SBTP: Fundamentals of Turns

Consider simple turns of a one-dimensional line on a flat, two-dimensional surface – what can be called pencil-and-paper curves. All such simple turns can be characterized with only two parameters, radius and angle. (Actual turns are also characterized by position and orientation; however, those do not change the geometry of the turn. We are examining not specific turns, but the shapes of turns.) Knowing these two parameters – radius and angle – and the values over which they can range is a powerful tool in analyzing these turns. We will now extend this method to analyze turns used in switchbacks.

A turn is a change in direction. The rate of that change – how fast the turn changes, or how "sharply" – can be measured in several ways; here it will be measured by the radius, or distance to the center of the arc corresponding to the turn.

A radius need not be constant. Highway and railway engineers often use so-called spiral turns which start at very large radius and then transition to smaller radius (tighter turn). Any kind of turn where a parameter varies is a compound turn. In trail work we will deal only with simple turns (where the parameters are constant), or composite turns (composed of segments of simple turns).

The angle of the turn (or turn angle) is the measure, in degrees, of how much the direction changes. (See "Turn Angle" diagram.) This is not the angle between the legs of the turn – that is the interior angle. The angle of the turn is the difference between your direction going into the turn and your direction coming out. The interior angle is how much further you would have to turn to have reversed the direction you came from. These two angles are supplemental, so given one the other can always be determined.

All simple pencil-and-paper curves can be characterized with just these two parameters; for our purposes any two such curves are identical if they have identical radius and angle. Note that location and orientation do not matter. (In reality, of course, proper location and orientation are very important. But here we are examining only the geometrical characteristics of turns, and these do not vary with location or orientation.)

As the context of switchbacks is slopes, two more parameters are required: the angle of the slope (in degrees), and the grade of the tread (in per cent grade). Note that these last three parameters (turn angle, slope, and grade) are correlated. The angle of the trail on the slope – more precisely, the angle of the intersection of the trail with the contour lines, or what might be called the contour angle – is determined by the relationship of slope and grade. When the traverse reverses (the definition of a switchback) the contour angle is doubled, and this turns out to be identical with the interior angle of the turn. Knowing any two of these parameters determines the third. Generally the slope is a given, the grade is specified, and if we really want to know the interior angle of the turn (or its supplement, the turn angle) that can be calculated.

Turn being shortened by use. [turn8245-3.jpg 38 KB]

This turn originally went all the way out to the bottom of the picture. Traffic has been wearing away the inside, making it shorter and steeper. The undercut edge on the upper side (left) from the original surface shows that several inches of tread has already been eroded away.

We need one more parameter, the width of the tread. Strictly speaking, we need the width of the bench upon which the tread is built, but switchback turns are usually so tight, and the propensity of the users to cut the inside corner so great, that we usually dispense with any inside margin, taking various measures to defend the inside (relative to the corner) edge. Protection of the outside (relative to the slope) edge may include a margin as has been discussed before (see Edges and Blowouts). But for analytical purposes we will assume that there are no margins, and therefore tread width and bench width are identical. We will also assume that the outside edge of the tread is at the surface of the slope, and that the tread (or bench) is a cut into the slope. ¹

(the turn angle being a dependent variable) it is possible to fully develop the geometry of turns on a slope. (To completely characterize the cross-section of the slope another parameter is needed, the angle of the cut-slope; however, this does not affect the geometry of the turn.) Each of these parameters can be considered a dimension in a "space" of all possible turn configurations. Any possible turn can be specified as a combination of specific values of these parameters. ² In a later section we will see output from a program that generates turns based on specified values of these parameters. ³

Geometrical complications.

At this point matters get complicated. (Which may be the simplest answer as to why switchbacks are so hard to build correctly.) For instance, to accommodate width a turn must make an incursion into the slope. (Or possibly an excursion out from the slope, implying fill, or perhaps a trestle.) Likewise for accommodating radius if grade would be violated on the fall-line. These have the effect of setting the end of the turn away from the surface of the slope. To avoid an excessively wide and endless excavation it is necessary to make a compensating offset to bring the tread back to the surface of the slope. Typically this means turning a little bit more than the nominal turn angle to point the turn back towards the surface of the slope, and then, when the surface is reached, an equal but opposite reverse turn to come back on course. (Which all makes the overall turn composite. See Offset Turns diagram.) The details of how to handle these complications will be discussed later.

Another complication arises from the "spiral staircase" effect: turns are steeper on the inside, and often much steeper. Failing to recognize this is perhaps the biggest single cause of switchback failure. If you measured "good enough" grade up the center of the turn (or, heaven forbid, the outside of the turn), the inside will certainly be steeper, and quite likely will violate Rule #2 (Do not exceed the maximum grade at any point). This is so important that it warrants a closer look, but first we need to digress and look at how to do some simple trail math.

