Punching Shear

A Lesson in Reinforced Concrete Design - Comments Welcome

1. Introduction

Consider a column delivering a downward onto on a slab of concrete. There load from the column will want to `punch through' the slab, pushing some concrete along with it into the soil, or air, or whatever lies below. Since concrete shears at 45 degree angles (diagonal tension), the `45 degree angle thing' will manifest itself as a 3-D `cone' or `pyramid' shape piece of concrete wanting to punch through.

Three common encounters of this (potential) phenomenon in structural concrete are:

1. A column or post tending to punch downward through a slab.
   
2. A column tending to punch through a big, thick slab footing.
   
3. The top of a column tending to break upward through a slab and the slab falling downward around around the column.
   

2. Strength Design Equations

PLAIN CONCRETE

Factored Strength ... φ V_n = φ 8/3 √ f'_c b_o h,

where,

... φ is the Strength Reduction Factor for Plain Concrete, 0.55,

V_n = is the nominal (`perfect world') punching shear strength,

√ f'_c = the specified 28-day compressive strength,

b_o = is the `perimeter' of an equivalent rectangular shaped cone whose vertical interest the presumed `cone' or `pyramid' at one-half the depth of the concrete slab, and

h = the depth of the slab.

Note: where the slab is cast on soil or earth the depth h is taken to be the physical depth minus 2 in.

So, what the equation above is doing for us is allowing us to not have to calculate the actual surface area of the diagonal-sided cone or pyramid but that of a `rectangular' surface (b_o h) instead.

The slab will be `strong enough' if the factored load tending to punch through the slab does not exceed the factored strength.

REINFORCED CONCRETE

Note: the following is for determining the shear strength of reinforced concrete. By this we mean that there is reinforcement present for flexure or temperature/shrinkage effects, but not necessarily any shear reinforcement. In fact, for all the punching shear applications I can think of, trying to put in reinforcement to resist this punching shear action would be ... well, weird.

Factored Strength is ... φ V_c = φ 4 b_o d √ f'_c,

where,

φ = 0.75 ...

V_c = `perfect world' strength of the concrete resisting punching shear

4 = a number, that accounts for the 3-D-ness and the equivalent vertical face cone used in the calcs.

b_o = the `perimeter' of an imaginary vertical surface of shearing concrete parallel to the load (perp to concrete face) at d/2 away from the perimeter of the boundary of bearing of the load,

d = effective depth (the depth of the reinforcement),

and ...

√ f'c = from before.

It appears that the higher Strength Reduction factor (0.75 vs. 0.55) and the higher coefficient (4 vs. 8/3) demonstrates the benefit of the reinforcement `holding' the concrete together (even though it is not counted as `shear' reinforcement).

Now let's look at the three common encounters where we might use these equations, listed above.

3. Column or Post Punching Through a Slab

Consider first the case of a column or post tending to punch through a slab, say a slab on grade. Such might be the case in a remodel of an existing structure where a bearing wall is being replaced by beam(s) and columns or posts. The slab may be either plain or reinforced, and it might take a bit of investigation to determine which (say, using a metal detector), if as-built plans are not available. And if the thickness of the slab is not known, drilling a hole or two should provided that information. If we can determine that the slab as-is is strong enough (won't punch through), then GREAT! If not, then we will need to saw cut the slab and put in a new footing for the column.

Note: when we look at column footings we will see that we can `count' the effect of soil pressure pushing back up. In the case of a column or post tending to punch through a slab on grade (not a slab footing) I do NOT count the soil pressure acting upward, as the concrete is much stiffer than the soil and the concrete would have to break before the soil directly underneath would pick up significant pressure. And if the concrete then breaks - we don't have a punching shear calculation to perform - but we better worry about the capacity of the underlying soil. So, for slabs on grade, I assume that there is just `air' underneath for the punching shear calculation.

As such, the factored load, V_u, acting to punch through the slab will be equal to the factored concentrated axial load from the column, P_u.

V_u = P_u,

Is V_u ≤ φ V_n (plain concrete) ... or ... is V_u ≤ φ V_c (reinforced concrete)?

If the answer is `yes', the `Good'! ... provide proper anchorage for the column and protection from moisture as appropriate, and use the slab as is (without cracking it in the installation process).

If `no', the cut the slab and install a proper reinforced slab footing with appropriate connection hardware ... OR ... consider a wider base for the column or post. Since b<sub>o</sub> is a `perimeter' a little bit of increase of base size goes a long way in increasing the punching strength of the concrete (increases the effective shearing resistance area).

4. Column Punching Trough a Footing

In this case we are considering the possibility of the column punching through a footing designed to carry the footing load on the soil. We follow the approach from before for footings in general:

1) make it wide enough so it doesn't fail the soil;

2) make it thick enough so it doesn't shear (beam or one-way shear);

3) provide reinforcement as needed so it doesn't break (in upward bending);

... and, now ...,

4) also make sure we don't punch through (punching, peripheral, or two-way shear).

In the case of a column footing the upward soil pressure acting upward on the bottom of the footing may be taken into account in providing some assistance to the concrete in resisting the punching action. (See Page 490 of the Ambrose text.) And it adds a bit of complexity to the design check. In my practice I generally do the design check not counting the soil, and if it doesn't check, then take into consideration the benefit of the soil pushing upward. I cannot recall where punching shear controlled the design of a reinforced concrete footing for a column, but `do the design check' anyway.

Note: in general we do not provide shear reinforcement in a footing for either beam shear or punching shear. (It would get `yucky'.)

So, in terms of equations,

V_u = P_u ... OR (if I want to go through the work) ... V_u = P_u - A _eff x σ _u,

where,

A eff = the contact area of soil acting upward to help counteract the concentrated load acting downward, and

σ_u = the factored soil pressure (P_u / A _footing).

You'll see in the Ambrose text the the A eff is taken to be that of the soil surface bounded by the `effective shearing surface' d/2 away from the column base. I might argue that the effective area could be taken out as far as `d' ... or even `h' (the intersection of the potential shearing cone with the bottom of the footing).

Then,

Is V_u ≤ φ V_c ? ... and so on.

5. Column Punching Upward Though a Slab Above.

This case is essentially the same as the first case, with actual `air'. If the design check isn't `good', then a `capital' (or some other means) may be used to increase the bearing area and thus the effective area of concrete providing punching shear resistance.

6. Conclusion

The punching shear thing is kind of cool. In the case of using an existing slab to resist a concentrated load it should go without saying (but I'll say it anyway) that if the slab is `cracked' (already), saw cut and put in a new footing.

The equations for beam shear and punching shear above are actually just a few of a whole `family' of equations for beam and punching (two-way) shear.

Examples follow.

Example 1 - Punching Shear in a Plain Concrete Slab.

Example 2 - Beam and Punching Shear in a Reinforced Concrete Footing for a Column.

7. References

Simplified Engineering for Architects and Builders, Ambrose, J. and P. Tripeny, 10^th edition, John Wiley & Sons, Hoboken, New Jersey.

Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.

Punching Shear Example 1, Jeff Filler, Associated Content.

Punching Shear Example 2, Jeff Filler, Associated Content.