Estimating Life Expectancies of Highway Assets, Volume 1: Guidebook (2012)

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35 In the research leading to development of this guide, various approaches were investigated for estimating life expectancy for a range of highway asset types. The potential methods were gleaned from current practice in not only highway engineering, but many fields that need to measure life expectancy. Methods were evaluated for their realism, policy sensitivity, data requirements, and appropriate precision for the quality of data available. Data sets were obtained from state DOTs to test and validate the methods. The statistical characteristics of the models, including goodness-of-fit and sensitivity to uncertainty, were important considerations. In this chapter, the best of the methods tested in the research are described in detail. In addi- tion to the criteria used in the research, some additional considerations in selecting methods for this chapter were: • Transparency (i.e., the ability for transportation agencies to thoroughly understand and rep- licate the models in their own applications and systems). • Applicability to all transportation agencies. • Focus on the estimation of life expectancy, separate from related applications such as deterio- ration modeling and lifecycle costing. Chapter 5 provides much more detail on deterioration and lifecycle cost. In Chapter 4, the analysis of asset deterioration is conducted only to the limited extent necessary in order to deter- mine life expectancy, thus keeping the methods as simple as possible. When an agency commits to the level of data collection and analysis necessary for life expectancy analysis, it can accom- plish much more by adding some additional detail and analysis to develop deterioration models. Chapter 5 addresses this consideration. Table 4-1 describes some of the basic decision-making criteria that can be used to select the model types that may have the best fit to a particular agency and application. In many cases, it may be worthwhile to try more than one type of model and compare the results in order to make a final decision on which form to implement. All of the models described in the table can be developed using a set of data about existing assets in order to quantify future behavior. They all require past condition and performance data, past preservation and replacement activity data, or both. If past replacement data are unavailable or are not indicative of future replacements, then it is necessary to have data that reliably show a condition threshold when replacement would normally be recommended or required. In other words, it is necessary to have a clear definition of end-of-life and reliable data to indicate when that end-of-life criterion was observed. If the data support it, the analyst can experiment with different definitions of end-of-life to investigate policy sensitivity. It is important at all times to ensure that the life expectancy or deterioration model is not biased by past maintenance, repair, and rehabilitation activity. When a model requires time-series data, C h a p t e r 4 Develop Foundation Tools: How to Compute Life Expectancy Models

36 estimating Life expectancies of highway assets Method of determining life expectancy When used, implications Section Wait for extreme events Replacement when required due to damage. In some cases historical records may provide guidance on the probability of future hazards. 3.2.1 Determine date of changes in standards Develop a plan for system-wide upgrades or replacements of affected assets. May drive the selection of life extension activities as a stop-gap in place of replacement for facilities that otherwise might be replaced earlier. 3.2.1 Determine date of changes in functional requirements (e.g., traffic or route changes) Once the date of the change in requirements is known, affected facilities may have a firm end-of-life. May drive the selection of life extension activities in place of replacement, for facilities that otherwise might be replaced earlier. 3.2.1 Life expectancy models (Chapter 4) Published data on life expectancy or replacement interval Used when it is impossible or uneconomical for the agency to develop its own data and models. Subject to substantial error, caused by unique site characteristics. At the network level, this may drive bulk procurement decisions. At the project level, it may determine individual asset replacements on an interval basis when condition data are unavailable. 4.1 Ordinary regression of age at replacement Used when replacement records are available and condition/performance data are not available. May be unreliable unless the reasons for historical replacements are known. At the network level, this may drive bulk procurement decisions. At the project level, it may set individual asset replacements on an interval basis when data are unavailable. 4.2.1 “Quick-and-dirty” Markov model Used when condition data are available and a condition threshold or state can be determined where replacement is commonly recommended or required. Recognizes just two states: failed and not-failed. The data set can be cross-sectional (does not have to follow each asset through its whole life) and must have pairs of inspections before and after a more or less uniform time interval (usually 1-2 years). At the network level, can be used to establish budgets and replacement quantities within a given time horizon. At the project level, replacement occurs when the failed state is observed. 4.2.2 Weibull survival probability model Similar to Markov model with the same applications but provides a better measure of the effects of age and uncertainty. Requires time series condition data (following each asset through its whole life to detect unreported repair activity) or knowledge of past life extension activity. Can be used to optimize the timing of blanket replacement projects (e.g., all the signs on a corridor). Includes Kaplan-Meier models. 4.2.3 Cox survival probability model Similar to the Weibull model but allows the effect of explanatory variables to be built right into the model (rather than developing separate Weibull models for separate classes of assets). Useful when explanatory variables are continuous, or when the size of the historical data set is too small to provide the desired resolution with Weibull models. 4.2.4 Machine learning Commercial “black box” applications to identify relationships among collected data items. Not addressed in this guide. 5.1.5 Deterioration models (Chapter 5) Ordinary regression of condition or performance as a deterioration model Requires continuous (i.e., not discrete) condition data in a time series. Used when uncertainty range is narrow or not relevant. Can indicate end-of-life when condition is forecast to pass a given threshold. May be used for programming of projects for constructed facilities, especially pavements. 5.1.1 Markov deterioration model Similar to the “quick-and-dirty” Markov model but more precise because it is used with more than two condition states. At the network level, can be used to establish budgets and quantities for replacement and life extension actions within a given time horizon. At the project level, replacement occurs when the failed state is observed. 5.1.2 Markov/Weibull hybrid deterioration model Similar to the Markov model, but provides more resolution on the onset of deterioration. Requires knowledge of past preventive maintenance activity. Used in the planning of preventive maintenance programs and for generating more accurate network-level condition forecasts. 5.1.3 Ordered probit deterioration model Provides a condition state-based deterioration model similar to the Markov model but quantifies the level of uncertainty and provides sensitivity to age and other explanatory variables for every condition state. Requires time series condition state inspection data or full knowledge of past work history on each asset and is relatively difficult to estimate. Provides maximum precision for network-level budgeting of life extension activities and replacement. 5.1.4 Table 4-1. Guidelines for selecting the most appropriate model type.

Develop Foundation tools: how to Compute Life expectancy Models 37 this also usually means that it is necessary to know for sure that no work was done during the asset’s life. When a model requires cross-sectional data in the form of inspection pairs, it is still necessary to know that no work was done between the two inspections in each pair. Often this has to be determined by looking for improvements in condition between inspections. As discussed in this guidebook, selection of a modeling technique can be made based on the general approach, nature of the dependent variable, preference for a probabilistic or determin- istic method, and data type and size. 4.1 Example Life Expectancy Models The research that contributed to the preparation of this guide quantified a set of life expec- tancy models to fit the data sets available to the researchers at the time of the study for various asset types. Table 4-2 summarizes the results, which are then described in the remaining parts of this section. These models reflect only specific agencies and might not be a good fit to other agencies. Before using these models, compare the characteristics of the source agencies and highway networks, including climatic conditions and operating practices, to make sure the models are suitable. The project Final Report contains detailed background information to help in this evaluation. 4.1.1 Culverts Culverts are most frequently provided as passages for water to flow across or along roadways. However, they may also be provided as means of passage for wildlife on low-volume roads. 4.1.1.1 Measuring Condition and Performance Markow (2007) and Wyant (2002) reported that most of the states have formal culvert inspec- tion programs. However, states differ in the types of data gathered and retained in databases, the frequency of inspection, and the sizes and types of culverts addressed (Figure 4-1). FHWA has published culvert inspection guidelines in Arnoult (1986) which provide backup guidance for NBI Item 62, Culvert Condition (FHWA 1995). The collection of this data item is mandatory for all culverts in the United States that are under roads, open to the public, and Asset type Typical life End-of-life* Method used Pipe culverts 87 years Age when 50% probability of failed state Weibull or Markov Box culverts 47 Age when 50% probability of failed state Markov Traffic signs 12 Age when 50% probability of failed state Markov Traffic signals 13 Historical replacement interval Weibull survival Roadway lighting 65 Historical replacement interval Weibull survival Pavement markings (1A Waterborne Yellow) 2.2 Age when retroreflectivity reaches 65 mcd/sq.m/lux (for yellow markings) Weibull survival Pavements (Resurfacing) 12 Age when IRI reaches 220 Markov See Table 4-23 for full bridge element life predictions. * for purposes of illustration only. Table 4-2. Summary of example models.

38 estimating Life expectancies of highway assets spanning at least 20 feet. Many agencies also collect the same data for smaller culverts, in some cases as small as 6 feet in diameter (Markow 2007). Table 4-3 shows the definitions that are used. In addition, more than 40 states use AASHTO CoRe Elements 240-243 (culverts made of unpainted steel, concrete, wood, and other materials, respectively) to describe the condition of culverts in more detail (AASHTO 2002, Thompson 2006). This level of detail is widely used for maintenance planning. It is usually collected for the same culverts that are subject to the agency’s routine NBI inspections, including those of less than 20 feet in span. However, culverts inspected Figure 4-1. Culverts of less than 20 feet in span are routinely inspected in many states (http://www2.dot. ca.gov/hq/oppd/dib/dib83-01-4.htm). NBI Item 62 – Culvert condition rating 9. No deficiencies. 8. No noticeable or noteworthy deficiencies which affect the condition of the culvert. Insignificant scrape marks caused by drift. 7. Shrinkage cracks, light scaling, and insignificant spalling which does not expose reinforcing steel. Insignificant damage caused by drift with no misalignment and not requiring corrective action. Some minor scouring has occurred near curtain walls, wingwalls, or pipes. Metal culverts have a smooth symmetrical curvature with superficial corrosion and no pitting. 6. Deterioration or initial disintegration, minor chloride contamination, cracking with some leaching, or spalls on concrete or masonry walls and slabs. Local minor scouring at curtain walls, wingwalls, or pipes. Metal culverts have a smooth curvature, non-symmetrical shape, significant corrosion or moderate pitting. 5. Moderate to major deterioration or disintegration, extensive cracking and leaching, or spalls on concrete or masonry walls and slabs. Minor settlement or misalignment. Noticeable scouring or erosion at curtain walls, wingwalls, or pipes. Metal culverts have significant distortion and deflection in one section, significant corrosion or deep pitting. 4. Large spalls, heavy scaling, wide cracks, considerable efflorescence, or opened construction joint permitting loss of backfill. Considerable settlement or misalignment. Considerable scouring or erosion at curtain walls, wingwalls or pipes. Metal culverts have significant distortion and deflection throughout, extensive corrosion or deep pitting. 3. Any condition described in Code 4 but which is excessive in scope. Severe movement or differential settlement of the segments, or loss of fill. Holes may exist in walls or slabs. Integral wingwalls nearly severed from culvert. Severe scour or erosion at curtain walls, wingwalls or pipes. Metal culverts have extreme distortion and deflection in one section, extensive corrosion, or deep pitting with scattered perforations. 2. Integral wingwalls are collapsed, severe settlement of roadway due to loss of fill. Section of culvert may have failed and can no longer support embankment. Complete undermining at curtain walls and pipes. Corrective action required to maintain traffic. Metal culverts have extreme distortion and deflection throughout with extensive perforations due to corrosion. 1. Bridge closed. Corrective action may put back in light service. 0. Bridge closed. Replacement necessary. Table 4-3. NBI culvert condition definitions (FHWA 1995).

Develop Foundation tools: how to Compute Life expectancy Models 39 by local agencies might not follow the state DOT’s procedures in this regard. Table 4-4 shows the definitions of the four condition states used for each type of culvert. The types of distresses that typically define culvert condition are summarized in AASHTO guid- ance (AASHTO 2006). Recently the definitions for all AASHTO elements were revised (AASHTO 2010). However, for culverts, the number of condition states and their general meaning did not change significantly enough to affect the life expectancy analysis. Models developed from histori- cal element inspection data should still be valid when the 2010 AASHTO Manual is implemented. Washington State DOT uses a culvert assessment system that is especially appropriate for smaller culverts. It rates groups of culverts by counting the percentage that are at least 50% filled with dirt and/or debris, on a scale of A-B-C-D-F, using the cutoffs of 2%, 5%, 10%, and 20% respectively (WSDOT 2008). There is no category E in the Washington system. A separate clas- sification is used for catch basins and inlets, with cutoff percentages of 3%, 7%, 15%, and 30% respectively. 4.1.1.2 End-of-Life Criteria Both the FHWA and AASHTO definitions are discrete scales where discrete choice models of life expectancy are appropriate, as described in Chapter 3. The recommended end-of-life 240 - Unpainted Steel Culvert 242 - Timber Culvert 1. The element shows little or no deterioration. Some discoloration or surface corrosion may exist but there is no metal pitting. There is little or no deterioration or separation of seams. 1. The timber and fasteners are in sound condition. 2. There may be minor to moderate corrosion and pitting, especially at the barrel invert. Little or no distortion exists. There may be minor deterioration and/or separation of seams. 2. There may be minor decay and weathering. Corrosion at fasteners and connections may have begun. There is little or no distortion and/or deflection. 3. Significant corrosion, deep pitting, or some holes in the invert may exist. Minor to moderate distortion and deflection may exist. Minor cracking or abrasion of the metal may exist. There may be considerable deterioration and/or separation of seams. 3. There may be significant decay, weathering, and warped or broken timbers. Significant decay and corrosion at fasteners and connections may be evident. Minor to moderate distortion of the culvert may exist. 4. Major corrosion, extreme pitting, or holes in the barrel may exist. Major distortion, deflection, or settlement may be evident. Major cracking or abrasion of the metal may exist. Major separation of seams may have occurred. 4. There may be major decay and many warped, broken, or missing timbers. There is major decay and corrosion at fasteners and connections. Major distortion or deflection of the culvert may exist. 241 - Reinforced Concrete Culvert 243 - Other Culvert 1. Superficial cracks and spalls may be present, but there is no exposed reinforcing or evidence of rebar corrosion. There is little or no deterioration or separation of joints. 1. There is little or no deterioration. Only surface defects are in evidence. There are no misalignment problems. 2. Deterioration, minor chloride contamination, minor abrasion, and minor cracking and/or leaching may have begun. There may be deterioration and separation of joints. 2. There may be minor deterioration, abrasion, cracking, and misalignment. 3. There may be moderate to major deterioration, abrasion, extensive cracking and/or leaching, and large areas of spalls. Minor to moderate distortion, settlement, or misalignment may have occurred. There may be considerable deterioration and separation of joints. 3. Moderate to major deterioration, abrasion, cracking, and/or minor to moderate distortion or deflection has occurred. 4. Major deterioration, abrasion, spalling, cracking, major distortion, deflection settlement, or misalignment of the barrel may be in evidence. Major separation of joints may have occurred. Holes may exist in floors and walls. 4. Major cracking, abrasion, distortion, deflection, settlement or misalignment, and/or major deterioration affecting structural integrity may have occurred. Table 4-4. AASHTO CoRe Element condition state definitions for culverts (AASHTO 1997).

40 estimating Life expectancies of highway assets condition for culverts is the age when there is a 50% probability of being in a condition state where replacement is normally recommended. Bridge management systems such as Pontis have built-in procedures that can estimate condition state transition times and life expectancy, using this definition, for any type of structural asset including culverts (Cambridge 2003, Thompson and Sobanjo 2010). These methods are in widespread use (Thompson 2006). The 50% probability threshold is a network-level criterion, appropriate for decisions about budgeting for example. States do not necessarily replace individual culverts at exactly this point in time. They may replace a culvert sooner when there is another justification besides condition (e.g., a need to widen the road), or they may delay replacement when insufficient funding is available or when preventive maintenance (e.g., flushing or patching) is a possibility for life extension. In other words, network-level and project-level end-of-life may differ. Federal policy determines a culvert to be structurally deficient, thereby eligible for replace- ment funding, if its NBI condition rating is 4 or below. However, for the purposes of this anal- ysis, it was assumed that a condition level of 3 is a more common threshold where culvert replacement is considered. For states using AASHTO CoRe Elements and Pontis, replacement is normally recommended by the lifecycle cost model when a sufficient percentage of the culvert reaches condition state 4. For consistency in the analysis, this percentage is 50% in the results provided here. Lifecycle cost analysis, however, may suggest a different percentage. 4.1.1.3 Life Extension Interventions About 25% of the states have preventive maintenance programs for culverts, as a means of life extension (Markow 2007). Chapter 5 describes methods to determine the potential increase in life expectancy, using models of deterioration and lifecycle cost. The examples in the current section assume the states’ normal preventive maintenance practices, which were not specified in the data set. 4.1.1.4 Published Life Expectancy Values Markow (2007) provides a table of asset life estimates developed from a survey of transporta- tion agencies. The number of responding agencies and the median estimate in years are repro- duced in Table 4-5. These estimates are primarily from expert judgment. 4.1.1.5 Example Analysis For this study, the model for pipe culverts was developed primarily from Pennsylvania data, with the addition of small amounts of data from Minnesota and Vermont. Given that not all Pipe culverts Box culverts Material Count Life Years Material Count Life Years Concrete 13 50 Reinforced concrete 15 50 Corrugated metal 16 35 Timber 3 30 Asphalt coated corrugated metal 5 50 Precast reinforced concrete 1 50 Small diameter plastic 7 50 Polyvinyl chloride 1 50 High-density polyethylene 1 50 Aluminum alloy 1 50 (Markow 2007) Table 4-5. Survey of life expectancy estimates for culverts.

Develop Foundation tools: how to Compute Life expectancy Models 41 states use the NBI or AASHTO inspection conventions, the researchers used a simpler scale consistent with the three states that contributed data: 0: Very poor or serious deterioration, warranting replacement 1: Poor condition 2: Fair; some wear but structurally sound 3: Excellent condition, like new In this scale, state 0 is assumed to be equivalent to an NBI condition rating of 3 or below, or an AASHTO CoRe Element condition state of 4. The researchers found the following variables to have a significant effect on life expectancy: • Material • Coating application • Type of inlet and outlet • Temperature • Precipitation • Freeze/thaw cycles • Soil corrosiveness For larger, box culverts, NBI data were utilized. Because of the existence of periodic inspections for large culverts, they are perfect candidates for either Weibull survival probability models or Markov models. A later section, “Developing Life Expectancy Models,” describes how to develop Weibull or Markov models. The researchers developed separate models for pipe culverts and box culverts, as follows. Pipe Culverts. A Weibull survival probability model, with regression used to predict the scaling parameter, was found to best fit the collected data having the following functional form: y gg1 1 0= − × ( )( )exp . α β where y1g is survival probability as a function of age g ≡ age at which the survival probability is sought, in years b = shape parameter = 1.064 and the scaling parameter is given by: α = ( + ∗exp . .4 754 0 215 1if metal culvert, 0otherwise average annual freeze/ ( ) − ∗0 009. thaw cycles 1if high soil corr ( ) − ∗.0 142 osiveness potential, 0otherwise( ) + ∗0 071. . 1if ditch inlet/outlet, 0otherwise( ) +0 097 0 098. ∗ ( ) + ∗ 1if coated, 0otherwise normal annual temperature in F n ( ) − ∗0 097. ormal annual precipitation in inches( )) The above results suggest that, in the given study area, pipe culverts in a warmer climate, hav- ing ditch inlet/outlets, made of a metal material type, and having protection coating have longer service lives. Areas having higher freeze/thaw cycles and precipitation were generally found to experience a shorter life for culverts. On average, the model calibrated to the collected data would suggest an average life of 87 years for pipe culverts (Figure 4-2).

42 estimating Life expectancies of highway assets Box Culverts. For the box culverts in the NBI database (see Section 4.1.8 for further details on NBI condition data), a Markov chain model was found to best describe the performance trends. The transition matrices (Table 4-6) were calibrated using the average deterioration curve, which was determined by regressing the age against the condition state. Multiple transi- tion matrices were developed, assuming homogenous deterioration rates within each age group. The modeling process yielded the survival curve in Figure 4-3. This curve can be interpreted to mean that box culverts are nearly certain to survive up to 30 years but are highly unlikely to survive beyond 54 years without maintenance or rehabilitation. On average, the applied dete- rioration curve suggests an average life of 47 years. 4.1.2 Traffic Signs Traffic signs are replaced for various reasons, including the need for, or accuracy of, the infor- mation on the sign; evolving standards for legibility, size, or location; physical condition and integrity; impact damage; and retroreflectivity (night visibility). When agencies become aware of a change in the need or the applicable standards, life expectancy becomes a deterministic pro- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 20 40 60 80 120100 140 160 180 200 Su rv iv al P ro ba bi lit y Age in Years Average Life 87 years Figure 4-2. Example life expectancy estimate of pipe culverts. Transition Probability Age Group P(9 8) P(8 7) P(7 6) P(6 5) P(5 4) P(4 3) 0-6 years 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 7-12 years 0.0856 0.0000 0.0000 0.0000 0.0000 0.0000 13-18 years 0.0555 0.1126 0.1202 0.0575 0.0000 0.0000 19-24 years 0.0279 0.0508 0.0855 0.2239 0.0813 0.0000 25-30 years 0.0433 0.0852 0.1158 0.1890 0.1088 0.0000 31-36 years 0.1820 0.1624 0.1308 0.0710 0.0787 0.0530 37-42 years 0.0892 0.2184 0.2762 0.2393 0.1391 0.1161 43-48 years 0.1282 0.1786 0.3031 0.5513 0.7880 0.5128 49-54 years 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Table 4-6. Example transition matrices of box culverts.

Develop Foundation tools: how to Compute Life expectancy Models 43 gramming decision. Therefore, the methods described in this guide focus on condition-based longevity in the absence of changes in the information or standards. The lifespan of sign sheeting (typically 10 to 15 years) is generally less than that of sign posts and much less than that of sign structures (typically 30 to 50 years) (Figure 4-4). Therefore, these components are not necessarily replaced simultaneously. 4.1.2.1 Measuring Condition and Performance Markow (2007) reported from a survey of the states, that more than 80% of respondents gather sign condition and performance data using visual inspections. Automated methods of measuring retroreflectivity have been under development, but their routine use is still relatively scarce (Markow 2007). Condition state language of the type used for culverts and bridges has not been developed for sign sheeting or posts, but it is becoming common for sign structures. Condition monitoring of sign sheeting and posts is typically performed by a drive-by assessment 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 5040 60 Su rv iv al P ro ba bi lit ie s Age in Years Average Life 47 years Figure 4-3. Example life expectancy estimate of box culverts. Figure 4-4. Traffic signs include sheeting, posts, and support structures (http://ops.fhwa.dot.gov/publications/ manag_demand_tis/travelinfo.htm).

44 estimating Life expectancies of highway assets during the day and at night. Condition monitoring of sign structures is increasingly done by bridge inspectors, often using hands-on procedures that look for fatigue cracking. FHWA has established minimum retroreflectivity standards, which are published in the Man- ual on Uniform Traffic Control Devices (MUTCD). Retroreflectivity is the ability of a sign to reflect the light from vehicle headlamps back to the driver’s eyes. It is measured in candelas per lux per square meter. Table 4-7 shows the standards (FHWA 2007). When inspections are conducted visually, FHWA recommends that the inspectors begin their nighttime shifts by viewing cali- bration signs under controlled conditions to improve the accuracy of judging retroreflectivity. Sign replacement is typically warranted when physical damage or loss of retroreflectivity ren- der the sign insufficiently legible (AASHTO 2006). Most often, in practice, legibility is a matter of judgment by field personnel. The types of damage typically noted are bullet holes, large dents, impact damage, dirt or sap accumulation, graffiti, vandalism, cracking, curling, pitting, edge lift- ing, blistering, color fading, weathering, and missing reflective material including missing letters. None of the releases of the AASHTO CoRe Element guides (AASHTO 1997, 2002, and 2010) have addressed sign structures. However, some of the states have developed analogous inspec- tion manuals. Table 4-8 shows the condition state language used by Colorado, and Table 4-9 shows the Florida language. 4.1.2.2 End-of-Life Criteria For the purpose of modeling life expectancy at the network level, the relevant end-of-life cri- terion for sign sheeting is the age when 50% of the signs in a given class or population become insufficiently legible or violate federal minimum retroreflectivity standards, thus requiring replacement. For sign structures, a 50% probability of condition state 5 in both the Colorado and Florida manuals would be appropriate, since those are the levels where the Pontis lifecycle cost analysis recommends replacement. Typically, at the project level, the end-of-life criterion would be the point where the individual sign violates minimum standards. For sign posts, the end-of-life criterion could be similar to that used for sign structures, even though none of the states have a routine inspection program for sign posts. Or more simply, the replacement criterion could be any set of conditions under which a maintenance engineer would recommend replacement. Sign color Additional criteria Sheeting Type (ASTM D4956-04) See note (1) Beaded Sheeting Prismatic Sheeting I II III III to X White on green Overhead W*; G 7 W*; G 15 W*; G 25 W 250; G 25 Ground-mounted W*; G 7 W 120; G 15 Black on yellow or black on orange See note (2) Y*; O* Y 50; O 50 See note (3) Y*; O* Y 75; O 75 White on red See note (4) W 35; R 7 Black on white W 50 1 The minimum maintained retroreflectivity levels shown in this table are in units of cd/lx/m2 measured at an observation angle of 0.2 ° and an entrance angle of -4.0 °. 2 For text and fine symbol signs measuring at least 1200 mm (48 inches) and for all sizes of bold symbol signs. 3 For text and fine symbol signs measuring less than 1200 mm (48 inches). 4 Minimum Sign Contrast Ratio 3:1 (white retroreflectivity ÷ red retroreflectivity). * This sheeting type should not be used for this color for this application. Table 4-7. Federal minimum retroreflectivity standards (FHWA 2007).

Develop Foundation tools: how to Compute Life expectancy Models 45 Because of mobilization and traffic control costs, there are economies of scale in replacing all signage along a roadway at the same time (blanket replacement). As a result, a lifecycle cost analysis may result in a shorter optimal life expectancy with fewer than 50% of the assets reach- ing the end-of-life criterion. This would be relevant to states that have blanket replacement policies or are considering implementing them. 4.1.2.3 Life Extension Interventions About half of the states have some sort of preventive maintenance program for signage (Markow 2007). Life extension activities include washing, at intervals from 1 to 5 years, and 620 –Steel Column 622- Concrete Column 1. There is little or no corrosion or misalignment of the member(s). Handhole covers and column caps are in place. 1. The unit shows no deterioration. There may be discoloration, efflorescence, and/or superficial cracking but without effect on strength and/or serviceability. 2. Surface rust, surface pitting, has formed or is forming. There may be minor collision damage that does not warrant addressing it in the traffic impact smart flag. Handhole covers or column caps are missing. 2. Minor cracks and spalls may be present but there is no exposed reinforcing or surface evidence of rebar corrosion. 3. Steel has measurable section loss due to corrosion but does not warrant structural analysis. There is moderate collision damage that warrants implementing the Traffic Impact Smart Flag. Standing water may be observed on the inside of the column. The column is out of plumb. 3. Some delamination and/or spalls may be present and some reinforcing may be exposed. Corrosion of rebar may be present but loss of section is incidental and does not significantly affect the strength and/or serviceability of the element. 4. Corrosion is advanced. Section loss, or collision damage, is sufficient to warrant structural analysis. 4. Advanced deterioration. Corrosion of reinforcement and/or loss of concrete section is sufficient to warrant analysis to ascertain the impact on the strength and/or serviceability of the element. 5. Deterioration is so severe that structural integrity is in doubt. A CIF notification is warranted. 5. Deterioration is so severe that the structural integrity of the column is in doubt. A CIF notification is warranted. 621- Prestressed Concrete Column 640 - Frame/Mast Arm 1. The unit shows no deterioration. There may be discoloration, efflorescence, and/or superficial cracking but without effect on strength and/or serviceability. 1. There is no evidence of active corrosion on metal. The paint system is sound and functioning as intended to protect the metal surface. Weathering steel is coating uniformly and is in excellent condition. 2. Minor cracks and spalls may be present and there may be exposed reinforcing but no evidence of corrosion. There is no exposure of the prestress system. 2. There is little or no active corrosion on the metal. Surface or freckled rust has formed or is forming. The paint system may be chalking, peeling, curling or showing other early evidence of paint system distress but there is no exposure of metal. 3. Some delamination and/or spalls may be present. There may be minor exposure but no deterioration of the prestress system. Corrosion of non-prestressed reinforcement may be present but loss of section is incidental and does not significantly affect the strength and/or serviceability of the element. 3. Corrosion is prevalent on the metal with 10% to 20% section loss. The paint system, if present, is no longer effective. 4. Delamination, spalls, and corrosion on non-prestressed reinforcement are prevalent. There may also be exposure and deterioration of the prestress system (manifested by loss of bond, broken strands or wire, failed anchorages, etc). There is sufficient concern to warrant an analysis to ascertain the impact on the strength and/or serviceability of the element. 4. Corrosion is prevalent on the metal with 20% to 30% section loss but does not warrant structural analysis of the element. 5. Deterioration is so severe that the structural integrity of the column is in doubt. A CIF notification is warranted. 5. Corrosion is advanced with section loss greater than 30%. The paint system, if present, has failed. Structural analysis is warranted to ascertain the impact on the ultimate strength and/or serviceability of the element. A CIF notification is required. Table 4-8. Colorado sign structure condition state definitions (LONCO 2007).

46 estimating Life expectancies of highway assets repairs to damaged posts and panels. For painted sign structures, painting is often performed as a preventive maintenance activity. Certain sign structures are subject to fatigue damage, for which the agency may have countermeasures. The data available to the researchers of the NCHRP Project 08-71 study did not distinguish which signs were subject to preventive maintenance programs. This would be a valuable topic for future research. Agencies having this type of maintenance history data could evaluate maintenance effectiveness using the methods in Chapter 5. 4.1.2.4 Published Life Expectancy Values Substantial data on life expectancy of signs, sign posts, and sign structures were gathered in Markow (2007) from a survey of transportation agencies and from a literature review. This information was determined primarily from expert judgment, with additional information taken from published state standards. The number of responding agencies and the median esti- mate in years are shown in Table 4-10. 4.1.2.5 Example Analysis The performance of traffic signs can be modeled using an appropriate performance indica- tor such as the retroreflectivity of the sign sheeting. Retroreflectivity is measured in units that represent a continuous variable. For this study, data from the National Transportation Product Evaluation Program (NTPEP) were used, which were gathered from various test sites located in 487 - Ov erlane Sign Structure Horizontal Member 488 - Ov erlane Sign Structure Vertical Member 1. There is no evidence of active corrosion and the coating sy stem is sound and functioning as intended to protect the metal surface. 1. There is no evidence of active corrosion and the coating sy stem is sound and functioning as intended to protect the metal surface. 2. There is little or no active corrosion. Surface corrosion has formed or is forming. The coating sy stem ma y be chalking, peeling, curling or showing other earl y evidence of paint sy stem distress but there is no exposure of metal. 2. There is little or no active corrosion. Surface corrosion has formed or is forming. The coating sy stem ma y be chalking, peeling, curling or showing other earl y evidence of paint sy stem distress but there is no exposure of metal. 3. Surface corrosion is prevalent. There ma y be exposed metal but there is no active corrosion which is causing loss of section. 3. Surface corrosion is prevalent. There ma y be exposed metal but there is no active corrosion which is causing loss of section. 4. Corrosion may be present but any section loss due to active corrosion does not ye t warrant structural review of the element. 4. Corrosion may be present but any section loss due to active corrosion does not ye t warrant structural review of the element. 5. Corrosion has caused section loss and is sufficient to warrant structural review to ascertain the impact on the ultimate strength and/or serviceability of the unit. 5. Corrosion has caused section loss and is sufficient to warrant structural review to ascertain the impact on the ultimate strength and/or serviceability of the unit. Table 4-9. Florida sign structure condition state definitions (Florida DOT 2010). Sign sheeting Sign posts Sign structures Ty pe Count Life Ty pe Count Life Ty pe Count Life All sheeting 17 10 Steel U-channel 10 15 Steel sign bridge 12 30 Aluminum 3 11 Steel square tube 10 15 Aluminum sign bridge 8 30 Viny l 2 6 Steel round tube 3 15 Overpass bridge mounting 1 50 Ty pes I-II Literature 5-7 Aluminum tube 1 10 Ty pes III-IV Literature 10-15 Wood 3 15 Ty pes V-X Literature 15-20 Structural steel beam 2 27.5 Table 4-10. Survey of life expectancy estimates for sign components (Markow 2007).

Develop Foundation tools: how to Compute Life expectancy Models 47 different states. To determine asset life, a Markov chain can be calibrated to estimate the tran- sition probability of traffic signs progressing from a subjective rating of “good” to “fair” and ultimately “poor.” Alternatives to sign sheeting retroreflectivity, such as physical deterioration of sign structure, lack of color/contrast of sign sheeting, and blistering, cracking and shrinkage of sign sheeting materials, can be duly assessed. The Markov model in Table 4-11 considers the “poor” stage as the end-of-life condition, while the “good” stage is the initial condition. The transition matrix was calibrated according to the average deterioration curve, based on a regression of asset age against condition state. The survival curve in Figure 4-5 suggests that the average life of traffic signs is about 12 years and that similar signs are unlikely to last beyond 30 years. 4.1.3 Traffic Signals Traffic signal systems and Intelligent Transportation Systems (ITS) provide traffic control and communication with drivers and vehicles. For asset management purposes, the systems are made up of signal heads, flashers, detectors, controllers, support structures, enclosures, com- munications equipment, and other electronic components. Traffic signal components are often replaced based on their condition but are replaced some- times based on improvements in technology. Signal heads and flashers contain lamps that are typically replaced on an interval basis (often 12 or 18 months), with long intervals for modern LED lamps (5 years or more). Often they are mounted on mast arm structures that are inspected by transportation agencies in the same manner as sign structures. To condition state: From condition state: Good Fair Poor Good 0.8949 0.1051 0 Fair 0 0.8277 0.1723 Poor 0 0 1.0000 Table 4-11. Example transition matrix for simple Markov model of traffic signs. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 5040 60 Su rv iv al P ro ba bi lit y Age in Years Average Life 12 years Figure 4-5. Example life expectancy estimate of traffic signs.

48 estimating Life expectancies of highway assets 4.1.3.1 Measuring Condition and Performance Agencies typically inspect key components annually and/or when relamping (Markow 2007). More than 70% of transportation agencies maintain an inventory of traffic signal components, and about ¹⁄³ of agencies maintain component condition data. There are no published standards for formal visual inspections of most signal components, except structural supports, so relatively informal methods, such as good-fair-poor, are often used. Traffic signal system repairs are often driven by operational requirements and become more frequent as the components age. This insight is behind the performance rating system used by Washington State DOT (WSDOT 2008). The system rates each signal system on a scale of A-B- C-D-F (omitting E), based on the frequency of repair. The repair frequencies corresponding to the letter grades are one per 2 years, one per year, two per year, three per year, and four per year, respectively. WSDOT has a similar scheme for ITS equipment. For poles, mast arms, and other structures that make up the structural support of traffic signal heads and flashers, many states perform routine inspections that are similar to their procedures for sign structures. The preceding section presents the definitions used by Colorado and Florida for this purpose. 4.1.3.2 End-of-Life Criteria For signal heads, flashers, detectors, controllers, communications equipment, and other elec- tronic components, an appropriate end-of-life condition would be a condition state so deterio- rated that no economical repair option is available or, as in the WSDOT case, an excessive repair frequency. This is separate from concerns about technological obsolescence, which would not be analyzed in the same way as deterioration. If an agency has developed replacement warrants based on condition, then these might form the basis of end-of-life criteria. For a population of traffic signals, life expectancy could be the age when there exists a 50% probability that a given asset needs to be replaced. For structural supports, the end-of-life condition would most appropriately correspond to condition state 5 in sign structure elements as presented for Colorado and Florida in the preced- ing section of this guide. Because of mobilization and traffic control costs and technological compatibility, there are economies of scale in replacing all signal equipment at an intersection, or even along a whole section of road, at the same time (blanket replacement). As a result, a lifecycle cost analysis may result in a shorter optimal life expectancy with fewer than 50% of the assets reaching the end- of-life criterion. 4.1.3.3 Life Extension Interventions About 50% of agencies have some form of preventive maintenance program for traffic sig- nals (Markow 2007). A significant portion is driven by operational problems noted by crews or the public. Repairs that are performed during or after inspections respond to damage that is observed, such as corrosion, loose connections, non-functioning components, damaged wiring or insulation, and accumulated debris. Typically, if such problems are not addressed, opera- tional failures may result. Given that most repair and rehabilitation activities are either driven by operational concerns or involve replacement of components, they are not considered life extension interventions for the purpose of this analysis (Harrison et al. 2004). 4.1.3.4 Published Life Expectancy Values Data on the life expectancy of traffic signal components were gathered in Markow (2007) from a survey of transportation agencies. This information was provided primarily from expert judgment. Table 4-12 summarizes the number of responding agencies and the median estimate in years for each component.

