A Helical Stairway Project Tom Farmer This is a real-world problem. It’s what I understand from a friendly exchange of several e-mails during a recent semester in which I was teaching a multivariable calculus class. The first message sent to me began with, “I am a carpenter with a problem,” and it turned out that the problem involved parameterizing lines and curves in space and finding intersections between lines and planes – exactly some of the topics in multivariable calculus. My students got a kick out of the developing story. By way of several messages back and forth, I learned about the plans for a 10-story hospital (now completed) that has an unconventional stairway system including, between each pair of floors, a free-standing, curved flight of stairs that follows an arc of a helix. The stairway system is an important visual element in this building. From the bottom floor, a visitor can look up and see the entire system including the 9 helical flights. Thus, these curved sections must look elegant as well as function properly. The entire system is enclosed within a quarter cylinder of radius 21 feet, and the stairway from one floor to the next has the following parts (see Figures 1 and 2): - a main landing adjoining the axis of the cylinder; then (rising clockwise around Figure 1), - a conventional flight of steps out from the axis to a landing bounded by the wall of the cylinder; then - the curved flight of steps that rise along a helical arc to another landing along the cylindrical wall; and, finally, - another straight flight of steps back toward the axis and rising to the main landing on the next floor. 1
This is a guide on how to design helical stairway which has became dominant nowadays
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A Helical Stairway Project
Tom Farmer
This is a real-world problem. It’s what I understand from a friendly
exchange of several e-mails during a recent semester in which I was teaching
a multivariable calculus class. The first message sent to me began with, “I
am a carpenter with a problem,” and it turned out that the problem involved
parameterizing lines and curves in space and finding intersections between
lines and planes – exactly some of the topics in multivariable calculus. My
students got a kick out of the developing story.
By way of several messages back and forth, I learned about the plans
for a 10-story hospital (now completed) that has an unconventional stairway
system including, between each pair of floors, a free-standing, curved flight
of stairs that follows an arc of a helix. The stairway system is an important
visual element in this building. From the bottom floor, a visitor can look up
and see the entire system including the 9 helical flights. Thus, these curved
sections must look elegant as well as function properly.
The entire system is enclosed within a quarter cylinder of radius 21 feet,
and the stairway from one floor to the next has the following parts (see
Figures 1 and 2):
- a main landing adjoining the axis of the cylinder; then (rising clockwise
around Figure 1),
- a conventional flight of steps out from the axis to a landing bounded
by the wall of the cylinder; then
- the curved flight of steps that rise along a helical arc to another landing
along the cylindrical wall; and, finally,
- another straight flight of steps back toward the axis and rising to the
main landing on the next floor.
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Figure 1: Stairway system.
2
Figure 2: One flight showing stringer and concrete beam.
3
Questions from the carpenter
In order to pour the concrete for the helical staircase, the carpenter envi-
sioned a temporary framework consisting of two main beams, AB and CD
in Figure 1, surmounted by a number of horizontal, radial beams that meet
the cylindrical surface along a curve. Upon this framework would be con-
structed the ramps of plywood and the forms (the temporary walls) that
would contain the concrete. A sideview of the finished product (Figure 2)
shows the zigzag outline of the steps on top of an 8-inch thick band that
forms the stringer supporting the steps. Below the stringer is a thicker con-
crete beam. Although the steps, the stringer, and the beam are all poured at
once, it is convenient to speak of them as separate objects. The ideal tempo-
rary framework is one that could be dismantled and reused, and this would
help to guarantee consistent results from one floor to the next. In fact, more
important than its structural role, the beam AB would serve as a template
in preparing to pour the staircase on each floor. The points A and B are
essentially the bottom and top points of the helical curve that the bottom
of the concrete beam is to follow. The points C and D are constructed from
A and B by drawing horizontal lines toward the axis of the cylinder and
measuring a distance equal to the long dimension of each step.
