Helical Handrail Chord Approximation
A helical handrail is to be fabricated for a 270° spiral stair around a central cylindrical core. The handrail centreline follows a helix of radius 1.50 m about the core axis and rises 3.60 m over the 270° turn. The rail is a 50 mm diameter steel tube, and the shop intends to bend it from straight lengths, checking the curve at posts. Because bending is expensive, the fabricator proposes to approximate the helix by straight chords between posts spaced at 90° intervals, with the posts themselves set on the true helix. The designer is asked to approve the shop drawing. The stair is in a public building, and the handrail must not deviate from the specified centreline by more than 10 mm in plan, because the clear width to the inner wall is tight. The fabricator argues that four chords over 270° will look smooth enough.
1. Calculate the true length of the handrail centreline and the slope angle of the helix. Show the development used.
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Unroll the cylinder of radius r = 1.50 m. The horizontal arc for 270° is s = rθ = 1.50 × (3π/2) = 7.069 m. The rise is h = 3.60 m. True length L = √(s² + h²) = √(7.069² + 3.60²) = √(49.97 + 12.96) = √62.93 = 7.93 m. The slope angle to the horizontal is α = tan⁻¹(h/s) = tan⁻¹(3.60/7.069) = 27.0°. On the development the helix is a straight line of length 7.93 m at 27.0° to the horizontal; that straight line is the set-out for a continuous bend, not for chords.
2. Evaluate the proposed 90° chord approximation. Calculate the maximum plan deviation from the true helix and compare with the 10 mm limit. Determine the maximum post spacing that would satisfy the limit.
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For a chord subtending Δθ at radius r, the maximum plan deviation, or sagitta, is r[1 − cos(Δθ/2)]. At 90°, Δθ/2 = 45°, so deviation = 1.50(1 − 0.7071) = 0.439 m = 439 mm, about 44 times the 10 mm limit. Four chords cannot be approved. For the limit: 1.50[1 − cos(Δθ/2)] ≤ 0.010, so cos(Δθ/2) ≥ 1 − 0.006667 = 0.993333, Δθ/2 ≤ 6.62°, and Δθ ≤ 13.2°. Thus posts or check points must be at about 13° spacing or closer, which means at least 21 intervals over 270°, if the rail is made from straight chords. That many joints is not a sensible handrail; the result points to continuous bending.
3. Recommend a fabrication strategy that meets the 10 mm plan tolerance and is buildable. Include one check the designer should require before approving the shop drawing.
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Specify continuous helical bending of the 50 mm tube on a CNC tube bender or a three-roll bender with a helical mandrel, using the developed straight-line length of 7.93 m and a plywood or MDF template of the plan curve. If segmentation is unavoidable, use no more than 10° per chord, which gives 5.7 mm deviation, with welded joints staggered away from the hand-contact zone and ground flush. Require a first-article check: set the rail on the true posts, measure plan deviation at mid-chord with a string or laser to the core, and verify that the rise over each 90° quadrant is 1.20 m. Also require the tube supplier's minimum bend radius to be checked against the 1.50 m centreline radius; a 50 mm tube may need a larger radius or induction bending. Do not approve the 90° chord scheme.