# Maintenance Modelling of Ceramic Claddings in Pitched Roofs Based on the Evaluation of Their In Situ Degradation Condition

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}. The remaining service life of the CCPR; the global costs over the claddings’ lifetime (considering inspection, maintenance, replacement and disposal costs); the claddings’ degradation condition and the number of replacements during the time horizon are used to evaluate the performance of the different maintenance strategies through a simplified multi-criteria analysis. The results show that the gains in performance, in terms of expected service life and durability, of the consideration of preventive maintenance actions (minor interventions or cleaning operations) outweigh the increase of the operation costs.

## 1. Introduction

## 2. Background

## 3. Methodology

#### 3.1. Maintenance Model

_{w}, which allows assessing the overall degradation condition of the cladding. The severity of degradation, S

_{w}, is computed by Equation (1), which relates the area affected by the existing anomalies and the reference area and ranges from 0% (no visual degradation) to 100% (generalised degradation).

_{w}is the severity of degradation of the cladding in %, k

_{n}the multiplying factor of anomaly n, considering its degradation condition (range from 0 to 4), k

_{a,n}the weight of the existing anomaly (k

_{a,n}R

^{+}), A

_{n}the area of cladding associated with the anomaly n in m

^{2}, A the total roof’s area in m

^{2}and ∑(k

_{max}) the sum of the multiplying factors for the highest degradation condition of each anomaly type. The type of anomalies, the k

_{n}and k

_{a,n}coefficient values, and the relationship between the severity of degradation, S

_{w}, and the degradation condition, C, for the CCPR are shown in Table 1.

#### 3.2. Deterioration Process

_{ij}is the probability of transition from degradation condition i to j, being i the degradation condition in the initial instant and j in the final instant, n the historical database size and k the number of intervals between inspections. In this methodology, Monte Carlo simulation is used to compute the probability of transition, p

_{ij}.

#### 3.3. Life Cycle Cost Analysis

_{inspection,t}is the accumulate costs associated with inspection over the time horizon, ∑C

_{maintenance,t}the accumulate costs associated with maintenance activities, t

_{h}the time horizon of the analysis and t the year of the intervention’s occurrence. Equations (4) and (5) are used to compute the present values of the inspection and maintenance activities [49].

_{inspection,t}and C

_{maintenance,t}are the present values of the inspection and the maintenance cost at the time of reference, respectively, C

_{inspection}and C

_{maintenance}the inspection and maintenance cost at time t, respectively, and υ, the real discount rate. Assuming a private sector environment, in this study, a 6% real discount rate is used [50]. Furthermore, the construction costs are not considered, since it is considered that the same type of CCPR is applied in all maintenance strategies analysed.

#### 3.4. Efficiency Index and Performance Indicator

_{annualised}the annualised operation cost of the maintenance strategy under analysis. The PI value ranges from 0 to +∞, and the higher the value, the more beneficial the cost/benefit ratio of the maintenance strategy is.

_{w}(t) dt is the area underneath the degradation profile (which represents the loss of performance of the CCPR over time), and 100∙t

_{h}is the area underneath the degradation profile when there is no degradation (utopian situation). This parameter varies from 0 (severity of degradation is always 100% or condition E) to 1 (severity of degradation is always 0% or condition A). Finally, the annualised cost is computed through Equation (8).

_{h}the time horizon of the analysis (equal for all maintenance strategies considered).

## 4. Case Study: Ceramic Claddings in Pitched Roofs (CCPR)

#### 4.1. Degradation Process

#### 4.1.1. Probabilistic Analysis

#### 4.1.2. Validation

#### 4.1.3. Service Life

#### 4.2. Maintenance Strategies, Assumptions and Costs

#### 4.2.1. Costs of the Interventions

- Minor level: the total area CCPR mainly affected by surface soiling, and accumulation of the debris (anomaly AE-E1) is cleaned. This action includes personal protective equipment (protection fall) installation and low-pressure water jet cleaning.
- Moderate level: the total area CCPR mainly affected by surface dirt, accumulation of debris and biological colonisation (anomaly AE-E1) is cleaned. This action includes personal protective equipment (protection fall) installation, low-pressure water jet cleaning and use of biocides to eliminate biological microorganisms.
- Extensive level: the total area CCPR extensively affected by surface dirt, accumulation of debris, biological colonisation and parasitic vegetation (anomaly AE-E1) is cleaned. This action includes personal protective equipment (protection fall) installation, low-pressure water jet cleaning, the use of biocides to eliminate biological microorganisms and the use of herbicides and manual removal of parasitic vegetation.

