Structural vulnerability index to the effects of landslides in masonry buildings based on fuzzy sets

The aim of this paper is to develop a simple model to assess structural vulnerability of buildings at a regional scale. This model was especially made for areas where current reports on damages of buildings do not exist. The method is based on the identification of the most influential characteristics associated with damage in building under the action of the effect effects of landslides. The evaluation of these characteristics was carried out by means of the determination of fourteen (14) parameters. A vulnerability degree and an importance value based on experts ́ opinions were assigned to each of the 14 parameters. Due to the imprecision and subjectivity of the opinions, data was processed by means of fuzzy logic techniques. Once each one of the parameters was qualified, the qualifications of the parameters were related to their value of importance by means of a fuzzy factorial. The result of this relationship, according to techniques based on fuzzy set, is the building vulnerability index.

Índice de vulnerabilidad estructural/Structural vulnerability index The accelerated urban development undergone by Colombian cities -due to a high population growth, lack of building design planning, economic housing shortage -has forced population to build in high risk areas, often underestimating nature disaster threats to housing construction projects and to population suffering losses and also leading to a higher death toll.
Consequently, it is important to work identifying the potential areas which would be affected by an effective natural disaster threat as well as to provide a better construction planning for future buildings construction, in order to achieve a less vulnerable city.This would only be possible if a vulnerability study for the cities is developed.
Generally, experience has demonstrated that action is taken after disasters have occurred.Most existing vulnerability, threads and risks studies due to landslides have been encouraged after tragedies have taken place in the past and this practice must be avoided.It is necessary to develop complete and fully supported studies, not conditioned to time limitations urging to deliver prompt results and proposing immediate measures, which are unaware of the real problem intended to solve.Therefore, effective caution and improvement measures will be only offered to the community with a clear planning on how buildings are and how they respond to landslide effects.
Therefore, the main purpose of this research is to introduce a new methodology to estimate building vulnerability of a given area to landslide effects.The model is based on the assessment of the most relevant characteristics of a masonry structure, which may affect its vulnerability faced to abovementioned effects.Quoted characteristics have been determined by selecting fourteen parameters.Parameters were classified in accordance to three or four quality standards, which were assigned with a specific vulnerability degree.Besides, each parameter was assigned with an importance value.These vulnerability degrees and weight value were identified by expert opinions.
Since the proposed model involves expert opinions and its further application is based on the responses to a form which classifies each parameter; it was decided to employ fuzzy sets techniques.Among fuzzy sets diffuse factorial was used, as a means of calculation for vulnerability index.Such index is the relation between vulnerability degree of each parameter and its importance value.

Vulnerability estimation models
Nowadays there are different proposed methodologies to estimate building vulnerability to landslide actions.However, a great deal employs historical data or they are based on the experience obtained from past events.Then, in areas lacking of past damages records, construction and adaptation of such methodologies is not direct.Therefore, it is important to develop vulnerability estimation models depending on the buildings own characteristics and the places were they will be implemented.Uzielli et al. (2008) introduce a quantitative vulnerability estimation method based on a probabilistic approach.They start by associating vulnerability to the lack of inherent capacity of the structure to preserve its physical integrity and functional capacity during the occurrence of a landslide effect and, they estimate that this mainly depends on the structure characteristic and its maintenance conditions.So they define vulnerability in function of structural typology and maintenance conditions.Leone et al. (1996) propose a model to evaluate damages due to landslide phenomena, based on damage matrixes built from statistics on past records.To do so, they introduce the concept of structure vulnerability in function of the structure type, age and its strength by assigning a value from 0 to 1. Finlay et al. (1997) depict an example of vulnerability estimation by assigning a value from 0 to 1 to vulnerability index based on historical records, without considering several components affecting vulnerability.Values are assigned in accordance with historical experience available from threat management.

