Brunelleschi’s Dome: A New Estimate of the Thrust and Stresses in the Underlying Piers

: The paper deals with the insurgence of the thrust, together with its valuation, in masonry domes, giving special attention to the Brunelleschi’s Dome in Florence. After a recalling of the kinematical approach in the context of the Heyman masonry model, the thrust of Brunelleschi’s Dome is estimated as the maximum of the set of all the kinematical ones. Comparisons are made with other valuations made by the usual, but less accurate, statical approach. The knowledge of the thrust allows an evaluation of the stresses acting in the supporting piers: their base sections are all compressed, with level stresses sufﬁciently low. This result shows the extraordinary conception of Filippo Brunelleschi’s Dome and the favorable design of the pillar sections and of the drum positioning, due, perhaps, to Arnolfo di Cambio or to the succeeding Masters.


Introduction
This article aims to study the statical behavior of Brunelleschi's Dome in Florence which, for its significance, since 1982 has been one of the UNESCO World Heritage sites. The research is developed by means of the limit analysis approach according to the simple no tension Heyman model [1] of the masonry material and in the context of the consequent mathematical formulation given in [2,3]. The first aim of this article is the thrust estimate of the Dome, required to assess the actual state of safety of the monument. This subject is strongly debated. Many studies have been performed by using numerical approaches [4][5][6][7][8][9][10][11][12] that require the use of models involving complex convergence analysis and heavy computational problems [5].
An alternative to these complex numerical approaches is the assessment of statics of the masonry dome according to the limit analysis, founded on the masonry Heyman no tension model [13]. In spite of the simplicity of this last approach, all the main aspects of the masonry dome's behavior are taken into account: the dome is divided into slices, in fact, with the development of meridian cracks due to the masonry weakness to sustain hoop stresses; the dome thrust takes place and radial settlement displacements occur in the drum; the dome adapts itself to the settlement and deforms with a mechanism activating the minimum thrust.
This complex picture of statical events occurring in masonry domes, with their passage from the initial membrane state of stress to that of the cracked behavior, is fully interpreted by the limit analysis, particularly in the context of the kinematical approach that directly employs the deformation modes and the mechanisms, in order to define, by means of the virtual work equation, the equilibrium of no tension structures.
The novelty of the paper lies in the application of the kinematical approach that directly pursues the insurgence of the thrust following the occurrence of the meridian cracking and the accompanying settlement mechanism of the Dome. In this context, the thrust is assessed by checking the maximum of all the kinematical thrusts, according to previous results obtained by the author [14,15]. This approach is not well known and rarely applied, At the cracking occurrence, the dome starts to widen and the circular courses have to slip one over the other. The friction strength produced by meridian compression Nφ opposes this slipping. Thus, the masonry's tensile strength is actually increased by the meridian compression. Humidity, which generally penetrates slowly into masonry, weakens the mortar and reduces this friction strength over time. These considerations explain why cracks generally occur only many years after the end of a dome's construction. The history of cracking of many famous vaults and domes bears this out. For example, both the domes of St Peter's in Rome and Santa Maria del Fiore in Florence began to crack at least fifty years, or more, after their completion [16] (p. 209).
The dome cracks when the tension stress in the hoop rings near the springing reach the limit value of the masonry's weak tensile strength. The initial membrane equilibrium is thereby lost and meridian cracks spread up the dome along a band much higher than that subjected to the hoop stresses in the initial membrane equilibrium. The cracked dome tends to open wide and breaks up into slices that behave as independent pairs of semiarches leaning on each other ( Figure 2). Cracking brings about a profound change in the dome's statics. The hoop stresses in the cracked zone vanish and stresses Nφ, before acting along the slices' center lines, are no longer able to ensure equilibrium. The pressure surface thus tilts towards the horizontal and deviates away from the central surface of the dome. Only a small compressed cap at At the cracking occurrence, the dome starts to widen and the circular courses have to slip one over the other. The friction strength produced by meridian compression N ϕ opposes this slipping. Thus, the masonry's tensile strength is actually increased by the meridian compression. Humidity, which generally penetrates slowly into masonry, weakens the mortar and reduces this friction strength over time. These considerations explain why cracks generally occur only many years after the end of a dome's construction. The history of cracking of many famous vaults and domes bears this out. For example, both the domes of St Peter's in Rome and Santa Maria del Fiore in Florence began to crack at least fifty years, or more, after their completion [16] (p. 209).
The dome cracks when the tension stress in the hoop rings near the springing reach the limit value of the masonry's weak tensile strength. The initial membrane equilibrium is thereby lost and meridian cracks spread up the dome along a band much higher than that subjected to the hoop stresses in the initial membrane equilibrium. The cracked dome tends to open wide and breaks up into slices that behave as independent pairs of semi-arches leaning on each other ( Figure 2).
The detected crack pattern is that of a revolution dome, due to Brunelleschi's circular arrangement of the brick courses, so all the eight webs transmit their thrusts equally to the underlying drum.

