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)

Additional moment due to slender column effect 277
(7.4-3)
Note that Cranston has replaced the actual column height l by the effective
height le (see Table 7.2-1) to allow for the effects of the various end
conditions in practice. Combining eqns (7.4-3) and (7.4-2),
eadd = 1 ; 50[*T[ 1 - 0.0035*] (7.4-4)
Before relating eqn (7.4-4) to current design practice, we shall carefully
define the following symbols:
le = the effective column height in the plane of bending;
h = the depth of the column section measured in the plane of
bending;
b = the width of the column section (i.e. the dimension of the
column section perpendicular to h);
b' = the smaller dimension of the column section (i.e. b' = h if
h < b; b' = b if h > b).
BS 8110 approximates eqn (7.4-4) to
1 [ le ] 2
au = 2000 b' h
= f3ah (7.4-5)
where au is BS 8110's symbol for the additional eccentricity eadd. au may
optionally be multiplied by a reduction factor K, which will be explained in
Section 7.5: Step 5. That is, au may be taken as f3aKh; the additional
moment Nau induced by the lateral deflection is therefore
Madd = Nf3aKh (7.4-6)
where f3a is as defined in eqn (7.4-5). Therefore, with reference to eqn
(7.4-1), the total moment is
Mt = Mi + Madd (7.4-7)
For the simple case illustrated in Fig. 7.4-1, the initial moment Mi to be
used in eqn (7.4-7) is simply the initial moment at an end of the column.
For the general case where the end moments are M 1 and M 2 (M 2 being the
larger), BS 8110: Clause 3.8.3.2 recommends that Mi be taken as
Mi = 0.6Mz + OAM1
for symmetrical bending (Fig. 7.4-4(a)) and
Mi = 0.6M2 - 0.4M1 but <1:0.4M 2
for asymmetrical bending, i.e. bending in double curvature (Fig.
7.4-4(b)). These two equations may be combined into one, namely:
Mi = OAM1 + 0.6Mz 2: 0.4M2 (7.4-8)
where M 1 = the smaller initial end moment (taken as negative if the
column is bent in double curvature); and