1. INTRODUCTION
⌅Maintenance
and preservation of the architectural heritage, with a high artistic
and cultural value, is one of the main concerns of today’s society (1(1)
Valente, M., Milani, G. (2018). Damage assessment and partial failure
mechanisms activation of historical masonry churches under seismic
actions: Three case studies in Mantua. Engineering Failure Analysis, 92: 495-519, doi: https://doi.org/10.1016/j.engfailanal.2018.06.017.
). In this regard, the use of numerical modeling techniques has become increasingly important over the lasts years (2(2)
Castellazzi, G., D’Altri, A. M., de Miranda, S., Ubertini, F. (2017).
An innovative numerical modeling strategy for the structural analysis of
historical monumental buildings. Engineering Structures, 132: 229-248, doi: https://doi.org/10.1016/j.engstruct.2016.11.032
, 3(3)
Roca, P., Cervera, M., Gariup, G., Pela’, L. (2010). Structural
analysis of masonry historical constructions. Classical and advanced
approaches. Archives of Computational Methods in Engineering, 17(3): 299-325, doi: https://doi.org/10.1007/s11831-010-9046-1.
), as they make it possible to reproduce and predict the structural behavior of existing buildings and hence prevent damage.
However,
the structural analysis of historic buildings is often a difficult task
due to the complexity of their geometry, the wide range of construction
techniques used, the high variability of the mechanical properties of
the traditional building materials and, frequently, the lack of
knowledge about the existing damage (4(4) Asteris, P.G. (2008). On the Structural Analysis and Seismic Protection of Historical Masonry Structures. The Open Construction and Building Technology Journal, 2(1): 124-133, doi: https://doi.org/10.2174/1874836800802010124.
, 5(5)
Ávila, F., Puertas, E., Gallego, R. (2020). Probabilistic reliability
assessment of existing masonry buildings: The church of San Justo y
Pastor. Engineering Structures, 223: 111160, doi: https://doi.org/10.1016/j.engstruct.2020.111160.
).
To overcome these difficulties and create a model of the building that
accurately represents its structural behavior, it is essential to carry
out an exhaustive investigation including all the historical
documentation regarding the considered building.
The development of these highly-detailed models requiring a thorough previous investigation and a careful and precise modeling process, with the consequent time cost, is particularly suitable for heritage buildings of special architectural or historical relevance and buildings with a high structural complexity. The Tower of Comares, at the Alhambra of Granada (Spain) meets both requirements.
The Tower of Comares (Figure 1)
is one of the best-known buildings within the monumental complex of the
Alhambra, which is currently one of the few preserved palatine cities
of the Islamic period, constituting the best example of Nasrid art in
its architecture and decorative aspects (6(6) UNESCO. (2020). Alhambra, Generalife and Albayzín, Granada. World Heritage List. https://whc.unesco.org/en/list/314/.
), being included in the UNESCO World Heritage List since 1984 (7(7) UNESCO. (1984). 8th session of the World Heritage Committee (CONF 004). https://whc.unesco.org/archive/1984/sc-84-conf004-9e.pdf.
).
The Tower of Comares was built between the 13th and 14th centuries, and
has undergone numerous modifications and alterations throughout its
history, resulting in a building of high structural complexity that
combines several materials and construction techniques.
Despite
its great relevance, there are only a few studies concerning the
structural evaluation of the Tower of Comares, most of them carried out
during the last decade of the 20th century. Astiz (8(8)
Astiz, M.A. (1998). The importance of earthquake - The case of the
Comares tower in the Alhambra palace of Granada. In P. Roca, J. L.
Gonzáles, E. Oñate, P. B. Lourenço (Eds.), Structural Analysis of Historical Constructions II (pp. 19-37). CIMNE. http://www.hms.civil.uminho.pt/sahc/1998/19.pdf.
)
developed a 3D model with very simplified geometry and one single
homogeneous and isotropic material and carried out both a static and a
seismic analysis; similar procedure was followed by Santos et al. (9(9)
Santos, A., Cuéllar, V., Martinez, J.M., Salinas, L. (1997).
Ground-structure interaction analysis of the Tower of Comares, Alhambra
of Granada, Spain. 14th International Conference on Soil Mechanics and
Foundation Engineering, 1025-1028. https://www.issmge.org/uploads/publications/1/31/1997_02_0065.pdf.
) but considering five materials (two different soils and three types of rammed earth). Most recently, Brazille Naulet et al. (10(10)
Brazille Naulet, V.C., Orihuela Uzal, A., García-Pulido, L.J. (2020).
La Torre de Comares en peligro de ruina. Afecciones de la estructura más
significativa de la Alhambra en los siglos XVI y XVII. In Navarro
Palazón y García-Pulido (Eds.), FORTMED2020 - Defensive Architecture of
the Mediterranean: Vol. X (pp. 21-28) https://doi.org/10.4995/fortmed2020.2020.11499.
) analyzed the external layout of the tower identifying the existing damage, but without performing a structural analysis.
