CHAPTER XV
DEVELOPMENTS BY SECTIONS
Problem 50
THE FRUSTUM OF A SCALENE CONE
89. The Frustum of a Scalene Cone.—Triangulation is the universal tool of the Sheet Metal Draftsman. Any surface capable of being developed can be developed by this method. However, in the case of the cone and the cylinder less laborious methods are available which are just as accurate. In many problems that cannot be classed as parallel line or tapering form developments, there is a shorter method known as Development by Sections. This method is generally employed in problems where solids can be cut into two equal parts by the cutting planes. Figure 290 shows a frustum of a scalene cone cut by a vertical plane in such a manner as to divide it into two equal parts. Figure 291 shows one of these halves placed so that the cutting plane assumes a horizontal position. If the semicircular ends (profiles) were divided into four equal parts, and perpendiculars dropped from each of these points to the base lines, a model cut along these hues would show the sections as pictured in Fig. 291. It is evident that hues 1-A and 5-E are true length hues, while C-4, 4-D, etc., are upper bases of trapezoids and must be developed. After the pattern has been developed, a cardboard model of Fig. 291 made from Fig. 292 and the diagram of sections, Fig. 293, should be constructed to aid in the visualisation of future problems. Figure 292 is a plan or top view of the object with half-profiles attached to each end. These half-profiles are divided into equal parts and extension lines carried to lines 1-5 and AE as shown. The divisions are then numbered and lettered. Before the base lines of the sections can be drawn in, the order of sections must be determined. The standard adopted for triangulation can still be adhered to and the order would read A-2, 2-B, B-3, 3-C, C-4, 4-D, and D-5. Since lines l-A and 5-E are true lengths they need not be mentioned in the order. Having determined the proper order, the base lines on the plan are now drawn in as shown in Fig. 292.
The diagram of sections. Fig. 293, is now constructed by drawing short horizontal lines equal in length to the several base lines of Fig. 292 and with numbers and letters that correspond to the order adopted. Perpendicular lines are erected at each end of these lines. Upon these perpendiculars are set off the lengths of the correspondingly numbered extension lines in the half-profiles. Attention is called to the fact that points 1 and 5 have no altitudes, since they fall on the horizontal plane upon which the entire figure
rests. Straight lines connecting the points established upon these perpendiculars are the true lengths from which the pattern is developed.
The pattern is started by drawing a Horizontal line such as line 1-A of Fig. 294. With A as a center and a radius equal to the hypotenuse of section A to 2, an arc is drawn bearing away from point 1. This is intersected by an arc drawn from point 1, with a radius equal to the distance 1-2 of the profile. This establishes point 2. With point 2 as a center and a radius equal to the upper base of section 2-B, an arc is drawn bearing away from point A. This is intersected at B by an arc drawn from point A, with a radius equal to distance AB of the profile. In this manner all the points of the pattern are located in the order previously adopted. Curved lines passing through these points give the half pattern for the entire frustum. Problem 51
CENTER OFFSET BOOT
The half-profiles are then drawn in their relative positions and divided into equal parts. It has been observed that in treating oval profiles the straight sides of the oval are never divided. The divisions of the curved portions are numbered and lettered and extension fines carried from each division to the miter lines. From this view the pattern of the round end and also the oval end of the fitting can be developed according to the rules given in Chapter III.
The center or transition piece of the fitting is developed by means of sections. Figure 296 shows this transition moved to one side in order to avoid a confusion of lines. The intersections of both miter lines are also transferred. Perpendiculars are erected at each intersection and the distances from points B, C, and D, to the center line of the half-profile, Fig. 295, set off on corresponding lines at one end of Fig. 296, and distances from points 2, 3, 4, 5, and 6 to the center line of the half-profile set off on corresponding perpendiculars at the other end of Fig. 296. Curved lines traced through these points give the true sections on the miter lines.
The diagram of sections, Fig. 299, is now constructed by drawing horizontal lines equal in length to the base lines in Fig. 296. Perpendiculars are erected at each end of these lines and lettered and numbered to correspond to the base lines. Upon these perpendiculars are set off distances equal to the length of correspondingly numbered and lettered lines in the true sections of Fig. 296. The lengths of straight lines connecting these points are the true lengths of the lines needed to develop the pattern. The section A to 2 has for its base line the true length line A to 1 of Fig. 296; therefore, both base and hypotenuse of this section are used in the development of the pattern. This is also true of section E to 7.
