The prismatic bar A B is made of a steel that is assumed to...

The prismatic bar $A B$ is made of a steel that is assumed to be elastoplastic and for which $E=200$ GPa. Knowing that the radius of curvature of the bar is $2.4 \mathrm{m}$ when a couple of moment $M=350 \mathrm{N} \cdot \mathrm{m}$ is applied as shown, determine (a) the yield strength of the steel. (b) the thickness of the elastic core of the bar.

Key Concepts

Elastoplastic Material Behavior

This concept describes materials that exhibit elastic behavior up to a certain stress level (the yield strength) and then undergo plastic deformation when the stress exceeds that level. In elastoplastic analysis, the material initially responds linearly but eventually yields and deforms permanently. Understanding this behavior is crucial when analyzing structures that might partially yield under high loads, such as bending in beams.

Yield Strength

Yield strength is the stress at which a material begins to deform plastically. In the context of bending, it determines the point where the outer fibers of a beam start to yield. This is key to designing structures for safety as it indicates the limit of linear-elastic behavior, beyond which permanent deformations occur.

Moment-Curvature Relationship

This principle relates the bending moment applied to a beam to the curvature of its deflection curve. Under purely elastic conditions, the curvature is proportional to the moment according to the material's elastic modulus and the moment of inertia of the cross-section. In elastoplastic analysis, modifications to this relationship must be made to account for the yielding of material, affecting how curvature develops past the elastic limit.

Elastic Core in Bending

The elastic core refers to the region within a beam’s cross-section that remains in the elastic, unyielded state despite the outer fibers undergoing plastic deformation. Determining the thickness of the elastic core is important in assessing how much of the structure is still capable of carrying additional load without permanent deformation, which plays a critical role in the performance and safety evaluation of bending members.

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