surveying and levelling 2

ahmadikhan 9,786 views 19 slides Feb 11, 2017
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About This Presentation

Ahmadi khan


Slide Content

Week No.2

Advance Engineering Surveying
Lecture No.2
B-Tech

By:
Engr. Shams Ul Islam
Lecturer, Civil Engg. Department
CECOS University Peshawar

Computation of Areas
(Irregular bounded fields)
The main objective of the surveying is to compute the areas and volumes.
Generally, the lands will be of irregular shaped polygons. There are formulae
readily available for regular polygons like, triangle, rectangle, square and other
polygons.
But for determining the areas of irregular polygons, different methods are used.
Earthwork computation is involved in the excavation of channels, digging of
trenches for laying underground pipelines, formation of bunds, earthen
embankments, digging farm ponds, land levelling and smoothening. In most of
the computation the cross sectional areas at different interval along the length of
the channels and embankments are first calculated and the volume of the
prismoids are obtained between successive cross section either by trapezoidal or
prismoidal formula.
2
Engr.Shams Ul Islam ([email protected])

Computation of Areas
(Irregular bounded fields)

Computation of areas is carried out by any one of the following
methods:

•Mid-ordinate method
•Average ordinate method
•Trapezoidal rule
•Simpson’s rule

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Engr.Shams Ul Islam ([email protected])

The mid-ordinate rule
In this method, the ordinates are measured at the mid-points of
each division and the area is calculated by the formula.
??????��� ∆=��� �� ��� ���������×�
����� �=��� ������ �������� �������� ��� ���������
��� ��������=
??????
1+??????
2
2

����� ??????
1,??????
2,??????
3,…..,??????
?????? ��� ��� �������� �� ���� �� ��� ��������



4
Engr.Shams Ul Islam ([email protected])

The mid-ordinate rule


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Engr.Shams Ul Islam ([email protected])

The mid-ordinate rule

The mid ordinate rule can also be in other way.
??????��� ∆=

1+ℎ
2+ℎ
3+⋯ℎ
??????
??????
×L
����� �
1,�
2,�
3,…..,�
?????? ��� ��� �������� �� ��� ��� ����� �� ���� ��������
�=��� ������ �� ��� ���� ����

�=��� ������ �� ����� ����� ���� ����� ��� ���� ���� �� �������

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Engr.Shams Ul Islam ([email protected])

Example
The following offsets were taken from a chain line of 60 m to an irregular
boundary line at an interval of 10 m, the offsets are:
0, 2.50, 3.50, 5.00, 4.60, 3.20, 0 m
Compute the area between the chain line, the irregular boundary line and the
end of offsets by mid ordinate rule



7
Engr.Shams Ul Islam ([email protected])

Solution
�
1=
0+2.5
2
=1.25
�
2=
2.5+2.5
2
=3.0
�
3=
3.5+5
2
=4.25
�
4=
5+4.60
2
=4.80
�
5=
4.60+3.20
2
=3.90
�
6=
3.20+0
2
=1.60
�=60 �
�=6
??????��� ∆=
ℎ1+ℎ2+ℎ3+⋯ℎ??????
??????
×L
??????=
1.25+3.0+4.25+4.80+3.90+1.60
6
×60=188 �
2


8
Engr.Shams Ul Islam ([email protected])

Average Ordinate Method
In this method, the ordinates are drawn and scaled at each of the points
of division of the base line and the area is calculated by the formula.
??????��� ∆=
��� �� ���������
�+1
�
??????��� ∆=
??????
1+??????
2+??????
3+⋯,+??????
??????
�+1
�
�=��� ������ �� ��� ���� ����
�=������ �� ����� ��������� ���� ����� ��� ���� ���� �� �������
����� �=��� ������ �������� �������� ��� ���������
����� ??????
1,??????
2,??????
3,…..,??????
?????? ��� ��� �������� �� ���� �� ��� ��������



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Engr.Shams Ul Islam ([email protected])

The Average ordinate method


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Engr.Shams Ul Islam ([email protected])

Trapezoidal Rule

This rule is more accurate than the first two ones. In this rule, boundaries between
the ends of ordinates are assumed to be straight. Thus the areas enclosed between
the base line and the irregular boundary line are considered as trapezoids.
Let
�=��� ������ �������� �������� ��� ��������,
�=������ �� ����� ��������� ���� ����� ��� ���� ���� �� �������
��� ??????
1,??????
2,??????
3,…..,??????
?????? ��� ��� �������� �� ���� �� ��� ��������
??????��� ∆=
??????
1+2??????
2+2??????
3+⋯,+2??????
??????−1+??????
??????
2
×
�
�

