Residual Stresses in Bending

During unloading, stress does not return to zero and may differ in sign from the loading phase, creating residual stress. The stress-strain relationship becomes linear during unloading, allowing elastic flexure formulas to apply. This contrasts with purely elastic materials, where stress retraces its original path.

Residual stresses are calculated using the superposition principle by combining two components: stresses from elastoplastic loading and stresses from elastic unloading with an opposite bending moment. Adding these stress distributions yields the final residual stress state at any point in the member.

When a sufficiently large bending moment exceeds the material's yield strength, the outer fibers undergo permanent deformation. This creates zones where material no longer obeys Hooke's Law, resulting in irreversible strain that persists after load removal. Understanding plastic deformations is essential for predicting member behavior.

The loading phase increases the bending moment from zero to maximum, causing elastic then plastic deformation. The unloading phase decreases the moment back to zero, following a linear stress-strain path that differs from the loading path, leaving residual stresses in the member.

Irreversible plastic strains induced during loading prevent the stress from retracing the original loading path. Instead, unloading follows a new linear path determined by elastic behavior, reflecting the permanent deformation that remains in the material after the load is removed.

Residual stresses persist in the member after unloading and can influence how structures respond to repeated loads. These stresses may lead to unexpected failures or altered structural performance, making accurate prediction and management critical for designing safer, more reliable structures.

Elastic flexure formulas apply during the unloading phase because the stress-strain relationship becomes linear again as the bending moment decreases. This linear behavior allows engineers to use standard elastic analysis methods to calculate stresses during unloading, simplifying residual stress determination.