Fuzzy logic applied to the visual inspection of existing buildings has been proposed in relation to simple structures. Isostatic structures are characterized by a unique and known collapse mechanism, which does not vary with geometry or load change. In this paper we apply fuzzy logic to visual inspection for complex structures such as hyperstatic ones in which the collapse mechanism depends not only on the geometry but also on the size and disposition of loads. The goal of this paper is to give relevant weight, in the fuzzy analysis, not only to the single expression of degradation, due to its localization within the element, but also to the structural element itself by assigning a different resistance to the various elements. The underlying aim of the proposed method is to manage, evaluate, and process all the information coming from visual inspections in order to realize a management information system for the evaluation of the safety level of even complex structures.
In recent years the need for a reliable evaluation of safety for existing buildings has become ever more necessary [
This request derives both from traumatic events that have caused great impact on the population and also from greater attention paid by public administrations in relation to the recovery of buildings and the need to preserve constructions holding a strategic and functional importance.
In order to formulate an objective judgment on the safety level [
Historical data collection.
Visual inspection.
In situ (nondestructive) tests.
Lab test (on properly chosen samples).
In situ (destructive) tests.
Through a comparative critical examination of all the information collected, a diagnosis on the degradation level of the structure under observation can be produced.
However, often the only available diagnosis instrument, in order to evaluate the vulnerability of a construction and decide if further damage evaluation with other tools is required, is represented by visual inspections, mainly because of the need to restore safety conditions in a short time.
Thus, visual inspections become the ruling practice in the management of maintenance, even when the number and importance of the constructions are significant.
The process of evaluation of degradation on the basis of the results of visual inspection is heavily affected by subjectivity. The staff in charge of the inspection write down on a safety assessment card a linguistic statement, which represents the subjective judgment for the degradation under examination. When relying only on visual inspection both the problems of dealing with different levels of expertise of the inspectors and the problems of handling subjective information on degradation raise this information, expressed by means of linguistic statements, which needs to be turned into objective and reliable assessments.
In order to use visual inspection as a robust and reliable instrument to evaluate the safety level of a construction it was decided to take advantage of the ability of fuzzy logic to treat uncertainty as expressed by linguistic judgments [
Thanks to the key role played by inference, where the technical knowledge of experts is taken into account, fuzzy logic is then able to output an objective evaluation of the safety level of a construction, based only on visual inspection.
This method provides a quick, low cost, and reliable way of assessing the evolution of degradation in a structure [
Fuzzy logic was introduced in 1965 [
According to the traditional definition of “set,” an element can belong to a set or not. According to the fuzzy set theory an element belongs to a set with a certain degree of membership [
Similarly, in classical logic a statement is evaluated as “true” or “false” while fuzzy logic assigns a value of partial truth. All this is particularly suitable for those situations where there can be no absolute certainty about a phenomenon. It is useful to specify that such uncertainty can be referred to the judgment of the operator in estimating the phenomenon or to the phenomenon itself.
There are three basic steps of a generic fuzzy process: fuzzification, inference, and defuzzification (Figure
Fuzzysystem architecture.
The fuzzification procedure consists in transforming the numerical value of the considered variable in its corresponding value of membership to the given fuzzy sets through the corresponding membership function.
The membership functions typically overlap so that values of the variable can partially belong to multiple fuzzy sets. The wider the area that overlaps, the more the uncertainty the system includes.
The procedure of inference involves the application of the rules of combination of fuzzy sets. Usually these are simple linguistic expressions, which are converted to mathematical formalism in the language of the “
The output is also a fuzzy membership value that can be used either “raw” as qualitative assessment or defuzzified as a real number, compatible with nonfuzzy approaches [
The aim of this research is to determine, through only the process of visual inspection, the probability of failure
The value of the safety assessment is well specified by Eurocode [
In this study we will refer to the Ultimate Limit State (ULS), which is the limit state associated with collapse or other similar forms of structural failure. The safety assessment is positive if
This inequality indicates the possibility that the state limit defined above is reached. According to Eurocode the probability
This value represents the limit value
Fuzzy logic applied to the visual inspection of existing buildings has been proposed in relation to simple structures such as isostatic structures [
In this paper we present the results of the application of fuzzy logic to hyperstatic structures.
In particular we chose as a case study the one shown in Figures
Plan view hyperstatic structure.
Hyperstatic structure.
The situation is more complex, because the collapse mechanism in hyperstatic structures is not unique and depends not only on the geometry but also on the size and disposition of loads [
The expression of degradation will therefore depend on the relevance of the structural element in respect to the whole structure.
It becomes necessary to assign not only the weight that a single expression of degradation has on safety but also the weight that the structural element has on the structure in its entirety, whether it is the beam or the column. It is essential to assign a different resistance to the various structural elements in order to understand which element might collapse sooner than the others.
For this purpose a nonlinear elastic analysis on the twodimensional structure type (Figure
The purpose of this study is not to perform a test, but to identify, through the formation of plastic hinges, the sections that reach the yield point and form the collapse mechanism [
The hypothesis of considering the twodimensional structure is valid, because the building in the chosen example is regular.
In order to verify the method, a seismic action was applied to the structure. The seismic action is characterized by the accelerometric horizontal component of the seismic motion. For the calculations, the software DRAIN2DX (DRAIN2DX: static and dynamic analysis of inelastic plane structures, software developed at the University of Illinois at UrbanaChampaign and provided by Professor Mark Aschheim; it is a finite element program for the seismic analysis of linear and nonlinear plane frames) was used. This software applies accelerograms of known seismic events such as the one relative to the earthquake which occurred in El Centro, Imperial Valley, California (USA), in 1940 whose diagram, as a function of time, will be shown later (Figure
See Tables
Structure geometry.
Length of the spans in the 

Length of the spans in the 

Floor height 

Slab thickness 

Section geometry.
Columns 

30 cm 

30 cm  

900 cm^{2}  

67500 cm^{4}  

747 cm^{2}  
Steel reinforcement columns on the ground floor 


Steel reinforcement columns on the first floor 




Beams 

30 cm 

50 cm  

186 cm  

26 cm  

6336 cm^{2}  

22 cm  

2238144 cm^{4}  

1680 cm^{2}  
Steel reinforcement in the tension zone 


Steel reinforcement in the compression zone 

Variable loads.
Intermediate floors: 



Top floor: 

Values loads.
Floors  Cover  First floor 

Distributed loads (KN/m)  
Dead weight floor  29.25  29.25 
Dead weight beam in the 
3.75  3.75 
Variables  1.35  6.48 
Total load  34.35  39.48 


Point loads (KN) exterior columns  
Dead weight beam in the 
16.87  16.87 
Upper column  0  3.37 
Lower column  3.37  3.37 

20.25  23.61 


Point loads (KN) interior columns  
Dead weight beam in the 
16.87  16.87 
Upper column  0  3.37 
Lower column  3.37  3.37 

20.25  23.61 
The structure is tested for medium ductility class. Rotations of the columns at foundation level are prevented (joint constraints). The horizontal elements are considered to be infinitely rigid; that is, it is assumed that there are no relative movements between the different points on the same plane. This hypothesis is valid given the nature of the slab, constructed in reinforced concrete.
The numbers of the nodes inserted in the data input of the program are shown in Figure
Types of loads applied to the structure are as follows:
Permanent loads:
Variable loads:
In order to determine the nodal masses, the following load combination was used:
For each floor, the following load distribution will be considered:
Distributed loads:
weight of the floor:
weight of the beam in the
variable load coverage:
variable load intermediate floors:
Point loads:
weight of the beam in the
weight of the upper column:
weight of the lower column:
The values are given in Table
It is assumed that the beams are fixed at their ends. Therefore the vertical reactions at the supports and the moments are equal to
Horizontal loads in the nodes are zero, while the vertical loads are given by the sum of the reactions
Nodal loads.
Accelerogram El Centro, Imperial Valley, California, in 1940.
Nodal load values for each floor are shown in Table
Vertical loads.
Floors  External nodes  Internal nodes  










Cover  20.25  85.87  106.12  71.56  20.25  171.75  192  0 
First floor  23.62  98.70  122.32  82.25  23.62  197.4  221.02  0 
Nodal masses are obtained by dividing the vertical loads concentrated at each node by the acceleration of gravity (
Based on the convection of signs adopted by the software DRAINDX forces pointing downwards are negative, while counterclockwise moments are positive.
The values of the nodal masses are given in Table
Nodal masses.
Floors  Node  Nodal masses 

Cover  3010  0.108 
3020  0.196  
3030  0.108  


First floor  2010  0.125 
2020  0.225  
2030  0.125 
The results obtained are shown below by applying the accelerogram corresponding to the earthquake which occurred in El Centro, Imperial Valley, California, in 1940 (Figure
The history of the development of plastic hinges in the structure under examination is shown in Figure
Development of plastic hinges.
This nonsimultaneous formation of plastic hinges involves significant plastic rotations in the hinges that were formed first, which allow the redistribution of moments between critical sections [
It can be noted in Figure
It is therefore clear how a possible reduction of the section of the lower columns, caused by any type of damage, will further weaken the structure and anticipate collapse.
The weight of columns of the ground floor is greater than the weight of the other elements of the structure. We could therefore build a hierarchy of elements following the story of formation of the plastic hinges. The problem is that this type of analysis can be performed only when the construction under inspection is perfectly known in terms of the geometry of the structure and sections and also of the applied loads.
At this point, it is necessary to give a priori weights to the different elements of the structure according to the regulations [
In this way, the weight that each structural element has in a hyperstatic structure may coincide with the optimal mechanism of development of plasticization.
Thinking in terms of weight, a greater weight to the columns should be assigned in respect to the beams and, within this differentiation, it is necessary to give greater weight to events related to shear degradation rather than to those related to bending. The remaining problem is therefore to assign to the columns a greater weight than to the beams, also taking into consideration that the columns of the ground floor must have greater weight than those of the first floor.
The aim of this study is to manipulate through fuzzy logic the subjective linguistic judgments expressed by an inspection staff on the visual signs of degradation in order to assess the current safety level of the inspected construction [
The method is consequently divided into two stages: the first consists in the insertion of data obtained by visual inspection on a standard form card (Inspection Card), eventually attaching to it photographs and videos [
Evaluation Card.
Structural element  Type of degradation  Linguistic judgment  Safety goal  Weight of the structural element 






The card consists of five columns. In the first column the structural element under inspection is indicated.
In the second column for each structural element the degradation expressions that are more relevant to the structural element itself are shown. In the third column the inspection staff will write down the assessments of gravity attributed to the different types of degradation. The fourth column shows the weights that each lexical judgment will have on the current safety assessments. The fifth indicates the weight that the structural element holds within the structure under consideration. It is important to stress that the card and the relationship between judgment and weight are constructed according to the expertise of a technically skilled team, while the third column is filled in during the visual inspection by the staff.
Our research proposes a manipulation of linguistic subjective judgments expressed by the inspection staff on the degradation of a structure using fuzzy logic. The goal is to assess the current safety level of the inspected construction.
The operator gives a linguistic judgment for each type of degradation
With each linguistic judgment
For each structural element
The linguistic variables represent our quality input (Figure
With each linguistic variable we associate a membership function
For both the judgments
Developments of the triangular membership functions.
The membership functions of linguistic judgments:
The membership functions
The membership functions of weight:
The membership functions of the weight that the structural element has within the structure:
In Table
Values
Membership functions 



Membership functions 




VS  —  0  0.1  VSS  7  6.5  — 
S  0  0.25  0.5  SS  7  5.75  4.5 
SLS  0.25  0.4  0.5  SLSS  5.75  5  4.5 
M  0.25  0.5  0.75  MS  5.75  4.5  3.25 
SLL  0.5  0.6  0.75  SLLS  4.5  4  3.25 
L  0.5  0.75  1  LS  4.5  3.25  2 
VL  0.9  1  —  VLS  —  2.25  2 
Values
Membership functions 




S  −0.25 

0.25 
M  −0.75  −0.5 

L  −1  −0.75 

The variation range of the domain of the weight
The proposed method was applied to the generic hyperstatic structure shown in Figure
It is assumed that the structure in question is in an advanced state of degradation and that the Evaluation Card is the one shown in Table
Degradation assessment card beams.
Structural element 
Type of degradation 
Linguistic judgement 
Safety goal 
Safety exponent 

Beam  Shear stress  Small  Medium  5.12 
Medium  Large  3.88  
Large  Very large  3.00  
Longitudinal stress  Small  Slightly small  5.62  
Medium  Medium  4.50  
Large  Large  3.25 
Degradation assessment card columns.
Structural element 
Type of degradation 
Linguistic judgement 
Safety goal 
Safety exponent 

Column  Reinforcement corrosion  Small  Medium  5.12 
Medium  Large  3.88  
Large  Very large  3.00  
Spalling  Small  Slightly small  5.62  
Medium  Medium  4.50  
Large  Large  3.25 
In the last column the safety factors obtained by varying the
The procedures suggested in the literature are two. The first, based on fuzzy sets theory [
The intersection operator initially combines the single judgment
Given two fuzzy sets, for example,
The intersection of two fuzzy sets
The fuzzy functions are represented by vectors
The combined effect of the judgment and the weight relative to each degradation was calculated (
In the proposed method the variation range of the exponent is assumed to be the domain of the security level. Since the safety measurement is positive if
It is assumed a priori that the hypothesis of the structure under examination is designed and constructed in conformity with the regulations.
In this case we assume the structure, in presence of the loads required by regulations, to be in a safe condition. Therefore, if we refer to the value of the safety measure usually denoted by
We need then to establish a fuzzy relation between the judgment of a single type of deterioration and the evaluation of the safety exponent, by adopting a fuzzy composition according to
We know the value of the fuzzy relation
With each linguistic variable used for the safety exponent
The membership functions of safety exponent:
The fuzzy relation
A fuzzy relation is a fuzzy set defined on multiple domains (multidimensional fuzzy set). Once expressed the fuzzy relations
Figure
The operation of combination (
The result of this composition (Figure
The result of the inference procedure gives the relation between the weight on the safety and the safety exponent itself; the result is expressed by a matrix
In Figure
Safety exponent
In order to evaluate the effect that the weight has on the value of the safety exponent it is necessary to assess its value by defuzzifying the result (Figure
In the literature [
Giving the values
The value of the defuzzified exponent obtained from the final fuzzification procedure of Cartesian product
Results columns.











Medium  Large  3.88  Large  3.25  3.16 

Medium  Medium  4.50  3.88  




Medium  Large  3.88  Medium  3.44  3.36 

Medium  Medium  4.50  4.13  




Small  Medium  5.12  Large  4.50  4.37 

Small  Slightly small  5.62  4.94  




Small  Medium  5.12  Medium  4.69  4.54 

Small  Slightly small  5.62  5.06  




Large  Very large  3.00  Large  2.50  2.31 

Large  Large  3.25  2.75  




Large  Very large  3.00  Medium  2.69  2.47 

Large  Large  3.25  2.88 
Results beams.











Small  Medium  5.12  Small  5.12  5.00 

Small  Slightly small  5.62  5.62  




Medium  Large  3.88  Small  3.88  3.87 

Small  Slightly small  5.62  5.62  




Medium  Very large  3.88  Small  3.88  3.79 

Medium  Large  4.50  4.50  




Medium  Large  3.88  Small  3.88  3.16 

Large  Large  3.25  3.25 
In Figure
Safety exponent achieved by giving
Safety exponent achieved by giving
Safety exponent achieved by giving
In Tables
Degradation assessment card middle beam and exponent and safety.
Type of degradation 
Linguistic judgement 
Safety goal 
Safety exponent 



Small 

Small 

5.75 
Medium 

Slightly small 

4.62  
Large 

Medium 

3.88  



Small 

Medium 

5.12 
Medium 

Slightly large 

4.38  
Large 

Large 

3.25 
In the last column of the table there is only one safety factor that takes into account the presence of the two expressions.
To obtain a single value an operation similar to what is done to assess the probability of failure in the case of a structural project was decided upon for which we have more causes of failure.
A chain mechanism is assumed, in series, in which the probability of failure of each event of degradation is identified with the probability of failure of a single link in the chain, and the probability of failure of the entire chain, in the hypothesis of independence of the results of each loop, takes the expression [
In Figure
The choice of the weights was made using the following criteria: firstly the weight of columns of the ground floor is greater than the weight of the other elements of the structure and secondly a greater weight to the columns in respect to the beams is assigned and, within this differentiation, it is necessary to give greater weight to events related to shear degradation rather than to those related to bending.
For the lower column E_{10102010} to both manifestations of degradation the linguistic subjective judgment medium was assigned obtaining, respectively,
The same linguistic judgment was given to the higher columns E20103010 and accordingly the same results
The difference lies in the fact that the weight associated with the element E_{10102010} is large, while the weight associated with the element E_{20103010} is medium. These weights change, according to the fuzzy logic method, the value of the safety exponent associated with each manifestation of degradation, resulting for the structural element E_{10102010} in weighed safety factors, respectively, equal to 3.25 and 3.88 (Table
It is clear then that both the weight large and the weight medium reduce the safety factor
The fuzzy procedure that relates the safety factor to the weight that the structural element has within the structure further reduces the coefficient within a range determined a priori by experts. In fact, attributing the weight large to both manifestations of degradation safety requirements are no longer satisfied in both cases, while attributing the weight medium safety condition is verified only in the case of the manifestation of degradation spalling.
In the case under study, to the elements E_{10202020} and E_{20202030} and for both degradation events the linguistic judgment small was assigned, obtaining, respectively,
The weight
Also in this case both the weights large and medium reduce the safety coefficient
In this case both E_{30103020} and E_{30203030} and for both manifestations of degradation the linguistic subjective judgment large was assigned obtaining, respectively,
The weight
Also in this case both the weight large and the weight medium reduce the value of
These values lead to an unsafe condition alert for the structural element. The weight
The weight
The attribution of weights medium and large to the structural element reduces the safety factor
The value of
The use of fuzzy logic has allowed an objective result to be obtained, that is, judgment, which is influenced not only by the weight of the manifestation of degradation but also by its localization within the structure. This method was applied to a bridge placed along a road, the SS 195 in Cagliari (Italy), suffering from forms of degradation related to the ULS. The aim of this study is to manipulate through fuzzy logic the subjective linguistic judgments expressed by the inspecting staff on the visual signs of degradation in order to assess the current safety level of the deck of the bridge [
In the procedure it was decided to give simple linguistic judgements on the level of degradation and not to make any measurements because often, as in the case under examination, the structural elements are not easy to reach. The presence of water under the bridge physically prevents the measurements of the damage, at least not without considerable cost.
The manifestations of degradation common to beams
The corrosion of the lower longitudinal bars, in this case, has different weight when localized at the middle rather than at the ends. Indeed, in a simply supported beam (static scheme of the present case) the kinematic collapse is unique and it occurs when a plastic hinge is formed in the middle. From here comes the need to assess the phenomenon of corrosion whereas the beam is divided into three parts: the centerline and the two left and right ends, consequently giving a different weight to the event itself, depending on its location.
Such manifestations could affect the bearing capacity of the structural element by reducing the resistant section and for this reason are defined as ULS [
Table
Degradation assessment card beam ends and exponent and safety.
Type of degradation 
Linguistic judgement 
Safety goal 
Safety exponent 



Small 

Small 

5.75 
Medium 

Slightly small 

4.62  
Large 

Medium 

3.88  



Small 

Slightly small 

5.62 
Medium 

Medium 

4.50  
Large 

Slightly large 

3.50 
With the application of the proposed method, the visual inspection is “translated” into an objective examination of the state of deterioration of the structures under investigation. The numerical evaluation of the safety factor
In this paper a procedure suitable for the manipulation and interpretation of subjective linguistic judgments was developed resulting from a visual inspection of a construction, in order to assess, in a more objective and reliable way, the safety level of the building under examination.
With the application of the proposed method, the visual inspection is “translated” into an objective examination of the state of deterioration of the structures under investigation. The numerical evaluation of the safety factor
All the information coming from the visual inspections is managed, evaluated, and processed, thus realizing a management information system based on the authors’ software.
Software is developed by the authors which implements their fuzzy logic procedure applied to lexical subjective judgements. The evolution of this procedure described in this paper not only considers the weight associated with the manifestation of degradation but also takes into consideration the weight that each element has on overall structural safety. This allows a specific assessment of detail and a broader assessment of the condition of the building as a whole taking into account the morphology of the structural system.
Interesting results were obtained in the applications described in the previous publications [
In this way, the costs resulting from this type of investigation are relatively low with no detriment on the reliability of the result which is reflected in the value of the safety factor
This information management system allows safety operators to sort the data resulting from visual inspection of the structures even when structural number is relevant. For each structure under study, the authors’ system returns the safety factor
The goal of the proposed method is to evaluate and manage all the information coming from visual inspections. In this way, experts can, through the evaluation of the safety factor
The authors declare that there is no conflict of interests regarding the publication of this paper.