The Design of Modern Steel Bridges - TEDI

138 The Design of Modern Steel Bridges
5.5.2 The concept of optimum rigidity of stiffeners
If the buckling stress s cr is plotted against the relative flexural rigidity g for a
stiffened plate with ‘open’-type stiffeners, it can be found that initially s cr
increases with g; but after g exceeds a certain value g*, there is no further
increase in s cr. For g < g* the buckling of the stiffened plate involves bending
of the stiffeners out of the plane of the plate as in mode (a) of Fig. 5.25, i.e. the
overall buckling mode; but for g > g* the plate panels between the stiffeners
and/or the boundaries buckle without any bending of the stiffeners, i.e. the
stiffeners form the nodal lines of the buckling of the plate panels, as in mode
(b) of Fig. 5.25, i.e. local buckling of plate panels. There is thus an optimum
rigidity g* of the stiffeners for the maximum possible value of the buckling
stress s cr of the whole stiffened panel (i.e. for overall buckling), which
coincides with the elastic critical buckling stress of the individual plate panels
of the stiffened plate (i.e. for local buckling). No further increase in the
buckling load is possible by increasing g beyond this optimum value.
The above concept of the optimum rigidity g* is thus based on the
fundamental concept of the elastic critical buckling phenomenon of both the
entire stiffened panel and the individual plate panels in it. As has been pointed
out earlier, this concept is truly valid only for residual-stress-free perfectly flat
plates with high yield stress; for a stiffened panel, in addition to this
requirement the stiffeners must also be perfectly straight and residual-stressfree.
In reality, just as plates have residual stresses and out-of-plane imperfections,
so also the stiffeners have initial out-of-straightness and/or twist and
residual stresses. As a result stiffeners tend to deflect even at low levels of
applied loading. In the overall buckling mode, i.e. g < g*, there is thus often no
critical value of loading at which sudden buckling of the stiffeners occurs.
Instead, as the applied load is gradually increased, the deflection of the
stiffeners continues to increase at a gradually faster rate until no further
increase in load can be resisted. This maximum load is usually less than the
theoretical elastic critical value, but there are cases where, due to post-buckling
reserve, the maximum load is higher than the latter.
In the local buckling mode, i.e. g > g*, because of their initial crookedness,
the stiffeners start deflecting even at low levels of loading and thus do not form
non-deflecting nodal lines for the local buckling of the individual plate panels;
as a consequence the elastic critical buckling load of the individual plate panels
is often not reached even when they have very low residual stress and out-ofplane
imperfections. The rigidity of the stiffeners has thus got to be n-times the
theoretical optimum value g* in order to ensure buckling in mode (b), i.e. local
buckling, of Fig. 5.25. The value of n depends upon the geometry of the
stiffened panel and the type of loading and is found to vary from 2.5 to 5. Thus,
according to the linear buckling theory of stiffened plates, either (i) stiffeners
are provided with n-times the optimum rigidity g* that will ensure that the
overall critical buckling stress is equal to the local critical buckling stress, or