A browser-based computational tool for the torsional analysis of circular, annular (hollow circular), rectangular and square cross-sections — including composite shafts built from concentric rings of different materials — with interactive stress-distribution drawing and 3D visualisation of the twisted (and warped) member.
Developed by Assoc. Prof. Rasim Temür · İstanbul University-Cerrahpaşa, Department of Civil Engineering
Part of the Vetin initiative for the digitisation of academic instruction tools.
The application is accessible directly through a web browser without requiring any software installation, user registration, or server-side processing. All computations are performed client-side.
Vetin Torsion is an open-source, web-based application developed for educational use in mechanics of materials curricula. It enables the interactive construction of shaft cross-sections — solid circles, rings (annuli), rectangles and squares — and computes the shear-stress distribution produced by a torsional moment.
Two distinct theories are involved, and the application keeps them apart. A circular cross-section is the one shape that remains plane under torsion, which is why the elementary formula τ = T·ρ/Ip is exact for it. Every non-circular section warps out of plane, and a rectangle must be solved with Saint-Venant torsion theory, in which neither the stress distribution nor the location of the maximum stress follows the circular intuition. Because the two rest on incompatible kinematic assumptions, the application refuses to combine them in one section rather than summing quantities that are not additive.
Beyond homogeneous sections, the application solves composite shafts consisting of concentric rings made of different materials (different shear moduli G). The stress diagram along a diameter, including the characteristic jumps at material interfaces, is drawn on the section in real time.
Key properties of the application:
- Operates entirely within the client browser; no server-side computation is required
- Functions offline as a Progressive Web App (PWA)
- User interface is localised in 33 languages
- Supports multi-material concentric sections (e.g. a steel core inside a brass sleeve)
- Closed-form and exact-series solutions throughout — no meshing, no numerical integration
- Source code is freely distributed under the MIT License
| Tool | Description |
|---|---|
| Circle | Add a solid circular region (drag from centre outwards) |
| Ring | Add an annular region with three clicks: the centre, then either of the two diameters, then the other (the larger becomes the outer radius) |
| Rectangle | Add a solid rectangular region by dragging corner to corner; hold Shift for a square. One rectangle per section (see the note under Thin-walled sections), and not combinable with circular parts |
| Profile (disabled) | Place a ready-made thin-walled profile. Clicking the tool opens a list — I, channel (U), Z, T, angle (L) or a closed box — and the chosen one is placed on the next click. It is generated from bf, bw, tf, tw, editable in the section list along with G; adding a further element by hand turns it into an ordinary assembly |
New circular parts automatically snap to the common centre of the existing section, so composite sections are always concentric — a requirement of the theory (see Assumptions). In Edit mode radii and rectangle edges can be adjusted by dragging handles, and the whole section can be repositioned. Each part's dimensions (ro, ri for circular parts; b, h for a rectangle) and its shear modulus G can also be edited numerically in the section list.
Section properties (exact, closed-form — no numerical integration):
- A — net cross-sectional area
- Ix, Iy, Ixy — centroidal second moments of area; for a circular or annular section the polar moment Ip is listed with them as well, where the reader can check it against Ip = Ix + Iy on the spot. The row is omitted for a rectangle, where the polar moment is not what governs torsion
- Ip (circular) or It (rectangular) — the torsion constant of the section
- Wt — torsional section modulus (Ip/ρmax for a circular section, α·a·b² for a rectangle); for a homogeneous section τmax = T/Wt, whereas in a composite shaft the stress follows from Gi·θ′·ρ and Wt remains a purely geometric quantity
Torsional response for an applied torque T, grouped into two panels — Kayma Gerilmeleri (shear stresses) and Deplasmanlar (displacements):
- τmax, and τmin (circular, at the bore) or τ2 (rectangular, at the midpoint of the short side)
- Per-material stresses τin, τout at the inner and outer radii of each ring
- ΣG·Ip (circular) or G·It (rectangular) — torsional rigidity of the section
- θ′ — rate of twist, the rotation per unit length (per metre)
- φ = θ′·L — relative rotation of the two ends of a bar of length L
L is entered in the Deplasmanlar panel and is the same bar length the 3D view extrudes: the two input fields mirror each other. Left empty it follows the section automatically (ten times the largest section dimension); typing a value pins it until the field is cleared again.
Angles are reported in radians by default — the natural unit of the theory, and the one in which θ′·L is a pure number — with a Radyan / Derece switch that converts both quantities to degrees. The choice is remembered between sessions and the 3D panel's true end-rotation readout follows it.
For a rectangular section the result panel switches its labels to the quantities that actually govern it: It, Wt, G·It and τ2, and it drops the Ip row from the moments-of-inertia list. The distinction matters: for a non-circular section the polar moment Ip is not the torsion constant, and displaying it would be misleading.
