Robustness has been appointed as a desirable property of systems in order to avoid significant indirect consequences caused by extreme events. In fact, robustness is usually related to global failure or collapse. Nevertheless, its definition is not consensual, having led researchers to propose several quantitative measures. In this work, an intuitive framework for the assessment of robustness as a probabilistic performance indicator of bridges to be applied either to ultimate limit states or service limit state is discussed. The research strategy consists in a review of the state-of-art of robustness of structures, aiming to identify the suitable quantitative procedures to assess system robustness and its contributing attributes. A non-linear finite element model of an existing pre-stressed concrete bridge was used to assess bridge structural performance when it is subjected to several damage scenarios. By means of a probabilistic approach, the load carrying capacity and structural safety are evaluated through advanced reliability analysis techniques.The authors would like to thank ISISE - Institute for Sustainability and Innovation in Structural Engineering (PEst-C/ECI/UI4029/2011 FCOM-01-0124-FEDER-022681), FCT Portuguese Scientific Foundation for the research grant PD/BD/113677/2015. This study also received funding from the European Union's Seventh Framework Programme for research, technological development and demonstration under grant agreement No. 606229. Also the collaboration and information provided by the professor Antonio Abel Henriques is gratefully acknowledged

Fernandes, J.

Guimarães, H.

Matos, José C.

