Page 300 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 28 1

The derivative in this expression may be obtained from (5.4),
- = [ $ (1 + E) - ax, $1 /( + 42,
aw a2u
a0
8x0
1
thus yielding the curvature (5.5) for pipes whose centreline may be extensible.
On the other hand, for cantilevered pipes whose centreline is assumed inextensible,
expressions (5.3) and (5.4) still hold, except that E = 0. In this case, s = XO, and hence
ae/axo becomes

(5.7)

Application of the inextensibility condition (5.1) leads to the following expression of the
curvature:

Alternatively, the curvature may also be defined as a vector:

where n is the normal unit vector which is always perpendicular to the tangent direction
of the pipe and r = (x, z) is the position vector along the pipe. Hence,

2
2 +($).
K=lZl=/($)
If the inextensibility condition is applied to the expression above, one obtains again
equation (5.8).
Note that for a curve defined by ~(x) (Eulerian description) rather than ~(s), one has
the familiar expression of curvature [e.g. Timoshenko (1955)],

Care must be taken as to which expression of K is used, depending on the physical
problem.

5.2.2 Hamilton's principle and energy expressions
A Hamiltonian derivation of the equations of motion is given here. It is based on the
same statement of Hamilton's principle as in Section 3.3.3, namely

(5.10)

%
where 2 is the Lagrangian of the system (2 = + % - y - q, and Vp being the
kinetic and potential energies associated with the pipe, and ?$ and @l$ the corresponding