Development of Surfaces
19
ENGINEERING GRAPHICS DR. I.P.S. AHUJA PROFESSOR IN MECHANICAL ENGINEERING PUNJABI UNIVERSITY, PATIALA DEVELOPMENT OF SURFACES
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Engineering Drawing
Transcript of Development of Surfaces
ENGINEERING GRAPHICS
DR. I.P.S. AHUJAPROFESSOR IN MECHANICAL ENGINEERING
PUNJABI UNIVERSITY, PATIALA
DEVELOPMENT OF SURFACES
Development is a graphical method of obtaining the surface area of a solid.
When a solid is opened out and its complete surface is laid on a plane, the surface of
the solid is said to be developed.
The figure thus obtained is called a development of the surfaces of the solid or simply
development. Development of the solid, when folded or rolled, gives the solid.
METHODS OF DEVELOPMENT
The following two methods are mainly employed to obtain the development of
surfaces of solids:
1. PARALLEL LINE DEVELOPMENT: This method is employed to develop the surfaces of
PRISMS AND CYLINDERS. Two parallel lines (called stretch-out lines) are drawn from
the two ends of the solids and the lateral faces are located between these lines.
2. RADIAL LINE DEVELOPMENT: This method is employed to develop the surfaces of
PYRAMIDS AND CONES. An arc of radius equal to the slant edge/generator is drawn
and the lateral faces/curved face are marked properly inside the arc.
DEVELOPMENT
D
H
D
SS
H
L
= RL
+ 3600
R=Base circle radius.L=Slant height.
L= Slant edge.S = Edge of base
L
S
S
H= Height S = Edge of base
H= Height D= base diameter
DEVELOPMENT OF LATERAL SURFACES OF DIFFERENT SOLIDS(LATERAL SURFACE IS THE SURFACE EXCLUDING TOP & BASE)
Prisms: No.of Rectangles
Cylinder: A RectangleCone: (Sector of circle) Pyramids: (No.of triangles)
Tetrahedron: Four Equilateral Triangles
All sides equal in length
Cube: Six Squares.
A
PARALLEL LINE
DEVELOPMENTRADIAL
LINE DEVELOPMENT
Fig. (a) shows the FV of a cube of side 50 mm. Develop all the faces of the cube.
PARALLEL LINE DEVELOPMENT
A cylinder of diameter 50 and height 60mm cut by an AIP inclined at 45o to HP, passing through left top corner of cylinder. Draw the development of truncated cylinder.
PARALLEL LINE DEVELOPMENT
11 98
6
532
12
A hexagonal prism, side of base 20mm and axis 50mm, rests with its base on HP such that one of its rectangular faces is parallel to VP. It is cut by a plane perpendicular to VP, inclined at 45o to HP and passing through right corner of the prism. Draw the sectional top view, and develop the lateral surface of the truncated prism.
DEVELOPMENT
e, 5f, 6
d, 4
c, 3b, 2
a, 1o, o1
o1’
o’
c’, e’
1’ 4’2’, 4’ 1 2 3 43’, 5’
b’, f’A
d’a’
5 6 1
B C D E F A
45o
20
X
20
Y
1314’
12’16’
11’
13’15’
11
12
15
11
16
14
Figure a) shows the TV and FV of a cone of base diameter 50 mm and slant height 75 mm. Draw the development of its curved surface.
RADIAL LINE DEVELOPMENT
Lθ = 2πr.
θ = 2π( r/ L)
θ (in degree) = 360( r/ L)
L= Slant Height
The radius of the sector will be equal to the slant height of the cone. The length of the arc will be equal to the circumference of the base of the cone, i.e., 2πr.
When the curved surface of a cone is opened and laid on a plane, it shows the shape of a sector. The included angle of the sector depends on the slant height, L and the radius of the base of the cone, r.
L
= RL
+ 3600
R= Base circle radius of coneL= Slant height of coneL1 = Slant height of cut part.
L1
DEVELOPMENT OF FRUSTUM OF CONE FRUSTUMS
ADEVELOPMENT OF LATERAL SURFACES OF DIFFERENT SOLIDS(LATERAL SURFACE IS THE SURFACE EXCLUDING TOP & BASE)
Figure a) shows the TV and FV of a cone of base diameter 50 mm and slant height 65 mm. Draw the development of its curved surface.
RADIAL LINE DEVELOPMENT
Approximate Method
Development
A cone of base diameter 40 mm and height 50 mm is kept on the ground on its base. An AIP inclined at 30° to the HP cuts the cone through the midpoint of its axis. Draw the development of truncated cone.
DEVELOPMENT OF TRUNCATED SOLIDS
Draw the lateral surface of funnel as shown in figure.
DEVELOPMENT
12
11
10
9
8
7
6
5
43
27
6
54
3
2
1
5 4 3 2 1
1
10 9 8 7 61 12 11
5 4 32
17
6
44
24
O
12
11109
87
65
43
2
1
54
32
1
76
54
3
2 1
109
8
7
6
1 12
11
Draw the lateral development of a pentagonal pyramid of base edge 30 mm and height 60 mm lying with its base on the HP. One of the edges of its base is parallel to the VP.
RADIAL LINE DEVELOPMENT
DEVELOPMENT
XY
41’3’2’
4
11
o’
o1’
2
3 4
1
A
B
3 4
O
1R = O41’ = TL
A pentagonal pyramid, side of base 30mm and height 52mm, stands with its base on HP and an edge of the base parallel to VP. It is cut by a plane perpendicular to VP, inclined at 40o to HP and passing through a point on the axis, 32mm above its base. Draw sectional top view, and develop the lateral surface of the truncated pyramid.
3
2
5
40o
o, o1
1’ 4’5’
TL
5
5
c1 ’
b1 ’a1 ’
e1 ’d1 ’
b’
c’d’
e’a’
32
41
52
A square pyramid, base 48x32mm and height 62mm, rests on its base on HP in such a way that its longer edge is parallel to VP. A cutting plane perpendicular to VP and inclined at 30o to HP bisects the axis of the solid. Draw sectional top view, and develop the lateral surface of the truncated pyramid.
DEVELOPMENT
X Y
2’, 1’
o, o1
TL
d1’
d1
c’, d’b’, a’
d
cb
a
30o
A
o
o1’
1
23
4
3’, 4’ 31’, 41’
21’, 11’
31
B
C D
A
1
2
34
O
1
R = od1’ = TL
A Cone base 54mm diameter and height 72mm, rests with its base on HP. A sectional plane perpendicular to HP, inclined at 25o to VP cuts the cone at a distance of 15 mm from the axis. Draw sectional front view, and develop the lateral surface of the remaining portion of the cone.
DEVELOPMENT
1
2
31
11107 8
O
R = O7’ = TL
12
9
1’ 2’ 12’
3’ 11’
4’ 10’
5’ 9’
6’ 8’
7’
o, o1
o’
Yo1’
72
a
a’
b
c’
ed
c
b’
e’
d’
X
Sectional Front View
f
f’
1
8
2
34
5
6
7
1110
9
12
25o
4
65
c1’
b1’
e1’
d1’
E
F
D C
B
A
TL
DEVELOPMENT
Draw the development of lateral surface of the truncated square prism as shown in the figure.
Draw the development of lateral surface of the truncated cylinder as shown in the figure.
DEVELOPMENT
Draw the development of lateral surface of the truncated cylinder as shown in the figure.
DEVELOPMENT