English

Universidade do Minho: RepositoriUM

1 INTRODUCTION Robustness of structures has been recognized over the years as interesting research topic due to the collapse of big structural systems in which the con-sequences were considered unacceptable concerning its initial damage. Indeed, several system failures have been occurring, such as: (1) Ronan Point Build-ing in 1968, London; (2) New World Hotel in 1985, Singapura; (3) The Highland towers in 1993, Kuala Lumpur; (4) Sampoong Department Store in 1995, Seul; (5) World Trade Center in 2001, New York; (6) The Torch Tower in 2015, Dubai and (7) The Adress Hotel in 2015, Dubai. Structural robustness began to be seriously stud-ied after the massive disaster of World Trade Centre collapse. Another reason for a renewed interest of robustness analysis were derived for failures due to unexpected loads, design errors, errors during execu-tion, unforeseen deterioration and poor maintenance (Canisius et al., 2007). In this context, a workshop carried by JCSS in collaboration with IABSE at the Building Research Establishment in London, UK (December 2005) gathered 50 experts, from research institutions, companies and government, to discuss issues related with robustness. The conclusions lead-ed to a general consensus that the present situation with regard to ensuring sufficient structural robust-ness through codes and standards was highly unsatis-factorily. As a consequence, a joint European project in Robustness was created, namely the COST action TU06010 – Robustness of Structures. The present work aims to develop a framework for the assessment of robustness as a performance indicator of bridges. In this way, a non-linear finite element model (FEM) of an existing pre-stressed concrete bridge, combined with probabilistic ap-proaches, is carried out in order to assess bridge’ safety. This paper is organized as follows: in chapter 2, general concepts and existing approaches regard-ing robustness are presented; probabilistic tech-niques to assess existing bridges are highlighted in chapter 3; later, theoretical concepts presented in previous chapters are applied in a case study. Final-ly, in the last chapter, conclusions are drawn and fu-ture developments are pointed out. 2 ROBUSTNESS 2.1 General Concepts In general, robustness can be defined as a capaci-ty that a certain structure has to support a certain amount of damage without occurring global col-lapse. Starossek and Haberland (Starossek & Haberland, 2010) presented several definitions of robustness in civil engineering domain. The same authors also discuss several terms related with ro-bustness, such as:  Exposure – possibility of a structure to be af-fected by a threat during its life-cycle;   Vulnerability – susceptibility of a structure to be damaged by an exposure;  Damage tolerance – ability of a structure to survive once it is damaged;  Redundancy – availability of alternative paths for a load to be transferred from a point of application to a point of resistance;  Ductility – ability of a structure to suffer plas-tic deformations without occurring rupture;  Reliability – ability of a structure to perform its intended function for a specific period of time under certain conditions.    Robustness of an existing pre-stressed concrete bridge Hugo Guimarães, João Fernandes & José C.Matos University of Minho, ISISE, Guimarães, Portugal ABSTRACT: Robustness has been appointed as a desirable property of systems in order to avoid signifi-cant indirect consequences caused by extreme events. In fact, robustness is usually related to global failure or collapse. Nevertheless, its definition is not consensual, having led researchers to propose several quantitative measures. In this work, an intuitive framework for the assessment of robustness as a probabilistic performance indicator of bridges to be applied either to ultimate limit states or service limit state is discussed. The research strategy consists in a review of the state-of-art of robustness of structures, aiming to identify the suitable quan-titative procedures to assess system robustness and its contributing attributes. A non-linear finite element model of an existing pre-stressed concrete bridge was used to assess bridge structural performance when it is subjected to several damage scenarios. By means of a probabilistic approach, the load carrying capacity and structural safety are evaluated through advanced reliability analysis techniques.    2.2 Robustness Measures Regarding the quantification of robustness, several approaches have been proposed by different re-searchers. These methodologies can be divided into the following levels of increasing complexity: de-terministic, probabilistic and risk-based way. Con-cerning deterministic approaches, the most relevant works are presented by Frangopol and Curley,1987 (Frangopol & Curley, 1987), Biondini and Restelli, 2008 (Biondini & Restelli, 2008), Starossek and Haberland,2011 (Starossek & Haberland, 2011) and Cavaco,2013  (Cavaco, 2013). According to a prob-abilistic perspective, additional insights are present-ed by Frangopol and Curley,1987 (Frangopol & Curley, 1987), Fu and Frangopol,1990 (Fu & Fran-gopol, 1990) Lind,1995 (Lind, 1995) and Goshn and Moses,1998 (Ghosn & Moses, 1998). Lastly, a com-plex and comprehensive framework to assess ro-bustness based on risk is presented by (Baker et al.,2008). 3 PROPOSED FRAMEWORK Despite this intense effort of the research communi-ty, both structural reliability analysis and robustness assessment require a comprehensive understanding of crucial topics, hindering their practical application in real situations. Indeed, the most complete ap-proach, namely, the risk-based robustness, usually surpasses the structural engineers scope, since mod-elling system consequences is not a trivial task. In the remaining approaches, handling with more than one hazard is also very limited. Besides that, ranges of existing robustness indexes still need to be nor-malized from 0 (null) to 1 (full robustness), facilitat-ing comprehension and comparison.  