Reasons of the Typical Cracking in Masonry Domes
Cracking is generally delayed in masonry domes. Cracks occur in a masonry arches as soon as stresses exceed the weak adhesion strength between the stones and the mortar. Compressive forces and shears continue to spread across the hinges that can form in the sections; the friction cannot oppose the development of cracks. On the contrary, in masonry domes, the friction increases their tensile strength, as Figure 1 shows. At the cracking occurrence, the dome starts to widen and the circular courses have to slip one over the other. The friction strength produced by meridian compression Nφ opposes this slipping. Thus, the masonry's tensile strength is actually increased by the meridian compression. Humidity, which generally penetrates slowly into masonry, weakens the mortar and reduces this friction strength over time. These considerations explain why cracks generally occur only many years after the end of a dome's construction. The history of cracking of many famous vaults and domes bears this out. For example, both the domes of St Peter's in Rome and Santa Maria del Fiore in Florence began to crack at least fifty years, or more, after their completion [16] (p. 209).
The dome cracks when the tension stress in the hoop rings near the springing reach the limit value of the masonry's weak tensile strength. The initial membrane equilibrium is thereby lost and meridian cracks spread up the dome along a band much higher than that subjected to the hoop stresses in the initial membrane equilibrium. The cracked dome tends to open wide and breaks up into slices that behave as independent pairs of semiarches leaning on each other ( Figure 2). Cracking brings about a profound change in the dome's statics. The hoop stresses in the cracked zone vanish and stresses Nφ, before acting along the slices' center lines, are no longer able to ensure equilibrium. The pressure surface thus tilts towards the horizontal and deviates away from the central surface of the dome. Only a small compressed cap at Cracking brings about a profound change in the dome's statics. The hoop stresses in the cracked zone vanish and stresses N ϕ , before acting along the slices' center lines, are no longer able to ensure equilibrium. The pressure surface thus tilts towards the horizontal and deviates away from the central surface of the dome. Only a small compressed cap at the top of each slice will be subjected to the thrusting action transmitted, by the other slices, all the way to the springers.
When cracks are barely perceptible, only elastic unloading of the dome rings has occurred. In the presence of numerous broader cracks, on the other hand, the widening of the drum will be the main factor responsible for the dome's deformation. Loaded by the dome's thrust, the drum deforms and slightly splays. Meridian cracks in the dome thus continue in the drum. A small widening in the dome span occurs. The dome slices adapt themselves to this deformation, freely following the spreading of the dome. They develop a settlement mechanism. This picture describes the occurrence of cracking as it happened in Brunelleschi's Dome.

Some Brief Examinations of the Statics of Masonry Structures
Many studies use advanced outlines of the geometry of the Dome but use questionable models of the material. The linear elastic model sometimes is assumed and the actual crack patterns are introduced in the initial geometry of the Dome without following the gradual crack propagation. Other studies use a constitutive model taken from the study of other materials, particularly of concrete.
In contrast to the large variety of approaches that, with different assumptions, try to analyze the statics of the Dome [4][5][6][7][8][9][10][11][12], there are the thrust assessments that move in the context of the limit analysis, founded on the masonry Heyman no tension model. We now recall the basic simple principles of the statics of masonry constructions, limited to the simpler case of unidimensional structures, to which the sliced cracked domes revert. The following constitutive assumptions were formulated by Heyman [1]: i. masonry is incapable of withstanding tensions ii. masonry has infinite compressive strength iii. elastic strains are negligible iv. sliding cannot occur because masonry has infinite shear strength In this framework, the single resistant cell of a masonry structural element, as, for example, the slices of a dome, is composed of two ideal rigid bricks, of height h, loaded by eccentric axial load and shear, as shown in Figure 3.
the top of each slice will be subjected to the thrusting action transmitted, by the other slices, all the way to the springers.
When cracks are barely perceptible, only elastic unloading of the dome rings has occurred. In the presence of numerous broader cracks, on the other hand, the widening of the drum will be the main factor responsible for the dome's deformation. Loaded by the dome's thrust, the drum deforms and slightly splays. Meridian cracks in the dome thus continue in the drum. A small widening in the dome span occurs. The dome slices adapt themselves to this deformation, freely following the spreading of the dome. They develop a settlement mechanism. This picture describes the occurrence of cracking as it happened in Brunelleschi's Dome.