Develop Foundation tools: how to Compute Life expectancy Models 49 Minnesota DOT noted that a life expectancy of 30 years is plausible for electronic components in the signal cabinet when a preventive maintenance program is in place. 4.1.3.5 Example Analysis The data collection aspect of this research suggests that few agencies track the deterioration of their traffic signals and flashers. However, agencies in Missouri, Oregon, and Pennsylvania were able to provide data on traffic controller deactivation intervals. With such data, an interval-based approach was used to develop the life expectancy models, and it was found that the following variables significantly affect the life expectancy of this asset type: • Temperature • Mounting structure • Wind speed • Roadway functional class • Control type A parametric model was developed for existing assets, assuming the control type served as a proxy for age. Merely installing a new signal of a certain control type does not cause life to be extended. Thus, a Weibull-distributed survival probability model can be developed for existing traffic signals as follows: y gg1 1 0= − × ( )( )exp . α β where y1g is survival probability as a function of age g ≡ age the survival probability is sought for in years b = the shape parameter, 1.415 and the scaling parameter is given by: α = ( − ∗exp . .9 343 0 101 average wind speed inmph average annual temperature i ( ) − ∗0 108. n F 1if pre-timed or semi-act ( ) + ∗.0 139 uated signal, 0otherwise 1if o ( ) − ∗0 288. n a city street, 0otherwise 1i ( ) − ∗0 583. f supported by amast arm,0otherwise( ) +0.352 ∗ 1if part of a closed loopor hardwire interconnected 1if fiber-op ( ) − ∗0 319. tic cables, 0otherwise( )) Structural components Controller sy stem components Signal displa y components Ty pe Count Life Ty pe Count Life Ty pe Count Life Tubular steel mast arm 14 20 Permanent loop detector 14 7.5 Incandescent lamps 15 1 Tubular aluminum mast arm 7 20 Non-invasive detector 12 10 Light-emitting diode lamps 18 6.5 Wood pole (and span wi re) 9 15 Traffic controller 18 15 Signal heads 15 20 Concrete pole (and span wire) 2 12.5 Traffic controller cabinet 17 15 Pedestrian display s 1 15 Steel pole (and span wi re) 9 20 Tw isted copper interconnect cable 11 20 Galvanized pole and span arm 1 >100 Fiber-optic cable 7 20 Table 4-12. Survey of life expectancy estimates for signal components (Markow 2007).

50 estimating Life expectancies of highway assets The example analysis suggests that pre-timed or semi-actuated traffic signals that were hard- wire interconnected or part of a closed loop tend to have longer service lives. On the other hand, signals located in warmer climates, areas with higher wind speeds, located on city streets, sup- ported by a mast arm, or with fiber-optic cables tended to have shorter service lives. On average, the calibrated model indicated an average life of 13 years (Figure 4-6). In the data provided by the agencies, there was no indication of the rationale for replacing a traffic signal controller. Therefore, it can be surmised that various factors besides physical deg- radation may have led to its replacement, such as the possible need to synchronize the timing of replacement of similar asset types. In this example application of life expectancy estimation techniques, physical deterioration was assumed to be the cause of replacement. However, in practice, agencies should discern the actual reason for replacement so that life expectancy can be estimated more reliably. 4.1.4 Roadway Lighting Roadway lighting provides safety, comfort, and aesthetic benefits to the public. However, agencies have had difficulty in developing routine condition assessment processes due to the large number of fixtures and relatively low cost of each fixture. This makes lighting a good can- didate for sample-based inspection. 4.1.4.1 Measuring Condition and Performance Most agencies have an inventory of roadway lighting, but few maintain a database of the con- dition of lighting components. Although lighting units are inspected annually by most agencies, the data resulting from such inspections are in the form of work orders for repairs that may be needed (Markow 2007). Thus, data for estimation of life expectancy are very scarce for most lighting components. Table 4-13 shows an example where the condition state concept used for culverts and sign structures has been applied to lighting. One area where data are more commonly available is high-mast light poles. Due to incidents where fatigue or corrosion has caused pole failure, many agencies have begun gathering high- mast light pole data as a part of the structure inspection program. As a result, data on the condi- tion of these assets are more readily available. Table 4-14 shows condition state language used in Florida to inspect high-mast light poles. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 5040 Su rv iv al P ro ba bi lit y Age in Years Average Life 13 years Figure 4-6. Example life expectancy estimate of traffic signal controllers.

Develop Foundation tools: how to Compute Life expectancy Models 51 703 – Lighting 1. Lighting standards and supports are properly anchored. There are no indications of fatigue damage. There are no missing or broken luminaires or exposed wires. 2. Lighting standards and supports are properly anchored. There are no indications of fatigue damage. There may be some missing or broken luminaires, but there are no exposed wires. 3. Lighting standards and supports are properly anchored. There may be some indications of fatigue damage. Luminaires may be missing or broken, but there are no exposed wires. 4. Lighting standards and supports may be improperly anchored. There may be indications of fatigue damage. Luminaires may be missing or broken, or there may be exposed wires. Table 4-13. Example of condition state language for lighting (Virginia). 495 - High-Mast Light Poles Metal Uncoated 498 - High-Mast Light Poles Other Material 1. There is little or no corrosion of the unpainted steel. The we athering steel is coated uniformly and remains in excellent condition. Oxide film is tightly adhered. 1. There is little or no deterioration. Surface defects onl y are in evidence. 2. Surface corrosion, surface pitting, has formed or is forming on the unpainted steel. The weathering steel has not corroded bey ond its design limits. Weathering steel color is yellow orange to light brown. Oxide film has a dust y to granular texture. 2. There ma y be minor deterioration, cracking and weathering. Mortar in joints ma y show minor deterioration. 3. The steel has measurable section loss due to corrosion but does not warrant structural review. Weathering steel is dark brown or black. Oxide film is flaking. 3. Moderate to major deterioration and cracking. Major deterioration of joints. 4. Corrosion is advanced. Oxide f ilm has a laminar texture with thin sheets of corrosion. Section loss is sufficient to warrant structural review to ascertain the impact on the ultimate strength and/or serviceability of either the element or the bridge. 4. Major deterioration, splitting, or cracking of materials may be affecting the structural capacity of the element. 497 - High-Mast Light Poles Galvanized (or Painted) 499 - High-Mast Light Pole Foundations 1. There is no evidence of active corrosion and the coating sy stem is sound and functioning as intended to protect the metal surface. 1. The element sh ow s little or no deterioration. There may be discoloration, efflorescence, and/or superficial cracking but wi thout affect on strength and/or serviceability . 2. There is little or no active corrosion. Surface corrosion has formed or is forming. The coating sy stem ma y be chalking, peeling, curling or showing other earl y evidence of paint sy stem distress but there is no exposure of metal. 2. Minor cracks and spalls may be present but there is no exposed reinforcing or surface evidence of rebar corrosion. 3. Surface corrosion is prevalent. There ma y be exposed metal but there is no active corrosion which is causing loss of section. 3. Some delamination and/or spalls ma y be present and some reinforcing ma y be exposed. Corrosion of rebar may be present but loss of section is incidental and does not significantly affect the strength and/or serviceability of either the element or the bridge. 4. Corrosion may be present but any section loss due to active corrosion does not ye t warrant structural review of the element. 4. Advanced deterioration. Corrosion of reinforcement and/or loss of concrete section and/or settlement or rotation of foundations are sufficient to warrant review to ascertain the effect on the strength and/or serviceability of either the element or the bridge. 5. Corrosion has caused section loss and is sufficient to warrant structural review to ascertain the impact on the ultimate strength and/or serviceability of the unit. Table 4-14. High-mast light pole condition states (Florida 2010).

52 estimating Life expectancies of highway assets 4.1.4.2 End-of-Life Criteria For electrical components and luminaires, an appropriate end-of-life condition would be a condition state so deteriorated that no economical repair option is available or, similar to Washington State’s treatment of traffic signals, an excessive repair or relamping frequency. This is separate from concerns about technological obsolescence, which would not be analyzed in the same way as deterioration. If an agency has developed replacement warrants based on condition, then these might form the basis of end-of-life criteria. For high-mast light poles, an appropriate end-of-life condition would be the worst-defined condition state in a visual inspection such as shown for Florida. For a population of lighting assets, the life expectancy would be the age when 50% of the population is in need of replacement according to these criteria. Lifecycle cost analysis may reduce the optimal percentage dramatically because of the mobilization and traffic control costs of lighting asset replacement. This is why the practice of group relamping is very common. Similar considerations apply to repairs and replacement. Agencies will normally tolerate a small number of failures before mobilizing to perform relamping and repair on a segment of road. However, if the failure rate becomes excessive, such that normal relamping intervals are insuf- ficient, then replacement may become economical even if most of the fixtures are still opera- tional. Thus the optimal life expectancy of a group of lights along a roadway may be less than the lifespan of the individual fixtures considered in isolation. 4.1.4.3 Life Extension Interventions Markow (2007) noted that life extension possibilities may exist for control cabinets and switchgear by means of cleaning, adjustment, and protection. Luminaires and lamps, however, rarely receive any sort of life extension action. Certain types of light poles can have their lives extended by painting. 4.1.4.4 Published Life Expectancy Values Data on the life expectancy of roadway lighting components were gathered in Markow (2007) from a survey of transportation agencies. This information is primarily from expert judgment. Table 4-15 summarizes the number of responding agencies and the median estimate in years, for each component. 4.1.4.5 Example Analysis Data from a relatively small sample of historical lighting fixtures’ deactivation records were obtained from Missouri for this part of the study. Due to the small size of the sample, the exam- ple herein uses a non-parametric Weibull probability model (Figure 4-7): Structural components Lamps Other components Ty pe Count Life Ty pe Count Life Ty pe Count Life Tubular steel 12 25 Incandescent 3 1 Ballast 9 7.5 Tubular aluminum 9 25 Mercury vapor 6 4 Photocells 11 5 Cast metal 2 22.5 High-pressure sodium 15 4 Control panels 7 20 Wood posts 2 32.5 Low-pressure sodium 3 4 Luminaires 2 16.25 High-mast or tower 11 30 Metal halide 9 3 Fluorescent 1 5 Table 4-15. Survey of life expectancy estimates for lighting components (Markow 2007).

Develop Foundation tools: how to Compute Life expectancy Models 53 y gg1 1 0= − × ( )( )exp . α β where y1g is survival probability as a function of age g ≡ age at which the survival probability is sought, in years b = shape parameter, 3.281 and a = scaling parameter, 71.788 On average, the fixtures in the dataset were predicted to survive 65 years. As is the case with traffic signals, the reason for replacement was not available in the dataset. Where an agency possesses data that have adequate observations involving recorded replacement reasons, a survival curve could be fitted for each replacement reason. With the likelihood of each replace- ment reason, a combined probability curve could be developed using basic probability theory as follows: P A B P A P B P A B∪( ) = ( )+ ( )− ∩( ) where Event A represents the probability of the asset life being reached due to reason A Event B represents the probability of the asset life being reached due to reason B. 4.1.5 Pavement Markings Pavement markings include the longitudinal lane, shoulder, and center lines; raised markers; and various symbols, guidance, and warning messages on the surface of the roadway. Because they are frequently in contact with tires, snowplows, precipitation, chemicals, and debris and are subject to direct sunlight, pavement markings deteriorate quickly. Yet they are extremely effective in facilitating safe and efficient travel (FHWA 1994). Replacement decisions are mostly condition-driven but can result from changes in requirements such as relocating lanes or recon- figuring intersections or changes in standards. The example provided for the life expectancy analysis focuses on condition-related replacement. 4.1.5.1 Measuring Condition and Performance Agencies typically try to calibrate their condition assessment of pavement markings with levels of safety or driver perception. The most common metric is retroreflectivity, the ability of the marking to reflect light from the headlights of a vehicle back to the driver’s eyes. Retroreflectivity degrades over time due to wear, ultraviolet and chemical attack, and accumulation of salt, dirt, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 20 40 60 80 100 Su rv iv al P ro ba bi lit y Age in Years Average Life 65 years Figure 4-7. Example life expectancy estimate of roadway lighting fixtures.

54 estimating Life expectancies of highway assets and debris. Most agencies assess retroreflectivity at least annually and at least visually, but, in some cases, use automated equipment. Agencies also assess the degree of missing or damaged markings and raised markers. WSDOT rates retroreflectivity on a scale of A-B-C-D-F (omitting E) using the cutoff values of 201, 165, 80, and 30 mcd/sq.m/lux, respectively. WSDOT assesses missing or damaged pavement markers on a section of road using the percentage cutoffs of 5%, 10%, 20%, and 30% respec- tively. For pavement markings such as stop bars, arrows, and crosswalks, WSDOT counts the percentage of these markings on a section of road that have at least 25% worn or missing. The cutoff percentages are 2%, 10%, 20%, and 40% (WSDOT 2008). FHWA has established recommended minimum retroreflectivity values for pavement mark- ings, optimized for aged asphalt pavements and passenger cars and maintained for in-service roads (Debaillon et al. 2007). These values are shown in Table 4-16. The recommendations apply to MUTCD-warranted center line and edge line pavement markings, including lane lines on Interstate highways and freeways, measured under dry conditions in accordance with the 30-m (98.4-ft) geometry described in ASTM E1710. The reduction factor recommended for raised reflective pavement markers (RRPMs) assumes that the RRPMs are in good working condition and that at least three of them are visible to nighttime drivers at any point along the road. On two-lane highways with RRPMs along the center line only, the reduction factor applies to both center lines and edge lines. Yellow lines, when new, have lower retroreflectivity than white lines. Since the two colors deteriorate at about the same rate, yellow pavement markings are seen in practice to have a shorter asset life. Some states compensate by establishing a replacement threshold for white markings that is 20% higher than for yellow (Markow 2007). 4.1.5.2 End-of-Life Criteria For the example analysis, the end-of-life criterion is the age when there is a 50% probabil- ity of reaching level F (using the Washington State definitions) or violating the federal rec- ommended minimum retroreflectivity levels. Most states make pavement marking decisions based on condition, rather than life expectancy, so the 50% level is appropriate for budgeting decisions. If life expectancy is to be used as the asset-level replacement criterion (without measuring actual retroreflectivity), then the probability threshold should be set lower. This change would yield a lower probability of violating the minimum standard and a shorter asset life. This is a case where effective performance measurement translates directly to life exten- sion and cost savings. 4.1.5.3 Life Extension Interventions Agencies commonly perform routine street cleaning to remove dirt, film, and debris from the road surface and improve the visibility of pavement markings. For the example analysis, data on the frequency of street cleaning were not available. Agencies that have this information Roadway marking configuration Without RRPMs With RRPMs <= 50 mph 55-65 mph >= 70 mph Fully-marked roadways 40 60 90 40 Roadways with center lines only 90 250 575 50 (Debaillon et al. 2007) Retroreflectivity measured in mcd/sq.m/lux. Recommendation applies to both white and yellow. Table 4-16. Recommended minimum in-service retroreflectivity of pavement markings.

Develop Foundation tools: how to Compute Life expectancy Models 55 can perform a lifecycle cost analysis, as in Chapter 5, to determine optimal cleaning intervals to maximize the life expectancy of pavement markings. 4.1.5.4 Published Life Expectancy Values Data on the life expectancy of pavement markings was gathered in Markow (2007) from a survey of transportation agencies. This information is primarily from expert judgment. Table 4-17 summarizes the number of responding agencies and the median estimate in years, for each type. The life expectancy of pavement markings can be sensitive to installation quality, winter chemical application, and snow removal practices. Some agencies install markings into a shal- low groove in the pavement to prolong the life expectancy. 4.1.5.5 Example Analysis The life expectancy of pavement markings varies with respect to different factors such as color and marking material type. The following example illustrates the Weibull-distributed survival probability model that was developed on the basis of “1A: 2-year Waterborne yellow markings” data from existing test decks of NTPEP. The skip-retroreflectivity value of 65 mcd/sq.m/lux was taken as the end-of-life performance threshold. y gg1 1 0= − × ( )( )exp . α β where y1g is survival probability as a function of age g ≡ the age at which the survival probability is sought, in months. b = shape parameter, 3.87 and the scaling parameter is given by α = ( − ∗exp . .1 1 0 58 Orientation 1if longitudinal, 0 if transverse Initial Ret ( ) − ∗0 01. roreflectivity value Road surface.− ∗0 29 type 1if asphalt, 0 if concrete( )) The percentiles of survival distribution can be plotted to give an indication of life expectancy. In this case, the plot suggests that 25% of the markings have an asset life of approximately 45 months or more, while 75% of the markings have an asset life of at least 18 months. On aver- age, the calibrated model indicates an average life of 26 months (Figure 4-8). The marking performance also can be rated using a discrete subjective rating process that may enable the modeler to apply alternative estimation methods such as Markov chains or ordered probit models. A rating scale may be more appropriate than the current continuous rating based on retroreflectivity only given that markings can deteriorate due to abrasion, lack of durability, and lack of contrast. Lane and edge striping Pa ve ment markers Ty pe Count Life Ty pe Count Life Ty pe Count Life Non-epoxy paint 22 1 yr Polyester 2 2.3 Ceramic 2 3 Epoxy paint 13 4 Tape 5 6 Raised 10 3 Thermoplastic 16 4 Thin thermo plastic 1 1-2 Recessed 6 2.5 Cold plastic 8 5 Preformed thermopl astic 1 3 Raised snowplowable 1 4 Table 4-17. Survey of life expectancy estimates for pavement markings (Markow 2007).

