Name: dropped-weight bending of a cantilever beam
Uploaded: 2024-05-23T13:00:12.000Z
Duration: 1 min 13 s
Channel: JoVE

Consider a cantilever beam fixed at one end, having a circular cross-section.
If an object is dropped on the free end of the cantilever from a height h, then the potential energy of the mass is converted into kinetic energy. This kinetic energy is then transferred to the cantilever due to impact loading.
The maximum strain energy at the fixed end is expressed in terms of the bending moment, where the bending moment at a distance x from the free end is written as a negative product of the weight of the object and distance x.
Integrating the strain energy equation and rearranging the terms expresses the maximum load.
The maximum stress occurring is proportional to the maximum load and inversely proportional to the moment of inertia of the cantilever.
Substituting the value of the maximum load in terms of strain energy and rewriting the moment of inertia in terms of the volume of the cantilever gives the expression for maximum stress in terms of the modulus of elasticity and the strain energy developed in the cantilever.

impact loading on a cantilever beam

The analysis of a cantilever beam with a circular cross-section subjected to impact loading at its free end illustrates the conversion of potential energy from a dropped object into kinetic energy, which is then absorbed by the beam as strain energy. This process is crucial for understanding how materials behave under dynamic loads, which is important in fields such as construction and aerospace.
When an object is dropped onto the free end of a cantilever, its potential energy due to gravity is transformed into kinetic energy at the point of impact. This energy causes the beam to bend, creating a bending moment that varies along the length of the cantilever. The strain energy, which is the energy stored due to this bending, reaches its maximum at the fixed end of the beam. Integrating the strain energy across the beam helps assess the maximum load the beam can withstand before failure.
<img alt="F_m=&#8730;(6U_mEI/L^3) formula; structural analysis; mechanical equilibrium equation." src="/files/ftp_upload/15687/15687_Equation_1.svg" style="text-align: center; height: 38px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
This maximum load is critical for determining the maximum stress experienced by the beam. Stress depends on both the maximum load and the geometric properties of the beam, specifically the moment of inertia, which involves its radius for a circular section.
<img alt="Static equilibrium; formula: &#963;&#8344;=&#8730;(24U&#8344;E/V); equations for material stress analysis." src="/files/ftp_upload/15687/15687_Equation_2.svg" style="text-align: center; height: 38px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Ultimately, understanding the maximum stress in terms of the material's modulus of elasticity and the developed strain energy is essential for designing structures that are capable of withstanding unexpected dynamic loads without failing.

Impact Loading on a Cantilever Beam

Consider a cantilever beam fixed at one end, having a circular cross-section.
If an object is dropped on the free end of the cantilever from a height h, ...

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dropped-weight bending of a cantilever beam

Dropped-weight bending of a cantilever beam shows how impact loading changes a beam’s shape and stress. The beam has a circular cross-section and is fixed at one end. An object is dropped on the free end from a height h, so its potential energy from gravity becomes kinetic energy at impact.

After the object hits the beam, the beam bends and develops a bending moment along its length. The bending is not uniform. The fixed end carries the highest strain energy, which is the energy stored in the beam because of deformation. That makes the fixed end the key point for checking the beam’s response.

The total strain energy is found by integrating across the beam. This helps determine the maximum load the beam can support before failure. That load is then used to find the maximum stress in the beam. For a circular cross-section, the stress depends on the load and on the moment of inertia, which depends on the beam radius.

This type of analysis helps show how materials respond to dynamic loads. It is important in construction and aerospace, where unexpected impacts can matter. The result also connects the material’s modulus of elasticity with the strain energy developed in the beam. Designers use that relationship to build structures that can withstand sudden loading without failing.

Dropped-Weight Bending of a Cantilever Beam

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Dropped-weight bending of a cantilever beam shows how impact loading changes a beam’s shape and stress. The beam has a circular cross-section and is fixed at one end. An object is dropped on the free end from a height h, so its potential energy from gravity becomes kinetic energy at impact. After the object hits the beam, the beam bends and develops a bending moment along its length. The bending is not uniform. The fixed end carries the highest strain energy, which is the energy stored in the...

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key terms and concepts

See how a dropped object loads a circular cantilever beam, creates strain energy at the fixed end, and helps determine maximum stress and failure load.

Dropped-Weight Bending of a Cantilever Beam | JoVE K12

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Dropped-Weight Bending of a Cantilever Beam

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