20.9:
Plastic Deformations
For pure bending, when maximum stress exceeds the yield strength of the member's material, it undergoes plastic deformation.
The strain at any point in the member is expressed in terms of maximum strain. For plastic deformations, the neutral axis of the member may not pass through the centroid of the member.
Here, the neutral axis of the member is located using an iterative method until the stress distribution curve is formed.
For members with a vertical and horizontal plane of symmetry and the same stress-strain relationship for both axes, the neutral axis coincides with the horizontal axis of symmetry. The stress distribution curve can be plotted using a specified maximum stress value.
Here, the bending moment of such a member is given in terms of the member's width and the stress at the distance y from the neutral axis. The ultimate value of the bending moment that causes a member to fail can be found experimentally.
This ultimate bending moment gives the corresponding maximum stress, known as the modulus of rupture.
20.9:
Plastic Deformations
It is essential to understand how structural members behave under plastic deformation when the bending stress exceeds the material's yield strength. This state of deformation permanently alters the shape of the member, in contrast to the linear elastic behavior observed before yielding. The strain at any point in the member is expressed in terms of maximum strain. Notably, the neutral axis, which coincides with the centroid during elastic bending, shifts away from the centroid under plastic conditions.
Locating the neutral axis in this altered state involves an iterative method, where the assumed position of the axis is adjusted until the stress distribution curve stabilizes. This process is particularly straightforward in members symmetric about vertical and horizontal planes, with identical stress-strain responses along these axes, allowing the neutral axis to align with the horizontal plane of symmetry.
The stress distribution for such symmetric members can be mapped using maximum stress values as a function of distance from the neutral axis. Experimentation determines the ultimate bending moment—the maximum moment which a member can endure before failure. This moment correlates to the modulus of rupture Ru, which is proportional to the Mu, which is the bending moment at which the member fails. Mu is a critical value representing the material's ultimate strength under bending stresses. Understanding these principles allows the design of structures that can withstand higher loads without catastrophic failure, enhancing safety and efficiency.