If AB is a straight beam (rather than having a curved profile) then it turns
out that the curve we get on the cylindrical wall is only an approximation of
a helix. This may not be obvious, but we work out the formulas below. So
here is the initial question raised in planning for this temporary framework:
What curved profile for the beam AB will cause the curve traced out on
the cylinder to be a true helix? A related question is: If a straight beam is
used instead of a beam with the proper curved profile, will the constructed
curve be a sufficiently good approximation of a helix? In other words, what
problems will stem from using this non-helical curve?
A primary consideration in building a stairway is that the rise (riser
height) and run (tread width) of each step must remain constant – if the
rise or run varies from one step to the next, then climbers might stumble.
4
A defining characteristic of a true helix is that it has constant slope along a
(right circular) cylinder, and this allows constant rise and run. On the other
hand, if the stairway were to follow a curve that is not quite a helix, then
the slope along the outer cylindrical surface would not be constant. Thus, if
the risers were held at constant height, then the treads would have to vary in
width. Similarly, if the treads were of constant width, then the risers would
have to vary in height. Either situation would mean trouble for the climbers.
Of course, even with a true helix, the tread width varies along the length
of the tread. Each tread in a helical section is bounded between the outer
cylinder of radius 21 feet and an inner cylinder of radius 16.5 feet (in this
application the tread length is 4.5 feet). The other boundaries of a tread are
along radial lines. Thus, two climbers walking side by side will face different
tread widths. But, as long as a person maintains a constant distance from the
axis of the cylinders, the tread width remains constant and climbing should
be no problem. By the way, as can be seen in Figure 2, the cylindrical walls
in this construction are imaginary and the stairway system is actually in an
open space alongside a cylindrical wall of glass.
Imagine that in preparing for a helical flight, we have drawn on the outer
cylindrical wall (supposing it existed as a solid wall) the outline of the steps
and the helical arc to be followed by the top surface of the stringer. The
outline is determined by the specifications of the rise and run of each step,
the desired number of steps in the flight, and the location of the lower landing.
But now let’s talk about the bottom surface of the concrete beam that is
to support the flight. Ideally, the bottom surface would follow a helix parallel
to the one formed by the steps. That’s why we would like for the beam AB to
have a profile that matches the helix. If, instead, AB were a straight beam,
then the bottom surface of the concrete beam would have the wrong shape.
Of course, even so, we could make the top surface of the stringer have the
correct shape simply by allowing the concrete beam or the stringer to have
varying thickness. Would that work?
Well, because the sides of all the stringers and the concrete beams are
5
visible from the stairwell, allowing them to have varying thickness turns
out to be unacceptable. This is so even though the necessary variation in
thickness is quite small (as we shall see).
A mathematical reply
The main goal is to discover the correct profile for a beam AB so that hori-
zontal rays from the axis of the cylinder and passing through points on the
top surface of the beam meet the cylindrical wall in a helix. We choose co-
ordinates so that the (vertical) z-axis is the axis of the cylinder with radius
R = 252 inches (21 feet) and let z = 0 represent ground level. Although the
actual stair system rises in the clockwise direction, we follow mathematical
convention in considering a helix that rises counter-clockwise. Also, let the x-
axis and y-axis be such that the first octant of space contains the entire stair-
way system as suggested by Figure 1. Then the coordinates of the points on
the desired helix for the first helical flight of stairs are x = R cos t, y = R sin t,
and z = ct, where t ∈ [α, β]. Here c is a constant determined by the slope
of the helix and [α, β] is the interval of polar rays that contains the helical
flight. The values of c, α, and β are derived from the building plans.
Any point P = (R cos t, R sin t, ct) of the helix uniquely determines a hor-
izontal line segment QP , where Q = (0, 0, ct) is on the axis of the cylinder.
Where does this line meet the vertical plane containing A and B? We can
take the direction vector of the line QP to be 〈cos t, sin t, 0〉, so using s as
the parameter, we find the line is given byx = s cos t
y = s sin t
z = ct.
(1)
Since A = (R cosα,R sinα, cα) and B = (R cos β,R sin β, cβ), the verti-
cal plane ΠAB containing these points has equation