Horizontal directional drilling
Minimum bend radius
A bore has three separate bend-radius rules — one for the drill string, one for steel product pipe, one for polyethylene — and they come from different failure mechanisms. On any job the rod’s limit and the product pipe’s limit both apply, they give different numbers, and the larger one governs the path.
What the number actually is
In HDD the working definition is the forward distance required for a drill string to make a 90-degree turn (Trenchless Technology, “Drill Master: Understanding Bend Radius”). That is the same number as the geometric radius, because a circular arc of radius R that turns through angle θ advances R sin θ in its original direction, and sin 90° = 1.
It is not the length of pipe consumed by the turn. That is the arc length, Rθ, which for 90° is πR/2 — 1.571 times the radius. A 108 ft bend radius therefore eats about 170 ft of drill string to turn a right angle, and the two figures get mixed up regularly.
Bending a tube to a radius stretches the outer wall and compresses the inner one. The outer-fibre strain (the fractional change in length of the material) is
and while the material stays elastic the stress (force per unit area in the wall) is that strain times the material’s modulus of elasticity, E:
This is the second term of Equation 16 in the Plastics Pipe Institute Handbook of PE Pipe, Chapter 12, where it is added to the axial pull stress. Everything below is that one equation, run with different values of E and different allowable stresses.
Limit one: the drill string
Rod makers publish a minimum bend radius per rod size. The figures below are Hunting’s for its HIWS1 rod, as listed in the HDD Drill Pipe Selection Guide, Underground Construction, December 2018. The last two columns are arithmetic on those figures, using E = 29 × 106 psi, the modulus used for steel line pipe in the PRCI design example (Huey, Hair and McLeod, 1996).
| Rod OD | Wall | Published min. radius | Radius ÷ OD | Outer-fibre bending stress |
|---|---|---|---|---|
| 1.32 in | 0.18 in | 57.3 ft | 521 | 27,800 psi |
| 1.66 in | 0.19 in | 95.5 ft | 690 | 21,000 psi |
| 1.90 in | 0.24 in | 98.7 ft | 623 | 23,300 psi |
| 2.06 in | 0.26 in | 101.9 ft | 594 | 24,400 psi |
| 2.38 in | 0.25–0.30 in | 108.2 ft | 546 | 26,600 psi |
| 2.63 in | 0.40 in | 145.0 ft | 662 | 21,900 psi |
| 2.88 in | 0.39 in | 168.7 ft | 703 | 20,600 psi |
| 3.50 in | 0.45–0.48 in | 197.4 ft | 677 | 21,400 psi |
The 2.38 in / 108.2 ft row is the rod the guide lists for machines in the Vermeer D24x40 and Ditch Witch JT20 to JT30 class. The guide does not give the steel grade of these small rods, so their stress cannot be set against a yield strength from it. It does give the grade for large drill pipe: Holly Pipe lists grade S-135 — 135,000 psi minimum yield — at a 292 ft minimum radius for 3.5 in pipe and 550 to 770 ft for 5 to 7⅝ in. Those radii hold the outer-fibre bending stress between 9,600 and 14,500 psi, 7 to 11 per cent of yield.
That margin is not timidity. The rod rotates while it is bent. A point on the wall is in tension on the outside of the bend and in compression half a turn later, so every revolution is one full reversed stress cycle. A rod turning at n revolutions per minute accumulates 60n cycles per hour while it sits in that curve. The governing limit is fatigue, not yield, and fatigue limits sit far below yield.
That is also why exceeding it does not announce itself. Trenchless Technology’s description is that “the useful life of the drill pipe decreases dramatically”, and that the damage is subtle and most likely will not be noticed at the time it is done. The rod usually breaks several jobs later, often on a simple bore, by which time the job that did the damage is long forgotten.
Limit two: steel product pipe
The planning rule for steel is 100 ft of bend radius per inch of pipe diameter (the same Trenchless Technology article; da Silva and co-authors, Rio Pipeline 2009, give it as 1,200 times the nominal diameter and note that it comes from established practice rather than theoretical analysis). Since 100 ft is 1,200 in, that is R = 1,200D. A 12 in pipe wants 1,200 ft of radius; a 3 in pipe wants 300 ft.
Put that into the strain equation and the rule stops being arbitrary:
σb = 29,000,000 / 2400 = 12,100 psi
That takes D as the outside diameter. The rule is stated per inch of nominal size, and in the smaller sizes the outside diameter is larger than the nominal one — 12.75 in for a 12 in line — so the real strain is higher by that ratio: 6 per cent for a 12 in line.
