How Can Strain Be Calculated from Force Displacement Data in a Cantilever Beam?

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Discussion Overview

The discussion revolves around calculating strain from force displacement data in a cantilever beam, specifically focusing on the maximum strain rather than using Hooke's Law. Participants explore various approaches and relationships between force, displacement, curvature, and strain in the context of a bending experiment involving a triangular cross-section beam.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks to calculate strain directly from force displacement data without using Hooke's Law.
  • Another participant notes that strain varies through the beam's cross-section and is related to curvature, indicating that strain is positive on the outside of the bend and negative on the inside.
  • A participant clarifies they are specifically looking for maximum strain and provides a formula for maximum stress based on the beam's geometry.
  • There is a proposed formula for maximum strain in terms of displacement and thickness, but the reasoning behind it is questioned.
  • One participant identifies that maximum curvature occurs at the fixed end of the beam but notes the difficulty in measuring curvature accurately.
  • Another participant attempts to express curvature in terms of displacement but expresses uncertainty about the correctness of their derived equation.
  • One participant suggests that maximum strain can be derived from maximum stress using Young's modulus and Hooke's Law, asserting that stresses and strains depend on applied loads rather than displacement.
  • Further steps are outlined to relate bending moment, displacement, and curvature, but these steps are not agreed upon as definitive.

Areas of Agreement / Disagreement

Participants express differing views on how to relate curvature, displacement, and strain. There is no consensus on the correct approach to derive strain from the given parameters, and multiple competing models and equations are presented.

Contextual Notes

Participants mention various assumptions and dependencies, such as the relationship between curvature and displacement, but these remain unresolved. The discussion also highlights the complexity of accurately measuring curvature in practical scenarios.

nomority
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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!
 
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nomority said:
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!
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
 
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.
 
nomority said:
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.

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

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

(2*displacement*thickness)/(Length^2)
, 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.
 
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.
 
nomority said:
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.
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
 

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