introduction
The finite element analysis is a numerical technique. In this method all the complexities
of the problems, like varying shape, boundary conditions and loads are maintained as
they are but the solutions obtained are approximate
The fast improvements in computer hardware technology and slashing of cost of
computers have boosted this method, since the computer is the basic need for the
application of this method
Some of the popular packages are STAAD-PRO, GT-STRUDEL, NASTRAN, NISA and
ANSYS. Using these packages one can analyse several complex structures.
The finite element analysis originated as a method of stress analysis in the design of
aircrafts. It started as an extension of matrix method of structural analysis. Today this
method is used not only for the analysis in solid mechanics, but even in the analysis of
fluid flow, heat transfer, electric and magnetic fields and many others.
Civil engineers use this method extensively for the analysis of beams, space frames,
plates, shells, folded plates, foundations, rock mechanics problems and seepage
analysis of fluid through porous media. Both static and dynamic problems can be
handled by finite element analysis.
various steps involved in the finite element
analysis are:
(i)Select suitable field variables and the elements.
(ii) Discritise the continua.
(iii) Select interpolation functions.
(iv) Find the element properties.
(v) Assemble element properties to get global properties.
(vi) Impose the boundary conditions.
(vii) Solve the system equations to get the nodal unknowns.
(viii) Make the additional calculations to get the required
values.
• Advantages of FEA
• 1. The physical properties, which are intractable and complex for
any closed bound solution, can be analyzed by this method.
• 2. It can take care of any geometry (may be regular or irregular).
• 3. It can take care of any boundary conditions.
• 4. Material anisotropy and non-homogeneity can be catered
without much difficulty.
• 5. It can take care of any type of loading conditions.
• 6. This method is superior to other approximate methods like
Galerkine and Rayleigh-Ritz methods.
• 7. In this method approximations are confined to small sub
domains.
• 8. In this method, the admissible functions are valid over the simple
domain and have nothing to do with boundary, however simple or
complex it may be.
• 9. Enable to computer programming.
• Disadvantages of FEA
• 1. Computational time involved in the solution
of the problem is high.
• 2. For fluid dynamics problems some other
methods of analysis may prove efficient than
the FEM.
• Limitations of FEA
• 1. Proper engineering judgment is to be
exercised to interpret results.
• 2. It requires large computer memory and
computational time to obtain intend results.
• 3. There are certain categories of problems
where other methods are more effective, e.g.,
fluid problems having boundaries at infinity
are better treated by the boundary element
method.
• Two dimensional elements
We need two dimensional elements to solve two dimensional problems.
Common two dimensional problems in stress analysis are plane stress,
plane strain and plate problems. Two dimensional elements often used is
three noded triangular element shown in It has the distinction of being
the first and most used element. These elements are known as Constant
Strain Triangles (CST) or Linear Displacement Triangles.
• Constant strain triangle(CST)
CST is the simplest 1st order triangular element used in the
finite element analysis which used to provide an approximate
solution in 2d domain it has 3node per element two degree of
freedom per node in all 6 degree of freedom per element.
We need two dimensional elements to solve two dimensional
problems. Common two dimensional problems in stress
analysis are plane stress, plane strain and plate problems. Two
dimensional elements often used is three noded triangular
element shown in It has the distinction of being the first and
most used element. These elements are known as Constant
Strain Triangles (CST) or Linear Displacement Triangles.
• Linear strain triangle
A six noded triangular element is known as linear strain
triangle (LST). It has six nodes three at vertices and three at
edges.
it has 12 degree of freedom
the displacement function of triangle is quadratic
LST is higher order triangular element Lst is also called as
quadratic displacement triangle.
Ten noded triangular elements are known as Quadratic Strain
Triangles (QST) or Cubic Displacement Triangles.
• Rectangular element
•
A simple but less used two dimensional element is the four noded
rectangular element whose sides are parallel to the global coordinate
systems. This systems is easy to construct automatically but it is not well
suited to approximate inclined boundaries.
Rectangular elements of higher order also can be used. Figure 4.6 shows a
family of Lagrange rectangle in which nodes are in the form of grid points.
Figure 4.7 shows the family of Serendipity rectangles which are having
nodes only along the external boundaries.
Coordinate Systems
The following terms are commonly referred in FEM
(i) Global coordinates
(ii) Local coordinates and
(iii) Natural coordinates.
However there is another term ‘generalized
coordinates’ used for defining a polynomial form
of interpolation function. This has nothing to do
with the ‘coordinates’ term used here to define
the location of points in the element.
Natural coordinate system
A natural coordinate system is a coordinate system which permits the
specification of a point within the element by a set of
dimensionless numbers, whose magnitude never exceeds unity. It is
obtained by assigning weightages to the nodal coordinates in
defining the coordinate of any point inside the element. Hence such
system has the property that ith coordinate has unit value at node i
of the element and zero value at all other nodes. The use of natural
coordinate system is advantages in assembling element properties
(stiffness matrices), since closed form integrations formulae are
available when the expressions are in natural coordinate systems.
• Natural coordinate in two dimension