Variability Of Structural Response Research Articles

Evaluation of the Inter-frequency Correlation of New Zealand CyberShake Crustal Earthquake Simulations

The inter-frequency correlation of ground-motion residuals is related to the width of peaks and troughs in the ground-motion spectra (either response spectra or Fourier amplitude spectra; FAS) and is therefore an essential component of ground-motion simulations for representing the variability of structural response. As such, this component of the simulations requires evaluation and validation when the intended application is seismic fragility and seismic risk. This article evaluates the CyberShake NZ [1] crustal earthquake ground-motion simulations for their inter-frequency correlation, including comparisons with an empirical model developed from a global catalogue of shallow crustal earthquakes in active tectonic regions, and with results from similar simulations (SCEC CyberShake; [2]). Compared with the empirical model, the CyberShake NZ simulations have a satisfactory level of total inter-frequency correlation between the frequencies 0.1 – 0.25 Hz. At frequencies above 0.25 Hz, the simulations have lower (statistically significant at 95% confidence level) total inter-frequency correlation than the empirical model and therefore require calibration. To calibrate the total correlation, it is useful to focus on the correlation of the residual components. The between-event residual correlations, physically related to source effects (e.g., stress drop) which drive ground motions over a broad frequency range, are low at frequencies greater than about 0.25 Hz. Modifications to the cross-correlation between source parameters in the kinematic rupture generator can improve the inter-frequency correlations in this range [3]. The between-site residual correlations, which represents the correlation between frequencies of the systematic site amplification deviations, are larger (statistically significant at 95% confidence level) than the empirical model for frequencies less than about 0.5 Hz. We postulate that this relates to the relative simplicity of site amplification methods in the simulations, which feature less variability than the amplification observed in the data. Additional insight would be gained from future evaluations accounting for repeatable path and basin effects, using simulations with refined or alternative seismic velocity models, and using simulations with a higher crossover frequency to deterministic methods (e.g., 1 Hz or higher).

The effect of random field parameter uncertainty on the response variability of composite structures

The accurate quantification of the random spatial variation of material properties at different scales is crucial for the systematic propagation of uncertainties through engineering models. In a previous work, the spatial variability of the apparent material properties of two-phase composites has been quantified in a Bayesian framework. This framework enables a consistent modeling of the statistical uncertainty in the parameters of the respective mesoscale random fields and also allows selecting the most plausible correlation model among different models belonging to the Matérn family. In this work, the most plausible random field model is employed in the context of uncertainty propagation of composite structures. Sample functions of the mesoscale random fields are generated using a covariance decomposition approach and the response variability of various composite structures is computed through Monte Carlo simulation. Parametric investigations are conducted to highlight the effect of the identified parameter uncertainty on structural response variability.

A hierarchy model for the uncertainty quantification of spatial variability in the constitutive parameters of concrete in structures

In engineering practice, the compressive strength of concrete of the same strength grade at different positions of a structure is usually treated either as perfectly correlated or as completely independent random variables. However, due to the common factors affecting the randomness, e.g., the mix proportion, placing, consolidating, curing, etc., such treatment is unreasonable and consequently the results may be misleading. In this paper, a hierarchy model is adopted to describe the variability of the constitutive parameters of concrete in a structure. In this model, the total variability of the constitutive parameters of concrete in a structure is separated according to the contributing factors on four hierarchies: the hierarchy of structure, of floor, of component and of representative volume element (RVE). By this model, the correlation between components and that between floors can be captured. In addition, based on a large number of test complete compressive stress–strain curves, the copula-based joint probability distribution of the constitutive parameters of concrete crossing multiple strength grades is given. Combining this joint distribution with the hierarchy model, the compressive constitutive parameters of concrete in a reinforced concrete (RC) structure can be obtained. Then the stochastic response analysis of a 6-story RC frame subjected to seismic ground motion is conducted. The results show that the variability of concrete on the hierarchy of floor has a non-negligible impact on the structural response, whereas the variability of concrete on the hierarchy of component and that of RVE may be ignored. In addition, it is found that only considering the variability of concrete on the hierarchy of structure will underestimate the variability of structural response and seriously overestimate the reliability of structure. On the contrary, considering the variability of concrete in a structure by the independent identically distribution is unreasonable and will lead to large errors in the variability of structural response and reliability.

