Seismic Vulnerability Indices of Facades of Colonial Houses in the Historic Center of Morelia, México

Abstract

Evaluating the seismic vulnerability of facades of historic masonry buildings is essential not only for their significant historical and heritage value, but also to evaluate the safety of this type of construction. This work applies a simplified methodology to assess the seismic vulnerability of the facade of masonry buildings in the historic center of Morelia, Michoacán, México. The historic center of Morelia was declared a World Cultural Heritage Site by UNESCO in 1991. On the facades, there is ornamentation with sculptural and vegetal decorative elements. The methodology involved conducting visual inspections to identify the location, type of structure, construction materials, doors, windows, balconies, cornices, ironwork, pediments, niches, and sculptures, among other characteristic elements of colonial architecture. The seismic demands were determined specifically for the city’s historic center based on a recent seismic hazard assessment of Morelia. Based on the methodology and the compiled database, characterized vulnerability indices were defined for the different damage scenarios that buildings may present. Results indicate that earthquakes with intensities greater than VIII on the Modified Mercalli scale risk collapsing heritage masonry buildings’ facades.

1. Introduction

The city of Morelia was founded on 18 May 1541, in the Guayangareo Valley, today the capital of Michoacán, with the original name of “third city of Michoacán”, the first being Tzintzuntzan and the second Pátzcuaro. The founder of the city, Viceroy Don Antonio de Mendoza, changed the name of the city to “Valladolid” in 1578, which was modified in 1828 to “Morelia” in honor of his son Don José María Morelos y Pavón, who was one of the main heroes of the Independence of México. The “third city” was founded as a city for the Spanish, unlike the “second city”, which was a pre-Hispanic settlement that was the seat of the Bishopric of Michoacán Don Vasco de Quiroga, and so these cities were rivals throughout the 16th century. With the death of Quiroga, the Bishopric moved to Valladolid, making the city the capital of Michoacán. The historic center has 1113 buildings cataloged as architectural monuments, of which 260 are recognized as being of most significant relevance, thanks to which UNESCO classified Morelia on the Heritage list on 12 December 1991. UNESCO declared that the city is characterized by its style described as “Morelian baroque” and by the originality of its local expressions reflected in the Aqueduct, the Metropolitan Cathedral, the Company Church complex, and the former Jesuit school, as well as on the facades and arcades of the corridors and patios of the Vallesoletana houses. They also pointed out that the diversity of styles ranges from architectural typologies of the late 16th century, where the appearance of a medieval fortress coexisted with Renaissance, Baroque, and Neoclassical elements, to the eclecticism and Frenchification of the Porfirio Díaz period [1].

The architecture of the buildings is distinguished by its exterior ornamentation, which is called “Morelian baroque”, where sculptural and vegetal decorative elements dominate the planes and lines of boards and moldings.

According to Ettinger [2]: “the layout of Morelia is considered a good example of Spanish urbanism in America due to its large city blocks, regular block patterns, and porticoed central plaza. The city center housed Spanish institutions and population, while indigenous neighborhoods were established peripherally”. Azevedo [3] describes: “The urban architectural complex that makes up the core of the city (the historic center) is the result of a historical process that began in the 16th century and continues to the present day”.

The facades of Morelia’s historical heritage go beyond being elements that delimit the internal space from the external; they testify the historical and cultural evolution of a society that provides identity and great artistic value. Therefore, its conservation and preservation are essential. When the materials’ properties present some pathology of degradation that frequently occurs on their facades, it is necessary to evaluate the possibility of rehabilitating the structure.

The facades of historic buildings are an image of urban landscapes and define outer and interior space. The facades include structural and non-structural elements that conform to heritage buildings. However, the facades are weak elements that can govern a historic building’s seismic vulnerability.

The houses of Morelia, built between the 17th and 19th centuries, have endured historical earthquakes, war conflicts, temporary coverings, rain, wind, changes in live loads due to building recycling, structural interventions, pollution, and lack of maintenance. Facades are often the most impacted structural element over the long lifespan of these structures. Several studies have been conducted in various countries to understand the structural behavior of the facades of historic buildings. Factors such as ground settlement, vibrations, irregular opening layouts, environmental effects, and masonry aggregates have deteriorated and increased seismic vulnerability in these buildings. Corlito et al. [4] used a simple procedure based on linear and non-linear kinematic approaches to assess the seismic capacity of masonry church facades in terms of spectral accelerations and displacement. Their findings, which were compared with the damage scenario observed in a population of churches analyzed by the research group in the aftermath of the 2009 L’Aquila earthquake, concluded that the procedure could help avoid time-consuming analyses and support the simulation of post-earthquake damage scenarios and strategies for seismic risk mitigation. Uzdil [5] studied a heritage building built in 1896 through linear, pushover, and kinematic limit analyses. The study aimed to select an adequate type of analysis to evaluate historic masonry buildings. The authors concluded that local wall behavior cannot be captured through linear analysis and can only be studied through kinematic analysis.

Bilgin et al. [6] analyzed the building damages caused by two earthquakes that hit Durrëss, Albania, in September and November 2019 with magnitudes of 5.6 and 6.4, respectively. The visual inspections exhibited deficiencies in current masonry buildings. They reported that the primary mechanism of collapse in these structures was tilting or overturning the structures’ longer sides, both within the plane of the walls and outward from the plane of the walls.

Prosperi et al. [7] and Korswagen et al. [8] used sensitivity analysis and FE non-linear models to explain crack-based damage in historical buildings. They showed that the position and crack width depended on the settlement and the length-to-height ratio (L/H). Long facades needed smaller differential settlements to develop visible damage than shorter facades, and vibrations with low peak ground velocities increased existing damage.

Morandini et al. [9] studied the effect of irregular facade opening areas on the lateral response of full-scale unreinforced masonry facades. The authors compared the Equivalent Frame Model [10] and the Applied Element Method [11] for two-level facades. They found that determining the frame topology and the effective length/height ratio of the structural members was challenging due to the irregular distribution of openings, and inaccurate definitions of deformable and rigid elements could significantly affect numerical predictions.

Pierdicca et al. [12] studied the Byzantine Stylite Tower in the Umm ar-Rasas archaeological site (Jordan). They performed a detailed 3D model of the tower for the seismic vulnerability assessment from aerial photogrammetric and historical investigation data. Several possible out-of-plane mechanisms involving the upper cell walls and portions of the tower facades were analyzed via kinematic analysis using the floor response spectra. The study revealed a new critical and risky scenario, which suggested a requirement for prompt intervention to ensure the safe conservation of the tower.

The city of Morelia is starting to show significant temperature variations due to global climate change, which could impact the expected behavior of the facades. Elyamani et al. [13] conducted a continuous dynamic monitoring campaign at the cathedral of Mallorca for 15 months between 2010 and 2012. Fluctuations in temperature between 10 °C and 30 °C caused an increase in modal frequencies from 10.4% to 18.5%. This led to the closure of cracks in stone masonry due to thermal expansion, resulting in improved overall building stiffness. This effect could be significant for the expected seismic performance of the structure in future earthquakes.

