SETK1213-03 - STATICS
SETK 1111-03 - ENGINEERING DRAWING
3D RIGID BODY
GROUP 4
TEAM MEMBERS:
LOUIS HONG (A23KT0122)
MOHANAD KHALED QASEM AL-SABRI (A23KT4002)
NUR NISA AMALIANA BINTI ZAHARI (A23KT0189)
ROHAIDA IDAYU (A23KT0211)
LECTURER(S):
DR. HAJAR ALIAS
DR. NUR HAFIZAH ABD HAMID
UNIVERSITI TEKNOLOGI MALAYSIA
1
TABLE OF CONTENTS
CHAPTER 1: INTRODUCTION.................................................................................................3
CHAPTER 2: AUTOCAD DRAWING........................................................................................5
CHAPTER 3: CALCULATION................................................................................................... 6
3.1 Calculation using (Pappus-Guldinus Theorem)................................................................... 6
3.2 Calculation of the Crescentic Prism...................................................................................10
3.2.1 Crescent face.............................................................................................................10
3.2.2 Inner circular area..................................................................................................... 12
3.2.3 Outer circular area.................................................................................................... 13
3.3 Calculation of the star prism.............................................................................................. 14
3.3.1 First Part....................................................................................................................14
3.3.2 Second Part............................................................................................................... 15
3.3.3 Third Part.................................................................................................................. 16
3.4 Material Cost......................................................................................................................17
3.4.1 Hemispheric dome.................................................................................................... 17
3.4.2 Supporting rod.......................................................................................................... 18
3.4.3 Star, Crescent and Small Sphere............................................................................... 18
3.4.4 Cylinder layers.......................................................................................................... 18
3.4.5 Coating...................................................................................................................... 19
3.4.6 Total cost for the structure........................................................................................ 19
CHAPTER 4: CONCLUSION................................................................................................... 20
References.....................................................................................................................................21
CHAPTER 1: INTRODUCTION
A mosque dome was selected as the rigid body of this project. This is because the
mosque dome as the basis of a 3D rigid body can provide the exploration of complex geometric
shapes and patterns for this project. Furthermore, it contains geometric shapes that allow
calculation of volume and surface area calculation using the Pappus-Guldinus theorem. The
chosen rigid body also contains five different shapes - cylinder, crescentic prism, star prism,
sphere, and hemisphere which complies with the restriction of this project in order to conduct a
numerical analysis. Through geometric analysis, various materials and the amount needed to
build the rigid body can be determined which leads to the quantification of the material cost.
Figure 1: Example of Mosque Dome
1.1 Objectives
The objectives of this project are:
1. To draw and design the 3D rigid body using AutoCAD software
2. To calculate the surface area and volume of rigid body
3. To calculate the material cost of the rigid body design
3
CHAPTER 2: AUTOCAD DRAWING
The creation and design of a 3D rigid body and its orthographic drawing involves the use
of AutoCAD software. Tools and features of said software are used to model and visualise the
geometry of the rigid body. Firstly, a sketch of the selected rigid body (mosque dome) was made
to create a visual depiction of the rigid body. Figure 2 and 3 shows the sketching of the mosque
dome.
Figure 2 : 2D sketch of the mosque dome
Figure 3: 3D sketch of the mosque dome
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The following stage of the project is the visualisation of the rigid body, which is the 3D
AutoCAD drawing. Figure 5 shows the isometric view of the mosque dome in AutoCAD
software while Figure 6 shows the orthographic view of the 3D rigid body.
Figure 5
Figure 6
5
CHAPTER 3: CALCULATION
3.1 Calculation using (Pappus-Guldinus Theorem)
Mathematical analysis is performed on the designed 3D rigid body by calculating its
surface area and volume based on the rigid body’s geometric properties. The calculation is done
using the Pappus-Guldinus theorem by rotating lines and surface area formed from sectioning of
half of the rigid body around the y-axis 360 degrees.
