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cross hairs also bisect the second tangent point; if so, the work is so far correct. Now set the vernier to the deflection angle for the first peg on the curve and drive that peg, then lay off the deflection angle for the second peg, and so on until all the pegs have been driven.
Check on Laying off the Deflection Angles. -Now lay off the deflection angle for the whole curve again; the cross hairs should again bisect the second tangent point, otherwise the instru ment has been shifted or some slip has been made.
Check on Chaining Round Curve.-Measure now from the last peg to the second tangent point or the end of the curve. This measurement should agree with the chainage of the second tangent point, as calculated from the length of the curve, otherwise the work is not correct, either the chaining of the tangents or the setting out or chaining round the curve being at fault, provided that there is no error in the calculations.
In practice, with fairly good chaining and on ordinary ground, the error in chainage should not be more than half a link on a long curve.
Final Error in Setting out Curve. When the deflection angle for the whole curve is set off the cross hairs should not be more than I or 2 in. at the outside off the second tangent point. In a long curve, in which the theodolite has to be shifted several times, 1 or 2 in. of error is excusable, and may be adjusted by altering slightly the last three or four pegs on the curve.*
The vernier should now be turned back again to zero, when the cross hairs should again bisect the intersection point and the back flag.
Continuing Work after Setting out Curve. To continue the work, set up the theodolite over the second tangent point, and direct the cross hairs on to the flag ahead on the second straight, turn the telescope over vertically and set off the deflection angle for the whole curve. The cross hairs should now bisect the first tangent point. This is as a check on the tangency of the second straight to the curve set out. The cross hairs are now
* These remarks as regards error apply only to field work of ordinary railways. In the case of curves in tunnels, &c., much greater accuracy is obtained by more accurate measurements, and by adopting special precautions in alignment (see Chapter VII.).
again directed on to the flag at the extremity of the second straight, the pegs on which are lined in at every chain in the same way as on the first straight until the second curve is reached, which is treated in exactly the same way as the first curve already described.
Chaining to be "through."-It is perhaps unnecessary to say that the chaining must be through or continuous from the commencement of the railway to its termination, the tangent points being put in at whatever odd points on the chainage they may occur, but the chain pegs must follow each other consecutively I chain apart whether tangent points intervene or not. In fact, the first peg on the curve may be measured, not from the tangent point, but from the last even chain peg behind it.
The formulæ already given are all that are necessary to set out a curve, but the surveyor should have a thorough knowledge of trigonometry and the properties of circular curves, as many problems may arise, such as inaccessible intersection or tangent points, obstacles in the curve itself, &c. &c. For the many field problems of this nature the reader is referred to one of the various engineers' field books which are devoted to these matters, and which usually contain in addition concise tables of logarithms, logarithmic sines and tangents, natural sines and tangents, deflection angles for curves, &c. &c. Fig. 172 will, however, illustrate the ordinary process of setting out and the principal properties of a circular curve required.
Shifting the Instrument forward to Intermediate Point on Curve.-When the curve is long and obstacles intervene so that the second tangent point cannot be seen from the first tangent point, we cannot check the accuracy of the work before beginning to set the pegs on the curve, by laying off the deflection angle for the whole curve to see whether we strike the second tangent point correctly. The theodolite has also to be shifted forward along the curve as the work proceeds, but the deflection angle for the whole curve should always be laid off as soon as the end of the curve is visible, to see that no error has occurred. As a rule, on a long curve the theodolite should be moved forward about every Io chains, otherwise the pegs will get out of line, especially on a sharp curve.
When it is necessary to shift the instrument forward the process
is as follows:-Having moved the theodolite forward, look back on the first tangent point and set off the deflection angle for the next chain peg, which is the total deflection angle for that peg, then proceed as before, laying off the angle for each peg in succession, all as originally calculated and entered in the field book opposite the chainage of each peg. This virtually amounts to laying out a tangent to the curve at the point at which the instrument is set up, and from that tangent laying off the deflection angle for the next chain ahead, only of course we do not stop to actually lay out the tangent to the curve, but continue to turn the instrument round to the angle of the next peg; in other words, for the first peg beyond the point to which the theodolite is shifted we have to lay off so much deflection angle behind for the part of the curve behind us plus the deflection angle for I chain forward. Beyond that we have to increase the angle by the deflection angle for chain for each peg in succession, all of which is already calculated and entered in the field book, as originally directed.
