Switchback Theory and Principles:

Geometrical Development of a Switchback.

Flagging a traverse that respects grade can be (as has been seen in the section on Measuring Grade) straightforward. And when the point is reached where the traverse needs to reverse – that is, to switchback – that, too, seems straightforward. But! complications quickly develop. In this section we will step through a geometrical development of a switchback – or rather, of the reversing turn by which a switchback is implemented. We will explore these difficulties and see how to deal with them through a series of views of computer generated surfaces. ¹

View 1: Flag line and axes [T01]

This view and the others that follow are of surfaces generated by a program according to specific values (such as was described in Fundamentals of Turns), and then "visualized" by another program with a choice of viewing positions. Here we are looking at a flat (and very boring) slope of 30 degrees upon which a grade of 15% has been flagged, with flags placed every 8' 4". ² At the point indicated by the red flag the flag line reverses. This is the vertex. For these views it is also the center of the coordinate system. (As has been mentioned before, any turn which differs only in location and orientation can be moved or translated to this "normal" location, without changing the geometry in any way.) Also shown here are the coordinate axes:

• red (not always readily visible, as the visualization software tends to bury it into the surface) is the horizontal (x) axis, and also the axis to which all contours here (not shown) are parallel.
• yellow is the vertical (y) axis.
• green is the horizontal depth of field (z) axis.

These axes will not always be seen level or vertical, as the viewing position will change as necessary to highlight points of interest. In general the basic surface will be the same as shown here.

View 2: Upper leg only. [T10]

Given a flag line for a switchback, we might start by cutting the upper leg. The angle of the slope was a given and the grade of the traverse was specified (see above), and only two more parameters are required: the width of the tread/bench (here a scaled 32"), and angle of the cut-slope (60°). Nothing complicated there, just plain, simple traverse work. So far, so good.

View 3: Lower leg undercuts upper leg. [T13]

Now let's cut the lower leg, ascending until it intercepts the upper leg at the vertex. Oops, there is a problem here: the lower leg has just undercut the upper leg (the red shaded volume). The tread now comes down the cut-slope, which is much too steep for that purpose. The underlying problem is that two treads have been brought together so they overlap without having first been brought to the same elevation. From which we might derive a rule for turns: Do not overlap (until at grade).

The flag line alone is a simple one-dimensional line that can (while steadily rising) execute a snappy about-face and continue on its way. All that was said before about describing turns with just four parameters still applies. ³ But once there is width, and differences of elevation that have to be brought together without exceeding grade, things get a lot more interesting. There is a zone of elevation mismatch which has to be dealt with some how, or got around. It is the same old core task of finding a route long enough to accommodate the rise at a proper grade, but here there is a complication in that any such route no longer has a simple mapping on the surface of the slope.

View 4: Split tread, no platform. [T16]

The naive solution is to split the tread. This is quite unsatisfactory, for several reasons. First, observe that while the upper and lower grades do not come together until at the vertex of the flag line (at the red flag), yet the turn is implicitly before (inside) the vertex: this turn cuts the switchback. Morever, as the treads converge they narrow, encouraging the users to use this shortcut. (We admonish and berate users that cut switchbacks; what should we say about trail – and trail builders – that force the cutting of switchbacks?) Second, what about that jump in elevation? Here it has been shown as a step (per Rule #3), but it is a pretty strange step. At one end the height is zero, but the other end (depending on the parameters) could be a foot or more, which most people will find uncomfortably high. ⁴

Turning inside of the vertex shortens the grade, making it too steep around the inside of the turn. If a step is not built, then users will be coming across the cut-slope between the upper and lower tread, which is simply too steep. This will cause it to erode into a ditch. The photo at the right shows a very typical situation: the rest of the traverse is okay (18% and 10% grades above and below the turn), but right at the corner the grade is 24 to 27% on the center-line of the tread, and 30 to 33% on the inside of the turn. The result is a trough six to eight inches deep (the yellow and red tape show the cross-section). This was quite unnecessary, as there was (and is) plenty of room at this location to extend the switchback for a full and proper turn.

View 5: Split tread, arrowhead platform. [T20]

So let us cut a platform (the yellow surface) that extends the tread past the vertex, so that the turn does not cut the switchback. Note that the platform that results is level not because that is a requirement, but as a consequent that as both treads have reached the same elevation there is no reason to go lower or higher. (Strictly speaking this not quite true, as there might be insloping or outsloping. But for analytical purposes we are ignoring that.) We now have a design that respects grade, although is still largely unsatisfactory.

