Hinge Formation in Overhang Simply Supported Beam 4

[willowman]

Taking my first steps into plastic analysis of steel beams. I have a problem whereby I have a simply supported beam with an overhanging cantilever section. I have identified that the plastic moment will be first reached at the location of the support B.

What is the interpretation of hinge formation at an existing roller support (which we model as already allowing rotation)? Is there another approach to take in this example.

 

[dik]

There is little to be gained by using plastic design for a mechanism that has one hinge. The advantage is when multiple hinges are formed allowing a re-distribution of loads before 'collapse' with added load capacity after initial yield. There is a slight improvement in that you can use the plastic section Z instead of the elastic section S. With cantilevers and columns I really don't like a 'tight' design... not much re-distribution.

It also depends on the backspan X, but again collapse is predicated on a single hinge forming. On a positive side, alternating loads have little effect.


Dik

 

[willowman]

Thanks Dik,

Just to confirm my understanding, how would you interpret a hinge forming at support B?

 

[r13]

I think a "plastic hinge" at point B represents a limit equilibrium condition, any addition load can cause unpredictable large deflection, which is unrecoverable, if not outright collapse.

 

[JoshPlumSE]

You can only use plastic design when the structure is capable of re-distributing load from a hing to other portions of the structure. Best example is a fixed - fixed beam. You load it until a plastic hings forms at the two fixed end supports. It can still take some more load, it just that the load no longer causes end moments and instead increases the mid-span moment. Once the mid-span moment reaches the plastic hinge level then the loading is complete (essentially).

The problem with your cantilever is that you cannot add any more load to the cantilever after the plastic hinge forms because the structure is unstable and will collapse. Or more accurately (as retired13 puts it), you get unpredictably large deflections.

 

[dik]

"I think a "plastic hinge" at point B represents a limit equilibrium condition, any addition load can cause unpredictable large deflection, which is unrecoverable, if not outright collapse."

On the positive side... if collapse doesn't occur, the next time the beam is loaded up to the same condition... it behaves elastically.



Dik

 

[dik]

Josh... in Canada you can design a cantilever using Zx, effectively you are designing it for a hinge condition... I generally build a little 'slop' into designs where there is no re-distribution.


Dik

 

[r13]

That's the point I can't get over it, when a section reached "yield", will it recovery to its original position upon unloading? If it can, why name the point "yield", shouldn't the point moves further down the stress-strain curve?

 

[Blackstar123]

What is the interpretation of hinge formation at an existing roller support (which we model as already allowing rotation)? Is there another approach to take in this example.

I think you are mistaking your internal roller to an end roller. Rotation is not allowed at internal roller. That's how the overhang acts like a cantilever because the internal roller resist moment due to continuity.

Once the plastic hinge is formed at the cantilever end, then overhang becomes unstable and we say that failure has occurred.

 

[r13]

Also note, it's been addressed before that the form factor of Z/S = 1.6 is valid for rectangular/square shapes only. For other shapes, one need to calculate Z, which would usually resulted in lower form factor (≅ 1.1 for wide flange beams, so FyZ ≅ 1.1FyS, for ∅ = 0.9,
∅Mn ≅ FyS).

 

[Blackstar123]

Retired,
Yeild point is basically the onset of plastic deformation. Once yeild point is crossed, the deformation becomes permanent and materail will retain some part of the deformation once load is removed.

 

[dik]

Retired13: "That's the point I can't get over it, when a section reached "yield", will it recovery to its original position upon unloading?"

Not normally, it will likely have a permanent 'bend' at the hinge point. When unloaded, it will have residual elastic stresses and that's why if it's reloaded, it will behave elastically up to the original point of yield. There are some computer programs that model plastic behaviour... I've not used any of them... mostly used for 'shakedown' for multistorey stuff. I've not used plastic design for anything over 2 stories... but it works well and I usually use stiffened end plates for splice connections accommodating both shear and moment. I should note that with load factors, steel sections do not normally go into the plastic range. At service loads they behave elastically.

"it's been addressed before that the form factor of Z/S = 1.6 is valid" Shape factor for square and rectangular sections is 1.5. bd^2/6 for elastic as opposed to bd^2/4 for plastic.

Blackstar123: The overhang does not have to become unstable... just bends...

The reason I'm posting a lot of info on plastic design is that I would like more engineers look into using it for future structures. My experience is that it is a myth that it costs 10% to 15% more. It usually has savings of that magnitude.

Dik

 

[Blackstar123]

Also note, it's been addressed before that the form factor of Z/S = 1.6 is valid for rectangular/square shapes only. For other shapes, one need to calculate Z, which would usually resulted in lower form factor (≅ 1.1 for wide flange beams, so FyZ ≅ 1.1FyS, for ∅ = 0.9,
∅Mn ≅ FyS).

You think its only a factor of 1.15, but trust me that makes a world of difference when you are trying to increase the flexure capacity of your section. Atleast, I provide every measure possible so that beam reaches that capacity.

There's a big difference in magnitude between S and Z of a section.

 

[r13]

Interesting. Thanks both of you.

 

[Blackstar123]

Dik, a cantilever with hinge support instead of fixed will be unstable. It will displace like a door and will not bend after plastic hinge is formed.

 

[r13]

Blackstar123,

I agree, the difference draws the line between strength design and elastic design. Been a ASD guy for life, sometimes very difficult to change thinking path. A lot to learn.

 

[dik]

Blackstar123: Yes if only a hinge, but there is generally other framing that keeps the 'noodle' from flopping in the wind even if provided by whatever is loading it. But you have to consider that... The web stiffeners generally provide stability to the cross section as it goes into the plastic state.


Dik

 

[dik]

Retired13: "Been a ASD guy for life, sometimes very difficult to change thinking path" When I went to university, we were the first year to be introduced to limit states design and only covered a smattering of ASD for historic reasons. Limit states eventually morphed into LRFD.




Dik

 

[r13]

Dik,

Yes. It's been a long time after conducted review on a project done by plastic design, pretty much learnt through self reading and helps from the cal, far from thorough understanding though. Thanks for the free refresh lesson

 

[JoshPlumSE]

Dik -

I don't consider designing to LRFD capacities (Mp = Z*Fy) to be "plastic design". To me, plastic design is only when you take into account redistributing of moments.

Not saying that you guys are using the wrong terms. Just trying to explain how we may be interpreting this term differently.