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Vehicle Climbing Lanes HIGHWAY RESEARCH BOARD Bulletin 104 Vehicle Climbing Lanes National Research Co HIGHWAY RESEARCH BOARD Officers and Members of the Executive Committee 1955 OFFICERS G. DONALD KENNEDY, Chairman K. B. WOODS, Vice Chairman FRED BUBGGRAF, Director ELMER M. WARD, Assistant Director Executive Committee C. D. CuRTiss, Commissioner^ Bureau of Public Roads A. E. JOHNSON, Executive Secretary, American Association of State Highway Officials LOUIS JORDAN, Executive Secretary, Division of Engineering and Industrial Research, National Research Council R. H. BALDOCK, State Highway Engineer, Oregon State Highway Commission PYKE JOHNSON, Consultant, Automotive Safety Foundation G. DONALD KENNEDY, President, Portland Cement Association O. L. KiPP, Assistant Commissioner and Chief Engineer, Minnesota Department of Highways BURTON W. MARSH, Director, Safety and Traffic Engineering Department, Ameri• can Automobile Association C. H. ScHOLER, Head, Applied Mechanics Department, Kansas State College REX M. WHITTON, Chief Engineer, Missouri State Highway Department K. B. WOODS, Director, Joint Highway Research Project, Purdue University Editorial Staff FRED BURGGRAF ELMER M. WARD WALTER J. MILLER 2101 Constitution Avenue Washington 25, D. C. The opinions and conclusions expressed in this publication are those of the authors and not necessarily those of the Highway Resoarch Board. HIGHWAY RESEARCH BOARD BuUetin 104 Vehicle Ctimbing Lanes PRESENTED AT THE Thirty-Fourth Annual Meeting January 11-14, 1955 1955 Washington, D. C. Department of Design T. E. Shelburne, Chairman Director of Research, Virginia Department of Highways, University of Virginia COMMITTEE ON GEOMETRIC fflGHWAY DESIGN D. W. Loutzenheiser, Chairman Acting Chief, Urban Highway Branch, Bureau of Public Roads R. H. Baldock, State Highway Engineer, Oregon State Highway Commission Warren James Cremean, Urban Projects Engineer, Ohio Department of Highways Ralph L, Fisher, Engineer of Design, New Jersey State Highway Department Fred W. Hurd, Yale Bureau of Highway Traffic, Strathcona Hall Emmett H. Karrer, Professor of Highway Engineering, Ohio State University Elmer R. Knight, Assistant Chief Highway Engineer, Illinois Division of Highways Harry C. Knudsen, Office, Chief of Engineers, Department of the Army O. K. Nermann, Chief, Traffic Operations Section, Highway Transport Research Branch, Bureau of Public Roads William S. Pollard, Jr., Assistant Professor of Civil Engineering, University of Illinois K. A. Stonex, General Motors Proving Ground, Milford Michigan Edward T. Telford, District Engineer, California Division of Highways, Los Angeles C. A. Weber, Assistant Chief Engineer, Michigan State Highway Depart• ment IV Contents SIMPLIFIED CLIMBING-LANE DESIGN THEORY AND ROAD-TEST RESULTS T. S. Huff and F. H. Scrivner 1 MOTOR-VEHICLE PERFORMANCE ON ASCENDING GRADES Robert E. Dunn 12 TRUCK CONGESTION ON UPHILL GRADES William E. Willey 21 Simplified Climbing-Lane Design Theory and Road-Test Results T. S. HUFF, Engineer of Road Design, and F. H. SCRIVNER, Supervising Research Engineer, Texas Highway Department A simplified theory of the motion of heavy vehicles on grades is presented. A set of speed-distance curves computed from the theory, based on values of maximum sustained speeds observed in Arizona, is given as the current basis for design of climbing lanes in Texas. Speed-distance curves representing the observed performance of a test vehicle on 11 grades are compared with the corresponding curves developed from the theory. Fair agreement was found, and it was concluded that the simplified theory is accurate enough for use in the design of climbing lanes. • CONSIDER a vehicle (Figure 1) of gross that although the net driving force must weight, W, travelling at a variable vel• satisfy Equation 1 involving the acceleration ocity, V, on a grade inclined at an angle, and the grade angle, it may also be ex• e, with the horizontal, the value of e pressed independently as some function of being taken as positive if the vehicle is as• velocity only, since each of its components cending and negative if it is descending. If is a function of velocity only. g represents the acceleration of gravity For example, if the truck operates at a and t the time, then, neglecting that part known maximum sustained velocity on any of the driving force required to impart grade, the numerical value of P/W corres• angular acceleration to rotating parts, we ponding to that velocity may be immediately may write the force equation, calculated from Equation 1, which in this case reduces to P/W = sin 9, and that Wdv P - W sin e, where P, magnitude of P/W will always exist at that g Ht velocity, at least approximately, regard• a variable, may be termed the net driving less of the value of the acceleration. In force acting on the vehicle. The above Figure 2 we have plotted