Page 462 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

434 SLENDER STRUCTURES AND AXIAL FLOW

All the foregoing applies to a single element. The next step is to assemble the global
equation of motion, which is similar to (6.77) but the associated vector now covers all
the nodes, j = 1,2, . . . , n; in the corresponding global matrices [Mi], [Ci] and [Ki] there
is partial superposition of the element matrices (6.86), since any node, except those at the
two ends of the pipe, is shared by two elements. This global equation is then converted
to a standard eigenvalue problem, from which the eigenfrequencies may be determined
and stability assessed.

(b) Out-of-plane motion

The variational statement used for the finite element model of out-of-plane motion is

(6.88)

where Sr$ and S$* are the variations in the out-of-plane transverse displacement and
twist, respectively, while Ao1($, +*) and Ao2(r$, +*) represent the left-hand sides of
equations (6.61) and (6.62). The subscript o stands for out-of-plane.
The solutions for r$ and $* are sought in the form

where [N2] and [N4] are two matrices of interpolation functions of the space coordinates
<, and {q0}" is the element-displacement vector for the out-of-plane motion. In this case,
a cubic interpolation model for q; and a linear one for $* can guarantee convergence.
Hence, proceeding as for in-plane motion, one eventually obtains

where

in which

(6.92)