Uncertainty Quantified Parametrically Homogenized Constitutive Models for Microstructure-Integrated Structural Simulations

This paper investigates the role of material microstructures in structural analysis and establishes the need for microstructure-integrated constitutive models in predicting structural response. A focus is on microstructure and temperature dependency of stresses and plastic strains in a structural panel under realistic loading conditions. Structural analysis is conducted using the recently developed uncertainty-quantified parametrically homogenized constitutive model (UQ-PHCM) for near-alpha Titanium alloys like Ti6242S. PHCMs exhibit explicit microstructural dependency and are developed from homogenization of crystal plasticity finite element simulation results with machine learning. Uncertainties due to model reduction, data sparsity and microstructural variability are accounted for in the model. Structural response with the UQ-PHCMs is compared to those predicted by isotropic elasticity and $${J}_2$$ plasticity models without explicit microstructure information. Parametric studies illustrate how different uncertainties in the UQ-PHCM framework propagate to the structural response variables. The results also show the relative contribution of different microstructural features to the propagated uncertainty in structural response variables. The studies establish the UQ-PHCM as an effective tool for reliable structural analysis with consequences in material design.

R2R-EU: Software for fragility fitting and evaluation of estimation uncertainty in seismic risk analysis

Analytical methods in performance-based earthquake engineering (PBEE) typically employ suites of ground motion records to run dynamic analysis of a structure's computer model. The sample size of ground motions used in this context, affects the estimation uncertainty that underlies the seismic risk metrics thus obtained. This article presents R2R-EU (record-to-record estimation uncertainty), which is a PBEE software tool that numerically implements various schemes for estimating structure-specific seismic fragility and for the quantification of the estimation uncertainty behind seismic risk estimates, emanating from record-to-record variability in structural response. The software accepts as input the results of structural dynamic analysis to a set of accelerograms and seismic hazard curves. Estimation uncertainty is quantified by providing statistics, such as mean and variance, of the estimators of the failure rate and the fragility parameters (where applicable) and possibly their distribution. The user can choose the analysis method among some resampling and/or simulation schemes belonging to the bootstrap family, the delta method and other solutions from probability and statistics theory.

Introducing a response-based duration metric and its correlation with structural damages

This study proposes a response-based parameter for strong motion duration which is computed for structures and is the total time they are nonlinear during an earthquake. Correlation between structural response and duration for structures, subjected to a set of spectrum matched ground motions, is employed to examine the efficiency of the proposed method. The spectral matching procedure ensures that the influence of amplitude and frequency content of motions on structural response variability is significantly removed. Four concrete building type systems are studied and correlation coefficients of structural response with the proposed duration definition are examined. Comparison of the proposed method with other existing definitions, the record-based and response-based metrics, shows about 15-20% improvement in the correlation values.

Sources of Variability in Structural Bending Response of Pediatric and Adult Human Ribs in Dynamic Frontal Impacts.

Despite safety advances, thoracic injuries in motor vehicle crashes remain a significant source of morbidity and mortality, and rib fractures are the most prevalent of thoracic injuries. The objective of this study was to explore sources of variation in rib structural properties in order to identify sources of differential risk of rib fracture between vehicle occupants. A hierarchical model was employed to quantify the effects of demographic differences and rib geometry on structural properties including stiffness, force, displacement, and energy at failure and yield. Three-hundred forty-seven mid-level ribs from 182 individual anatomical donors were dynamically (~2 m/s) tested to failure in a simplified bending scenario mimicking a frontal thoracic impact. Individuals ranged in age from 4 - 108 years (mean 53 ± 23 years) and included 59 females and 123 males of diverse body sizes. Age, sex, body size, aBMD, whole rib geometry and cross-sectional geometry were explored as predictors of rib structural properties. Measures of cross-sectional rib size (Tt.Ar), bone quantity (Ct.Ar), and bone distribution (Z) generally explained more variation than any other predictors, and were further improved when normalized by rib length (e.g., robustness and WBSI). Cortical thickness (Ct.Th) was not found to be a useful predictor. Rib level predictors performed better than individual level predictors. These findings moderately explain differential risk for rib fracture and with additional exploration of the rib's role in thoracic response, may be able contribute to ATD and HBM development and alterations in addition to improvements to thoracic injury criteria and scaling methods.

