4. The appearance of the highway or street is enhanced by the application of spirals. Their use avoids the noticeable breaks at the beginning and ending of circular curves as shown. for example, in Figure III-15. which are made more prominent by super elevation runoff.
Generally, the Euler spiral. which is also known as the clothed, is used. The radius varies from infinity at the tangent end of the spiral to the radius of the circular arc at the circular curve end. By definition the radius at any point of the spiral varies inversely with the distance measured along the spiral. In the case of a combining spiral connecting two circular curves having different radii. there is an initial radius rather than infinite value.
Length of spiral. The following equation, developed in 1909 by Shortt (23) for gradual attainment of centripetal acceleration on railroad track curves. is he basic expression used by some for computing minimum length of a spiral:
where: L = minimum length of spiral, m:
V = speed, km/h;
R = curve radius, m; and
C = rate of increase of centripetal acceleration, m/s.
The factories an empirical value indicating the comfort and safety involved. The value of C = 1 generally is accepted for railroad operation, but values ranging from 1 to 3 have been used for highways. Sometimes this formula is modified In taking into account the effect of super elevation, which .results in much shorter lengths. Highways do not appear to need as much precision as is obtained Irom computing the length of spiral by this formula or its modification. A more practical control for the length of spiral is that in which it equals the length required for super elevation runoff.
runoff should be effected uniformly over a length adequate for the likely operating speeds. To be pleasing in appearance the runoff pavement edges should not be distorted as the driver views them.
Some agencies employ the spiral and use its length in which to make the desired change in cross slope. One agency believes that the length of spiral should be based on a 4 s minimum maneuver time at design speed. Other agencies do not employ the spiral but empirically designate proportional lengths of tangent and circular curve for the same purpose. In either case, as far as can be determined, the length of roadway to effect the super elevation runoff should be the same for the same rate of super elevation and radius of curvature. The spiral simulates the natural turning path of a vehicle. On unsprayed curves the average vehicle tends to traverse a similar transitioned path within the limits of the traveled way.
Review of current design practice indicates that the appearance aspect of super elevation runoff largely governs the length. Spiral lengths as determined otherwise often are shorter than that determined for general appearance," so that spiral formula values-give way to long empirical runoff values. A number of agencies have established one or more control runoff lengths within a range of about 30 to 200 m, but there is no universally accepted empirical basis, considering all likely traveled way widths. In one widely used empirical expression the required length is indicated in terms of the slope of the outside edge of traveled way relative to the centerline profile.
Current practice indicates that for appearance and comfort the length of super elevation runoff should not exceed a longitudinal slope (edge compared to centerline of a two-lane highway) of 1:200. In other words, when considering a two-lane highway with plane sections, the difference in longitudinal gardenia between the edge of traveled way profile and its centerline profile should not exceed 0.5 percent.
In another source (24) the same 1:200 slope is used for a design speed of 80 km/h and higher. Where design speeds are less than 80 km/h, greater relative slopes are used. To reflect the importance of the higher design speed and to harmonize with the flatter curving elements, both horizontal and vertical, it appears logical to extrapolate the changing relative slope to the higher design speeds, as given in Table III-13.
The maximum relative ؟؟؟ between profiles of the edges of two-lane traveled ways are double those given in Table III-13.
Length of runoff on this basis is directly proportional to the total super elevation, which is the product or the lane width and superelevaton rate. However, there are certain minimum runoff lengths which should be provided for reasons of general appearance and to avoid undesirably abrupt edge-of-pavement profiles. These minimum values approximate the distance traveled in 2 s at the design speed, Table III-14 gives values for two-lane highways with 3.0- and 3.6-m
On a purely empirical basis it is concluded that minimum-design super elevation runoff lengths for highways wider than two lanes should be as follows:
Three-lane traveled ways, 1.2 times the corresponding length for two-lane traveled ways;
Four-lane undivided traveled ways, 1.5 times the corresponding length for two-lane traveled ways and Six-lane undivided traveled ways, 2.0 times the corresponding length for two-lane traveled ways.
The four-lane lengths shown in Tables II1-7 to III-11 are determined on this empirical basis. Proper design attention to obtain smooth edge profiles and to avoid distorted appearances may suggest lengths greater than these minimums. Runoff lengths for divided highways are discussed in the section, Runoff With Medians.