Some Brief Examinations of the Statics of Masonry Structures
Many studies use advanced outlines of the geometry of the Dome but use questionable models of the material. The linear elastic model sometimes is assumed and the actual crack patterns are introduced in the initial geometry of the Dome without following the gradual crack propagation. Other studies use a constitutive model taken from the study of other materials, particularly of concrete.
In contrast to the large variety of approaches that, with different assumptions, try to analyze the statics of the Dome [4][5][6][7][8][9][10][11][12], there are the thrust assessments that move in the context of the limit analysis, founded on the masonry Heyman no tension model. We now recall the basic simple principles of the statics of masonry constructions, limited to the simpler case of unidimensional structures, to which the sliced cracked domes revert. The following constitutive assumptions were formulated by Heyman [1]: i. masonry is incapable of withstanding tensions ii. masonry has infinite compressive strength iii. elastic strains are negligible iv. sliding cannot occur because masonry has infinite shear strength In this framework, the single resistant cell of a masonry structural element, as, for example, the slices of a dome, is composed of two ideal rigid bricks, of height h, loaded by eccentric axial load and shear, as shown in Figure 3.
will respect the inequality Assumption iv prohibits sliding. Consequently, the shear force T is involved only in the equilibrium between the two ideal bricks. Any strain of the unit resistant cell will be either zero or the detachment strain will respect the inequality Assumption iv prohibits sliding. Consequently, the shear force T is involved only in the equilibrium between the two ideal bricks. Any strain of the unit resistant cell will be either zero or the detachment strain where the third strain component is zero because, according to assumption iv, no sliding can occur. Admissible stress and strain vectors are shown in Figure 4.
where the third strain component is zero because, according to assumption iv, no sliding can occur. Admissible stress and strain vectors are shown in Figure 4. The set of all allowable stresses and strains Σ and E are, respectively, denoted by V and Y. The direct consequence of the pressing and expanding features of Σ and E is the following fundamental inequality [2,3] A hinge in the ideal cell takes place when the strains sketched in Figure 4 occur. When, in a masonry structure, a sufficient number of hinges occurs, the arch, or any other structure, can become deformed according to a displacement field u, called the mechanism. The set of all the mechanisms is denoted by M.
The admissible equilibrium of masonry structures is governed by the principle of virtual work that takes, for unidimensional masonry elements, the typical form expressing the equality between internal and external virtual work [16] (pp. 79-82) , coupled to the admissibility condition on the stresses Σ and the virtual strains δE(δu) , 0

Settlement State. Allowable Equilibrium
Let us consider a masonry structure, for instance, an arch or a dome, under the loads g, representative of the weight. The structure is in equilibrium at the configuration Ci under the loads g and the corresponding internal stresses Σi. A slight settlement of the external constraints of the structure now takes place, as shown, in the case of the arch, in Figure 5. The arch changes its pressure line and adapts itself to the settlement deformation with a mechanism and follows the displacements of its springers. Let Cs be the The set of all allowable stresses and strains Σ and E are, respectively, denoted by V and Y. The direct consequence of the pressing and expanding features of Σ and E is the following fundamental inequality [2,3] A hinge in the ideal cell takes place when the strains sketched in Figure 4 occur. When, in a masonry structure, a sufficient number of hinges occurs, the arch, or any other structure, can become deformed according to a displacement field u, called the mechanism. The set of all the mechanisms is denoted by M.
The admissible equilibrium of masonry structures is governed by the principle of virtual work that takes, for unidimensional masonry elements, the typical form expressing the equality between internal and external virtual work [16] (pp. 79-82) coupled to the admissibility condition on the stresses Σ and the virtual strains δE(δu)

Settlement State. Allowable Equilibrium
Let us consider a masonry structure, for instance, an arch or a dome, under the loads g, representative of the weight. The structure is in equilibrium at the configuration C i under the loads g and the corresponding internal stresses Σ i . A slight settlement of the external constraints of the structure now takes place, as shown, in the case of the arch, in Figure 5.
where the third strain component is zero because, according to assumption iv, no sliding can occur. Admissible stress and strain vectors are shown in Figure 4. The set of all allowable stresses and strains Σ and E are, respectively, denoted by V and Y. The direct consequence of the pressing and expanding features of Σ and E is the following fundamental inequality [2,3] A hinge in the ideal cell takes place when the strains sketched in Figure 4 occur. When, in a masonry structure, a sufficient number of hinges occurs, the arch, or any other structure, can become deformed according to a displacement field u, called the mechanism. The set of all the mechanisms is denoted by M.
The admissible equilibrium of masonry structures is governed by the principle of virtual work that takes, for unidimensional masonry elements, the typical form expressing the equality between internal and external virtual work [16] (pp. 79-82) , coupled to the admissibility condition on the stresses Σ and the virtual strains δE(δu) , 0