Several studies have evaluated the vulnerability of masonry building facades. Calvi et al. [14] provided an overview of significant contributions to vulnerability assessment over the past 30 years. Ferreira et al. [15] developed a seismic vulnerability index applied to over 600 building facades in the historic city center of Coimbra, Portugal. The results indicated the most relevant parameters in the seismic response of building facades and their significance by quantifying each parameter within the assessment methodology and a vulnerability function for masonry facade walls. The parameter weights were adjusted based on the post-seismic damage observation of the 2009 Abruzzo earthquake.

Jaimes et al. [16] and Colonna et al. [17] evaluated the risk of the out-of-plane mechanism of church masonry facades through linear and nonlinear dynamic analyses in Mexico and Italy, respectively. Both works developed fragility curves related to the facades’ slenderness ratio (width/height). They found a significant and positive correlation between the risk of the out-of-plane mechanism of masonry facades and the facades’ slenderness ratio. Colonna et al. [17] pointed out the importance of the masonry quality on the seismic response in a detailed numerical model and through nonlinear kinematic analysis of a historic masonry building.

Khattak et al. [18] evaluated the seismic vulnerability of 363 unreinforced masonry facade walls for the Australian state of Queensland, built before 1940. The results were presented through maps created using an open Geographical Information System (GIS) tool. Empirical vulnerability curves were built, and they found that many buildings were associated with critically high vulnerability indices, signaling potential for undergoing damage grades of 3 or higher when subjected to macroseismic intensities of VIII or higher. They found that many of the assessed buildings are potentially vulnerable to earthquakes of low to moderate intensities, requiring further investigation using more detailed tools.

Fontenele et al. [19] introduced a vulnerability assessment procedure that uses qualitative and quantitative parameters. These parameters depend on analyzing general information about a structure, including its geometrical characteristics, construction technology, and material properties. The procedure was applied to the facades of the main historic centers of Aracati and Sobral in the state of Ceará, Brazil. The method is based on the seismic vulnerability index developed in Italy by the GNDT-SSN [20]. The authors concluded that half of the evaluated facades of the historic center of Aracati would be exposed to severe damage, and some would collapse. The historic center of Sobral showed D2 (moderate) and D3 (substantial to heavy) damage for the worst-case scenario.

The objective of this study is to contribute to the assessment of the state of conservation of the facades of colonial houses in the historic center of Morelia through a simplified methodology that allows estimating a seismic vulnerability index. Contrary to other studies, the expected value of the Modified Mercalli Intensity was determined based on a seismic hazard assessment that quantifies the expected response in the city’s historic center. The diagnosis begins with this analysis and, in future studies, the development of strategies to mitigate the seismic risk of the facades during seismic events of great intensity.

Based on the results of this study and the identification of the most vulnerable facades, future studies could monitor the structural health of Morelia’s historic buildings and re-evaluate the seismic vulnerability using more detailed analysis, such as the methodologies described in previous paragraphs. According to Rossi and Bournas [21], structural health monitoring (SHM) is an essential and crucial tool in preserving a historic building and planning its maintenance. Rossi and Bournas [21] wrote a paper review of this topic for historic buildings, and Zhihang et al. [22] discussed the advantages and disadvantages of current bridge methodologies. The data recovery of incomplete sensor measurements for monitoring structural health was addressed by Jianwei et al. [23].

2. Materials and Methods

Assessing the structural vulnerability of historic building facades is crucial for evaluating the safety of heritage structures and establishing intervention and management strategies. If we start from the premise that historical buildings are aged, non-engineered structures and do not have a continuous maintenance program, their structural vulnerability could be high. The methodology to achieve the evaluation objective must consider the purpose, technology, available information, and applicability [19]. In general, the methods for a simplified assessment are based on qualitative analysis, which is helpful for a preliminary verification stage that should be completed, when possible, with a quantitative method.

Various methodologies, including simplified, intermediate, and detailed approaches, exist for assessing seismic damage and the vulnerability of structures. A simplified method, in general, is based on visual inspections, and part of its efficacy relies on the expertise of an engineer who is highly skilled and trained in the area. The main advantage is the simplicity of conducting the assessment. Although it is a subjective methodology, the seismic damage assessment is compelling since it does not require any instrument to make the assessment. Other techniques, such as vibration-based damage identification [24,25], are frequently based on identifying modal characteristics of the structure, and might not be the best option because of the specialized equipment required to make the vibration measurements. However, vibration-based methodologies have the advantage of including a quantitative analysis to assess the damage.

More refined methodologies included structural health monitoring (SHM) for assessing and detecting damage to civil infrastructure. SHM is based on experimental testing, system identification, data acquisition and management, and long-term environmental and operational conditions testing to locate and quantify structural system damage. Detailed methodologies involve not only SHM and structural damage detection but also a finite element model of the structure, which could be very specialized depending on the quality and amount of information known on the materials and structural model of the system used for better damage identification. Despite this methodology providing quantitative information closest to the state of damage, the evaluation of seismic vulnerability on a larger scale must begin with the application of simplified methodologies, which lead to the application of intermediate methodologies for a smaller group of structures and finally to the application of a detailed methodology for an even smaller group of structures. This underscores the need for different methodologies based on the evaluation scale.

The choice of analysis methodology depends on various factors, such as the number of cases to be addressed. As the number of buildings to be considered increases, less detailed methodologies are employed. The methodologies found in the literature can be categorized into four groups: direct, indirect, conventional, and hybrid methodologies [26]. The first approach involves assessing the parameters of the structures, such as the type of construction, structural characteristics, and building materials. The vulnerability is estimated by correlating the observed damage with seismic intensity using vulnerability functions and probability matrices. These are determined based on the damage caused by earthquakes in previous seismic events. The second approach involves using a numerical model and non-linear static or dynamic analysis to assess the seismic impact on the building. The extent of damage is assessed by calculating displacements or drift demands. Indirect techniques, on the other hand, rely on assessing a vulnerability index and establishing a connection between damage and seismic intensity using statistical data obtained from post-earthquake databases. Conventional methods follow poorly defined rules known as heuristic methods, where a vulnerability index is assumed without considering the level of damage. This type of methodology is used to compare similar types of buildings calibrated by experts’ opinions. Hybrid techniques use a combination of the three methods described. The methodology chosen for this study considers the lack of research on the seismic vulnerability of facades in the historic center of Morelia, limited access to information about the structural and material properties of the buildings, and the large number of structures that need to be analyzed. Therefore, at this stage, it was decided to initiate the analysis using an indirect methodology to identify the most vulnerable facades. Subsequently, more refined methods would be employed, incorporating a numerical model of the structures.

