Punching Shear Example 2

EXAMPLE 2 - PUNCHING SHEAR INVESTIGATION FOR COLUMN FOOTING

GIVEN: a column with base dimensions 6 in. x 6 in. delivers a factored load of 29,000 lb to a footing. The footing dimensions are 44 in. x 44 in. x 12 in. thick. The footing is to be constructed of 3000 psi concrete and will have flexural reinforcement placed 3 in. clear from the bottom.

INVESTIGATE: 1) Beam Shear; and, 2) Punching Shear

APPROACH: placing the reinforcement at 3 in. clear from the bottom should give us an `average' effective depth of about 8.5 in. (12 in. minus, say, 3.5 in.). Let's do it ...

CALCULATIONS:

1) Beam Shear

An easy way to investigate beam shear for a footing is to just take the total load and divide by two for the shear force on each `toe' acting as an upside down cantilever beam. As such,

FACTORED LOAD

V_u = P_u /2 = 14,500 lb.

FACTORED STRENGTH

φ V_c = φ 2 bd √ f'_c,

... where,

φ = 0.75 for shear in reinforced concrete,

V_c = nominal shear strength of the concrete,

... b = the whole footing width, 44 in., in this example,

... and d = the effective depth, 8.5 in. ...

So,

φ V_c = 0.75 (2) (44 in.)(8.5 in.)√ 3000 psi = ...

φ V_c = 30,700 lb.

Since V_u = 14,500 lb ≤ φ V_c = 30,700 lb ... GOOD!

In many textbooks you'll see the authors taking the load as that applied out a distance `d' and beyond from the face of the column, effectively taking into account the upward soil pressure `helping' resist the downward load. Fine. But in many cases it's not even necessary. (Or, I could turn this around and say, " ... hey, if we are willing to split hairs as described above, we could perhaps justify a thinner footing." Yeah, I could.)

2. Punching Shear

First let's see if the concrete is able to resist the punching shear tendency without `help' from the soil pushing back up (right under the load point). In this case the punching shear load will equal P_u.

So,

FACTORED LOAD

V_u = P_u = 29,000 lb.

FACTORED STRENGTH

φ V_c = φ 4 b_o d √ f'_c ...

... b_o = 4 x [6 in. + 2 x (½ of 8.5 in.)] = 4 x 14.5 in. = 58 in. (See Page 490 in the Ambrose text.)

φ V_c = 0.75 (4) 58 in. (8.5 in.) √3000 psi = 81,000 lb. Huge.

Since V_u = 29,000 lb ≤ φ V_c = 81,000 lb ... Good! (Way good!)

Let's go ahead and `count' the effect of the soil pressure acting upward (helping to resist the 29,000 lb downward load). We don't need to in this case, but at least you'll have something to follow if you ever need to do it in the future.

Approach:

1. Add in the weight of the footing.
   
2. Determine the factored soil pressure, σ_u.
   
3. By free-body analysis ... V_u = P_u - σ _u A _eff.
   

Okay, then ...

The weight of the footing is 44 /12 ft x 44/12 ft x 12/12 ft x 150 pcf = 2017 lb.

The total factored load on the soil is 29,000 lb + 1.2 (2017 lb) = 31,420 lb.

The factored soil pressure, σ_u , is ... 31,420 lb / (44/12 ft x 44/12 ft) = 31,420 lb / 13.44 sq ft = 2338 psf.

The effective area for the soil acting upward ... (projected area of the effective punching shear volume) ...

A _{eff, inside} = 14.5 /12 ft x 14.5 / 12 ft = 1.46 sq. ft.

So, V_u = P_u - σ _u A _eff = 31,420 lb - 2338 psf (1.46 sq ft) = 31,420 lb - 3414 lb = 28,000 lb.

Since V_u = 28,000 lb ≤ φ V_c = 81,000 lb ... Good! (Still way good!)

And a lot of work for not too much difference.

Another way that some authors illustrate this calc is to basically draw a free-body diagram of the concrete outside the punching pyramid, where, then, the equal and opposite V_u is equal to σ _u A _eff ... where now the A _eff = A _{eff, outside} = A - A _{eff, inside}.

Let's do it, just for kicks and giggles, and to make sure it checks!

A _{eff, outside} = A - A _{eff, inside} = 13.44 sq ft = 1.46 sq ft = 11.98 sq ft.

V_u = σ _u A _eff = 2338 psf x 11.98 sq ft = 28,000 lb ... the same thing!

Again, I argue that if one is going to count the upward soil pressure, why stop at distance d/2 from the face of the load? ... why not go out to `d' ... or even the full thickness `h'? (I mean, look! ... it's customary to go way out to `d' for beam shear!)

References

Punching Shear, Jeff Filler, Associated Content

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