“Primer” is my personal reference notebook. Building codes and design guides are often dense and voluminous. These auto-didactic notes help me collect, organize, condense technical topics I’m learning as an engineer. I like to think of ‘primer’ as an extension to my brain whose storage capacity is sadly all too limited.
Please understand I made these for myself. Reader discretion is advised. No warranty is expressed or implied by me on the validity of the information presented herein.
This is my study note which summarizes the content of Paul A. Cuomo’s “Surveying Principles for Civil Engineers” textbook. I highly recommened getting a physical copy, as it would be prudent for you to have it during the exam. Amazon Link
1.0 California Surveying Law
1.1 Types of Surveys
Cartographic - rough hand sketches
Boundary - property lines
Geodetic - planetary wide perspective. Shape of the earth.
Hydrographic - underwater land characteristic (existing body of water)
Topographic - land elevation and objects. Conducted fairly early on prior to construction documents
Route - highways, pipes (parallel and perpendicular measurements)
Construction - field control during construction with stakes
As-built - conducted after construction for record-keeping (final survey)
Control - establish location of benchmarks
1.2 Survey Party
Party chief - team leader
Instrument person - often note keeper as well
Rod person - holds rod plumb
Chain persons - pair stretching tape between them
Stake setter - drives stakes into ground per instrument or rod person
1.3 California Law
Professional Land Surveyors Act
Defines licensure requirements and roles of a surveyor
Professional Engineers Act
Defines licensure requirements for civil engineer and their allowed role
Subdivision Map Act
Defines procedure for boundary surveys (property lines) and how they are recorded by local municipality
Parcel map = large parcel of land divided into (4) or less divisions
Subdivision Map = large parcel of land divided into (5) or more divisions. AKA final map.
Tentative Map - before developers submit their plans. Preliminary maps
Licensed surveyor and Civil Engineer licensed BEFORE 1982 can do everything
Civil engineer registered after 1982 CAN perform:
contour surveys (aerial survey, topographic)
survey of fixed work
tunnel survey
tentative maps
Civil engineer registered after 1982 CANNOT perform:
geodetic surveying (taking into account Earth’s curvature)
cadastral surveying (records of property lines)
survey for land division
determine position of anything that establishes a property line
Basically any sort of benchmark setting
2.0 Horizontal Measurements - Distance
2.1 Basic Trigonometry
Vertical angle = angle with respect to horizontal line (\(\alpha\)). Think of it as a projection of our vector onto the horizontal line.
\[H = S \cos \alpha\]
Zenith angle - angle with respect to vertical line (\(z\)). Note you can use angle greater than 90 degrees directly because sin(z) = sin(90+z)
\[H = S \sin z\]
Figure 1: Horizontal Measurement Trigonometry
2.2 Taping and Chaining
Taping and chaining involves two people measuring horizontal distance with Gunter’s chain (66 ft) or a steel tape. Try to measure horizontal distance directly without using any trigonometry (use a plumb bob for leveling)
Here are some useful unit conversions:
Figure 2: Common Unit Conversion
1 m = 3.28 ft
1 mile = 5280 ft
1 hectare = 2.472 acre
1 acre = 43560 ft^2
1 cubic yard = 27 ft^3
There are three types of tapes: 1.) add tape, 2.) cut tape, and graduated tape. Their main difference is how the last feet is measured and calibrated (see figure below for an illustration. The rear chain-person holds the roll of tape. The forward chain-person holds near the zero mark.
Figure 3: Different Type of Tape
Temperature correction can be applied with the following equation. Let c = the correction amount. To = standard temperature = 68o>F unless otherwise noted.
If T > To, the tape is hot and expanded which reads smaller, c is positive, real distance is actually larger
If T < To, the tape is cold and contracted which reads bigger, c is negative, real distance is actually smaller
Tape Sag Adjustment (always negative - actual length is shorter)
w = weight of tape (lbs/ft or kg/m)
P = pull force (lbs or kg)
Don’t forget to divide by length (lbs/ft units!)
\[C_s = \frac{w^2L^3}{24P^2}\]
Tape Pull Adjustment
Po = standardized pull tension (usually around 15 lbs)
A = cross section area of tape
E = elastic modulus of steel (30 000 000 psi)
\[C_p = \frac{(P-P_o)L}{EA}\]
When tape is measured at an incline. Make sure to correct for S, before calculating H
Pay attention to units for the pull adjustment! Denominator can stay in inches because in^2 and psi cancels out
For distance measurements exceeding tape length, need to add the adjustment from all measurements. For example, 450 ft, we will have (4) adjustment with 100 ft and (1) adjustment with 50 ft
2.3 Stadia Measurement
Measuring distance with a lens w/ cross-hair and a rod. Distance can be indirectly deduced by reading the gap between cross-hair. Usually K = 100 ft and C is 0 ft unless otherwise noted.