In this sense, herein, a reliability-based robustness assessment framework is introduced, seeking to combine the merits of existing knowledge, in order to obtain a new robustness index to be applied at two performance levels: structural behavior at ultimate or service limit states.  The proposed robustness index aims to depict the structural performance by assessing a wise selection of four different attributes or performance indicators according to a predefined goal. In this approach, ro-bustness is computed as equal to the area of a quad-rilateral, whose sides' lengths represent a selected performance indicator. With the aim of assessing the structural performance under degradation phenome-na, four hazardous events are studied. Hence, for each scenario a probabilistic performance indicator, namely, comparison of reliability indexes corre-sponding to undamaged and damage situations is computed.     In the light of life-cycle perspective, those perfor-mance indicators are time-dependent, so that they can be computed to predict performance decay. This approach can easily achieve a quantitative measure to support road manager’s decisions regarding the al-location of resources for investments and mainte-nance, as well as the mitigation of the consequences. In fact, each performance indicator can be weighted according to a qualitative risk matrix, taking to ac-count indirect consequences of failure. With regard to structural reliability, since the ex-pected probability of failure is low, crude Monte Carlo requires a large number of numerical simula-tions in order to solve the convolution integral. To tackle this, the performance limit function can be approximated by the so-called meta-models, namely, quadratic response surfaces, polynomial chaos, and so on. Herein, quadratic response surfaces (RS), which are able to efficiently cope with highly non-linear relations between inputs and outputs, are used.  To do so, an adaptive procedure based on Monte Carlo realizations inspired on schemes proposed by Bucher and Bourgund (Bucher & Bourgund, 1990) and also Rajashekhar and Ellingwood (Rajashekhar & Ellingwood, 1993) is accomplished. In this ap-proach, a stepwise regression, which combines for-ward and backward regression methods to select the most important terms according to their statistical significance, is used to minimize the approximation error. This RS is built based on an initial experi-mental design (ED) referred to search domain, a Monte Carlo sample, whose realizations are dis-persed with a given parameter from mean value ac-cording to their bias.    Both design point coordinates and probability of failure are computed through the first reliability method (FORM). Regarding the following steps, new sampling points are added to enrich the ED around the design point. The procedure is stopped when a convergence criterion is satisfied, which is based on reliability index relative error tolerance be-tween consecutives iterations. In this procedure, the limit state function can be defined according to prob-lem definition. Herein, a performance limit function based on the difference of resisting and acting loads,      XSXRXG  , is accomplished. 4 CASE STUDY 4.1 Description of the structure The case study concerns a highway overpass ideal-ized as a three-span pre-stressed concrete continuous rigid-frame, whose spans’ length are 18, 27.8 and 18 m, respectively. This viaduct is composed by precast girder elements supported by the abutments and the piers. A longitudinal view of the bridge is shown in figure 1.  The deck’s cross-section comprises three precast I shape 1.5 m deep beams spaced by 3 m and cast-in-place 0.20 m thick and 8.9 m width reinforced con-crete slab. The piers are monolithically connected to the deck and fixed to footings. Both 15 m height piers have a rectangular cross section with 2,40 per 0,80 meters.   Figure 1 – Longitudinal view of the bridge  4.2 Probabilisitic model The probabilistic values for the mechanical proper-ties of the materials as well as the applied loads are presented in table 1. All the probabilistic parameters are represented by their mean values and coefficient of variation (CoV). A correlation for the material was considered too. It was considered correlation be-tween compressive strength and Young modulus (r=0,90), correlation between compressive strength and the tensile strength (r=0,70) and yielding strength and ultimate strength of pre-stressing steel (r=0,80). For the load scenario, the dead loads were divided into: (1) the initial dead loads before concreting cast-in-place slab, Gi; (2) additional dead loads after con-creting remaining parts, Ga. For traffic loads, load model 1(LM1), provided by Eurocode(CEN, 1991), was considered with: (1) uniformly distributed load (UDL),q; (2) tandem system including two axles (TS), Q.   4.3 Damage Scenarios In this section, idealized damage scenarios are for-mulated assuming a degradation phenomenon and soil instability at footings. Knowing that bridge pre-sents a critical cross section with low capacity of re-distributing bending moments, the main goal is to analyze the ability to cope with additional damage scenarios. In fact, although the structure presents a significant reserve capacity, it does not present a ductile rupture since there is no strands yielding. With this, considered scenarios were the following: (i) loss of cross section in the tendons; (ii) deteriora-tion of the precast concrete; (iii) soil settlement at pier due to lack of bearing capacity; (iv) combina-tion of the first and second damage scenarios. 4.4 Numerical model As mentioned, a non-linear analysis through DIANA software was carried out in order to assess the per-formance indicators of the bridges at ultimate limit state. Regarding the type of analysis, a 2D non-linear structural analysis was performed with class III beam elements based on Mindlin-Reissner theory with in-cremental load steps until failure. The adopted method to solve the non-linear problem was the Modified Newton-Raphson method, at the first itera-tion of each step. Concerning materials constitutive laws, a total strain fixed crack model was adopted, in which for tensile behavior a linear ultimate strain based was imple-mented and an ideal behavior for concrete compres-sion. Regarding the reinforcing steel and pre-stressing steel, a tri-linear diagram was adopted. In figure 2, a numerical model through DIANA software is presented.             