Modelling the service life of rendered facades using fuzzy systems

Automation in Construction 51 (2015) 1–7

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Automation in Construction journal homepage: www.elsevier.com/locate/autcon

Modelling the service life of rendered facades using fuzzy systems S.M. Vieira a, A. Silva b,⁎, J.M.C. Sousa a, J. de Brito c, P.L. Gaspar d a

IDMEC/LAETA, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal IST–Universidade de Lisboa, Av. Rovisco Pais, Lisbon 1049-001, Portugal c Department of Civil Engineering and Architecture, IST–Universidade de Lisboa, Av. Rovisco Pais, Lisbon 1049-001, Portugal d Faculty of Architecture–Universidade de Lisboa, R. Sá Nogueira, Pólo Universitário, Alto da Ajuda, Lisbon 1349-055, Portugal b

a r t i c l e

i n f o

Article history: Received 5 March 2014 Received in revised form 23 September 2014 Accepted 12 December 2014 Available online xxxx Keywords: Service life prediction Rendered facades Fuzzy model

a b s t r a c t The knowledge of the service life and durability of building components is paramount to sustainable analysis and decision making since it allows a more rational management of the maintenance of building and provides data for life cycle analysis procedures. Nevertheless, predicting the service life of a building or its components is a complex process with which a number of variables are associated. The main difficulties associated with service life prediction are related to the complexity of the degradation phenomena and to the lack of understanding of degradation factors and mechanisms. This paper aims at establishing a model for the service life prediction of rendered facades using a Takagi–Sugeno fuzzy model. The models proposed include the variables that influence the degradation of rendered facades (render age, render type, building height, facade orientation, exposure to damp, and facade protection level). In this study, the degradation condition of 100 case studies located in Portugal is analyzed based only on in situ visual inspections. The proposed models are able to describe appropriately the degradation of rendered facades and to predict the service life of the sample analyzed. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The durability of constructions is essential to the quality of everyone's life and is a critical component of the social and economic stability of contemporary societies [31]. In fact, the built environment represents approximately 50% of the wealth of most European countries [17]. In general, an attitude of “build and let decay” is adopted in construction [10], which results on the loss of building's performance. In Portugal, the building stock is relatively recent, when compared to other European countries, but it shows clear signs of degradation [15]. This situation is due to the natural process of ageing of materials and components combined with the scarcity of resources (due to social and economic conditions) and an incipient culture of maintenance. Building's life cycle comprises three phases: design, construction, and operation (or service phase, which represents 95% of the life time of a building) [18]; as soon as they are built, buildings begin the process of decay [23]. The progressive degradation of the building stock manifests itself in higher levels over time, translating the inability to fulfill their users' needs and expectations. In this context, the study of durability and the prediction of buildings' service life are increasingly important. The performance of external coatings plays a fundamental role in the performance of buildings. The cladding is the most exterior layer of the ⁎ Corresponding author. E-mail addresses: [email protected] (S.M. Vieira), [email protected] (A. Silva), [email protected] (J.M.C. Sousa), [email protected] (J. de Brito), [email protected] (P.L. Gaspar).

http://dx.doi.org/10.1016/j.autcon.2014.12.011 0926-5805/© 2014 Elsevier B.V. All rights reserved.

building and is directly exposed to agents causing degradation. It is therefore more prone to have defects, with direct consequences on the quality of urban space, on users' comfort, and on maintenance and repair costs [11]. In Portugal renders are the most common coating type [6], and according to the 2011 Census [21], they represent around 62% of existing solutions. Rendered facades contribute to the water tightness of the wall system (primary function) and furthermore to the aesthetic performance of the building (secondary function), having a determining influence on the buildings durability [5,28]. This paper proposes a methodology to estimate the global degradation and service life of rendered facades based on fuzzy models. Fuzzy models have been successfully applied in several areas [26] and are a powerful tool to solve problems related to construction. The proposed models use variables that contribute to explaining the degradation phenomena of rendered facades. To that effect, field data on the durability of rendered facades were collected by assessing the degradation condition of 100 rendered facades, located in the Lisbon area, Portugal.

