INELASTIC BUCKLING THEORY
Introduction
The Euler formula derived by Leonhard Euler in 1744 (Euler, 1759) for the elastic critical
buckling load of a slender column is the earliest engineering design formula (Johnston, 1983).
This formula is based on Hooke’s law; therefore, the Euler formula is applicable for a very
slender column that fails by buckling when the material is still linear-elastic. On the other hand,
very short columns fail by yielding and crushing the material, and hence their strength depends
solely on the ultimate compressive strength of the material (Huang et al., 2011). Between the
regions of short and long columns, there is a range of intermediate slenderness ratios too small
for elastic stability to govern and too large for strength considerations alone to control such an
intermediate-length column fails by inelastic buckling (Gere & Timoshenko, 1991). Instability
of the column occurs, but the maximum stress exceeds the proportional limit.
Objective
This report aims to study the inelastic column buckling theories proposed by different
researchers over the years and present the related experimental validations to the proposed
theories.
Historical review of inelastic buckling theories
In 1889, Considère proposed that the elastic modulus
used in the Euler’s equation to be replaced by an effective
modulus, 𝐸𝑒𝑓𝑓 after performing 32 column tests. He
stated that the effective modulus should be somewhere
between the elastic modulus 𝐸 and the tangent modulus
𝐸𝑡 (Figure 1). Considère (Considere, 1891) and Engesser
(Engesser, 1889) independently suggested that column
strength in the inelastic range might be obtained simply
by the substitution of 𝐸𝑡 in the place of Euler formula.
Thus, for a pinned end and perfectly straight column with
small deformations, the critical load can be expressed as:
Figure 1. Typical compressive
stress-strain curve (Salmon &
Johnson, 1996)
(1)
𝜋 2 𝐸𝑡 𝐼
𝑙2
This expression is generally referred to as the tangent modulus load. The tangent modulus
depends on the concurrent strain levels, which vary through the cross-section and may even be
subjected to elastic strain reversal on the convex side of the member due to bending.
𝐹𝑡 =
In 1910, Von Karman (Kármán, 1910) derived explicit expressions based on the “reduced
modulus” for both the rectangular and the idealized H-section column (Johnston, 1983). This
theory assumes that the axial load remains constant when the column bends marginally at the
critical load, which also accounts for the strain reversal that was neglected in the tangent
modulus theory. The tangent modulus governs the stress-strain relationship in the concave side,
𝐸𝑡 while the Young’s modulus, 𝐸 governs the relation in the convex side when the buckling
occurs. An effective modulus (also termed as reduced modulus, 𝐸𝑟 ), which lies between 𝐸 and
𝐸𝑡 is introduced to replace 𝐸𝑡 in the tangent-modulus formula. Thus, for a pinned end and
perfectly straight column, the critical load can be expressed as:
𝜋 2 𝐸𝑟 𝐼
𝑙2
The reduced modulus for the rectangular section is expressed as:
𝐹𝑟 =
𝐸𝑟 =
4𝐸𝐸𝑡
(2)
(3)
2
(√𝐸 + √𝐸𝑡 )
The reduced modulus of an 𝐼 section, with two flanges of the equal-area connected by a web
of negligible thickness, is expressed as:
2𝐸𝐸𝑡
(4)
𝐸 + 𝐸𝑡
The tangent-modulus theory is relatively simpler and independent of the shape of the crosssection. It gives a conservative value of the critical load and is widely used in practice. Shanley
(Shanley, 1946) stated that the reduced-modulus (or double-modulus) theory is not correct for
predicting the maximum load up to which a perfect column will remain straight. The column
can undergo bending simultaneously with increasing axial load. The tangent modulus varies
across the cross-section and no more constant as larger finite deformation is considered. Under
such conditions, bending without introducing any strain reversal can occur, upon which the
reduced-modulus theory depends. Also, the column will begin to bend as soon as the axial load
exceeds the value predicted by the tangent-modulus (Engesser) theory. Shanley (Shanley,
1947) states that the Engesser load could be exceeded but that the reduced-modulus load could
not be reached, as shown in Figure 2.
𝐸𝑟 =
Figure 2: Load-deflection curves given by various column buckling theories (Huang et al., 2011)
In 1939 Osgood and Holt (Osgood & Holt, 1939) conducted the experiment and stated that the
curve based on the tangent modulus of elasticity shows good agreement with the data. The
curve based on the double-modulus theory lies somewhat above the test results in the region of
plastic action of the material, as shown in Figure 3.
Figure 3: Column theories and test data (Osgood & Holt, 1939)
Shanley (Shanley, 1947) adopted a model consisting of a two-legged hinged column which is
assumed to be rigid elements. A deformable ‘unit’ cell forms the hinge, which is made of two
small axial elements and is negligible compared to the column length. This arrangement
simplifies the problem as the load is estimated based on the stresses/strains in these axial
elements. The axial load can be expressed as:
𝐴𝐸𝑡
1
(1 +
)
(5)
1 𝑘+1
𝑙
+
2𝑑 𝑘 − 1
Where d is the lateral deflection, 𝑘 = 𝐸 ⁄𝐸𝑡 , and 𝐹𝑡 (Engesser) = 𝐴𝐸𝑡 ⁄𝑙, the column load
increases with an increase in the value of 𝑑, and the increment depends on the value of 𝑘. The
assumptions made in the reduced modulus theory is used to determine the axial load, 𝐹, which
is expressed as:
𝐹=
𝐴𝐸𝑡
𝑘−1
(6)
(1 +
)
𝑙
𝑘+1
Eq. (6) can also be derived from Eq. (5) with 𝑑 → ∞ , i.e., for large lateral deflection giving
the limiting value for the column load (as per reduced modulus theory). The actual limiting
value depends on the variation of Et with the strain. Thus, all the previous equations can be
derived from Eq. (6), i.e., Euler’s equation (k=1), tangent modulus equation (d=0), and reduced
modulus equation (𝑑 → ∞).
𝐹=
Conclusion
Researchers have proposed various theories based on inelastic buckling. The following
conclusions can be made for the inelastic buckling of the column:
• The tangent modulus equation (Engesser equation) provides the load at which
the initially perfectly straight column starts to bend.
• The axial column load increases with the increase in the lateral deflection and
reaches a maximum value is between the Engesser (tangent modulus) load and
reduced modulus load.
• The decrease in the tangent modulus with the increase in the strain plays limits
the column load's value exceeding the tangent modulus load.
• The tangent modulus load should be used to determine the strength of members
as it gives value slightly on the conservative side.
References
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Procédés de Construction, Libraire Polytechnique, Paris, 3, 371.
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Hannover, 35(4), 455–562.
Euler, L. (1759). Sur la force des colonnes. Memoires de l’academie Des Sciences de Berlin,
13, 252–282.
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Science+Business Media Dordrecht (3rd ed.).
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and
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