View 6: Arrowhead platform widened. [T22]

The geometry here is almost identical to that in the previous view, except that the platform was widened by extending it to edge of the slope. This is a significant improvement, as switchback turns are where hikers frequently pass, and so some extra room is needed. ⁵ Also, a six-inch diameter "log" was added to protect the extremely vulnerable edge of the upper ramp. ⁶ This illustrates the constriction of splitting the tread: the nominally 32 inch tread gets squeezed into ten inches.

Something else to note is that the level platform (in yellow) is slightly beyond, and slightly higher than, the flag line vertex (marked by flag). This is because an ascending (or descending) grade cannot be parallel with the contour line (by definition, as then it would be level), and therefore the line of the cross-section, which is perpendicular to the direction of the tread, cannot be perpendicular to the contour line. So when the outside edge of the lower tread reaches the vertex the inside edge (which is at the same elevation) is slightly behind. At that point the upper tread has not yet reached the vertex, or the elevation of the vertex. Before meeting they both have to go a little further, which means slightly past the vertex, and therefore slightly higher. If the landing had been built at the vertex – more significantly, at the elevation of the vertex – this lagging of the inside edge would have effectively pushed the vertex back for the upper tread, thereby shortening the run, and increasing the grade. ⁷

This shape is getting better, and is representative of a great many real world turns. But the approaches are still too narrow (forcing users onto the edges), and too close together to prevent shortcutting. In some variations of this shape the trail builders simply grade a shortcut across the small cut-slope. This shortcut reduces the steepness of cut-slope, but the graded portion has reduced the length of the run, and thus over-steepened the grade. It is precisely the kind of problem anticipated in the discussion of View 4, and violates the principle of Respect Grade. The result is a turn that erodes into a ditch, with headward erosion propagating up the trail. (If the grade is low enough the erosion may be slow, and the ditch shallow, but if it exceeds the critical grade it will eventually happen.) This leads to a characteristic peculiarity: the tip of the original constructed tread is often left behind as a raised platform just outside of the turn.

These short comings are so patently obvious (well, sometimes) that there is a nearly automatic tendency to install curb logs on the upper and lower edges. But this is usually futile: any thing substantial enough to avoid getting kicked loose leaves hardly any tread width for use. The "obvious" solution is to then widen the lower tread, which then undercuts the upper tread, which is in turn cut back further, thereby swinging the ramp into the slope. This will be seen in the form of the next view.

View 7: Full width tread, upper ramp swings in. [T30]

Here we have the upper and lower approaches cut to full width, the upper tread being cut deeper into the slope to accommodate the lower tread. This is actually a fairly decent, workable design. But it has a challenge.

As we have already seen, the approaches cannot come together (overlap) until they are at the same elevation. As a consequence, the platform must be double wide, accommodating each approach separately. This also means cutting into the slope twice as far, with a cut-slope twice as high, and four times as much excavation. I suspect this is why trail-builders have often been resistant to building full-width platforms: 1) they don't like all that cut-slope, and 2) they are intimidated by the excavation.

The additional cut-slope is unavoidable. In the oblique triangle formed by the cross-section of tread width (depth of excavation) and the given angles of slope and cut-slope, the height of the cut-slope is fixed (a relationship described by the Law of Sines). The best that can be done is to extend the turn outwards, which essentially hides some of the cut-slope below the trail. But remember that half-bench construction is a bad idea. If you build out, you must build a proper wall. (Excavation will be discussed further on.)

Right: one of my favorite real-world zero-radius turns. (But not quite zero: there is effectively eight inches of radius because of the thickness of the logs.) Aside from a curved slope, a lower angle of slope (and therefore a lower cut-slope), and the two steps, this is exactly the type of turn depicted in View 7. It could have been yet another turn constructed too casually, too short, and too steep, but someone had enough sense to push it out a little bit, even though a small wall had to be built. And the not quite a crib wall arrangement in the center seems to be very effective in discouraging short-cutting. This is on the Mount Si (WA) trail.

Right: swinging a ramp.

