ACI 318 Punching Shear - Reinforcement Contribution (Vs)

[Jacob-William]

Hi all,

I'm doing some punching shear checks for a series of columns acting on a 1.0m base slab for a deep underground metro station.

The contribution of stirrups or links is given by S11.4.7.2 in ACI318-11(M), or equivalent 22.6.7.2 in ACI318-19(M) etc.

As the applied shear stress (vu) is taken around the critical perimeter (bo * d), and the concrete contribution to a slabs resistance (vc) is also around this critical perimeter, the shear stress that the stirrups or links must provide is also calculated over the critical area:

ɸvs = vu - ɸvc
ɸVs = ɸvs * (bo * d) {total shear force sitrrups or links needs to provide}

Therefore:

Total Area of Stirrups Required, Av = ɸVs / ɸfyt {where fyt is yield stress of stirrup or link)
Total Area of Stirrups Required, Av = [ɸvs * (bo * d)] / ɸfyt - {Eq. 1}

Up to this point this makes sense to me - and is almost the same as what is specified in the ACI code.

In order to get the above equation I have derived (Eq. 1) to what ACI specifies (Eq. 2) you have to multiply by (s/d), where s is the spacing of longitudinal reinforcement i.e. spacing of perimeter between stirrups or links perpendicular to face of column considered (see below):

Total Area of Stirrups Required, Av = [ɸvs * (bo * d) * s]/ [ɸfyt * d] = [ɸvs * bo * s]/ [ɸfyt] -{Eq. 2 - d terms cancel}

I can rationalize what the use of this factor is trying to do by accounting for the number of 'perimeters' within the critical area d/2 from the face of the column. I.e. if d/2 = 300mm and s = 150mm then you have 2 perimeters within d/2 and therefore you need half the reinforcement so a factor of 0.5 is applied to Av.

However this would require multiplying by a factor of s/0.5d and not a factor of s/d.

I am probably missing something obvious here as I usually design to Eurocodes and ACI is new to me -but I would appreciate if anyone could tell me why a factor of s/d is used instead of s/0.5d?

Thanks in advance!

 

[KootK]

I can rationalize what the use of this factor is trying to do by accounting for the number of 'perimeters' within the critical area d/2 from the face of the column.

My understanding is that the provision is considering a single perimeter and that [s/d] is simply counting the number of reinforcement legs likely to be crossing that perimeter.

 

[r13]

I think the relationship is derived from Vs = Av*fyt*d/s.

 

[Jacob-William]

Thank you both for your responses.

@KootK I also think this is considering the amount of reinforcement in the critical area (d/2) from face - however with what I have derived above I still don't see how this equation should have s/d instead of s/0.5d when you consider it from first principles? (i.e. spacing of stirrups or ties within critical spacing from face 0.5?).

 

[KootK]

@KootK I also think this is considering the amount of reinforcement in the critical area (d/2) from face

It's the amount of reinforcing expected to cross the anticipated shear crack for any punching shear perimeter considered.

however with what I have derived above I still don't see how this equation should have s/d instead of s/0.5d when you consider it from first principles?

Taken as I've explained it above, it couldn't be any more "first principle". The number of stirrups you'll get across a 45 degree crack will be about the depth of the member divided by the spacing of the reinforcing.

 

[r13]

I think you need to check your formulation, which seems false.

ɸvs = vu - ɸvc (EQ1)
ɸVs = ɸvs * (bo * d) (EQ2) {total shear force sitrrups or links needs to provide}

For vs = fy(d/s), vu = Vu/b₂d, and vc = Vc/b₂d; in which b₂ = 4sqr(fc')b_o, can you demonstrate the validity of EQ1, and how to get to EQ2 from there - in terms of s/d maybe?

 

[fugeeo]

If to recall it, Section 11.11 has several types of punching shear reinforcement, depicted in Figures.
Thus, the first stirrup can be no further than d/2 from the first punching zone.
The next punching zone starts at the next stirrup, and then s = d, unless s< d is provided.
Thus, verifying for each outwards perimeter, you get the worst case.
Then if you have overturning, you verify the overturning rotation.