As shown three decades ago, in situations where the initial stresses before buckling are not negligible compared to the elastic moduli, the geometrical dependence of the tangential moduli on the initial stresses must be taken into account in stability analysis, and the stability or bifurcation criteria have different forms for tangential moduli associated with different choices of the finite strain measure. So it has appeared paradoxical that, for sandwich columns, different but equally plausible assumptions yield different formulas, Engesser’s and Haringx’ formulas, even though the axial stress in the skins is negligible compared to the axial elastic modulus of the skins and the axial stress in the core is negligible compared to the shear modulus of the core. This apparent paradox is explained by variational energy analysis. It is shown that the shear stiffness of a sandwich column, provided by the core, generally depends on the axial force carried by the skins if that force is not negligible compared to the shear stiffness of the column (if the column is short). The Engesser-type, Haringx-type, and other possible formulas associated with different finite strain measures are all, in principle, equivalent, although a different shear stiffness of the core, depending linearly on the applied axial load, must be used for each. The Haringx-type formula, however, is most convenient because it represents the only case in which the shear modulus of the core can be considered to be independent of the axial force in the skins and to be equal to the shear modulus measured in simple shear tests (e.g., torsional test). Extensions of the analysis further show that Haringx’s formula is preferable for a highly orthotropic composite because a constant shear modulus of the soft matrix can be used for calculating the shear stiffness of the column, and further confirm that Haringx’s buckling formula with a constant shear stiffness is appropriate for helical springs and built-up columns (laced or battened).

1.
Biot, M. A., 1965, Mechanics of Incremental Deformations. John Wiley and Sons, New York.
2.
Bazˇant, Z. P., and Cedolin, L., 1991, Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories, Oxford University Press, New York (and 2nd updated Ed., Dover, New York, 2002).
3.
Bazˇant
,
Z. P.
,
1968
, “
Conditions of Deformation Instability of a Continuum and Their Application to Thick Slabs and a Half Space
” (in Czech, with English summary),
Stavebnı´cky Cˇasopis (SAV, Bratislava)
,
16
, pp.
48
64
.
4.
Bazˇant
,
Z. P.
,
1971
, “
A Correlation Study of Incremental Deformations and Stability of Continuous Bodies
,”
ASME J. Appl. Mech.
,
38
, pp.
919
928
.
5.
Goodier
,
J. N.
, and
Hsu
,
C. S.
,
1954
, “
Nonsinusoidal Buckling Modes of Sandwich Plates
,”
J. Aeronaut. Sci.
,
21
, pp.
525
532
.
6.
Plantema, F. J., 1966, Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates and Shells, John Wiley and Sons, New York.
7.
Allen, H. G., 1969, Analysis and Design of Sandwich Panels, Pergamon Press, Oxford, UK.
8.
Kovarˇı´k, V., and Sˇlapa´k, P., 1973, Stability and Vibrations of Sandwich Plates, Academia, Prague (in Czech).
9.
Michiharu
,
O.
,
1976
, “
Antisymmteric and Symmetric Buckling of Sandwich Columns Under Compressive Loads
,”
Trans., Jap. Soc. of Aeronaut. Space Sci.
,
19
, pp.
163
178
.
10.
Chong, K. P., Wang, K. A., and Griffith, G. R., 1979, “Analysis of Continuous Sandwich Panels in Building Systems,” Building and Environment 44.
11.
Frostig
,
Y.
, and
Baruch
,
M.
,
1993
, “
Buckling of Simply Supported Sandwich Beams With Transversely Flexible Core—A High Order Theory
,”
J. Eng. Mech.
119
, pp.
955
972
.
12.
Engesser
,
F.
,
1889
, “
,”
Zentralblatt des Bauverwaltung.
,
11
, p.
483
486
.
13.
Engesser
,
F.
,
1889
, “
,”
Z. Architekten und Ing. Verein zu Hannover
,
35
, p.
455
455
.
14.
Engesser
,
F.
,
1891
, “
,”
Zentralblatt der Bauverwaltung
,
11
, pp.
483
486
.
15.
Haringx, J. A., 1942, “On the Buckling and Lateral Rigidity of Helical Springs,” Proc., Konink. Ned. Akad. Wetenschap., 45, p. 533.
16.
Haringx, J. A., 1948–1949, Phillips Research Reports, Vols. 3–4, Phillips Research Laboratories, Eindhoven.
17.
Kardomateas
,
G. A.
,
1995
, “
Three Dimensional Elasticity Solution for the Buckling of Transversely Isotropic Rods: The Euler Load Revisited
,”
ASME J. Appl. Mech.
