Elsevier

Structural Safety

Volume 84, May 2020, 101939
Structural Safety

Updating structural reliability efficiently using load measurement

https://doi.org/10.1016/j.strusafe.2020.101939Get rights and content

Highlights

  • A method for updating structural reliability using measured loading is developed.

  • Its first order implementation is shown to be computationally efficient and accurate.

  • The method can be applied to a diverse spectrum of multi-parameter structural problems.

Abstract

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 structural reliability allows monitoring and maintenance planning to be done from a risk optimal perspective. Existing 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 reliability method, this study develops and validates a first order reliability updating approach which is computationally efficient. The resulting formulation is shown to be applicable to reliability problems tractably considered using the first order reliability method. This method is illustrated for two example structures: a reinforced concrete beam forming part of a highway bridge to which traffic loading data is applied, and a granular embankment forming a seawall on a shoreline mining operation for which the phreatic surface level is monitored.

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 isg1=R-S,for which the requirement that g10 implies RS. 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, g1=0 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 (g=0) 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 g=0 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.

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