Large-area mapping of Canadian boreal forest cover, height, biomass and other structural attributes using Landsat composites and lidar plots

Passive optical remotely sensed images such as those from the Landsat satellites enable the development of spatially comprehensive, well-calibrated reflectance measures that support large-area mapping. In recent years, as an alternative to field plot data, the use of Light Detection and Ranging (lidar) acquisitions for calibration and validation purposes in combination with such satellite reflectance data to model a range of forest structural response variables has become well established. In this research, we use a predictive modeling approach to map forest structural attributes over the ~552millionha boreal forest of Canada. For model calibration and independent validation we utilize airborne lidar-derived measurements of forest vertical structure (known as lidar plots) obtained in 2010 via a >25,000km transect-based national survey. Models were developed linking the lidar plot structural variables to wall-to-wall 30-m spatial resolution surface reflectance composites derived from Landsat Thematic Mapper and Enhanced Thematic Mapper Plus imagery. Spectral indices extracted from the composites, disturbance information (years since disturbance and type), as well as geographic position and topographic variables (i.e., elevation, slope, radiation, etc.) were considered as predictor variables. A nearest neighbor imputation approach based on the Random Forest framework was used to predict a total of 10 forest structural attributes. The model was developed and validated on >80,000 lidar plots, with R2 values ranging from 0.49 to 0.61 for key response variables such as canopy cover, stand height, basal area, stem volume, and aboveground biomass. Additionally, a predictor variable importance analysis confirmed that spectral indices, elevation, and geographic coordinates were key sources of information, ultimately offering an improved understanding of the driving variables for large-area forest structure modeling. This study demonstrates the integration of airborne lidar and Landsat-derived reflectance products to generate detailed and spatially extensive maps of forest structure. The methods are portable to map other attributes of interest (based upon calibration data) through access to Landsat or other appropriate optical remotely-sensed data sources, thereby offering unique opportunities for science, monitoring, and reporting programs.

Quantification of the amplitude variability of the ground motion in Argostoli, Greece. Variability of linear and non-linear structural response of a single degree of freedom system.

The term “spatial variability of seismic ground motions” denotes the differences in the amplitude and phase content of seismic motions. The effect of such spatial variability on the structural response is still an open issue. In-situ experiments may be helpful in order to answer the questions regarding both the quantification of the spatial variability of the ground motion within the dimensions of a structure as well as the effect on its dynamic response. The goal of the present study is to quantify the variability of the seismic ground motion accelerations in the shallow sedimentary basin of Argostoli, Greece, and thereafter to identify its effect on the linear and non-linear elasto-plastic response of a single degree of freedom system in terms of spectral displacements. Around 400 earthquakes are used, recorded by the 21-element very dense seismological array deployed in Argostoli with inter-station spacing ranging from 5 to 160 meters. The seismic motion variability, evaluated in terms of spectral accelerations, is found to be significant and to increase with inter-station distance and frequency. Thereafter, the amplitude variability in terms of spectral displacements, which is indeed the linear response of a single degree of freedom (SDOF) system with various fundamental periods, is compared with the amplitude variability of a SDOF with non-linear elasto-plastic response. The variability of the maximum top displacement of the linear single degree of freedom system is estimated to be on average 12% with larger variabilities to be observed within two narrow frequency ranges (between 1.5 and 1.7 Hz and between 3 and 4 Hz). Such high variabilities are caused by locally edge-generated diffracted surface waves. The non-linear perfectly elasto-platic structural response of the SDOF system shows that although the variability has the same trends as in the case of linear response, it is almost constantly increased by 5%.