The length of tangent run out is determined by the amount of adverse cross slope to be removed and the rate at which it is removed. This rate of removal should preferably be the same as the rate used to effect the super elevation runoff. However, where the super elevation is less than maximum the length of run out may be less as discussed in the following section on 'Methods of Attaining Super elevation.
Location with respect to end of curve. In alignment design with spirals the super elevation runoff is effected over the whole of the transition curve. The length of runoff is the spiral length with the tangent to spiral (TS) at the beginning and the spiral to curve (SC) at the end. The change uncross slope begins by removing the adverse cross slope from the lane or lanes on the outside of the curse on a length of tangent just ahead of TS (the tangent run out). (See Figure III-16.) Between the. TS and SC (the super elevation runoff) the traveled way is rotated to reach the full super elevation at the SC. This procedure is reversed on leaving the curve. By this design the whole of the circular curve has full super elevation.
In design of curves without spirals the super elevation runoff also is considered to be that length beyond the tangent run out. Empirical methods are employed to locale the super elevation runoff length with respect to the point of curvature (PC). No method for division between the tangent and the circular curve can be completely rationalized. With full super elevation attained at the PC, the runoff lies entirely on the approach tangent, where theoretically no super elevation is needed. At the other extreme, placement of the runoff entirely on the circular curve results in a portion of the curve having less than
the desired .amount of super elevation. Most agencies that do not use spirals design with part of the runoff length on the tangent and part on the curve. In this compromise design the tangent receives some but not the maximum
unneeded super elevation, and the end portion of the circular curve receives some what less than the needed super elevation. Currant desire practice is to place approximately two-thirds of the runoff on the tangent approach and • one-thir d"oh the curve.
In general, theoretical considerations favor the practice of placing a larger proportion of the runoff length on the approach tangent rather than on the circular curve. With the resultant super elevation on the tangent the driver may have to steer in a direction opposite to the direction of the curve ahead to stay in line, but the maximum side friction developed, which is equal to the rate of applied super elevation, is at all times below the rate of side friction considered comfortable. A vehicle traveling the design speed on the minimum radius curve (with maximum rate of super elevation) develops the maximum side friction considered safe and comfortable. To apply rates of super elevation less than the maximum at any point on the curve means that vehicles traveling at the design speed develop side friction factors in excess of the allowable minimum. While the side friction factor developed on the tangent is undesirable, ;he development on curves of friction factors greatly in excess of the design basis results in a worse condition.
The resulting side friction factor depends on the actual vehicle path of travel. Regardless of the super elevation runoff, some form of transition path of travel can be expected. This actual transition path usually begins well back on the tangent and ends beyond the beginning of the circular curve, depending on the speed, sharpness of curvature, width available, and effect of other traflc. What might appear to be an undesirable cross slope on the tangent actually compensates for the curvilinear path of the vehicle. Also, what can be considered lack of super elevation at the beginning of the circular curve proper is compensated for by the vehicle's traveling a curvilinear path that is flatter than the roadway circular arc.
It is evident that in alignment design without spirals the placement of the length of runoff with respect to the PC cannot be determined exactly from available practice and information. In general, design with 50 to 100 percent of the length of super elevation runoff on the tangent can be considered as suitable. For a more precise design control it is concluded that from 60 to 80 percent of the length of runoff preferably should be located on the tangent at curves without spirals.
Methods of attaining super elevation. Change in cross slope should be effected with edge-of-traveled way profiles that are rounded to smooth-flowing lines. The methods of changing cross slope are most conveniently discussed in terms of straight-line relations and controls, but it is emphasized that these straight-line profiles with angular breaks are to be rounded in the finished design as later discussed.
Four specific methods of profile design are practiced in attaining super elevation: (1) revolving a traveled way with normal cross slopes about the centerline profile. (2) revolving a traveled way with normal cross slopes about the inside-edge profile. (3) revolving a traveled way with normal cross slopes about the out side-edge profile, and (4) revolving a straight cross-slope traveled way about the outside edge profile. Figure III-16 illustrates these four methods diagrammatically. The centerline profile for Figures III-16A, III-16B or III-16C or the outside-edge profiles for Figure III-16D, drawn as a horizontal line, represents the calculated profile, which may be a tangent, a vertical curve, or a combination of the two. The small cross sections at the bottom of each diagram indicate the traveled way cross slope condition at the lettered points.