\[distance = KI + C\]
I is the crosshair reading as illustrated below:
Figure 4: Stadia Cross Hair
Very rare for reading to be level, more often you the scope is slanted and need to adjust measurements. Referring to the figure below:
Figure 5: Inclined Stadia Measurement
Reading line AB = I, which is longer than actual because slanted scope
Convert to the effective rod intercept by first determining the vertical angle. Note how the vertical angle corresponds in the figure above (\(\alpha\))
\[I_1 = I \cos(\alpha)\]
Now the actual slanted distance (S) can be determined as we’ve shown before:
\[S = K I_1 + c = KI \cos(\theta) + c\]
Finally let’s convert to horizontal and vertical distance. Note vertical distance is to center of rod reading.
\[H = S \cos(\alpha) = (KI \cos(\alpha) + c) \cos(\alpha)\]
\[V = S \sin(\alpha) = (KI \cos(\alpha) + c) \sin(\alpha)\]
2.4 Electronic Distance Measuring (EDM) and Total Station
EDM is essentially the high-tech version of stadia that is superior all around. Distance is measured by measuring wavelength of light as it is reflected. Occasionally, some error adjustment is required:
Constant prizm offset. Reduce distanced measured by 30 mm to 40 mm (because of internal reflector distance)
Varying atmospheric pressure adjustment (measured in ppm)
\[c = L_{measured} \times \frac{ppm}{1000000}\]
Total station is the even more advanced version that allows for measurement of all distances and angles. No need to take notes when you have a total station.
3.0 Vertical Measurements - Elevations
3.1 Conceptual Stuff
All elevation is measured with respect to a datum. In the US, that datum is based on the Mean Sea Level 1929 Adjustment (MSL29) and was updated to NGVD 88
Here are some commonly used terminologies:
leveling circuit - our instrument which has a telescope for rod readings. Often rotation is restricted to only vertical axis (i.e. always horizontal)
(RR) rod reading
(HI) instrument height - usually refers to elevation at sightline. But may also refer to height of equipment itself
(BS) backsight - RR at the previous point of measurement. associated with plus (+)
(FS) foresight - RR at the next point of measurement. associated with minus (-)
(BM) benchmark - a location where we know the exact elevation (serves as control and error correction)
(TBM) temporary benchmark - not official, usually used by individuals for temporary purposes
(TP) turning point - point whose elevation is determined prior to moving the level
Our instrument does not have to be placed in line of FS and BS (rotating 180 degree) each time. But our equipment should always be set up with equal distance to BS and FS to minimize curvature, collimation, and refraction errors. Having equal distance means the error could balance out.Curvature error arises because the surface of Earth is curved, but our line of sight is straight.
If the above is not possible (say that places our instrument in the middle of a river), then use reciprocal leveling
Figure 6: Leveling Instrument
Referring to the figure above, the procedure for leveling is as follows:
We know elevation at point X is 100 ft
From RR of BS, we read 4.5 ft
=> 100 + 4.5 = 104.5 ft is the elevation of HI
\[HI = elev_{BS} + BS\]
Now rotate to the FS, we read RR of FS as 1.5 ft. Subtract this value to get elevation of FS.
=> 104.5 - 1.5 = 103 ft is the elevation at Y
\[elev_{FS} = HI - FS\]
Move instrument, FS becomes the new BS, repeat until done
Precise Leveling - high order of accuracy. Used to furnish low-order work. Covers large area. More precise equipment is used.
3.2 Indirect Leveling With Stadia
Use stadia as we have seen in the last chapter, but calculate “V” this time. Then add/subtract by given elevation
Often we need to adjust by add/subtracting rod and instrument height:
3.3 Leveling Rod
Rods can extend up to 13 ft, with 0 at the base. There are two types:
self-reading rod - Philadelphia rod (most popular)
self-reducing rod - Lenker rod. No notes need to be taken. Set zero at HI. Value increases from top to bottom
Figure 7: Type of Leveling Rod
3.4 Direct Leveling Note Example
Here is a sample note taken from the field. Note the sequence of calculation and how BS is associated with (+), and FS is associated with (-). Also note the summation check at the end to ensure no adjustment is needed.
Negative (-) error means too short. If the summation above does not calculate out, we need to smear the error in the “adjusted elevation column” as shown above.
