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Abstract:
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Many finite elements used in structural analysis possess deficiencies like shear
locking, incompressibility locking, poor stress predictions within the element domain,
violent stress oscillation, poor convergence etc. An approach that can probably
overcome many of these problems would be to consider elements in which the
assumed displacement functions satisfy the equations of stress field equilibrium. In
this method, the finite element will not only have nodal equilibrium of forces, but also
have inner stress field equilibrium. The displacement interpolation functions inside
each individual element are truncated polynomial solutions of differential equations.
Such elements are likely to give better solutions than the existing elements.In this thesis, a new family of finite elements in which the assumed displacement
function satisfies the differential equations of stress field equilibrium is proposed. A
general procedure for constructing the displacement functions and use of these
functions in the generation of elemental stiffness matrices has been developed. The
approach to develop field equilibrium elements is quite general and various elements
to analyse different types of structures can be formulated from corresponding stress
field equilibrium equations. Using this procedure, a nine node quadrilateral element
SFCNQ for plane stress analysis, a sixteen node solid element SFCSS for three
dimensional stress analysis and a four node quadrilateral element SFCFP for plate
bending problems have been formulated.For implementing these elements, computer programs based on modular concepts
have been developed. Numerical investigations on the performance of these elements
have been carried out through standard test problems for validation purpose.
Comparisons involving theoretical closed form solutions as well as results obtained
with existing finite elements have also been made. It is found that the new elements
perform well in all the situations considered. Solutions in all the cases converge
correctly to the exact values. In many cases, convergence is faster when compared
with other existing finite elements. The behaviour of field consistent elements would
definitely generate a great deal of interest amongst the users of the finite elements. |