F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Problems 289
refers to a particular initial crookedness e0 = e1 sinHzll. However, any
arbitrary initial crookedness may be represented by a Fourier series:
. Hz . 2Jr:z . 3Hz
e0 = e1siDT + e2 siD- 1 - + e3 siD- 1 - + · · ·
Using the same technique that you used for Problem 7.6, show that for this
general case the total transverse deflection is
e1 . HZ e2 . 2Jr:z
eo+ eadd = 1- aESIDT + 1- (aE/22)siD-le3
. 3Hz en . nHz
+ 1 - (aE/32)siD-l- + · · · 1- (aE/n2)siD-l- +
Comments on Problem 7.7 (May be omitted during first reading)
(a) The Fourier series solution is of wide application, as illustrated by the
following two examples.
First, consider an eccentrically loaded column (Fig. 7.4-3). The
eccentricity e may be thought of as an initial crookedness of uniform
magnitude ( = e) along the entire column length, and represented by a
Fourier series. The result of Problem 7. 7 becomes directly applicable.
Next consider a column subjected to an axial load N and end
moments Mi (Fig. 7.4-1(a)). The transverse deflection e 0 due to Mi
acting alone is of course easily calculated. The total transverse
deflection due to N and Mi may then be determined by considering
the load N acting (in the absence of Mi) on a column having an initial
crookedness e 0 . By expressing e 0 as a Fourier series, the result of
Problem 7. 7 becomes directly applicable:
"' en . nHZ
eo + eadd = LJ 1 _ (aE/n2) SID -~-
Note that (e0 + eadd) here has the same meaning as eadd in Fig.
7.4-1(a).
(b) The Fourier series solution for e0 + eadd tends to converge rapidly. In
many practical applications, a good estimate is obtained by just
taking the first term:
el . HZ
eo + eadd = 1 SID -1
- aE
e1 I
= 1 - . aE
at z = - 2
Therefore, referring to Fig. 7.4-1(a), the total moment is,
approximately:
M = M· + Net
t • 1- aE
Interested readers are referred to Section 9.4 of Reference 6, which
shows that this simple equation forms the basis of the well-known
Perry-Robertson formula in structural steelwork design: