MEZNÍ STAVY TRVANLIVOSTI MODELOVÁNÍ A ROZMĚR ČASU LIMIT STATES FOR DURABILITY DESIGN MODELLING AND THE TIME FORMAT Dita Matesová 1, Florentina Pernica 2, Břetislav Teplý 3 Abstract Durability limit states (DLS) are recognised as a new category of limit states (LS) by some new international documents which are under development recently (fib Model Code, ISO). First, the basis of the structural design according to DLS is described showing the service life format and the limit state format. Second, the alternatives of simplified LS for RC structures are discussed in more details. Some software tolls are presented briefly as well as some illustrative examples of application. 1 Introduction Limit states (LS) are the basic approach utilized for structural design all over the world in the last several decades see also the current codes of practice, e.g. [1, 2]. The Ultimate Limit State (ULS) and the Serviceability Limit State (SLS) are traditionally distinguished. The general condition for the probability of failure P f reads: Pf = PA ( B) < Pd (1) where A = action effect, B = barrier and P d = the design (acceptable) probability value. Generally, both A and B (and hence the P f ) are time dependent, which has not been considered for common cases of ULS or SLS in praxis very frequently. Two exceptions have to be mentioned: (i) the fatigue limit states in this case (presumably ULS) the time domain is usually transformed into a number of load cycles; (ii) the creep and shrinkage effects time dependent special issues that concern the concrete structures mostly and do not usually effect ULS. Let us note that the reliability approach covers safety (ULS), serviceability (SLS) and durability see [1]. Durability is related to the design working life. In the context of performance-based approaches, sustainability consideration and the whole life costing the time is the decisive variable and the durability issues are pronounced. These facts have gained a considerable attention recently which is reflected also in standardisation activities: the ISO/WD 13823 [3] and fib-model Code 2005 [4]. Both these documents are based on probabilistic approaches, are under the development by the international bodies currently and will introduce (or enhance) the design of structures for durability i.e. time-dependent LS approach and service live consideration. The utilization of design for durability may bring pronounced economical and sustainability impacts. A broad application is, unfortunately, prevented still by insufficient dissemination of basic ideas, relevant knowledge or experimental evidence 1 Ing. Dita Matesová, Brno University of Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, Veveří 95, 602 00 Brno, matesova.d@fce.vutbr.cz 2 Ing. Florentina Pernica, Brno University of Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, Veveří 95, 602 00 Brno, florentina.pernica@dlh.de 3 Prof. Ing. Břetislav Teplý, CSc., Brno University of Technology, Faculty of Civil Engineering, Institute of Chemistry, Žižkova 17, 602 00 Brno, teply.b@fce.vutbr.cz 1
and by a lack of simple, user friendly and efficient design instruments (software and others). The prescriptive approach of current standards (Eurocodes) does not allow directly a design focused on a specific service life and/or a specific level of reliability which would require the dealing with the inherent uncertainties in material and technological and environmental characteristics while assessing the service life of a structure. Such tasks necessarily require the utilization of stochastic approaches, analytical models of degradation effects and also simulation techniques, all based on an experimental evidence and relevant observations of structures in real conditions. The goal of the present paper is twofold: (i) to show the basic formulas for DLS and discuss the durability design/assessment of RC structures; (ii) to present briefly some practical software tools which may serve for: - the durability design of RC structures; - helping a client to specify or satisfy the serviceability criteria; - differentiation of reliability. 2 Time dependent limit states Two safety formats for service life design (or durability design) may be considered: Service life format and Limit state format [4]. The service life format consists of considering a design service life t D of the component or structure and assessing the predicted service life t PS which shall meet or exceed the design life, i.e.: tps td (2) Within the probability framework: the probability of failure P f is determined and compared to a specified design (target, acceptable) probability P d : { ( ) } P ( t ) = P t X, t t P (3) f D PS i D d t PS is predicted value modelled as a function of basic variables X i (i = 1, 2,..., n) and time t; n is the number of input parameters involved in the modell in question. In the case where a structural component is protected against degrading agents (e.g. a concrete cover of reinforcement, a coating of steel or others), t PS can be determined as a sum of two service-life predictors: tps = tstart + texp (4) where t start is the time to the initiation of degradation and t exp is the service life after the initiation of degradation (often called propagation period). Basically, the variables X i are random variables (or random fields; some of them might be treated as deterministic values in certain situations). The basic requirement of limit state format for the ULS during the design life t D is: ( ) S( t ) R t D (5) D where R(t D ) is the resistance capacity of the structural component at the design life t D and represents the barrier B. S(t D ) represents a cumulative degradation of the component at the design life t D or in other words an action A of environmental load. Both, R(t) and S(t) may be modeled as a function of basic variables X i and t. The LS condition in Equation (5) is ensured by checking that: 2
