About this calculator
This tool solves the inverse tightening problem: instead of asking what torque produces a chosen preload, it asks what is the largest torque and preload this bolt can take before the combined stress reaches a chosen limit? You set the target Von Mises equivalent stress as a fraction α of the yield strength Rp0.2 — 100% corresponds to tightening to yield — and the calculator iterates on the tensile stress until the tension-plus-torsion combination exactly meets that target. The result is the maximum admissible tightening torque MA, the corresponding preload Fi, the loosening torque, and the stress relaxation that occurs after the wrench is removed.
If you already know the preload you want and simply need the assembly torque for it, use the forward-solving Tightening Torque Calculator instead. The two tools share the same thread geometry, material data, and friction model, so results are directly comparable.
Theory: solving for the preload limit
During tightening the bolt shank carries tensile stress σt from the preload and torsional shear stress τ from the thread friction torque, combined by the Von Mises criterion:
Because τ itself depends on the preload through the thread torque, the equation cannot be rearranged in closed form. The calculator solves it by bisection: it adjusts σt until the resulting equivalent stress matches the target within numerical precision, using the same relations as the forward calculator:
Two additional outputs distinguish this calculator from the forward version:
- Loosening thread torque. Evaluated with the friction term acting against the lead angle: MG,loosen = Fi (d2/2)(tan λ − μ/cos 30°)/(1 + μ tan λ/cos 30°). A negative value means the thread is self-locking — it will not unwind without an actively applied loosening torque.
- Stress relaxation. The torsional stress fades shortly after tightening through micro-slip, so the equivalent stress drops from σe to σt. The relaxation (σe − σt)/σe quantifies the share of the tightening stress that disappears in service — the physical justification for tightening to yield.
Standards used
- ISO 724 / DIN 13 T1 — metric coarse thread geometry (d2, d3, lead angle).
- EN ISO 898-1 — strength classes 3.6–12.9 with nominal Rm and Rp0.2, and the tensile stress area At.
- The 90–100% utilisation targets follow the practice of VDI 2230 for torque- and yield-controlled tightening.
Worked example: M12, class 8.8, tightening to yield
M12 coarse (P = 1.75 mm), class 8.8 (Rp0.2 = 640 MPa), lightly oiled (μthread = μhead = 0.12), target σe = 100% of yield = 640 MPa.
- The bisection converges to a tensile stress of σt = 498.0 MPa — i.e. the preload may only use 77.8% of yield in pure tension, because torsion consumes the rest of the stress budget.
- Preload: Fi = 498.0 × 84.27 mm² = 41.97 kN; load at yield F0.2 = 53.93 kN, leaving a reserve of 11.96 kN.
- Torques: MG = 43.59 N·m, MWD = 39.28 N·m, so the maximum tightening torque is MA = 82.9 N·m.
- Check: τ = 232.1 MPa gives σe = √(498² + 3 × 232.1²) = 640.0 MPa = Rp0.2. Utilisation exactly 100%.
- After tightening the torsion relaxes: stress relaxation = (640 − 498)/640 = 22.2%. The loosening thread torque is −19.76 N·m — negative, so the thread is self-locking.
Compare with the forward calculator: tightening the same bolt to a plain 90% tensile preload would demand σe ≈ 740 MPa — well beyond yield. Solving the inverse problem is what makes the true limit visible.
Assumptions and limitations
- Coarse-pitch ISO metric threads M6–M42; constant friction during tightening.
- The yield criterion is applied to the nominal stress state in the thread root section; local notch stresses are not modelled (they are accounted for implicitly in standard fatigue data).
- Head friction radius 0.65 d assumes standard hexagon heads or plain washers.
- Tightening exactly to yield in production requires yield- or angle-controlled tools; with a plain torque wrench, friction scatter can push individual bolts past yield. Keep margin accordingly.
Frequently asked questions
What does “tightening to yield” mean?
Setting the torque so the combined tension-plus-torsion stress reaches 100% of Rp0.2 during tightening — the maximum preload a given bolt can deliver. Because the torsional share (typically 20–25%) relaxes afterwards, the bolt is not permanently at yield in service.
Why is the loosening torque negative?
A negative loosening thread torque means the thread is self-locking: friction outweighs the lead-angle drive and the joint cannot unwind on its own. Practically all metric coarse threads with μ > 0.05 are statically self-locking — though vibration and transverse slip can still loosen them, hence securing elements in dynamic joints.
What does the stress relaxation percentage mean?
It is (σe − σt)/σe — the share of the tightening stress that disappears once the torsional component fades after the wrench is removed. Around 20–25% for typical friction values.
How is this different from the tightening torque calculator?
The tightening torque calculator works forward from a chosen tensile stress to a torque. This one works backward from an allowable equivalent stress to the maximum torque and preload, solving the coupled tension–torsion equation iteratively.
What utilisation should I target?
VDI 2230 practice: 90% of Rp0.2 for torque-controlled tightening (margin for friction scatter and embedding), 100% only with yield- or angle-controlled tools that detect the yield point directly.