Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

5.4 Strength of compression members in practice
The highly idealized straight form assumed for the struts considered so far
cannot be achieved in practice. Members are never perfectly straight and they can
never be loaded exactly at the centroid of the cross section. Deviations from the ideal
elastic plastic behaviour defined by Fig. 5 are encountered due to strain hardening at
high strains and the absence of clearly defined yield point in some steel. Moreover,
residual stresses locked-in during the process of rolling also provide an added
complexity.

Thus the three components, which contribute to a reduction in the actual strength
of columns (compared with the predictions from the “ideal” column curve) are

(i) Initial imperfection or initial bow.
(ii) Eccentricity of application of loads.
(iii) Residual stresses locked into the cross section.
5.4.1 The effect of initial out-of-straightness

Fig 5.7 Pin -ended strut with initial imperfection
A pin-ended strut having an initial imperfection and acted upon by a gradually
increasing axial load is shown in Fig 5.7. As soon as the load is applied, the member
experiences a bending moment at every cross section, which in turn causes a bending

Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

deformation. For simplicity of calculations, it is usual to assume the initial shape of the
column defined by

y 0 = a 0 sin

πx
l

(5.8)

where ao is the maximum imperfection at the centre, where x = l / 2. Other
initial shapes are, of course, possible, but the half sine-wave assumed above
corresponding to the lowest mode shape, represents the greatest influence on the
actual behaviour, and hence is adequate.

Provided the material remains elastic, it is possible to show that the applied
force, P, enhances the initial deflection at every point along the length of the column by
a multiplier factor, given by

MF =

1
⎛ P ⎞
1− ⎜
⎟
⎝ Pcr ⎠

(5.9)

The deflection will tend to infinity, as P is tends to Pcr as shown by curve-A, in
Fig. 5.8. However the column will fail at a lower load Pf when the deflection becomes
large enough. The corresponding stress is denoted as ff

Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

Fig 5.8 Theoretical and actual load deflection response of a strut with initial
imperfection
If a large number of imperfect columns are tested to failure, and the data points
representing the values of the mean stress at failure plotted against the slenderness (λ)
values, the resulting lower bound curve would be similar to the curve shown in Fig. 5.9.

Fig 5.9 Strength curves for strut with initial imperfection
For very stocky members, the initial out of straightness – which is more of a
function of length than of cross sectional dimensions – has a very negligible effect and
the failure is at plastic squash load. For a very slender member, the lower bound curve
is close to the elastic critical stress (fcr) curve. At intermediate values of slenderness the
effect of initial out of straightness is very marked and the lower bound curve is
significantly below the fy line and fcr line.

5.4.2 The effect of eccentricity of applied loading
As has already been pointed out, it is impossible to ensure that the load is
applied at the exact centroid of the column. Fig. 5.10 shows a straight column with a
small eccentricity (e) in the applied loading. The applied load (P) induces a bending
moment (P.e) at every cross section. This would cause the column to deflect laterally,
in a manner similar to the initially deformed member discussed previously. Once again
the greatest compressive stress will occur at the concave face of the column at a
Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

section midway along its length. The load-deflection response for purely elastic and
elastic-plastic behaviour is similar to those described in Fig. 5.8 except that the
deflection is zero at zero load.

Fig 5.10 Strength curves for eccentrically loaded columns
The form of the lower bound strength curve obtained by allowing for eccentricity
is shown in Fig. 5.10. The only difference between this curve and that given in Fig. 5.9
is that the load carrying capacity is reduced (for stocky members) even for low values of
λ.

5.4.3 The effect of residual stress
As a consequence of the differential heating and cooling in the rolling and
forming processes, there will always be inherent residual stresses.

A simple

explanation for this phenomenon follows. Consider a billet during the rolling process
when it is shaped into an I section. As the hot billet shown in Fig. 5.11(a) is passed
successively through a series of rollers, the shapes shown in 5.11(b), (c) and (d) are
gradually obtained. The outstands (b-b) cool off earlier, before the thicker inner
elements (a-a) cool down.

Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

Fig 5.11 Various stages of rolling a steel grider
As one part of the cross section (b-b) cools off, it tends to shrink first but
continues to remain an integral part of the rest of the cross section. Eventually the
thicker element (a) also cools off and shrinks. As these elements remain composite
with the edge elements, the differential shrinkage induces compression at the outer
edges (b). But as the cross section is in equilibrium – these stresses have to be
balanced by tensile stresses at inner location (a). These stress called residual stresses,
can sometimes be very high and reach upto yield stress.

Fig. 5.12 Distribution of residual stresses
Consider a short compression member (called a “stub column”, having a residual
stress distribution as shown in Fig. 5.12. When this cross section is subjected to an
applied uniform compressive stress ( fa ) the stress distribution across the cross section
can be obtained by superposing the applied stress over the residual stress fr, provided

Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

the total stress nowhere reaches yield, the section continues to deform elastically.
Under incremental loading, the flange tips will yield first when [(fa + fr) = fy]. Under
further loading, yielding will spread inwards and eventually the web will also yield.
When fa = fy, the entire section will have yielded and the column will get squashed.

Only in a very stocky column (i.e. one with a very low slenderness) the residual
stress causes premature yielding in the manner just described. The mean stress at
failure will be fy, i.e. failure load is not affected by the residual stress. A very slender
strut will fail by buckling, i.e. fcr<< fy. For struts having intermediate slenderness, the
premature yielding at the tips reduces the effective bending stiffness of the column; in
this case, the column will buckle elastically at a load below the elastic critical load and
the plastic squash load. The column strength curve will thus be as shown in Fig. 5.13.

Notice the difference between the buckling strength and the plastic squash load
is most pronounced when
λ = l / r = π (E / fy) 1/2

Fig 5.13 Strength curve for columns having residual stress

Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

5.4.4 The effect of strain-hardening and the absence of clearly
defined yield point
If the material of the column shows strain harderning after an yield platean, the
onset of first yield will not be affected, but the collapse load may be increased.
Designers tend to ignore the effect of strain hardening which in fact provides an
additional margin of safety.
High strength steels generally have stress-strain curves without a clear yield
point. At stresses above the limit of proportionality (fp), the material behaviour is non
linear and on unloading and reloading the material is linear-elastic. Most high strength
structural steels have an ultimate stress beyond which the curve becomes more or less
horizontal. Some steels do not have a plastic plateau and exhibit strain-hardening
throughout the inelastic range. In such cases, the yield stress is generally taken as the
0.2% proof stress, for purposes of computation.

5.4.5 The effect of all features taken together
In practice, a loaded column may experience most, if not all, of the effects listed
above i.e. out of straightness, eccentricity of loading, residual stresses and lack of
clearly defined yield point and strain hardening occurring simultaneously. Only strain
hardening tends to raise the column strengths, particularly at low slenderness values.
All other effects lower the column strength values for all or part of the slenderness ratio
range.

When all the effects are put together, the resulting column strength curve is
generally of the form shown in Fig. 5.14. The beneficial effect of strain hardening at low
slenderness values is generally more than adequate to provide compensation for any
loss of strength due to small, accidental eccentricities in loading. Although the column
strength can exceed the value obtained from the yield strength (fy ), for purposes of
structural design, the column strength curve is generally considered as having a cut off
at fy, to avoid large plastic compressive deformation. Since it is impossible to quantify
the variations in geometric imperfections, accidental eccentricity, residual stresses and
Indian Institute of Technology Madras

Design of Steel Structures

Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

material properties, it is impossible to calculate with certainty, the greatest reduction in
strength they might produce in practice. Thus for design purposes, it may be impossible
to draw a true lower bound column strength curve. A commonly employed method is to
construct a curve on the basis of specified survival probability. (For example, over 98%
of the columns to which the column curve relates, can be expected - on a statistical
basis – to survive at applied loads equal to those given by the curve). All design codes
provide column curves based on this philosophy. Thus a lower band curve (Fig 5.14) or
a family of such curves is used in design.

Fig. 5.14 Column strength curves for struts used in practice

Indian Institute of Technology Madras