Settlement State. Allowable Equilibrium
Let us consider a masonry structure, for instance, an arch or a dome, under the loads g, representative of the weight. The structure is in equilibrium at the configuration Ci under the loads g and the corresponding internal stresses Σi. A slight settlement of the external constraints of the structure now takes place, as shown, in the case of the arch, in Figure 5. The arch changes its pressure line and adapts itself to the settlement deformation with a mechanism and follows the displacements of its springers. Let Cs be the The arch changes its pressure line and adapts itself to the settlement deformation with a mechanism and follows the displacements of its springers. Let C s be the configuration of the arch after the settlement. The displacement field that moves the arch from C i to C s is the settlement mechanism v s .
Changes of internal stresses occur in the change from C i to C s . The initial stresses Σ i , allowable and in equilibrium with the loads g, change into Σ s For instance, in the case of the settled arch, the pressure line changes from π i to π s . Consequently, the horizontal component of the reaction of the settled constraint changes from µ i r to µ s r.
We can release the structure, removing the settled constraint, by applying the corresponding reaction of the canceled constraint ( Figure 6). Making reference to the released structure, let M* be the set of all the mechanisms of the structure with the removed constraint.
The structure follows the settlement displacement without any opposition. In the case of the above settled arch, the pressure curve π s will pass through the hinges of the settlement mechanism v s . No work of the internal stresses Σ S will be made for the settlement strains. Thus, at the settlement state (8) showing that, in the settled structure, the pressure line passes through the hinges of the settlement mechanism. In the released structure, the external loads are the loads g and the reactions µ s r, if r is the reaction corresponding to a unitary value of the multiplier µ. The released structure is in an allowable equilibrium state. Consequently, the following equation of the principle of virtual works holds: for any mechanism δu of the released structure. Consequently, from (9), taking δu = v S, because of (8), we get The weight g pushes against the settled constraint. These loads g thus make positive work along the settlement mechanism v s .
On the contrary, the thrust µ s r is resisting and, according to (10) and (11), we get Conditions (8) from Ci to Cs is the settlement mechanism vs.
Changes of internal stresses occur in the change from Ci to Cs. The initial stresses Σi, allowable and in equilibrium with the loads g, change into s Σ (7) For instance, in the case of the settled arch, the pressure line changes from πi to πs. Consequently, the horizontal component of the reaction of the settled constraint changes from μir to μsr.
We can release the structure, removing the settled constraint, by applying the corresponding reaction of the canceled constraint ( Figure 6). Making reference to the released structure, let M* be the set of all the mechanisms of the structure with the removed constraint.
The structure follows the settlement displacement without any opposition. In the case of the above settled arch, the pressure curve πs will pass through the hinges of the settlement mechanism vs. No work of the internal stresses ΣS will be made for the settlement strains. Thus, at the settlement state (8) showing that, in the settled structure, the pressure line passes through the hinges of the settlement mechanism. In the released structure, the external loads are the loads g and the reactions μsr, if r is the reaction corresponding to a unitary value of the multiplier μ. The released structure is in an allowable equilibrium state. Consequently, the following equation of the principle of virtual works holds: for any mechanism δu of the released structure. Consequently, from (9), taking δu = vS, because of (8) The weight g pushes against the settled constraint. These loads g thus make positive work along the settlement mechanism vs. , > 0 s < > g v (11) On the contrary, the thrust μsr is resisting and, according to (10) and (11), we get , 0 s s μ < > < r v (12) Conditions (8)-(12) define the settlement state.

Minimum Thrust
Let us consider the released structure, loaded also by the reaction of the settled constraint. Let us choose any stress Σ, statically admissible, i.e., compressive and in equilibrium with the loads. The statically allowable thrust µr(Σ) (13) will correspond to the chosen statically allowable stress Σ. It is well known that the actual thrust in the settled state is the minimum among all the statically admissible thrusts (13): This statement, universally accepted, extends the well-known property of the masonry settled arch to any structure. In the state of minimum thrust, the pressure line of the arch, that takes the maximum slope at the springers, corresponds to an arch having minimum span and maximum sag, as shown by Coulomb [26] and Heyman [27]. The extension of this statement to any structure with a settling constraint was proven in [14,15], transferring these concepts into a mathematical framework.
In this context, the minimum thrust of a dome is generally obtained using the statical approach, by constructing sequences of funicular polygons of the loads and searching among them for the allowable pressure line with the maximum slope at the springers. Alternatively, the minimum thrust, or more generally, the minimum reaction of the settled constraint, can be obtained as follows.

