Simulate Example: Chan2 (Classic Monte Carlo)

This page documents a simulate run in plain Monte Carlo mode (without importance sampling).

Note: Table values are rounded to 4 significant figures for readability. Very small/large values use scientific notation. Refer to the Excel/JSON result files for full precision.

Run Context

Problem module:

• problems/problems2/Chan2Problem.py

Recorded result set:

• results/2026-03-16/10-52-46/Chan2-51baa.xlsx
• results/2026-03-16/10-52-46/Chan2-51baa.json
• results/2026-03-16/10-52-46/Chan2-51baa.py
• results/2026-03-16/10-52-46/Chan2-51baa.pickle
• results/2026-03-16/10-52-46/Chan2-51baa.pdf
• results/2026-03-16/10-52-46/profile-51baa.yaml

Profile and run mode from saved profile:

• Profile used: default
• run_type: simulate
• include_mc: true
• mc_with_is: false

For results-folder and filename conventions, see CLI Result Files.

Equivalent command shape:

reliafy simulate <profile>

The simulate command without -i runs plain (crude) Monte Carlo only.

Profile Customization

This example uses the default profile, but simulate behavior is configurable. See Profile Options Reference.

• Monte Carlo controls: reliability_options.mc_n, mc_max_cv, mc_seed, mc_remove_oob.
• Report toggles: reporting_options.save_plots_to_pdf, save_plots_to_pickle, save_excel_summary.

Problem File Used

Source: Chan, C. L. and Low, B. K., "Practical second-order reliability analysis applied to foundation engineering," International Journal for Numerical and Analytical Methods in Geomechanics, 2012, vol. 36, no. 11, p. 1387–1409, Problem 2, p. 1397. →

Chan2Problem.py defines:

• Stochastic variables: cp, fp, g, Ph, Pv — all Normal
• Correlation: full 5 × 5 matrix defined using the cor key (see note below)
• Deterministic variables: B = 5.0, L = 25.0, D = 1.8, h = 2.5
• LSFreturnsGradient: False, LSFreturnsHessian: False, LSFreturnsLandR: True
• Limit state returns load ($P_v / B'$) and resistance (bearing-capacity formula)

Using cor vs cor_list

Chan2Problem.py specifies the correlation matrix as a full n × n list of lists under the cor key. This is an alternative to cor_list, which accepts a flat list of [var1, var2, value] pairs. If neither cor nor cor_list is supplied, Reliafy assumes zero correlation for all variables. Either key may be used; the full matrix form is convenient when many off-diagonal correlations are non-zero.

"StochasticVariables": {
    "name": ["cp", "fp", "g", "Ph", "Pv"],
    "type": ["normal", "normal", "normal", "normal", "normal"],
    "cor": [
        [1.0, -0.5,  0.0,  0.0, 0.0],
        [-0.5, 1.0,  0.5,  0.0, 0.0],
        [0.0,  0.5,  1.0,  0.0, 0.0],
        [0.0,  0.0,  0.0,  1.0, 0.5],
        [0.0,  0.0,  0.0,  0.5, 1.0],
    ],
    "mean": [15.0, 25.0, 20.0, 400.0, 800.0],
    "std":  [4.5,  5.0,  2.0,  40.0,  80.0],
},

Because LSFreturnsLandR: True, the CLI generates a Load and Resistance Histogram in addition to the per-variable and LSF histograms.

Extracted Results Worksheet Tables

The tables below are transcribed from the Results worksheet in Chan2-51baa.xlsx.

Header Information

Field	Value
Problem	Chan2
Request ID	9614c856e5bb41f4b96bd9b1d4051baa
Run time	00 min 25.62 sec

Deterministic Variables

var_name	value
B	5
L	25
D	1.8
h	2.5

Stochastic Variables Definition

var_name	var_type	mean	std	param1	param2
cp	Normal	15	4.5	15	4.5
fp	Normal	25	5	25	5
g	Normal	20	2	20	2
Ph	Normal	400	40	400	40
Pv	Normal	800	80	800	80

Monte Carlo Results

beta	pf	cv	max_cv	size	%_removed	cycles	auto_size	mc_with_is
1.5409	0.06167	0.002252	0.05	3,000,000	0.000733	3	True	False

Monte Carlo Variable Statistics and Correlations

The sampled statistics confirm the input distributions and correlation structure.

var_name	mean	std	%_oob	cor(cp)	cor(fp)	cor(g)	cor(Ph)	cor(Pv)
cp	14.9993	4.4999	0	1.0000	−0.4999	0.0003	0.0003	0.0008
fp	24.9972	4.9994	0	−0.4999	1.0000	0.5003	−0.0008	0.0000
g	19.9997	2.0012	0	0.0003	0.5003	1.0000	0.0001	0.0008
Ph	400.035	39.985	0	0.0003	−0.0008	0.0001	1.0000	0.5006
Pv	799.993	79.962	0	0.0008	0.0000	0.0008	0.5006	1.0000

Notes Reported by Reliafy

1. Validation: Stochastic variables definition and limit state function validation required 1 function call.
2. Monte Carlo: Completed 3 cycles with 1.00e+06 samples per cycle.
3. Monte Carlo: Detected 22 NaN values in the limit state function out of 3.00e+06 samples. Review the list of code warnings and update the limit state function to address their source.

Interpretation Snapshot

• beta = 1.541, pf = 6.17% — a relatively low reliability index for a bearing-capacity problem with correlated soil parameters.
• The coefficient of variation (cv = 0.00225) is well below max_cv = 0.05, indicating high MC precision with 3 million samples.
• The sampled correlation matrix closely matches the targets: cor(cp, fp) ≈ −0.500, cor(fp, g) ≈ 0.500, cor(Ph, Pv) ≈ 0.501.
• A small number of NaN results (22 out of 3,000,000) were detected. These typically arise from geometric degeneration in the bearing-capacity formula (e.g., B' ≤ 0 due to large eccentricity). They do not invalidate the result at this sample size.

Generated Figures

The PDF result file for this run is saved as results/2026-03-16/10-52-46/Chan2-51baa.pdf. That PDF is composed of vector-based pages, rendered below at 2× resolution from each page.

Figure 1: MC Histogram — cp

Figure 2: MC Histogram — fp

Figure 3: MC Histogram — g

Figure 4: MC Histogram — Ph

Figure 5: MC Histogram — Pv

Figure 6: Histogram of Limit State Function Values

Figure 7: Load and Resistance Histogram

Generated because LSFreturnsLandR: True in the problem file.