Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 306

C

N M

m

M

N 1 + = 1

1

+ = N 1 - N / N M

b , y , Rd cr , y pl

N 1- N / N M

b , y , Rd cr , y pl

M / M

1.18 (- N / N ) 1

=

≤

pl

y

Analytical solutions

Analytical solutions

1.0 1.0

M

N

L
i ---- =40
y

0.5 0.5 N m =1

80
120

M

0
-

L
i ---- =120
y

y y

0 0

0 0.5 1.0 0 0.5 1.0

M/M

M/M pl

pl

(a) Effect of slenderness (b) Effect of end moment ratio

Figure 7.11 Interaction formulae.

( L / i y → 0), and of some members which are not bent in single curvature (i.e.
β m > 0), as shown, for example, in Figure 7.11b.
Equation 7.21 represents the interaction between buckling and bending which
determinestheresistanceofmoreslendermembers.Ifthecaseofequalandoppo-
siteendmomentsisconsideredfirst( β m =− 1and C m = 1),thenitwillbeseenthat
equation7.21includesthesameapproximateamplificationfactor1 /( 1 − N / N cr , y )
as does equation 7.7 for the maximum moments in elastic beam-columns. It is of
a similar form to the first yield condition of equation 7.13, except that a conver-
siontoultimateresistancehasbeenmadebyusingtheultimateresistance N b , y , Rd ,
and moment M pl , y instead of the squash load N y and the yield moment M y .These
substitutions ensure that this interaction formula gives the same limit predictions
for concentrically loaded columns ( M = 0) and for beams ( N = 0) as do the
treatments given in Chapters 3 and 5. The accuracy of equation 7.21 for beam-
columns with equal and opposite end moments ( β m =− 1) is demonstrated in
Figure 7.11a.
The effects of unequal end moments ( β m > − 1) on the ultimate resistances of
beam-columns are allowed for approximately in equation 7.21 by the coefficient
C m whichconvertstheunequalendmoments M and β m M intoequivalentequaland
oppositeendmoments C m M .Althoughthisconversionisforin-planebehaviour,it
isremarkablysimilartotheconversionusedforbeamswithunequalendmoments
which buckle out of the plane of bending (see Section 6.2.1.2). The accuracy of
equation 7.21 for beam-columns with unequal end moments is demonstrated in