Probabilistic design and management of environmentally sustainable repair and rehabilitation of reinforced concrete structures

Cement & Concrete Composites 47 (2014) 19–31

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Probabilistic design and management of environmentally sustainable repair and rehabilitation of reinforced concrete structures Michael D. Lepech a,⇑, Mette Geiker b, Henrik Stang c a

Department of Civil and Environmental Engineering, Stanford University, Room 285B, Yang and Yamazaki Energy and Environment Building, Stanford, CA 94305-4020, USA Department of Structural Engineering, Norwegian University of Science and Technology, Trondheim, Norway c Department of Civil Engineering, Technical University of Denmark, Lyngby, Denmark b

a r t i c l e

i n f o

Article history: Received 15 October 2012 Received in revised form 27 June 2013 Accepted 9 October 2013 Available online 17 October 2013 Keywords: Sustainability design Life cycle assessment Service life Concrete repair Life cycle management

a b s t r a c t This paper presents a probabilistic sustainability design framework for the design of concrete repairs and rehabilitations intended to achieve targeted improvements in quantitative sustainability indicators. The framework consists of service life prediction models combining deterioration mechanisms with limit states and life cycle assessment models for measuring the impact of a repair or rehabilitation. Both types of models (service life or LCA) are formulated stochastically so that the time to repair and the accumulated sustainability impact are described by probability density functions. This leads to a probabilistic calculation of cumulative impacts throughout the structure’s service life, from initial repair to functional obsolescence (end of life). The methods discussed are in accordance with sustainability design requirements within the 2010 fib Model Code. A case study is presented which computes the probability that reinforced concrete repair strategies using thicker concrete cover will meet future greenhouse gas emission reduction targets proposed by the UN Intergovernmental Panel on Climate Change. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In 1992, the United Nations Framework Convention on Climate Change was adopted to ‘‘stabilize greenhouse gas concentrations at a level that would prevent dangerous anthropogenic interference with the climate’’ and scientists recommended capping atmospheric CO2 concentrations below 550 ppm [1]. For large societal systems, such as transportation and energy production, many strategies have been proposed to meet these goals in the next 50 years including widespread adoption of nuclear energy, drastic reductions in automobile fuel consumption, and geologic carbon sequestration [2]. Comprising a major part of these strategies, civil infrastructure built of reinforced concrete lies at the nexus of two major sustainability challenges; emissions from transportation and construction materials, in particular, cement production. Transportation comprises 30% of US CO2 emissions [3], while Portland cement production emits approximately 5% of global anthropogenic CO2 emissions [4]. Collectively, the design, operation, maintenance, rehabilitation, and retirement of reinforced concrete transportation infrastructure, and concrete structures at a whole, represent a large opportunity and challenge in realizing the United Nations’ vision of sustainable development.

⇑ Corresponding author. Tel.: +1 650 724 9459; fax: +1 650 723 7514. E-mail address: [email protected] (M.D. Lepech). 0958-9465/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cemconcomp.2013.10.009

Recognizing this opportunity, frameworks and guidelines have been introduced to aid transportation planners and structural engineers in the design and management of more sustainable infrastructure. It is helpful to classify these into three subsets as proposed by Gowri [5]; (1) knowledge-based methods, (2) rating schema, and (3) performance-based tools. It is also useful to distinguish among approaches that are regional, local, or project specific. Knowledge-based tools are manuals, guidelines, and deemed-to-satisfy design recommendations including, for example, Green Playbook [6] in the US, and the Green Infrastructure Planning Guide [7] in the EU. These approaches can span from regional guidelines for greenspace management to project-specific design recommendations. The second subset, rating schema, includes design checklists, frameworks, and calculators used to quantify an infrastructure’s sustainability profile. There are a number of infrastructure rating systems currently in use including CEEQUAL [8] in the UK, Greenroads [9] in Washington State in the US, I-LAST [10] in Illinois State in the US, GreenLITES [11] in New York State in the US, the Australian Green Infrastructure Council Rating Tool [12], LEED for Neighborhood Development (ND) [13] in the US, BREEAM Communities [14] in the UK, and CASBEE Urban Development (UD) [15] in Japan. Similar to guidelines and recommendations, these range from regional to project-specific in scope. The third subset, performance-based tools, include life cycle assessment methods and material flow analyses, along with performance simulation tools for calculating total energy

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(fuel) consumption and environmental impacts associated with infrastructure systems. These would include LCA software tools such as SimaPro [16] and GaBi [17], along with governance and research models such as those developed by Ramaswami et al. [18,19] (regional scope), Kendall et al. [20] (project scope), and Zhang et al. [21–24] (project and local scope). In addition to sustainability-focused tools, infrastructure management tools focus on life cycle cost management of infrastructure systems. Such methods include structural reliability-based methods as proposed by Frangopol et al. [25] and Markovian-based decision processes as used by PONTIS [26] and BRIDGIT [27] in the United States and the Finnra Bridge Management System in Finland [28]. A comprehensive history and review of highway management systems is provided by Markow [29]. Challenges exist when implementing each subset of sustainable infrastructure design tools. In the case of knowledge-based tools, sustainable development becomes defined by the criteria used to recognize it; e.g., ‘‘limited construction material transportation distance’’ or ‘‘purchase of renewable energy for construction site use’’ [30]. These tools use a criteria definition for sustainability of infrastructure. These are not, however, formal logic definitions that are based on ecosystem carrying capacity. Thus, the problem with these concepts of sustainable infrastructure is the fundamental ex post facto nature of sustainability, i.e., today’s developments can only be judged as sustainable from far in the future with little evidence of causality if appropriate environmental limits are not used. Having sustainability framed in such long, ambiguous time frames using ambiguous metrics, there is little incentive for infrastructure owners, designers, managers, and users to proactively participate in the design, use, and management of more sustainable infrastructure. In the case of rating schema and checklists this criteria definition of sustainability (e.g. a higher rating is equivalent to sustainability) remains. Due to the discrete nature of rating schema they also limit optimization of resources and effective managerial tradeoff decisions between sustainability goals and other project objectives such as project budget. The discrete (non-continuous) nature of these schemes also prevents calibration with comprehensive sustainability assessment methods [31]. Responding to these challenges, the use of performance-based tools that incorporate continuous (non-discrete) impact variables (subset three within the Gowri taxonomy) for the repair and rehabilitation of civil infrastructure is a preferred solution. Moreover, these tools incorporate environmental, social, and economic impacts over the full infrastructure life cycle from extraction of raw construction materials to end of life. To overcome the criteria definition of sustainability and allow for tradeoffs between sustainability and other multi-objective goals, comprehensive quantitative sustainability metrics for measurement and comparison of infrastructure repair and rehabilitation designs should be adopted. These can then be coupled with probabilistic approaches that are translatable to rational design procedures that recognize uncertainty in infrastructure design, construction, and operation. Such probabilistic approaches are the hallmark of current civil engineering design theories (e.g. AISC-LRFD in the United States [32], ACI318 in the United States [33], Eurocode 2 in Europe [34], etc.) and should remain central in sustainability design. Inherent within probabilistic methods are limit state functions that relate actual performance to acceptable or targeted performance [35]. Thus far, such notions of limit state design have been limited in sustainable infrastructure initiatives across knowledgebased, rating-based, and performance-based methods. The lack of limit state analysis in sustainability design is partly due to the perceived similarity between life cycle economic cost (LCC) methods and life cycle economic, social, and environmental sustainability assessment (LCA) methods. In the case of infrastructure, design

and management methods that minimize economic costs while maintaining minimum structural safety requirements (i.e. PONTIS, BRIDGIT, etc.) prima facie meet ‘‘economic sustainability’’ goals by minimizing economic cost or maximizing economic value. As such, these methods represent the best possible economic outcome and are thus carried out until replaced by better methods or technologies in an ongoing cycle of innovation and economy efficiency improvement. Unlike economic costs, minimization of environmental impacts, as measured through environmental midpoint indicators, may not ultimately prove sustainable. Different from economic systems, natural ecosystems have a finite carrying capacity as expressed by Arrow et al. [36], Daily and Ehrlich [37], and others. Therefore, environmental impacts associated with infrastructure design and management strategies that are environmentally sustainable must comply with local, regional, and global natural ecosystem carrying capacities. While minimization of environmental impacts of infrastructure moves closer toward environmental sustainability of infrastructure systems, it does not necessarily meet ecosystem carrying capacity constraints. Only when evaluated against a sciencebased limit state can efforts of environmental impact reduction be assessed as environmentally sustainable. Similar arguments could apply to meeting fundamental social metrics of sustainability that include equity, literacy rates, etc. Responding to the different design approach needed when addressing environmental sustainability rather than economic cost minimization, the Federation International du Beton (fib) incorporated an environmental design and construction requirement into the 2010 fib Model Code [38]. Specifically, Section 3.4 of the code ‘‘Design Principles for Sustainability’’ requires concrete designers, engineers, and contractors to account for environmental impacts, social, impacts, and aesthetics through the application of life cycle concepts that begin in the design phase and continue until the final removal or reuse of concrete infrastructure. Environmental impacts suggested for consideration include urban air pollution, hazardous substances, global warming potential, waste material production, and resource consumption. While Section 3.4 introduces the concept of designing for sustainability, Section 7.10 of the fib 2010 Model Code attempts to implement it within a probabilistic framework. Within that framework the designer must first set the environmental performance requirements, R, of the infrastructure system. Second, the environmental impact performance, S, must be measured using ISO 14040-series compliant life cycle assessment methods. Finally, the designer should evaluate the performance by testing the limit state whether R (environmental performance requirement) is greater than S (environmental performance). Unfortunately for designers, no tools, methods, or guidelines are provided to set appropriate environmental performance requirements, measure environmental impact performance, or evaluate the limit state function. This paper presents a probabilistic sustainability design framework for the design of concrete repairs and rehabilitations intended to achieve targeted improvements in quantitative environmental sustainability indicators (hereto referred to as ‘‘the sustainability design framework’’). A combined environmental, societal and economic assessment is not described, but should be allowed for when used by designers and decision makers. The sustainability design framework is in line with fib 2010 Model Code recommendations using probabilistic-based approaches that allow for rational decision-making among specific concrete design alternatives based on their economic costs, the likelihood that they will meet internationally accepted environmental performance requirements, and the collective risk borne by not meeting future emission reduction and environmental performance targets. Additionally, it addresses the fundamentally problematic ‘‘criteria definition of sustainability’’ present within guidelines and rating

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schema and allows for tradeoffs among various objectives in design, use, and maintenance of a complex, long-lasting reinforced concrete infrastructure component.

2. Sustainability design and management framework The use of probabilistic methods for the design of infrastructure system components, including bridges, tunnels, and viaducts is well documented. Beginning with the work of ICE and ASCE committees in 1955 and 1956, respectively [39,40], along with seminal work by Freudenthal [41] the use of reliability methods to design for acceptable structural safety is a central concept of modern structural engineering. Likewise, the probabilistic characterization of resistance and load, compared using a limit state function, serves as the basis of this sustainability design framework. By adopting probabilistic methods, this sustainability design framework promotes design for sustainability as a rigorous, quantifiable effort, on par with design for structural reliability. Much of the sustainability design framework is based on the probabilistic design for service life proposed by the DuraCrete project [42], the 2006 fib Model Code for Service Life Design of Reinforced Concrete [43], and the 2010 fib Model Code [38]. Within the fib Model Codes, a probabilistic approach for durability design is taken in which environmental loads are characterized using probabilistic distributions, as is the time dependent resistance of the structure. This time dependent resistance is a result of concrete deterioration mechanisms including carbonation-induced corrosion, chloride-induced corrosion, and freeze-thaw attack. End of life is defined by the probability of load exceeding resistance reaching an unacceptable level. The sustainability design framework adopts a similar probabilistic approach to the design of repairs and rehabilitations of sustainable concrete infrastructure. When considering repair and rehabilitations, evaluation begins with measurement of the cumulative impact of the repair and rehabilitation timeline from the first repair event up to the time of functional obsolescence. This is shown in Fig. 1. Cumulative impact is expressed as midpoint environmental indicators such as global warming potential (kg CO2 equivalents), polluted water produced (L), solid waste generated (kg), or total primary energy consumed (MJ), along with social indicators such as public health statistics. Environmental indicators of sustainability will be the focus of this paper. Captured within the vertical segments of the cumulative impact in Fig. 1, the time at which any repair is made (trj) is probabilistically characterized based on reaching a service life limit state

Fig. 1. Probabilistic envelope of cumulative impact of infrastructure from initial repair (tr1) to functional obsolescence (tfo).

