ULS design question 1

[breaking_point]

Hey everyone.

I have a question regarding ultimate limit state design. Using this method, we calculate member capacities based on their maximum plastic capacities. We then select member parameters based on the factored loads and materials, so that the materials do not reach failure during their lifetime. However, it is possible that during the members life-cycle, it could go beyond yield under certain loading but not reach the ultimate failure. Are we concerned about this? Is it a concern that the members will have residual strain in them after going beyond yield?

 

[jayrod12]

I don't believe you quite have this correct.

ULS means you are comparing factored loading to reduced material properties. For example you factor your loads up based on the prescribed code, in my neck of the woods it is 1.25*DL + 1.5*SL, and you compare that against the steel structure that uses 0.9*Fy. Whether you get to use the plastic section modulus, or the elastic section modulus depends on the class of section you are checking. The fact that you are using limit states design, compared to ASD doesn't factor in.

The loading factors in Limit states design don't change regardless of the material you are designing with, the material strength reduction factors change with each differing material, e.g. steel is 0.9, reinforcing steel is 0.85, concrete is 0.65, wood depends on which failure mode you are checking but is in the 0.8 range.

 

[KootK]

However, it is possible that during the members life-cycle, it could go beyond yield under certain loading but not reach the ultimate failure.

Yup, partial section yield at least.

Is it a concern that the members will have residual strain in them after going beyond yield?

Folks don't normally worry about this. But yeah, once a structural element sees ultimate, it may well no longer be serviceable.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[breaking_point]

@jayrpd12, My apologies, I don't think I explained myself clearly in my first post. I understand the principle you outlined in your post. However, I was asking whether it is of concern that it is possible that section undergo loading that would take it beyond the yield point but within the ultimate capacity, as the section would develop some (possibly a small amount) of plastic behavior an therefore irrecoverable strain.

@KootK, thanks for your explanation! I guess upon unloading the state of the material would travel back down the stress-strain line, but would have a different linear portion of the curve because it now has residual strain from the plastic behaviour. But when the member is reloaded it would travel up the line to the point on the stress-strain line it was previously loaded and beyond (if the loading is higher). Is my understanding correct? I guess it would have been easier to explain this graphically!

 

[breaking_point]

@KootK, forgive my crude drawing (from MS Paint haha!) but I thought this might illustrate my thoughts better to see if I am on the right track. If the moment rotation capacity of a section is plotted in black, with (phi)Mp representing the ultimate capacity we design it for (assuming section 1 stocky beam etc.). The yellow line represents some loading condition that the material may endure it's lifetime that causes it to go beyond yield but under ultimate capacity. Upon unloading, it would go down a new path back to an unloaded condition, and upon reloading it would travel up this path again (green line).

I understand that the ULS principle is to use factored loads against reduced material capacity to limit the possibility of this happening.

 

[KootK]

The graph looks about right for many kinds of materials and members.

I understand that the ULS principle is to use factored loads against reduced material capacity to limit the possibility of this happening.

I don't know that's the case really. I think that's it's straight up probability on loads not being exceeded and capacities not being lacking based on 5th percentile kind of stuff. Any way that this prevents plasticity is just a coincidence I think. Of course, our serviceability checks help.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[dik]

it is possible that during the members life-cycle, it could go beyond yield under certain loading but not reach the ultimate failure

Often... look into 'shake down' for plastic analysis...

A classic simple example is a fixed end beam with a UDL. This can be loaded until ql^2/12. This is the elastic limit load and the mid span moment is ql^2/24. Loading it further, the supports enter a plastic state (Mp) and the beam can be loaded further until the mid span is ql^2/12. This is the plastic limit load. At this point you have a mechanism, and the beam is said to have failed.

If a UDL is applied up to the point where the midspan moment is beyond the elastic moment until it almost reaches the plastic moment. We'll call this load ql and the moment Ml where Ml < Mp. Moment resistance for elastic behaviour is based on Sx while for plastic behaviour is based on Zx. Plastic sections have to be a little more robust to maintain stability of the section through large deformations and should be Class I sections or equivalent.

The beam can be unloaded. There are residual flexural stresses in the beam from the load ql. The beam can then be re-loaded to ql and it behaves elastically until it is loaded beyond ql.

The residual stresses, noted above, give rise to 'shake down' failure, which is a study unto its own... Hope I haven't confused you too much.

Dik

 

[breaking_point]

Hi Dik. That was very well explained (and I was able to follow it!), thank you. I will take a look into shake down behaviour and see if I can further my understanding a little more.

 

[KootK]

Just last week I was reading about shakedown in relation to designing wit flexible moment connections. Here if you're interested: Link

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[dik]

Thanks KootK... good info

Dik

 

[dik]

OP: FYI

 

[breaking_point]

Dik, Kootk, thanks for the literature. I'll read through them this weekend! Cheers.

 

[CCox]

One time in my career I performed a plastic analysis. I had to because it was an existing beam and overstressed when considered through elastic analysis. The whole concept of plastic analysis is to take advantage of plastic hinges (nonlinear behavior) forming in beams that can allow the hinges to occur and remain stable. If for example, you had a beam that was fixed on one end and a roller on the other, that beam would form two plastic hinges to collapse. The first would probably form at the fixed support and the second would form at the point load (if that was the loading condition).

I think I remember a rule of thumb somewhere where the moment capacity is increased by around 15% when a plastic analysis is performed. The member must be indeterminate to have a plastic hinge form and still remain stable. Because of the large deformations, the member must be compact and have adequate bracing.