Figure 7: Half of the mosque with the dimensions
Vertical Lines
r bar
Figure 8: Vertical Lines locations
Length
Surface Area formed
from revolution
Line 1
75 m
20 m
2 x pi x 75 x 20
Line 2
65 m
20 m
2 x pi x 65 x 20
6
Line 3
60 m
10 m
2 x pi x 60 x 20
Line 4
0.3 m
6m
2 x pi x 0.3 x 6
● Surface Area formed from revolution of Line 1 = 9424.8 m²
● Surface Area formed from revolution of Line 2 = 8168.1 m²
● Surface Area formed from revolution of Line 3 = 3769.9 m²
● Surface Area formed from revolution of Line 4 = 11.31 m²
Figure 9: Areas of Rectangles
Rectangles
r bar
Area as a rectangle
Volume formed from
revolution
Rectangle 1
37.5 m
75 x 20 = 1500 m²
2 x pi x 37.5 x 1500
7
Rectangle 2
32.5 m
65 x 20 = 1300 m²
2 x pi x 32.5 x1300
Rectangle 3
30 m
60 x 10 = 600 m²
2 x pi x 30 x 600
Rectangle 4
0.15 m
0.3 x 6 = 1.8 m²
2 x pi x 0.15 x 1.8
● Volume formed from revolution of Rectangle 1 = 353000 m³
● Volume formed from revolution of Rectangle 2 = 265000 m³
● Volume formed from revolution of Rectangle 3 = 113000 m³
● Volume formed from revolution of Rectangle 4 = 1.696 m³
Figure 10: Vertical Lines locations
Horizontal Lines
r bar
Length
Surface Area formed
from revolution
Line 5
70 m
10 m
2 x pi x 70 x 10
Line 6
62.5 m
5m
2 x pi x 62.5 x 5
8
Line 7
55 m
10 m
2 x pi x 55 x 10
● Surface Area formed from revolution of Line 5 = 4398.2 m²
● Surface Area formed from revolution of Line 6 = 1963.5 m²
● Surface Area formed from revolution of Line 7 = 3455.8 m²
Arcs
r bar
Length
Surface Area formed from
revolution
Quarter circle arc
( 2 x 50) / pi
( pi x 50) x 5
2 x pi x (100/ pi) x 250 x
pi
Semicircle arc
( 2 x 3) / pi
pi x 3
2 x pi x (6/pi) x pi x 3
● Surface Area formed from revolution of quarter circle arc = 15707.96 m²
● Surface Area formed from revolution of semicircle arc = 113.1 m²
Circles
r bar
Area
Volume formed from
revolution
Quarter circle
( 4 x 50 /(3 x pi)
( pi x 50²) / 4
2 x pi x ((200)/3 x pi) x (625 /
pi)
Semicircle
( 4 x 3) / (3 x pi)
(pi x 3²) / 2
2 x pi x (4 x pi) x (4.5 x pi )
● Volume formed from revolution of Quarter circle = 261800 m³
● Volume formed from revolution of Semicircle = 113.1 m³
9
3.2 Calculation of the Crescentic Prism
3.2.1 Crescent face
Figure 11: Crescent face
The design of the crescent face of the crescentic prism is obtained by superimposing two
circles of different sizes in a way so that the centre of the smaller circle (small circle) with a
radius of 4.5m is 1m away from the centre of the larger circle (big circle) with a radius of 5m.
The area of the crescent can then be calculated by subtracting the area of the big circle from that
of the small circle. However, due to the fact that the area of the small circle is not fully enclosed
by the big circle, the area of a smaller leftover crescent (shaded in yellow) needs to be accounted
for by adding it back to the difference of circle areas. The calculation can be summarised as so:
crescent area = area of big circle - area of small circle + yellow shaded area.
Full calculation steps are shown below;
1. Notations:
● CA: Crescent Area
● BC: Big Circle Area
● SC: Small Circle Area
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● YS: Yellow Shaded Area
● SS: Segment of the Small Circle
● SB: Segment of the Big Circle
2. Formulas:
● SS=0.5 x 4.5² x ( (131/180)*pi)) - sin(131))
● SB = 0.5 x 5² x ((110/180)*pi)) - sin (110))
● YS= SS - SB
● BC = pi x 5²
● SC = pi x 4.5²
● CA=BC−SC+YS
3. Calculations:
● YS = 3.26 m²
● BC = pi x 5²
● SC = pi x 4.5²
● CA = 78.84 - 63.62 + 3026 = 18.18 m²
Total Crescent Area (Both Faces):
● Total Crescent Area = 18.18 x 2 = 36.36 m²
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3.2.2 Inner circular area
Figure 12: Inner circular face
The inner circular area of the crescentic prism is calculated by multiplying a fraction of
the circumference of the small circle by the thickness of the prism.
