Torsion_(mechanics) - Wikiparency

# Torsion (mechanics) **Torsion** is the twisting of a member about its own axis under an applied [[Torque|torque]]. The deformation it produces is pure [[Shear_stress|shear]]: one cross-section rotates relative to the next, no fibre changes length in a circular shaft, and the stress that resists the load acts tangentially, in the plane of the cut, rather than normal to it. Two results describe almost everything a designer needs. The shear stress grows linearly from zero on the axis to a maximum at the surface, `τ = T r / J`, and the total angle the free end turns through is `φ = T L / (G J)`, with `G` the [[Shear_modulus|shear modulus]] and `J` the [[Second_polar_moment_of_area|second polar moment of area]] of the section. Both results turn on `J`, and `J` grows as the fourth power of radius. That single fact organises the subject. Material close to the axis moves very little and carries almost no stress, so it earns almost nothing; material at the surface travels furthest and carries the peak stress. A hollow shaft of the same mass as a solid one is therefore stiffer and less stressed, and the gain is large rather than marginal. The same fourth-power scaling explains why a shaft that must be stiff in twist grows in diameter rather than in wall thickness, and why the open and closed versions of the same thin-walled tube differ in torsional stiffness by a factor of hundreds. A circular section is the one case in which plane cross-sections stay plane. Every other shape warps — points on a cut face move along the axis as well as around it — and the polar second moment of area stops being the right quantity. It is replaced by the [[Torsion_constant|torsion constant]], always smaller, and by formulas that are fitted or solved numerically rather than derived in a line. Torsion in that general setting is a classical problem of [[Solid_mechanics|solid mechanics]], and the difference between the exact answer and the engineering approximation is itself a subject, treated in the sections on certified relations below. The framework microsim *Torsion: a twisted shaft turns a straight line into a helix* is built on the circular case. The reader twists a shaft with a torque `T`, watches a line drawn along its surface wind into a helix while the free end turns through `φ`, reads the shear stress growing from zero on the axis to `τ = T r / J` at the surface, and then swaps the solid shaft for a hollow one of the same mass to see how little the material near the axis was doing. ## Properties The circular shaft can be worked out from one kinematic assumption: as the shaft twists, each cross-section rotates rigidly about the axis, so plane sections remain plane and radii remain straight. A point at radius `r` on a shaft of length `L` whose end has turned through `φ` has therefore slid a distance `rφ` around the axis relative to the fixed end, giving a shear [[Strain_(mechanics)|strain]] `γ = r φ / L` that is zero on the axis and greatest at the surface. [[Hooke's_law|Hooke's law]] in shear, `τ = G γ`, converts that into a stress with the same linear profile, and requiring the stresses on the cut face to sum to the applied torque, `T = ∫ τ r dA`, fixes the constant: `τ = T r / J` with `J = ∫ r² dA`, the polar second moment of area, equal to `π r⁴ / 2` for a solid round shaft.[^hibbeler] Rearranging the same three relations gives the twist, `φ = T L / (G J)`, and the torsional stiffness of the shaft as a spring, `k_t = G J / L`. The shear modulus is not an independent number for an isotropic material: `G = E / (2 (1 + ν))`, so a steel of `E = 207 GPa` and `ν = 0.30` has `G = 79.6 GPa`.[^mat][^up12-3] The microsim's default shaft puts all of this together. A 1020 steel shaft of 20 mm radius and 1 m length under `T = 500 N·m` has `J = π (0.02)⁴ / 2 = 2.51 × 10⁻⁷ m⁴`, or 25.1 cm⁴; the surface shear stress is `500 × 0.02 / 2.51 × 10⁻⁷ = 39.8 MPa`; the free end turns through `0.0250 rad`, which is 1.43 degrees; and the shaft's torsional stiffness is 20.0 kN·m per radian.[^sim] Against a shear yield stress of `S_y/√3 = 170 MPa` — the von Mises value the sim computes — that is a factor of safety of 4.3 on yield.[^sim] What the reader sees is the geometry of the twist made visible. A straight line scribed along the surface of the shaft becomes a [[Helix|helix]] of pitch angle `arctan(rφ/L)`, and because that angle is what shear strain means, the helix is a direct picture of `γ` at the surface. The drawing is not to scale: the sim exaggerates the twist by a factor of 25 and clamps it at 300 degrees of drawn rotation, and the shaft's drawn length and radius are chosen for legibility rather than proportion. Both are labelled ILLUSTRATIVE on the sim's own note sheet, and the clamped state is captioned when it happens.