# Strain of a Cantilever beam

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.

## Answers and Replies

JBA
Science Advisor
Gold Member
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?

JBA
Science Advisor
Gold Member
My D = deflection, sorry, I should have used y.

PhanthomJay
Science Advisor
Homework Helper
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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?

PhanthomJay
Science Advisor
Homework Helper
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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.

JBA
Science Advisor
Gold Member
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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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 lenght of the beam.

Thanks for all the answers

Nidum
Science Advisor
Gold Member
@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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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.

Nidum
Science Advisor
Gold Member
Please post a diagram as requested in post #11 .

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