Structure-specific selection of earthquake ground motions for the reliable design and assessment of structures

A decision support process is presented to accommodate selecting and scaling of earthquake motions as required for the time domain analysis of structures. Code-compatible suites of seismic motions are provided being, at the same time, prequalified through a multi-criterion approach to induce response parameters with reduced variability. The latter is imperative to increase the reliability of the average response values, normally required for the code-prescribed design verification of structures. Structural attributes like the dynamic characteristics as well as criteria related to variability of seismic motions and their compliance with a target spectrum are quantified through a newly introduced index, δ sv–sc , which aims to prioritize motions suites for response history analysis. To demonstrate the applicability of the procedure presented, the structural model of a multi-story building was subjected to numerous suites of motions that were highly ranked according to both the proposed approach (δ sv–sc ) and the conventional one (δ conv ), that is commonly used for earthquake records selection and scaling. The findings from numerous linear response history analyses reveal the superiority of the proposed multi-criterion approach, as it extensively reduces the intra-suite structural response variability and consequently, increases the reliability of the design values. The relation between the target reliability in assessing structural response and the size of the suite of motions selected was also investigated, further demonstrating the efficiency of the proposed selection procedure to achieve higher response reliability levels with smaller samples of ground motion.

A unified response surface framework for the interval and stochastic finite element analysis of structures with uncertain parameters

In this paper, finite element analysis of structures with uncertain properties is addressed within both a probabilistic and non-probabilistic framework. Specifically, uncertainties affecting structural parameters are modelled either as interval or random variables. In both cases, uncertainty propagation analysis is performed by applying a ratio of polynomial response surface which enables to derive approximate closed-form expressions of the main descriptors of interval and random response variability by requiring just a few deterministic analyses at selected sampling points. A unified response surface framework for interval and stochastic finite element analysis is thus developed which allows comparisons of structural response variability under different uncertainty models. Numerical results focusing on the comparison between the interval and stochastic response of structures with uncertain Young’s modulus are presented.

Seismic fragility curves based on the probability density evolution method

A seismic fragility curve that shows the probability of failure of a structure in function of a seismic intensity, for example peak ground acceleration (PGA), is a powerful tool for the evaluation of the seismic vulnerability of the structures in nuclear engineering and civil engineering. The common assumption of existing approaches is that the fragility curve is a cumulative probability log-normal function. In this paper, we propose a new technique for construction of seismic fragility curves by numerical simulation using the Probability Density Evolution Method (PDEM). From the joint probability density function between structural response and random variables of a system and/or excitations, seismic fragility curves can be derived without the log-normal assumption. The validation of the proposed technique is performed on two numerical examples.

Entropy-based sensitivity analysis of global seismic demand of concrete structures

Abstract This study presents a global sensitivity analysis approach based on entropy to investigate the influence of excitation and structural parameters on the engineering demand parameters. The sensitivity of the nonlinear dynamic structural response using synthetic earthquake ground motions to major uncertain sources, propagation paths and site variables of the simulated ground motion, and physical characteristics of the structures is investigated using an entropy-based sensitivity index as a measure of importance to determine which variables are most significant. The results show that the uncertainties of ground-motion variables are more significant than those in the structural properties. The greatest contributor to the variability in the seismic demand is the uncertainty in earthquake source parameters. Our analysis also revealed that viscous damping is the most important structural source of variability in seismic structural demands. Structural dynamic analysis due to simulated excitation opens the door for the wider use of seismological theory to understand the relationship between the structural response and seismological variables.