This study presents the development process of a finite element method (FEM) numerical model of the Tower of Comares at the Alhambra of Granada (Spain). This process begins with a thorough historical and architectural investigation and leads to the creation of a detailed 3D model of the structure that is subjected to a static analysis in order to verify the accurateness of its behavior. The structural complexity of the building, together with its great historical and artistic significance, makes it necessary to elaborate a highly-detailed numerical model in order to successfully carry out its structural analysis and so be able to ensure its preservation in safety conditions. The model developed in the present study aims to be the basis for future works regarding dynamic analysis and reliability assessment of such a relevant heritage building.
2. THE TOWER OF COMARES: HISTORICAL CONTEXT
⌅The
origins of the Tower of Comares date back to the 13th century, when the
enlargement of the walled perimeter of the Alhambra carried out by the
emir Muhammad I (1237-1273), also known as Alhamar, and his successor
Muhammad II (1273-1302) already included a small military tower,
probably in Almohad style, at the location of the current Tower of
Comares (11-13(11) Astiz, M.A. (1997). Estudio de la seguridad estructural de la Torre de Comares. Cuadernos de La Alhambra, 33-34: 115-132. https://www.alhambra-patronato.es/ria/bitstream/handle/10514/14152/8 encriptado.pdf?sequence=1.
(12) Gallego y Burín, A. (1961). Granada. Guía artística e histórica de la ciudad. Ed. Fundación Rodriguez Acosta.
(13)
Pavon Maldonado, B. (1975). Estudios sobre la Alhambra I: la Alcazaba,
el Palacio de Abencerrajes, los accesos a la Casa Real Vieja. El Palacio de Comares, el Partal. Cuadernos de La Alhambra, Anejo I.
).
Remains of this primitive tower can still be found in the basement of
the present tower, as the great hardness of the material made it very
difficult to demolish, so the following constructions were developed on
the existing ones.
Afterwards, around 1314, Ismail I built a first
palace in the same location, but it was demolished by Yusuf I between
1333 and 1354 to build, using rammed earth (RE) construction technique,
which is considered the main structure of the current Tower of Comares (14(14) Gómez Moreno y González, M. (1966). Granada en el siglo XIII. Cuadernos de La Alhambra, 2: 38-40.
). The works were finished by Muhammad V, who also added the court, the portico and the Hall of the Boat (Sala de la Barca).
The architectural design of the tower is characteristic of the Nasrid
architecture, where it is frequent to observe a contrast between the
sumptuousness of the inside, typical of a palace, and the sobriety of
the outside, typical of a defensive tower placed in the walled perimeter
of a fortress (15(15)
Cruces Blanco, E., Galera Andreu, P. A. (2001). Las torres de la
Alhambra, población y ocupación del espacio: Informes de Juan de Orea
(1572). Cuadernos de La Alhambra, 37: 41-58. https://www.alhambra-patronato.es/ria/handle/10514/14184.
).
During
the following years, there were only minor alterations in the tower.
The most relevant event affecting the tower during the 16th century
occurred in 1590, when an explosion in a nearby gunpowder factory
affected the Tower of Comares causing severe damage to windows, floor
slabs, partition walls and external walls (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
). The repairs, made with brick masonry, were not enough to solve the structural damages (17(17)
Archivo del Patronato de la Alhambra y Generalife. (1590). L-6-27.
Declaración del maestro mayor de las obras reales, Juan de la Vega, de
las obras y reparos en la casa real y asesorías, con motivo de fuego.
, 18(18) Casares López, M. (1973). Documentos sobre la Torre de Comares (1686). Cuadernos de La Alhambra, 9: 53-66.
).
The
structural pathologies present in the tower got exacerbated over the
next decades due to the lack of maintenance and the execution of several
inadequate alterations (10(10)
Brazille Naulet, V.C., Orihuela Uzal, A., García-Pulido, L.J. (2020).
La Torre de Comares en peligro de ruina. Afecciones de la estructura más
significativa de la Alhambra en los siglos XVI y XVII. In Navarro
Palazón y García-Pulido (Eds.), FORTMED2020 - Defensive Architecture of
the Mediterranean: Vol. X (pp. 21-28) https://doi.org/10.4995/fortmed2020.2020.11499.
).
Thus, during the first third of the 17th century, a number of reforms
were made in the basements, causing the collapse of the north wall of
the central vault of the primitive tower (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
), which was still preserved, and the opening of a hole in the north wall of the new dome, dating back to Yusuf I (19(19) Orihuela Uzal, A. (1996). Casas y palacios nazaríes. Siglos XIII-XV. Granada: Fundación El Legado Andalusí-Lunwerg.
).
Also in the first half of the 17th century, wooden braces were placed
in the south wall to ensure its stability and to resist the lateral
forces resulting from the heavy dome that closed the tower. This wooden
braces were later replaced by new ones made of iron (8(8)
Astiz, M.A. (1998). The importance of earthquake - The case of the
Comares tower in the Alhambra palace of Granada. In P. Roca, J. L.
Gonzáles, E. Oñate, P. B. Lourenço (Eds.), Structural Analysis of Historical Constructions II (pp. 19-37). CIMNE. http://www.hms.civil.uminho.pt/sahc/1998/19.pdf.
).