- Minor level: cleaning operation of moderate level: repair and/or replacement up to 15% of the cladding affected by peeling/flaking/exfoliation (anomaly AE-F1), replacement up to 5% of the cladding affected by cracking/fracture (anomaly AE-F2), repair up to 5% of the cladding affected by detachment/cladding release (anomaly AE-F3) and repair up to 13% of the cladding affected by misalignment of the cladding (anomaly AE-S2), including replacement of the rafters/framing (tiles supporting system) in a suitable area (when necessary).
- Moderate level: cleaning operation of moderate level: repair and/or replacement up to 23% of the cladding affected by peeling/flaking/exfoliation (anomaly AE-F1), replacement up to 8% of the cladding affected by cracking/fracture (anomaly AE-F2), repair or replacement up to 5% of the cladding affected by detachment/cladding release (anomaly AE-F3) and repair up to 19% of the cladding affected by misalignment of the cladding (anomaly AE-S2), including replacement of the rafters/framing in a suitable area (when necessary).
- Extensive level: cleaning operation of moderate level: repair and/or replacement up to 30% of the cladding affected by peeling/flaking/exfoliation (anomaly AE-F1), replacement up to 10% of the cladding affected by cracking/fracture (anomaly AE-F2), replacement up to 5% of the cladding affected by detachment/cladding release (anomaly AE-F3) and repair up to 25% of the cladding affected by misalignment of the cladding (anomaly AE-S2), including replacement of the rafters/framing in a suitable area (when necessary).

#### 4.2.2. Impacts of the Interventions

^{2}(Table 5). This action is applied for a cladding in condition C, allowing improving, with a probability of 70.8%; the cladding’s condition to condition A, with a probability of 24.0%, and to condition B, with a probability of 5.2%, causes no significant improvement (i.e., maintains condition C).

#### 4.2.3. Inspection Frequency

#### 4.2.4. Constraints

#### 4.3. Maintenance Strategies Comparison

_{w}, is reached, occurring later in the time horizon and the impact on the mean severity of degradation, S

_{w}, over time is greater. In other words, MS3 is the one that presents a better mean severity of degradation, S

_{w}, over time. However, if the operations costs associated with the three maintenance strategies are analysed (Figure 4), EM3 is the alternative with the highest maintenance costs, while MS1 is with the lowest maintenance costs. For example, at year 150, MS3 presents a life cycle cost of 74.14 €/m

^{2}(23% of the cost is spent in inspections), while, for MS1, a life cycle cost of 19.78 €/m

^{2}is estimated (87% of the cost is spent in inspections). These results are expected, since a number of interventions have a directed relationship with the maintenance costs (Figure 5), since the increase of the number of interventions implies the increase of the maintenance costs.

_{w}= 100%), while, in MS1, a total replacement of the cladding is carried out when the cladding achieves condition D, which allows improving the cladding to degradation condition A (or S

_{w}= 0%). Instead, an analysis of the instant in which the first minor intervention occurs is analysed, revealing that cleaning operations postpone its need. For example, for MS2 and MS3, the minor interventions’ cumulative distribution functions are compared in Figure 6b. Considering MS2, at year 12, the cladding that has a probability of 90% has already been intervened at least once. For MS3, this probability is delayed to year 17. However, if the analysis is expanded to the instant of the second minor intervention, in year 21, for MS2, the cladding presents a probability of 90% that has already been intervened at least twice, while, for MS3, this probability is postponed to year 31.

#### 4.4. Inspection Frequency Sensitivity Analysis

^{2}(55% of the cost is spent in inspections) while, for a two-year inspection frequency, a life cycle cost of 34.75 €/m

^{2}is obtained (24% of the cost is spent in inspections).

## 5. Multi-Criteria Analysis

_{i}is the global ranking of the option i, λ

_{j}the weight of criterion j and x

_{ij}the standardised classification of option i, according to criterion j. Since the different criterion studied have different scales, a standardisation by interval is implemented in this study, using Equation (10) for an increased order of preference and Equation (11) for a decreased order of preference [47].