2. Background The degradation process of building materials is a stochastic phenomenon [20], i.e., it involves large variations and uncertainties. In fact, even for identical sets of initial conditions, degradation mechanisms and performance requirements, there may result many possible outcomes with different levels of probability [25]. Moreover, degradation agents and their possible effects on building materials depend on

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numerous agents and their combinations, as well as possible synergistic effects on the deterioration processes [2,22]. Gaspar and de Brito [7] refer that the overall degradation level of any construction element can usually be assessed by means of a quantitative index that portrays its global performance. This degradation severity index is obtained as the ratio between the extent of the facade degradation, weighted as a function of the degradation level and the severity of the defects, also referred to as “condition,” and a reference area, equivalent to the maximum theoretical extent of the degradation for the facade in question, as shown in Eq. (1).

Sw ¼


 Σ An  kn  ka;n

ð1Þ

Ak

where Sw is the normalized severity of degradation of the facade, as a percentage; An is the area of coating affected by a defect, in m2; kn is the defects of n multiplying factor, as a function of its condition (between 1 and 4); ka,n is the weighting coefficient corresponding to the relative importance of each defect (ka,n R+) (if no instructions are provided, it is assumed ka,n = 1); k is the weighting factor equal to the highest condition level; and A is the total area of the cladding, in m2. The weighting coefficients corresponding to the relative importance of each defect are obtained based on the current market cost of repair techniques associated with the various anomalies as a ratio of the cost of executing a new render [9]. These weighting coefficients for the sample studied are shown in Table 1. The degradation severity index is measured based on the defects ascertained, identified during the field work. This indicator takes into account both the degraded area of the coating, affected by the various defects, and the severity level of the defects. The classification system adopted allows benchmarking various facades, from the most favorable condition level (no visible degradation—condition A) to the least favorable one (extensive degradation or loss of functionality—condition E), as shown in Table 2. 2.1. Field work The sample analyzed is composed by 100 facades located in buildings from the XX century used for dwelling, commerce and services. In this study, the age of the cladding can generally be defined as the period between the last overall repair of the cladding and the inspection date. The sample comprises case studies randomly chosen, located in the Lisbon area, Portugal. The analysis of the age of the case studies shows that the buildings inspected have ages comprised between 0 and 60 years. The age distribution is relatively uniform, with one to three buildings per year. Eighty-six percent of the sample is less than 40 years old. Further breakdown of results shows that (a) 27% of the sample is less than 5 years old, (b) 24% of the sample is between 5 and 10 years old, and (c) 24% of the sample is between 10 and 20 years old. The assessment of the degradation condition of the facades is based on in situ visual inspections. For this reason, it was not determined the exact composition of the renders in each case study of the sample. Due to this limitation, the renderings were simply classified in four groups in terms of mortar type [8]: (i) renderings with mixed binders, made of a mix of cement and lime, usually applied in several coats over the substrate; (ii) current cement renderings, made essentially of cement, applied in several coats over the substrate, with a smooth or

rough finish; (iii) renderings with crushed marble, similar to current cement renderings, which contain coarser aggregates that confer a rough finishing; and (iv) single-layer premixed renderings, rich in cement and usually manufactured off site, which are applied in a single layer. In the sample analyzed, current cement renderings correspond to 60% of the sample, followed by renderings with mixed binders (21%), single-layer renderings (13%), and renderings with crushed marble (representing no more than 6% of the cases). Regarding the height of the building, three categories were adopted: (i) low-rise buildings (13% of the sample), with one or two storeys and heights below 9 m; (ii) medium-high buildings (50% of the case studies), with three or four storeys and heights ranging between 9 and 14 m; and (iii) tall buildings (37% of the sample), with five or more storeys, covering all cases over 14 m high. Concerning the facades orientation, Gaspar [8] refers that the most aggressive directions in Portugal are usually north—because greater humidity is combined with fewer hours of sunshine—and west—because of strong solar exposure leading to temperatures that may affect the walls. In the sample analyzed, it is possible to identify a relatively homogenous distribution of the case studies in all of directions, except for northwest, with only three case studies in this direction. At a macroclimatic level, it can be said that the sample analyzed is located in a warm temperate climate zone with dry summer, subjected to the influence of the proximity of the sea, with the classification “CSa” in the Köppen–Geiger system [13]. The categories adopted for exposure to damp were as follows: (i) favorable, for buildings located in urban context, at more than 3 km from the sea and without the influence of marine strong winds; (ii) normal, for buildings located in urban context but located near to the coastal areas (less than 3 km from the sea); and (iii) unfavorable, for buildings located in the first line of the coastal areas, under the influence of strong winds that carry moisture and soluble salts. Sixty-six percent of the sample analyzed corresponds to favorable exposure to damp; only 11% of the facades are subjected to unfavorable conditions and 23% have normal exposure to damp. The protection level of facades is related with rain incidence, the height of the building and the density of occupation of the soil in the area analyzed. In this study, three categories were defined for this variable: (i) with protection, for low-rise buildings, in densely populated areas, protected from the trade winds by other buildings, adjacent hills or vegetation; (ii) current situation, for medium-high buildings, in typically populated urban areas, protected from the trade winds by other buildings, adjacent hills or vegetation; and (iii) without protection, for buildings with more than 4 storeys or in open country or crossroads. The data obtained shows a predominance of situations without protection (43% of the cases), followed by facades with protection (31%) and facades in normal situations (26%). 3. Fuzzy modelling Fuzzy modelling has been successfully used to solve complex problems in different areas. Nowadays, there are various studies that address the application of fuzzy theory to solve problems related with civil and construction engineering, such as analysis of the uncertainty of geotechnical models of failure mechanisms and its implications on construction management [4], the evaluation of stakeholders' satisfaction in publicfunded major infrastructure and construction projects [16], and ranking

