Cantilever Slab Deflection

[TTJT]

Dear All,

I have a 6'-6" cantilever slab that is supporting a 12 foot tall masonry wall at the end of the cantilever. Strength is not a problem but I'm concerned about the deflections particularly long term. The backspan of the cantilever slab is a 17'x17' two way slab supported on beams and walls. How do I account for the impact of the backspan on the deflections. It seems to me that analyzing the slab as a beam with a 17 foot backspan is too conservative.

Thanks

 

[msquared48]

Well, I don't see why you would not use the whole 17 foot back span since it is there to use structurally. Just put reasonable alternating live loads on it to get your moment envelope and design the top and bottom steel accordingly.

Mike McCann
MMC Engineering

 

[TTJT]

Thanks Mike,

Since the backspan on the slab is a two way slab, I thought it would help with the rotation around the support and reduce the overall deflection. Analyzing the 17' backspan as a one way spanning member seemed conservative.

 

[hokie66]

Yes, you are correct. The two way action in the backspan will restrict that deflection, thus will restrict the cantilever deflection.

 

[TTJT]

Thanks Hokie,

Is there a design procedure to account for this if these calculations are being done by hand?

 

[hokie66]

I don't know about an actual manual design procedure, but I would approach it by setting the following boundary conditions.

1. I would assume all 4 supporting walls/beams around the back span stay level, i.e. at the same elevation.
2. Reinforce the supported wall so that no deflection occurs parallel to the wall. That way, the cantilever deflection has to be constant.
3. Now just calculate/estimate deflection of the cantilever assuming it is an encastre cantilever. Apply whatever long term estimates you would normally use. Some here will tell you this calculation can be done with great accuracy, but there are so many things during construction which can affect deflection, I take it all with a grain of salt.
4. Be conservative and add maybe 20% for the contribution due to the upward deflection of the two way internal span.

 

[rapt]

I would be careful making assumptions on this. The back span length and loading acting in the direction being considered are critical in determining the effect of the back span on the cantilever deflections.

If the back span is treated as a one way slab, then all of its loading is considered to deflect the back span downwards, which creates a rotation at the cantilever support thus reducing the cantilever deflection. The extra cracking induced by the load in the one way assumption will increase this effect.

If the back span is actually a two way slab then only part of the loading will be acting in the direction of the cantilever and will reduce the back span deflection and its effect on the cantilever deflection. If the back span is square with the same boundary conditions in both direction, only half of the loading will be acting in that direction. This will reduce the elastic deflection but also the amount of cracking in that direction and will significantly increase the cantilever deflection compared to the one way back span slab assumption.

 

[hokie66]

I can't see any reason, cracking or otherwise, that the cantilever deflection with a two way backspan slab would be greater than with a one way backspan. Assuming the slab depth remains the same, that is. Maybe that is not what you mean.

 

[rapt]

In the two way back span with beam/wall supports, half of the load is carried in each direction by the slab. Thus only half of the load is acting in the direction of the design with the cantilever.

In a one way slab, all of the load would be carried by the slab in this direction.

So the back span in the two way case deflects far less than it would in the one way case. Forgetting cracking and long term effects, the deflection in the one way case would be double.

The more deflection you get in the back span, the larger the rotation caused at the cantilever support by the back span deflection. This rotation reduces the cantilever deflection, so the larger deflection in the one way back span case causes more rotation at the support giving a smaller cantilever deflection.

 

[hokie66]

That assumes that the back span is actually loaded, other than self weight. With a 12' tall masonry wall on the end of the cantilever, I imagine the backspan deflecting upward.

 

[TTJT]

The way I thought of it is that a two way backspan is "stiffer" than a one way span and would deflect less. The lesser the backspan deflects, the small the rotation about the support and thus less deflection at the tip. I'm thinking though that the cantilever will deflect more than if it was an escastre beam.

 

[Ingenuity]

I would check deflections based upon slab SW on back span (neglect LL here), with SW on cantilever, full LL on cantilever, and DL of masonry wall at tip of cantilever. Boundary conditions assuming that the back span is one-way behavior, which would result in the max 'upwards' defection of the backspan, so therefore the maximum downward defection at the tip of the cantilever.

For the OP case with 2-way back span of 17' x17' (1:1 aspect ratio) the actual upwards deflection of the backspan would be considerably less (about 3 times) when you compare plate bending with span aspect ratio of 1:1 vs infinity (infinity being the special plate bending case of 5/384 for a strip).

Cracking and LT deflections need to be encountered for.

 

[rapt]

Hokie,

Yes, the case of back span loaded only with SW should be considered. That is why RAPT now automatically does alternate span deflection cases, much to the dismay of some designers who do not like to see the bigger deflection estimates and would like me to remove the option! This will increase the cantilever deflection.

The SW still distributes the same way. Without the point load, with a span/cantilever of about 3, the span would be dominating normally if one way, but probably not if 2 way. Adding the point load will change this.

But it does not matter if the cantilever deflection dominates or not, the effect of the back span on the overall cantilever deflection will still be underestimated if the back span is considered as one way if the actual slab is two way.

TTJT
Same result. Less back span deflection so less rotation at the cantilever support so more cantilever deflection.

 

[hokie66]

A lot of double talk here. But anyway, I think TTJT understands how it will behave, but as to a design procedure done by hand or otherwise, quite a few judgments/assumptions are needed. The edges ot the cantilever backspan will be much stiffer than the middle part because of the wall supports under. In addition, if the wall above on the end of the cantilever is considered to keep the slab level along that boundary, the edges would take more of the load than the central part.

 

[Ingenuity]

rapt, I am a bit confused with your words too...so with a 'picture being worth a thousand words', do you agree that for the OP's case that MODEL 2 (3:1 aspect ratio, or effectively infinity, so therefore 1-way behavior) is going to be a conservative-based assumption for the real behavior of the CANTILEVER tip deflection, of the actual structure that is a 1:1 aspect 2-way back span with cantilever (MODEL 1)?

 

[TTJT]

To perhaps clarify the geometry. The cantilever width is 14' and there is at least one bay on each side of the two way backspan so that all the edges are continuous.

 

[ahmedhegazi]

Lets go to the basics ..
-Deflection in concrete depends on stiffness of the section on consider
-Back span stiffness defiently affects deflection od the cantilever.

Assumptions shall be based on knowing if the slab is assumed 1 way what will be the actual bending moment compared to Mcrack and the same for two ways actions , if the slab will crack we talk about reduced back span stiffness

 

[hokie66]

With the cantilever only 14' wide instead of 17', the point I was trying to make about the edges being stiffer than the middle is less relevant. The slab would perform much better if it were full width.

 

[asixth]

You should also considered patterned loading as well. I generally find that end spans and cantilevers are very sensitive to patterned load conditions.

I am not an advocate of patterned loadings. I feel that the probability of live loads is remote enough as it is without the consideration of having the most unfavourable live load pattern.