Page 319 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

306 Introduction to the Finite Element Method Chap. 10

Variable properties. One simple approach to problems of variable prop­
erties is to use many elements of short length. The variation of mass or stiffness
over each element is then small and can be neglected. The problem then becomes
one of constant mass and stiffness for each element that simplifies the problem
considerably because these terms can be placed outside of the integrals. Of course,
the larger numbers of elements will lead to equations of larger DOF.

10.2 STIFFNESS AND MASS FOR THE BEAM ELEMENT

Beam stiffness. If the ends of the element are rigidly connected to the
adjoining structure instead of being pinned, the element will act like a beam with
moments and lateral forces acting at the joints. In general, the relative axial
displacement U2 —U\ will be small compared to the lateral displacement v of the

beam and can be assumed to be zero. When axial forces as well as beam forces and
moments must be considered, it is a simple matter to make additions to the beam
stiffness matrix, as we show later.
The local coordinates for the beam element are the lateral displacements and
rotations at the two ends. We consider only the planar structure in this discussion,
so that each joint will have a lateral displacement v and a rotation 6, resulting in
four coordinates, i\,6x ^2’^2* positive sense of these coordinates is
arbitrary, but for computer bookkeeping purposes, the diagram of Fig. 10.2-1 is the
one accepted by most structural engineers. Positive senses of the forces and
moments also follow the same diagram.

K E l ) Figure 10.2-1. Positive sense of
beam displacement and forces.

^,{X)

e.-

Figure 10.2-2.