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Steel Designers' Manual - 6th Edition (2003)
514 Members with compression and moments
Fl
PC
M1
corresponding
value of M Mc
(a) (b)
Fig. 18.3 Concept of interaction diagrams for combined loading: (a) two-dimensional, (b)
three-dimensional
is possible, the member slenderness and the cross-sectional shape. Design methods
for beam-columns must therefore seek to balance the conflicting requirements of
rigour, which would try to adjust the form of the design boundary of Fig. 18.3 to
reflect the influence of each of these factors, and simplicity. However, some appreciation
of the role of each factor is necessary if even the simplest design approach
is to be properly appreciated and applied.
The importance of member slenderness may be appreciated readily with reference
to the two-dimensional example illustrated in Fig. 18.4. The member is loaded
by compression plus equal and opposite end moments and is assumed to respond
simply by deflecting in the plane of the applied loading. Under the action of the
applied moments bending occurs, leading to a lateral deflection v. The moment at
any point within the length comprises two components: a constant primary moment
M due to the applied end moments plus a secondary moment Fv due to the axial
load F acting through the lateral deflection v. Summing the effects of compression
and bending gives
FP / + M / M =10 .
(18.1)
c max c
in which Pc and Mc are the resistance as a strut and a beam respectively and Mmax
is the total moment.
Analysis of beam-column problems shows that Mmax may be closely approximated
by
Mmax = M/ ( 1 -F/
Pcr)
(18.2)