The Kinematical Approach
This approach is founded on the following: Theorem 1. The settlement factor µ s is the maximum of the kinematical multipliers λ(v) for any displacement v varying in the set M* of all the mechanisms [14,15].
Proof. Let us consider in the released structure of any displacement mechanism v ∈ M * that can represent the occurred settlement. Consequently, according to (11), along the chosen displacement < g, v > > 0 Let us consider the kinematical multiplier λ of the reaction r(v) of the settled constraint, satisfying the equilibrium along v < g, v > +λ < r, v>= 0 that is The quantity λr(v) is defined as the kinematical reaction of the settled constraint. Consequently, according to (16) and (17), The so defined thrust is thus associated with the chosen settlement mechanism v. Let us assume δu = v in the virtual Equation (9) expressing the allowable equilibrium in the settlement state between the external loads g and µ s r and the internal stresses Σ s . We get Subtracting (17) from (21) we get and, taking into account (6), Consequently and, taking into account (19), The kinematical multiplier λ(v) is never larger than the actual settlement multiplier settlement µ s , the minimum among all the statically allowable stress states Σ. This µ s is the maximum of the kinematical multipliers λ(v) for any displacement v varying in the set of all the mechanisms M* [14,15]. Thus,

The Minimum Thrust in the Dome Estimated with the Kinematical Approach
The occurrence of thrust slices the masonry dome, due to the development of meridian cracks. The thrust deforms the underlying support structures, firstly, the drum, yielding a slight increase in the dome span. Slices follow this deformation and, as described earlier, mobilize a thrust that is the minimum among all the statically allowable ones. Figure 7 sketches the outline of a generic mechanism v of a dome slice whose base undergoes a slight broadening ∆.
The point O indicates the position at the intrados of the internal hinge. Figure 7 shows all the quantities involved. The settlement mechanism is represented by the outwards horizontal settling ∆(v) associated with the chosen mechanism v. The minimum thrust of the dome is evaluated by applying the kinematic procedure. The unknown thrust, that is, the minimum of all statically allowable thrusts, is obtained [16] (p. 286) as the maximum of all the kinematical thrusts where v is any settlement mechanism. The work done by loads g along the displacement v is given by  if D* and xi indicate, respectively, the horizontal distances of the hinge O and of the force Gi from the vertical alignment T, as shown in Figure 7. Finally, we have where H* is the vertical distance between the internal hinge O and the extrados of the top section, at the joining of the slice with the lantern. The vertical displacements vi of the application points of loads Gi, representing the weights of the various wedges, are then given by if D* and x i indicate, respectively, the horizontal distances of the hinge O and of the force G i from the vertical alignment T, as shown in Figure 7. Finally, we have where H* is the vertical distance between the internal hinge O and the extrados of the top section, at the joining of the slice with the lantern. The vertical displacements v i of the This approach has been applied by the author to evaluate the thrust of St Peter's in Rome [28,29].