When it is necessary to move the theodolite forward a second time we have to look back on the furthest back point of the curve which it is possible to see. Suppose this is 10 chains back, then for the next peg we have to lay off the deflection angle for II chains, i.e., 10 chains behind and 1 chain in front. We have then to add to this the deflection angle for 1 chain for each peg in succession. It will be seen that in this case fresh calculation of the deflection angles is necessary, these being no longer continuous round the whole curve; but the angles which have to be calculated are merely the deflection angles for even numbers of chains, with the exception of the fraction of a chain from the last chain peg on the curve to the second tangent point.
Curve to Left.-When the curve curves to the right hand the deflection angles are laid off in a right-handed direction round the graduations of the theodolite. When, however, the curve curves to the left hand the angles have to be laid off in a left-handed direction or contrary to the way in which the theodolite is graduated. In this case we must either subtract each deflection angle from 360° and set the vernier to that angle, or we must use the vernier backwards and read off each angle directly. To do this, set the end of the vernier to 360°, not the zero as is usual, the zero being regarded as the end of the vernier for the time being. The degrees
are then counted off in the left-handed direction, for instance 10° will be at 350°, and the minutes are read off the vernier in its reversed position. With a little practice it is almost as easy to use the vernier one way as the other.
Transition Curves.-The object of a transition curve is to make a gradual change from a straight line to a curve. As on every curve the rails must have a certain amount of "cant" to counteract the centrifugal force of the train, the transition curve also serves to put the cant on the rails gradually. In addition to this there is the "widening of the gauge on curves"; this also may be gradually done on the transition curve. In the absence of transition curves both the cant and the widening of the gauge are usually effected on the straight, so that there is the full cant and extra width of gauge at the commencement and throughout the whole of the curve. It is obviously a very incorrect proceeding to put cant and extra width of gauge on a straight, while equally obviously the change from a straight to a curve ought to be made gradually, the radius of curvature decreasing from infinity at the commencement of the curve of adjustment to the same radius as the circular curve at its termination.
For these reasons it is the universal practice to put in curves of adjustment on the Continent, and the practice is also general in the United States and South America. Of late years they have been adopted to a small extent by English engineers in home work, and their universal use on railways in this country will no doubt not be long delayed.
Various transition curves have been proposed, among them being the cubic parabola and the compound transition curve. The latter formed of successive circular arcs increasing in curvature a certain amount for each chord.
Froude's Curve of Adjustment.*-The best practical form of transition curve is Froude's curve of adjustment, which approaches nearly a cubic parabola and is readily calculated and laid out on the ground.
To allow of the cant being gradually applied with a gradient of
* Minutes of Proceedings, Institution of Civil Engineers, vol. cxxxiii., "The Field Practice of Laying Out Transition Curves," by Mr John Robinson, M. Inst. C. E.
I in 300, which is the usual gradient and that recommended by Froude, the length of the curve of adjustment will be 300 x cant. In Fig. 173, let FI, LI be two straights to be united by a circular curve with transition curves, and let the intersection angle 1= central angle of curve, be measured as usual. Let AB represent the ordinary circular curve, having its tangent points at A and B. The method of introducing transition curves is to "shift" the curve AB to CD, AB and CD being concentric; the ordinary curve AB must be therefore considered as being of a radius = proposed radius of
circular curve plus the "shift" AC. Let R = radius of curve CD, then radius of curve AB = R+S where S = shift.
The length of the tangents AI, BI is then given by
AIBI (R+S) tan 1.
To get s we have length of curve of adjustment = 300 × cant = / say. Then
The distances IA and IB are now to be measured along the