Page 87 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

70 SLENDER STRUCTURES AND AXIAL FLOW

k
L b
Z

Figure 3.5 (a) The Eulerian coordinate system (x, z) and the Lagrangian one (XO, ZO) = (XO, 0) in
which the xo-axis is superposed on the x-axis, showing the deflection of a point Po = Po(&, 0)
to P(x, z) and the definition of u and w; (b) diagram used for the derivation of the inextensibility
condition.

Two further points should be made: (i) whenever Lagrangian coordinates are used, they
are used for pipe motions only, not for the fluid; (ii) it is customary to use a curvilinear
coordinate s, along the length of the pipe, as shown in Figure 3.5(a) - especially useful
if the pipe is considered to be inextensible.
The second concept of importance to be discussed in this section is that of inex-
tensibility. For pipes supported as in Figure 3.1(b,c) for instance, where no deflection-
dependent axial forces come into play, one may clearly consider the pipe to be inexten-
sible, i.e. the length of its centreline to remain constant during oscillation. However, in
the case of a pipe with positively supported ends [Figure 3.l(a)], i.e. with no axial sliding
permitted, lateral deflection may occur only if the pipe is extensible.
Consider contiguous points P and Q of the deflected pipe, originally (in the undeflected
state) at PO and Qo, as in Figure 3.5(b). Then,

(W2 = (sx)2 + (sz>2, (8s0l2 = (6xo)2 + (6zo)2 = (sxo)2,
from which one may write

(3.13)

If the pipe is inextensible, 6s = 6s0 by definition, and the condition of inextensibility
may be expressed as (g)2+(g)
=I.
2
(3.14)

The inextensibility condition may also be expressed in terms of the displacements (u, w);
by invoking (3.12),