Surveying Fundamentals for the FE Exam
Surveying provides the spatial data needed for engineering design and construction. On the FE Civil exam, surveying questions test your ability to work with distances, angles, elevations, and coordinates. These problems are computational and formulaic, making them very manageable with preparation.
Distance Measurement
Distances can be measured directly (taping, electronic distance measurement) or indirectly (stadia, triangulation). Key considerations include:
- Horizontal distance: The distance measured along a horizontal plane. Most engineering calculations use horizontal distances.
- Slope distance: The distance measured along the slope. Must be corrected to horizontal: H = S * cos(alpha), where S is slope distance and alpha is the vertical angle.
- Tape corrections: Include temperature correction, tension correction, and sag correction.
Angle Measurement
- Horizontal angles: Measured in the horizontal plane between two lines of sight.
- Vertical angles: Measured from the horizontal plane (zenith angles measured from directly overhead).
- Bearings: Angles measured from north or south toward east or west (e.g., N 45 degrees E, S 30 degrees W).
- Azimuths: Angles measured clockwise from north, ranging from 0 to 360 degrees.
Converting Between Bearings and Azimuths
- NE quadrant: Azimuth = bearing angle
- SE quadrant: Azimuth = 180 - bearing angle
- SW quadrant: Azimuth = 180 + bearing angle
- NW quadrant: Azimuth = 360 - bearing angle
Leveling
Leveling determines elevation differences between points. The basic method uses a level instrument and a leveling rod:
- Backsight (BS): Rod reading on a point of known elevation.
- Foresight (FS): Rod reading on the point whose elevation is being determined.
- Height of instrument (HI): HI = elevation of known point + BS
- Elevation of new point: Elevation = HI - FS
For a leveling circuit (returning to the starting point), the sum of backsights minus the sum of foresights should equal the known elevation difference. Any discrepancy is the closure error, which should be within acceptable limits.
Traverse Calculations
A traverse is a series of connected survey lines forming a path. Two types:
- Open traverse: Does not close back to the starting point. Cannot be checked for accuracy.
- Closed traverse: Returns to the starting point or connects to another known point. Allows error detection and adjustment.
Latitude and Departure
For each traverse leg:
- Latitude = L * cos(bearing) (north-south component)
- Departure = L * sin(bearing) (east-west component)
For a closed traverse, the sum of latitudes and the sum of departures should each equal zero (or match the known closure). The actual errors are the closure in latitude and closure in departure.
Error of Closure
The linear error of closure is: E = sqrt(error_lat^2 + error_dep^2)
The relative precision is: 1 : (total traverse length / E). A typical acceptable precision for engineering surveys is 1:5,000 or better.
Area Calculations
The area enclosed by a closed traverse can be calculated using the coordinate method (also called the shoelace formula):
2A = sum of (x_i * y_(i+1) - x_(i+1) * y_i) for all vertices
This method works for any polygon shape and is the most common approach on the FE exam.
Coordinate Geometry (COGO)
Coordinate geometry calculations are used to determine positions, distances, and angles from coordinate data:
- Distance between two points: D = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Bearing from point 1 to point 2: tan(bearing) = (x2 - x1) / (y2 - y1). Adjust the quadrant based on the signs of the differences.
- Inverse problem: Given two coordinates, find the distance and bearing between them.
- Forward problem: Given a starting coordinate, distance, and bearing, find the ending coordinate.
Topographic Mapping
- Contour lines: Connect points of equal elevation. Closer spacing indicates steeper terrain.
- Contour interval: The elevation difference between adjacent contour lines.
- Interpolation: Estimate elevations between contour lines using linear interpolation.
Earthwork Volumes
Earthwork volumes (cut and fill) are calculated using cross-sectional areas:
- Average end area method: V = (A1 + A2) * L / 2, where A1 and A2 are cross-sectional areas at each end and L is the distance between them.
- Prismoidal formula: V = L * (A1 + 4*Am + A2) / 6, where Am is the area at the midpoint. More accurate than the average end area method.
Surveying problems on the FE exam are straightforward once you know the formulas and procedures. Practice traverse calculations and leveling problems until they become routine.