Page 452 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

424 SLENDER STRUCTURES AND AXIAL FLOW

According to the generalization of the Euler-Bernoulli beam theory, the stress couples
Ax, A,, Az in the beam when bent and twisted from the state expressed by K,, K:, to to
that expressed by K, K’, t* are given by
=
Ax = EZ(K - K,), dy = H(K’ - K:), dZ GJ(t* - to). (6.30)
In the discretization to be introduced later, the pipe is divided into a series of constant
curvature elements; i.e. within a given element,

- 0, (6.31)
=
as
where the number of elements required will depend on the shape of the pipe centreline
and the accuracy desired.
Combining equations (6.2), (6.3) and (6.30) yields

(6.32)
Then, substituting equations (6.32) into (6.22) and (6.23), and neglecting the higher-
order terms, one obtains

Qx=-El (6.33)

(6.34)

By adding equations (6.19), (6.20) and (6.21) to (6.27), (6.28) and (6.29), respectively,
one may obtain the equations of motion of the system, which no longer depend on the
reaction forces R,, R, and Rzo between the pipe and the fluid. Then, utilizing equa-
tions (6.6), (6.15) and (6.32)-(6.34) and neglecting higher order terms, the governing
equations of motion for the dynamical system may be obtained, namely:

a2u
+(m+M+M,)- =o, (6.35)
at2

+
+
a2v a2v av a2v
- (GYO + Gfyo) + MU - 2MU - c - + (WZ + M +Ma)- = 0,
as2 atas at at2
(6.36)