Deflection of beam after reaching elastic limit

[k.udhay]

Hi,

When we find out the deflection of beam, the factors considered are its geometry (moment of inertia) and young's modulus (E) of the material. As per text Hook's law 'E' is constant only till the elastic limit of the material. Assuming that the stress induced crosses the elastic limit, 'E' is no more valid. How do we take this effect in the calculation, if the stress is above elastic limit? Thanks.

 

[timthereaper]

In engineering, normally you don't. There's not a really good way to factor that into your formulas, so you should design the beam (or alter the loading condition) to something that won't induce a stress over the yield stress of the beam.

After the yield point, it starts to become nonlinear and there isn't a good way to account for that without really taking multiple things into account. In essence, you're cold working the beam by yielding it and making it harder, but only in certain regions. The more you cold work it, the harder and more brittle it becomes in those spots. After a while, cracks are induced and the beam catastrophically fails. The stress-strain curves for a specific material can help you determine how it will generally behave, but then it's also entirely dependent on how your sample was manufactured and any heat treatment it received. As well, if there are any inclusions in your specific batch of material, it could also affect its behavior.

So, short answer: we try not to take that into consideration. If you feel you need to do so, consult the stress-strain diagrams or do your own sample testing.

 

[AlephZero]

In general, you just have a statics or dynamics problem with nonlinear material behavior, and you have to solve it numerically.

A special case is for elastic - perfectly plastic materials, where the stress strain curve is modeled as two straight line segments. See http://en.wikipedia.org/wiki/Plastic_hinge

 

[k.udhay]

In engineering, normally you don't. There's not a really good way to factor that into your formulas, so you should design the beam (or alter the loading condition) to something that won't induce a stress over the yield stress of the beam.

After the yield point, it starts to become nonlinear and there isn't a good way to account for that without really taking multiple things into account. In essence, you're cold working the beam by yielding it and making it harder, but only in certain regions. The more you cold work it, the harder and more brittle it becomes in those spots. After a while, cracks are induced and the beam catastrophically fails. The stress-strain curves for a specific material can help you determine how it will generally behave, but then it's also entirely dependent on how your sample was manufactured and any heat treatment it received. As well, if there are any inclusions in your specific batch of material, it could also affect its behavior.

So, short answer: we try not to take that into consideration. If you feel you need to do so, consult the stress-strain diagrams or do your own sample testing.

Thank you, Tim. So, it's safe to keep the component within its yield limit.

 

[k.udhay]

In general, you just have a statics or dynamics problem with nonlinear material behavior, and you have to solve it numerically.

A special case is for elastic - perfectly plastic materials, where the stress strain curve is modeled as two straight line segments. See http://en.wikipedia.org/wiki/Plastic_hinge

Thanks, Aleph. :)