The applied methodology in this study is called the vulnerability index method [14,15]. It is based on four data groups considering geometric information, material parameters, connection with other elements, and non-structural elements on the facade. Thirteen parameters are assessed and classified into four vulnerability classes: A, B, C, and D, organized from excellent to unfavorable. The numerical values of these categories are 0, 5, 20, and 50, respectively. This classification is linked to the facade damage observed during visual building inspections. Class A indicates no damage detected, Class B is superficial damage, Class C relates to damage detected inside the masonry, and Class D shows facades with a critical reduction in material resistance. Each parameter has a specific weight as a function of its importance on the seismic vulnerability of the facade. Multiplying each category value by its weight gives the partial vulnerability index. The sum of all partial indices leads to the global vulnerability index (Iv).

The vulnerability index method has been adapted and calibrated to the specific conditions of various cities in Europe, Africa, South America, and North America [14,15,19,27,28,29]. As a result, differences in the weights assigned to each parameter can be observed. In Mexico, this approach was adapted for Atlixco, Puebla [28] following the damage caused by the 19 September 2017 earthquake. The historic center of Puebla has buildings and materials similar to those in the historic downtown of Morelia. Consequently, the parameters and weights used to evaluate the seismic vulnerability of Morelia’s facades were based on the ones reported by Ramírez et al. [28], as summarized in

Table 1. Weights of parameters for quantifying the vulnerability index.

Parameter ID	Weight pi
P1	0.50
P2	0.50
P3	0.50
P4	0.50
P5	0.25
P6	2.00
P7	2.00
P8	0.25
P9	2.00
P10	0.50
P11	2.00
P12	0.50
P13	−2.00

.

Consequently, the parameters used to evaluate the seismic vulnerability of Morelia’s facades were: P1 Facade geometry, P2 Wall slenderness, P3, Area of openings, P4 Misalignment of openings, P5 Interaction between facades, P6 Masonry quality, P7 State of conservation, P8 Replacement of flooring system, P9 Connection to orthogonal walls, P10 Connection to horizontal diaphragms, P11 Impulsive nature of the roofing system, P12 Elements connected to the facade, and P13 Improving elements. The thirteen parameters are grouped into four categories [28] related to potential facade failure mechanisms. Group 1, facade geometry, opening, and interaction, relates to the effects of parameters associated with wall slenderness, number of floors, and the influence of openings on the concentration of tensile stresses. This group quantifies parameters P1–P5 (

Table 2. Definition of vulnerability classes for Group 1.

Parameter P1
A	$H/B<0.4$
B	$0.4\le H/B<0.6$
C	$0.6\le H/B<1.0$
D	$H/B\ge 1.0$
Parameter P2
A	$H/s<9$
B	$9\le H/s<15$
C	$15\le H/s<20$
D	$H/B\ge 20$
Parameter P3
A	Area of openings less than 20%
B	Area of openings less than 35%
C	Area of openings less than 60%
D	Area of openings larger than 60%
Parameter P4
A	Regular dimensions and aligned in height.
B	Regular or irregular dimensions are horizontally misaligned in more than ½ of their height.
C	Regular or irregular dimensions are vertically misaligned in more than ½ of their height.
D	Relatively large openings at any level of the construction.
Parameter P5
A	In the middle of two facades of the same height.
B	Contiguous to a facade of the same height and a shorter one.
C	Contiguous to two shorter facades.
D	Contiguous to only one, shorter facade.

).

Parameter P5, which considers the interaction between facades, depends on the height of the building under evaluation and the neighboring buildings. When they are of the same height, class A is assigned. In a simplified way, this parameter considers the effect of pounding between neighboring buildings, which various authors have shown is important for the seismic behavior of buildings. Cole et al. [30] described methodologies for analytically modeling pounding between buildings. The authors showed that in cases where buildings have similar heights and no joints exist between them, as occurs in the facades of the historic center of Morelia, the pounding is transmitted from building to building similarly to Newton’s cradle, generating the major movement to the furthest away structure that lacks a neighboring building, which causes minor damage to the central buildings.

In other studies, simplified methods have been proposed to consider pounding in numerical models [31], and protection systems have also been suggested when the height of the slab of a building coincides with an intermediate zone of a column of the neighboring building [32]. Torsion is another effect that, combined with pounding, can cause significant damage to buildings [33]. Most analytical and experimental studies have focused on structures formed by reinforced concrete frames with beams and columns, as Elgammal et al. [34] presented in a state-of-the-art review. These authors also emphasize the contradictory results of various studies, showing the need to study this topic further. There are few studies of pounding analysis between historical structures; Zhang et al. [35] analyzed this effect on typical unreinforced masonry buildings in Canada using the Distinct Element Method and a pushover analysis. The results showed that the maximum shear force value of the facades depended on the number of adjacent structures. However, the conclusions were limited by the type of analysis performed.

Group 2, materials and conservation (

Table 3. Definition of vulnerability classes for Group 2.

Parameter P6
A	Brick or stone masonry.
B	Brick or stone masonry and adobe.
C	✓ Adobe, rounded stone and clay brick. ✓ Stone masonry and brick masonry. ✓ Rounded stone, volcanic stone, and clay brick. ✓ Adobe, round stone, and clay brick.
D	Adobe.
Parameter P7
A	Walls in good condition with no visible damage.
B	Small and non-generalized cracks (<0.5 mm width). Signs of decay due to humidity in masonry and timber elements. Cracks in coatings that do not penetrate the support.
C	Generalized cracks (2–3 mm width). Regular state of conservation of the walls. Serious deformations in the staircases and floor diaphragms; diagonal cracks in internal walls, and cracks in the middle of openings.
D	Walls with serious decay and cracking, even if not generalized. Decays that evidence a loss of strength. Cracks in sensitive locations, such as corners (e.g., due to disconnection between walls). Damage in the interfaces between roofs and walls, displacement of floor diaphragms. Rotting and decay of timber beams. Out-of-plane deformations of the walls.
Parameter P8
A	$0\le P<25\%$
B	$25\le P<50\%$
C	$50\le P<75\%$
D	$75\le P<100\%$

), pertains to the quality and conservation status of materials (P6–P7) and the percentage of replacement of the original flooring system (P8).

The connection efficiency parameters for Group 3 (

Table 4. Definition of vulnerability classes for Group 3.

