Resisting system and failure modes of masonry domes

Transcription

Engineering Failure Analysis 44 (2014) 315–337
Contents lists available at ScienceDirect
Engineering Failure Analysis
journal homepage: www.elsevier.com/locate/engfailanal
Resisting system and failure modes of masonry domes
Paolo Foraboschi
Dipartimento Architettura Costruzione Conservazione, Università IUAV di Venezia, Venice, Italy
a r t i c l e
i n f o
Article history:
Received 25 October 2013
Received in revised form 1 May 2014
Accepted 6 May 2014
Available online 23 May 2014
Keywords:
Brunelleschi’s dome
Dome’s failure
Force-resisting system
Palladio’s dome
Vasari’s dome
a b s t r a c t
The research synthesized in this paper focused on ultimate strength, structural safety
assessment, and collapse of masonry domes. Activity was directed at analyzing the
relationships between safety factor and geometry, and carrying out research targeted at
reducing the incidence and severity of structural failures in cultural buildings. This paper
shows that the resisting system of a masonry dome is not the two-dimensional shell,
but a one-dimensional mechanism that derives from the splitting of the shell and drum.
The resisting system, whose geometry depends on the dome shape and brick or stone pattern, may include the lantern and/or the masonry constructions around the drum.
Well-known domes taken from architectural cultural heritage are used to exemplify the
pivotal role of geometry and construction techniques in providing ultimate strength. These
examples also show the importance of considering the architectural design of the time, in
structural analyses.
The results found in the paper suggest how to provide masonry domes with adequate
safety, using the minimal level of structural intervention; in particular, without altering
the way the building reacts to applied loads. Hence, the paper helps understand how to
reduce the amount of structural work, which, in turn, guarantees conservation and restoration, as well as safeguarding.
The conclusions are devoted to analyzing which observations are valid for seismic
assessment and how the other observations have to be modified for seismic actions.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
This paper is devoted to masonry domes, with special reference to those of buildings deemed important to the culture—
especially history and architecture—of an area, i.e. cultural buildings and architectural heritage.
In order to take advantage of architectural heritage, every building must undergo safety assessment, though cultural
buildings have survived for centuries. In fact, on one hand masonry degradation due to aging has marginal structural effects,
but on the other hand a building may have suffered from structural damage [1–5] (earthquakes [6–13], overloadings
[14–18], soil or foundation failure [2,6,7,11,14,19]). Moreover, safety demand is higher than in the past, and longevity does
not ensure that safety of the building complies with modern codes [1,3,6–8,10,19,20].
Taking advantage of architectural heritage also requires some cultural buildings to undergo change of occupancy or
remodeling, because the modern use of historical buildings is different from the original use.
These two needs must satisfy the requirements of safeguarding, renovation, and conservation, which however are in
conflict with each other.
E-mail address: [email protected]
http://dx.doi.org/10.1016/j.engfailanal.2014.05.005
1350-6307/Ó 2014 Elsevier Ltd. All rights reserved.
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P. Foraboschi / Engineering Failure Analysis 44 (2014) 315–337
Safeguarding, which consists of protecting human lives and cultural heritage, aims at prolonging the life of historical constructions to the utmost. Thus, when structural safety results to be inadequate, safeguarding calls for a structural intervention [11,15,20–23].
Renovation, which consists of modifying the building or part thereof, aims at meeting the present occupants’ needs. Thus,
when the original architecture is not compatible with the present use, renovation calls for an architectural intervention
[5,16,18].
Conservation, which consists of preserving the cultural aspects of the architectural heritage, aims at retaining
authenticity of buildings’ architectural identity. Thus, conservation calls for interventions that do not alter the building or
part thereof, i.e. that guarantee integrity.
Integrity includes those visual aspects and physical elements that make up the appearance of a building, and that are significant to its cultural value. In particular, integrity includes the overall shape of the building, its materials, decorative details,
interior spaces, and features, as well as the various aspects of its site and environment (character-defining features).
Nevertheless, integrity is not limited to only this, but it also includes the structure and craftsmanship; to maintain the
original structural behavior of the building and the historical construction technique is mandatory for guaranteeing integrity.
Accordingly, integrity requires that, after the intervention, the way the building reacts to applied loads is equal to the way it
reacted before the intervention. An intervention that modifies the original structural behavior alters the cultural building,
compromising its integrity. After such an intervention, the building loses authenticity.
Hence, making use of cultural heritage requires a balance between safeguarding, renovation, and conservation. Balance
means that the cultural buildings have to be subjected only to interventions that are necessary to guarantee adequate safety
and functionality (minimal intervention). Moreover, balance means that an intervention has not to modify the original structural behavior.
