THE STRUCTURAL ANALYSIS OF POLYGONAL MASONRY DOMES. THE CASE OF BRUNELLESCHI'S DOME IN FLORENCE

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DOMES AND CUPOLAS — Vol. 2 - N.2 - 2015 (00-00)

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The Structural analysis of polygonal masonry domes.

The case of Brunelleschi’s dome in Florence

Giacomo Tempesta

Department of Architecture, University of Florence, Italy, E-mail: giacomo.tempesta@unifi.it

Michele Paradiso

Department of Architecture, University of Florence, Italy E-mail: michele. paradiso@unifi.it

Stefano Galassi

Department of Architecture, University of Florence, Italy E-mail: stefanogalassi@hotmail.it

Eva Pieroni

Department of Architecture, University of Florence, Italy E-mail: betulla.dt@alice.it

The paper deals with the structural analysis of polygonal masonry domes by taking into account the thickness. A numerical procedure is presented through which the actual behaviour of the material is considered in the analy-sis. The structure is modelled as a discrete system of rigid blocks linked through elastic mortar layers. No-tension behaviour of the material is assumed to be totally concentrated in the mortar joints located between the adjacent blocks. Such a joint can therefore be assumed as a unilateral elastic contact constraint. The solution is achieved by a step by step algorithm in which the starting solution, relative to the standard material (linear elastic and bilateral), is subsequently corrected according to the actual material behaviour. The numerical procedure can be applied to the analysis of any type of polygonal masonry domes, with a spherical or pointed shape, subject to self weight loads, complete or with hole and lantern. As a particular case of analysis, some interesting remarks dealing with the case of Brunelleschi’s Dome are presented.

Keywords: masonry dome, no-tension analysis, numerical method

Intoduction

The study of masonry domes has been the object of

several research projects starting from some eighteenth

century memoirs of the Academiae Royale des

Scienc-es in Paris. One of the first aspects of the problem to

be examined, was the search for the optimal shape to

be given to a masonry dome. Pierre Bougure (Bouguer

1734), while answering the question of what shape a

masonry dome, subjected toself-weight, should have,

proposed for the first time the equation of the

funic-ular meridian. He stated that in order to have

equi-librium, the meridian must have the same shape of

the curve that represents the funicular of the loads

which are relative to a slice of dome. Subsequently,

a similar solution to the same problem was found by

Charles Bossut[2], Lorenzo Mascheroni[3], and

Gi-useppe Venturoli[4]. All the solutions suggested by

these authors have a common characteristic: the dome

is considered, de facto, as a one-dimensional behaviour

structure, composed of a series of distinct segments

or slices or “lunes”, wider at the base and tapering to

zero at the crown, placed in mutual contact with each

other but without any interactions among them. In

conclusion, since the equilibrium of each slice is

in-vestigated separately, if it can be shown that each

ele-ment of the sliced structure is stable, then it is argued

that the original structure must be stable. Under these

assumptions the analysis of masonry domes does not

present any difference compared with the analysis of

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Tempesta, Paradiso, Galassi, Pieroni

86

86 masonry arches. It is interesting to note that the fact

of not considering any action among the slices of the

dome, corresponding to the hypothesis of zero hoop

stresses, means that the problem lies, actually, in the

field of modern limit analysis, as applied by

J.Hey-man (HeyJ.Hey-man 1967) in his fundamental studies on

masonry structures. Moreover, in his introductory

as-sumptions, Heyman himself makes explicit reference

to the eighteen-century model, considering that it was

still perfectly suitable to deal with the general solution

of the problem. The model of limit analysis proposed

by Heyman in any way differs from the analysis

per-formed by Poleni (Poleni 1734) or the Three

Mathe-maticians (Le Seur et al.1742) for evaluating the

sta-bility of the dome of St.Peter. The similarity between

the “settore solido” of Poleni and the “orange slice” of

Heyman is very clear.

Masonry dome. An appropriate three-dimensional

finite element modelling

Following a typically eighteenth century idea, let us

consider the general problem of a masonry structure

consisting of rigid blocks linked through elastic

mor-tar layers. In such a model the no-tension behaviour

of the material is totally supposed to be

concentrat-ed in the mortar joint locatconcentrat-ed between two adjacent

blocks. Such a joint, can therefore, be assumed as a

unilateral elastic contact constraint.

In particular, the mortar joint can be idealized, in a

Drucker’s way, through an interface device consisting of

a set of elastic links, orthogonal to the contact surface,

capable of transmitting only compressive forces between

the blocks, and additional links, parallel to the interface,

through which the shear forces can be transmitted. The

behaviour of the orthogonal links is assumed to be

uni-lateral and linear elastic, whereas for the parallel ones

further hypotheses can be added in order to specify

either the shear strength and to calibrate, for instance,

the influence of the friction between the blocks, or a

bilateral rigid behaviour totally capable of preventing

sliding (Fig. 1). In practice a reasonably low number of

orthogonal bars is sufficient to describe, with significant

expressiveness, the behaviour of the joint and to clearly

appraise the location and depth of possible cracks.

In the case of a masonry dome the structure is modelled

by a set of discrete three-dimensional rigid elements which

represent single or multiple blocks of stone. The element

of the shell, considering the actual thickness, is cut out by

two meridian planes and two sloping planes perpendicular

to the generating curve of the middle surface of the shell.

