Correlating Displacement and Strain in a Cantilever Beam System

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Discussion Overview

The discussion revolves around correlating displacement and strain in a cantilever beam system, particularly focusing on how to derive an expression for strain based on the vertical displacement of the beam. Participants explore theoretical and mathematical aspects of beam deflection and strain under load, with references to integration and differential equations in MATLAB.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant seeks to derive a correlation between beam displacement and strain, noting the displacement formula for a cantilever beam.
  • Another participant suggests using the relationship between stress and strain, proposing a method to integrate stress over the beam length to find total strain.
  • Clarifications are made regarding the definition of variables, particularly the use of 'D' for deflection.
  • Concerns are raised about using the deflection at the free end of the beam, with suggestions to consider the general deflection along the beam length.
  • Participants discuss the need to derive equations for stress and strain, with references to bending stress and its dependence on the moment and beam geometry.
  • One participant emphasizes the importance of drawing a moment diagram to understand the variation of moment, stress, and strain along the beam.
  • There is a request for clarification on whether the problem pertains to static deflection or vibration, indicating potential confusion about the concept of strain.
  • A participant expresses the need for a function that relates strain to vertical displacement in the context of a vibrating beam.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to derive strain from displacement, with some advocating for integration of stress along the beam and others questioning the assumptions made about deflection and strain. The discussion remains unresolved, with multiple competing perspectives on the methodology.

Contextual Notes

Participants note the complexity of the problem, including the need for clear definitions of variables and the importance of considering the entire beam length rather than just the free end. There are indications of missing assumptions regarding the nature of the problem (static vs. dynamic) and the relationship between displacement and strain.

Who May Find This Useful

This discussion may be useful for individuals interested in structural engineering, mechanics of materials, or those working with cantilever beam systems in theoretical or applied contexts.

Gerrgegeorge
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Hi,

Well i have a system with a excited (in y axis) cantilever beam. I struggling to obtain a expression that gives the strain based on the dislocation y.

I know that the displacement of the beam is given by:

Ya=PL^3/(3EI)
but how i make a correlation between this and the strain of the surface of the beam?

Resuming, i am integrating a system, in this system i have the displacement of the beam in y-axis (X(1) in my program) but i want the strain of the surface of the beam.
 
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Strain = Stress/E, so take the two equations and resolve them into one equation for Stress as a function of L and D, with D known, integrate that equation for L from 0 to L and divide the result by E to obtain the resulting total strain in the outer fibers of the beam. + for the tension side and - for compression side.
 
Sorry, but In this case what is the D?
 
Well, i have made:
Stress=(ya*3E*I)/(A*L^3)

Strain=Stress/E

Strain=(ya*3*I)/(L^3*A)

Where:
A= Area
I=Inertia
E=young modulus

My question now is, as i am integrating this system (Matlab, ode45) i need to derive this equation, no?
 
My D = deflection, sorry, I should have used y.
 
You are using the deflection at the free end of the cantilever, where the strain is 0. You should instead be using the general deflection along the entire beam length as a function of x. Also, stress in bending is not P/A, it is Mc/I at the outer fibers, where M at any point is Px (where x is 0 at the free end).
 
PhantomJay, i have tried here:

X(1)=(P*(L^3))/(3*E*I) %%%Equation of the deflection, where x(1) is the variation of the deflection in time

P=(X(1)*3*E*I)/(L^3)

Substituing in the equation

Stress=Mc/I

Strain=Stress/E

Strain=(X(1)*3*I*c*x)/(I*L^3)

Is that correct?
 
No. What do you mean that x is variation of deflection in time? Are you trying to find the strain at the outer fibers of the beam at some distance x from the the free end? And as a function of the displacement y at that distance? I am not sure why. You are also getting hung up by looking at the displacement at the free end and not as a function of the length x along the beam. As mentioned, bending strain is bending stress/E. And bending stress at outer fibers is Mc/I. So strain is Mc/EI, and since M is Px, strain is Pxc/EI, positive at top fibers and negative at bottom fibers.
 
Gerrgegeorge,
You need to draw a moment diagram for your beam under load/deflection. When you do this, you will see that the magnitudes of the moment, stress and strain are continuously variable along the length of the beam. (Assuming you are working with a cantilever beam the moment will be zero at the free end of the beam and linearly increase to its maximum at the base connection of the beam.) This is the reason that, for any given beam load/deflection, it is necessary to integrate the stress along the beam to determine the total accumulated strain in the top and bottom surfaces of the beam at any given deflection.
 
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  • #10
Sorry for the incorrect use of some constants,

the deflection is Ya=X(1) and do not have relationship with the x who is based in the length of the beam.

Thanks for all the answers
 
  • #11
@Gerrgegeorge .

(1) Your first posting does not make it clear whether this is a static deflection problem or a vibration problem .

(2) I think that you may not properly understand what strain is .

Please explain again what you are trying to do . Use simple words rather than technical terms and include a diagram .
 
Last edited:
  • #12
In a simple form, i am integrating some differential equations in matlab.

One of this equations need the strain of the beam (in this case). As the beam is vibrating i have the information of the displacement, who varies in time (vibration).

In this question i want a function who gives the strain based in the vertical displacement of the cantilever beam (the information that i have).

Sorry for the bad explanation.
 
  • #13
Please post a diagram as requested in post #11 .
 
Last edited:

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