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Probabilistic Design and Reliability | Module 10 of 10

Engineering Reliability Case Study and Capstone

The capstone integrates the course: define the component, model uncertainty, estimate failure probability, improve the design, and report residual risk without pretending the model knows more than it does.

Readiness check

Tick only what you can do closed-notes before using this module for design decisions.

Define a component function and failure mode.
Write at least one performance function.
Run a reproducible Python simulation.
Interpret failure probability with consequence.
Explain model assumptions and limitations.

0 or 1 weak itemsContinue with the capstone.

2 weak itemsReview Monte Carlo in Module 4 and robust design in Module 8.

3 or more weak itemsRevisit Modules 1 through 4 before attempting the final project.

The core idea

A defensible reliability case study is a chain of evidence: function, failure modes, uncertain variables, probability models, limit states, simulation checks, sensitivity, redesign, and residual risk.

g_strength = S_y - sigma_bendingg_deflection = delta_allow - deltareport: P_f, convergence, drivers, assumptions, residual risk

The capstone uses a cantilever bracket because it is mechanically familiar and feasible without commercial software. The bracket supports a load at a known distance. The design must avoid yielding and excessive deflection. The uncertain variables include load, yield strength, modulus, width, thickness, and lever arm. The learner justifies distributions, runs a Monte Carlo analysis, checks convergence, identifies influential variables, redesigns by changing dimensions or material, and compares deterministic and probabilistic decisions. The final report must be honest. It should say what was modeled, what was not modeled, which variables control risk, how the result was checked, and what residual risk remains.

The skill works when: the failure event, units, assumptions, distributions, and checks are visible.

The skill breaks down when: a probability is reported without the mechanical model and evidence boundary.

The concept. A defensible reliability case study is a chain of evidence: function, failure modes, uncertain variables, probability models, limit states, simulation checks, sensitivity, redesign, and residual risk.

The skills, taught in order

Scope the engineering decision

State the component, function, mission, failure modes, and consequence before coding.

Build the first model

Use simple mechanics so every equation can be checked by hand.

Run and verify simulation

Use seeds, units, convergence, and independent checks.

Improve design with evidence

Change variables that reduce failure probability or sensitivity, not just the variables that are easy to change.

Write the reliability argument

Report probability, assumptions, limitations, residual risk, and next evidence needed.

Engineering connection: use these skills to turn variability into a design decision, not just a plot.

Worked example 1: capstone performance functions

A rectangular cantilever bracket of width b and thickness h carries tip load P at length L. Write limit states for bending yield and deflection.

ProblemA rectangular cantilever bracket of width b and thickness h carries tip load P at length L. Write limit states for bending yield and deflection.
AssumptionsSmall deflection Euler-Bernoulli beam model, rectangular section, fixed end, tip load, elastic behavior for deflection.
Modelsigma = 6 P L / (b h^2)delta = 4 P L^3 / (E b h^3)g_y = S_y - sigmag_d = delta_allow - delta
SolveThe yield limit state is positive when yield strength exceeds bending stress. The deflection limit state is positive when allowable deflection exceeds predicted tip deflection.
CheckUnits: P in N, L, b, h in mm, E and S_y in N/mm^2. Then stress is MPa and deflection is mm.
ConclusionThe capstone uses two failure events: yielding and excessive deflection.

Result. The capstone uses two failure events: yielding and excessive deflection.

Worked example 2: probabilistic redesign decision

A baseline bracket fails the deflection requirement more often than the yield requirement. Name two redesign moves and what each changes.

ProblemA baseline bracket fails the deflection requirement more often than the yield requirement. Name two redesign moves and what each changes.
AssumptionsThe beam model is adequate for screening.
Modeldelta proportional to 1 / h^3sigma proportional to 1 / h^2
SolveIncreasing thickness h strongly reduces both deflection and stress, especially deflection. Increasing width b reduces both linearly. A higher modulus material reduces deflection but not yield stress unless strength also changes.
CheckIf deflection dominates, thickness is usually a powerful variable, but manufacturability, mass, packaging, and stress concentration still need review.
ConclusionThe redesign should target the dominant failure mode and influential variables.

Result. The redesign should target the dominant failure mode and influential variables.

Python activity

Engineering question: Complete a bracket reliability capstone with baseline and redesigned thickness.

