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Finite Element Method Introduction
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Truss Examples
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Beam Examples
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U Shaped Beam
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Centrifuge 400 mm dia 3000 revolutions per
minute
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• The problems shown before can not be solved
  using by conventional method of Mechanics of
  solids Or Mechanics of Materials
• Numerical methods are used to these types of
  problems.

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• Mechanics of Materials approach
• Using experiment, experience and intuition we
  decide how the structure deforms.
• Deformation field yields strain field
• With an elastic law the strain field yields to
  stress field.
• Finally equations of statics relate stresses to
  applied load

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• Theory of elasticity or continuum mechanics
  approach
• In this method we must satisfy simultaneously
• governing differential equations of equilibrium
  at every point
• Continuity of displacement field and
• loading and supporting conditions

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The finite element method is one of the most
important developments in numerical analysis. It
is versatile and powerful for solving linear as
well as nonlinear complex problems. The method
was developed in the 1950's by engineers as an
outgrowth of the so-called matrix method (Argyris
1955, 1966, Argyris and Kelsey 1960) for
systematically analyzing complex structures
containing a large number of components. Over the
years, the finite element method has spread to
applications in all fields of engineering,
science and medicine.
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The finite element method establishes approximate
solutions in terms of unknown parameters in
sub-regions called "elements" and then deduces an
approximate solution for the whole domain by
enforcing certain relations among the solutions
of all elements.
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For structural analysis the procedure is to
express the relations between the displacements
and internal forces at the selected nodal points
of individual structural components in the form
of a system of algebraic equations. The unknowns
for the equations are nodal displacements, nodal
internal forces, or both. The system of equations
is written most conveniently in matrix notation,
and the solution of these equations is obtained
efficiently by high-speed computation.
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The foundations for many finite element methods
have been established. Formulations of methods
with variational principles or principles of
virtual work or minimization of total potential
were developed.
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The accuracy of the approximate solution depends
on the smoothness of the field variable. When a
physical problem is analyzed using the finite
element method, it involves approximation in the
geometry as well as local solutions. The errors
can be estimated from the approximate solutions.
If the errors are large, then the finite element
model is refined by improving the geometrical
approximation and/or the representation of the
local solutions. The new model is re-analyzed
until the estimated errors fall below the
specified limits.
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Numerous finite-element computer programs are
commercially available for specific or general
applications to name a few, DYNA3D, ANSYS,
ABACUS, ADINA, STRUDL and NASTRAN.
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The finite element method has been extensively
applied to complicated problems. The description
of the problem itself can be very involved and
prone to errors, A new field has evolved
specifically to ease the burden of preparing the
complex input information This includes
techniques for automatic mesh generation,
graphics for input PRE-PROCESSORS
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• The success of the finite element method lies
  largely in the development of efficient pre- and
  post-processors, and algorithms for solving large
  systems of equations.
• The pre-processor enables users to describe
  efficiently and in a relatively error free manner
  a complex problem in terms of its geometry,
  configuration, material properties, loading
  conditions, etc., through the input.
• The post-processor handles the voluminous output
  making it easily understandable through
  interactive graphic, tables, charts, and
  summaries.
• The pre- and post-processors also enable users to
  examine at ease any specific location of the
  domain or aspect of the solution prior to,
  during, or after the analysis.

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BASIC APPROACH Turner et al. (1956) in their
celebrated paper first applied the displacement
method to plane stress problems. They divided a
structure into triangular and rectangular
sub-domains called "elements" and designated the
vertices of the elements as nodes. The behavior
of each element was represented by the nodal
displacements and an element-stiffness matrix
that related the forces at the nodal points of
the element to the nodal displacements.
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They assembled the element-stiffness matrices to
form a system of algebraic equations and solved
them using high-speed computers. There are two
key aspects in the finite element method. One is
the determination of the element stiffness
matrices. The other is the assembly of
element-stiffness matrices to form a system of
algebraic equations, called the global system,
for the selected unknowns.
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The former involves the local approximation for
establishing the relations between element nodal
forces and nodal displacements. The latter
enforces the appropriate relations between the
nodal quantities of adjacent elements at the
inter-element boundaries.
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The finite element method approximates the field
variables (displacements, temperature etc) within
an element by inter-poIating their values at the
nodes by shape or interpolation functions. The
interpolated functions approximate the behaviors
of the element through an integral formulation.
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The nodal values are parameters, called
degrees-of freedom or generalized coordinates, to
be determined. These two names are used
interchangeably. One may also include the nodal
values of the spatial derivatives of the field
variables for this purpose, especially when
dealing with higher order differential equations.
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The local solution of a field variable must
satisfy certain continuity requirements within
the elements and along the common boundaries of
adjacent elements. This is to assure that the
finite-element solution will converge to the
exact solution as the size of elements reduces to
zero as the number of degrees-of-freedom per
element increases to infinity.
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