ACI 318 10.7.1 3

[jreit]

Hello everyone.

I had a question regarding the definition of deep beams. I searched previous threads but was unable to find the answer.
ACI clearly states that beams with a span length of less than or equal to 4 times the member depth (ln <= 4h) will be classified as deep beams.
Does this apply to cantilever beams also? I have seen designers use ln <= 2h for cantilevered beams but cannot find a reference to justify this.
And if it true, could anyone explain why the stress distribution can be assumed as linear in cantilever beams for ln > 2h but for simply supported beams it is ln > 4h?

Thank you in advance.

 

[TXStructural]

The provisions are mostly due to the assumption that design of deep beams is dominated by shear rather than flexure. While mostly ignored, the three-dimensional nature of compression struts and large forces therein gets addressed by the additional reinforcement in deep beam provisions.

As far as cantilevers, the high stresses need to be similarly addressed, although I can't say that I've seen it applied this way very often. A relatively deep cantilever may actually function like a corbel, maybe? At the very least, there would be a significant D (disturbed) region at the fixed end. If you are designing on the margins of safety, consider using more confinement. If, on the other hand, the depth is just a function of making the cantilever look like the adjacent beam (typically for architectural or formwork/constructibility considerations) and there isn't that much demand, I would not go overboard.

A quick Google:

http://www.sciencedirect.com/science/article/pii/S0045794998001278

 

[jreit]

Thanks TX.

 

[rapt]

Australian code says 1.5 for cantilevers, 3 for simply supported beams and 4 for continuous!

TX, a corbel IS a cantilever deep beam!

 

[WillisV]

The 2h for cantilevers vs. 4h for beams is just recognizing that for simple beams assuming centered loading the struts will form like this \/ and for cantilevers you only get one strut like this \ so to maintain approximately the same type strut distribution in a cantilever as in a beam, the cantilever is half as long. Similar to how deflection limits are written as 2L/360 vs L/360 for cantilevers vs. beams.

 

[KootK]

A more helpful definition of a "deep beam" is a beam in which the shear span is less that 2.0. And the shear span is defined as the distance from the point of maximum shear to the point of zero shear (I think -- it's been a while). For uniform loads, this would mean L/d = 4 for simple span and L/d = 2 for cantilevers (assuming uniform loads for each). This would also mesh well with Rapt's comments, with a little more conservatism thrown in.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[jreit]

That does make sense now.
I don't have any experience with the strut and tie model so it was hard to visualize that without the shear span explanation.
TX, I wasn't able to get access to that article so I'll have to find another source, but your example of the corbel helped because it was the element being designed.
ACI Section 11.8 outlines the shear span-to-depth ratio as < 2 for corbels and brackets so that's where it was coming from.
It's an interesting question though because does that suggest if there was a cantilever beam that was not a corbel or bracket, the span-to-depth ratio of 4 would apply?
Thanks for all your help everyone.

 

[KootK]

A span to depth limit of two, rather than four, would apply to a non-corbel cantilever beam I believe.

Here's another way to look at it that might be helpful:

1) Most codes limit the angle that a viable strut can make with its ties. It varies a bit from place to place but the angle usually amounts to a run/rise ratio of about two.

2) You are in deep beam territory if the aspect ratio of your element is such that you can construct a "one triangle" strut and tie model with viable struts that run directly from your applied load to your supports(s). If a strut and tie model (STM) like this is possible, it will likely be the stiffest load path available.

For a simple span beam with a point load at the centre, your one triangle STM is an isosceles triangle with struts connecting the load and the two supports. Since the minimum rise/run for the struts is 0.5, this leads to a span/depth of 4.0.

For a cantilevered beam with a point at the tip, your one triangle STM is a right triangle with a single strut that connects the applied load and the underside of the beam where it meets its support. This leads to a span/depth ratio of 2.0 with a viably oriented strut.

Before strut and tie analogies became popular, the same phenomenon was discussed as arching. Obviously, there's a limit to how shallow an arch can be just like there's a limit to how shallow a strut can be. Think of a simple span beam as an arch in the shape of a frown. Think of a cantilever beam as an arch in the shape of half a smile.



The greatest trick that bond stress ever pulled was convincing the world it didn't exist.