Route Surveying - Earthworks
Route Surveying - Earthworks
Route Surveying - Earthworks
EARTHWORKS
Intended Learning Outcomes
1. Identify the methods in
calculating the area and volume
of cross sectional area for
earthworks
Intended Learning Outcomes
Excavation
Earthworks
Embankment
Earthworks
Volume
Volume of Earthworks
𝐿
𝑉𝑒𝑛𝑑 = 𝐴1 + 𝐴2
2
𝐿
𝑉𝑝 = 𝐴1 + 4𝐴𝑚 + 𝐴2
6
Prismoidal Formula
Volume of Earthworks
𝑉𝑝 = 𝑉𝑒𝑛𝑑 − 𝐶𝑣
Prismoidal Formula
Volume of Earthworks
𝐿
𝐶𝑣 = 𝑐2 − 𝑐1 𝑑2 − 𝑑1
12
Prismoidal Correction
Earthworks
Area
AREA
COMPUTATIONS
Area Computations
Counting Squares
Method
Area Computations
Geometric
Method
Geometric Method
𝑏ℎ
𝐴=
2
Area of a Triangle
Geometric Method
2𝐴 = 𝑏ℎ
𝑏1 + 𝑏2
𝐴=ℎ
2
Area of a Trapezoid
Geometric Method
2𝐴 = ℎ 𝑏1 + 𝑏2
5.7 x 4.0
= 22.8m 2
2A of Triangle AJK
Geometric Method
5.2 x (5.7+6.0)
= 60.8m 2
2A of Triangle ABJ
Geometric Method
9.3 x (5.2+4.2)
= 87.4m 2
2A of Trapezoid BCIJ
Geometric Method
15.0 x (4.2+4.7)
= 133.5m 2
2A of Trapezoid CDHI
Geometric Method
4.8 x (4.7+1.5)
= 29.8m 2
2A of Trapezoid DEGH
Geometric Method
1.5 x 4.5
= 6.8m 2
2A of Triangle EFG
Geometric Method
341.1/2
= 170.5m 2
elevations
(latitudes)
Double Meridian Distance (DMD)
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
A–B +0.1 -5.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
A–B +0.1 -5.0
B–C -1.1 -15.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
A–B +0.1 -5.0
B–C -1.1 -15.0
C–D +1.9 -14.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0 4.0
E–F +1.5 +30.0 38.0
F–I -1.5 +30.0 98.0
I–H -1.8 +7.0 135.0
H–G -0.9 -22.0 120.0
G–A +0.8 -15.0 83.0
A–B +0.1 -5.0 63.0
B–C -1.1 -15.0 43.0
C–D +1.9 -14.0 14.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0 4.0 4.0
E–F +1.5 +30.0 38.0 57.0
F–I -1.5 +30.0 98.0 -147.0
I–H -1.8 +7.0 135.0 -243.0
H–G -0.9 -22.0 120.0 -108.0
G–A +0.8 -15.0 83.0 66.4
A–B +0.1 -5.0 63.0 6.3
B–C -1.1 -15.0 43.0 -47.3
C–D +1.9 -14.0 14.0 26.6
Total 385.0
Double Area:
Double Meridian Distance
385m /2
2
= 192.50m 2
Three-Level
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
EXAMPLE
#3
Three Level Section
Example #3
Station 4+120
8.00 0 9.20
+2.00 +3.20 +2.80
Station 4+160
9.50 0 10.70
+3.00 +2.60 +3.80
𝑐1 𝑏
𝐴𝑟𝑒𝑎1 = 8 + 9.2 + 2 + 2.8
2 2
3.2 10
𝐴𝑟𝑒𝑎1 = 8 + 9.2 + 2 + 2.8
2 2
3.2 10
𝐴𝑟𝑒𝑎1 = 8 + 9.2 + 2 + 2.8
2 2
𝟐
𝐴𝑟𝑒𝑎1 = 𝟑𝟗. 𝟓𝟐𝒎
2.6 10
𝐴𝑟𝑒𝑎2 = 9.5 + 10.7 + 3 + 3.8
2 2
2.6 10
𝐴𝑟𝑒𝑎2 = 9.5 + 10.7 + 3 + 3.8
2 2
𝟐
𝐴𝑟𝑒𝑎2 = 𝟒𝟑. 𝟐𝟔𝒎
40
𝑉𝑒𝑛𝑑 = 39.52 + 43.26
