Problem #1: We solved the cantilever beam problem with a tip load using finite element method
(FEM) in the class.
Now discretize the beam using two elements (note that in the example problem we used only one
element). Consider the following set of loading and boundary conditions:
Case A: (35 points)
Load: Apply the same magnitude of load (i.e. 1000 N) at the center of the beam (note in the
example problem it was applied at the tip end.
Boundary conditions: Fix both the ends of the beam
Case B: (35 points)
Load: Apply uniformly distributed load of 2000 N/m on the entire length of the beam (recall
discussion on consistent nodal load).
Boundary conditions: Fix both the ends of the beam (same as Case A)
Solve for the unknown displacements (translation and rotation) and reactions. Compare the
displacement at the center of the beam from FEA to theoretical solutions (refer to any Solid
Mechanics or Structural Analysis text books) for the two cases.
Note: Use the same material and geometry properties as in the example
Problem #2: A linear (four-noded) quadrilateral element is shown in the figure below.
The shape functions as given by
( )( ) ( )( )
( )( ) ( )( )
1 2
3 4
1 1 1 1, 1 1 4 4
1 1 1 1, 1 1 4 4
N N
N N
ξη ξη
ξη ξη
=− − =+ −
=+ + =− +
Show that the element satisfies the Kronecker delta property and completeness. (30 points
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