Analyze Example: AT610 (-s -m)

This page documents a full analysis example driven by a custom problem file and includes the extracted tables from the generated Excel summary.

Note: Table values are rounded to 4 significant figures for readability. Very small/large values use scientific notation. Refer to the linked Excel/JSON artifacts for full precision. Note: Advanced solver diagnostics (for example lsf_mult and bnds_mult Lagrange multipliers) are listed in the Excel/JSON artifacts and are intentionally omitted from the tables below for readability.

Run Context

Problem module:

• problems/AT610Problem.py

Recorded result set:

• results/2026-03-15/14-26-51/AT610-5abe5.xlsx
• results/2026-03-15/14-26-51/AT610-5abe5.json
• results/2026-03-15/14-26-51/profile-5abe5.yaml

For your own runs, use the same naming pattern under a different timestamped folder:

• results/<YYYY-MM-DD>/<HH-MM-SS>/<ProblemName>-<suffix>.xlsx
• results/<YYYY-MM-DD>/<HH-MM-SS>/<ProblemName>-<suffix>.json
• results/<YYYY-MM-DD>/<HH-MM-SS>/profile-<suffix>.yaml

Note: <YYYY-MM-DD>/<HH-MM-SS> and <suffix> are generated per run and will differ on your machine.

Profile and run mode from saved profile:

• Profile used: default
• run_type: analyze
• include_sorm: true
• include_mc: true
• mc_with_is: false

Results path/ID convention used in this example:

• Results are saved under results/<YYYY-MM-DD>/<HH-MM-SS>/ (date and time subfolders).
• Each run gets a new random request ID, so the exact ID changes every time the analysis is run.
• Output files include the last 5 digits of that request ID in their names.

Equivalent command shape:

python -m reliafy analyze <profile> -s -m

Option meaning in this command:

• -s enables SORM (second-order reliability analysis).
• -m enables Monte Carlo simulation.

Profile Customization

This example uses the default profile, but these options are configurable. See ../profile-reference.md.

• SORM behavior: reliability_options.sor_method, sor_approximation, sor_fit_method, sor_fdm.
• Monte Carlo behavior: reliability_options.mc_n, mc_max_cv, mc_seed, mc_remove_oob.
• Run toggles in analyze mode: run_configuration.include_sorm, include_mc, mc_with_is.

Problem File Used

Source: Ang, A. H-S. and Tang, W. H., Probability Concepts in Engineering Planning and Design, Vol. II, Wiley, 1990, p. 368, Problem 6.10.

AT610Problem.py defines:

• Stochastic variables: Y, Z, M
• Distributions: lognormal, lognormal, gumbelmax
• Correlation: corr(Y, Z) = 0.4
• Limit state:
• R = Y * Z
• L = M
• g = R - L

Extracted Results Worksheet Tables

The tables below are transcribed from the Results worksheet in AT610-5abe5.xlsx.

Header Information

Field	Value
Problem	AT610
Request ID	726e0fa551e94093aa14f99b3d55abe5
Run time	00 min 01.58 sec

FORM Results

| beta | pf | beta_count | hbeta_count | lsf_count | glsf_count | hlsf_count | nit | min_tries | min_method | lsf_mult | |---:|---:|---:|---:|---:|---:|---:|---:|---:|---| | 2.664 | 0.003857 | 15 | 15 | 7 | 6 | 6 | 10 | 1 | tr_interior_point |

SORM Results

beta	pf	lsf_class	sor_method	sor_approx	sor_fit_meth	lsf_count
2.649	0.004035	Hyperboloid	SOSPA_H	Taylor2	AnalyticAll	0

SORM LSF Fit Parameters

sospa_a	sospa_b	sospa_c
319.3	-102.9	-67.66
	1.933e-12	1.552e-14
	-295.0	18.55

SOSPA Diagnostics

at_mean	ts	wh	uh	E3	E4	E5
False	-0.005182	-2.446	-3.848	-1.533	3.633	-10.82

Stochastic Variable Inputs and FORM Failure Point Outputs

Input definitions (var_type, moments/parameters, and bounds):

var_name	var_type	mean	std	param1	param2	lb	ub	truncated
Y	LogNormal	40	5	3.681	0.1245	0	inf	False
Z	LogNormal	50	2.5	3.911	0.04997	0	inf	False
M	GumbelMax	1000	200	910.0	155.9	-inf	inf	False

FORM output values at the failure point (x, z, u, alpha, importance):

var_name	x	z	u	alpha	importance
Y	33.78	-1.294	-1.294	-0.4857	-0.4307
Z	47.75	-0.8947	-0.4097	-0.1538	-0.1728
M	1613	2.293	2.293	0.8605	0.8858

User-Defined Correlation Inputs

var 1	var 2	cor_x	cor_z
Y	Z	0.4	0.4013

Monte Carlo Results

beta	pf	cv	max_cv	size	%_removed	cycles	auto_size	mc_with_is
2.664	0.003857	0.02934	0.05	300000	0	3	True	False

Monte Carlo Variable Statistics and Correlations

var_name	mean	std	%_oob	cor_x row	Y	Z	M
Y	40.01	5.009	0	Y	1.0	0.3967	-0.002751
Z	50.0	2.503	0	Z	0.3967	1.0	-0.00192
M	1001	200.4	0	M	-0.002751	-0.00192	1.0

Notes Reported by Reliafy

1. Validation: Stochastic variables definition and limit state function validation required 8 function calls.
2. Validation: SORM sor_fit_method was changed from null to AnalyticAll because the LSF returns both gradient and Hessian and is compatible with the other reliability options.
3. Validation: Validation of the limit state function's analytic gradient and Hessian required 34 function calls.
4. FORM: Active bound multipliers reported for Y, Z, and M. If large relative to Beta (2.664), try truncating the statistical distributions and check if the reliability index changes significantly.
5. Monte Carlo: Completed 3 cycles with 1.00e+05 samples per cycle.

Interpretation Snapshot

• FORM and SORM are close (beta ~ 2.664 vs 2.649), which is typically expected for a smooth, mildly nonlinear limit state.
• Monte Carlo estimate (beta ~ 2.664) is consistent with FORM/SORM and has acceptable precision (cv ~ 0.0293 < 0.05).
• The sampled correlation matrix reproduces the intended Y-Z correlation (target 0.4, observed 0.397).

Generated Figures

The PDF result file for this run is saved as results/2026-03-15/14-26-51/AT610-5abe5.pdf. That PDF is composed of vector-based pages rather than embedded raster images, so the figures below are rendered from each PDF page and saved under docs/problem-authoring/images/.

Figure 1: Importance Factors

Figure 2: SOSPA Fit Functions

Figure 3: Monte Carlo Histogram for Y

Figure 4: Monte Carlo Histogram for Z

Figure 5: Monte Carlo Histogram for M

Figure 6: Histogram of Limit State Function Values

Figure 7: Histogram of Load and Resistance Values

Reproducing This Example

1. Ensure AT610Problem.py exists in problems/.
2. Run an analyze profile with SORM and MC enabled.
3. Open the timestamped folder under results/<date>/<time>/.
4. Use *.xlsx (Results worksheet) for tabular summaries and *.json for machine-readable values.

For this run specifically:

• Folder path: results/2026-03-15/14-26-51/
• Request ID: 726e0fa551e94093aa14f99b3d55abe5
• Last-5 suffix used in file names: 5abe5