The bifurcation of equilibrium of a compressed transversely isotropic bar is investigated by using a three-dimensional elasticity formulation. In this manner, an assessment of the thickness effects can be accurately performed. For isotropic rods of circular cross-section, the bifurcation value of the compressive force turns out to coincide with the Euler critical load for values of the length-over-radius ratio approximately greater than 15. The elasticity approach predicts always a lower (than the Euler value) critical load for isotropic bodies; the two examples of transversely isotropic bodies considered show also a lower critical load in comparison with the Euler value based on the axial modulus, and the reduction is larger than the one corresponding to isotropic rods with the same length over radius ratio. However, for the isotropic material, both Timoshenko’s formulas for transverse shear correction are conservative; i.e., they predict a lower critical load than the elasticity solution. For a generally transversely isotropic material only the first Timoshenko shear correction formula proved to be a conservative estimate in all cases considered. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one and therefore, a more precise estimate of the transverse shear effects. Furthermore, by performing a series expansion of the terms of the resulting characteristic equation from the elasticity formulation for the isotropic case, the Euler load is proven to be the solution in the first approximation; consideration of the second approximation gives a direct expression for the correction to the Euler load, therefore defining a new, revised, yet simple formula for column buckling. Finally, the examination of a rod with different end conditions, namely a pinned-pinned rod, shows that the thickness effects depend also on the end fixity.

1.
Abramowitz, M., and Stegun, I. A., 1964, Handbook of Mathematical Functions, Applied Mathematics Series Vol. 55, Washington National Bureau of Standards (reprinted 1968 by Dover Publications, New York, Sect. 9.8).
2.
Brush, D. O., and Almroth, B. O., 1975, Buckling of Bars, Plates, and Shells, McGraw-Hill, New York.
3.
Ciarlet, P. G., 1988, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, North Holland, Amsterdam.
4.
Danielson
D. A.
, and
Simmonds
J. G.
,
1969
, “
Accurate Buckling Equations for Arbitrary and Cylindrical Elastic Shells
,”
International Journal of Engineering Science
, Vol.
7
, pp.
459
468
.
5.
Elliott
H. A.
,
1948
, “
Three-Dimensional Stress Distribution in Hexagonal Aeolotropic Crystals
,”
Proceedings of the Cambridge Philosophical Society
, Vol.
44
, pp.
522
533
.
6.
Euler, L., 1744, 1933, De Curvis Elasticis, Vol. 20, No. 58, P. 1, Nov. 1933, Bruges, Belgium (English translation of the book “Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes,” 1744, Lausanne).
7.
Flu¨gge, W., 1960, Stresses in Shells, Springer, pp. 426–432.
8.
Horgan
C. O.
,
1989
, “
Recent Developments Concerning Saint-Venant’s Principle: An Update
,”
ASME Applied Mechanics Reviews
, Vol.
42
, No.
11
, pp.
295
303
.
9.
Kardomateas
G. A.
,
1993
a, “
Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
60
, pp.
195
202
.
10.
Kardomateas
G. A.
,
1993
b, “
Stability Loss in Thick Transversely Isotropic Cylindrical Shells Under Axial Compression
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
60
, pp.
506
513
.
11.
Kardomateas
G. A.
,
1995
, “
Bifurcation of Equilibrium in Thick Orthotropic Cylindrical Shells Under Axial Compression
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
62
, pp.
43
52
.
12.
Kardomateas
G. A.
, and
Chung
C. B.
,
1994
, “
Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure Based on Non-Planar Equilibrium Modes
,”
International Journal of Solids and Structures
, Vol.
31
, No.
16
, pp.
2195
2210
.
13.
Kumar
A.
, and
Niyogi
B. B.
,
1982
, “
Bifurcations in Axially Compressed Thick Elastic Tubes
,”
International Journal of Engineering Science
, Vol.
20
, No.
12
, pp.
1311
1324
.
14.
Ogden
R. W.
,
1972
, “
Large Deformation Isotropic Elasticity: on the Correlation of Theory and Experiment for Compressible Rubberlike Solids
,”
Proceedings of the Royal Society London
, Vol.
A328
, pp.
567
583
.
15.
Simitses, G. J., 1986, An Introduction to the Elastic Stability of Structures, Krieger.
16.
Timoshenko, S. P., and Gere, J. M., 1961, Theory of Elastic Stability, McGraw-Hill, New York.
This content is only available via PDF.