Let us consider, therefore, the general problem of a

ma-sonry dome consisting of n three dimensional shell rigid

elements linked through m unilateral elastic contact

in-terfaces. (Fig. 2).

Fig. 1. Behaviour of the joints: rigid-cracking behaviour of the links orthogonal to the interface (a) and tangential (b); elastic-cracking behaviour of the links orthogonal to the interface (c) or (d) and tan-gential (e).

the slices of the dome, corresponding to the hypothesis of zero hoop stresses, means that the problem lies, actually, in the field of modern limit analysis, as applied by J. Heyman (Heyman 1967) in his fundamental studies on masonry structures. Moreover, in his introductory assumptions, Heyman himself makes explicit reference to the eighteen-century model, considering that it was still perfectly suitable to deal with the general solution of the problem. The model of limit analysis proposed by Heyman in nothing differs from the analysis performed by Poleni (Poleni 1734) or the Three

Mathematicians (Le Seur et al. 1742) for evaluating the stability of the dome of St. Peter. The

similarity between the “settore solido” of Poleni and the “orange slice” of Heyman is very clear. 2. MASONRY DOME. AN APPROPRIATE THREE-DIMENSIONAL FINITE

ELEMENT MODELLING

Following a typically eighteenth century idea, let us consider the general problem of a masonry structure consisting of rigid blocks linked through elastic mortar layers. In such a model the no-tension behaviour of the material is totally is supposed concentrated in the mortar joint located in between two adjacent blocks. Such a joint can therefore be assumed as an unilateral elastic contact constraint. In particular, the mortar joint can be idealized, in a Drucker’s way, through an interface device consisting of a set of elastic links, orthogonal to the contact surface, capable of transmitting only compressive forces between the blocks, and additional links, parallel to the interface, through which the shear forces can be transmitted. The behaviour of the orthogonal links is assumed unilateral linear elastic, whereas for the parallel ones further hypotheses can be added in order to specify either the shear strength and to calibrate, for instance, the influence of the friction between the blocks, or a bilateral rigid behaviour totally capable to prevent the sliding. In practice a reasonably low number of orthogonal bars is enough to describe with significant expressiveness the behaviour of the joint and to appraise clearly the location ad depth of possible cracks.

Figura 1 - Modello di comportamento per i giunti di interfaccia: giunto rigido-fragile in direzione normale (a) e

tangenziale (b) – giunto elastico fessurante in direzione normale (c) o (d) e tangenziale (e). In the case of a masonry dome the structure is modelled by a set of discrete three-dimensional rigid elements which represent single or multiple blocks of stone. The element of shell, considering the actual thickness, is cut out by two meridian planes and two sloping planes perpendicular to the generating curve of the middle surface of the shell.

Let’s consider, therefore, the general problem of a masonry dome consisting of n three dimensional shell rigid elements linked through m unilateral elastic contact interfaces. (Fig. ).

Fig. 2. Internal forces and loads acting on the joint for each element

Figure 2: Internal forces and loads acting on the joint for each element

Assuming the structure subjected to the action of external loads represented by the vectors ∈

F ℜ

6n

,

the problem can be expressed through a system of equilibrium and elastic-kinematical equations,

whose variables, those which correspond to the unilateral links in the interface model, are subject to

inequalities:







=

+

=

+

δ

KX

x

A

F

AX

T

0

sub













≥

⋅

≤

∑

=

0

1

δ

k j nj t n

X

f

X

(1)

In the previous form (1) ∈

X ℜ

km

indicates the unknown vector of internal forces located on the

interface joints, where

X includes the components orthogonal to the contact surface of the joint and

_n

t

X includes the shear components; x ∈ℜ

6n

represents the unknown vector of components of

displacement related to the centroids of the elements; ∈

K ℜ

km×km

is the diagonal stiffness matrix of

the contact constraints.; ∈

δ

ℜ

km

indicates the unknown vector whose components are internal

distortions which need for obtaining a solution capable of satisfying both the equilibrium equations,

while respecting the sign conditions, and the elastic-kinematical compatibility of the actual reacting

structure. On this subject, it is convenient to distinguish, within the vector

δ

, two types of entities,

assuming for the former, related to the equilibrium aspects, the notation

δ

1

and for the latter, related

to the compatibility ones, the notation

δ

2

. Notice that the value k depends on the number of contact

constraints chosen to characterize the interface device and defines the degree of statically

indeterminacy of the structure.

Of course the system of equations (1) could have no solution under the sign conditions expressed in

the first inequality; in such a case it means that the structure cannot be equilibrated under the given

system of the external actions. In this case there is no vector ∈

X ℜ

km

which satisfies,

simultaneously, the n

6 equations and the km inequalities.

