Developing the wall patterns

Chapter 5 turned the trough’s four walls into four part cones and gave each its pattern radii and angle. This chapter draws them. The end walls are ordinary cone patterns, bands of a sector with their apex within reach of a long trammel. The side walls are cone patterns too, but their apex is nearly ten metres away, so their arcs are marked by offsets from straight chords. Both carry radial division lines for forming and checking, a seam lap on one edge and a rim allowance on their top edge.

This chapter sets out both patterns with every number needed to mark them, adds the allowances, explains how the patterns are checked before cutting, and ends with a side pattern cut with straight edges because its curves were too gentle to see.

The end pattern

This is an illustration. Each end wall’s pattern is a band between two arcs centred on an apex: the rim arc at 1,824.8 mm radius and the bottom arc at 1,368.6 mm, spanning 20.36°. The shop marks it like this.

1. Strike a centre line and mark the apex at one end of it, on a scrap plate clamped beside the sheet if the sheet is not long enough.
2. Swing the rim arc at 1,824.8 mm and the bottom arc at 1,368.6 mm across the centre line.
3. Set out the band’s two radial edges, 10.18° either side of the centre line. Without a large protractor, use the chords: the rim arc’s ends are 645.1 mm apart, and the bottom arc’s 483.8 mm.
4. Check that each radial edge, from bottom arc to rim arc, measures 456.2 mm, the slant height.
5. Divide the band into six equal parts by radial lines from the apex, each 3.39° apart, stepping 108.1 mm along the rim arc.

The rim arc rises 28.7 mm above its chord at the middle, and the bottom arc 21.5 mm above its own: enough curvature to see clearly and to check with a template.

Figure 1Pattern dimensions for the trough's walls, net

Dimension	End wall	Side wall
Rim arc radius	1,824.8 mm	9,580.4 mm
Bottom arc radius	1,368.6 mm	9,124.1 mm
Pattern angle	20.36°	9.23°
Rim chord and rise	645.1 and 28.7 mm	1,541.7 and 31.1 mm
Bottom chord and rise	483.8 and 21.5 mm	1,468.3 and 29.6 mm
Rim and bottom arc lengths	648.5 and 486.4 mm	1,543.4 and 1,469.9 mm
Division lines	6 spaces of 3.39°	4 spaces of 2.31°

The side pattern by offsets

Each side wall’s pattern is a band between arcs of 9,580.4 mm and 9,124.1 mm radius, spanning only 9.23°. Its apex, nearly 9.6 metres from the rim edge, is off the bench and probably out of the shop. The arcs are marked instead by offsets from their chords.

The rim arc’s chord is 1,541.7 mm, and the arc rises above it by an amount that varies along the chord. Measured from one end of the chord, the offsets square to it are:

• at 192.7 mm, an eighth of the way along: 13.6 mm;
• at 385.4 mm, a quarter: 23.3 mm;
• at 578.1 mm, three eighths: 29.1 mm;
• at 770.8 mm, the middle: 31.1 mm;

and the same in reverse for the other half. The bottom arc is marked the same way from its own chord of 1,468.3 mm, rising 29.6 mm at the middle. The two chords are parallel and 454.8 mm apart, the slant height less the difference between the arcs’ rises, and the band’s radial edges join the chords’ ends.

A flexible batten bent through each arc’s nine points draws the arc. The method is exact at the points and accurate to well under a millimetre between them for curves this gentle.

Figure 2Offsets from the chord to the side wall's rim arc

At 192.7 mm13.6 mm

At 385.4 mm23.3 mm

At 578.1 mm29.1 mm

At 770.8 mm, middle31.1 mm

Division lines on the side pattern

The side pattern is divided into four parts by radial lines, 2.31° apart at the apex. With the apex out of reach, each division line is marked by its two ends: on the rim arc, at steps of 385.8 mm along the arc, and on the bottom arc, at steps of 367.5 mm. Joining the two ends of each pair gives a line that runs towards the distant apex, which is the generator the forming and the checks rely on.

A division line drawn square to the rim chord instead of towards the apex is nearly right on the side pattern, because its lines lean only slightly, but it is wrong by a few millimetres at the band’s ends, exactly where the side wall meets the end wall at the seam. The two-ended method avoids the guess.