3.5 Profile or Cross-Section Leveling
Involves obtaining elevation along a perpendicular line (used to determine a cross-section). The surveying note will now have a new column for side shots (SS)
\[elev_{SS} = HI - SS\]
Figure 9: Profile Leveling
Note we can skip the intermediate SS when calculating TP1. Cannot skip if there is a BS recording
3.6 Reciprocal Leveling
Occurs when instrument cannot be placed equi-distance between FS and BS (such as in the middle of a river). We are essentially averaging out the error.
Calculate ELEV with instrument 1, calculate ELEV with instrument 2. Average the results.
Figure 10: Reciprocal Leveling
4.0 Angle Measurements
4.1 Conceptual Stuff
Three types of angle measurements. Internal and exterior angle are shown below. Deflection angle is how much the equipment has rotated.
Figure 11: Types of Angle Measurements
4.2 Bearing
Bearing = angle with respect to 4 quadrants. The reference meridian is usually North. The format of a bearing measurement is usually in the form of (North or South)(rotate toward)(West or East)(e.g. N35o>E)
Interestingly, these quadrants are different from standard mathematics
Reverse bearing = The bearing that is parallel but points in the opposite direction is called a
Read questions carefully:
\[Bearing_{BA} \neq Bearing_{AB}\]
Figure 12: Bearing Angles
The reference meridian can be of three types:
True - parallel to line drawn from north pole to south pole. Same everywhere on Earth at any point in time
Magnetic - taken by compass. Not equal to true meridian! Magnetic pole is constantly changing and depends on where you are on Earth
Grid - An imaginary grid is superimposed on to a map. Angle references this grid
Magnetic declination is the difference between True meridian and Magnetic meridian
Agonic line = Line where magnetic meridian = true meridian
Isogonic line = The “topographic” line that shows the same magnetic declination is called
declination = angle from TRUE to MAG
\[TRUE = MAG + DECLINATION\]
Figure 13: Isogonic Contours
4.3 Azimuth
Azimuth is a measurement that starts North and rotates CLOCKWISE ranging from 0 to 360 degrees.
Figure 14: Azimuth
Same as above, Azimuth can either be 1.) true, 2.) magnetic, or 3.) grid.
A common mistake is thinking 90 degrees = 180 degrees…
4.4 Traverses
Traversing is the task of relating points through angles and distances. There is Open Traverse and Closed Traverse
Figure 15: Open and Closed Traverse
For closed traverse, the internal angles must add up to the following where n = number of legs (e.g. triangle = 3)
It is impossible to measure things exactly without error. Sources of error can be systematic or random. Here is a list of potential sources:
Procedure shortcomings
Care exercised by surveyor
Equipment calibration frequency
Weather
Level of Accuracy is the measure of uncertainty regarding the survey overall. For any survey, you can exercise an extreme amount of care. In order to establish uniformity and prevent oversurveying (and reduce cost), Caltrans Survey Manual provides detailed specifications for classification of accuracy
Figure 16: Caltrans Accuracy Standards
Higher order survey is more accurate, but also take longer and is more costly.
Accuracy => how close your observed value is to the true value
Precision => how close together your repeated measurements are to each other (could be precise but inaccurate)
5.1 Data and Error
The figure below shows some example angle measurements.
Figure 17: Error in Measurements
Mean of the measurements can be calculated as shown:
\[\mu = \frac{\sum X_i}{N}\]
Where \(X_i\) is the sum of all measurements, N is the number of measurements
For multiple angle measurements, simply average the seconds if degree and minute is unchanged.
Otherwise, add degree, minute, second separately, then use calculator function to automatically aggregate them. In the example above => 1600 degrees, 240 minutes, 579 seconds, then convert => (1604,9,39)
Residual is the difference between each measurement and the mean (note that residual can be negative). Also sometimes referred to as deviation.
\[r = X_i - \mu\]
Standard Deviation is a measure of dispersion of data (how flat the bell curve is). One standard deviation of the mean value contains 68.27% of the sample. Variance is defined as \(\sigma^2\)
\[\sigma = \pm \sqrt{\frac{\sum r^2}{N-1}}\]
Figure 18: Standard Deviation
Standard Error of the Mean is a measure of uncertainty regarding the mean and is calculated as a function of sample size and standard deviation. If our sample is the entire population, then the standard error of the mean is zero. Smaller sample size = more uncertainty, and vice versa.