56 estimating Life expectancies of highway assets 4.1.6 Curbs, Gutters, and Sidewalks Curb and sidewalk replacement is often driven by functional stimulus such as changes in requirements, changes in land use, urban betterment projects, or related roadway projects such as widening. Condition-related replacement can occur when movement or deterioration cause the asset to exceed a level-of-service standard for accessibility, driven by concern for lawsuits or compliance with the Americans with Disabilities Act (ADA). In residential areas, aesthetics can also play a significant role in the decision to replace assets of these types. 4.1.6.1 Measuring Condition and Performance Condition assessment of sidewalks occurs very infrequently, if at all. Most agencies in a recent survey assessed sidewalk condition less often than once every 2 years. Portland, Oregon, for example, with a relatively mature asset management program, performs sidewalk assessments on a 20-year cycle (Markow 2007). Typically, in many agencies, complaints trigger an inspection, at which time the sidewalk may be evaluated using LOS standards. The sidewalk is replaced if it fails the standards. 4.1.6.2 End-of-Life Criteria An appropriate network-level end-of-life criterion is the age at which there is a 50% chance that a sidewalk inspection will fail the level-of-service standards over an extensive length. The project-level criterion would be the actual violation of standards over an extensive length. 4.1.6.3 Life Extension Interventions For isolated cracks or slab movement, agencies have several life extension options available, including crack sealing, mudjacking, tree root removal, drainage improvements, and planing or filling of projections and tripping hazards. Given that both the costs and benefits of these activi- ties are low, life extension decisions are typically made using engineering judgment. Duration .20 .40 .60 .80 1.00 .00 5 10 15 20 25 30 35 400 Estimated Survival Function for LTIME Su rv iv al P ro ba bi lit y Average Life 26 months Figure 4-8. Example life expectancy estimate of 1A: 2-yr Water-Based Yellow Pavement Marking.

Develop Foundation tools: how to Compute Life expectancy Models 57 4.1.6.4 Published Life Expectancy Values Data on the life expectancy of curbs and sidewalks were gathered in Markow (2007) from a survey of transportation agencies. This information is primarily from expert judgment. Table 4-18 summarizes the number of responding agencies and the median estimate in years, for each type. 4.1.6.5 Example Analysis The New York State Department of Transportation (NYSDOT) is one of the few agencies that has developed basic models for bridge sidewalk fascia deterioration. The agency uses a sidewalk condition rating (CR) on a scale of 0 (worst) to 7 (best). NYSDOT developed the following dete- rioration model for concrete bridge sidewalks (Agrawal and Kawaguchi 2009): CR E- Age E- Age E-= − ∗( )+ ∗( ) − ∗7 0 698 1 0 190 3 0 4 62. . . Age( )3 Assuming this deterioration function and an end-of-life criterion of CR = 2, the life of side- walk fascia design, on the basis of the collected data, is 90 years. The New York study provides similar deterioration curves for other bridge-related elements. 4.1.7 Pavements Pavements represent the most extensive and expensive asset type in larger transportation agencies. Pavement management systems provide modeling of deterioration and life expectancy, sensitive to the factors that are important to each agency. Such models may distinguish rigid, flexible, and granular traveled surfaces for various categories of traffic and subgrade charac- teristics. The models may also address shoulders, curbs and sidewalks, medians, barriers, and markings. The wearing surface of a pavement may be replaced separately from the full-depth pavement structure so the surface typically has a shorter life expectancy. 4.1.7.1 Measuring Condition and Performance Transportation agencies separately measure several aspects of pavement condition, which separately or together may determine the life. Typical quantities measured are • Roughness. Typically using IRI, a measure of deviation from a smooth surface, in inches per mile; or the older PSR, a subjective measure on a scale of 0 to 5. IRI is almost universally used as the most direct measure of the public perception of pavements. • Distress. Depending on the type of pavement, the typical distresses are rutting, transverse cracking, fatigue cracking, longitudinal cracking, map/block/alligator cracking, raveling, faulting, spalling, bleeding, and flushing. In a recent survey of 55 transportation agencies (mostly state DOTs), it was found that each of these distresses is quantified by more than half of the respondents, usually on an annual basis (Flintsch and McGhee 2009). • Structural capacity. A measure of the ability of the pavement structure to carry loads. Only 16% of the respondents in the Flintsch survey routinely gather this information network-wide, but 71% gather it for specific pavement segments as part of project design. • Friction. A measure of safety, the ability of the pavement to support strong braking of vehicles without skidding. The Flintsch survey showed that 34% of respondents gather this informa- tion network-wide, and 55% gather it on a project-level basis. Of the above measures, structural capacity may be the most direct determinant of life expec- tancy. However, structural capacity data are relatively expensive to collect routinely, and few agencies do so. Among the various distresses, rutting and faulting have the most direct correla- tion to life expectancy, but any of the distresses can limit life extension possibilities.

58 estimating Life expectancies of highway assets In pavement management systems, it is common to combine various distresses into a com- posite pavement condition rating (PCR) (sometimes called Pavement Quality Index or a state- specific name) as a more convenient measure of structural condition. Each agency has its own way of calculating PCR, sensitive to its own management concerns. In some agencies, roughness, structural capacity, and/or friction may be included in the PCR. Very often, but not always, PCR is on a scale of 0-100 with 100 being like-new condition (Flintsch and McGhee 2009). Another approach, which works for multiple pavement distresses, is to add the lane-feet of any type of distress and divide by the lane-miles in a section of road. Like PCR, this quantity can be discretized into service levels. Washington State uses this measure and divides it into intervals characterized by letter grades A-B-C-D-F (omitting E). The cutoff levels, in lane-feet of distress per lane-mile, are 500, 1000, 2500, and 5000 (WSDOT 2008). Pavement management systems typically contain deterioration models. The deterioration of various distresses might be analyzed separately and then later combined to yield a forecast of PCR. Alternatively, the agency may compute PCR first and develop a single deterioration model for PCR. Usually these models are developed as deterministic regression equations, but Markov- ian models are also used by a few agencies. 4.1.7.2 End-of-Life Criteria For life expectancy analysis, the important part of the deterioration model is the point where each condition measure reaches a minimum tolerable condition (MTC). At this point, the model assumes that pavement must either be replaced or must receive some kind of life exten- sion action. If there are separate deterioration models for separate distresses, then the first one to reach the MTC determines the end-of-life (Figure 4-9, left side). As discussed in Chapter 3, knowledge of the variability in age of the end-of-life is also impor- tant because it reveals how much of a population of pavement segments will reach their end- of-life within a given time frame. In a Markovian deterioration model or other probabilistic model, this variability is easily determined because the model computes the probability distribu- Figure 4-9. MTC and uncertainty. Sidew alks Curbs Corners (urban areas) Ty pe Count Life Ty pe Count Life Ty pe Count Life Concrete 7 25 Concrete 7 20 Concrete curb s 6 20 Asphalt 5 10 Asphalt 2 10 Granite curbs 1 20 Brick or block 2 20 Granite block 1 20 Concrete ramp 4 20 Gravel, crushed rock 1 10 Stone/brick ramp 2 20 Table 4-18. Survey of life expectancy estimates for sidewalks and curbs (Markow 2007).

Develop Foundation tools: how to Compute Life expectancy Models 59 tion directly. For the more common deterministic models, it is important to have a measure of regression error in the vicinity of the point where the MTC is reached (Figure 4-9, right side). Few pavement management systems provide this information. Regardless of the deterioration model used, it is possible to work directly with historical pave- ment condition data to reach life expectancy in a simpler, more direct way. This starts with discretizing the range of PCR into two ranges, failed and not-failed. As a variation, the separate distresses could each be discretized in this way, with the pavement overall considered to have failed if any one of the separate measures has failed (Figure 4-10). Frequently, in practice, pavement life is expressed as the age when the pavement is considered to need wearing surface replacement, rather than full-depth replacement. Both definitions are useful, but the results of course will differ substantially. For wearing surface life, typical end- of-life thresholds are PCR=70 (Boyer 1999, naturally depending on how PCR is defined by the agency); PSR=2.5 (CTC 2004); and IRI=170 (FHWA 2008). Full-depth life would be indicated by levels of rutting, faulting, or structural capacity that indicate that mere surface replacement would not be sufficiently effective. Also, in practice, stud- ies for specific transportation agencies express a longer term lifespan in terms of the total life of the original pavement plus the next three or four overlays (CTC 2004). 4.1.7.3 Life Extension Interventions Certain routine maintenance actions, if performed consistently, can extend the life of pave- ments. These actions include crack sealing, surface sealing, spall patching, and drainage main- tenance. Deficiencies in roughness, certain distresses, and friction can often be corrected, at least temporarily, using life extension actions. In addition, replacement of the wearing surface is often performed as a life extension activity for the full-depth pavement structure. Chapter 5 introduces some of the concepts of life extension, using deterioration and lifecycle cost models. When estimating pavement life expectancy from historical data, it is important to know the types of routine maintenance and repair/rehabilitation actions that have been performed during each road segment’s history. In many agencies, this information is missing or very difficult to use. Without this knowledge, a typical life expectancy can still be estimated, but it will not have reliable sensitivity to changes in maintenance policy, making it less useful for many common applications. 4.1.7.4 Published Life Expectancy Values Existing literature is inconsistent about pavement life expectancy, apparently because the states differ in their construction methods, material specifications, maintenance decision- making, performance measurement, traffic characteristics, soils, and climate (CTC 2004). Figure 4-10. Multi-scale end-of-life criterion.

60 estimating Life expectancies of highway assets Published values of age at first overlay for asphalt concrete pavements range from 11 to 20 years; and for reinforced concrete pavements, from 20 to 34 years. The full-depth pavement life for both types of pavements is typically quoted at about 50 years; however, there is little published evidence behind these numbers. 4.1.7.5 Example Analysis Data for analyzing pavement life and pavement treatment life were collected from the Long- Term Pavement Performance (LTPP) database and from two state DOTs. Data for new asphalt pavements were from the General Pavement Study—1 (GPS-1) of the LTPP database. Pavement sections in the GPS experiment included those with a hot-mix asphalt concrete (HMAC) surface layer with or without other HMAC layers (total HMAC layers thickness ~ 4–8 inches), placed over a granular base. The life of flexible pavement rehabilitation treatments was modeled using data from the Specific Performance Study # 5 (SPS-5) of LTPP’s western region. SPS-5 has nine test sections in each participating state, and the requisite data were obtained for all the sections at all five states in the SHRP-LTPP western regions. Data included test site location, rehabilitation year, condition (in terms of IRI), climate, and treatment characteristics (e.g., thickness of new layer, level of surface preparation, and mix type). Life of New Asphaltic Concrete Pavements. A non-parametric survival analysis (Kaplan- Meier method) was conducted to estimate the actual probability of survival of the flexible pave- ment sections in relation to pavement age. For purposes of illustration, it was considered that a pavement section has failed when IRI>150. Having chosen this threshold value, the estimated life represents the age at which the pavement section will need its first rehabilitation treatment. The survival curve for the GPS-1 pavement sections is shown in Figure 4-11. The figure suggests that the average life of an asphaltic pavement is approximately 25 years. An age-based model was developed to determine the life of different rehabilitation techniques in the LTPP SPS-5 study. The number of observations is 493 and the resulting model is ln . . . .IRI AGE LTHICK( ) = + ∗( ) − ∗( ) −0 035 0 049 0 12 0 19 0 522∗( ) =SPREP R; . where ln(IRI) = the natural log of IRI of a treated pavement section in given year in m/km; AGE = Time elapsed since the rehabilitation treatment, in years; LTHICK = Indicator variable for thickness of the rehabilitation treatment (1 if 5 inches and 0 if 2 inches); SPREP = Indicator variable for surface preparation of rehabilitation treatment (1 if intensive and 0 if minimal). Life of Functional AC (Asphalt Concrete) Overlay Treatment. Functional AC overlay is a common rehabilitation treatment for AC pavements. The following model was developed using data from Interstates in a mid-western state in the United States. Using these data, it was deter- mined that the best regression model for functional AC overlay performance is IRI e PRE IRI AGE T= − + × ( )+ × − × ×1 37 2 18 0 3 10 5. . log _ . RAADT PRECIP R+ × =0 03 2 0 59. , . where PRE_IRI = IRI before the implementation of the treatment; AGE = Treatment age; TRAADT = Truck annual average daily traffic; PRECIP = Annual average precipitation. This makes AGE the subject of the equation and, assuming that when IRI reaches the thresh- old value, treatment age can be found which is equal to the treatment life, tSL.

Develop Foundation tools: how to Compute Life expectancy Models 61 The functional AC overlay average life can be estimated in years. For instance, using the aver- age values in the model, the following result was obtained: t IRI Avg PRE SL Threshold IRI = ( )+ − ×ln . . log1 37 2 18 ( )[ ]− × ( ) × × ( ) =− 0 03 0 3 10 5 . . Avg PRECIP Avg TRAADT 16 The functional AC overlay average service life was estimated at 16 years. In this illustration, the average values of the independent variables were used to estimate the average life. Life of Resurfacing Treatment on Flexible Pavement. Data from Washington State were used to model the performance of resurfacing on existing flexible pavements. The per- formance indicator, IRI was used to categorize the pavements into five groups—‘very good’ (5) for IRI=<60, ‘good’ (4) for 60<IRI<94, ‘fair’ (3) for 94<IRI<170, ‘mediocre’ (2) for 170<IRI<220, and ‘poor’ for IRI=>220. The end-of-life was defined as the time when IRI equals 220. A simple Markov chain model was developed, with a transition matrix as shown in Table 4-19. The model was calibrated according to the average deterioration curve, a quadratic function of the average ages in each condition state. The resulting survival curve in Figure 4-12 suggests that the resurfacing treatment has a median life of 12 years. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 45 Pr ob ab ili ty o f S ur vi va l Age (years) Figure 4-11. Survival curve (K-M) for rehabilitation treatments of asphaltic concrete pavements. To condition state: From condition state: 5 4 3 2 1 5 0.8176 0.1824 0 0 0 4 0 0.7408 0.2592 0 0 3 0 0 0.6230 0.3770 0 2 0 0 0 0.4361 0.5639 1 0 0 0 0 1.0000 Table 4-19. Markov model of pavement resurfacing.

62 estimating Life expectancies of highway assets 4.1.8 Bridges Bridges consist of a collection of separate components, each with its own life expectancy. Based on site characteristics, design considerations, and market conditions, bridge designers attempt to minimize the cost of providing a given crossing for a period of 50 to 100 years. With such a long design lifespan, the end of a bridge’s actual life is often shaped more by land use, economic conditions, climate change, and service standards than by material deterioration. Over a bridge’s long life, its individual components undergo traffic, weather, floods, earth- quakes, collisions, movement, and fatigue, and eventually need to be replaced. At the end of a bridge’s life, it may have little left of its original structure with the exception of the foundation. Certain bridge elements are designed to take the most punishment and are intended to be replaced at relatively frequent intervals, protecting the larger and more expensive components to prolong their lives. These protective elements include expansion joints, coating systems, deck wearing surfaces, cathodic protection systems, bearings, drainage systems, pile jackets, fenders, and slope protection. The protective elements are of special concern in life expectancy analysis. 4.1.8.1 Measuring Condition and Performance Bridges in the United States are routinely inspected, in most states on a 2-year interval, accord- ing to two sets of standards: • The “Federal NBI Standards” were created in the early 1970s based on a Congressional mandate to provide a continuous national picture of the conditions and performance of the nation’s bridges, mainly from a perspective of functionality and safety (FHWA 1995). Table 4-20 shows the definitions of the three NBI data items describing bridge condition. • The “AASHTO Guide for Commonly-Recognized (CoRe) Structural Elements” was created in 1992 as a basis for states to describe bridge element condition at an appropriate level of detail for maintenance management (AASHTO 1997, 2002, and 2010). Table 4-21 lists the structural elements addressed by the AASHTO guide. Table 4-22 shows selected examples of condition state descriptions used by bridge inspectors to classify bridge elements. All states are required to provide NBI data to FHWA each year, generally for all bridges and culverts over 20 feet in span that are open to the public, regardless of ownership. Forty-five states currently collect AASHTO CoRe Element data, at least for state-owned bridges. Many states gather NBI and/or AASHTO CoRe Element data for other structures where they are not 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 5040 Su rv iv al P ro ba bi lit y Age in Years Average Life 12 years Figure 4-12. Example life expectancy estimate of pavements treated with resurfacing.

Develop Foundation tools: how to Compute Life expectancy Models 63 mandated, including non-bridge structures and bridges or culverts of less than 20 feet in span. Forty of the states use AASHTO’s Pontis Bridge Management System to manage and use NBI and CoRe Element data (Thompson 2006). 4.1.8.2 End-of-Life Criteria Bridges generally can qualify for federal funding for replacement if any one of the three NBI condition ratings is 4 or below. Because of funding scarcity, pre-construction activities, or related road network plans, agencies may allow a bridge to remain in condition level 4, or even condition level 3, for many years before replacing the structure. There also are often life extension opportu- nities at these condition levels that would improve condition for some period of time. The NBI condition level definitions generally are not concerned with bridge maintenance and do not address the important protective elements listed above. As a result, the most relevant life expectancy issues of expansion joints, coating systems, wearing surfaces, and other shorter-lived bridge components cannot be addressed with NBI data. Most of the agencies that collect AASHTO CoRe Element data use AASHTO’s Pontis Bridge Management System to perform lifecycle cost analysis of bridge elements (Thompson 2006). In most cases, the worst-defined condition state of each element is the optimal level for element replacement. As a result, the CoRe Element language provides useful end-of-life definitions. It is convenient to define end-of-life of an element as the age when there is a 50% chance of a given unit of the element to be in its worst-defined condition state. A more sophisticated lifecycle cost analysis may indicate a different probability level. For a bridge as a whole, the definition of end-of-life is trickier. End-of-life could be defined as the age when 50% of all the elements of the bridge (perhaps on a cost-weighted basis) are in their worst-defined condition states. To account for the many life extension opportunities, bridge end-of-life could alternatively be defined as the age when replacement has a lower lifecycle cost than any other preservation strategy. In both cases, it would be assumed that no additional preservation actions are taken in the meantime. For a bridge under a proactive maintenance National Bridge Inventory condition data items: 58 – Deck condition 59 – Superstructure condition 60 – Substructure condition 9. EXCELLENT CONDITION 8. VERY GOOD CONDITION - no problems noted. 7. GOOD CONDITION - some minor problems. 6. SATISFACTORY CONDITION - structural elements show some minor deterioration. 5. FAIR CONDITION - all primary structural elements are sound but may have minor section loss, cracking, spalling or scour. 4. POOR CONDITION - advanced section loss, deterioration, spalling or scour. 3. SERIOUS CONDITION - loss of section, deterioration, spalling, or scour have seriously affected primary structural components. Local failures are possible. Fatigue cracks in steel or shear cracks in concrete may be present. 2. CRITICAL CONDITION - advanced deterioration of primary structural elements. Fatigue cracks in steel or shear cracks in concrete may be present or scour may have removed substructure support. Unless closely monitored, it may be necessary to close the bridge until corrective action is taken. 1. "IMMINENT" FAILURE CONDITION - major deterioration or section loss present in critical structural components or obvious vertical or horizontal movement affecting structure stability. Bridge is closed to traffic but corrective action may put back in light service. 0. FAILED CONDITION - out of service - beyond corrective action. Table 4-20. NBI condition data items.