Table 1 Relative weights of the anomalies in rendered facades (ka,n). Degradation level

Stain

1 2 3 4

ka,n = 0.12 ka,n = 0.53 ka,n = 0.53 ka,n = 0.53

Cracking 2.50 €/m2 11.50 €/m2 11.50 €/m2 11.50 €/m2

ka,n = 0.95 ka,n = 0.95 ka,n = 1.12 ka,n = 1.53

Loss of adherence 20.50 €/m2 20.50 €/m2 24.00 €/m2 33.00 €/m2

ka,n = 1.53 ka,n = 1.53 ka,n = 1.53 ka,n = 1.53

33.00 €/m2 33.00 €/m2 33.00 €/m2 33.00 €/m2

S.M. Vieira et al. / Automation in Construction 51 (2015) 1–7

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Table 2 Proposed classification of degradation condition of renderings. Example

Physical and visual assessment

Severity of degradation

Condition level

Complete mortar surface with no deterioration. Surface even and uniform. No visible cracking or cracking ≤ 0.1 mm. Uniform color and no dirt. No detachment of elements.

b1%

Level A

Non-uniform mortar surface with likelihood of hollow localized areas determined by percussion, but no signs of detachment. Small cracking (0.25 mm to 1.0 mm) in localized areas. Changed in the general color of the surface. Eventual presence of microorganisms.

1% to 5%

Level B

Localized detachments or perforations of the mortar. Hollow sound when tapped. Detachments only in the socle. Easily visible cracking (1.0 mm to 2.0 mm). Dark patches of damp and dirt, often with microorganisms and algae.

5% to 15%

Level C

Incomplete mortar surface due to detachments and falling of mortar patches. Wide or extensive cracking (≥2 mm). Very dark patches with probable presence of algae.

15% to 30%

Level D

Incomplete mortar surface due to detachments and falling of mortar patches. Wide or extensive cracking (≥2 mm). Very dark patches with probable presence of algae.

N30%

Level E

of the life cycle sustainability performance of different pavement alternatives [14]. Fuzzy modelling using real measures of the system variables is a tool that allows an approximation of nonlinear systems when there is no prior knowledge on the structure and system dynamics or when it is only partially known [29]. Usually, fuzzy modelling follows three steps: structure identification, parameter estimation, and model validation. One of the important advantages of fuzzy modelling is that it combines numerical accuracy with transparency in the form of linguistic rules. Hence, fuzzy models take an intermediate place between numerical and symbolic models. Thus, fuzzy models can be considered “gray box” and transparent [1] models since they describe relations by means of if–then rules. In computational terms, fuzzy models are flexible mathematical structures that are known to be universal function approximators. Usually, the achieved fuzzy model has better performance and accuracy than classical linear models.

