Yield Line Analysis Of Concrete Slab, .

[jmmstr]

Homework Statement

We were given many different slab sizes, shapes etc and asked to analyse them using Work = Energy equilibrium checks to find q, the maximum load per m^2 the slab can take.

The first part was to show the fracture lines to use for analysis but I am stuck on one of them.
I have included the image below to show some examples of what I mean and the one I am stuck on.

Any help would be appreciated as always!

To clarify, the triangle one with the star above it is the one I am stuck on, the others I have shown working for.

 

[pongo38]

If you take the diagonal line to be a beam strip, same width as the column, encastre at the top, continuous over the column, and simply supported at the lower right corner, could you sketch the deflected shape of that beam? Does that not give you a clue as to where the initiation points are for yield lines? In trying to decide whether a trial YL pattern is valid, I have found it helpful to draw the elevations of the deflected (collapsing) slab from the x and y directions.

 

[pongo38]

Second reply: In the case of the other patterns, you haven't shown dimensions, and I assume you can cope with those, but it is worth noting that in the case of the rectangle with two columns and one SS edge, the position of the red node in the middle is uncertain without more data. Although the problem CAN be solved by calculus to find the optimal position (which carries the least load), the practical way is to do it by trial and error. There are several variables including the positions of the green nodes and the position of the red node. The problem is even more intractable if the moments of resistance about the x and y axes are different. I would initially assume that the slab was isotropic and try specific fixed values of the red and green nodes. Incidentally, with other ratios for the edge lengths of the rectangle, that red node can be outside the slab.

 

[jmmstr]

Hi there,

thanks for the help, no dimensions have been shown for any so it is pretty much trial and error for this part at the moment, I think its more to show that you can show why a certain failure can occur.

Using the information you have provided I assume that the yield lines span from each corner and meet at some point within the triangle itself (depending on dimensions).

This was my initial thought as the fourth example shows that all the green lines must meet and then the yield lines tend towards the intersections of them. The example I am stuck on will therefore cross at each corner.

 

[pongo38]

Well, your hypothesis is correct and is similar to the case where the column is replaced by a simply supported edge. Don't forget the hogging moments along the left edge, as well as the interior sagging lines. However, did you know that yield line analysis is an upper bound method. this means that any yield line pattern you postulate and calculate the value for is an upper bound on the actual collapse load. It is also known as the unsafe theorem. This means you ignore other possible yield line patterns at your peril. However, they are used, and give good results. I gave you a clue to the two other patterns I can see, both of which might be more critical. But you didn't respond concerning the edge 'beam' question.

 

[pongo38]

Another way to help visualise this is to make a rough model from cardboard. Support it as shown in the figure. Even applying point loads with your finger will indicate the location of yield line patterns worth exploring.

 

[jmmstr]

Hi there,

thanks again, yes our tutor has told us that it can be used but you must be careful in identifying all possible failures and not just designing for one as it may not be the worst case.

I have made a quick model triangle and put blutack under for the column and wedged the other two between books, I found another failure to be between the column as shown in rough sketch attached.

 

[pongo38]

Nearly right. can you see that the column is preventing the failure you have drawn. What is happening geometrically along the line joining the column to the right angled corner?

 

[jmmstr]

Hi there,

I raised the model higher and it seems that a line forms at the 90 degree corner and then branches out into the two lines I have drawn. Is this correct?

Sorry I mean that it forms at a 45 degree angle out of the right angle and makes a "Y" shape.

 

[pongo38]

No. Not correct. If you draw an elevation of the diagonal edge, from the "North-East" looking "South-West" and draw a view of your collapsing slab according to the lines you gave in post #7, I hope you will see why it needs modifying.

 

[jmmstr]

Hi there,

it cannot fail over the column unless there's also a yield line at this point from the right angled corner to the column.

This would be a dotted line like that which would occur at the fixed support (hogging)

 

[pongo38]

correct. That's exactly it. But you end up with two triangular panels which each is trying to collapse. In practice only one of these will be critical, and you can evaluate each one with number crunching on the work equation. I think you can tell by inspection which triangle is likely to be critical because it requires less work to collapse it. It depends exactly where the column is.