Allowances

The net patterns are the walls’ mean surfaces. Each needs three allowances before cutting.

• Rim. The rim is rolled outward over a 10 mm rod. A three-quarter wrap of the sheet’s mean line round the rod uses about 0.75 × π × (10 + 1.6) = 27.3 mm of sheet. This is an illustration; the shop’s trial strip settles it at 27 mm, added outward from the rim arc, as a band of constant width following the arc.
• Seam lap. Each vertical seam is a 20 mm lap, riveted and sealed. The lap is added to one radial edge of each piece, parallel to that edge, not square to the arcs.
• Bottom. The bottom joint is made on the bottom plate’s flange, Chapter 7’s choice, so the wall’s bottom edge needs no allowance beyond a clean cut.

The lap and the rim allowance overlap at the top corner of each piece, where the rim will be rolled over the lap. The corner is notched there, as Chapter 7 shows, so that the rim does not have to wrap four layers of sheet.

Figure 3Marking a side wall pattern

1. 1Strike the rim chord, 1,541.7 mma straight datum on the sheet
2. 2Plot the rim arc's offsets13.6, 23.3, 29.1, 31.1 mm and back
3. 3Strike the bottom chord parallel, and plot its offsetsrising 29.6 mm at the middle
4. 4Join the arcs' ends with the radial edgeseach 456.2 mm along the wall
5. 5Mark division lines by their two ends385.8 mm steps on the rim, 367.5 on the bottom
6. 6Add the rim band and the seam lap27 mm and 20 mm
7. 7Check arc lengths and radial edges1,543.4, 1,469.9 and 456.2 mm

Marking the rim band on a gentle arc

The rim allowance is a band 27 mm wide outside the rim arc, and its outer edge is another arc from the same apex, 27 mm larger in radius. On the end pattern it is swung with the trammel at 1,824.8 + 27 = 1,851.8 mm. On the side pattern, with no apex to swing from, it is marked by measuring 27 mm outward from the rim arc at each of the nine offset points, square to the arc, which for a curve this gentle is square to the chord to within a fraction of a degree. The band’s outer edge then has the same shape as the rim arc, and the rolled rim will be the same height all round.

How accurate the marking must be

Small marking errors add up around an oval. If each of the four pieces’ rim arcs is marked 1 mm long, the trough’s rim girth is 4 mm long, and the rim rod of Chapter 3, cut to the correct length, leaves a 4 mm gap at its joint or forces the rim into a slight wave. If the radial edges are 1 mm long, the seams overlap unevenly from rim to bottom.

The shop’s target is ±0.5 mm on every arc length and radial edge, which keeps the girth within ±2 mm overall and the seams even. That is achievable with a steel tape, a flexible rule along the arcs and a fine scriber, and it is the reason for marking a single checked template rather than marking every piece afresh.

Hard templates for a batch

The running case is a batch of troughs, not one, and marking every piece from chords and offsets would be slow and would invite drift. The shop marks one pattern of each kind carefully, checks it, and cuts it from thin sheet as a hard template: one end template and one side template, each with its division lines scribed, its allowances included and its checks written on it.

Every wall piece in the batch is then scribed round the template, and the template is checked against its written numbers, arc lengths, radial edges and rises, at the start of each day’s cutting. A template that has been dropped and bent, or whose edge has worn from scribing, fails the check before it spoils a piece.

For a batch cut on a CNC table, the cutting file plays the same role, and the same checks are made on the first piece cut from it each time the program is loaded.

Patterns from CAD

A CAD model of the trough can produce the same patterns. Modelled as a lofted sheet metal body between the rim oval and the bottom oval, with the bottom struck from the same centres, the wall is a ruled surface made of the four part cones, and the program’s unfold gives four flat pieces. Each should match this chapter’s numbers: arc radii, angles, chords and rises.