OR
??????��� ∆=
�
2
×??????
1+2??????
2+2??????
3+⋯,+2??????
??????−1+??????
??????
OR
??????��� ∆=�×
??????
1+??????
??????
2
+??????
2+??????
3+⋯,+??????
??????−1

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Engr.Shams Ul Islam ([email protected])

Simpson’s Rule
In this rule, the boundaries between the ends of ordinates are assumed to
form an arc of parabola. Hence Simpson’s rule is some times called as
parabolic rule. Refer to figure:



12
Engr.Shams Ul Islam ([email protected])

Continue..
??????
1,??????
2,??????
3 are the consecutive coordinates
�=��� ������ �������� ������� ��� ���������

A��� ??????����= ���� �� �����??????��� ??????���+ ���� �� ������� �����

??????��� �� �����??????���=
??????
1+??????
3
2
×??????�
??????��� �� �����??????���=
??????
1+??????
3
2
×2�
??????��� �� �������=
2
3
×??????��� �� �������������� ����
=
2
3
����
=
2
3
×��×2�
����� ��=??????
2−(
??????1+??????3
2
) So
??????��� �� �������=
2
3
×??????
2−(
??????
1+??????
3
2
)×2�


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Engr.Shams Ul Islam ([email protected])

Continue..
So the area between the first Two divisions,
??????��� ∆=
??????
1+??????
3
2
×2�+
2
3
×??????
2−(
??????
1+??????
3
2
)×2�
=??????
1+??????
3�+
4�
3
2??????
2−??????
1−??????
3
2

=??????
1�+??????
3�+
4�
6
2??????
2−??????
1−??????
3
=??????
1�+??????
3�+
2�
3
2??????
2−??????
1−??????
3
=
3??????
1�+3??????
3�+4??????
2�−2??????
1�−2??????
3�
3

=
3??????
1�−2??????
1�+4??????
2�+3??????
3�−2??????
3�
3

=
??????
1�+4??????
2�+??????
3�
3

=
�
�
×??????
�+�??????
�+??????
�



14
Engr.Shams Ul Islam ([email protected])

Continue..
Similarly, the area of next two divisions
=
�
�
×??????
�+�??????
�+??????
�

����� ??????���=
�
3
×??????
1+4??????
2+2??????
3+4??????
4+⋯…+2??????
??????−2+4??????
??????−1+??????
??????
OR
����� ??????���=
�
3
×??????
1+??????
??????)+4(??????
2+??????
4+⋯)+2(??????
3+??????
5+⋯

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Engr.Shams Ul Islam ([email protected])

Comparison of Trapezoidal rule with Simpson’s Rule

Trapezoidal rule

The boundary between the
ordinates is considered to be
straight


There is no limitation. It can be
applied for any number of
ordinates

It gives an approximate result




Simpson’s Rule

 The boundary between the
ordinates is considered to be
an arc of a parabola


To apply this rule, the
number of ordinates must be
odd

It gives a more accurate
result

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Engr.Shams Ul Islam ([email protected])

Problem
The following offsets were taken at 15 m intervals from a survey line to an
irregular boundary line.
3.50,4.30, 6.75, 5.25, 7.50, 8.80, 7.90, 6.40, 4.40, 3.25 m
Calculate the area enclosed between the survey line, the irregular boundary line,
and the offsets, by:
(1) The trapezoidal rule (2) Simpson’s rule


17
Engr.Shams Ul Islam ([email protected])

Continue..
By Using Trapezoidal Rule
??????��� ∆=�
??????
1+??????
??????
2
+??????
2+??????
3+⋯,+??????
??????−1
??????��� ∆=�
??????
1+??????
10
2
+??????
2+??????
3+⋯,+??????
9
??????��� ∆=15×
3.50+3.25
2
+4.30+6.75+5.25+7.50+8.80+7.90+6.40+4.40

????????????�?????? ∆=���.����
�







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Engr.Shams Ul Islam ([email protected])

Continue..
By Using Simpson’s Rule
����� ??????���=
�
3
×??????
1+??????
??????)+4(??????
2+??????
4+⋯)+2(??????
3+??????
5+⋯
??????��� (∆1)=
15
3
×3.50+4.40)+44.30+5.25+8.80+6.40+2(6.75+7.50+7.9
=756.00�
2

??????��� ∆2=
??????
9+??????
10
2
�=
4.40+3.25
2
×15=57.38�
2

????????????????????????� ????????????�??????=∆�+∆�
????????????????????????� ????????????�??????=���.��+��.��=���.���
�

19
Engr.Shams Ul Islam ([email protected])
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