Parameter P9
A	The facade is well connected to the orthogonal walls, diaphragms, and roofing structures by means of metallic perimetral straps, tie rods, and well-locked masonry. These connections exist in all orthogonal connections.
B	Masonry is well-locked in the corners, with no signs of fragility. Conditions of grade A exist in some corners.
C	There is no good connection between the facade and the orthogonal walls. However, there are no signs of fragility. Conditions of Grade B exist in some corners.
D	The facade is deformed and at an elevated risk of imminent failure (cracked according to mechanism patterns): Detachments and cracking in corners and orthogonal interfaces.
Parameter P10
A	$e\ge 75\%$
B	$e\ge 75\% \mathrm{with}\mathrm{decays}$ * $50\%\le e<75\%$
C	$50\%\le e<75\%$ with decays * $25\%\le e<50\%$
D	$25\%\le e<50\% \mathrm{with}\mathrm{decauys}$ * $e<25\%$
Parameter P11
	Impulsive nature	Perimetral straps/beam	State of conservation
A	Non-impulsive	At least one of both	Good
B	Non-impulsive Non-impulsive Slightly impulsive	At least one of both At least one of both None of them	Bad Good Good
C	Non-impulsive Non-impulsive Slightly impulsive Slightly impulsive Impulsive	At least one of both None of them At least one of both None of them At least one of both	Very bad Bad Bad Good Good
D	Non-impulsive Slightly impulsive Slightly impulsive Impulsive Impulsive	None of them At least one of both None of them At least one of both None of them	Very bad Very bad Bad or very bad Bad or very bad Indifferent

* Weakness at the regions of contact with the vertical structure, signs of deformation, rotting or shrinkage, or lack of safety for circulation.

) depend on the connections among the facades with perpendicular walls (P9), floor diaphragms (P10), and roofing structures (P11) involving out-of-plane collapse mechanisms.

Group 4, non-structural elements and conservation (

Table 5. Definition of vulnerability classes for Group 4.

Parameter P12
A	There are no elements attached to the façade.
B	Some light elements, such as lights, signals, light roofs, etc.
C	Medium weight elements attached to the facade: equipment, air-conditioned devices, etc.
D	Existence of balconies or other heavy elements connected to the façade.
Parameter P13
A	There are no external elements.
B	Presence of exterior stairs, arches, giants, etc.
C	Explicit reinforcement through the use of reinforced plaster.
D	Strengthening actions by the means of elements such as buttresses or similar.

), considers the attached elements to the facade that may contribute to wall overturning (P12). It also examines beneficial elements that could reduce the possibility of out-of-plane failures.

The vulnerability index method was developed as a simplified methodology to evaluate the seismic vulnerability of buildings. It was later adapted to quantify the vulnerability of facades. Unlike complete construction, facades are particularly susceptible to damage due to environmental effects that sometimes increase their seismic vulnerability. Therefore, this study proposes to conduct a detailed evaluation of the facade’s state of conservation to select the category (A–D) of parameter P7. As shown in Table 3, the original methodology determines it by only observing possible fissures and cracks in the facade. The proposal is focused on assessing the condition of the building’s exterior wall based on the criteria outlined in the below section on facade pathology.

Facade Pathology

Various hazards such as climate, earthquakes, wind, rain, and anthropogenic factors intervene in the pathology of the facades of historic buildings. Some environmental variables include precipitation, relative humidity, diffuse solar radiation, ambient temperature, wind direction, and wind speed. Pathologies cause deterioration that, over time, reduces structural elements’ capacity and their useful life. Among the most common pathologies that facades present are the following:

1. Humidity. Relative humidity is the amount of water in the air. The humidity on the facade may be caused by high relative humidity, direct rain action, or water capillarity problems in the subsoil due to rain leaks. Lack of maintenance sometimes leads to leaks from the building’s pipes and water stagnation under the facades’ walls.

2. Erosion. Wind action can cause erosion of building facades. It is important to study wind direction, speed, and temperature gradients.

3. Dirt. Rough surfaces in rock promote the accumulation of suspended particles and sometimes water stains. Wind gusts can carry dust and smoke from vehicles, hitting facades and depositing dirt.

4. Settlement. Soil settlements usually generate diagonal cracking in the facades, reducing their stiffness and seismic capacity.

5. Cracking. Cracking on facades can have diverse origins. Sometimes, it occurs due to soil settlements in the site, and other times, it may be due to a more general subsidence problem in the region. Moreover, the combination of rain cycles and solar radiation can cause the material to expand and contract, favoring the opening of fissures and cracks. After an earthquake, diagonal cracks can appear on the facades.

6. Fissures. Fissures are narrow openings often caused by temperature changes, causing facade materials to expand and contract.

7. Stone spalling. This process can begin with the cracking of the wall and later evolve into spalling. It is frequently related to moisture infiltration caused by improper water drainage from the gutter on top of the walls.

8. Out of plumb. Facades can lose their vertical alignment due to soil settlement, which is more common during the rainy season. Some soils expand and contract based on the amount of water they hold, and seismic movements can cause problems with a facade’s vertical alignment.

9. Chemical injuries. Chemical injuries cause damage to building exteriors, as shown by efflorescence, oxidation, and corrosion. When moisture is present on the facade, it can lead to mortar deterioration, efflorescence, and fungus growth. Mortar deterioration leads to loss of joints, allowing moisture infiltration and consequently, more significant damage. Efflorescence is the deposit of mineral salts that occurs more easily on unprotected or uncoated facades, and over time, mold and moss can proliferate. Fungus growth is often found in cracked areas, which can promote the growth of the cracks.

10. Loss of joints. Joint loss reduces the strength capacity of walls and depends on the quality of the mortar and its age. It is also a function of environmental actions such as changes in temperature, wind speed, and rainfall.

This study proposes evaluating the ten previous parameters of possible facade pathologies to determine the state of conservation. These parameters are evaluated according to their area of affectation and normalized by the total facade area ($A_{Facade})$. Based on this, a state index (I_s) is estimated according to Equation (1) that depends on the relationship of the damaged area ($A_{Pathology-i}$) exposed by the facade, multiplied by a weight assigned ($W_i$) according to the importance of each parameter. I_s is defined in the range of 0 to 100.

$$I_s={\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^{11}{A}_{Pathology-i}\times W_i}{A_{Facade}}}\times 100$$

(1)

The authors selected the weight of each parameter based on their assessment of each pathology’s impact on the facade’s deterioration. Subsequent studies could evaluate and, where appropriate, update the weights based on experimental results of the behavior of walls with these pathologies.

Table 6. Facade pathologies and assigned weights $\left(W_i\right)$.

$\mathit{i}$	Pathology	${\mathit{W}}_{\mathit{i}}$
1	Humidity	0.025
2	Erosion	0.100
3	Dirt	0.025
4	Settlement	0.150
5	Cracking	0.200
6	Fissures	0.050
7	Stone spalling	0.075
8	Out of plumb	0.150
9	Chemical injuries	0.100
10	Loss of joints	0.125

shows the weights ($W_i$) assumed in this study. Values in the range of 0 to 10 indicate category A for the P7 parameter, in the range of 10 to 30 indicate category B, in the range of 30 to 60 indicate category C, and finally, those in the range of 61 to 100 indicate category D. In this way, the facade would be classified not only as a function of fissures and cracks (as in the original method proposal) but also as a function of the area of this pathology to the total facade area. With this proposal, it is accepted that the deterioration comes not only from the cracking of the facade (which in this new proposal is given a weight of 25%), but also from the nine remaining parameters.

