properties of the element itself (specifically, on
the geometric properties of its cross-section
and on the modulus of elasticity of the mater-
ial). For a particular amount of curvature (i.e.
of bending strain), the size of the restoring
couple generated by the bending stress is
always the same, but the size of the disturbing
couple caused by the eccentricity depends on
the magnitude of the compressive load.
Compressive elements are therefore potentially
unstable internally depending on the relation-
ship between these couples. The bending type
failure which results from this type of instabil-
ity is known as buckling.
The critical factor in determining the suscep-
tibility of a particular element to buckling, is
the magnitude of the applied compressive load
(Fig. A2.3). If this is small the disturbing couple
will also be small, even if the eccentricity is
large, and the restoring couple will increase at
a faster rate than the disturbing couple if some
sideways-acting external agency destroys the
original straight alignment of the element. The
system is stable because the restoring couple
will always be able to return the element to its
original straight condition when the external
agency is removed. If the compressive load is
large, however, this will produce a disturbing
couple which is greater than the restoring
couple at all levels of curvature and the
element will be unstable because any external
agency which introduces a small amount of
curvature will precipitate a progressive increase
in curvature until buckling failure occurs. For a
particular compressive element the magnitude
of the compressive load at which instability
develops is known as the critical load [Pcr].
The analysis of buckling is one of the classic
problems of structural design and many
mathematicians and engineers have investi-
gated methods for predicting the critical loads
of structural components. Perhaps the best
known of these is due to the eighteenth-
century Swiss mathematician Leonhard Euler
and, although the formula which Euler derived
for the calculation of critical loads is not
suitable for most practical designs, it is never-
theless described here because the study of it
provides a good introduction to the factors on
which the stability of compressive elements
depends.Eider's analysis of buckling
Euler's analysis, which is described in detail in
Pippard and Baker, The Analysis of Engineering
Structures (4th Edition, Edward Arnold, London,
1984), is a theoretical investigation of a perfect
strut, that is of a compressive element which is
perfectly straight initially and in which no
eccentricity is present, either in the element
itself or in the application of the load. It yields
the following formula for the critical load of a
perfect strut with hinged end connections,Pcr = π^2 EI/L^2 (A2.7)where Pcr = the Euler critical load
E = the modulus of elasticity of the
material
/ = the second moment of area of
the cross-section of the strut
L = the length of the strutThe Euler critical load for an ideal strut is
equivalent to the buckling strength of a real
strut. In the case of the ideal strut, a curvature
must be deliberately introduced to cause
buckling failure and the compressive load
concerned must be greater than the critical
load. Eccentricity is always present in a real
strut, however, and so if a compressive load
greater than the critical load is applied to it
the strut will automatically fail by buckling.
It will be seen that Euler identified the
slendemess of a compressive element as the
most significant factor which determines the
critical load, with slenderness being defined, in
the most basic version of the formula as L^2 /I. The
more slender the element, i.e. the higher the
slenderness ratio, the smaller is the critical load.
It is significant that the quantity by which the
'thickness' of the element is judged in the deter-
mination of its slenderness is the second
moment of area of the cross-section (/). This is,
of course, a measure of the bending perform-
ance of the cross-section and its use in this
context of compressive stability is not surprising
because buckling is a bending phenomenon.Appendix 2253