Saturday, October 2, 2010

GENERAL DESCRIPTION OF FINITE ELEMENT METHOD

In the finite element method ,the actual continuum or body of matter like solid ,liquid or gas is represented as an assemblage of sub-divisions called finite elements.These elements are considered to be interconnected at specified joints which are called ‘nodal points’ or nodes .The nodes usually lie on the elements boundaries where adjacent elements are considered to be connected .Since the actual variation of the field variable (displacements,stress,temperature,pressure,velocity) inside the continuum is not known ,we assume that ,the variation of the field variable inside a finite element can be approximated by a simple function .These approximating functions (are called interpolating models) are defined in terms of field variables at the nodes .When field equations for the whole continuum are written ,the new unknowns will be the nodal values of the field variable.
By solving the field equations, which are generally in the form of matrix equation, the nodal values of the field variables will be known. Once these are known, the approximating functions define the field variables throughout the assemblage of elements.
The solution of a general continuum problem by the FEM always follows an orderly step by step process as follows: -
STEP I: DESCRITISATION OF THE STRUCTURE
The continuum is separated by imaginary lines of surfaces into a number of finite elements .The number, type, size and the arrangements of the elements have to be decided.
STEP 2: SELECTION OF A PROPER INTERPOLATION OR DISPLACEMENT MODEL
Since the displacement solution of a complex structure under any specified load conditions cannot be predicted exactly, we assume some suitable solution within element to approximate the unknown solution.( The assumed solution must be simple from computational point of view.)
Step III : DERIVATION OF ELEMENT STIFFNESS MATRICES AND LOAD VECTORS
From the assumed displacement model, the stiffness matrix [ ke] and the load vector [ pe ], of element ‘e’ are to be derived by using either equilibrium conditions or a suitable variational principle.
Step IV : ASSEMBLAGE OF ELEMENT EQUATIONS TO OBTAIN THE OVERALL EQUILIBRIUM EQUATIONS
Since the structure is composed of several finite elements , the individual element stiffness matrices and load vectors are to be assembled in suitable manner and overall equilibrium equations have to be formulated as ,
[ K ] = p
where ,
[k] = assembled stiffness matrix ,
 = Vector of nodal displacements ,
T = Vector of nodal forces for the complete structure.
Step V : SOLUTION FOR THE UNKNOWN NODAL DISPLACEMENTS
The overall equilibrium equations have to be modified to account for the boundary conditions of the problem. After the incorporation of the boundary condition , the equilibrium equation can be expressed as :
[ k ]  = P
Step VI : COMPUTATION OF ELEMENT STRAINS AND STRESSES
From the known nodal displacements  , element strains and stresses can be computed using equation of solid or structural mechanics.
The application of above 6 steps can be illustrated with the help of stepper bar problem.
In an equilibrium problem,we need to find the steady state displacement or stressed distribution if it is a solid mechanics problem,temperature or heat flux distribution if it is heat transfer problem and pressure or velocity distribution if it is fluid mechanics problem.

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