Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 323

providedtheratioofendmomentsislessthanzero(i.e.atriangular,orlesssevere,
bending moment diagram).
Aworkedexampleforcheckingthebiaxialbendingofabeam-columnisgivenin
Section7.7.5.FurtherworkedexamplesoftheapplicationoftheEC3beam-column
formulae are given in [30] and [32].

7.5 Appendix - in-plane behaviour of elastic
beam-columns

7.5.1 Straight beam-columns

The bending moment in the beam-column shown in Figure 7.3 is the sum of the
moment [ M − M ( 1 + β m ) x / L ] due to the end moments and reactions and the
moment Nw due to the deflection w . Thus the differential equation of bending is
obtained by equating this bending moment to the internal moment of resistance
− EI y d 2 w / d x 2 , whence

− EI y d 2 w

d x 2 = M − M ( 1 + β m ) x

L + Nw .

(7.73)

The solution of this equation which satisfies the support conditions ( 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

, (7.1)

where

EI y = π 2

µ 2 = N

N
N cr , y .

(7.2)

L 2

Thebendingmomentdistributioncanbeobtainedbysubstitutingequation7.1into
equation 7.73. The maximum bending moment can be determined by solving the
condition

d ( − EI y d 2 w / d x 2 )
d x

= 0

for the position x max of the maximum moment, whence

tan µ x max =− β m cosec µ L − cot µ L . (7.74)

The value of x max is positive while

N
N cr , y ,

β m < − cos π