Bi-level dominance GA for minimum weight and maximum feasibility robustness of composite structures

In the proposed approach the robust design optimization (RDO) of composite structures is addressed as a bi-objective optimization problem with following objective functions: (1) the weight of the structure which optimality is associated with performance robustness; and (2) the variability of structural response which is associated with feasibility robustness of design constraints. The determinant of the variance–covariance matrix of the response is adopted for feasibility robustness assessment, being the sensitivities calculated by the adjoint variable method. The design and uncertainty rules are controlled by the following classes of variables and parameters: the deterministic design variables, the random design variables, and the random parameters.To solve the RDO of composite structures an evolutionary algorithm, denoted by Bi-level Dominance Multi-Objective Genetic Algorithm (MOGA-2D) is proposed. The Pareto front is built using a hierarchical structure where evolution is based on the exchange data between two populations: a small population using local dominance and elitism and an enlarged population to store the non-dominated solutions. The numerical tests show the capabilities of the approach. Although the optimal Pareto front establishes the trade-off between performance and robustness, knowledge on the importance of each uncertainty source can help the designer to make a decision on design space.

Structural response variability under spatially dependent uncertainty: Stochastic versus interval model

This paper deals with structural response variability under spatially varying uncertainties represented using both probabilistic and non-probabilistic models. Attention is focused on Euler–Bernoulli beams with uncertain flexibility subjected to deterministic static loads. Within the probabilistic framework, the uncertain flexibility is modeled as a random field and the well-established concept of variability response function is adopted to derive spectral- and probability-distribution-free upper bounds on the response variability. Within the non-probabilistic context, the uncertain property is represented resorting to a recently proposed interval field model able to quantify the dependency between adjacent values of an interval uncertainty that cannot differ as much as values that are further apart. For statically determinate beams, the interval displacement field and the associated bounds are evaluated analytically by using a Green’s function formulation. Conversely, for statically indeterminate beams, approximate explicit expressions of the bounds of the interval response are derived by applying a finite difference based procedure.Numerical results present consistent comparisons between response variability under random and interval uncertainty for both statically determinate and indeterminate beams.

Comparing of the effects of scaled and real earthquake records on structural response

Time history analyses have been preferred commonly in earthquake engineering area to determine earthquake performances of structures in recent years. Advances in computer technology and structural analysis have led to common usage of time history analyses. Eurocode 8 allows the use of real earthquake records as an input for linear and nonlinear time history analyses of structures. However, real earthquake records with the desired characteristics sometimes may not be found, for example depending on soil classes, in this case artificial and synthetic earthquake records can be used for seismic analyses rather than real records. Selected earthquake records should be scaled to a code design spectrum to reduce record to record variability in structural responses of considered structures. So, scaling of earthquake records is one of the most important procedures of time history analyses. In this paper, four real earthquake records are scaled to Eurocode 8 design spectrums by using SESCAP (Selection and Scaling Program) based on time domain scaling method and developed by using MATLAB, GUI software, and then scaled and real earthquake records are used for linear time history analyses of a six-storied building. This building is modeled as spatial by SAP2000 software. The objectives of this study are to put basic procedures and criteria of selecting and scaling earthquake records in a nutshell, and to compare the effects of scaled earthquake records on structural response with the effects of real earthquake records on structural response in terms of record to record variability of structural response. Seismic analysis results of building show that record to record variability of structural response caused by scaled earthquake records are fewer than ones caused by real earthquake records.

Variance decomposition of the seismic response of structures

The reliability of structures is dependent on the variability in random characteristics of strong ground motion. In this paper, the influence of the variability of ground motion variables on the variability of structural response has been accessed by sensitivity analysis. The Variance-Based Sensitivity Analysis (VBSA) is considered using the Monte Carlo based procedure, for computing the set of first order sensitivity indices of variables to estimate the contribution that each uncertain input makes to variance in the model output. The variables that have a dominant influence on the uncertainty of structural response are identified. The analytical results show that the formulation of dynamic structural response in the frequency domain, based on the seismological information of excitation, can be used in some countries, such as Iran, with lack of recorded strong motion.