64 estimating Life expectancies of highway assets program, it is conceivable that asset life could be extended far beyond its design life, until fatigue, functional requirements, or natural or man-made hazards finally bring its life to an end. 4.1.8.3 Life Extension Interventions Bridge life extension activities can occur at any point in a structure’s life. Bridge washing and concrete sealing can occur even on new bridges. Some of the most cost-effective life extension options occur with bridges in mid-life, when opportunities arise to keep protective systems such as expansion joints, paint, wearing surfaces, and bearings in good repair. During the life of AASHTO Commonly-Recognized (CoRe) Structural Elements 12 - Concrete Deck - Bare 156 - Timber Floor Beam 13 - Concrete Deck - Unprotected w/ AC Overlay 160 - Unpainted Steel Pin and/or Pin and Hanger Assembly 14 - Concrete Deck - Protected w/ AC Overlay 161 - Painted Steel Pin and/or Pin and Hanger Assembly 18 - Concrete Deck - Protected w/ Thin Overlay 201 - Unpainted Steel Column or Pile Extension 22 - Concrete Deck - Protected w/ Rigid Overlay 202 - Painted Steel Column or Pile Extension 26 - Concrete Deck - Protected w/ Coated Bars 204 - P/S Conc Column or Pile Extension 27 - Concrete Deck - Protected w/ Cathodic System 205 - Reinforced Conc Column or Pile Extension 28 - Steel Deck - Open Grid 206 - Timber Column or Pile Extension 29 - Steel Deck - Concrete Filled Grid 210 - Reinforced Conc Pier Wall 30 - Steel Deck - Corrugated/Orthotropic/Etc. 211 - Other Material Pier Wall 31 - Timber Deck - Bare 215 - Reinforced Conc Abutment 32 - Timber Deck - w/ AC Overlay 216 - Timber Abutment 38 - Concrete Slab - Bare 217 - Other Material Abutment 39 - Concrete Slab - Unprotected w/ AC Overlay 220 - Reinforced Conc Submerged Pile Cap/Footing 40 - Concrete Slab - Protected w/ AC Overlay 225 - Unpainted Steel Submerged Pile 44 - Concrete Slab - Protected w/ Thin Overlay 226 - P/S Conc Submerged Pile 48 - Concrete Slab - Protected w/ Rigid Overlay 227 - Reinforced Conc Submerged Pile 52 - Concrete Slab - Protected w/ Coated Bars 228 - Timber Submerged Pile 53 - Concrete Slab - Protected w/ Cathodic System 230 - Unpainted Steel Cap 54 - Timber Slab 231 - Painted Steel Cap 55 - Timber Slab - w/ AC Overlay 233 - P/S Conc Cap 101 - Unpainted Steel Closed Web/Box Girder 234 - Reinforced Conc Cap 102 - Painted Steel Closed Web/Box Girder 235 - Timber Cap 104 - P/S Conc Closed Web/Box Girder 240 - Unpainted Steel Culvert 105 - Reinforced Concrete Closed Webs/Box Girder 241 - Reinforced Concrete Culvert 106 - Unpainted Steel Open Girder/Beam 242 - Timber Culvert 107 - Painted Steel Open Girder/Beam 243 - Other Culvert 109 - P/S Conc Open Girder/Beam 300 - Strip Seal Expansion Joint 110 - Reinforced Conc Open Girder/Beam 301 - Pourable Joint Seal 111 - Timber Open Girder/Beam 302 - Compression Joint Seal 112 - Unpainted Steel Stringer 303 - Assembly Joint/Seal (modular) 113 - Painted Steel Stringer 304 - Open Expansion Joint 115 - P/S Conc Stringer 310 - Elastomeric Bearing 116 - Reinforced Conc Stringer 311 - Moveable Bearing (roller, sliding, etc.) 117 - Timber Stringer 312 - Enclosed/Concealed Bearing 120 - Unpainted Steel Bottom Chord Thru Truss 313 - Fixed Bearing 121 - Painted Steel Bottom Chord Thru Truss 314 - Pot Bearing 125 - Unpainted Steel Thru Truss (excl. bottom chord) 315 - Disk Bearing 126 - Painted Steel Thru Truss (excl. bottom chord) 320 - P/S Concrete Approach Slab w/ or w-o/AC Only 130 - Unpainted Steel Deck Truss 321 - Reinforced Conc Approach Slab w/ or w/o AC Only 131 - Painted Steel Deck Truss 330 - Metal Bridge Railing - Uncoated 135 - Timber Truss/Arch 331 - Reinforced Conc Bridge Railing 140 - Unpainted Steel Arch 332 - Timber Bridge Railing 141 - Painted Steel Arch 333 - Other Bridge Railing 143 - P/S Conc Arch 334 - Metal Bridge Railing - Coated 144 - Reinforced Conc Arch 356 - Steel Fatigue 145 - Other Arch 357 - Pack Rust 146 - Cable - Uncoated (not embedded in concrete) 358 - Deck Cracking 147 - Cable - Coated (not embedded in concrete) 359 - Soffit of Concrete Deck or Slab 151 - Unpainted Steel Floor Beam 360 - Settlement 152 - Painted Steel Floor Beam 361 - Scour 154 - P/S Conc Floor Beam 362 - Traffic Impact 155 - Reinforced Conc Floor Beam 363 - Section Loss Table 4-21. AASHTO CoRe Elements.

Develop Foundation tools: how to Compute Life expectancy Models 65 13 - Concrete Deck - Unprotected w/ AC Overlay 107 - Painted Steel Open Girder/Beam 1. The surfacing on the deck has no patched areas and there are no potholes in the surfacing. 1. There is no evidence of active corrosion, and the paint system is sound and functioning as intended to protect the metal surface. 2. Patched areas and/or potholes or impending potholes exist. Their combined area is 10% or less of the total deck area. 2. There is little or no active corrosion. Surface or freckled rust has formed or is forming. The paint system may be chalking, peeling, curling, or showing other early evidence of paint system distress, but there is no exposure of metal. 3. Patched areas and/or potholes or impending potholes exist. Their combined area is more than 10% but 25% or less of the total deck area. 3. Surface or freckled rust is prevalent. There may be exposed metal, but there is no active corrosion which is causing loss of section. 4. Patched areas and/or potholes or impending potholes exist. Their combined area is more than 25% but less than 50% of the total deck area. 4. Corrosion may be present but any section loss due to active corrosion does not yet warrant structural analysis of either the element or the bridge. 5. Patched areas and/or potholes or impending potholes exist. Their combined area is 50% or more of the total deck area. 5. Corrosion has caused section loss and is sufficient to warrant structural analysis to ascertain the impact on the ultimate strength and/or serviceability of either the element or the bridge. 106 - Unpainted Steel Open Girder/Beam 111 - Timber Open Girder/Beam 1. There is little or no corrosion of the unpainted steel. The weathering steel is coated uniformly and remains in excellent condition. Oxide film is tightly adhered. 1. Investigation indicates no decay. There may be superficial cracks, splits, and checks having no effect on strength or serviceability. 2. Surface rust or surface pitting has formed or is forming on the unpainted steel. The weathering steel has not corroded beyond design limits. Weathering steel color is yellow orange to light brown. Oxide film has a dusty to granular texture. 2. Decay, insect/marine borer infestation, abrasion, splitting, cracking, checking, or crushing may exist but none is sufficiently advanced to affect strength or serviceability of the element. 3. Steel has measurable section loss due to corrosion but does not warrant structural analysis. Weathering steel is dark brown or black. Oxide film is flaking. 3. Decay, insect/marine borer infestation, abrasion, splitting, cracking, or crushing has produced loss of strength or deflection of the element but not of a sufficient magnitude to affect the serviceability of the bridge. 4. Corrosion is advanced. Oxide film has a laminar texture with thin sheets of rust. Section loss is sufficient to warrant structural analysis to ascertain the impact on the ultimate strength and/or serviceability of either the element or the bridge. 4. Deterioration is advanced. Decay, insect/marine borer infestation, abrasion, splits, cracks, or crushing has produced loss of strength or deflection that affects the serviceability of the bridge. 109 - P/S Conc Open Girder/Beam 110 - Reinforced Conc Open Girder/Beam 1. The element shows little or no deterioration. There may be discoloration, efflorescence, and/or superficial cracking but without effect on strength and/or serviceability. 1. The element shows little or no deterioration. There may be discoloration, efflorescence, and/or superficial cracking but without effect on strength and/or serviceability. 2. Minor cracks and spalls may be present, and there may be exposed reinforcing with no evidence of corrosion. There is no exposure of the prestress system. 2. Minor cracks and spalls may be present, but there is no exposed reinforcing or surface evidence of rebar corrosion. 3. Some delamination and/or spalls may be present. There may be minor exposure but no deterioration of the prestress system. Corrosion of non-prestressed reinforcement may be present, but loss of section is incidental and does not significantly affect the strength and/or serviceability of either the element or the bridge. 3. Some delamination and/or spalls may be present and some reinforcing may be exposed. Corrosion of rebar may be present, but loss of section is incidental and does not significantly affect the strength and/or serviceability of either the element or the bridge. 4. Delamination, spalls, and corrosion of non-prestressed reinforcement are prevalent. There may also be exposure and deterioration of the prestress system (manifested by loss of bond, broken strands or wire, failed anchorages, etc). There is sufficient concern to warrant an analysis to ascertain the impact on the strength and/or serviceability of either the element or the bridge. 4. Deterioration is advanced. Corrosion of reinforcement and/or loss of concrete section are sufficient to warrant analysis to ascertain the impact on the strength and/or serviceability of either the element or the bridge. Table 4-22. Example AASHTO CoRe Element condition states. (continued on next page)

66 estimating Life expectancies of highway assets a bridge, its deck may be entirely replaced two or more times. It is often possible to replace the entire superstructure. Concrete rehabilitation activities and slope protection on the substructure can keep it in service for a very long time. Bridge management systems, with their thorough deterioration models and lifecycle costing capabilities, are necessary for finding the best life extension opportunities. 4.1.8.4 Published Life Expectancy Values There are no authoritative published sources of life expectancy estimates for bridges, other than those concerned with design life. However, many states have now collected 12 years’ or more of CoRe Element data, enough to develop reliable life expectancy estimates. The Pontis Bridge Man- agement System has a built-in process, described in Chapter 5, to generate Markovian transition probabilities from inspection data (Cambridge 2003). Life expectancy estimates can be readily generated from Markovian transition probability matrices using the methods described later in this chapter. 4.1.8.5 Example Analysis A 2010 study for Florida DOT (Thompson and Sobanjo 2010) used the one-step method described in Chapter 5 to estimate Markovian transition probabilities for groups of bridge and non-bridge elements in the Florida inventory. The bridge elements use the CoRe Element condi- tion rating system described above. Table 4-23 presents the resulting life expectancy estimates for all of the bridge and non-bridge elements. From these estimates, it can be seen that cross-sectional methods such as Markovian models are capable of providing life expectancy estimates for very long-lived facilities. In Florida’s inventory, the concrete elements in particular enter the worst condition state, where replacement may be warranted, very infrequently. This fact leads to life expectancies of hundreds of years in some cases. Given that Florida has more than 19,000 structures and biennial inspections covering 14 years of history, the sample sizes used in these estimates range from 547 to 47,725 inspection pairs. Concrete elements have the largest sample sizes because they are the most common material used in Florida’s inventory. Florida’s results, in a relatively benign environment where deicing chemicals are not used, are not necessarily indicative of other states. An FHWA study of Pontis deterioration models across the 300 - Strip Seal Expansion Joint 311 - Moveable Bearing (roller, sliding, etc.) 1. The element shows minimal deterioration. There is no leakage at any point along the joint. Gland is secure and has no defects. Debris in joint is not causing any problems. The adjacent deck and/or header are sound. 1. The element shows little or no deterioration. The paint system, if present, is sound and functioning as intended to protect the metal. The bearing has minimal debris and corrosion. Vertical and horizontal alignments are within limits. Bearing support member is sound. Any lubrication system is functioning properly. 2. Signs of seepage along the joint may be present. The gland may be punctured, ripped, or partially pulled out of the extrusion. Significant debris is in all or part of the joint. Minor spalls in the deck and/or header may be present, adjacent to the joint. 2. The paint system, if present, may show moderate to heavy corrosion with some pitting but still functions as intended. The assemblies may have moved enough to cause minor cracking in the supporting concrete. Debris buildup is affecting bearing movement. Bearing alignment is still tolerable. 3. Signs or observance of leakage along the joint may be present. The gland may have failed from abrasion or tearing. The gland has pulled out of the extrusion. Major spalls may be present in the deck and/or header adjacent to the joint. 3. There is advanced corrosion with section loss. There may be loss of section of the supporting member sufficient to warrant supplemental supports or load restrictions. Bearing alignment may be beyond tolerable limits. Shear keys may have failed. The lubrication system, if any, may have failed. Table 4-22. (Continued).

Develop Foundation tools: how to Compute Life expectancy Models 67 Element ty pe Life (y rs) Element ty pe Life (y rs) A1- Concrete deck 146 G1- Reinforced concrete culverts 208 A2- Concrete slab 98 G2- Metal and other culverts 91 A3- Prestressed concrete slab 174 H1- Channel 66 A4- Steel deck 37 I1- Pile jacket w/o cathodic protection 63 A5- Timber deck/slab 41 I2- Pile jacket with cathodic protection 150 A6- Approach slabs 83 I3- Fender/dolphin/bulkhead/seawall 60 B1- Strip Seal expansion joint 67 I4- Reinforced conc slope protection 99 B2- Pourable joint seal 23 I5- Timber slope protection 260 B3- Compression joint seal 21 I 6- Ot her (incl asphalt) slope protection 71 B4- Assembly joi nt/seal 34 I7- Drainage sy stem 17 B5- Open expansion joint 58 I7- Drainage sy stem (coated) 17 B6- Other expansion joint 92 J1- Uncoated metal wall 95 C1- Uncoated metal rail 84 J2- Reinforced concrete wall 158 C2- Coated metal rail 45 J3- Timber wall 61 C3- Reinforced concrete railing 163 J4- Other (incl masonry) wall 62 C4- Timber railing 26 J5- Mechanically stabilized earth wall 119 C5- Other railing 62 K1- Sign structures/hi-mast light poles 51 D1- Unpainted steel super/substructure 46 K1- Sign str/hi-mast light poles (coated) 99 D2- Painted girder/floorbeam/cable/p&h 99 L1- Moveable bridge mechanical 73 D3- Painted steel stringer 323 L2- Moveable bri dge brakes 25 D4- Painted steel truss bottom 51 L3- Moveable bri dge motors 34 D5- Painted steel truss/arch top 189 L4- Moveable bridge hy draulic pow er 48 D6- Prestressed concrete superstr 335 L5- Moveable bridge pipe and conduit 37 D7- Reinforced concrete superstructure 80 L6- Moveable bri dge structure 38 D8- Timber superstructure 92 L7- Moveable bri dge locks 31 E1- Elastomeric bearings 393 L8- Moveable bridge live load items 32 E2- Metal bearings 72 L9- Moveable bridge cw/trunion/track 124 F1- Painted steel substructure 32 M1- Moveable bridge electronics 70 F2- Prestressed column/pile/cap 142 M2- Moveable bridge submarine cable 22 F3- Reinforced concrete column/pile 200 M3- Moveable bridge control console 31 F5- Reinforced c oncrete abutment 656 M4- Moveable bridge navigational lights 23 F6- Reinforced concrete cap 428 M5- Moveable bridge operator facilities 59 F7- Pile cap/footing 116 M6- Moveable bridge misc equipment 13 F8- Timber substructure 58 M7- Moveable bridge barriers/gates 37 M8- Moveable bridge traffic signals 41 Table 4-23. Florida bridge and non-bridge element life expectancies (Thompson and Sobanjo 2010). nation (Thompson 2007) found that a state with a very severe winter environment, such as Maine, can have bridge element life expectancies that are only half those of Florida. In warm very dry regions, such as southern California, life expectancy may be more than twice as long as in Florida. 4.1.9 Other Asset Types Although not within the scope of this guide, there are several other highway asset types for which a life expectancy analysis is appropriate and for which the methods described in this guide could be used: • Paved and unpaved ditches and swales • Storm detention ponds • Dams • Fences • Landscaping

68 estimating Life expectancies of highway assets • Retaining walls • Sound barriers • Guiderails and impact attenuators • Rest area facilities • Tunnels • Weigh stations • Maintenance facilities • Highway agency vehicles and equipment 4.1.10 Summary Estimates From the literature, wide ranges in asset life were found, with estimates varying by material/ design type, end-of-life threshold applied, climatic conditions, and levels of applied mainte- nance. Typical values by asset class were found to be overall bridge life equal to 50–60 years, bridge deck life equal to 25–45 years, culvert life equal to 30–50 years, traffic sign life equal to 10–20 years, pavement markings life equal to 1–5 years, traffic signal life equal to 15–20 years, and roadway lighting life equal to 25–30 years. 4.2 Developing Life Expectancy Models When not from published sources, the method of developing life expectancy models depends very much on the kind of data available. The most significant considerations are as follows: • Availability of data on past replacement actions; • Availability of data on past life extension actions; • Availability of relevant inventory, condition, and performance data on existing assets; • Availability of relevant inventory, condition, and performance data on assets that no longer exist because they were replaced; • Availability of a time series of past observations of condition and performance, preferably evenly spaced in time; • Consistency of data collection definitions and processes over time; • Quality of the existing models and judgment, including research literature that can be helpful in selecting an appropriate model form; and • Degree to which the available data are representative of the population whose life expectancy is desired. The final point is especially challenging because construction methods, materials, and utili- zation change over time. Even if the agency has quality data about its historical infrastructure, newer facilities may have different performance characteristics. Thus, it may be necessary to make adjustments based on laboratory data or judgment. Another important consideration that interacts with data availability is the type of policy sensitivity desired. A model based on actual replacement activities may correspond with a com- monly understood concept of life expectancy, but the data set may contain assets replaced for various reasons that might not be representative of future assets or future policies under con- sideration (Figure 4-13). One way to respond to the diversity of most real-life data sets is to try to separate the popu- lation into groups, according to the reasons for replacement and the types of actions that may have been taken. These sorts of historical data are often very difficult to find and interpret successfully.

