Yield line method VS finite element for flat slabs 7

[Gus14]

Quoting the practical yield line design book published by the concrete center, " flat slabs on a rectangular grid of columns are essentially one-way continuous slabs in two directions and as such are analyzed and designed separately in both directions. "

I found their example a bit confusing, as they only studied one direction,
Did they ignore the other direction due to symmetry? as the same moments will occur in the other direction

I studied two models of flat slabs. Model 1, yield line moment results were about 10 percent larger than the finite element results.

Model 1,

https://files.engineering.com/getfi...-c93b-4664-9ba9-6c3c26d25e8e&file=Model_1.pdf

In model 2, yield line moment results were about 15 percent larger than the finite element results.

Model 2,

https://files.engineering.com/getfi...-d18a-4e05-8bf2-d148c8af1e73&file=model_2.pdf

Am I applying the yield line method correctly?

*Sidenotes:
1. Regarding top reinforcement, I understand that a middle strip will attract some of the moments. I ignored it to be more conservative for the column strip. The middle strip reinforcement will be provided appropriately later.

2. I assumed 0.35 * the positive moment along the slab edges because simple support is impossible to achieve. The columns will attract negative moments, as observed in the finite element model.

 

[centondollar]

Did they ignore the other direction due to symmetry? as the same moments will occur in the other direction

The classical plate model causes bending in two directions and a twisting moment at each point of the plate, and twisting moments can be very significant for non-symmetrical grids or column supports. Designing slabs as "essentially one-way continuous slabs" (it should say "beams") is not based on mechanics and merely a simplification from the days when frame programs and hand-calculations were the only tools available to most engineers.

Am I applying the yield line method correctly?

Designing a slab as a unit-width beam in orthogonal directions is not applying the yield-line model (it is just elastic beam analysis in two directions), but merely oversimplifying the problem in a way that is not always conservative.

Finite element models for slabs are based on plate theory, which is not compatible with yield line theory or so-called "strip methods" where a slab is designed as a beam bending in two orthogonal directions. One should expect the solutions to differ.

 

[Gus14]

Thank you centondollar, for replying

Designing slabs as "essentially one-way continuous slabs" (it should say "beams") is not based on mechanics and merely a simplification from the days when frame programs and hand-calculations were the only tools available to most engineers.

I agree with you, it is just designing beams.

Designing a slab as a unit-width beam in orthogonal directions is not applying the yield-line model (it is just elastic beam analysis in two directions), but merely oversimplifying the problem in a way that is not always conservative.

Yeah, I agree with you again. This is why I was confused. I think the "true" yield line method moments would be unsafe for flat slabs.

 

[centondollar]

Well, that method is based on finding a mechanism which produces the minimum collapse load. If such a mechanism is found (accurately minimizing work done by a mechanism at collapse implies finding the collapse load), the solution is theoretically completely safe.

The same can not be said for designing slabs as a collection of orthogonal beams, since it lacks basis in mechanics.

 

[dik]

I've use yield line plate theory with conc slabs (plates) and it's valid. If reinforcing is the same in each direction, so it's isotropically reinforced, it simplifies it a bit. Also lower strength concrete behaves a little more plastically than higher strength concrete. One of my first encounters with an AHJ was about 50 years back where I used plastic design methods to design a 'waffle' flat slab using plastic design... the one line of columns was offset half a bay dimension across the full line of columns.

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[Gus14]

If such a mechanism is found (accurately minimizing work done by a mechanism at collapse implies finding the collapse load), the solution is theoretically completely safe.

I disagree, For example, I reanalyzed Model 1 ( as attached ) using what one could assume reasonably true yield lines and moments were 35 percent less than finite element moments. Although, I did not redistribute any negative moments to the column strip. I think this is why the book requires designing flat slabs as a collection of orthogonal beams, even if we had an irregular column layout.

Quoting page 88, "Notwithstanding the need to check the folding plate failures, flat slabs supported on an irregular grid of columns is most easily dealt with using the Work Method of analysis"

Model 1 using reasonable yield lines pattern:

https://files.engineering.com/getfi...afd-b710-726f8f50d4f0&file=Model_1_review.pdf

I've use yield line plate theory with conc slabs (plates) and it's valid. If reinforcing is the same in each direction, so it's isotropically reinforced, it simplifies it a bit

I agree with you dik, but only if done according to the concrete center. I think it will even be highly conservative too.

 

[dik]

There was no concrete center 50 years ago... and it's been two decades since I've last used yield line for concrete.

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[271828]

YLA results in the collapse load, which should be higher than the load that would cause the slab to plastic hinge in one location. Thus, I would expect its results to be less conservative than the other methods.

However, one has to find the mechanism with the least strength. Any chance you're not looking at the weakest mechanism?

 

[Gus14]

There was no concrete center 50 years ago... and it's been two decades since I've last used yield line.

Then you have probably used your engineering judgment and arrived at a similar procedure to get the maximum moments. I am only studying this method to understand what to expect from the finite element.

However, one has to find the mechanism with the least strength. Any chance you're not looking at the weakest mechanism?

I believe I have chosen a reasonable mechanism. I think a 35 percent difference in moments is not less conservative, it's a failure. This is even without taking the columns' negative moments into account, which should decrease the positive moments even more.

The problem with choosing the weakest mechanism is that it needs to be reasonable. Designing flat slabs as a collection of orthogonal beams is not, which is a point of view I share with centondollar. But still, in some cases, such as model 1 and model 2, it gives the right moments.
In other cases, however, applying reasonable yield lines would give similar moments to finite elements depending on the columns' layout.
That's why I agree with the book requiring both patterns to be checked.

