Updating structural reliability efficiently using load measurement
Introduction
Risk plays a principal role in the balance of safety and economy at the core of structural and geostructural engineering. Understood as the probability-weighted equivalent financial losses that would result from failure, it allows the aggregate effects of good design and maintenance practices to be expressed in financial terms that are useful in managing maintenance of infrastructure assets.
In the design of new structures, this balance is captured in an approximate sense by means of the target reliability embodied in the calibration of design standards [e.g.[1], [2], [3]]. Existing structures are commonly assessed using these same techniques and standard calibrations, despite the fact that there is notable potential for reducing the uncertainty about performance limits through measurements of response to loading [4], [5] and comparison of these values to the design model predictions [6].
Knowledge of the change in reliability with time allows maintenance planning to be done from a risk optimal perspective [7], and is also of value when a structure is repurposed for a use in a different risk class [2], [4], [5].
The classification of uncertainty as epistemic versus aleatory [8], [9], [10], suggests a key perspective on time dependence in structural reliability problems. Loading parameters are predominantly aleatory and time varying, with uncertainty stemming from the inability to predict the future. Material parameters are mostly epistemic and spatially varying, with the possibility of reducing uncertainty by more representative sampling strategies.
To the extent that material deterioration can be accounted for, the lack of time-dependent variability in material parameters allows the lower bound on structural resistance capacity implied by its ability to resist a severe loading event observed in the past to be applied at the present time, albeit with a degree of uncertainty stemming from factors such as measurement error and the description of deterioration. Such a loading event can take the form of a controlled proof load test [e.g. [11], [12]], or occur during the service life of the structure [e.g. [13], [14]].
The first order reliability method (FORM) [15], [16] has been extensively applied to obtain reliability estimates for many common structural and geostructural design problems, and forms an integral part of standards calibration [1], [2]. The source of the utility and economy of the method is its simple geometrical formulation, by which it effectively reduces a many-dimensional integration problem to a one-dimensional standard normal probability determination.
Previous work on updating of structural reliability for existing structures using the lower bound on resistance capacity implied by severe loading or proof load-testing rely on a Bayesian updating formulation, in which the distribution of the structural resistance is truncated at the limit implied by the observed load [17], [18]. Updated reliability values are then obtained either by integration of the posterior distribution function [18], [19], [20], [21], or by updating the failure probability using the probability of the implied resistance bound conditional on the structure surviving the loading event [11], [22].
In the former approach, the irregular distribution shape due to truncation, coupled with the fact that typical structural and geostructural design problems involve resistance terms with multiple parameters with a variety of non-normal distribution types, results in a strongly non-linear limit state function in standard normal space [17]. As a result, the updated reliability index cannot be directly determined using FORM, and instead needs to be obtained by numerical integration via a variant of the Monte-Carlo technique [e.g. [23]]. Similarly, the latter approach requires evaluation of the conditional bound probability, again requiring Monte-Carlo integration.
This paper develops an efficient methodology for updating structural reliability following observations of severe loading. The method builds on the geometrical simplicity of FORM, reducing the multi-dimensional reliability updating problem to a two-dimensional combination of two related FORM solutions. The theoretical basis for this first order reliability updating method is developed in the following section, followed by validation and assessment of its performance for limit state functions with curvature at the design point. The method is then implemented for two example structures: firstly, a reinforced concrete beam forming part of a highway bridge with weigh-in-motion traffic loading measurements, and secondly a granular seawall embankment with piezometric records of the time variation in the phreatic surface.
Section snippets
Geometrical interpretation of reliability updating
Consider a simple performance function expressing the balance between the resistance capacity R and an applied load effect S. That isfor which the requirement that implies . Load effect S is the result of a time-varying random process; R reflects a function of inherent properties of the structure and material, known only from a set of sample characterisations.
In a reliability context, represents a single limit state function delimiting the domain of parameter values
Validation
Integration for the updated probability of failure via the linearised limit state function in Eq. (7) is based directly on FORM, and assumes the limit state function () to be linear in standard normal space. Although this is almost never truly the case for real structural problems, FORM remains tractable because the part of the parameter domain where the difference between and its linear approximation at the design point becomes significant, fall predominantly in regions of very low
Weigh-in-motion traffic loading applied to reinforced concrete bridge
As a first example implementation, recorded weigh-in-motion data, in the form of point loads corresponding to the measured axle loads of individual vehicles, are applied to the moment influence line of a single supported beam. As simplification, it is assumed that the beam is supporting a single traffic lane and that a single vehicle governs the moment response. The data is obtained from the Roosboom measurement station in the left lane of the N3 toll route connecting Durban and Johannesburg
Discussion
In general, accurate reliability update values would require numerical integration of Eq. (5) or Eq. (19). Direct numerical integration [17], [18], [20] becomes prohibitive for multidimensional problems, so that such evaluations generally require the use of Monte-Carlo techniques [19], [35], [36], which still represent a significant undertaking when numerical simulations are required to evaluate the performance function. Similarly, Bayesian updating of the probability of failure [11], [22]
Conclusions
The observation of an existing structure supporting a particular maximal load provides a direct constraint on the possible range of values its resistance capacity may take. The implied update of the structural reliability allows monitoring and maintenance planning to be done from a risk optimal perspective. Proof load-based reliability updating techniques require multiple numerical computations which are often too cumbersome for routine use.
By building on the assumptions of the first order
Acknowledgements
Reliability analysis computations for the two example structures were performed using the UQLab reliability analysis library [39]. Colin Caprani and two anonymous reviewers provided valuable input towards improving the manuscript.
References (40)
- et al.
Structural health monitoring and reliability estimation: long span truss bridge application with environmental monitoring data
Eng Struct
(2008) - et al.
Proof load testing for bridge assessment and upgrading
Eng Struct
(2000) Reliability updating with equality information
Probab Eng Mech
(2011)- et al.
Estimation of small failure probabilities in high dimensions by subset simulation
Probab Eng Mech
(2001) - et al.
Partial factors for loads due to special vehicles on road bridges
Eng Struct
(2016) - et al.
Verification of existing reinforced concrete bridges using the semi-probabilistic approach
Eng Struct
(2013) - et al.
Reliability updating in geotechnical engineering including spatial variability of soil
Comput Geotech
(2012) - et al.
Bayesian updating of slope reliability in spatially variable soils with in-situ measurements
Eng Geol
(2018) - et al.
Reliability-based design and its complementary role to Eurocode 7 design approach
Comput Geotech
(2015) Eurocode: Basis of structural design
A framework for reliability-based system assessment based on structural health monitoring
Struct Infrastruct Eng
Risk-informed condition assessment of civil infrastructure: state of practice and research issues
Struct Infrastruct Eng
Probability concepts in engineering planning and design: Volume II decision, risk, and reliability
On the treatment of uncertainties and probabilities in engineering decision analysis
J Offshore Mech Arct Eng
Aleatory or epistemic? Does it matter?
Struct Saf
Traffic characteristics and bridge loading in South Africa
J South Afr Inst Civil Eng
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