However let us suppose that the system (1) in consistent. In such a case the general solution

N

X

=

₀

+

, that is able to satisfy the equilibrium problem and the first of the two inequalities, can

Assuming the structure subjected to the action of external loads represented by the vectors ∈

F ℜ

6n

,

the problem can be expressed through a system of equilibrium and elastic-kinematical equations,

whose variables, those which correspond to the unilateral links in the interface model, are subject to

inequalities:







=

+

=

+

δ

KX

x

A

F

AX

T

0

sub













≥

⋅

≤

∑

=

0

1

δ

k j nj t n

X

f

X

(1)

In the previous form (1) ∈

X ℜ

km

indicates the unknown vector of internal forces located on the

interface joints, where

X includes the components orthogonal to the contact surface of the joint and

_n

t

X includes the shear components; x ∈ℜ

6n

represents the unknown vector of components of

displacement related to the centroids of the elements; ∈

K ℜ

km×km

is the diagonal stiffness matrix of

the contact constraints.; ∈

δ

ℜ

km

indicates the unknown vector whose components are internal

distortions which need for obtaining a solution capable of satisfying both the equilibrium equations,

while respecting the sign conditions, and the elastic-kinematical compatibility of the actual reacting

structure. On this subject, it is convenient to distinguish, within the vector

δ

, two types of entities,

assuming for the former, related to the equilibrium aspects, the notation

δ

1

and for the latter, related

to the compatibility ones, the notation

δ

2

. Notice that the value k depends on the number of contact

constraints chosen to characterize the interface device and defines the degree of statically

indeterminacy of the structure.

Of course the system of equations (1) could have no solution under the sign conditions expressed in

the first inequality; in such a case it means that the structure cannot be equilibrated under the given

system of the external actions. In this case there is no vector ∈

X ℜ

km

which satisfies,

simultaneously, the n

6 equations and the km inequalities.

However let us suppose that the system (1) in consistent. In such a case the general solution

N

X

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Structural analysis of polygonal masonry domes. The case of Brunelleschi’s dome in Florence

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Assuming the structure subjected to the action of

ex-ternal loads represented by the vectors F ∈ ℜ

6n

_{, the}

problem can be expressed through a system of

equilibri-um and elastic-kinematical equations, which variables,

those that correspond to the unilateral links in the

inter-face model, are subject to inequalities:

the two inequalities, can be obtained assuming, as the

initial solution X

₀

, which is relative to the bilateral linear

elastic behaviour of the contact constraints:

Assuming the structure subjected to the action of external loads represented by the vectors ∈F ℜ6n, the problem can be expressed through a system of equilibrium and elastic-kinematical equations, whose variables, those which correspond to the unilateral links in the interface model, are subject to inequalities:    = + = + δ KX x A F AX T 0 sub             ≥ ⋅ ≤ ≤

∑

= 0 0 1 δ k j nj t n X f X X (1)

In the previous form (1) ∈X ℜkm indicates the unknown vector of internal forces located on the interface joints, where X includes the components orthogonal to the contact surface of the joint and _n

t

X includes the shear components; x ∈ℜ6n represents the unknown vector of components of displacement related to the centroids of the elements; ∈K ℜkm×km_{is the diagonal stiffness matrix of} the contact constraints.; ∈δ ℜkm indicates the unknown vector whose components are internal distortions which need for obtaining a solution capable of satisfying both the equilibrium equations, while respecting the sign conditions, and the elastic-kinematical compatibility of the actual reacting structure. On this subject, it is convenient to distinguish, within the vector δ, two types of entities, assuming for the former, related to the equilibrium aspects, the notation δ1 and for the latter, related

to the compatibility ones, the notation δ2. Notice that the value k depends on the number of contact

constraints chosen to characterize the interface device and defines the degree of statically indeterminacy of the structure.

Of course the system of equations (1) could have no solution under the sign conditions expressed in the first inequality; in such a case it means that the structure cannot be equilibrated under the given system of the external actions. In this case there is no vector ∈X ℜkm which satisfies, simultaneously, the n6 equations and the km inequalities.

However let us suppose that the system (1) in consistent. In such a case the general solution

N

X X

X = ₀+ , that is able to satisfy the equilibrium problem and the first of the two inequalities, can

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be obtained assuming, as initial solution X₀, that is relative to the bilateral linear elastic behaviour of the contact constraints:

F A AK A K X₀ ₌ −1 T( −1 T)−1 ₍₂₎

Such an initial solution is then modified through the vector:

1 1 1 1 1 ₍ ₎ ₎ ( − ⋅δ = ⋅δ = I K− A AK− A − A C X_N T T (3)

which, added toX , satisfies the first of the (1) while respecting the sign conditions. ₀

Computing the Moore-Penrose generalized inverse of C_i, it is easily possible to evaluate the vector

i i

i CX0

1 =−

δ , where X₀_i are the components of the interactions those in the joint do not respect the inequalities conditions of (1). If the solution of the unilateral problem exists, the vector solution which satisfies simultaneously the equilibrium equations and the first and the second inequality of (1), assumes the form :

          = c t XX X 0

where for each joint we have: X_c <0and

∑

= ⋅ ≤ k j nj t f X X 1 (4) Since the final vector X is different from the first elastic vector solution X₀, it cannot satisfy the kinematical compatibility expressed through the second set of equations in the system (1). A very easy way to build up again such a compatibility is to consider the second set of equations in the system (1) in the form: c c c T c cA A K X A x₌₋( )−1 ₍₅₎

which represents the vector of the displacements of the centroids of the elements only due to the actual reacting structure. Finally we can determine the vector δ2, so that the compatibility of the second of

the (1) is already reached : δ₂ =A_iTx .

