Slope and grade: drains, ramps, stairs and roofs

Water runs downhill, people climb stairs, wheelchairs roll up ramps and snow slides off roofs. Each of these depends on a slope being right: steep enough to work, gentle enough to be safe, and the same from one end to the other. On Kingsway Tower, Maya sets the slope on every horizontal drain, the concrete crew forms the ramp down to the parking levels, the carpenters lay out temporary and permanent stairs, and the roofers check the pitch on the mechanical penthouse roof. They describe slope in different ways, but the arithmetic underneath is the same rise over run you met in lessons 2 and 7.

This lesson uses metric for the Canadian examples, but you will hear plumbers in Hamilton talk about a quarter inch per foot just as often as 2 percent. Being able to move between the descriptions is part of the skill.

Four ways to write a slope

Slope is always the vertical change divided by the horizontal distance. What differs is how the number is written:

To convert, write the slope as a fraction of rise over run and do the arithmetic. A slope of 1 in 50 is 1/50 = 0.02, which is 2 percent. In imperial, 0.02 times 12 in is 0.24 in per foot, very close to a quarter inch (0.25 in) per foot. That is why plumbers treat 2 percent and a quarter inch per foot as the same thing for everyday work, though strictly a quarter inch per foot is 2.08 percent.

Drain slope and fall

Gravity drains must slope enough to carry water and waste without leaving solids behind, but not so steeply that the water runs away from the solids. The plumbing code sets minimum slopes by pipe size. In the National Plumbing Code and the Ontario Building Code, small horizontal drains are commonly run at 1 in 50 (2 percent), and larger building drains and sewers may be permitted at 1 in 100 (1 percent). Always confirm the required slope on the drawings and in the code that applies; this course gives the arithmetic, not the code rules.

The quantity a plumber needs most is the fall, the total drop over the length of the run. It is the run times the slope.

Working the other way is just as common: given the room you have, what slope can you get? If a drain has to run 9 m but can only drop 120 mm before it hits a beam, the slope is 120 divided by 9000, which is 0.0133 or 1.33 percent. If the code needs 2 percent, the pipe has to be rerouted, the beam penetrated (only with the engineer's approval) or a different route found. Doing this check on paper during layout saves tearing out pipe later.

Civil drawings give elevations in decimal metres or decimal feet, not in feet and inches. Lesson 2's conversions between decimal feet and inches are essential when a plumber with an inch tape has to hit an elevation given in decimal feet.

Ramps

A ramp's slope controls whether people using wheelchairs, walkers or carts can use it safely. Building codes and accessibility standards limit the slope of accessible ramps; a ratio of 1 in 12 is a widely used maximum, and some codes and standards call for gentler slopes, along with level landings at set intervals and handrails. Vehicle ramps in a parkade, like the one down to Kingsway Tower's parking levels, are much steeper, often in the range of 10 to 15 percent with transitions at the top and bottom.

Stairs

A stair is a slope broken into steps. The vertical part of each step is the riser (or rise) and the horizontal part is the tread (or run, or going). Two kinds of numbers control stair layout: the total rise between floors, which is fixed by the building, and the individual riser and tread sizes, which the code limits and the designer chooses. The rule that matters most for safety is that every riser in a flight must be the same height, and every tread the same depth, within a small tolerance. People's feet learn the rhythm of a stair in the first few steps; one odd step is a trip hazard.

The National Building Code and the Ontario Building Code set maximum riser heights and minimum tread depths that differ for stairs inside houses and for public stairs like those in Kingsway Tower. For buildings like Kingsway Tower, risers are limited to roughly 180 mm and treads must be at least about 280 mm; the drawings and the applicable code give the exact values.

To lay out a stair, divide the total rise by a trial riser height and round up to a whole number of risers. Then divide the total rise by that whole number to get the actual riser height. A straight flight has one fewer tread than risers, because the top riser steps onto the upper floor.

The odd brackets mean round up to the next whole number. Rounding up makes the risers a little shorter than the maximum, which keeps them within the code.

A traditional comfort check is that twice the riser plus the tread should fall between about 600 mm and 650 mm. For the example, \(2 \times 170.6 + 280 = 621\) mm, comfortably inside the range. Carpenters measure the total rise on site rather than trusting the drawing, because slabs and finishes are never exactly at their design elevations; a stair built to the drawing can end up with a short or tall top step.

Roof pitch

Roof slopes in North America are usually written as pitch: the rise in inches for every 12 inches of run, such as 4 in 12 or 6/12. Even on metric drawings, roofers often use pitch, and metric drawings may give the same slope as a ratio or an angle. A low-slope roof like the membrane roof on Kingsway Tower may slope as little as 1 or 2 percent toward its drains, while a steep shingled roof might be 8/12 or more.

The last column is the Pythagorean theorem applied once and stored. A common rafter on a 6/12 roof with a horizontal run of 4.0 m (or 13 ft) is \(4.0 \times 1.118 = 4.47\) m (or \(13 \times 1.118 = 14.53\) ft) long along its top edge, before the overhang and the ridge allowance. Roofing material is estimated the same way: the plan area of the roof, from the drawings, multiplied by the factor for the pitch gives the actual surface area to cover.

A stair calculator in code

This program lays out a straight stair from the total rise, the maximum riser and the chosen tread, and checks the comfort rule.

import math

def stair(total_rise, max_riser, tread):
    n = math.ceil(total_rise / max_riser)      # round UP to a whole number of risers
    riser = total_rise / n
    treads = n - 1
    run = treads * tread
    comfort = 2 * riser + tread
    return n, riser, treads, run, comfort

for rise in [2900, 3200, 4100]:
    n, r, t, run, c = stair(rise, 180, 280)
    print(f"total rise {rise:7.1f}: {n} risers of {r:6.2f}, {t} treads, run {run:7.1f}, 2R+T {c:6.1f} mm")

total rise  2900.0: 17 risers of 170.59, 16 treads, run  4480.0, 2R+T  621.2 mm
total rise  3200.0: 18 risers of 177.78, 17 treads, run  4760.0, 2R+T  635.6 mm
total rise  4100.0: 23 risers of 178.26, 22 treads, run  6160.0, 2R+T  636.5 mm

math.ceil is the round-up function. Try changing the maximum riser or the tread to see how the number of steps and the total run change; this is exactly the trade-off a designer makes when a stair has to fit a fixed opening in the floor.