Draft – Comments Welcome
Shear, Not Sheer
Now we will look at the `shearing’ of a beam, typically near a support. Recall that the approach of `Strength Design’ is to investigate how a structural member fails, and then design so that it doesn’t. When concrete members fail in shear they tend to crack in so-called diagonal tension cracks. These cracks are generally at 45 degree angles to the beam axis in regions of high shear (and low moment).
The 45 degree angle comes about as the result of pure shear being simultaneously felt as pure tension and pure compression at 45 degree angles to the `pure’ shear. Since concrete is weak in tension it will actually fail in tension (diagonal tension) when subject to a shearing load. These cracks may start at the bottom of the beam at the face of the support and angle upward.
If the beam has no reinforcement, the development of a diagonal tension crack will lead directly to the beam breaking apart, catastrophically … `breaking’! Not good. Reinforcement will `hold the beam together’ with the onset of diagonal tension cracking.
Reinforcement related to shear is as follows:
1) Tension (and compression) reinforcement. Generally we don’t directly count the reinforcement, except that with this reinforcement we can use the general equations for reinforced concrete (over the equations for plain concrete) … which will give us higher strength values. And I suspect that the higher Code strength values are real-life reflections of actual increased strength, or safety, or both.
2) Shear Reinforcement. Where / when we use it, shear reinforcement adds directly to the shear strength of the beam. In many applications shear reinforcement will not be necessary. In many applications shear reinforcement would be a nightmare. And in many applications shear reinforcement is indeed necessary.
We will cover shear reinforcement in another lesson (here); in this lesson we will cover shear in concrete beams (and perhaps other members) with flexural tension reinforcement.
2. Shear Strength
Ref.: ACI 318 Chapter 11
Our approach to shear is similar to the other situations we have looked at. We make sure that the factored load at any section does not exceed the factored strength at the section. In equation form,
… Vu ≤ φ (Vc + Vs) …
Vu = factored shear load,
Vc = shear strength provided by the concrete,
Vs = shear strength provided by shear reinforcement (e.g., stirrups), and
φ = strength reduction factor for shear in reinforced concrete.
Shear reinforcement is required where,
Vu > φ Vc … (obviously),
Vu > 1/2 of φ Vc … in flexural members, except for reinforced slabs and footings, concrete joist construction (as defined by the Code), and beams with depths total depths not greater than 10 in., 2.5 times the thickness of the flange, or ½ the width of the web, whichever is greatest.
… Vc = 2 b d √f’c …
Where Vc = the shear strength provided by the concrete,
b is the member width … in the case of joist or T-beam construction b = bw the width of the web,
d = the effective depth (distance from the compressive face to centroid of the longitudinal tension reinforcement),
and f’c is our familiar 28-day specified compressive strength,
and where the strength reduction factor for shear in reinforced concrete is … φ = 0.75.
If the design check at any section is not satisfied, then,
- increase the section size (b or d or both), or
- add or increase shear reinforcement, or
- do approved alternate or more detailed calculations (not covered here).
Note: the above equation will underestimate strength in sections subject to significant axial compression and OVER-estimate strength in sections subject to axial tension. In sections subject to significant axial tension the use of shear reinforcement should be considered for carrying the full shear load.
3. Shear Load
The Code permits us to omit distributed loads applied to the top of the beam within distance `d’ of the support where the support is from below and produces a `compressive’ reaction. This makes sense in that such loads would be `within’ the diagonal tension crack and thus not contribute to shearing the beam.
We saw an upside down version of this when we looked at footings.
4. Example – Shearing a Stem Wall with a Concentrated Load
Let’s consider a 24 in. tall stem wall, 8 in. thick, reinforced as a `wall’ (ACI 318 Chapter 14). Concrete compressive strength is given to be 3000 psi, and let’s say a column delivers a service load P = 10k D and 20k L. Let’s see if the stem wall is strong enough to prevent the concentrated load from shearing through the wall. And let’s not count the benefit of any soil pressure acting upward. If the wall does not shear, then the whole wall will carry the load and an individual footing for the column may not be necessary.
Pu = 1.2 (10k) + 1.6 (20k) = 44 k
To break the stem wall the column would have to shear two sections of the wall.
So, Vu = Pu / 2 = 22k
… φ Vc = φ 2 b d √f’c …
Assuming reinforcement is detailed no farther than 3 in. from the bottom of the wall, d = 24 – 3 = 21 in.
… φ Vc = 0.75 (2) (8 in.)(21 in.) √3000 psi = 13,800 lb (say 13.8 k).
Though the member is greater than 10 in. deep, it is not really a flexural member, so (the `1/2′ is not used) …
Is Vu = 22k ≤ 1/2 φ (Vc) = 1/2 of 13.8k = ? … No, not good.
1) Some designers detail a big footing under the wall to accommodate the column load. I don’t like this approach because it requires the wall to shear to load the footing. Sheared walls don’t look all that great.
2) I generally detail shear reinforcement in the wall `under’ the concentrated load going both directions to points where the upward action of the reactive soil pressure relieves enough shear so that the shear reinforcement is not necessary.
Note that the load used in this example is a pretty huge load for, say, a residence. Many of the concentrated loads encountered in residential and light commercial construction are much lighter than this and will not require shear reinforcement of the stem wall.
5. Example – Shearing a Non-Deep Beam
Example is … here.
6. Example – Shearing a Footing
Example of Beam Shear of a Square Column Footing … here.
Shear Reinforcement in Reinforced Concrete Beams, Jeff Filler, Associated Content.
Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.
Axial Strength of a Reinforced Concrete Wall, Jeff Filler, Associated Content.
Strength of a Reinforced Concrete Beam, Jeff Filler, Associated Content.
Big Column Footing Part 1, Jeff Filler, Associated Content