Page 313 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

294 SLENDER STRUCTURES AND AXIAL FLOW

nonlinear force T is added to (5.57), giving rise to
I”
- [g z”dx] z” (5.59)

Because of the extensibility assumption, this equation cannot be compared with any of the
previous ones. However, it should be mentioned that the strain is approximated to second
order only, which does not fulfill the order considerations discussed in Section 5.2.1.
Additionally, Ch’ng and Dowel1 also consider a nonlinear relationship for the curvature.
They use expression (5.9) for the curvature K and the elastic strain energy

and obtain additional terms, -El (3 z” z”” + 12z’z”z”’ + 3~’’~). It is seen that expres-
sion (5.60) is not fully consistent with the strain energy derived by Stoker (1968), because
the pipe is implicitly assumed to be extensible (E # 0). Moreover, it is not obvious how
the Eulerian description can be used with the energy method to derive nonlinear equations.
Therefore, comparison cannot be made with other versions of the governing equations.

5.2.9 Comparison with other equations for pipes with fixed ends

In this section, two papers are discussed, representative of all the derivations for pipes
fixed at both ends. Again, a standardization of the notation has been undertaken.

(a) Thurman and Mote‘s work
Thurman & Mote (1969b) were mainly concerned with the oscillations of bands of moving
materials. They consider an axially-moving strip, simply-supported at its ends, in order to
show how the axial motion could significantly reduce the applicability of linear analysis.
This work is then extended to deal with pipes conveying fluid. The centreline being exten-
sible, nonlinearities are associated with the axial elongation and the extension-induced
tension in the tube. Therefore, the strain and the tension become

To J
E=-+ (1 +u’)~ +w’~ - 1, T = To +EA (J(1 +u’)~ +w’~ - 1). (5.61)
EA
Since a linear moment-curvature relationship and a linear approximation for the velocities
are considered, the equations of motion obtained are
EZ w”” - (To - M U2)w’’ + 2 M U W’ + (m + M)G
= (EA - To) ( ;w” W” + U’ W” + u’’ w’) , (5.62)

M U - EA U” = (EA - To) W’ w”.
These are actually a simplified set of equations (5.36a,b). The differences come
from the assumptions made: (i) no gravity forces, (ii) steady flow velocity, (iii) linear
moment-curvature relationship, (iv) simple approximation of the fluid velocity; on the
basis of these assumptions, the equations derived are correct.