Calculating Self Weight of Structural Concrete

Outline

1. Joist / Beam Self-Weight
2. Slab-Joist System Self-Weight
3. Equivalent `Area' Weights
4. Weight per Thickness
5. Summary

1. Joist / Beam Self Weight

Let's consider a concrete beam 5 in. wide and 9 in. deep (tall) reinforced with 1 - # 5 bar. For beams we generally describe the self-weight in terms of weight per foot (of beam) or `line-load'. The weight of a foot of the beam described will be made up of the weight of the concrete in (a foot of the) beam and the weight of the steel in (a foot of) the beam.

First let's deal with the steel. Steel weights 490 lb per cubic foot (pcf). The number of cubic feet of steel in the above beam is ...

Volume = A_s x L

A_s = pi d ² / 4 = pi (5/8 in.)² / 4 = 0.31 in.² ...

We'll need it in ft² ... 0.31 in.² / (144 in.²/ft²) = 0.00213 ft².

So, volume of steel is ... 0.00213 ft² x 1 ft = 0.00213 ft³.

The weight of the steel is ...

W = spec. weight x Vol = 490 pcf x 0.00213 ft³ = 1.04 lb.

Now let's tackle the concrete.

Plain normal-weight concrete weighs ... 145 pcf.

The volume of the concrete and steel composite is ...

Volume = A x L

where A = 5 x 9 in.² = 45 in.² = 0.3125 ft² ...

So, the volume of a 1 ft long piece is ... 0.3125 x 1 = 0.3125 ft³.

Taking out the volume of the steel rebar, ...

Volume of concrete (only) = 0.3125 - 0.00213 = 0.3104 ft³.

The weight of the concrete (only) is ...

W = 145 pcf x 0.3104 ft³ = 45 lb (per foot)

Thus, the weight of the composite, per foot, is ...

45 lb + 1.0 lb = 46 lb (per foot).

Weight (or load) per foot of beam we generally denote by ... `w'

Thus, for this beam,

w _self-weight = 46 plf.

In this particular example the weight of the steel accounts for about 2 percent of the total weight. Usually what we do with reinforced concrete is use kind of an `average' value of 150 pcf for normal weight concrete with `normal' amounts of steel (usually not more than a few percent by volume). This makes our calculations easier ... we just use the gross dimensions of the beam ...

So, ...

Vol = A _gross x L = 0.3125 ft² x 1 ft = 0.3125 ft³ (per foot)

And now the weight (per foot) is 150 pcf x 0.3125 ft³ = 46.9 lb (per foot).

So, if we use the `150' pcf average value, we over-estimate the weight of the beam in this example, but by just a bit.

It turns out that the 150 pcf represents concrete that is about 1.5% steel by volume. Our beam it turns out is only a bit over half a percent steel by volume. The value of 150 pcf is probably suitable for most joists and beams, maybe a bit low for heavily reinforced columns.

Before we move on, let's make another simplification.

From the above we can see that

w = weight / length = (spec weight x Vol ) / L = spec weight x A x L / L = spec weight x A.

So, here's our shortcut formula,

w _self-weight = spec weight x A ...

giving us the self-weight per foot of length of the member.

2. Self Weight of a Slab-Joist System

Now let's consider something slightly more complicated ... a slab-joist system (see illustration) where we have a 5 in. slab poured monolithically with 4 in. x 8 in. joists. (The self weight calculation we will do would be the same if the joist and slab were poured separately, but the system would be way different structurally). And since the joists are 24 in. apart, let's do the calculation on a 24 in. wide basis.

w _self-weight = spec weight _slab A _slab + spec weight _joist A _joist ...

Let's use spec weight = 150 pcf for both slab and joist ...

and getting our `areas' into ft² ...

w _self-weight = 150 pcf (5/12 x 24/12 ft²) + 150 pcf (4/12 x 8/12 ft²) = ...

... 150 pcf (0.833 ft² + 0.222 ft²) = 150 pcf (1.056 ft²) =

w _self-weight = 158 plf.

That was easy.

So, a 24-inch wide strip of slab-joist composite weighs 158 lb per foot.

And we did this calculation assuming `normal' amounts of rebar (though we didn't deal with exactly how much).

3. Equivalent Area Weights

Sometimes what we like to do with systems like the one above is talk about `area weights' ... so many pounds per square foot of floor.

It's easy! ... Let's get it by basic principles, and then by equation.

First, from above, our strip that is 2 ft wide weighs 158 lb per foot. So, this 158 lb is for 2 x 1 sq ft of floor. So, our `area' load (weight) is ... 158 lb / 2 sq ft = 79 psf.

If we want a formula ...

area load = w / tributary width ... or w / spacing.

So,

`area load' is in, say psf (pounds per square foot)

and

`w' (a.k.a. a `line load') is in, say, plf (pounds per lineal foot)

and, tributary width = the width of the system, or whatever (ft) `contributing' the area load.

So,

area load = line load / tributary width = 158 plf / 2 ft = 79 psf ... same thing.

This slab-joist floor system weighs 79 psf ...

And if we wanted the self-weight of the entire floor system, we'd add the weight of insulation, ceiling finish, floor finish, etc. ... (also commonly expressed in area load (psf) values).

4. Weight per Thickness

Sometimes we like to deal with weights of things on a `per inch of thickness' basis.

In terms of an `area' weight per thickness, for plain concrete, we would have ...

145 lb per cubic foot, divided by 12 in. per ft of thickness = 12.1 psf per in.

So, normal weight concrete weighs about 12 psf per inch of thickness.

If we have, for example, a 3-1/2 in. thick slab, then it weighs (per square foot) ...

... 12 psf / in. x 3.5 in. = 42 psf.

Cool.

5. Summary

We take normal weight concrete to weigh 145 lb per cubic foot (pcf) and reinforced (normal weight) concrete to weigh 150 pcf.

The value of 150 pcf takes into account `normal' amounts of rebar, generally a few percent, by volume.

And since we didn't take the `exact' amount of rebar into account in our calculations, the numbers don't change if we change the amount (or rebar) a little.

And we looked at some useful equations ...

Weight = W = spec weight x Vol ... where ... Volume = A x L ... where A is the cross-section area, and L is length.

w _self-weight = spec weight x A

w = area load x tributary width, or area load = w / tributary width.

And concrete weighs about 12 lb per square foot per inch of thickness.

By now it should be obvious that some of these numbers are less exact than they look. I mean, look! - we didn't account for small rounding of corners, either.

We are so ready to roll!