Curves of Clay: Bóvedas del Bajío

Curves of Clay: Bóvedas del Bajío
ALFONSO RAMÍREZ PONCE*
RAFAEL RAMÍREZ MELÉNDEZ†
Introduction
Since the beginning of time, man has had to confront the world around him in order
to survive. To this end, he has had to create a vital second skin, thereby transcending his
biological skin. This second skin has come to be termed Architecture.
The building of this second skin was begun with Man’s dreams and out of the raw
materials which nature provided for him. Buildings made of materials such as stone,
wood, cane, clay, and brick are to be found in different regions of the world throughout
Man’s history. The specific techniques employed in these buildings form an integral part
of our cultures and our building traditions.
The comprehensive knowledge of materials and their corresponding building
techniques has become a vital necessity, given the backdrop of an ever increasing demand
for living space, particularly housing. Moreover, it has become vitally important to
rationalise the building process in order to achieve the lowest possible cost.
Figs. 1 and 2. Constructing the bóvedas del Bajío
*
†
Nacional University of Mexico, Faculty of Architecture, C.U., MEXICO D.F., [email protected]
Pompeu Fabra University, Technology Department, Ocata 1, Barcelona, SPAIN, [email protected]
Alfonso Ramírez Ponce and Rafael Ramírez Meléndez
This paper aims to describe, analyse, and formalise some of the fundamental
properties of a popular construction technique for building brick vaults without any use
of framework or any additional reinforcements whatsoever. The technique is of collective
invention in Mexico dating back to the nineteenth century. Brick vaults, made only out
of several pieces of clay and the intuition and skillful hands of craftsmen, apart from
offering an economical solution even to this day to the housing problem, possess both an
architectonic and mathematical beauty.
The paper is organized as follows: background; a description and examples of the
technique we have termed “the leaning brick”; a mathematical formalization of the
surface generated by this type of construction technique; and some conclusions and
indications of areas of future research.
Background
Throughout history, there have been many different techniques for building covers
with brick. According to their structural characteristics, the covers may be divided in two
main groups: covers in which the brick works only as a final encasement, and covers in
which the brick is also part of the supporting structure. This second group may be further
divided into “layered” covers and “leaning” covers. Our technique is inside this last
subgroup.
Historically, the most ancient are the Nubic vaults of adobe, in southern Egypt which
were built at least 3,300 years ago. One example of this which can still be seen today is in
the Rameses funereal centre, in the Valley of the Kings, on the banks of the Nile opposite
the city of Luxor. Later, around the third century, we have the vaults built in Persia.
Later still, in the tenth century, are the incorrectly named “false Mayan arches”, built
with limestone in Yucatan (Figs. 3, 4).
Figs. 3 and 4. Examples of “false” Mayan arches
The misnomer is due to the fact that these structures are not really arches; the stones
in the Mayan structures do not transmit their loads from the top to the base, but are
rather simply superimposed on the stones underneath with a small salient part (more or
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Curves of Clay: Bóvedos del Bajío
less a fifth) coming out. This results in a structure forming a steep “A” shape. Thus, the
last stone placed on top of the structure which joins the two inclined planes is not a
keystone but a simple lid. Generally, this kind of structure had a limited depth and was
used to cover thresholds, doors or transition spaces. However, a structure with several
metres of depth is structurally a vault and may cover an inhabitable space. Thus, the
Mayan structures are really vaults initially formed by inclined flat surfaces as in the
cuadrángulo de las Monjas (Quadrangle of the Nuns) in Uxmal, and later forming curved
surfaces as in Labná. Finally, we have the bóvedas del Bajío (vaults from the Bajío) in
central Mexico. These vaults and the Nubic vaults mentioned earlier are based on the
same basic principle, reinforced bricks or bricks slightly tilted and leaning on one another.
They are however quite different both in terms of the type of brick used—adobe bricks in
Nubia and small bits of baked brick in Mexico, and limestone in Egypt—and also in the
way the bricks lean on one another (Figs. 5a and 5b).
