DEVELOPMENT OF SURFACES OF SOLIDS.

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1 DEVELOPMENT OF SURFACES OF SOLIDS.
MEANING:- ASSUME OBJECT HOLLOW AND MADE-UP OF THIN SHEET. CUT OPEN IT FROM ONE SIDE AND UNFOLD THE SHEET COMPLETELY. THEN THE SHAPE OF THAT UNFOLDED SHEET IS CALLED DEVELOPMENT OF LATERLAL SUEFACES OF THAT OBJECT OR SOLID. LATERLAL SURFACE IS THE SURFACE EXCLUDING SOLID’S TOP & BASE. ENGINEERING APLICATION: THERE ARE SO MANY PRODUCTS OR OBJECTS WHICH ARE DIFFICULT TO MANUFACTURE BY CONVENTIONAL MANUFACTURING PROCESSES, BECAUSE OF THEIR SHAPES AND SIZES. THOSE ARE FABRICATED IN SHEET METAL INDUSTRY BY USING DEVELOPMENT TECHNIQUE. THERE IS A VAST RANGE OF SUCH OBJECTS. EXAMPLES:- Boiler Shells & chimneys, Pressure Vessels, Shovels, Trays, Boxes & Cartons, Feeding Hoppers, Large Pipe sections, Body & Parts of automotives, Ships, Aeroplanes and many more. WHAT IS OUR OBJECTIVE IN THIS TOPIC ? To learn methods of development of surfaces of different solids, their sections and frustums. 1. Development is different drawing than PROJECTIONS. 2. It is a shape showing AREA, means it’s a 2-D plain drawing. 3. Hence all dimensions of it must be TRUE dimensions. 4. As it is representing shape of an un-folded sheet, no edges can remain hidden And hence DOTTED LINES are never shown on development. But before going ahead, note following Important points. Study illustrations given on next page carefully.

2 = Development of lateral surfaces of different solids.
(Lateral surface is the surface excluding top & base) Cylinder: A Rectangle Pyramids: (No.of triangles) Cone: (Sector of circle) L S H D S L D  H= Height D= base diameter  = R L + 3600 R=Base circle radius. L=Slant height. Prisms: No.of Rectangles L= Slant edge. S = Edge of base S H H= Height S = Edge of base Cube: Six Squares. Tetrahedron: Four Equilateral Triangles All sides equal in length

3 FRUSTUMS = STUDY NEXT NINE PROBLEMS OF SECTIONS & DEVELOPMENT
DEVELOPMENT OF FRUSTUM OF CONE DEVELOPMENT OF FRUSTUM OF SQUARE PYRAMID Base side L Top side L L1 L1   = R L + 3600 R= Base circle radius of cone L= Slant height of cone L1 = Slant height of cut part. L= Slant edge of pyramid L1 = Slant edge of cut part. STUDY NEXT NINE PROBLEMS OF SECTIONS & DEVELOPMENT

4 Problem 1: A pentagonal prism , 30 mm base side & 50 mm axis
is standing on Hp on it’s base whose one side is perpendicular to Vp. It is cut by a section plane 450 inclined to Hp, through mid point of axis. Draw Fv, sec.Tv & sec. Side view. Also draw true shape of section and Development of surface of remaining solid. Solution Steps:for sectional views: Draw three views of standing prism. Locate sec.plane in Fv as described. Project points where edges are getting Cut on Tv & Sv as shown in illustration. Join those points in sequence and show Section lines in it. Make remaining part of solid dark. A B C E D TRUE SHAPE X1 Y1 A B C D E a” b” c” d” e” a’ b’ e’ c’ d’ X Y a e d b c DEVELOPMENT For True Shape: Draw x1y1 // to sec. plane Draw projectors on it from cut points. Mark distances of points of Sectioned part from Tv, on above projectors from x1y1 and join in sequence. Draw section lines in it. It is required true shape. For Development: Draw development of entire solid. Name from cut-open edge I.e. A. in sequence as shown. Mark the cut points on respective edges. Join them in sequence in st. lines. Make existing parts dev.dark.

5 Problem 2: A cone, 50 mm base diameter and 70 mm axis is
standing on it’s base on Hp. It cut by a section plane 450 inclined to Hp through base end of end generator.Draw projections, sectional views, true shape of section and development of surfaces of remaining solid. Solution Steps:for sectional views: Draw three views of standing cone. Locate sec.plane in Fv as described. Project points where generators are getting Cut on Tv & Sv as shown in illustration.Join those points in sequence and show Section lines in it. Make remaining part of solid dark. TRUE SHAPE OF SECTION X1 Y1 A SECTIONAL S.V o’ B SECTION PLANE DEVELOPMENT C D E a’ b’ d’ e’ c’ g’ f’ h’ X g” h”f” a”e” b”d” c” Y F h a b c d e g f For True Shape: Draw x1y1 // to sec. plane Draw projectors on it from cut points. Mark distances of points of Sectioned part from Tv, on above projectors from x1y1 and join in sequence. Draw section lines in it. It is required true shape. G For Development: Draw development of entire solid. Name from cut-open edge i.e. A. in sequence as shown.Mark the cut points on respective edges. Join them in sequence in curvature. Make existing parts dev.dark. H A SECTIONAL T.V