Assumptions and units: Use the units shown in the code comments. Keep the random seed fixed while learning so the result is reproducible.

from math import pi
from random import Random

def bracket_run(thickness_mean, seed=101, n=200_000):
    rng = Random(seed)
    yield_failures = 0
    deflection_failures = 0
    any_failures = 0
    sigma_sum = 0.0
    delta_sum = 0.0
    for _ in range(n):
        P = rng.gauss(900.0, 90.0)       # N
        L = rng.gauss(120.0, 2.0)        # mm
        b = rng.gauss(30.0, 0.25)        # mm
        h = rng.gauss(thickness_mean, 0.12)  # mm
        E = rng.gauss(205_000.0, 6_000.0)    # MPa
        Sy = rng.gauss(280.0, 18.0)          # MPa
        sigma = 6.0 * P * L / (b * h**2)
        delta = 4.0 * P * L**3 / (E * b * h**3)
        g_y = Sy - sigma
        g_d = 1.5 - delta
        yield_failed = g_y <= 0.0
        deflection_failed = g_d <= 0.0
        if yield_failed:
            yield_failures += 1
        if deflection_failed:
            deflection_failures += 1
        if yield_failed or deflection_failed:
            any_failures += 1
        sigma_sum += sigma
        delta_sum += delta
    return yield_failures / n, deflection_failures / n, any_failures / n, sigma_sum / n, delta_sum / n

for h in [6.0, 6.5, 7.0]:
    result = bracket_run(h)
    print("h mean", h, "pf_y", round(result[0], 5), "pf_d", round(result[1], 5), "pf_any", round(result[2], 5), "mean sigma", round(result[3], 1), "mean delta", round(result[4], 3))

Numerical checks: Deflection probability should generally fall rapidly as thickness increases because h appears cubed in the denominator.

Limitations: The capstone omits local stress concentration, fatigue, fastener flexibility, bracket fillets, manufacturing defects, and validation test data.

Misconceptions and diagnostics

Mistake	Diagnostic question	Correction
A complete report only needs the final probability.	Ask: where are the function, assumptions, convergence, drivers, and residual risk?	Correct the assumption, then rerun the calculation or rewrite the report.
A capstone simulation replaces testing.	Ask: what test or simpler calculation could challenge the model?	Correct the assumption, then rerun the calculation or rewrite the report.
The lowest probability design is automatically best.	Ask: what mass, cost, packaging, manufacturability, and consequence trade-offs remain?	Correct the assumption, then rerun the calculation or rewrite the report.
A credible result can hide unsupported distributions.	Ask: what evidence supports each probability model?	Correct the assumption, then rerun the calculation or rewrite the report.

Practice ladder

Recognize

List the two limit states used in the bracket capstone.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Calculate

If thickness increases by 10 percent, estimate the deflection change using the h^3 relationship.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Diagnose

A final report has no convergence check. Explain why the probability estimate is incomplete.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Design

Write a capstone abstract that states component, failure modes, method, key result, and residual risk.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Working with AI, and proving it yourself

Useful AI support

Ask for alternative explanations of terminology, candidate code tests, or a checklist of assumptions to verify.

Ask it to challenge a derivation, then prove the result with units, limiting cases, and an independent calculation.

Do not let AI invent material distributions, fabricate failure data, claim standards compliance, or make final safety decisions.

Portfolio evidence

Create one short reliability note for this module. It must include the engineering question, assumptions, input definitions, equations, calculation, checks, interpretation, and limitations.

Verification required: dimensional analysis, a benchmark case, convergence or sample-size check, and a plain-language residual-risk statement.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

What are the required capstone elements?

Function, failure modes, uncertain variables, probability models, limit states, propagation, failure probability, convergence, sensitivity, redesign, and residual risk.

Why use a simple beam model?

It is transparent and checkable, which is appropriate for a first reliability capstone.

What must code include for reproducibility?

Seed, inputs, units, equations, and enough detail to rerun.

Why compare deterministic and probabilistic decisions?

They can recommend different designs when variability matters.

What should the final paragraph of the report include?

Assumptions, limitations, residual risk, and next evidence needed.

TodayFinish the review and first two practice levels.

+1 dayRebuild one worked example from a blank page.

+3 daysChange one assumption and predict the effect.

+7 daysConnect this module to the capstone bracket.

+30 daysReuse the method in a new mechanical component.

References and source discipline

This course synthesizes original MechCompass explanations from established reliability engineering sources. Confirm current editions and standards before using them for formal design approval.

Topic in this module	Reference direction
Reliability-based mechanical design workflow	Le, Reliability-Based Mechanical Design, Volumes 1 and 2
Engineering reliability case studies	Haldar and Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design
System and structural reliability reporting	Der Kiureghian, Structural and System Reliability