Figure III- I6A illustrates the method where a traveled way with normal cross slopes is revolved about the centerline profile. This general method is the most widely used in design because the required change in elevation of the edge of the traveled way is made with less distortion than with the other methods. The usual
calculated centerline profile is the baseline, and one-half of the required elevation change is made at each edge.
Figure III-I6B illustrates the method where the traveled way is revolved about the inside-edge profile. In this case the inside-edge profile is determined as a line parallel to the calculated centerline profile. One-half of the required change in cross slope is made by raising the centerline profile with respect to the inside traveled way edge and the other half by raising the outside traveled way edge an equal amount with respect to the centerline profile. The third method involves similar genome tries, where the traveled way with normal cross slopes is revolved about the outside-edge profile, as shown in Figure III-16C, except that the change is effected below the outside-edge profile instead of above the inside-edge profile.
The fourth method, as shown in Figure III-I6D, revolves a traveled way having a typical straight cross-slope instead of the typical cross section illustrated for the first three methods. The traveled way is shown revolved about the outside-edge profile in Figure III-16D because this point is most often used for evolvement of two-lane one-way roadways, with profile along the median edge of traveled way. The traveled way could also be revolved about the lane division or about the inside-edge profile similar to Figures III-16A or III-I6B. For the straight cross slope, the outside-edge profile is calculated, and the required change in cross slope is made by lowering the inside edge, as shown in Figure III-16D.
The design controls for attaining superelevation are nearly the same for all four of the methods. Cross section A at one end of the tangent run out is a normal or straight cross slope section. At cross section B, the other end of the tangent run out and the beginning of the spiral or length of runoff, the lane or lanes on the outside of the curve are made horizontal with the centerline profile for m wide. These are the lengths for curves with maximum superelevation. Where there is less than manicure superelevation, tangent run out lengths will be longer if the same relative slope as for superelevation runoff are retained. It is desirable that these relative slopes be retained but where this is not possible the run out lengths should be at least equal to those required for a curve with maximum superelevation where the same relative slopes for tangent runoff and run out are retained. On curves where the normal parabolic or circular traveled way cross sections are not retained, the normal cross section should be changed to straight cross slopes in the tangent run out length.
At cross section C the traveled way is a plane, super elevated at the normal straight cross slope rate. Between cross sections B and C for Figures III-16A, III-16B,and III-16C, the outside lane or lanes change from a level condition to one of superelevation at the normal cross slope rate, which rate is retained on the inner lanes. There is no change between cross sections B and C for Figure III- 16D. Between cross sections C and E the roadway section is revolved to the full rate of superelevation. The rate of cross slope at any intermediate point, cross section D, is proportional to the distance from cross section C.
Considering the infinite number of profile arrangements and in recognition of such specific problems as drainage, avoidance of critical grades, aesthetics, and fining the roadway to the ground, the adoption of any specific axis of rotation, i.e.. making a choice between methods A, B, C, or D in Figure III, cannot be recommended. To obtain the most pleasing and functional results, each runoff section should be considered an individual problem. In practice, any longitudinal profile line for the axis of revolution may be the most adaptable for the problem at hand. In any case a smooth-edge profile is desired as discussed in the following section. In an overall sense, the method of rotation about the centerline usually is the most adaptable. For example, in Figure III- 16A, the change in longitudinal grade required for each profile with a relative slope of, say, 1:200 to the centerline, is 0.5 percent at point E. In Figure II1-16B, no change is required in the direction of the inside edge of traveled way profile, whereas double this amount, or 1.0 percent, is required in the outside edge of traveled way profile at point E.
The method shown in Figure III-16B is preferable to the other three in cases where the lower edge profile is a major control, as for drainage. With uniform profile conditions its use results in the greatest distortion of the upper edge profile. Where the overall appearance is to be emphasized, the methods of Figures 1II-16C and III-16D are advantageous in that the upper-edge profile — the edge most noticeable to drivers — retains the smoothness of the control profile. The shape and direction of the centerline profile may determine the preferred method for attaining superelevation in the first three methods.
