5. The curve of equilibrium and the civil constructions of the eighteenth century

After the Bourbon dynasty's ascendancy to the Spanish throne (1700), Catholic diplomatic and military families of Irish and Scottish origin emigrated under royal protection, preserving their status. O'Connor family was installed in Benicarló in the eighteenth century, and they associated with the McDonnells in the wine export business.

The Bourbon dynasty, which established itself in Spain in 1700 with King Philip V (1683–1746), created the Army Corps of Engineers by the Royal Decree of 17 April 1711. Several Irish families moved to eastern Spain in the mid-seventeenth century. Patrick White Limerick, a trader in agricultural products and wine, and the

• d3 describes the ratio between the length of the semimajor axis and the radius

• p1 describes the ratio between the semimajor axis and the minor axis.

The ovals used in the layout of the gunpowder magazines are thus used as a reference for purposes of comparison with the cellar's layout. The layout of [O1, O2, O3] is based on a ratio between d3 and e2 of [0.39:0.36:0.50].

In addition, the layout of each oval is compared with a catenary that has the same rise and span, which is drawn using InnerSoft software. According to the results, the inner surface defined between the corresponding geometric shape and

the maximum distance between geometric shapes and the arch's span ranges from 2.14 to 3.44%. From these data, we can conclude that the approximation made by the engineers by drawing ovals in the three projects considered is sufficiently precise for the drawing scale used, between E: 1:90 and E: 1:70. Finally, the curves are compared with an elipse with the same rise and span. The obtained shape is

). Furthermore, the ratio between

• p2 describes the ratio between the center-to-center distance on the minor axis and the distance from the center of the semimajor axis to the

of the oval's minor arc.

Oval method applied to the projects for the gunpowder magazines.

the springline is different (1.33 m<sup>2</sup> vs. 0.98 m<sup>2</sup>

clearly not coincident with the curves of the projects.

minor axis.

Figure 15.

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44

O'Connor family settled in Benicarló in 1749. Gaspar White and the O'Gorman family were based in Alicante, whereas the Lilikells, the Tuppers, and Henry O'Shea lived in Valencia. Against this backdrop, the O'Connor family built a Carlón wine cellar in Benicarló in 1757 [45] (Figure 17).

parameters, for arches a(2–8) the average span calculated is e1a(2–8) = 9.72 m, and the

Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century

It seems that the measurement units used for the construction of the cellar were the toise (194.90 cm) and the toise foot (32.48 cm). Arches a(1–8) have an average span of 29.92 toise feet (9.71 m), with an error of 0.02 m for 30 feet (5 toises, 9.74 m). The rise of arch a1 is 17.92 feet (5.82 m), i.e., there is an error of 0.02 m in 18 feet (5.84 m), which are 3 toises (Figure 19). Arches a(2–8) have a rise of 16.80 toise feet (5.46 m), i.e., there is an error of 0.06 m in 17 feet (5.52 m). The arches are 0.60 m in width (1 + 10/12 feet). Regarding the outside measurements, the nave is 41.50 feet (13.48 m) wide and 92.36 feet (30 m) long, and the arches' abutments are structures measuring 5 + 9/12 feet. The inside length of the cellar is 43.01 m, i.e., 132 + 5/12 toise feet. The enclosure wall on the façade is 2 feet thick; thus, the span-

A metrological analysis of the arches in Benicarló's cellar reveals that the eight arches show the same metric relations, i.e., a 5 toise span and a 3 toise rise. The dimension of the catenary arch a1 are 30 18 feet (exactly 5 3 toises). The dimensions of the elliptical or oval-shaped arches a(2–8) are 30 17 feet. If we follow the hypothesis that the minor axis (either the ellipse minor axis or the oval minor axis) is 1 foot below the impost, then the geometric relation of arches a(2–8) is also

A statistical analysis is now performed on each of the eight arches a(1–8) to determine the difference between the shape of the arches built and the shapes of reference: ellipse (E), catenary (C), and ovals [O1, O2, O3]. The following values

The mean deviation has a spread of only 0.03 m, which is approximately 0.31% of the arch's span, making it difficult to conclude whether it is a catenary or an oval. The determining feature is that arch a1 has an angle of incidence on the springline

average rise is e2a(2–8) = 5.46 m (Figure 18).