[^sim] Every number in the readout, by contrast, is computed from the equations above at the true values. *Try: set the torque to 500 N·m on the 20 mm steel shaft and read τ = 39.8 MPa and φ = 1.43°, then drag the radius from 20 mm down to 10 mm without touching the torque — the stress goes up eightfold and the twist sixteenfold, because τ follows `1/r³` and φ follows `1/r⁴`.* ## Torsion constants for common cross-sections For a hollow round shaft the polar second moment is the difference of two fourth powers, `J = π (r_o⁴ − r_i⁴) / 2`, and the peak stress is still `T r_o / J` at the outer surface. Because the integrand is `r²` weighted by an area that itself grows with `r`, the bore removes very little `J` while removing a great deal of mass. The cleanest way to see it is to compare at constant mass. Hollowing a solid shaft of radius `r` to a bore ratio `k = r_i/r_o` at the same area requires `r_o = r / √(1 − k²)`, and the stiffness ratio then collapses to `J_hollow / J_solid = (1 + k²) / (1 − k²)`. At `k = 0.8` that is 4.56. The microsim's default shaft, hollowed this way, grows from 20 mm radius to 33.3 mm outer with a 26.7 mm bore; its `J` rises from 25.1 cm⁴ to 114 cm⁴, and the peak shear stress falls from 39.8 MPa to 14.6 MPa for the same 500 N·m.[^sim] The same shaft, the same steel, the same weight: four and a half times stiffer and a third of the stress. The `r_i/r_o = 0.8` wall ratio is a design choice the sim states on its sheet rather than a physical constant; pushing it toward 1 improves the ratio without limit on paper and runs into local wall buckling and manufacturing limits in practice. For any section that is not a circle, the polar second moment of area overstates the stiffness, because the section warps out of plane and so cannot develop the shear pattern the circular derivation assumed. The quantity that belongs in `φ = T L / (G K)` is the torsion constant `K`, and `K < I_p` always. The gap is not small. A 2:1 rectangular bar of exactly the same area as the 20 mm round shaft measures 50.1 mm by 25.1 mm; its polar second moment is 32.9 cm⁴, but its torsion constant is 18.1 cm⁴, a ratio of 0.55.[^sim] The worst case is a thin-walled section that is open rather than closed. A closed tube of mean radius `R` and wall `t` has `J ≈ 2 π R³ t`; slit that same tube lengthwise and it becomes an open strip of width `2πR`, with `K ≈ (2 π R) t³ / 3`. The ratio is `3 R²/t²` — for a 60 mm tube with a 3 mm wall, a factor of 300. Cutting a slot along a tube costs almost nothing in bending and nearly all of its torsional stiffness, which is why a closed box section is chosen wherever a member must resist twist and why the [[Shear_flow|shear flow]] round a single closed cell is the first thing a designer of such a section computes. Tabulated values for the common shapes are collected in the [[List_of_second_moments_of_area|list of second moments of area]]. *Try: open the `Second_polar_moment_of_area` variant, which starts on the same-mass hollow shaft with the bore and end ring visible and reads J = 114 cm⁴ in place of the stiffness — sweep the radius and watch J climb as `r⁴` while the τ(r) chart shows the hollow shaft's short high-radius segment sitting at 14.6 MPa under the solid shaft's 39.8 MPa line.* ### Exact and engineering formulas The exact treatment is Saint-Venant's. Allow each cross-section to warp by an unknown function of the in-plane coordinates, impose equilibrium and the traction-free condition on the lateral surface, and the problem reduces to a [[Boundary_value_problem|boundary value problem]] on the cross-section — Laplace's equation for the warping function, or Poisson's equation for the [[Stress_functions|stress function]] whose contours are the lines of shear flow and whose volume is proportional to `K`. Closed-form solutions exist for the circle, the ellipse and a handful of other shapes; the rectangle requires an infinite series; a general section requires a numerical solution, today usually by the [[Finite_element_method|finite element method]]. The same formulation explains the end effects: the simple results hold away from the ends and from any restraint on warping, which is [[Saint-Venant's_principle|Saint-Venant's principle]] in its original setting. Engineering practice replaces the series by fitted coefficients. For a rectangle of long side `a` and short side `b` the torsion constant is written `K = c₁ a b³`, with `c₁` rising from 0.141 for a square through 0.229 at 2:1 toward 1/3 for a long thin strip — the 0.229 used above.