Structural Analysis

Introduction

This section describes the nonlinear time-history analysis (NLTHA) workflow using modified cloud analysis on multi-degree-of-freedom (MDOF) stick-and-mass 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 stick-and-mass models with essential structural properties

  2. Nonlinear Time-History Analysis: Simulate dynamic response under ground motion records

  3. Modified Cloud Analysis: Fit probabilistic seismic demand models to analysis results accounting for non-collapse and collapse cases

Modified Cloud Analysis Methodology

Modified Cloud Analysis (MCA) is a method used to assess the fragility of structures under seismic events. It involves performing NLTHA using a set of “natural” recorded ground motions without scaling them.

The EDP–IM relationship is first expressed as a power law:

\[EDP = a \cdot IM^b\]

Applying a logarithmic transformation yields a linear regression model:

\[\ln(EDP) = \ln(a) + b \ln(IM)\]

where \(\ln(a)\) and \(b\) 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:

\[\beta_{EDP|IM} \approx \sqrt{\frac{\sum_{i=1}^{n} \left[\ln(EDP_i) - \ln(a \cdot IM_i^b)\right]^2}{n - 2}}\]

where n is the number of non-collapse ground-motion records.

The approach also provides a systematic framework for estimating seismic fragility functions while explicitly accounting for structural collapse cases through Logistic regression and probabilistic combination.

In NLTHA, numerical non-convergence or excessive response may indicate structural collapse, commonly defined by exceeding a collapse threshold \(EDP \ge EDP_C\). 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):

\[P(DS | IM) = P(DS | NC, IM) \cdot P(NC | IM) + P(DS | C, IM) \cdot P(C | IM)\]

Since exceeding any damage state is guaranteed given collapse (\(P(DS | C, IM) = 1\)), the expression simplifies to:

\[P(DS | IM) = P(DS | NC, IM) \cdot [1 - P(C | IM)] + P(C | IM)\]

where:

  • \(P(DS | NC, IM)\) is the fragility derived from cloud regression using only non-collapse data

  • \(P(C | IM)\) is the probability of collapse

The probability of collapse is estimated using Logistic regression:

\[P(C | IM) = \frac{1}{1 + \exp\left[-(\alpha_0 + \alpha_1 \ln(IM))\right]}\]

where \(\alpha_0\) and \(\alpha_1\) are regression coefficients fitted to collapse/non-collapse outcomes.

MCA 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.

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

Intensity Measures

Intensity measures include:

  • Peak Ground Acceleration (PGA) for low-rise structures

  • Spectral Accelerations, SA(0.3s), SA(0.6s), SA(1.0s) for various building periods

References

  • Jalayer, F., De Risi, R. & Manfredi, G. Bayesian Cloud Analysis: efficient structural fragility assessment using linear regression. Bull Earthquake Eng 13, 1183–1203 (2015). https://doi.org/10.1007/s10518-014-9692-z

  • Jalayer F, Ebrahimian H, Miano A, Manfredi G, Sezen H. Analytical fragility assessment using unscaled ground motion records. Earthquake Engng Struct Dyn. 2017;46:2639–2663. https://doi.org/10.1002/eqe.2922

  • Cornell C.A. et al. (2002). Probabilistic Basis for 2000 SAC Federal Emergency Management Agency Steel Moment Frame Guidelines. Journal of Structural Engineering.