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corresponding to a reduction in materials quality or structural performance beyond which is unacceptable to the owner or not conforming with the appropriate structural design code. The probabilistic time between repairs (trj+1  trj) is based on the chosen repair strategy, the quality of the repair work, the variable nature of exposure and load conditions, limit state, th presence of multiple deterioration mechanisms, etc. The stochastic time to repair for each repair is shown as a Gaussian-type distribution for illustrative purposes. The cumulative impact of the repair and rehabilitation timeline is defined as

I¼

X irj ðtrj Þ

ð1Þ

j

where I is the cumulative impact and irj(trj) is the impact due to the jth repair event taking place at time trj, as measured using ISO 14040 series life cycle assessment methods [44]. Metrics of environmental impact are based on standardized environmental impact assessment midpoint indicators within the United States’ TRACI [45] or the Netherland’s ReCiPe [46] protocols. Within such protocols, environmental impacts can include climate change, acidification, land use, energy consumption, and toxicity indicators. Impacts accrued due to a given repair event can vary as a function of time. Examples of time-variant impacts would be the use of cathodic protection which consumes energy resources following installation, or repairs which cause changes in use patterns such as bridge deck overlays which increase surface roughness and decrease vehicle fuel economy. In addition to the probabilistic determination of the time of future repairs and maintenance, the amount of impact associated with each repair is also probabilistic in nature. This is also shown in Fig. 1. The impact associated with a given repair, irj, can vary due to uncertainty in repair and rehabilitation construction processes eventually used on site, uncertainty in the supply chain of materials, uncertainty in the effects on infrastructure users (e.g. how many automobiles are disrupted by construction activities), etc. This uncertainty is modeled stochastically, analogous to that for repair timeline predictions. Once again, stochastic impact for each repair is shown as a Gaussian-type distribution for illustrative purposes. Combining the probabilistic models for both the life cycle repair timeline (trj) and the amount of impact (irj), a probabilistic envelope can be constructed for the entire infrastructure service life from the time of the first repair (tr1) up to the time of functional obsolescence (tfo). This envelope can be constructed by plotting numerous stochastic sequences of possible times of repair (trj) for a structure against the corresponding stochastic cumulative impact (irj) for each repair in the sequence. The resulting plot provides a field envelope of possible life cycle repair timelines with corresponding accrued environmental impacts. Based on the boundaries of this envelope (shown in Fig. 1 using dashes), an aggregated probabilistic assessment for cumulative impact at any time, t, for the concrete infrastructure can be constructed. The distribution of this aggregated envelope is shown schematically in Fig. 1 as a Gaussian-type distribution for the time at functional obsolescence, tfo. The probabilistic construction of an impact envelope meets the fib 2010 Model Code requirement for probabilistically establishing the environmental sustainability performance of the structure over its life cycle, S. However, identification of an environmental sustainability performance requirement, R, and a check of the limit state of ‘‘sustainability’’ has not been completed. This evaluation of sustainability is addressed through the use of science-based ecosystem carrying capacities. The definition of ‘‘sustainability’’ can be very broad and highly subjective. For infrastructure and other forms of built environment,

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the Brundtland Commission definition is often adopted which states, ‘‘Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs’’ [47]. While this definition can be lauded for its comprehensive and far-reaching sustainability goals, this definition serves limited use as an operational design guideline, target, or sustainability limit state. Therefore, the proposed sustainability design framework adopts the concept of triple-bottom-line sustainability measurement comprising social, environmental, and economic sustainability metrics. Within this paper, it focuses on environmental sustainability through the reduction of environmental impact midpoint indicators such as global warming potential (CO2 equivalents), acidification potential (H+ mol equivalents), and similar indicators as specified by ISO 14040 series life cycle impact assessment standards. Along with the definition of metrics, appropriate sustainability limit states must be set. As mentioned previously, a simple reduction of environmental impact fails to recognize requirements of ecosystem carrying capacity, therefore a more comprehensive sustainability limit state is needed. Powell et al. note the difficulty in determining such limit states and avoiding criteria definitions of sustainability [48]. To insulate infrastructure designers from subjective decisions, limit states for meeting broadly defined sustainable development are drawn from science-based targets such as those proposed by the Intergovernmental Panel on Climate Change (IPCC) for ongoing reductions in global greenhouse gas emissions [49]. In this regard, the decision of an appropriate sustainability target is left to ecologists, climate scientists, and policymakers. The measurement of impacts and design for impact reduction is left to infrastructure engineers and planners. Environmental sustainability targets are often expressed as a reduction from baseline or business as usual (BAU) emissions. To achieve reductions in environmental impact midpoint indictors in line with such targets, an alternative repair and rehabilitation scenario can be designed to meet reductions as compared to a status quo design. The comparison of the two proposed scenarios (status quo and a more sustainable alternative) is shown in Fig. 2. Based on this, the level of impact reduction using an alternative repair timeline versus the status quo repair timeline can be quantified at any time in the future and associated with a given level of confidence for actually realizing that cumulative reduction. Such reductions could be annualized to meet annual reduction targets

in the future. The probability of failing to meet a reduction goal by implementing the alternative design viewed as the overlap between these two envelopes and the failure probability, Pf(t), over the life cycle is shown in Fig. 2. This probability of failure would be computed similar to that shown in Eq. (2).