\[\sigma_m = \pm \frac{\sigma}{\sqrt{N}}\]
In the example above, we had 16 measurements and the standard error of the mean is \(\pm 1\)”, which means our average value is about 1” plus or minus from the true value. If we made 1,000,000 measurements, we would get even closer to the true value
This is what we usually referred to when we say we are uncertain to \(\pm x\)”
5.2 Groups of Measurements
Say you have a traverse with 4 courses. Each course you took 20 measurements. In this case:
Number of groups = 4
N of each group = 20
The total standard deviation of the entire group is calculated as:
Example 1: A party chief and his chainman can read the tape end to \(\pm 0.01\) ft with a 100 ft tape. What is the uncertainty in a 1000 ft distance measurement? Note that each tape reading occurs “twice” (one at each end)
Note that if \(n = 1\), then \(\sigma_m = \sigma = \pm 0.01\) ft
But since the tape reading rely on two readings, the standard deviation needs to be modified slightly: \(\sigma_m = \pm 0.01\) ft and \(n = 2\), thus \(\sigma = \sigma_m \sqrt{n} = \pm 0.014\) ft
For a series of 10 grouped measurements: \(\sigma_{sum} = \sqrt{0.014^2 \times 10} = \pm 0.04\) ft
Example 2: Three distance segments measured had uncertainty of +-0.05 ft. Another four distance segments measured had uncertainty of +- 0.1 ft. What is the total uncertainty in the seven segments?
Example 3: Error of computing an area. Let L and W be the length of width of a rectangle
\[error = \pm \sqrt{L^2 e_L^2 + W^2 e_W^2}\]
6.0 Traverses
6.1 Conceptual Stuff
Traverse is simply measuring distance and angles. Here are some definitions that you should know:
bearings - horizontal angle measured
headings - horizontal distance measured
Course or Leg - course or leg of a traverse is described by a distance and an angle. The angle may be provided in bearing or azimuth
Figure 19: Course or Leg
Departure - change in X-direction
Latitude - change in Y-direction
Figure 20: Departure and Latitude
Here is a tip for quicker angle calculation. Think of cosine as projection. Since bearing is always provided with respect to vertical axis (N-S), the following always holds true:
\[latitude = \Delta Y = L \cos(\alpha)\]
\[departure = \Delta X = L \sin(\alpha)\]
Also make sure you assign the correct sign(+,-). If using azimuth, the sign is automatically taken care of.
Given departure and latitude, the overall course length can be calculated using Pythagorean Theorem (easiest), or any trig ratios (sin/cos/tan). This is called inversing the course.
\[L = \sqrt{\Delta X^2 + \Delta Y^2}\]
6.2 Closed Traverse
Closed traverse involves several courses that ultimately return to the starting point. The algebraic sum of departure and latitude should be zero. But this will never be the case no matter how careful the surveys performed. Error correction will need to be performed.
Figure 21: Closed Traverse Notes Example
The procedure for balancing a closed traverse is lengthy and involves 1.) first balancing bearing angle with interior angle 2.) then balancing departure and latitude, 3.) then balancing bearing angle again along with length based on adjusted dep. and lat. The procedure is as follows:
First balance the interior angles \((n-2)\times 180^o\)
Revise bearing of each course based on adjusted interior angle (adjusted bearing)
Compute departure and latitude based on adjusted bearing (uncorrected dep. & lat. based on adjusted bearing in the previous step)
Compute error in departure (dx) and latitude (dy)
Adjust departure and latitude using compass rule (adjusted dept. & lat.)
Finally recalculate length and bearing angle with the adjusted departure and latitude (length and bearing recalculated)
There are three methods for error adjustment of closed traverses:
transit rule - adjustment in proportion to latitude. Use when angular measurement is considered more reliable
compass rule - adjustment in proportion to leg distance. Use when angular and linear measurements have equal reliability. Regarded as superior to transit rule.
Often, a coordinate (x,y) is assigned to the starting point. From the departure and latitude measurements, the coordinate of every other point can be determined. The process is simple but pay attention to signs! And:
\[departure = \Delta X\]
\[latitude = \Delta Y\]
6.5 Enclosed Area - DMD Method
Closed traverses must be balanced before calculating its area. The most common method is called Double Meridian Distance (DMD) method. Area is calculated by summing the product of latitude and DMD for each course.
\[A = \frac{\sum lat_i \times DMD_i}{2}\]
Reference meridian (a vertical line) is drawn at the most westerly point
Meridian Distance (MD) is the distance between this vertical line, to the mid point of any course. Refer to the figure below
Make sure to calculate DMD for each course separately to avoid having error propagate through your calculation
Divide by 43560 to convert from ft2 to acres
Figure 24: DMD Method for Calculating Enclosed Area
6.6 Enclosed Area - Criss Cross Method
Another more numerically convenient way of calculating enclosed area is the Criss Cross Method. But we need to get the grid (x,y) coordinates first before we can use this method. An example of this method is shown below.