{ ( ) ( ) } P ( t ) = P R t S t 0 P (6) f D D D d Either the resistance R or the load action S or both can be time-dependent quantities. Thus the failure probability is also a time dependent quantity. The basic requirement for the LS during the design life of the component t D is: ( D ) Slim S t (7) where S lim represents the limit value. The Equation (7) is again ensured by checking the condition (6) and interchanging R(t D ) S lim. In some design strategies it might be advisable to use a conservatively defined limit state by simplifying the Equation (4) into the form: t PS = t (8) start This kind of limit state precedes the occurrence of other SLS or ULS and represents a simplified limit state (SiLS) intending a prevention of deterioration to start; such LS are based on the initiation of deterioration and are incorporated in both [3] and [4]. Some examples are mentioned in the following paragraph. Note: the assessment of t PS, R(t D ), S(t D ) and P f (t D ) requires the utilization of suitable and well verified models of degradation processes. 3 Simplified limit states Some examples of relevant LS equations follow: Concrete structures When considering the durability of reinforced concrete structures, the corrosion of reinforcement is the dominating effect. In the context of corrosion the following limit states can be recognized: (i) depassivation of reinforcement due to carbonation or chloride ion penetration and hence the possible reinforcement corrosion initiation. By using of the relevant limit state conditions (6) and considering the Equations (7, 8) follow: Pf( td) = P{ a xc( td) 0} Pd (9) or P ( t ) = P C C t 0 P (10) { ( ) } f D cr a D d where a is the thickness of concrete cover, x c is the depth of carbonated zone (measured from the concrete surface), C cr is the critical chloride concentration and C a the chloride concentration at the depth a. The formula (9) expresses simply the limiting condition or time when the carbonation front reaches the reinforcement surface and the reinforcement is depassivated and Equation (10) describes the condition of reinforcement depassivation due to chloride ions ingress. Note: a and C cr stand for B and x c or C a for A. (ii) Cracking due to rust products expansion and consequent stress occurrence in the conrete surrounding the rebar. The limit state condition may read: Pf( td) = P{ r ra( td) 0} Pd (11) or P ( t ) = P w w t 0 P (12) { ( ) } f D cr a D d 3
where r = maximal corrosion inducing an increase of rebar radius which can be accommodated, r a = an increase of rebar radius due to corrosion, w cr = crack width limit and w a = actual crack width at time t D. (iii) Delamination of concrete cover. Limit condition may be similar to (11) or (12) considering different limiting values. (iv) Decrease of the effective reinforcement area - leading to an excessive deformation, loss of bearing capacity and finally to collapse. States (i) and (ii) belong to the SiLS category of limit states, (iii) might fall into one of all categories depending on the location and level of degradation and (iv) lies within ULS (load bearing capacity) or SLS (deformation capacity) categories. For SiLS also the area or % of affected part of the structure/member might be decisive for description of the acceptable limit of deterioration. Note: The probabilities (6) and (9 12) describe events which could occur at a time t D - on the contrary to the probability of an event during the time period 0<t<t D. The former type is used for the sake of simplicity and is based on the presumption of degradation being continuous and monotonous processes. Steel structures (i) degradation of coating due to environmental or operational effects a conservative limit state, equation (8) SiLS or (ii) decrease of the effective cross-sectional area leading to an excessive deformation, loss of bearing capacity and finally to collapse SLS or ULS. 4 Software tools 4.1 FreetD FreetD is an associated product of the multipurpose probabilistic software for statistical, sensitivity and reliability analysis of engineering problems Freet (Feasible Reliability Engineering Tool) [8], is based on efficient reliability techniques and has been developed by the team of authors [9]. Freet can be utilized in two modes: as a stand-alone multipurpose program for any user-defined problem and as a module integrated with software ATENA by Červenka Consulting. In this fold we are going to refer to the first mode of utilizaton. The tools of the present version of Freet are divided into three parts: stochastic model including statistical correlation, sampling, and assessment. For generation of realizations from random variables the