Pf ¼ P

  Iold ðt G Þ  Inew ðt G Þ  GðtG Þ  0 Iold ðtG Þ

ð2Þ

where Pf is the probability of not meeting the target reduction in environmental midpoint indicator, Iold(tG) is the cumulative impact of the status quo construction/repair strategy, Inew(tG) is the cumulative impact of the alternative construction/repair strategy, G is the target (or goal) reduction in environmental midpoint indicators recommended by policy, and tG is the time in the future at which the goal reduction should be achieved. The development of probabilistic envelopes for the comparison of cumulative sustainability impacts requires the integration of a number of models. The sustainability design framework and the integration of applicable models for probabilistic design of sustainable concrete infrastructure is shown in Fig. 3(a and b). Fig. 3(a) shows the general model integration that couples service life modeling with life cycle assessment to evaluate and compare the sustainability of alternative repair and maintenance designs. Fig. 3(b) details this by showing the connections between the service life model, life cycle inventory model, life cycle impact assessment model, and overall life cycle assessment. The service life model begins with definition of the material and structural state, n, and implements a material deterioration model subject to a set of environmental loads, to arrive at a new material and structural state, n + 1. This new state is compared to material or structural limit state models to determine if a repair needs to be designed and construction. If no repair is needed, the service life model is run iteratively until the limit state function fails. If a repair is needed, an appropriate repair is designed and constructed. The associated life cycle inventory models are then used to determine the impact of this construction. The time at which this repair construction is implemented is also input into the life cycle assessment model. From these Fig. 3(a and b), the dual nature of the sustainability design framework is clear (modeling of material and structural condition combined with modeling of life cycle performance). Also clear is the highly connected nature of these two fields, such that individual design, construction, and management decisions heavily influence many parts of the life cycle model. Ultimately, the use of a comprehensive life cycle assessment model can be used to effectively guide the design, construction, and management of infrastructure repair and rehabilitation through an iterative approach. Such an approach would begin with proposing potentially sustainable repair scenarios, followed by analytical modeling of future cumulative impacts using predictive service life models and consequential life cycle assessments, and ending with iterative improvement of proposed repair scenarios until the likelihood of the cumulative environmental impact meeting sustainability targets is acceptable to the designer.

3. Modeling approaches

Fig. 2. Probabilistic envelopes for cumulative impact from first repair to functional obsolescence for status quo (black) and alternative (gray) sustainable concrete infrastructure designs. Failure probability of not meeting reduction targets (Pf) is shown as a function of time.

Many of the models shown in Fig. 3 currently exist and only require integration within a probabilistic framework to implement this design approach. As such, the framework is constructed using a ‘‘plug-and-play’’ structure in which designers can select among many probabilistic service life models or probabilistic life cycle environmental impact approaches for implementation.

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Fig. 3. (a) General and (b) detailed sustainability design framework and model integration for life cycle assessment and management of repaired sustainable concrete infrastructure.

3.1. Probabilistic material deterioration models and limit states Material deterioration models serve as the basis for prediction of structural service life. A number of material deterioration models have been proposed for reinforced concrete structures. These models can be empirical or theoretical in nature and include deterioration mechanisms of chloride-induced corrosion of reinforcing steel, carbonation-induced corrosion of reinforcing steel, and freeze-thaw damage of concrete. Not to be considered exhaustive, this list could include other deterioration models and phenomena. Moreover, the models that are included in this discussion are only intended to serve as practical examples taken from the literature (e.g. 2010 fib Model Code) and do not necessarily reflect either the state of the art or the preferred modeling approach for a given structure or environmental exposure. Chloride-induced corrosion of reinforcing steel is one of the most common causes of reinforced concrete material degradation.

Numerous models have been proposed to capture the multi-stage process of chloride ion transport, depassivation, and active corrosion of reinforcing steel as proposed by Tutti [50]. Often, the first stage of deterioration (chloride ion transport through concrete cover) is modeled using Fick’s Second Law of Diffusion shown in the following equation. 2

dCðx; tÞ d Cðx; tÞ ¼ Dc 2 dt dx

ð3Þ

where C(x, t) is the chloride ion concentration at distance, x from concrete surface after time, t, and Dc is the effective chloride diffusion coefficient of the concrete. Solving this fundamental transport equation, the 2006 fib Model Code for Service Life Design of Reinforced Concrete probabilistically defined time to chloride-induced depassivation of reinforced concrete as Eq. (4) [43]

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2 C crit

3

6 7 d  Dx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi7 ¼ C 0 þ ðC S;Dx  C 0 Þ6   41  erf r 5   n be T 1  T 1 DRMC;0 k tt0 t 2 ref

real

ð4Þ where Ccrit is the critical chloride concentration for depassivation in weight % of cement, C0 is the initial chloride content of the concrete in weight % of cement, Cs,Dx is the chloride content at a depth Dx at a certain point in time, T0, in weight % of cement, d is the concrete cover in millimeters, Dx is the depth of concrete convection zone in millimeters, be is a regression variable in Kelvin, Tref is a standard test temperature in Kelvin, Treal is the temperature of the structural element or ambient air temperature in Kelvin, DRMC,0 is the chloride migration coefficient in m2/year, kt is a transfer parameter, t0 is a reference point in time in years, t is the time in years, a is an aging exponent, erf is the error function. For reinforced concrete structures not exposed to chlorides, probabilistic time dependent models for carbonation-induced corrosion of reinforcing steel and freeze-thaw deterioration are also available in the 2006 fib Model Code for Service Life Design of Reinforced Concrete [43]. 3.2. Probabilistic structural deterioration models and limit states Structural deterioration models serve as a link between structural failure criteria and prediction of structural service life. Structural failure criteria can be expressed as an ultimate or serviceability limit state. A number of structural deterioration models have been proposed for reinforced concrete structures. Theoretical structural deterioration models rely on upscaling of probabilistic material deterioration models to determine time-dependent structural load carrying capacity, reliability indices, time-dependent structural deformations, and crack width evolution over time. Markovian structural deterioration models rely on past performance of large populations of existing structures exposed to aggressive environments to probabilistically characterize population behavior over time. Numerous researchers have looked at the effect of material deterioration, particularly reinforcement corrosion, on the timedependent load capacity and reliability of reinforced concrete structures. Such research includes that by Vu and Stewart [51], Stewart and Rosokowsky [52], Frangopol et al. [53], and Val et al. [54]. Through upscaling of material deterioration models, these studies have found that the corrosion of reinforcing steel can lead to significant reductions in reliability over time in reinforced concrete members and structural systems. Using Fickian ion transport models and corrosion-induced cracking models proposed by Liu and Weyers [55] and Val and Melchers [56], Rao et al. [57] recently modeled the probabilistic change in structural fragility of bridge columns over time when subject to seismic loads. The decrease in structural capacity due to steel reinforcement corrosion and mass loss in a reinforced concrete column is shown in Fig. 4. Fig. 4(a) shows the structural push-over capacity of an uncorroded reinforced concrete column. Fig. 4(b) shows the structural pushover capacity of a reinforced concrete column with corrosion-induced mass loss of 9.4% in the longitudinal reinforcing steel. A comparison between analytical modeling results and experimental results from Rao et al. is also shown in Fig. 4. Coupled numerical material degradation and structural deterioration models have also been created, such as those by Maekawa et al. [58]. However, a probabilistic formulation of these approaches has yet to be formulated. Unconnected with material deterioration models, structural deterioration models have been created based on probabilistic characterization of structural damage states over time for large