[^roark] The peak shear stress, which for a rectangle occurs at the middle of the long side rather than at the corners, has its own fitted form, `τ_max = T (3a + 1.8b) / (a² b²)`. For the equal-area 2:1 bar under 500 N·m this gives 61.9 MPa, against 39.8 MPa in the round shaft of the same weight:[^sim] a rectangular bar is both softer and more highly stressed than the circle it replaces. Thin-walled forms have their own short formulas — `K = Σ b t³ / 3` summed over the rectangles of an open section, and Bredt's `K = 4 A_m² / ∮ (ds/t)` for a single closed cell — and these are the formulas the reliability question below is asked about. *Try: open the `Torsion_constant` variant, which replaces the round shaft with the equal-area 2:1 rectangular bar — read K = 18.1 cm⁴ against I_p = 32.9 cm⁴ on the sheet and watch the diamond marking the rectangle's peak shear at 61.9 MPa sit well above the round shaft's τ(r) line on the same chart.* ## Certified primal–dual method Because the torsion constant of a general cross-section is the solution of a boundary value problem rather than a formula, every tabulated coefficient in the previous section is an approximation whose error is not stated. A certified method is one that returns not a single number but an interval guaranteed to contain the true value. The classical route to such an interval is a dual pair of variational principles: the displacement, or primal, formulation makes the torsional rigidity the minimum of one functional over admissible warping fields, while the stress-function, or dual, formulation makes the same rigidity the maximum of a complementary functional over admissible stress fields. Any admissible trial field of either kind therefore yields a one-sided bound, and a trial pair brackets the answer, with the width of the bracket serving as a certificate of accuracy rather than an estimate of it. This section of the pair article surveys that literature and the relations it has been used to establish; the treatment here states the principle and links on to [[Stress_functions|stress functions]] and the [[Finite_element_method|finite element method]], which is where such bounds are computed in practice.[^certified] ### Examples of certified relations A certified relation is an inequality between the torsion constant and quantities that are easy to compute — the area, the polar second moment, the perimeter, the inradius — that has been proved rather than fitted, together with the cases in which it is sharp. Relations of that shape are what make a table of coefficients auditable: given one, a designer can bound the error of an interpolated coefficient without re-solving the cross-section. The pair article collects specific examples of such relations; this article does not restate them, because the individual results could not be attributed to a primary source at the time of writing, and the house rule is to leave the gap visible rather than to fill it.[^certified] ### Reliability of commonly used relations The practical question behind the whole method is how much trust the short formulas deserve. Most of them are good where they are used and poor where they are extrapolated. The rectangle's `c₁` coefficients are interpolations of a series solution and are accurate to better than a per cent over the tabulated aspect ratios; the thin-strip limit `K = b t³ / 3` is an asymptote approached slowly, so it understates `K` for stubby rectangles; Bredt's single-cell formula assumes the shear flow is uniform through the wall and degrades as the wall thickens relative to the cell. The one place where an unexamined formula does real damage is the open-versus-closed distinction: a section that a drawing shows as a tube but that is in fact slit, or joined by a seam that does not transmit shear, is softer in torsion by the factor `3R²/t²` computed above, and no correction to a coefficient will recover it. The reliability of a relation, in other words, is dominated by whether the right relation was chosen, which is the argument for a method that certifies its own error.