The pattern, Fig. 300, is started by drawing a vertical line
equal in length to line A-1 of Fig. 296. With point A as a center and a radius equal to the hypotenuse of section A to 2, an arc is drawn bearing away from point 1. This is intersected by an arc drawn from point 1 with a radius equal to line 1-2 of the true section. Fig. 296. This establishes point 2. With point 2 as a center and a radius equal to the upper base of section 2 to B, an arc is drawn bearing away from point A. This is intersected by an arc drawn from point A with a radius equal to distance AB of the true section, Fig. 296, thereby establishing point B. In this manner all of the points of the pattern are fixed and the curved and straight lines of the pattern drawn in. Care should be observed with regard to these items:
(a) The spacing between the points of the pattern must be taken from corresponding spaces in the true sections.
(b) Distances 1-2 and 7-6 are greater than any of the other spaces and must be connected by straight lines.
(c) A chisel point must be used in the compass to assure fine lines without which the necessary accuracy cannot be attained. The whole pattern may be produced by copying on the other side of line A-1 the half which has already been drawn.
The pattern may be checked for accuracy by ascertaining whether or not the angle E-7-6 is a right angle. If this angle is of more or less than 90°, the pattern is incorrect. In this connection it should be observed that a slight error in any measurement will throw the whole pattern "out of true." Problem 52
SQUARE TO ROUND SPLIT HEADER
91. Square to Round Split Header.—This type of fitting is often used where a fan with a rectangular outlet must supply two round pipes running in different directions.
This problem presents a case wherein development both by triangulation and by sections may be employed in order to obtain the pattern.
The pattern is in reality two square to round transitions mitering upon each other. Where the two transitions come together a miter line is produced. A true section on this miter line must be developed. The practice of assuming a section, common to some drafting rooms, results in a more or less distorted fitting, according to the experience the draftsman has had in designing such fittings. The workman in forming and assembling the fitting has difficulty in that he must compel the assembly to take an unnatural shape.
The plan, Fig. 301, is first drawn according to the dimensions taken at the job. The profiles are then divided into the same number of equal spaces. After the order of triangulation has been determined, base lines are carried to the corners M and N of the rectangle.
The elevation is drawn in its proper location by means of extension lines carried from the plan. The miter line is drawn in both plan and elevation and should pass through the intersections of the base lines, Fig. 301, at points A, B, C, D, and E; it should also pass through the elevation of the elements, Fig. 302, at points F, G, and H.
A true section on this miter line should now be developed by drawing a horizontal line equal in length to twice the distance AE of Fig. 301. The center point of this line. Fig. 303, should be lettered A, and a perpendicular center line erected. Upon each side of this center, points B, C, D, and E should be located exactly as they appear on the miter line of Fig. 301. Perpendiculars are now erected at each of these points. Point B in Fig. 301 falls on line N-1, and point H falls on line N-1 of Fig. 302; therefore, the perpendicular height of point H from the base line of Fig. 302 should be set off on perpendicular B of Fig. 303. For the same reasons point G should be located on perpendicular C and point F on perpendicular D. A curved line traced through these points gives a true section on the miter line.
Because of the intersections of the miter line it will now be necessary to revise the order of triangulation originally adopted when the fitting was considered as two separate square to rounds. The original and revised orders are as follows:
| Original Order. | Revised Order. |
| N to 1 | N to B and 1 |
| N to 2 | N to C and 2 |
| N to 3 | 3 to M |
| M to 4 | M to 4 |
| M to 5 | M to 5 |
| 5 to K | 5 to K |
As will readily be seen the revision is based on the fact that lines N-l, N-2, and N-3 of Fig. 301 cross the miter line at points B, C, and D.
The diagram of triangles, Fig. 304, is now constructed by drawing horizontal lines equal in length to the base lines in Fig. 301. Perpendicular lines are erected at each end of these lines and also at points B, C, and D. Since both the upper and the lower planes of the fitting are parallel there will be but one altitude to the triangles. This altitude is shown in Fig. 302 and should be placed on perpendiculars 1, 2, 3, M, 4, 5, and K of Fig. 304. The hypotenuses of these triangles may now be drawn and points H, G, and F located by the intersection of perpendiculars B, C, and D with the respective hypotenuses.