_{ij}is the classification of option i according to criterion j, and min X

_{ij}and max X

_{ij}are, respectively, the minimum and maximum value of criterion j.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Shohet, I.M. Key performance indicators for strategic healthcare facilities maintenance. J. Constr. Eng. Manag.
**2006**, 132, 345–352. [Google Scholar] [CrossRef] - Lind, H.; Muying, H. Building maintenance strategies: Planning under uncertainty. Prop. Manag.
**2012**, 30, 14–28. [Google Scholar] [CrossRef] - Silva, A.; de Brito, J. Do we need a buildings’ inspection, diagnosis and service life prediction software? J. Build. Eng.
**2019**, 22, 335–348. [Google Scholar] [CrossRef] - Falorca, J.F. Main functions for building maintenance management: An outline application. Int. J. Build. Pathol. Adapt.
**2019**, 37, 490–509. [Google Scholar] [CrossRef] - Eurostat. Available online: https://ec.europa.eu/eurostat/statistics-explained/index.php/Construction_production_(volume)_index_overview (accessed on 15 July 2020).
- Shohet, I.M.; Puterman, M.; Gilboa, E. Deterioration patterns of building cladding components for maintenance management. Constr. Manag. Econ.
**2002**, 20, 305–314. [Google Scholar] [CrossRef] - Mobley, R.K. Maintenance Engineering Handbook, 8th ed.; McGraw-Hill Professional: New York, NY, USA, 2014. [Google Scholar]
- Dann, N.; Cantell, T. Maintenance: From philosophy to practice. J. Archit. Conserv.
**2005**, 11, 42–54. [Google Scholar] [CrossRef] - Chan, D.W. Sustainable building maintenance for safer and healthier cities: Effective strategies for implementing the Mandatory Building Inspection Scheme (MBIS) in Hong Kong. J. Build. Eng.
**2019**, 24. [Google Scholar] [CrossRef] - Ashworth, A. Estimating the life expectancies of building components in life-cycle costing calculations. Struct. Surv.
**1996**, 14, 4–8. [Google Scholar] [CrossRef] - Sherwin, D. A review of overall models for maintenance management. J. Qual. Maint. Eng.
**2000**, 6, 138–164. [Google Scholar] [CrossRef] [Green Version] - Forster, A.M.; Kayan, B. Maintenance for historic buildings: A current perspective. Struct. Surv.
**2009**, 27, 210–229. [Google Scholar] [CrossRef] - Ruparathna, R.; Hewage, K.; Sadiq, R. Multi-period maintenance planning for public buildings: A risk based approach for climate conscious operation. J. Clean. Prod.
**2018**, 170, 1338–1353. [Google Scholar] [CrossRef] - BS 8298-1. Code of Practice for the Design and Installation of Natural Stone Cladding and Linin—Part 1: General; British Standards Institution (BSI): London, UK, 2010. [Google Scholar]
- Sandak, A.; Sandak, J.; Brzezicki, M.; Riggio, M. Bio-Based Building Skin; Springer International Publishing: Gateway East, Singapore, 2019. [Google Scholar]
- Garcez, N.; Lopes, N.; de Brito, J.; Silvestre, J. System of inspection, diagnosis and repair of external claddings of pitched roofs. Constr. Build. Mater.
**2012**, 35, 1034–1044. [Google Scholar] [CrossRef] - Ilozor, B.D.; Okoroh, M.I.; Egbu, C.E. Understanding residential house defects in Australia from the State of Victoria. Build. Environ.
**2004**, 39, 327–337. [Google Scholar] [CrossRef] - Watt, D.S. Building Pathology: Principles and Practice, 2nd ed.; Blackwell Publishing Ltd.: Oxford, UK, 2009. [Google Scholar]
- Yazdani, N.; Dowgul, R.W.; Manzur, T. Deficiency analysis of coastal buildings toward storm damage reduction. J. Perform. Constr. Facil.
**2010**, 24, 128–137. [Google Scholar] [CrossRef] - Gullbrekken, L.; Kvande, T.; Jelle, B.P.; Time, B. Norwegian pitched roof defects. Buildings
**2016**, 6, 24. [Google Scholar] [CrossRef] [Green Version] - Morgado, J.; Flores-Colen, I.; de Brito, J.; Silva, A. Maintenance planning of pitched roofs in current buildings. J. Constr. Eng. Manag.
**2017**, 143. [Google Scholar] [CrossRef] - Kyle, B.R.; Kalinger, P. Service life prediction of roof systems by reliability-based analysis. In Proceedings of the 4th International Symposium on Roofing Technology, Gaithersburg, MD, USA, 17–19 September 1997. [Google Scholar]
- Garcez, N.; Lopes, N.; de Brito, J.; Sá, G. Pathology, diagnosis and repair of pitched roofs with ceramic tiles: Statistical characterisation and lessons learned from inspections. Constr. Build. Mater.
