# Residual Stresses in Bending

JoVE Core
Mechanical Engineering
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JoVE Core Mechanical Engineering
Residual Stresses in Bending
##### Video precedente20.11: Plastic Deformations of Members with a Single Plane of Symmetry

If a sufficiently large bending moment is applied to an elastoplastic member, plastic zones will develop in the member.

Now, if the bending moment is reduced to zero, the stress and strain reduction can be plotted as a straight line on the stress-strain plot.

Applying the bending moment from zero to its maximum value is called the loading phase, and the decrease in the bending moment from its maximum value to zero is called the unloading phase.

During the unloading phase, the stress will not be zero at any given point and may not have the same sign as during the loading phase, resulting in a residual stress.

The stress and strain have a linear relationship during the unloading phase, and elastic flexure formulas can be applied.

The residual stresses can be calculated using the superposition principle. First, the stresses due to the bending moment during the loading phase, corresponding to the elastoplastic nature, and the stresses due to the opposite bending moment during the unloading phase, corresponding to the elastic nature, are added to get the residual stress.

## Residual Stresses in Bending

In the study of elastoplastic members subjected to bending moments, understanding the loading and unloading phases is crucial for assessing material behavior and structural integrity. During the loading phase, as the bending moment increases, the material initially responds elastically, adhering to Hooke's Law, where stress is directly proportional to strain. When the load exceeds the yield strength, plastic deformation occurs, resulting in permanent strain and deformation that remains even after the load is removed.

The unloading phase begins as the bending moment decreases to zero. Unlike purely elastic materials, the stress in an elastoplastic member during unloading does not retrace the original loading path but instead follows a new, linear path. This change indicates the presence of residual stresses, which persist due to the irreversible plastic strains induced during loading.

These residual stresses are calculated using the superposition principle, combining stresses from the elastoplastic loading phase with those from the elastic unloading phase.

During unloading, the stress-strain relationship becomes linear again, allowing for the use of elastic flexure formulas. This phase is critical for engineering applications, as residual stresses can influence structural behavior under repeated loads, potentially leading to unexpected failures. Understanding these dynamics enables the design of safer, more reliable structures by accurately predicting and managing the stresses that arise from complex loading scenarios.