Compression members - The Behaviour and Design of Steel Structures to EC3 - page 70

(b) Unbraced members

(a) Braced members

N N N N N N

M M M M M M

M M M

N N N N N N

−

( )

( ) ( ) ( ) ( ) ( )

3 EI N 6 EI N

2 2

2 EI N 4 EI N

π N / 4 N π N / N

2 EI EI

1 − 1 − 1 − 1 −

cr , L cr

N

, L

− −

L N L N L 4 N L 2 N , L cr , L cr

L 1 − N / N L 1 4 N /

cr
N

, L , L cr

cr , L cr , L

(while N < ) cr , L

Figure 3.19 Approximate stiffnesses of restraining members.

Similar analyses may be made of braced restraining members under other end
conditions, and some approximate solutions for the stiffnesses of these are sum-
marised in Figure 3.19. These approximations have accuracies comparable with
that of equation 3.38.
When a restraining member is unbraced, its ends sway ∆ as shown in
Figure3.20a,andrestoringendmoments M arerequiredtomaintainequilibrium.
Such a member which receives equal end moments is analysed in Section 3.10.3,
where it is shown that its stiffness is given by

2 EI
L

π
2 N cr , L tan π

N N

N cr , L

α =− (3.39)

2

when the axial load N is compressive, and by

2 EI
L

π
2 N cr , L tanh π

N N

N cr , L

α = (3.40)

2

when the axial load N is tensile. These relationships are shown in Figure 3.20b,
anditcanbeseenthatthestiffness α decreasesasthetensileload N decreases,and
becomesnegativeastheloadchangestocompressive.AlsoshowninFigure3.20b
is the approximation

(π 2 / 4 )( N / N cr , L )
( 1 − N / N cr , L ) ,

α =− 2 EI
L

(3.41)