Torque and Bending: How Force Affects Bending Length

[disgradius]

If I were to bend a cantilever by applying a certain amount of force at one end, would the distance it bends in a circle be linearly related to the torque resulting from the force? If not, would there be a measurable difference from being linear if I were to bend a, say, 0.5m long object with less than 20N? Thanks in advance! :D

 

[Cyrus]

Its going to matter what the modulus of the material is, and what is the stress in relation to the yield strength. How is the beam supported, where are the loads, and a whole bunch of other factors.

 

[FredGarvin]

For very small deflections, you can feel safe in assuming a linear relationship. However, there will be a fast approaching point that your deflection will be considered "large" in terms of classical beam theory in which case the problem become non-linear.

In classical beam theory, for small deflections, the equation for the maximum deflection of a cantilevered beam with the load applied to the very end is

$$\delta_{max}=\frac{PL^3}{3EI}$$

The deflection is linear with P, the load. This changes for a distributed load.

To determine what is a large deflection, the strains would have to be calculated. It does not take much to leave the realm of linear responses though.