Here is an elegant way of calculating an exact easement curve
On the early railways it was perfectly adequate to locate the line as a series of tangents joined by flat circular curves. With low speeds, it was unnecessary to superelevate the outer rail to make a comfortable passage of a curve. After a while, traffic would straighten out the first part of the curve a little, sharpening the middle, and this would do quite well.
When speeds increased above about 30 mph, the centrifugal force became uncomfortable to passengers, and the outer rail was subjected to considerable lateral pressure. Superelevation would counteract both of these tendencies, at least approximately. The necessary amount of superelevation could not be introduced suddenly, so a ramp of about 100 ft or so was used, shared between the tangent and the curve. The lateral forces would be even more effective in straightening out the entrance to the curve at these speeds, and an approximate transition curve would result. The location was still done with tangents and circles. Superelevation, incidentally, is a matter of comfort, not safety, since a train is a long way from overturning on any curve that is negotiated without inducing terror.
What is called for is a gradual decrease in radius of curvature R concomitant with the elevation of the outer rail, so that the transition to the circular curve is smooth. This length of track in which the radius of curvature decreases from infinity to the radius of the circular curve is called the transition spiral. A spiral in which the superelevation, and therefore the curvature, increases linearly with distance along the spiral has been found to be completely satisfactory.
How such a spiral is determined mathematically is shown in the Figure. Taking the origin at the T.S., the point where the tangent ends and the spiral begins, the coordinates of a point on the spiral a distance s from the T.S. are x and y. The angle ψ is the angle of the tangent to the spiral at (x,y). The first line shows that ψ is proportional to the square of the distance s from the T.S. Now we can express x and y in terms of integrals, in fact taking ψ as a parameter. These are the Fresnel integrals familiar in physical optics, which are tabulated and can be computed easily. They can also be expressed as integrals of Bessel functions of order 1/2, and this can be worked into expressions for x and y as infinite series of Bessel functions.
To get the series, we use the recurrence relation 2Jn' = Jn-1 - Jn+1 repeatedly, to express J1/2 in terms of an infinite series of derivatives of the Bessel functions, which can then be integrated at once. Jn(x) goes to zero as n approaches infinity, for any x, and the convergence is fairly rapid. There is no round-off error when using the recurrence relations in this way, so the method is safe. The results are:
bx = (π/2)1/2(J1/2 + J5/2 + J9/2 + ...)
by = (π/2)1/2(J3/2 + J7/2 + J11/2 + ...)
These elegant results, due to Lommel, are given by F. E. Relton in his text on Bessel Functions. They are, of course, not used by civil engineers for this purpose, but are still practical ways for evaluating the Fresnel integrals in this region. The argument of the Bessel functions is s/2R, usually a rather small number, and the leading term in Jn is xn, so not many terms are needed for an accurate result.
An approximation to the spiral, found by keeping only the leading terms in the expansion of the sine and cosine, is the cubic spiral y = b2x3/3. This spiral was actually used as a practical transition curve, since it can be calculated without advanced mathematics. It has the conveninent property that the deflection angle from the tangent to any point on the spiral is one-third of ψ = (bs)2, and so proportional to the square of the length s. It is easy to lay out such a spiral, since the calculations can almost be done in the head. There are other somewhat more accurate approximate spirals as well, which give satisfactory results. With modern computing power, it is not much more difficult to use the exact spiral, although there is no special virtue in doing so.
In American railway practice, distances are expressed in stations of 100 ft, the length of a standard steel surveying tape. Decimal feet are always used. Sta 43 + 17.80 is 4317.80 ft from Sta 0. Curves are laid out by chords, not by arc distances. There is a short chord to the first even station, then full 100 ft stations until a second short chord is necessary. On tangents and circular curves, even stations are usually sufficient to guide construction. The reason for this pratice is to avoid fiddly fractions depending on calculations and theory, and to record the measurements in the same way as they are most conveniently laid out in the field. There is very little difference with measurements made along arcs, but the field work and notes are simpler.
It is necessary that a spiral be laid out very accurately, since the line is not straight or circular, but a unique shape. For this reason, it is customary to divide the spiral into ten equal chords, which are smaller than the standard 100 ft station, and which give sufficient points on the ground to construct the spiral accurately. The difference between chord and arc measurements is quite small in this case, and is usually neglected. Modern computing power makes a precise solution easy, if that is desired. Tables exist from which adequate spirals can be computed easily. There are many little tricks and properties that make introducing spirals fairly easy.
When a transit was used to lay out the spiral, it was set up at the T.S., and a backsight along the tangent (or a foresight to the vertex, if that point was set) gave a reference direction. Then the points on the spiral were set up by deflections from the tangent, the chainmen setting the successive points by chords, directed by the transitman to the proper deflections. The intersections are always quite good. The final point on the spiral was the S.C., where the circular curve began. The transit could then be moved to this point, a backsight taken on the T.S., and the circular curve laid out in the normal manner from there to the C.S., where the spiral at the other end of the curve began. This spiral was best laid out backwards from the S.T., where the next tangent began. There are many methods of carrying out this work, and this is only one possibility. The aim is always to make as few transit setups as necessary, to have long backsights, and to use only good intersections to fix points.
The method of calculating the spiral gives the offsets y directly, and the spiral could also be laid out by this method, intersecting the chord distances with the offsets y. This method can be used without a transit, and can give reasonable results if the T.S. and S.C. are already fixed, and it is only necessary to connect them with the spiral. It depends, of course, on alignment by eye.
The superelevation can be made by elevating the outer rail only, or by depressing the inner rail at the same time to keep the centre of the track at grade. Since superelevation does not exceed 8 inches, the difference is quite small and largely inconsequential. It is probably better to add ballast on the outside than to take it from the inside. The required amount of superelevation h = V2s/gR, where g is the acceleration of gravity, R the radius of the curve, s the track gauge, and V the speed. For 8" superelevation on a 1° curve (R = 5730 ft or 87 chains), V is about 110 mph. This is the equilibrium or balancing speed on the curve; the maximum speed can be somewhat greater. [Note: curves in American practice are specified by the angle at the centre subtended by a chord of 100 ft, or 20 metres in metric work. If this is a round number, the transit work in laying out the curve is simplified.]
The necessity of spirals in highway geometric design is doubtful, since the vehicles can make their own adjustments to curvature and superelevation, speeds and weights are lower, and beginning the superelevation on the tangent probably helps the cars to steer into the curve anyway. In icy or wet weather, superelevation may be something of a hazard as well.
Composed by J. B. Calvert
Created 3 July 2000