Fig. 5. a (left) Nubic vaults; b (right) elongated brick vault
In Nubic vaults the bricks lean against a wall which is higher than the other
supporting walls. In the Mexican vaults the supports are the smaller sides (in the case of
an elongated dome) or the corners—literally just the points—(for square-shaped vaults);
this shall be shown in the corresponding diagrams. In Mexico, this technique dates from
the second half of the nineteenth century and there are two possible sites of origin: San
Juan del Rio in the state of Queretaro and Lagos de Moreno in Jalisco. These structures
have been constructed since the nineteenth century and are the object of our research.
The “Leaning Brick” Technique
The “leaning brick” is a popular construction technique which is at the same time
millenary and modern. This technique is used for building roofs and covers with bricks
without any framework or any kind of external support, making it a very economic way
of covering space. Moreover it can be used between floors in a housing block or to cover
an open area such as a terrace. It is a technique which can be learnt by professional
builders as well as self-taught builders. It is an ingenious technique, as will be illustrated
later, not invented by architects or engineers, but instead the fruit of common knowledge,
all too often ignored or disregarded by professionals and academics. Hence the reason
that it is not taught in colleges and is therefore in decline. The technique allows spaces of
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Alfonso Ramírez Ponce and Rafael Ramírez Meléndez
up to ten metres wide to be covered and is ideal for the vast majority of architectural
spaces, especially living spaces in individual or collective housing, and spaces destined for
educational or public service use.
The technique’s low cost is based on three underlying conditions. The first, as
mentioned earlier, is that no scaffolding or additional supports are required whilst the
cover is being built. Secondly, low cost materials are used, such as the common handmade
clay brick or, alternatively, wet clay brick commonly known as adobe, or an earth-cement
brick in proportions of one to ten. Lastly, the labour-to-time ratio is highly efficient, on
average just two hours manpower are needed to complete one square metre of cover.
Given our “build to finish” concept, that means that the vault’s lower section would be
completed. The technique does not even need additional iron or concrete reinforcements,
just clay bricks and building expertise.
The brick used, called cuña, measures 5x10x20 cm (1000cm3) (Fig. 6a). It has a
resistance that fluctuates between 60 and 75kg/cm2 and an approximate weight of 1.5 Kg.
In the case that a brick of these dimensions cannot be obtained or if it is too costly to
manufacture it, the standard wall brick can also be used, either whole or cut in half; in
Mexico, the standard wall brick’s dimensions are 6x12x24 cm. (Fig. 6b).
Fig. 6. a (left) the cuña ; b (right) the standard Mexican brick
The mortar used is a mix of chalk, cement, and sand, similar to that used for walls.
This low level of resistance means that it can be cut in half manually with the builder’s
trowel (an important requirement for the timely building of vaults). A skilled craftsman
with the aid of a helper is able to achieve up to seven or eight square metres per day. In
other words, each square metre of vault takes two hours work, a figure which is three or
four times lower than the man hours necessary to complete a concrete cover. It is
important to emphasise this point as this kind of technique is often criticised for being
too artesanal, overlooking the fact that to build a concrete cover requires three or four
times more hours/man per square metre.
The Process
When applying this technique, generally the vaults are built to cover a flat area limited
by a rectangular or square horizontal perimeter. When the spaces to be covered do not
take one of the aforementioned forms, then they are forcibly “regulated”.
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Curves of Clay: Bóvedos del Bajío
Fig. 7. Vaults built to cover a flat area limited
by a rectangular or square perimeter
Fig. 8 (above); Fig. 9 (below, left and right)
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Alfonso Ramírez Ponce and Rafael Ramírez Meléndez
Fig 10. Examples of vaults. The covers can be regular surfaces, generally rectangles of different
proportions and squares. However, any regular or irregular polygon can also be covered with the
system by subdividing the space in small sections.
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Curves of Clay: Bóvedos del Bajío
In other words, if the area to be covered is L-shaped, then the craftsman or architect
requests for an intermediate beam to be placed, such that the L is now subdivided into
two rectangular shapes. Twelve years ago when we began our experiments, we always
based them on the conditions of the internal space, i.e., the space to be lived in, and as a
consequence we changed the vault’s horizontality and its enclosing perimeter line.
The perimeter of the spherical sections we build can be regular or irregular. Moreover
the lines that make it up can be straight, curved or mixed lines and also horizontal or
inclined. These lines are the surface’s directives. Occasionally the directives are concrete
sections and at other times they are commercial metal angles. On the other hand, the lines
of brick which compose the surface have different dimensions and as they move on the
surface they become its generators (Figs. 8 and 9).
