Physics-Informed Sensitivity Analysis for Enhanced Structural Health Assessment: Test-Case for a Mixed Steel-Concrete Bridge

This paper presents a high-fidelity physics-based numerical model of a mixed steel-concrete bridge at the University of the Bundeswehr Munich, utilizing sensitivity analysis within a structural health monitoring framework to identify influential parameters, quantify uncertainty, and enhance prognostic capabilities for aging infrastructure.

The Gist

Imagine a bridge not just as a pile of steel and concrete, but as a giant, complex musical instrument. Just like a guitar string vibrates differently depending on its thickness, tension, and the wood it's attached to, a bridge "sings" at specific frequencies when wind or traffic hits it.

This paper is about building a super-accurate virtual twin of a real bridge located at a university in Germany and then playing a game of "What If?" to figure out which parts of that bridge matter most when it starts to vibrate.

Here is the breakdown of their work in everyday terms:

1. The Virtual Twin (The Digital Shadow)

The researchers built a highly detailed computer model of a real, two-span bridge made of mixed steel and concrete. Think of this model as a "digital shadow" or a video game replica of the real thing.

• Why build it? Real bridges are hard to test. You can't easily change the steel bolts or the concrete strength on a real bridge without breaking it. In the computer, they can tweak anything instantly.
• The Goal: To create a "physics-informed" system. This means the computer doesn't just guess; it follows the actual laws of physics to predict how the bridge behaves.

2. The "What If?" Game (Sensitivity Analysis)

Once they had the virtual bridge, they asked: "If we change one tiny thing, how much does the bridge's 'song' (its vibration) change?"

They picked three specific "knobs" to turn on their virtual bridge:

1. Concrete Stiffness ($E_c$): How hard or soft is the concrete deck? (Like changing the wood on a guitar).
2. Bolt Size ($d_b$): How thick are the screws holding the concrete to the steel? (Like tightening or loosening the tuning pegs).
3. Joint Stiffness ($k_{eq}$): How well do the concrete slabs talk to each other? (Like the glue between guitar frets).

They didn't just turn these knobs once; they turned them thousands of times with random variations to see which one caused the biggest change in the bridge's vibration.

3. The Magic Math (The "Recipe" for Answers)

To do this without waiting years for the computer to finish, they used a special mathematical trick called Generalized Polynomial Chaos Expansion (gPCE).

• The Analogy: Imagine you want to know how a cake tastes if you change the sugar, flour, and eggs. Instead of baking 10,000 cakes, you bake a few, taste them, and use a smart formula to predict the taste of every possible combination instantly.
• The Result: This math allowed them to calculate Sobol' indices. Think of these indices as a "blame score." If the bridge's vibration changes by 100%, how much of that change is the concrete's fault? How much is the bolts' fault?

4. What They Found (The Surprising Results)

When they looked at the "blame scores," they found some very clear patterns:

• The Bolts are the Boss: For most of the bridge's vibrations (like bending side-to-side or up-and-down), the size of the bolts was the single most important factor. If you get the bolt size wrong in your model, your prediction is wrong.
• Concrete Matters (But Only Sometimes): For the bridge's twisting motion (torsion), the stiffness of the concrete was the most important factor. However, for one specific vibration mode (Mode 5), the concrete stiffness didn't matter at all.
• The Joints are Invisible: Surprisingly, the stiffness of the connections between the concrete slabs (the "glue" or springs) had almost zero effect on how the bridge vibrated. It was as if those joints didn't exist for the purpose of vibration.

5. Why This Matters

The paper concludes that by using this "virtual twin" and the "blame score" math, engineers can:

• Stop guessing: They know exactly which parts of a bridge need the most careful monitoring.
• Save money: They don't need to waste sensors on parts that don't matter (like the concrete joints in this specific case).
• Build better models: They know where to focus their engineering attention to make the computer model match reality perfectly.

In short, they turned a complex bridge into a manageable puzzle, figured out which pieces actually move the needle, and proved that sometimes, the smallest screws (bolts) matter more than the biggest blocks (concrete).

Technical Summary

Technical Summary: Physics-Informed Sensitivity Analysis for Enhanced Structural Health Assessment

Problem Statement
Civil infrastructure, particularly bridges, faces escalating challenges due to material aging, increasing traffic loads, and climate-induced extreme weather. Effective Structural Health Monitoring (SHM) frameworks are required to assess structural performance under uncertainty. While high-fidelity physics-based Finite Element (FE) models serve as crucial "digital shadows" or "digital twins" for decision-making, their predictive capability relies heavily on the accuracy of input parameters. A significant gap exists in understanding how uncertainties in specific structural parameters—such as material properties and geometric details—propagate to the dynamic response of the structure. Without quantifying these sensitivities, monitoring campaigns may be inefficient, and model-based decisions may lack the necessary physical rigor.