1. Notations:
● Circumference of Small Circle: The circumference of the small circle on the crescent
● Fraction: Fraction of the circumference on the crescent
● Area of Small Circle: The area of the small circle on the crescent
2. Formulas:
● The Circumference of the small circle = pi * r * 2
● Fraction = ((360 -theta of unwanted circumference) / 360)
● Circumference of the small circle on the crescent = pi x r x 2 x fraction
● Area of the face = length(circumference) * width
3. Calculations:
● The Circumference of the small circle = pi x 4.5 x 2
● Fraction = ((360 -131) / 360)
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● Circumference of the small circle on the crescent = pi x 4.5 x 2 x 0.636
● Area of the face = 17.99 x 0.4 = 7.196 m²
3.2.3 Outer circular area
Figure 13: Outer circular face
The outer circular area of the crescentic prism is calculated by multiplying a fraction of
the circumference of the small circle by the thickness of the prism.
1. Notations:
● Circumference of Big Circle: The circumference of the big circle on the crescent
● Fraction: Fraction of the circumference on the crescent
● Area of Big Circle face : The area of the big circle face on the crescent
2. Formulas:
● The Circumference of the small circle = pi x r x 2
● Fraction = ((360-110) / 360)
● Circumference of the small circle on the crescent = pi x r x 2 *Fraction
● Area of the face = length(circumference) * width
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3. Calculations:
● The Circumference of the big circle = pi x 5 x 2
● Fraction = 0.694
● Circumference of the big circle on the crescent = pi x 5 x 2 x 0.694
● Area of the face = 21.82 x 0.4 = 8.728 m²
● Surface Area of the whole Crescent = 36.36 + 7.196 + 8.728= 52.28 m²
● Volume of the crescent = face of the crescent * thickness = 18.18 x 0.4 = 7.27 m³
3.3 Calculation of the star prism
3.3.1 First Part
Figure 14: Star face (dimensions in mm)
The calculation of the star face of the prism is done by sectioning into 2 shapes - 5
triangles and 1 regular pentagon.
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3.3.2 Second Part
Calculation of Star Face Area:
1. Definitions:
● Area of Triangle: Area of an individual triangle
● Total Area of Triangles: Total area of the five triangles
● Area of Regular Pentagon: Area of the pentagon
● Total Area of the Face: Total area of the star face
● Total Star Area (Both Faces): Total area of both similar faces of the star
2. Formulas:
● Area of triangle = ½ x base x height
● Total Area of Triangles = Area of triangle x 5
● Area of regular pentagon = ¼ (5x(5+ 2 x (5)^(½)) ^ (½)) x side
● Total Area of the Face = Total Area of Triangles + Area of Pentagon
● Total Star Area (Both Faces) = Total Area of the Face * 2
3. Calculations:
● Area of triangle = ½ x 1.76179 x 3.28755 = 2.896 m²
● Total Area of Triangles = 5 x 2.896 = 14.48 m²
● Area of pentagon = ¼ (5 x (5+2 x (5)^(½)) ^ (½)) x 1.76179 = 5.34 m²
● Total Area of the Face = 14.48 + 5.34 = 19.82 m²
● Total Star Area (Both Faces) = 19.82 x 2 = 39.64 m²
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3.3.3 Third Part
Figure 15: Rectangular faces on the star
The sides of the star prism are made of 10 rectangles
Calculation of sides (10 Rectangles):
1. Definitions:
● Area of Rectangle: Area of an individual rectangle
● Total Area of Rectangles: Total area of the ten rectangles
2. Formulas:
● Area of rectangle = Length x width
● Total Area of Rectangles = Area of rectangle x 10
3. Calculations:
● Area of Rectangle = 3.4 x 0.4 = 1.36 m²
● Total Area of Rectangles = 1.36 x 10 = 13.6 m²
● Surface Area of the whole star prism = 39.64 + 13.6 = 53.24 m²
● Volume of the star prism = surface area of star face * thickness = 19.82 x 0.4 = 7.93 m³
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3.4 Material Cost
In this section, the determination of material used and the associated costs for
manufacturing the designed rigid body are going to be discussed and calculated.
3.4.1 Hemispheric dome
Material: Low-E glass roof
Justification: Protection against UV light, Low-E glass, like thermal glass, may deflect the
majority of UV radiation, preventing it from passing through.
Cost: RM 55 / ft²
Total surface area of hemisphere= 15707.96 m²
1m = 3.2808 ft
15707.96 m² x (3.2808 ft)² = 15707.96 m² x 10.76365 ft²
1
m²
1
1
= 169074.984 ft²
= 169074.984 ft² x RM 55
= RM 9 299 124.00
3.4.2 Supporting rod
Material: Steel
Type: SCM440 Steel round bar
Size: Diameter : 0.23 m
Length : 6.0m
Cost: RM 1724.83
3.4.3 Star, Crescent and Small Sphere
Material : Steel plate
Justification: Steel plates are popular due to their corrosion and wear resistance, and their
thickness range is significantly broader than that of regular steel sheets. Steel plates can thus be
used in a wide range of applications requiring superstructure frameworks and unbreakable
toughness.