In
the year 1644, the master builder of the Alhambra, Bartolomé Fernández
Lechuga, expressed his concern about the structural state of the Tower
of Comares. In order to avoid its collapse, the windows at the lower
floor were filled in, and the demolition of the battlements and the
upper vault was proposed to reduce weight. Taking into account that the
current battlements are made of brick masonry, instead of rammed earth,
it seems reasonable to think that they were actually removed (10(10)
Brazille Naulet, V.C., Orihuela Uzal, A., García-Pulido, L.J. (2020).
La Torre de Comares en peligro de ruina. Afecciones de la estructura más
significativa de la Alhambra en los siglos XVI y XVII. In Navarro
Palazón y García-Pulido (Eds.), FORTMED2020 - Defensive Architecture of
the Mediterranean: Vol. X (pp. 21-28) https://doi.org/10.4995/fortmed2020.2020.11499.
).
Over
the next years, several alterations and repairs were done in the tower
due to its precarious structural conditions. Between 1671 and 1674, the
underpinning of the south wall and main entrance of the tower was
required (20(20)
Galera Mendoza, M.E. (2014). Arquitectos y maestros de obras en la
Alhambra (siglos XVI-XVIII): artífices de cantería, albañilería, yesería
y forja. Granada: Comares, Universidad de Granada.
),
but this measure was not enough to avoid its partial collapse. In view
of this situation, in 1688 the preservation works were resumed under the
direction of Juan de Rueda, beginning with the substitution of the
original vault, made of bricks, by a wooden truss covered with tiles (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
).
This new pyramid hip roof, which remained erect until 1931, was lighter
than the previous one and stood above the profile of the tower. The
works also included the underpinning of the tower and the reinforcement
with Alfacar travertine of the spaces between the lower windows (10(10)
Brazille Naulet, V.C., Orihuela Uzal, A., García-Pulido, L.J. (2020).
La Torre de Comares en peligro de ruina. Afecciones de la estructura más
significativa de la Alhambra en los siglos XVI y XVII. In Navarro
Palazón y García-Pulido (Eds.), FORTMED2020 - Defensive Architecture of
the Mediterranean: Vol. X (pp. 21-28) https://doi.org/10.4995/fortmed2020.2020.11499.
). These interventions significantly improved the structural integrity of the tower.
Later
on, in the 18th century, a report of the royal master builder in 1734
noted the existence of a large crack on the tower walls, from the ground
floor level to the top of the building, as a consequence of a previous
earthquake. During the following years, more underpinnings were made in
the tower, being worth to mention the one on the northeast corner in
1791 (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
, 21(21)
de Paula Valladar, F. (1890). El incendio de la Alhambra. Continuación
de la «Novísisma Guía de Granada», Granada: Vda. e Hijos de P. V.
Sabatel. http://www.bibliotecavirtualdeandalucia.es/catalogo/es/catalogo_imagenes/grupo.cmd?path=1008857n.
) and another one in 1814 at the part of the tower in contact with the perimeter wall.
In 1822, another seism caused damages in the main room of the tower, the so-called Chamber of the Ambassadors (Salón de los Embajadores), been necessary to tighten the armature of the roof from its basement (22(22)
Archivo del Patronato de la Alhambra y Generalife (1838). L-228 año
1838. Oficio de haber sido aprobados los presupuestos de obras de la
Torre de Comares.
). These works started in 1837.
Between
1853 and 1857, some other maintenance works were carried out,
concerning the restoration of the vault of the Chamber of the
Ambassadors by placing new wooden ribs and an iron compression ring with
clamps at the top part of the structure. The works also included the
reparation of the lateral arches of the main chamber and some
improvements in the staircase at the southwest corner of the tower (14(14) Gómez Moreno y González, M. (1966). Granada en el siglo XIII. Cuadernos de La Alhambra, 2: 38-40.
, 16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
).
In 1917, the Conservation Plan of the Alhambra (23(23) Velázquez Bosco, R. (1917). Plan de Conservación de la Alhambra. https://www.alhambra-patronato.es/ria/handle/10514/14222.
)
was established, including in its eighth section the maintenance works
to be carried out in the Tower of Comares as a matter of urgency. In
1923, the architect Leopoldo Torres Balbás was designated as chief
conservator of the Alhambra and, two years later, he presented a first
restoration project concerning the recovery of the passage between the
Hall of the Boat and the Chamber of the Ambassadors (24(24)
Vílchez Vílchez, C.T. (1985). La disposición musulmana del Patio de la
Reja de la Alhambra de Granada. Memoria de excavación. Cuadernos de
Arte, XVII, 353-380. https://revistaseug.ugr.es/index.php/caug/article/view/11080.
). Within this project, probably also the north wall of the passage was rebuilt.
Between
the years 1930 and 1931, relevant restoration works were carried out by
Torres Balbás, including the filling of the cracks in the north wall of
the Chamber of the Ambassadors and the staircase. The pyramid hip
wooden roof built in the 17th century was dismantled and replaced by a
reinforced concrete slab placed on steel joist resting on a steel I-beam
(16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
).
These interventions during the first half of the 20th century can be
considered as the last significant structural alterations carried out in
the Tower of Comares.
3. MATERIALS AND METHODS
⌅3.1. Materials characterization
⌅The evolution of the tower throughout history and the many additions and interventions have led to a great heterogeneity in the materials and construction techniques present in the building. This section aims to identify the existing materials and their position within the structure and to determine their mechanical properties.
