the instrument work and measurement absolutely correct, it will not close by 0.8 foot. 73. The length of a curve, in terms of 100-foot stations, as measured along 100-foot chords, may be at once found by dividing the total angle (C) at the centre, in degrees, by the degree of the curve. Thus if L: = true length of curve, L = C (nearly), (5) = So that if the 40°, the length where I angle of intersection. (See Eq. 7.) angle subtended at the centre of a 10° curve = of the curve along the chords 400 feet; and this method, on account of its simplicity, is that usually adopted on railroad work for the measurement of curves. But the true length of the curve will of course be greater than this in the same ratio as the arc AFB in Fig. 19 exceeds the 100-foot chord AB. Now the angle at the centre of a circle which is subtended by an arc equal to the radius equals so that the true length of a curve is given by the equation L400.507 feet,-not 400 feet, as in the example above. Had this 10° curve been set out with corrected 50-foot chords, it would have measured (along the chords) 400.38 feet. Table IV gives the length of arcs of various curves subtended by 100-foot chords, from which the true length of a curve may be at once found. 74. Before proceeding to the more practical problems in connection with the setting out of curves in the field, it will be well to consider a few of the more important equations which form the groundwork on which these problems are built up. First, as regards the nomenclature of the various parts, as shown in Fig. 20. 75. Now because Aa and Ab are tangents to the curve at a and b, therefore ОaA and ObA must each equal 90°, and the angle a Ab at the apex must equal 180° C; therefore Again, in the triangle bod, since the angle at b = 90° therefore the Tangential Deflect.-Angle for a 100-foot chord D (8) And if in this we substitute the value for R given in Equation 2, this becomes And by combining Equations 11 and 13, we obtain (13) Again, by combining Equations 13 and 16, we obtain (15) (16) (17) The above equations can readily be followed by referring to Secs. 231 and 232. The following table may be of assistance in selecting quickly the equations required. Thus, suppose we have T and Y given, and want R; we see at once that Equation 15 will give us I; and then, by Equation 9, we can obtain R. PROBLEMS IN SIMPLE CURVES. 76. To lay out a curve by deflection-angles.-In Fig. 20 we have already seen (Eq. 8) that the angle Abd D but suppose we measure off another 100-foot chord de: then dbe D also = (since boe= 2D, which makes Obe 90° — D). 2 Similarly, we might show that for any number of consecutive 100-foot chords the total deflection-angle would, for each one, But though the Total Deflection-angle from the tangent is proportional to the number of full stations when these are the only points given on the curve, as we have already seen in the case of 50-foot subchords, if we insert intermediate stations without correcting the lengths of the subchords, the degree of the curve increases at once. = In order to find the corrected length of any subchord we may proceed thus: In Fig. 21 let ab represent a hundredfoot chord, then the angle abc D; and let represent any subdivision of it corresponding with the length of any uncorrected subchord: then the corrected length I will be given by Equation 16, when a FIG. 21. Y being the corrected length of the nominal subchord . In ordinary work, except where a sharp curve is run continuously throughout with subchords, we may ignore this correction. Not taking the correction into account, the deflection for any subchord is to D as the length of the subchord is to 100 feet; so that for any subchord we have |
