⚡ electrical engineering

Structural Reliability of a Cessna 172 Main Wing Spar under Dynamic Excitation Using Monte Carlo Simulation

This paper presents a probabilistic reliability assessment of a Cessna 172 main wing spar under uncertain aerodynamic and harmonic loads using Monte Carlo simulation, revealing that resonance significantly impacts failure probability and identifying the second and third vibration modes as the most critical excitation bands.

Original authors: Marcelo Araujo da Silva, Ivan Pedroso, Jean Vasco

Published 2026-09-04

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CC BY 4.0

Original authors: Marcelo Araujo da Silva, Ivan Pedroso, Jean Vasco

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Every aircraft wing is a marvel of engineering, designed to hold the weight of the plane against the invisible but powerful push of the air. Inside that wing lies a critical backbone called a spar, a long, strong beam that carries the load from the wingtip to the fuselage. While engineers can calculate how much weight a wing should hold, the real world is rarely so predictable. The air does not push with a steady, unchanging force, and the materials used to build the wing are not perfectly uniform. Furthermore, the wing vibrates as it flies, reacting to the rhythm of the air and the engine. When these vibrations match the natural rhythm of the wing itself, a phenomenon known as resonance occurs, causing the structure to shake with far greater intensity than the wind alone would suggest. Understanding how these uncertainties combine to stress a wing is vital for safety, as it determines the margin between a structure that holds firm and one that might fail.

Researchers at the Federal University of ABC in Brazil set out to explore this delicate balance using a specific, well-known aircraft: the Cessna 172. They focused their attention on the main wing spar of this plane, treating it as a long beam fixed at one end and free at the other. To understand how this beam behaves under real-world chaos, they did not rely on a single, perfect calculation. Instead, they built a digital model that accounts for the small, inevitable variations found in every real object. They considered that the thickness of the metal, the width of the flanges, and the strength of the aluminum alloy might vary slightly from one wing to the next. They also accounted for the fact that the weight of the air pushing against the wing and the frequency of the vibrations shaking it are not fixed numbers but ranges of possibilities.

The team used a powerful computational method called Monte Carlo simulation to run thousands of virtual flight tests. In each test, the computer randomly selected a specific set of dimensions, material strengths, and vibration frequencies within the realistic ranges they had defined. It then calculated the stress on the wing for that specific combination of conditions. By repeating this process one hundred thousand times, they created a vast picture of how the wing would perform across a wide spectrum of possibilities. They were particularly interested in what happens when the shaking frequency of the air matches the natural frequency of the wing, a condition that can dramatically amplify the stress on the metal.

The results of this massive digital experiment revealed that the safety of the wing is not a single number but a story that changes depending on the frequency of the vibration. When the shaking occurs at most frequencies, the wing remains remarkably safe, with a very low chance of failure. However, the story changes drastically when the vibration frequency aligns with the wing's natural rhythms. The researchers found that the second and third modes of vibration, which correspond to specific ways the wing bends and twists, create the most dangerous conditions. In these specific frequency bands, the stress on the wing spikes, and the probability of failure rises significantly compared to other frequencies.

When the researchers looked at the entire range of possible vibrations, from zero to two hundred cycles per second, they found an overall failure probability of 0.852 percent. This translates to a reliability index of 3.38, a measure that indicates the structure is generally safe under the assumptions they used. The study highlighted that while the wing is robust, its safety is heavily dependent on avoiding those critical resonance zones. The analysis showed that the most critical bands were the 40–60 Hz and 120–140 Hz ranges, where the likelihood of the wing failing was highest. These bands correspond to the mean natural frequencies of approximately 46.81 Hz and 131.09 Hz, where resonance effects are most pronounced. Outside of these specific ranges, the wing performed with a high degree of confidence.

This work does not claim to solve every problem of wing safety, as it simplified the complex interactions of the entire aircraft structure into a single beam model. It did not account for the fatigue that builds up over years of flying or the complex ways the wing skin and ribs interact with the spar. However, it provides a clear and practical framework for understanding how uncertainty and vibration combine to affect structural integrity. By isolating the effect of random variations and resonance, the study offers a way to quantify the safety margin of aerospace structures when the inputs are not fixed but fluid. The findings suggest that for a Cessna 172 wing, the greatest risk comes not from the average flight conditions, but from the specific, rhythmic shaking that pushes the structure into its most vulnerable state.