values of P/W equation may be rewritten in the form computed in this way against correspond• ing values of the velocity, v, from basic P 1 dv sin e data supplied mainly by Willey (1.) in 1950, W=gc[f ^ (1) and applying to an average heavy vehicle The net driving force is the total traction operating on mountain grades in Arizona. exerted by the driving wheels against the The points plotted in Figure 2 are con• road surface, less wind resistance and road nected by straight lines to form a con• surface resistance. Again neglecting in- tinuous graph of P/W versus v. Each ertial resistance to angular acceleration, straight line segment extending from, say, it follows that if the truck is always operated Vn to v^ may be represented by an at the highest possible speed and always equation bf^the form. within the range of engine speed recom• P /W = av + b mended by the manufacturer, then the total (2) driving force must be expressible, at least where varies within the interval, Vjj approximately, as a single-valued function to n + 1> and a and b are constant of the velocity only. Air resistance in within the same interval. still air is usually considered to be a func• From Equations 1 and 2 we may form a tion of the velocity only, and we shall as• third equation, not containing P/W ex• sume that no wind exists. We shall also plicitly, which becomes the general motion assume that the type and roughness of the equation for the vehicle, as follows: pavement do not change and, therefore, that the road surface resistance may be dv - gav + g (sin 6 b) = 0 (3) taken as constant, or at most as a function of velocity only. We therefore conclude where v is restricted to the velocity in- terval, to v^+j, and a and b are interval to v^+i- Thus, during the constant in the same interval. time t, the velocity changes from VQ to V, the vehicle travels a distance x, and the ratio, net driving force to gross weight, changes m value from (avQ + b) to (av + b). (The logarithm is taken to the base, e). Pa net driving forci {|b>) grosi Mfight (Ibt) In using Equations 4 for calculating the vihielB It OBCtntfing distance traveled or time consumed by a vehicle while it changes velocity over an Mass X Acctlaralion > Forct, interval greater than that for which a and "fSt-p- b are constant, it is necessary to compute WMrB w = vtlocilr {lt/s»),and the increments of distance and time cor• t ' limm (Mc) responding to each subinterval of the type, (Tha additionol driving force raquirtd lo oeeelarota rotating pans II noglocttd ) Vn to v'n_ ' 'n to ^n+l. and Vn+i, to v, and to add these increments in order to Figure 1. obtain total distance and total time. MAXIMUM SUSTAINED SPEEDS NUMERICAL VALUES USED IN PLJOTTING GRAPH FOR USE IN Psrcent Sine = v EQUATIONS (91 ,( Max 6rad« P/w (fWllc) le II9I5 Ellimottd - 0096049 11919 T 06983 - 0067991 12926 6 05990 10 27 - 0093993 094811 5 04994 19 20 - 0022699 0796S0 4 03997 17 60 - 00r5594 067419 5 02999 24 00 -0010267 094631 02000 J3 73 - 00090909 037039 00000 73 33 NOTE TU .olun of 0 oiul D opply batnan Iht nlociliti indicated Far tROmplt, b. 079690 in tha valeeity ronga liam 13 20 la 1760 ft / neluiKa • William E Willa>;uiailll Trach Spaada: ROADS a STREETS,JAN 1990,P92 vdt /sec) Figure 2. Graph of Equation 2. We now denote the position of the vehicle TEXAS DESIGN METHOD at any instant by its coordinate, x, meas• ured in the direction of motion from a sta• Figure 3 shows speed-distance curves tionary point on the grade behind the truck. computed from Equations 4, the value of We also stipulate that x = t = 0 when v = the constants, a and b, having been taken V(j, and that the grade angle, 6, is con• from Figure 2. By interpolating between stant. Then the solution of Equation 3 may these curves, one may determine the ap• be written in the following form suitable proximate speed, in the range from 0 to for the construction of speed-distance and 47 miles per hour, of Willey's (j.) aver• time-distance curves: age heavy vehicle at any point on any series of successive grades ranging between X = (sin e - b) t minus 7 percent and plus 7 percent pro• (4) vided the speed at one point on the series of grades is known. The upper limit of wherein t = 1_ log/av + b - sin e 47 mph. was selected because that figure ag \avo + b sin e J was the average speed of trucks on ap• proximately level grades in Texas. and both v and v, pre restricted to the EXAMPLE OF USE OF CURVES WARRANTS FOR CLIMBING LANES iDashed Imai on graph indicot* itapt taken in CLASS B HIGHWAYS Providt climbing lont ond parking shouldtr finding propdr location for climbing lont ihawn CLASS C HIGHWAYS — OlSirable treotmsnt lama os for CLASS 8 HIGHWAYS i&=2sa; on ikoten I Minimum treotmcnl convarl shoulder to climbing lone _30M PH ond SPEED-DISTANCE CURVES CLASS D HIGHWAYS Moki studies to determine feosibility of converting occeleroting -Traniitions.
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