- Colour-coded materials (each part drawn in its own material colour). Each of the three themes — light, dark and blueprint — carries its own palette rather than reusing the light pastels, which washed out on a dark canvas and clashed with blueprint's blue paper. The palettes are anchored to the 3D view: a theme's first material colour is the 3D bar colour, and the 2D canvas background is the 3D scene background, so the same section reads the same in both panels
- Stress map in 3D — the same colour field is painted onto the extruded member, driven by the same toggle and the same scale, with its own colour bar beside the 3D view. It is applied as vertex colours written before the twist deformation, so the colours travel with the section as it rotates. Because a vertex colour is only visible where a vertex exists, switching the map on also subdivides the section outline and replaces the extruded end caps with a subdivided surface — an unsubdivided rectangle has vertices only at its four corners, where τ = 0, and its whole lateral face would come out blue
- Stress map — Gerilme Haritası fills the cross-section with a colour field: every point is tinted by the magnitude |τ| there, blue at the smallest value rising through cyan, green and yellow to red at the largest, with a labelled colour bar beside the drawing. It shows the whole two-dimensional distribution at once — the saddle-shaped field of a rectangle, the radial gradient of a shaft, and the step at a material interface in a composite, which a line diagram can only report at two points. The field is exact, not interpolated from samples: circular sections use τ = Gi·θ′·ρ and rectangles the same Saint-Venant series that produces τmak. The scale runs between the true extremes of the section, so a hollow shaft starts at τmin at the bore rather than at zero. It is off by default — the arrow diagram below is the default view — and the two are independent toggles that can also be shown together. Two controls shape the scale. Renk Ölçeği chooses the range: Otomatik spans the section's own τmin…τmax, so the colours read the shape of the distribution and do not change with the torque (τ and τmak scale together); Sabit spans 0…τref for a reference stress you enter, so the colours track the absolute magnitude — raise the torque and the section moves from blue towards red, saturating at red once τref is exceeded, which makes a comparison against an allowable stress immediate. Renk dağılımı γ bends the ramp itself (t → tγ): γ < 1 separates the low-stress region, γ > 1 the high-stress one, with the ends fixed. The colour bar is drawn through the same curve, so it always shows the mapping actually in use, while its tick labels stay linear in stress. Both settings are remembered between sessions
- Circular sections — the shear-stress diagram is drawn over the full vertical diameter on an opaque background, so it reads as a block laid over the section. Since τ is tangential, the arrows are horizontal and reverse across the centre (antisymmetric). The envelope is linear within each material and jumps at material interfaces; both sides of a jump are labelled, τmak and τmin appear at both ends of the diameter, and the envelope continues as a dashed line across the bore of a hollow section
- Rectangular sections — the diagram is drawn along both centroidal axes with tangential (perpendicular) arrows: τmak at the midpoints of the long sides, τ2 at the midpoints of the short sides, and zero at the corners. The envelope follows the exact series profile, which is not linear
- Rectangular sections, diagram path — Diagram along: Axes / Diagonal / All chooses where the diagram is drawn. Diagonal replaces the two symmetry axes with a single diagonal; All shows the horizontal axis, the vertical axis and the diagonal together in one figure. A diagonal is not a symmetry axis (except in a square), and it makes a point the axis diagrams cannot: τ vanishes at both ends of the diagonal — at the centroid and at the corner — with a single maximum in between, worth ≈0.46·τmak for a square rising to ≈0.60·τmak at 4:1. Ordinates are the stress magnitude |τ|, computed from the exact stress field at each point and plotted normal to the diagonal in the usual diagram convention. Note that the arrow length carries the magnitude while its direction only marks the sense of twist: the true τ vector is normal to the diagonal only in a square, and leans towards the long-side direction as the rectangle lengthens (≈15° off the diagonal at 4:1). Every mode uses the same scale, so switching between them compares the three paths directly. Because the distribution is antisymmetric about the centroid, a single-path mode draws both lobes of its line; All draws one lobe each — one horizontal, one vertical, one diagonal — and drops the corner τ = 0 label, which would otherwise collide with the axis label on a square (the envelope still visibly returns to the baseline there)
- Torque Mb — a compact circular arrow near the centroid, turning in the same sense as the shear-stress arrows (the τ distribution equilibrates this torque), with its gap left open along the radius dimension arrows; the sense follows the sign of T
- Radius dimensioning — an arrow from the centre to each circle, labelled R for a solid part and Rd / Ri for a ring (part number appended in composite sections; a circle shared by two parts is dimensioned once). Rectangles are dimensioned by their overall b × h
- Centroidal axes, centroid marker, dimension lines, part borders — each toggleable
- Verification of the thin-walled formulation: element areas and second moments checked against hand calculation (including the non-zero Ixy of Z and L), the open-section J against (1/3)Σb t³ and against the exact Saint-Venant series in the thin-strip limit (within 5 % at b/t = 40), and the closed box against Bredt–Batho term by term (Am, ∮ds/t, q, τ = q/t, Wt)
- SVG export of the section drawing (the stress map travels with it as an embedded image, the colour bar as vector rectangles); JSON save/load of the project (format v2.1)
An interactive 3D view (Three.js/WebGL) extrudes the section along the member axis with per-material colours, wireframe/edge toggles and fullscreen support. Dragging with the left mouse button pans, the right button orbits, and the middle button zooms (as does the scroll wheel); on touch, one finger orbits and pinch zooms.
