## grade, curve, intersection, tangent, vertical, change and equal

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The difficulty of adjusting the grades at an intersection is considerably increased if the two streets do not intersect at right angles. It is impossible to formulate any general rule, since each case must be decided according to the local conditions; and since close observation and good judgment are required to secure a reasonably satisfactory adjustment.* Notice that if either street has a grade and is carried past the intersection nominally unchanged, the area between the four curb corners and that immediately adjacent will be a warped surface. For example, in Fig. 88, if the street S has a descent as indicated and the street W is level, and the unchanged crowns of the street intersect at C, the area marked w must be raised to carry the upper side of the street W over the intersection, and the portions marked v must be raised to carry the street S over the lower side of the street W. If the grade of either street is small. this adjustment can be made by "warping in," or "bon ing in " the surface for a short distance by the eye.

## Vertical Curves at Grade Intersection.

It is frequently claimed that the grade should be carried straight through from street intersection to street intersection, i. e., that the grade should not be broken in the block. Apparently the reason for this prac tice is the claim that a break of grade between streets is unsightly. As usually put in, the angle of intersection is simply rounded off a little by eye; and if the change of grade is considerable, the appearance is not good. A change of grade in the block is nowise different from a change at the street intersection, except that the former is a little niore conspicuous. For both appearance and the comfort of the traffic, wherever there is considerable change of grade the two grade lines should be connected by a vertical curve; and if this is properly done, a break of grade in the block or elsewhere is unobjectionable. A vertical curve should be in serted at a change of grade either of the pavement or of the curb.

By breaking grade in the block, it is possible to fit the grade line more closely to the natural surface, and thereby to decrease the cost of construction, to lessen the damage to abutting prop erty, and to improve the general appearance of the street.

A parabola is the best form for a vertical curve and is most easily put in. In Fig. 89, A B and A C represent two grade lines meeting in the apex A, joined by the vertical parabola B C, which is tangent to the straight grade line at B and C. The curve may be located by measuring ordinates vertically below the points 1, 2, 3, etc. The tangent distances A B and A C are equal. D E is equal to the rise in half the length of the curve, i. e., from B to A; and E C is equal to the fall in the second half, i. e., from A to C. If n represents the number of equidistant points to be established on the curve (including the second tangent point, C), then the ordinate at the first point The ordinate at any other point is equal to x times the square of the number of equal divisions between B and that point; that is, the ordinate from 2 is 4x, from 3 is 9x, from 4 is 16x, from 5 is 25x, and from 6 is 36x. In actual work, the grade elevation of the points 1, 2, 3, etc., are to be worked out in the usual manner; from these eleva tions subtract the ordinates as computed above, and the remainder is the grade elevation of the respective points on the parabola B C.

The agreement of the elevation of the last point on the curve, 6 in Fig. 89, with the point C on the tangent, checks the work of com puting the elevations.

If the second tangent, A C, is level, EC the above value for x is 0; and if the second tangent has an up grade, E C is minus, and the numerator=D E-E C. If the first tangent is level, DE = 0; and if the first tangent has a down grade, DE is minus, and the numerator = E C - D E. The principles deduced for Fig. 89 are equally true, if that diagram be turned upside down.

To secure the best results, there should be 15 feet of curve for each per cent of change of grade, although 10 feet per degree will give fair results. Long vertical curves make a graceful street. The effect of any proposed curve in lowering (or raising) the apex can be judged of beforehand by remembering that the distance from the apex A, Fig. 89, to the curve is equal to half of the differ ence in elevation between A and the mean of the elevations of B and C.

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