Monte Carlo method and Latin Hypercube Sampling Metod [10] are implemented. As a result following parameters can be obtained from Freet: statistical characteristics of a resulting parameter, sensitivity analysis and estimation of reliability index β and theoretical failure probability P f. The module FreetD has been developed by implementing of a number of degradation models for reinforced concrete structures. Three different categories of degradation models are included: models for carbonation, chloride ingress and reinforcement corrosion. Implemented degradation models may serve directly for durability assessment of structures (in the form of SiLS mostly) like the assessment of service life and the level of reliability measure. The user may create different limit conditions, e.g.: carbonation depth = concrete cover thickness cf. Equations (9). The examples of FreetD module are presented in Figs. 1 and 2. 4
Fig. 1: FreetD input of parameters to Fib model [4] for estimation of chloride concentration in concrete in time. Fig. 2: FreetD result of stochastic computation by using of inputs in Fig. 1. 5
4.2 RC-LifeTime Based on the condition (9), using the model for carbonation process, considering the input data as random variables and performing the stochastic simulation technique; a web page RC-LifeTime, freely accessible on http://rc-lifetime.stm.fce.vutbr.cz/ has been recently introduced by the authors [5] and its effectiveness shown e.g. in [7]. The RC-LifeTime offers two following options for reinforced concrete structures assessment: (i) Service Life Assessment provides the evaluation of service life t PS and its statistical characteristics based on the equality condition of carbonation depth and concrete cover. Optionally, the service life corresponding to the target value of reliability index β is provided. (ii) Concrete Cover Assessment provides the statistical evaluation of a designed concrete cover from the point of carbonation process. The reliability index β relevant to required concrete cover may be gained. Included analytical model for concretes from OPC originally developed by Papadakis et al. [6] (see also the Introduction part in RC-LifeTime) in its deterministic form has been enhanced in two points: a) by incorporating a new function for the effect of humidity; b) by randomization of the modelled process. Figs. 3 and 4 show illustrative examples of inputs and numerical outputs for option (ii) of RC-LifeTime. From the table on Fig. 4 is evident that the case with the resulting reliability level of β 1.5 is satisfied up to service life of 35 years approximately. 5 Conclusions The consideration of time format in limit states is presented in this paper with special focus on simplified durability limit states. Software tools for concrete structure design or assessment are briefly introduced. However, several issues deserve an extra consideration (and dissemination!): (i) following the idea of live cycle costing (LCC) the design service life has to be determined (or agreed) by the client; (ii) consequently the appropriate limit states and relevant reliability level (in the form of the reliability index β) have to be determined/satisfied. 6
Fig. 3: Input data Fig. 4.: Output data (shown only in the numerical form). 7
Acknowledgement This outcome has been achieved with the financial support of the Ministry of Education, Youth and Sports, project No. 1M680470001, within activities of the CIDEAS research centre. References [1] EN 1990. BASIS OF STRUCTURAL DESIGN (EUROPEAN STANDARD, 2002) OR ČSN EN 1990 [2] EN 1992-1-1. DESIGN OF CONCRETE STRUCTURES PART 1.1: GENERAL RULES AND RULES FOR BUILDINGS ( EUROCODE 2, 2003) [3] ISO/WD 13823. GENERAL PRINCIPLES ON THE DESIGN OF STRUCTURES FOR DURABILITY (CURRENTLY UNDER DEVELOPMENT), ISO TC 98/SC2/WG10 [4] 2005FIB TG 5.6. FIB MODEL CODE FOR SERVICE LIFE DESIGN (DRAFT) PROPOSAL FOR FUTURE FIB MODEL CODE, 2005 [5] Keršner, Z., Rovnaníková, P., Teplý, B. and Novák, D. 2004. DESIGN FOR DURABILITY: AN INTERACTIVE TOOL FOR RC STRUCTURES, PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON LIFE CYCLE ASSESSMENT, BEHAVIOUR AND PROPERTIES OF CONCRETE STRUCTURES LC 2004 (BRNO, CZECH REPUBLIC, 2004) 172-182 [6] Papadakis, V.G., Fardis, M.N. and Vayenas, C.G. EFFECT OF COMPOSITION, ENVIRONMENTAL FACTORS AND CEMENT-LIME MORTAR COATING ON CONCRETE CARBONATION, MATERIALS AND STRUCTURES 25 (149) (1992), 293 304 [7] Teplý, B., Keršner, Z., Rovnaník, P. and Chromá, M. DURABILITY VS. RELIABILITY OF RC STRUCTURES, PROCEEDINGS OF OF THE 10 TH INTERNATIONAL CONFERENCE ON DURABILITY OF BUILDING MATERIALS AND COMPONENTS 10DBMC (CD ROM, PAPER NO. TT4-42) (LYON, FRANCE, 2005) 6 PAGES [8] Novák, D., Vořechovský, M., Rusina, R. SMALL-SAMPLE PROBABILISTIC ASSESSMENT SOFTWARE FREET. PROCEEDINGS 9 TH INTERNATIONAL CONFERENCE ON APPLICATIONS OF STATISTICS AND PROBABILITY IN CIVIL ENGINEERING ICASP 9 (MILLPRESS ROTTERDAM, SAN FRANCISCO, USA, 2003) 91 96 [9] Novák, D., Vořechovský, M., Rusina, R. FREET V.1.3 PROGRAM DOCUMENTATION, USER S AND THEORY GUIDES, 2006 [10] Vořechovský, M., Novák, D. STATISTICAL CORRELATION IN STRATIFIED SAMPLING. IN 9TH INT. CONF. ON APPLICATIONS OF STATISTICS AND PROBABILITY IN CIVIL ENGINEERING ICASP 9, SAN FRANCISCO, USA, ROTTERDAM MILLPRESS, 2003, PP.119-124. 8