Cutting length of spiral stirrups (bored piles / circular columns): the Pythagorean development formula, the centreline-diameter basis, and the setting of horizontal ends, hooks and laps

Topic Positioning

The cutting-length computation of spiral stirrups in cylindrical members (bored and hand-dug piles, circular columns, tubular columns), and the detailing basis of each parameter (centreline diameter, pitch, horizontal ends, hooks, laps).

Core Conclusions

1. The length formula (geometric development)

One turn of spiral, cut and straightened, is a right triangle: one leg the horizontal projection of one turn - the circumference pi D - the other the pitch p (the clear vertical distance between adjacent turns); the hypotenuse is the true one-turn length.

1. One-turn true length: l1 = root of [(pi D)2 + p2]
2. One spiral bar totals: L = sum over i of (Hi / pi) x root of [(pi D)2 + pi2] + L(horizontal ends) + L(hooks) + L(laps)
   • summed segment by segment: the dense and plain zones each take their own height Hi and pitch pi, computed then added; a pile / column with dense and ordinary segments must never merge into one pitch.
3. Stirrup centreline winding diameter: D = member diameter d - 2 x cover c + stirrup diameter dstirrup (valid where cover runs from main-bar skin to member face). Where the drawing notes cover to the stirrup skin, D = d - 2c - dstirrup. Verify against the shop drawings / albums - the most error-prone point of the computation.
4. Quick per-metre cage content (table use): l = (1000 / p) x root of [(pi D)2 + p2] + pi d / 2 (mm per m, d the stirrup diameter) - general for cylindrical members.
5. Mass conversion: W = L x unit theoretical mass, quick g about 0.00617 x d2 (kg/m, d the nominal diameter in mm) - d8 = 0.395 kg/m.

2. Basis of each detailing parameter

1. Horizontal ends: spirals start and end with horizontal turns of not less than one and a half turns (about three turns over both ends, i.e. about 3 pi D) - DBJ 52/T 088-2018 clause 6.4.2 item 2 sub 2).
2. Hook angle and straight tail (GB 50666-2011 clause 5.3.6; GB 50204-2015 clause 5.3.3):
   • General members: stirrup hooks bend not under 90 degrees, the straight tail after bending not under 5 diameters of the stirrup bar;
   • Members with seismic defence or special design: bend not under 135 degrees, tail not under 10 diameters, GB 50666-2011 clause 5.3.6 adding "and the larger of 75 mm" (seismic governed by max(10d, 75 mm));
   • Ties serving as single-leg stirrups in compound beam / column stirrups or as web ties between beam side bars: both hooks not under 135 degrees; as ties in shear walls and slabs, one hook 135 and the other 90 degrees allowed, tails not under 5 diameters of the tie bar.
3. One 135-degree hook addition: derived on bend diameter 2.5d and tail 10d, trade convention 11.9d (stirrups only, never longitudinal bars); two hooks on seismic members total 2 x 11.9d.
4. Laps: spiral stirrup laps not less than la or laE and not less than 300 mm, hooking the longitudinal bars - DBJ 52/T 088-2018 clause 6.4.2 item 2 sub 4).
5. Parameters setting pitch and dense-zone height:
   • Pile spirals: diameter not under 6 mm, pitch preferably 200-300 mm; for piles under heavy lateral load or lateral seismic action, and where main bars count in compression capacity, the spirals densify within 5d (5 pile diameters) below the top at pitch not over 100 mm; through liquefiable strata, densify; where spirals are counted as load-bearing, detailing per GB 50010 (DBJ 52/T 088-2018 clause 6.4.2; DB/T 29-20-2017 clause 10.2.6; JGJ/T 419-2018 clause 4.3.4);
   • Stiffening rings: cages over 4 m carry a welded stiffening ring of not less than 12 mm bar every 2 m (DBJ 52/T 088-2018); JGJ/T 419-2018 clause 4.3.4 sets ring spacing not over 2.0 m and diameter not under 14 mm;
   • Circular-column stirrups (GB 50010-2010 clause 9.3.2): laps not less than the anchorage of clause 8.3.1, ends hooked 135 degrees with tails not under 5d (d the stirrup bar); where the total longitudinal steel ratio exceeds 3%, tails not under 10d (d the smallest longitudinal bar).
6. Albums: the G101 series (cast-in-place frames, shear walls, beams, slabs) give the spiral stirrup DETAILING, not the length formula - the formula derives from the geometry above (geometric derivation is not a code clause). The commentary to DBJ 52/T 088-2018 likewise: "the spiral stirrup requirements follow the G101 albums, helping cure common defects."

3. Worked example

d800 bored pile, 20 m long, spiral d8 at 100 (dense) / 200 (plain), dense zone 4 m, cover 50 mm:

• Centreline diameter D = 0.8 - 0.1 + 0.008 = 0.708 m, circumference pi D = 2.224 m;
• Dense zone: one turn root of (2.2242 + 0.12) = 2.226 m, 40 turns, developed 89.06 m;
• Plain zone: one turn root of (2.2242 + 0.22) = 2.233 m, 80 turns, developed 178.66 m;
• Horizontal ends 2 x 1.5 x 2.224 = 6.673 m; hooks 2 x 11.9 x 0.008 = 0.190 m;
• Total developed length about 274.58 m; at 0.395 kg/m about 108.4 kg per pile.

Applicability Boundaries

1. The formula serves spiral stirrups of cylindrical members; rectangular column and beam stirrups compute by girth plus hooks and laps - not by this formula.
2. The parameters are DETAILING FLOORS: seismic-rated members (members with seismic grades in the drawings and standards) and members with special design (torsion or combined states; columns whose total longitudinal ratio exceeds 3%) take the stricter values per the drawings and notes.
3. The basis of D must be checked against the drawing's cover datum (main-bar-skin versus stirrup-skin) - a wrong datum shifts the content systematically.
4. Dense-zone extent (5d below pile top, liquefiable strata, main bars in compression), ring spacing and diameter differ slightly across standards (rings 12 mm in DBJ 52, 14 mm in JGJ/T 419) - take the project location's rules and design notes.
5. 11.9d derives on bend diameter 2.5d and tail 10d; a different design bend diameter re-derives it.
6. Ties with 135-degree hooks both ends may be fabricated one end 135 and the other 90, the 90 closed to 135 on site after fitting (workmanship allows).

Further Reading

• GB 50666-2011 Code for Construction of Concrete Structures, clause 5.3.6 and commentary (stirrup and tie end hooks);
• GB 50204-2015 Code for Acceptance of Construction Quality of Concrete Structures, clause 5.3.3 (hook acceptance: 3 pieces per shift per bar type per machine, by measurement);
• GB 50010-2010 Code for Design of Concrete Structures, clause 9.3.2 (column stirrup diameter, spacing, closed form and 135-degree hooks);
• JGJ/T 419-2018 Technical Standard for Long-Spiral Drilled and Grouted Piles, clause 4.3.4 (spirals, stiffening rings, 5d densification);
• DBJ 52/T 088-2018 Technical Regulation for Building Pile Foundations in Guizhou, clause 6.4.2 and table 6.4.2 (hand-dug pile cage tolerances, spiral detailing);
• DB/T 29-20-2017 Technical Code for Geotechnical Engineering in Tianjin, clause 10.2.6 (pile stirrup provision);
• G101 series albums (plan-representation rules and detailing for cast-in-place frames, shear walls, beams and slabs), column details;
• Unit rebar masses per GB 50666-2011 annex B table B.0.1 (same table as GB 50010-2010 annex A).