Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 305

7.2.3.2 Elastic-plastic resistances of straight beam-columns

The resistance of an initially straight beam-column with residual stresses may
be found by analysing numerically its non-linear elastic-plastic behaviour and
determiningitsmaximumresistance[1].Someoftheseresistances[2,3]areshown
as interaction plots in Figure 7.10. Figure 7.10a shows how the ultimate load N
and the end moment M vary with the major axis slenderness ratio L / i y for beam-
columns with equal and opposite end moments ( β m =− 1), while Figure 7.10b
shows how these vary with the end moment ratio β m for a slenderness ratio of
L / i y = 60.
It has been proposed that these ultimate resistances can be simply and closely
approximated by using the values of M and N which satisfy the interaction
equations of both equation 7.17 and

N
N b , y , Rd +

C m
( 1 − N / N cr , y )

M
M pl , y ≤ 1

(7.21)

in which N b , y , Rd is the ultimate resistance of a concentrically loaded column (the
beam-column with M = 0) which fails by deflecting about the major axis, N cr , y
is the major axis elastic buckling load of this concentrically loaded column, and
C m is given by

C m = 0.6 − 0.4 β m ≥ 0.4. (7.22)

Equation 7.17 ensures that the reduced plastic moment M pl , r is not exceeded
by the end moment M , and represents the resistances of very short members

1.0 1.0

L/i y

M

N

M

N

m

0

M

N

1.0

20
40

L N

M

M

0.5

N

N

0.5 0.5

N

0

M

60

- 0.5

y y

N

80
100
120

M

- 1.0

y y L

i ---- =60
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 7.10 Ultimate resistance interaction curves.