Unlocking Bridge Resilience: Why Energy and Residuals Matter in Quake Design

Hey there! Let me tell you about something super important in making sure our bridges can stand tall, even when the ground decides to do a little jig. We’re talking about seismic design, specifically for those big, continuous concrete box-girder bridges you see everywhere, especially in places like California. It’s not just about making them strong; it’s about making them *resilient*.

For ages, we’ve designed structures to handle earthquakes by thinking about forces – like pushing or pulling. It’s a bit like designing a wall to resist a certain wind pressure. But earthquakes are way more complex! They don’t just push once; they shake, rattle, and roll, often causing structures to bend and deform way beyond their initial elastic limits. Traditional methods, while ensuring basic stability, don’t really give us a clear picture of the *actual* damage accumulating during these wild rides.

This is where performance-based design steps in. It’s a much cooler approach because it focuses on how the structure *performs* under different earthquake intensities. Instead of just saying “it won’t fall down,” we can start asking questions like “how much will it sway?” or “can we still use it after the quake?” Most performance-based methods lean heavily on measuring peak displacement – the maximum amount a structure moves. And yeah, that’s better than just forces, but it still misses a big piece of the puzzle.

Think about bending a paperclip back and forth. The peak bend might be the same each time, but the *cumulative* effect of all that bending eventually breaks it, right? Structures in an earthquake are similar. Repeated inelastic deformations – that bending and yielding – cause damage over time. Peak displacement doesn’t capture this cumulative effect.

Enter Energy-Based Design and Residual Demands

This is precisely why we’re getting excited about energy-based design (EBD). Instead of just looking at forces or peak wiggles, EBD considers the *energy* an earthquake shoves into a structure and how the structure absorbs or dissipates that energy. It’s a much more holistic metric, combining both force and displacement over the entire duration of the shaking. This is key for understanding cumulative damage, which is tied directly to how a structure’s ability to dissipate energy through inelastic cycles (its hysteresis loops) deteriorates. If a structure can dissipate the energy input from the quake, it’s in a much better place.

Accurately estimating this seismic energy demand is super important for EBD. Folks have been working on formulas for decades to figure out things like hysteretic energy (Eh), which is the energy dissipated through inelastic deformation. They’ve looked at everything from simple models to the characteristics of the ground motion itself.

But here’s a snag: a lot of this foundational work focused on simple, single-degree-of-freedom systems. Real-world structures, like our multi-span continuous concrete box-girder bridges, are way more complex! Investigating hysteretic energy in critical bridge parts, like the columns, hasn’t been explored as much as it should be.

And then there are *residual demands*. These are the permanent deformations left behind after the shaking stops. Imagine a bridge column that’s supposed to be vertical but is now leaning a bit. That’s residual drift. Or a bearing pad that’s shifted permanently. These residual demands are absolutely critical for figuring out if a bridge is still functional after an earthquake without needing massive, immediate repairs. We’ve seen significant residual drifts cause major problems after big quakes like Northridge or Kobe. Most studies on residual demands have focused on columns, often looking at new materials or rehab methods, and usually for quakes happening far away (far-field). Near-field quakes, especially those with “pulses,” can be a different beast entirely.

Connecting the Dots: Demands and Intensity Measures

Performance-based design relies heavily on something called Intensity Measures (IMs). These are single values that try to capture the “intensity” or “damageability” of a ground motion. Think of Peak Ground Acceleration (PGA) or Peak Ground Velocity (PGV) – those are common IMs. Finding the *right* IMs that correlate well with structural demands is crucial for building reliable probabilistic seismic demand models (PSDM). A PSDM basically gives you the probability of a certain level of damage (demand) happening given a certain earthquake intensity (IM).

Lots of research has looked at which IMs work best for bridges, often focusing on displacement. Some studies found velocity-related IMs work better than acceleration or displacement ones for certain bridge types. For continuous concrete box-girder bridges specifically, velocity spectrum intensity (VSIH) and PGVH have been suggested as good candidates, especially for near-field quakes. More recent work even looked at energy-related demands and found VSIH was good for predicting column hysteretic energy (Eh Column) and total hysteretic energy (ΣEh).

