STRUCTURAL DESIGN FOR ARCHITECTURE

same as the actual length of the hinge-ended
element which has the same buckling strength
and which is identical to it in every other
respect. Thus the effective length of an
element with hinged ends is the same as its
actual length. That of an element with fixed
ends is approximately 0.5 times its actual
length and that of an element with one end
fixed and the other completely free is twice its
actual length (Fig. A2.7).

Limitations of the Euler formula
Because the assumptions on which it is based
are not strictly valid, the Euler formula does
not predict accurately the critical load of real
elements. In the Euler analysis it is assumed
that the critical load value is achieved while
the stress in the element is within the elastic
range of the material and that instability is due
to the inability of the bending stress to resist
the bending moment which is caused by the
eccentricity which is present. It is not neces-
sary for the stress in the material to pass the
elastic limit for this to be possible. The elastic
limit of the material would of course eventually
be exceeded when the element failed but is
not exceeded in the initial stages of the failure.
The element does not fail therefore due to
yielding of the most heavily stressed part, but
because the disturbing couple is greater than
the restoring couple while the stress is within
the elastic range. The system is unstable, in
other words. The phenomenon is known as
elastic buckling and it occurs in real structures
only if they are very slender.
In most real structures the failure mecha-
nism is slightly different because when a real
element is subjected to an increasing amount
of compressive load the maximum stress in the
cross-section, which of course is the sum of
the axial stress and the compressive bending
stress, usually becomes greater than the yield
stress of the material before the Euler critical
load is reached (an indication of the relation-
ship between 'ideal' and 'real' behaviour is
given in Fig. A2.5). This causes a sudden
increase in the lateral deflection which initi-
ates buckling when the load is less than the
Euler critical value. Most real elements there-

Fig. A2.7 The concept of effective length. The effective

length of a compression element depends on the end

conditions. Effective lengths for different end conditions

are illustrated here. (Note that these are theoretical values

The values which are used in practice are different for

different materials and will be found in design codes.)

fore fail at load levels which are smaller than

the Euler critical load and the less slender the

element the greater is the discrepancy between

its true failure load and the failure load which

is predicted by the Euler formula. The extent of

the discrepancy also depends on the type of

structure of which the element forms part. The

type of material which is used is a particularly

influential factor - the behaviour of a masonry

pier, for example, is significantly different from

that of a steel column.

In practice, therefore, while the design of

compressive elements is based on procedures

which are similar to the one which has been

outlined in connection with the Euler formula,

the precise details of the procedures are differ-

ent for different structural materials. The

sequence of operations is broadly the same as

the one given above; a cyclic process is used to

arrive at a suitable size and shape for the 257

Appendix 2

Both ends

hinged

Both ends

fixed

L 0.5L

0.7L

Points of contraflexure

One end hinged,

one end fixed

One end fixed,

one end free

2L