Volume: concrete, fill and excavation

Volume is the amount of space a solid or liquid occupies. In the trades it decides how much concrete to order, how much soil leaves a trench and how much has to come back, how much water a piping system holds when it is filled for a test, and how big a tank or sump must be. Volume mistakes are expensive: a concrete truck that arrives short leaves a cold joint in a slab, and one that arrives with too much leaves the contractor paying for concrete that goes back to the plant.

At Kingsway Tower, Maya's crew pours small concrete housekeeping pads for the domestic water heaters and booster pumps in the basement mechanical room, digs and backfills a trench for the storm connection outside, and fills the new risers with water for pressure testing. Each job is a volume calculation. Concrete in Canada is ordered by the cubic metre.

Volume of prisms and cylinders

A prism is a solid with the same cross-section all along its length: a slab, a beam, a wall, a footing, a trench. Its volume is the area of the cross-section times the length. A rectangular block is the simplest case: length times width times depth. A cylinder is a prism with a circular cross-section, such as a pier, a round column, a pipe or a tank.

As with area, all three dimensions must be in the same unit before you multiply. The trap with slabs is that length and width are given in metres or feet but thickness in millimetres or inches. Convert the thickness first: 100 mm is 0.100 m; 4 in is 4/12 = 0.333 ft.

Concrete

Concrete is ordered in cubic metres, usually to the nearest half or quarter cubic metre depending on the supplier. Ready-mix trucks commonly carry up to about 8 to 10 m³, and small loads often carry an extra charge. Every concrete order should include an allowance for waste, spillage, uneven subgrade and form bulging, typically 5 to 10 percent depending on the job.

When concrete is mixed from bags, each bag yields a stated volume. Divide the volume needed by the yield per bag and round up. The bag label is the authority, because yields vary by product and bag size.

Excavation and backfill

Soil does not keep a fixed volume. When it is dug, it loosens and takes up more space; this increase is called swell. When it is placed back and compacted, it squeezes into less space than the loose material, and sometimes less than it occupied in the ground; this is shrinkage. Earthwork volumes are therefore described in three states: bank (in place, undisturbed), loose (after digging, in a truck or a pile) and compacted (placed and compacted in a fill).

Swell depends on the soil. Sand and gravel swell less; clay swells more. Figures from about 10 to 40 percent are common, with around 25 percent a reasonable planning number for ordinary soils when no better information is available. The geotechnical report or the site superintendent will give the figure to use on a given job.

Backfill volume is the trench volume minus the volume taken up by the pipe and its bedding. For small pipes in a large trench the pipe hardly matters; for large pipes it does. Remember that the trench in this example has vertical sides only because a trench box protects the workers. A sloped or benched excavation, as required where workers enter without a protective system, removes far more soil. The volume of a trench with sloped sides is its trapezoid cross-section area, from lesson 4, times its length.

Liquid volume in pipes and tanks

Plumbers and pipefitters need liquid volumes to fill systems for testing, to size expansion tanks and to calculate how much glycol or chemical treatment a closed loop needs. The volume of a pipe is the area of its inside diameter times its length. Since one cubic metre holds 1,000 litres, one litre is 1,000,000 mm³, and the arithmetic is straightforward in metric.

Inside diameter matters. The nominal size of a pipe or tube is not its inside diameter, and the inside diameter of the same nominal size varies with the type of pipe and its wall thickness. Look up the actual inside diameter in the manufacturer's table when the volume needs to be accurate, such as when dosing chemicals.

Volume units and conversions

Volume factors are length factors cubed. A cubic yard is 3 x 3 x 3 = 27 cubic feet. A cubic metre is 1.0936 x 1.0936 x 1.0936 = 1.308 cubic yards. That is why a cubic metre of concrete is noticeably more than a cubic yard, about 31 percent more.

A concrete order in code

This program works out a concrete order for several pours, adds waste and rounds up to the supplier's increment. It uses the ceiling function, which always rounds up, the rule for purchases.

import math

pours = [  # name, length, width, thickness (m)
    ("heater pad", 2.40, 1.20, 0.100),
    ("pump pad", 1.50, 1.00, 0.100),
    ("electrical room slab", 6.0, 4.5, 0.150),
]
WASTE = 0.10
STEP = 0.25          # supplier increment in m3

total = 0.0
for name, l, w, t in pours:
    v = l * w * t 
    total += v
    print(f"{name:22s} {v:6.2f} m3")
order = math.ceil(total * (1 + WASTE) / STEP) * STEP
print(f"{'total':22s} {total:6.2f} m3; with waste {total * (1 + WASTE):.2f}; order {order:.2f} m3")

heater pad               0.29 m3
pump pad                 0.15 m3
electrical room slab     4.05 m3
total                    4.49 m3; with waste 4.94; order 5.00 m3

Ordering all three pours together on one truck saves the small-load charges, provided the forms are ready at the same time. That is a scheduling decision for the foreman, but the math tells him what is at stake.

From volume to weight: density

A volume calculation often leads straight to a weight question. Will the truck carry it? Can the floor support it? Can the crane lift the bucket? The link is density, the mass of one unit of volume.

Footings, columns and stairs

Concrete on a building comes in shapes other than slabs. The same rule, area of the cross-section times length, handles nearly all of them.

• A strip footing is a long rectangular prism: width times depth times total length around the building.
• A round column or sonotube is a cylinder: pi times radius squared times height. Use the inside diameter of the form.
• A pad footing is a rectangular prism under each column; multiply one pad by the number of pads.
• Poured concrete stairs are a prism whose cross-section is the saw-tooth shape of the stair seen from the side: the triangles of the steps on top of the sloping slab underneath. Find the side area, then multiply by the stair width.

Two habits keep concrete numbers honest. First, sketch the shape and write every dimension in the same unit before multiplying; most concrete errors come from mixing millimetres and metres, or inches and feet, in one calculation. Second, compare the answer with something you know. A typical ready-mix truck carries somewhere around 8 to 10 cubic metres (about 10 to 12 cubic yards). If your column pour comes out as three truckloads, or as a wheelbarrow, recheck the units before calling the supplier.