VERTICAL CURVES - 14071

familiar enough with field notes to preclude a com- plete discussion of everything shown in these notes. You should notice, however, that the stations are entered in reverse order (bottom to top). In this manner the data is presented as it appears in the field when you are sighting ahead on the line. This same practice applies to the sketch shown on the right-hand page of the field notes. For information about other situations involving inaccessible points or the uses of external and middle ordinate distance, spiral transitions, and other types of horizontal curves, study books such as those men- tioned at the beginning of this chapter. VERTICAL CURVES In addition to horizontal curves that go to the right or left, roads also have vertical curves that go up or down. Vertical curves at a crest or the top of a hill are called summit curves, or oververticals. Vertical curves at the bottom of a hill or dip are called sag curves, or underverticals. GRADES Vertical curves are used to connect stretches of road that go up or down at a constant slope. These lines of constant slope are called grade tangents (fig. 11- 13). The rate of slope is called the gradient, or simply the grade. (Do not confuse this use of the term grade with other meanings, such as the design Figure 11-13.—A vertical curve. elevation of a finished surface at a given point or the actual elevation of the existing ground at a given point.) Grades that ascend in the direction of the stationing are designated as plus; those that descend in the direction of the stationing are designated as minus. Grades are measured in terms of percent; that is, the number of feet of rise or fall in a 100-foot horizontal stretch of the road. After the location of a road has been determined and the necessary fieldwork has been obtained, the engineer designs or fixes (sets) the grades. A number of factors are considered, including the intended use and importance of the road and the existing topography. If a road is too steep, the comfort and safety of the users and fuel consumption of the vehicles will be adversely affected; therefore, the design criteria will specify maximum grades. Typical maximum grades are a 4-percent desired maximum and a 6-percent absolute maximum for a primary road. (The 6 percent means, as indicated before, a 6-foot rise for each 100 feet ahead on the road.) For a secondary road or a major street, the maximum grades might be a 5-percent desired and an 8-percent absolute maximum; and for a tertiary road or a secondary street, an 8-percent desired and a 10-percent (or perhaps a 12-percent) absolute maximum. Conditions may sometimes demand that grades or ramps, driveways, or short access streets go as high as 20 percent. The engineer must also consider minimum grades. A street with curb and gutter must have enough fall so that the storm water will drain to the inlets; 0.5 percent is a typical minimum grade for curb and gutter (that is, 1/2 foot minimum fall for each 100 feet ahead). For roads with side ditches, the desired minimum grade might be 1 percent; but since ditches may slope at a grade different from the pavement, a road may be designed with a zero-percent grade. Zero-percent grades are not unusual, particularly through plains or tidewater areas. Another factor considered in designing the finished profile of a road is the earthwork balance; that is, the grades should be set so that all the soil cut off of the hills may be economically hauled to fill in the low areas. In the design of urban streets, the best use of the building sites next to the street will generally be more important than seeking an earthwork balance. COMPUTING VERTICAL CURVES As you have learned earlier, the horizontal curves used in highway work are generally the arcs of circles. But vertical curves are usually parabolic. The 11-12