. The material properties
inside an element are assumed to be constant. The elements may be 1D elements, 2D elements or 3D
elements. The physical object can be modeled by choosing appropriate element such as frame
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Fig. 1.1.2 Finite difference equation at node i

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