Page 314 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

10

Introduction to the

Finite Element

Method

In Chapter 6, we were able to determine the stiffness matrix of simple frame
struetures by eonsidering the structure as an assemblage of structural elements.
With the forces and moments at the ends of the elements known from structural
theory, the joints between the elements were matched for compatibility of dis­
placements and the forces and moments at the joints were established by imposing
the condition of equilibrium.
In the finite element method, the same procedure is followed, but in a more
systematic way for computer calculation. Although structures with few elements
can be analyzed simply by the method outlined in Chapter 6, the “ bookkeeping”
for a large structure of many elements would soon overcome the patience of the
analyst. In the finite element method, element coordinates and forces are trans­
formed into global coordinates and the stiffness matrix of the entire structure is
presented in a global system of common orientation.
The accuracy obtainable from the finite element method depends on being
able to duplicate the vibration mode shapes. Using only one finite element
between structural joints or corners gives good results for the first lowest mode
because the static deflection curve is a good approximation to the lowest dynamic
mode shape. For higher modes, several elements are necessary between structural
joints. This leads to large matrices for which the computer program CHOLJAC of
Chapter 8 will be essential in solving for the eigenvalues and eigenvectors of the
system.
This chapter introduces the reader to the basic ideas of the finite element
method and also includes the development of the corresponding mass matrix to
complete the equations of motion for the dynamic problem. Only structural
elements for the axial and beam elements are discussed here. For the treatment of
plates and shells, the reader is referred to other texts.

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