To build or even consider a turn properly it is necessary to be able to calculate the length of the turn. As was shown, in practice this can be so simple as to be a trivial calculation.

Consider a spiral staircase (see image). It has distinct steps rather than a smooth ramp, and in the example here is much steeper than a turn built in soil could ever be. But for all that it clearly illustrates the key problem with a turn, of the inside getting too steep. The explanation is simple: although each step has the same rise from inside to outside, the inside has less – and in a tight turn, very much less – run. Grade being inversely proportional to run, as the run reduces to zero the grade approaches verticality.

The result is that the inside of the turn can rise only so far as it has sufficient run, and the outside of the turn, which has plenty of run, has to wait for the inside. (Much like the case of a marching band, albeit inverted, where the guy on the inside of the turn can't step off in the new direction until the guy on the outside has finished the turn.) This is the principle of "inside corner rules": a turn can rise only as fast as the inside.

Now consider the special case of a zero-radius turn. If the radius is zero, then the length of the arc is zero. (Any value of pi times any value of radius times zero is – zero.) If the length – run – is zero, then the allowable rise is also zero. Imagine two sections of grade connected by a wedge-shaped landing with an inside radius of zero. Once the grade hits that landing it is absolutely constrained to that elevation until the turn is complete and both sides of the tread are free to rise. While it is always possible to insert a landing into a traverse (topography permitting), a zero-radius turn is the only case where such a landing must be level.

Although switchback turns are often conceived as zero-radius turns, reality is rarely so tight. Barriers (such as the common "telephone pole on posts") are often built with corners that are essentially zero radius, but even there people tend to stand off from the structure some several inches, so when they do turn a corner there is an effective minimum radius of four to six inches. Where switchbacks are built with minimum separation it is common to put a largish log or rock at the vertex of the turn. To avoid being displaced these need to be as big and heavy as possible, which is typically 8 to 10 inches in diameter. Allow another three inches for wall clearance, and there is easily 6 to 8 inches of radius. Assuming a 120° turn, there is 12 to 16 inches of run. At a 12% grade this would permit only an inch or so of rise, which is hardly enough to count. But the point is not how much, but that even in this tightest of turns there is just enough rise so that the "platform" is not level.

It is much better to have an inner radius of 24" or so on a turn. ⁴ This provides over four feet of run on a 120° turn, which can easily provide eight inches of rise – the equivalent of a step. When grade is tight on a switchback and more run is needed, the first option is to lengthen the switchback (push the turn out). The second option is to make a larger radius turn. (And if the grade is still too steep, remember Rule #3: Build a step!)

Defending the inside of the turn.

A well defended turn. [Turn4-66.jpg 137 KB ]

A well defended turn! This was built by serious bikers for serious biking. I believe the hat is from the last person who tried to shortcut this turn.

Turn and picture courtesy of Sierra Butte Trails Stewardship.

Also effective in preventing shortening.

The spiral staircase effect – the dramatic increase of grade on the inside of the turn as radius is shortened – can be pernicious. Always keep in mind Rule #4, that grade must be respected at every point. This applies especially to the inside of a turn. Even if you carefully measure, design, and construct a turn to maintain grade, any shortening of the inside radius will increase grade, and increase erosion. Especially on sharp turns with a radius of only a foot or two. And such shortening will happen (at least on trails built in soil), as users tend to push against the inside of a turn. (It really is easier to swing wide, but we seem to have a cultural mindset that shortcuts must always be taken.)

To avoid the shortening and eventual erosion of a turn (as was seen in the picture above) it is absolutely vital to: defend the inside radius!!! This usually means no margin on the inside of the turn ⁵ and hardening the inside of the turn with rock. A pile of rock as substantial as seen in the picture here is not usually needed; even a few large (200+ lbs.) and well-settled rocks can be effective. Even a large log, well and securely placed, can be effective (see pictures). Or even a large hole.

The focus in this section has been on the fundamentals of turns; practical tips for avoiding failure of switchbacks will be discussed in the tips section.

Summary.

(the turn angle being a dependent variable). Turns have geometrical complications, such as the spiral staircase effect, which must be accommodated to avoid excessive grade. In particular, the grade of a turn is constrained by the length of the inside corner ("inside corner rules"); the length and grade of a turn must always be measured on the inside of the turn. And suitable measures must be taken to defend the inner radius from being shortened, which would lead to excessive grade.

Grade can be ameliorated by extending the switchback (pushing the turn out) or increasing the inside radius of the turn.

Switchback Theory and Principles:

Fundamentals of Turns.

The basic parameters.

Geometrical complications.

Defending the inside of the turn.

Summary.

Notes.