Compression members - The Behaviour and Design of Steel Structures to EC3 - page 54

3.2.2 Bending of members with initial curvature

Real structural members are not perfectly straight, but have small initial curva-
tures,asshowninFigure3.2a.Thebucklingbehaviourofthehypotheticalstraight
members discussed in Section 3.2.1 must therefore be interpreted as the limit-
ing behaviour of real members with infinitesimally small initial curvatures. The
initial curvature of the real member causes it to bend from the commencement
of application of the axial load, and this increases the maximum stress in the
member.
If the initial curvature is such that

v 0 = δ 0 sin π x / L , (3.7)

then the deflection of member is given by

v = δ sin π x / L , (3.8)

where

δ
δ 0 =

N / N cr
1 − N / N cr ,

(3.9)

as shown in Section 3.8.2. The variation of the dimensionless central deflection
δ/δ 0 is shown in Figure 3.2b, and it can be seen that deflection begins at the
commencement of loading and increases rapidly as the elastic buckling load N cr
is approached.
The simple load-deflection relationship of equation 3.9 is the basis of the
Southwell plot technique for extrapolating the elastic buckling load from experi-
mental measurements. If equation 3.9 is rearranged as

N = 1

δ

N cr δ + δ 0 N cr ,

(3.10)

thenthelinearrelationbetween δ/ N and δ showninFigure3.5isobtained.Thus,
if a straight line is drawn which best fits the points determined from experimen-
tal measurements of N and δ , the reciprocal of the slope of this line gives an
experimental estimate of the buckling load N cr . An estimate of the magnitude
δ 0 of the initial crookedness can also be determined from the intercept on the
horizontal axis.
As the deflections v increase with the load N , so also do the bending moments
andthestresses.ItisshowninSection3.8.2thatthelimitingaxialload N L atwhich
the compression member first yields (due to a combination of axial plus bending