The components of the vector δ₂ ≠0 give the position and width of the cracks located in the mortar joints.

3. A FIRST NUMERICAL APPLICATIONS

The numerical procedure described in the previous paragraph can been applied to the analysis of two types of axial-symmetrical masonry domes provided of thickness, respectively hemispherical ad ogival, subject to self weight loads, with three different conditions at the crown: complete, with hole and lantern. In any case, with assigned radius R, the thickness s has been considered as a constant. According to axial symmetry of the structure the internal shear forces acting along the rings and on the planes of the meridians become zero. The stiffness in the contact devices, which simulates the behaviour of the mortar layers, has been considered uniform. The procedure allows us to define the limit configuration of equilibrium and, correspondently, the collapse mechanism that is due to the formation of cracks along the meridian and parallel interface sections. With reference to the limit configuration of equilibrium the minimum thickness of the dome is pointed out and the corresponding ratio Rs / is performed.

Afterwards an interesting comparison is presented between the numerical method described here and two different solutions of the same cases (Table 1).

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F A AK A K X₀ ₌ −1 T( −1 T)−1 ₍₂₎

1 1 1 1 1 ₍ ₎ ₎ ( − ⋅δ = ⋅δ = I K− A AK− A − A C X_N T T (3)

i i

i CX0

1 =−

          = c t X X

X 0 where for each joint we have: X_c <0and

∑

3. A FIRST NUMERICAL APPLICATIONS

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In the previous form (1) X ∈ ℜ

km

_{indicates the}

un-known vector of internal forces located on the interface

joints, where X

_n

_{includes the components orthogonal}

to the contact surface of the joint and X

_t

includes the

shear components; x ∈ ℜ represents the unknown

vector of components of the displacements related to

the centroids of the elements ; K ∈ ℜ

km x km

_{; is the}

diago-nal stiffness matrix of the contact constraints.; δ ∈ ℜ

km

indicates the unknown vector whose components are

internal distortions which need for obtaining a solution

capable of satisfying both the equilibrium equations,

while respecting the sign conditions, and the

elastic-ki-nematical compatibility of the actual reacting structure.

On this subject, it is convenient to distinguish, within

the vector δ , two types of entities, assuming for the

former, related to the equilibrium aspects, the notation

δ

₁

and for the latter, related to the compatibility ones,

the notation δ

₂

. Notice that the value k depends on the

number of contact constraints chosen to characterize

the interface device and states the degree of statically

indeterminacy of the structure.

Of course the system of equations (1) could have no

solution under the sign conditions expressed in the first

inequality; in such a case it means that the structure

cannot be equilibrated under the given system of the

external actions. In this case there is no vector X ∈ ℜ

km

which simultaneously satisfies, the 6n equations and the

km inequalities.

However let us suppose that the system (1) is consistent.

In such a case, the general solution

X = X

₀

+ X

_N

, that is

able to satisfy the equilibrium problem and the first of

F A AK A K X₀ ₌ −1 T( −1 T)−1 ₍₂₎

1 1 1 1 1 ₍ ₎ ₎ ( − ⋅δ = ⋅δ = I K− A AK− A − A C X_N T T (3)

i i

i C X0

1 =−

          = c t X X

∑

the (1) is already reached : δ₂ = A_iTx .

3. A FIRST NUMERICAL APPLICATIONS

Such an initial solution is then modified through the

vector:

F A AK A K X₀ −1 T( −1 T)−1 = (2)

1 1 1 1 1 ₍ ₎ ₎ ( − ⋅δ = ⋅δ = I K− A AK− A − A C X_N T T (3)

i i

i C X0

1 =−

          = c t XX

∑

= ⋅ ≤ k j nj t f X X 1 (4) Since the final vector X is different from the first elastic vector solution X₀, it cannot satisfy the kinematical compatibility expressed through the second set of equations in the system (1). A very easy way to build up again such a compatibility is to consider the second set of equations in the system (1) in the form: c c c T c cA A K X A x₌₋( )−1 (5) which represents the vector of the displacements of the centroids of the elements only due to the actual reacting structure. Finally we can determine the vector δ2, so that the compatibility of the second of

3. A FIRST NUMERICAL APPLICATIONS

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which, added to X

₀

, satisfies the first equation of the

system (1) while respecting the sign conditions.

Com-puting the Moore-Penrose generalized inverse of C

_i

, it

is easily possible to evaluate the vector δ

_1i

= −C

_i

X

_0i

,

where X

_0i

are the components of the interactions that,

in the joint, do not respect the inequalities of (1). If

the solution of the unilateral problem exists, the vector

solution, which satisfies simultaneously the equilibrium

equations and the first and the second inequality of (1),

assumes the form:

F A AK A K X₀ −1 T( −1 T)−1 = (2)

1 1 1 1 1 ₍ ₎ ₎ ( − ⋅δ = ⋅δ = I K− A AK− A − A C X_N T T (3)

i i

i C X0

1 =−

          = c t XX

∑

= ⋅ ≤ k j nj t f X X 1 (4) Since the final vector X is different from the first elastic vector solution X₀, it cannot satisfy the kinematical compatibility expressed through the second set of equations in the system (1). A very easy way to build up again such a compatibility is to consider the second set of equations in the system (1) in the form: c c c T c cA A K X A x₌₋( )−1 (5) which represents the vector of the displacements of the centroids of the elements only due to the actual reacting structure. Finally we can determine the vector δ2, so that the compatibility of the second of

the (1) is already reached : δ₂ = A_iTx .