A balance is often not reached in Italy, which is regrettable since Italy has a huge number of cultural buildings of any sort,
and is home to the greatest number of UNESCO World heritage sites. Despite the fact that only few countries in the world can
boast a collection of cultural buildings as great in magnitude as Italy can, the Italian structural codes devoted to assessment
of existing buildings [24,25] tend to unbalance the equilibrium in favor of safeguarding, which means that conservation is
penalized and renovation is scarcely considered.
1.1. Aims and objectives of the research
Highlighted, are the new requirements for existing masonry structures that have been adopted by a number of codes of the
new generation [26], in particular in Italy [24,25]. These codes consider only the least common denominator in the structural
behavior of masonry constructions, which is the load-carrying capacity due to the masses. In so doing, codes ignore architect’s
initial design intentions and disregard specific construction techniques and devices, which lead to a substantial underestimation of the safety factor.
These requirements are suitable for new masonry buildings, where steel bars and beams reinforce the masonry elements,
but are inappropriate for cultural buildings, which are made of unreinforced masonry. Unacceptable cultural and economical
penalty could be imposed to architectural heritage should assessment allow only for the load-carrying capacity provided by the
masses.
Thus, there is an acute need for a new code approach that allows architectural heritage to be assessed differently from
new buildings. With reference to masonry domes, on one hand there are a great number of consolidated research findings
(too many to be cited individually), but on the other hand they have not been incorporated into practice.
Considering this, activity was directed at carrying out research that advances the state-of-the-practice on masonry
domes. This paper provides material to identify and assess the actual resisting system of masonry domes, which is different
from that recognized by the codes.
2. Review of the structural behavior of the masonry arch
In masonry structures, typically, compression stresses are drastically lower than crushing strength [3,8,13–16,20,22,
27–30]. Thus, crushing is not a significant mode of failure for masonry structures, excluding columns made of poor masonry
or subjected to moisture and salt crystallization.
In masonry structures, tension stresses, including the principal stresses due to shear stresses, reach tension strength in
numerous points. Thus, cracking of masonry is a common phenomenon [2,3,7,15,19,20,23,30–32]. Under the higher loads, thus,
masonry is cracked and tension stresses are redundant. Therefore, the ultimate load is carried by compression and shear stresses only. It follows that failure occurs when a load causes the masonry structure to turn into an unstable mechanism, rather than
when a load induces excessive stresses into the masonry.
At the ultimate, a masonry structure behaves as an assemblage of rigid blocks, which passes from a stable to an unstable
state as the load increases and surpasses the ultimate load. Thus, the load-carrying capacity of a masonry structure fundamentally depends on the points that the loads are applied to and on the shape of the mechanism that the structure converts into.
Conversely, a masonry structure does not primarily fail due to a lack of material strength; however, compression strength
and tension strength influence the load-carrying capacity. Masonry compression strength influences the distance of each rotation pin from the edge of the cross-section. In masonry arches and domes, however, this distance is almost always negligible.
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Masonry tension strength influences how the structure splits into blocks. For some types of masonry structures, which include
the domes, the resisting system strongly depends on tension strength.
In domes, hence, masonry compression strength has a marginal role, while masonry tension strength has a pivotal role.
Unfortunately, codes allow for masonry compression strength (to evaluate the aforementioned distance of the pin), but do
not allow for masonry tension strength.
Considering that masonry tension strength does not depend on the materials (Section 4), the load-carrying capacity of
arches and domes do not depend on the mechanical characteristics of bricks (or stones) and mortar that the structure is made
of.
The failure modes can be grouped into rotating and translating shapes. The rotating shapes consist of rigid blocks joined
by hinges placed at the vertexes of each rigid block; the translating shapes, of a part that slides with respect to the other part.
In the case of masonry arches and domes, the failure modes can also be grouped into shapes with fixed springing sections
(Figs. 1 and 2) or with the springing sections that translate laterally (Figs. 3 and 4). Contrary to the former shapes, the latter
shapes involve the abutments. Figs. 1–4 show symmetric structures and loads. In actual fact, real structures and real loads
are not symmetric. Thus, the mechanisms are not symmetric; so, each shape has one hinge less. On one hand, this does not
modify the theoretical developments; on the other hand, however, symmetric failure modes allow the presentation to be
more expressive.
2.1. No-tension friction assumption for safety assessment
The ultimate behavior of masonry structures is typically analyzed under the no-tension friction assumption: The material
has no tension strength, infinite compression strength, and shear strength equal to the normal force multiplied by the friction coefficient of masonry ([4,12,13,16,22,29,35]).
