Page 89 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

72 SLENDER STRUCTURES AND AXIAL FLOW

4
8.)
-
dS
(a) (b)
Figure 3.6 (a) Forces acting on an element 6s of the fluid; (b) forces and moments on the
corresponding element of the pipe.

The fluid element of Figure 3.6(a) is subjected to: (i) pressure forces, where the pressure
p = p(s, t) because of frictional losses, and p is measured above the ambient pressure;
(ii) reaction forces of the pipe on the fluid normal to the fluid element, F as, and tangential
to it, qS as, associated with the wall-shear stress q; (iii) gravity forces Mg 6s in the x-
direction. Applying Newton’s second law in the x- and z-directions, while keeping in
mind the small-deflection approximation, yields

(3.18)

(3.19)

where afx and afz are the accelerations of the fluid element in the x- and z-direction,
respectively. Similarly, for the pipe element of Figure 3.6(b) one obtains

aT aw
-+qqS+mg-F- =0, (3.20)
ax ax
aQ aw aw
T-
+~S--C-=~LZ,,,,
-+F+- :x ( Z) (3.21)
ax ax at
(3.22)

where T is the longitudinal tension, Q the transverse shear force, and A the bending
moment; moreover, the pipe is subjected to internal dissipation of the Kelvin-Voigt type
(e.g. Shames 1964; Meirovitch 1967; Snowdon 1968), thus following a stress-strain
(a, E) relationship of the form a = EE + E*(ds/dt), and also to damping due to friction