3. A FIRST NUMERICAL APPLICATIONS

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where for each joint:

Since the final vector X is different from the first

elas-tic vector solution X

₀

, it cannot satisfy the kinematical

compatibility expressed through the second set of

equa-tions in the system (1). A very easy way to again build

up such a compatibility is to consider the second set of

equations in system (1) in the form:

which represents the vector of the displacements of the

centroids of the elements only relative to the actual

re-acting structure. Finally, one can determine the vector

δ

₂

, in such a way that the compatibility of the second of

(4)

Tempesta, Paradiso, Galassi, Pieroni

88 The components of the vector δ

₂

≠ 0 give the position

and width of the cracks located in the mortar joints.

First numerical applications

The numerical procedure described in the previous

paragraph can be applied to the analysis of two types

of axial-symmetrical masonry domes, respectively

hemi-spherical and ogival, subject to self weight loads, with

three different conditions at the crown: complete, with

hole and lantern. In any case, with assigned radius R,

the thickness s has been considered as a constant.

Ac-cording to the axial symmetry of the structure the

inter-nal shear forces acting along the rings and on the planes

of the meridians become zero. The stiffness in the

con-tact devices, which simulates the behaviour of the

mor-tar layers, has been considered uniform. The procedure

allows one to define the limit configuration of

equilibri-um and, correspondently, the collapse mechanism that

is due to the formation of cracks along the meridian and

parallel interface sections. With reference to the limit

configuration of equilibrium the minimum thickness of

the dome is identified and the corresponding ratio s / R

is performed.

Afterwards an interesting comparison is presented

be-tween the numerical method described here and two

different solutions of the same cases (Table 1).

The first solution concerns Heyman’s hypothesis, also

used by Hoppenheim, based on the assumption that the

resultant hoop stress between contiguous lunes of the

dome is zero. This means that one considers the dome

as a one-dimensional behaviour structure, to be seen as

Table 1: Comparison between different approaches.

Heyman’s hypothesis

 _{limit thickness} Method presented here  _{limit thickness}  _{last compressive ring}

complete  _{0.044 R}  0.043 R  ϕ = 29.59°

Hemispherical dome

hole and lantern  _{0.050 R}  0.044 R  ϕ = 24.00°

complete  _{0.085 R} _ 0.050 R ϕ = 48.00°

Ogival dome

hole and lantern  0.032 R _ 0.022 R ϕ = 40.00°

The first solution is concerned with the Heyman’s hypothesis, used by Hoppenheim too, based on the assumption that the hoop stress resultant between contiguous lunes of the dome is zero. This means to consider the dome as a one-dimensional behaviour structure, to be seen as composed of a series of distinct segments or slices or “lunes”, wider at the base and tapering to zero at the crown, placed in mutual contact with each other but without any interactions among them: of course such a solution must be understood as limit solution.

4. BRUNELLESCHI’S DOME IN FLORENCE

The numerical procedure described above has been applied to the analysis of the Dome of S. Maria del Fiore. The first aspect to tackle was that of the definition of the overall geometrical model of the structure. The Dome, formed from two shells divided by a space of about 1.2 m, is made up of eight lunes that represent segments of an elliptical cylinder. For better understand the spatial location of the volume of the building see the figures shown below (fig 3) .

Figure 3 - Axonometric view of the general geometry of the segment of dome relating to the elliptical cylinder. The geometrical model adopted for the analysis is based the studies, published in 1977, by Salvatore Di Pasquale, for the contents of which, refer for brevity to the specific bibliography. The principal aspect is that every laying bed of bricks is lying on a conic surface, easily obtaining by the intersection between the cylindrical segment of the dome, with elliptical directrix, and the surface of ideal cone, defined by rotating e generatrix line with a variable angle and a vertex, whose level moves progressively upwards, located on the axis of the dome (fig. 4 - 5).

Tab. 1: Comparison between different approaches.

composed of a series of distinct segments or slices or

“lunes”, wider at the base and tapering to zero at the

crown, placed in mutual contact with each other but

without any interactions among them: of course, such a

solution must be considered as limit solution.

Brunelleschi’s dome in Florence

The numerical procedure described above has been

ap-plied to the analysis of the Dome of Santa Maria del

Fiore. The first aspect to tackle was to define the overall

geometrical model of the structure. The Dome, formed

by two shells divided by a space of approximately 1.2

m, is made up of eight lunes that represent segments of

an elliptical cylinder. To better understand the spatial

location of the volume of the building see figure 3.