[^certified] ## Sample calculation A shaft transmitting `P = 20 kW` at 1500 rpm carries a torque `T = P / ω`, with `ω = 1500 × 2π/60 = 157.1 rad/s`, so `T = 127.3 N·m`. Sizing it on strength first: for 1020 steel the shear yield stress is `S_y/√3 = 170 MPa`, and taking a factor of safety of 2 gives an allowable of 85 MPa. Setting `T r / J = τ_allow` with `J/r = π d³/16` gives `d = (16 T / (π τ_allow))^(1/3) = 19.7 mm`, so a 20 mm shaft is the first standard size that works: at that diameter the surface stress is 81.1 MPa and the factor of safety on shear yield is 2.1.[^mat] Strength is not usually what decides. Checking the same 20 mm shaft for twist, `φ/L = T/(G J) = 127.3 / (79.6 × 10⁹ × 1.57 × 10⁻⁸) = 0.102 rad/m`, which is 5.83 degrees per metre. A machine shaft is commonly held to about one degree of twist per metre of length so that driven elements stay in phase; against that limit the 20 mm shaft is off by a factor of six. Sizing on stiffness instead, `J = T / (G × 0.01745) = 9.16 × 10⁻⁸ m⁴`, which needs `d = 31.1 mm`. The diameter rises by 58 per cent and `J`, going as `d⁴`, rises by a factor of 6.2, so the shaft ends up loafing at a shear stress of about 22 MPa — a factor of nearly eight on yield — purely to meet a deflection limit. That is the general pattern, and it follows directly from the two governing equations. Stress scales as `1/d³` and twist as `1/d⁴`, so as a shaft gets longer or a stiffness requirement gets tighter, the twist limit overtakes the strength limit and sets the size. Short, heavily loaded shafts are strength-governed and are checked against `τ`; long ones, and any shaft in a [[Power_transmission|power transmission]] where timing matters, are stiffness-governed and are checked against `φ`. The same arithmetic run in reverse is how a [[Shaft_(mechanical_engineering)|shaft]] is hollowed: replacing the 31 mm solid section with a same-`J` tube at `r_i/r_o = 0.8` keeps the stiffness while removing about half the metal. ## Failure modes A shaft in pure torsion is in a state of pure shear, and pure shear is tension and compression of equal magnitude on planes at 45 degrees to the axis — the fact [[Mohr's_circle|Mohr's circle]] makes geometric, since the stress point sits on the vertical axis of the circle and the principal points lie a quarter turn away on it.[^mohr] Which of those the material minds decides how it breaks. A ductile steel yields on the plane of maximum shear, so a ductile shaft twisted to destruction fails on a flat transverse face. A brittle material such as cast iron or chalk fails on the plane of maximum tension, which in torsion is a 45-degree helix, so a brittle shaft breaks along a clean helical surface. The two fracture surfaces are a direct read-out of whether the material's limit is shear or tension, and the distinction between [[Ductility|ductile]] and [[Brittleness|brittle]] behaviour is nowhere easier to demonstrate. Yield in shear is predicted at `τ_y = S_y/√3 = 0.577 S_y` by the von Mises criterion and at `τ_y = S_y/2` by Tresca; the sim uses the von Mises value, 170 MPa for 1020 steel, and draws it as a limit line across the stress chart.[^sim][^mat] Two further modes matter in practice and neither is a strength limit. A thin-walled tube under torque can buckle before it yields, folding into a helical wave pattern at a critical torque set by the wall-to-radius ratio and the elastic modulus, in the same imperfection-sensitive way described under [[Buckling|buckling]]. And a shaft that never approaches yield under steady load can still fail in [[Fatigue_(material)|fatigue]] under a fluctuating one, almost always starting at a [[Stress_concentration|stress raiser]] — a shoulder fillet, a keyway, a cross-hole — where the local shear stress is two or three times the nominal `T r / J`. A third mode is peculiar to open sections: if warping is restrained, at a built-in end or by a cross-member, the section develops axial stresses that the Saint-Venant solution does not contain, and a thin-walled open member can be limited by those rather than by shear. ## Torsional resonator A shaft with an inertia on the end of it is a torsional spring–mass oscillator, and its natural frequency follows directly from the stiffness derived above: `ω_n = √(k_t / I)` with `k_t = G J / L`. The microsim's default shaft, at 20.0 kN·m per radian, carrying a steel disc 200 mm in diameter and 20 mm thick — a mass of 4.93 kg and a moment of inertia of `½ m R² = 0.0247 kg·m²` — resonates at `√(20000/0.0247)/2π = 143 Hz`.