# Deriving Strain in Cantilever Beam with Known Deflection

• randall016
In summary, to find the strain at the base of a cantilever beam with a known deflection, one needs to use the moment curvature relationship and find the point load, P, for that 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?

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.

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

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.

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.

## 1. How do you calculate strain in a cantilever beam with known deflection?

The strain in a cantilever beam with known deflection can be calculated using the formula: strain = deflection / beam length. This formula assumes that the beam is linearly elastic and has a constant cross-sectional area.

## 2. What is the significance of deriving strain in a cantilever beam?

Deriving strain in a cantilever beam allows us to understand the deformation and stress distribution in the beam. This information is crucial in designing and analyzing structures to ensure they can withstand the expected loads and forces.

## 3. Can strain be negative in a cantilever beam with known deflection?

Yes, strain can be negative in a cantilever beam with known deflection. Negative strain indicates compression, while positive strain indicates tension.

## 4. How does the deflection of a cantilever beam affect the strain?

The deflection of a cantilever beam directly affects the strain. As the deflection increases, the strain also increases. This relationship is captured by Hooke's Law, which states that the strain is directly proportional to the applied stress.

## 5. Are there any assumptions made when deriving strain in a cantilever beam with known deflection?

Yes, there are a few assumptions made when deriving strain in a cantilever beam with known deflection. These include assuming that the beam is linearly elastic, has a constant cross-sectional area, and that the deflection is small compared to the beam length.

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