5.3 Flexural buckling with in-plane shear
5.3.5 Demonstration of the buckling modes with shear
The critical force formulae derived above will be discussed and illustrated in this Section. The presented figures belong to the major-axis flexural buckling of an IPE400 steel column, but it is to emphasize that in this Section only the buckling behaviour and tendencies are presented and the formulae are discussed in general (therefore, the actual numerical values are not important at all). The length of the column is changing in an extremely wide range so that the differences between the various options would be more visible. Here, only the 4 consistent options are discussed.
5.3.5.1 Option nnn
In case of nnn option only one single critical force exists, as illustrated in Figure 5.6. The critical force and also the deformation is combined from two components: the classical shear-free flexural mode and the transverse-only shear mode. The formula, see Eq. (5.47) is a well-known interaction formula, well-known as Dunkerley-type summation formula. (More precisely, it is an example when the resulting buckling mode can exactly be calculated in accordance with the Föppl-Papkovich theorem, see [9/5].)
As it can clearly be seen from Figure 5.6, and can also be concluded from the critical force formula, for greater buckling lengths the shear-free mode is dominant, therefore, the critical force is practically identical to that calculated from Euler-formula. On the other hand, for small lengths the pure shear mode is dominant, that is why a (practically horizontal) shear plateau is found in the buckling force diagram. In Figure 5.6 (as well in subsequent similar figures) these modes are illustrated, which are, therefore, not pure modes, but dominant modes for the given section of the buckling curve.
As far as the bending term is concerned, there is a clear difference between (5.47) and the Euler-formula, namely the (1-2) term in the denominator. This is clearly the consequence of the applied shell model and the global buckling definition which does not allow transverse extension/shortening of the strips (x = 0), as already discussed in Section 5.2.
It is also to mention that the derived solution for the nnn option is identical to the classical beam-model-based formula in [8/5], or also to the one (as a special case) in [11/5], if (but only if) the bending rigidity is calculated with neglecting the own plate inertias (i.e., the bt3/12 terms), and the Poisson’s ratio is set to zero to avoid the artificial increase of rigidity.
Figure 5.6: Illustration of critical forces and buckling modes in nnn option
5.3.5.2 Option nyy
In case of nyy option only one single critical force exists, as illustrated in Figure 5.7. The critical force and also the deformation is combined from three components: (i) the classical shear-free (primary) flexural mode, (ii) the transverse-only shear mode, and (iii) a secondary bending mode. The first two modes are identical to those found in option nnn. As far as primary and secondary bending modes are concerned, both are pure bending modes, however, in the primary mode the flexural rigidity is calculated with neglecting the own plate inertias (i.e., the bt3/12 terms), while in the secondary mode only the own plate inertias are considered in the rigidity. In most of practical thin-walled sections the secondary terms are small and are frequently neglected.
The formula of Eq (5.48) is a combination of two summation formulae: the primary bending mode and the pure shear mode is combined according to Dunkerley summation formula, while the secondary bending term is simply added to the combined effect of the two others, as it is the case in Southwell summation, see [9/5]. The secondary bending mode has little effect unless the column is extremely short (Figure 5.7), or the shear rigidity is very low.
It might be interesting to mention that if the shear rigidity is infinitely large, the formula tends to the classical Euler-formula. It is still slightly different from option nnn, however, since in option nyy the classical solution includes bending rigidity with considering the own plate inertias (i.e., includes the effect of secondary rigidity).
Figure 5.7: Illustration of critical forces and buckling modes in nyy option 5.3.5.3 Option ynn
In case of ynn option there are two or three critical forces. One is Fa which belongs to so-called axial mode. The other one or two critical values can be calculated by solving the equation from Eq. (5.49):
, 0
, ,
, ,
,
2 ,
X r sZ
a Z s r X Z s r X a
r
X F F
F F F F
F F F
F F
If none of FX,r and Fs,Z is zero, the equation has two distinct real positive roots. However, if any of them zero (which might be a practical case), there is only one critical solution.
(5.50)
Let us explore the tendencies at extreme length values. If L 0, then 1FX,r 0, which leads to two critical forces as follows:
a ynn
X
cr F
F , and Fcrynn,X Fs,Z
In case of any regular material Fs,Z < Fa, therefore the smaller critical force tends to Fs,Z as L approaches zero.
In case of large L-s it is fair to assume that 1 Fa and 1Fs,Z are negligible small compared to
r
FX,
1 , from which two critical forces can be found:
r X ynn
X
cr F
F , , and
r X
a Z ynn s
X
cr F
F F F
, ,
,
It means that in the ynn option four buckling modes are involved, though only two appear in the mode that belongs to the lowest eigen-value, as shown in Figure 5.8.
Figure 5.8: Illustration of critical forces and buckling modes in ynn option 5.3.5.4 Option yyy
In case of yyy option, three critical forces can always be found. Altogether 5 buckling modes are involved, i.e., all the 5 modes that appear in other options. The modes and tendencies are demonstrated in Figure 5.9. As it can be seen from the figure (and can also be proved mathematically), all the three critical forces tend to Fa as L approaches zero. On the other hand, three distinct critical forces are found for large L-s, namely (in ascending order):
r X yyy
X
cr F
F , , , Fcryyy,X Fa, and
r X
a Z yyy s
X
cr F
F F F
, ,
,
(5.51)
(5.52)
(5.53)
Figure 5.9: Illustration of critical forces and buckling modes in yyy option