[^sim] Because `k_t` scales as `d⁴/L`, a torsional natural frequency is extremely sensitive to shaft diameter and length, which is why [[Torsional_vibration|torsional vibration]] is designed around rather than damped out: a [[Crankshaft|crankshaft]] or a turbine-generator shaft line is laid out so that its torsional modes fall clear of the harmonics of the firing or electrical excitation, and a tuned damper is added where they cannot be. The same modes are what a [[Steam_turbine|steam turbine]]'s shaft line is checked against before it is coupled to an [[Electric_generator|electric generator]]. Made deliberately soft, the same oscillator becomes an instrument. A long, fine fibre has a torsion constant so small that a minute torque produces a measurable rotation, and the period of the resulting oscillation calibrates it. Charles-Augustin de [[Charles-Augustin_de_Coulomb|Coulomb]] built such a balance in the 1780s and used it both to establish the law of torsion of wires and to measure electrostatic force; [[Henry_Cavendish|Henry Cavendish]] used one in 1797 and 1798 to weigh the Earth, reporting a mean density about 5.48 times that of water from the deflection of a 6-foot torsion rod carrying two 2-inch lead balls.[^cavendish] Modern torsional resonators run the same principle at the opposite end of the scale: quartz and silicon [[MEMS|micromachined]] torsional resonators serve as the sensing elements of a [[Vibrating_structure_gyroscope|vibrating structure gyroscope]] and of scanning micromirrors, while an oscillatory [[Rheometer|rheometer]] drives a sample in torsion and reads the in-phase and out-of-phase components of the response as the storage and loss parts of a [[Dynamic_modulus|dynamic modulus]], the standard measurement of [[Viscoelasticity|viscoelasticity]]. In every case the design quantity is the same `G J / L` computed for the shaft above, and the quality factor of the resonance — how many cycles it rings for — is what the instrument's resolution is built on. ## Minnesota *This section is specific to Wikitube.* The largest torques in [[Minnesota]] turn slowly. A utility-scale [[Wind_turbine|wind turbine]] takes its power from a rotor turning at roughly a tenth the speed of a car engine, and since `T = P / ω`, low speed at high power means enormous torque in the low-speed shaft between the hub and the gearbox. Researchers at the University of Minnesota used a 2.5-MW machine for a well-known field experiment, seeding the flow with falling snow during a winter storm to make the tip and trailing-sheet vortices in its wake visible over a sampling area about 36 m square — the first in-situ visualisation of the large-scale flow structures behind a utility-scale turbine.[^hong2014] The machine belongs to the Eolos wind energy research station operated by the university, a Minnesota field facility built for exactly this kind of full-scale measurement.[^eolos] The torsion arithmetic for a machine of that size follows from the article's own equations. At 2.5 MW and a rotor speed of the order of 15 rpm — a figure assumed here as typical for a rotor of that rating rather than taken from the measurement — the angular velocity is 1.571 rad/s and the shaft torque is `2.5 × 10⁶ / 1.571 = 1.59 MN·m`, some twelve thousand times the 127 N·m of the machine shaft sized in the sample calculation above. Carrying that at the same 85 MPa allowable would need a solid shaft 457 mm in diameter; the same-strength hollow shaft at `r_i/r_o = 0.8` is 545 mm across the outside and uses about half the steel, which is why the main shaft of a large turbine is a hollow forging rather than a bar. The fourth-power rule that the microsim demonstrates on a 40 mm shaft is the same rule that sets the size of the one at [[Buffalo_Ridge|Buffalo Ridge]], and the penalty for getting it wrong scales with it. ## See also - [[Torsion_constant]] - [[Second_polar_moment_of_area]] - [[Shear_stress]] - [[Shear_modulus]] - [[Torsion_spring]] - [[Torsional_vibration]] - [[List_of_second_moments_of_area]] - [[Structural_rigidity]] - [[Torsion_siege_engine]] - [[Stress_(mechanics)]] (section 1) - [[Bending]] (section 4) - [[Buckling]] (section 5) - [[Gear]] (section 9) - [[Flywheel]] (section 13) ## References [^hibbeler]: The circular-shaft results used throughout — `τ = T r / J`, `φ = T L / (G J)`, `J = π r⁴ / 2`, and the kinematic assumption that plane sections remain plane and radii remain straight — in the form given by Hibbeler, *Mechanics of Materials*, chapter 5 ("Torsion"), which is the form the microsim computes and the form named in the portal's sim spec. Standard textbook mechanics; no page pinned for this article. [^up12-3]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. §12.3 "Stress, Strain, and Elastic Modulus," pp. 594–599 — shear stress, shear strain