For API 5L grade X-52 — the grade number is the specified minimum yield strength (SMYS) in ksi, so 52,000 psi — 12,100 psi is 23 per cent of yield. The rest of the wall’s capacity is reserved for the pull. HDD design practice checks the two together as an interaction ratio: the tensile stress divided by 0.9 SMYS, plus the bending stress divided by an allowable bending stress, must not exceed 1. The bending allowable, adapted from API RP 2A-WSD for offshore tubular members, is 0.75 SMYS for thick-walled pipe and lower as the wall gets thinner relative to the diameter (da Silva and co-authors, above). On thick-walled X-52 the rule’s 12,100 psi is 31 per cent of that allowance, and more on a thinner wall, spent before the pull starts. Spend the wall on curvature and there is less of it left for tension at the moment tension is highest.
Unlike the drill rod, the product pipe does not rotate, so this is a static limit and there is no fatigue cycle counting to do.
Limit three: polyethylene product pipe
PE is quoted as a bend ratio, R ÷ OD. The long-term values below are Table 1 of Performance Pipe Technical Note PP 819-TN, Field Bending of DriscoPlex® Pipe. DR is the dimension ratio, outside diameter divided by minimum wall thickness — a higher DR is a thinner wall.
| DR | Minimum long-term bend ratio |
|---|---|
| 7, 7.3, 9 | 20 × OD |
| 11, 13.5 | 25 × OD |
| 17, 21 | 27 × OD |
| 26 | 34 × OD |
| 32.5 | 42 × OD |
| 41 | 52 × OD |
| Fitting or flange present in the bend | 100 × OD |
Two adjustments matter for a bore. First, PP 819-TN states that in the bore itself the bend ratio “should not be less than two times the value shown in Table 1”, because the pipe is under high tension at the same time. A DR 11 pipe therefore wants 50 × OD downhole rather than 25. Second, the worked example in the PPI HDD chapter applies a separate ring-stability floor: the radius of curvature should exceed 40 times the pipe outside diameter to prevent ring collapse — 80 ft for a 24 in pipe.
The mechanism here is not strain. PP 819-TN is explicit: polyethylene’s strain capacity is generally safe at a bend ratio of 20, and the real limit is that longitudinal bending ovalises the pipe in the ring direction, which reduces its resistance to kinking — local buckling of the wall. Thicker walls resist ovality better, which is why lower DR bends tighter, and PE’s modulus falls with time under load, which is why a temporary bend is allowed tighter than a permanent one.
The strain figures make the material difference concrete. At 50 × OD the outer fibre of a PE pipe is at 1 per cent strain; at the 40 × OD ring-collapse floor, 1.25 per cent. Steel at 1,200 × D is at 0.042 per cent. PE tolerates twenty-four to thirty times the curvature, and then fails by buckling rather than by yielding.
Which one governs
Usually the drill string. The PPI chapter says the permitted bending radius of the drill rod more often controls the curvature, so significant bending stresses do not occur in the pipe. Performance Pipe puts it plainly: “The amount of permissible curvature in the bore itself is generally limited by the drill stem’s bending capability and not the polyethylene pipe,” noting that the 5 in stem used with a 36 in pipe generally has at least a 700 ft bend radius.
The worked river crossing in PPI Chapter 12 shows the margin. A 24 in DR 11 PE4710 pipe on an exit curve of 1,021 ft radius is at 511 × OD — nearly thirteen times the ring-collapse floor of 80 ft. Its bending strain is 9.79 × 10−4 and its bending stress, against a 12-hour modulus of 63,000 psi, is 61.7 psi. Negligible. That radius comes straight out of the example’s chosen depth and exit angle (bore path geometry); the pipe was never close to limiting it.
For planning, take the larger of the rod limit and the product-pipe limit — for PE, the larger of the in-bore bend ratio and the 40 × OD floor — and design to that.
Check a bore
Pick the rod and the product pipe. The calculator applies the rules above and reports which one governs. Enter the radius you plan to drill and it checks that radius against every limit and gives the strain and stress it puts on the pipe and the rod.
What exceeding it costs
- Drill string: cumulative fatigue damage that shortens the rod’s life and is not visible when it is done.
- Steel product pipe: bending stress adds to pull tension on the outer fibre, and past yield the pipe takes a permanent set. Tight bends and dog-legs also raise the pull, through the reaction forces needed to force a stiff pipe round them — the stiffness term in the PRCI method.
- PE product pipe: ovality, then kinking — and higher tension and ovality both reduce the pipe’s collapse resistance against the external mud head, at the moment it is under maximum tension.
- Every case: extra curvature — dog-legs and over-corrections — adds angle turned, and the capstan term multiplies tension by eμθ, where θ is that extra angle in radians. That is usually how the problem is discovered: the pull needs more force than planned.
BorePlanner Coming soon checks this on an Android phone. It finds the tightest radius of curvature along the planned path and flags the station where it falls below the minimum radius you set. Picking a rod size fills in a starting estimate; replace it with the rod maker’s published radius, or with the product-pipe limit when that is larger.