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Finite Element Methods
Assignment No. 3
Problem 3.12:
Consider the displacement model of a triangular bending element given in
Problem 3.11. Determine whether the convergence requirements of Section 3.6 are
satisfied by this model.
Note: The expression for the functional I (potential energy) of a plate in bending
is given by
Where p is the distributed transverse load per unit area, D is the flexural rigidity,
v is the Poisson's ratio, and A is the surface area of the plate.
Solution:
Given Displacement model is,
And convergence requirements are
The field variable must be continuous within the elements. Since polynomials are inherently continuous, so it satisfies.
a. All uniform states of the field variable 0 and its partial derivatives up to the
highest order appearing in the functional must have representation in the
interpolation polynomial. b. If the body is subdivided into smaller and smaller elements, the partial
derivatives of the field variable up to the highest order appearing in the function
approach a constant value within each element.
The field variable o and its partial derivatives up to one order less than the highest
order derivative appearing in the function must be continuous at element boundaries or interfaces.
As the given interpolation function is polynomial, so first condition for
convergence is satisfied.
For second condition we have,
2 2 2 2 4 5 3 7 82
w x y x xy y x
And
2
2 2 4 6 7 2 8
w x y x
2 2 2 3 5 2 6 3 4 8 2 3 9
w x y y x xy y y
2
2 2 6 2 8 6 9
w x y y
2 2 8
w
x y
3
3 6 9
w
y
…….. (a)
4
4 0
w
y
3
3 6 7
w
x
……… (b)
4
4 0
w
x
All the partial derivatives are the representation of the interpolation model, so
second condition is satisfied as the partial derivative again generate the polynomial. The
smaller division of the elements gives constant value, (see equation ‘a’ and ‘b’) so 3
rd
