Parameters¶
| Field | Value |
|---|---|
| Author | Antoni Dudij, Maksim Feldmann — RWTH Aachen |
| Status | Review |
| Last Updated | 2026-03-01 |
TL;DR¶
This document defines every parameter that enters the Bayesian inference loop: the two or three calibrated unknowns per model, their prior distributions in both YAML-config and normalised-MCMC form, the sign conventions for loads and deflections, and the full parametric study grid from \(L/h = 5\) to \(L/h = 100\) with empirical log Bayes factors.
1. Calibrated Parameters¶
The pipeline estimates material properties from synthetic displacement and strain sensor data. The two models differ by one parameter.
Euler-Bernoulli — 2 free parameters:
| Parameter | Symbol | Nominal Value | Role |
|---|---|---|---|
| Elastic modulus | \(E\) | 210 GPa | Controls bending stiffness; deflection \(\propto 1/E\) |
| Observation noise | \(\sigma\) | ~0.05% of signal | Gaussian measurement uncertainty |
Timoshenko — 3 free parameters:
| Parameter | Symbol | Nominal Value | Role |
|---|---|---|---|
| Elastic modulus | \(E\) | 210 GPa | Bending stiffness |
| Poisson's ratio | \(\nu\) | 0.3 | Determines shear modulus \(G = E / (2(1+\nu))\) |
| Observation noise | \(\sigma\) | ~0.05% of signal | Gaussian measurement uncertainty |
Timoshenko carries one extra parameter (\(\nu\)). The bridge sampling marginal likelihood naturally penalises this extra complexity when \(\nu\) does not improve the fit to data — Bayesian Occam's razor in action (see bayesian-glossary.md).
Calibration Observable Types¶
Two types of sensor data can drive calibration, selectable via data_type in calibrate():
| Data Type | Observable | Forward Model | Typical Use |
|---|---|---|---|
displacement (default) |
\(w(x_i)\) | Analytical deflection formula | Primary calibration |
strain |
\(\varepsilon(x_i)\) | \(\varepsilon = -(h/2) \cdot P(L-x)/(EI)\) | Surface strain gauge setups |
Both EB and Timoshenko share the same bending-strain formula because shear deformation does not produce axial strain in the Timoshenko model.
2. Prior Distributions¶
Priors are defined in configs/default_config.yaml (physical units) but all MCMC sampling occurs in a normalised coordinate system where every quantity is \(\mathcal{O}(1)\).
| Parameter | Config Definition | Normalised MCMC Prior | Scale Factor |
|---|---|---|---|
| \(E\) | LogNormal(\(\mu=26.07\), \(\sigma=0.05\)) | \(\mathcal{N}(1.0,\ 0.05)\) | \(E_{scale} = 210 \times 10^9\) Pa |
| \(\nu\) | \(\mathcal{N}(0.3,\ 0.03)\) | \(\mathcal{N}(0.3,\ 0.03)\) | (dimensionless, no scaling) |
| \(\sigma\) | HalfNormal(\(\sigma=10^{-6}\)) | HalfNormal(\(1.0\)) | \(w_{scale} = \max\lvert\mathbf{w}_{obs}\rvert\) |
The \(\sigma\) prior is tight relative to the signal-to-noise ratio of the synthetic data, which reflects the known noise level of the simulator (`noise_fraction = 5 \times 10^{-4}$). Using a looser prior would not change the MAP estimate materially but would increase bridge sampling variance.
Posterior Recovery¶
Across all aspect ratios studied, the posterior for \(E\) consistently recovers the true value with: - \(\hat{R} = 1.002\)–\(1.003\) (well converged) - \(\text{ESS} = 1{,}250\)–\(1{,}650\) (ample independent samples) - Posterior mean within 0.5% of 210 GPa
3. Sign Conventions¶
These conventions are enforced consistently across base_beam.py, calibration.py, and all test fixtures.
