Probabilistic Load Model Research Articles

Avaliação probabilística da carga variável de projeto para garagens brasileiras de veículos leves

Abstract Design loads for parking garages should be reviewed every few years due to changes in fleet characteristics and the impact of design loads on carbon emissions by the built environment. Specifically for Brazil, the authors are unaware of existing studies to justify the design values and corresponding exceedance probabilities stated in NBR 6120:2019 - Design Loads for Structures. In this paper, a simplified probabilistic model for live loads in parking garages is presented. A set of updated statistics characterizing the gross curb weight of the fleet in circulation was obtained based on technical specifications and sales reports between 2003 and 2022. These statistics and the probabilistic load model are employed to derive the equivalent uniform design load corresponding to a 30% probability of being exceeded in 50 years, according to the definition stated in NBR 6120:2019 and NBR 8681:2003. The results provide support for a significant reduction of the current design load for light-duty vehicle parking garages, from the current 3.0 kN/m2 to at least 2.5 kN/m2. Such a reduction has a significant impact on new building construction costs and carbon emissions and would not compromise the structural safety of parking garages built in Brazil.

Failure probability estimation of dynamic systems employing relaxed power spectral density functions with dependent frequency modeling and sampling

This work addresses the critical task of accurately estimating failure probabilities in dynamic systems by utilizing a probabilistic load model based on a set of data with similar characteristics, namely the relaxed power spectral density (PSD) function. A major drawback of the relaxed PSD function is the lack of dependency between frequencies, which leads to unrealistic PSD functions being sampled, resulting in an unfavorable effect on the failure probability estimation. In this work, this limitation is addressed by various methods of modeling the dependency, including the incorporation of statistical quantities such as the correlation present in the data set. Specifically, a novel technique is proposed, incorporating probabilistic dependencies between different frequencies for sampling representative PSD functions, thereby enhancing the realism of load representation. By accounting for the dependencies between frequencies, the relaxed PSD function enhances the precision of failure probability estimates, opening the opportunity for a more robust and accurate reliability assessment under uncertainty. The effectiveness and accuracy of the proposed approach is demonstrated through numerical examples, showcasing its ability to provide reliable failure probability estimates in dynamic systems.

A critical review of probabilistic live load models for buildings: Models, surveys, Eurocode statistics and reliability-based calibration

The appropriate assignment of gravity live loads is of fundamental importance in establishing a design base for safe and economical structures. This paper presents a critical review of the probabilistic modelling of live loads in buildings, providing a historical overview of the theoretical models found in the literature, as well as the load surveys that provided the empirical evidence for their development. As a general rule, these models divide the live load into sustained and extraordinary components, represented as a Poisson square wave and spike processes, respectively, with the differences lying in the underlying hypotheses and process parameters. A comparison of different models is presented, and it is shown that the model parameters currently in use in background documents on the reliability of the Eurocodes do not agree well with survey data, leading to an overestimation of extreme loads. While the Eurocodes lack clear specification regarding the exceedance probability for design loads, this study demonstrates, using a model with modified parameters, that the office design load corresponds to an approximate 27% exceedance probability over 50 years for a reference area of 20 m2. The impact of employing this model on the reliability-based calibration of the Eurocode partial safety factors for loads is also examined, and it is found that the average reliability falls somewhat below the prescribed 50-year target reliability index of 3.8.

Failure analysis and structural reliability of unreinforced masonry veneer walls: Influence of wall tie corrosion