The foundations of
the building, whose construction began during the early Nasrid period,
are made of RE with a very hard clayey matrix with river stones,
resulting in a mixture with a very high strength. The walls of the main
body of the tower of the 14th century are also made of RE, using a
mixture of fine gravel, sand, ferruginous clay and a significant
percentage of lime (around 20 %), with a particle size of the coarser
materials smaller than the one used in the foundations (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
, 25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
, 26(26) Valverde-Espinosa, I., Ontiveros-Ortega, E., Sebastián-Pardo, E. (1997). El tapial de las murallas de Granada. Re. Revista de Edificación, 26: 58-63. https://revistas.unav.edu/index.php/revista-de-edificacion/article/view/34878/30040.
).
According to the studies developed by González Limón et al. (25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
), the RE present in the Tower of Comares has an average bulk density equal to 2250 kg/m3 and an average pore size of 0.0174 μm. The same authors carried out
uniaxial compression test, according to standard UNE 83.304:84, on ten
RE samples extracted from the tower, obtaining a compressive strength (fc) equal to 2.45 MPa for the material of the walls and 7.85 MPa for the foundations (9(9)
Santos, A., Cuéllar, V., Martinez, J.M., Salinas, L. (1997).
Ground-structure interaction analysis of the Tower of Comares, Alhambra
of Granada, Spain. 14th International Conference on Soil Mechanics and
Foundation Engineering, 1025-1028. https://www.issmge.org/uploads/publications/1/31/1997_02_0065.pdf.
, 25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
). Both values, especially the one obtained at the foundations, are much higher than the ones typically obtained for RE (27(27)
Ávila, F., Puertas, E., Gallego, R. (2021). Characterization of the
mechanical and physical properties of unstabilized rammed earth: A
review. Construction and Building Materials, 270: 121435, doi: https://doi.org/10.1016/j.conbuildmat.2020.121435.
). The reason could be the great compaction reached after centuries of load bearing.
RE has very low tensile strength (ft),
in this case it was measured equal to 0.29 MPa for the walls and 0.74
MPa for the foundations. Via uniaxial compression tests, also the
elastic modulus (E), 0.92 GPa for the walls and 6.18 GPa for the
foundations, and the Poisson’s ratio (ν), 0.3 for the walls and 0.2 for
the foundations, were obtained by (25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
).
Another
material with a significant presence in the structure of the tower is
brick masonry. The bricks are made of fired clay and joined together
with mud and lime mortar (25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
). This masonry was used in combination with rammed earth in the walls of the vaults at the lower basements (24(24)
Vílchez Vílchez, C.T. (1985). La disposición musulmana del Patio de la
Reja de la Alhambra de Granada. Memoria de excavación. Cuadernos de
Arte, XVII, 353-380. https://revistaseug.ugr.es/index.php/caug/article/view/11080.
),
in pilasters at the corners of the tower, between the arches of the
tower of the 14th century and lining the north, east and west walls
under the pavement level of the Chamber of the Ambassadors.
According to the aforementioned study (25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
), the brick masonry has a density of 1440 kg/m3.
The compressive strength is equal to 14.71 MPa for the bricks and
between 1 and 2 MPa for the mortar. From these two values it is possible
to obtain a characteristic compressive strength of the masonry equal to
3.92 MPa according to standard NBE FL-90 and 3.88 MPa according to
Eurocode 6 (EC-6). Considering again those standards, the tensile
strength of the masonry is equal to 0.19 MPa (NBE FL-90) or 0.39 MPa
(EC-6). The elastic modulus is between 2.94 and 3.73 GPa according to
NBE FL-90 and 3.88 GPa if one considers the EC-2. The Poisson’s ratio is
considered equal to 0.25. To develop the numerical model, the most
unfavorable values of the material properties were considered, as shown
in Table 1.
The
current structure of the tower still includes some of the underpinnings
and insertions made of Alfacar travertine, although some of them were
removed during the restoration works carried out by Torres Balbás in the
first half of the 20th century. This travertine, obtained from a quarry
at Alfacar (Granada), is a carbonate rock that has been frequently used
as a construction material within the region of eastern Andalusia
(Spain) due to its great strength and durability (28(28)
Suarez, F.J., Bravo, R. (2014). Historical and probabilistic structural
analysis of the Royal ditch aqueduct in the Alhambra (Granada). Journal of Cultural Heritage, 15(5), 499-510, doi: https://doi.org/10.1016/j.culher.2013.11.010.
, 29(29)
Urosevic, M., Sebastián Pardo, E., Ruiz-Agudo, E., Cardell, C. (2011).
Physical properties of carbonate rocks used as a modern and historic
construction material in Eastern Andalusia, Spain. Materiales de
Construcción, 61(301): 93-114. doi: https://doi.org/10.3989/mc.2010.53809.
).
It is mainly composed of calcite (> 84 %) and quartz (ca. 14 %),
with presence of dolomite (< 5 %) and feldspar (<1 %) (29(29)
Urosevic, M., Sebastián Pardo, E., Ruiz-Agudo, E., Cardell, C. (2011).