Methodology
The study employs a high-fidelity, physics-based numerical model of a full-scale, two-span mixed steel-concrete bridge located at the University of the Bundeswehr Munich. The model, originally coded in ANSYS APDL and ported to PyAnsys for automation, utilizes shell elements for steel and concrete components and beam elements with distributed springs to simulate bolted connections and inter-segment stress transfer.

To quantify the influence of input uncertainties on the model's output, the authors implement a Sensitivity Analysis (SA) framework based on the Generalized Polynomial Chaos Expansion (gPCE) method. The process involves:

1. Input Characterization: Three uncertain parameters are identified and modeled as log-normal random variables:
   • $E_c$: Young's modulus of the concrete segments.
   • $d_b$: Diameter of the screw stem representing the stiffness of the bolted connection.
   • $k_{eq}$: Stiffness of distributed springs modeling the connection between concrete slab segments.
2. Output Selection: The model output consists of four eigenfrequencies corresponding to specific vibration modes: principal in-plane bending ($f_1$), first out-of-plane bending ($f_5$), principal torsional ($f_6$), and second out-of-plane bending ($f_{10}$).
3. Surrogate Modeling: A non-intrusive regression method is used to construct a polynomial surrogate of the FE model. Hermite polynomials are employed, assuming normally distributed inputs after a transformation from the log-normal space.
4. Sensitivity Quantification: Sobol' indices are derived analytically from the polynomial coefficients. This approach decomposes the output variance into contributions from individual input variables (first-order indices) and their interactions (higher-order indices), offering a computationally efficient alternative to Monte Carlo methods.

Key Contributions

• Development of a Virtual Replica: The creation of a validated, high-fidelity FE model acting as a digital shadow of a real-world test bridge, integrated with state-of-the-art data science workflows via PyAnsys.
• Application of gPCE to SHM: The demonstration of using gPCE to analytically compute Sobol' indices for structural dynamics, providing a rigorous foundation for uncertainty quantification in physics-informed models.
• Identification of Critical Parameters: The systematic evaluation of how specific modeling assumptions (e.g., bolt stiffness vs. concrete stiffness) influence the dynamic response of a composite bridge.

Results
The sensitivity analysis yielded distinct insights regarding the influence of the three input parameters on the four vibration modes:

• Dominance of Bolt Stiffness: For the in-plane bending mode ($f_1$), the first out-of-plane bending mode ($f_5$), and the secondary out-of-plane bending mode ($f_{10}$), the stiffness of the bolted connection ($d_b$) is the most influential parameter, accounting for the vast majority of the output variance (e.g., ~84% for $f_1$ and ~98% for $f_5$).
• Torsional Mode Sensitivity: In contrast, the principal torsional mode ($f_6$) is primarily driven by the Young's modulus of the concrete ($E_c$), which accounts for approximately 63% of the variance, while bolt stiffness contributes roughly 36%.
• Negligible Influence of Joint Stiffness: The stiffness of the distributed springs ($k_{eq}$), representing the connection between concrete segments, exhibited negligible influence across all analyzed modes. The Sobol' indices for this parameter were effectively zero, suggesting that the current modeling approach for inter-segment stress transfer has minimal impact on the global dynamic response.
• Model Behavior: Response surfaces confirmed that the models are highly sensitive to variations in $d_b$ and, to a lesser extent, $E_c$, while remaining almost entirely independent of $k_{eq}$.

Significance and Claims
The paper posits that integrating Sobol' sensitivity indices with high-fidelity numerical models provides a robust foundation for physics-informed decision-making in infrastructure management. The primary significance lies in the ability to:

• Optimize Monitoring Strategies: By identifying which parameters drive structural response variability, monitoring campaigns can be better designed, focusing sensor networks on the most critical quantities.
• Guide Model Refinement: The identification of negligible parameters (like $k_{eq}$) allows for model simplification, while highlighting dominant parameters (like $d_b$) directs attention toward more accurate modeling approaches for those specific components.
• Support Proactive Maintenance: The framework contributes to the development of more robust digital twin systems, enabling proactive maintenance and risk management strategies for critical infrastructure.

The authors conclude that while the study focuses on a single test case, the proposed framework offers a scalable approach that could eventually be extended to population-based analyses of similar structures, thereby supporting a holistic management of critical infrastructure networks.