Compression members - The Behaviour and Design of Steel Structures to EC3 - page 68

∆

N

M 2

M 1 + M 2 + N
L

∆

N

2

2

v

EI, L

x

1 1

M 1 + M 2 + N
L

∆

N

M 1

N

(a) Frame (b) Member

Figure 3.17 Buckling of a member of a rigid frame.

exert total elastic restraining moments M 1 , M 2 which oppose buckling and which
are proportional to the end rotations θ 1 , θ 2 of the compression member.Thus

M 1 =− α 1 θ 1
M 2 =− α 2 θ 2

(3.33)

in which

α 1 = α , (3.34)

1

where α is the stiffness of any adjacent member connected to the end 1 of the
compression member, and α 2 is similarly defined. The stiffness α of an adjacent
member depends not only on its length L and flexural rigidity EI but also on its
support conditions and on the magnitude of any axial load transmitted by it.
The particular case of a braced restraining member, which acts as if simply
supported at both ends as shown in Figure 3.18a, and which provides equal and
opposite end moments M and has an axial force N , is analysed in Section 3.10.2,
where it is shown that when the axial load N is compressive, the stiffness α is
given by

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

α = 2 EI
L

(3.35)

where

N cr , L = π 2 EI / L 2 , (3.36)