Physical properties of carbonate rocks used as a modern and historic
construction material in Eastern Andalusia, Spain. Materiales de
Construcción, 61(301): 93-114. doi: https://doi.org/10.3989/mc.2010.53809.
).
The studies undertaken by Suarez and Bravo (28(28)
Suarez, F.J., Bravo, R. (2014). Historical and probabilistic structural
analysis of the Royal ditch aqueduct in the Alhambra (Granada). Journal of Cultural Heritage, 15(5), 499-510, doi: https://doi.org/10.1016/j.culher.2013.11.010.
)
at the aqueduct of the Royal Ditch of the Alhambra, made of the same
material, indicate a density of the Alfacar travertine equal to 2140
kg/m3, an elastic modulus of 23.67 GPa and a Poisson’s ratio of 0.35. Other study considering the same rock (5(5)
Ávila, F., Puertas, E., Gallego, R. (2020). Probabilistic reliability
assessment of existing masonry buildings: The church of San Justo y
Pastor. Engineering Structures, 223: 111160, doi: https://doi.org/10.1016/j.engstruct.2020.111160.
)
obtained similar results for the elastic modulus and Poisson’s ratio,
and defined a compressive strength of 29.4 MPa and a tensile strength of
1 MPa.
The beams and reinforcements in the concrete slab
introduced by Torres Balbás in 1932 are made of carbon steel. Taking
into account the characteristics of the steel used in construction in
Spain until 1960, the mechanical properties of this material are
considered equivalent to category A7 (carbon steel) as defined in
standard ASTM A36. To define the mechanical properties of the concrete
used in that slab, reference is made to the first Spanish standard
regarding structural concrete (30(30) Instrucción definitiva para el proyecto de ejecución de obras de hormigón, (1944).
),
approved in 1944 but written in 1939, so it is reasonable to consider
that the concrete used by Torres Balbás in 1932 was similar to the one
indicated in this standard.
Together with the steel found in the beams and reinforcements, the other metallic material present in the tower is the wrought iron used for the braces that help supporting the south wall between the fifth and the sixth floor of the tower. The mechanical characteristics of this material are also shown in Table 1.
| Material | Ref. | ρ [kg/m3] | fc [MPa] | ft [MPa] | E [GPa] | ν [-] |
|---|---|---|---|---|---|---|
| RE (walls) | (9(9)
Santos, A., Cuéllar, V., Martinez, J.M., Salinas, L. (1997).
Ground-structure interaction analysis of the Tower of Comares, Alhambra
of Granada, Spain. 14th International Conference on Soil Mechanics and
Foundation Engineering, 1025-1028. https://www.issmge.org/uploads/publications/1/31/1997_02_0065.pdf. , 25(25) González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A. (1997). Estudio de los materiales y de las fábricas de la Torre de Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104. ) |
2250 | 2.45 | 0.30 | 0.92 | 0.30 |
| RE (foundations) | (9(9)
Santos, A., Cuéllar, V., Martinez, J.M., Salinas, L. (1997).
Ground-structure interaction analysis of the Tower of Comares, Alhambra
of Granada, Spain. 14th International Conference on Soil Mechanics and
Foundation Engineering, 1025-1028. https://www.issmge.org/uploads/publications/1/31/1997_02_0065.pdf. , 25(25) González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A. (1997). Estudio de los materiales y de las fábricas de la Torre de Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104. ) |
2250 | 7.85 | 0.75 | 6.18 | 0.20 |
| Brick masonry | (25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104. ) |
1440 | 3.88 | 0.19 | 2.94 | 0.25 |
| Travertine | (5(5)
Ávila, F., Puertas, E., Gallego, R. (2020). Probabilistic reliability
assessment of existing masonry buildings: The church of San Justo y
Pastor. Engineering Structures, 223: 111160, doi: https://doi.org/10.1016/j.engstruct.2020.111160. , 28(28) Suarez, F.J., Bravo, R. (2014). Historical and probabilistic structural analysis of the Royal ditch aqueduct in the Alhambra (Granada). Journal of Cultural Heritage, 15(5), 499-510, doi: https://doi.org/10.1016/j.culher.2013.11.010. ) |
2140 | 29.40 | 1.00 | 23.67 | 0.35 |
| Cedar wood | (31(31)
Rodríguez Rodríguez, A. (2012). Influencias de los barnices incoloros
en las propiedades superficiales de las maderas de construcción.
Universidad Politécnica de Madrid. http://oa.upm.es/14842/. ) |
560 | 55.00 | 140.00 | 9.20 | 0.15 |
| Pine wood | (31(31)
Rodríguez Rodríguez, A. (2012). Influencias de los barnices incoloros
en las propiedades superficiales de las maderas de construcción.
Universidad Politécnica de Madrid. http://oa.upm.es/14842/. ) |
520 | 43.00 | 114.00 | 7.50 | 0.15 |
| Carbon steel | (32(32) A36/A36M-19 (2019). Standard Specification for Carbon Structural Steel. doi: https://doi.org/10.1520/A0036_A0036M-19. ) |
780 | - | 400.00 | 196.00 | 0.30 |
| Wrought iron | (8(8)
Astiz, M.A. (1998). The importance of earthquake - The case of the
Comares tower in the Alhambra palace of Granada. In P. Roca, J. L.
