Geology and Surveying
70380
(Part B - Surveying)
Volumes and DTMs
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
Volume of a Regular Box
V=a×b×c
c
a
b
Volume of a Pyramid
abh
V
3
h
b
a
Volume of a Frustum

h
V  A1  A2  A1 A2
3
h

Perpendicular
height between A1
and A2
A1 and A2 are parallel
Volume of a Wedge
h = Vertical Height
a
hb
2a  d 
V
6
If d=a
h
b
d
hb
V 
a
2
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
A1  A2
V  l
2
Trapezoidal Rule for Volumes
• Prismoidal Formula
• End Area Formula
Combine several prismoid formulae.
A2  A3
A3  A4
A1  A2
V  l
l
l
 ....
2
2
2
l
V   A1  2 A2  2 A3  ..... 2 An 1  An 
2
Where n is the last cross-section
Convenient since any number of prismoids
or cross sections can be used
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
l
V   A1  4 A2  2 A3  4 A4  2 A5  ..... 2 An  2  4 An 1  An 
3
• Odd number of cross sections at equal
distance apart
• If even number, calculate last prismoid
independently
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
Grid Leveling
h2
h3
h4
h1
a
b
ab
h1  h2  h3  h4 
V
4
Grid Leveling
h2
h3
h4
h1
b
h5
a
h6
b
a
b
ab
2h1  h2  h3  2h4  h5  h6 
V
4
Grid Leveling
h2
h3
h4
h1
h8
b
h5
a
h6
b
b
h7
a
b
ab
2h1  h2  h3  3h4  h5  2h6  h7  h8 
V
4
Grid Leveling
General Formula
  h appe aring once 




ab  2 h appe aring tw ice 


V
4  3 h appe aring thre e tim e s 


 4 h appe aring four tim e s 


Appearing once = corners of grid
Appearing twice = sides of grid
Appearing three times = only irregular shapes
Appearing four times = internal points
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
Example of Contours
Levels for Contours and DTM
Levels can be taken by several means
including:
1. Grid Leveling (pre-marked grid)
2. Spot Leveling (Chainage and offset)
3. Spot Leveling (H angle and distance)
4. Spot Leveling (GPS)
Much better handled by a fully automated
surveying sustem – Total Station and
data collector or RTK GPS
Earthworks Quantities from
Contours
CI
V   A1  4 A2  2 A3  4 A4  2 A5   Area A 5 to top
3
Digital Terrain Models
DTM = Digital Terrain Model
DEM = Digital Elevation Model
Used by CADD packages
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
Points – Spot Levels
Triangles
Contours from Triangles
Contours from Triangles
Contours from Triangles
Triangles and Contours
Contours
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.
Without Breakline Strings
101.0
101.0
102.0
101.0
102.0
102.0
101 m Contour
100.0
100.0
100.0
With Breakline Strings
101.0
101.0
102.0
Breakline
101.0
102.0
102.0
101 m Contour
100.0
100.0
100.0
Breaklines
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)
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
Self Study
• Read Module 21
• Do self assessment Questions
Questions?