The mortar is placed in such a way that the inferior part of the bricks is enveloped in
mortar whilst gaps are left in the upper part of the bricks. This is so that when the vault is
covered from above the mortar penetrates into the brick joints. Only two people work on
the vault, the bovedero (the vault builder) and his helper. The latter takes care of
rendering and cleaning the interior of the vault as the work progresses.
This, it seems to us, is a very unique and intelligent construction technique. Rather
than confronting and fighting gravity, it assumes immediate defeat. But it is thanks to the
technique’s surrender, and other assisting factors such as its light weight—that of a small
brick—that it gains its stability and its vaulted form.
Within the process there are three key characteristics of the technique. Firstly, the
bricks are placed one on top of the other in continual succession. Second, the bricks to be
supported need to be small and light (completely the opposite of those large and heavy
bricks whose purpose it is to support). With small reductions in its dimensions, the brick
goes from 1728cm3 to just 1000cm3 and it weighs only 60% of a wall brick. Third, the
vault brick, unlike a wall brick, is used dry (wall bricks are wetted before being used) to
increase the adhesion. The mortar is made up of cement, chalk, and sand in proportions
of 1:1:8 or 1:1:10 (Figs. 10 -12).
Fig. 11. Another possible vault configuration
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Alfonso Ramírez Ponce and Rafael Ramírez Meléndez
Fig. 12. Examples of vaults and
configurations.
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Curves of Clay: Bóvedos del Bajío
Mathematical Analysis
Background. Generally we think of a function as a correspondence which associates to
each element of a set X, an element and only one element of another set Y. The set X is
called the domain of the function. Typically, the domain of such functions is the set of
points of the X-axis. These functions are generally called functions of one real variable. It
is easy to extend this idea of a function to functions of two or more real variables: a
function of two real variables is a function whose domain X is a set of points in the xy
plane. If we call f such function, its value at point (x,y) is a real number denoted by f(x,y).
It is easy to imagine how such a function can represent a practical situation in
architecture. For instance, consider a situation in which we are interested in determining
the height of a ceiling of a room. We may represent the height of the ceiling at a certain
point by f(x,y) where the domain of the function is the set of all points (x,y) which
correspond to points in the room´s floor.
Surface integrals. A surface integral can be imagined as the equivalent in two dimensions
to the linear integral, a surface being the region of integration instead of a curve. There
are three main representations of a surface. One is the implicit representation which
considers a surface as a set of points (x,y,z) that satisfies an equation of the form F(x,y,z) =
0. Sometimes it is possible to isolate one of the variables in the equation, e.g. z, from the
other two, e.g. x, y. When this is possible, we can obtain an explicit representation of the
form z = f(x,y). There exists a third surface representation method which is more useful
for the study of surfaces. The parametric representation uses three equations to express x,
y, and z as a function of two parameters u and v:
x = X (u, v ) , y = Y (u, v ) , z = Z(u, v )
Applying these equations, the surface defined by the points (x,y,z) is the image of a
two-dimension connected set T defined by the points (u,v). In order to combine these
three equations, a vector r is introduced which connects the origin (0,0,0) and a generic
point (x,y,z) on the surface. This results in the so-called vector equation of the form:
r (u, v ) = X (u, v )i + Y (u, v )j + Z(u, v )k where (u, v ) ∈ T
It turns out that the area of a (parametric) surface S, denoted by a(S), is determined by
the double integral
a (S) =
∫∫ δr δu × δr δv
du dv
T
Thus, in order to calculate the area of S, it is necessary to firstly calculate the
fundamental vector product δr δu × δr δv and then integrate its length in the region T.
