Compression members - The Behaviour and Design of Steel Structures to EC3 - page 53

The elastic buckling load N cr and the elastic buckling stress

σ cr = N cr / A (3.3)

can be expressed in terms of the geometrical slenderness ratio L / i by

N cr = σ cr A = π 2 EA
( L / i ) 2

(3.4)

in which i = √ ( I / A ) is the radius of gyration (which can be determined for a
number of sections by using Figure 5.6). The buckling load varies inversely as
the square of the slenderness ratio L / i , as shown in Figure 3.4, in which the
dimensionlessbucklingload N cr / N y isplottedagainstthegeneralisedslenderness
ratio

N y
N cr = σ cr = L

f y f y

π 2 E

λ = (3.5)

i

in which

N y = Af y (3.6)

is the squash load. If the material ceases to be linear elastic at the yield stress f y ,

then the above analysis is only valid for λ = √ ( N y / N cr ) = √ ( f y /σ cr ) ≥ 1. This
limit is equivalent to a slenderness ratio L / i of approximately 85 for a material
with a yield stress f y of 275 N/mm 2 .

1.2

Complete
yielding

1.0

0.8

Elastic buckling N cr / N y
(equations 3.4 and 3.5)

0.6

0.4

First yield N L / N y
due to initial curvature
(equations 3.11-3.14)

0.2

0

0 0.5 1.0 1.5 2.0 2.5

Generalised slenderness = N /

y N cr

Figure 3.4 Buckling and yielding of compression members.