Correlating Displacement and Strain in a Cantilever Beam System

In summary, you are integrating a system to determine the strain of a beam. 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).
  • #1
Gerrgegeorge
8
0
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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  • #2
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.
 
  • #3
Sorry, but In this case what is the D?
 
  • #4
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?
 
  • #5
My D = deflection, sorry, I should have used y.
 
  • #6
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).
 
  • #7
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?
 
  • #8
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.
 
  • #9
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 .
 
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  • #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 .
 
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1. What is the definition of strain in a cantilever beam?

Strain is a measure of the deformation or elongation of a material in response to external forces. In a cantilever beam, strain is the amount of bending or stretching that occurs along the length of the beam due to applied loads.

2. How is strain calculated in a cantilever beam?

The strain in a cantilever beam can be calculated using the formula ε = δ/L, where ε is the strain, δ is the deflection of the beam, and L is the original length of the beam. This formula is based on the assumption that the strain is directly proportional to the deflection of the beam.

3. What factors affect the strain of a cantilever beam?

The strain of a cantilever beam is affected by the type of material, the dimensions and shape of the beam, and the magnitude and location of the applied load. Other factors that can impact strain include temperature changes, material properties such as Young's modulus, and the presence of any cracks or defects in the beam.

4. How does strain impact the strength of a cantilever beam?

The strain in a cantilever beam is directly related to the stress that the beam is experiencing. If the strain exceeds the material's yield point, the beam will experience permanent deformation or failure. Therefore, monitoring and controlling the strain in a cantilever beam is crucial for ensuring its structural integrity and strength.

5. How can the strain of a cantilever beam be measured?

The strain of a cantilever beam can be measured using strain gauges, which are small sensors that can be attached to the surface of the beam. These gauges detect the changes in electrical resistance caused by the strain in the beam and convert it into a measurable strain value. Other methods of measuring strain include using mechanical dial gauges or optical techniques such as moiré interferometry.

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