Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 324

and the maximum moment obtained by substituting equation 7.74 into equation
7.73 is

+

2 -
.

0.5

N
N cr , y + cot π

N
N cr , y

, 1 +

M max = M β m cosec π . (7.3)

When β m > − cos π √ ( N / N cr , y ) , x max is negative, in which case the maximum
moment in the beam-column is the end moment M at x = 0, whence

M max = M , (7.4)

For beam-columns which are bent in single curvature by equal and opposite
endmoments (β m =− 1 ) , equation7.1forthedeflectedshapesimplifies, andthe
central deflection δ max can be expressed non-dimensionally as

8 /π 2
N / N cr , y

δ max
δ

sec π
2

N
N cr , y − 1

= (7.5)

inwhich δ = ML 2 / 8 EI y isthevalueof δ max when N = 0.Themaximummoment
occurs at the centre of the beam-column, and is given by

M max = M sec π
2

N
N cr , y .

(7.6)

Equations 7.5 and 7.6 are plotted in Figure 7.4.

7.5.2 Beam-columns with initial curvature

If the beam-column shown in Figure 7.3 is not straight, but has an initial
crookedness given by

w 0 = δ 0 sin π x / L ,

then the differential equation of bending becomes

− EI y d 2 w

d x 2 = M − M ( 1 + β m ) x L + N ( w + w 0 ) .

(7.75)

The solution of this equation which satisfies the support conditions of ( w ) 0 =
( w ) L = 0is

w = ( M / N ) [ cos µ x − (β m cosec µ L + cot µ L ) sin µ x − 1
+ ( 1 + β m ) x / L ]+[ (µ L /π) 2 / { 1 − (µ L /π) 2 }] δ 0 sin π x / L , (7.76)