and the shear modulus, and the isotropic relation between the elastic constants. https://openstax.org/details/books/university-physics-volume-1 (Portal Book 077). [^mat]: `solid.mech.MATERIALS` and `solid.mech.ELASTIC` in the Wikitube framework library, rows `steel_1020` (`E = 207 GPa`, `S_y = 295 MPa`) and `al_6061_T6` (`E = 68.9 GPa`), with Poisson's ratio 0.30 for iron and 0.33 for aluminium, transcribed from Callister, *Materials Science and Engineering: An Introduction*, Appendix B. `G` is computed by `design.stress.shearModulus(E, ν) = E/(2(1+ν))`, giving 79.6 GPa for the steel used here. [^sim]: Wikitube engineering-portal run, September 18, 2026: sim spec `specs/sims/Torsion_(mechanics).json` and the job E-C build report in `plans/reports_engrun_2026-09-18.md`, with the See-also variants `Second_polar_moment_of_area` and `Torsion_constant`. The report's hand check reproduces every figure used here — `G = 207/(2 × 1.30) = 79.6 GPa`, `J = 25.1 cm⁴`, `τ = 39.8 MPa`, `φ = 0.0250 rad = 1.43°`, `GJ/L = 20.0 kN·m/rad`; the same-mass tube at `r_i/r_o = 0.8` giving `J × 4.56 = 114 cm⁴` and `τ × 0.366 = 14.6 MPa`; the shear yield line `τ_y = 170 MPa` from `design.stress.vonMises`; and the equal-area 2:1 rectangle at `K = 18.1 cm⁴` against `I_p = 32.9 cm⁴`. It also records the ILLUSTRATIVE items named in the prose: the twist drawn at ×25 and clamped at 300 degrees, the shaft drawn not to scale, and the `r_i/r_o = 0.8` wall ratio as a stated design choice rather than a physical constant. [^roark]: The fitted rectangular-section coefficients (`c₁` = 0.141 at 1:1, 0.229 at 2:1, → 1/3 for a long strip) and the maximum-shear form `τ_max = T (3a + 1.8b) / (a² b²)` as given in Roark's *Formulas for Stress and Strain*, the torsion table. The Wikitube framework library has no rectangular shear-stress function, so the `Torsion_constant` variant carries this form in its own code append; the gap is logged as a LIB GAP in the job E-C build report. (Edition and table number not re-checked for this article.) [^certified]: *Citation needed.* The complementary primal–dual variational principles that bracket the torsional rigidity are classical, but the specific certified relations and the reliability assessment that the pair article's section reports could not be traced to a primary source from this container at the time of writing. Searched: the pair article's own reference list was not retrievable at the pinned revision under the API rate limit in force during this run, and no DOI was confirmed. What would settle it: the DOIs of the papers cited in the pair's "Certified primal–dual method" section at revision 1373283689, which should be resolved and cited directly before this article moves from `drafted` to `validated`. [^mohr]: The construction that reads a pure shear state as equal tension and compression on planes at 45 degrees is Mohr's circle in the Hibbeler sign convention, as built by `design.stress.principal` and `design.stress.mohrPoint` and taught by the portal's own stress sim (engineering-portal section 1, job E-A build report). Standard; no page pinned. [^cavendish]: Cavendish, Henry (1798). "Experiments to determine the density of the earth." *Philosophical Transactions of the Royal Society of London* 88: 469–526. https://doi.org/10.1098/rstl.1798.0022 (The apparatus is a torsion balance; the mean density reported in the paper's conclusion is about 5.48 times that of water. Coulomb's torsion-balance work of the 1780s is cited here from the standard histories of the instrument, without a page.) [^hong2014]: Hong, Jiarong; Toloui, Mostafa; Chamorro, Leonardo P.; Guala, Michele; Howard, Kevin; Riley, Sean; Tucker, James; Sotiropoulos, Fotis (June 24, 2014). "Natural snowfall reveals large-scale flow structures in the wake of a 2.5-MW wind turbine." *Nature Communications* 5: 4216. https://doi.org/10.1038/ncomms5216 [^eolos]: Eolos Wind Energy Research Consortium, University of Minnesota. "Who We Are" — a Minnesota-based wind energy research group whose work includes field-scale demonstration at its own wind research station. https://eolos.umn.edu/  **Microsim — three.js (Wikitube framework):** *Torsion (mechanics)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Torsion_(mechanics).html" data-title="Torsion (mechanics)"></div> *Built from `MICROSIM_GUIDE/specs/sims/Torsion_(mechanics).json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Torsion_(mechanics)) : [Wikitube](https://en.wikitube.io/wiki/Torsion_(mechanics)) - skeleton pinned to revision 1373283689 (2026-09-18).