Picture-in-picture preview — loading a model or drawing the first shape on an empty canvas pops up a small 3D preview in the bottom-right corner of the section-drawing panel, without switching the full layout to split view. It shares the same 3D scene as the full view (only its size and position change, so no second render context is created), and the camera re-frames the member for the box it is moving into — the small preview and the full panel have very different proportions, and a camera left at the other one's distance would not fit, and carries two controls in its top-right corner: a restore button that opens the 3D panel at its normal, full application size, and a close button that dismisses only the small preview. It reappears the next time a model is loaded or a new section is started from a blank canvas.
Orientation is controlled by an AutoCAD-style ViewCube in the corner of the 3D panel: the cube always shows the current camera direction, clicking one of its faces swings the camera to that view, and dragging the cube orbits freely. Six labelled buttons below it (front, back, left, right, top and isometric) do the same for the axis views. A view change only rotates the camera — the zoom level and the panned centre are preserved, as in CAD; Fit All is what re-frames the model. Looking straight down the axis keeps the same left-right orientation as the front view, so the member does not appear to spin when switching between them.
The camera re-fits itself only when the bar geometry changes. Changing the torque rebuilds the deformed bar, and the twisted shape's bounding box grows with it, so an unconditional auto-fit made the view zoom in and out on every step of the torque slider — the load is what the user is watching, and it was the one thing the camera would not hold still for. The torque is therefore excluded from the fit trigger: dragging the slider leaves the viewpoint, and any zoom the user has set, exactly where it was. Fit All still re-fits on demand.
Fit All frames everything the scene draws — bar, edge lines and the two end axis triads — for the direction the camera is currently looking from, and it resets the pan as well as the zoom, so a model dragged off the panel comes back. The distance is solved from the eight corners of the bounding box projected onto the camera's own screen axes rather than from the largest edge alone: the largest edge is the bar's length, which says nothing about how much of the screen the bar covers when you look down its axis, and on a short bar it put the camera inside the model. The margin is taken out of the field of view instead of multiplying the distance, because scaling the distance pushes only the near end of a long bar back and leaves the view half empty. The zoom limit and the far clipping plane follow the fit distance, so a 30 m member is framed rather than clipped away.
Member length — the drawn length defaults to ten times the largest section dimension, which keeps the proportions of a slender bar whatever the section size. It is a default, not a constraint: any value entered in the field is kept from then on (down to zero), and clearing the field returns to the automatic length. Changing it re-frames the camera, since the bar's geometry has changed. The same length also drives the relative end rotation φ = θ′·L in the results panel, so it can be set from either field and is saved with the project once it has been pinned.
Deformed shape — the bar can be drawn in its twisted state: each cross-section is rotated by φ(z) = θ′·z, and a rectangular section additionally warps out of plane by w = θ′·ψ(x, y) — a circular section does not warp (ψ ≡ 0), which is exactly why the elementary formula holds for it. The warping function ψ is the exact series solution of ∇²ψ = 0 with the free-surface condition.
Warping display — the out-of-plane deformation is a separate option, off by default, with its own exaggeration factor, and it is only offered for the section family that actually warps: for a circular or annular section the control is disabled and says why (ψ ≡ 0). Switched on, both end faces are drawn as a subdivided saddle surface carrying a grid, so the shape of ψ can be read directly, and the same axial displacement is applied to the whole body — under uniform torsion warping does not vary along the bar, so every fibre simply shifts along the axis by its own w, and the transverse contours pick the saddle up at every station. The end faces need that dedicated surface because the extruded lid is triangulated from the section outline alone: for a rectangle that is four corners and two triangles, which cannot show a saddle. While warping is drawn, the lid is hidden and the section outline is subdivided so that the lateral surface ends on exactly the same curve as the saddle — with only four corner points the lateral surface would end on a straight chord and tear away from it.