This assumption allows safety to be assessed with the path of the resultants of the compression stresses through the
masonry, called line of thrust [36] (Figs. 5 and 6). In fact, the no-tension assumption implies that each point of the thrust
line has a distance from the axis of the structure that is equal to the ratio between the couple and the normal force in
the section. This ratio is called eccentricity, e (Fig. 5). The thrust line is a viable means for assessing safety of the masonry
arch [28,29,35].
Given the arch, there are infinite thrust lines in equilibrium with a given load. The real thrust line is the one that respects
the two boundary conditions at the springing—namely, the horizontal thrust, Hs, and the bending moment. Since the vertical
reaction is statically determinate, the springing bending moment can be expressed by the eccentricity of the springing normal force, es (Fig. 5). However, the lower bound theorem allows safety assessment to ignore Hs and es, since ultimate strength
analysis does not require the real thrust line [37].
If a line of thrust can be found for the arch which is in equilibrium with the external loading (including own weight),
which lies everywhere within the masonry of the arch, whose tangent lies everywhere within the friction angle, and whose
springing thrust does not cause the failure of the abutments, then the structure is stable. Using the symbols represented in
Fig. 5, the above statement can be synthesized by the following formulas:
eðuÞ <
t
2
06u6u
hðuÞ 6 arctgðlÞ
Hs 6 F ab
0 6u6u
ð1Þ
ð2Þ
ð3Þ
Fig. 1. Masonry arch and abutments. Kinematic mechanism that does not involve the abutments; the springing sections are fixed and the crown moves
downwards. In the figure, both the structure and load are symmetric. In reality, structures and loads are not symmetric. Thus, the hinge h5 does not form
and the hinge and h3 is not exactly at the crown; moreover, h2 and h4 are not symmetric. The actual form is that of Fig. 7.
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Fig. 2. Kinematic mechanism that does not involve the abutments; the springing sections are fixed and the crown moves upwards. In the figure, both the
structure and load are symmetric. In reality, structures and loads are not symmetric. Thus, the hinge h5 does not form and h3 is not exactly at the crown;
moreover, h2 and h4 are not symmetric (Fig. 7).
Fig. 3. Kinematic mechanism that involves the abutments; the springing sections and abutments open, while the crown moves downwards. The aspect
ratio of the abutments is greater than 0.5, and therefore the sliding mechanism of the abutment cannot occur. In the figure, both the structure and load are
symmetric. In real conditions, however, the hinge h5 does not form and h3 is not exactly at the crown; moreover, h2 and h4 are not symmetric.
is the maximum angular coordinate u (polar angle, which in Figs. 5 and 6 is p), l is the masonry friction coefficient,
where u
and Fab is the force at the common boundary between arch and abutment that causes the failure of the abutment. A thrust
line that satisfies (1–3) is called compatible.
The (1–3) prove that safety assessment does not require the real locus of the centroid of the compression stresses as they
flow from section to section, but a locus in each section that may guarantee the equilibrium between the external load and
the internal actions. According to (1–3), in fact, Hs and es can be chosen arbitrarily, and these values remain unknown, apart
from the ultimate condition [37].
The line of thrust can also be applied ignoring the ultimate thrust of the abutment Fab. The arch and abutments are able to
bear the load if a thrust line is placed within the masonry thickness of both the arch and abutments, and within the friction
angle [28,29,35–40].
If it is impossible to place a thrust line totally inside the thickness t of the arch (Fig. 6a), the structure is unstable (the sole
arch or the arch and abutments). The failure mode is a rotation mechanism, which may (Figs. 3 and 4) or may not (Figs. 1 and
2) involve the abutments. In fact, a stress resultant N that acts outside the cross-section of the arch can be equilibrated only
by an internal tension force, F, together with the internal compression force C in the masonry cross-section (Fig. 6a). If also
the tension force F acts at the edge of the cross-section (e.g., external reinforcement [15,21,22,39,41]):
Nðua 6 u 6 ub Þ ¼ CðuÞ FðuÞ
2 e N ðua 6 u 6 ub Þ ¼ t ½CðuÞ þ FðuÞ
ð4Þ
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Fig. 4. Kinematic mechanism that involves the abutments; the springing sections and the abutments close, while the crown moves upwards.
Fig. 5. Masonry arch of thickness t. The figure shows the line containing all the points where the stress resultants act at every section of the arch (thrust
line). This thrust line lies within the thickness of the arch. In fact, these points, which are identified by the angular coordinate u, have eccentricity e less than
t/2 for any u. Moreover, the tangents h (slopes of the thrust line) lie everywhere within the friction angle. This thrust line was constructed using es, which
lies within the masonry, and the springing lateral thrust Hs. If the abutment can withstand Hs transmitted by the springing section, the arch with this thrust
line can resist the load (stable arch), under the assumption that crushing strength is infinite, tension strength is zero, and shear strength is dictated by
friction.
