A bar in tension does not only get longer, it also gets thinner. Calculate the lateral strain εq = −ν·εl and the absolute diameter or width change Δd = εq·d₀ from longitudinal strain, Poisson's ratio and the initial dimension – or conversely Poisson's ratio from measured deformations.
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A bar in tension does not only get longer, it also gets thinner. Calculate the lateral strain εq = −ν·εl and the absolute diameter or width change Δd = εq·d₀ from longitudinal strain, Poisson's ratio and the initial dimension – or conversely Poisson's ratio from measured deformations.
A round steel bar (ν = 0.3) with d₀ = 20 mm is stretched longitudinally by εl = 0.001 (= 0.1 % = 1,000 µm/m). The lateral strain is εq = −0.3·0.001 = −0.0003, and the diameter decreases by Δd = −0.0003·20 mm = −0.006 mm = −6 µm. Conversely: measuring a 6 µm diameter reduction at 1,000 µm/m longitudinal strain gives ν = 0.3 – which is how Poisson's ratio is actually determined in a tensile test.
Valid for homogeneous, isotropic, linear-elastic materials in a uniaxial stress state, i.e. a bar loaded only along its axis and free to deform transversely. Under multiaxial loading the lateral strains from all stress components superpose; the full Hooke's law must then be used. Restrained lateral contraction (for instance inside an interference fit or under a press plate), yielding above the elastic limit where Poisson's ratio migrates towards 0.5, anisotropic materials such as fibre composites and wood, and thermal strains are all excluded.
A bar in tension does not only get longer, it simultaneously gets thinner. This calculator determines that lateral strain and the resulting absolute diameter or width change – and can be rearranged to obtain Poisson's ratio itself from measured deformations.
In a tensile test a material is observed to contract perpendicular to the loading direction. This effect is called transverse contraction. The magnitude of the lateral strain is proportional to the longitudinal strain; the dimensionless proportionality factor is Poisson's ratio ν. Thus εq = −ν·εl, where the minus sign expresses the reversal of sign: extension in one direction means shortening in the other two. Multiplied by the initial dimension d₀ this gives the absolute change Δd = εq·d₀.
Like a piece of modelling clay pulled between your hands: the longer it gets, the thinner. Exactly the same happens in metal, only on a scale of micrometres instead of millimetres – and unlike clay it springs fully back when unloaded.
εq = −ν · εl; Δd = εq · d₀ = −ν · εl · d₀
Lateral strain: εq = −ν·εlTransverse change: Δd = εq·d₀ = −ν·εl·d₀Poisson's ratio from measurements: ν = −Δd/(εl·d₀)Longitudinal strain from stress: εl = σ/EConstant volume as the limiting case: ν = 0.5| Symbol / input | Meaning |
|---|---|
| Transverse dimension change Δd | The absolute change of the transverse dimension: negative in tension (the bar gets thinner), positive in compression (it gets thicker). For a round bar this is the diameter change, for a flat bar the width or thickness change. Typical magnitude in the elastic range: a few micrometres, which is why µm is the default unit. |
| Poisson's ratio ν | Poisson's ratio as a positive decimal: it states what fraction of the longitudinal strain appears as lateral strain with the opposite sign. Source: material tables – most metals have ν ≈ 0.3, concrete ≈ 0.2, rubber close to 0.5, cork near 0. The minus sign is already in the formula and must not be entered again. At ν = 0.5 the volume stays constant; larger values are physically impossible for isotropic materials. |
| Longitudinal strain εl | The strain in the load direction as a fraction, not a percentage: 0.001 equals 0.1 % or 1,000 µm/m. Positive in tension, negative in compression. Source: a strain gauge, an extensometer, or εl = σ/E from stress and elastic modulus. It must lie in the linear-elastic range, i.e. below the yield strength. |
| Initial transverse dimension d₀ | The unloaded transverse dimension the lateral strain refers to: the initial diameter of a round bar, or the initial width or thickness of a flat bar. For non-square sections, width and thickness must be calculated separately, because both experience the same lateral strain but yield different absolute changes. |
ν is Poisson's ratio as a positive decimal from a material table (metals ≈ 0.3, concrete ≈ 0.2, rubber close to 0.5). The minus sign is already in the formula. εl is the longitudinal strain as a fraction, positive in tension and negative in compression; it comes from a strain gauge, an extensometer, or from εl = σ/E. d₀ is the unloaded transverse dimension, i.e. the initial diameter of a round bar or the initial width or thickness of a flat bar.