**2012**, 36, 807–819. [Google Scholar] [CrossRef] - Ferreira, C.; Neves, L.C.; Silva, A.; de Brito, J. Stochastic maintenance models for ceramic claddings. Struct. Infrastruct. Eng.
**2020**, 16, 247–265. [Google Scholar] [CrossRef] - Meijer, F.; Itard, L.; Sunikka-Blank, M. Comparing European residential building stocks: Performance, renovation and policy opportunities. Build. Res. Inf.
**2009**, 37, 533–551. [Google Scholar] [CrossRef] - Da Silva, J.M.; Vicente, R. Pathology and defect of building façades and roofs: A state-of-the-art report on building pathology. In Building Pathology; de Freitas, V.P., Ed.; Conseil International du Bâtiment: Rotterdam, The Netherlands, 2013; pp. 123–131. [Google Scholar]
- INE. Censos 2011 Resultados Definitivos—Portugal; Instituto Nacional de Estatística: Lisbon, Portugal, 2012. [Google Scholar]
- Zhang, Y.; Vidakovic, B.; Augenbroe, G. Uncertainty analysis in using Markov chain model to predict roof life cycle performance. In Proceedings of the 10DBMC International Conference on Durability of Building Materials and Components, Lyon, France, 17–19 April 2005. [Google Scholar]
- Alaimo, G.; Accurso, F. The methods for the durability evaluation of pitched roof. In CIB W80 WG3—Test Methods for Service Life Prediction—State of the Art Report on Accelerated Laboratory Test Procedures and Correlation between Laboratory Tests and Service Life Data; Daniotti, B., Re Cecconi, F., Eds.; Conseil International du Bâtiment: Rotterdam, The Netherlands, 2010; pp. 3–13. [Google Scholar]
- ISO 15686-1. Buildings and Constructed Assets—Service Life Planning—Part 2: Service Life Prediction Procedures; International Organization for Standardization (ISO): Geneva, Switzerland, 2012. [Google Scholar]
- Rudbeck, C. Service life of building envelope components: Making it operational in economical assessment. Constr. Build. Mater.
**2002**, 16, 83–89. [Google Scholar] [CrossRef] - Ramos, R.; Silva, A.; de Brito, J.; Gaspar, P.L. Methodology for the service life prediction of ceramic claddings in pitched roofs. Constr. Build. Mater.
**2018**, 166, 386–399. [Google Scholar] [CrossRef] - Gaspar, P.L.; de Brito, J. Limit states and service life of cement renders on façades. J. Mater. Civ. Eng.
**2011**, 23, 1396–1404. [Google Scholar] [CrossRef] - Flores-Colen, I.; de Brito, J.; Freitas, V. Discussion of criteria for prioritization of predictive maintenance of building façades: Survey of 30 experts. J. Perform. Constr. Facil.
**2010**, 24, 337–344. [Google Scholar] [CrossRef] - Flores-Colen, I.; de Brito, J. A systematic approach for maintenance budgeting of buildings façades based on predictive and preventive strategies. Constr. Build. Mater.
**2010**, 24, 1718–1729. [Google Scholar] [CrossRef] - Flores-Colen, I.; de Brito, J. Discussion of proactive maintenance strategies in façades’ coatings of social housing. J. Build. Apprais.
**2010**, 5, 223–240. [Google Scholar] [CrossRef] - Madureira, S.; Flores-Colen, I.; de Brito, J.; Pereira, C. Maintenance planning of façades in current buildings. Constr. Build. Mater.
**2017**, 147, 790–802. [Google Scholar] [CrossRef] - Ferreira, C.; Silva, A.; de Brito, J.; Dias, I.S.; Flores-Colen, I. Definition of a condition-based model for natural stone claddings. J. Build. Eng.
**2020**. [Google Scholar] [CrossRef] - Molloy, M.K. Performance analysis using stochastic Petri nets. IEEE Trans. Comput.
**1982**, 913–917. [Google Scholar] [CrossRef] - Butt, A.A.; Shahin, M.Y.; Feighan, K.J.; Carpenter, S.H. Pavement performance prediction model using the Markov process. Transp. Res. Rec.
**1987**, 1123, 12–19. [Google Scholar] - Hawk, H.; Small, E.P. The BRIDGIT bridge management system. Struct. Eng. Int.
**1998**, 8, 309–314. [Google Scholar] [CrossRef] - Thompson, P.D.; Small, E.P.; Johnson, M.; Marshall, A.R. The Pontis bridge management system. Struct. Eng. Int.
**1998**, 8, 303–308. [Google Scholar] [CrossRef] - Ortiz-García, J.J.; Costello, S.B.; Snaith, M.S. Derivation of transition probability matrices for pavement deterioration modeling. J. Transp. Eng.
**2006**, 132, 141–161. [Google Scholar] [CrossRef] - McDuling, J.; Horak, E.; Cloete, C. Service life prediction beyond the ‘factor method’. In Proceedings of the 11DBMC International Conference on Durability of Building Materials and Components, Istanbul, Turkey, 11–14 May 2008. [Google Scholar]
- Caleyo, F.; Velázquez, J.C.; Valor, A.; Hallen, J.M. Markov chain modelling of pitting corrosion in underground pipelines. Corros. Sci.
