1. Geology and Surveying 70380 (Part B - Surveying) Volumes and DTMs1

2. Objectives In this lecture we will look at:
General volume calculations from field information
Trapezoidal Rule for volumes
Simpson’s Rule for volumes
Forming DTMs

3. Volume of a Regular Box
V = a × b × c c a b

4. Volume of a Pyramid h b a

5. Volume of a Frustum A1 h A2 A1 and A2 are parallel
Perpendicular height between A1 and A2 h A2
A1 and A2 are parallel

6. Volume of a Wedge
h = Vertical Height a b d h
If d=a

7. Volume of a Prismoid
A Prismoid is a solid having for its ends any two parallel plane figures, and having plane sides. l A2 A1

8. Trapezoidal Rule for Volumes
Prismoidal Formula
End Area Formula
Combine several prismoid formulae.
Where n is the last cross-section
Convenient since any number of prismoids or cross sections can be used

9. Simpson’s Rule for Volume
Follow similar arguments to Trapezoidal Formula to extend Simpson’s Rule for areas and develop Simpson’s Rule for volumes
Odd number of cross sections at equal distance apart
If even number, calculate last prismoid independently

10. Earthworks Volume
Areas and volumes are included on Cross Sections and Longitudinal Sections
Generally economy suggests a “balance” of cut and fill after topsoil stripping
Mass Haul is also a major consideration to contractors
Study example on page 21.17

11. Grid Leveling h2 h3 h4 h1 a b

12. Grid Leveling h2 h3 h4 h1 b a h5 h6 b a b

13. Grid Leveling h2 h3 h4 h1 h8 b a h5 h6 h7 b a b b

14. Grid Leveling General Formula Appearing once = corners of grid
Appearing twice = sides of grid
Appearing three times = only irregular shapes
Appearing four times = internal points

15. Contours
Contouring is the geographic representation of land forms (shape of land surface)
A contour is a line of constant elevation
Therefore the vertical interval between contours (or contour interval - CI) is constant
Therefore the distance between contours indicates the steepness of grade

16. Example of Contours

17. Levels for Contours and DTM
Levels can be taken by several means including:
Grid Leveling (pre-marked grid)
Spot Leveling (Chainage and offset)
Spot Leveling (H angle and distance)
Spot Leveling (GPS)
Much better handled by a fully automated surveying sustem – Total Station and data collector or RTK GPS

18. Earthworks Quantities from Contours

19. Digital Terrain Models
DTM = Digital Terrain Model
DEM = Digital Elevation Model
Used by CADD packages

20. Contouring Models Usually a points modeler (with strings)
Surface represented by E, N and RL of a number of grid or random points
Surface must be defined (usually be triangles – triangular plates)
This assumes a smooth surface between points – Caution!
Triangles are then contoured

21. Points – Spot Levels

22. Triangles

23. Contours from Triangles

24. Contours from Triangles

25. Contours from Triangles

26. Triangles and Contours

27. Contours

28. String Lines Model represented by points and string lines
Strings used to join common points
Strings also used to define changes of grade (COGs)
These COGs are also known as “Breaklines”
Every string becomes the side of a triangle if that string is a breakline.

29. Without Breakline Strings
100.0
101.0
102.0
101 m Contour

30. With Breakline Strings
101.0
101.0
101.0
Breakline
102.0
102.0
102.0
101 m Contour
100.0
100.0
100.0

31. Breaklines

32. DTMs for Volume Calculation
Volumes from prisms
Volumes from cross-sections
Volumes from triangles
Make decision on most accurate for each application – then check by alternate method
Volumes can be from a datum surface, or between two surfaces (eg Design surface – Natural Surface)

33. Summary In this lecture we investigated:
General volume calculations from field information
Trapezoidal Rule for volumes
Simpson’s Rule for volumes
Forming DTMs
Significance of Breaklines
CADD applications and volume calcs

34. Self Study
Read Module 21
Do self assessment Questions

35. Questions?