bridge populations. These methods apply Markovian processes in the assignment of transitional probabilities in which an individual structure transitions from damage state to damage state. Mentioned previously, these methods are used by PONTIS [26] and BRIDGIT [27] bridge management systems in the United States and the Finnra Bridge Management System in Finland [28]. The end of structural service life is computed probabilistically as the time from initial construction or rehabilitation to reaching an unacceptably high damage state. 3.3. Probabilistic evaluation of environmental midpoint indicators Variation in life cycle assessment models accrues from a number of life cycle assessment (LCA) modeling components. These include (1) variation in quantities of materials actually consumed, equipment actually used, and processes actually carried out to complete concrete construction and repair activities; (2) uncertainty in quantity and type of emissions and wastes associated with those materials and processes; and (3) uncertainty in the environmental impact assessment that links specific wastes and emissions to environmental midpoint indicators such as those within the EPA TRACI or EcoIndicator protocols. These three sources are analogous to classical definitions of uncertainty stemming from parameter uncertainty, scenario uncertainty, and modeling uncertainty, respectively [59]. Lloyd and Ries carried out an extensive survey of life cycle assessment studies that adopt a variety of methods to estimate uncertainty propagation [59]. They note that stochastic modeling, scenario modeling, fuzzy data sets, interval calculations, and analytical uncertainty propagation were all used with varying success. Moreover, Lloyd and Ries point out that life cycle assessment researchers disagree on the appropriateness of propagation techniques with some favoring fuzzy sets, some favoring interval calculations, others implementing analytical uncertainty propagation, and many using a combination of these techniques. Research is ongoing in the stochastic characterization of life cycle impacts of basic construction materials and construction processes. Ali and Lepech, in cooperation with the American Composite Manufacturer Association (ACMA) recently developed a probabilistic life cycle inventory for glass fiber reinforced polymer composites produced in North America [60]. In the context of this infrastructure applications, such composites can be used in reinforced concrete structural repair and strengthening applications. The Ali and Lepech model was developed by aggregating the variation in glass-polymer material composition from various upstream material suppliers along with variation in the constituent makeup of the finished composite. A pedigree matrix was used to obtain geometric standard deviations for the impact assessment of composite materials based on modeling uncertainty. The inventory for the composites was developed as part of the Ali and Lepech analysis, so specific uncertainty levels were assigned in the application of the pedigree matrix resulting in probabilistic characterization such as those shown in Fig. 5. 3.4. Service life and life cycle assessment model integration Due to the unknown nature of the distributions that characterize the life cycle inventory and the time to end of service life, the integration of these two models relies heavily on numerical Monte Carlo methods to create a two dimensional probability field for the time-dependent cumulative impact. As shown in Fig. 1, the life cycle of a reinforced concrete structure is comprised of a series of repair and rehabilitation events. In the absence of an accumulating deterioration model, the time of future repairs is modeled as a Markovian chain of independent, recurring, identical repair, rehabilitation, and deterioration processes according to Eq. (5). The

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Fig. 4. Reinforced concrete column push-over hysteresis response comparing (a) uncorroded column and (b) corroded column with 9.4% mass loss of longitudinal reinforcing steel. Experimental results (dashed) and analytical reinforced concrete deterioration models are also shown for comparison. [57].

duration of any repair work is considered to be insignificant when compared to the duration of material or structural deterioration processes.

Pðtnþ1 ¼ xjt n ¼ yÞ ¼ Pðt n ¼ xjt n1 ¼ yÞ

ð5Þ

where P is the probability that the deterioration process will last a given time, tn+1 is the time from the most recent failure event, n, to the next failure event, n + 1, tn is the time from the second most recent failure event, n  1, to the most recent failure event, n, tn1 is the time from the third most recent failure event, n  2, to the second most recent failure event, n  1, x is a random probability value, and y is a random probability value. The time of a given future repair or construction event is computed based on the times of all previous repair events for that individual life cycle scenario and the probabilistic service life model. This is shown in the following equation.

Tn ¼

n X

ft i ðSi ¼ Ri Þg

ð6Þ

i¼1

where Tn is the cumulative age of the structure at the time of the nth deterioration failure, ti is the service life of the ith deterioration cycle, Si is the probabilistic load value of the ith deterioration cycle, Ri is the probabilistic resistance value of the ith deterioration cycle. 3.5. Environmental sustainability limit state evaluation Numerous governments and policy-makers have proposed environmental impact or emission reduction targets of global pollutants to achieve environmental sustainability. These take the form of reductions for each of 1–15 environmental midpoint indicators. The most well known of these targets were set for substances with ozone depleting potential (ODP) and substances

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Fig. 6. Global atmospheric carbon stabilization target scenarios [49].

Fig. 5. Probabilistic impact characterization of glass fiber reinforced unsaturated polyester composite produced in North America in terms of (a) greenhouse gas emissions and (b) carcinogens per kilogram of composite produced [60].

with global warming potential (GWP). The Montreal Protocol was the first international agreement put in place to achieve global reduction of ozone depleting atmospheric emissions [61]. More recently, the Kyoto Protocol has aimed to duplicate the success of Montreal in global reduction of greenhouse gas emissions [62]. Following the Vienna Convention of 1985, the Montreal Protocol proposed phase out schedules for chlorofluorocarbons (CFC) and halons, which were proven to be responsible for stratospheric depletion of ozone gas. The Montreal Protocol was amended in London (1990), Copenhagen (1992), and Montreal (1997) to include CCl4, CH3CCl3, methyl bromides, hydrobromylfluorocarbons, and hydrochlorofluorocarbons along with accelerating the phaseout schedule for each of these emission compounds. To date, the binding, multinational nature of the Montreal Protocol has effectively reduced planetary emission of substances with ozone depletion potential. Within the United Nations Framework Convention on Climate Change (UNFCCC), the Kyoto Protocol is a binding, multinational proposed reduction plan for the atmospheric emission of substances which have been shown to cause global climate change. The protocol calls for global emission reduction of four greenhouses gases (GHGs); carbon dioxide, methane, nitrous oxide, and sulfur hexafluoride, along with two groups of gases; hydrofluorocarbons and perfluorocarbons. To date, binding, multinational commitments to the Kyoto Protocol include 191 nations. Each of these policies proposes a reduction timeline for annual emission of targeted substances with the ultimate goal of achieving an influx of emissions that the biosphere can sustainably accommodate. Ultimately, these targets achieve rigorous environmental sustainability (the ability of the biosphere to survive in perpetuity) for the case of each emission. However, the reduction timeline has been set corresponding to technologically feasible or politically acceptable timetables. Such efforts strike a balance between the ability of the biosphere to accommodate emissions and the economic and social ramifications associated with reducing anthropogenic emission of targeted substances. In addition to the multinational Kyoto and Montreal Protocols, various governments, policy-makers, non-governmental-organizations (NGOs), and environmental groups have proposed reduction targets and goals based on scientifically acceptable rationale, political acceptability, technological feasibility, or social equity. The

