perhitungan-volume-dan-dasar-dasarnya

Week 10
Prof. Dr. Ergin TARI
Assist. Prof. Dr. Himmet KARAMAN
JDF211E COURSE - ISTANBUL TECHNICAL UNIVERSITY - DEPARTMENT OF GEOMATICS ENGINEERING

ISTANBUL TECHNICAL UNIVERSITY - DEPARTMENT OF GEOMATICSENGINEERING
Information for Users

The following slides are compiled from;



The references given for the course,



The course notes of the lecturers from all around the
world,



Notes and slides published in the world wide web without
restrictions.

2
Class Presentations for Surveying II (JDF211E) Course by E. TARI, H.KARAMAN

Information for Students

These presentations are compiled from the previous
versions of the Surveying II course slides which were
created by Prof. Dr. Muhammed Sahin and Prof. Dr.

Ergin Tarı between the years of 1998 and 2008.



The update process of these presentations will
continue, and will never end.



The responsibilities of the students for the exams will be
from the presentations, applications and practices
covered during the course.

3

Profile (1)

The outline produced where the plane of a vertical
section intersects the surface of ground; e.g., the
longitudinal profile of a stream, or the profile of a
coast or hill. Syn.: topographic profile



A graph or drawing that shows the variation of one
property such as elevation or gravity, usually as
ordinate, with respect to another property, such as
distance.

4

Profile (2)

Cross section of a region of cylindrical folds drawn
perpendicular to the fold axes.



A vertical section of a water table or other
potentiometric surface, or of a body of surface water.



A drawing used in civil engineering to show a vertical
section of the ground along a surveyed line or
graded work.

5

Profile Leveling (1)

The process of determining the elevations of a series
of points at measured intervals along a line such as
the centerline of a projected highway or railway.

6

Profile Leveling is an application of
Differential Leveling

Elevations are determined in the same manner.



The same definitions define the concepts and terms
involved.


The same types of mistakes and errors are possible.



An arithmetic check(difference between BS reading
and FS readings) should always be done.



A closure check should be done if the profile
line runs between bench marks which is the
desired case.

7


On root surveys for highways or pipelines, elevations
are required at every 25 m station;


at angle points (points marking changes in direction);



at breaks in the ground surface slope; and



at critical points such as roads, bridges and culverts.







When plotted, these elevations show a profile – a
line depicting ground elevations at a vertical section
along a survey line.

8


For most of the engineering projects, profiles are
taken along the center line.



Profiles were usually plotted on a special paper,
called “milimetric paper”, of course, when the
computers and plotters did not exist.

9

Profile
10

Cross Section Leveling (1)

Cross sections are lines of levels or short profiles
made perpendicular to the center line of the project.
Cross sections are usually taken at regular intervals
and at sudden changes in the center-line profile.

11

Cross Section Leveling (2)

The cross sections must extend a sufficient distance on
each side of the center line to provide a view of the
surrounding terrain.



Rod readings should be taken at equal intervals on both
sides of the center line and at significant changes in the
terrain.



Field notes for a cross section should include an
elevation or difference in elevation from the center line
horizontal distance from the center line

12

Cross Section within the Profile Leveling

At each profile point, a cross-section leveling is
performed.



The cross-section line is perpendicular to the profile, and
has a 50 meter length: 25 m on the left and 25 m on the
right side of the profile, depending on the project
requirements.


Rod readings are secured at all breaks in the ground

surface. Cross-section leveling
25.00
10.50
Profile
0.00
leveling
1+00
1+2
5
1
+
5
0
1+7
5
12.50
25.02
13

Profile Leveling Sketch
14

Cross Section Leveling Computations
Distance from Rod Collimation Height of
Profile Point Readings (m) Height Point
BS FS (m) (m)
20.00 2.72 582.79 580.07
18.62 3.04 3.04 582.79 579.75
16.45 1.36 1.36 582.79 581.43
10.60 3.42 0.50 582.79 582.29
5.00 3.75 3.75 585.71 581.96
0.00 1.00 3.02 583.69 582.69
3.42 0.84 0.84 583.69 582.85
6.30 2.70 2.70 583.69 580.99
12.26 3.82 1.11 586.40 582.58
18.00 3.75 3.75 586.40 582.65
20.00 3.03 586.40 583.37
15

Surface Leveling (1)

Another method, surface leveling, is used for an area
which has a smooth (or flat) topography.



In this technique, the area is divided into rectangular
blocks (grids or the smallest geometrical figure) as in
the following figure.

