Page 299 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

280 SLENDER STRUCTURES AND AXIAL FLOW

Figure 5.l(a,b). Then, as a material point of the pipe moves, its displacement may be
defined by u = x - xo and w = z - zo for planar motions, where (u, w) may be expressed
fully in either set of coordinates. Similarly, other quantities, such as the deformation gradi-
ents and strain tensors, can also be expressed in either set of coordinates. In infinitesimal
deformation theory, which is the basis of the linear derivation, the distinction between
Lagrangian and Eulerian strains disappears (Eringen 1987); however, this distinction must
absolutely be made when nonlinear relationships are sought.
For a slender pipe with its initially undeformed state along the xo-axis, undergoing
motions in the (XO, ZO) plane, we have zo = 0, so that w E z. Here the Lagrangian repre-
sentation is chosen, so that the position and deformation of any point of the pipe is
expressed in terms of xu, i.e. the position of that point in the undeformed state.
However, an exception is made in the case of a cantilevered pipe, the centreline of
which may be assumed to be inextensible, in which case the coordinate s, measured
along the centreline, is introduced (Figure 5.1), and all physical quantities, including the
equations of motion, may be expressed in terms of (s, t).
The condition of inextensibility has already been defined in Section 3.3.1. Briefly, it
states that the distance 6s between two contiguous points P and Q, originally located at PO
and Qo and 6s0 apart, satisfy 6s = 6s0 = 6x0; whereupon two forms of the inextensibility
condition may be obtained, equations (3.14) and (3. lS), repeated here for convenience:

(”) + (2) (1 + 2) (E) = 1.
+
1,
=
ax0 8x0 ax0
For pipes fixed at both ends, 6x0 and 6s are not identically equal, but they are related
via equation (3.16), which may be expressed as

where E is the axial strain along the centreline. If E = 0, the second of (5.1) is retrieved.
An exact expression for the curvature, K, is useful in the derivations that follow, and is
hence derived next. Depending on the choice of the coordinate system and the assumptions
concerning the inextensibility of the pipe, the expression for K differs. Let 8 be the angle
between the position of the pipe and the xo-axis, and s the curvilinear coordinate along
the pipe [Figure S.l(b)]. For a pipe undergoing planar motion, extensible or inextensible,
the curvature is given by
ae
K= -. (5.3)
as
For simply-supported pipes, 8 is defined by

In terms of the xo-coordinate, equation (5.3) becomes

ae ax, 1 a8
/(=--=-- (5.5)
axo as 1 +E ax;