INELASTIC BUCKLING THEORY Ravi Shanker 2011CES3024

Structural Engineering

Abstract: Structural members subjected to axial compressive loads may fail in a manner that depends on their geometrical properties rather than their material properties. A long slender structural member, when subjected to axial compressive load, may suddenly bow with large lateral displacements. Such a failure is known as buckling. Theoretically, buckling is caused by a bifurcation in the solution to the equations of static equilibrium. At a certain stage under an increasing load, further load is able to be sustained in one of two states of equilibrium: an un-deformed state or a laterally-deformed state. The linear elastic analysis is valid for slender columns and the Euler load represents the correct buckling load of such members. But when it comes to the relatively short columns, where before reaching the critical load material crosses its proportional limit, then the inelastic buckling analysis comes in picture. In this report we are going to briefly discuss about the inelastic analysis and its methods. Introduction: Inelastic buckling includes those buckling phenomenon during which before buckling failure, the proportional limit of the material is exceeded somewhere within the cross section. For usual material properties, generally this will be the case for a column that has a length less than 20 to 25 times its diameter. Such columns generally buckle inelastically, i.e. permanent deformations will occur upon reaching the critical buckling load. If residual stresses are a factor, inelastic buckling can occur when the compressive stress due to applied load plus the local residual compressive stress locally exceeds the material proportional limit, a condition often reached at the tips or corners of wide flange steel columns. In 1889, Considere and Engesser concluded that Euler’s formula was valid only for slender columns. They suggest that in order to apply Euler’s formula to short columns, the constant modulus E should be replaced by an effective modulus that depends upon the magnitude of stress at buckling. According to Engesser, the tangent modulus is the correct effective modulus for inelastic column buckling. However, Considere suggested that as the column begins to bend at the critical load there is possibility that the stress on the concave side increases in accordance with the tangent modulus and that the stress on the convex side decreases in accordance with the young’s modulus. This line of reasoning is the basis of the Double modulus theory. According to the Double modulus theory, the effective modulus is a function of both the tangent modulus and elastic modulus. For the next 30 years this theory was accepted as the correct theory for inelastic buckling. Then in 1947, Shanley re-examined the behaviour mechanism of inelastic buckling and concluded that the tangent modulus and not the double modulus is the correct effective modulus. The double modulus theory is based on assumption that the axial load remains constant as the column passes from straight to slightly bent configuration at the critical load. Due to this assumption only, bending necessarily make a decrease in strain on the convex side of the member while strain on the concave side are increasing. Shanley pointed out that it is possible for the axial load to increase instead of remaining constant as the column begins to bend, and that no strain reversal need therefore take place at any point in the cross section. If there is no strain reversal, the tangent modulus governs the behaviour of all fibres in the member at buckling. The tangent modulus theory leads to a lower buckling load than the double modulus theory and agrees better with the test results than the latter. It has therefore been accepted by the most engineers as the correct theory of inelastic buckling.

Double/Reduced Modulus Theory:- The analysis involves the following assumptions:

1. The column is initially perfectly straight and concentrically loaded. 2. Both ends of the member are hinged. 3. The deformations are small. 4. Plane sections before bending remains plane after bending.

In this theory, the loading path O-B-A as shown in Fig. 1 is assumed. That is, an axial force P >Py is applied first. Keeping P constant, a small lateral disturbance 훿Q and thus a small bending moment ∆M is applied next. When the disturbance force 훿Q is removed, a bent equilibrium position will be attained if the axial force P is equal to the reduced-modulus load Pr.

Fig 1 Loading path during buckling

As the axial force P is kept constant.

∆푃 = 훾 = 0 Then, the reduced-modulus is obtained from

퐸 =퐸퐼 + 퐸 퐼

퐼

Where Er = Reduced modulus Et = Tangent modulus

Tangent Modulus Theory:

In this theory, the assumptions made in the double modulus theory are retained. However, one assumption that the axial load remains constant as the column passes from the straight to a slightly bent position of equilibrium, no longer applies. Instead it assumes that the axial load increases during the transition from the straight to the slightly bent position.

Strictly speaking, this theory applies only to members whose non-linear stress- strain curve are elastic so that the loading and unloading path are identical as shown in fig 2 .

Straight form Bent form

Fig 2 Tangent Modulus Theory

Consider a column that is initially straight and remains straight until the axial load P equals the critical load. The column then moves from the straight position to slightly bent configuration, and axial load increases from P to P+∆P. It is assumed that P is large enough, relative to the bending moment at any section, so that the stress at all points in the member increases as bending takes place. Since deformation beyond the critical load are assumed to be infinitesimally small, the increase in stress ∆휎 that occurs during bending is very small compared to the critical stress 휎cr , and Et corresponding to 휎cr can be assumed to govern the increase in stress at all point in the member.

푃 =휋 퐸 퐼푙

The above expression is generally referred as tangent modulus load.

Shanley Theory:

Although the tangent modulus theory appears to be invalid for the inelastic material careful experimentation shows that it leads to more accurate prediction than the apparently rigorous reduced modulus theory. This paradox was resolved by the Shanley, who reasoned that the tangent modulus theory is valid when buckling deflections are accompanied by the simultaneous increases in the applied load of sufficient magnitude to prevent strain reversal, as shown in fig below.

P

P

Shanley conducted very careful experiments on small aluminum columns. He found that lateral deflection started very near the theoretical tangent modulus load and the load capacity increased with increasing lateral deflections. The column axial load capacity never reached the calculated reduced or double modulus load. This leads to the following conclusions:

1. An initially straight column will begin to bend as soon as the tangent modulus load is exceeded. 2. The maximum value of axial load lies somewhere between the tangent modulus load and the reduced

modulus load. 3. Strain reversal occurs as soon as the bending deformations are finite.

Conclusion: Thus it can be concluded that the tangent modulus load is very close to the maximum load that an inelastic column can support. The difference between the tangent modulus and the double modulus theory can be summarized as follows: the double modulus theory assumes that the axial load remains constant as the column moves from the straight to a slightly bent position, at critical load. Hence the compressive stresses increases according to Et on the concave side of the member and decreases according to E on the convex side. However in the tangent modulus theory, the load is assumed to increase during the transition to the bent form. There is no stress reversal anywhere in the member, and the increase in stress is governed by Et at all points in the cross section.

The difference between Shanley's theory and the tangent-modulus theory are not significant enough to justify a much more complicated formula in practical applications. This is the reason why many design formulas are based on the overly-conservative tangent-modulus theory.

Reference:

1. Chajes, Alexander, principle of structural stability ,prentice hall 2. Trahair, N.S. , flexure torsional buckling of structure,first edition 1993, pg 275 E&FN SPON 3. Iyengar, NGR , structural stability of columns and plates,EWP 4. en.wikipedia.org/wiki/Buckling 5. http://www.efunda.com/formulae/solid_mechanics/columns/inelastic.cfm