Because warping is a separate option it can also be shown on its own, with the twist switched off: a straight prismatic bar whose end faces are nevertheless not plane.
Two axis triads are drawn at the centre of the free end face. The coloured one is attached to the cross-section: it rotates with it by exactly the same φ(L) = k·θ′·L used to draw the deformed body, so the twist is readable as an angle rather than inferred from the surface. The second triad is the fixed end's frame, carried to the same origin and drawn in grey tones — it does not rotate, so the angle between the two triads is the visualised end rotation. The grey reference appears only while there is a twist to compare against; with zero torque, or with the deformed shape switched off, the two would coincide exactly and only the section triad is drawn.
Real twist angles are far too small to see (thousandths of a degree), so the shape is drawn with an exaggeration factor, exactly as in finite-element post-processing. The factor is deliberately independent of the applied torque — it is calibrated from the section's own rigidity so that a 1 kNm reference torque would show ≈25° of end rotation. The drawn twist is therefore directly proportional to T: doubling the torque doubles the visible twist. (An auto-factor recomputed from the current torque would cancel it out and leave the picture unchanged whatever the load.) Very large torques are clamped to keep the model readable, and the label reports it. The true end rotation θ′·L is always reported alongside, and the factor can be overridden manually.
Warping carries its own factor for the same reason it needs its own option. The twist of a slender bar accumulates over the length, while warping does not: the true wmax is of the order of a ten-thousandth of the section size, so at the factor that makes the twist visible the saddle is still invisible. Its factor is calibrated the same way — from a 1 kNm reference torque, so the drawn warping stays proportional to T — but to a target of ≈8 % of √(w·h), and it too can be overridden or clamped. The two components are therefore not drawn to a common scale; the panel reports the true wmax = |θ′|·max|ψ| in millimetres so the real magnitude is never lost.
Because a circular bar's outer surface is a surface of revolution, twisting it leaves the silhouette unchanged. Longitudinal reference lines — straight before deformation, helical after — are drawn on the surface (at the corners of a rectangle) so that the twist is visible for every section type.
The lateral surface is outlined with transverse cross-section contours (toggleable, drawn at any torque including zero) rather than mesh edges: on a twisted surface the two triangles of each quad no longer share a normal, so an edge detector reports the triangulation diagonals instead of real edges. The contours are generated from the section outline itself, so they follow the twisted — and, for a rectangle, warped — surface exactly; with no torque they are simply straight cross-sections along the bar. Contours and longitudinal lines are drawn in the section's own border colour.
For the same reason the shading uses crease-angle normal smoothing instead of per-triangle normals: normals are averaged between neighbouring faces whose angle stays under a threshold, so the twisted surface shades continuously while the 90° corners of a rectangle and the section/lateral boundary stay sharp. Without it the mesh triangles show up as alternating tones.
| Symbol | Meaning | Unit in the interface |
|---|---|---|
| T | applied torsional moment (torque) | kNm |
| G | shear modulus of a material | GPa |
| θ′ | rate of twist (twist per unit length) | rad/mm internally, reported in rad/m or °/m |
| φ | relative rotation of the bar ends, φ = θ′·L | rad or ° |
| L | bar length | mm |
| τ | shear stress | MPa |
| ρ | radial distance from the axis (circular sections) | mm |
| ro, ri | outer / inner radius of a ring | mm |
| b, h | width and height of a rectangle | mm |
| a, b | long and short side of a rectangle (theory), q = a/b ≥ 1 | mm |
| Ip | polar second moment of area | mm⁴ |
| It | torsion constant (Saint-Venant) | mm⁴ |
| Wt | torsional section modulus, τmax = T/Wt | mm³ |
| φ | Prandtl stress function | — |
| ψ | warping function, w = θ′·ψ | mm² |
| α, β, γ | rectangle coefficients: τmax = T/(α·a·b²), It = β·a·b³, τ2 = γ·τmax | — |
The application solves uniform (pure) Saint-Venant torsion of a prismatic bar: a straight member of constant cross-section loaded only by equal and opposite torques at its ends, free to warp along its entire length. This is the classical problem treated in mechanics-of-materials and theory-of-elasticity courses, and it is the reference case against which more complex situations (warping restraint, variable section, combined loading) are compared.
- Linear elastic material. Hooke's law in shear, τ = G·γ, holds throughout. No yielding, no non-linearity, no unloading history. All results therefore scale linearly with T.
- Homogeneous and isotropic material within each part. A part is described by a single shear modulus G; the material has no directional dependence.
- Prismatic member. The cross-section is constant along the member axis; there are no tapers, holes, notches, keyways or fillets.
