Strength of a Reinforced Concrete Beam

Outline

1. Introduction

2. Our Beam

3. Load Expressions

4. Shear Strength

5. Bending Strength

6. Loads that Break the Beam

7. Safe Loads

8. Conclusion

9. References

1. Introduction

Now let's add some reinforcement to our `Make and Break' (lab) beam. We found that it without reinforcement it wasn't very strong, and it certainly isn't safe. We'll see when we break these beams that without reinforcement they break at relatively small (really small) loads, and they break suddenly, catastrophically, without warning. One second you are standing happily on the beam; the next second you and the beam are on the floor below. As such, we always put reinforcement in concrete beams. And not just any reinforcement - we put in reinforcing steel bars (re-bar). Fiber reinforcement, for example, we have found in previous lab beams, will increase the rupture strength of the concrete a bit, but it (the concrete) still ruptures at a relatively low flexural tension stress, and it still ruptures suddenly.

2. Our Beam

Let's add 2 - # 3 (Grade 60) bars, centered 2 in. from the bottom face of the beam. The beam is the one we have already been investigating (we left off ... here). Actually, we won't put reinforcement in the plain concrete beam - we'll put reinforcement in one just like it, and test both.

3. Load Expressions

The beam will weigh more, with reinforcement. Our lesson on 'Calculating Self-Weight of Concrete' (here) shows us how to calculate the weight exactly. In that lesson, however, we also saw that for most of our calculations we simply use an `average' weight of concrete with reinforcement of 150 pcf, instead of 145 pcf, for plain. So, we need to adjust our load equations a bit to accommodate a slightly heavier beam. From our earlier lessons (here and here),

V _max = V _{sw @ d/2} + V _{applied ω @ d} + V _{from P} ... (I might have used slightly different symbols ...)

For the self weight part ...

... ω _self-weight = γ A = 150 pcf (5/12 ft x 11/12 ft) = 57.3 plf (slightly heavier).

And, so,

V _max = 57.3 plf (3 ft - 2.5/12 ft) + ω (3 ft - 5/12 ft) + P/2.

V _max = 160 lb + 2.58 ω + P/2.

For Bending Moment,

M _max = M _{sw, simple span} - M _{sw overhang} + M _{applied ω} + M _{from P}.

M _max = 57.3 plf (6 ft)² /8 - 57.3 plf (0.67 ft)² / 2 + ω (6 ft)² / 8 + P (6 ft)/4 ...

M _max = 258 lb-ft - 12.7 lb-ft + 4.5 ω ft² + 1.5 P ft = ...

M _max = 245 lb-ft + 4.5 ω ft² + 1.5 P ft.

Factored load expressions ...

V _{max, u} = 1.2 (160 lb) + 1.6 (2.58 ω) + (1.6) P/2 =

V _{max, u} = 192 lb + 4.13 ω + 0.8 P.

M _{max, u} = (1.2) 245 lb-ft + (1.6) 4.5 ω ft² + (1.6) 1.5 P ft.

M _{max, u} = 294 lb-ft + 7.2 ω ft² + 2.4 P ft.

4. Shear Strength

For a reinforced concrete beam the shear strength is,

... V_c = 2 b d √f '_c .

Where,

V_c = the shear strength provided by the concrete (which gets added to the shear strength of shear reinforcement, but we don't have any shear reinforcement in this example, only flexural reinforcement),

... b = width = 11 in. in this example,

... d = effective depth = from compression face down to center of rebar, in our example 5 in. - 2 in. = 3 in., and

... f '_c = our specified compressive strength, for which, for now, we are using 3500 psi.

So,

V_c = 2 (11 in.) (3 in.)√3500 psi =

Vc = 3905 lb.

Note that this is less than the value calculated for a plain beam. Generally the strength of the concrete `calcs out' stronger for a reinforced beam, even when the reinforcement is only flexural reinforcement. The main reason it is less in this case is because we use a different depth, 3 in. instead of 5 in.). And the reason for that is ... even though flexural reinforcement better `holds the beam together' (even for shear) ... it does so at the `expense' of the concrete below the flexural reinforcement.

5. Bending Strength

Our bending strength of the reinforced concrete beam (plank) is calculated as follows ...

