tangents, and pegs put in temporarily at c and D, making AC and BD each = s. The transition curves will bisect AC and BD at E and K. The points of commencement of the transition curves are now to be marked by three pegs at F and L, the distances AF and BL 1 being made = 2, i.e., half length of transition curve. (This is on supposition that AF and BL measured on the tangents do not appreciably differ from EF and KL measured on the curves, which is in practice the case.) The terminations of the transition curves and their points of junction with the circular curve are at G and H, EG and KH being ι each == 2 These points are fixed by means of the offsets xc and x'H measured at right angles to the tangents FI, li. These offsets AG and x'H are each = 48, i.e., 4 times the shift. They are also equal to the offset xa to the circular curve + the shift s or to chord Aa2 + S LI. Setting out Transition Curves.-The transition curves are set off by means of offsets measured square off the tangents FI, These offsets are proportional to the cubes of their distances from the commencement of the transition curve, and having the offset XG = 48 at a distance / from F the offset at any other point is ι at a distance S easily calculated. from F, and calculate the others from this. should then be = 4S. Having thus set off the transition curves by means of their calculated offsets from the tangents, set ranging rods in the lines FI, L'I', these lines being parallel to the original tangents and at a distance from them = shift. The circular curve GH may now be set out in the usual manner, by setting up the theodolite at c and working off the tangent F'I'. On laying off the deflection angle for the chord CG the point found should coincide with G as previously fixed by measuring the offset XG from FI. Similarly the termination of the curve should coincide with H as fixed by measuring x'H from LI. Otherwise take the offset AE R 2 2 The last offset at x The length of curve CD = circular measure of angle IX R. Total length of curve from F to L=GH + 2/. Or total length of curves from F to L = (circular measure These lengths of curves are on supposition that EG and CG are equal. In practice they will not differ appreciably. The portions CG and DH of the circular curve are of course not staked out, and the temporary pegs at C and D are removed, after setting out the curve GH. The radius of the above transition curve at G and H is equal to R, the radius of the circular curve. "Cant.”—The length of the transition curve above described will be proportional to the cant, which again depends on the radius of curve and speed of train. The following formula gives the cant: v2 Cant in inches = gauge in feet 1.25 R where v = velocity in miles per hour, R = radius of curve in feet. For a 10 chain curve on 4 ft. 8 in. gauge at 40 miles per hour the cant is about 9 in.; at 1 in 300 this gives a length of transition curve of 225 ft. For curves of greater radius the cant is less, and length of transition curve therefore less. The length of transition curve is, however, often made greater for curves of larger radius. When sharp curves follow each other closely there is sometimes little room to get in the transition curves, but in practice transition curves from 180 to 400 ft. long will usually cover all cases. Curves above about 40 chains radius do not usually require transition curves, the small amount of cant and widening of gauge being effected on the straight. Transition Curves between Reversed Curves.-When there is not room to get in a piece of straight between two reverse curves, the change of cant may be readily effected by uniting the two reversed curves by a transition curve as shown in Fig. 174. Let ABC be two reverse curves reversing at M. Shift each curve the proper amount to GF and DE. Calculate the length of transition curve, which will be in this case 300 x sum of cants. The middle point of the transition curve will bisect the total shift DF at M, and the ends of the transition curve will be at H and K, MK and мH being each = half length of transition curve. Set off the curve Kм by offsets from KF, making each offset proportional to the cube of its distance from K, and the curve Hм similarly. The final centre line is then GKMHE. Vertical Curves at Changes of Gradient. For uniting steep gradients parabolic vertical curves should be used. Fig. 175 shows a ready method of calculating the levels of a parabolic vertical curve. Let EF be the datum line on the section, and AB, BC the gradients to be united by the parabolic vertical curve A Datum D Fig. 174.-Transition Curve between Reversed Curves. d B e D B A E b, C F Fig. 175. Vertical Curve at Change of Gradient. AC. Produce one of the gradients to D, and measure the offset DC at the proposed termination of the vertical curve. All the other offsets from A to D, as db, ec, &c., may be readily calculated, as they are simply proportional to the square of their distances from a measured along the datum line, i.e., to (Fb1)2, (Fc1)2, &c. From these offsets the levels of the points b, c, &c., may be calculated and figured on the section as usual. Setting out of Work during Construction: Transferring Tangent Points.-The first thing the engineer in charge of the setting out of work on a railway must do is to carefully transfer the tangent points of the curves. In doing this he must calculate on the amount of cutting or bank at the tangent point in question, so as to transfer the points to beyond the slope of the cutting or bank, where they will not be disturbed by the work. The best way to transfer the tangent points is to drive in two pegs just beyond where the top of the slope of the cutting or the bottom of the slope of the bank will come, and on the same side of the railway. Then measure with a steel tape the exact distance of each of these pegs from the tangent point. It is also advisable to put in a third peg on the other side of the railway and in line with the tangent point and one of the first two pegs. Thus in Fig. 176 having the pegs a, b, and the distances ap, bp, the point P may be fixed by these measurements in the bottom of the cutting or on the top of the bank after the original tangent point has been excavated away or buried in the bank. The third peg c serves as a check on the measurements, or comes in useful if either of the other pegs is lost. Lining in between a and c is also of assistance in refixing the tangent point. The object of the above is to get the centre line pegged out again for the rail laying, &c., after the cuttings and embankments have been made and the original centre line pegs excavated or buried, and also for the alignment during their construction. Without the means of relocating the tangent points this might be a somewhat troublesome business. Setting out Fence Widths.-The fence width is usually pegged out at every chain, and at any intermediate points necessary. A list of fence widths should be made out. They are calculated as follows:-Suppose the depth of cutting is 10 ft. and the formation * width 28 ft., the slope being 1 to 1; then for fence width we have 14 ft. half formation width + 10 ft. slope +6 ft. allowance from top of slope to fence = 30 ft. fence width, i.e., from centre line of railway to fence is 30 ft. This supposes the ground to be level across. When the ground is irregular or sloping transversely the fence widths may be scaled off the cross sections. The fence widths are measured from the centre line pegs before the work of excavation or embankment is commenced, and pegs are driven in, one on each side of the railway, at every chain, to which the fence is erected. Levelling in Tops and Bottoms of Slopes. - The slopes of cuttings are usually worked up from the bottom and trimmed off to the correct slope by the foreman with a batter rule and plumb bob. In banks the material is usually tipped and left to find its own natural slope, being afterwards trimmed off with the batter rule, working down from the top. In deep cuttings or high banks, however, it is sometimes necessary to peg out the top or bottom of the slope, and the engineer may occasionally be called upon to do this. When the cross sections are reliable and the ground is not very irregular, possibly the widths from the centre line to the top or bottom of the slope may be scaled off the cross sections. The best way is, however, to level them in. As this is a tentative process, it often presents some little difficulty to the beginner. The procedure is as follows:-Take the depth of cutting or height of bank on the centre line from the longitudinal section at the given chainage. From this calculate the distance from the centre line to top or bottom of slope, which, on the supposition that the ground is level, will be half formation width plus slope. If now the ground slopes transversely to the centre line, add to or deduct from this distance to allow for the slope of the ground as much as may be judged by estimating the rise in the ground by the eye. Now direct the staff-holder to hold the staff at that distance. (Another chainman to hold the tape on the centre line is required in addition to the staff-holder.) Read now the level of the ground at "Formation" is the bottom of cutting or top of bank. |