Figure 25: Criss Cross Method for Calculating Enclosed Area
A couple of COMMON ERRORS when applying criss-cross method
Repeat the first point!
Notice that “y” is on the first column
Remember to divide by 2
Translate all points to the top right quadrant before computing
7.0 Topography
7.1 Conceptual Stuff
Topography - process of mapping elevations and physical features of a site in plan view. Also sometimes include annotations of natural or man-made features
Horizontal Control - known point of line such as property line, centerline of highway, streets, etc.
Vertical Control - known elevation/datum. Usually use the National Vertical Geodetic Datum (NVGD 1929) or the North American Vertical Datum of 1988 (NAVD 88)
Four methods:
Cross-section method - great for roads
Grid method - great for contour
Radial method - great for single point of interest
Radiation method - fastest
There is also photogrammetry which involve aerial photos. We will discuss this further in future chapters.
7.2 Cross-Section Method
Mostly used for roads. Cross sections for elevations are taken at right angles (offset lines). Refer to the figure below.
Figure 26: Cross Section Method
Baseline (horizontal control) is usually the proposed centerline of a road or highway. Features and other point of interest are fully described in terms of:
vertical (its elevation)
horizontal (station along baseline + offset distance to feature)
7.3 Grid Method
Method involves superimposing a grid onto an area of interest (divided into series of square regions). Refer to the figure below.
Figure 27: Grid Method
Determine elevation at every intersection point. Then enter the data into a software to generate contour plot
7.4 Radial Topographic Survey
Method commonly used to map a special point of interest. Start at a known point/elevation. Use stadia or total station an traverse to point of interest.
7.5 Reading Contour Maps
Contour maps show lines of equal elevation
The direction of a contour at any point is at right angles to the direction of steepest slope
Contour lines cannot cross each other
Slope (or gradient) can be determined by dividing elevation change by horizontal distance:
\[slope = \frac{\Delta elev}{\Delta x}\]
Profile is kind of like a section cut on the contour:
Figure 28: Contour Profile
The figure below shows some of the common features of contour plot that you should recognize.
Figure 29: Common Contour Features
8.0 Route Surveying
8.1 Conceptual Stuff
Route surveying usually involve two types of curves:
Horizontal curves - curves in direction (e.g. ramps of highway or bridge)
mathematically circular
straight portions are called tangents
Vertical curves - changes in elevation and slope
mathematically equal-tangent-parabolic
curve can be sag or crest
8.2 Horizontal Curves
Equation and elements of a circular horizontal curve is shown below:
Figure 30: Horizontal Curve
Some other equations:
Area enclosed within “E” region shaped like a boomerang:
\[A = R^2 (\tan(I/2) - I(\frac{\pi}{360}))\]
Area of polygon inscribed by R and T:
\[A = \frac{R}{2} \times T \times 2T\]
Area of curve sector:
\[A = \frac{I}{360} \pi R^2\]
Finding station of EC from PI
\[sta_{EC} = sta_{PI} - T + L\]
Note we CANNOT do sta_PI + T because length of curve is not equal to 2T!
Notations:
C = long chord length from start to finish (straight line)
D = degree of curvature
E = external distance
I (or \(\Delta\)) = interior angle, intersection angle
I/2 = full deflection angle for the long chord
L = length of curve
M = middle ordinate
PC (or TC or BC) = START. point of curvature, tangent-to-curve, beginning-of-curve
PI (or V) = point of intersection, vertex, intersection of tangents
POC = any point on the curve
PT (or CT or EC) = END. point of tangency, curve-to-tangent, end of curve
R = radius of curve
T = tangent distance. Semi-tangent. (see diagram)
Degree of Curvature
Degree of curvature (D) is a measure of rate of change of rotation. Usually it is stated mathematically as 1 over radius of curvature:
\[\kappa = \frac{1}{\rho}\]
However, we like to express curvature as “degree per station” rather than per feet. Note how there is a direct relationship between D and R (knowing one automatically reveals the other)
For railroad projects with very large radius, the chord basis may be preferred:
\[sin(\frac{D}{2}) = \frac{50 ft}{R}\]
Deflection Angle
In the process of laying out a road, construction stakes must be installed at BC and EC and along the road at full stations (sometimes half if turn is sharp).
Deflection angle is defined as the angle between the tangent and a chord (see figure below for clarification)
Figure 31: Deflection Angle
highlighted area can be calculated as area of sector minus the triangle
Another useful parameter is deflection angle per foot. To get the deflection angle at any station, simply multiply this value by the distance from BC to the point under consideration:
\[\delta /ft = \frac{I/2}{L}\]
From this exercise, we can sight the entire curve at BC by specifying how much to turn, and the corresponding chord distance.
