Page 464 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

436 SLENDER STRUCTURES AND AXIAL FLOW

6.3.2 Analysis for extensible pipes

Because the static displacements appear explicitly in the equations of motion, the static
equilibrium is determined first, and then the stability of motions about this position.

(a) Determination of the static equilibrium
In order to discretize the equations governing the static deformations, the following vari-
ational statement is utilized

(6.100)
j= 1

where Sqp and 6q; are the variations in the steady-state displacements q: and q:, while
Afl(g:, q;) and qz) represent the left-hand sides of equations (6.69) and (6.70),
respectively; n and <, are as in the foregoing.
The solutions for and qi are sought in the form

where [Nle] and [N3e] are two matrices of interpolation functions of the space coordinate
<, and is the element in-plane displacement vector. It may be shown that a cubic
interpolation model for qy and linear interpolation for qi can guarantee convergence of
the finite element scheme. Thus, one can proceed in the same manner as for the out-
of-plane motion in the inextensible case to obtain a matrix equation governing the static
equilibrium of an element as follows:t

[Kp]"(q;)" = (Fpje, (6.102)

where
=
[qe ([Aol-')T [{[M + @([1151 + [1l5lT> + 02[1*1)
+ de([I81 - @([[I21 + [112JT) + @2[111)
+ (npo -I- z2)([191 4- @([I221 - [In]) - 02[141) (6.103)
-
- hE2{[151 + [I101 + @([116l + [I141 - [1171) - @2[1i,l)] [&I-',
-
(FpJe = [@(npo + z2){F1} - hZ2(@{F2) + {F3)) + (F4)] .

For the integrals [ZI], (Fl}, etc. and [A,,], see Appendix K.
In deriving equation (6.102) it has been assumed that the pressure in the external fluid
is constant, while the internal pressure varies linearly along the centreline. Thus

np = nplo - ZE2(, (6.104)

+It is recalled that, for an inextensible pipe, in-plane motion involves sixth-order derivatives in <, and hence
the corresponding shape functions are not useful here. However, the shape functions for out-of-plune motions
of the inextensible pipe may be used for this analysis, since the maximum orders of partial derivatives match;
hence the appearance of [A,] in equations (6.103).