United Nations Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4) proposes one such set of targets [49]. As shown in Fig. 6, a set of global annual reduction scenarios for carbon dioxide equivalent emissions are proposed which correspond to expected mean global temperature changes for each scenario. Six global scenarios, labeled I through VI, are presented in Fig. 6 and each scenario correlates to an expected stabilization concentration of atmospheric CO2-equivalents ranging from 445 to 490 ppm for Scenario I to 855–1130 ppm for Scenario VI. The dashed lines in Fig. 6 show the 80th percentile range of recent greenhouse gas emissions scenarios published since the Intergovernmental Panel on Climate Change (IPCC) Special Report on Emissions Scenarios (post-SRES) [63]. Based on Fig. 6, to achieve a stabilized atmospheric carbonequivalent concentration of 490–535 ppm (Scenario II), a 30–60% reduction in annual carbon-equivalent emissions is needed by Year 2050 (Year 2000 baseline). At these emission levels, a global average temperature increase of 2.4–2.8 °C is expected (steady state) along with a global sea level rise (due to thermal expansion only) of 0.5–1.7 m [49]. Additional reductions are needed thereafter. Based on such reduction targets and timelines for a selected emission, the likelihood of not attaining or falling short of this reduction for any given year is computed using a shifted emission probability density function. An illustration of the general case of shifting the probability density function is shown in Fig. 7. Analogous to the probability of failure for structural systems, the probability that the cumulative impacts of the alternate concrete repair timeline do not meet targeted reductions in cumulative impacts as compared to the status quo cumulative impacts can be envisioned as the overlap between the alternative cumulative impact distribution and the shifted reduction target distribution. By evaluating the environmental sustainability of alternative concrete repairs and rehabilitations in this way, traditional hypothesis tests can be used to evaluate claims of environmental sustainability. Alternate hypotheses can be tested against the null, in which case the potential of a new, more sustainable, infrastructure repair methodology to meet future emission reduction targets can be confirmed or refuted within defined confidence limits and included in rational cost-benefit analyses or multi-objective design decision-making.

4. Sustainability design framework demonstration 4.1. Probabilistic evaluation of life cycle environmental impacts To demonstrate the application of the sustainability design framework, a hypothetical concrete repair timeline on a bridge

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Fig. 7. Cumulative impact distribution probability density functions at functional obsolescence (Year tfo) for the status quo infrastructure, alternate infrastructure, and a reduction target based on status quo impacts.

superstructure is modeled from first repair to functional obsolescence. Two thicknesses of trial repairs are considered. The trial repairs selected for demonstrating the sustainability design framework were (1) a 40 mm deep concrete cover replacement and (2) a 80 mm deep cover replacement. Both of these are expected to be constructed in Year 2011. Selected information on the supply chain and impact of repair activities is summarized in Stang et al. [64]. Traffic was assumed to be uninterrupted during the completion of these repairs due to their location on the superstructure outside of the live traffic lanes. Therefore, impacts associated with construction congestion-related traffic emissions do not occur in the case study but are an important part of the sustainability design framework. To determine the life cycle impacts of the repair activities, a life cycle inventory and impact assessment of the materials, processes, and procedures used was constructed in compliance with ISO 14040 series standards. The main sources for this data were primary data from the contractors, product marketing materials for construction materials, personal safety and hygiene sheets (MSDS), and commercial life cycle inventory datasets. The repair was comprised of two major construction activities, previously detailed in Stang et al. [63], and include (A) hydrodemolition of concrete cover material, and (B) shotcreting of replacement concrete material. For each of these steps the commercial products used, the equipment needed, and the transportation associated with bringing the materials to the site were catalogued. The impact due to hydrodemolition, IA, is computed as the sum total of impacts per m2 of repair associated with the amount of water consumed (N[60, 3] kg/m2), movement of potable water (N[20, 2] km), waste disposal of the concrete (N[0.04, 0.01] m3/ m2), and impacts associated with operation of the hydrodemolition equipment. The hydrodemolition equipment includes an air compressor (250 cubic feet per minute capacity) (U[36, 83] kW), a hydrodemolition machine (U[250, 750] kW), and a front-end loader (N[3.2, 0.37] m3/h). Probabilistic productivity rates and equipment needs were determined from RS Means Construction Cost Data [65], industry literature, and contractor interviews. The impact due to shotreting, IB, is computed as the sum total of impacts per m2 of repair associated with the amount of shotcrete consumed (N[varies, 0.25 lm] m3/m2), movement of shotcrete (N[1400, 140] km), waste disposal of rebound (N[5% of shotcrete consumed, 0.25 lm] m3/m2), and impacts associated with the operation of shotcreting equipment. The shotcreting equipment includes an air compressor (250 cubic feet per minute capacity) (U[36, 83] kW), a concrete pump (U[30, 105] kW), and a shotcrete rig (N[2.3, 0.35] m3/h). The concrete is made of cement (U[222, 672] km/m3), water (0.5mcement kg/m3), and aggregate (1.0  pcement  pwater). Probabilistic productivity rates and equipment needs were determined from RS Means Construction Cost Data [65], industry literature, and contractor interviews. Monte Carlo analysis was carried out to determine the magnitude and shape of the distributions for 10 environmental impact midpoint indicators for each of the repair activities. Fig. 8 shows

