Deriving Strain in Cantilever Beam with Known Deflection

[randall016]

I am trying to derive the strain at the base of a cantilever beam with a known deflection. I know the bending stress is equal to Mc/I, so the strain is Mc/IE, where c is the distance from the neutral axis. For a point load ,P, the strain would then be PL/IE. Since the deflection is known I need to find the point load ,P, for that deflection. From the moment curvature relationship I can derive the deflection,y, to be equal to -PL^3/3EI. Solving for P and substituting it into the previous equation I can solve for the strain to be equal to -3yc/L^2.

This seems to be a roundabout way to solve for the strain. Is there a cleaner way to derive this relationship?

 

[PhanthomJay]

Strain can be positive or negative, positive at the outer fibers in tension and negative at the outer fibers in compression. Otherwise, your equation is very straightforward, provided you note, however, that the strain so derived is the strain at the outer fibers at the base of a cantilever under a point load P applied at the free end , where y is the deflection at that free end under the point load P. It is a very specific formula for a very specific case. The formula strain = Mc/EI is much more general.

 

[randall016]

So my approach is correct? Is there a better way to derive this relationship?

 

[PhanthomJay]

Since for whatever reason you want to find max strain as a function of max deflection, your approach is as good as it gets, since you have already calculated the deflection.

 

[randall016]

Alright thanks and the reason is for designing a displacement transducer. I know the working range so I want to optimize the strain in order to get the largest signal/noise ratio with a strain gauge.