DOI: http://dx.doi.org/10.5772/intechopen.87060

to-arch ratio is 5.75/30 feet (Figure 20).

are calculated for 29 points on each cellar arch:

b.The angle of incidence on the springline

Transversal section of the O'Connor cellar in Benicarló (1757).

a. The average and maximum deviation of these 29 points

30 18 toise feet.

Figure 18.

47

The wine cellar's construction, using diaphragm arches, is very similar to the gunpowder magazine projects built by the Army Corps of Engineers in a neighbouring geographical area. These include the project by Carlos Beranger [MPD, 06, 169] for Benimàmet (1751) and the one by Juan Bautista French (1756) for Peñíscola [MPD, 07, 208]. In these projects, as opposed to the O'Connor cellar, the arch abutment is on the outside of the building. Nonetheless, there is a subsequent project by Antonio López Sopeña [MPD, 28, 027] for A Coruña (1774), in which he uses diaphragm arches with inside abutments similars to those used in the Benicarló's wine cellar.

The O'Connors family Benicarló's building was built in 1757. The geometric study of the cellar arches is based on the topographical survey conducted with a laser scanner. The Carlón wine cellar has a rectangular floor plan; its inside measures are 12.42 m in width and 43.01 m in length. In the nave, there are eight diaphragm arches, each having a single two-piece offset-jointed ring and an average depth of 0.60 m. The arches are made of solid ceramic bricks (measuring 0.37 0.18 0.04 m), and they rest on a limestone base that was brought from Santa Magdalena de Polpís. The top of this base determines the springline of the ceramic arch. The arch's abutment and the outside walls are made of ordinary uneven masonry. The formal characteristics of the arches are different: arch a1 has a clear span of e1a1 = 9.65 m and a rise of e2a1 = 5.82 m, whereas the other seven arches can be grouped together. Their span is within a range of e1a(2–8) = [9.76–9.69 m], similar to arch a1, but their rise significantly differs from the first arch, within a range of e2a(2–8) = [5.46–5.45 m]. All of the arches share a special feature: they do not have a vertical tangent on the stone base. The angle of incidence (α) of these arches with respect to both vertical sides, left (αa)and right (αb), has the following values: αa.a1 = 13.94° and αb.a1 = 8.97° in arch a1 and αa.a(2–8) = [4.49°–1.80°] and αb.a(2–8) = [7.38°–2.58°] in arches a(2–8) (Figure 8). By statistically analysing the

Figure 17. Floor plan and cross section of the O'Connor cellar in Benicarló (1757).

parameters, for arches a(2–8) the average span calculated is e1a(2–8) = 9.72 m, and the average rise is e2a(2–8) = 5.46 m (Figure 18).

It seems that the measurement units used for the construction of the cellar were the toise (194.90 cm) and the toise foot (32.48 cm). Arches a(1–8) have an average span of 29.92 toise feet (9.71 m), with an error of 0.02 m for 30 feet (5 toises, 9.74 m). The rise of arch a1 is 17.92 feet (5.82 m), i.e., there is an error of 0.02 m in 18 feet (5.84 m), which are 3 toises (Figure 19). Arches a(2–8) have a rise of 16.80 toise feet (5.46 m), i.e., there is an error of 0.06 m in 17 feet (5.52 m). The arches are 0.60 m in width (1 + 10/12 feet). Regarding the outside measurements, the nave is 41.50 feet (13.48 m) wide and 92.36 feet (30 m) long, and the arches' abutments are structures measuring 5 + 9/12 feet. The inside length of the cellar is 43.01 m, i.e., 132 + 5/12 toise feet. The enclosure wall on the façade is 2 feet thick; thus, the spanto-arch ratio is 5.75/30 feet (Figure 20).

