CHAPTER 8 DEVELOPMENTS

8.21 Developments

A development is a flat representation or pattern that when folded together

creates a 3-D object. Sheet-metal construction is the most common

application for developments and intersections. The resulting flat pattern

gives the true size of each connected area of the form so that the part or

structure can be fabricated.

8.22 Terminology The following terminology describes objects and concepts used in

developments and intersections:

A ruled surface is one that may be generated by sweeping a straight line,

called the generatrix, along a path, which may be straight or curved. Any

position of the generatrix is an element of the surface. A face may be a

plane, a single-curved surface, or a warped surface.

A plane is a ruled surface that is generated by a line, one point of which

moves along a straight path while the generatrix remains parallel to its

original position.

A single-curved surface is a developable ruled surface; that is, it can be

unrolled to coincide with a plane. Examples are the cylinder and the cone.

A warped surface is a ruled surface that is not developable. Some

examples are shown in Figure 8.20. No two adjacent positions of the

generatrix lie in a flat plane. Warped surfaces cannot he unrolled or

unfolded to lie flat.

A warped surface is a ruled surface that is not developable. Some examples

are shown in Figure 8.20. No two adjacent positions of the generatrix lie in a

flat plane. Warped surfaces cannot he unrolled or unfolded to lie flat.

A double-curved surface is generated by a curved line and has no straight-

line elements. A surface generated by revolving a curved line about a

straight line in the plane of the curve is called a double-curved surface

of revolution. Common examples are the sphere, torus, ellipsoid, and

hyperboloid.

A developable surface may be unfolded or unrolled to lie flat. Surfaces

composed of single-curved surfaces, of planes, or of combinations of these

types are developable. Warped surfaces and double-curved surfaces are

not directly developable.

8.22 Solids Polyhedra are solids which are bounded entirely by plane surfaces - for

exam pie cubes, pyramids, and prisms. Convex solids are ones that have

no dents. Convex solids with all equal faces are called regular polyhedra.

Examples of regular solids are shown in Figure 8.22.

A solid generated by revolving a plane figure about an axis in the

plane of figure is a solid of revolution.

Solids bounded by warped surfaces have no name – for example,

the screw thread.

8.25 Developments

The development of a surface is that surface laid out on a plane.

Practical applications of developments occur in sheetmetal work,

stone cutting, pattern making, packaging, and package design.

Single-curved surfaces and the surfaces of polyhedra can he

developed. Developments for warped surfaces and double-curved

surfaces can only be approximated.

It is common practice to draw development layouts with the inside

surfaces up. In this way, all folded and other markings are related

directly to inside measurements, which are the important dimensions

in all ducts, pipes, tanks, and other vessels.

The direction of development (i.e., unfolding or unrolling) must be

perpendicular to the intersection and is called stretch-out line.

8.26 Hems and Joints for Sheetmetal and Other Materials

Figure below shows a wide variety of hems and joints used in

fabricating sheet-metal parts and other items. Hems are used to

eliminate the raw edge as well as to stiffen the material. Joints and

seams may be made for sheet-metal by bending, welding, riveting, and

soldering and for package materials by gluing and stapling.

You must add materials for hems and joints to the layout or

development. The amount of material to add depends on the thickness

of the material and production equipment.

8.27 Developing a Prism

Figure below shows the intersection of a plane and a prism in the auxiliary

view as true size.

Step by Step 8.4 Developing a Prism Parallel Line Development

8.28 Developing a Cylinder

Figure below shows the intersection of a plane and a cylinder in the

auxiliary view as true size.

8.29 More Examples of the Developments

Radial Line Development

8.30 Transition Piece

A transition piece is one that connects two differently shaped, differently

sized, or skewed-position openings. In most cases, transition pieces are

composed of plane surfaces and conical surfaces, as shown in Figure

8.33. You will learn about developing conical surfaces by triangulation

next. Triangulation can also be used to develop, approximately, certain

warped surfaces. Transition pieces are used extensively in air

conditioning, heating, ventilating, and similar construction.

8.31 Triangulation Triangulation is simply a method of dividing a surface into a number of

triangles and transferring them to the development. To find the development

of an oblique cone by triangulation, divide the base of the cone in the top view into any

number of equal parts and draw an element at each division point, as shown in Figure 8.34.

Find the true length of each element. If the divisions of; the base are comparatively small,

the lengths of the chords may be used in the development to represent the lengths of the

respective arcs. Since the development is symmetrical, it is necessary to lay out only half

the development, as shown.