Figure 2 – Numerical Model of the bridge 4.5 Obtained Results  Before performing a probabilistic analysis, in order to obtain the reliability index, firstly, a deterministic analysis was made. In figure 3, a load-deflection curve is presented. The mechanism of failure were a three hinge plastic formation with the yielding of the steel denoting a ductile behavior. The maximum load factor obtained was 5.60. For a probabilistic approach, a reliability index, for damaged and un-damaged situation, was obtained. In table 2, the ob-tained reliability indexes can be seen.                   Figure 3 – Load-deflection curve  Table 1 – Material Properties  Description Random Variable Notation Distr. Type Mean Value Coefficient of Variance  Cast in-situ  C30/37 Concrete compres-sive strength fc Normal 30 MPa 12% Concrete tensile strength fct Lognormal 2.90 MPa 20% Concrete elasticity modulus Ec Normal 33 GPa 8% Precast girder C45/55 Concrete compres-sive strength fc Normal 45 MPa 9% Concrete tensile strength fct Lognormal 3,70 MPa 20% Concrete elasticity modulus Ec Normal 36 GPa 8% A500 Steel yielding strength fsy Normal 560 MPa 5.35% Steel ultimate strength fsu Normal 650 MPa 6.0% Steel ultimate strain εsu Normal 10,11% 15% A1670/1860 Strands yielding strength fpy Normal 1737 MPa 2.5% Strands ultimate strength fpu Normal 1935 MPa 2.5% Strands ultimate strain εpu Normal 5.00% 8% Pre-stress Force Prestressing Stress (span 1) P1 Normal 1046 MPa 1.5% Prestressing Stress (span 1) P2 Normal 1087 MPa 1.5% Prestressing Stress (span 3) P3 Normal 1046 MPa 1.5% Loads Self-weight Gi Normal 25,75kN/m3 8% Additional dead loads Ga Normal 26,25kN/m3 10% Traffic loads Q Gumbel 500kN (TS) 10% 37,75kN/m(UDL)         Table 2- Obtained reliability indexes for the considered scenarios Intact  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Analysing the obtained reliability indexes, it can be seen that for an intact structure the obtained value is extremely high. This is due to its redundancy, the large amount of steel that the bridge is composed and by the fact that the pre-stress steel commands the structure at ultimate limit state. This can be clearly seen when analyzing the result obtained for the first scenario. With a loss of cross section in pre-stressing steel, there is a considerable decrease of the obtained reliability index. In the other way round, for the scenario 2, deterioration of concrete, the influ-ence is almost null. The same happens for scenario 3, the pier settlement, once this is an imposed dis-placement being the effect on ultimate limit state analysis is almost null. The scenario 4 presents the lower value of the reliability index once it is a con-jugation of scenarios 1 and 2. 5 CONCLUSION  A reliability-based robustness assessment framework to evaluate bridge’s safety is introduced. The main point of the procedure is to facilitate the understand-ing regarding the vulnerability of different hazardous events.   Herein, the performance of a reinforced concrete bridge under several degradation phenomena is stud-ied. Indeed, this paper presents some preliminary studies concerning reliability analysis. The main goal is to facilitate the understanding of some attrib-utes regarding robustness, aiming to propose a versa-tile framework to evaluate robustness according to a choice of key performance indicators. Regarding re-liability analysis, used approach intends to reduce computational time and also to reproduce an explicit limit state function avoiding overfitting and dimin-ishing approximation error. In fact, this methodology can be improved by introducing some features: i) use of pseudo random-generators to populate region of failure; ii) establishing cross-validation procedures; iii) considering model error as random variable; iv) bootstrap sampling to estimate boundaries of proba-bility of failure. Finally, the application of these framework with additional improvements is to be applied in a near future.   6 ACKNOWLEDGMENTS The authors would like to thank ISISE – Institute for Sustainability and Innovation in Structural Engineer-ing (PEst-C/ECI/UI4029/2011 FCOM-01-0124-FEDER-022681), FCT– Portuguese Scientific Foun-dation for the research grant PD/BD/113677/2015. This study also received funding from the European Union’s Seventh Framework Programme for re-search, technological development and demonstra-tion under grant agreement No. 606229. Also the collaboration and information provided by the pro-fessor António Abel Henriques is gratefully acknowledged.   REFERENCES Uwe Starossek and Marco Haberland. Disproportionate collapse: terminology and procedures. Journal of Performance of Constructed Facilities, 24(6):2010. Dan M Frangopol and James P Curley. Effects of damage and redundancy on structural reliability. Journal of Structural Engineering, 113(7):1987. Uwe Starossek and Marco Haberland. Approaches to measures of structural robustness. Structure and Infrastructure Engineering, 7(7-8):2011. Eduardo Cavaco. Robustness of corroded bridges, PhD thesis, Universidade Nova de Lisboa,. 2013. Gongkang Fu and Dan M Frangopol. Balancing weight, system reliability and redundancy in a multiobjective optimization framework. Structural Safety, 7(2):1990. Niels C Lind. A measure of vulnerability and damage tolerance. Reliability Engineering & System Safety, 48(1):1995. Michel Ghosn and Fred Moses. Redundancy in highway bridge superstructures. 1998. CG Bucher and U Bourgund. A fast and efficient response surface approach for structural reliability problems. Structural safety, 7(1):1990. Malur R Rajashekhar and Bruce R Ellingwood. A new look at the response surface approach for reliability analysis. Structural safety, 12(3):1993. CEN. Eurocode 1: Actions on structures, Part 2: Traffic loads on bridges. Brussels: European Standard 1991.             

Robustness of an existing pre-stressed concrete bridge

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Robustness of an existing pre-stressed concrete bridge

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