3.1. Fuzzy identification The system to be identified can be represented as a multi-input nonlinear model: y = f(x), where x is a vector obtained from input data. In this case, the vector x, for each sample k, can be obtained from the inputs of the process (Eq. (2)). xðkÞ ¼ ½x1 ðkÞ; x2 ðkÞ; …; xn ðkÞ

ð2Þ

The parameter n is the number of available variables, and k = 1…N, where N is the number of samples. For the estimation of the expected life, the model can be represented by Eq. (3). ^ðkÞ ¼ f ðxðkÞÞ y

ð3Þ

The vector x is called the regressor and the predicted outputs ŷ the regressand.

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In this work, Takagi–Sugeno (TS) fuzzy models [27] were used, which consist of fuzzy rules where each rule describes a local inputoutput relation. When TS fuzzy systems are used, each discriminant function consists of rules of the following type:


 ℕ μ Ai j x jk ¼ proj j nþ1 ðμ ik Þ

Rule Ri: If x1 is Ai1 and … and xn is Ain then yi(x) = fi(x), i = 1, 2,…, C where fi is the consequent function for rule Ri and C is the number of rules. The degree of activation of the ith rule is given by Eq. (4): n

βi ¼ ∏ μ Ai j ðxÞ

ð4Þ

j¼1

where μ Ai j ðxÞ : ℝ → [0, 1] is the membership function of the fuzzy set Aij. The model output, ŷ, is computed by aggregating the individual rules contribution, as represented in Eq. (5). XC ^ðxÞ ¼ y

β f ðxÞ i¼1 i i X C β i¼1 i

ð5Þ

In Takagi–Sugeno fuzzy models, the discriminant function is defined as shown in Eq. (6). f i ðxÞ ¼ ai x þ bi ; i ¼ 1; 2; …; C

degree of the data object zk in cluster i. One-dimensional fuzzy sets Aij are obtained from the multidimensional fuzzy sets defined point-wise in the ith row of the partition matrix by projections onto the space of the input variables xj (Eq. (10)).

ð6Þ

ð10Þ

Where proj is the point-wise projection operator [12]. The pointwise defined fuzzy set Aij is approximated by suitable parametric functions in order to compute μAij (xj) for any value of xj. The consequent parameters for each rule are obtained as a weighted ordinary least-square estimate. Let θTi = fi, Xe denote the matrix [X, 1], and Wi denote a diagonal matrix in ℝN × N with the degree of activation, βi(xk), as its kth diagonal element. Assuming that the columns of Xe are linearly independent and βi(xk) N 0 for 1 ≤ k ≤ N, the weighted leastsquares solution of y = Xe θ + ε becomes in Eq. (11): h i−1 T T θi ¼ X e W i X e Xe W iy

ð11Þ

Rule bases constructed from clusters can be redundant due to the fact that the rules defined using the multidimensional antecedents are overlapping in one or more dimensions. A possible approach to solve this problem is to reduce the number of features of the model [30]. 3.2. Fuzzy model applied to service life prediction of rendered facades

Given N available input–output data pairs (xk, yk), the n-dimensional pattern matrix X = [x1,…, xn]T, and the corresponding output vector y = [y1,…, yN]T is constructed. The number of rules C, the antecedent fuzzy sets Aij, and the consequent parameters ai and bi are determined in this step, by means of fuzzy clustering in the product space of the input and output variables [26]. Hence, the data set Z to be clustered is composed by X and y (Eq. (7)): Z ¼ ½X; y

T

ð7Þ

Given the data Z and the number of clusters C, several fuzzy clustering algorithms can be used. This paper uses the fuzzy C-means (FCM) [3], clustering algorithm to compute the fuzzy partition matrix U. The matrix Z provides a description of the system in terms of its local characteristic behavior in regions of the data identified by the clustering algorithm, and each cluster defines a rule. The FCM algorithm uses the Euclidean distance to measure the distance between data point xj and cluster centre vi and is denoted by dij, which is a squared inner-product distance given by Eq. (8).

