# Pure bending of a perfectly elastic cantilever beam

• Shivam Sinha

#### 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
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.

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.
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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.