Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 299

1.0

1.0

0.8

m M

N

0.6

0.4

M max / M
/ ( = 1)
(0.6 - 0.4 )/(1 - N / N cr,y ) 1.0

N

0.2

m

max -

M

m

≥

0

0 1 2 3 4 5 6 7 8

M max / M or max /

Figure 7.4 Maximum moments and deflections in elastic beam-columns.

in which δ = ML 2 / 8 EI y is the value of δ max when N = 0, and the maximum
moment by

M max = M sec π
2

N
N cr , y .

(7.6)

These two equations are shown non-dimensionally in Figure 7.4. It can be seen
that they may be approximated by using a factor 1 /( 1 − N / N cr , y ) to amplify the
deflection δ and the moment M due to the applied moments alone ( N = 0). This
sameapproximationwasusedinSection3.6.3toestimatethereducedstiffnessof
an axially loaded restraining member.
For beam-columns with unequal end moments ( β m > − 1), the value of M max
may be approximated by using

M max
M

C m
1 − N / N cr , y ≥ 1.0

= (7.7)

in which

C m = 0.6 − 0.4 β m .

(7.8)

TheseapproximationsarealsoshowninFigure7.4. Itcanbeseenthattheyare
generally conservative for β m > 0, and only a little unconservative for β m < 0.