A metrological analysis of the arches in Benicarló's cellar reveals that the eight arches show the same metric relations, i.e., a 5 toise span and a 3 toise rise. The dimension of the catenary arch a1 are 30 18 feet (exactly 5 3 toises). The dimensions of the elliptical or oval-shaped arches a(2–8) are 30 17 feet. If we follow the hypothesis that the minor axis (either the ellipse minor axis or the oval minor axis) is 1 foot below the impost, then the geometric relation of arches a(2–8) is also 30 18 toise feet.

A statistical analysis is now performed on each of the eight arches a(1–8) to determine the difference between the shape of the arches built and the shapes of reference: ellipse (E), catenary (C), and ovals [O1, O2, O3]. The following values are calculated for 29 points on each cellar arch:

a. The average and maximum deviation of these 29 points

b.The angle of incidence on the springline

The mean deviation has a spread of only 0.03 m, which is approximately 0.31% of the arch's span, making it difficult to conclude whether it is a catenary or an oval. The determining feature is that arch a1 has an angle of incidence on the springline

Figure 18. Transversal section of the O'Connor cellar in Benicarló (1757).

O'Connor family settled in Benicarló in 1749. Gaspar White and the O'Gorman family were based in Alicante, whereas the Lilikells, the Tuppers, and Henry O'Shea lived in Valencia. Against this backdrop, the O'Connor family built a Carlón wine

The wine cellar's construction, using diaphragm arches, is very similar to the

The O'Connors family Benicarló's building was built in 1757. The geometric study of the cellar arches is based on the topographical survey conducted with a laser scanner. The Carlón wine cellar has a rectangular floor plan; its inside measures are 12.42 m in width and 43.01 m in length. In the nave, there are eight diaphragm arches, each having a single two-piece offset-jointed ring and an average

depth of 0.60 m. The arches are made of solid ceramic bricks (measuring

0.37 0.18 0.04 m), and they rest on a limestone base that was brought from Santa Magdalena de Polpís. The top of this base determines the springline of the ceramic arch. The arch's abutment and the outside walls are made of ordinary uneven masonry. The formal characteristics of the arches are different: arch a1 has a clear span of e1a1 = 9.65 m and a rise of e2a1 = 5.82 m, whereas the other seven arches can be grouped together. Their span is within a range of e1a(2–8) = [9.76–9.69 m], similar to arch a1, but their rise significantly differs from the first arch, within a range of e2a(2–8) = [5.46–5.45 m]. All of the arches share a special feature: they do not have a vertical tangent on the stone base. The angle of incidence (α) of these arches with respect to both vertical sides, left (αa)and right (αb), has the following values: αa.a1 = 13.94° and αb.a1 = 8.97° in arch a1 and αa.a(2–8) = [4.49°–1.80°] and αb.a(2–8) = [7.38°–2.58°] in arches a(2–8) (Figure 8). By statistically analysing the

gunpowder magazine projects built by the Army Corps of Engineers in a neighbouring geographical area. These include the project by Carlos Beranger [MPD, 06, 169] for Benimàmet (1751) and the one by Juan Bautista French (1756) for Peñíscola [MPD, 07, 208]. In these projects, as opposed to the O'Connor cellar, the arch abutment is on the outside of the building. Nonetheless, there is a subsequent project by Antonio López Sopeña [MPD, 28, 027] for A Coruña (1774), in which he uses diaphragm arches with inside abutments similars to those used in the

cellar in Benicarló in 1757 [45] (Figure 17).

Benicarló's wine cellar.

Military Engineering

Figure 17.

46

Floor plan and cross section of the O'Connor cellar in Benicarló (1757).