T
 2 di j x j ; vi ¼ x j −vi A x j −vi

ð8Þ

where A is a norm-inducing matrix. The clustering criterion is expressed via a cost function called C-means functional, which has to be minimized in order to obtain optimal cluster solutions (Eq. (9)): C X N

 X m 2 μ i j di j x j ; vi J f X; U f ; V ¼

ð9Þ

i¼1 J¼1

The C × N matrix Uf = (μij) is called a fuzzy partition matrix. Parameter m ∈[1, ∞[is the weighting exponent that determines the degree of “fuzziness” of the clustering. FCM clustering requires the definition of two parameters: (i) the number of clusters (which translates into the number of fuzzy rules) and (ii) the degree of fuzziness of the clustering m, which is the weighting exponent of the clustering algorithm [3]. The fuzzy sets in the antecedent of the rules are obtained from the partition matrix U, whose ikth element μik ∈ [0, 1] is the membership

Using the methodology described in the previous sections, in this study, fuzzy models were used to define a model to estimate the degradation severity of rendered facades. The models were generated from 75 data samples and tested in 25 samples. The termination tolerance of the clustering algorithm was 0.01. To evaluate the performance of each of the models proposed, the following statistical parameters are used: mean percentage error normalized with respect to the maximum (ε), normalized mean square error (NMSE), and Pearson correlation coefficient (r) (Eqs. (12)–(16)). Furthermore, the models were also tested for the maximum normalized percentage error (εmax.), and the percentage of patterns where ε is greater than x = 5%, x = 10%, x = 20%, and x = 30% (PPε N x).   zp −t p  :100 % ε p ¼  t max  ε¼

ð12Þ

P 1X ε P p¼1 p

ð13Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P
2 u1 X MSE ¼ t z −t P p¼1 p p

NMSE ¼

ð14Þ

P MSE 1X ; t¼ t P p¼1 p t

ð15Þ


 X
P zp −z t p −t 1X p zp r ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 X
2 ; z ¼ P X
p¼1 z −z t −t p p p p

ð16Þ

Table 3 Consequent parameters. Rule

u1

u2

u3

u4

Offset

1 2

1.31 · 10−2 1.83 · 10−2

−6.65 · 10−3 1.08 · 10−1

−3.22 · 10−2 −7.61 · 10−2

4.73 · 10−3 1.32 · 10−2

6.51 · 10−2 −1.87 · 10−1

S.M. Vieira et al. / Automation in Construction 51 (2015) 1–7 Table 4 Cluster centers. Clusters

u1

u2

u3

u4

1 2

9.47 41.2

1.35 1.70

2.10 1.88

0.185 −0.241

where P is the number of patterns, tp is the observed value, zp is the output of the model, and tmax is the maximum observed value. Initially two models are tested. In the first model, the degradation severity (Sw) was a function of four variables: render age, render type, building height, and facade orientation. The second model includes two more variables: exposure to damp and facade protection level. The choice of the variables included in the models is based on previous knowledge concerning the degradation of rendered facades. A study conducted by Silva et al. [25], using the same sample as this study, has shown that the most influential factors to describe the degradation of rendered facades are the six variables listed above. Additionally, a sensitivity analysis was performed in order to identify other variables that could possibly have a significant capacity to explain the variability of the degradation phenomena of rendered facades. However, none of the other variables analyzed shows statistical significance to justify their inclusion in the models proposed. A second analysis was performed removing 15 potential outliers from the original sample. These case studies were removed because they had relatively low ages (less than 5 years), with extremely small values of the degradation severity index (three case studies had a degradation index of 0% at age 0, which is redundant for the model). These cases can skew the results obtained due to the inherent uncertainty of the evolution of the phenomenon of degradation and the difficulty to model this reality at a very early stage. Sometimes the degradation phenomenon changes its intensity over time—it is felt slowly in the early years of life (between year 0 and year 5), accelerating near the end of the service life of building components. After removing these case studies, two new models were created (like in the analysis with the whole sample); the first with four variables (render age, render type, building height, and facade orientation); and the second with six variables (render age, render type, building height, facade orientation, exposure to damp, and facade protection level).

