Understanding the Equation for Measuring Strain on a Cantilever Beam

[JSBeckton]

I have a lab where we used the following equation to theroretically measure the strain on a cantilever beam heald stationary at the left end. Measuring back from the right side its length is L2 and the length from the free (right) end to the point where the strain gagues were was L1

(fixed end)________________________(free end)

L2=whole length
L1=length from free end to measured point.
t=thichness

strain=(3L1)(t)(deformation)
______ 2(L2)^3

I know the equation for strain is deformation over L2 but can't seem to wrap my head around how they derived this equation.

Any help would be greatly appreciated, thanks in advance.

 

[Mapes]

The end deformation of a cantilever loaded by force P is

$$\delta=\frac{PL^3}{3EI}$$

The strain on the surface of a beam in bending mode is

$$\epsilon=\frac{My}{EI}=\frac{Mt}{2EI}$$

where the moment M is

$$M=L_1P$$

Put these together and you'll have the equation you were trying to derive.

 

[JSBeckton]

Thanks a lot, can't believe I didn't see that!

 

[JSBeckton]

DO you know where I can find a derivation of the deformation equation?

Thanks

 

[Andy Resnick]

Lots of places. Try:

online course work:
http://www.clarkson.edu/class/es22201/
(chapter 9 is beam bending)

Mathematics applied to continuum mechanics, Segel
Theory of elasticity, Landau and Lif****z (vol. 7)

The original equation is a 4-th order inhomogeneous differential expression. After simplifying (thin rod, equilibrium deformation, etc), the equation is easily solved.

 

[JSBeckton]

I got it, thanks.