Civil Engineering Reference
In-Depth Information
(a)
An approximate method
The combination of the basic critical loads may be taken into account by using
the Föppl-Papkovich theorem. According to the theorem, if a structure is
characterized by
n
stiffness parameters, the critical load can be approximated
by combining the
n
corresponding part critical loads [Tarnai, 1999]. Each part
critical load belongs to a case where all but one stiffness parameters are assumed
to be infinitely great. In applying the theorem to 3-dimensional buckling, the
combined critical load can be obtained from
(3.20)
where
N
cr,X
, N
cr,Y
and
N
cr,
ϕ
are the basic critical loads defined by formulae (3.11) and
(3.16). The advantage of formula (3.20) is that it is easy to use and it is always
conservative. However, its use in certain cases can lead to considerably uneconomical
structural solutions, as the error of the formula can be as great as 67%. The more
sophisticated and only slightly more complicated exact solution is given in the next
section.
(b)
The exact method
The exact solution of the simultaneous differential equations (3.1) to (3.3)
leads to the determinant
(3.21)
which defines the coupling of the basic modes and which can be used for the calculation
of the critical load of the combined sway-torsional buckling if the basic critical loads
(
N
cr,X
, N
cr,Y
and
N
cr,
ϕ
) are known. The expansion of the determinant results in a cubic
equation in the form
N
3
+
a
2
N
2
+
a
l
N
-
a
0
=0,
(3.22)
where the coefficients are
