INTEGRATION OF AREAS

undefined
 
INTEGRATION OF AREAS
 
MODULE 1 UNIT 1
 
Rawle Russell
 
1
 
BUILDING & MECHANICAL ENGINEERING
DRAWING
Rawle Russell
undefined
 
Objectives
 
At the end of this session students will be able to:
Generally
determine the centroid of a plane figure by
integration of areas.
 
Specifically:
construct two derived figures from a given figure;
find graphically the area of a plane figure;
apply simple calculations to find the centroid, 1
st
Moment of Area about XX, and 2
nd
 Moment of
Area about the centroid;
 
Rawle Russell
 
2
undefined
 
Rawle Russell
 
3
 
For the purpose of this lesson, we will look specifically at specific objective 1:
 
Construct two derived figures from a
given figure
 
 
OBJECTIVES
undefined
 
APPLICATION OF TOPIC
 
This topic is useful when finding the centre of
mass of a geometric object, these objects can
be 2 & 3 – Dimensional.
The center of mass is a point at which all the
mass of the object may be theoretically
considered to be concentrated for design
purposes.
In addition, it is used to properly balance a
structure or retaining wall.
 
Rawle Russell
 
4
undefined
 
PREREQUISTE KNOWLEDGE
 
Ability to setup the CAD environment: Units, Limits,
 
Layers, Leader line, Dimension Style, Point Size, Object
 
Snap mode.
Ability to manipulate the draw toolbar.
Ability to manipulate the modify toolbar.
Knowledge of finding the area of basic shapes: Square,
Rectangle, Circle, Segment, Sector, Trapezium.
Ability to find the centroid of plane figure.
Understand the Laws of Exponential.
 
Rawle Russell
 
5
undefined
 
OVERVIEW
 
To determine the centroid by integration of areas
the following must be done:
 
1.
Drawing the given figure;
2.
Creating a 1
st
 derived & 2
nd
 derived shape;
3.
Finding the areas of the original, 1
st
 & 2
nd
 derived
shapes;
4.
Applying three formulae –
Centroid
1
st
 Moment of Area
2
nd
 Moment of Area
 
Rawle Russell
 
6
undefined
 
Step 1: Construct the plane figure
 
Use the following measurements and construct the
given figure.
 
Rawle Russell
 
7
undefined
 
Step 2:
 
Draw a horizontal axis at
the base of the shape to
represent the X-X Axis.
 
Note: it is very common to
use the XX Axis to find the
centroid of a plane figure
when using the integration
of area method.
 
Creating the 1
st
 Derived Shape
 
Rawle Russell
 
8
undefined
 
Divide the shape into a number of
parts (equal or unequal).
The division lines will be drawn
parallel to the X-X axis.
NOTE:
It is recommended  that wherever the
shape changes direction a division
line should be drawn through that
point. portion in the shape.
 
Step 3:  Creating the 1
st
 Derived Shape
 
Rawle Russell
 
9
undefined
 
Rawle Russell
 
10
 
Step 3:  Creating the 1
st
 Derived Shape
 
 
Please note, this is for a greater
level of accuracy when
constructing the derived shapes
and finding the areas of the three
shapes.
 
Note well, the lower portion in the
shape has a different spacing
than the upper
undefined
 
In this step we are going to create the 1
st
 derived shape.
1.
Choose a point on the base of the shape at (a).
2.
Point ‘
a
’ could be anywhere along the base and is
called the pole.
Note: 
The midpoint is commonly used.
 
Secondly, identify where the division lines intersect the
shape (
b
, 
c
,
 d
, & 
e
).
 
See the diagram on the next slide.
 
Step 4:  Creating the 1
st
 Derived Shape
 
Rawle Russell
 
11
undefined
 
Rawle Russell
 
12
undefined
 
Step 5: Creating the 1
st
 Derived Shape
 
This step requires a series of vertical and
angled lines. The procedure is as
follows:
A line is drawn from point ‘
b
vertically and perpendicular to
the highest parallel.
Then another line is drawn from
that point to the pole ‘
a
’.
The first  point for the 1st
derived figure is establish when
the line drawn from the highest
parallel to point ‘a’ intersects
the first division line at ‘b’.
 
 
Rawle Russell
 
13
undefined
 
  Step 6: Creating the 1
st
 Derived Shape
 
This step follows the previous
step, this time starting at
point ‘
c
’ and going through
the same procedure.
For instance:
A line is drawn from point ‘
c
vertically to the pole ‘
a
’,
establish a point on the second
division line.
Follow this same procedure for
points ‘
d
’ and ‘
e
’.
 
Rawle Russell
 
14
undefined
 
Step 7:  Creating the 1
st
 Derived Shape
 
Follow the same procedure as Step 5 from points ‘d’
and ‘e’.
 
