# Cantilever Beam Failure as a function of Force Applied to Free End

1. Nov 4, 2012

### seuss

Hello,

First post, go easy.

Trying to do some basic calculations on how to find when a steel cantilever beam will yield when a load is applied to the free end of the beam. I have attached my best MS Paint diagram of the problem. Assume the left end is clamped and essentially a fixed end. Failure is when the "beam" permanently deforms (yields)

E - Modulus of Elasticity 200 GPa 29000 ksi
G - Shear Modulus 80 GPa
I - (1/12)(25mm)(4mm)^3 133.33 mm^4
L - 60 mm

I have done typical axial stress-strain problems in undergrad, and I have done bending moment, shear, deflection, diagrams of beams, but never analyzed a beam in this way for failure.

I think if calculate the shear stress at O as a function of the load applied at the other end, L, and find when that shear exceeds the shear stress for yielding, that might be the answer I'm looking for. BUT, how do I determine the yield point? I know it's going to be very clear once I get the problem framed correctly, but I'm not connecting all the dots.
Any validity to this line of thinking?

This isn't a homework problem, it's for a project I'm working on.
The application is a latch securing between a door and a frame with by a rotating arm (cam).
Need some preliminary numbers. Help me get started! Thanks!

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2. Nov 4, 2012

### SteamKing

Staff Emeritus
What makes you think the beam is going to fail in shear before the bending stress
at the clamped end reached yield?

3. Nov 4, 2012

### AlephZero

For beam bending, the usual way to exceed the yield stress is the axial stress along the length of the beam, not the shear stress.

If y is the distance from the neutral axis you have $\sigma/y = M/I$. So the maximum stress will be furthest frmo the axis, where the bending moment is biggest - i.e. at the fixed end.

You get the yield stress by looking up the material properties for the particular grade of steel you have. Young's modulus doesn't vary much between different grades of steel, but the yield stress varies a lot. A conservative number at the low end of the range would be about 150 MPa or 10 tsi, but high strength steels might be nearer 750 MPa.

4. Nov 4, 2012

### seuss

I see. It's really the bending stress causing the deformation. Back to the drawing board.

5. Nov 5, 2012

### Studiot

Be aware that the figure you have drawn looks like a wide beam or even a plate so simple engineering beam theory will not be applicable without corrections.