Compression members - The Behaviour and Design of Steel Structures to EC3 - page 81

in which
Φ = 1 + η + λ 2
2

(3.12)

and

λ =

N y / N cr .

(3.5)

3.9 Appendix - inelastic compression members

3.9.1 Reduced modulus theory of buckling

The reduced modulus buckling load N cr , r of a rectangular section compression
member which buckles in the y direction (see Figure 3.26a) can be determined
fromthebendingstrainandstressdistributions,whicharerelatedtothecurvature
κ( =− d 2 v / d x 2 ) and the moduli E and E t as shown in Figure 3.26b and c. The
positionofthelineofzerobendingstresscanbefoundbyusingtheconditionthat
the axial force remains constant during buckling, from which it follows that the
force resultant of the bending stresses must be zero, so that
1

2 db c E t b c κ = 1 2 db t Eb t κ ,

or

√ E

b c
b =

√

E + √ E t .

Axial strain b

c

d

E t
E

b c

b

z

b t

(b) Strain
distribution

y

b
t

(a) Rectangular cross-section

Axial stress N / A E t b c

b c

b t

(c) Stress
distribution

Eb t

Figure 3.26 Reduced modulus buckling of a rectangular section member.