16


Rod readings are 1 2 4 5
performed at each corner
of the rectangle (1, 2,
...,17). Level

The length between 1
6 7 8 9
and 2 should not be more
than 20 m. 10 11 12 13

Once setting up the level,
the operator should read Level
as many points as
possible. 14 15 16 17
17

Mark No Height (m)
1 17.06
2 17.48
3 17.63
4 17.37
5 17.70
6 17.96
7 17.58
8 18.01
9 18.25
The volume of excavation
in triangle 124 will be
A/3 (d1 + d2 +d4)
18

Area Computation
GAUSS AREA COMPUTATION
Mark No X (m) Y (m) Difference
1 X1 Y1
2 X2 Y2 (X3-X1)*Y2 = a
3 X3 Y3 (X4-X2)*Y3 = b
4 X4 Y4 (X1-X3)*Y4 = c
1 X1 Y1 (X2-X4)*Y1 = d
2 X2 Y2
Area F = 0.5 * (a+b+c+d)
19

Trapezoidal Area Computation
Computing the irregular area
AXYZCBA is done by
approximating the area by a series
ofequally spacedtrapezia,
The area of the first trapezoid is given by;
measuring these either in the field
or off a plan, and then computing
O1  O2
A1  L the area of each of these.
2
where L is the constant distance along the traverse line between offsets O1 and O2
The total area is
AT = A1 + A2 + A3 + A4 + A5
AT =L[(O1 + O2) + (O2 + O3) + (O3 + O4) + (O4 + O5) + (O5 + O6)]/2
A 
L

O
1
O
n O
2
 O
3
 O
4
 O  or A 
L O  O
n
 2 Others
2 n 2 1 
20

Simpson’s Area Computation
By assuming that each two adjacent sub-areas are a single bounded parabola
rather than each sub-area being a trapezoid
For the area contained between 01 and 03;
A = Trapezoid (abdea) + parabolic area (agefa)
A = (01 + 03)L + 2/3(area bounded by parabola)
A = (01 + 03)L + 2/3 x 2L[02 - (01 +
03)/2] A = L[01 + 402 + 03]/3
For the area AXYZCBA
A = L[(O1 + On) + 2(O3 + O5 + On-2) + 4(O2 + O4 + On-1)]/3
A = [S(1st + last offset) + 2S(odd offsets) + 4S(even offsets)] S=L/3
21

Comparison of the Methods (1)

In the trapezoidal formula, the resulting area is generally
less than the true area. The accuracy of the area will
depend on the number of offsets (and therefore the
distance between them) and the degree of irregularity of
the boundary. Of course the more irregular the
boundary the more offsets should measured; this will
demand a compromise between the time spent
gathering the data and the required accuracy.

22

Comparison of the Methods (2)

The Simpson’s formula is more accurate but has the
disadvantage that n must be odd. In this case it is not
possible to directly compute the total area AXYZCBA.
Instead the area AXYBA is computed using Simpson's
Rule and the additional area BYZCB must be computed
separately. This could have been avoided if the
irregular area had been originally subdivided into an
odd number of sub-areas.

23

Volume Calculations from Cross Section
Areas (1)

Several successive cross sections are situated at
equal distances, d, along a fixed direction. Then,

V = d(A1 + A2)/2 + d(A2 + A3)/2 + d(A3 + A4)/2 + ........ + d(An-1 + An)/2
V = d[A1 + 2A2 + 2A3 + 2A4 + ......... + 2An-1 + An]/2
24

Volume Calculations from Cross Section
Areas (2)
V = [First area + last area + 2S(all remaining areas)]

Called End Area formula may be applied to any
number of cross sections equally spaced along a
straight line.

25

Digital Terrain Models - DTM (1)

With the advent of the computer it became possible to
process large data sets to compute a volume.



This had not been previously possible because of the
large amount of computing involved.


The mathematics is not complex but most tedious.



So Digital Terrain Models (DTMs) gained in acceptance,
to the point where they are now the most frequently
used method.

26


The basic theory is that points are
located (X, Y, Z) on the terrain to
define the surface (usually at
changes of grade).

Each point is connected to
neighboring points in a unique
manner so that a series of
triangles is formed that entirely
covers the surface.



As shown in the figure each of
these right triangular prisms is a
simple solution to an individual
volume, their sums being the total
volume between the surface and a
datum plane.

27

The use of DTMs is also a
very convenient way to
compute and plot
contours, cross sections,
long sections, surface
profiles and plans for
complex surfaces.
Various commercial
packages are available
beside the free ones such
as GRASS, GMT, etc...
28

DTM Examples (1)
29

DTM Examples (2)
30

DTM Examples (3)
31

DTM Examples (4)
32

33

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