Now, while single IMs are simple, they often miss the nuance of complex ground motions, particularly those pulse-like ones in near-field areas. These pulses pack a punch and can cause repeated, damaging cycles. Using *multiple* IMs can give a more complete picture, incorporating different aspects like velocity, acceleration, duration, and frequency content. This multi-IM approach *should* lead to more reliable PSDMs, but it hasn’t been fully explored, especially for energy-related and residual demands. Plus, how do residual demands relate to energy and displacement demands? That’s a gap we needed to fill.

This is where our study comes in! We wanted to develop multi-parameter PSDMs specifically for column hysteretic energy and residual drift ratio in these MSCC-BG bridges. By bringing in multiple IMs and using some smart regression techniques, we aimed to build a more robust framework for assessing seismic demands and improving vulnerability evaluations for these critical structures.

Globally and locally, the consequences of earthquakes are devastating. In seismically active spots like near-fault zones in California, the unique challenge of pulse-like motions means conventional single-IM models might underestimate the cumulative damage to key components like columns and bearings. This could mean designs aren’t sufficient, leaving bridges more vulnerable when the big one hits. We really need these more robust, multi-IM approaches for performance-based design.

Single-IM PSDMs, introduced way back in 2002, are easy to use, but they can fall short, especially with those tricky near-field, pulse-like quakes. They just can’t capture the full complexity. Multi-IM approaches try to fix this by adding more parameters, which can boost accuracy, but you have to be careful about choosing the right ones so you don’t make the model unstable. Lately, AI techniques like neural networks are showing promise in capturing these complex relationships, often leading to better accuracy, but they need lots of data and can be harder to understand. So, choosing the right modeling strategy is a balancing act between simplicity, data needs, and how well it truly reflects the structure’s behavior.

Our Deep Dive: Bridges, Quakes, and Data

So, what did we do? First, we looked closely at the multi-span continuous concrete box-girder bridge type. It’s super common in California – about 37% of bridges there! We focused on configurations with two, three, and four columns, typical of the three-span designs from the 1971–1990 era, which are known to have some design vulnerabilities. We created three distinct bridge models in the OpenSees software, varying things like span length and deck dimensions based on existing research, and incorporating realistic material properties.

These models were pretty detailed. We used 3D finite elements, accounting for how materials behave nonlinearly (like concrete cracking or steel yielding) and how the geometry changes under load (P-Delta effects). We modeled the columns with fiber sections to capture plastification, included rigid links to the deck, and used appropriate damping. We simulated the concrete and steel behavior using established material models. The deck was modeled as elastic, which is usually okay as it stays elastic in quakes, and we added extra mass to account for things like railings.

The foundation system included pile-supported footings, modeled with springs at the column bases (or pinned for multiple columns). Seat-type abutments, common for these bridges, were modeled to capture soil resistance (passive and active) and the potential for pounding between the deck and the back wall using special gap elements. Elastomeric bearings, where the deck rests on the abutments, were modeled as elastic-perfectly-plastic elements, and we included shear keys with hysteretic behavior. We even included a small gap (19.05 mm) at the abutment back wall to simulate pounding, which can cause damage.

For the earthquake part, we gathered a dataset of 164 near-fault ground motions (with three components) from the PEER NGA-West database. These included both pulse-like and non-pulse records, with magnitudes from 4.7 to 7.9, mostly on soil types C and D. To push the structures into significant inelastic deformation, we doubled the acceleration of all records, giving us a total of 328 ground motions to analyze.

We looked at a whopping thirty-six Intensity Measures (IMs)! These included measures from both horizontal and vertical ground motion components. We classified them as non-structure-specific (like PGA, PGV) or structure-specific (derived from response spectra, like spectral acceleration). We also broke them down by type: acceleration, velocity, displacement, and time-related. We used specialized software to calculate these IMs for every single ground motion.

Then came the demands – the structural responses we cared about. We looked at twenty-eight different demands across three categories: displacement-based, energy-related, and residual. Displacement-based demands are what you’d expect – things like column drift ratio, deck displacement, abutment displacement, and deck unseating (when the deck slides off its support). Energy-related demands included total seismic input energy (Ei), damping energy (Ed), elastic strain energy (Ese), kinetic energy (Ek), and crucially, hysteretic energy (Eh) for both individual components (like columns) and the whole structure (ΣEh). Measuring these energy terms, especially the member-based ones, required a customized version of OpenSees, which we used. Residual demands were the permanent ones after the shaking stopped and vibrations died down – specifically, column residual drift ratio and bearing pads’ residual displacement.

