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7.2 Finite Element Method 123

7.2 Finite Element Method

7.2.1 Finite Element Equations

The finite element equations for the truss, beam, plate
and solid elements can be derived by applying the method of
weighted residuals to their differential equations. Detailed deriva-
tion can be found in many finite element textbooks including the
one written by the same author. The derived finite element
equations are in the same form of,
    M       K   
F
M
K
where   is the element mass matrix;   is the element stiffness
F
matrix;   is the element load vector;    is the element vector

containing nodal unknowns; and    is the element vector contain-
ing nodal accelerations.
After assembling all element equations together and
applying the boundary conditions, solutions of the element nodal
unknowns    at different times can be determined using the
method of: (a) modal superposition, and (b) recurrence relations.
The modal superposition method involves determination of the
eigenvalues and eigenvectors as the first step. The recurrence
relations method employs the finite difference approximation to

transform the acceleration vector    into the nodal unknown
vector    . Details of these two methods are omitted herein for
brevity. They can be found in many advanced finite element
method books including the book written by the author.

7.2.2 Element Types
Truss, beam, plate and solid elements are presented in
the preceding chapters. With their element interpolation functions,
K
the corresponding element stiffness matrix   and element load
M
vector  can be derived. The mass matrix   that arises in
F
this chapter for analysis of vibration problems is in the form of an
integral over element domain  as,