Post Punching Through a Concrete Slab

Situation

Now let's consider a `remodel' or some other situation where a `new' post is going to bear on and `existing' concrete slab. The slab is 4 in. thick and reinforced with welded wire fabric (WWF) presumably at mid-depth. The slab is founded (again, presumably) on decent bearing material, and isn't cracked (at least not near where we want to put the post). The load to be carried by the post is 2100 lb Live and 900 lb Dead (as it arrives to the slab). The question is: do we need to install a new footing for the post (saw cutting through the existing, digging down, etc.), or is the existing slab adequate to carry this new load? The post is a wood 4 x 4, which has actual dimensions of 3.5 in. x 3.5 in. It would be nice if the existing slab is adequate. Well, let's see.

Approach

This is a `punching' shear problem; we want to see if the post will `punch through' the existing slab. We will assume that the downward load P is resisted entirely by the shear resistance of the concrete. By this I mean that we will not count any upward soil pressure. In the case of investigating punching through an existing slab, this is appropriate, as to actually `load' the soil directly under the post any significant amount, we would have to crack the concrete, and that is the very thing we are hoping to avoid. So,

... V = P.

Where

P = the (service) load from the post, and

V = the shear (load), for equilibrium.

Unless we have documentation indicating otherwise, I will assume a concrete compressive strength of 2500 psi (the min. allowed compressive strength).

Allowable Stress Design Approach

Current practice with reinforced concrete is to use the Strength Design (SD) approach. However, in some foundation calcs, especially where I want to see if, for example, `I am even close', I will use Allowable Stress Design (ASD). And, if I am close, and need to follow up with SD, I do so. However, heretofore we have been talking `Service' loads, in our (course) conversation (in contrast to `factored' loads), and so in that regard ASD is nice, in that it utilizes our Service (un-factored) design loads. So, in the ASD approach, we will calculate the applied punching shear under design load, and make sure it is less than the allowable punching shear stress.

... f_v = V / b_o d ≤ F_v ... ?

Where

... f_v = applied shear stress at design (Service) load,

V = applied design (Service) shear,

b_o = the `perimeter' of an imaginary vertical shearing surface d/2 away from the face of the post (or post base),

... d is the effective depth, in this case the depth to the rebar, and

F_v = 2 √ f'c ... for punching shear (sometimes called `peripheral' shear, and at other times called two-way shear).

So,

V = 900 lb + 2100 lb = 3000 lb.

... b_o = 4 (d/2 + a + d/2), where a = the width of the `face' of the post (or post base) = ...

... b_o = 4 (2 in./2 + 3.5 in. + 2 in./2) = 4 (5.5 in.) = 22 in.

... f_v = V / (b_o d) = 3000 lb / ( 22 in. x 2 in.) = 68 psi.

F_v = 2 √ f'c = 2 √ 2500 psi = 2 (50 psi) = 100 psi.

Is f_v = 68 psi ≤ F_v = 100 psi? ... YES! ... Good!

Or, cast in terms of a Unity Check ...

Is f_v / F_v = 68 psi / 100 psi = 0.68 ≤ 1.00? ... Yes!

Strength Design

In the SD approach we make sure that Factored Load (on the whole section, or surface) does not exceed Factored Strength (of the section, or surface); cast in equation form,

Is V_u ≤ φ V_c ?

FACTORED LOAD

In generic form,

U = 1.2 D + 1.6 L.

So, P_u = 1.2 (900 lb) + 1.6 (2100 lb) = 4440 lb.

For equilibrium,

V_u= P_u = 4440 lb.

FACTORED STRENGTH

... φ = 0.75 for shear in reinforced concrete, and

V_c = 4 b_o d √ f'c for punching shear,

where b_o , d, etc. are the same as for ASD.

So,

... φ V_c= 0.75 (4) 22 in. (2 in.) √ f'c = 6600 lb.

Is V_u = 4440 lb ≤ φ V_c = 6600 lb?

YES!

In terms of a Unity Check? ...

Is 4440 / 6600 = 0.67 ≤ 1.00? Yes.

Wow, ... the Unity Checks are almost identical. This is not always the case. The fact that they are so close is comforting in the sense of using ASD as at least a first order check. But the checks will `vary' in closeness depending on how Dead and Live (non-dead) split out. ASD essentially assigns only a single split of D and L, whereas in Strength Design the D and L are split specifically for the situation at hand.

NOTE: and, even though the Unity Checks are nearly identical, which, we would hope they would be, the two `checks' are philosophically quite different. In ASD we are making sure that the loads expected in service are way below the failure condition. (By `way' I mean some failure or ultimate state divided by a factor of safety.) In Strength Design we take a look at limits (strengths) of the materials, factor them down with strength reduction factors, and make sure that they are never exceeded by expected service loads, factored up for legitimate uncertainties (in the loads).

Discussion

The above calculations are only as good as they represent actual conditions. If the concrete appears degraded, it may be that the strength is low, and the value of 2500 is not suitable. The reinforcement may be at mid-depth, or not. Or, it may be at an average of mid-depth, but varying somewhat. (Anyone who has unrolled WWF and tried to hold it in place during a pour knows what I am talking about.) The depth may need to be verified. The minimum slab depth is 3.5 inches, which could be placed using nominal 4 in. dimension lumber (3.5 in. actual). And, as I mentioned, if the area of interest is cracked, why bother (with the calcs).

If the design checks `don't work out' (or if the slab is already cracked), then we have several options:

1. cut the slab and place a new reinforced footing, or
2. use a post/column base that spreads the shear out (by increasing b_o) enough so that the design check does work out.

Summary

Some useful equations you may take away from this are:

ASD

f_v = V / (b_o d ) ...

F_v = 2 √ f'c ...

b_o = d/2 + a + d/2 = a + d

SD

V_c = 4 b_o d √ f'c for punching shear in reinforced concrete,

... and b_o is the same as for ASD.

Finally, make sure the post (wood) is properly protected from moisture in the concrete, and that the post is anchored against horizontal movement at the slab.

References

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