We ran 3D nonlinear time-history simulations for all three bridge models under all 328 ground motions. That’s a lot of shaking! For each simulation, we recorded the maximum value of each demand parameter.

What We Found: Correlations and Pulse Power

First off, we looked at the correlations between all these demands and IMs. We visualized this in 3D, which was pretty cool, showing how different IMs relate to different demands. We found that displacement-based and energy-related demands correlated strongly with horizontal IMs. But interestingly, some displacement-based and residual demands, like column axial load ratio and bearing pad residual displacement, also showed significant links to *vertical* IMs. Among all the IMs, velocity- and acceleration-related ones had the strongest correlations across the board. Structure-specific IMs like VSIH and HIH generally performed better than non-structure-specific ones. This reinforced our idea that using multiple IMs that capture different aspects of ground motion is the way to go for better predictions.

One of the most striking findings was the *huge* difference in demands caused by pulse-like versus non-pulse ground motions. We compared the average values of all demands for the pulse-like records versus the non-pulse ones. The results were eye-opening!

For residual demands, the average column residual drift ratio jumped from 0.15% for non-pulse to a whopping 0.62% for pulse-like quakes – a four-fold increase! The average residual displacement of the bearing pads went from 23 mm to 70 mm – a three-fold increase. These are significant permanent deformations that would severely impact post-earthquake functionality and require major repairs. This clearly shows that current design codes *must* account for the effects of pulse-like motions.

Energy-related demands also saw massive increases. The average values of total hysteretic energy (ΣEh), seismic input energy (Ei), column hysteretic energy (Eh Column), and maximum kinetic energy (Max Ek) nearly *doubled* under pulse-like quakes compared to non-pulse ones. For instance, average column hysteretic energy went from 4,700 kN.m to 9,142 kN.m. Max kinetic energy and total hysteretic energy were the most affected.

Column-related demands weren’t spared either. The column drift ratio (peak displacement) increased by 2.1 times under pulse-like motions. Axial load ratio and shear ratio increased too, but less dramatically (1.3 times). This confirms that peak column drift is particularly sensitive to these pulses.

Other displacement-based demands, like shear key displacement, transverse deck displacement, deck unseating, and passive abutment displacement, also saw significant increases (1.9 to 2.3 times) due to pulse characteristics. Passive abutment displacement was the most affected among these, increasing 2.3 times on average.

It’s clear: pulse-like ground motions significantly amplify seismic demands across the board – energy, displacement, and residual. Ignoring them means potentially underestimating damage and vulnerability.

We also looked at how the demands relate to *each other*. We found strong correlations between energy-related and displacement-based demands, especially for column hysteretic energy, which correlated well with column curvature ductility and deck unseating (key indicators of bridge failure). Residual demands also showed strong links to displacement-related demands like column curvature ductility, column residual drift ratio, bearing pad residual displacement, and passive abutment displacement. Moderate to strong correlations were also seen between residual demands and energy-related demands, confirming that the energy dumped into the structure influences how much it deforms permanently.

Building the Multi-Variable PSDM

Given the importance of columns as primary load-carrying elements and their connection to the deck, we focused on developing multi-variable PSDMs for two critical demands: column hysteretic energy (Eh Column) and column residual drift ratio (RDRCol). We wanted to see if using multiple IMs could improve prediction accuracy compared to single-IM models.

We used a two-step regression approach: Lasso and Stepwise regression. Why Lasso? Because we had 36 potential IMs, and many might be correlated (multicollinearity). Lasso is great at selecting the most important variables by shrinking the coefficients of less influential ones towards zero. It helps simplify the model and makes it more interpretable. After Lasso gave us a reduced set of promising IMs, we used Stepwise regression (specifically, a forward selection algorithm) to refine the model further, ensuring only statistically significant IMs were included. This process helped us avoid overfitting and build a robust model. We considered other methods like Ridge regression or PCR, but Lasso and Stepwise together gave us the best balance of variable selection, robustness, and interpretability for engineering use.