For the forward calculation, enter ν and d₀ from material data and the drawing, and either measure the longitudinal strain or compute it from εl = σ/E. The result Δd is negative in tension and positive in compression. For the reverse calculation – determining ν in a tensile test – enter the measured transverse change with its sign together with the measured longitudinal strain and select ν as the target. For non-square sections, calculate width and thickness separately: the lateral strain is the same, the absolute changes are not.
A round steel bar with ν = 0.3 and d₀ = 20 mm is stretched longitudinally by εl = 0.001, i.e. 0.1 % or 1,000 µm/m. The lateral strain is then εq = −0.3·0.001 = −0.0003, and the diameter decreases by Δd = −0.0003·20 mm = −0.006 mm = −6 µm. For comparison: at the same strain a 200 mm long bar extends by +0.2 mm, more than thirty times as much. Worked backwards, a measured diameter reduction of 6 µm at 1,000 µm/m longitudinal strain gives exactly ν = 0.3 – which is how Poisson's ratio is actually determined in a tensile test.
For metals the lateral strain is always markedly smaller than the longitudinal strain because ν ≈ 0.3. Absolute transverse changes therefore stay in the micrometre range and are detectable only with a dial gauge, strain gauges or optical measurement. Transverse contraction becomes decisive where it is restrained: if a part cannot deform freely sideways – inside an interference fit or under a large press plate – additional transverse stresses arise that change the loading considerably compared with the uniaxial case.
The calculation runs in metres internally. The transverse change Δd defaults to micrometres because it is very small in the elastic range; mm, cm, m or inches work equally well. d₀ is usually entered in mm. Longitudinal and lateral strain are dimensionless: 0.001 equals 0.1 % or 1,000 µm/m – enter it as a decimal, not as a percentage and not in µm/m.
Evaluating tensile tests and determining Poisson's ratio; assessing diameter changes on tension and compression members, bolts and shafts; fit and assembly calculations where joint pressure changes through lateral strain; designing strain-gauge arrangements that deliberately capture longitudinal and transverse strain; plausibility-checking FE results.
Valid for homogeneous, isotropic, linear-elastic materials in a uniaxial stress state, i.e. a bar loaded only along its axis and free to deform transversely. Under multiaxial loading the lateral strains of all stress components superpose and the full Hooke's law must be used. Restrained lateral contraction, yielding above the elastic limit (where the effective Poisson's ratio migrates towards 0.5, because plastic deformation does not change volume), anisotropic materials such as fibre composites and wood, and thermal strains are not covered.
Common mistake: Most commonly ν is entered with a negative sign although the minus is already in the formula – the result then carries the wrong sign. Equally widespread is entering strain as a percentage (0.1 instead of 0.001) or in µm/m (1000 instead of 0.001), which is wrong by a factor of 100 or 10⁶. A conceptual error is confusing transverse contraction with the reduction of area measured after fracture in a tensile test: that necking is a plastic quantity and has nothing to do with the elastic Poisson's ratio.
Evaluating tensile tests and determining Poisson's ratio; assessing diameter changes on tension and compression members, bolts and shafts; fit and assembly calculations where joint pressure changes through lateral strain; designing strain-gauge arrangements that deliberately capture longitudinal and transverse strain; plausibility-checking FE results.
ν is Poisson's ratio as a positive decimal from a material table (metals ≈ 0.3, concrete ≈ 0.2, rubber close to 0.5). The minus sign is already in the formula. εl is the longitudinal strain as a fraction, positive in tension and negative in compression; it comes from a strain gauge, an extensometer, or from εl = σ/E. d₀ is the unloaded transverse dimension, i.e. the initial diameter of a round bar or the initial width or thickness of a flat bar.
Valid for homogeneous, isotropic, linear-elastic materials in a uniaxial stress state, i.e. a bar loaded only along its axis and free to deform transversely. Under multiaxial loading the lateral strains of all stress components superpose and the full Hooke's law must be used. Restrained lateral contraction, yielding above the elastic limit (where the effective Poisson's ratio migrates towards 0.5, because plastic deformation does not change volume), anisotropic materials such as fibre composites and wood, and thermal strains are not covered.