**2009**, 51, 2197–2207. [Google Scholar] [CrossRef] - Silva, A.; de Brito, J.; Gaspar, P.L. Methodologies for Service Life Prediction of Buildings: With a Focus on Façade Claddings; Springer International Publishing: Zurich, Switzerland, 2016. [Google Scholar]
- Sánchez-Silva, M.; Klutke, G.A. Reliability and Life-Cycle Analysis of Deteriorating Systems; Springer International Publishing: Zurich, Switzerland, 2016. [Google Scholar]
- Kalbfleisch, J.D.; Lawless, J.F. The analysis of panel data under a Markov assumption. J. Am. Stat. Assoc.
**1985**, 80, 863–871. [Google Scholar] [CrossRef] - Frangopol, D.M.; Kallen, M.J.; van Noortwijk, J.M. Probabilistic models for lifecycle performance of deteriorating structures: Review and future directions. Prog. Struct. Eng. Mater.
**2004**, 6, 197–212. [Google Scholar] [CrossRef] - Langdon, D. Life Cycle Costing (LCC) as a Contribution to Sustainable Construction. Guidance on the Use of the LCC Methodology and Its Application in Public Procurement; Davis Langdon Management Consulting: London, UK, 2007. [Google Scholar]
- Housing Association Property Mutual (HAPM). Component Life Manual; E&FN Spon: London, UK, 1999. [Google Scholar]
- RICS. Building Maintenance: Strategy, Planning and Performance; Royal Institution of Chartered Surveyors (RICS): London, UK, 2000. [Google Scholar]
- Bauer, M.; Lair, J.; Wetzel, C. Predictive Model for Future Deterioration; INVESTIMMO Project, Final Technical Report; European Commission: Brussels, Belgium, 2004. [Google Scholar]
- NRCA. The Green roof Systems Manual; National Roofing Contractors Association (NRCA): Rosemont, IL, USA, 2007. [Google Scholar]
- National Association of Home Builders (NAHB). Bank of America Home Equity Study of Life Expectancy of Home Components; Economics Group: Washington, DC, USA, 2007. [Google Scholar]
- Madrigal, L.O.; Lanzarote, B.S.; Bretones, J.M.F. Proposed method of estimating the service life of building envelope. Rev. Constr.
**2015**, 14, 60–68. [Google Scholar] - Bechthold, M.; Kane, A.; King, N. Ceramic Material Systems: In Architecture and Interior Design; Birkhäuser: Basel, Switzerland, 2015. [Google Scholar]
- Thai-Ker, L.; Chung-Wan, W. Challenges of external wall tiling in Singapore. In Proceedings of the Qualicer 2006: IX World Congress on Ceramic Tile Quality, Castellón, Spain, 12–15 February 2006. [Google Scholar]
- Raposo, S.; de Brito, J.; Fonseca, M. Planned preventive maintenance activities: Analysis of guidance documents. In Durability of Buildings Materials and Components; de Freitas, V.P., Delgado, J.M.P.Q., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; pp. 35–60. [Google Scholar]
- CYPE Price Generator. Available online: http://www.geradordeprecos.info/ (accessed on 24 April 2020).
- Ferreira, C.; Silva, A.; de Brito, J.; Dias, I.S.; Flores-Colen, I. The impact of imperfect maintenance actions on the degradation of buildings’ envelope components. J. Build. Eng.
**2020**, 33. [Google Scholar] [CrossRef] - Bana e Costa, C.A. Três convicções fundamentais na prática do apoio à decisão. Soc. Bras. Pesqui. Oper.
**1993**, 13, 9–20. [Google Scholar] - Dodgson, J.S.; Spackman, M.; Pearman, A.; Phillips, L.D. Multi-Criteria Analysis: A manual; Department for Communities and Local Government: London, UK, 2009.
- Cochran, J.K.; Chen, H.N. Fuzzy multi-criteria selection of object-oriented simulation software for production system analysis. Comput. Oper. Res.
**2005**, 32, 153–168. [Google Scholar] [CrossRef]