6 X Y Use similar steps for sec.views & true shape.
Note the steps to locate Points 1, 2 , 5, 6 in sec.Fv: Those are transferred to 1st TV, then to 1st Fv and Then on 2nd Fv. Problem 4: A hexagonal prism. 30 mm base side & 55 mm axis is lying on Hp on it’s rect.face with axis // to Vp. It is cut by a section plane normal to Hp and 300 inclined to Vp bisecting axis. Draw sec. Views, true shape & development. b’ f’ a’ e’ c’ d’ SECTIONAL F.V. Use similar steps for sec.views & true shape. NOTE: for development, always cut open object from From an edge in the boundary of the view in which sec.plane appears as a line. Here it is Tv and in boundary, there is c1 edge.Hence it is opened from c and named C,D,E,F,A,B,C. 1 2 3 4 5 6 7 8 b’ f’ a’ e’ c’ d’ X Y a b c d e f a1 d1 b1 e1 c1 f1 A.V.P300 inclined to Vp Through mid-point of axis. 1,2 3, , ,6 AS SECTION PLANE IS IN T.V., CUT OPEN FROM BOUNDRY EDGE C1 FOR DEVELOPMENT. X1 Y1 C D E F A B 2 1 8 7 6 5 4 3 TRUE SHAPE OF SECTION DEVELOPMENT

7 Problem 5:A solid composed of a half-cone and half- hexagonal pyramid is
shown in figure.It is cut by a section plane 450 inclined to Hp, passing through mid-point of axis.Draw F.v., sectional T.v.,true shape of section and development of remaining part of the solid. ( take radius of cone and each side of hexagon 30mm long and axis 70mm.) 7 1 6 5 4 3 2 TRUE SHAPE X1 Y1 Note: Fv & TV 8f two solids sandwiched Section lines style in both: Development of half cone & half pyramid: O’ A B C D E F G O DEVELOPMENT 1’ 2’ 3’ 4’ 5’ 6’ 7’ F.V. 4 5 3 6 2 7 1 X Y d’e’ c’f’ g’b’ a’ a b c d e f g 7 1 5 4 3 2 6 SECTIONAL TOP VIEW.

8 TO DRAW A CURVE ON PRINCIPAL VIEWS FROM DEVELOPMENT.
Problem 8: A half cone of 50 mm base diameter, 70 mm axis, is standing on it’s half base on HP with it’s flat face parallel and nearer to VP.An inextensible string is wound round it’s surface from one point of base circle and brought back to the same point.If the string is of shortest length, find it and show it on the projections of the cone. TO DRAW A CURVE ON PRINCIPAL VIEWS FROM DEVELOPMENT. Concept: A string wound from a point up to the same Point, of shortest length Must appear st. line on it’s Development. Solution steps: Hence draw development, Name it as usual and join A to A This is shortest Length of that string. Further steps are as usual. On dev. Name the points of Intersections of this line with Different generators.Bring Those on Fv & Tv and join by smooth curves. Draw 4’ a’ part of string dotted As it is on back side of cone. a’ b’ c’ d’ o’ e’ A B C D E O 2 3 4 1 1’ 2’ 3’ 4’ X Y a b c d o e 1 2 3 4

9 It’s a construction of curve Helix of one turn on cone:
Problem 9: A particle which is initially on base circle of a cone, standing on Hp, moves upwards and reaches apex in one complete turn around the cone. Draw it’s path on projections of cone as well as on it’s development. Take base circle diameter 50 mm and axis 70 mm long. It’s a construction of curve Helix of one turn on cone: Draw Fv & Tv & dev.as usual On all form generators & name. Construction of curve Helix:: Show 8 generators on both views Divide axis also in same parts. Draw horizontal lines from those points on both end generators. 1’ is a point where first horizontal Line & gen. b’o’ intersect. 2’ is a point where second horiz. Line & gen. c’o’ intersect. In this way locate all points on Fv. Project all on Tv.Join in curvature. For Development: Then taking each points true Distance From resp.generator from apex, Mark on development & join. o’ 1’ 2’ 3’ 4’ 5’ 6’ 7’ DEVELOPMENT HELIX CURVE A B C D E F G H O 1 2 3 4 5 6 7 X Y e’ a’ b’ d’ c’ g’ f’ h’ h a b c d e g f O 1 2 3 4 6 5 7