Three everyday mechanics tools: springs, axial stress, bending and torsion
Pick the tool that matches your problem. Spring: force at a deflection from a rate (Hooke’s law), the deflection a target force needs, or the rate hidden in two measurements, with preload and stored energy handled openly. Axial stress and strain: stress from force over area, strain and elongation from a chosen modulus, and a safety factor against a yield strength you supply. Bending and torsion: surface stress in a rectangular or round section, torsional shear in a round shaft, and, where a round shaft carries bending and torsion together, the von Mises equivalent for that defined surface stress state. Every assumption prints with its result, because in stress work the assumptions are half the answer.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Stress results are only as good as the load and section assumptions behind them; safety-critical design belongs with a qualified engineer and the governing codes.
What these tools do
The spring tool runs Hooke’s law in the three directions workshops actually ask: force from rate and deflection (adding any preload, and reporting the stored energy), deflection from a target force (net of the same preload), and the rate itself from two measured points, which is how unmarked springs get identified. The axial tool divides force by area for stress, divides stress by modulus for strain, scales strain by length for elongation, and, given a yield strength, states the safety factor plainly. The bending and torsion tool computes section properties (second moment of area for a rectangle or a circle, polar moment for the shaft), surface stresses from moment and torque, and the von Mises equivalent where a round shaft’s surface carries both at once, labelled as the ductile-material equivalent stress for exactly that stress state, not a universal verdict on failure.
How to use them
- Springs: measure deflection from the free state unless preload is stated. Installed springs start pre-compressed; the preload field exists so working force is not understated by exactly the amount everyone forgets.
- Axial: stress is force over the smallest honest area. Threads, holes and necks shrink the section that matters; use the net area at the critical point, not the bar’s catalogue area.
- Bending: depth is squared, width is not. Doubling a beam’s depth quarters its bending stress (and eighths its deflection in the Beam tool); doubling width only halves stress. Orientation errors here are the commonest stress mistakes in home engineering.
- Torsion belongs to round sections in this tool. Rectangular torsion follows different, harsher rules and is deliberately not approximated by the round formula.
The formulas
- Spring: F = preload + k × x; x = (F – preload) / k; k = (F2 – F1) / (x2 – x1); energy = ½ × k × x².
- Axial: stress σ = F / A; strain ε = σ / E; elongation = ε × L; safety factor = yield / σ.
- Bending: σ = M × c / I, with I = b × h³ / 12 (rectangle) or π × d⁴ / 64 (round), c = half the depth or diameter.
- Torsion (round): τ = T × r / J, J = π × d⁴ / 32. von Mises for the round-shaft surface state: σ’ = √( σ² + 3 × τ² ).
Worked examples
The spring. Rate 10 N/mm at 25 mm deflection: force 250.00 N; with a 50 N preload installed, the working force is 300.00 N; energy stored at that deflection is 3.13 J. Ask the inverse question (what deflection gives a 300 N target against the same preload?) and the answer returns 25.00 mm, the round trip every spring buyer should check before ordering.
The tie bar. 1,000 N over 50 mm² is 20.00 MPa; in steel that is 100.00 microstrain, half a millimetre of stretch in a 5 m bar, and against a 250 MPa yield the safety factor is 12.50. Numbers that comfortable mean the design question moves elsewhere (connections, buckling in compression, fatigue), which is a finding, not a failure of the tool.
The bent and twisted shaft. A rectangular 20 × 40 mm section under 500 N·m: I = 106,666.67 mm⁴ and surface stress 93.75 MPa. A 20 mm round shaft under 100 N·m of torque: J = 15,707.96 mm⁴ and surface shear 63.66 MPa. Put bending and torsion on one round shaft and the von Mises row combines them into the single equivalent a ductile yield check consumes.
Reading your results
Stress without a comparison is trivia: read it against yield (through the safety factor), against fatigue limits where loads repeat, and against the stress raisers this page cannot see (fillets, keyways, threads multiply local stress two to threefold). Read spring energy as a safety quantity as much as a design one: stored joules are what a released spring spends. And read strain where fit matters: a part can be comfortably strong and still stretch enough to break the machine’s alignment.
Limitations
- Linear elasticity only. Past yield, springs that coil-bind, and sections that buckle all leave the model; the formulas say nothing once the material stops springing back.
- No stress concentrations, no fatigue, no buckling. Each is a discipline of its own; the Beam and Truss Calculator handles beam deflection, and nothing here certifies a design.
- von Mises is scoped. It assesses ductile yielding under the stated surface stress state; brittle materials, hydrostatic states and multiaxial loadings beyond bending plus torsion need their own criteria, and the brief for this page says so in as many words.
- Spring rate is treated as constant. Progressive and dual-rate springs, and rates that shift near coil bind, break the straight line Hooke draws; the two-measurement mode reveals a spring that refuses to sit on one line.
Frequently asked questions
Why does preload change the force but not the rate?
Preload is where the spring’s working line starts, not how steep it is: the spring still gains k newtons per millimetre, it simply begins the journey already carrying the installed load. Confusing the two understates working force by the preload and overstates available travel by the preload’s deflection, which is why the form separates them and why the deflection-mode calculation subtracts preload before dividing by the rate.
What safety factor should I aim for?
There is no honest universal number: it prices uncertainty. Well-known static loads on ductile material with verified properties live at 1.5 to 2 in professional practice; guessed loads, dynamic service, brittle material or consequences that hurt people push factors to 4 and beyond, and codes fix the figure for regulated work. The calculator reports the factor you have, transparently, and the limitations explain why blessing it is a judgement it will not make for you.
Why is the von Mises figure not just the bigger stress?
Because yielding responds to the distortion in the material, not to one component alone: a surface carrying bending tension and torsional shear yields sooner than either suggests in isolation, and the von Mises combination (square root of σ² + 3τ² for this state) is the empirically validated way to fold them into one equivalent tensile stress for ductile metals. Where only one acts, it reduces to that stress, which the results demonstrate rather than assert.
Can this check a spring I found in a drawer?
Yes, that is exactly the two-measurement mode: hang two known weights (or press it on a scale at two measured compressions), enter both points, and read the rate, then use the force mode to predict its behaviour in your application. A spring whose two segments imply different rates is progressive, damaged or bottoming, and each of those is worth knowing before it is trusted.
Related calculators
Method: Hooke’s law with preload; axial stress, strain and safety factor; bending and torsion section formulas with von Mises for the defined round-shaft surface state. Last reviewed: 14 October 2026.