Compression members - The Behaviour and Design of Steel Structures to EC3 - page 71

Exact
equation 3.40

4

M

N

2

Tension Compression

4 3 2 1 1

0

Dimensionless axial load N / N cr,L

L

-2

Approx.
equation 3.41

Exact
equation 3.39

x

-4

N

M

-6

(a) Unbraced member

(b) Stiffness

Figure 3.20 Stiffness of an unbraced member.

whichiscloseandconservativewhentheaxialload N iscompressive, andwhich
errsonthesafesidewhentheaxialload N istensile.ItcanbeseenfromFigure3.20b
that the stiffness of the member is negative when it is subjected to compression,
so that it disturbs the buckling member, rather than restraining it.
Similaranalysesmaybemadeofunbracedrestrainingmemberswithotherend
conditions, and another approximate solution is given in Figure 3.19.This has an
accuracy comparable with that of equation 3.41.

3.6.4 Buckling of braced members with end restraints

The buckling of an end-restrained compression member 1-2 is analysed in
Section 3.10.4, where it is shown that if the member is braced so that it cannot
sway (∆ = 0 ) , then its elastic buckling load N cr can be expressed in the gen-
eral form of equation 3.23 when the effective length ratio k cr = L cr / L is the
solution of

π
k cr

2 γ 1 + γ 2

2

1 − π

γ 1 γ 2
4 k cr cot π

+ tan (π/ 2 k cr )
π/ 2 k cr

+ = 1, (3.42)

k cr

where the relative stiffness of the braced member at its end 1 is

γ 1 = ( 2 EI / L ) 12 , (3.43)

α

1