Model parameters
The proposed model is based on the assessment of 14 building parameters and their closest surroundings, which are considered of major influence on a masonry building vulnerability faced to landslide effects.Each one of them were assigned with three or four quality standards (A, B, C y D), as follows: 1. Building placed on a gradient lower than 30° or beyond a slope higher than 30°, fulfilling required isolation distances already established.
Índice de vulnerabilidad estructural/Structural vulnerability index 2. Building placed on a hill top with land gradient between 30° and 45°, on a slope lower than 5 meters high, but counting with a rigid-structure-retaining wall, which height is equal or higher than the slope total height.3. Building placed under the base or slope foundation, with a total height lower than 5 meters, counting with a rigid-structure-retaining wall, which height is equal or higher than the slope total height.

B. Not fulfilling A either C.
C. Building having one of the following characteristics: 1. Building placed on the slope with natural gradient lower than 30° or out of its scope, but not fulfilling required isolation distances 2. Built on cross sections 3. Placed on a cross section of a natural land gradient higher than 30°º 4. Placed on a back-filled hillside Parameter 6 -Foundation location A. Foundation is located at the same benchmark.Lack of unbalanced pressure due to back-filling.
B. Maximum differences among foundation benchmarks are lower than 1 meter, with lack of unbalanced pressure due to backfill or; because the foundation is located at the same benchmark but there is presence of unbalanced pressure due to backfill.C. Maximum differences among foundation benchmarks are lower than 1 meter, with presence of unbalanced pressure due to backfill.D. Maximum differences among foundation benchmarks are higher than 1 meter

Parameter12 -Handling of waste materials
A. There is an adequate waste collection system B. Waste collection system is not appropriate or waste material is directly dumped to the hillside.
Parameter 13 -Surrounding conservation conditions A. The area has clean drainage systems.There are no miss-matched joints in the sewage system or leakage in the supply piping either.Water leakage is not observed on the soil.

B. Does not meet A either C.
C. The area counts with a clean drainage system.Furthermore, there is at least one of the following aspects: a.There are broken joints alongside the sewage system b.There are leakages in the supply piping c.There are visible leakages on the soil Parameter 14 -Building Preservation Conditions A. Walls in good conditions, non visible crackings.
C. Walls having medium size crackings, ranging from 2 up to 3 millimeters width.Building that does not have crackings but it is characterized by a poor masonry preservation conditions.
D. Walls have a major material damage or have crackings higher than 3 millimeters width.

Definition of the vulnerability index model
In this paper vulnerability is quantitatively defined as the susceptibility of the elements to suffer damages due to landslides effects.The proposed model is a complete description of required information to obtain a quantitative expression of vulnerability, called Vulnerability Index.This index relates parameters vulnerability degrees mostly affecting building vulnerability and their importance values in the total vulnerability estimation.Selected parameters are associated in accordance with the structures own characteristics, their location and surrounding factors.Besides, the input associated to structural aspects can be direct or indirect, see Figure 1.The model is based on the estimation of a total vulnerability index (IV T ), which is the input of four estimated partial vulnerability indexes.One is associated with direct aspects (IVE D ), other with indirect structural aspects (IV EI ), other with building locations (IV U ) and finally another associated with surroundings aspects (IV E ).Each one of them has a range from 0 (non vulnerable) to 1 (completely vulnerable).See Figure 1.
Each index is estimated from qualifications of different quality conditions on selected parameters (A, B, C and D) named K I .The importance values of each parameter the total vulnerability determination of the building are named W i .And values K I and W i were analyzed under fuzzy sets because of their definition ambiguity (Zadeh, 1965;Chakroborty and Kikuchi, 1990;Chao and Ayyub, 1995;Maldonado 2007).Diffuse arithmetic (Bandemer y Gottwald, 1996;Tee, et. al., 1988) was employed to relate qualifications of each parameter (K I ) and their respective importance values (W i ).The proposed method to combine diffuse -of uneven importance or weights -was diffuse factorial, which was used to determine vulnerability index, expressed as: Where I.V. is the vulnerability index; i is i-th of n vulnerability factor (each one defined in a range of [0.1]);K i is the vulnerability degree of i parameter category i; W i is the importance associated to parameter i in relation to the other parameters and; n is the number of parameters in the model.
Therefore, each index will be determined as follows: Parameters associated to this index are structural system, quality and structural strength. (1) (2) (3) (4) Esperanza Maldonado Rondón, Gustavo Chio Cho This index will include parameters associated with height lay out, type of foundation, retaining walls and preservation conditions.This index will consider parameters associated with location, foundation level and the type of soil where building is constructed.This index considers the parameters related to building surroundings, which are directly related to whose affectation can actually damage vulnerability.Consequently, parameters associated with this group are: site vegetation, drainage system, waste material control and surrounding preservation.
In order to estimate total vulnerability the higher value among direct structural vulnerability index IV ED and, location index VU are considered in the first place.Then the remaining vulnerability indexes are accounted and they are averaged out.Finally, these values are replaced in Equation 7.
Total vulnerability index will always be lower or equal to 1.

