22.4:
Plastic Deformations
A cantilever beam, when subjected to a load at its free end, develops the maximum bending moment at its fixed end.
Until the bending moment surpasses the beam's elastic limit, the normal stress remains under the yield strength. Beyond this point, yielding is initiated from the fixed end.
The boundary between the elastic and plastic zones can be obtained by measuring the half-thickness of the elastic core.
When the load reaches a certain threshold, the section near the fixed end becomes fully plastic, forming a plastic hinge. This hinge represents the maximum load the beam can support.
In a partially plastic section, the normal stresses on the faces are uniformly distributed and equal to the yield strength.
The shearing force on the lower face is zero, resulting in a zero average value of the shearing stress. The vertical shear is distributed entirely over the elastic portion of the cross-section.
As the area of the elastic portion decreases, the maximum shearing stress increases and eventually reaches the yield strength, contributing to the ultimate failure of the beam.
22.4:
Plastic Deformations
Plastic deformation represents a fundamental concept in materials science, which explains the irreversible change in the shape of a material when it experiences stress beyond its elastic capability. This phenomenon is important in structural engineering, especially in designing and analyzing cantilever beams—structures that are securely fixed at one end and bear loads at the opposite end. When these beams are subjected to loads within their elastic range, they will return to their original shape when the load is removed, demonstrating the property of elasticity. However, surpassing this elastic boundary leads to plastic deformation, initiating at the fixed end due to the maximum bending moment and manifesting as a permanent bend known as a 'plastic hinge.'
As the loading increases, the plastic hinge progressively extends toward the beam's free end, marking the transition from reversible to irreversible deformation. This shift also alters the stress distribution within the beam from a uniform pattern, directly proportional to the applied load, to a more complex arrangement which defines the material's yield point. Continued loading will result in the beam's fracture.
Understanding the concept of plastic deformation is vital when designing structures that must be both safe and economically viable. It is crucial to ensure these structures can withstand their anticipated loads.