 _{limit thickness}  _{limit thickness}  _{last compressive ring}

complete  0.044 R  0.043 R  ϕ = 29.59° Hemispherical dome

hole and lantern  _{0.050 R}  0.044 R  ϕ = 24.00°

complete  _{0.085 R}  0.050 R  ϕ = 48.00° Ogival dome

hole and lantern  0.032 R _ 0.022 R _{ϕ = 40.00°}

4. BRUNELLESCHI’S DOME IN FLORENCE

Figure 3 - Axonometric view of the general geometry of the segment of dome relating to the elliptical cylinder.

The geometrical model adopted for the analysis is based the studies, published in 1977, by Salvatore Di Pasquale, for the contents of which, refer for brevity to the specific bibliography. The principal aspect is that every laying bed of bricks is lying on a conic surface, easily obtaining by the intersection between the cylindrical segment of the dome, with elliptical directrix, and the surface of ideal cone, defined by rotating e generatrix line with a variable angle and a vertex, whose level moves progressively upwards, located on the axis of the dome (fig. 4 - 5).

Table 1: Comparison between different approaches.

Heyman’s hypothesis

 limit thickness Method presented here  limit thickness

 last compressive ring complete  0.044 R _ 0.043 R _{ϕ = 29.59°}

Hemispherical dome

hole and lantern  0.050 R _ 0.044 R _{ϕ = 24.00°}

complete  0.085 R _ 0.050 R _{ϕ = 48.00°}

Ogival dome _{hole and lantern} _

0.032 R _ 0.022 R _{ϕ = 40.00°}

4. BRUNELLESCHI’S DOME IN FLORENCE

Figure 3 - Axonometric view of the general geometry of the segment of dome relating to the elliptical cylinder. The geometrical model adopted for the analysis is based the studies, published in 1977, by Salvatore Di Pasquale, for the contents of which, refer for brevity to the specific bibliography. The principal aspect is that every laying bed of bricks is lying on a conic surface, easily obtaining by the intersection between the cylindrical segment of the dome, with elliptical directrix, and the surface of ideal cone, defined by rotating e generatrix line with a variable angle and a vertex, whose level moves progressively upwards, located on the axis of the dome (fig. 4 - 5).

Fig. 3 Axonometric view of the general geometry of the segment of dome relating to the elliptical cylinder.

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₈₉

The geometrical model adopted for the analysis is

based on the studies, published in 1977, by Salvatore

Di Pasquale; for the contents refer to the bibliography.

The principal aspect is that every laying bed of the bricks

is lying on a conic surface, and is easily indentified by

the intersection between the cylindrical segment of the

dome, with elliptical directrix, and the surface of ideal

cones, defined by rotating a generatrix line with a

vari-able angle and a vertex, whose level moves progressively

upwards, located on the axis of the dome (fig. 4 - 5).

Figure 4 - Brunelleschi’s Dome. Geometry and structure.

Figure 4 - Brunelleschi’s Dome. Geometry and structure. Figure 4 - Brunelleschi’s Dome. Geometry and structure.

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Figure 5 - Axonometric view of geometrical model of conic laying bed of the bricks.

Below are some mathematical notations for the analytic reading of the problem:

a

= 1

C = radius of curvature of the big corner rib (

₁

36

5

4 ₌

=

D

a

m )

ϖ

cos

⋅

=

a

b

(where

8 π

ϖ

=

)

β

sen

a

h

=

⋅

(

z of P on the big corner rib – with

0 ≤

β

≤

62 °

)

β

tg

a

k

_{= 8}

3 ⋅

( z of the vertex V of the cone)

)

8

3 (cos −

=

a

β

r

( radius of the circle at the base of the generic cone)

Equation of the cone (

x ,

,

y

z

) :

(

)

(

)

2

0

2 2 2 2 2

=

−

+

k

h

k

z

r

y

r

x

₍₆₎

Let us define a orthogonal plane ( y

x, ) through the axis z as a function of

ϕ

(

8

8 π

β

π

≤

−

)

ϕ

tg

x

y

=

⋅

(7)

Substituting in (6) the equation above becomes:

(

)

(

)

2

0

2 2 2 2 2 2

=

−

⋅

+

k

h

k

z

r

tg

x

ϕ

r

The equation of the elliptical cylinder is:

2 2 2 2

8

3 a

c

ac

x

z

=













−

+

with

c

=

cos

ϖ

(8)

5. NUMERICAL REMARKS

The structure of the Dome has been analyzed using the numerical model presented above. The

numerical tests have focused on the general stability of the Monument despite the presence of

inevitable and manifest cracks along the meridians in the middle of the lune and on the big corner ribs.

The numerical results, obtained under no-tension hypothesis for the material, confirmed in any case,

the actual behaviour of the structure.

Figure 5 - Axonometric view of geometrical model of conic laying bed of the bricks.