| Quantity | Convention |
|---|---|
| Applied load \(P\) | Positive = downward |
| Deflection \(w\) | Negative = downward (positive \(P\) → negative \(w\)) |
| \(y\)-coordinate | Positive = downward from neutral axis |
| Bending strain | \(\varepsilon = -y \cdot M(x)/(EI)\); tension at bottom face for positive moment |
| \(\ln B_{EB/Timo}\) | \(\ln p(\mathbf{y}\mid M_{EB}) - \ln p(\mathbf{y}\mid M_{Timo})\) |
| Negative \(\ln B\) | Favours Timoshenko |
| Positive \(\ln B\) | Favours Euler-Bernoulli |
| \(\lvert\ln B\rvert < 0.5\) in transition zone (\(L/h \approx 15\)–\(19\)) | Inconclusive — use Euler-Bernoulli (simpler model) as default |
| \(\lvert\ln B\rvert \approx 0\) for \(L/h \geq 20\) | Euler-Bernoulli — shear negligible; log BF near zero is physically expected |
4. Prior Distribution Types¶
| Distribution | Applied To | Rationale |
|---|---|---|
| \(\mathcal{N}(1.0,\ 0.05)\) | \(E\) (normalised) | Symmetric; 5% relative uncertainty around steel nominal |
| \(\mathcal{N}(0.3,\ 0.03)\) | \(\nu\) (Timoshenko) | Steel \(\nu \in [0.25, 0.35]\); tight prior limits Occam penalty |
| HalfNormal\((1.0)\) | \(\sigma\) (normalised) | Forces positive noise; concentrates mass near zero |
| \(\mathcal{N}(\hat{w},\ \sigma^2)\) | Likelihood | Gaussian sensor noise model |
5. Parametric Study Grid¶
Fixed parameters: \(L = 1.0\) m, \(b = 0.1\) m, \(P = 1{,}000\) N, \(E = 210\) GPa, \(\nu = 0.3\), \(\kappa = 5/6\). Varied: \(L/h\), which fixes beam height as \(h = L/(L/h)\).
| \(L/h\) | \(h\) [m] | Beam Type | \(\ln B_{EB/Timo}\) | Recommendation |
|---|---|---|---|---|
| 5 | 0.200 | Very thick | −11.11 | Timoshenko |
| 8 | 0.125 | Thick | −7.68 | Timoshenko |
| 10 | 0.100 | Moderate | −4.32 | Timoshenko |
| 12 | 0.083 | Moderate | −3.91 | Timoshenko |
| 15 | 0.067 | Transition zone | −2.45 | Timoshenko |
| 20 | 0.050 | Slender | +0.39 | Euler-Bernoulli |
| 30 | 0.033 | Slender | +0.08 | Euler-Bernoulli |
| 50 | 0.020 | Very slender | +0.06 | Euler-Bernoulli |
| 60 | 0.017 | Very slender | −0.06 | Euler-Bernoulli |
| 70 | 0.014 | Very slender | +0.18 | Euler-Bernoulli |
| 100 | 0.010 | Very slender | −0.02 | Euler-Bernoulli |
Transition point: \(L/h \approx 19.2\) (linear interpolation of the \(\ln B = 0\) zero-crossing between \(L/h = 15\) and \(L/h = 20\)).
The study grid is denser in the \(L/h = 10\)–\(20\) region to resolve the transition accurately. For slender beams (\(L/h \geq 20\)) the shear term \(PL/(\kappa GA)\) is physically negligible relative to the bending term \(PL^3/(3EI)\), and both models predict essentially identical deflections. The resulting \(|\ln B| \approx 0\) is the expected physical outcome — not a sign of ambiguity. Because the models are equivalent in this regime, Occam's razor (encoded in the marginal likelihood) selects the simpler Euler-Bernoulli model. The small fluctuations in sign seen at \(L/h = 50\)–\(60\) are MCMC sampling noise, not evidence of Timoshenko being preferred. Inconclusive as a label applies only in the narrow transition zone (\(L/h \approx 15\)–\(19\)) where the shear contribution is non-trivial but not yet dominant.
6. Frequency Analysis Parameters¶
The pipeline also computes analytical natural frequencies for both theories as a physics-based cross-check. The EB natural frequency for cantilever mode \(n\) is:
where \(\lambda_n\) are the eigenvalues of the clamped-free boundary condition (\(\lambda_1 = 1.875\), \(\lambda_2 = 4.694\), …).
Timoshenko frequencies include shear and rotary inertia corrections that reduce predicted values compared to EB, with the discrepancy growing for thick beams and higher modes. Results are written to outputs/reports/frequency_analysis.txt by FrequencyBasedModelSelector in hyperparameter_optimization.py.
References¶
- Kass, R. E., & Raftery, A. E. (1995). Bayes Factors. Journal of the American Statistical Association, 90(430), 773–795.
- Timoshenko, S. P. (1921). On the correction factor for shear of the differential equation for transverse vibrations of bars. Philosophical Magazine, 41, 744–746.