CHAPTER XIV
TRIANGULATION OF TRANSITION PIECES
Problem 48
TRANSITION BETWEEN A SQUARE PIPE AND THE SECOND PIECE OF AN ELBOW
87. Transition between a Square Pipe and the Second Piece of an Elbow.—In order to save height the sheet metal draftsman is often compelled to design a transition between a square or a rectangular pipe and the second piece of a round pipe elbow. In other words, the transition takes the place of the first piece of the elbow.
In order to accomplish this purpose it is necessary to incline the plane of the top to that of the base of the transition. Figure 278 shows the elevation of the small end of a two-piece 45° elbow with the transition attached in its proper position. In drawing this view care must be taken to get the true miter line according to the rules laid down in Chapter III.
Directly above the small end of the elbow a half-profile is drawn and divided into eight equal parts. The divisions are numbered as shown, and extension lines carried down through the elevation until they meet the miter line. Numbers are placed on the miter line to correspond to the numbering of the half-profile.
As has already been pointed out in previous chapters, whenever a cylinder is cut by an inclined plane, the section on that cutting plane is an ellipse. In order to obtain the proper spacing on the pattern a true section on the miter line must be developed in the following manner. Line 1-9 of Fig. 279 is an extension of line 1-9 of the half-profile. Upon this line should be placed the exact spacing of the miter line, and perpendiculars erected at each point. These perpendiculars are intersected by extension lines brought over from correspondingly numbered points in the half-profile. A curved line traced through the intersections thus obtained gives a true section on line 1-9 (miter line) of the elevation.
From each intersection of the miter line of Fig. 278 extension lines are dropped vertically for an indefinite distance. The profile of the square base of the transition is next drawn in its proper position as shown by A, B, C, D of Fig. 280.
The horizontal center line, EF, locates the center of the circle which is also the profile of the round pipe. The extension lines from the miter line of Fig. 278 should divide this circle into equal spaces. If they fail to do so, an error in drawing has been made which should be corrected before proceeding further. The order of development must now be decided upon. The
triangles naturally divide into two groups, Group A being those triangles having their base lines starting from point A, and Group B starting from point B. Besides these triangles there is a starting line which is the hypotenuse of a right triangle upon the base line E-1 and a finishing line upon base line 9–F.
The diagrams of triangles are now constructed by drawing short horizontal lines equal in length to their respective base lines with corresponding numbers and letters. Perpendiculars are erected at one end of each of these horizontals. Since the plane of the top of the transition is inclined, the altitudes of these triangles vary. This variation is shown in Fig. 278 where the altitudes of various triangles are plainly marked. In determining the altitude of any point it should be remembered that the altitude is always the perpendicular distance between the plane of the base and the point in question. These altitudes should be placed on the proper perpendiculars, and in this connection it may be noted that the altitude always changes with the number; that is, wherever the number 2 occurs the altitude of 2 as shown in Fig. 278 must be used. The hypotenuses of the several triangles are now drawn.
The pattern development is started by drawing a horizontal line equal in length to the side DA of Fig. 280. Upon this line the center point E should be placed. A perpendicular line is erected at point E equal in length to the hypotenuse of triangle E-1. This establishes point 1, and the distance from point A to point 1 should correspond exactly in length to the hypotenuse of triangle A-1.
Since the center line EF of Fig. 280 divides the figure into two equal parts the pattern can be developed on each side of line E-1 of Fig. 283 simultaneously. The experienced draftsman always takes advantage of this fact when a whole pattern is to be developed. The line D-1 in Fig. 283 is next drawn, and when a distance is laid off from point A a like distance is also laid off from point D. With point A as a center and a radius equal to the hypotenuse of triangle A-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 Fig. 279. In like manner points 3, 4, and 5 are established, but it must be remembered that the distances between figures must be taken each time from the true section, Fig. 279. With point 5 as a center and a radius equal to the hypotenuse of triangle 5-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 side A-B of Fig. 280. This establishes point B.
