Compression members - The Behaviour and Design of Steel Structures to EC3 - page 69

Approx.
equation 3.38

N

4

Exact
equation 3.35

M

2

Exact
equation 3.37

Tension Compression

3 3 2 2 1 1 2 3

0

Dimensionless axial load N / N cr,L

L

v

-2

x

M

-4

N

-6

(a) Braced member

(b) Stiffness

Figure 3.18 Stiffness of a braced member.

and by

(π/ 2 ) N / N cr , L
tanh (π/ 2 ) N / N cr , L

α = 2 EI
L

(3.37)

when the axial load N is tensile. These relationships are shown in Figure 3.18b,
and it can be seen that the stiffness decreases almost linearly from 2 EI / L to zero
asthecompressiveaxialloadincreasesfromzeroto N cr , L ,andthatthestiffnessis
negative when the axial load exceeds N cr , L . In this case the adjacent member no
longer restrains the buckling member, but disturbs it. When the axial load causes
tension, the stiffness is increased above the value 2 EI / L .
Also shown in Figure 3.18b is the simple approximation

α = 2 EI
L

N
N cr , L

1 − . (3.38)

Theterm ( 1 − N / N cr , L ) inthisequationisthereciprocaloftheamplificationfac-
tor, which expresses the fact that the first-order rotations ML /2 EI associated with
the end moments M are amplified by the compressive axial load N to ( ML /2 EI )/
(1 − N / N cr , L ). The approximation provided by equation 3.38 is close and con-
servative in the range 0 < N / N cr , L < 1, but errs on the unsafe side and with
increasing error as the axial load N increases away from this range.