 

[Celt83]

I would not expect yield line analysis to produce nearly the same design moments as an elastic Finite Element Analysis.

If you’ve not considered the negative moments around supports in a flat slab yield line model then your model is likely not accurate.

This software package can perform the Yield Line Analysis: Link

 

[271828]

I would not expect yield line analysis to produce nearly the same design moments as an elastic Finite Element Analysis.

Me neither.

Very cool link. I had not seen that before.

...I think a 35 percent difference in moments is not less conservative, it's a failure. ...

I'm not sure what you mean by "it's a failure." Do you mean the YLA methodology is failing because its result is 35% different from one of the other analyses that you trust more? If so, then that seems to point at the answer: toss the YLA and use the one you trust more. LOL

 

[GregLocock]

YLA is an upper bound because it assumes a particular failure mechanism. At uni we did many experiments and had use a variable mechanism (ie moving the hinge lines analytically) to get the lowest failure loads.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[dik]

The link has a software product called 'Ring'... I didn't look into it, but the screenshot appears to be a masonry package. The University of Sheffield had a masonry program called 'Ring'. Is this the same? and are these guys marketting things for the U of Sheffield? or just capitalising on their efforts? Do you know?

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[centondollar]

I disagree, For example, I reanalyzed Model 1 ( as attached ) using what one could assume reasonably true yield lines and moments were 35 percent less than finite element moments. Although, I did not redistribute any negative moments to the column strip.

You obviously did not assume the correct yield lines and thus did not find the minimum energy state for collapse. That looks more like a solution for a point load with two opposite clamped ends. Supports on corner columns only makes finding the correct shape difficult, and if the slab continues over columns into other bays, the task may as well be impossible.

Also keep in mind that FE solutions are not necessarily correct at sharp discontinuities (column modelled as a point support on the plate midsurface), and that some plate elements perform worse than others.

I think this is why the book requires designing flat slabs as a collection of orthogonal beams, even if we had an irregular column layout.

That recommendation is a relic from the days where only frame analysis programs were available to an engineer. A plate does not act as a collection of orthogonal beams - twisting "Mxy" and the incorporation of Poisson's ratio is required to model a plate with any semblance of accuracy.

 

[dik]

The key to any FEM model is how it is modelled and how it is interpreted... either can be 'way out of whack'... another childhood experience...

-----*****-----
So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

 

[Gus14]

Thank you, everyone, for having this discussion, happy holidays.

 

[ThomasH]

Yield Line Method is an Upper Bound Method. That means that the result is either correct or the calculated load is to high. It means that you approach the correct solution from the "unsafe" side as you optimize the yield lines or failure mechanism. The optimal yield line is the solution that requires the least energy. Hence, can support the lowest load.

When I went to Uni the opposite, the Lower Bound Method that was used for comparison, was Strip Method Design (by Hillerborg). A simple example is a rectangular plate with supports on all four sides. You choose the distribution of the loading for the two perpendicular directions, two "beam directions" can be a good description. And if you are successful in optimizing the load distribution you get a economic solution. But as long as you support 100% of the load you approach the optimal solution from the "safe" side.

Finite Element Analysis is usually the linear elastic solution for the slabs moment distribution. But if you use nonlinear material models you can probably do somehing similar to yield line method. You can do a lot with FEM but the usual linear solution is, in my experience, not considered as either Upper nor Lower Bound Method.

 

[Gus14]

Thank you, Thomas, for replying. I plan on learning how to apply the strip method for flat slabs too soon.

 

[centondollar]

When I went to Uni the opposite, the Lower Bound Method that was used for comparison, was Strip Method Design (by Hillerborg). A simple example is a rectangular plate with supports on all four sides. You choose the distribution of the loading for the two perpendicular directions, two "beam directions" can be a good description. And if you are successful in optimizing the load distribution you get a economic solution. But as long as you support 100% of the load you approach the optimal solution from the "safe" side.

The Strip Design Method is not a lower bound method in the sense that it is not based on mechanics (no twisting moments and Poisson's effect) and requires ad-hoc reasoning; it is also seldom applied with the flexibility method (unknowns are forces or moments, with displacements and rotations solved afterwards), which is a lower-bound method unlike e.g. the FEM.

Finite Element Analysis is usually the linear elastic solution for the slabs moment distribution. But if you use nonlinear material models you can probably do somehing similar to yield line method. You can do a lot with FEM but the usual linear solution is, in my experience, not considered as either Upper nor Lower Bound Method.

The Finite Element Method provides an upper bound method by definition, since the field of primary variables (displacement and rotation in structural mechanics plate problems) is approximated by locally (in each "element") defined sums of polynomials weighted by the unknown nodal variables. The "real" displacement field is infinitely smooth in a non-trivial problem, while the polynomial approximation is not, and the consequence is that the FEM overestimates stiffness and underestimates the magnitude of primary variables. Taking derivatives to define secondary variables (moment, shear etc.) further increases error for those variables.

 

[ThomasH]

@centondollar
I would say that the strip method for design is a lower bound method in the sense that it is always safe. It approaches the correct solution from the "safe" side. The reason that Hillerborg "invented" the method was that the yield line method was considerad as unsafe. There was a request for an alternative. Some short info:

https://www.concrete.org/publications/internationalconcreteabstractsportal/m/details/id/5536

As for the Finite Element Method, I did a quick Google search and one of the first hits was a text starting with "It is well known that the finite element method (FEM) provides a lower bound in energy norm for the exact solution to elasticity problems." I can't say that I am too surprised and I won't go into any indepth discussion about this .

https://link.springer.com/chapter/10.1007/978-3-540-75999-7_13