The first step in the evaluation consists of visually inspecting the facades to analyze the actual damage condition by identifying the presence of pathologies. A technical sheet containing all the information required for evaluating the I_s parameter (Equation (1)) was developed. The ten sections of the technical sheet are: (1) Location, (2) Identification, (3) Geographic location, (4) Graphic documentation, (5) Location sketch, (6) Building description, (7) Building Pathology, (8) Visual assessment, (9) State index, and (10) Observations. Section (6) (description of the building) captures the characteristics of the facade, including all its elements, such as walls, doors, windows, arches, columns, pediments, gargoyles, and niches. Section (7) (building pathology) gathers information about the facade pathology (Table 6) and the causes reported by the observer. Section (8) (visual evaluation) presents a photograph of the facade where all damaged areas must be identified. Finally, all the collected information is quantified in Section (9) to compute the state index.

Once the $I_s$ index for the facades is assessed, the category of P7 parameter is determined. After that, the other parameters are quantified to obtain the global vulnerability index. The vulnerability index $I_v$ has to be related to the seismic hazard according to the method implemented by Ramírez [28] based on the seismic vulnerability index method developed in Italy [11]. Recently, this methodology was implemented in Mexico to evaluate heritage buildings [28]. The authors considered historical buildings affected by the 19 September 2017 earthquake in the states of Oaxaca, Chiapas, and Puebla. Hence, the expressions proposed in [29] were adapted for the Mexican buildings, leading to Equations (2)–(4). The expressions presented in the methodology aim to measure the expected levels of damage under several seismic scenarios based on the European Macro Seismic Intensity ($I$) and the expected ductility demand ($Q$) on the structure. The normalized vulnerability index (V) is defined with Equation (2). Equation (4) allows for estimating the average value of the degree of damage ${\mu }_D$ [28]. This study uses the Modified Mercalli Intensity (MMI), and the relationship between MMI and European Macro Seismic Intensity ($I$) was established according to Musson et al. [36]. The ductility factor was assumed to be $Q$ = 1.0, as established in the Complementary Technical Norm for the Seismic Design of Buildings in Mexico City [37]. This value corresponds to unreinforced masonry structures built with stone bricks. Finally, f(V,I) is a correction factor for I ≤ 7.

$$V=0.592+0.0057I_v$$

(2)

$$f\left(V, I\right)=\{\begin{array}{c}{e}^{V/2\left(I-7\right)} I\le 7\\ \\ 1 I>7\end{array}$$

(3)

$${\mu }_D=2.5+\left[3\ast tanh\left({\displaystyle \frac{I+6.25V-12.7}{Q}}\right)\right]f\left(V,I\right); 0\le {\mu }_D\le 5$$

(4)

The average value of the degree of damage ${\mu }_D$ [19,28] ranges from 0 to 5, indicating no damage to collapse.

Table 7. Description of the degree of damage parameter.

${\mathit{\mu }}_{\mathit{D}}$	Damage Grade	Description
1	No damage	No damage or the presence of very localized hairline cracking
2	Moderate damage	Cracking around door and window openings; localized detachments of the wall coverings (plasters, tiles, etc.)
3	Substantial and extensive damage	Opening of large diagonal cracks; significant cracking of parapets; masonry walls may exhibit visible separation from diaphragms; generalized plaster detachments
4	Severe and heavy damage	Facade walls with large areas of openings have suffered extensive cracking. Partial collapse of the facade (shear cracking, disaggregation, etc.)
5	Collapse	Total in-plane or out-of-plane failure of the faced wall

describes how to identify the level of damage.

3. Seismic Vulnerability

UNESCO recognizes the city center of Morelia as a historical and cultural heritage site. The interest in evaluating the seismic vulnerability in this area is based on Morelia’s geographic location, which is classified as a high seismic risk zone [38]. Seismic movements are relatively frequent in the town. The urban layout of the city is, according to Azevedo [3], as follows: “The system of streets and public free spaces define the urban fabric of the historic city, which presents as its central core the majestic building of the cathedral located between two open spaces, the Plaza de Armas and the current Plaza Melchor Ocampo. The ecclesiastical preponderance of Valladolid-Morelia, as the Capital of Bishopric of Michoacán, constructed large religious factories, in front of which plazas and small squares were formed”. Figure 1 shows a map of the city’s historic center area, where the facades of 40 residential houses facing north, south, east, and west directions were evaluated.

Figure 2 displays the location of the buildings with red dots and some representative images of the building facades.

The methodology proposed in this study involves, as a first step, carrying out visual inspections to identify the location, construction materials, doors, windows, balconies, cornices, ironwork, pediments, niches, and sculptures, among other characteristic elements of colonial architecture, pathologies of the facades, geometric characteristics, openings, connections with other components, and attached elements on the facade. The buildings were chosen based on the most representative facades in the historic center of Morelia and their orientation, aiming for an equal number of facades facing north, south, east, and west. All information was captured in technical sheets. Figure 3 shows the facade pathologies found during the visual inspection of the buildings.

The mechanical injuries that most contributed to damage were dilatation and loss of joint, whereas the chemical injuries that mainly affected the material were efflorescence, followed by corrosion and oxidation, and the physical injuries caused by rain, deposit, and wind.

Most of the facades in the study were built with irregular quarry walls, and very few with aqua-dry ashlar blocks, because construction with ashlar blocks was expensive. Ashlar blocks were only used on the facades of relatively important buildings, mainly on the facades of houses built for Spanish or Creole families. The erosion of joints is a common problem in both types of walls, caused by rain, wind, and expansion. This issue has been growing due to the use of mortars made from gray cement and sand instead of the traditional hydrated lime and sand mix used in the constructions in the past. This leads to further deterioration of the outer layers of the quarry walls due to the incompatibility between the materials, in addition to altering the property’s authenticity. In other cases, the walls have been plastered using the same gray cement-based mortars, leading to damage due to the incompatibility of materials in larger areas of the wall. Such damage resulting from material incompatibility that weakens the outer layers of the facade wall can reduce the cross-section width and, consequently, its in-plane shear resistance.

3.1. Seismic Demand

Morelia is affected by the movements of several tectonic plates in Mexico’s north, west, and center. The earthquake sources affecting the town are interplate faults, originating on the Pacific coast, and hypocenters at depths lower than 45 km. Into the continent, there are intraplate earthquakes with a focus at depths greater than 45 km. The Trans-Mexican volcanic belt crosses the central portion of the country, and in this region, shallow crustal seismic events occur with epicenters that can be close to Morelia [38]. The most frequent earthquakes originate on the coast. These seismic movements dissipate most of the energy accumulated on the border of the tectonic plates and are highly destructive; despite that, some towns located in the country’s center at medium distances from the subduction zone are not always affected by this type of earthquake, which promotes a climate of confidence among the population about the seismic risk of the towns. Unfortunately, other seismic faults, with less frequent occurrence processes, can govern the seismic hazard of these cities and eventually produce significant structural damage. These seismic sources, intraplate and crustal faults, are closer to many central interior towns in the country. Historical earthquakes in these faults have caused significant damage and substantial economic losses.