Corrosion appears to be a very common problem in historical and aged structures with steel ties and anchorages. A potential rapid deterioration of the wall tie due to the corrosion leading to a premature structural failure poses a serious threat to the built heritage infrastructures’ safety. This paper describes an overview of the probabilistic failure analysis of the unreinforced masonry (URM) veneer wall system with flexible backup under uniformly distributed out-of-plane loadings. Moreover, a framework is proposed to consider the wall tie corrosion in the stochastic finite element analysis (FEA) while estimating the structural reliability. A probabilistic experimental study where 18 full-scale URM veneer wall systems with theoretically identical geometries and properties were tested under inward and outward lateral loading. Wall failure statistics along with the probabilistic characterisation of veneer wall constituent materials including mortar, wall tie and timber were accomplished. A stochastic computational model was then developed which combines the FEA and Monte Carlo simulation to evaluate the failure progression and system peak load (veneer capacity) while considering the spatial variability of veneer wall material properties. Two scenarios, with and without wall tie corrosion, are considered, and the probabilistic characterisation of the veneer capacities for these scenarios under inward and outward out-of-plane loading is also reported. The model error statistics were combined with the probabilistic load models to determine the reliability index corresponding to the Australian Standard for Masonry Structures AS 3700. Annual reliabilities are compared to target reliabilities recommended by ISO 2394. Capacity reduction factor and design minimum tie strength consideration for corroded wall tie is also discussed.

Structural reliability and partial safety factor assessment of unreinforced masonry in vertical bending

ABSTRACT This paper focuses on a structural reliability-based assessment of clay brick unreinforced masonry (URM) walls subjected to uniformly distributed out-of-plane loads in one-way vertical bending. Stochastic models combining finite element analysis (FEA) and Monte Carlo simulations (MCS) are used to account for spatial variability of the flexural tensile bond strength when estimating the wall failure loads. The strength of URM walls is known to be influenced by the flexural tensile bond strength, which is subject to high spatial variability as batching, workmanship, and environmental exposure alter the strength of this bond. For this assessment, single skin walls have been considered with bond strength statistics seen in typical construction. The model error statistics available for similar walls are combined with the results of the spatial stochastic FEA and probabilistic load models to determine the reliability index corresponding to the Australian Standard for Masonry Structures AS 3700 design of members in vertical bending. It was found that existing levels of reliability exceed target reliabilities, and the capacity reduction factor can be increased from 0.60 to 0.65 for URM walls in one-way vertical bending while still providing an acceptable level of reliability. A sensitivity analysis showed this finding to be robust.

The influence of ASCE 7–16 wind load provisions on a vulnerability model of Florida residential construction

The Florida Public Hurricane Loss Model is a probabilistic risk model designed to estimate wind induced insurance losses to residential infrastructure in Florida. In the physical damage estimation module, Monte Carlo simulation is employed to generate damage matrices as a function of wind speed by comparing probabilistic building component capacities to wind loads. These damage matrices are input to a repair cost algorithm, ultimately producing damage ratios as a function of wind speed for different building classes. It is required that the inventory of residential building models reflect the history of construction practice within Florida. Differing eras of construction are represented by developing component-wise options for probabilistic capacities and load paths, resulting in a suite of hundreds of model variants. Regarding loads, the probabilistic wind load model is based on a directionalized adaptation of ASCE 7, and its representation in the model is kept current with the version adopted by the Florida Building Code (FBC). In 2020 the Florida Building Code adopted by reference ASCE 7–16, which contains important changes to the wind-load provisions. These changes have implications for both the loading and resistance aspects of the physical damage model, thus influencing insured loss projections. This study discusses the development and implementation of these changes within the model and investigates the impacts on the modeled vulnerability of residential structures in Florida. Comparing vulnerability estimates of structures designed before and after the FBC adoption of ASCE 7–16 provide insight into the potential effectiveness of such changes to improve community resilience.

Thermal fatigue load models for fatigue design of steel box girder bridges

Coupling with traffic loads, thermal loads can also contribute to the fatigue damage cumulation of bridges. Referring to traffic and wind fatigue load models, the concepts of two kinds of probabilistic thermal fatigue load models for the first system of steel box girder bridges were proposed. Specifically, the basic forms of the two models are 1) equivalent vertical temperature gradient models based on the damage equivalence rule and 2) the effect ratio curves based on a statistical analysis of thermal actions. Then, two design probabilistic thermal fatigue load models following the given procedures are set up using an extended dataset based on a 350-day long-term in-suit temperature monitoring, and parametric analysis on three damage equivalence factors including geometric features, design life, and degree of indeterminacy was conducted. To validate the proposed models and related approaches, two kinds of fatigue details in a case steel box girder bridge were evaluated. The results show that thermal effects on fatigue damage cumulation are significant and should be taken into consideration, and the proposed thermal fatigue load models are applicable for fatigue design and evaluation.

