Problem 4: FEA - BEAM (20 Points)

Consider the beam fixed at one end as shown in the diagram below. It is divided into 2 elements of

equal lengths. The moment of inertia of the beam is I, elastic modulus is E and the mass of the beam

is M. The total length of the beam is L. There is a distributed load A (N/m) over element 2 and a

moment B (N-m) at the free end.

L/2

L/2

a) Draw a schematic diagram of the 2 elements with the loads and moments distributed along the

nodes. Indicate the degree of freedom at each node.

b) Write down the elemental stiffness matrix for each element.

c) Assemble the element stiffness matrix into a global stiffness matrix.

d) Write down the displacement vector and force vector. Substitute the relevant boundary

conditions and loads.

e) Find out the reduced stiffness matrix and reduced force vector.

f) Consider the elemental mass matrix given below where m and I are the mass and length of the

element respectively.

B

[1 0 0

m0 1²/12 0

1

20 0

Lo

0 0 1²/12]

Assemble the global mass matrix using the elemental mass matrix [M] given above.

g) Set up the eigenvalue problem to solve for the eigenvalues of the beam.

[M]

=

0

0

0