In the west of the country, four seismic zones [39] within the North American plate are associated with the evolution of the Trans-Mexican volcanic belt (cortical earthquakes). The area between Morelia and Los Azufres defines the western sector of the Trans-Mexican volcanic belt fault system. Some faults exceed 25 km in length, and the scarp height is less than 100 m [40]. In the Morelia region, the faults can be associated with the segments of the Acambay-Morelia fault system known as “La Paloma”, “Cerritos”, and “Coíntzio”. Bayona et al. [41] conducted a seismic hazard study incorporating instrumental and historical data from the Trans-Mexican volcanic belt. For a return period of 500 years, the authors found expected values of maximum ground acceleration in Morelia in the interval of 112 to 176 gals. However, this study does not incorporate the local failures reported by Garduño et al. [42].

Garduño and Escamilla [43] reported 34 seismic records from 1786 to 1899 close to Morelia. The 19 June 1858 earthquake was initially assumed to have originated from an intraplate source [44]. However, another study attributed it to a crustal origin within the Trans-Mexican Volcanic Belt, with an epicenter much closer to Morelia [38]. This earthquake caused significant damage in Morelia, with an estimated intensity of IX on the Modified Mercalli Intensity (MMI) Scale [45,46]. In Morelia, it was felt in a very violent way. The houses and buildings suffered greatly, including the Cathedral and the Company of San Agustín. Figure 4 shows the MMI isoseismals of the 19 June 1858 and 19 September 1985 earthquakes. The shape and distribution of the curves of the 1858 event suggest an epicenter near Morelia and a seismic source different from those that originated on the Pacific coast. Maximum intensities occurred in Morelia and its surroundings.

According to Hernández et al. [47], the 14 March 1979 earthquake (Ms = 6.1) caused damage to the SITE Vasco de Quiroga Hospital building and residential buildings. The earthquake on 18 April 2014 (Ms = 7.2) caused structural and non-structural damage in the Regional General Hospital, HGR N1. Additionally, it resulted in a longitudinal crack along the intrados of the crown in the continuous barrel vault of the temple of San Agustín. This crack bifurcated transversally over the high choir and descended vertically along the two longitudinal walls of the nave. During the 19 September 2017 seismic event (Mw = 7.1), 11 educational centers suffered moderate structural damage, and 569 presented slight structural damage in Michoacan. In Morelia, at least ten educational centers suffered structural damage. Religious structures were the only damaged buildings in the city’s center in some of these earthquakes, and no damage was reported to the other buildings.

Recently [48], a probabilistic seismic hazard study was carried out for Morelia, where all seismic sources that could affect the city were incorporated. Shallow crustal events and interplate and intraplate seismic sources were considered in the study. Additionally, two local seismic sources were incorporated into the analysis. Figure 5a shows the contribution of each seismic source in the uniform hazard spectrum for the stiff ground of Morelia with a return period of Tr = 475 years. The seismic hazard spectrum of shallow crustal earthquakes includes local and outside seismic sources. The most significant seismic sources for the seismic hazard in the short-period region (historical buildings) correspond to intraplate and cortical faults. On the other hand, for medium- and long-period structures, interplate tremors have a more substantial contribution. Morelia has an accelerograph network of eleven instruments located in the north, south, east, and west of the city. One seismic station is situated in the historic center. Based on the spectral ratios between the stiff ground zone of the town and the historic center, the uniform hazard spectra of the historic center were determined for Tr = 100 years (service condition), Tr = 475 years (design condition), and Tr = 1000 years (rare earthquake), shown in Figure 5b. The maximum ground acceleration was 0.11 g, 0.25 g, and 0.34 g, respectively. In short periods, the maximum spectral acceleration demand reached a value of 0.26 g, 0.55 g, and 0.75 g for Tr = 100, 475, and 1000 years, respectively. The exceedance rate in the historic center was computed by taking the exceedance rate obtained for stiff soil, based on the seismic hazard assessment, and multiplying it by the spectral ratio between both seismic stations. A lognormal probability density was assigned to the spectral ratio to consider uncertainty in these analyses. Once the historic center’s exceedance rate, including site effects, was determined, the uniform hazard spectra in Figure 5b were calculated.

Linkimer [49] proposed PGA-MMI relationships using seismic events recorded in Costa Rica and also described other proposals based on a literature review. PGA of 0.11 g, 0.25 g, and 0.34 g would lead to expected values of the Modified Mercalli intensity in the range of VI and IX. According to this proposal, for the expected PGA values of Tr = 100, 475, and 1000 years, MMI intensities in Morelia would be again in the range of VI to IX.

Based on the seismic hazard assessment of Morelia and the history of earthquake occurrence in the zone of study, it is expected that the facades of the historic center of Morelia could be subjected to earthquakes that lead to MMI values larger than VI. To evaluate the seismic vulnerability of the facades, the vulnerability index, and the expected damage state, two values of MMI were selected, VI and IX. In addition to being expected MMI values for return periods of 100 and 1000 years, these values were the estimated values in Morelia for the 19 September 1985 and 19 June 1858 earthquakes, respectively.

Building supports can be fixed when the structure rests on stiff soils or rocks. However, in other types of soils, the effects of soil–structure interaction (SSI) can be significant and sometimes beneficial or detrimental [50]. According to Dutta [51], low-rise stiff buildings can increase the shear force at the base due to the SSI effect, which is reduced when the soil hardness and the building’s number of stories are increased. Two main approaches are used to incorporate soil–structure interaction (SSI): direct methods and indirect methods (substructure approach). In the direct method, a combined model of the building and soil is created to establish the dynamic equilibrium equations. The soil is discretized using Finite Element Methods or Boundary Element Methods, and it can be modeled with linear or nonlinear behavior [52]. This approach directly includes inertial and kinematic effects in the analyses [53,54]. In indirect methods, the SSI is included in two stages: transfer functions (for inertial interaction) and impedance functions (for kinematic interaction). The kinematic interaction determines the effective movement of the soil when seismic waves arrive on the structure’s foundation. Inertial interaction is modeled by incorporating springs and dampers for each degree of freedom of the foundation [55,56].

Avilés [57] reported that soil flexibility and a structure’s nonlinear behavior can be advantageous for slender structures with a fundamental period longer than the site period. However, they also found that there can be detrimental effects in cases where the site period is longer than the fundamental period of the structure. After comparing direct and indirect methods [58], the authors found that the indirect approach increased the inter-story drift demands in the structures. Raheem [59] investigated buildings supported by moment-resisting frames on flexible soils and found that SSI increased displacement demands, especially in lower and upper stories. The inclusion of SSI leads to an increase in second-order effects, as noted by Ghandil [60]. These effects result in higher drift ratios and are particularly significant in tall buildings due to foundation rocking. Carbonari [61] also observed increased drift ratios in the lower stories of buildings. From a practical perspective, the indirect method is more commonly used and is included in regulations, as is the case in Mexico. However, when the structure is highly important, is situated on flexible soils, and soil non-linearity is significant, direct methods should be used to account for soil–structure interaction (SSI). A more detailed review and analysis of SSI can be found in the literature [50,62,63].