Mid- and long-term strategy based on electric vehicle charging unpredictability and ownership estimation

Predicting the charging load of electric vehicles (EVs) is critical for the safe and reliable operation of the distribution network. Analyzing an EV's random charging characteristics and the uncertainty associated with its development scale are important to accurate prediction of its charging load. For this reason, we proposed a seminal method for predicting EV charging load based on stochastic uncertainty analysis. This included not only a probabilistic load model for describing the stochastic characteristics of the EV charging, but also an ownership forecasting model for estimating the EV development scale. EVs are classified into four categories based on their intended use: electric buses, electric taxis, private EVs, and official EVs. The corresponding load calculation model was developed by analyzing the charging behavior of various EVs. Simultaneously, the improved grey model method (IGMM) based on the Fourier residual correction is used to accurately forecast EV ownership. Finally, the scientific method of Monte Carlo simulation(MCS) was used to estimate the charging load demand of EVs. This method was used in Wuhan that has a lot of potential for EV production. As compared to the basic grey model method (BGMM), the IGMM outlined in this work can triple the prediction effect. Due to the large-scale charging of EVs, Wuhan's maximum daily total load would rise to 15,532.9 MW on working days and 15,475.5 MW on rest days in 2025. Additionally, the total load curves on working days and rest days will show a new peak load with the value of 14751.3 MW and 14787.2 MW at 14:01, resulting in an increase of 13.56% and 13.83% respectively in the basic daily load stage. As a result, it is necessary for grid operators to build adequate capacity to meet EV charging demands, while developing rational and orderly charging strategies to avoid the emergence of new load peaks.

Possibilistic uncertainty assessment in the presence of optimally integrated solar PV-DG and probabilistic load model in distribution network

To integrate network load and line uncertainties in the radial distribution network (RDN), the probabilistic and possibilistic method has been applied. The load uncertainty is considered to vary as Gaussian distribution function whereas line uncertainty is varied at a fixed proportion. A voltage stability index is proposed to assign solar PV-DG optimally followed by application of PSO technique to determine the optimal power rating of DG. Standard IEEE 33- and 69-bus RDN are considered for the analysis. The impact of various uncertainties in the presence of optimally integrated solar PV-DG has been carried out on 69-bus network. The results obtained are superior to fuzzy-arithmetic algorithm. Faster convergence characteristic is obtained and analyzed at different degree of belongingness and realistic load models. The narrower interval width indicates that the observed results are numerically stable. To improve network performance, the technique takes into account long-term changes in the load profile during the planning stage. The significant drop in network power losses, upgraded bus voltage profile and noteworthy energy loss savings are observed due to the introduction of renewable DG. The results are also statistically verified.

Forecasting voltage harmonic distortion in residential distribution networks using smart meter data

This paper introduces a methodology to forecast voltage total harmonic distortion (THD) at low voltage busbars of residential distribution feeders based on the data provided by a limited number of smart meters. The methodology provides relevant power quality indices to system operators using only the existing monitoring infrastructure required for demand response operation. Different algorithms for voltage THD forecasting are implemented, including artificial neural networks, and their performance is tested and compared. The necessary coverage of smart meters for the acceptable accuracy of the estimated THD is also established. The estimation algorithms are validated considering probabilistic demand load model developed based on typical harmonic injections of household devices obtained from measurements and using a typical European low voltage test-feeder with 471 residential consumers.

