Page 552 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

Appendix J

Detailed Analysis for the

Derivation of the Equations of

Motion of Chapter 6

J.1 RELATIONSHIP BETWEEN (XO, yo, 10) AND (x, y, Z)
The derivation of this relationship is given by Love (1927; Chapter XXI) for the analysis
of the ‘Small deformation of naturally curved rods’. The detailed derivation, specifically
for the curved pipe problem, may be found in Van (1 986; Appendix A). Here, only some
definitions and the final result are given.
Let us define a so-called Frenet-Serret reference frame (XO, yo, ZO) centered at Po,
consisting of the principal axes of the undeformed cross-section of the pipe, zo being
tangent to the centreline (Figure 6.1); also, a so-called flexure-torsion reference frame
(x, y, z) associated with the deformed centreline. Further, let the unit vectors associated
with the (XO, yo, zo) and (x, y, z) systems be (ex,,, eye, e,,) and (ex, ey. e,), respectively.
The initial curvature is defined by K, and K; and the initial twist by to; for the initially
planar [in the (XO, ZO) plane], untwisted pipe, these are

K, = 0, K: = l/Ro, 7, = 0. (J.1)

After deformation, point PO moves to P through displacements ti, v and w, referred to the
(XO, yo, zo) system, as shown in Figure 6.1. The angle between xo and x is ~, which is
the angle of rotation about the z-axis of a plane section at Po due to deformation.
The centreline strain is given by

where s is the curvilinear coordinate along e, referred to the (XO, yo, ZO) system. Since
K, = 0, if the centreline is inextensible, then clearly

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