Below are some mathematical notations for the analytic reading of the problem:

a

= 1

C = radius of curvature of the big corner rib (

₁

36

5

4 ₌

=

D

a

m )

ϖ

cos

⋅

=

a

b

(where

8 π

ϖ

=

)

β

sen

a

h

=

⋅

(

z

of P on the big corner rib – with

0 ≤

β

≤

62 °

)

β

tg

a

k

_{= 8}

3 ⋅

( z of the vertex V of the cone)

)

8

3 (cos −

=

a

β

r

( radius of the circle at the base of the generic cone)

Equation of the cone (

x ,

,

y

z

) :

(

)

(

)

2

0

2 2 2 2 2

=

−

+

k

h

k

z

r

y

r

x

₍₆₎

Let us define a orthogonal plane ( y

x, ) through the axis z as a function of

ϕ

(

8

8 β

π

_≤

−

)

ϕ

tg

x

y

=

⋅

(7)

Substituting in (6) the equation above becomes:

(

)

(

)

2

0

2 2 2 2 2 2

=

−

⋅

+

k

h

k

z

r

tg

x

ϕ

r

The equation of the elliptical cylinder is:

2 2 2 2

8

3 a

c

ac

x

z

=













−

+

with

c

=

cos

ϖ

(8)

5. NUMERICAL REMARKS

The structure of the Dome has been analyzed using the numerical model presented above. The

numerical tests have focused on the general stability of the Monument despite the presence of

inevitable and manifest cracks along the meridians in the middle of the lune and on the big corner ribs.

The numerical results, obtained under no-tension hypothesis for the material, confirmed in any case,

the actual behaviour of the structure.

Fig. 5. Axonometric view of geometrical model of conic laying bed of the bricks. Figure 5 - Axonometric view of geometrical model of conic laying bed of the bricks.

Below are some mathematical notations for the analytic reading of the problem:

a

= 1

C = radius of curvature of the big corner rib (

₁

36

5

4 ₌

=

D

a

m )

ϖ

cos

⋅

=

a

b

(where

8 π

ϖ

=

)

β

sen

a

h

=

⋅

(

z of P on the big corner rib – with

0 ≤

β

≤

62 °

)

β

tg

a

k

_{= 8}

3 ⋅

( z of the vertex V of the cone)

)

8

3 (cos −

=

a

β

r

( radius of the circle at the base of the generic cone)

Equation of the cone (

x ,

,

y

z

) :

(

)

(

)

2

0

2 2 2 2 2

=

−

+

k

h

k

z

r

y

r

x

₍₆₎

Let us define a orthogonal plane ( y

x, ) through the axis z as a function of

ϕ

(

8

8 β

π

≤

−

)

ϕ

tg

x

y

=

⋅

(7)

Substituting in (6) the equation above becomes:

(

)

(

)

2

0

2 2 2 2 2 2

=

−

⋅

+

k

h

k

z

r

tg

x

ϕ

r

The equation of the elliptical cylinder is:

2 2 2 2

8

3 a

c

ac

x

z

=













−

+

with

c

=

cos

ϖ

(8)

5. NUMERICAL REMARKS

The structure of the Dome has been analyzed using the numerical model presented above. The

numerical tests have focused on the general stability of the Monument despite the presence of

inevitable and manifest cracks along the meridians in the middle of the lune and on the big corner ribs.

The numerical results, obtained under no-tension hypothesis for the material, confirmed in any case,

the actual behaviour of the structure.

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₉₁

Numerical remarks

The structure of the Dome has been analyzed using the

numerical model presented above. The numerical tests

have focused on the general stability of the Monument

despite the presence of inevitable and manifest cracks

along the meridians in the middle of the lune and on

the big corner ribs (Figures 7 and 8). The numerical

re-sults, obtained under the no-tension hypothesis for the

material, confirmed in every case, the actual behaviour

of the structure (Table 2).

In Figure 6 a series of synthesized images of the various

models tested is shown. In the construction of the

geo-metrical shape the conical laying beds of the bricks has

been taken into account: this has influenced the mesh of

the finite element model.

Below are shown a series of synthesized images of the various models tested. In the construction of the geometrical shape has taken into account of the conical laying beds of the bricks: this has influenced the mesh of the finite element model.

Figure 6 - Views of geometrical model and mesh.

Figure 7 – Axonometric view. Location and spreading of cracks. Below are shown a series of synthesized images of the various models tested. In the construction of

the geometrical shape has taken into account of the conical laying beds of the bricks: this has influenced the mesh of the finite element model.

Figure 7 – Axonometric view. Location and spreading of cracks.

Below are shown a series of synthesized images of the various models tested. In the construction of the geometrical shape has taken into account of the conical laying beds of the bricks: this has influenced the mesh of the finite element model.

Figure 7 – Axonometric view. Location and spreading of cracks. Fig. 6 . Views of the geometrical model and mesh.

Fig.7. Axonometric view. Location and spreading of cracks.

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Figure 8 - Location and spreading of cracks. Cross section on corner big rib (left) and on the middle meridian of

the lune (right).

Following Di Pasquale’s idea, a significant test was performed by modeling the structure in order to obtain, within the actual structure, a circular dome that, while keeping the ribs, has a possible thickness of only 48 cm (fig. 9). The Dome is still stable also in this case (fig. 10)

Table 2 : Values of compressive stress at the base of the Dome.

Maximum compressive stress (MPa) at the base of the Dome

Corner big rib Medium rib Middle of the lune

0.79 0.78 0.47

Figure 9 – Ideal circular dome within the actual thickness.

Fig. 8. Location and spreading of cracks. Cross section on corner big rib (left) and on the middle meridian of the lune (right).

Figure 8 - Location and spreading of cracks. Cross section on corner big rib (left) and on the middle meridian of

the lune (right).

