CIVIL ENGINEERING FORMULAS

182 CHAPTER SEVEN

whereM,F, and Kare distances in miles, thousands of feet, and kilometers,
respectively, from the point of tangency to the earth.
Refractioncauses light rays that pass through the earth’s atmosphere to
bend toward the earth’s surface. For horizontal sights, the average angular dis-
placement (like the sun’s diameter) is about 32 min. The displacement Rf, ft, or
Rm, m, is given approximately by

(7.19)

(7.20)

To obtain the combined effect of refraction and curvature of the earth, subtract
RffromCforRmfromCm.
Borrow-pit or cross-section levelingproduces elevations at the corners of
squares or rectangles with sides that are dependent on the area to be covered,
type of terrain, and accuracy desired. For example, sides may be 10, 20, 40,
50, or 100 ft (3.048, 6.09, 12.19, 15.24, or 30.48 m). Contours can be located
readily, but topographic features, not so well. Quantities of material to be
excavated or filled are computed, in yd^3 (m^3 ), by selecting a grade elevation or
final ground elevation, computing elevation differences for the corners, and
substituting in

(7.21)

wherennumber of times a particular corner enters as part of a division block
xdifference in ground and grade elevation for each corner, ft (m)
Aarea of each block, ft^2 (m^2 )

VERTICAL CONTROL

The NGS provides vertical control for all types of surveys. NGS furnishes
descriptions and elevations of bench marks on request. As given in “Standards
and Specifications for GeodeticControl Networks,” Federal Geodetic Control
Committee, the relative accuracy C, mm, required between directly connected
bench marks for the three orders of leveling is

First order: for Class I and for Class II (7.22)

Second order: for Class I and for Class II (7.23)

Third order: (7.24)

whereKis the distance between bench marks, km.

C2.0 2 K

C1.0 2 K 1.3 2 K

C0.5 2 K 0.7 2 K

Q

nxA

108

Rm0.011K^2

Rf0.093M^2 0.0033F^2