From B as a center and with the several hypotenuses of Group B triangles as radii, points 6, 7, 8, and 9 are established in the same manner as were points 2, 3, 4, and 5.
With point 9 as a center and a radius equal to the hypotenuse of triangle 9 to F, an arc is drawn bearing away from point B. This is intersected by an arc drawn from point B with a radius equal to line BF of Fig. 280. This establishes point F.
A curved line is now drawn through points 1 to 9. Straight lines are drawn connecting points F, B, and A. If both sides of the pattern have been worked simultaneously, the whole pattern has been developed and locks may now be added as shown.
The straight lines representing the square base are treated according to the type of joint adopted for the system of piping, of which this "fitting" is a part. Problem 49
TRANSITION BETWEEN AN OVAL PIPE AND THE SECOND PIECE OF AN ELBOW
88. Transition between an Oval Pipe and the Second Piece of an Elbow.—Figure 284 is constructed by first drawing an elevation of the required elbow according to the directions given in Chapter III. The first piece of the elbow is then erased and the elevation of the transition added. A half-profile is then drawn adjacent to the small end of the elbow and divided into eight equal parts. These divisions are numbered as shown, and extension lines are carried through the elevation until they meet the first miter fine, where corresponding numbers are placed at each intersection.
From the intersections of the miter fine vertical extension lines are dropped. These extension lines are crossed by the horizontal center line AK as shown in Fig. 286. About this center line the plan, Fig. 286, is now drawn. The vertical extension lines divide the circumference of the circle into equal parts. One-half of the oval profile may now be equally divided, although the straight line EF may be considered as one space.
A true section on the miter line should now be developed as shown by Fig. 285, in the following manner. Extension lines are carried horizontally from points 1 and 9 of Fig. 284, and a new line 1-9 is drawn parallel to line 1-9 of Fig. 284. The exact spacings of the miter line are transferred to this line, and perpendiculars are erected at each point with numbers to correspond. Upon each of these perpendiculars and on each side of the line 1-9, a distance is laid off equal to the distance from a correspondingly numbered point in the half-profile to the center line 1-9 of the half-profile. A curved line traced through the intersections thus obtained is a true section on miter line 1-9 of Fig. 284.
Before the base lines for the triangles can be drawn in Fig. 286, the order of triangulation must be determined. In a transition of this kind the elements of the surface must alternate between the upper and lower bases in order to have sufficient data with which to develop the pattern. A standard order of triangulation is given below, together with the altitude for each triangle.
The base lines may now be drawn in Fig. 287 according to this order. 
| Triangles. | Altitudes. | Triangles. | Altitudes. | Triangles. | Altitudes. |
| A to 1 | 1 | D to 4 | 4 | G to 7 | 7 |
| 1 to B | 1 | 4 to E | 4 | 7 to H | 7 |
| B to 2 | 2 | E to 5 | 5 | H to 8 | 8 |
| 2 to C | 2 | 5 to F | 5 | 8 to J | 8 |
| C to 3 | 3 | F to 6 | 6 | J to 9 | 9 |
| 3 to D | 3 | 6 to G | 6 | 9 to K | 9 |
The diagrams of triangles, Figs. 287 and 288, are now constructed by drawing short horizontal lines equal to the base lines of Fig. 286 with letters and numbers to correspond. Upon perpendiculars erected at one end of each of these lines the proper altitude is placed. The true lengths of the altitudes are plainly marked in Fig. 284.
The pattern development is started by drawing a straight line and setting off upon it a distance equal to the hypotenuse of triangle A to 1. Next in order comes triangle 1-B, so with point 1, Fig. 289, as a center and with a radius equal to the hypotenuse of triangle 1-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 line AB of Fig. 286.
With B as a center and a radius equal to the hypotenuse of triangle B-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 Fig. 285, thereby establishing point 2 of the pattern. In this manner the entire half pattern is developed by following the order of triangulation, taking the spaces between the figures from Fig. 285, and the spaces between the letters from Fig. 286. Should the draftsman require a whole pattern, he would work both ways from the center line A-1, as the development progressed.