Tangent Offset
Tangent offset can be calculated via the following
Figure 33: Tangent Offsets
8.3 Vertical Curves
Vertical curves are used to transition between grades. They can be symmetric or asymmetric. However, point V is always equidistant to PVC and PVT
Figure 34: Vertical Curves
Notations:
BVC, PVC = beginning of curve
EVC, PVT = end of curve
g1 = starting grade (%)
g2 = ending grade (%)
L = HORIZONTAL length of curve
M = middle ordinate (usually very small)
V, PI, PVI = vertex point
Since the grade typically very small. The chord distance (BVC to V and V to EVC) is essentially equal to the length of the curve.
Elevation of every point on the curve can be defined using a parabolic equation. A parabolic curve is completely defined by two grades and a horizontal length.
NOTE: r can be negative as is the case in crest curves.
If elevation at PVI is given, can simply use the straight grade lines to back-calculate elevation at BVC or EVC. Length to BVC or EVC is usually taken to be equidistant. Remember station is in horizontal distance.
\[y_{BVC} = y_{PVI} + g_1 L_1\]
If the curve is not symmetric, you may need to solve the length of curve on either side of PVI first:
Figure 35: Solving for BVC and EVC Location Given Different Slopes
A spreadsheet table could be set up to organize calculation:
Compound Horizontal Curve - two horizontal curve of different radius that lie on the same tangent. Analyze as two separate curves. All previous equations apply
Reverse Horizontal Curve - same as above but the second curve flips curvature
Spiral Horizontal Curve - Used to produce gradual transition from tangent to circular
TS = tangent to spiral
LS = length of spiral
SC = spiral to circle
\(\theta_s\) = interior angle of spiral
D = curvature of circle
Length of spiral recommended to be a function of approach speed of vehicles. V is in mph, R is in feet
\[LS = \frac{1.6 (V)^3}{R}\]
LS can also be calculated as:
\[LS = \frac{100 \theta_s}{D/2}\]
Deflection angle for setting out the spiral can be calculated as:
\[\alpha = \frac{\theta_s}{3} \times (L/LS)^2\]
Asymmetrical Vertical Curve - tangents are not equal. Analyze as two separate curves joined at the point of compound vertical curve (PCVC). Refer to page 67 of Cuomo text for more information and equations.
8.5 Estimating Earth Work
There are two common methods for estimating amount of soil to be cut or filled between stations. Remember to divide by 27 to convert to cubic yards.
Figure 37: Earthwork Calculation Cross Section Area
Conservative and quick method. Usually over-estimates the soil. Essentially average the two end areas multiply by length.
\[V = \frac{A_1 + A_2}{2} \times L \; [ft^3]\]
In some cases, one of the end area could be 0 which results in a pyramid rather than a prism. Use this formula instead if that is the case:
\[V = \frac{A_{base}}{3} \times L \; [ft^3]\]
Prismoidal Method
Figure 38: Prismoidal Method
If the ground is very irregular, or the two end areas differ drastically, the prismoidal method could be better. Essentially we interpolate a cross-section between the two ends, and a weighted average is taken. Usually produces smaller volume than average-end method.
\[A = c (\frac{d_L + d_R}{2}) + b (\frac{h_L + h_R}{4})\]
The formula above is a simplified express from the following sequence:
Effect of Shrinkage
Sometimes effect of shrinkage and swell are considered in earthwork calculations.
(1) Bank volume = natural state
(2) Loose volume = volume when its in the trucks, disturbed and loose
(3) Compacted volume = volume after compaction
Swell = (2) / (1)
Shrinkage = (3) / (1)
Mass Diagram
Mass diagrams are often produced to best plan hauling schedules.
The grade is represented as y(x), then the mass ordinate is the integration of y(x) that shows the cumulative volume. Let the width of a profile slice be b(x):
\[V = \int_{0}^{x} b(x) y(x) dx\]
9.0 Construction Staking
9.1 Stake and Laths
Stakes are placed by surveyor to guide contractors in laying out the construction project.
A topographic survey must be performed prior to staking to establish horizontal and vertical datum
In some cases, construction staking is critically important:
Determining earthwork quantities. Need to ensure correct vertical measurement. Very expensive mistake…
Determining elevation for drainage devices. Mistake in staking for concrete gutter (which only has 0.25% slope) could lead to inadequate drainage
Stakes (Ginneys) are usually small 1” by 2” spikes that is drilled into the ground.
Laths (Ground stakes) are the actual poles containing information. They are about 2” by 36” in dimension.