Develop Foundation tools: how to Compute Life expectancy Models 69 Moreover, if the goal is to quantify asset longevity in the absence of extenuating circum- stances, then it is often more useful to work with condition data directly and quantify the length of the deterioration curve, regardless of whether or not the asset was replaced exactly at the end of the curve. Historical condition data are often easier to find, especially for assets that are still in service and have not yet been replaced. Most of the examples given earlier in this chapter are based on this perspective. As Chapter 5 will show, many of the useful applications of life expectancy analysis involve life- cycle costing and a comparison of design and life extension alternatives. For these applications, it is important to try to separate the effect of simple deterioration, deterioration under preventive maintenance, and the beneficial effects of specific actions of interest. In practice, it is often easier and more useful to model these effects separately and combine them later to simulate possible future policies. 4.2.1 Ordinary Regression of Age At Replacement If the goal is a direct model of age at replacement, one approach is to develop a regression model with age at replacement as the dependent variable. Possible sources of data are as follows: • A contract management system or maintenance management system which provides the age or year of construction of the asset that was taken out of service. • Records of asset demolition, combined with archived inventory records for the demolished assets. There would need to be a way of associating records in the two databases; for example, a common identifier or description. • Archived inventory records that directly indicate the date the asset was taken out of service. • If new assets carry the same identification number or location tag as the assets they replace, then a time series of condition might show a sudden improvement that pinpoints the time of replacement. Figure 4-13. Difficulties in using historical replacement data.

70 estimating Life expectancies of highway assets The simplest possible model would be a model which does not have any explanatory variables (Table 4-24). In other words, simply make a list of all the replacement ages of the assets and compute the average. In this example, the table on the left-hand side contains a list of culverts, along with the age at which each culvert was replaced. The table on the right shows the average replacement age for each district and the standard deviation. The ability to calculate separate averages for each district is useful if this reflects different conditions of climate, topography, or soils, all of which could affect life expectancy. In a real analysis, it would be necessary to have a longer list of culverts, at least 30 in each district, in order to obtain statistically reliable results. If the number of data points available is substantially larger, it would be possible to divide up the model more finely if desired, to make it sensitive to more variables that might affect culvert life expectancy. For example, separate aver- ages could be computed for different soil types. In that case, each separate category would need at least 30 data points. The standard deviation is useful for describing how certain the estimate of life expectancy may be, when applied to a future set of culverts. That the average replacement age in District 1 was 50.75 does not mean that all future culverts will fail at the exact age of 50 years and 9 months. Some will fail sooner, some later; and the standard deviation is an estimate of how much sooner or later. Table 4-24 shows the formulas for computing standard deviation. If the data set is a complete list of all the culverts replaced, then the formula for population standard deviation should be used. If the list is a random sample, use the sample standard deviation formula. When develop- ing an application in a programming language such as Visual Basic or C#, it will be necessary to write computer code for these formulas. Table 4-24, like all the examples in this guide, can be found in a Microsoft Excel spread- sheet file available on line. A table and a graph showing the probability of replacement for each possible age of a culvert also can be found in the spreadsheet for this example. This is computed directly from the average and standard deviation, under the assumption that the List of culverts with age at replacement Average and standard deviation of age at replacement District Culvert Replace- Deviation Square of District Number of Average Population Sample name identifier ment age from avg deviation name culverts age StDev (1) StDev (2) District CulvertID ReplAge Deviation SqDev District Count AvgAge PopStDev SamStDev D1 195451 55 4.25 18.0625 D1 4 50.75 3.03 3.50 D1 185701 52 1.25 1.5625 D2 6 43.50 3.20 3.51 D1 137132 47 -3.75 14.0625 D3 5 40.60 3.01 3.36 D1 194845 49 -1.75 3.0625 D2 268014 42 -1.50 2.2500 Average age at replacement D2 205563 47 3.50 12.2500 a is culvert age, N is number of culverts D2 261619 41 -2.50 6.2500 D2 275579 48 4.50 20.2500 Population standard deviation D2 226692 39 -4.50 20.2500 (use if list is whole population) D2 278272 44 0.50 0.2500 D3 352904 46 5.40 29.1600 D3 372275 41 0.40 0.1600 Sample standard deviation D3 326486 37 -3.60 12.9600 (use if list is a random sample) D3 306439 39 -1.60 2.5600 s is an estimate of D3 314958 40 -0.60 0.3600 = = N i iaN a 1 1 ( ) = −= N i i aaN 1 21σ ( ) = − − = N i i aaN s 1 2 1 1 Table 4-24. Average age at replacement.

Develop Foundation tools: how to Compute Life expectancy Models 71 variation in replacement age is shaped like the normal distribution. Figure 4-14 shows the graph for District D1. In order to compute the probability of replacement at any given age, the formula for a normal distribution was used. This formula is Pr expob a a = − −( )    1 2 2 2 2σ pi σ where a is the age (horizontal axis) and s is the standard deviation. This formula can be used as an estimate of the fraction of culverts that will need to be replaced each year (labeled “This year” on the graph). To determine how many culverts will need to be replaced in the next 10 years, the most accurate approach is to use the cumulative normal distribution, which computes the total area under the normal distribution up to a given time. Although this distribution does not have an easy formula, there is an approximation that is just as good for practical purposes. Cum ob z z z kz kz Pr exp= − − × + +     +  1 2 1 4 1 12 2 2 pi     = −( ) =z a a k 2 0 140012 σ . The value of k is a mathematical constant and is the same for any age or type of asset. The frac- tion just before the radical, z, divided by the absolute value of z, serves only to change the sign of the square-root term so the formula works equally well before or after the average replacement age (Note: If this analysis is performed in a Microsoft Excel spreadsheet, the function NORMDIST can be used in place of this large formula for CumProb, and gives a more precise result. An example worksheet accompanying this report compares the two methods.) If a family of culverts, all installed at the same time, are now 40 years old, the number likely to be replaced in the next 10 years can be computed from Pr Pr Prob Cum ob Cum ob= ( )− ( )50 40 In other words, compute the cumulative probability before age 50, and subtract the cumu- lative probability before age 40 (the current age), to arrive at the estimate, which in this case is about 40%. Even though the average age at replacement is 50.75 years, and it is now only year 40, about 40% of the culverts probably will need to be replaced within the next 10 years, in this example. This is just another example of why it’s important to measure uncertainty in life expectancy analysis. It is useful to develop a model that has causal factors or that at least distinguishes different asset characteristics. The feasibility of this will depend, of course, on whether the distinguishing characteristics of the assets are available in the data. Two ways of doing this are • Partitioning. The data set can be divided into groups according to one or more classification variables, as was done in Table 4-24 for districts. Then, simple averaging or a regression model can be developed separately for each group. • Linear or non-linear regression. This process develops a mathematical model to compute life expectancy as a function of one or more explanatory variables (Table 4-25). Linear regression models can be developed using regression as described in the following paragraphs. Certain types of non-linear models can also be developed in this way. For more complex non-linear models, software can be used to perform maximum likelihood estimation.

72 estimating Life expectancies of highway assets Table 4-25 uses the same culverts as in Table 4-24. The only difference in the data set is that barrel length (in feet) has been included as an additional explanatory variable. The analyst believes that longer culverts are more likely to be damaged by debris washing through them and less likely to be thoroughly cleaned by the agency’s routine annual flushing, hence a shorter life expectancy. Regression variables should not be added unless the analyst has a credible intuitive reason why such variables should be significant. As in the previous example, the analyst believes “district” should be significant because it reflects different conditions of climate, topography, or soils. Because district is a categorical variable, it cannot be used directly in a regression model. A way around this is to create “dummy variables” to represent the separate districts. So the variable Dist1 is 1 if the culvert is in District 1, and 0 oth- erwise. Dist2, similarly, is 1 if in District 2, 0 otherwise. There is no Dist3 variable. This is because Dist3 would be mutually correlated with Dist1 and Dist2. In fact, it can easily be computed from Dist1 and Dist2. In a regression model, all of the variables must be independent of each other. Some software packages check for such situations; others do not. List of culverts with age at replacement Average and standard deviation of repl age District Culvert Replace- 1 if 1 if Barrel Predict Devi- Sq of District Number of Average Population name identifier ment age D1 D2 length age ation Devn name culverts age StDev District CulvertID ReplAge Dist1 Dist2 Length Pred Devn SqDev District Count AvgAge PopStDev D1 195451 55 1 0 20 52.50 2.50 6.27 D1 4 50.75 2.28 D1 185701 52 1 0 36 51.02 0.98 0.95 D2 6 43.50 2.87 D1 137132 47 1 0 40 50.68 -3.68 13.56 D3 5 40.60 2.20 D1 194845 49 1 0 62 48.80 0.20 0.04 D2 268014 42 0 1 48 45.65 -3.65 13.29 Regression results D2 205563 47 0 1 59 44.62 2.38 5.68 R-squared 0.75 D2 261619 41 0 1 86 42.30 -1.30 1.68 Variable Coeffi- Standard t-Statistic D2 275579 48 0 1 77 43.03 4.97 24.69 cient error D2 226692 39 0 1 100 40.99 -1.99 3.95 Intercept 49.02 3.77 13.01 D2 278272 44 0 1 62 44.42 -0.42 0.18 Dist1 5.22 2.85 1.83 D3 352904 46 0 0 48 44.80 1.20 1.44 Dist2 0.85 1.97 0.43 D3 372275 41 0 0 106 39.65 1.35 1.82 Length -0.09 0.04 -2.38 D3 326486 37 0 0 86 41.37 -4.37 19.11 D3 306439 39 0 0 120 38.44 0.56 0.32 D3 314958 40 0 0 116 38.74 1.26 1.58 Table 4-25. Regression of age at replacement. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Pr ob ab ili ty Age Cumulative This year Average Figure 4-14. Graph of replacement probability, from the data in Table 4-24.

Develop Foundation tools: how to Compute Life expectancy Models 73 In order to use Microsoft Excel’s linear regression capability, it is necessary to make sure it is installed. On the Data ribbon in Microsoft Excel 2007, check for “Data Analysis” in the “Analysis” section on the right side of the Data ribbon (Figure 4-15). If it is not present, do the following: 1. Click the Microsoft Office button (in the upper left corner of Figure 4-15) and then click “Excel Options.” 2. Click the “Add-Ins” tab on the left side of the window (Figure 4-16). 3. In the pick list labeled “Manage” in the bottom center of the Add-Ins window, choose “Excel Add-Ins,” then click “Go . . .” 4. Another dialog box will appear (Figure 4-17), which should list “Analysis ToolPack” as one of its choices. Check the box next to it. If “Analysis ToolPack” does not appear in the list, you may need to click “Browse . . .” and search for it. At this point you may also want to check “Solver Add-in” since this will be used in later examples in this guide. Then click OK. 5. If you are prompted to install the Analysis ToolPack, click “Yes” and proceed with installing it, according to the program’s instructions. 6. At this point, the Analysis ToolPack should appear on the Data ribbon as in Figure 4-15. With the Analysis ToolPack ready to use, click the “Data Analysis” button to start the regres- sion process. A menu of analysis types will be shown, where you should choose “Regression” and click “OK” as in Figure 4-18. When the example in the accompanying Microsoft Excel file was created, its linear regres- sion options were set up as in Figure 4-19. The “Input Y Range” should be the data set column containing the variable that you are trying to estimate, in this case the age at replacement (ReplAge). Include the column label in the range. “Input X Range” is a group of columns containing the explanatory variables for the model. It includes the columns Dist1, Dist2, and Length. “Output Range” should point to the upper left cell in an area of the worksheet that does not contain any other information because the regression procedure will overwrite these cells with the results. Click “OK” to run the regression. The results are placed in the worksheet and, from there, can be moved or reformatted as desired. The most important results are reported in the lower right table in Table 4-25. An R-squared value of 0.75 is quite good; even 0.5 is often acceptable when the data set has few good explanatory variables. The t-Statistic column shows the performance of the individual explanatory variables. If the absolute value is at least 1.5 or 2.0, then the variable is considered to be a strong contributor to the model. A smaller t-Statistic might be acceptable, however, if the variable contributes to the intuitive sensibility of the model or if it is necessary for using the model. Because a great many factors can influence deterioration, and only a few of these are ever measured, it is best to keep the number of variables minimal and just use the strongest and most necessary ones. If the R-squared value or t-Statistics are small, and there are no explanatory variables that improve them, this means that the regression method is not adding much value compared with the simple average computed in the previous example. Figure 4-15. The Data ribbon showing “Data Analysis” button.

74 estimating Life expectancies of highway assets Figure 4-16. Manage Office add-ins. Figure 4-17. Add-ins dialog.

Develop Foundation tools: how to Compute Life expectancy Models 75 Based on the results reported in this example, the predicted life expectancy of a culvert is computed from the following equation: á = + × + × − ×49 02 5 22 1 0 85 2 09. . . .Dist Dist Length Consistent with the input data, the age is in years and the length is in feet. The results are con- sistent with the previous example, in that District 1 and District 2 both have longer life expectan- cies than District 3. The effect of length is as the analyst expected. The negative coefficient means that longer lengths have shorter lifespans. Using this regression formula, the predicted replacement age estimates are placed in Table 4-25 at the beginning of this example (column Pred), for comparison with the actual values (column ReplAge). Standard deviation can be computed from this information in exactly the same way as for simple averaging, using the predicted value instead of the average. It can be seen that the new estimates are generally closer than the estimates obtained from simple averaging. The upper-right table shows smaller standard deviations, which means that the addition of barrel length as an explanatory variable improved the precision of the model. Figure 4-18. Choosing Regression. Figure 4-19. Launching the regression process.

76 estimating Life expectancies of highway assets For the purposes of programming, the method of simple averaging in the preceding example is still the most straightforward way of determining the needed level of investment in each dis- trict within any given time frame. The addition of the length variable improves the quality of forecasts for individual culverts, but it does not change the amount of variability within each district, assuming each district has about the same variability of culvert barrel lengths. What the regression model does provide is the accurate computation of priority and schedule for replace- ment of each individual culvert and a better indication of which culverts (namely, the longest ones) will be needing replacement within the 10-year program. In research studies that have developed regression models of replacement age, sample sizes of at least 100 have usually been sufficient for models having up to five or six explanatory variables. There is rarely any need to have more explanatory variables than six. This of course does not mean that every model with at least 100 data points is good. If the explanatory variables are weak or if they are moderately correlated with each other (rather than completely uncorrelated, which is desired), then larger data sets are likely to be needed. It is often useful to partition a regression model; for example, making a separate model for each district or functional class. In this case, each of the sub-models needs to have a sufficient sample size. One of the pitfalls of using regression models for life expectancy is the possibility of bias due to an effect called “censoring.” The regression model is developed from past replacements and gives an average age at replacement. This is not necessarily the same thing as life expectancy, however, because some of the assets that should be in the data set have unknown replacement dates in the future. These replacement dates are hidden, or “censored” from the analyst. Figure 4-20 shows this. The left side of the figure depicts a list of assets having various procurement and disposal dates. At the time of the analysis, many of these assets are still in service so they have unknown disposal dates in the future. On the right side, a typical normal probability distribution of replacement age is shown. If the full population is used for analysis, then among the assets procured more recently than the typical asset lifespan, some will have failed and some will still be in service. A data set that contains all of the historical replacements from this population will have too many early replacements and not enough late replacements. As a result, the right side of the normal probability distribution is cut off. In this situation, the average computed from this data set will be biased toward a shorter life expectancy than the true value. One possible solution to this problem is to limit the data set to older assets, those that were procured so long ago that they are almost certain to have been replaced. This time interval can be determined by starting from the published life expectancy estimates and adding a safety allowance; or by using a time interval that is longer than all, or nearly all (for example, 95%), of the life spans in the data set. Only assets put in service before the start of this time interval would be used in the analysis. Of course, this approach has problems which might make it difficult to follow. Older data usually are of lower quality so the precision or confidence level of the results may be reduced. Figure 4-20. Censoring of time series data.