**Figure 1.**Illustrative examples of the degradation condition of the pitched roofs analysed. From top to bottom: Condition (

**A**)—S

_{w}≤ 1%; Condition (

**B**)—1% < S

_{w}≤ 6%; Condition (

**C**)—6% < S

_{w}≤ 20%; Condition (

**D**)—20% < S

_{w}≤ 50%; and none of the elements analysed are in condition E.

**Figure 2.**Degradation process of the ceramic claddings in pitched roofs (CCPR) without the impact of maintenance activities: (

**a**) mean degradation profile overlapped by the fieldwork data and (

**b**) cumulative distribution functions.

**Figure 3.**Degradation profiles of the three maintenance strategies and the situation without maintenance.

**Figure 4.**Cumulative costs (inspection, maintenance and total) profiles of the three maintenance strategies.

**Figure 6.**Cumulative distribution functions associated with the: (

**a**) total replacement and (

**b**) minor intervention.

**Figure 8.**Cumulative cost (inspection, maintenance and total) profiles of the three inspection frequencies.

**Figure 9.**Cumulative distribution functions associated with the total replacement for the three inspection frequencies.

**Table 1.**Pitched roofs’ ceramic claddings classification system (data adapted from [32]).

Degradation Condition | Anomalies | k_{a,n} | % Area Affected | Severity of Degradation [%] | |
---|---|---|---|---|---|

A (k_{n} = 0) | No visible degradation | - | - | S_{w} ≤ 1 | |

B (k_{n} = 1) | Aesthetic | Development of parasitic vegetation/biological colonisation | 0.55 | ≤10 | 1 < S_{w} ≤ 6 |

Staining, change of colour or of brightness of the tiles | 0.20 | ≤20 | |||

Functional | Peeling/flaking/exfoliation | 1.00 | ≤10 | ||

Structural | Misalignment of the cladding | 1.00 | ≤10 | ||

C (k_{n} = 2) | Aesthetic | Development of parasitic vegetation/biological colonisation | 0.55 | >10 and ≤30 | 6 < S_{w} ≤ 20 |

Staining, change of colour or of brightness of the tiles | 0.20 | >20 and ≤50 | |||

Functional | Peeling/flaking/exfoliation | 1.00 | >10 and ≤30 | ||

Cracking/fracture | 1.00 | ≤10 | |||

Detachment/cladding release | 0.20 | ≤5 | |||

Structural | Pronounced cladding deformation | 1.30 | ≤25 | ||

Misalignment of the cladding | 1.00 | >10 and ≤25 | |||

D (k_{n} = 3) | Aesthetic | Development of parasitic vegetation/biological colonisation | 0.55 | >30 and ≤50 | 20 < S_{w} ≤ 50 |

Staining, change of colour or of brightness of the tiles | 0.20 | >50 | |||

Functional | Peeling/flaking/exfoliation | 1.00 | >30 and ≤50 | ||

Cracking/fracture | 1.00 | >10 and ≤30 | |||

Detachment/cladding release | 0.20 | >5 and ≤10 | |||

Structural | Pronounced cladding deformation | 1.30 | >25 and ≤50 | ||

Misalignment of the cladding | 1.00 | >25 and ≤50 | |||

E (k_{n} = 4) | Aesthetic | Development of parasitic vegetation/biological colonisation | 0.55 | >50 | S_{w} > 50 |

Functional | Peeling/flaking/exfoliation | 1.00 | >50 | ||

Cracking/fracture | 1.00 | >30 | |||

Detachment/cladding release | 0.20 | >10 | |||

Structural | Pronounced cladding deformation | 1.30 | >50 | ||

Misalignment of the cladding | 1.00 | >50 |

Sojourn Time in Each Degradation Condition | Exponential | Weibull | Lognormal | |
---|---|---|---|---|

Mean time–T (years) | T_{A} | 2.7 | 2.8 | 2.7 |

T_{B} | 5.1 | 4.5 | 4.6 | |

T_{C} | 148.2 | 67.9 | 91.6 | |

T_{D} | 7.19 × 10^{4} | 68.1 | 1.87 × 10^{8} | |

Standard deviation–SD (years) | SD_{A} | 2.7 | 1.2 | 1.5 |

SD_{B} | 5.1 | 2.6 | 3.4 | |

SD_{C} | 148.2 | 31.2 | 75.9 | |

SD_{D} | 7.19 × 10^{4} | 3.1 | 5.33 × 10^{9} | |

- log L | 76.61 | 68.41 | 70.13 |

**Table 3.**Predicted and observed ceramic claddings in pitched roofs (CCPR) in the different degradation conditions, as well as the respective relative error.

