Compression members - The Behaviour and Design of Steel Structures to EC3 - page 79

This equation states that for equilibrium, the internal moment of resistance
EI (d 2 v /d x 2 ) must exactly balance the external disturbing moment − N cr v at any
pointalongthelengthofthemember.Whenthisequationissatisfiedatallpoints,
the displaced position is one of equilibrium.
Thesolutionofequation3.58whichsatisfiestheboundaryconditionatthelower
end that ( v ) 0 = 0is
v = δ sin π x

k cr L ,

where
1
k cr =

N cr
π 2 EI / L 2 ,

and δ is an undetermined constant. The boundary condition at the upper end that
( v ) L = 0 is satisfied when either

δ = 0
v = 0

(3.59)

or k cr = 1 / n in which n is an integer, so that

N cr = n 2 π 2 EI / L 2 , (3.60)

v = δ sin n π x / L . (3.61)

The first solution (equations 3.59) defines the straight stable equilibrium position
which is valid for all loads N less than the lowest value of N cr , as shown in
Figure 3.2b. The second solution (equations 3.60 and 3.61) defines the buckling
loads N cr atwhichdisplacedequilibriumpositionscanexist.Thissolutiondoesnot
determine the magnitude δ of the central deflection, as indicated in Figure 3.2b.
The lowest buckling load is the most important, and this occurs when n = 1,
so that

N cr = π 2 EI / L 2 ,

(3.2)

v = δ sin π x / L . (3.1)

3.8.2 Bending of members with initial curvature

The bending of the compression member with initial curvature shown in
Figure 3.2a can be analysed by considering the differential equilibrium equation

EI d 2 v

d x 2 =− N ( v + v 0 ) ,

(3.62)

whichisobtainedfromequation3.58forastraightmemberbyaddingtheadditional
bending moment − Nv 0 induced by the initial curvature.