Algorithm used for the calculation of vulnerability index
Calculation of each vulnerability index, including total index, was developed by means of the use of fuzzy sets, as follows.1) Translation of qualification variables of fourteen-quality conditions parameters into fuzzy sets is made by means of a definition of the corresponding membership functions based on expert opinions.
2) Conversion of importance variables of each parameter into fuzzy sets by employing membership functions based on expert opinions. (

Employed linguistic variables
For the definition of vulnerability degrees of this model´s parameters and importance values obtained from fuzzy sets, it was necessary to accurately define the linguistic variables to describe different degrees of vulnerability and importance values.Such linguistic variables were employed to characterize each quality condition assigned to each parameter and; to define the importance of each parameter in the total structural vulnerability estimation.Therefore, natural linguistic variables selected to characterize vulnerability degrees were vey low, low, medium, high or very high.Each variable was represented by fuzzy sets as follows.
• Where (x) indicates x membership degree; defining x as the universe of building vulnerability in fuzzy sets.
The graphical representation of membership functions of linguistic variables employed in this research are shown in Figure 2.

Quality and importance values of the model
Membership functions of quality conditions and created importance values of parameters to be analyzed by the proposed model are presented in Tables 1 and 2.

Model application method
For the application of this model, it is necessary to make an inventory including the buildings of the area under study.Such inventory consists of gathering the necessary information to assign a quality standard to each fourteen parameters of the model.Afterwards the information is stored by a SIG, which includes the vulnerability index calculation (UIS and CDMB, 2011).Finally by means of SIG data processing, a vulnerability mapping is obtained for the area under study.Such mapping identifies vulnerability degree of each building in the area.

Model application
Vulnerability index model was applied on a sector in Bucaramanga city, located in an area called El Porvenir, which has 1.110 houses having squared masonry in every storey that corresponds to 54.7%, while the remaining 45.3% are squared structures in the first floor, but the following floors have square or semi-square structures.Most buildings are two-story buildings (54.5%) and the remaining ones are: three-storey buildings (24.5%), one-storey (20%) and even four-storey buildings (1%).They were built in the 80´s and 90´s, according to antiseismic regulations in force at that time.One-storey houses were delivered to owners who had the possibility to build a second floor.Over the years, most of them were enlarged into two and three storey houses, without following proper construction design patterns.The area is located on a slope and some houses are quite close to the slope gradient edge.
Then the inventory was made and respective vulnerability mapping calculations were obtained, which are presented in Figure 3.

Índice de vulnerabilidad vulnerability index
The vulnerability index mapping shows that 19.73% of houses are qualified with a very low vulnerability index; these are one or two-storey houses with a structural system in every storey.Then 25.95% of buildings were qualified with low vulnerability index.36.49% of them were registered with a medium vulnerability index.A high vulnerability index was assigned to 16.22% and finally 1.62% with a very high vulnerability index in the whole sample.The latter are located in the neighborhood south-east sector and the high degree of vulnerability is explained by their proximity to the slope.

