3. Common statistical distributions in crash CAE
In crash CAE, every scatter source you decide to model — sheet metal thickness, yield stress, spot-weld nugget diameter, foam density, airbag inflator mass flow, barrier impact velocity — must be assigned a probability distribution before sampling can begin. The distribution is not a cosmetic choice: it determines the tails, and in crashworthiness the tails are exactly where the interesting failures live. A 99th-percentile thick, strong rail may fail to fold; a 1st-percentile thin rail may tear at a bead. This section covers the five distributions that account for the overwhelming majority of practical parameter models, what each one physically represents, how to fit it from data or engineering judgement, and how to sanity-check the result before it enters a DOE or Monte Carlo campaign.
Normal: symmetric scatter around a nominal
The normal distribution is the default for parameters produced by an additive, many-small-cause error mechanism: gage R&T variation in a stamped panel, springback compensation residual, or a material property reported as mean ± standard deviation by a supplier. It is defined by location μ and scale σ, and its support is the whole real line — which is its main hazard in crash CAE. A thickness parameter modelled as N(1.20, 0.05) mm will occasionally sample 1.05 mm or 1.35 mm, values that may be physically impossible for the coil in question. For parameters with a hard physical floor (thickness, density, yield stress), truncate the normal at a defensible bound or switch to a lognormal.
A practical fitting workflow for a normal parameter: collect at least 30 measurements from the same process stream, compute the sample mean and standard deviation, then verify normality with a normal probability plot rather than a single p-value. If the plotted points bow upward at the high end, the data are right-skewed and a lognormal is the better model. In the CAE deck, the parameter is then written as a mean and coefficient of variation (CoV = σ/μ), because CoV transfers more reliably between grades and suppliers than absolute σ.
Lognormal: positive, right-skewed quantities
When a quantity is strictly positive and its scatter arises from multiplicative effects, the lognormal is the natural model. Typical crash CAE candidates: ultimate tensile strength and elongation of AHSS grades, rupture strain of weld nuggets, fracture-initiation strain from a calibration campaign, and manufacturing-induced thickness variation across a coil. If X ~ Lognormal(μ_ln, σ_ln), then ln X is normal, and the median is exp(μ_ln) while the mean is exp(μ_ln + σ_ln²/2). This distinction matters: a supplier quoting a mean strength of 980 MPa with CoV 0.06 corresponds to μ_ln = ln(980) − 0.5·σ_ln² with σ_ln ≈ 0.0599, giving a median near 976 MPa. Using the mean directly as μ_ln overstates the median and biases every subsequent percentile.
Fitting from data is straightforward: take natural logs of the observations, fit a normal to the logged values, and back-transform. Always report the fitted parameters in both spaces so a reviewer can reproduce the sampling. A quick check: on a lognormal probability plot the data should be linear; curvature suggests a mixture — for example, two coil suppliers with different means — and a mixture model or a supplier-indexed categorical factor is then more honest than a single wide lognormal.
Weibull: strength, life, and weakest-link behaviour
The Weibull distribution is the standard model for brittle and quasi-brittle failure, and it appears in crash CAE wherever failure is governed by the weakest link: cast aluminium shock-tower brackets, die-cast nodes, glass, and bonded joint strength. Its two-parameter form has shape β and scale η; β < 1 indicates decreasing failure rate (infant mortality, e.g. casting porosity), β ≈ 1 is memoryless (random defects), and β > 2–4 describes wear-out or strength-limited populations. For material strength, β between 5 and 20 is common, producing a fairly tight, left-skewed-in-the-lower-tail distribution.
Weibull parameters are rarely available from a supplier datasheet, so they are usually estimated from a small test series. With n specimens ranked in ascending failure stress σ_(i), use the median-rank estimator F_i = (i − 0.3)/(n + 0.4), then regress ln(−ln(1 − F_i)) on ln σ_(i). The slope is β and the intercept gives η. With n = 10 this estimate is coarse — β may carry a factor-of-two uncertainty — so document the sample size and treat the fitted β as a modelling assumption, not a material constant. In the CAE workflow, the sampled strength is applied as a scaling factor on the calibrated failure strain or on the LS-DYNA material card's failure parameter, not as a re-calibration of the whole card.