Degradation Condition | A | B | C | D | E | |
---|---|---|---|---|---|---|

Observed | 13 | 13 | 96 | 24 | 0 | |

Predicted | Exponential | 11.0 | 14.1 | 96.8 | 24.0 | 0.0 |

Weibull | 12.8 | 13.1 | 97.1 | 23.0 | 0.0 | |

Lognormal | 12.9 | 12.4 | 97.8 | 22.9 | 0.0 | |

Relative error (%) | Exponential | 15.1 | 8.4 | 0.9 | 0.1 | - |

Weibull | 1.5 | 0.5 | 1.2 | 4.2 | - | |

Lognormal | 0.8 | 4.3 | 1.8 | 4.6 | - |

**Table 4.**Predicted and observed CCPR in the different degradation conditions, as well as the respective relative error obtained by Markov chains (MC) and the Petri net (PN) with transitions exponentially distributed.

Degradation Condition | Observed | Predicted | Relative Error | |||
---|---|---|---|---|---|---|

MC | PN | MC (%) | PN (%) | MC/PN | ||

A | 13 | 11.4 | 11.0 | 12.0 | 15.1 | 1.0 |

B | 13 | 14.5 | 14.1 | 11.7 | 8.4 | 1.0 |

C | 96 | 89.8 | 96.8 | 6.5 | 0.9 | 0.9 |

D | 24 | 24.7 | 24.0 | 2.8 | 0.1 | 1.0 |

E | 0 | 5.6 | 0.0 | - | - | - |

Interventions | Application Zone | Cost (Year 0) [€/m^{2}] | |
---|---|---|---|

Inspections | All | 1.03 | |

Cleaning operations | Minor | B | 12.45 |

Moderate | 14.00 | ||

Extensive | 15.77 | ||

Minor interventions | Minor | C | 20.80 |

Moderate | 24.14 | ||

Extensive | 27.10 | ||

Total replacement | D, E | 59.12 |

Interventions | Application Zone | Impact of the Interventions (Probability of Transition to Conditions A (P _{A}), B (P_{B}) or C (P_{C})) | |||
---|---|---|---|---|---|

P_{A} (%) | P_{B} (%) | P_{C} (%) | |||

Inspections | All | - | - | - | |

Cleaning operations | Minor | B | 61.5 | 38.5 | - |

Moderate | |||||

Extensive | |||||

Minor interventions | Minor | C | 69.8 | 25.0 | 5.2 |

Moderate | 70.8 | 24.0 | 5.2 | ||

Extensive | 70.8 | 24.0 | 5.2 | ||

Total replacement | D, E | 100 | - | - |

Maintenance Strategy | Without Maintenance | MS1, 1-Year | MS2, Moderate, 1-Year | MS3, Moderate, 1-Year | |
---|---|---|---|---|---|

Mean time of permanence in each condition | tp_{A} (years) | 2.33 | 7.50 | 16.36 | 35.97 |

tp_{B} (years) | 4.53 | 11.46 | 28.28 | 26.55 | |

tp_{C} (years) | 67.63 | 131.04 | 105.35 | 87.48 | |

tp_{D} (years) | 58.73 | 0.00 | 0.00 | 0.00 | |

tp_{E} (years) | 16.78 | 0.00 | 0.00 | 0.00 | |

EI (-) | 0.69 | 0.88 | 0.90 | 0.92 | |

C_{annualised} (€/m^{2}) | - | 0.13 | 0.30 | 0.49 | |

PI (m^{2}/€) | - | 6.70 | 3.02 | 1.86 |

Maintenance Strategy | MS2, Moderate, 0.5-Years | MS2, Moderate, 1-Year | MS2, Moderate, 2-Years |
---|---|---|---|

tp_{A} (years) | 15.47 | 16.36 | 16.10 |

tp_{B} (years) | 28.32 | 28.28 | 27.93 |

tp_{C} (years) | 106.21 | 105.35 | 105.39 |

tp_{D} (years) | 0.00 | 0.00 | 0.59 |

tp_{E} (years) | 0.00 | 0.00 | 0.00 |

EI (-) | 0.90 | 0.90 | 0.90 |

C_{annualised} (€/m^{2}) | 0.42 | 0.30 | 0.23 |

PI (m^{2}/€) | 2.14 | 3.02 | 3.89 |

Efficiency Index (Criterion 1) | Maintenance Cost (Criterion 2) | Number of Total Replacements (Criterion 3) | Total Rating | Standardised Total Rating | |
---|---|---|---|---|---|

