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

Class Presentations for Surveying II (JDF211E) Course by E. TARI, H.KARAMAN

Information for Users

2

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.

ISTANBUL TECHNICAL UNIVERSITY - DEPARTMENT OF GEOMATICS ENGINEERING


Information for Students

3

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.



Profile (1)


4

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.


Profile (2)


5

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.


Profile Leveling (1)


6

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.


Profile Leveling is an application of

Differential Leveling


7

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.




8

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.




9

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.


Profile


10


Cross Section Leveling (1)


11

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.


Cross Section Leveling (2)


12

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


Cross Section within the Profile Leveling


13

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.

Profileleveling

Cross-section leveling

1+

00

1+

25

1+

50

1+

75

0.00

10.50

25.00

12.50

25.02


Profile Leveling Sketch


14


Cross Section Leveling Computations


15

Distance from

Profile Point

Rod

Readings (m)

Collimation

Height

Height of

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


Surface Leveling (1)


16

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.




17

Rod readings are

performed at each corner

of the rectangle (1, 2,

...,17).

The length between 1

and 2 should not be more

than 20 m.

Once setting up the level,

the operator should read

as many points as

possible.

Level

Level

1 2 4 5

6 7 8 9

10 11 12 13

14 15 16 17




18

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)


Area Computation


19

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)


Trapezoidal Area Computation


20

Computing the irregular area

AXYZCBA is done by

approximating the area by a series

of equally spaced trapezia,

measuring these either in the field

or off a plan, and then computing

the area of each of these.

The area of the first trapezoid is given by;

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

OthersOO

LAorOOOO

OOLA nn

n 222

14321

2

211

OOLA


Simpson’s Area Computation


21

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

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

For the area AXYZCBA


Comparison of the Methods (1)


22

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.


Comparison of the Methods (2)


23

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.


Volume Calculations from Cross Section

Areas (1)


24

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


Volume Calculations from Cross Section

Areas (2)


25

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.


Digital Terrain Models - DTM (1)


26

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.




27

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.




28

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...


DTM Examples (1)


29


DTM Examples (2)


30


DTM Examples (3)


31


DTM Examples (4)


32



33

Perhitungan Volume Dan Dasar-dasarnya

Documents