However, when Eqs. (4) provide a nonzero value of F, Eq. (1) is violated. Nevertheless, according to Eqs. (4), the nonzero
value of F is necessary for the rotational equilibrium. Hence, (1) guarantees the rotational equilibrium of the structure and
parts thereof, within the no-tension framework.
If it is impossible to place the slopes of the thrust line of the arch totally within the friction angle (Fig. 6a), the arch is
unstable. The failure mode is the sliding mechanism of a part of the arch with respect to the other part. In fact, a resultant
force that lies outside the friction angle, which is arctg(l), can be equilibrated only by a cohesive shear force, Sch, together
with the friction force due to the normal force N. With reference to the normal force N and the slope h at the point ua of
Fig. 6a:
Nðua Þ tg½hðua Þ ¼ Nðua Þ l þ Sch
ð5Þ
However, when Eq. (5) provides a nonzero value of Sch, Eq. (2) is violated. Nevertheless, according to Eq. (5), the nonzero
value of Sch is necessary for the translational equilibrium. Hence, (2) guarantees the translation equilibrium of the structure
and parts thereof, within the friction framework.
In a masonry arch, however, a thrust line that satisfies (1) also satisfies (2), unless the arch is unrealistically thick (Figs. 5
and 6). Thus, the typical failure modes of the masonry arch are the rotating mechanisms, while the sliding mechanism is
atypical. Hereafter, (2) is not considered any longer.
If it is impossible to place a thrust line with springing thrust Hs < Fab, totally within the thickness t of the arch, the structure is unstable. The failure mode is the rotation mechanism that causes the springing sections and abutments to open
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Fig. 6. Two different thrust lines constructed for each arch. Each thrust line, identified by the labels 1 and 2 respectively, is in equilibrium with the applied
load, including the initial conditions. (a) Thrust line 1 is contained entirely within the masonry and therefore it satisfies inequality (1); moreover, it also
satisfies inequality (2). If the springing lateral thrust associated with thrust line 1 satisfies inequality (3), the arch bears the load and so it is stable.
Conversely, thrust line 2 passes outside the masonry; in particular, it violates inequality (1) from the point ea of coordinate ua to the point eb of coordinate
ub. Moreover, it does not lie within the friction angle too; in particular, the tangent violates inequality (2) around the points ea and eb. Thrust line 2 proves
neither that the arch is stable nor that the arch is unstable. (b) Two thrust lines constructed starting from the same point of the springing section. Thrust line
1 uses the springing thrust Hs1; thrust line 2 uses the springing thrust Hs2. Since both e1 and e2 are lower than t/2 for 0 6 u 6 p, either thrust line guarantees
the equilibrium of the arch under the applied load. Generally, none of them is the real thrust line, since the starting point and lateral thrust at the springing
are not real (unless by an unrealistic coincidence). However, safety assessment does not require the real thrust line.
Fig. 7. Virtual work done by the applied load and weight over the displacement produced by the mechanism in the figure. The weights Pj whose centroid
moves downwards and the forces Fi applied to the boundaries that move downwards belong to the external loading system; and, conversely, the weights Pj
whose centroid moves upwards and the forces Fi applied to the boundaries that move upwards belong to the resisting system. The loads applied to
boundaries that do not move, do not participate in the ultimate limit state of the structure.
(Fig. 3). Theoretically, the opening of the springing sections can be caused by the shear failure of the abutment. In this case,
the kinematic of the abutment is the sliding rather than the rotation. However, the sliding mechanism of the abutment lies at
the bottom level of the strength hierarchy only if the aspect ratio of the abutment (i.e., the ratio of height of the abutment to
width of the abutment) is less than 2.0, which is uncommon in masonry arches and domes. Therefore, the sliding mode of
failure of the abutment is ignored hereafter. Hence, (3) ensures the rotation and translation equilibrium of the abutments.
The kinematically admissible displacements include the failure modes with the springing sections that close (Fig. 4),
which is important for the arch but irrelevant for the dome (Section 3).
Ultimately, to demonstrate that the arch will stand as a structure, it is necessary and sufficient to find at least one thrust
line that satisfies (1) and (3), which, in turn, satisfies (2) automatically.
Thrust lines constructed starting from different values of Hs are different from each other (Fig. 6b). Nevertheless, every
compatible thrust line proves that the load does not cause the structure to collapse.
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Eqs. (1) and (3) prove that the middle-third rule [29] is wrong. To place the thrust line within the middle-third has no
meaning, since the thrust line that lies within the middle-third is not the real one, but one of the infinite compatible thrust
lines. That a masonry arch is uncracked, however, is pointless, since the ultimate behavior implies cracks.