Figure 39: Stake and Lath
Fundamentally, the goal of construction staking is simple. Present information on a stick in the most concise and terse way possible. You should be familiar with some short-hands but in general just know there are often deviations from common practice and this isn’t exactly a science.
Some possible information shown on a stake (not exhaustive at all)
Station or line - horizontal location of the stake (e.g. 25+00, property line)
Offset - some stakes must be offset from the actual location because it would otherwise impede construction (e.g. 3.00 to curb face, 10 to cl pipe)
Cut (C) or fill (F) - indicates whether to excavate or fill or keep as is (grade) (e.g. C 1.75 to top of curb, F 1.96 to top of wall)
Elevation - Elevation at top of stake is marked on the lath (e.g. elev = 123.76)
9.2 Types of Stakes
Slope Stakes
Slope stakes - used to indicate earthwork involved in creating a nice slope. Here are some example interpretations:
Figure 40: Reading Slope Stakes
Some definitions:
2:1 => 2 Horizontal : 1 Vertical
1/2 => 1 Vertical / 2 Horizontal
40% => 40 Vertical / 100 Horizontal
Toe - bottom of slope
Top - top of slope
Catch Point - where slope intersect with existing ground
Hinge Point - Upper or lower limit of the slope, usually shoulder of a road (see next figure for illustration)
Slope Ratio - HORIZONTAL:VERTICAL
RPSS - reference point slope stake. The offset stake
Figure 41: Some Slope Definitions
Other Types of Stakes
Contour stakes - Used to demarcate contour lines
Rough grade stakes - Used to control dirt grade. Not very important and is only good to nearest 0.10 ft
Blue tops - Stakes set at a predetermined elevation. Also called “finished grade stakes”
Building layout stakes - Offset stakes used to demarcate building footprint
Sewer or storm drain stakes - Offset stake to demarcate sewer or storm drain lines. Usually offset by about 10 ft. Otherwise could be damaged or moved during construction
Wall stakes - Offset stake used to demarcate retaining wall
Curb stakes - Offset stake for curbs. Usually smaller offset (3 ft) and placed fairly close to each other (25 ft maybe closer for tighter bends). They are a drainage device so care must be taken
Catch basin (manhole) stakes - Offset stakes. Again must take extreme care because it is a drainage device and portions of it is cast monolithically with the curb
10.0 Photogrammetry
10.1 Aerial Photogrammetry
Aerial photogrammetry involves taking measurements of site using photos taken on an airplane. Result is a topographic map with contours, elevations, and features.
Quick but not as precise
Does not work in wooded area
Since multiple photographs must be taken, they need to be stitched together (known as a stereo photo)with a stereoplotter
Therefore, photos must overlap a specific amount. Overlapping photographs are called models
In order have proper map scale. A series of control network should be established on the ground.
AERIAL PHOTOGRAMMETRY: Minimum of (3) vertical control not in a straight line, and (2) horizontal control in overlap. Suggest using (5) each for redundancy
GPS control survey: 3H minimum spread out (4 for kinetic GPS), 2V minimum on opposite ends.
Control point is known as a monument and should be large and clear. Usually a large cross painted on the ground. Length of cross hair should be at least 1/1000. So for a 1:7200 scale => 7.2 ft
Figure 42: Control Point Cross Hair
Alternatively, can avoid control point by using existing landmarks such as curbs, sidewalks, prominent rocks, etc. This is called post-marking.
Contours can be generated using elevation models and interpolation
10.2 Photo Scale
Scales are expressed in two forms. Note that the two are equivalent. Convert feet to inches to go between the two.
1 in = 250 ft
1:3000
REMEMBER to multiply by 12. For example:
\[\frac{6 in}{7900 ft} = 1 in : 1317 ft = 1 in : 15800 in = 1:15800\]
Scale of the photo (S) is dependent on two factors:
Camera focal length (F)
Flight altitude (H)
Figure 43: Photo Scale in Aerial Photogrammetry
For most project, camera focal length is fixed (typically 6 in focal length with 93 degrees field of view). Thus the variable is flight height (H).
To stitch two photos together, an object must show up in at least two photos (i.e. overlap). How much overlap should be provided? The common practice is to use the neat model which provides:
A intervalometer is used to automatically trigger the camera at pre-determined intervals.