The pattern, Fig. 305, is started with a horizontal line equal to line N-O of Fig. 301. With N and as centers and a radius equal to the hypotenuse of triangle N to 1, intersecting arcs are drawn above the line, thereby locating point 1. Since both sides of the fitting are equal the pattern may be developed from points N and O simultaneously. The remainder of the pattern is developed exactly as was the square to round transition of Chapter XIII.
After the entire pattern has been developed the miter cut is developed as follows: Point E is located on line MN of Fig. 305 exactly as it appears on line MN of Fig. 301. The hypotenuse of triangle A to D is placed on line A-3 of Fig. 305, thereby locating point F. Placing the hypotenuse of triangle A to C on line A-2 of Fig. 305 locates point G, and the hypotenuse of triangle A to B placed on line A-l locates point H.
The miter cut is drawn with straight lines between points E and F and curved lines connecting points F, G, and H of Fig. 305. This completes the pattern except for locks and riveting laps.
In case the round pipes are of unequal diameter the order of triangulation is altered somewhat, but the general method of procedure remains the same. The true section on the miter line is developed by considering the fitting as a transition between the round pipe of larger diameter and the whole of the rectangular base. This section is used for the development of both pieces and the order would read:
| Order. | Proper Altitudes. | Spaces obtained from. | ||
| Triangles | K to 5 | True altitude of all triangles as shown in Fig. 301 | Circular profile in Fig. 301 | |
| 5 to M | True altitude of all triangles ss shown in Fig. 301 | Base Rectangle in Fig. 301 | ||
| M to 4 | True altitude of all triangles as shown in Fig. 302 | Circular profile in Fig. 301 | ||
| M to 3 | True altitude of all triangles as shown in Fig. 302 | Circular profile in Fig. 301 | ||
| 3 to E | True altitude of all triangles ss shown in Fig. 302 | Base Rectangle in Fig. 301 | ||
| Sections | 3 to D | True altitude and alt. of F, Fig. 303 | E to F in true section, Fig. 303 | |
| D to 2 | True altitude and alt. of F, Fig. 303 | 3 to 2 in true section, Fig. 301 | ||
| 2 to G | True altitude and alt. of G, Fig. 303 | F to G in true section, Fig. 303 | ||
| G to 1 | True altitude and alt. of G, Fig. 303 | 2 to 1 in true section, Fig. 301 | ||
| 1 to B | True altitude and alt. of H, Fig. 303 | G to H in true section, Fig. 301 | ||
| 1 to A | True altitude and alt. of H, Fig. 303 | H to A in true section, Fig. 301 | ||
It is required that the pattern be redeveloped according to the above order and the results compared with the first development.
92. Related Mathematics on Split Headers.—The question as to how large the branch pipes can or should be made often arises in fan or blower design. The following factors enter into the consideration of such questions :
(a) What percentage of the total volume available must be delivered in a given direction?
(b) How many and what kinds of machines are to be served by the branch pipes? (c) Losses caused by friction, and the effects upon static and velocity pressures caused by changes in cross-sectional area of the duct.
Item (c) is a matter that largely concerns the engineer although the sheet metal worker would do well to have some understanding of these things.
Items (a) and (b), however, are matters of common arithmetic and the following problems are based on them.
Problem 52A.—The fan outlet measures 19"×35½". One branch of a split header is equal in area to 69 per cent of the area of the outlet. If both branch pipes are round, what are their diametrs?
Problem 52B.—A 20" pipe carries 75 per cent of the air from a split header. What is the diameter of the other round pipe?
Problem 52C.—A split header has one 15" and one 18" branch. What will be the dimensions of the rectangular opening 19" wide that will accommodate these two branches?
Problem 52D.—On one side of a fan the machines to be served require four 6", three 4", and two 3" pipes, while those on the other side require six 8", three 6", and two 4" pipes. What will be the area of the branch pipes that are needed to serve these machines? (Hint: Loss of head not considered.)