M_n = bd² R where

R = ρ f_y (1 - 0.59 ρ f_y / f '_c),

and

... ρ = A_s/(b d).

We have picked the smallest rebar size commercially available, # 3, which is 3/8 in. in diameter.

A single bar has a cross section area, a_s, of 0.11 in.², so A_s = 0.22 in.².

Using Gr. 60 reinforcement, for which f_y = 60,000 psi, so,

... ρ = 0.22 in.² / (11 in. x 3 in.) = 0.0067

R = 373 psi,

and,

M_n = (11 in.)(3 in.)² 373 psi

M_n = 36,930 lb-in ... or ... 3078 lb-ft.

This `moment strength' is developed in the `ultimate' condition where steel has yielded and concrete is crushing in the compression zone.

6. Loads that `Exactly' Break the Beam

Let's calculate the load that will `exactly' break the beam in shear; we'll do that by setting our shear load expression equal to our shear strength ... let's do it for load P.

V _max = 160 lb + P/2 = 3905 lb.

P = 7490 lb. (Sometimes I'll denote this as ... P _v ... the P that shears the beam.)

We could similarly calculate the uniform load that shears the beam.

Now let's calculate the P that (`exactly') breaks the beam in flexure.

M _max = 245 lb-ft + 1.5 P ft = 3078 lb-ft

P = 1889 lb. (We could call it P_ult ... the P that yields the steel in `ultimate' flexure condition.)

Note that the P that broke the plain concrete beam in flexure was 595 lb. By adding two little pieces of rebar we have increase the strength three-fold over the plain concrete beam. But even more important, we have made the beam WAY SAFER. With the plain concrete beam, once we hit, say, 596 lb, the beam breaks apart suddenly and ends up in pieces on the floor below. With the reinforced concrete beam, once we hit 596 lb, basically nothing happens. The steel will start to yield a little. The beam will have some cracks, but the cracks will be small. Under more and more load, we eventually yield the steel, and these cracks begin to open up a bit. The beam is `failing' (steel yielding), but it is doing so SAFELY. It is still in one piece; and it has not `come down'. As we apply more load to the beam, the load value will increase some more, and the cracks will open up more (as the steel yields more). The beam will get more and more `bent', but it will still be in one piece. Eventually we will meet the ultimate condition where we crush the concrete in the compression zone. As it deforms more and more, it will essentially `hinge' at the mid-span. We may indeed never be able to get it to break in half. We'll probably half to haul it out of the lab still in one piece (albeit quite bent one piece).

Note that P_ult is less than P_v ... it will `yield' before it breaks in shear.

We could also calculate the `exact' uniform load to break the beam.

I'll leave that up for your exercise. We do not have an apparatus in the lab, however, to test our numbers, and I don't think we will be able to get enough students on the beam to test it, either.

7. Safe Loads

To determine the `safe' loads on the beam we need to bring in our safety factors. Our load expressions already have the factors of safety for the `load' part; now we need to bring in our `strength reduction factors'. For reinforced concrete the φ values are 0.90 for flexure governed by steel yielding and 0.75 for shear.

So, for the `safe' load P that shears the beam (call it P _{safe, v}) ...

Let V_u = 192 lb + 0.8 P = φ V_c = 0.75 (3905 lb)

P = P _{safe, v} = 3421 lb.

The safe load P for bending, is obtained ...

M _{max, u} = 294 lb-ft + 2.4 P ft = 0.9 (3078 lb-ft) ... gives

P _{safe, bending} = 1032 lb.

So, flexure controls, ... we could say that ...

The safe load P for the beam is ... 1032 lb.

Note that in the case where we are controlled by flexure the overall factor of safety is about two.

8. Conclusion

Now we have calculated the loads that will break the beam with some reinforcement in it. Actually, I have calculated concentrated loads (P) at mid-span only (also considering, of course, self weight). By adding a couple small pieces of reinforcement we have made the beam stronger, and way safer. Also, we now have a `legal' beam, as plain concrete is not allowed for beams in building construction.

9. References

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

Shear and Moment Diagrams for a Simple Beam, Associated Content.

Shear and Moment Diagrams for a Simple Beam, Part 2, Associated Content.

Breaking a Plain Concrete Beam, Associated Content.

Calculating Self-Weight of Structural Concrete, Associated Content.