Design of smooth profiles for traveled way edges In the diagrammatic
profiles of Figure III-16 the tangent profile control lines result in angular breaks at cross sections A, C, and E. For general appearance and safety, these breaks should be rounded in final design by insertion of vertical curves. With the method of Figure III-16A, usually short vertical curves are required. Even where the maximum relative slopes are used (minimum length of runoff), the length of vertical curve required to con form to the 0.67 percent break at the 50-km/h design speed and 0.4 percent break at the I20-km/h design speed need not be great. Where the traveled way is revolved about an edge, these grade breaks are doubled to 1.3 percent for the 50-km/h design speed and to 0.8 percent for the 120-kni/ h design speed. Greater lengths of vertical curves obviously are needed in these cases. Positive controls cannot be cited for the lengths of vertical curves at the breaks in the diagrammatic profiles. For an approximate guide, however, the minimum vertical curve length in meters can be used as numerically equal to the design speed in kilometers per hour. As the general profile condition may determine, the greater lengths should be used where possible.
Several design procedures are followed by different agencies in the development of profiles for runoff sections. Some compute the edge profiles on the straight-line basis of Figure III-16 and insert vertical curves as eye adjustments in the field. Other agencies specify minimum lengths of vertical curves at the breaks for edge profiles. Some employ a selected reverse vertical curve for the entire transition section, with resultant computed edge profiles. This method is laborious when the edge vertical curves are superimposed on a centerline vertical curve. It does provide essential controls to the designer and should yield uniformity of results.
Several agencies use a design procedure in which the runoff profiles are determined graphically. The method essentially is one of spline-line development. In this method the centerline or other base profile, which usually is computed, is plotted on an appropriate vertical scale. Superelevation control points are in the form of the break points shown in Figure III-16. Then by means of a spine, curve template, ship curve, or circular curve, smooth-flowing lines are drawn to approximate the straight-line controls. The natural bending of the spline, or if curve templates are used, the attainment of smooth-flowing profiles without marked distortion, almost always will satisfy :he requirements for minimum smoothing. Once the edge profiles — and lane profiles if required — are drawn in the proper relation to one another, elevations can be read for stations, half stations, quarter stations, or otherwise as necessary for construction control.
An important advantage of the graphical or spine-line method is the infinite study possibilities it affords the designer. Alternate profile solutions can be developed with a minimum expenditure of time. The net result is a design that is well suited to the particular control conditions. The engineering design labor required for this procedure is minimal. These several advantages make this method preferable over the methods of developing profile details for runoff sections.
Runoff with medians. In the design of divided highways, streets, and parkways, the inclusion of a median in the cross section alters somewhat the superelevation runoff treatment. Depending on the width of median and its cross section, there are three general cases for superelevation runoff design:
Case I —The whole of the traveled way. including the median, is super elevated as a plane section.
Case II — The median is held in a horizontal plane and the two traveled ways are rotated separately around the median edges or. where applicable around the inside gutter lines.
Case III — The two traveled ways are separately treated for runoff with a resultant variable difference in elevation at the median edges.
Case 1 necessarily is limited to narrow medians and moderate superelevation rates to avoid substantial differences in elevation of the extreme traveled way edges because of the median tilt. For the most part the method of rotation about the median centerline is used. Diagrammatic profile controls will be similar to those in Figure III-16A except for the two median edges, which will appear as profiles only slightly removed from the centerline.
Case II applies for any width median but has most application with medians of intermediate width to about 10 m. By holding the median edges level the difference in elevation of the extreme traveled way edges is limited to that of the superelevation. Although this type of cross section may be used with wide medians, especially in flat terrain, its general use is limited to medians not wide enough to favor the Case III treatment. Runoff design for Case II usually has the median-edge profiles as the control. One traveled way is rotated about its lower edge and the other about its higher edge. The diagrammatic profile controls are those shown in Figures III-I6B,III-16C, and III-16D, the centerline grade control being the same for the two traveled ways.
The Case III design is preferable on pther-than-narrow medians in that the differences in elevation of the extreme traveled way edges are minimized by a compensating slope across the median. This design may be used on a median of intermediate width by use of a sharply sloped section, but a fairly wide median is usually necessary to develop left shoulder areas and the desired gentle slopes between. With medians of about 12m or more in width. it is possible to design separately the profiles and superelevation runoff for the two roadways, a tie-in control being at the median edges. Accordingly, the rotation can be made by the method otherwise considered appropriate, i.e.. any of the methods in Figure III-16.
