# Strain in cantilever beam

1. Aug 11, 2014

### nomority

Hi everyone. I feel there should be a simple answer to this but I can't seem to find anything on this.

So I have a simple cantilever beam, supported at one side and loaded at the free end. I have the force displacement data and can easily calculate the stress.

However, for the strain I do not want to use Hookes Law, but instead calculate the strain from the force displacement data. Any hints?

Thanks!

2. Aug 11, 2014

### Staff: Mentor

The strain varies through the cross section of the beam. It is equal to the distance from the neutral axis times the curvature. So if you know the curvature at any location, you know the strain variation through the thickness. The strain is positive on the outside of the bend, and negative on the inside of the bend. Of course, it also varies with distance along the beam.

Chet

3. Aug 11, 2014

### nomority

Apologies, I should have stated I am looking for the maximum strain.

I have got the full force-displacement data of a bend experiment. I know the geometry of the beam (triangular cross section) so can calculate the maximum stress at any time as (M*y)/I, which for the triangular cross-section is equal to (24*Force*Length)/(width*thickness^2).

In a code I have access to it states that the maximum strain at any time would be equal to (2*displacement*thickness)/(Length^2), but I can't figure out why this would be the case.

4. Aug 11, 2014

### Staff: Mentor

Where does the maximum curvature occur, and, in terms of the displacement , what is that curvature?

Chet

5. Aug 11, 2014

### nomority

Thanks for your reply. Maximum curvature is at the fixed end. However, I can't measure the curvature to any degree of accuracy, as such for the sake of this problem it is not available.

6. Aug 11, 2014

### Staff: Mentor

What I meant was, analytically , what is the curvature in terms of the displacement ?

7. Aug 11, 2014

### nomority

I'm not sure how I could describe the curvature in terms of the parameters I have.

If I was to reverse engineer the equation for strain I have

, and assume that strain is equal to y/R (as found online), I would have an expression for the curvature of L^2/(6*displacement). This doesn't seem correct to me though.

8. Aug 11, 2014

### AlephZero

If you know the maximum stress, you can get the maximum strain using Young's modulus and Hooke's law.

The stresses and strains in the beam are statically determinate. They only depend on the applied loads, not on the displacement of the beam.

9. Aug 11, 2014

### Staff: Mentor

Step 1: Express the bending moment M at the built-in end in terms of the load F.
Step 2: What is your equation for the displacement in terms of the load F.
Step 3: Combine these relationships to get the bending moment in terms of the displacement.
Step 4: Determine the curvature from the bending moment
Step 5: Determine the curvature as a function of the displacement
Step 6: Determine the strain from the curvature

chet