Another significant point: Swinging a section of tread away from the vertex makes it extend past the vertex. This is shown in the adjoining diagram. Point V is the vertex of the flag line (in red), marking the outside edge of the lower tread. Point VI is the extension of V into the slope after cutting the width of the tread. If the outside edge of the upper tread followed the flag line to V it would be undercut (as was seen in View 3). Something needs to be moved. In this case, the upper tread is rotated at point B (opposite point A) into the slope to clear VI. This moves A to A' (the wedge B-A-A' becoming the level platform of a zero-radius turn), and the corner that was at V is now at V'.

It is tempting to think that the tread can be cut from A' to VI. This would be incorrect, as it would be steeper than the grade flagged. The length of A'-VI is between, and therefore shorter than, the run of A-V, but the rise is just as great, so it follows that the grade would be steeper. The effect of swinging a section of tread is that the corner is swung out (e.g., from V to V'), effectively extending it past the vertex. Failure to accommodate this in the construction will result in steepened grade.

The length of the extension needed depends on the parameters (of slope, grade, and width), and could range from about a foot to several feet. The easiest way to handle this is to split the difference between the upper and lower grade, horizontally and vertically, essentially pushing the platform out a little bit and up a little bit – exactly as was seen in Views 5 and 6, and will be seen in View 9. ⁸

Another solution will be shown later.

Note that in View 7, and the in diagram above, the point where the tread changes direction – swings – is a zero radius turn. (The turn could be given some radius, but that would be more complicated.) As was previously explained under Turns, the inside of a turn rules, and if the inside has zero radius it also has zero rise. That is, the turn has a level platform or landing (indicated here in yellow). Now some of these turning platforms are quite small (e.g., see View 9), even picayune, and it might be wondered if we should be bothered with a feature only a few inches across that use will quickly blur out of existence. Of course not. These landings are shown here exactly, but in practice the bottom of the upper tread comes up slightly (easing the grade), and the top of the lower tread comes down slightly (also easing the grade), for a grade that eases as it goes around the longer run of the outside of the turn. This is perfectly acceptable. As long as the inside of the turn – which has the shortest radius, and therefore shortest circumference, and therefore the shortest run – does not exceed the maximum permitted grade, then all of the other points on the turn will be okay because they have a longer radius, and therefore more run, and an easier grade.

View 8: Full width tread, lower ramp swings out. [T34]

Here is essentially the same turn as above, except that the turn goes outward of the slope rather than inward. (And one ramp is again slightly longer, for the same reason as before.)

This design is especially appealing in that the cut material is used directly as fill, and in roughly equal proportions – what surveyors call "balanced cut and fill". But do not skip the necessary wall! If you are in an area with a lot of suitable rock conveniently available, this may be fine. Or you may find some kind of timber wall suitable. If not, then it may be easier to simply do the full excavation, and haul the fill off to some place where it can be used. In practice, if I don't have a sufficiency of suitable rock I will drag in the biggest log I can find to make what is essentially a small wall, pushing the turn outwards slightly. And I can pretty nearly always use the excess excavated material somewhere nearby. Even if there is a sufficiency of rock, the effort and time of building a proper wall may exceed that of doing a full excavation, so in the end it may just come down to aesthetics and personal preference.

Simple, full excavation will generally be used here, as the analysis is simpler. If you learn to design switchback turns as described here, it will be a straight-forward adaptation to move them outward a bit to reduce the excavation required. In practice the excavation required need not be as challenging is suggested in these views, as with thorough reconnaissance and astute planning it is often possible to take advantage of a natural landing or hollow to greatly reduce the work needed.

Right: A real-world instance of a turn just like that shown in View 7 (above). In this case measures to prevent shortcutting are not needed because, hey, it's carved out of rock! But don't try doing this in soil, as it will quickly become a disaster. This is on the Little Si (WA) trail.

One difference between this and the views shown here: no cut-slope across the end of the platform. For simplicity of analysis the views shown here are all done on a simple, flat slope, which is actually the worst case. In the real world slopes are neither flat nor simple, and in this case a location was found where the slope turned away, so there was nothing to excavate. (And fortunately there was no need to extend the turn further out.) Slopes tend to be complex. That is not a problem – that is a realm of opportunities!

View 9: Both ramps swung equally. [T38]

Both turning in and turning out have challenges, and sometimes it is useful to do a little of both. Here we see both the upper and lower treads turning a little bit. As was discussed above, this makes them both a little longer, pushing the platform out a little past the vertex (still marked by the red flag), and little bit higher.