The no-tension friction model also allows the safety of masonry arches to be assessed considering that, at collapse, the
arch consists of rigid bodies connected to one another by hinges placed at the boundaries (pinning mechanism). The virtual
work theorem proves that a masonry arch under the weights P and the forces F collapses if and only if, for at least a mechanism (Fig. 7) [33]:
N
N
X
X
F i dFi þ
Pj dPj > 0
i¼1
ð6Þ
j¼1
where N is the number of rigid blocks, Fi is the force applied to the i-th block, Pj is the own weight of the j-th block, and d are
the virtual displacements produced by the mechanism; in particular, dFi of the application point of Fi in the direction of Fi, and
dPj of the j-th centroid in the vertical direction. A vertical force is positive if it points down; a vertical displacement is positive
if it is directed downwards. A horizontal force is positive if it points to the right (left); a horizontal displacement is positive if
it is directed to the right (left).
2.2. No-tension friction assumption for ultimate load assessment
The no-tension friction assumption implies that a masonry arch is an internally statically unstable structure, whose failure is not due to lack of material strength but to equilibrium only.
Since e, h, and Fab depend on the application points but not on the level of the total load, the safety factor of a masonry
arch depends on the distribution but not on the level of the total load. Thus, the sum of own weight, dead loads, and live
loads cannot express the safety factor of the masonry arch.
Conversely, the safety factor of the masonry arch is measured by the ratio between the ultimate value and design value of
the imposed loads; the imposed loads consist of the live loads, but they also have to include some dead loads, when they can
change during the service life.
The ultimate load is the higher level of the imposed load for which it is possible to place the thrust line within the arch
and abutments, independently of the distribution of the imposed load.
Consequently, there is only one thrust line in equilibrium with the ultimate load, which therefore is the real thrust line.
From the thrust line perspective, hence, the ultimate limit state represents an exception.
At the ultimate, the thrust line is tangent to the extrados and intrados of the arch in either 3 or 4 points:
Hs < F ab ! eðuk Þ ¼
t
2
for k ¼ 1; . . . ; 4
ð7Þ
Hs P F ab ! eðuk Þ ¼
t
2
for k ¼ 1; . . . ; 3
ð8Þ
Contrary to what shown in Figs. 1–4, Eqs. (7) and (8) consider that the real structures and loads are never symmetric.
Accordingly, a mechanism always has four hinges, which can be located either only in the arch or three in the arch and
one in an abutment. Eqs. (7) and (8) show the role of cracking in masonry. Any thrust line with e(u) 6 t/6, which is a necessary (but not sufficient) condition for masonry to be uncracked, equilibrates a load that is inferior to the ultimate load.
Hence, the same phenomenon seen from the displacement perspective is cracking and seen from the force perspective is
transmitting the higher levels of bending moment.
The ultimate load of a masonry arch is also the minimum level of the imposed load that triggers a rotation mechanism in
the structure and/or its supports. The ultimate load can be expressed by the minimum multiplier k of the imposed load F for
which a mechanism and a distribution of the imposed load F satisfy the following equation, which derives from (6):
N
N
X
X
k F i dFi þ
Pj dPj ¼ 0
i¼1
ð9Þ
j¼1
In (9), as the structure moves through the virtual displacement, the application point of Fi undergoes the displacement dFi
in the direction of the force (in general Fi may also have a horizontal component), and the centroid of the weight Pj undergoes
the vertical displacements dPj (Fig. 5).
The minimum k has to be found by applying (9) to all the kinematically admissible mechanisms and using all the possible
application points of Fi. Hence, the assemblage of rigid blocks that Eq. (9) must be applied to, may or may not involve the
abutments. In the former case (Fig. 3), the sum of the N forces Pj is equal to the weight of the arch plus either the abutments
or parts thereof. In the latter case (Figs. 1 and 2), the sum of the N forces Pj is equal to the weight of the sole arch.
Eq. (9) shows that the load-carrying capacity of a masonry arch is dictated by the weakest mechanism, which forms when
the most severe distribution of the imposed load reaches a certain value (ultimate load).
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3. Structural behavior of the masonry dome
The loads (including own weight) induce meridian (arch) and parallel (circumferential or hoop) stresses in a dome
[35,40]. The meridian stresses are always compressions. The parallel stresses are tensions in the lower part of the dome
and compressions in the upper part (except for the atypical case of domes with flat crown and heavy lantern).
The maximum parallel tension stress is usually at the springing and very often it reaches masonry tension strength even
under the dead load. Hence, masonry domes are typically cracked, and cracking occurs immediately after the construction
(sometimes during the construction). Cracking initiates at the springing and propagates vertically downwards into the drum
and upwards toward the crown, sometimes arresting at the haunches while at other times approaching the crown
[3,4,7,13,29,30,41–43].