- Uniform torsion with free warping. The torque is constant along the bar and warping is unrestrained at every section, so no axial (warping normal) stresses arise: σz = 0 and the only non-zero stresses are the two transverse shear components τzx, τzy. This is what distinguishes Saint-Venant torsion from non-uniform (Vlasov) torsion, where restrained warping produces a bimoment and axial stresses.
- Constant rate of twist. Because T, G and the section are constant, the rate of twist θ′ (rotation per unit length) is the same at every section, and the total rotation of a bar of length L is θ′·L.
- Small displacements and small rotations. Geometric linearity; equilibrium is written on the undeformed geometry. The 3D deformed shape is an exaggerated visualisation, not a large-displacement analysis.
- Pure torsion only. No axial force, bending moment or transverse shear is considered, so no interaction or combined-stress check is performed.
- Saint-Venant's principle. The solution describes the state away from the ends; the local disturbance where the torque is actually introduced, and any support detail, is outside the model.
- Torque applied about the centroidal axis. All sections treated here (circle, annulus, rectangle) are doubly symmetric, so the shear centre coincides with the centroid and the twist axis is the centroidal axis.
For a circular or annular section — and only for these — the assumed kinematics are exact:
- Cross-sections remain plane and radii remain straight; each section rotates as a rigid disc about the axis. The warping function vanishes identically (ψ ≡ 0).
- Compatibility: the shear strain at radius ρ is θ′·ρ, varying linearly from zero at the axis.
- Constitutive law: τ(ρ) = G·θ′·ρ.
- Equilibrium: T = ∬ τ·ρ dA.
For a homogeneous section this yields the classical result
τ(ρ) = T·ρ / Ip, τmax = T·ρmax / Ip = T / Wt, θ′ = T / (G·Ip)
with section properties evaluated in closed form for each part (ri = 0 for a solid circle):
- Ai = π (ro² − ri²)
- Ix,i = Iy,i = (π/4)(ro⁴ − ri⁴)
- Ip,i = Ji = (π/2)(ro⁴ − ri⁴)
Additional assumptions:
- Concentric parts. All rings share a common centre, so a single twist axis exists. The application enforces this when parts are drawn and reports an error otherwise.
- Perfect bond at the interfaces. No slip occurs between adjacent materials, so the displacement — and therefore the shear strain — is continuous across an interface.
- Parts do not overlap. Material regions are radially disjoint; overlapping geometry is rejected.
The rings twist together, so the rate of twist is common to all materials:
- Compatibility: the shear strain at radius ρ is θ′·ρ in every material
- Equilibrium: T = θ′ · Σ (Gi · Ip,i) → θ′ = T / Σ (Gi · Ip,i)
- Stress in material i: τi(ρ) = Gi · θ′ · ρ
Two consequences are worth emphasising, and both are drawn by the application:
- The strain is continuous across an interface but the stress is not: it jumps in proportion to the ratio of the shear moduli. Two different values therefore exist at the same radius, one for each material.
- τmax need not occur at the outer surface. If an inner material is considerably stiffer, the largest stress may occur at an internal interface; the application searches all band edges rather than assuming the outermost one.
For a single material these expressions reduce to the classical τ = T·ρ/Ip.
A non-circular section warps: points of the cross-section displace along the member axis by w = θ′·ψ(x, y). Assumptions 1–9 above still hold, but the plane-section kinematics do not, and consequently:
- τ = T·ρ/Ip is invalid;
- the maximum stress does not occur at the point farthest from the centroid — it occurs at the midpoint of the long side, which is the boundary point nearest the axis;
- the shear stress at the corners is zero, because at a free surface the shear stress must be tangent to the boundary and a corner belongs to two mutually perpendicular faces.
The problem is solved by Saint-Venant's semi-inverse method through the Prandtl stress function φ:
∇²φ = −2·G·θ′ inside the section, φ = 0 on the boundary,
τzx = ∂φ/∂y, τzy = −∂φ/∂x, T = 2∬ φ dA
Equivalently, in terms of the warping function, ∇²ψ = 0 with the free-surface condition ∂ψ/∂n = y·nx − x·ny. Both forms are used in the application: φ supplies the stresses and the torsion constant, ψ supplies the out-of-plane deformation drawn in the 3D view.