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Cost: RM 16.00 per plate
Size: Length :0.1524m
Width 0.1524m
Thickness: 0.003m
Surface area for a plate of steel = 2(0.1524 x 0.1524) + 2(0.1524 x 0.003) + 2(0.1524 x 0.003)
= 0.04828 m²
Total surface area for star, crescent and mini sphere = 53.28 + 52.28 + 113.1
= 218.67 m²
= 218.67 m²
0.04828 m²
= 4529 plates of steel x RM 16.00
= RM 72464
3.4.4 Cylinder layers
Material: Clay Brick
Justification: Brick is non-combustible because its basic constituent is clay, which is heated to
roughly 1000 degrees Celsius. This means it won't burn in a bushfire. Brick is such a robust and
long-lasting building material. Brick veneer wall assemblies regulate moisture better than other
exterior material wall systems.
Size: Length: 215 mm
Width: 100 m
Height: 67mm
Price for a brick = RM 1.20
Surface area for a brick = 2(215 x 100) + 2(215 x 67) +2(100 x 67)
= 85 110 mm²
= 0.08511 m²
Total surface area of 3 cylinder layers = 9424.8 + 8168.1 + 376.99
= 21362.8 m²
Brick needed = 21362.8 m²
18
0.08511 m²
= 251 002 bricks needed
Cost for bricks = 251 002 x RM 1.20
= RM 301 202.40
3.4.5 Coating
Paint: Aluminium
Justification: Aluminium provides enhanced protection. Aluminium paint functions as a
dependable barrier, protecting surfaces from corrosive elements and weather. It produces a
protective coating that aids in the prevention of corrosion and deterioration, hence extending the
life of the coated material.
Cost: RM 143.00 per 5L
Coverage = 11m²
Surface area for a tin of paint = 5 x 11
= 55m²
Total surface area excluded the hemisphere = 47 117.5 - 15 707.96
=31 409.54 m²
Amount of paint needed = 31409.54 m²
55m²
Amount of tins of paint needed = 571.08 litres ÷ 5
= 114.2
= 114 tins of paint needed
Cost for 114 tins of paint = 114 x RM 143.00
= RM 16 302
3.4.6 Total cost for the structure
Total Cost = RM 9 299 124 + RM 16 302 + RM 72 464 + RM 301 202.40 + RM 1 724.83
= RM 9 595 117.23
Therefore, the total cost for the mosque dome is RM RM 9 595 117.23
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CHAPTER 4: CONCLUSION
In summary, the design and analysis of the 3D rigid body have provided valuable insights
into the structural characteristics and practicability for the real-world applications henceforth.
The chosen rigid body was successfully designed from the utilisation of AutoCAD software and
has optimised the access to the visualisation for the calculations of its volume and surface area.
The calculations have offered a precise aid for the material cost configuration of the selected
rigid body. Material costs for the rigid body was roughly calculated, which provides a conception
of cost if such rigid body is to be actually constructed. Therefore, all the objectives from this 3D
rigid body project were successfully accomplished.
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References
1. Dome of Masjid al-Aqsa - Madain Project (en). (2021). Madainproject.com.
https://madainproject.com/dome_of_al_aqsa
2. BYJUS. “Area of a Pentagon Formula | Area of regular pentagon | Solved Example”..
https://byjus.com/area-of-a-pentagon-formula/
3. BYJUS. “Area of Segment of a Circle (Formula, Theorems & Examples)”.
https://byjus.com/maths/area-segment-circle/‌
4. Aluminium Paint: Enhancing Protection and Aesthetics — Civil Engineering Profile.
(2023, August 4). Civil Engineering Profile. https://civilengpro.com/aluminium-paint/
5. What is Low-E Glass & Does it Make Windows Energy Efficient? - ~. (2023).
https://www.stanekwindows.com/what-is-low-e-glass-and-does-it-make-windows-more-e
nergy-efficient.aspx
6. The Brickery. (2023, September 24). Benefits of clay bricks - The Brickery.
https://thebrickery.co.nz/meet-the-bricks/benefits-of-clay-bricks/
7. Australian Steel Institute - Advantages of Steel Construction. (2023).
https://www.steel.org.au/what-we-do/focus-areas/steel-in-architecture/advantages-of-steel
-construction/
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