5

The four different models were tested and the above-mentioned statistical parameters were used to evaluate them. In this paper, only the parameters and formulation of the model with the best performance are shown. The model selected was the one with four variables and no outliers. In the selection of the best model the parsimony criteria was used, i.e., since the model with four and six variables have similar performance—based on the comparison of the relevant statistical indicators (a more detailed analysis is presented in the section of results and discussion), the model with fewest variables was chosen. It also seemed more logical to use a model without outliers since simple linear regression techniques showed that these case studies affected the average evolution of the degradation condition of facades over time. The fuzzy rules describing the local input-output relation are presented in Eqs. (17) and (18). Rule 1: If u1 is A11 and u2 is A12 and u3 is A13 and u4 is A14 then, −2

y1 ðkÞ ¼ 1:31  10

þ4:73  10

u1 −6:65  10

−3

−3

−2

u2 −3:22  10

u3

ð17Þ

−2

u4 þ 6:51  10

Rule 2: If u1 is A21 and u2 is A22 and u3 is A23 and u4 is A24 then, −2

y2 ðkÞ ¼ 1:83  10

þ1:32  10

u1 þ 1:08  10

−2

−1

−2

u2 −7:61  10

u3

ð18Þ

−1

u4 −1:87  10

where u1 is the render age, u2 is the render type, u3 is the building height, and u4 is the facade orientation. Table 3 presents the consequent parameters of the model and Table 4 the clusters centers. The membership functions for each of the four input variables are shown in Fig. 1. The membership functions obtained from cluster 1 are represented by the dashed lines, and the membership functions obtained from cluster 2 are represented by the continuous lines. 4. Results and discussion All the proposed models were tested using the test dataset which represents 25% of the samples. Table 5 presents the performance of

Fig. 1. Membership functions for the four fuzzy model variables.

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S.M. Vieira et al. / Automation in Construction 51 (2015) 1–7

the four Takagi–Sugeno fuzzy models. In general, all models have a similar performance, with correlation r ranging from 0.92 to 0.94. The models built without the outliers have a higher mean percentage error, and this can be explained by the lower number of case studies used to train the model (15% less). However, the difference in the mean percentage error is not significant. This means that the presence of these outliers in the data does not affect the training of the fuzzy model. Fuzzy models will only be affected by outliers when these fall significantly far from the neighbor case studies. These results show the potential of the fuzzy models since they can cope with the inherent uncertainty of real data, where the measurements are prone to errors and variability. Although fuzzy models can cope with uncertainty, if the model is trained without outliers, the result will be less biased. Thus, the chosen model is number 3 (4 variables without outliers). Between the model with 4 variables and the model with 6 variables, the difference in performance is not significant, and therefore there is no added value from increasing the model complexity, adding 2 variables. The analysis of the parameters of the chosen model, which are presented in Fig. 1 and Table 4, can give some clues about which variables most influence the degradation severity, and in what way. For “Render Age,” it is observed that the clustering clearly divides the data between newer and older buildings. The membership function resulting from cluster one shows that buildings with less than 30 years strongly belong to cluster one (membership degree over 0.5) and will positively influence the output value of rule number one (Table 4). The membership function for cluster two shows that buildings with more than 25 years strongly belong to cluster two (membership degree over 0.5) and will positively influence the output value of rule number two of the fuzzy model (Table 4). For “Render Type,” only cluster number two can discriminate the different types of render. The render type “lime-cement renderings” (number 2) has a higher positive contribution to the output of rule number two of the fuzzy model. In rule number one, although all the types of render give a contribution to the output of the rule, it is a small contribution since the consequent parameter associated with “u2” is small (10-3) (Table 4). For “Building Height,” the membership function associated with rule one has high values of membership for all the three categories but is maximum only for category “tall buildings,” which indicates that the higher the building the higher the influence on the degradation severity. In this case, the consequent parameter for “u3” in Table 4 is negative, indicating a decrease in degradation severity with building height. This should not be the case, and if one looks at the average value of the building age for each building height category (small, medium, high), the “high” category has a much lower average, 8.5 years compared with 19.5 and 19 years for categories “small” and “medium,” respectively. This means that the number of older high buildings is very low and for that reason the fuzzy model cannot learn the influence of the building height for older buildings in the degradation severity. For “Orientation,” as for building age, there is a good separation of variable values for membership function one and two. Membership function one, although it has a low influence on the output of rule number one (consequent parameter is 10-3), for orientations N, NE, and NW (2, 3 and 4 respectively), will increase the rule output, thus increasing

Table 5 Performance measures of the four first order Takagi–Sugeno fuzzy models. Model