Following Di Pasquale’s idea, a significant test was performed by modeling the structure in order to

obtain, within the actual structure, a circular dome that, while keeping the ribs, has a possible

thickness of only 48 cm (fig. 9). The Dome is still stable also in this case (fig. 10)

Table 2 : Values of compressive stress at the base of the Dome.

Maximum compressive stress (MPa) at the base of the Dome

Corner big rib

Medium rib

Middle of the lune

0.79

0.78

0.47 Figure 9 – Ideal circular dome within the actual thickness.

Table 2: Values of compressive stress at the base of the Dome.

Figure 8 - Location and spreading of cracks. Cross section on corner big rib (left) and on the middle meridian of

the lune (right).

Following Di Pasquale’s idea, a significant test was performed by modeling the structure in order to

obtain, within the actual structure, a circular dome that, while keeping the ribs, has a possible

thickness of only 48 cm (fig. 9). The Dome is still stable also in this case (fig. 10)

Table 2 : Values of compressive stress at the base of the Dome.

Maximum compressive stress (MPa) at the base of the Dome

Corner big rib

Medium rib

Middle of the lune

0.79

0.78

0.47

(9)

₉₃

Following the idea of Di Pasquale, a significant test was

performed by modeling the structure in order to obtain,

within the actual structure, a circular dome that, while

keeping the ribs, has a possible thickness of only 48 cm

(fig. 9). The Dome is still stable also in this case (fig. 10)

References

Bouguer, P. (1734). Sur les lignes courbes qui sont propres à former les voûtes en dôme, Memoires. Academiae Royale des Sciences, 149-166.

Poleni, G., (1734). Memorie istoriche della Gran Cupola del tem-pio Vaticano, Padova.

Le Seur, T., Jacquier, F., and Boscovich, R.G., (1742). Parere di Tre Matematici Sopra i danni, che si sono trovati nella cupola di San Pietro sul fine dell’anno MDCCXLII. Dato per ordine di Nos-tro Signore Papa Benedetto XIV.

Bossut, C. (1778). Nouvelles Recherches sur l’equilibre des voûtes en dôme, Memoires. Academiae Royale des Sciences, 587-596.

Figure 10 – Axonometric view. Location and spreading of cracks.

REFERENCES

Bouguer, P. (1734). Sur les lignes courbes qui sont propres à former les voûtes en dôme, Memoires. Academiae Royale des Sciences, 149-166.

Poleni, G., (1734). Memorie istoriche della Gran Cupola del tempio Vaticano, Padova.

Le Seur, T., Jacquier, F., and Boscovich, R., G., (1742). Parere di Tre Matematici Sopra i danni, che

si sono trovati nella cupola di San Pietro sul fine dell'anno MDCCXLII. Dato per ordine di Nostro Signore Papa Benedetto XIV.

Bossut, C. (1778). Nouvelles Recherches sur l'equilibre des voûtes en dôme, Memoires. Academiae Royale des Sciences, 587-596.

Mascheroni, L. (1785). Nuove ricerche sull'equilibrio delle volte, Bergamo. Venturoli, G. (1883). Elementi di meccanica, Napoli, (I ed. 1806).

Heyman, J., (1967). “On shell solution for masonry domes”, International Journal of Sol.ids and

Structures, 3, 227-241.

Oppenheim, I.,J., Gunaratnam, D.,J., and Allen, R.,H., (1989). “Limit state analysis of masonry domes”, ASCE Journal of Structural Engineering, 115, 868-882.

Di Pasquale, S., (2002). Brunelleschi. La costruzione della Cupola di Santa Maria del Fiore. Marsilio Editore, Venezia.

Fig. 10. Axonometric view. Location and spreading of cracks.

Mascheroni, L. (1785). Nuove ricerche sull’equilibrio delle volte, Ber-gamo.

Venturoli, G. (1883). Elementi di meccanica, Napoli, (I ed. 1806). Heyman, J., (1967). “On shell solution for masonry domes”, International Journal of Solids and Structures, 3, 227-241. Oppenheim, I.J., Gunaratnam, D.J., and Allen, R.H., (1989). “Limit state analysis of masonry domes”, ASCE Journal of Struc-tural Engineering, 115, 868-882.

Di Pasquale, S., (2002). Brunelleschi. La costruzione della Cupo-la di Santa Maria del Fiore. Marsilio Editore, Venezia.

Paradiso M., and Tempesta, G., (2003). “Limit Analysis of Ma-sonry Domes. A Numerical Method”, XVI Congresso AIMETA di Meccanica Teorica e Applicata, Ferrara.

Paradiso, M., Pieroni, E., and Tempesta, G., (2006). “Masonry domes. topical nineteenth century graphical solutions and new approaches, Wondermasonry Workshop on Design for Rehabilita-tion of Masonry Structures, Modellazione e Progetto di Interventi sul Costruito in Muratura, Firenze.

Rowland J. Mainstone, R., J., (1977). “Le origini strutturali del-la cupodel-la di Santa Maria del Fiore”, Atti del Convegno Brunelles-chiano, Firenze.

Corazzi, R., Conti, G., (2011). Il segreto della Cupola del Brunel-leschi a Firenze, Angelo Pontecorboli Editore, Firenze.

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