Between turning all the way in or all the way out is a continuous range of possibilities, of which the one shown here is only a middle value. As mentioned above, I sometimes drag in the biggest log I can find for the outside wall, and then move the turn out as far as can be accommodated by that wall. For low angle slopes this is usually a reasonable compromise. For higher angle slopes you should carefully consider the trade-offs.

One of the problems not yet addressed is that having the upper and lower tread adjacent invites shortcutting (therefore the "log" ⁹), and leaves no room for a substantial barrier. As one means for deterring shortcutting, some authorities recommend increasing the grade at a turn by 10 or 15 percent. This is quite ineffective, as it only provides a more enticing jump. ¹⁰ It is more effective to separate approaches horizontally, and use the space in between for barriers that are physically and psychologically more effective.

View 10: "Full fill". [T24, flipped]

Just for laughs, here is a simple "full fill": no excavation, even for the traverse. Well, that is not quite true. Something like this would require massive excavation somewhere else to provide the fill needed. And continuous walls, as those fill slopes are much too steep to last, or even to build, without substantial support. (This view is actually the bottom side of the surface in View 7, flipped over by means of the visualization software.)

View 11: Cross-wise ramp with 'square' turn. [T30]

In the previous views one approach ramp or the other needed to be extended into the platform. In this view the needed extension has been applied as a ramp placed cross-wise or perpendicular to the slope (flag line), thus increasing the horizontal separation of the approaches. Pushing the upper approach deeper into the slope also results in a berm or bulwark on the inside of the turn.¹¹ This is immensely effective in deterring shortcutting: the would be shortcutter is no longer on the edge of an enticing jump or slide, but has to jump up a foot or so, and then may face a greater depth of visual and physical barriers.

If space is available to push the ramp out, it could be level. It could even be widened, thereby becoming the "level platform" at the heart of some definitions of "switchback". However, it need not be level, and indeed, if space is tight it may be useful to make grade on all the run available. Note that the two turns, of approximately 90 degrees, are zero-radius, and therefore must be level.

The central problem addressed by the last several views – how to incorporate the extension required when part of the tread is swung away from the flag line – can, as we have seen, be tackled in several ways. In the simplistic slope presented here, which lacks any complicating features, it is easy to simply push the platform out a bit. But if this is impossible (perhaps due to a large boulder?), then going deeper into the slope with a cross-wise ramp as shown here would more attractive. It all depends on the circumstances. In a tight, difficult location flexibility of design may be needed. However, resist the temptation to be flexible in respect of specifications and principles. If you cannot maintain grade then you will need to build steps. (Or even ladders, in extreme cases.) Do not skimp or cheat just because the situation is tight! You may think that it doesn't matter, or even that it can't be helped. But it does matter (as I have explained elsewhere), and the point of all this demonstration is to show you how to grapple with these difficult situations.

View 12: Cross-wise ramp becoming round. [T32]

People do not turn in sharp (short radius) turns (bikes and horses even less so). Large changes of direction (e.g., 90 degree turns) in a short space are also irksome. People generally graze the inside vertex of the turn, but at a somewhat larger radius, pulling away from the side. In this view we take advantage of this to push vertex out a little bit. This has the effect of breaking two 90 degree turns into three turns. Even though each individual turn is still zero-radius (and therefore the arc of each turn must be level), making each turn smaller makes them more gradual, gentler, smoother to negotiate. The overall turn is also becoming gentler as it gets split into multiple arcs. As it is split into more parts it is becoming circular, with a definite radius. ¹²

(It would be nice to show a circular turn here, but the software gets too complicated. It should be easy enough to extrapolate from this view what a truly circular turn would look like.)

A similar process works on all these wedge-like level turning platforms (shown in yellow). In the analysis I have tried to be meticulous in showing the effects of turns on grade, and especially that turns with zero radius (and therefore zero run) on the inside of the turn cannot make grade. But in practice we do not make a sort of pleated surface, but grade it into a smooth, continuous grade. As long as the grade is measured on the inside of the turn, and does not exceed the critical grade, the turn can become a smoothly ascending ramp.

