Page 534 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

504 SLENDER STRUCTURES AND AXIAL FLOW

where the material derivative is defined as in (5.30). By taking into account the force due
to (To - P) and the extensibility of the pipe, the force Q may be expressed as

From expression (5.34), the axial force Q1 is

while the shear force Q2, perpendicular to Q1 (see Figure G.l), is given by

(G. 10)

where n is the unit vector normal to t. As the effect of rotatory motion is neglected, the
moment due to bending has a contribution only in the n direction. Moreover, the moment
in its scalar form simply becomes

(G. 11)

Therefore, decomposing Q along t and n, one obtains

EZ 3%
Q = Q1+ Q2 = (To - P +EA&)t - - - (G.12)
n.
1 +E ax;
By decomposing these two components along the x- and z-directions, recalling the expres-
sions of the accelerations obtained in (5.18), extending the results of (5.30), and intro-
ducing again the angle 8, one obtains
a a 8% D2(xo + u)
h+M)g+ -@I cos 6) - - (Q2 sin Q) = rn - +M , (G.13a)
axo ax0 at2 Dt2
a a a2w D2 w
-(Ql sin Q) + -((e2 cos Q) = rn - +M - (G.13b)
8x0 8x0 at2 Dt2 ’
where sin8 and cos Q are defined by equations (5.4).
Here, an order of magnitude analysis is useful, so as to simplify the algebra as much
as possible. The first equation (in the xo-direction) is of second order, and the second (in
the z-direction) of third order. Hence, all the terms have to be exact up to third order. For
example,

sin Q = w’ (1 - u’ - iw”) + 0(t4),

cos 6 = 1 - +W’2 + 0(€4), E = u’ + iW’2 + O(E4).