Develop Foundation tools: how to Compute Life expectancy Models 77 Also, certain assets are so long-lived that it may be impossible to exclude enough of them. For example, the typical life span of a bridge currently in service may be 50 years, and the analyst might judge that 70 years gives enough of a safety margin to include 95% of all bridge lifespans. The agency might have relatively few records concerning bridges built so long ago though, and the oldest databases of bridge condition in the United States go back only about 40 years. As a result, correcting for one bias might cause other biases. Because of these issues, the ordinary regression approach might not work well for long-lived assets where the censoring problem arises. Fortunately, there are better alternatives, which are discussed in the following sections. 4.2.2 Markov Model In the previous section, one of the simplest possible approaches to computing life expectancy was to compute the average age of all demolished assets in a data set. Unfortunately, data issues may make this method impractical or inaccurate in many cases. There is another very simple method, the Markov model. In exchange for accepting a few simplifying assumptions, the Markov model avoids a great many of the data quality and censoring problems that plague regression models. The Markov model adopts a totally different perspective from regression models. The first important characteristic of a Markov model is that it defines end-of-life in terms of condition, rather than action. The full range of possible conditions of an asset is divided into a small num- ber of condition states. Many of the examples given in earlier sections of this guide used condi- tion rating schemes based on condition states. Two prominent examples are the Washington State Maintenance Accountability Process (WSDOT 2008) and the AASHTO CoRe Structural Elements Process (AASHTO 1997, 2002, and 2010). To use a condition state rating scheme in a Markov model of life expectancy, first define “failed” as the worst of the defined condition states. This does not necessarily mean that a struc- ture literally fell down or even that its condition is interfering with traffic. It may mean that an asset in the worst condition state is a strong candidate for replacement. It might also be a strong candidate for a life extension action such as rehabilitation. There can be any number of additional condition states besides “failed.” In the simplest case, there might be just one additional state, “not-failed.” The WSDOT process consistently uses five states, and the AASHTO CoRe Elements process uses four. If condition data are gathered using visual inspection techniques, it may be difficult to discern more than three or four states reliably. The ability to discern more condition states can produce a more precise and accurate model if the data can be gathered accurately (Table 4-26). 107 - Painted Steel Open Girder/Beam 1. There is no evidence of active corrosion, and the paint system is sound and functioning as intended to protect the metal surface. 2. There is little or no active corrosion. Surface or freckled rust has formed or is forming. The paint system may be chalking, peeling, curling, or showing other early evidence of paint system distress, but there is no exposure of metal. 3. Surface or freckled rust is prevalent. There may be exposed metal but there is no active corrosion which is causing loss of section. 4. Corrosion may be present but any section loss due to active corrosion does not yet warrant structural analysis of either the element or the bridge. 5. Corrosion has caused section loss and is sufficient to warrant structural analysis to ascertain the impact on the ultimate strength and/or serviceability of either the element or the bridge. “Failed” state “Almost failed” state Table 4-26. Condition states for a Markov model of life expectancy.

78 estimating Life expectancies of highway assets When the condition of an asset is determined, the entire asset might be classified in one of the condition states. Alternatively, the quantity of the asset (e.g., feet of culvert) might be divided among the states. For example, an inspector might assess a 100-foot long steel beam and decide that 10 feet are in state 5, 20 feet in state 4, and the remainder in state 1. Any population of assets (e.g., 100,000 feet of steel girder on 150 different bridges) can also be described by the percent in each condition state. Building on this discrete condition state concept, the Markov model makes a few additional assumptions: • Condition is determined on a regular interval, such as once a year. • Over any single interval, a unit of the asset either remains in the same condition state or jumps to one of the other states. No in-between states are observed. • The probability of jumping from any one state to any other state is a constant. The first two of these assumptions usually are dictated by routine data collection practices so they are easy to accept. The third one, often called the “memoryless assumption,” requires more thought however. Because of the memoryless assumption, a Markov deterioration model always looks like Table 4-27. If a piece of steel girder is in condition state 1 this year, then next year there is (for this example) a 95.3% chance it will still be in state 1. If there are 100,000 feet of steel girder in state 1 now, then next year 95,300 feet will still be in state 1, 4,600 feet will be in state 2, 100 feet in state 3, and none in states 4 or 5. Each row of the table sums to 100%. The numbers in the body of Table 4-27 are called “transition probabilities,” because they are the probabilities of making each possible state transition. The matrix describes what happens in one year, but it is easy to compute the transition probabilities for any number of years into the future by multiplying the matrix by itself that many times (Table 4-28). So the condition of the inventory of assets deteriorates steadily over time and obviously varies with age. However, the transition probabilities themselves are constant: they don’t change as the asset gets older, and are not affected by anything that may have happened to the facility in the past. The only variation that is allowed is an improvement in condition if an action is taken this year. This is what is meant by the “memoryless assumption.” Because future predictions of condition are made by using matrix multiplication, it is possible to start with an asset that is entirely in state 1, and repeatedly multiply by the transition prob- ability matrix until the fraction in the failed state finally reaches 50%. Doing that would simulate the years of the asset’s life until half of them have failed, thus giving an estimate of the typical life expectancy of the asset, which is flagged in Table 4-28 as 40 years. Probability of each condition state one year later (%) 1 2 3 4 5 Co nd iti on s ta te n o w 1 95.3 4.6 0.1 0 0 2 0 93.2 3.9 1.9 1.0 3 0 0 89.4 7.3 3.3 4 0 0 0 82.8 17.2 5 0 0 0 0 100 Table 4-27. Example Markov deterioration model.

Develop Foundation tools: how to Compute Life expectancy Models 79 The methods for developing Markov deterioration models are described in Chapter 5. But even without going through the process of deterioration modeling, there is a simpler, quick- and-easy way of estimating life expectancy using the ideas behind the Markov model. It proceeds through these steps (Table 4-29): 1. Starting from a list of past condition state inspections, collapse the states into just two: failed and not-failed. For example, if traffic signals are rated on a four-state scale, and a particular intersection was inspected in 2007 with 25% of signal heads in state 1, 25% in state 2, 25% in state 3, and 25% in state 4 (the “failed” state), then count this inspection as 75% not-failed and 25% failed. 2. Group the inspections of each facility into pairs, each with an interval of one year. (Other intervals are also possible, as described in the final step below.) So each pair describes the condition before and after a one-year period. 3. Remove from the pairs list any pairs that are believed to have received life extension work. This determination might be based on maintenance records if available or might be based on Markov transition probability matrix State State probability in one year Today Probability of state k next year: for all k 1 95.3 4.6 0.1 0.0 0.0 2 0 93.2 3.9 1.9 1.0 j is the condition state this year and x is the fraction in state j 3 0 0 89.4 7.3 3.3 p is the transition probability from j to k 4 0 0 0 82.8 17.2 5 0 0 0 0 100 Future condition forecasts Percent by condition state Percent by condition state Year 0 100 0 0 0.0 0.0 25 30.0 28.1 10.1 6.8 25.0 1 95.3 4.6 0.1 0.0 0.0 26 28.6 27.6 10.1 6.9 26.8 2 90.8 8.7 0.4 0.1 0.0 27 27.3 27.0 10.2 7.0 28.6 3 86.6 12.3 0.8 0.3 0.2 28 26.0 26.4 10.2 7.1 30.4 4 82.5 15.4 1.2 0.5 0.4 29 24.8 25.8 10.1 7.1 32.2 5 78.6 18.2 1.8 0.8 0.6 30 23.6 25.2 10.1 7.1 34.0 6 74.9 20.5 2.4 1.1 1.0 31 22.5 24.6 10.0 7.1 35.8 7 71.4 22.6 3.0 1.5 1.5 32 21.4 23.9 9.9 7.1 37.6 8 68.0 24.3 3.6 1.9 2.1 33 20.4 23.3 9.8 7.0 39.4 9 64.8 25.8 4.3 2.3 2.8 34 19.5 22.6 9.7 7.0 41.2 10 61.8 27.0 4.9 2.7 3.6 35 18.5 22.0 9.6 6.9 42.9 11 58.9 28.0 5.5 3.1 4.5 36 17.7 21.4 9.5 6.9 44.7 12 56.1 28.8 6.1 3.5 5.5 37 16.8 20.7 9.3 6.8 46.4 13 53.5 29.5 6.6 3.9 6.6 38 16.1 20.1 9.1 6.7 48.0 14 51.0 29.9 7.1 4.3 7.7 39 15.3 19.5 9.0 6.6 49.7 15 48.6 30.2 7.6 4.6 9.0 40 14.6 18.8 8.8 6.5 51.3 << Median life expectancy 16 46.3 30.4 8.0 5.0 10.4 41 13.9 18.2 8.6 6.4 52.9 17 44.1 30.5 8.4 5.3 11.8 42 13.2 17.6 8.4 6.2 54.5 18 42.0 30.4 8.7 5.5 13.3 43 12.6 17.0 8.2 6.1 56.0 19 40.1 30.3 9.0 5.8 14.8 44 12.0 16.5 8.0 6.0 57.5 20 38.2 30.1 9.3 6.0 16.4 45 11.5 15.9 7.8 5.9 58.9 21 36.4 29.8 9.5 6.3 18.1 46 10.9 15.3 7.6 5.7 60.4 22 34.7 29.4 9.7 6.4 19.7 47 10.4 14.8 7.4 5.6 61.8 23 33.0 29.0 9.9 6.6 21.5 48 9.9 14.3 7.2 5.5 63.1 24 31.5 28.6 10.0 6.7 23.2 49 9.5 13.8 7.0 5.3 64.4 25 30.0 28.1 10.1 6.8 25.0 50 9.0 13.3 6.8 5.2 65.7 = j jkjk pxy1 2 3 4 5 1 2 3 4 5 Year 1 2 3 4 5 Table 4-28. Markov model prediction.

80 estimating Life expectancies of highway assets improvement in condition (i.e., where the percent not-failed increased from before to after). These signal installations probably received some kind of life extension or replacement activity. 4. Over the entire list of inspection pairs, compute the average percent in failed and not-failed for the before case, and again for the after case. This is a measure of condition for the inven- tory as a whole, comparing before and after any typical one-year period when no action was taken. 5. Compute the probability of remaining in the non-failed state as the non-failed percent after, divided by the non-failed percent before. Call this the “same-state” probability. The deterio- ration probability then is one minus the same-state probability. 6. Based on the matrix algebra described above, the median life expectancy is readily computed as: t pjj = ( ) ( ) log . log 0 5 where t is the median life expectancy and pjj is the same-state probability. 7. If the 50% threshold of the failed state is too high (for example, if planning a blanket replace- ment project for an asset type where failure creates a hazard to the public), simply replace 0.5 with the desired threshold in this formula, such as 5%. If the inspection interval is something other than 1 year (it must be of some uniform length), then t is expressed in terms of intervals and can be converted to years. For example, if the inspection interval is 2 years, then multiply t by 2 in order to express life expectancy in years. This procedure is just a special case of the “one-step method” for the Markov deterioration models described in Chapter 5. Even though the method is quite rough, it may be appropriate for data sets that also are very rough, especially when the condition is only described in terms of pass/fail in the first place. The method is especially valuable because it makes efficient use of small data sets in order to develop separate models for subsets of the inventory, such as wire- mounted versus pole-mounted signal heads or components from different manufacturers or with different features. Thus, it is a very practical and useful solution for many types of assets. Original inspection data Step 1 Step 2 - Inspection pairs Step 3 Inter- Not Inter- Not Not Work section Year 1 2 3 4 failed Failed section Year failed Failed Year failed Failed done INT001 2007 25 25 25 25 75 25 INT001 2007 75 25 2008 100 0 Delete INT001 2008 80 20 0 0 100 0 INT001 2008 100 0 2009 100 0 INT001 2009 70 20 5 5 95 5 INT001 2009 100 0 2010 95 5 INT001 2010 60 15 15 10 90 10 INT002 2008 100 0 2009 95 5 INT002 2008 75 10 15 0 100 0 INT002 2009 95 5 2010 80 20 INT002 2009 70 15 10 5 95 5 INT003 2006 100 0 2007 100 0 INT002 2010 60 10 10 20 80 20 INT003 2007 100 0 2008 100 0 INT003 2006 100 0 0 0 100 0 INT003 2008 100 0 2009 90 10 INT003 2007 90 10 0 0 100 0 INT003 2009 90 10 2010 85 15 INT003 2008 75 15 10 0 100 0 INT004 2008 80 20 2009 100 0 Delete INT003 2009 65 15 10 10 90 10 INT004 2009 100 0 2010 100 0 INT003 2010 50 25 10 15 85 15 INT004 2008 30 30 20 20 80 20 Step 4 - Average condition before and after INT004 2009 100 0 0 0 100 0 All Before 98.33 1.667 After 93.89 6.111 INT004 2010 90 10 0 0 100 0 Step 5 - Transition probs Step 6 Not Median failed Failed Life Not-failed 95.48 4.52 years Failed 0 100 14.99 Table 4-29. Quick-and-dirty Markov life expectancy.

Develop Foundation tools: how to Compute Life expectancy Models 81 4.2.3 Weibull Survival Probability Model The Markov model described in the preceding section is simple, but for certain applications it may be too simple. The memoryless assumption is often viewed as a weakness because it implies that the rate of deterioration does not increase with age. Consider a galvanized steel guardrail, for example. As long as the metal coating on the rail is solid, the rail will deteriorate slowly. However, if the coating starts to break down due to chemi- cal attack (e.g., from deicing salts), contact with moving objects, and age, it begins to expose the underlying steel. The steel deteriorates at a faster rate as the effectiveness of the coating declines. This problem can be addressed with a more detailed visual inspection, such as what is com- mon on bridge rails; but an agency may not want to make a data collection investment of that magnitude. Perhaps the agency rates guardrail condition using a video log so technicians are only able to discern pass/fail condition states when viewing the video in the office. Fortunately, it is not too difficult to add age dependency to the Markov model, making it into what is called a “Weibull survival probability” model. Weibull models are useful as deterioration models, an application discussed in Chapter 5, but they are also useful for the simpler purpose of life expectancy estimation. The Weibull curve has the following functional form: y gg1 1 0= − × ( )( )exp . α β where y1g is the probability of the not-failed state at age g, if no intervening maintenance action is taken between year 0 and year g; b is the shaping parameter, which determines the initial slowing effect on deterioration (e.g., when the galvanized coating is performing well); and a is the scaling parameter, calculated as α β= ( ) t ln2 1 where t is the median life expectancy from the Markov model as calculated in the preceding section. Figure 4-21 shows the form of the Weibull curve, for four different values of the shaping parameter b, with t = 20. A shaping parameter of 1 is mathematically equivalent to a Markov model (also known as an exponential distribution), where the transition probability does not 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 Age of element (years) Pr ob ab ili ty o f s ta te 1 Markov (Beta=1) Beta=2 Beta=4 Beta=8 Figure 4-21. Examples of the Weibull survival probability model.

82 estimating Life expectancies of highway assets vary with age. Higher shaping parameters slow the initial rate of deterioration, which then accel- erates as the facility gets older. Note that all the curves intersect in 20 years at a probability of 0.5, since the median transition time is the same in all cases. It is important to note that the Weibull model does not change the Markov median life expec- tancy and is not necessary if median life expectancy is the only result desired from the analy- sis. Where the Weibull model helps is in the calculation of uncertainty in life expectancy. As the shaping parameter increases, the range of uncertainty narrows. In Figure 4-21, the Markov model, after 10 years, has a 70% survival probability; in other words, 30% of the inventory will need to be replaced during a 10-year program period. However, if the shaping parameter is 8, the survival probability after 10 years is nearly 100%, with little or no replacement funding needed. The shaping parameter can be determined using a statistical procedure called “maximum likelihood estimation,” which is a structured trial-and-error procedure to experiment with dif- ferent values of beta until the best fit to the data is found. (Note: The trial-and-error can be orchestrated by Microsoft Excel’s Solver module or can be done manually by inputting the pos- sible values in a spreadsheet.) To develop the Weibull model, perform all of the steps described in the preceding section for the Markov model, with the following enhancements: • When forming pairs in Step 2, keep track of the age of the asset at the time of the second inspection in each pair. • When filtering pairs in Step 3, keep track of the pairs that are removed. After completing the calculation of Markov model life expectancy, remove from the data set not only the pairs where work may have been done, but also remove all subsequent pairs for those assets. Because the Weibull model is a time-series analysis, it is necessary to have inspection data for ages at least up to the Markov median life expectancy. The analysis works best on assets where it is unusual to perform life extension work before the median life expectancy is reached. Table 4-30 shows a list of road segments with data on their traffic signs. In the example agency, signs are inspected on a pass/fail basis every 2 years. The pass/fail criterion is a level-of-service standard based on retroreflectivity and damage. Each segment of road has a group of signs, which is characterized by the fraction satisfying the level-of-service standard. This lends itself to a relatively low-cost drive-by visual process of rating sign condition. It is desired to estimate a model of the fraction of signs that pass the standards as a function of age. For this model, the only required data for each segment of road are the age (assuming all signs on the segment were installed at the same time) and the fraction that passed. The procedure for estimating the model is called “maximum likelihood estimation.” This is an iterative process that starts with an initial educated guess and then uses a systematic trial-and- error process to improve on the guess. The guesses are directed by the objective of maximizing the likelihood that the estimated parameters are the correct ones. On the right-hand side of the spreadsheet, the median life expectancy and shaping parameter are initially provided by the analyst as educated guesses, perhaps based on published life expec- tancy estimates. For the example, it would make sense to use initial values of 10 years for life expectancy, 2.0 as the shaping parameter, and 0.01 as the standard deviation. In most cases the initial values would not affect the results, as long as they are reasonable. The prediction equation is y g T g = − ×( )( ) = ( )exp ln1 2 1α α β β where yg is the fraction predicted to pass at age g; a is the scaling parameter; b is the shaping parameter; and T is the median life expectancy.

