Plastic Deformations

When a cantilever beam surpasses its elastic limit under loading, yielding initiates at the fixed end where the maximum bending moment occurs. The beam transitions from reversible elastic deformation to permanent plastic deformation. Normal stresses exceed the yield strength, and a plastic hinge forms, representing the maximum load the beam can support before failure.

In a partially plastic section, normal stresses on the faces are uniformly distributed and equal to the yield strength. The vertical shear is distributed entirely over the elastic portion of the cross-section. As the elastic area decreases with increased loading, maximum shearing stress increases and eventually reaches the yield strength, contributing to ultimate beam failure.

A plastic hinge forms when a cantilever beam's section near the fixed end becomes fully plastic under sufficient loading. This hinge represents the maximum load capacity the beam can support. It marks the transition point where the beam shifts from elastic behavior to permanent deformation, and continued loading results in fracture.

Yielding begins at the fixed end because this location experiences the maximum bending moment when the cantilever beam is loaded at its free end. The concentrated stress at the fixed end causes normal stresses to exceed the yield strength first, initiating plastic deformation before other sections of the beam.

The boundary between elastic and plastic zones can be obtained by measuring the half-thickness of the elastic core. This measurement indicates where normal stresses transition from below the yield strength to equal the yield strength. As loading increases, this elastic core shrinks, and the plastic zone expands toward the beam's free end.

Shearing stress becomes critical as the elastic portion of a beam's cross-section decreases under loading. Maximum shearing stress increases as the elastic area shrinks and eventually reaches the yield strength. This shearing stress contribution accelerates the ultimate failure of the beam when combined with bending stresses.

Understanding plastic deformation is vital for designing structures that are both safe and economically viable. Engineers must ensure structures can withstand anticipated loads while accounting for the transition from reversible elastic behavior to irreversible plastic deformation. This knowledge prevents unexpected failures and optimizes material usage in load-bearing applications.