Conical Hopper Pattern: Radial-Line Development and Method Selection
A sheet-metal shop must fabricate a conical hopper in 2 mm mild steel sheet. The finished outside dimensions are: bottom diameter 400 mm, top diameter 200 mm, and a vertical height of 300 mm between the two parallel faces. The cone is rolled from a single flat pattern and closed with a lap joint along one radial edge, with a 25 mm lap. The foreman asks the trainee to lay the pattern out directly on the sheet using radial-line development, and to state the sector radius, the sector angle and the blank size, so that blanks can be cut from 1250 × 2500 mm sheets without excessive waste. Two other jobs arrive the same week: a 90° segmental cone elbow made from the same cone, and a square-to-round transition from a 300 × 300 mm square duct to a Ø250 mm round duct over a height of 350 mm. The trainee proposes to use the same radial-line method for all three.
1. Compute the sector radius, the sector angle and the inner arc radius for the hopper pattern, and check the result. Then estimate how the blanks nest on a 1250 × 2500 mm sheet.
Read the worked answer
Step 1 - slant height of the frustum L = √((R − r)² + h²) = √((200 − 100)² + 300²) = √(10 000 + 90 000) = √100 000 = 316.2 mm
Step 2 - slant height of the full cone (apex to base) By similar triangles, S = L × R/(R − r) = 316.2 × 200/100 = 632.5 mm This S is the sector radius of the development.
Step 3 - sector angle Outer arc length must equal the base circumference: π × 400 = 1256.6 mm. θ = arc / S = 1256.6 / 632.5 = 1.987 rad = 113.8°
Step 4 - inner arc radius S − L = 632.5 − 316.2 = 316.3 mm; check: 316.3 × 1.987 = 628.5 mm ≈ π × 200 = 628.3 mm ✔
Step 5 - blank envelope and nesting Width = 2S sin(θ/2) = 2 × 632.5 × sin 56.9° = 1060 mm; depth = S(1 − cos 56.9°) = 287 mm. Adding the 25 mm lap and about 3 mm trim gives roughly 1085 × 290 mm per blank. On a 1250 × 2500 sheet, one row of blanks fits across the width (1085 < 1250) and about eight rows fit along the length (8 × 290 = 2320 < 2500), especially when successive sectors are turned apex-to-apex to interlock the curved edges.
2. For the 90° segmental elbow, explain why the simple sector of the hopper cannot simply be reused, and outline the correct development procedure.
Read the worked answer
A plain frustum is cut by two planes perpendicular to its axis, so every element (apex-to-base line) has the same true length and the development is bounded by two concentric arcs. In a segmental elbow each piece is cut by an oblique plane, so the elements have different lengths and the joint line is a curve, not an arc.
Correct procedure (radial line with true lengths)
- Draw the elevation of one segment and divide the base circle in the plan into 12 equal parts.
- Project each division point up to the oblique joint line in the elevation.
- Rotate each of those points about the apex into a generator that is parallel to the VP; that generator shows the true length of that element.
- Develop the full cone sector as before, then set the true length of each element out along its own radial line, measuring from the apex.
- Join the points with a smooth curve to get the shaped edge; repeat for the second cut.
Build-up of a 90° elbow Use three pieces: two end pieces (one cut perpendicular, one oblique - each half a segment) and one middle piece cut obliquely at both ends. Every piece is developed by the same divided-circle method; the two end pieces are identical mirror images, so only one pattern is needed. Add the lap/stitch allowance at every joint, not just at the longitudinal seam.
3. The square-to-round transition is also offered to the trainee. State whether the radial-line method applies, and give the correct method with its key construction step.
Read the worked answer
No - radial-line development does not apply. The surface is not a cone or pyramid that turns about a single axis; it has no common apex. It is made of four triangular flat plates (the corners) blended with four oblique cone segments (the sides), so elements do not radiate from one point. Treating it as a cone would produce a pattern that cannot be rolled to shape.
Correct method: triangulation
- Draw plan and elevation. Divide the round end into 12 equal parts and number the points; label the four square corners (e.g. 1, 4, 7, 10).
- Join each circle division to the two adjacent square corners, so the surface is broken into a series of adjacent triangles.
- Find the true length of every triangle side: for a line joining a corner to a circle point, TL = √(plan length² + height²). The plan length is measured on the plan between the two points; height is the transition height, 350 mm.
- Lay out the triangles in order on the sheet, starting from a known corner edge, using the three true lengths of each triangle (side-side-side).
- Check closure: the developed round-end points must give the correct arc length (about π × 250 / 12 between successive points) before cutting. Add seam and lap allowances on the two mating edges.