Develop Foundation tools: how to Compute Life expectancy Models 83 List of biennial traffic sign inspections Year Age Actual Predict Markov Square of Square of Road of of fraction fraction fraction deviation deviation Log segment insp signs passing passing passing act-pred act-mean likelihood Segment Year Age PassPredicted Markovq DevPred DevMean LogLike Coeff Value RS00001 1994 0 1.00 1.000 1.000 0.0000 0.0976 1.584 Median years 9.88 RS00001 1996 2 1.00 0.966 0.869 0.0012 0.0976 1.496 Shaping param 1.87 RS00001 1998 4 0.99 0.880 0.755 0.0121 0.0914 0.682 Std deviation 0.0819 RS00001 2000 6 0.95 0.761 0.657 0.0356 0.0688 -1.071 Sum LogLike 49.852 RS00001 2002 8 0.89 0.627 0.571 0.0692 0.0410 -3.577 RS00001 2004 10 0.62 0.492 0.496 0.0163 0.0046 0.369 Scaling param 12.025 RS00001 2006 12 0.43 0.369 0.431 0.0037 0.0664 1.309 Markov scaling 14.259 RS00001 2008 14 0.31 0.265 0.375 0.0020 0.1426 1.431 RS00001 2010 16 0.19 0.182 0.326 0.0001 0.2476 1.579 Mean passing 0.6876 RS00002 1998 0 1.00 1.000 1.000 0.0000 0.0976 1.584 SSE 0.3083 RS00002 2000 2 0.96 0.966 0.869 0.0000 0.0742 1.581 SST 3.2848 RS00002 2002 4 0.88 0.880 0.755 0.0000 0.0370 1.584 R-squared 0.9061 RS00002 2004 6 0.73 0.761 0.657 0.0010 0.0018 1.510 RS00002 2006 8 0.64 0.627 0.571 0.0002 0.0023 1.571 RS00002 2008 10 0.51 0.492 0.496 0.0003 0.0315 1.561 RS00002 2010 12 0.42 0.369 0.431 0.0026 0.0716 1.392 RS00003 1996 0 1.00 1.000 1.000 0.0000 0.0976 1.584 RS00003 1998 2 0.97 0.966 0.869 0.0000 0.0797 1.582 RS00003 2000 4 0.91 0.880 0.755 0.0009 0.0495 1.517 RS00003 2002 6 0.71 0.761 0.657 0.0026 0.0005 1.387 RS00003 2004 8 0.58 0.627 0.571 0.0022 0.0116 1.419 RS00003 2006 10 0.41 0.492 0.496 0.0068 0.0771 1.077 RS00003 2008 12 0.34 0.369 0.431 0.0009 0.1208 1.520 RS00003 2010 14 0.21 0.265 0.375 0.0030 0.2281 1.360 RS00004 1998 0 1.00 1.000 1.000 0.0000 0.0976 1.584 RS00004 2000 2 0.95 0.966 0.869 0.0002 0.0688 1.565 RS00004 2002 4 0.87 0.880 0.755 0.0001 0.0333 1.576 RS00004 2004 6 0.73 0.761 0.657 0.0010 0.0018 1.510 RS00004 2006 8 0.54 0.627 0.571 0.0076 0.0218 1.019 RS00004 2008 10 0.44 0.492 0.496 0.0027 0.0613 1.379 RS00004 2010 12 0.31 0.369 0.431 0.0035 0.1426 1.322 RS00005 1996 0 1.00 1.000 1.000 0.0000 0.0976 1.584 RS00005 1998 2 1.00 0.966 0.869 0.0012 0.0976 1.496 RS00005 2000 4 0.91 0.880 0.755 0.0009 0.0495 1.517 RS00005 2002 6 0.83 0.761 0.657 0.0047 0.0203 1.232 RS00005 2004 8 0.71 0.627 0.571 0.0069 0.0005 1.070 RS00005 2006 10 0.51 0.492 0.496 0.0003 0.0315 1.561 RS00005 2008 12 0.46 0.369 0.431 0.0082 0.0518 0.970 RS00005 2010 14 0.33 0.265 0.375 0.0043 0.1279 1.266 RS00006 1998 0 1.00 1.000 1.000 0.0000 0.0976 1.584 RS00006 2000 2 0.95 0.966 0.869 0.0002 0.0688 1.565 RS00006 2002 4 0.79 0.880 0.755 0.0081 0.0105 0.979 RS00006 2004 6 0.61 0.761 0.657 0.0229 0.0060 -0.125 RS00006 2006 8 0.43 0.627 0.571 0.0388 0.0664 -1.311 RS00006 2008 10 0.32 0.492 0.496 0.0297 0.1351 -0.634 RS00006 2010 12 0.29 0.369 0.431 0.0063 0.1581 1.115 Table 4-30. Weibull survival probability model for signs.

84 estimating Life expectancies of highway assets The value of T can be determined using the Markov model described in the previous example. For this example, however, it is determined using maximum likelihood estimation at the same time as the shaping parameter. The Weibull model gives the same results as the Markov model if the shaping parameter is 1.0. This is shown in the Markov column of the spreadsheet. To assist with further computations, the spreadsheet has a column showing the square of the deviation between actual and predicted, calculated as SqDev ed Pass edictedPr Pr= −( )2 Also shown is the square of the deviation between actual and mean, calculated as SqDevMean Pass MeanPas g= −( )sin 2 The maximum likelihood procedure tries to find values of median life expectancy and shaping parameter that maximize the value of a “log likelihood function,” which is just a measure of how likely the parameters are to be the correct ones that explain the observed data. The likelihood function is a formula chosen to converge quickly on the best solution, in order to make the pro- cedure as fast as possible. This formula is LogLikelihood Sq= − × ( )− × ( )− ×0 5 2 0 5 0 52. ln . ln .pi σ Dev edPr σ2( ) The standard deviation s is determined iteratively by the estimation procedure, based on the choices for life expectancy and shaping parameter. The sum of log likelihood over all the data points is shown in the upper-right table of the example, just below the parameters to be estimated. As a more familiar measure of goodness-of-fit, the example spreadsheet also computes R-squared, using the formula R SqDev ed SqDevMean i i 2 1= − ∑ ∑ Pr This has the same interpretation as in linear regression. It is an estimate of how much of the variability in the dependent variable (fraction that passed) is explained by the model. It can be used to compare different versions of the model, to see which one has the best fit to the data. Microsoft Excel’s Solver module is used in order to drive the trial-and-error process of find- ing the best values of life expectancy and shaping parameter. The Solver module appears on the Data ribbon in Microsoft Excel 2007. See the linear regression example above for instructions on how to ensure that the Solver is installed. Click the Solver button, and complete the Solver dialog box as shown in Figure 4-22. The target cell is the cell containing the sum of the log likelihood function. This is the quantity to be maximized. The “By Changing Cells” range is the range containing the cells whose values are to be estimated. It consists of three cells in this example: Median years (life expectancy), Shaping parameter, and Standard deviation. The constraints set a maximum and minimum value on the shaping parameter, which are included just to prevent the model from finding nonsensical values of the shaping parameter. Click the “Solve” button to perform the estimation procedure. Microsoft Excel will present the results and ask whether to keep them. The example above shows the final values of the parameters. The main difference between the Weibull survival probability model and the Markov model is the ability to include age as an explanatory variable. Figure 4-23 shows the effect.

Develop Foundation tools: how to Compute Life expectancy Models 85 It can be seen in the graph that the Weibull survival probability model is a better fit to the data than the Markov model. Under the Markov model, the R-squared value is only 0.8081, and under the survival probability model, it is 0.9061. The survival probability model has the same life expectancy as the Markov model, with a 50% probability of failure after 9.88 years; but there is less uncertainty in life expectancy: after 6 years, the Markov model predicts that 65.7% of the signs will pass, while the survival probability model predicts that 76.1% will pass. The Weibull model gives both a more accurate and more precise indication of when sign replacement will be needed. For data sets where censoring is an issue (where it is not possible to use a database of retired assets to estimate the model), there are advanced techniques to correct for censoring bias. See Dodson (2006) for an extensive set of methods and examples. Just like the Markov model, the survival probability model does not accommodate explana- tory variables, but it is efficient in its use of data. Reliable models can be constructed with as few as 20 data points, provided the data set is carefully constructed to be representative of the popu- lation (Dodson 2006). When there is a need for explanatory variables, one simple approach is to partition the data set into subsets of the asset inventory distinguished by categorical data values, such as by district or climate zone. Figure 4-22. Microsoft Excel 2007 Solver dialog box. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 5 10 15 Pr ob ab ili ty o f p as si ng Age Pass Predicted Markov Figure 4-23. Comparing actual data (Pass) with Weibull model (Predicted) and Markov.

86 estimating Life expectancies of highway assets For continuous explanatory variables, another approach is to use a linear multivariate model for the scaling parameter, as was done in several of the examples presented earlier in this chapter. The same maximum likelihood estimation technique then can be used for estimation of this model. Alternatively, a somewhat more elaborate model called a Cox model can be used, which follows. 4.2.4 Cox Survival Probability Model The Cox proportional hazard model is very similar to a Weibull survival probability model, but it incorporates a multiplier to the survival probability to account for explanatory variables. The full equation for the Cox model is y g b X b X b Xg n n1 1 1 2 21 0= − ×( )( ) × + + +( )exp . expα β  where y1g is the probability of the not-failed state at age g, if no intervening maintenance action is taken between year 0 and year g; b is the shaping parameter; and a is the scaling parameter, cal- culated as for the Weibull model. The variables Xn are explanatory variables such as traffic volume or location. They can be continuous variables or 0/1 flags. The coefficients bn are determined by linear regression or can be estimated at the same time as the Weibull shaping parameter using Microsoft Excel’s Solver. The multiplier can shift the survival probability either upward or down- ward. If all of the explanatory variables are zero, then the multiplier has no effect. Table 4-31 uses the same data as Table 4-30, but includes explanatory variables for sun expo- sure and plywood backing. The spreadsheet model for estimating the Cox regression coefficients is very similar to the one used for the previous example, except for the use of the Cox equation and the additional explanatory variables. The results are shown on the right side of the table. It can be seen in these results that the life expectancy estimate increased by a small amount, to 10.39 years. Also, the model is a better fit to the data, with an R-squared value of 0.9373. By taking advantage of additional data about the signs, it was possible to improve the quality of the model. 4.3 Validating and Refining Models It is considered good practice in statistical analysis to divide the data set of inspection data into two subsets, one for model estimation and one for validation. The predictive models are devel- oped using the first data set, then tested on the second data set to see if they produce accurate results (i.e., to check if their life expectancy estimates are correct). If the validation results are not “close enough,” it might mean an error in the model development process. Typical causes of such errors might be • Sample sizes that are too small. The Markov model typically needs a sample size of 100 inspec- tion pairs or more. The Weibull and Cox models might need 200 or more for a realistic set of explanatory variables. If the model is partitioned, then each separate model needs to have a sufficient sample size. • Too many explanatory variables. It is unusual for more than three or four explanatory vari- ables to have a beneficial effect on the Cox model. After that, what appears to be a gain in performance might just be accidental correlation with randomness in the data. The ordinary regression model might be able to use five or six variables, but usually less are needed. • Explanatory variables correlated with each other. If a model has both ADT and number of lanes as variables, for example, there is a good chance that the relationship between these two quantities will harm the performance of the model. • Lack of variability in the data. If a data set has 1,000 inspection pairs, but they are all identical, then the model likely would not produce useful results.

Develop Foundation tools: how to Compute Life expectancy Models 87 • Lack of movement. If none of the inspection pairs show any deterioration, then the models would not work. • Lack of population. If a condition state has no quantity entered, in the before case or the after case, then the model would not work. • Lack of intuitive sense. In a regression model, it is easy to input every possible variable, just to see the results. Unfortunately, this could very likely produce misleading results. Only use variables that make intuitive sense. List of biennial traffic sign inspections Year Age Sun Ply Actual Predict Markov Square of Square of Road of of expo- wood fraction fraction fraction deviation deviation Log segment insp signs sure back passing passing passing act-pred act-mean likelihood Segment Year Age Sun Wood PassPredicted MarkovqDevPred DevMean LogLike Coeff Value RS00001 1994 0 0.57 -1 1.00 1.000 1.000 0.0000 0.0976 1.785 Median years 10.39 RS00001 1996 2 0.57 -1 1.00 0.970 0.875 0.0009 0.0976 1.687 Shaping param 1.89 RS00001 1998 4 0.57 -1 0.99 0.894 0.766 0.0091 0.0914 0.767 Sunshine coef 0.18 RS00001 2000 6 0.57 -1 0.95 0.787 0.670 0.0265 0.0688 -1.178 Plyw ood coef 0.13 RS00001 2002 8 0.57 -1 0.89 0.663 0.587 0.0516 0.0410 -3.976 Std deviation 0.0669 RS00001 2004 10 0.57 -1 0.62 0.535 0.513 0.0072 0.0046 0.987 Sum LogLike 59.120 RS00001 2006 12 0.57 -1 0.43 0.416 0.449 0.0002 0.0664 1.763 RS00001 2008 14 0.57 -1 0.31 0.312 0.393 0.0000 0.1426 1.785 Scaling param 12.618 RS00001 2010 16 0.57 -1 0.19 0.226 0.344 0.0013 0.2476 1.638 Markov scaling 14.994 RS00002 1998 0 0.69 -1 1.00 1.000 1.000 0.0000 0.0976 1.785 RS00002 2000 2 0.69 -1 0.96 0.970 0.875 0.0001 0.0742 1.775 Mean passing 0.6876 RS00002 2002 4 0.69 -1 0.88 0.892 0.766 0.0001 0.0370 1.769 SSE 0.2060 RS00002 2004 6 0.69 -1 0.73 0.782 0.670 0.0027 0.0018 1.480 SST 3.2848 RS00002 2006 8 0.69 -1 0.64 0.655 0.587 0.0002 0.0023 1.759 R-squared 0.9373 RS00002 2008 10 0.69 -1 0.51 0.525 0.513 0.0002 0.0315 1.760 RS00002 2010 12 0.69 -1 0.42 0.403 0.449 0.0003 0.0716 1.753 RS00003 1996 0 0.59 -1 1.00 1.000 1.000 0.0000 0.0976 1.785 RS00003 1998 2 0.59 -1 0.97 0.970 0.875 0.0000 0.0797 1.785 RS00003 2000 4 0.59 -1 0.91 0.894 0.766 0.0003 0.0495 1.757 RS00003 2002 6 0.59 -1 0.71 0.786 0.670 0.0058 0.0005 1.136 RS00003 2004 8 0.59 -1 0.58 0.662 0.587 0.0067 0.0116 1.043 RS00003 2006 10 0.59 -1 0.41 0.534 0.513 0.0153 0.0771 0.077 RS00003 2008 12 0.59 -1 0.34 0.414 0.449 0.0054 0.1208 1.177 RS00003 2010 14 0.59 -1 0.21 0.309 0.393 0.0098 0.2281 0.687 RS00004 1998 0 0.69 1 1.00 1.000 1.000 0.0000 0.0976 1.785 RS00004 2000 2 0.69 1 0.95 0.961 0.875 0.0001 0.0688 1.772 RS00004 2002 4 0.69 1 0.87 0.861 0.766 0.0001 0.0333 1.776 RS00004 2004 6 0.69 1 0.73 0.719 0.670 0.0001 0.0018 1.773 RS00004 2006 8 0.69 1 0.54 0.556 0.587 0.0002 0.0218 1.758 RS00004 2008 10 0.69 1 0.44 0.388 0.513 0.0027 0.0613 1.479 RS00004 2010 12 0.69 1 0.31 0.230 0.449 0.0064 0.1426 1.073 RS00005 1996 0 0.69 -1 1.00 1.000 1.000 0.0000 0.0976 1.785 RS00005 1998 2 0.69 -1 1.00 0.970 0.875 0.0009 0.0976 1.682 RS00005 2000 4 0.69 -1 0.91 0.892 0.766 0.0003 0.0495 1.750 RS00005 2002 6 0.69 -1 0.83 0.782 0.670 0.0023 0.0203 1.531 RS00005 2004 8 0.69 -1 0.71 0.655 0.587 0.0030 0.0005 1.451 RS00005 2006 10 0.69 -1 0.51 0.525 0.513 0.0002 0.0315 1.760 RS00005 2008 12 0.69 -1 0.46 0.403 0.449 0.0033 0.0518 1.422 RS00005 2010 14 0.69 -1 0.33 0.296 0.393 0.0011 0.1279 1.659 RS00006 1998 0 0.68 1 1.00 1.000 1.000 0.0000 0.0976 1.785 RS00006 2000 2 0.68 1 0.95 0.961 0.875 0.0001 0.0688 1.772 RS00006 2002 4 0.68 1 0.79 0.861 0.766 0.0051 0.0105 1.220 RS00006 2004 6 0.68 1 0.61 0.720 0.670 0.0121 0.0060 0.438 RS00006 2006 8 0.68 1 0.43 0.556 0.587 0.0160 0.0664 0.003 RS00006 2008 10 0.68 1 0.32 0.389 0.513 0.0047 0.1351 1.257 RS00006 2010 12 0.68 1 0.29 0.232 0.449 0.0034 0.1581 1.404 Table 4-31. Cox regression model for signs.

88 estimating Life expectancies of highway assets A good way to determine whether a life expectancy model will work in practice is to start with a quick-and-easy version of the model, and then build it into a prototype of the envisioned application. Microsoft Excel is a good way to do this because development and refinement in Microsoft Excel can be done very quickly. This exercise will help the analyst see all the way through the problem, from raw data to finished product. This experience often leads to design changes that vastly improve the product. To visualize the accuracy of the model, a common technique is validation plots. These plots vary by the type of model calibrated. To validate the developed techniques, it is recommended to randomly split the data set in two, with one set used for calibration and the other purely for validation. The assessment of the model is then typically evaluated by plotting the predictions of the validation set using the calibrated model to the observed value from the validation set (Figure 4-24). The closer the data points to the straight line, the better the fit. The proportion of the data set used for validation is subject to expert opinion. A typical proportion may be 75% calibration, 25% validation. However if there is a lack of available data for calibration, a greater percentage may be used for calibration. Conversely, for large data sets a much smaller percentage may be appropriate for calibration. When validating probabilistic models, a similar technique can be applied by comparing survival curves to non-parametric estimates such as the Kaplan-Meier estimate or non-homogenous Markov chain. Statistical analysis is part science and part art, with a lot of opportunity for creativity and a lot of room for error. To ensure that the results of the life expectancy analysis are intuitive and can be implemented, it is helpful to solicit advice from experienced modelers and users of these models. The presence of outlying data is an important issue that requires careful treatment and may warrant the solicitation of expert opinion. The presence of outliers may greatly influence model parameters and may result in models with low goodness-of-fit measures. On the other hand, outright exclusion of outliers may lead to the unintended suppression of important infor- mation which may adversely affect the model reliability. 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Pr ed ic te d Se rv ic e Li fe Actual Service Life Figure 4-24. Example validation plot.