The effect of material and ground motion uncertainty on the seismic vulnerability curves of RC structure

The study investigates how uncertainties in earthquake ground motion characteristics and structural material properties affect seismic vulnerability curves for reinforced concrete (RC) buildings. Vulnerability curves relate measures of shaking intensity (e.g., PGA, PGV, spectral ordinates) to the probability of reaching or exceeding defined performance limit states (serviceability, damage control, collapse prevention). Such curves are central to regional seismic risk and loss estimation, alongside hazard maps, inventory data, and integration/visualization tools. The paper aims to quantify the relative influence of ground motion variability versus material randomness, and to assess the impact of analysis choices such as ground-motion selection and scaling, duration definition, damping assumptions, limit-state definitions, and statistical treatment of simulation outcomes.

Vulnerability curves are derived using different data sources and methods, each with advantages and limitations: empirical (observational post-earthquake data; realistic but region- and sample-specific, often sparse and biased toward low damage), judgmental (expert opinion; comprehensive but subjective), analytical (simulation-based; generalizable but sensitive to modeling assumptions and computationally intensive), and hybrid (combining observational, analytical, and expert data). Prior analytical approaches often simplified demand estimation using SDOF representations, inelastic spectra with capacity spectrum methods, or default variability (e.g., HAZUS), which may neglect higher modes, hysteretic damping, and local failure limit states, potentially biasing results. Increasing computational power and improved nonlinear analysis tools reduce these limitations, motivating the present use of detailed inelastic time-history analyses.

- Reference structure: A three-story ordinary moment resisting concrete frame (OMRCF), non-seismically detailed, originally designed and tested experimentally (1/3-scale) for gravity loads (ACI 318-89, Grade 40 steel, specified concrete strength 24 MPa). Plan: 3 and 4 bays in E-W and N-S, 5.5 m bay width, 3.7 m story height, total height 11 m. Representative of medium-rise, limited-ductility RC frames typical of Mid-America and similar regions. - Analysis platform: ZEUS-NL for nonlinear response-history analysis, modeling material and geometric nonlinearity with fiber sections and Eulerian geometric nonlinearity. Hysteretic behavior includes stiffness/strength degradation. Pushover (conventional/adaptive), eigenvalue, and dynamic analyses available. - Model verification: Eigen periods (analytical) 0.898 s, 0.305 s, 0.200 s compared with snap-back test (scaled to prototype) 0.932 s, 0.307 s, 0.206 s (3–4% difference). Time-history comparisons at PGA 0.20g and 0.30g (Taft) showed very good agreement; at 0.05g less agreement due to uncertain small-amplitude damping and initial cracking. For medium/high motions, only hysteretic damping assumed (no additional viscous damping) to avoid non-conservative results. - Uncertainties considered (aleatory): • Material properties: Concrete in-place mean strength taken as 1.4× specified (mean 33.6 MPa) with COV 18.6%, normal distribution; steel yield strength mean 337 MPa with COV 10.7%, normal distribution; steel modulus deterministic 201,327 MPa. • Input motions: Nine ground-motion sets. - Three recorded sets categorized by PGA/PGV (a/v) ratio: low (a/v < 0.8 g/ms−1), intermediate (0.8–1.2), high (>1.2), with distinct average spectra. - Six artificial sets for Memphis, TN (Mississippi Embayment): Lowlands L-1, L-2, L-3 and Uplands U-1, U-2, U-3, each with 10 motions generated via bedrock synthesis and equivalent-linear site response with randomized Vs profiles; scenarios span magnitudes/distances (sample PGA/PGV/PGD provided). - Random variable sampling: For a/v sets, full combination of 10 concrete × 10 steel strengths (100 unique frames). For artificial Lowlands sets, 50 random pairs; for Uplands sets, 100 random pairs. Full Monte Carlo across all motions and scale levels without reducing sample size. - Ground-motion processing: • Significant duration defined between 0.5% and 95% of cumulative Arias intensity (to avoid unrealistic initial pulses and account for skewed energy content), rather than traditional 5–95%. • Scaling: Motions scaled in PGA from 0.05g to up to ~0.5g in 0.05g steps, with ranges adjusted per set based on capacity spectrum estimates to capture collapse onset (e.g., low a/v sets collapsed near ~0.2g; high a/v required higher PGA). - Limit states (derived specifically for the prototype via adaptive pushover and sectional responses): • Serviceability: first yielding of steel → ISD = 0.57%. • Damage control: attainment of maximum element strength → ISD = 1.2%. • Collapse prevention: confined concrete strain reaching 0.01 → ISD = 2.3%. Interstory drift (ISD) used as the global damage measure for all stories; local element damage not explicitly modeled. - Simulation scale and computation: For a/v sets alone, ~23,000 nonlinear time-history analyses (~456 CPU hours on Pentium IV 2.65 GHz). For artificial sets, a mass-simulation environment with 32 parallel processes on an IBM p690 supercomputer was used. - Statistical treatment: Maximum ISD per analysis assumed lognormal for stable responses. Collapsed cases (ISD ≥ 2.3% or numerical instability under gravity) excluded from parametric fit but incorporated via total probability: P(ISD > limit | E1)·P(E1) + 1·P(E2), where E1=stable, E2=collapse. This yields conservative exceedance estimates. Linear interpolation of empirical points used for comparisons; regression can be applied for software implementation.