At the start of most projects, the designers must provide a desired map scale (S) and contour interval (CI). Tighter CI usually means more detail is desired. The ratio between flying altitude and CI is known as C-factor
\[\mbox{C-factor} = \frac{H}{CI}\]
Or CI and C-factors are given and you need to find the flying height. Then knowing the focal length, we can calculate the photo scale.
\[H = CI \times \mbox{C-factor}\]
\[S = \frac{f}{H}\]
There is another factor called the D-factor which is just the ratio of photo scale to map scale. Most plotters have a D factor between 5 to 8.
\[\mbox{map scale} = \frac{S}{\mbox{D-factor}}\]
10.4 Flight Planning
Figure 45: Flight Planning
In the project area shown above, set up required flight information given the client wants 4 ft contours. The plotter will have a C-factor of 1500. Camera will have focal length of 6 in:
Determine flying height (H)
\[H = CI \times \mbox{C-factor} = (4)(1500) = 6000 ft\]
Photogrammetrist will plan a control scheme, which is the ideal location to place monuments.
Figure 46: Placement of Control Points
The flying height we calculated before (H) is the height above mean terrain (AMT). Another useful height to calculate is the height above sea level (ASL):
\[ASL = AMT + \mbox{elevation of mean terrain from sea level}\]
The following standards of accuracy must be met when using aerial photogrammetry: (note: compare scale by just using the decimal representation)
Vertical:
Contour - half contour interval error in not more than 10% of points tested
Spot elevation - 90% of all spot elevation shall be accurate to within 1/4 of contour interval
Horizontal:
Scale larger than 1/2000 (e.g. 1/100), 1/30 of an inch error in not more than 10% of points tested
Scale smaller than 1/2000 (e.g. 1/10000), 1/50 of an inch error in not more than 10% of points tested
Note (1:100) > (1:12000). Calculate fraction. Don’t just look at denominator
Larger Scale = More Zoom!
Small Scale (e.g. 1:12000)
Map of USA
Large Scale (e.g. 1:100)
Map of San Francisco
Appendix: Miscellaneous Stuff
Angle Conversion
Degree, Minute, Seconds Formula. Use built-in calculator function instead of calculating manually
\[decimal = degree + minute/60 + second/3600\]
To convert the other way, for example 58.666 degrees:
Robotic total station - tracks prism automatically and can be operated by one person
Real Time Kinematic GPS - Can be operated by one person via rover receiver. Commonly used for construction surveying. Static GPS is more accurate but too slow (about 5 min per point). Static has 0.02 ft accuracy whereas RTK has 0.1 ft accuracy.
CORS point - permanently recording GPS station
GPS Errors. Weather cannot impact GPS precision but atmospheric conditions does. Refraction and electromagnetic interference does! Dilution of precision occurs if satellites are too close to each other.
GIS data type: raster = img, vector = coordinates, metadata = data about the data
Digital orthophoto = photo image in which every pixel is in its correct map position
Aerial Triangulation = process of determining the ground coordinates of artificial points marked on the emulsion of an aerial photo
Planimetric Map = plan view of any drawing. Does not contain any elevation information
Standards
Lambert Conformal Conic - projection of California state plane system
US survey foot was introduced in the North American Datum of 1983 (NAD83) because the typical foot was not accurate enough in the NAD 1927. 3937/1200 is more accurate than 10 ft = 3.048 m. In the past, the baseline was based on Gunter Chains.
Metes and Bounds Description = legal description involving complete perimeter of a piece of land (bearing and length)
Legal definition of property = depends on intent of the parties to a conveyance
NVGD 29 vertical datum (old) was adjusted to fit 26 mean sea level stations. Used prior to 1991
NAVD 88 vertical datum (new) held one point fixed – Father Point Rimouski. Fixed to a tidal benchmark in Quebec. Supersedes NVGD 29
NAD adjustments were purely horizontal and does not affect stationing which is relative
Underground Utilities
Potholing is the most accurate way of finding underground utilities. Conduct field investigation first. Potholing usually occurs later on
Underground Service Alert (Dig Alert) - a non-profit that will help identify location of underground utilities with spray paint. Not depth information though
A pipe has three different elevations:
Invert elevation = bottom inner
obvert elevation = top inner
crown elevation = top outer
Miscellaneous
Map scale of 0.99 means 100 ft indicated is actually 100/0.99 = 101 ft
Vertical control accuracy listed from most accurate to least:
Global regional geodynamics
Basic framework
Secondary network
General area
Local control
For short distances, tape is preferred over total station.
Trigonometric leveling does not have high precision compared to BS/FS because of refraction in the atmosphere
Invar strip - material used to achieve highest precision in leveling for tapes and rods
Transits are old instruments used to measure horizontal distance and vertical angle, but read from a vernier scale (caliper)
Double transit - (double centering) best method for extending a straight line with theodolite (inverting a transit)
Two-peg test - is used to determine mislevelment of line of sight of the leveling device
3-Wire Leveling - very precise. Basically differential leveling w/ 3 readings for BS and FS
Angular resolution looking through a scope can be calculated as: