Page 532 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

Appendix G

Newtonian Derivation of the

Nonlinear Equations of Motion of

a Pipe Conveying Fluid

G.l CANTILEVERED PIPE
This derivation is based on the work of Lundgren et al. (1979) but here it is developed
further to lead to a single equation of motion, better suited for later analysis.

Figure G.1 Free-body diagram of an element of the pipe; gravity and flow-velocity-dependent
forces are not included for clarity.

Consider an element of the pipe of length 6s (Figure G.l). Let Q and M represent the
resultant force and bending moment on the left cross-section, and Q + 6Q and M + 6M
on the right cross-section. A force balance leads to
aQ a2 r DZr
- + (rn +M)gi = rn - +M -,
as at2 Dt2

where t is the unit vector tangential to the pipe centreline.

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