# Members Made of Elastoplastic Material

JoVE 核
Mechanical Engineering

JoVE 核 Mechanical Engineering

### Next Video20.11: Plastic Deformations of Members with a Single Plane of Symmetry

Consider a member made up of an elastoplastic material with a rectangular cross-section. Within the elastic limit, the stress distribution across the section is linear.

As the bending moment increases, the maximum stress in the member increases. The maximum bending moment is observed when the deformation in the member remains fully elastic.

The maximum elastic moment of the section can be calculated by substituting the ratio of the moment of inertia and length of the section.

With a further increase in bending moment, the plastic deformation takes place uniformly but with an opposing magnitude of stresses in the upper and lower zones. The elastic cores are present within plastic zones, and the stress for elastic cores varies linearly with thickness.

The bending moment corresponding to the stress within the elastic cores can be estimated analytically using the maximum elastic moment equation.

As the bending moment is increased further to the limiting value, the deformation becomes fully plastic, and the plastic moment of the member is expressed in terms of the maximum elastic moment.

## Members Made of Elastoplastic Material

The behavior of elastoplastic materials under bending stresses, particularly in structural members with rectangular cross-sections, is crucial for predicting material responses and understanding failure modes. Initially, when a bending moment is applied, the stress distribution across the section follows Hooke's Law and is linear and elastic. This distribution means the stress increases from the neutral axis to the maximum at the outer fibers, up to the elastic limit.

As the bending moment surpasses this initial elastic phase, the outer fibers begin to yield while the inner fibers remain elastic. This transition is marked by plastic zones forming at the top and bottom of the section, with an elastic core that continues to exhibit linear stress variation within a reduced thickness. During this phase, the bending moment can still be analyzed by adapting the initial calculations for elastic stress distribution to account for the reduced effective area.

The final stage in the bending response occurs when the deformation across the entire cross-section becomes fully plastic, known as the plastic moment. This moment is the maximum the section can sustain and is significantly higher than the elastic limit. The plastic moment is calculated assuming a uniform stress distribution at the yield stress across the entire section.