ENGINEERS' ANNUAL CONTAINING THE PROCEEDINGS OF THE MICHIGAN PUBLISHED BY THE SOCIETY F. HODGMAN, SECRETARY, CLIMAX, MICH. COPYRIGHTED 1894 BY THE MICHIGAN ENGINEERING SOCIETY PRESS OF ROBERT SMITH & CO. LANSING, MICH. MICHIGAN ENGINEERING SOCIETY. OFFICERS FOR 1894. E. W. MUENSCHER, President MANISTEE, MICH.. GEO. S. PIERSON, Vice President. . KALAMAZOO, MICH. DIRECTORS. JOHN J. GRANVILLE, Saginaw, DORR SKEELS, Grand Rapids, J. B. DAVIS, Ann Arbor. MICHIGAN ENGINEERING SOCIETY. ANNUAL CONVENTION, 1894. PAPERS AND DISCUSSIONS. ANSWERS TO ENGINEERING QUERIES. E. W. MUENSCHER. It was the lot of the writer to enter upon the practice of engineering at a time when the work of the profession was not so divided up into specialties, as it has since become, and a large proportion of the aspirants of that day were obliged to take such employment as they could get, passing from one branch of work to another as opportunity offered or necessity required, and thus acquiring rather an extended but somewhat superficial acquaintance with several branches than a profound knowledge of any one. During a practice of this character, now extending over a period of more than thirty years, during which he was initiated as back flagman on a railroad survey, passed to the degree of city engineer, and raised to the sublime degree of a county surveyor the writer has had in his employ many bright and promising young men, some of whom have since risen to responsible and important positions. Some of these young men, as well as others who have never been in his employ, have referred to the writer some of the troublesome questions which have arisen in their own practice. The substance of the replies which have been given to some of these questions is given in this paper, in the hope that it may be of service to some of the younger members of this society, who may possibly have to deal with some of these questions themselves. First—Application of transition curves to race tracks. A gentleman writes from the Province of Ontario that he has been trying the method of laying out transition curves described in the "Annuals" of this society for 1891 and 1892 in easing off the circular curves of a regulation race track, one mile in length, and finds that it does not fit precisely. The track without easements would consist of two parallel straight stretches 1,320 feet long, and 840 feet 4 inches apart, connected by two semicircular arcs each 1,320 feet long. He says, "The radius of the circular curve being 420' 2'', D= 13° 40.2', and deflection angle for 200' taper =4° 33'. Setting up instrument at O (P. C. C.) (and deflecting 9° 06' from chord A O to get tangent), I commenced putting in the circular curve, using the radius 420' 2'', but chords of 25 feet instead of 100; the deflection for 25 feet being 1° 42' 16''; but, instead of keeping the same distance from the circular curve (if no taper had been used), I kept approaching the true circular curve, and when nearing the second P. C. C. the two curves coincided, which is contrary to what I was led to expect from reading your paper. I then ran one-half the curve from each end, and found myself out only six inches when I met in the middle at C, which error I distributed among three stations on each side. I found the lengths of the curves from P. C. C. to P. C. C. to be 1,112.4 and 1,112.7 feet respectively, instead of 1,119.4, as it should have been for a circular curve--using chords of 25 feet, necessitating the lengthening of the tangents." The following was the substance of the reply to this letter: The method of laying out easement curves described in the Michigan Engineers' Annuals for 1891 and 1892 was intended for use in ordinary cases of railroad curves of moderate degrees of curvature, where the variations from exact mathematical precision would be so small as to be immaterial, and where extreme nicety of alignment is not necessary. I have never had occasion to use it in laying out a race track, and, in fact, hardly anticipated its application to so extreme a case. Since, however, your letter has called my attention to the matter, I think it will be found that there need be no difficulty in applying the general method even in this case. To do this, however, will require a little preliminary calculation, and it will be necessary also to always bear in mind three important practical points. First, That even the most favorable natural surface of ground makes a rough drawing board, and that to lay out a semicircle of 13° 40' curve from one end by chords and deflection angles, and hit the tack at the other end exactly, is a feat that requires an excellent instrument and the utmost care on the part of even expert transitmen and tapemen. In fact it is very seldom accomplished. chord Second, That in an arc of given radius the degree of curvature (the D of the field books) is not precisely the same for chords of different lengths, as, for example, 100 and 25 feet. Thus in your case of a semicircle with radius 420.17 feet, if 100 feet chords are used D=13° 40′ almost exactly, but if 25 feet chords are used D-3° 24' 33'', since Sin D= SinD= and 3° 24′ 33'' × 4 : Third, That the length of the curve will be different if measured on the arc from what it is when measured on the chords, and that its length will also vary if measured by chords of different lengths. 13° 38' 12''. 2X radius, This is an important point, because in race tracks the measurement must always be reckoned on the arc, while in railroad work we deal only with chords, and the precise length of the arc is of little moment. Thus in your case the length of the semicircular arc should be 1,320 feet, but, since there are 10800 and 648000'' in 180°, the length by chords of 100 feet would be 180° 13° 40' × 100' 10800' 180° ×100'1317.07', or nearly three feet less than on the arc, and by chords of 25 feet 3° 24' 33' X25'1319.97 feet, showing that the difference between the arc and the 25 feet chords is inappreciable. X 25': 648000'' Probably all three of these facts concurred in causing the discrepancies in measurement which you mention, and will always tend to produce a similar result. The first difficulty can only be overcome by extreme care at the transit and the tape, and the second and third by using chords of 25 feet with the precise value of D for chords of that length. Let us now apply these considerations to the case in point. |