The prismatic rod shown is made of a steel that is assumed...

The prismatic rod shown is made of a steel that is assumed to be elastoplastic with $E=200 \mathrm{GPa}$ and $\sigma_{Y}=280 \mathrm{MPa}$. Knowing that couples $\mathbf{M}$ and $\mathbf{M}$ ' of moment $525 \mathrm{N} \cdot \mathrm{m}$ are applied and maintained about axes parallel to the $y$ axis, determine (a) the thickness of the elastic core, $(b)$ the radius of curvature of the bar.

Key Concepts

Elastoplastic Material Behavior

This concept describes materials that exhibit a linear elastic response up to a certain yield point, after which they deform plastically with a constant stress. In structural analysis, especially when dealing with metals like steel, understanding elastoplastic behavior is critical for predicting how the material will respond under high loads that exceed the elastic limit.

Pure Bending and Beam Theory

Under pure bending, the beam’s cross-section undergoes a linearly varying strain distribution, with one side in tension and the opposite in compression. The theory enables the establishment of the relationship between applied bending moments, curvature, and the resulting stresses across the section, forming the basis for analyzing and designing beams.

Plastic Hinge Formation and Elastic Core

When a beam is subjected to bending moments that exceed the yield moment of its material, plastic regions develop at the outer fibers while a central elastic core remains where stresses are below the yield limit. Determining the thickness of the elastic core involves analyzing the stress distribution and equilibrium conditions, which is essential for understanding the partial plastic behavior of the beam.

Moment-Curvature Relationship

This relationship links the applied bending moment to the curvature of a beam while taking into account the material’s stiffness and geometric properties. In the elastic range, it is defined by the product of the modulus of elasticity and the second moment of area (EI). In the elastoplastic regime, the bending moment and curvature relationship is modified to account for yielding effects, influencing the calculation of the radius of curvature.

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