Name: cantilever beam yield and failure
Uploaded: 2024-05-23T12:59:50.000Z
Duration: 1 min 19 s
Channel: JoVE

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&#39;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.

plastic deformations

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&#8212;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 &#39;plastic hinge.&#39;
As the loading increases, the plastic hinge progressively extends toward the beam&#39;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&#160; the material&#39;s yield point. Continued loading will result in the beam&#39;s fracture.
Understanding the concept of plastic deformation is vital when designing&#160; structures that must be both&#160; safe and&#160; economically viable. It is crucial to ensure these structures can withstand their anticipated loads.

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 ...

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JoVE Core

Mechanical Engineering

Unit 1

Shearing Stresses in Beams and Thin-Walled Members

shearing-stresses-in-beams-and-thin-walled-members

KEY TERMS AND CONCEPTS

cantilever beam yield and failure

Cantilever beams show how a material bends, yields, and can eventually fail under load. A cantilever beam is fixed firmly at one end and loaded at the other end. When the load stays within the elastic range, the beam returns to its original shape after the load is removed. That reversible behavior is called elasticity.

If the load goes beyond the elastic limit, the beam begins plastic deformation. Plastic deformation means the shape change is permanent. In a cantilever beam, this change starts at the fixed end because the bending moment is greatest there. The bent region at that point is called a plastic hinge.

As the load increases, the plastic hinge spreads toward the free end of the beam. This marks the shift from reversible deformation to irreversible deformation. The stress pattern in the beam also changes. It no longer stays uniform and directly proportional to the applied load, and this change helps define the yield point of the material.

If loading continues, the beam will fracture. Knowing how cantilever beams behave under stress is important in structural engineering. It helps engineers design structures that are safe and still economical. These structures must be able to handle the loads they are expected to carry.

Cantilever Beam Yield and Failure

shear on the horizontal face of a beam element

shear-on-the-horizontal-face-of-a-beam-element

Beam Shear Flow from Bending Moments

Beam shear flow comes from the horizontal shear that acts on a prismatic beam element. The element is also acted on by vertical shear forces and normal forces. To study this behavior, the beam is divided into upper and lower sections by the neutral axis. The force balance on the small element is written with an equilibrium equation. That step reveals the horizontal shearing force. This force is tied to the bending moments and to the cross-section’s first moment of inertia. The same result is...

distribution of stresses in a narrow rectangular beam

distribution-of-stresses-in-a-narrow-rectangular-beam

Shearing Stress Pattern in a Narrow Beam

Shearing stress in a narrow rectangular beam is found by studying a small elemental section of the beam. The average shearing stress on that face is calculated by dividing the shear by the surface area. This gives a simple way to describe how the load is spread through the beam. The shearing stresses on the beam&amp;#39;s transverse and horizontal planes match each other. That means the stress pattern is consistent in the upper region of the beam. Shearing stress is also absent at the top and...

shearing stresses in a beam: problem solving

shearing-stresses-in-a-beam-problem-solving

Beam Shear Stress Analysis and Safety Check

Beam shear stress analysis shows how loads create internal stress in a cantilever beam with a rectangular cross-section. The beam in this problem carries both a distributed load and a point load. The goal is to find the shearing stress at key points and check whether the beam stays within a safe limit. The first step is to identify all loads acting on the beam. Next, the reactions at the fixed end are found using equilibrium equations. The vertical reaction combines the distributed load and...

Cantilever beams show how a material bends, yields, and can eventually fail under load. A cantilever beam is fixed firmly at one end and loaded at the other end. When the load stays within the elastic range, the beam returns to its original shape after the load is removed. That reversible behavior is called elasticity. If the load goes beyond the elastic limit, the beam begins plastic deformation. Plastic deformation means the shape change is permanent. In a cantilever beam, this change starts...

unsymmetric loading of thin-walled members

unsymmetric-loading-of-thin-walled-members

Shear Center in Thin-Walled Members

Thin-walled members with non-symmetrical cross-sections are widely used in engineering structures because they save material while still providing strength. Their shape helps with structural integrity, but it also makes load analysis more complex. When these members are loaded in an unsymmetrical way, the stress is not spread evenly. Unsymmetrical loading can make a member bend and twist at the same time. That combined action can lead to deformation or even structural failure. Engineers must...

unsymmetric loading of thin-walled members: problem solving

unsymmetric-loading-of-thin-walled-members-problem-solving

Finding the Shear Center of a Channel Section

Finding the shear center of a channel section starts with the member’s shear force and moments of inertia. A shear center is the point where a load must act so the section bends without twisting. For a channel with uniform thickness, height, and width, the distance to this point can be calculated from the section geometry. The first step is to compute the shear flow at a set distance from the end of the flange. Shear flow is the force per unit length along the wall of the section. It is found...

key terms and concepts

Learn how a cantilever beam bends under load, forms a plastic hinge at the fixed end, reaches yield, and can fracture if loading continues.

Cantilever Beam Yield and Failure | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

Cantilever Beam Yield and Failure

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