Compression members - The Behaviour and Design of Steel Structures to EC3 - page 82

The moment of resistance of the section E r I ( d 2 v / d x 2 ) is equal to the moment
resultant of the bending stresses, so that

E r I d 2 v

d x 2 =− db c 2 E t b c κ 2 b c 3 − db t 2 Eb t κ 2 b t 3

or

E r I d 2 v

d x 2 =− 4 b c E t

db 3
12 κ . b 2

The reduced modulus of elasticity E r is therefore given by

4 EE t
( √ E + √ E t ) 2 .

E r = (3.16)

3.9.2 Buckling of members with residual stresses

Arectangularsectionmemberwithasimplifiedresidualstressdistributionisshown
inFigure3.10.Thesectionfirstyieldsattheedges z =± d / 2ataload N = 0.5 N y ,
and yielding then spreads through the section as the load approaches the squash
load N y .Ifthedepthoftheelasticcoreis d e ,thentheflexuralrigidityforbending
about the z axis is

d e
d

( EI ) t = EI

,

where I = b 3 d / 12, and the axial force is

1 − 1
2

d e
d 2

N = N y .

Bycombiningthesetworelationships,theeffectiveflexuralrigidityofthepartially
yielded section can be written as

( EI ) t = EI [ 2 ( 1 − N / N y ) ] 1 / 2 . (3.18)

Thus the axial load at buckling N t is given by

N cr , t

N cr =[ 2 ( 1 − N cr , t / N y ) ] 1 / 2 ,

which can be rearranged as

N cr
N y

2

N cr , t
N y

{[ 1 + 2 ( N y / N cr ) 2 ] 1 / 2 − 1 } . = (3.19)