With a the long side, b the short side and q = a/b ≥ 1:
- Torsion constant: It = β·a·b³ (≠ Ip)
- Rate of twist: θ′ = T / (G·It)
- Maximum stress, at the midpoint of the long side: τmax = T / (α·a·b²) = T/Wt
- Stress at the midpoint of the short side: τ2 = γ·τmax
The coefficients α, β, γ are evaluated from the exact series solution rather than interpolated from a table. The hyperbolic terms are rewritten so that the sums converge exponentially, and a handful of terms suffice at any aspect ratio:
- β = 1/3 − (64/π⁵)(1/q) · Σn odd tanh(nπq/2)/n⁵
- τmax = k₁·G·θ′·b, with k₁ = 1 − (8/π²) Σn odd 1/(n² cosh(nπq/2))
- α = β/k₁, and γ from the corresponding series at the short-side midpoint
Limiting cases reproduce the familiar results: for a square, α = 0.208, β = 0.141 and γ = 1 (all four mid-sides equally stressed); for a thin strip (q → ∞), α, β → 1/3 and γ → 0.742, i.e. It → a·b³/3 and τmax → 3T/(a·b²).
Away from the two centroidal axes the closed-form mid-side profiles no longer apply, so the full stress field is evaluated from φ directly — this is what the diagonal diagram plots:
τzx = Gθ′ [ −2y + (8h/π²) Σn odd (−1)(n−1)/2/n² · cosh(nπx/h)/cosh(nπw/2h) · sin(nπy/h) ]
τzy = Gθ′ (8h/π²) Σn odd (−1)(n−1)/2/n² · sinh(nπx/h)/cosh(nπw/2h) · cos(nπy/h)
with x, y measured from the centroid and the shorter side taken along y so that the series decay exponentially. Evaluated at the mid-sides these reduce exactly to k₁·Gθ′·b and k₂·Gθ′·b, and the field is antisymmetric about the centroid, τ(−P) = −τ(P) — which is why the diagonal ordinates fall on opposite sides of the baseline in the two halves.
Currently disabled in the interface. The thin-walled profile feature is switched off behind a single flag (
PROFILE_UI_ENABLEDinscript.js): the Profil tool is hidden and a section again holds at most one rectangle. The implementation below is intact and still runs — a project file containing a multi-element section loads and computes as described — and setting the flag back totruerestores the tools. The rest of this section documents the feature as it behaves when enabled.
A thin-walled member is idealised as an assembly of narrow rectangular walls. The section is built by adding elements: each rectangle drawn (or generated by a ready-made profile) is one wall, and the assembly is analysed as a whole. Because elements may be drawn overlapping, the area and the second moments are taken from an exact decomposition of the union — all element edges are collected into a grid whose every cell lies wholly inside or wholly outside, so overlapping material is counted once and the result stays closed-form.
Whether the section is open or closed is read off the geometry rather than declared: a flood fill over that same grid looks for a void the outside cannot reach. Bredt–Batho is solved for a single rectangular cell, so anything else — two cells, an L-shaped void — is reported instead of being silently approximated. All elements must share one shear modulus; a multi-material thin-walled profile is a different problem and is refused rather than answered with the first element's G. The two topologies obey different laws, and the difference is the single most important fact about torsion of such members.
Open profiles (I, channel, Z, T, angle) behave as the sum of their walls. Each wall carries the thin-rectangle solution, so with bi the wall mid-line length and ti its thickness:
J = (1/3) Σ bi·ti³, θ′ = T/(G·J), τi(n) = 2·G·θ′·n
with bi and ti the long and short side of the element as drawn. The junction where two walls meet belongs to the element that covers it, so it is counted once; against the mid-line convention this is slightly conservative (≈1 % for a typical I).
The shear stress is linear across the thickness: zero on the wall mid-line, greatest on the two surfaces where τi = G·θ′·ti. The largest stress is therefore in the thickest wall, and τmax = T·tmax/J. Stiffening from the fillets at the junctions is neglected (the usual η factor is taken as 1), so J is a lower bound.
Because the assembly is what is analysed, a shape outside the ready-made list — a hat, a stiffened plate, an unequal angle — is obtained simply by drawing its walls.
Closed profiles (box) carry a shear flow q that is constant around the perimeter (Bredt–Batho). With Am the area enclosed by the wall mid-line:
q = T/(2·Am), τi = q/ti, J = 4·Am²/∮(ds/t)
Here τ is constant through the thickness and the largest stress is in the thinnest wall — the opposite of the open case. Closing a section is dramatic: for the same outside dimensions a box is two orders of magnitude stiffer in torsion than the corresponding channel (≈200× for the default 100×200×10×7 profile), and its peak stress is smaller in the same proportion. The colour map shows both facts at once — the gradient across each wall of an open profile, the flat colour of each wall of a closed one.
Why the two families are never mixed. Circular and rectangular torsion rest on incompatible kinematic assumptions — plane sections versus warping — and their stiffnesses are not additive across such a boundary. A section therefore holds either concentric circular parts or a single rectangle; the application blocks the combination both while drawing and while computing.