Thrust Estimate of Brunelleschi's Dome
The thrust of Brunelleschi's Dome has been evaluated with regard to a Dome slice equal to 1/8 of the entire Dome, referring to the single web supported directly by the underlying pier. The outer and the inner shells forming the Dome are strictly connected by ribs and the horizontal arches, so that the vault, despite its complexity, can be accurately described as behaving like a single solid structure.
Making reference to Figure 8 that shows the octagonal horizontal section of the dome, we notice that the meridian sections of the Dome change with the position of the vertical sectioning planes so that the thrust, corresponding to the various slices, will change along the drum. The thrust, on the unit length of the average circle at Dome springing, will thus exhibit the undulating pattern sketched out in Figure 8, in which the middle sections of the webs correspond to the dotted lines. The computation of the thrust of the dome is made by firstly dividing the slice into 31 voussoirs and determining their weights and the positions of their centers with respect to a definite coordinate system, as shown in Table 1. The unit weight of the masonry voussoirs has been assumed to be equal to 1.85 t/mc. The first voussoir defined by the number 1 corresponds to the voussoir at the crown. The corresponding share of the weight of the lantern is reported in the last row of Table 1.
According to the foregoing calculations, the total weight Gs of the slice, evaluated up to the height of the extrados of the drum and taken as 1/8 of the whole dome, including the share of the lantern, equals 3575 t.
The maximum kinematic thrust Skin(v) is found by trials, varying the position of the internal hinge O along the intrados curve of the slice. Four positions, denoted by 1, 2, 3 and 4 (Figure 9), have been chosen to place the internal hinge at the intratrados of the dome.. In Figure 9, the pressure curve, corresponding to the calculation of the minimum thrust performed in [16] (pp. 260-261) by means of the statical approach, is also sketched. We can recognize the approximation of this evaluation due to graphical imperfection in the construction of the sequence of funicular polygons. The computation of the thrust of the dome is made by firstly dividing the slice into 31 voussoirs and determining their weights and the positions of their centers with respect to a definite coordinate system, as shown in Table 1. The unit weight of the masonry voussoirs has been assumed to be equal to 1.85 t/mc. The first voussoir defined by the number 1 corresponds to the voussoir at the crown. The corresponding share of the weight of the lantern is reported in the last row of Table 1.
According to the foregoing calculations, the total weight G s of the slice, evaluated up to the height of the extrados of the drum and taken as 1/8 of the whole dome, including the share of the lantern, equals 3575 t.
The maximum kinematic thrust S kin (v) is found by trials, varying the position of the internal hinge O along the intrados curve of the slice. Four positions, denoted by 1, 2, 3 and 4 ( Figure 9), have been chosen to place the internal hinge at the intratrados of the dome. In Figure 9, the pressure curve, corresponding to the calculation of the minimum thrust performed in [16] (pp. 260-261) by means of the statical approach, is also sketched. We can recognize the approximation of this evaluation due to graphical imperfection in the construction of the sequence of funicular polygons.
The maximum kinematic thrust Skin(v) is found by trials, varying the position of the internal hinge O along the intrados curve of the slice. Four positions, denoted by 1, 2, 3 and 4 (Figure 9), have been chosen to place the internal hinge at the intratrados of the dome.. In Figure 9, the pressure curve, corresponding to the calculation of the minimum thrust performed in [16] (pp. 260-261) by means of the statical approach, is also sketched. We can recognize the approximation of this evaluation due to graphical imperfection in the construction of the sequence of funicular polygons.   The maximum kinematic thrust S kin (v) is found by varying the position of the internal hinge O along the intrados curve of the slice.
In the following section, four different mechanisms, numbered from (1) to (4), are considered and the corresponding kinematic thrust S kin (v) is evaluated.
Quantities D* i and H* i (i = 1, 2, 3, 4) are, respectively, the distances of the internal hinges (1, 2, 3, 4) of the assumed mechanisms from the vertical and the horizontal alignments passing through the top A of the slice (Figure 9). Table 2 develops the calculation of the kinematical thrusts according to the various settlement mechanisms corresponding to the chosen four positions of the internal hinge O.
Other choices of the positions of the internal hinges give lower values. The maximum of the kinematical thrust in the slice reaches a value of 470.8 t, corresponding to position 3 of the hinge O. With a previous calculation of the thrust performed by means of the statical approach, i.e., with the construction of a sequence of funicular polygons, a value of 400 t was obtained in [16] (pp. 260-261). This last value, because of the approximations inherent the funicular polygon layout, can be considered less accurate than the value of 470.8 t obtained by the kinematical approach.
In conclusion, the value of 470.8 t can be considered the actual value of the minimum thrust conveyed by a single slice equal to 1/8 of the Dome.
Averaging the lengths of the crown boundaries in which the drum is inscribed, assuming a circumferential length at the Dome springing equal to π·0.5 × (27.08 × 2 + 20.77 × 2) = 150.32 m, the length of the base slice is 150.32/8 = 18.79 m and the unit minimum thrust for single meter of the circumferential length of the Dome springing circle is 470.8/18.79 = 25 t/mL. This value is remarkably less than the thrusts of St Peter's dome in Rome that reaches a value of about 30 t/mL [29] (pp. 981-990).

Loads Conveyed to the Pillar
The four pillars sustain all the weight of the dome. The pillars are directly charged at a height of 59 m by the weight of the webs directly overlying them. The weight of the webs overlying the arches are conveyed to the pillars following the stress flow shown in Figure 10 and reaches them at a height of 39 m. The vertical load V 52 stands for the weight of the web directly above the pillar, including the weight of the lantern slice. Thus, summing the weights G i of the ashlars composing the slice, according to the values given in Table 1, The vertical load V52 stands for the weight of the web directly above the pillar, including the weight of the lantern slice. Thus, summing the weights Gi of the ashlars composing the slice, according to the values given in Table 1, The following loads have to be added to V52: The weight Ghaw of the two halves of the adjacent webs overlying the arches.

Gthw = 3575 t
This weight is conveyed to the pillar at this height because it is deviated by the presence of "eyes" in the drum (Figure 10). The diameter of the "eye" is equal to 5.8 m.
The  The following loads have to be added to V 52 : The weight G haw of the two halves of the adjacent webs overlying the arches.
This weight is conveyed to the pillar at this height because it is deviated by the presence of "eyes" in the drum (Figure 10). The diameter of the "eye" is equal to 5.8 m.
The weight G usd of the upper segment of the drum directly overlying the pillar. The following loads have to be added to V 39 : The weight G thd of the two halves of the adjacent segments of the upper drum, conveyed by the adjacent pointed arches The weight V aa of the two halves of the two adjacent arches. This weight V aa can be evaluated by making reference to a simplified scheme of the geometry of the arch. With reference to Figure 11, the two components V 1 and V 2 of the arch give V 1 = 10.00 · 7.5 · 0.5 · 4.18 · 2.2 = 345 t d 1 = 3.33 m This weight Vaa can be evaluated by making reference to a simplified scheme of the geometry of the arch. With reference to Figure 11, the two components V1 and V2 of the arch give The pointed arches are loaded, in addition to their weight, by the weight of the segment of the drum directly overlying them. The presence of these loads activates the thrust in these pointed arches. They are inserted between the four large pillars supporting the dome. No cracks have been detected at the intrados of these arches, as described in [5] (Figure 7, p. 413).
The assessment of the thrust in these pointed arches is difficult to evaluate. The presence of the minimum thrust state in the arches has to be excluded; their thrust will be included between their minimum and maximum values. In the case of the minimum thrust, the arm h* of the thrust couple has a length of 11.5 m. We assume a lower length of h* equal to h* 9.25 m, as shown in Figure 11.
With reference to Figure 11, the two components V1 and V2 of the arch weight are

Stresses at the Pillar Base
The analysis concerns the two absidal piers even if analogous results can be obtained for the other two piers flanking the nave. At the pillar head (h = 28.0 m), All the various dimensions of the section have been taken from documents and drawings and processed via a computer. The hollow section is shown in Figure 13.

2.
At a height of 28.00 m The vector sum S' va28 of the thrusts S a28 of the two adjacent pointed arches. The thrust S a28 .
The pointed arches are loaded, in addition to their weight, by the weight of the segment of the drum directly overlying them. The presence of these loads activates the thrust in these pointed arches. They are inserted between the four large pillars supporting the dome. No cracks have been detected at the intrados of these arches, as described in [5] (Figure 7, p. 413).
The assessment of the thrust in these pointed arches is difficult to evaluate. The presence of the minimum thrust state in the arches has to be excluded; their thrust will be included between their minimum and maximum values. In the case of the minimum thrust, the arm h* of the thrust couple has a length of 11.5 m. We assume a lower length of h* equal to h* 9.25 m, as shown in Figure 11.
With reference to Figure 11, the two components V 1 and V 2 of the arch weight are .60 + 322 · 5.00 + 345 · 3.33 and the thrust of the pointed arch is S a28 = 1 9. 25 (4940 + 1610 + 1148) = 832t and acts at a height of 28 m from ground level. The vector sum S' va28 is thus S va28 = 832x √ 2 = 1176 t In conclusion, while at a height of 52.00 m, the resultant radial thrust delivered by webs is S Rw52 = 470.8 + √ 2 470.8 = 1136 t, at a height of 28 m, the resultant thrust S' vRa28 of the pointed arches spanning between the arches is S' vRa28 = 1176 t.

Stresses at the Pillar Base
The analysis concerns the two absidal piers even if analogous results can be obtained for the other two piers flanking the nave. At the pillar head (h = 28.0 m), The load N 28 is directed along the vertical axis passing through the center of the trapeze section of the single side segment of the octagonal crown. This section, part of the whole cross section of the pillar and sustaining a side of the drum, directly carries the load N 28 . The area of this section equals about 1 ⁄2 (20.67 + 17.21)0.5 = 79.2 m 2 and the corresponding compression stress is 12,652/79.2 = 16 kg/cm 2 .

Section of the pillar
The pillar cross section is hollow in the center, up to a height of 20.0 m, while it is solid from a height of 20.00 m up to 28.00 m.
The masonry structure of the pillar is made up of about 2.0 m thick stone facings, tightly bound to a rubble and mortar core.
All the various dimensions of the section have been taken from documents and drawings and processed via a computer. The hollow section is shown in Figure 13.
The area A of the base section (hollow) and the distances of the center from its external sides ( Figure 13)  The area A of the base section (hollow) and the distances of the center from its external sides (Figure 13   The moment of inertia J x of the section with respect to the central axis x, orthogonal to the symmetry axis y, is: J x = 3.98 · 10 11 cm 4 .
The distances of the core points from the section center C are