Conclusions
Facing the current need of conducting vulnerability studies due to several nature threads, it is a common practice to apply the proposed models on other contexts and to build own models.By applying existing models created from other locations data, where structural characteristics are different, it exposes us to the risk of using an inadequate model and consequently reaching wrong results.On the other hand, creating an own building vulnerability model, for different cities in our nation, demands accurate information about damages in past events or structural experimental behavior records, which are often not available.Therefore, data collection and information analysis due to landslides must be broadened.Hence, the research on building structural behavior shall be strengthened.While this process takes place, it is necessary to develop new models, by using the existing resources.Therefore, the alternative of using expert opinions for these kinds of studies is quite appropriate.
In the case of cities vulnerability such as Bucaramanga, where there are no damage records on masonry buildings either experimental information, the application of the proposed model proved to be simple and practical; therefore, its use is appropriate for studies at great scale.It is expected that new models will be developed based on further information.

Esperanza Maldonado Rondón, Gustavo Chio Cho
The proposed methodology is based on the evaluation of parameters that affect at a great extent the building vulnerability faced to landslide effects, which correspond to structural system, quality of strength system, foundation position, type of soil, building location, retaining elements availability, drainage systems, area vegetation, height lay out and preservation conditions of the building and surroundings.They may be qualitative or quantitative parameters, but they are defined based on general information of the building.
The parameters chosen for the definition of this model and its respective classifications correspond to buildings´ own environmental conditions.Hence, the proposed model is well applicable to cities having similar characteristics to the ones already studied.Besides, vulnerability degree of each parameters classifications and its importance value were defined by means of the use of fuzzy sets theory, which makes the model applicable on places not counting with damage information records either experimental information on study interest areas.
Based on the relation between vulnerability degree of different parameters and their respective importance value, a vulnerability index was defined.This index qualifies structure vulnerability as very low, low, medium, high or very high, by associating it with a numeric value and a representation of response ambiguity.
The method is greatly applicable to high-scale studies, where a great amount of buildings are to be assed by using the minimum possible information.In this way, it allows the comparison among different zones to detect the most vulnerable ones, thus becoming a useful tool for seismic mitigation plans and land-use planning.
The proposed model has been included in the vulnerability construction functions stated by the Planning support project for natural sources and man-induced risks -handling and care -in the jurisdiction area of CDMB (UIS and CDMB, 2011).

Esperanza
Masonry in every floor plan B. Reinforced masonry in every floor plan C. Squared masonry not in every floor plan D. Masonry which only has supporting beams, without columns, columns without supporting beams E. Non-squared masonry in every floor plant Parameter 2 -quality of load-bearing structure A. High quality masonry, with uniform pieces properly bonded B. High quality masonry, with not quite uniform pieces but properly bonded C. Low quality masonry with uniform and non-uniform pieces, properly bonded D. High and low quality masonry, with non-uniform pieces or improperly bonded Parameter 3 -Structural strength A. Building with a value of < 1. (a is the capacitydemand relation to movements in the building foundation) B. Building with a value of 0.6 = <1.C. Building with a value of 0.4 = < 0.6.D. Building with a value of < 0.4.Parameter 4 -Height lay outA.Building with -M/M <10% or only one storey building ( M is the maximum mass variation between two storey, being (-) for decrease and (+) for increase; M is the mass of lower storey.Bshall meet one of the following conditions:

Figura 1 .
Figura 1. Esquema de la definición del modelo del índice de vulnerabilidad Figure1.Definition lay out of vulnerability index model Figura 2. Funciones de pertenencia de las variables lingüísticas muy baja, baja, medio, alta y muy alta Figure 2. Membership functions of linguistic variables vey low, low, medium, high or very high

Table 1 .
Funciones de pertenencia para los valores de calidad de cada uno de los parámetros, K i Membership functions for quality values of each K i , parameter # se refiere al número del parámetro # refers to parameter number Tabla 2. Funciones de pertenencia para los valores de importancia, W i

Table 2 .
Membership functions for W i importance values # se refiere al número del parámetro # refers to parameter number Índice de vulnerabilidad estructural/Structural vulnerability index Figura 3. Mapa de vulnerabilidad de la zona de estudio Figure 3. mapping of the area under study Esperanza Maldonado Gustavo Chio Cho