,
62
, pp.
346
355
.
18.
Kardomateas
,
G. A.
, and
Dancila
,
D. S.
,
1997
, “
Buckling of Moderately Thick Orthotropic Columns: Comparison of an Elasticity Solution With the Euler and Engesser/Haringx/Timoshenko Formulas
,”
Int. J. Solids Struct.
,
34
(
3
), pp.
341
357
.
19.
Kardomateas
,
G. A.
,
2001
, “
Elasticity Solutions for Sandwich Orthotropic Cylindrical Shell Under External Pressure, Internal Pressure and Axial Force
,”
AIAA J.
,
39
(
4
), pp.
713
719
.
20.
Kardomateas, G. A., 2001, “Three-Dimensional Elasticity Solutions for the Buckling of Sandwich Columns,” ASME Intern. Mechanical Engrg. Congress, pp. 1–6.
21.
Kardomateas, G. A., and Huang, H., 2002, “Buckling and Initial Postbuckling Behavior of Sandwich Beams Including Transverse Shear,” AIAA J., in press.
22.
Kardomateas, G. A., Simitses, G. J., Shen, L., and Li, R., 2002, “Buckling of Sandwich Wide Columns,” Int. J. Non-Linear Mech., (special issue on “Nonlinear Stability of Structures”), in press.
23.
Simitses, G. J., and Shen, L., 2000, “Static and Dynamic Buckling of Sandwich Columns,” Mechanics of Sandwich Structures, Y. D. S. Rajapakse et al., eds., ASME, New York, AD-Vol. 62/AMD-Vol. 245, pp. 41–50.
24.
Timoshenko, S. P., and Gere, J. M., 1961, Theory of Elastic Stability, McGraw-Hill, New York, pp. 135, 142.
25.
Bazˇant
,
Z. P.
,
1992
, discussion of “
Stability of Built-up Columns
,” by A. Gjelsvik,
J. Eng. Mech.
,
118
(
6
), pp.
1279
1281
.
26.
Bazˇant
,
Z. P.
,
1993
, discussion of “
Use of Engineering Strain and Trefftz Theory in Buckling of Columns
,” by C. M. Wang and W. A. M. Alwis
ASME J. Appl. Mech.
,
119
(
12
), pp.
2536
2537
.
27.
Ziegler
,
F.
,
1982
, “
Arguments for and Against Engesser’s Buckling Formulas
,”
Ingenieur-Archiv
,
52
, pp.
105
113
.
28.
Reissner
,
E.
,
1972
, “
On One-Dimensional Finite-Strain Beam Theory: The Plane Problem
,”
J. of Applied Mathematics and Physics
,
23
, pp.
795
804
.
29.
Reissner
,
E.
,
1982
, “
Some Remarks on the Problem of Column Buckling
,”
Ingenieur-Archiv
,
52
, pp.
115
119
.
30.
Simo
,
J. C.
, and
Kelly
,
J. M.
,
1984
, “
The Analysis of Multilayer Elastomeric Bearings
,”
ASME J. Appl. Mech.
,
51
, pp.
256
262
.
31.
Simo
,
J. C.
,
,
K. D.
, and
Taylor
,
R. L.
,
1984
, “
Numerical Formulation of Elasto-Viscoplastic Response of Beams Accounting for the Effect of Shear
,”
Comput. Methods Appl. Mech. Eng.
,
42
, pp.
301
330
.
32.
Gjelsvik
,
A.
,
1991
, “
Stability of Built-Up Columns
,”
J. Eng. Mech.
,
117
(
6
), pp.
1331
1345
.
33.
Wang
,
C. M.
, and
Alwis
,
W. A. M.
,
1992
, “
Use of Engineering Strain and Trefftz Theory in Buckling of Columns
,”
J. Eng. Mech.
,
118
(
10
), pp.
2135
2140
.
34.
Attard, M. A., 2002, draft of a manuscript on “Finite Strain Beam Theory,” University of New South Wales, Australia (private communication to Bazˇant, 2002).
35.
Buckle
,
I.
,
Nagarajaiah
,
S.
, and
Ferell
,
K.
,
2002
, “
Stability of Elastomeric Isolation Bearings: Experimental Study
,”
J. Eng. Mech.
,
128
(
1
), pp.
3
11
.
36.
Timoshenko
,
S. P.
,
1921
, “
On the Correction for Shear in the Differential Equation of Transverse Vibrations of Prismatic Bars
,”
Philos. Mag.
,
21
, p.
747
747
.