Rawle Russell
 
15
undefined
 
Step 8: Creating the 1
st
 Derived Shape
 
Connect the points
using the spline
command to form half
of the derived shape.
 
Rawle Russell
 
16
undefined
 
Step 9:
 
1
st
 Derived
Shape
 
Mirror the curve to
show the actual
size of the 1
st
derived shape.
 
Original Shape
 
1
st
 Derived Shape
 
Rawle Russell
 
17
undefined
 
1
st
 Derived Shape
 
This is what the 1
st
Derived Shape will look
like.
 
Rawle Russell
 
18
undefined
 
 Step 10: Creating the 2
nd
 Derived Shape
 
To draw the 2
nd
 derived
shape you have to work
with the 1
st
 derived shape,
therefore, copy the 1
st
derived shape with the
same division lines as
shown.
 
Rawle Russell
 
19
undefined
 
Step 11: Creating the 2
nd
 Derived Shape
 
The same steps used to create the
1
st
 derived shape will be used to
draw the 2
nd
 derived shape.
Note: The 1
st
 derived shape will be
used.
The previous slide asked us to
copy the 1
st
 derived shape
with the division lines.
Number the division lines
where they intersect the
shape, such as, ‘
f
’, ‘
g
’, ‘
h
’ &
j
’.
 
Rawle Russell
 
20
undefined
 
Step 12: Creating the 2
nd
 Derived Shape
 
Follow the same
procedures as of Step 5.
For example:
Draw a line from ‘
f
vertically upwards and
perpendicular to the top
parallel line, then down to
the pole at ‘
a
’.
At the intersection of the
corresponding division line is
the first point (green) for the
second derived shape.
 
Rawle Russell
 
21
undefined
 
Follow the same procedures for
points ‘
g
’, ‘
h
’ & ‘
j
’ to obtain the other
points for the 2
nd
 derived curve
.
 
Step 13: Creating the 2
nd
 Derived Shape
 
Points ‘
h
’ & ‘
j
’ are obtained in
the figure below.
 
Rawle Russell
 
22
undefined
 
 Step 14: Creating the 2
nd
 Derived Shape
 
Connect the points using the
spline command to form half
of the 2
nd
 derived shape.
 
.
 
Rawle Russell
 
23
undefined
 
Rawle Russell
 
24
 
Mirror the curve to show the actual size of the 2nd
derived shape
undefined
 
2
nd
 Derived Shape
 
This is what the 2
nd
 Derived Shape will look
like.
 
Rawle Russell
 
25
undefined
 
Original
Shape
 
1
st
 Derived
Shape
 
2
nd
 Derived
Shape
 
Rawle Russell
 
26
undefined
 
Rawle Russell
 
27
 
ASSIGNMENT
 
Reproduce the given figure using
the dimensions given. You are
required to Follow the instructions
in the lesson to produce the:
i.
1
st
 derived figure
ii.
2
nd
 derived figure
undefined
 
SOLUTION
 
Your solutions for the 1
st
derived and 2
nd
derived figures should
look like those
produced on Slides:
26 & 27
 
 
 
Did you get that??
Slide Note
Embed
Share

This module focuses on determining centroids of plane figures using integration of areas. Students will learn to construct derived figures, find areas graphically, and apply calculations for centroids and moments of area. The lesson emphasizes the practical application of finding the center of mass for design and structural balance purposes.

  • Engineering
  • Drawing
  • Integration
  • Centroids
  • Mechanical

Uploaded on Mar 27, 2024 | 0 Views


Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

E N D

Presentation Transcript


  1. 1 INTEGRATION OF AREAS BUILDING & MECHANICAL ENGINEERING DRAWING MODULE 1 UNIT 1 Rawle Russell

  2. Objectives 2 At the end of this session students will be able to: Generally determine the centroid of a plane figure by integration of areas. Specifically: construct two derived figures from a given figure; find graphically the area of a plane figure; apply simple calculations to find the centroid, 1st Moment of Area about XX, and 2nd Moment of Area about the centroid; Rawle Russell

  3. OBJECTIVES 3 For the purpose of this lesson, we will look specifically at specific objective 1: Construct two derived figures from a given figure Rawle Russell

  4. APPLICATION OF TOPIC 4 This topic is useful when finding the centre of mass of a geometric object, these objects can be 2 & 3 Dimensional. The center of mass is a point at which all the mass of the object may be theoretically considered to be concentrated for design purposes. In addition, it is used to properly balance a structure or retaining wall. Rawle Russell

  5. PREREQUISTE KNOWLEDGE 5 Ability to setup the CAD environment: Units, Limits, Layers, Leader line, Dimension Style, Point Size, Object Snap mode. Ability to manipulate the draw toolbar. Ability to manipulate the modify toolbar. Knowledge of finding the area of basic shapes: Square, Rectangle, Circle, Segment, Sector, Trapezium. Ability to find the centroid of plane figure. Understand the Laws of Exponential. Rawle Russell

