Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 298

The bending moment in the beam-column is the sum of M − M ( 1 + β m ) x / L
due to the end moments and shears, and Nw due to the deflection w , as shown
in Figure 7.3b. It is shown in Section 7.5.1 that the deflected shape of the beam-
column is given by

w = M
N

cos µ x − (β m cosec µ L + cot µ L ) sin µ x − 1 + ( 1 + β m ) x
L

,

(7.1)

where

EI y = π 2

µ 2 = N

N
N cr , y .

(7.2)

L 2

As the axial force N approaches the buckling load N cr , y , the value of µ L
approaches π , and the values of cosec µ L , cot µ L , and therefore of w approach
infinity, as indicated by curve 3 in Figure 7.2. This behaviour is similar to that of
a compression member with geometrical imperfections (see Section 3.2.2). It is
alsoshowninSection7.5.1thatthemaximummoment M max inthebeam-column
is given by

+
, 1 +

2 -
.

0.5

N
N cr , y + cot π

N
N cr , y

M max = M

β m cosec π

, (7.3)

when β m < − cos π √ ( N / N cr , y ) and the point of maximum moment lies in the
span, and is given by the end moment M , that is

M max = M , (7.4)

when β m ≥− cos π √ ( N / N cr , y ) .
Thevariationsof M max / M withtheaxialloadratio N / N cr , y andtheendmoment
ratio β m are shown in Figure 7.4. In general, M max remains equal to M for low
valuesof N / N cr , y butincreaseslater.Thevalueof N / N cr , y atwhich M max begins
to increase above M is lowest for β m =− 1 (uniform bending), and increases
with increasing β m . Once M max departs from M , it increases slowly at first, then
more rapidly, and reaches very high values as N approaches the column buckling
load N cr , y .
For beam-columns which are bent in single curvature by equal and oppo-
site end moments ( β m =− 1), the maximum deflection δ max is given by (see
Section 7.5.1)

8 /π 2
N / N cr , y

δ max
δ

sec π
2

N
N cr , y − 1

= (7.5)