Sometimes δr δu × δr δv is expressed as
δr δu × δr δv = δ(Y , Z ) δ(u, v )i + δ(Z, X ) δ(u, v )j + δ(X , Y ) δ(u, v )k
in which case we have
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Alfonso Ramírez Ponce and Rafael Ramírez Meléndez
a (S) =
2
2
2
∫∫ (δ(Y , Z) δ(u, v )) + (δ(Z, X ) δ(u, v )) + (δ(X, Y ) δ(u, v )) du dv
T
If the surface S is explicitly given by an equation of the form z = f(x,y), x and y can be
used as parameters and the fundamental vector product is
δr δu × δr δv = 1 + (δf δx )2 + (δf δy )2 dx dy
and the integral for calculating the surface of the area takes the form
a (S)
∫∫
1 + (δf δx )2 + (δf δy )2 dx dy
T
where the region T is the projection of S on the xy plane.
The Vault Surface as a Mathematical Function
Vault constructed on a square perimeter. The first step towards a mathematical analysis of
the vaults we described is to formalize their surface as a function of two real variables.
Consider for simplicity, a vault of height 1 constructed on the perimeter of a square of 2
by 2 (the generalization to a height and rectangular perimeter of arbitrary dimensions is
straightforward). The resulting vault would look like the one in figure 13.
Fig. 13
For any point p=(x,y) in the xy plane, in particular for any point inside the area
delimited by the perimeter of the square of 2 by 2, we have:
Fig. 14 (left) tan θ = y x . Fig. 15 (right)
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Curves of Clay: Bóvedos del Bajío
Thus,
θ = arctan(y x )
Also, if we denote X =
(x 2 + y 2 ) , then
X 2 + z 2 = D2
where z represents the point in which the line perpendicular to the plane xy and passing
by the point p = (x,y), intersects the curve defined by the vault (see figure 3). That is, z =
f(x,y) where f is the function of two real variables that we are seeking. D is the distance
from the point (x,y,z) to the origin. Thus z, representing the expression we are interested
in, is defined by the equation
z = D2 − X 2
D can be defined as a function of X and θ. Let g(X, θ) be such a function:
g (X , θ ) = 1 + 2 − 1 X 2 tan θ .
Fig. 16 is a graphical representation of g(X, θ):
(
)
Fig. 16.
Thus, z =
(D2 − X 2 ) becomes
2
z = ⎡(1 + ( 2 − 1)(x 2 + y 2 )(y x )) − (x 2 + y 2 )⎤
⎢⎣
⎥⎦
if x ≠ 0
and given that tan θ = y/x and X = x 2 + y 2 ,
z = f ( x, y ) =
( (
⎡ 1+
⎢⎣
)(
)
) (
)
2
2 − 1 x 2 + y 2 (y x ) − x 2 + y 2 ⎤ if x ≠ 0
⎥⎦
(1 − y 2 ) if x = 0
In order to calculate a(S), the area of the surface defined by the vault, we need to
evaluate the double integral
a (S) =
2
∫∫ (δf δx ) + (δf δy ) dx dy
T
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Alfonso Ramírez Ponce and Rafael Ramírez Meléndez
where f is defined as above and the region T is the projection of S on the xy plane, i.e.,
the square of 2 by 2 on which the vault is built.
Vault constructed as a section of a sphere. Now consider again for simplicity, a spherical
vault with a radius of 1 constructed on four arches (the generalization to a vault with
arbitrary radius is straightforward). The resulting vault would look like the one in Fig.17.
Fig. 17.
Being a spherical vault, here we already know the two-real-variables function which
formalize its surface:
(
)
z = f (x, y ) = 1 − x 2 − y 2
In order to calculate the exact vault surface we simply have to calculate the surface of
the semi-sphere on the square region on top of which the vault is constructed (see figure
4). This results in
a (S) =
2
2
∫∫ (1 + (δf δx ) + (δf δy ) ) dx dy
T
where f is as above and the square region T is the projection of S on the xy plane.
Conclusions
We have described a popular construction technique for building brick vaults, without
any use of scaffolding and or additional reinforcements. The technique, a collective
invention in Mexico dating back to the nineteenth century, offers an economical solution
even to this day to the housing problem. As a first step towards a mathematical analysis of
the vaults we have formalized their surface as a function of two real variables. We have
explored two cases: the surface of a vault constructed on a rectangular perimeter, and the
surface of a vault constructed on four circular arcs resulting in a section of a sphere. We
also have indicated how to calculate the area of the surface in both cases. In the near
future, we plan to contrast our mathematical results with measurements in existent vaults
and also provide a formalisation of the different patterns formed by the bricks on the
vaults surface.
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