Concrete slab design for shear - ACI 318-05

[RHTPE]

In ACI 318-05 equation 11-5 (11.3.2.1) uses ρw (reinforcement ratio). For typical one-way slabs, shear is highest near the face of the supports. When determining Vc, equation 11-5 can provide a slightly higher value.

My question for the group - Is ρw the ratio of top steel to concrete cross-sectional area, bottom steel to concrete cross-sectional area, or top and bottom steel to concrete cross-sectional area? I would be inclined to think that is top steel area only since this equation is affected by Mu, which, at the supports, is resisted primarily by the top steel.

Anyone care to weigh in?


Ralph
Structures Consulting
Northeast USA

 

[SethGuthrie]

As is the tension steel, so just the top bars in your case.

 

[hokie66]

Seth is correct. The effective depth is always to the extreme compressive fibre, so the steel giving pw is on the tension side.

 

[Archie264]

Hmmm...does this merit further examination? <==(That’s a question, not a challenge; I’m asking to learn.)

I agree that ρw refers to tension steel ratio (as it is, indeed, so defined, explicitly.) And yet two things make me wonder:

(1) In Notes on ACI 318-08, as published by the Portland Cement Association, Example 12.4, they specifically use bottom steel in this scenario. Granted, it’s for the design of a continuous concrete joist, not a slab, per se, but the principle should be the same. Granted also that they mislabeled the equation as 11-3 but it is clearly 11-5. And we can't rule out that they are simply wrong in this case.

(2) Wang & Salmon (4th Ed.) have the following commentary:

Note…that the reinforcement ratio, ρw = As/(bwd) is used in the ACI Code formula, where bw is the web with for a T-Section rather than the flange width. The ACI Code defines bw as “web width.” For tapered webs such a definition is unclear. In general, when the web is subject to flexural tension the “average web width” should be used as bw. However, when the web is subject to flexureal compression (as for negative moment regions) the use of average web witdth may be unsafe. For such negative moment reginons the use of a value lower than the average, perhaps the minimum, web width is more appropriate.

The value of Vud/Mu shall not be taken greater than 1.0 except when axial compression is present, which has the effect of limiting Vc at and near the points of inflection.

Continuous Beams. The application of [Eq. 11-5] for continuous beams has recently been subject to question. The compression-strut action on a continuous beam is shown in Fig. 5.5.2* The analogy in Fig. 5.4.3** that M/V equals the shear span a implies that at the point of zero moment there is a support to accommodate a compression-strut [See Fig. 5.4.5(a)***]. For a continuous beam as in Fig. 5.5.2, the distances a1 and a2 are analogous to a in Fig. 5.4.3; however the is no support to take a compression-strut reaction at the inflection point. The actual strut would relate to a longer length a1 + a2. Thus Ferguson**** recommends using d/(a1 + a2) for concentrated loads and .25 for uniform load, respectively, as the effective depth to shear-span ration in place of Vud/Mu in [Eq. 11-5]. Alternatively, he recommends using the simplified constant value of Vc for continuous beams.

If in doubt about all of this note the punch line of using the simplified version of Vc. However, that rather dodges the point of the original question.

* Fig. 5.5.2: Sorry, I can’t provide a sketch but a1 is the horizontal projection of the compressive region in negative bending to the point of inflection and a2 is the horizontal projection of the compressive region in positive bending to the point of inflection.

**Fig. 5.4.3 is a simply supported beam with two point loads, with Mmax = Va.

***Fig. 5.4.5(a) is a sketch of a deep beam undergoing strut-and-tie action.

****Phil M. Ferguson. Reinforced Concrete Fundamentals (4th Ed.)

 

[RHTPE]

Thanks for the input. I used only the tension reinforcing in my check so hopefully it will be conservative.


Ralph
Structures Consulting
Northeast USA