MS1, 1-year | 0.00 | 1.00 | 0.00 | 1.00 | 0.00 |

MS2, Moderate, 1 year | 0.54 | 0.54 | 0.63 | 1.70 | 0.70 |

MS3, Moderate, 1 year | 1.00 | 0.00 | 1.00 | 2.00 | 1.00 |

MS2, Moderate, 0.5 years | 0.51 | 0.20 | 0.59 | 1.30 | 0.30 |

MS2, Moderate, 2 years | 0.50 | 0.72 | 0.66 | 1.89 | 0.89 |

Weights (%) | 33.3 | 33.3 | 33.3 |

Efficiency Index (Criterion 1) | Maintenance Cost (Criterion 2) | Number of Total Replacements (Criterion 3) | Total Rating | Standardised Total Rating | |
---|---|---|---|---|---|

MS1, 1 year | 0.00 | 1.00 | 0.00 | 1.00 | 0.00 |

MS2, Moderate, 1 year | 0.54 | 0.54 | 0.63 | 2.24 | 0.62 |

MS3, Moderate, 1 year | 1.00 | 0.00 | 1.00 | 3.00 | 1.00 |

MS2, Moderate, 0.5 years | 0.51 | 0.20 | 0.59 | 1.82 | 0.41 |

MS2, Moderate, 2 years | 0.50 | 0.72 | 0.66 | 2.38 | 0.69 |

Weights (%) | 50 | 25 | 25 |

Efficiency Index (Criterion 1) | Maintenance Cost (Criterion 2) | Number of Total Replacements (Criterion 3) | Total Rating | Standardised Total Rating | |
---|---|---|---|---|---|

MS1, 1 year | 0.00 | 1.00 | 0.00 | 2.00 | 0.45 |

MS2, Moderate, 1 year | 0.54 | 0.54 | 0.63 | 2.24 | 0.66 |

MS3, Moderate, 1 year | 1.00 | 0.00 | 1.00 | 2.00 | 0.45 |

MS2, Moderate, 0.5 years | 0.51 | 0.20 | 0.59 | 1.50 | 0.00 |

MS2, Moderate, 2 years | 0.50 | 0.72 | 0.66 | 2.61 | 1.00 |

Weights (%) | 25 | 50 | 25 |

Efficiency Index (Criterion 1) | Maintenance Cost (Criterion 2) | Number of Total Replacements (Criterion 3) | Total Rating | Standardised Total Rating | |
---|---|---|---|---|---|

MS1, 1 year | 0.00 | 1.00 | 0.00 | 1.00 | 0.00 |

MS2, Moderate, 1 year | 0.54 | 0.54 | 0.63 | 2.33 | 0.66 |

MS3, Moderate, 1 year | 1.00 | 0.00 | 1.00 | 3.00 | 1.00 |

MS2, Moderate, 0.5 years | 0.51 | 0.20 | 0.59 | 1.89 | 0.44 |

MS2, Moderate, 2 years | 0.50 | 0.72 | 0.66 | 2.55 | 0.77 |

Weights (%) | 25 | 25 | 50 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ferreira, C.; Silva, A.; Brito, J.d.; S. Dias, I.; Flores-Colen, I.
Maintenance Modelling of Ceramic Claddings in Pitched Roofs Based on the Evaluation of Their In Situ Degradation Condition. *Infrastructures* **2020**, *5*, 77.
https://doi.org/10.3390/infrastructures5090077

**AMA Style**

Ferreira C, Silva A, Brito Jd, S. Dias I, Flores-Colen I.
Maintenance Modelling of Ceramic Claddings in Pitched Roofs Based on the Evaluation of Their In Situ Degradation Condition. *Infrastructures*. 2020; 5(9):77.
https://doi.org/10.3390/infrastructures5090077

**Chicago/Turabian Style**

Ferreira, Cláudia, Ana Silva, Jorge de Brito, Ilídio S. Dias, and Inês Flores-Colen.
2020. "Maintenance Modelling of Ceramic Claddings in Pitched Roofs Based on the Evaluation of Their In Situ Degradation Condition" *Infrastructures* 5, no. 9: 77.
https://doi.org/10.3390/infrastructures5090077