Superelevation runoff lengths for four- and six-lane undivided highways have been shown as 1.5 and 2 times, respectively, the lengths for two-lane highways. For Case I designs of divided highways the length of runoff properly should be increased in the proportion to the total width, including the median. Because Case I applies mainly to narrow medians, the added length usually will
be insignificant. With medians of the order of 1 to 3 m wide, any increase may well be ignored.
Under Case II conditions with narrow medians in a horizontal plane, the runoff lengths should be the same as those for undivided highways, as shown in Tables III-7 to III-11 for four-lane highways. This length applies to highways with medians about 4 m or less in width. However, with medians about 12 mor more in width, the two-lane values should he used for the one-way roadways because the extreme traveled way edges are at least 24 m apart and independent of each other. Values for the one way roadways of six-lane highways when, separated by a wide median should be 1.2 times the two-lane values of Tables III-7 to III-11 The one-way traveled ways of highways with medians between 4 and 12 m might be designed on the basis of either the two-lane or multilane suggested lengths.
With Case III cross sections, the median generally will be 12 m or more in width, and the two-lane values for length of runoff are applicable for one-way roadways of four-lane divided highways. The values for the one-way roadways of six-lane divided highways should be somewhat greater. In situations where the median is less than about 12 m, the runoff length will be determined indeed same manner as for Case II.
Divided highways warrant a greater refinement in design and greater attention to appearance than two-lane highways do because they serve much greater traffic volumes and because the cost of refinements is insignificant compared with the cost of construction. Accordingly, the indicated values for length of runoff should be considered manicures, and an effort should be made to use yet longer values. Likewise, there should be emphasis on the development of smooth-flowing traveled way-edge profiles of the type obtained by spine-line design methods.
Design for Low-Speed Urban Streets
As previously discussed, the maximum allowable side friction factor for use in the design of horizontal curves is the point at which the centrifugal force causes the driver to experience a feeling of discomfort when driving a curve at a certain design speed. Figure III-5 summarized the test results of side friction factors developed on curves at these apparent limits of comfort.-Use of the solid line in FigureIII-8 and method 5 was recommended for distributing e and fin the design of all rural high ways and high-speed urban streets. Method 2 is recommended lord the design of horizontal curves on low-speed urban streets where, through conditioning, drivers have developed a higher threshold of discomfort. By this method, none of the centrifugal force is counteracted by superelevation so Ion;: as the side friction factor is less than the maximum assumed for design for the radius of the curve and the design speed. For sharper curves f remains at the maximum and e is used in direct proportion to the continued increase in curvature until reaches . The recommended design values for f that are applicable to max low-speed urban streets are shown in Figure 111-17 as a solid line superimposed on the analysis curves from. Figure I1I-5. They are based on a tolerable degree of discomfort and provide a reasonable margin of safety against skidding under normal driving conditions in the urban. environment. These values vary with the design speed from 0.31 at 30 km/h to about 0.19 at 60 km/h, 60 km/h being the upper limit for low speed established in the design speed discussion of Chapter 11. Although superelevation is advantageous for traffic operations, various factors often combine to make its use impractical in many built-up areas. Such factors include wide pavement areas. need to meet the grade of adjacent properly. surface drainage considerations, and frequency of cross streets, alleys and driveways. Therefore, horizontal curves on low-speed streets in urban areas are frequently designed without superelevation, counteracting the centrifugal force-solely with side friction. On these curves for traffic entering a curve to the left the normal cross slope is an adverse or negative superelevation, but with flat curves the resultant friction required to counteract both the centrifugal force and the negative superelevation is small. However, on successively sharper curves for the same design speed, the minimum radius or sharpest curve without superelevation is reached when the side friction factor developed to counteract centrifugal force and adverse cross slope reaches the maximum allowable value based on safety and comfort considerations. For travel on sharper curves, superelevation is needed. The maximum superelevation rate of zero in Table 111-15 establishes the minimum radius for each speed below which superelevation is not provided on local streets in residential and commercial areas but should be considered industrial areas or other streets where operating speeds will be higher. A maximum superelevation rate of 4.0 percent or 6.0 percent is commonly used. The maximum curvature for a given design speed is defined for low-speed urban streets when both the maximum superelevation rate and the maximum allowable side friction factors are utilized.
Maximum Comfortable Speed on Horizontal Curves
Figure III-18 and Table III-15, for low speed urban streets, are derived from the simplified curve formula:
Figure III-18 has been prepared by using the recommenced values of for