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the probability density function for global warming potential per square meter of repair work performed (kg CO2-eq/m2). Similar charts have been developed for other environmental impact midpoint indicators according to Ecoindicator 99 impact assessment protocols (Goedkoop et al., [46]) including ozone depletion (kg CFC-11-eq/m2), acidification (kg SO2-eq/m2), eutrophication (kg PO4-eq/m2), heavy metals (kg Pb/m2), carcinogens (kg B(a)P/m2), summer smog (kg C2H4/m2), winter smog (kg SPM/m2), primary energy consumption (MJ LHV/m2), and solid waste generation (kg/m2). As seen in Fig. 8 the 40 mm thick cover replacement has an average global warming potential impact per square meter of repair of 81 CO2-eq/m2 with a standard deviation of 12 CO2-eq/m2. The 80 mm thick repair requires additional shotcrete material and has an average global warming potential impact per square meter of repair of 124 CO2-eq/m2 with a standard deviation of 14 CO2-eq/m2. The fib Model Code was chosen as the basis for probabilistic modeling of concrete repair service life [43]. Specifically for this case study, chloride-induced corrosion of reinforcement was assumed to be the mode of failure. Depassivation of the reinforcing steel was assumed to the limit state. The limit state is characterized in Equation (4). Probabilistic variables used for Eq. (4) are characterized in Table 1. A probabilistic model for time to depassivation of reinforcing steel was constructed for each of the two cover replacements. The histogram for this model, developed using Monte Carlo

Fig. 8. Probabilistic global warming potential impact per square meter of (a) 40 mm and (b) 80 mm cover replacement.

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Table 1 Case study probabilistic environmental load and material performance characteristics. (Model distribution parameters are taken from the 2006 fib Model Code). Variable

Unit

Distribution

Parameters

Ccrit

wt.% of cement wt.% of cement wt.% of cement lm mm – K K m2/year – year –

Beta

l = 0.6, r = 0.15, min = 0.2,

Deterministic

max = 2.0 0.0

Normal

l = 6.0, r = 0.5

C0 CS,Dx

DX d be Tref Treal DRMC,0 kt t0 a

Deterministic Normal Normal Deterministic Normal Normal Deterministic Deterministic Beta

0.0

l = varies, r = 10 l = 4800, r = 700 293.0

l = 279, r = 11 l = 6.9  1012, r = 1.4  1012 1.0 0.767 l = 0.6, r = 0.15, min = 0.0, max = 1.0

analysis (100,000 runs), is shown in Fig. 9. As seen, the depassivation phenomenon for the parameters given follows a lognormal distribution. Combining the probabilistic life cycle inventory model for one square meter of repair work with a probabilistic model of 1,000,000 service life timelines, a probabilistic envelope of cumulative impact versus time was numerically constructed (shown conceptually in the dashed lines of Fig. 2). This numerical construction is shown in Fig. 10(a and b) for the 40 mm and 80 mm cover replacement scenarios, respectively. Each horizontal grouping of similarly colored points in Fig. 10(a) and b) represents the probabilistic time of occurrence and probabilistic impact of one repair cycle, with each point representing one repair within a bridge life cycle Monte Carlo run. Together, the horizontal rows represent the series of sequential repairs conducted over a 100-year bridge life cycle. In any future year, the cumulative impact of the repairs is a function of both the number of repairs and impact of repairs already completed. As shown in Fig. 10(c), a probabilistic distribution of the cumulative impact of repairs at any time in the future can be constructed (shown in Years 20, 60 and 100 of the 80 mm cover replacement timeline as a Gaussian distribution for illustrative purposes) by vertically slicing through the population. Within this case study the functional unit selected for analysis is one square meter of repair. Thus, only one square meter of repair work is assumed to be deteriorating and then repaired following each depassivation event. As needed, these impact values can be scaled to reflect the repair of multiples of square meters in an actual structural repair application. As seen in Fig. 10, most scenarios experience their first cover replacement event, and the accrual of the associated cover replacement impact, early in the structural service life. This is expected as a result of the shape of the distribution in Fig. 9. However, some scenarios do not experience their first cover replacement until years 90 through 100, at which time they accrue their first repair-related environmental impacts at this old age. Following the first cover replacement event, the performance of repairs and the accrual of impact become more distributed over the population. By the time four, five, or more cover replacement events are required for a given structural life cycle, the time at which the next event will happen on a random structure in the bridge population is nearly equally likely for any year. Fig. 10 illustrates the probabilistic envelope of cumulative global warming potential impact due to cover replacement activities from the time of first cover replacement to year 100 of the life cycle. However, this representation of the impacts is of little use for evaluation and design. Therefore, the probabilistic impact

Fig. 9. Histogram for time to depassivation of (a) 40 mm cover replacement and (b) 80 mm cover replacement.

confidence envelopes are shown for the 40 mm and 80 mm cover replacement scenarios in Fig. 11(a and b), respectively. As was seen in Fig. 8, the mean global warming potential per square of 80 mm cover replacement was 35% higher than for the 40 mm cover replacement. However, as seen in Fig. 11, when accounting for the greater durability of the thicker cover replacement the cumulative impact over the life cycle is reduced. Over a 100-year life cycle, the average 80 mm cover replacement has approximately a 60% lower global warming potential as compared to the 40 mm cover replacement.

4.2. Evaluation of sustainability limit state For this hypothetical case study, the global greenhouse gas emission target suggested by the UN Intergovernmental Panel on Climate Change is adopted as an environmental sustainability limit state for greenhouse gas emissions. As mentioned previously, to achieve an acceptable, stabilized atmospheric carbon-equivalent concentration of 490–535 ppm, a 30–60% reduction in annual carbon-equivalent emissions is needed by Year 2050 as compared with a Year 2000 baseline. Assuming, for this case, that concrete material and construction technologies have not changed since Year 2000 (therefore no progress toward climate change mitigation has been made), and assuming a linear reduction from Year 2000 to Year 2050 in annual emissions, a cumulative reduction in greenhouse gas emissions from present (Year 2011) to Year 2050 of 38% corresponds to IPCC guidelines for annual greenhouse gas reductions. Based on this timeline, the probability of failure of the proposed 80 mm cover replacement timeline to meet the Year 2050 reduction goal is 31%. Selecting an appropriate probability of failure for meeting or missing an environmental sustainability target is outside the scope of this paper. This is a topic that should be addressed by physical scientists, engineers, policy-makers, economists, and social scientists. Some guidance may be taken from structural building codes which design for one failure in 100,000 for typical structures [43]. Yet given the uncertain consequences associated with missing global, regional, or local environmental impact targets, adopting failure probabilities similar to those for structural failures will likely result in unnecessary overdesign. Unarguably, this is a topic in need of future study. As an alternative to failure probability, traditional hypothesis tests can be used to evaluate claims of sustainability. Alternate hypotheses can be tested against the null, in which case the potential of a concrete infrastructure repair methodology to meet future

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Fig. 10. Probabilistic cumulative global warming potential throughout the structure life cycle for a population 1,000,000 bridges with (a) 40 mm and (b) 80 mm cover replacement and (c) 80 mm cover replacement with probabilistic distributions overlaid.

