Compression members - The Behaviour and Design of Steel Structures to EC3 - page 80

If the initial curvature of the member is such that

v 0 = δ 0 sin π x / L , (3.7)

then the solution of equation 3.62 which satisfies the boundary conditions
( v ) 0, L = 0 is the deflected shape

v = δ sin π x / L , (3.8)

where
δ
δ 0 =

N / N cr
1 − N / N cr .

(3.9)

Themaximummomentinthecompressionmemberis N (δ + δ 0 ) ,andsothemax-
imum bending stress is N (δ + δ 0 )/ W el , where W el is the elastic section modulus.
Thus the maximum total stress is
σ max = N

A + N (δ + δ 0 )

.

W el

Iftheelasticlimitistakenastheyieldstress f y ,thenthelimitingaxialload N L for
which the above elastic analysis is valid is given by

N L = N y − N L (δ + δ 0 ) A
W el

, (3.63)

where N y = Af y is the squash load. By writing W el = 2 I / b , in which b is the
member width, equation 3.63 becomes

N L
( 1 − N L / N cr ) ,
which can be solved for the dimensionless limiting load N L / N y as

N L = N y − δ 0 b
2 i 2

1 + (1 + η ) N cr / N y
2

1 / 2
, (3.64)

1 + ( 1 + η) N cr / N y
2

2

N L
N y =

− N cr
N y

−

where
η = δ 0 b

2 i 2 .

(3.13)

Alternatively, equation 3.64 can be rearranged to give the dimensionless limiting
load N L / N y as

N L
N y =

1

(3.11)

Φ 2 − λ 2 Φ +