# Multi-stage/multi-section reinforced concrete slab design

In pre-cast bridges, one often lays beams over the columns and then places thin pre-cast "pre-slabs" over the transversal spans between the beams. These slabs already contain all the requisite positive transversal (between-beams) steel reinforcement. All other reinforcements (positive longitudinal, negative transversal and longitudinal) are then placed and In-situ concrete is poured over the "pre-slabs" up until the requisite thickness. Steel framing contained within the "pre-slabs" certifies that the pre-cast and in situ concrete work together under ultimate loads. Here's an example:

So, how does one calculate the slab's positive transversal reinforcement? The steel must be stressed so as to create a force couple with the resultant of the concrete's compression.

Now, the "pre-slab" needs to be able to withstand a load equal to its own self-weight and that of the wet in situ concrete. If this were calculated by traditional methods, a certain steel area $A_{s,1}$ would be obtained.

The additional loads (paving, moving loads, etc) are then applied as well once the in situ concrete has hardened. These loads are therefore resisted by the force couple composed of compression of the in situ concrete and tension of the reinforcing steel. If this were also calculated by traditional means, a steel area $A_{s,2}$ would be found.

Now, is the resultant necessary steel equal to $A_{s,total} = A_{s,1}+A_{s,2}$? I was initially concerned that there might be an interference in the compression regions, but given that the pre-slabs are usually less than half the total slab thickness (in the example above, a 7-cm pre-slab supports 13 cm of in situ concrete, for a total slab thickness of 20 cm), I don't believe that'd be a problem.

That being said, I'm not entirely certain of this conclusion and would like confirmation or correction on this matter. This summation of areas means that all of the steel would be stressed below their ultimate capacity in the first stage (such that $A_{s,total}\sigma = A_{s,1}f_{yd}$) and then would reach the ultimate stress under final loads (such that $A_{s,total}\left(f_{yd} - \sigma\right) = A_{s,2}f_{yd}$). Is this fine or is there some behavior I'm not taking into consideration?

Obviously, then there's the case where the compression zones of the different stages interfere one with the other, but I'll leave that for a future question.

EDIT: Here's an example of a pre-slab blueprint, including the steel reinforcement. This case is for the pre-slab including a cantilever, which explains the gap: that's where the beam's transversal steel joins with the in situ concrete. The units are centimeters. The "criss-cross" in the cross-section is a steel truss.

This site describes pre-slabs thusly (according to my rough French translation): "The pre-slab is layer 6 to 8 cm thick, creating a sort of base, whereupon once installed on the site, one must add a compression layer." It also has the following picture of a real pre-slab, where the trusses responsible for certifying that the pre-cast and in situ concrete layers work in unison are clearly visible:

• All of that sounds about right. Were you just wanting confirmation, or is there some finer question that I missed? – hazzey Jan 6 '16 at 1:27
• @hazzey, Mostly confirmation. This question's been bugging me for a long time and I finally sat down to think through it. I went through quite a bit of flexure-design math before figuring out that this seems like a somewhat trivial solution. Fearing it was too trivial, I felt the need to confer with a wider audience. – Wasabi Jan 6 '16 at 2:10

I'm inferring that you're in the UK and this is a real design rather than some theoretic statics exercise. I've not come across pre-slabs before. They're not permanent form work are they?

Unfortunately your solution is somewhat trivial :-( Top slab design can quite involved, and the load analysis certainly is. There's Eurocode 2, BS5400 and it's offshoot, BD44. These will give you the detailed processes to go through. If these pre-slabs are meant to be structural I suggest referring to the codes for how they are to be incorporated into the load bearing design. Those bureaucrats love to include this kind of stuff.

Don't forget that the top surface of the slab passing over the beams might be in compression, so you'll need bottom steel to resist the moments. In this situation a pre-slab would have no effect and have to be ignored. This all depends on the type of analysis you perform, and agreement with the client for your said methods.

Speaking from ignorance vis a vis pre-slabs, I'm still surprised that you only need to consider the wet weight of concrete for it's steel design. This is negligible. You also need to consider construction loads, and I would have expected that there would be a standard loading applied to allow for this. Some paddy is going to want to drive a cherry picker onto the un poured deck. Remember that bridges that fall down, usually fall down during construction when loads are in flux and difficult to quantify.

Bending is only one design criterion affecting the amount of steel. There will also be flexural shear, punching shear, cracking, fire resistance and minimum bar spacings. It might take you a week to design the slab, excluding the load analysis.

• I'm actually in Brazil, but our codes are mostly quite similar to Eurocode 2. However, our codes don't comment on this subject (here, our bureaucrats are too lazy to include this kind of stuff). The pre-slabs are not permanent form-work. They are pre-cast reinforced concrete slabs which serve a structural purpose. I'm trying to find a good example to add to my question to show how they are reinforced in order to ensure they work together with the wet concrete. And you're right: there should be additional loads as well, just forgot about them when writing the question. – Wasabi Feb 3 '16 at 9:59
• Also, I don't quite understand your third paragraph "top surface of the slab... might be in compression..." Over the beams there are no pre-slabs, it's all in-situ. The pre-slabs do have protruding steel, though, but that's transversal to the bridge. Do you mean bending in the longitudinal direction (the direction of traffic flow)? For longitudinal bending all the steel reinforcement is placed in the in situ concrete, so the pre-slab is ignored, yes. Regarding the other design criteria, you are also correct, I just didn't want to broaden the question too much. – Wasabi Feb 3 '16 at 10:28
• @Wasabi Err, no I meant orthogonally to the traffic flow. You photo is different to your original cross section as the main beams are much closer together. Depending on your analysis, you might find that as an inner beam deflects downwards due to a loading edge, it drags down the adjacent beam. This will cause hogging within the slab at right angles to the traffic flow. It also hogs due to loads between beams as the slab is continuous over many beams. It's magnitude will depend on whether the slab design is static or a finite element type. Sorry about the UK comment :-) – Paul Uszak Feb 5 '16 at 2:20