[αa.a1 = 13.94°, αb.a1 = 8.97°]. Because the catenary's angle of incidence is 19.38°, the geometric shape that most closely resembles the arch is the catenary.

curves used by the Bourbon engineers to layout the projects for the gunpowder magazines are very similar to the cellar's arches. Nevertheless, the angle of incidence on the springline tends not to have a vertical tangent, which is a fundamental

feature of both the ellipse and the oval in the arches considered here. In the springline of these arches, the angle of incidence ranges between αa .a5 = 1.54° and

Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century

DOI: http://dx.doi.org/10.5772/intechopen.87060

αb .a5 = 7.38°.

Figure 20.

49

Geometrical analysis of the arches 2–8 in the O'Connor cellar (1757).

Conversely, the statistical analysis of the remaining seven arches a(2–8) shows that the geometric shape that they most resemble is the ellipse, with an average deviation ranging between 0.001 and 0.015 m. The range for oval-shaped arches is 0.006–0.186 m (arco apaynelado or arco carpanel according to Tosca, i.e., threecentred arch or basket-handle arch; or anse de panier according to Bélidor), so the

Figure 19. Geometrical analysis of the arch 1 in the O'Connor cellar (1757).

### Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century DOI: http://dx.doi.org/10.5772/intechopen.87060

curves used by the Bourbon engineers to layout the projects for the gunpowder magazines are very similar to the cellar's arches. Nevertheless, the angle of incidence on the springline tends not to have a vertical tangent, which is a fundamental feature of both the ellipse and the oval in the arches considered here. In the springline of these arches, the angle of incidence ranges between αa .a5 = 1.54° and αb .a5 = 7.38°.

Figure 20. Geometrical analysis of the arches 2–8 in the O'Connor cellar (1757).

[αa.a1 = 13.94°, αb.a1 = 8.97°]. Because the catenary's angle of incidence is 19.38°, the

Conversely, the statistical analysis of the remaining seven arches a(2–8) shows that the geometric shape that they most resemble is the ellipse, with an average deviation ranging between 0.001 and 0.015 m. The range for oval-shaped arches is 0.006–0.186 m (arco apaynelado or arco carpanel according to Tosca, i.e., threecentred arch or basket-handle arch; or anse de panier according to Bélidor), so the

geometric shape that most closely resembles the arch is the catenary.

Military Engineering

Figure 19.

48

Geometrical analysis of the arch 1 in the O'Connor cellar (1757).

Thus, arch a1 resembles a catenary arch, whereas the other seven arches a2–<sup>8</sup> tend to be ellipses. These seven arches do not have a vertical tangent on the springline because their horizontal axis has been moved 1 foot below the arch's springline. As defined by Frézier (1738), the shape of the catenary has the following essential property: the vertical line which is tangent to the curve at the springline does not form a right angle with the horizontal plane. Therefore, geometrically, the catenary can be understood as any curve that does not have a vertical tangent at its springline. This is what happens in the springline of St. Paul's dome in London [11], which was designed by Christopher Wren in collaboration with Robert Hooke [46]. Otherwise, it should be noted that from a mechanical perspective, catenary arches are an optimal solution to build masonry arches, since the material has very low tensile strength.

first 17 courses on the other side), the ring is 0.36 m wide. On the remaining seven arches, the ring is 0.60 m wide (just like the arch's depth). It is clear that less ceramic material is necessary for the construction of arch a1 than for the other seven

The assessment of some drawings of gunpowder warehouses, found in the collection of Mapas planos y Dibujos (MPD) of the General Archive of Simancas (Archivo General de Simancas, AGS) (AGS 2014), has revealed the use of the chain theory in Miguel Marín's projects for Barcelona (1731) and Tortosa (1733) and Juan de la Feriére ones in A Coruña (1736). A built evidence has also been found: the Carlón wine cellars in Benicarló, built by the O'Connors family from Ireland (1757). The analysis of these examples proved the theory of the chain arrival to Spain during the first half of the eighteenth century. However, 50 years before Antoni Gaudí, Catholic families emigrating from Scotland and Ireland already initiated some of the catenary's form mathematical theory in some practical uses, a theory that begun to be taught at the Mathematics Academy of Barcelona in 1720.

