Page 496 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

Appendix B

Analytical Evaluation of bsr, csr

and dsr

The method - or at least a method - for the analytical evaluation of the constants defined
by equation (3.87) is illustrated here, first for bsr.
Let us re-write

which, after successive integration by parts, yields

the last integrand may be written as @;A:& which leads to

1
- Psr = [l;+r4s - 4;4: + 4:4: - ~ ~ Y I I O . (B.2)
~
This can be evaluated for any particular set of the standard boundary conditions. Thus, for
acantileveredpipe, &(l) = 2(-1)r, 4:(1) = 4:(1) = 0, and@,(O) = &(O) = 0, @:(O) =
2h:, and similarly for 4s (Bishop & Johnson 1960; Blevins 1979). Hence, after some
manipulation, equation (B.2) gives

For r = s, this clearly gives
brr = 2. (B.4)
Working in a similar manner, the other entries of Table 3.1 may be determined - at
least for r # s. For r = s, however, some of the expressions obtained with r # s become
singular and have to be determined in another way. An example is cSr which is zero for
pinned (simply-supported) ends and

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