Two other points to note. First, what appears to be a drainage ditch along side the turn is not a requirement. While it might be a good idea, it was added here to avoid some geometrical complications that arise in the interaction of different grades, slopes, and radii. Second, note that this turn cuts inside of the vertex of the flag line (marked by the red flag). This is acceptable because the turn has added additional run to keep the grade within spec. Indeed, this can be a very useful technique if for some reason the vertex cannot be extended sufficiently to make grade – just make a wide radius turn. This is essentially adding the needed run cross-wise (perpendicular to the slope).

One final detail. The turn as so far developed swings well into the surface of the slope. This is shown in the diagram at right, where the lower leg (at the bottom of the figure) comes up on the inside of the flag line (red dashed line), then swings around until it is parallel to the upper flag line. But now it is offset well into the the slope, and continuing on this line of extension (dashed blue line) would require substantial excavation. To avoid this the tread needs to be brought back to the surface of the slope (i.e., to the flag line). This is done with a pair of offset turns (in green). The first is an extension of the main turn, and heads the turn towards the flag line. The second, of equal but opposite turn angle, restores the intended line of travel when the flag line has been reached.

If the sum of the angle of the main turn and the angle of the offset turn equals 180 degrees (i.e., the angles are supplemental), then the section between the offset turns will be parallel to the other leg, and offset from the other leg by twice the radius of the main turn. If the total turn is more than 180 degrees than the tread is converging on the other leg. (This might be necessary if the topography requires it, but the reduced horizontal separation will mandate greater effort in preventing shortcutting.) If the sum is less than 180 degrees than the legs are diverging (leaving the turn); this is generally the situation in the views here.

Here is another view of two of the switchbacks at Berthoud Pass (CO). The turn in the foreground does a 180° turn, and the upper and lower approaches run parallel for a while. At the next turn the situation is tighter, and the legs come closer together. They both do a slight reverse curve before doing the main turn, which has some what more than 180° of curvature.

Photo by Gregg Gargan (CDOT)

The turn angle of the offset turns is also the angle at which the oblique section between the two turns crosses from the line of extension to the flag line. As excavation will be maximal along the line of extension, it might be thought that there is a premium on leaving the line of extension as quickly as possible, which is to say with as large an angle as is feasible. However, a smaller angle results in a shorter length (taking the hypotenuse or diagonal of a triangle is shorter than following the other two sides), which also reduces the excavation required. (Not good if you need run to eat up rise, but here we are considering the amount of excavation required.) The net result is so small that it really makes little difference what the offset angle is. At the end of the main turn you can pretty much select what ever angle you prefer for reaching the flag line, and then turn that much more.

The offset turns are necessarily equal in how far they turn, but need not be equal in radius. In these views the second offset turn – also known as the reverse turn – has been shown as zero-radius, whereas in the diagram above it had a large radius. In practice I have found that the turn angles involved tend to be so slight – perhaps ten or fifteen degrees – and the rate of turn so slow – effectively a large radius of several feet – that the reverse turn can be resolved casually like a typical meander.

The canonical turn

This completes the geometrical development of a workable reversing turn from a simple flag line. While any form of a reversing turn can be transformed into any other form through a smooth transformation of various parameters, the canonical form developed here has the following general characteristics:

• Each part respects grade, not exceeding the specified maximum grade.
• No part of the tread overlaps any other part.
• Sections join only when at the same elevation, or joined by a step.
• Either or both of the legs are swung away from the flag line to avoid an overlap, and extended as necessary to maintain grade.
• The full specified width is maintained all around the turn.
• The turn is past the vertex of the flag line (i.e., no shortcutting).
• The turn is circular (or a multi-sided polygon approaching circular).
• The turn has a non-zero radius (even if only a few inches).
• There is typically (but not necessarily) a raised berm inside the turn.
• The offset angle is generally less than the interior angle of the turn.

One of my favorite turns. There is 13% grade along the wall, 15% grade on the upper part, and two steps (one not visible in this picture). In the approximately 50 feet of trail seen here there is eight feet of rise.

The original trail (and flag line) went right down the middle of the picture, where the large branch is. Effective closure of old trail and shortcuts warrants a long essay in itself.

Summary

Generating a satisfactory turn of a two-dimensional tread from a reversed ("switchbacked") one-dimensional flag line presents some challenges; naive approaches lead to failure. A key challenge is to avoid overlapping the upper and lower legs until they are at the same elevation. This requires that one or the other or both be swung away from the flag line, and this generally requires an extension of the leg past the vertex. While this can be done by simply moving the turn out from the vertex, it is better to use a circular turn that also creates greater horizontal separation of the legs. If necessary the turn should be pushed past the vertex, and the radius increased, to ensure sufficient run to maintain proper grade. These considerations lead to the form of a turn as described above. If the situation is very tight then extra care should be taken in measurement, design, and construction to ensure a satisfactory and sustainable turn.

