The bar having a diameter of 2 in. is fixed connected at its...

The bar having a diameter of 2 in. is fixed connected at its ends and supports the axial load $\mathbf{P}$. If the material is elastic perfectly plastic as shown by the stressstrain diagram, determine the smallest load $P$ needed to cause segment $C B$ to yield. If this load is released, determine the permanent displacement of point $C$

Key Concepts

Elastic Perfectly Plastic Material Behavior

This concept describes a material whose stress–strain relationship is linear (elastic) up to a critical yield stress, after which the material deforms plastically at a constant stress regardless of further strain. In engineering analyses, this idealization is used to delineate between recoverable (elastic) strains and permanent (plastic) strains, and it underpins many limit load and residual deformation calculations.

Yield Load Calculation

Determining the load at which yielding begins involves relating the applied axial load to the resultant stress in the member’s cross?section and comparing it to the yield stress. In axial loading, the yield load is typically calculated from the cross-sectional area and the material’s yield strength. This concept is fundamental when assessing the onset of non?linear behavior in a structure subjected to an increasing load.

Residual Plastic Deformation

Residual or permanent plastic deformation is the irreversible strain that remains in a structure after an applied load is removed. In structures with elastic perfectly plastic behavior, once a segment yields, unloading does not return the part of the strain that has exceeded the elastic limit, leading to permanent displacement. This concept is crucial in designing structural components to ensure serviceability even after plastic deformations occur.

Axial Loading in Fixed-Connected Bars

The analysis of axial loading in members with fixed boundary conditions involves understanding how constraints influence stress and deformation distributions in the member. Fixed connections impose conditions that prevent rigid body motion, leading to non?uniform strain distributions when the load is applied. Such analyses help determine critical load levels for yielding in specific segments and predict residual displacements after unloading.

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