Gonzáles, E. Oñate, P. B. Lourenço (Eds.), Structural Analysis of Historical Constructions II (pp. 19-37). CIMNE. http://www.hms.civil.uminho.pt/sahc/1998/19.pdf. ) |
6920 | - | 390.00 | 190.00 | 0.25 |
| Concrete | (30(30) Instrucción definitiva para el proyecto de ejecución de obras de hormigón, (1944). ) |
2200 | 16.00 | 1.30 | 27.00 | 0.20 |
Also
wood is present as a construction material in the Tower of Comares.
Cedar wood was used for the vault covering the Chamber of the
Ambassadors, while pine wood was chosen for the beams and boards
conforming the floor slabs, most of them in the south wall. To define
their mechanical properties, the values offered by Rodríguez Rodríguez (31(31)
Rodríguez Rodríguez, A. (2012). Influencias de los barnices incoloros
en las propiedades superficiales de las maderas de construcción.
Universidad Politécnica de Madrid. http://oa.upm.es/14842/.
) for Atlas cedar and stone pine have been considered, as shown in Table 1.
3.2. Analysis of geometry
⌅A thorough analysis to define the geometry of the building is essential to develop an accurate model and to obtain reliable results. Due to the geometrical complexity of the Tower of Comares, the present study aims to create a highly-detailed 3D model of the structure that can represent with precision its mechanical behavior.
To define the geometry of
the building, diverse references have been consulted and analyzed. The
most exhaustive and accurate geometrical description of the tower,
together with the one developed by Jiménez Martín in 1996 (35(35) Jiménez Martín, A. (1996). Jardín con plantas (y alzados) de papel. In A. Almagro Gorbea (Ed.), Arquitectura en Al-Andalus: documentos para el siglo XXI (pp. 205-284). Lunwerg Editores. https://hdl.handle.net/11441/97094.
), was carried out by Torres Balbás between 1927 and 1932 (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
)
on the occasion of the restoration project that was developed during
the 1930s. Also some drawings by Juan de Rueda for the restoration
project carried out in 1688 are still preserved. An example of the
blueprints and drawings from these two projects are shown in Figure 2. The geometrical information obtained from literature was complemented with some measurements in-situ.
) and West-East vertical section (right) (34(34) Rueda Alcántara, J. (1686). Dibujo de la Torre de Comares de La Alhambra de Granada para su restauración. Archivo General de Simancas. http://www.mcu.es/ccbae/es/consulta/registro.cmd?id=179875.
).
Externally, the Tower of Comares is a 16.75 × 19.15 m rectangular prism with a height of 22.60 m above the floor level of the Chamber of the Ambassadors (1.30 m-tall battlements included) and between 6 and 26 m below the ground level, foundations included. The basements of the tower consist of several vaulted tunnels covered with brick masonry, among which stands out the Hall of the Nymphs.
The main body of the tower has 2.5 to 3.0 m-thick walls and is dominated by the Chamber of the Ambassadors, also referred to as Hall of Comares or Throne Room, which is considered the most majestic room of the whole palace. The room has a 11.30 m-side square plan and is 18.20 m high, crowned by the Cedar wood dome. A staircase at the southwest corner gives access to the upper rooms, most of them placed at the south side of the tower.
This
south wall has, at the level of the fifth and sixth floors, the wrought
iron braces placed in the 17th century to contain the pressure
generated by the wooden roof. These braces have square section of side
2.5 cm and 2 m free span (8(8)
Astiz, M.A. (1998). The importance of earthquake - The case of the
Comares tower in the Alhambra palace of Granada. In P. Roca, J. L.
Gonzáles, E. Oñate, P. B. Lourenço (Eds.), Structural Analysis of Historical Constructions II (pp. 19-37). CIMNE. http://www.hms.civil.uminho.pt/sahc/1998/19.pdf.
).
The
tower is topped by the roof constructed in 1932, composed of an 8
cm-thick reinforced concrete slab resting on I-joists that rest, in
turn, on a 12.30 m-long I-beam. This main beam is embedded in the east
and west walls of the tower, which are the best preserved ones (16(16)
Torres Balbás, L. (1932). Proyecto de obras complementarias de la Torre
de Comares. Archivo del Patronato de la Alhambra y Generalife.
).
As a result of the architectural and geometrical analysis, the geometry of the building, used to create the model, can be defined as shown in Figure 3.
).
3.3. FEM modeling analysis
⌅Considering the information obtained from the historical and geometrical analysis, it was possible to create a three-dimensional model of the Tower of Comares. This model was generated as a combination between several sub-models defined according to the characteristics of the construction materials and the building period of each part of the structure. In this regard, the following sub-models were identified:
-
RE foundations.
-
Primitive tower from the 13th century, made of RE and whose walls are now composing part of the foundations and the ramparts passing under the tower.
-
Basements under the Chamber of the Ambassadors, made of RE.
-
Brick masonry vaults of the tunnels and Hall of the Nymphs in the basements of the tower.
-
RE walls at the main body of the tower.
-
Brick masonry at walls, arches and battlements.
-
Alfacar travertine insertions and underpinning.
