Geology and Surveying 70380 (Part B - Surveying) Volumes and DTMs 1

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 h b a

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

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

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

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

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

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

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

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

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

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: 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

Earthworks Quantities from Contours

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 100.0 101.0 102.0 101 m Contour

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

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?