Being explicit about the boundaries of the model is part of using it correctly:
- Warping restraint (non-uniform / Vlasov torsion) — no bimoment, no warping normal stresses, no torsional buckling
- Plasticity, creep, fatigue, temperature effects — the response is linear elastic throughout
- Stress concentrations — fillets, keyways, holes, re-entrant corners and other geometric discontinuities
- End and support effects — only the Saint-Venant region is described
- Combined loading — no axial, bending or shear interaction; no principal-stress or yield-criterion check (the state is pure shear, so principal stresses of ±τ act at 45° to the axis)
- Hollow rectangular (box) sections — these require thin-walled Bredt–Batho theory (It = 4Am²/∮(ds/t)), which is a different formulation and is out of scope
- Open thin-walled profiles (L, T, I sections built from rectangles, It ≈ Σ hibi³/3) and arbitrary polygonal sections
- Composite (multi-material) rectangles — no elementary solution exists for a warping composite section
- Multi-cell closed sections — a single rectangular cell is assumed, so one shear flow; anything else is reported rather than approximated
- Multi-material thin-walled assemblies — all elements must share one G
- Junction stiffening in open profiles — the fillet correction factor η is taken as 1, so J is slightly conservative for rolled shapes
- Non-concentric circular assemblies — a single twist axis is assumed
- The 3D deformed shape is a scaled visualisation, not a stress plot or a large-displacement analysis
The implementation is checked against independent references rather than against itself:
- Circular and composite sections use closed-form expressions; the composite results are verified against hand calculations and against the limiting single-material case.
- Rectangle coefficients reproduce the published Timoshenko/Roark tables across the full range of aspect ratios (α, β to ±0.0015 and γ to ±0.0025), satisfy the square-symmetry condition γ = 1 exactly, converge to the thin-strip limits, and agree with an independent finite-difference solution of ∇²φ = −2Gθ′ to within 0.02 %.
- The stress field used by the diagonal diagram is checked against an independent finite-difference solution of the Prandtl equation over the whole section (worst-case deviation below 0.1 % of τmax at aspect ratios from 1:1 to 4:1), reproduces k₁ and k₂ exactly at the mid-sides, and returns zero at the centroid and at the corners.
- The warping function ψ is verified by confirming ∇²ψ = 0, by checking the free-surface boundary condition, and — the decisive test — by recovering the torsion constant from it, It = ∬ (x² + y² + x·∂ψ/∂y − y·∂ψ/∂x) dA, which matches β·a·b³ to better than 0.01 % for common aspect ratios and 0.05 % for a 10:1 strip.
- The warped 3D surface is read back out of the generated geometry and checked against ψ's own properties: zero on the centroidal axes, antisymmetric across them, identical at both ends (uniform torsion), proportional to T, and — for a 10:1 strip — matching the classical thin-rectangle limit w → −θ′·x·y away from the short edges. Its boundary is verified to coincide vertex-for-vertex with the lateral surface, so no gap can open between them.
- The drawing, interaction and file-format logic is covered by an automated test suite that exercises the application's own functions.
The user interface is localised in 33 languages, selectable at runtime and persisted via localStorage:
| Code | Language | Code | Language | Code | Language |
|---|---|---|---|---|---|
tr |
🇹🇷 Turkish | en |
🇬🇧 English | de |
🇩🇪 German |
fr |
🇫🇷 French | es |
🇪🇸 Spanish | it |
🇮🇹 Italian |
pt |
🇧🇷 Portuguese | ru |
🇷🇺 Russian | ro |
🇷🇴 Romanian |
bg |
🇧🇬 Bulgarian | el |
🇬🇷 Greek | sl |
🇸🇮 Slovenian |
sq |
🇦🇱 Albanian | hy |
🇦🇲 Armenian | ka |
🇬🇪 Georgian |
he |
🇮🇱 Hebrew | ar |
🇸🇦 Arabic | fa |
🇮🇷 Persian |
ur |
🇵🇰 Urdu | hi |
🇮🇳 Hindi | bn |
🇧🇩 Bengali |
ne |
🇳🇵 Nepali | dz |
🇧🇹 Dzongkha | my |
🇲🇲 Burmese |
th |
🇹🇭 Thai | id |
🇮🇩 Indonesian | tl |
🇵🇭 Filipino |
zh |
🇨🇳 Chinese | ja |
🇯🇵 Japanese | ko |
🇰🇷 Korean |
uz |
🇺🇿 Uzbek | tg |
🇹🇯 Tajik | ky |
🇰🇬 Kyrgyz |
| Technology | Role |
|---|---|
| HTML5 / CSS3 / JavaScript (ES6+) | Core application architecture |
| HTML5 Canvas API | Section drawing and stress visualisation |
| Three.js (WebGL) | Interactive 3D member visualisation |
| SVG | Vector export of section drawings |
| Service Worker API | Offline caching and PWA functionality |
| Web App Manifest | Home screen installation support |
| localStorage API | Persistence of user preferences (language, theme) |
torsion/
├── index.html # Application entry point and HTML shell
├── manifest.json # PWA manifest descriptor
├── sw.js # Service Worker (offline caching)
│
├── script.js # Torsion computations, canvas drawing, UI logic
├── script3d.js # Three.js 3D visualisation and deformed shape
│
├── translations.js # Localisation string repository (33 languages)
│
├── style.css # Base styles, themes (light / dark / blueprint)
│
├── logo.svg # Application logotype
├── icon.svg # Source vector icon
├── IUC.svg # İstanbul University-Cerrahpaşa logo
├── icon-192.png # PWA icon (192 × 192 px)
└── icon-512.png # PWA icon (512 × 512 px)
As the application comprises static files only, it may be served locally with any HTTP server:
# Python 3 — built-in HTTP server
python -m http.server 8000
# Node.js — via npx
npx serve .Navigate to http://localhost:8000 in a web browser to launch the application. On PWA-capable browsers the application may be installed to the device home screen for offline use.