-
Cedar wood main dome.
-
Pine wood beams and floor slabs.
-
Wrought iron braces in the south wall.
-
Reinforced concrete slab at the roof.
-
Steel joists and I-beam supporting the roof.
Figure 4 shows the 3D model of the building, identifying the parts and materials that compose it.
This
geometry was introduced in a FEM software in order to perform the
structural analysis. For the purposes of the present study, only the
elastic material properties were included. The behavior of the materials
was represented via macro-modeling, a common procedure when analyzing
large structural members or full structures, as it requires lower
computing times and offers an adequate approach for the characterization
of the structural response (3(3)
Roca, P., Cervera, M., Gariup, G., Pela’, L. (2010). Structural
analysis of masonry historical constructions. Classical and advanced
approaches. Archives of Computational Methods in Engineering, 17(3): 299-325, doi: https://doi.org/10.1007/s11831-010-9046-1.
, 5(5)
Ávila, F., Puertas, E., Gallego, R. (2020). Probabilistic reliability
assessment of existing masonry buildings: The church of San Justo y
Pastor. Engineering Structures, 223: 111160, doi: https://doi.org/10.1016/j.engstruct.2020.111160.
).
The parts of the model were meshed mainly with linear 4-nodes tetrahedral elements, in order to properly fit with the complex geometry of the building. The model has a total of 480128 elements, 143888 nodes and 431664 degrees of freedom. The mesh size was defined computing the first three natural frequencies using meshes with increasing element density until reaching convergence.
Once the FEM model was defined, it was
first subjected to a full modal analysis including the first ten modes
of vibration, in order to validate its behavior, and then to linear
static analysis. The loads considered for the static analysis were the
self-weight of the building, live load, snow load and wind load,
according to the values indicated by the Spanish Building Code
(CTE-DB-SE-AE) (36(36)
CTE DB SE-AE. Código Técnico de la Edificación, Documento Básico de
Seguridad Estructural - Acciones en la edificación, (2009). https://www.codigotecnico.org/images/stories/pdf/seguridadEstructural/DBSE-AE.pdf.
) for the city of Granada, as shown in Table 2.
The live load was considered applied on the passable zones (corridors,
basements, rooms and walkable roof) and the snow load was applied on the
roof and other exterior horizontal surfaces. The static wind load
includes a pressure that was applied on the north and west walls,
oriented to the exterior of the fortress and therefore more exposed to
the action of the wind, and a suction effect applied on the opposite
facades. Regarding the boundary conditions, the foundations of the tower
were considered fixed in the hillside where it is placed.
).
| Load | Value [kN/m²] |
|---|---|
| Self-weight | (depending on the material) |
| Live load | 2.00 |
| Snow | 0.50 |
| Wind (pressure) | 1.00 |
| Wind (suction) | 0.50 |
4. RESULTS AND DISCUSSION
⌅4.1. Modal analysis
⌅The first ten modes of vibration and natural frequencies of the Tower of Comares obtained from the modal analysis are shown in Figure 5. First ten modes of vibration and natural frequencies of the tower. The modal analysis is a useful tool to have a first validation of the correctness of the model and to forecast its seismic response.
If
we attend to the first mode of vibration, a first natural frequency of
1.97 Hz was obtained. This value is very close to the 2 Hz determined by
Astiz (11(11) Astiz, M.A. (1997). Estudio de la seguridad estructural de la Torre de Comares. Cuadernos de La Alhambra, 33-34: 115-132. https://www.alhambra-patronato.es/ria/bitstream/handle/10514/14152/8 encriptado.pdf?sequence=1.
) using a model of the tower with simplified geometry and homogeneous material, and calculated by Santos et al. (9(9)
Santos, A., Cuéllar, V., Martinez, J.M., Salinas, L. (1997).
Ground-structure interaction analysis of the Tower of Comares, Alhambra
of Granada, Spain. 14th International Conference on Soil Mechanics and
Foundation Engineering, 1025-1028. https://www.issmge.org/uploads/publications/1/31/1997_02_0065.pdf.
)
by the installation of seismographs at two levels of the tower. The
result is also in agreement with the approximation that can be
calculated applying the expression proposed by the Spanish seismic code
(NCSE-02) (37(37) NCSE-02 Norma de Construcción Sismorresistente: Parte general y edificación, (2009). https://www.mitma.gob.es/recursos_mfom/0820200.pdf.
) for brick masonry structures:
where fF [Hz] is the fundamental frequency of vibration of the structure, L [m] is the length of the plan of the building in the direction of the vibration, and H [m] is the height of the building. Considering the total height of the tower (H = 47.30 m) and repeating the calculation for both main directions (L = 16.75 m and L = 19.15 m), an average value of 1.98 Hz is obtained for the first natural frequency of the tower.
According to these results, it is possible to conclude that the outputs of the modal analysis support the accuracy in the behavior of the developed FEM model.
4.2. Static analysis
⌅The results from the static analysis of the building under gravity and wind loads show that the greatest displacements (Figure 6) occur at the top of the north wall, towards the exterior of the fortress. This is due to the slanted foundations adapting to the shape of hillside where the tower is placed. However, the obtained deformations are very low, with maximum displacements lower than 17 mm, showing the high stiffness of the structure.