2
𝟑
𝑉𝑒𝑛𝑑 = 𝟏𝟔𝟓𝟓. 𝟔𝒎
Station 4+160
9.50 0 10.70
+3.00 +2.60 +3.80
Three Level Section
Example #3
Station 4+140
Horizontal Distance
8.00+9.50 0+0 9.20+10.70
2 2 2
Vertical Distance
2.00+3.00 3.20+2.60 2.80+3.80
2 2 2
Three Level Section
Example #3
Station 4+140
8.75 0 9.95
+2.50 +2.90 +3.30
2.9 10
𝐴𝑟𝑒𝑎𝑚 = 8.75 + 9.95 + 2.5 + 3.3
2 2
2.9 10
𝐴𝑟𝑒𝑎𝑚 = 8.75 + 9.95 + 2.5 + 3.3
2 2
𝟐
𝐴𝑟𝑒𝑎𝑚 = 𝟒𝟏. 𝟔𝟐𝒎
𝐿
𝑉𝑝 = 𝐴1 + 4𝐴𝑚 + 𝐴2
2
40
𝑉𝑝 = 39.52 + 4 41.62 + 43.26
2
40
𝑉𝑝 = 39.52 + 4 41.62 + 43.26
2
𝟑
𝑉𝑒𝑛𝑑 = 𝟏𝟔𝟔𝟏. 𝟕𝟑𝒎
𝐿
𝐶𝑣 = 𝑐2 − 𝑐1 𝑑2 − 𝑑1
12
40
𝐶𝑣 = 2.6 − 3.2 20.2 − 17.2
12
40
𝐶𝑣 = 2.6 − 3.2 20.2 − 17.2
12
𝐶𝑣 = −𝟔𝒎𝟑
Three Level Section
Area Computations
Special Case
Three-Level Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
Area Computations
Five-Level
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
EXAMPLE
#4
Five Level Section
Example #4
Station 7+460
+0.32 +0.60 +1.92 +1.52 +1.32
4.14 3.50 0 3.50 6.14
Station 7+500
2.70 3.20 1.60 2.0 2.40
8.90 3.50 0 3.50 8.30
Area1
1 −3.5 −4.14 −3.5 0 3.5 6.14 3.5
2 0 0.32 0.60 1.92 1.52 1.32 0
Area1 = 12.63m 2
Area2
1 −3.5 −8.9 −3.5 0 3.5 8.3 3.5
2 0 2.7 3.2 1.6 2 2.4 0
Area1 = 28.14m 2
𝐿
𝑉𝑒𝑛𝑑 = 𝐴1 + 𝐴2
2
40𝑚
𝑉𝑒𝑛𝑑 = 12.63 + 28.14
2
Irregular
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
Area Computations
Levelled
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
CUT AND FILL
SECTIONS
EXAMPLE
#5
Cut and Fill Section
Example #5
Station 3+000
5.48 0 5.00
+0.98 +3.05 +0.50
Station 3+060
6.76 0 4.63
+0.98 −1.22 −0.42
5.05 9
𝐴𝑟𝑒𝑎1 = 5.48 + 5 + 0.98 + 0.5
2 2
5.05 9
𝐴𝑟𝑒𝑎1 = 5.48 + 5 + 0.98 + 0.5
2 2
𝟐
𝐴𝑟𝑒𝑎1 = 𝟏𝟗. 𝟑𝟏𝒎
1.22 8
𝐴𝑟𝑒𝑎2 = 6.76 + 4.63 + 1.84 + 0.42
2 2
1.22 8
𝐴𝑟𝑒𝑎2 = 6.76 + 4.63 + 1.84 + 0.42
2 2
𝟐
𝐴𝑟𝑒𝑎2 = 𝟏𝟏. 𝟒𝟕𝒎
42.86
𝑉𝑐𝑢𝑡 = 19.31
2
𝟑
𝑉𝑐𝑢𝑡 = 𝟒𝟏𝟑. 𝟖𝟏𝒎
60 − 42.86
𝑉𝑓𝑖𝑙𝑙 = 0 + 11.47
2
𝟑
𝑉𝑓𝑖𝑙𝑙 = 𝟗𝟖. 𝟑𝟎𝒎
1.20
𝐺𝑟𝑎𝑑𝑒𝑟𝑜𝑎𝑑 =
60
𝐺𝑟𝑎𝑑𝑒𝑟𝑜𝑎𝑑 = 𝟎. 𝟐𝟎
𝑥 60 − 𝑥
=
1.4 2.8
𝑥 = 𝟐𝟎𝒎
𝑦 2.8
=
30 40
𝑦 = 𝟐. 𝟏𝒎
𝐴
𝑉𝑝𝑖𝑡 = ℎ1 + 2 ℎ2 + 3 ℎ3 + 4 ℎ4
4
Elevation (h1)
Borrow Pit
Elevation (h2)
Borrow Pit
Elevation (h3)
Borrow Pit
Elevation (h4)
Borrow Pit
𝐴
𝑉𝑝𝑖𝑡 = ℎ1 + 2 ℎ2 + 3 ℎ3 + 4 ℎ4
4
E F G H
I J K L
M N O P
A = 3.3m B = 7.9m C = 10.8m D = 6.8m
ℎ1 = 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑡 𝐴 𝐷 𝑀 𝑃
ℎ1 = 11.2m
ℎ2 = 𝑒𝑙𝑒𝑣. 𝑎𝑡 𝐵 𝐶 𝐻 𝐿 𝑂 𝑁 𝐼 𝐸
ℎ2 = 24.8m
ℎ3 = 𝑁𝑂𝑁𝐸
ℎ4 = 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑡 𝐹 𝐺 𝐾 𝐽
ℎ4 = 19.4m
𝐴
𝑉𝑝𝑖𝑡 = ℎ1 + 2 ℎ2 + 3 ℎ3 + 4 ℎ4
4
2
30
𝑉𝑝𝑖𝑡 = 11.2 + 2 24.8 + 4 19.4
4
2
30
𝑉𝑝𝑖𝑡 = 11.2 + 2 24.8 + 4 19.4
4
𝑉𝑝𝑖𝑡 = 𝟑𝟏𝟏𝟒𝟎𝒎𝟑