Interval Arithmetic Power Flow Analysis of Radial Distribution System with Probabilistic Load Model and Distributed Generation

Interval Arithmetic Power Flow Analysis of Radial Distribution System with Probabilistic Load Model and Distributed Generation

Load Model Identification using Steady-state Measurements for Power System Control

Power system control requires verified models of all its elements: generators, transformers, transmission lines, and power system loads. To identify load model parameters, the measurement data are used. The paper discusses the possibility of using steady-state measurements instead staged field test data. This requires taking into account an unobvious effect-the network response. The network response is due to the probabilistic nature of the power and voltage changes and the correlation between them. The network response effect is demonstrated using a probabilistic load model and a simple power supply scheme.

External blast load factors for dome structures based on reliability

Dome structures have been used extensively for industrial, residential, and military infrastructure. Therefore, it is necessary to understand the damage risk potential for such structures for blast-resistant design considerations. This paper investigates the effect of blast load variability on the design value and the structural dynamic response. Therefore, the sources of uncertainty in the external blast load on dome structures were discussed firstly. Then based on the probabilistic blast load model for the dome, the rationality of a deterministic mass-increase safety method was assessed. It was found that previous deterministic design method cannot provide a consistent and sound assurance factor or reliability index on the entire dome roof. In addition, it was also proved that the assurance-based load method fails to ensure compliance with structural safety design standards on the dome roof when compared with the reliability-based blast method. A sensitivity analysis on the probabilistic blast load was conducted, and the results indicate that stand-off distance and explosive mass both act as dominant sources to influence the mean and variability of blast load. Therefore, based on the Latin hypercube sampling method, a reliability-based external blast load factor technique was proposed. This technique was further used to estimate structural damage levels of a single-layer reticulated dome under different reliability requirements, associated with a low, medium, and high level of protection grades for a specific explosion scenario, and it indicated that this technique can be useful in the building design to achieve a higher structural anti-explosion capacity. This study herein can serve as a reference for the calculation method of designed blast load.

Reliability analysis of planar steel trusses based on p-box models

Introduction. The development of probabilistic approaches to the assessment of mechanical safety of bearing structural elements is one of the most relevant areas of research in the construction industry. In this research, probabilistic methods are developed to perform the reliability analysis of steel truss elements using the p-box (probability box) approach. This approach ensures a more conservative (interval-based) reliability assessment made within the framework of attaining practical objectives of the reliability analysis of planar trusses and their elements. The truss is analyzed as a provisional sequential mechanical system (in the language of the theory of reliability) consisting of elements that represent reliability values for each individual bar and truss node in terms of all criteria of limit states.
 Materials and methods. The co-authors suggest using p-blocks consisting of two boundary distribution functions designated for modeling random variables in the mathematical models of limit states performed within the framework of the truss reliability analysis instead of independent true functions of the probability distribution of random variables. Boundary distribution functions produce a probability distribution domain in which a true distribution function of a random variable is located. However this function is unknown in advance due to the aleatory and epistemic uncertainty. The choice of a p-block for modeling a random variable will depend on the type and amount of statistical information about the random variable.
 Results. The probabilistic snow load model and the numerical simulation of tests of steel samples of truss rods are employed to show that p-box models are optimal for modeling random variables to solve numerous practical problems of the probabilistic assessment of reliability of structural elements. The proposed p-box snow load model is based on the Gumbel distribution. The mathematical model used to perform the reliability analysis of planar steel truss elements is proposed. The co-authors provide calculation formulas to assess the reliability of a truss element for different types of p-blocks used to describe random variables depending on the amount of statistical data available.
 Conclusions. The application of statistically unsubstantiated hypotheses for choosing the probability distribution law or assessing the parameters of the probability distribution of a random variable leads to erroneous assessments of the reliability of structural elements, including trusses. P-boxes ensure a more careful reliability assessment of a structural element, but at the same time this assessment is less informative, as it is presented in the form of an interval. A more accurate reliability interval requires interval-based assessments of distribution parameters or types of p-boxes applied to mathematical models of the limit state, which entails an increase in the economic and labor costs of the statistical data.