Regarding the stresses generated in the structure (Figure 7), it is possible to observe that the RE walls conforming the main body of the tower are subjected to limited stresses (lower than 2 MPa), and the maximum values are reached at the foundations and near the arches of the windows at the Chamber of the Ambassadors.
Stress concentration is found at the brick masonry arches and, especially, at the travertine underpinnings at the corners of the tower, although theses stresses are much lower than the strength of the materials. Also a slight stress concentration is observed at the basements under the south wall, but these stresses are still limited, indicating the correctness of the intervention carried out by Torres Balbás removing the ashlars in that position and allowing the connection between the staircase and the oratory. Also the execution of the concrete slab seems appropriate, as the beams show stresses much lower than their limit load and they are able to transfer that load to the walls without generating excessive stresses to the RE.
The iron braces in the
south wall are subjected to tensile stresses much higher than the rest
of the structure (up to ca. 18 MPa), as shown in Figure 8,
proving the essential role that these elements are playing even today
in the global stability of the tower, as indicated by previous studies (8(8)
Astiz, M.A. (1998). The importance of earthquake - The case of the
Comares tower in the Alhambra palace of Granada. In P. Roca, J. L.
Gonzáles, E. Oñate, P. B. Lourenço (Eds.), Structural Analysis of Historical Constructions II (pp. 19-37). CIMNE. http://www.hms.civil.uminho.pt/sahc/1998/19.pdf.
, 11(11) Astiz, M.A. (1997). Estudio de la seguridad estructural de la Torre de Comares. Cuadernos de La Alhambra, 33-34: 115-132. https://www.alhambra-patronato.es/ria/bitstream/handle/10514/14152/8 encriptado.pdf?sequence=1.
).
This stress state, however, is still much lower than the tensile
strength of the material, and might be substantially increased when
subjecting the structure to a seismic load, which implies significant
horizontal forces to be absorbed by the iron rods.
These
results make it possible to understand the structural behavior of the
Tower of Comares under load conditions that are common during its
lifetime, as a result of a long history of interventions and restoration
works. The behavior shown by the detailed model developed in this study
is in agreement with the results obtained by several authors that have
previously analyzed the structural state of the tower (8-10(8)
Astiz, M.A. (1998). The importance of earthquake - The case of the
Comares tower in the Alhambra palace of Granada. In P. Roca, J. L.
Gonzáles, E. Oñate, P. B. Lourenço (Eds.), Structural Analysis of Historical Constructions II (pp. 19-37). CIMNE. http://www.hms.civil.uminho.pt/sahc/1998/19.pdf.
(9)
Santos, A., Cuéllar, V., Martinez, J.M., Salinas, L. (1997).
Ground-structure interaction analysis of the Tower of Comares, Alhambra
of Granada, Spain. 14th International Conference on Soil Mechanics and
Foundation Engineering, 1025-1028. https://www.issmge.org/uploads/publications/1/31/1997_02_0065.pdf.
(10)
Brazille Naulet, V.C., Orihuela Uzal, A., García-Pulido, L.J. (2020).
La Torre de Comares en peligro de ruina. Afecciones de la estructura más
significativa de la Alhambra en los siglos XVI y XVII. In Navarro
Palazón y García-Pulido (Eds.), FORTMED2020 - Defensive Architecture of
the Mediterranean: Vol. X (pp. 21-28) https://doi.org/10.4995/fortmed2020.2020.11499.
, 25(25)
González Limón, T., Álvarez de Buergo, M., de las Casas Gómez, A.
(1997). Estudio de los materiales y de las fábricas de la Torre de
Comares de la Alhambra. Cuadernos de La Alhambra, 33-34: 95-104.
).
5. CONCLUSIONS
⌅This work presents a historic and architectural study of a building with a high heritage value, the Tower of Comares at the Alhambra of Granada (Spain), with the aim of creating a numerical FEM model to analyze its structural behavior.
A thorough literature review made it possible to define the geometry and mechanical properties of the diverse elements, construction techniques and materials that compose the structure. With this information, it was possible to develop a highly-detailed three-dimensional FEM model of the tower, reflecting its geometry with a high level of precision. This historical and architectural analysis is essential to create accurate models of heritage buildings that properly represent their structural behavior; and it is also a useful tool to identify which parts of the geometry could be simplified or not included in the model without affecting its structural response in case the computational cost is unacceptably high.
The model is subjected to a modal analysis and a static analysis including gravity and wind loads. The results from the modal analysis, in agreement with previous studies, show a natural frequency of the tower slightly lower than 2 Hz. The static analysis allows to understand the structural behavior of the building under normal load conditions, indicating that the rammed earth making up the majority of the structure is subjected to limited stresses, while the main stresses are located at the brick masonry arches, travertine underpinnings and wrought iron braces. The displacements in the tower under these load conditions are also very limited, showing the high stiffness of the building.
It is worth to mention that the model developed in the present study includes only the elastic mechanical properties of the construction materials, as they are enough for the static evaluation performed. This study, therefore, aims to be the basis for future studies including the plastic behavior of the materials and regarding the dynamic evaluation of the tower oriented to the assessment of its structural vulnerability and reliability.