ε

εmax

PPε N 5%

PPε N 10%

PPε N 20%

PPε N 30%

r

4 variables 6 variables 4 variables no outliers 6 variables no outliers

3.52% 3.22% 4.05%

10.06% 9.60% 10.97%

24.0% 20.0% 23.8%

4.0% 0% 4.8%

0% 0% 0%

0% 0% 0%

0.92 0.94 0.93

3.79%

10.60%

23.8%

4.8%

0%

0%

0.94

the degradation severity. Orientations S, SE, and SW (− 2, − 3, and − 4, respectively) have a higher membership degree for membership function number two, and as the values are negative the rule output will decrease. The four Takagi–Sugeno fuzzy models were able to provide an estimated service life of rendered facades, as shown in Table 6. The values of estimated service life were obtained for a maximum accepted degradation level of 20%. The values obtained are coherent; in fact, the average estimated service life was around 18 years, for the four fuzzy models, with relatively low values of standard deviation. Furthermore, the results obtained agree with previous studies. Based on the same sample, Silva et al. [25] obtained an estimated service life of 17.5 years for a service life prediction model based on artificial neural networks (with variables render age, render type, exposure to damp, and facade protection level). Both methods (ANNs and fuzzy systems) are capable of modelling extremely complex nonlinear functions; however, in the literature, neural networks are associated to the concept of “black box” due to the difficulty of understanding how the network operates (ANNs use thousands synaptic weights that are not subject of a logical interpretation). On the other hand, fuzzy systems are seen as a “gray box” or transparent model, since they describe relationships by means of if–then rules, using in the models concepts used in human communication (more understandable to a non-specialized stakeholder). A study performed by Shohet and Paciuk [24] using an empirical method (based on a simple regression analysis) leads to an estimated service life of 15 years (with a range between 12 and 19 years), using a more stringent performance criteria; the same study reveals an estimated service life of 23 years (with a range between 19 and 27 years) for a lower level of performance requirements. Simple regression techniques are simple tools that can be easily implemented by any user. Fuzzy systems are much more complex than regression techniques but can deal with imprecise and vague data and with the uncertainty associated with the degradation phenomenon. The results obtained in the present study are also confirmed by an extensive survey of the deterioration of over 900 building elements in the UK, from which durability rankings were identified and a relationship was established between the latter and estimated service life. According to Mayer and Bourke [19], the durability ranking for external renders is of 15 years—a figure used by Insurance companies—and the estimated reference service life for renders is of 18 years.

5. Conclusion The degradation of building components is a complex phenomenon that depends on various variables and that seriously affects the built stocks. Models for service life prediction are thus necessary. In this paper, a Takagi–Sugeno fuzzy model was used to predict the service life of rendered facades. The proposed model allows evaluating the degradation condition of the rendered facades over time. Beyond that, this methodology allows to estimate the service life of each case study within the sample of 100 rendered facades analyzed. In this study, it is shown that the proposed models appropriately expresses the degradation condition of this type of coating. The method yielded an average service life of rendered facades of around 18 years. This value seems realistic and is coherent with the literature related with this thematic. Furthermore, it is possible to improve this service life prediction method. The definition of the maximum admissible level of degradation and the relationship between degradation severity and condition (shown in Table 2) are both prone to some subjectivity and could therefore be adapted to other perspectives and thus lead to different results. This model can also be adapted to circumstances prevailing in other countries and other types of coatings. Furthermore, the model can improve, by improving the sample (it is possible to add more case studies anytime in the process of modelling).

S.M. Vieira et al. / Automation in Construction 51 (2015) 1–7

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Table 6 Statistical indicators concerning the estimated service life determined using the four Takagi–Sugeno fuzzy models. Model

4 variables

6 variables

4 variables no outliers

6 variables no outliers

Average reference service life Median reference service life Maximum reference service life Minimum reference service life Standard deviation of the reference service life 95% confidence interval

18.39 years 18.04 years 26.06 years 16.13 years 1.86 years ±0.73 years

18.18 years 17.88 years 19.68 years 16.88 years 0.87 years ±0.34 years

18.85 years 18.60 years 26.70 years 15.40 years 2.29 years ±0.90 years

18.68 years 18.60 years 20.70 years 16.20 years 1.33 years ±0.52 years

The reference service life was estimated admitting 20% as the maximum acceptable degradation level. In other words, it is admitted that cases whose degradation severity is higher than 20% have reached the end of their service life.

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