Stresses at the pillar base section
Axial load: The axial load N oo at the pillar base is obtained by summing the various vertical forces transmitted with the weight of the pillar: N oo = V 28 + W pill = 12,652 + 15,528 = 28,180 t Eccentricities of the various vertical loads conveyed to the pillar with respect to the center C of its base section: A force of V 28 = 12,652 t is conveyed through the side segment of the octagonal drum section, at a distance of d 28 = 1 ⁄2 4.18 m = 2.09 m from the internal boundary of the crown.
The weight of the pillar W pill = 15,528 t passes through the center of the hollow section. Thus, the following equation gives the eccentricity e oo of the resultant force N oo : 12,652·4.00 = 28,180·e oo and e oo = 1.8 m. The eccentric load N oo = 28,306 t acts outwards at the distance e oo = 1.8 m from the center C of the section (Figure 13).
Bending moments: Vertical cracks cross all of the body of the drum, as shown in Figure 7 on page 413 of [5]. The drum has lost its hooping function. Brunelleschi concentrically inserted in the dome, at the level of first, second and third galleries, three girdling belts of "pietra di macigno", a strong sandstone, cramped by iron brackets.
A wood chain, made of wood lintels, connected by iron clamps, was also inserted over the first stone chain. All these chains are very deformable and can sustain only negligible shares of the dome thrust. Consequently, both the resultant radial thrust S Rw52 conveyed by webs at a height of 52 m and the resultant radial thrust S Ra28 conveyed by the arches at a height of 28 m push against the underlying pillar. The moment at the pillar base due to the thrusts is: M oo = 1136.8 × 52 + 1176 × 28 = 92,146 tm and is directed outside.

Stresses in the pillar base section
The composition of the bending moment M oo with the eccentric load N oo = 28,180 t yields a resultant eccentric load N R having a total eccentricity e R = 1.82 − 92,146/28,180 = 1.82 − 3.27 = −1.45 m outside ( Figure 13). The pier section is wholly compressed. The maximum and minimum compression stresses are: The stress analysis reported in [5] (p. 421) gives a lower value of 15.4 kg/cmq for the maximum compression stress in the pillar section at ground level. This difference can be explained considering that, in the model [5], the supporting structure of the dome also takes into account the structure of the side apses. The compressive stresses are sufficiently moderate to be sustained safely by the masonry, built with blocks of sandstone (pietra forte) and good mortar.
The expansion of the base of the dome, due to the occurrence of the two major cracks (type A in [5]) of a width of about 6 cm, involves a radial displacement of the dome base of about 6.0 cm/π = 1.9 cm. This displacement can be in full agreement with the visco-elastic flexure of the piers and of their foundations.
It is worth further remarking that the slow creep deformations occurring in the piers and in their foundations could explain the detected slow increase over time of the width of the major cracks in the webs, estimated, on average, to be about 5.5 mm/century [22] (p. 50).

Conclusions and Results
The estimate of the thrust of Brunelleschi's Dome in Florence and of the consequent stresses in the sustaining piers is the subject of the paper. After a brief description of the Dome and its damage state, the paper first explained the reasons for the insurgence of thrust in rotational masonry domes. The same occurred in Brunelleschi's Dome. The detected crack patterns reveal a rotational behavior of the dome, in spite of its external appearance of a segmental cloister vault, due to the clever constructional devices of Brunelleschi.
Useful in this context is the application of the kinematical approach, first studied by the author [14,15] in the framework of the Heyman model of masonry behavior. The basic principles of this approach are first recalled and then applied to the assessment of the thrust of Brunelleschi's Dome. The thrust has been evaluated by making reference to a Dome slice equal to 1/8 of the entire dome, a single web that is directly supported by the underlying pier. The minimum thrust of the slice has thus been obtained as the maximum of all the kinematical ones. This maximum reaches a value of 470.8 t, corresponding to 25 t/mL for a meter of the dome springing circle. This value is larger than the value of 21.3 t/mL obtained by applying the statical approach to the Dome, a less accurate procedure using the construction of sequences of funicular polygons.
The base section of the underlying pillars is eccentrically compressed, and the level of stresses is sufficiently low. Small rotations of the basement of the piers, due to the development of limited creep deformation of the underlying soil, the gravel of the River Arno, can account for the detected very slow increase over time of the width of the major cracks in the dome webs.
Over the centuries, the dome has largely maintained a stable configuration and stands as one of the most extraordinary examples of the soundness and beauty of Renaissance architecture. This result is due to the extraordinary dome design of Filippo Brunelleschi and to the smart plan of the pillar section, with the favorable positioning of the drum, perhaps due to Arnolfo di Cambio or to the succeeding Masters, such as Talenti, Lapo di Ghini and Neri di Fioravanti.