Page 459 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CURVED PIPES CONVEYING FLUID 43 1

pipes of uniform curvature as to S- or a-shaped pipes - without having to determine
the equivalent of new comparison functions for each case. According to this method, the
continuum is subdivided into a number of elements, at the edges of which there are one
or more nodes (one for a structure such as this, in which spatial variations involve only
one coordinate, 0. The equations of motion are satisfied at the nodes rigorously and
elsewhere approximately via interpolation functions. A variational formulation ensures
a systematic minimization of the error for the number of elements utilized. The use of
the finite element method has now become routine, and hence the uninitiated reader is
referred to one of several texts on the subject, e.g. Zienkiewicz & Cheung (1968), Desai
& Abel (1972), Becker et al. (1984) or Zienkiewicz & Taylor (1989).
The details of the particular form of finite element analysis employed here may be
found in Van (1986). However, different forms may be employed, and this is partly the
reason for not presenting the minutiae of the analysis.

6.3.1 Analysis for inextensible pipes

(a) In-plane motion
The pipe is discretized into n elements. The initial curvature of a particular element is
constant, although it can vary from element to element. The variational statement used
for the finite element discretization is

(6.75)

where 87; is an arbitrary variational displacement, Ai(q;) represents the left-hand side of
equation (6.60) and is the length of the jth element. The subscript i in Ai stands for
‘in-plane’ .
The longitudinal displacement q; may be expressed in the form

11; = [N31{qiIe, (6.76)
where [N3] is a matrix of interpolation functions at the space coordinate {, and {si)“
is the element displacement vector, of appropriate dimension and dependent only on
time; the superscript e stands for ‘element’. Hence, by using (6.75) and (6.76), integrating
by parts, and applying the boundary conditions - thereby eliminating the integrated-out
components - one obtains the discretized equation

[M,Ie{ijile + [DiIe{4iI“ + [KiIe(qiJe = (01, (6.77)

where