When relevant, the soil–structure interaction modifies the structure’s dynamic properties and the expected movement at the site. To evaluate its possible effect in the buildings of the historic center of Morelia, the seismic regulations of Mexico City [37] were used. NTC-2017 [37] divided the soil–structure interaction into kinematic and inertial effects. The kinematic interaction that modifies the movement on the free surface due to the presence of the foundation can be disregarded when $T_s/{\tau }_v>12$ and $D/r<0.5$, where $T_s$ is the fundamental period of the structure, $D$ is the depth of foundation, $r=\sqrt{A/\pi }$, ${\tau }_v=D/V_s$, $A$ is the area of the foundation, and $V_s$ is the shear wave velocity. In historic buildings, the depth of foundation is usually around 10% of the height of the building. The foundation area corresponds to the width of the walls, in this case 0.65 m for their length in plan. For typical historic buildings in Morelia, a depth of foundation in the range of 0.6 to 1.0 m and foundation areas in the range of 50 to 80 m² were considered. According to a geotechnical study carried out in the historic center [64], a shear wave velocity of $V_S=400 \mathrm{m}/\mathrm{s}$ and a dominant period of $T_s=0.24 \mathrm{s}$ were estimated. Using these parameters, the ratio $T_s/{\tau }_v$ is always greater than 12 and $D/r$ is always less than 0.5, so the kinematic interaction is not relevant. On the other hand, the inertial interaction can be disregarded when $(T_e/T_s)(H_s/H_e)>2.5$. $T_e$ and $H_e$ are the fundamental period and the building height, respectively, and $H_s$ is the depth of the stiff soil. In this case, for the buildings’ heights in the historic center (5 to 6 m) and their estimated periods (0.1–0.15 s), the dominant period of the soil $T_s=0.24 \mathrm{s}$ and the soil depth $H_s=40 \mathrm{m}$. Thus, the multiplication of the quotients of periods and heights are always greater than 2.5, so the inertial interaction is not relevant either.

3.2. Seismic Vulnerability Indices of Heritage Buildings’ Facades

Information about the facade pathology was collected and documented in technical sheets following visual inspections of the buildings and measuring the areas affected by each pathology. For instance, Figure 6 illustrates one of the identification cards used to gather the information.

Table 8. Pathologies found on the heritage buildings’ facades in Morelia.

Facade 1	Facade 2	Facade 3	Facade 4
Facade 5	Facade 6	Facade 7	Facade 8
Facade 9	Facade 10	Facade11	Facade 12
Facade 13	Facade 14	Facade 15	Facade 16
Facade 17	Facade 18	Facade 19	Facade 20
Facade 21	Facade 22	Facade 23	Facade 24
Facade 25	Facade 26	Facade 27	Facade 28
Facade 29	Facade 30	Facade 31	Facade 32
Facade 33	Facade 34	Facade 35	Facade 36
Facade 37	Facade 38	Facade 39	Facade 40

displays the images of all facades evaluated, identifying the pathology according to the symbology defined in Figure 6.

The vulnerability index of each facade was used to evaluate the seismic vulnerability through the average value of the degree of damage index (μ_D) for the expected seismic intensities in Morelia. μ_D takes values in the range of 0 to 5, indicating no damage (≤1) to collapse (5). As previously mentioned, the expected degree of damage was evaluated for VI and IX MMI.

Using Equations (1)–(4), the vulnerability index (Iv), the normalized damage index (V), and the degree of damage index μ_D were assessed for the MMI values and the selected facades. The results of the effects that earthquakes of great intensity could have on the facades are summarized and reported in

Table 9. Degree of damage indices μ_D for heritage building facades in Morelia city.

Facade	${\mathit{I}}_{\mathit{v}}$	$\mathit{V}$	${\mathit{\mu }}_{\mathit{D}}$		Facade	${\mathit{I}}_{\mathit{v}}$	$\mathit{V}$	${\mathit{\mu }}_{\mathit{D}}$
			MMI = VI	MMI = IX				MMI = VI	MMI = IX
1	22.895	0.723	0.462	4.607	21	18.158	0.706	0.419	4.209
2	21.316	0.714	0.447	4.422	22	18.158	0.696	0.419	4.209
3	24.474	0.732	0.447	4.607	23	35.526	0.795	0.605	5.058
4	21.316	0.714	0.447	4.422	24	32.368	0.678	0.564	4.956
5	21.316	0.714	0.447	4.422	25	19.737	0.705	0.433	4.319
6	19.737	0.705	0.433	4.319	26	21.316	0.714	0.447	4.422
7	19.737	0.705	0.433	4.319	27	19.737	0.703	0.433	4.319
8	19.737	0.705	0.433	4.319	28	19.737	0.705	0.433	4.319
9	21.316	0.714	0.447	4.422	29	38.684	0.813	0.652	5.142
10	24.474	0.732	0.477	4.607	30	18.158	0.706	0.419	4.209
11	19.737	0.705	0.433	4.319	31	22.895	0.697	0.462	4.518
12	21.316	0.714	0.447	4.422	32	22.895	0.723	0.462	4.518
13	18.158	0.706	0.419	4.209	33	22.895	0.723	0.462	4.518
14	30.789	0.681	0.545	4.898	34	21.316	0.714	0.447	4.422
15	21.316	0.714	0.447	4.422	35	22.895	0.723	0.462	4.518
16	21.316	0.714	0.447	4.422	36	16.053	0.711	0.401	4.050
17	19.737	0.705	0.433	4.319	37	19.737	0.705	0.433	4.319
18	19.737	0.705	0.433	4.319	38	20.789	0.711	0.442	4.389
19	21.316	0.714	0.447	4.422	39	19.737	0.703	0.433	4.319
20	21.316	0.714	0.447	4.422	40	41.842	0.660	0.704	5.210

. The facades’ vulnerability index was found in the interval 16.05–41.84, with a mean value of 22.85 and a coefficient of variation of 0.24. The damage indices μ_D varied in the range of 0.401 (Grade 1: no damage) to 0.704 (Grade 1: no damage) under the seismic intensity of VI, and in the range of 4.050 (Grade 4: severe and heavy damage) to 5.21 (Grade 5: collapse) for a seismic intensity of IX. The findings indicate that the facades of the buildings in the historic center have a high vulnerability to earthquakes with an intensity of IX. Such intensity can be caused by a strong earthquake originating from intraplate or crustal faults with an epicenter near Morelia. In this scenario, some facades could potentially reach the collapse limit state.

