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Steel Designers' Manual - 6th Edition (2003)
518 Members with compression and moments
moments about both principal axes, the member’s response is a three-dimensional
one involving bending about both axes combined with twisting. This leads to a
complex analytical problem which cannot really be solved in such a way that it
provides a direct indication of the type of interaction formulae that might be used
as a basis for design. A practical view of the problem, however, suggests that some
form of combination of the two previous cases might be suitable, provided any proposal
were properly checked against data obtained from tests and reliable analyses.
This leads to two possibilities: combining the acceptable moments Mx and My that
can safely be combined separately with the axial load F obtained by solving Equations
(18.3) and (18.4) to give
M / M + M / M =10 .
(18.5)
x ax y cy
in which Max and May are the solutions of Equations (18.3) and (18.4); or simply
adding the minor axis bending effect to Equation (18.4) as
FP / cy + Mx/ ( 1-FP / crx) Mb + My/ ( 1-FP / cry) Mcy
= 1. 0
(18.6)
Although the first of these two approaches does not lead to such a seemingly
straightforward end result as the second, it has the advantage that interaction about
both axes may be treated separately, and so leads to a more logical treatment of
cases for which major axis bending does not lead to a minor axis failure as for a rectangular
tube in which Mb = Mcx, and Pcx and Pcy are likely to be much closer than
is the case for a UB. Similarly for members with different effective lengths for the
two planes, for example, due to intermediate bracing acting in the weaker plane
only, the ability to treat in-plane and out-of-plane response separately and to
combine the weaker with minor axis bending leads to a more rational result.
The foregoing discussion has deliberately been conducted in rather general terms,
the main intention being to illustrate those principles on which beam-column design
should be based. Collecting them together:
(1) interaction between different load components must be recognized; merely
summing the separate components can lead to unsafe results
(2) interaction tends to be more pronounced as member slenderness increases
(3) different forms of response are possible depending on the form of the applied
loading.
Having identified these three principles it is comparatively easy to recognize their
inclusion in the design procedures of BS 5950 and BS 5400.
18.3 Effect of moment gradient loading
Returning to the comparatively simple in-plane case, Fig. 18.8 illustrates the patterns
of primary and secondary moments in a pair of members subject to unequal
end moments that produce either single- or double-curvature bending. For the first