Parallel tension stresses exist only until they crack the masonry. Hence, vertical cracking transforms the dome from a
two-dimensional to a one-dimensional structure. Considering that the vertical cracks are typical, the masonry dome is a shell
only from the architectural point of view, while it is a system of arches from the structural point of view.
Stability of masonry arches is governed by two parameters—namely, the thickness-to-span ratio, and the lateral springing
thrust [33,37]. Considering that the cracked dome behaves as a masonry arch, it follows that stability of masonry domes is
governed by these two parameters.
The thickness of a masonry dome is that of the arches it splits into, due to vertical cracking. Thus, the thickness-to-span
ratio (first parameters) does not refer to the thickness of the shell but it depends on the shape of the curved elements that the
shell splits into.
In a masonry dome, the abutment coincides with a block that the drum is split into, due to vertical cracking. Therefore,
the lateral springing thrust (second parameters) is the horizontal thrust transmitted by the cracked dome to the cracked
drum through the parallel with the lowest latitude.
There is a minimum thickness-to-span ratio for any masonry arch [33,37]. Thus, a masonry dome is stable only if the
thickness of the arches that it splits into is greater than the minimum thickness for the span.
If, conversely, the thickness-to-span ratio is less than the minimum, the dome is unstable. In this case, there are two possible failure modes. In both of them, the springing parallel is fixed (no lateral displacement); in the first one, the crown
moves downwards (Fig. 1), while in the second one, the crown moves upwards (Fig. 2).
The minimum thickness-to-span ratio and the failure mode depend on the shape of the dome and the brick pattern. Safety
assessment and design of the structural intervention require prior knowledge of the minimum thickness-to-span ratio and
the mode that dictates the failure.
The lateral springing thrust of an arch has to lie within a range of values that depends on the geometry and loading. The
lower bound of this range is called ‘‘lower thrust’’ [37]. Thus, a masonry dome is stable only if the lateral thrust transmitted
through the springing parallel is greater than the lower thrust (and less than the upper bound). Considering also the lower
bound theorem, a masonry dome is stable only if the lower thrust is inferior to the horizontal thrust that causes the failure of
the drum.
If, conversely, the lower thrust is greater than the maximum lateral thrust that the drum can bear, the dome and the drum
are unstable. In this case, the mode of failure is demonstrated by the opening of the drum and springing parallel together
with the downward translation of the crown (Fig. 3).
The drum, even if buttressed, is not a thrust structure. Hence, the dome cannot be subjected to a springing thrust greater
than the upper bound of the range of possible lateral thrusts. Consequently, the thrust cannot trigger the failure mode by
which the drum and springing parallel close, while the crown translates upwards (Fig. 4).
The lower thrust depends on the shape of the dome and the brick (stone) pattern. The lower thrust of the dome needs to
be known for the safety assessment and design of the structural intervention.
3.1. Failure mode hierarchy for the masonry dome
The ultimate behavior of a masonry dome can be analyzed more easily by using the virtual work theorem than the thrust
line. To this end, (9) has to be written for the arches that result from the splitting of the dome and to include the lantern and
the structures around the drum (if they exist).
Let M denote the number of curved elements that the dome and drum split into or will split into (i.e., M is the number of
meridian cracks). Eq. (9) turns into:
PL dPL þ
M X
N
M X
N
X
X
k F i dFi þ
Pj dPj ¼ Lv b
k¼1 i¼1
ð10Þ
k¼1 j¼1
where PL is the weight of the lantern at the crown of the dome, and dPj is the virtual vertical displacement of the crown due to
the infinitesimal movement of the considered mechanism. A vertical force is positive if it points down; a vertical displacement is positive if it is directed downwards; and vice versa. A horizontal force is positive if it points to the right (left); a horizontal displacement is positive if it is directed to the right (left). The term Lvb is the virtual work done by the buttressing
action of the structures around the drum or by circumferential belts, due to the virtual displacement of the mechanism.
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Eq. (10) has to be applied to all the kinematically admissible mechanisms of the dome split into arches. For each mechanism, all the possible application points of the force F have to be considered in (10). The mechanism and load distribution
that provide the minimum value of k are the failure mode and the load-carrying capacity of the dome, respectively.
When (10) is applied to a mechanism that does not involve the drum or if the drum is neither buttressed nor strengthened
with belts, Lvb = 0. When (10) is applied to a mechanism that involves the drum, the weights Pj include pieces of drum and
the forces Fi may be applied to the drum.