Uniform: bounded ignorance and deliberate design variables
The uniform distribution is correct in two distinct situations. First, when a parameter is genuinely bounded with no preferred value inside the bounds — for example, a fixture locating tolerance of ±0.5 mm where the true offset is equally likely anywhere in the band, or an impact velocity specified as 50 ± 1 km/h for a regulatory matrix. Second, when the analyst deliberately wants equal weight across a design space, as in a Latin hypercube study over gauge options. Uniform is a statement of maximum ignorance within bounds, not a claim that the physics is flat; if you have supplier data showing a central tendency, using uniform overstates the tails and inflates the predicted failure probability.
For a uniform parameter on [a, b], the mean is (a+b)/2 and the standard deviation is (b−a)/√12. That factor of √12 is worth remembering: a ±0.5 mm tolerance band corresponds to σ ≈ 0.29 mm, noticeably larger than the σ ≈ 0.17 mm a ±0.5 mm normal truncation would imply. Choosing between them changes predicted scrap rates, so record the rationale in the model documentation.
Beta: proportions, fractions, and bounded calibration factors
The beta distribution lives on [0, 1] and is the right tool for quantities that are naturally fractions: the ratio of actual to nominal weld nugget diameter, a calibration multiplier on failure strain, the fraction of a crush initiator's intended depth actually formed, or a porosity volume fraction. Its two shape parameters, α and β, give it enormous flexibility — U-shaped when both are below 1, bell-shaped when both exceed 1, skewed in either direction otherwise. This flexibility is also the risk: beta can fit almost anything, so it must be anchored to data or to a defensible engineering interval, not tuned until the output looks acceptable.
A defensible construction when only an interval and a most-likely value are known: map the interval to [0, 1], set the mode at the mapped most-likely value m, and choose a concentration κ = α + β that reflects confidence (κ ≈ 6 for weak evidence, 20+ for a well-populated test series). Then α = 1 + κm and β = 1 + κ(1 − m). For a calibration multiplier believed to lie between 0.85 and 1.15 with a mode at 1.00, the mapped mode is 0.5 and with κ = 10 the parameters are α = β = 6 — a symmetric, moderately concentrated prior that will be dominated by data once a calibration campaign exists.
From distribution choice to a defensible sampling setup
The table below is a working template you can fill in before a Monte Carlo or DOE run. It forces the two decisions that most often go undocumented: where the parameters came from, and what the tails are expected to do.
| Parameter | Distribution | Fitted / assumed parameters | Basis | Tail check |
|---|---|---|---|---|
| Front rail gauge, mm | Normal, truncated [1.10, 1.30] | μ = 1.20, σ = 0.025 | n = 42 coil measurements | No sample below 1.10 after truncation |
| B-pillar UTS, MPa | Lognormal | μ_ln = 6.887, σ_ln = 0.060 | Supplier CoV 0.06, mean 980 MPa | P99 ≈ 1120 MPa, within grade spec |
| Cast shock-tower strength factor | Weibull | β = 8.5, η = 1.02 | n = 10 coupon tests, median-rank fit | P01 ≈ 0.87; flag β uncertainty |
| Barrier velocity, km/h | Uniform | a = 49, b = 51 | Regulatory test band | σ = 0.58 km/h |
| Failure-strain calibration multiplier | Beta | α = 6, β = 6 on [0.85, 1.15] | Engineering interval, mode 1.00 | P(<0.90) ≈ 4%; revisit after calibration |
Two further practical points close the loop. First, the choice of distribution changes the required sample size: a Weibull with β = 8.5 concentrates mass tightly, so 100 Latin hypercube samples resolve its body well but say little about the 1st percentile; a uniform parameter needs far fewer samples for the same tail resolution. Second, record the distribution, parameters, basis and truncation in the model's input deck header or a companion parameter file, so that a reviewer rerunning the study two months later reproduces the same population. In crash CAE the distribution is an engineering claim about the tails of a physical population — treat it with the same rigour as the material card itself.