Keywords: dynamic buckling, shear deformation, rotary inertia, column.

In this paper, the critical load of columns has been investigated by the consideration of shear deformation and rotary inertia. Such an analysis is essential for performing proper evaluation of buckling load of columns made of composite materials and thick columns that are shear sensitive. Both static and dynamic buckling of columns is investigated. It is shown that the shear deformation effect always tends to reduce the buckling load. In this way, ignoring shear deformation in the case of thick columns or columns made of shear sensitive materials would result in unsafe designs. The effects of these modifications reduce considerably the more slender the column becomes. Furthermore, the shear deformation effect is observed to be the same for the static and dynamic loadings, at least for simply supported columns. The consideration of rotary inertia proved that for simply supported columns this phenomenon has no effect on the buckling load.

Instability is defined as a phenomenon, when a small increase in the applied axial force or pulse, results in a relatively large increase of the displacement response (bounded response changes to an unbounded one). The critical buckling load of perfect columns subjected to static axial loading is usually calculated by the application of the Euler-Bernoulli theory. In this model, the effects of shear deformation and rotary inertia are not included.

Generally, the Euler-Bernoulli theory is believed to present sufficiently accurate results for metallic columns with solid sections. This is especially true for static loadings on columns having large slenderness ratios. However, for cases where the E/G ratio is large (composite materials) the section is hollow, or the loading is dynamic in nature, modifications to the Euler-Bernoulli theory should be considered. Williams and Wittrick[1] have shown that buckling and vibration problems of columns are closely related, even though the first one is known to be essentially static and the second one inherently dynamic. They both result in the solution of eigenvalue problems. The eigenvalues being natural frequencies in vibration and load factors in buckling problems.

Amba-Rao[2] is one of the many authors who have proved that the increase of compressive axial load on a column reduces its natural frequency of vibration. Eventually, at the critical buckling load, the column would eventually have a zero natural frequency of vibration.

After years of studying pure static buckling problems, the investigation of dynamic buckling has attracted more attention in recent years. A comprehensive and valuable monograph on the subject of dynamic pulse buckling is that of Lindberg and Florence [3]. In this reference, one can find analyses and solutions to the dynamic buckling of various structures.

For thick columns, and also when higher modes are important, the shear deformation and rotary inertia effects become important, both in buckling and vibration. However, so far, these effects have been mostly studied in vibration and less in buckling. One may, however, indicate Banerjee and Williams[4] as one of few works on this subject.

The relatively greater importance attached to vibration problems when including shear deformation and rotary inertia is justified, because in buckling problems usually only the first critical buckling load of the structure is of practical importance whereas in vibration problems higher natural frequencies and modes are often required. However, there are cases with large E/G (like composites columns) where the sensitivity to shear deformation is large, even in the first mode. Therefore, a comprehensive study seems to be justifiable.

Ari-Gur and Elishakoff[5] have used a numerical finite difference procedure to show that for isotropic columns the static classic theory predicts accurately the dynamic buckling strength, as well.

The main object of the present paper is the evaluation of the individual and combined effect of shear deformation and rotary inertia effects in static and dynamic cases. The combined effect of these two phenomena has also been investigated. The method of solution is analytical and the stability analysis in the dynamic range is performed by the Routh-Hurwitz stability criterion.

1

F.W. Williams and W.H. Wittrick, "Exact buckling and frequency calculations surveyed", Journal of Structural Engineering, American Society of Civil Engineers, 109, 169-187, 1983. doi:10.1061/(ASCE)0733-9445(1983)109:1(169)

2

C.L. Amba-Rao, "Effect of end conditions on the lateral frequencies of uniform straight columns", Journal of the Acoustical Society of America," 42, 900-901, 1967. doi:10.1121/1.1910667

3

H.E. Lindberg and A.L. Florence, "Dynamic Pulse Buckling", Martinus Nijhoff Publishers, 1987.

4

J.R. Banerjee and F.W. Williams, "The Effect of Shear Deformation on the Critical Buckling of Columns," Journal of Sound and Vibration, 174(5), 607- 616, 1994. doi:10.1006/jsvi.1994.1297

5

J. Ari-Gur and I. Elishakoff, "Dynamic Instability of a Transversely Isotropic Column Subjected to a Compression Pulse" Computers & Structures, 62(5), 811-815, 1997. doi:10.1016/S0045-7949(96)00295-7

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