The variability of deformation demand with ground motion intensity

The use of Nonlinear Dynamic Analyses is coupled to the demands produced by earthquakes of different intensities. The uncertainties associated with the demands are highly dependent on the variable adopted as the intensity measure (IM). This generates the need to compare different IMs and in particular the dispersion of the demand measure in relation to each IM. For this purpose correlations between maximum inter-story drift demand of nine reinforced concrete Multiple Degree of Freedom tridimensional structures and a number of widely used ground motion intensity parameters have been investigated. To determine the optimum parameter to be used as the IM, two additional intensity parameters are analyzed, the spectral acceleration corresponding to the cracked and non-linear period of each structure, Sa(Tcrack) and Sa(T∗), respectively. The results showed that these two IMs have the best correlation with the deformation demand and their use produces a lower variability of structural response.

Progressive Incremental Dynamic Analysis for First-Mode Dominated Structures

Incremental dynamic analysis (IDA) is a widely used method for assessing structural performance under earthquake excitations. It enables direct evaluation of the record-to-record variability in structural response through a set of ground-motion records. If the number of ground-motion records is large then, the method becomes computationally demanding. To facilitate its practical application, a precedence list of ground-motion records has been introduced, aiming at selecting the most representative ground-motion records for IDA analysis. In progressive IDA analysis, the IDA curves are computed progressively, starting from the first ground-motion record in the precedence list. After an acceptable tolerance has been achieved, the analysis is terminated. This approach may significantly reduce the computational effort for first-mode-dominated structures, since the seismic response can be computed only for a certain number of ground-motion records from the precedence list to achieve an acceptable level of confidence in the prediction of the summarized (16th, 50th, and 84th fractiles) IDA curves. The proposed implementation of incremental dynamic analysis, which is demonstrated using an example of a 4-story reinforced concrete frame, can also be used for the selection of ground-motion records from a very large set of records, provided that all the records properly represent the seismic scenario for a given site.

A New Approach to Multibody Model Development: Pedestrian Lower Extremity

Objective: The goal of this study was to develop a mathematical model of the 50th percentile male lower extremity capable of predicting injury risk and simulating the kinetic and kinematic response of the pedestrian lower extremity under vehicle impact loading. Methods: The hip-to-foot multibody model was developed for the MADYMO software platform using exterior and interior geometry and inertial properties from a detailed finite element model (FEM) of the human lower extremity and stiffness and failure tolerance data from the literature. The leg and thigh models' structural and contact parameters were simultaneously optimized to validate model response in simulations replicating previous dynamic bending experiments. The aggregate model's full-scale kinematic response was verified by comparing 3-D local (knee bending angles) and global (linear accelerations and velocities) frame leg and thigh kinematics from vehicle impact simulations with data generated from seven vehicle–pedestrian (PMHS) impact experiments. Results: By optimizing contact and structural response variables, the applied moment vs. deflection response of the leg and thigh showed excellent correlation with the experimental corridor averages in component-level bending simulations. The full-scale kinematic response of the 50th percentile male model showed good correlation with the PMHS response data in both the rate of valgus knee bending (∼ 3 degress/ms) and in the timing and magnitude of the peak thigh and leg accelerations (250 g and 400 g). Additionally, as a result of vehicle interaction, both the model and the experiments showed that the thigh and leg are initially accelerated upward (100 g) and downward (100 g), respectively, and then downward (60 g) and upward (100 g), respectively. The model also predicted a valgus knee injury and a tibia fracture similar to those seen in the PMHS. Conclusions: The use of a facet surface model of the lower extremity skin and simultaneous optimization of the model's structural response and contact parameters resulted in a model capable of accurately predicting the detailed kinematic response of the lower extremity under vehicle impact loading at 40 km/h. The model can be scaled to represent varying pedestrian anthropometries and can assess the risks associated with sustaining the most common pedestrian injuries. As a vehicle design tool, the model can be used to optimize front-end designs, or it can be used in combination with a detailed FEM to reduce the vast design space prior to FE simulations. Additionally, the model can be used as a tool to study pedestrian impact kinematics, real-world case reconstructions, or particular vehicle countermeasures.