3.2. Failure mode hierarchy for the masonry dome
In masonry domes, the top level of the strength hierarchy is occupied by the translation mechanism (Section 2.1). Accordingly, the shear action does not interest the masonry domes [33,37]. Thus, the bottom levels in the strength hierarchy are
occupied by the rotation mechanisms.
The strength hierarchy of the rotation mechanisms depends on the lateral strength of the drum, which, in turn, depends
on the thickness of the drum and on the presence of buttresses or circumferential strengthening elements (steel tie-rods or
timber belts).
If the drum is thick, buttressed or strengthened, the rotation mechanisms with fixed springing lie at the lowest levels in
the hierarchy. More specifically, the lowest level is occupied by the mechanism with the crown that moves downwards
(Fig. 1) in the case of hemispherical domes, and that moves upwards (Fig. 2) in the case of domical domes.
If the drum is thin and neither buttressed nor strengthened, the mechanism with the drum and springing parallel that
open while the crown moves downwards lies at the lowest levels in the hierarchy (Fig. 3).
4. New approach for safety assessment of masonry domes
Section 3 has demonstrated how to analyze the masonry dome in the framework of the masonry arch. However, Eq. (10)
can be applied only to domes with meridian cracks, i.e. to a dome split into arches. In order to assess the safety of the domes
not yet cracked, and to interpret the splitting of the cracked domes, it is necessary to predict how an uncracked masonry
shell will split into arches.
4.1. From the two-dimensional masonry shell to a system of arches
The thickness t of the arches that barrel, cross or groin masonry vaults split into, almost never depends on the position of
the cracks. Thus, the minimum k obtained by applying (9) to whatever system of arches that a structure can split into is the
load-carrying capacity. The mechanism associated with k is the resisting system.
Conversely, the thickness t of the arches into which the domes split depends on the position of the meridian cracks. Thus,
the minimum k obtained by applying (10) to a possible system of arches into which a dome can split is not necessarily the
load-carrying capacity, and the mechanism associated with k is not necessarily the resisting system.
The position of the meridian cracks depends on tension stresses and strength. Thus, the no-tension assumption (Section 2)
has to be replaced by an assumption that allows the actual crack position to be considered or predicted.
Masonry tension strength at the material level is very low, but at the level of the brick or stone pattern may be considerable, and indeed it was considered by past architects and master masons. In fact, sometimes, the brick patterns were
designed and laid to maximize the interaction between the masonry units. As a result, in masonry domes, tension stresses
are one order of magnitude greater than the tension strength of the mortar, but are of the same order of magnitude of the
tension strength of the brick (stone) pattern. The tension strength of a brick or a stone is greater than the tension strength of
the pattern; thus, the former does not dictate the capacity of carrying tension forces.
During a relative rotation, crack opening is not resisted by interlocking and friction; in this case, thus, tension strength
does not provide any contribution to the load-carrying capacity. During a relative translation, conversely, crack opening
may be resisted by interlocking and friction; in this case, thus, tension strength provides considerable contributions.
The splitting of the dome can be governed by adding to Eq. (10) a term that allows for the virtual work done by the tension stresses provided by the masonry unit pattern, Lvc. During a virtual relative translation, the stress profile by which the
brick (stone) pattern resists the opening of a crack is uniform; its value is equal to the tension strength, fmt, and this tension
stress is smeared onto the area that is involved in the relative translation, whose shape is rectangular.
Lv c ¼
M
X
bk t k dk fmt
ð11Þ
k¼1
where M is the number of meridian sections that undergo relative horizontal translation d. Moreover, bk and tk are the length
and the thickness, respectively, of the k-th area that is involved in the relative translation. If the thickness of the dome does
not taper, tk is the same for all the areas (i.e., tk does not depend on k).
4.2. Small-tension-strength assumption
Eq. (11) has to be inserted into Eq. (10):
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PL dPL þ
M X
N
M X
N
M
X
X
X
k F i dFi þ
Pj dPj ¼ Lv b þ
b t dk fmt
k¼1 i¼1
k¼1 j¼1
ð12Þ
k¼1
Eq. (12) was applied to several brickwork domes that exhibited meridian cracks. This analysis adopted fmt = 0.01 N/mm2
for the crack openings resisted by interlocking and friction, and fmt = 0 in the other cases. Although 0.01 N/mm2 is a very low
value, this analysis proved that cracks nevertheless form where the tension strength is zero. In these cases, tension strength
has a direct role in the ultimate behavior of the dome, since it forces the dome to split where crack opening is not resisted by
interlocking and friction. But the ultimate behavior of the dome split into arches does not depend on tension strength. Thus,
the load-carrying capacity can be predicted by using (10), since Lvc = 0 for the dome split into arches.