Fig. 11. Cumulative global warming potential impact 90% and 95% confidence intervals for (a) 40 mm and (b) 80 mm cover replacements.

emission reduction targets can be confirmed or refuted within defined confidence limits and included in rational cost-benefit analyses or multi-objective design decision-making. Additionally, claims that one repair technology, plan, or material is ‘‘more sustainable’’ than another can be analyzed by testing the difference in mean, cumulative, life cycle environmental impacts associated with each repair and rehabilitation timeline. 4.3. Modeling discussion The adoption of one-dimensional transport of chloride ions as the deterioration mode and electrochemical depassivation of reinforcing steel as the failure limit state are simplistic models upon which to base decade-long repair and rehabilitation decisions associated with environmental sustainability. Research in the development and validation of fundamental models for determining reinforced concrete service life is needed to support the sustainability design framework and reduce uncertainty captured within this approach. As new models are introduced, their application within the sustainability design framework is encouraged to advance the field of design for sustainability. Rather than seeing the limitations of current models as a limitation to such design, it is

an opportunity for the broader application of fundamental engineering damage mechanics, deterioration modeling, and service life modeling research to the field of sustainability engineering. 5. Conclusion Reinforced concrete infrastructure, in particular transportation infrastructure, lies at the nexus of two major sources of our global sustainability challenge; large amounts of emissions from both vehicles and construction materials production. Therefore, significant opportunities exist in the reduction of environmental impacts associated with concrete infrastructure repair, rehabilitation, and use. In line with recently proposed probabilistic sustainable design requirements in the fib 2010 Model Code, the sustainability design framework presented in this paper integrates probabilistic life cycle assessment with probabilistic reinforced concrete service life models to provide a foundation for more rational design, construction, and operation of concrete infrastructure that meets science-based environmental sustainability targets. The proposed sustainability design framework leads the development and application of probabilistic service life modeling of reinforced concrete structures, the creation of probabilistic life cycle assessment

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models, the application of policy-based and science-based sustainability targets to civil engineering practice, and the formulation of sustainability design as a probabilistic design problem. Specifically within this study, two alternative concrete cover replacement repair timelines were compared. It was found that by using an 80 mm cover replacement versus a 40 mm cover replacement over the repair and rehabilitation life of a concrete structure, there is approximately a 30% probability of failing to meet climate change reduction targets for Year 2050 as proposed by the Intergovernmental Panel on Climate Change (IPCC). A great deal of research still remains in the development and validation of the methods and tools introduced in this work to facilitate implementation into the broader reinforced concrete design community. Among many areas, significant work remains in the probabilistic assessment of combined environmental, societal and economic indicators, which are required in the fib 2010 Model Code as part of design for sustainable reinforced concrete structures. Acknowledgements The authors would like to thank the gracious support of the Nordic Innovation Center (NICe Project 08190 SR) and the Stanford University Terman Faculty Fellowship. References [1] United Nations Framework on Climate Change. United Nations. New York, NY; 1992. [2] Pacala S, Socolow R. Stabilization wedges: solving the climate problem for the next 50 years with current technologies. Science 2004;305:968–72. [3] US Environmental Protection Agency. Inventory of US Greenhouse gas emissions and sinks: 1990–2006. US EPA, Washington DC, April 15, 2008. [4] van Oss H, Padovani A. Cement manufacture and the environment. Part II: Environmental challenges and opportunities. J Ind Ecol 2003;7(1):93–126. [5] Gowri K. Desktop tools for sustainable design. ASHRAE J 2005;47(1):42–6. [6] Green play book. [accessed 25.07.11]. [7] Green infrastructure planning guide. [accessed 25.07.11]. [8] CEEQUAL. [accessed 25.07.11]. [9] Greenroads. [accessed 25.07.11]. [10] I-LAST. [accessed 26.07.11]. [11] GreenLITES. [accessed 26.07.11]. [12] Austrialian green infrastructure council rating tool. [accessed 26.07.11]. [13] LEED for neighborhood development. [accessed 26.07.11]. [14] BREEAM communities. [accessed 25.07.11]. [15] CASBEE urban development. [accessed 26.07.11]. [16] Pré. Introduction to LCA with SimaPro 7. Netherlands; 2010. [17] PE International. GaBi 4 Manual. Germany; 2006. [18] Hillman T, Ramaswami A. Greenhouse gas emission footprints and energy use metrics for eight US cities. Environ Sci Technol 2010;44:1902–10. [19] Ramaswami A, Hillman T, Janson B, Reiner M, Thomas G. A demand-centered hybrid lifecycle methodology for city-scale greenhouse gas inventories. Environ Sci Technol 2008;42(17):6456–61. [20] Kendall A, Keoleian G, Lepech M. Materials design for sustainability through life cycle modeling of engineered cementitious composites. Mater Struct 2008;41(6):1117–31. [21] Zhang H, Keoleian GA, Lepech M. An integrated life cycle assessment and life cycle analysis model for pavement overlay systems. First international symposium on life-cycle civil engineering. Varenna, Lake Como, Italy, June 10–14, 2008. [22] Zhang H, Lepech M, Keoleian G, Qian S, Li V. Dynamic life cycle modeling of pavement overlay systems: capturing the impacts of users, construction, and roadway deterioration. J Infrastruct Syst 2010;16(4):299–309. [23] Zhang H, Keoleian GA, Lepech MD, Kendall A. Life cycle optimization of pavement overlay systems. J Infrastruct Syst 2010;16(4):310–22. [24] Zhang, Han. Sustainable pavement asset management based on life cycle models and optimization methods. Doctoral dissertation, University of Michigan, Ann Arbor; 2009. p. 1–125. [25] Frangopol DM, Kong JS, Gharaibeh ES. Reliability-based life cycle management of highway bridges. J Comput Civil Eng 2001;15(1):27–34. [26] Thompson PD, Small EP, Johnson M, Marshall AR. The Pontis bridge management system. Struct Eng Int 1998;8(4):303–8.

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