- Ground motion variability dominates: Vulnerability curves differ markedly across ground-motion sets, with discrepancies growing for higher damage states and at higher intensities up to collapse. Non-monotonic trends can appear at large PGA due to collapse statistics and data handling. - Material variability is secondary: Concrete and steel property randomness contributes much less to ISD variability than input motion characteristics. Concrete strength influences response at low PGA (via stiffness/period changes), whereas steel yield strength has little effect at low PGA; at high PGA, materials add variability but remain far less influential than ground motion. - Sample size effects: Increasing the number of frames with randomized materials has minimal impact on the coefficient of variation of ISD at low PGA. A variance relationship shows that when material effects are small, adding frames primarily reduces variance associated with ground motion sampling. - Statistical modeling matters: Treating collapse explicitly and adopting lognormal fits for stable responses avoids biased underestimation of exceedance probabilities at high PGA that occurs if unstable (collapsed) drifts are included in parametric statistics. - Damping assumption: Using identical low-amplitude viscous damping at all intensities could yield non-conservative vulnerability at medium/high motions; assuming only hysteretic damping is more appropriate for inelastic ranges. - Model verification: Analytical eigen periods closely matched experimental (e.g., first mode 0.898 s vs 0.932 s experimental), and mid/high-intensity time histories matched well, supporting the reliability of the simulation framework. Quantitative examples: ~23,000 analyses for three a/v sets; concrete mean 33.6 MPa (COV 18.6%), steel yield mean 337 MPa (COV 10.7%); limit-state ISDs 0.57%, 1.2%, 2.3%.

The research question—how material and ground-motion uncertainties influence seismic vulnerability curves—is addressed by controlled simulation on a verified RC frame model with explicit uncertainty treatment. Findings show that, for a regular, low-ductility, medium-rise RC frame, the selection and scaling of input motions, their frequency content (captured via a/v categorization), duration, and site effects govern the resulting fragility more than the typical randomness in concrete and steel strengths. This underscores that ground-motion set curation (including scenario relevance and site-response consistency) is critical. Material variability modestly affects response, primarily through stiffness (concrete) at low intensities; steel yield strength has negligible effect before yielding. Incorporating collapse via total probability and using lognormal fits for stable responses yield more reliable exceedance probabilities. Overall, the methodology and results promote rigorous nonlinear time-history analysis with careful statistical treatment and structure-specific limit states to produce realistic vulnerability functions.

- Concrete ultimate strength affects response at low intensities through stiffness-period interaction; steel yield strength has little influence at low intensities. - At higher intensities, material variability contributes to response scatter, but is much smaller than the variability induced by ground-motion characteristics. - Ground-motion selection and scaling have a predominant effect on vulnerability curves; meticulous curation is essential. - Explicit consideration of collapse and appropriate statistical modeling (lognormal for stable responses with total probability for collapse) prevents bias at higher intensities. - The analysis framework (verified model, extensive inelastic dynamic simulations, structure-specific limit states) supports robust conclusions. Future research should extend to irregular structures, systems with spatially variable/un-correlated material properties and alternative damage measures (e.g., component-level damage, joint/connection failures), and broaden hazard scenarios and site conditions to enhance generalizability.

- Results pertain to a specific three-story, regular, non-seismically detailed OMRCF; generalization to other RC typologies, heights, or irregular configurations is limited. - Damage is measured only via interstory drift; local element failures (e.g., joint shear, bar slip) are not modeled. - Only aleatory uncertainties in materials and input motions are considered; epistemic uncertainties (modeling choices, record selection bias, scaling methods) are not quantified. - Damping is assumed hysteretic only at medium/high intensities; low-amplitude damping uncertainties could affect low-PGA results (potential conservatism acknowledged). - Limit states are structure-specific and may not transfer directly to different RC systems. - Ground-motion sets, while diverse (recorded a/v classes and Memphis artificial motions), cannot encompass all possible hazard characteristics.