BUCKLING OF COLUMNS BASIC AND TUTORIALS

Columns are compression members whose cross-sectional dimensions are relatively small compared with their length in the direction of the compressive force. Failure of such members occurs because of instability when a certain axial load Pc (called critical or Euler load) is equated or exceeded. The member may bend, or buckle, suddenly and collapse.

Hence the strength P of a column is not determined by the unit stress (P = Aƒ) but by the maximum load it can carry without becoming unstable. The condition of instability is characterized by disproportionately large increases in lateral deformation with slight increase in axial load. Instability may occur in slender columns before the unit stress reaches the elastic limit.

Stable Equilibrium
Consider, for example, an axially loaded column with ends unrestrained against rotation, shown in Fig. 5.43. If the member is initially perfectly straight, it will remain straight as long as the load P is less than the critical load Pc.

If a small transverse force is applied, the column will deflect, but it will return to the straight position when this force is removed. Thus, when P is less than Pc, internal and external forces are in stable equilibrium.


Unstable Equilibrium
If P = Pc and a small transverse force is applied, the column again will deflect, but this time, when the force is removed, the column will remain in the bent position (dashed line in Fig. 5.43).

The equation of this elastic curve can be obtained from Eq. (5.62):
EI d2y/dx2 = -pCY


in which E modulus of elasticity
I = least moment of inertia
y = deflection of the bent member from the straight position at a distance
x = from one end

SHANLEY'S THEORY OF INELASTIC BUCKLING OF STEEL MEMBERS BASIC AND TUTORIALS

SHANLEY'S THEORY OF INELASTIC BUCKLING OF STEEL MEMBERS BASIC INFORMATION
What Is Shanley's Theory of Inelastic Buckling Of Steel Members?



Although the tangent modulus theory appears to be invalid for inelastic materials, careful experiments have shown that it leads to more accurate predictions than the apparently rigorous reduced modulus theory.

This paradox was resolved by Shanley [1], who reasoned that the tangent modulus theory is valid when buckling is accompanied by a simultaneous increase in the applied load (see Figure 3.8) of sufficient magnitude to prevent strain reversal in the member.

When this happens, all the bending stresses and strains are related by the tangent modulus of elasticity Et , the initial modulus E does not feature, and so the buckling load is equal to the tangent modulus value Ncr,t .

As the lateral deflection of the member increases as shown in Figure 3.8, the tangent modulus Et decreases (see Figure 3.6b) because of the increased axial and bending strains, and the post-buckling curve approaches a maximum load Nmax which defines the ultimate resistance of the member.

Also shown in Figure 3.8 is a post-buckling curve which commences at the reduced modulus load Ncr,r (at which buckling can take place without any increase in the load). The tangent modulus load Ncr,t is the lowest load at which buckling can begin, and the reduced modulus load Ncr,r is the highest load for which the member can remain straight.

It is theoretically possible for buckling to begin at any load between Ncr,t and Ncr,r . It can be seen that not only is the tangent modulus load more easily calculated, but it also provides a conservative estimate of the member resistance, and is in closer agreement with experimental results than the reduced modulus load.

For these reasons, the tangent modulus theory of inelastic buckling has gained wide acceptance.