Notes.

1. These surfaces are not simply illustrations; they can also be viewed as demonstrations of a mathematical type. E.g., that there is connectivity of the different pieces – i.e., no holes – is a demonstration that the shape is geometrically valid, and not mere artistic license. I also do a lot of checking to ensure that each tread segment does not exceed the specified grade.
2. The distance of 8' 4" being a magical value, as was described in the section on Measuring Grade.
3. Parameterizing a one-dimensional turn on a slope starts to get tricky because having one axis not perpendicular to the surface leads into some perspective problems. The basic relationships are the same, but the measurements get skewed.
4. I confess that in my early days I did build a few switchbacks like that. We seemed to have an idea that having flagged the "end" – more or less – of a switchback, it was good enough to come within waving distance before turning.
5. Many hikers do not notice on-coming traffic until they reach the turn. In other cases children are allowed to run ahead "to the next turn". And in some cases the turn provides a vista of interest, or just a convenient place to rest a moment. For whatever reason, it is usually good to widen the turn.
6. For sure, it does look like a concrete parking curb. Which is just about what is needed to avoid getting kicked aside.
7. In this regard View 4 was a bit disingenuous. The bottom of the triangle of the end piece, where both treads end, is perpendicular to the contour. And close to horizontal, but not quite. A close examination would show that it rises slightly. This is because it is not perpendicular to the tread, which is to say, not parallel to the contour across the tread. But do not mistake this seeming cross-slope for out-slope; it is entirely an effect of perspective. The fall-line is straight down the tread, not a bit across it. If water was poured at the point where the treads meet it would not go across the tread, not even obliquely, but straight down the tread. There is no cross-slope here.
8. If the elevation or location of the platform is critical, then the method of running a flag line up to the right spot and simply reversing will be insufficient; it will be necessary to carefully measure and mark the entire turn to ensure it will fit.
9. The log also hides a sticky detail: as the upper ramp gets pushed to the left (because the grade to "make" kept going down, even when the tread did not) it is effectively raised. It is only an inch or two, but scanting this would make the grade too steep at some point, violating Rule #2. So this log is not just a barrier, it is also a curb log, holding back some of the tread.
10. The intention is apparently to increase the vertical separation between the approaches. But if you already have a 15 percent grade, and then increase it by an additional 10 or 15 percent (which mostly likely will violate Rule #1), the vertical separation four feet back from the vertex is only two or two and a half feet. To some folks, and many kids, this is more of a challenge than a deterrent. Meanwhile, the grade probably exceeds the critical grade, and in a few years you will lose this turn.
11. Leaving a berm has been criticised as impeding out-slope drainage. For sure, this theory, with its emphasis on respecting grade, does not specifically require a berm, and it is a simple matter to remove it. But in the particular case shown here, how will you prevent shortcutting? If you do not then install a fence or other barriers, people will shortcut, cutting a trough between the ramps, and eventually resulting in failure of the trail. As to the drainage, recall that a central tenet of this theory is that water is not the cause of erosion, but only its agent; erosion is caused by excessive grade. The implication (which I have observed) is that if the grade is not excessive a little water flow will not hurt it. If you are not comfortable with that, or if you have a situation where there water flow may be problem, then out-sloping the upper ramp only dumps the water onto the lower ramp, which is probably not a good idea. The standard solution is to in-slope the upper ramp, perhaps adding a small drainage ditch to carry water around the turn.
12. This process shrinks the perimeter of the turn slightly, so the length or radius of the turn may need to be increased slightly. It may be useful to note that with a "radius" of, say, 3, the length of the quarter turn (or quarter of the total perimeter) of a square, the equivalent quarter of an octagon, and the length of a quarter arc of a circle are in the proportion of 6 to 5 to a shade over 4.7. For analytical purposes it may be useful to calculate such lengths exactly, but for practical purposes they are so slight that this shrinkage can be accommodated, and an adequate grade ensured, by adding just a couple extra inches to the radius. Do keep in mind that the inside of the turn may need to be armored to keep it from shrinking.