Probabilistic blast load model for domes under external surface burst explosions

The increasing threat of terrorist attacks and the risk of accidental explosions has raised the alarm on structural design, and consequently, the dome structure as the classical building configuration model for public buildings requires safe and reliable defence capabilities against blast loads for the duration of its lifespan. Some deterministic blast load models for specific blast loading scenarios are unable to take the risk of damage into consideration. Therefore, the variability and uncertainty of blast loads need to be studied and quantified. In this study, a probabilistic blast load model is proposed based on a series of repeated field trials. The repeatability of explosive mass and size, detonator type and location, stand-off distance, field condition, data acquisition device and the structure were ensured to realise consistency and standardisation. The experimental results reveal a high degree of variability and uncertainty in the impulse, reflected overpressure and decay coefficient distributed on the studied dome under the effect of external surface explosions. The decay coefficient of a blast wave on the dome has a large variability with a coefficient of variation of 55% when the impulse is less than 8%, which demonstrates that the impulse distributed on the dome has higher precision than other blast load parameters. Furthermore, statistical analysis shows that the normal distribution is the best-fit probability blast load model in an anti-explosion protection design. Furthermore, the accuracy of some blast load predictive methods was also assessed. It was found that the semi-empirical CONWEP air blast code is not an appropriate blast load model for the dome. However, the Arbitrary Lagrangian-Eulerian model is a safer and relatively more accurate method for describing the distribution and variation characteristics of a blast wave distributed on the dome, which can serve as an approximate mean blast load.

Probabilistic models for blast parameters and fragility estimates of steel columns subject to blast loads

This paper proposes a probabilistic framework to predict the failure probabilities of steel columns subject to blast loads. The framework considers the uncertainties in the blast phenomenon, the demands imposed on the column, and the capacities of the column for the limit states of flexure, and global buckling. As part of the work, we propose four probabilistic blast load models. For different types of explosives and atmospheric conditions, two models predict the incident and reflected peak pressure generated by the explosion and two models predict the incident and reflected positive time duration of the blast wave. The models are probabilistic to capture the associated uncertainties, including variations in the atmospheric conditions, the inherent variability in the blast load data even for identical experimental conditions, and model error. The blast load models are used to predict the structural demands (maximum internal moment and deflection) imposed by the blast on a column. The demand models are combined with strain-rate dependent capacity models for flexure and global buckling to estimate the conditional probability of failure (or fragility) of a steel column for given scaled distance. As an example, fragility estimates for different columns representative of typical columns in steel frames are developed. The results highlight the importance of the explosive weight and column axial load on the failure probabilities.

Review of Current Practice in Probabilistic Structural Fire Engineering: Permanent and Live Load Modelling

Probabilistic analysis is receiving increased attention from fire engineers, assessment bodies and researchers. It is however often unclear which probabilistic models are appropriate for the analysis. For example, in probabilistic structural fire engineering, the models used to describe the permanent and live loads differ widely between studies. Through a literature review, it is observed that these diverging load models are based on surveys conducted between 1893 and 1976 and that widely adopted assumptions, such as the rule for combining permanent and live loads into the total load effect, are commonly adopted based on precedent. The diverging current models however relate to mostly the same underlying datasets and basic methodologies. Differences can be attributed (largely) to specific assumptions in different background papers, which have become consolidated through repeated use in research papers and adoption in background documents to codes. By reviewing the studies underlying currently applied probabilistic load models in structural fire engineering, a consolidated probabilistic load model is proposed in this paper. It is concluded that the total load effect is ideally described by KE·(G + Q), with KE the model uncertainty for the load effect, G the permanent load, and Q the imposed load. The model uncertainty KE can be described by a lognormal distribution with mean equal to unity and coefficient of variation (COV) of 0.10. The permanent load is preferably modelled by a normal distribution with mean equal to the nominal permanent load, and a COV which can either be assessed on a project specific basis, or can be set to 0.10 for a first assessment. For common occupancies (office, residential), the live load is preferably modelled by a Gamma distribution. The mean live load can be taken as 0.2 times the nominal, and the live load COV can be taken as 0.60 for large load areas (> 200 m2) and 0.95 for smaller load areas (< 100 m2). Comparison between the failure probabilities of steel and concrete columns subject to fire, considering the proposed consolidated model and two currently commonly used models, indicates that relative differences of the probability of failure can be in the order of 10%. Live load models for evacuation routes and warehouses require specific study and are outside the scope of the review.