Technical Summary: Structural Reliability of a Cessna 172 Main Wing Spar under Dynamic Excitation Using Monte Carlo Simulation

Problem Statement
Aircraft wing spars are primary load-carrying components responsible for transferring aerodynamic lift from the wing to the fuselage. Their structural performance is inherently subject to uncertainties arising from geometry, material properties, aerodynamic loading, and dynamic excitation. This paper addresses the need to quantify the margin between structural capacity and demand for a Cessna 172 Skyhawk main wing spar. Specifically, the study investigates how uncertain excitation frequencies and resonance phenomena influence the probability of failure under dynamic loading conditions.

Methodology
The research employs a probabilistic framework combining a simplified structural model with Monte Carlo simulation (MCS).

Structural Idealization: The spar is modeled as a cantilever Euler-Bernoulli beam with an I-shaped cross-section made of 2024-T3 aluminum alloy. The model simplifies the structure by neglecting interactions with ribs, skin panels, and secondary spars, focusing on the primary longitudinal element fixed at the fuselage and free at the wing tip.
Loading and Internal Forces: The aerodynamic load is assumed to follow an elliptical lift distribution along the half-span, scaled by a reference distributed load. Internal forces (shear, bending, and torsional moments) are derived from this distribution, with the critical section identified at the wing root.
Probabilistic Model: The uncertainty model defines a random vector containing geometric dimensions, material properties (Young's modulus, yield stress, density), aerodynamic load parameters, and excitation frequency.
- Geometric, material, and load variables are treated as independent Gaussian random variables.
- The excitation frequency is modeled as an independent uniform random variable within prescribed frequency bands.
- All variables are assumed statistically independent due to the lack of experimental covariance data.
Dynamic Analysis: Natural frequencies are approximated using the Euler-Bernoulli cantilever beam expression. Dynamic amplification is calculated using a modal frequency response approach with a damping ratio of 0.06. To prevent non-physical simultaneous resonance, an averaged modal amplification factor is applied to the stress components.
Failure Criterion: The limit-state function is defined as the difference between the material yield stress and the equivalent Von Mises stress (incorporating dynamic amplification). Failure occurs when this function is negative.
Simulation: A Monte Carlo simulation with $N = 100,000$ samples is conducted. The analysis covers ten narrow-band excitation intervals (20 Hz each from 0 to 200 Hz) and one broad-band simulation (0 to 200 Hz).

Key Contributions
The paper provides a condensed probabilistic assessment framework that integrates simplified structural mechanics with stochastic simulation to evaluate aerospace component reliability. Key technical contributions include:

Definition of a Stochastic Model: Explicitly characterizing the uncertainty of geometric, material, and loading parameters for a Cessna 172 spar, distinguishing between Gaussian structural parameters and uniform excitation frequencies.
Resonance Sensitivity Analysis: Demonstrating how failure probability varies across different excitation frequency bands, specifically identifying the critical impact of the second and third vibration modes.
Broad-Band Reliability Estimation: Providing a comprehensive reliability index and failure probability for a wide operational frequency spectrum (0–200 Hz), offering a holistic view of structural safety under dynamic uncertainty.

Results
The simulation results highlight a strong correlation between excitation frequency proximity to natural frequencies and structural reliability:

Resonance Effects: The second natural frequency (approx. 46.81 Hz) and the third natural frequency (approx. 131.09 Hz) define the most critical excitation bands.
- In the 40–60 Hz band (encompassing the second mode), the failure probability ( $P_f$ ) peaks at 2.668% with a reliability index ( $\beta$ ) of 2.31.
- In the 120–140 Hz band (encompassing the third mode), $P_f$ is 1.995% with $\beta = 2.33$ .
Off-Resonance Behavior: Frequency bands far from natural frequencies exhibit significantly lower failure probabilities (e.g., $P_f < 0.01\%$ for bands above 160 Hz) and higher reliability indices.
Broad-Band Analysis: The simulation over the full 0–200 Hz range yielded a failure probability of 0.852% and a reliability index of 3.38. The mean Von Mises stress in this broad-band scenario was 60.67 MPa.

Significance and Claims
The authors claim that the study provides a practical approach for the preliminary reliability assessment of aerospace structures subjected to dynamic uncertainties. The work underscores that structural reliability is strongly governed by the relationship between excitation and natural frequencies, with resonance being the dominant factor in failure probability. While the model neglects complex factors such as aeroelastic coupling, fatigue, and detailed structural interactions, the framework successfully identifies critical resonance regions and offers a quantitative measure of safety margins. The broad-band analysis is presented as a method to obtain an overall structural reliability measure, complementing narrow-band analyses that identify specific critical zones.

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