Ring Taper & Key Draw — segmental lining

Ring taper and cruciform joint check

Universal tapered ring built by rolling each ring a whole number of clock positions. The taper delivers curve capability; the roll must also keep the radial joints of consecutive rings apart.

1 Visualisation

Figure 1 — Developed elevation, one ring (unrolled at the extrados, viewed from outside)

key segment S1ordinary segmentradial jointbolt position Pnspear bolt / dowel

Unrolled at the extrados and viewed from outside the ring. The strip is drawn at the nominal width B: the ±T taper is only about 1.3 % of the width and would be less than a line thickness at this scale, so it is reported numerically instead of being drawn. Radial joints are inclined ι in the developed plane, so each ordinary segment is a parallelogram; the key S1 is a trapezoid whose flanks splay by ±α, widening it at the leading edge by B tan α on each flank. Numbering runs S1 for the key then anticlockwise S2, S3 … so the two segments meeting at the invert are the highest numbers. P1 is the circumferential bolt on the tunnel setting-out point.

Figure 2 — Two consecutive rings developed — joint stagger and cruciform check

segment lengthjoint staggerstagger below tolerance — cruciformkey

The cruciform check drawn the way it is dimensioned on a ring build drawing. Both rings are unrolled on the same angular axis; the upper ring is the one just erected, rolled c clock positions relative to the ring behind it. The dimension chain measures the arc distance at the extrados between every consecutive pair of radial joints taken from the two rings, so the short dimensions are the joint staggers and the long ones the segment lengths between them. Any stagger below the tolerance ε is flagged red — two radial joints that close up form a cruciform, a four-way joint intersection with no continuous load path across it.

Figure 3 — Axial view, both rings

Figure 4 — Buildable roll positions

Figure 4 evaluates every one of the N clock positions and plots the smallest joint stagger each one produces, so the buildable rolls can be read straight off. Bars below the tolerance line are cruciform positions and must not be used.

2 Calculation

Key draw

Key draw is the axial distance the key must be held back from the leading edge of the ring so that it can be pushed radially into the gap between the counter-key segments without fouling them. It governs the TBM ram stroke.

1 Visualisation

Figure 5 — Key insertion, developed plan on the intrados

Trailing edge held back

key, clearskey, foulscounter-key segmentkey draw KD

Plan on the intrados, looking radially outwards, with the key shown part-inserted. Because the key flanks and the faces of the gap taper at the same rate, the clearance on each flank stays constant along the whole overlap — it does not vary with how far the key has entered. That is why a single clearance check is rigorous: setting the flank clearance equal to g / cos α′ recovers x_max exactly. Slide the key in to see the clearance close up; past x_max the outline turns red.

Figure 6 — Ram stroke demand

Figure 7 — Sensitivity of key draw

Figure 7 is the reason α and g matter. Both curves are steep, so the surplus ram stroke shown in Figure 6 can be consumed by a modest change in either assumption. Confirm both against the segment detail drawing before this check is relied on.