The application opens on a model chooser: start a new model, open a saved file, or pick one of the ready examples. The examples are chosen to show different things rather than different dimensions — the efficiency of a hollow shaft, the stress jump at a material interface in a composite, the case where τmax falls inside the section because the core is the stiffer material, and the rectangular family from a square to a 2:1 section (warping).
Each card's picture is an SVG generated from the model itself, so it cannot drift out of step with what clicking it produces; the colours come from the same material palette as the canvas. A don't show again checkbox remembers the choice, and the language can be switched from the same screen — the card names and captions are localised along with the rest of the dialog.
- Draw the cross-section — Use the Circle tool for a solid shaft, or the Ring tool for a hollow shaft: click the centre, then one diameter, then the other — the ring is created in one go (press Esc to cancel a half-finished ring). Add further concentric rings for composite sections; they snap to the common centre automatically. For a rectangular or square shaft use the Rectangle tool (drag corner to corner, Shift for a square) — one rectangle per section, and it cannot be combined with circular parts.
- Assign materials and dimensions — In the Kesitler (Sections) list, set each part's shear modulus G (GPa). Dimensions can be edited numerically there as well: rd / ri for circular parts, b / h for a rectangle.
- Apply the torque — Enter the torsional moment T (kNm) or drag the slider beneath it; the section drawing, the results and the 3D deformed shape follow live (double-click the slider to return to zero). Negative values reverse the twist direction. Typing a value beyond the slider's range widens it.
- Read the results — τmax and τmin (or τ2 for a rectangle) and the per-material interface stresses appear under Kayma Gerilmeleri; Ip / It and Wt under the section-property panel; and ΣG·Ip / G·It with the rate of twist θ′ and the relative end rotation φ = θ′·L under Deplasmanlar. All of them update instantly. Set the bar length L there, and switch the angle unit between radians (default) and degrees.
- Read the diagram — Gerilme Dağılımı is on by default and draws the classical ordinate diagram. For a circular section it is drawn over the full vertical diameter, with the tangential arrows reversing across the centre and jumping at material interfaces. For a rectangle it is drawn along both centroidal axes, with the maximum at the midpoints of the long sides and zero at the corners.
- Switch on the stress map (optional) — Gerilme Haritası fills the cross-section with the colour field of |τ|: blue where the stress is smallest, red where it is largest, with the scale shown beside the drawing. In a composite shaft the colour steps at the material interface; in a rectangle the red bands sit at the midpoints of the long sides and the corners stay blue. It can be shown together with the diagram.
- Study the deformation — Switch on the 3D view to see the twisted member. For a rectangle, switching on Çarpılma (warping) — off by default, so the bar starts out as a plain prism — additionally draws the end sections as the saddle surface they really become; it has its own exaggeration factor, and it can be shown with the twist switched off. The factors, the true end rotation and the true maximum warping are reported in the 3D settings; the end rotation is the same φ as in the Deplasmanlar panel, in the same unit.
- Export — Export the drawing as SVG, or save/load the project as JSON.
This software is distributed under the MIT License.
Full license terms are available in the LICENSE file.
Assoc. Prof. Rasim Temür
Department of Civil Engineering
İstanbul University-Cerrahpaşa
🌐 rasimtemur.com
Vetin is a collection of open-source, browser-based computational tools developed for use in civil and structural engineering education. Additional tools within the Vetin ecosystem are accessible at rasimtemur.com/vetin.
Developed in support of engineering education.
MIT License ·
İstanbul University-Cerrahpaşa