Two things can go wrong. If the bottom oval in the model was drawn with its own end radius rather than offset from the top, the loft is not made of cones, and the unfold, though it will still produce flat shapes, will put creases where the shop expects smooth seams. And if the program’s flattening uses a K-factor for rolled cones that differs from the mean line, the arc lengths shift slightly. Comparing the unfolded side piece’s rim arc with 1,543.4 mm, and its radial edge with 456.2 mm, catches both.

Checking a pattern before cutting

Three checks on each pattern catch nearly every marking error.

1. Arc lengths. Measured along the curve with a flexible rule, the rim arc should be 648.5 mm on an end pattern and 1,543.4 mm on a side pattern; the bottom arcs 486.4 and 1,469.9 mm. These are the wall’s girths, from Chapter 3, and they must match the rim and the bottom.
2. Radial edges. Each edge, from bottom arc to rim arc, should be 456.2 mm, the slant height, on every pattern. That is the check that would have caught Chapter 5’s error.
3. Rises. Each arc’s rise above its chord should match the table: 28.7 and 21.5 mm on an end pattern, 31.1 and 29.6 mm on a side pattern.

A pattern that passes all three will form into a wall that meets its neighbours, the rim and the bottom.

The error: a side pattern cut straight

This is an illustration. Looking at the side pattern’s numbers, a fabricator reasoned that an arc rising only 31 mm over 1.5 metres was “near enough straight”, and cut the side pieces as trapezoids with straight rim and bottom edges, 1,543.4 mm along the rim and 1,469.9 mm along the bottom.

On a cone, a straight edge is not a level edge. The rim of a correctly cut side piece is an arc on the pattern, and that arc becomes a level line when the piece is formed. Cut straight, the piece’s rim edge was 31.1 mm short of the correct arc at its middle, measured along the wall, which put the middle of the rim about 30.7 mm lower than its ends once formed. The bottom edge had a similar error the other way, standing proud of the bottom plate at its middle.

The fault showed at assembly, when the side pieces’ rims dipped between the seams and their bottoms would not sit on the bottom’s flange. Both side pieces were recut from offset-marked patterns. The dip had been measured simply: with the trough’s bottom on a level floor, the rim’s height above the floor was read at the two seams and at the middle of each side, giving 450 mm at the seams and about 419 mm at the middle. That five-minute check, made on the first trough of every batch, now appears on the shop’s assembly sheet, because a rim that is not level is the most visible sign of a pattern error on any flared vessel. The two scrapped side pieces and a morning’s assembly would have paid for several hard templates. The lesson was the one this chapter is built on: a cone pattern’s edges are arcs, however gentle, and the rise of 31 mm is precisely the error made by ignoring it.

Figure 4Pattern checks before cutting the walls

• Do: Rim and bottom arcs drawn as arcs, by trammel or offsetsrises 28.7 / 21.5 and 31.1 / 29.6 mm
• Do: Arc lengths equal the wall's girths648.5 / 486.4 and 1,543.4 / 1,469.9 mm
• Do: Radial edges equal the slant height456.2 mm on every pattern
• Do: Division lines run towards the apexmarked by both ends
• Do: Seam lap parallel to the radial edge, rim band following the arc20 and 27 mm
• Avoid: Side edges cut straight because the curve is gentlerim 30.7 mm low at the middle

Exercise: the smaller trough’s end pattern

The smaller trough from earlier exercises has end radii of 200 mm at the rim and 150 mm at the bottom, and the same 9.46° flare. Find its end pattern’s radii, angle, rim chord and rise, and its rim arc length.

Worked answer.

• Pattern radii: 200 ÷ 0.1644 = 1,216.6 mm and 150 ÷ 0.1644 = 912.4 mm; they differ by about 304.1 mm, the smaller trough’s slant height, √(300² + 50²).
• Pattern angle: the same 20.36°, because the plan angle and the flare angle are unchanged.
• Rim chord: 2 × 1,216.6 × sin 10.18° = 430.1 mm; rise 1,216.6 × (1 − cos 10.18°) = 19.2 mm.
• Rim arc length: 1,216.6 × 20.36° in radians = 432.4 mm, two thirds of the running case’s 648.5 mm.

The smaller trough’s patterns are the running case’s scaled by two thirds, angles unchanged. A shop that makes several sizes of the same proportions can keep one set of angles and scale every length.