This paper addresses the introduction of the concept of the catenary arch in Spain before the nineteenth century. After an exhaustive review of the theoretical framework, some cases are assessed. The aim of the research is to find out if the mechanical concept of the chain was used by the Spanish military engineers and by the exiled English engineers, who built several wine cellars in Spain. Thus, we intend to determine whether there is any geometrical relationship between the layout of several gunpowder magazines made by Spanish military engineers in the 1730s and the construction of a civil building—the Carlón wine cellar in Benicarló

The assessments of the gunpowder warehouses by Miguel Marín for Barcelona (1731) and Tortosa (1733) and by Juan de la Feriére y Valentín in A Coruña (1736) are only a mere 4.05% of the projects analysed. However, they prove the intention to lay out the vault as a catenary. These authors knew that in a catenary the tensility in the shape of a hanging chain has the same compression values in the inverted geometrical figure. These engineers had a vast knowledge of the mechanical principles of the modern theory for masonry. From a scientific perspective, catenary vaults are the most interesting because they introduce the principles established by Hooke (1676). Both the arches of gunpowder magazines and the arches a(2–8) of Benicarló were laid out using the geometrical construction of an oval. Otherwise, the location of the horizontal axis of the ovals under the springline reveals the application of one of the characteristics of the catenary. This causes that the vertical line which is tangent to the curve in the springing does not form a right angle with

the horizontal, so they used the chain's theory in the layout of the projects.

Formally, if the distance between the axes and the springline of the arch is small, then the angle of incidence has a minimum influence on the thrust and the line of pressure. Otherwise, the location of the axis under the springline reveals the intention to minimize stresses in this point and in the neighbouring areas, even though

Although there is no evidence of the construction of the gunpowder warehouses, it is possible to confirm the use of catenary arches in the construction of the Carlón cellars of the O'Connor in Benicarló (1757). There are significant differences between the measures of the arches of the gunpowder magazines (maximum span: 22 feet; maximum rise: 14 feet) and the arches of the Benicarló cellar (span: 30 feet; rise: between 17 and 18 "toise" feet, until the springline). In addition, the span-to-rise ratio

6. Conclusion. The origin of the catenary arch in Spain

Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century

(1757)—in which catenary arches may have been used.

the final mechanical influence is small.

51

elliptical arches a(2–8) (Figure 21).

DOI: http://dx.doi.org/10.5772/intechopen.87060

Finally, from the construction point of view, the catenary shape can be approximated using other geometric forms such as ovals or ellipses, under the condition that there is not a vertical tangent at the springline. The catenary shape forms a barycentric axis, which minimizes the tensions on a linear element that is subject to only vertical loads. In the arch, the inverted catenary shape prevents the appearance of stresses other than compression stresses.

Thus, there are two hypotheses regarding the construction of the wine cellar. The first one is that the construction work was started from the inside toward the façade; thus, arches a(2–8) were constructed before the catenary arch a1. The second hypothesis is that the construction work began with arch a1. According to the second hypothesis, there is also a difference between both types of arches: on the first brick courses from the springline of arch a1 (the first 9 courses on 1 side and the

Figure 21. Springing of arches no. 1 and no. 2 in the O'Connor cellar.

#### Scientific Knowledge of Spanish Military Engineers in the Seventeenth Century DOI: http://dx.doi.org/10.5772/intechopen.87060

first 17 courses on the other side), the ring is 0.36 m wide. On the remaining seven arches, the ring is 0.60 m wide (just like the arch's depth). It is clear that less ceramic material is necessary for the construction of arch a1 than for the other seven elliptical arches a(2–8) (Figure 21).
