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.
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.
