# Pure bending of a perfectly elastic cantilever beam

• Shivam Sinha
In summary: The natural frequency is the frequency at which the beam oscillates about its equilibrium point, which is also the frequency at which the beam will deflect the most. If the force is linearly increased from 0 to F over time T, then maintained at force F, the beam will be steady (no oscillation) at position X.
Shivam Sinha
Hi,

My question is:

If a constant load is applied at the free end of a perfectly elastic cantilever beam, would the beam oscillate about a mean position? Or would it eventually come to rest at the equilibrium position? There are no damping effects.

Thank you.

If the constant load is applied slowly, it won’t oscillate at all. Only after a seismic or other weather related event or vibration induced motion will it oscillate about its loaded equilibrium point. Also, if the constant load is suddenly imparted to the free end, or dropped from a height, it will also oscillate about the equilibrium point ...forever if there is no damping forces, but in real world, eventually come to rest at equilibrium position. just like a spring , with a spring constant that depends on the material and geometric properties of the beam.

Randy Beikmann and Shivam Sinha
PhanthomJay said:
If the constant load is applied slowly, it won’t oscillate at all. Only after a seismic or other weather related event or vibration induced motion will it oscillate about its loaded equilibrium point. Also, if the constant load is suddenly imparted to the free end, or dropped from a height, it will also oscillate about the equilibrium point ...forever if there is no damping forces, but in real world, eventually come to rest at equilibrium position. just like a spring , with a spring constant that depends on the material and geometric properties of the beam.
Thank you! This is the answer I was looking for.

Just one question: You said it won't oscillate if the load is applied slowly. When you say that, do you mean that the load starts from zero and gradually increases to the maximum value?

Also, how slow should the load be applied? I believe that if the load is increased at a finite rate, there would still be oscillations.

Define load. Is it a constant force, or are you hanging a constant mass from the end.

Shivam Sinha said:
Thank you! This is the answer I was looking for.

Just one question: You said it won't oscillate if the load is applied slowly. When you say that, do you mean that the load starts from zero and gradually increases to the maximum value?
Yes, exactly. It might be an object that you hold in your hands while placing it on the beam, then gradually transfer more of the load onto the beam by lessening the grip of your hands.
.
Also, how slow should the load be applied? I believe that if the load is increased at a finite rate, there would still be oscillations.
apply it slowly enough so that the final release of your grip occurs at the full load equilibrium point

If the cantilever beam has mass and stiffness, it has a natural frequency. If a force F (just a force, no mass) is slowly applied, the beam will slowly deflect to a position X. If a constant force F (just a force, no mass) is suddenly applied to the tip of the beam, the beam will oscillate about position X at its natural frequency. The positive peak deflection will be 2X, the negative peak deflection will be zero. With zero damping, the oscillation will continue forever. Keep in mind that theoretical physics beams can have zero damping, while real world beams will always have some damping.

If the period of the natural frequency is T, and the force is linearly increased from 0 to F over time T, then maintained at force F, the beam will be steady (no oscillation) at position X.

## 1. What is pure bending of a perfectly elastic cantilever beam?

Pure bending of a perfectly elastic cantilever beam is a type of loading on a beam where the external forces act perpendicular to the axis of the beam, causing a uniform bending moment along the length of the beam. This type of loading is idealized and assumes that the beam is made of a perfectly elastic material, meaning it can withstand deformation without any permanent changes to its shape.

## 2. What is the equation for calculating the bending moment in pure bending?

The equation for calculating the bending moment in pure bending is M = -EI(d^2y/dx^2), where M is the bending moment, E is the modulus of elasticity of the beam, I is the area moment of inertia, y is the deflection of the beam, and x is the distance along the beam's length.

## 3. How does the maximum bending stress in a cantilever beam occur in pure bending?

In pure bending, the maximum bending stress occurs at the neutral axis, which is the axis of the beam that experiences no change in length due to the applied bending moment. This is because the fibers on the neutral axis are not stretched or compressed, while fibers above or below the neutral axis experience tension or compression, respectively.

## 4. What are the boundary conditions for a perfectly elastic cantilever beam in pure bending?

The boundary conditions for a perfectly elastic cantilever beam in pure bending include a fixed or clamped end at the point of attachment to the supporting structure, and a free end where there is no external force or moment acting. This means that the beam is able to rotate and deflect at the free end.

## 5. How does the bending moment vary along the length of a cantilever beam in pure bending?

In pure bending, the bending moment is constant along the length of the beam, assuming the external forces are also constant. This means that the beam will experience a uniform deflection and a linear variation in bending stress along its length. The maximum bending moment occurs at the fixed end, and the bending moment gradually decreases as it approaches the free end of the beam.

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