We transformed both the IMs and the demands into logarithmic space, which is common for PSDMs, and standardized them so they were on a similar scale. We used a standard 10-fold cross-validation on 80% of our data (leaving 20% for testing) to find the best parameters for the Lasso model. This involved iterating through different penalty values (lambda) and seeing which one minimized the prediction error (Mean Squared Error). Lasso identified thirteen IMs as potential candidates with non-zero coefficients.

Since thirteen IMs are still a lot for a simple model, we fed these into the Stepwise regression. Stepwise picked the best subset based on statistical criteria like RMSE, R-squared (adjusted), AIC, and BIC, which measure how well the model fits the data and penalize complexity. We also checked the Variance Inflation Factor (VIF) to make sure the selected IMs weren’t too correlated with each other, which could mess up the results.

The stepwise procedure selected six IMs for the final multi-variable PSDM equations for the case considering all ground motions. Interestingly, the specific IMs selected differed when we ran the analysis separately for just pulse-like or non-pulse ground motions, highlighting the unique characteristics of each type.

We ended up with empirical equations (in logarithmic space) that predict column hysteretic energy and residual drift ratio based on the selected IMs. We confirmed that the selected IMs had low VIF values (no multicollinearity issues) and that the models had good statistical fits (low AIC/BIC, reasonable R-squared). We also checked that the prediction errors followed a normal distribution, which is important for the model’s validity, using tests like the Kolmogorov–Smirnov test.

The Payoff: Modest Improvement, Big Implications

So, how did our multi-variable PSDMs perform? The logarithmic equations showed satisfactory accuracy overall, with an average R-squared of 83% and an RMSE of 0.77. Compared to previous work on displacement-based demands, the RMSE was a bit higher for energy and residual demands, suggesting more inherent variability in these parameters from one earthquake record to the next.

But here’s the key: compared to using just a *single* optimal IM (like VSIH, which was previously identified as a good predictor), our multi-parameter PSDMs *did* show improved prediction accuracy. The adjusted R-squared, for example, improved by about 14% compared to single-IM models. While the improvement for column hysteretic energy and residual drift ratio was described as “modest,” it’s still an important step forward. Even small gains in prediction accuracy can make a difference in seismic design and evaluation.

This study really hammers home a few things:
* Using multiple IMs *does* improve prediction accuracy for PSDMs, even if the gains are sometimes modest for specific demands like hysteretic energy and residual drift. It helps capture the complexity of ground motion better.
* Energy-based demands (like hysteretic energy) and residual demands are strongly linked to displacement-based demands. Considering cumulative damage and permanent deformation is crucial.
* Residual demands are vital indicators of post-earthquake functionality and need to be included in vulnerability assessments.
* Velocity and acceleration-related IMs seem particularly effective for predicting demands in these types of bridges.
* Pulse-like ground motions are game-changers. They significantly amplify energy, displacement, and residual demands. Designs *must* account for them, especially in near-field areas.

Looking Ahead

Our proposed PSDM equations are applicable to similar continuous concrete box-girder bridges, but like any research, there are limitations. We used simplified foundation models, assumed the deck stayed elastic, focused only on near-field quakes, and used a uniform ground motion excitation across the bridge supports (not accounting for spatial variations, which can matter for long bridges). Also, our models were based on bridges from a specific design era (1971-1990), so applying them to newer or very different bridge types would require more work.

Future research could definitely expand on this. We could look at other bridge types, include more detailed soil-structure interaction, consider deck damage, analyze far-field quakes, and incorporate spatially varying ground motions. Adding uncertainties in material properties and geometry would also make the models even more realistic.

Ultimately, we hope findings like these can influence seismic design codes. Codes could start recommending multi-IM approaches, perhaps highlighting IMs like VSI. They should definitely incorporate residual demands into performance objectives – saying not just “it won’t collapse,” but “it will be usable with minimal permanent deformation.” Pulse-like motions need specific guidance, maybe requiring analyses with scaled pulse records. And bringing energy-based indicators alongside traditional displacement limits would ensure cumulative damage is properly considered.

By doing this, we can give engineers better tools to design more resilient bridges that can truly withstand the complex shaking of real earthquakes, especially in those vulnerable near-field zones. It’s all about building a safer future, one shake at a time.

Source: Springer