Probabilistic Multiple Pedestrian Walking Force Model including Pedestrian Inter‐ and Intrasubject Variabilities

A probabilistic walking load model that accounts for inter‐ and intrasubject variabilities has been developed to generate synthetic vertical load waveforms induced by pedestrians. The mathematical model is based on a comprehensive database of continuously recorded pedestrian walking forces on an instrumented treadmill, having a wide range of walking frequencies. The proposed model is able to replicate temporal and spectral features of real walking forces, which is a significant advantage over conventional Fourier series models. The load model results in more realistic force time histories than previous models, since it incorporates significant components of the spectra that are omitted in Fourier series approaches. The proposed mathematical model can be implemented in vibration serviceability assessment of civil engineering structures, such as building floors and footbridges, to estimate more realistically dynamic structural responses due to people walking.

A Comprehensive Probabilistic Model of Traffic Loads based on Weigh-in-Motion Data for Applications to Bridge Structures

In various efforts to assure safety and serviceability of a bridge structure throughout its lifetime, it is essential to accurately estimate the traffic load effects. Although traffic loads involve large uncertainties and can vary significantly with site-specific traffic environments, bridge design codes and maintenance strategies do not utilize a probabilistic model that can reflect the actual environments and uncertainties of the target bridge. Rapid developments of weigh-in-motion (WIM) technologies now make it possible to collect various types of data describing the characteristics of vehicles and traffic patterns. Based on actual WIM data collected in South Korea, this paper develops a comprehensive probabilistic model describing the characteristics of vehicles and traffic flow so that the traffic load effects of a target bridge can be assessed using a Monte Carlo simulation approach. To describe the characteristics of vehicles and traffic flow in the WIM data, several important random variables are first identified. These key random variables are then incorporated into a comprehensive probabilistic model based on fitted probability distributions and theories of transportation engineering. The developed model is successfully verified by comparing the daily maximum total loads estimated using actual WIM data with those estimated using artificial WIM data generated from the model. Furthermore, bridge traffic load effects, e.g., moment and tension, are estimated using the influence lines of an actual cable-stayed bridge in South Korea (the Incheon Bridge) and are compared with those from the live load model of a design code. Finally, a brief parametric study is performed to explore the possibility that a probabilistic model developed by the proposed approach can be used as a generic probabilistic traffic load model capable of estimating the site-specific traffic loads through customizations based on partial measurements and available information regarding the target bridge.

Limit mechanisms for ice loads on inclined structures: Local crushing

This paper focuses on mechanisms that limit the sea ice loads on offshore structures. It introduces a probabilistic limit load model, which can be used to analyze peak ice load events and to estimate the maximum peak ice load values on a wide, inclined, offshore structure. The model is based on simple mechanical principles, and it accounts for a mixed-mode ice failure process that includes buckling and local crushing of ice. The model development is based on observations on two-dimensional combined finite-discrete element method simulations on the ice-structure interaction process. The paper also presents a numerical limit load algorithm, which is an extension of the probabilistic limit load model and capable of yielding a large number of stochastic peak ice load values. The algorithm is compared to simulation-based and full-scale observations. Analyzing peak ice load events is challenging as sea ice goes through a complex mixed-mode failure process during such events. The algorithm is an effective tool for this analysis, and it shows that distinguishing between the buckling and local crushing failure is virtually impossible if the only data available from a peak load event is the value of the peak ice load. The algorithm shows potential in improving estimates of maximum peak ice load values on offshore structures.