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Steel Designers' Manual - 6th Edition (2003)
Mmax
Effect of moment gradient loading 519
Fig. 18.8 Primary and secondary moments 1: (a) single curvature, (b) double curvature
case, the point of maximum combined moment occurs near mid-height where
secondary bending effects are greatest. On the other hand, for double-curvature
bending the two individual maxima occur at quite different locations, and for the
case illustrated, in which the secondary moments have deliberately been shown as
small, the point of absolute maximum moment is at the top. Had larger secondary
moments been shown, as is the case in Fig. 18.9, then the point of maximum moment
moves down slightly but is still far from that of the single-curvature case.
Theoretical and experimental studies of steel beam-columns constrained to
respond in-plane and subject to different moment gradients, as represented by the
factor of the ratio of the numerically smaller end moment to the numerically larger
end moment (b), show clearly that, when all other parameters are held constant,
failure loads tend to increase as b is varied from +1 (uniform single-curvature
bending) to -1 (uniform double-curvature bending). Figure 18.10 illustrates the
point in the form of a set of interaction curves. Clearly, if all beam-column designs
were to be based upon the b =+1 case, safe but rather conservative designs would
result.
Figure 18.10 also shows how for high moments the curves for b π+1 tend to merge
into a single line corresponding to the condition in which the more heavily stressed
end of the member controls design. Reference to Figs 18.8 and 18.9 illustrates the
point. The left-hand and lower parts of Fig. 18.10 correspond to situations in which
Fig. 18.9 controls, while the right-hand and upper parts represent failure at one end.
The two cases are sometimes referred to as ‘stability’ and ‘strength’ failure respectively.
While this may offend the purists, for in both cases the limiting condition is
one of exhausting the cross-sectional capacity, but at different locations within the
member length, it does, nonetheless, serve to draw attention to the principal difference.
Also shown in Fig. 18.10 is a line corresponding to the cross-sectional inter-
(b)