Figure 7 shows boxplots of the average degree of damage for MMI of VI and IX. Strong motions with the characteristics of the 19 September 1985 earthquake (MMI = VI) could generate no damage in most cases, slight or moderate damage in a few cases, and substantial damage in even fewer cases. On the other hand, earthquakes with the characteristics of the seismic event of 19 June 1858 (MMI = IX) could cause severe damage or collapse to most of the facades of the historic center of Morelia. This analysis showed that in very few cases, the facades could be in a slightly damaged state. The results showed that the facades of the buildings in the city center of Morelia are particularly vulnerable to strong motion from crustal and intraplate seismic sources.

Equation (4) determines the average damage grade based on the expected seismic intensity, the facades’ seismic vulnerability, and the ductility demand. To gain a deeper insight into the expected damage to the facades in various seismic scenarios, we computed vulnerability curves for the average seismic vulnerability index of the facade group, and the average value plus/minus one and two standard deviations.

Several studies [15,19,27,28,65] have used ductility values ($Q$) of 1, 2, and 3 for historic temples, buildings’ facades, and structures. Some have selected the $Q$ value after calibrating vulnerability functions with earthquake damage. The Mexican code recommends that this type of construction must use a value of $Q$ = 1.0. To assess the impact of this parameter, vulnerability curves were calculated for the three different $Q$ values found in the published works. Figure 8 displays the expected damage levels of the facades based on the mean vulnerability index $I_v$ for three ductility values (1, 2, and 3), along with the mean value plus/minus one to two standard deviations.

The abscissas axis displays the seismic intensity, and the ordinate axis presents the mean damage grade (${\mu }_D$). The $Q$ value determines the vulnerability curve slope, which is relevant to the expected damage. The mean damage grade rapidly changes from no damage to severe damage and collapse as the seismic intensity increases from 6 to 9 when using $Q$ = 1. If the more dispersed model is considered with an intensity value of 9, the facade will experience extensive to heavy damage and collapse. For $Q$ = 2 and intensities of 6 and 9, the facade would have no damage in the first case and close to extensive damage in the second case. By increasing $Q$ to 3, the expected damage for high intensities is even lower. While intensity 6 keeps the facade without damage, intensity 9 places it again between extensive and severe damage closer to the first case. Although achieving a ductility of $Q$ = 3 in stone walls, such as those of the facades of Morelia, is highly unlikely, it is also unclear if this value could be slightly higher than $Q$ = 1, the recommended value in Mexico. Therefore, this parameter should be studied in greater detail in subsequent studies, preferably with experimental analyses. Higher $Q$ values imply attaining larger intensity values without reaching the collapse scenario. For example, a damage grade of 4 (severe damage) would require an intensity value of 8.7, 9.2, and 9.9 for $Q$ = 1, $Q$ = 2, and $Q$= 3, respectively.

Figure 8 displays the mean damage grades for the average vulnerability index $I_V$ of the city’s facades. To observe the behavior of the curves for the extreme vulnerability index values, Figure 9 shows the mean damage grades to compare the effect of the three $Q$ values as a function of the minimum, average, and maximum vulnerability indices of the group of facades evaluated.

The three curves intersect at ${\mu }_D=2.5$ independently of the $I_V$ value. However, they cross at different intensities; for $I_{Vmin}$, the curves join at an intensity of I = 8.4, whereas this value is I = 7.4 in the $I_{Vmax}$ case. The high slope of $Q$ = 1 and low slope of $Q$ = 3 increase the mean damage grade with the increase in $Q$ in the left zone of the intersection point. In contrast, the opposite behavior is observed in the right region of the curves’ intersection.

Finally, the facades facing north and south and the facades facing east and west were analyzed to verify if the preferential direction of the wind in Morelia causes more significant deterioration in one direction or another. According to the Weather Spark website [66], the wind has a predominant north–south direction in the city from October to August. This wind direction impacts the facades that face east and west. The results showed that the average degree of damage for the east-west facades was 12% higher than that of the north-south facades. However, further research is required, and the impact of wind-driven rain on the facades should also be analyzed.

4. Discussion

This study proposes a simplified methodology to evaluate the seismic vulnerability of colonial house facades in the historic center of Morelia, Mexico. The methodology emphasizes the importance of accurately assessing the condition of the building’s facades. It recognizes the significance of facade deterioration in reducing their seismic capacity. The state of conservation is determined by the ratio of the damaged area to the total area of the facade, assessed by qualifying ten parameters. Subsequently, the seismic vulnerability index is evaluated using the expected seismic intensity on the Modified Mercalli scale at the site.

The study revealed that most facade pathologies were caused by rain-induced humidity, wind erosion, and dirt accumulation, leading to fissures, cracks, and stone spalling. Facades facing east and west showed slightly more deterioration than those facing north and south.

The visual inspections showed that facades have deteriorated, primarily from physical factors such as rain and wind and chemical factors like efflorescence. This damage has resulted in the loss of joints in several walls. On the other hand, facades were practically unaffected by ground settlements, and minor damage was found due to capillarity.

After considering historical seismic activity in Morelia and a recent seismic hazard assessment, it was determined that the city center could experience Modified Mercalli intensities VI and IX for return periods of 100–1000 years. The study also evaluated the seismic vulnerability indices and expected damage levels for these intensities. An MMI of VI poses a low seismic risk, while intensity IX poses a high risk to the facades. For intensity VI, the expected damage is minimal, but for intensity IX, it could lead to the partial or total collapse of the facades.

It is important to note that large earthquakes caused by subduction do not produce the most significant seismic risk to the facades of the historic center of Morelia. However, seismic events within the earth’s crust at intermediate and shallow depths significantly increase the vulnerability of existing facades, even putting their stability at risk in future earthquakes.

5. Conclusions

The analysis of the evaluated set of facades in Morelia is summarized in the following conclusions:

➢

The study identified several highly vulnerable facades that could be at risk during large-magnitude earthquakes in Mexico. These facades should undergo an intermediate methodology to evaluate their seismic vulnerability, and if necessary, a detailed methodology should follow. Subsequently, a rehabilitation process should be carried out to mitigate the seismic risk.

➢

The simplified proposal allows for evaluating the seismic vulnerability index of building facades. The methodology is simple and quick to implement. It includes assessing facade pathologies and then determining the expected level of damage based on Modified Mercalli intensity. However, it does not involve any vibration measurement to determine the dynamic properties of the facades. This limitation can be addressed by using more detailed methodologies.

➢

Future studies should investigate the impact of wind direction on the town and the facade pathologies caused by wind-driven rain. The ductility factor is another parameter of the simplified methodology that needs to be examined in these studies.

➢

This study introduces the initial method for assessing the seismic vulnerability of facades in the historic center. It aims to develop a thorough evaluation process for the city’s facades, involving a more detailed analysis than the simplified method.