=================== Structural Analysis =================== Introduction ------------ This section describes the nonlinear time-history analysis (NLTHA) workflow using cloud analysis on multi-degree-of-freedom (MDOF) structural models. By combining functions for MDOF calibration, modelling and dynamic analysis, the framework enables setup, execution, and post-processing of structural responses under earthquake loading. The main components include: 1. **MDOF Model Construction**: Define and assemble MDOF models with essential structural properties 2. **Nonlinear Time-History Analysis**: Simulate dynamic response under ground motion records 3. **Cloud Analysis**: Fit probabilistic seismic demand models to analysis results Cloud Analysis Methodology -------------------------- **Cloud Analysis (CA)** is a method used to assess the fragility of structures under seismic events. It involves performing nonlinear dynamic analyses using a set of "natural" recorded ground motions **without scaling them**, and then applying simple linear regression in the logarithmic space of structural response. Classical Cloud Analysis ~~~~~~~~~~~~~~~~~~~~~~~~ The EDP–IM relationship is first expressed as a power law: .. math:: EDP = a \cdot IM^b Applying a logarithmic transformation yields a linear regression model: .. math:: \ln(EDP) = \ln(a) + b \ln(IM) where :math:`\ln(a)` and :math:`b` are the regression intercept and slope, estimated via least-squares fitting. The record-to-record uncertainty, expressed as the logarithmic standard deviation of the EDP conditioned on the IM, is given by: .. math:: \beta_{EDP|IM} \approx \sqrt{\frac{\sum_{i=1}^{n} \left[\ln(EDP_i) - \ln(a \cdot IM_i^b)\right]^2}{n - 2}} where n is the number of non-collapse ground-motion records. Modified Cloud Analysis (MCA) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The **Modified Cloud Analysis (MCA)** approach provides a systematic framework for estimating seismic fragility functions while explicitly **accounting for structural collapse cases through Logistic regression and probabilistic combination**. In nonlinear dynamic analyses, numerical non-convergence or excessive response may indicate structural collapse, commonly defined by exceeding a collapse threshold :math:`EDP \ge EDP_C`. Such cases cannot be directly included in classical cloud regression. Using the Total Probability Theorem, the probability of exceeding a damage state DS at a given intensity level is decomposed into two mutually exclusive events—collapse (C) and no-collapse (NC): .. math:: P(DS | IM) = P(DS | NC, IM) \cdot P(NC | IM) + P(DS | C, IM) \cdot P(C | IM) Since exceeding any damage state is guaranteed given collapse (:math:`P(DS | C, IM) = 1`), the expression simplifies to: .. math:: P(DS | IM) = P(DS | NC, IM) \cdot [1 - P(C | IM)] + P(C | IM) where: * :math:`P(DS | NC, IM)` is the fragility derived from cloud regression using only non-collapse data * :math:`P(C | IM)` is the probability of collapse The probability of collapse is estimated using Logistic regression: .. math:: P(C | IM) = \frac{1}{1 + \exp\left[-(\alpha_0 + \alpha_1 \ln(IM))\right]} where :math:`\alpha_0` and :math:`\alpha_1` are regression coefficients fitted to collapse/non-collapse outcomes. Statistical Stabilisation via Bootstrapping ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Modified Cloud Analysis is often based on a limited number of ground-motion records, which can lead to numerical instability and sensitivity to outliers. **Bootstrapping** is employed to improve robustness: 1. **Resampling**: Generate N bootstrap datasets (e.g., N = 200) by random sampling with replacement from the original dataset 2. **Estimation**: For each bootstrap sample, perform cloud regression on non-collapse records and Logistic regression for collapse probability 3. **Aggregation**: Compute the mean probability of exceedance across all bootstrap realisations Benefits of bootstrapping include: * Reducing the influence of individual extreme ground motions * Producing smoother transitions in fragility curves near collapse-dominated regions * Enabling estimation of epistemic uncertainty through percentile bounds (e.g., 16th and 84th) Interactive Viewer ------------------ Use the interactive viewer below to explore structural analysis results, including cloud analysis and demand profiles from nonlinear response history analyses. .. raw:: html

Demand Profiles

Peak storey drift and peak floor acceleration profiles from 700 ground-motion records.

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Cloud Analysis

Modified Cloud Analysis (MCA) with bootstrapped regressions.

Derived Fragility Functions

Fragility functions derived from cloud analysis.

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Engineering Demand Parameters ----------------------------- The analysis extracts key engineering demand parameters (EDPs): **Peak Storey Drift (PSD)** Maximum interstorey drift ratio across all storeys, strongly correlated with structural damage **Peak Floor Acceleration (PFA)** Maximum absolute floor acceleration, critical for non-structural components and contents **Peak Floor Displacement** Maximum floor displacement relative to ground Ground-Motion Records --------------------- A comprehensive ground-motion dataset of 700 records spanning various tectonic regimes is used: * **Active Shallow Crust**: NGA-West database * **Stable Continental**: NGA-East database * **Subduction Zones**: NGA-Sub database * **Additional sources**: KiK-Net, Engineering Strong Motion Database Intensity measures include: * Peak Ground Acceleration (PGA) for low-rise structures * Spectral Acceleration SA(0.3s), SA(0.6s), SA(1.0s) for various building periods * Average Spectral Acceleration (AvgSA) for improved efficiency References ---------- * Jalayer F. et al. (2015). Bayesian Cloud Analysis: Efficient structural fragility assessment using linear regression. Bulletin of Earthquake Engineering. * Jalayer F. et al. (2017). Analytical fragility assessment using unscaled ground motion records. Earthquake Engineering & Structural Dynamics. * Cornell C.A. et al. (2002). Probabilistic Basis for 2000 SAC Federal Emergency Management Agency Steel Moment Frame Guidelines. Journal of Structural Engineering.