  6. OVERVIEW 6 To determine the centroid by integration of areas the following must be done: 1. Drawing the given figure; 2. Creating a 1st derived & 2nd derived shape; 3. Finding the areas of the original, 1st & 2nd derived shapes; 4. Applying three formulae Centroid 1st Moment of Area 2nd Moment of Area Rawle Russell

  7. Step 1: Construct the plane figure Use the following measurements and construct the given figure. 7 Rawle Russell

  8. Step 2: 8 Creating the 1st Derived Shape Draw a horizontal axis at the base of the shape to represent the X-X Axis. Note: it is very common to use the XX Axis to find the centroid of a plane figure when using the integration of area method. Rawle Russell

  9. Step 3: Creating the 1st Derived Shape 9 Divide the shape into a number of parts (equal or unequal). The division lines will be drawn parallel to the X-X axis. NOTE: It is recommended that wherever the shape changes direction a division line should be drawn through that point. portion in the shape. Rawle Russell

  10. Step 3: Creating the 1st Derived Shape 10 Please note, this is for a greater level of accuracy when constructing the derived shapes and finding the areas of the three shapes. Note well, the lower portion in the shape has a different spacing than the upper Rawle Russell

  11. 11 Step 4: Creating the 1st Derived Shape In this step we are going to create the 1st derived shape. 1. Choose a point on the base of the shape at (a). 2. Point a could be anywhere along the base and is called the pole. Note: The midpoint is commonly used. Secondly, identify where the division lines intersect the shape (b, c, d, & e). See the diagram on the next slide. Rawle Russell

  12. 12 Rawle Russell

  13. Step 5: Creating the 1st Derived Shape This step requires a series of vertical and angled lines. The procedure is as follows: A line is drawn from point b vertically and perpendicular to the highest parallel. Then another line is drawn from that point to the pole a . The first point for the 1st derived figure is establish when the line drawn from the highest parallel to point a intersects the first division line at b . 13 Rawle Russell

  14. Step 6: Creating the 1st Derived Shape 14 This step follows the previous step, this time starting at point c and going through the same procedure. For instance: A line is drawn from point c vertically to the pole a , establish a point on the second division line. Follow this same procedure for points d and e . Rawle Russell

  15. Step 7: Creating the 1st Derived Shape 15 Follow the same procedure as Step 5 from points d and e . Rawle Russell

  16. Step 8: Creating the 1st Derived Shape 16 Connect the points using the spline command to form half of the derived shape. Rawle Russell

  17. Step 9: 1st Derived Shape Mirror the curve to show the actual size of the 1st derived shape. 17 Original Shape 1st Derived Shape Rawle Russell

  18. 1st Derived Shape 18 This is what the 1st Derived Shape will look like. Rawle Russell

  19. Step 10: Creating the 2nd Derived Shape 19 To draw the 2nd derived shape you have to work with the 1st derived shape, therefore, copy the 1st derived shape with the same division lines as shown. Rawle Russell

  20. Step 11: Creating the 2nd Derived Shape 20 The same steps used to create the 1st derived shape will be used to draw the 2nd derived shape. Note: The 1st derived shape will be used. The previous slide asked us to copy the 1st derived shape with the division lines. Number the division lines where they intersect the shape, such as, f , g , h & j . Rawle Russell

  21. Step 12: Creating the 2nd Derived Shape 21 Follow the same procedures as of Step 5. For example: Draw a line from f vertically upwards and perpendicular to the top parallel line, then down to the pole at a . At the intersection of the corresponding division line is the first point (green) for the second derived shape. Rawle Russell

  22. Step 13: Creating the 2nd Derived Shape 22 Follow the same procedures for points g , h & j to obtain the other points for the 2nd derived curve. Points h & j are obtained in the figure below. Rawle Russell

  23. Step 14: Creating the 2nd Derived Shape 23 . Connect the points using the spline command to form half of the 2nd derived shape. Rawle Russell

  24. 24 Mirror the curve to show the actual size of the 2nd derived shape Rawle Russell

  25. 2nd Derived Shape 25 This is what the 2nd Derived Shape will look like. Rawle Russell

  26. Original Shape 1st Derived Shape 2nd Derived Shape 26 Rawle Russell

  27. 27 ASSIGNMENT Reproduce the given figure using the dimensions given. You are required to Follow the instructions in the lesson to produce the: i. 1st derived figure ii. 2nd derived figure Rawle Russell

  28. SOLUTION Your solutions for the 1st derived and 2nd derived figures should look like those produced on Slides: 26 & 27 Did you get that??

Related


More Related Content

giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#giItT1WQy@!-/#