Compression members - The Behaviour and Design of Steel Structures to EC3 - page 73

then its elastic buckling load N cr can be expressed in the general form of
equation 3.23 when the effective length ratio k cr = L cr / L is the solution of

γ 1 γ 2 (π/ k cr ) 2 − 36
6 (γ 1 + γ 2 )

= π

k cr cot π k cr ,

(3.46)

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

γ 1 = ( 6 EI / L ) 12 (3.47)

α

1

in which the summation

1 α is for all the other members at end 1, and γ 2 is
similarly defined.The relative stiffnesses may also be expressed as

( 6 EI / L ) 12
1.5
1

γ 1
1.5 + γ 1

k 1 =

α + ( 6 EI / L ) 12 =

(3.48)

and a similar definition of k 2 . Values of the effective length ratio L cr / L which
satisfy equation 3.46 are presented in chart form in Figure 3.21b.
In using the chart of Figure 3.21b, the stiffness factors k may be calculated
fromequation3.48byusingthestiffnessapproximationsofFigure3.19.Thusfor
braced restraining members of the third type shown in Figure 3.19a,

( I / L ) 12

1.5
1

k 1 =

( I / L )( 1 − N / 4 N cr , L ) + ( I / L ) 12 .

(3.49)

3.6.6 Bracing stiffness required for a braced member

The elastic buckling load of an unbraced compression member may be increased
substantially by providing a translational bracing system which effectively pre-
vents sway. The bracing system need not be completely rigid, but may be elastic
as shown in Figure 3.22, provided its stiffness α exceeds a certain minimum
value α L . It is shown in Section 3.10.6 that the minimum value for a pin-ended
compression member is

α L = π 2 EI / L 3 . (3.50)

This conclusion can be extended to compression members with rotational end
restraints, and it can be shown that if the sway bracing stiffness α is greater than

α L = N cr / L , (3.51)

where N cr is the elastic buckling load for the braced mode, then the member is
effectively braced against sway.