In many domes, however, whatever meridian crack that can open is resisted by interlocking and friction, because the
dome does not exhibit any discontinuity in tension strength; therefore, no zone of the dome has marginal tension strength,
i.e. masonry tension strength has a uniform distribution over the dome and its value is everywhere not small. In order to
analyze how these domes split into arches, a second analysis was carried out. This analysis calibrated the value of fmt in
Eq. (12) against the actual cracks observed in the analyzed domes. The result is that fmt = 0.05 N/mm2 well represents the
tension strength provided by interlocking and friction.
5. The structural role of the lantern
A dome is crowned by a lantern through which daylight is admitted into the interior and which can be uses as a belvedere. However, the lantern may have a significant structural role too.
If the virtual work performed by the weight of the lantern over the kinematic of a binding mechanism is positive, the lantern belongs to the external load system; conversely, if the virtual work is negative, the lantern belongs to the load resisting
system (Fig. 7).
5.1. Hemispherical masonry dome
During the binding mechanisms of the hemispherical masonry dome (i.e., those shown in Figs. 1 and 3), the crown always
translates downwards. Thus, the virtual work done by the lantern, LvL, is:
Lv L ¼ PL dPL > 0
ð13Þ
Let qj denote the weight of the j-th block of unitary thickness (rather than thickness t). The virtual work done by the shell
with unitary thickness, lv1(t = 1) over the mechanism that does not involve the drum (with fixed springing) while the crown
translates downwards is (Fig. 1):
lv 1 ðt ¼ 1Þ ¼
N
X
qj dPj < 0
ð14Þ
j¼1
Considering that the weight Pj of the j-th block is Pj = tqj, the combination of (13) and (14) provides a necessary condition
to prevent the triggering of the failure mechanism of Fig. 1:
P L dPL
Lv L þ t lv 1 6 0 ! t > PN
j¼1 qj dPj
ð15Þ
Inequality (15) represents a necessary condition for stability of the dome, but not sufficient. If (14) is not respected, the
dome collapses, demonstrating the failure mode by which the crown translates downwards while the springing parallel and
the drum does not move (Fig. 1).
The thickness t of the shell not only has to satisfy (14), but it has to exceed the right member as much as necessary so that
Eq. (10) provides a k greater than the value associated with the design live loads.
The virtual work done by the shells with unitary thickness lv1 during the mechanism that involves the drum (with the
springing parallel the opens), while the crown translates downwards is (Fig. 3):
lv 1 ðt ¼ 1Þ ¼
N
X
qj dPj > 0
ð16Þ
j¼1
The virtual work done by the drum during this mechanism, LvD, is negative (which is obvious, since the other contributions are positive). Thus, LvD has to counterbalance the positive contributions of lantern and shell:
Lv D P t N
X
qj dPj þ PL dPL
ð17Þ
j¼1
In order to satisfy (17), the drum has to have an adequate thickness; if it is not thick enough, it has to be buttressed or
strengthened with belts.
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Ultimately, the heavier the lantern, the heavier the shell has to be to prevent the mechanism with fixed springing (Fig. 1).
Moreover, the heavier the lantern and the shell, the greater the strength of the drum has to be in order to prevent the opening of the springing and drum (Fig. 3). Thus, a hemispherical masonry dome with a heavy lantern needs that the drum is
thick or strengthened (by buttressing structures or belts). On the contrary, a hemispherical masonry dome with a lightweight
lantern may be stable also if the drum is not particularly thick and not strengthened.
5.2. Domical masonry dome
During the binding mechanisms of the domical masonry dome (i.e., those shown in Figs. 2 and 3), the crown may translate
downwards or upwards:
Lv L ¼ PL dPL < 0
ð18-aÞ
Lv L ¼ PL dPL > 0
ð18-bÞ
where Eq. (18-a) describes the failure mode that does not involve the drum, (springing parallel fixed; Fig. 2), and Eq. (18-b)
describes the failure mode that involves the drum (springing parallel that opens; Fig. 3).
The virtual work done by the domical shell of unitary thickness for the mechanism that does not involve the drum is
(Fig. 2):
lv 1 ðt ¼ 1Þ ¼
N
X
qj dPj < 0
ð19Þ
j¼1
The combination of (18-a) and (19) proves that the mechanism with fixed springing cannot occur in the masonry domical
dome with constant thickness, whatever the weight of the lantern.
In numerous domical domes, however, the shell tapers in thickness over the span, from the springing to the crown. If the
crown is thin, Eq. (19) does not hold true any longer, and the virtual work done by the shell, Lvs, is positive. In this case, the
weight PL of the lantern has to be great enough to counterbalance the positive virtual work done by the shell for the mechanism with fixed springing parallel (Fig. 2):
Lv L þ Lv s < 0 ! PL dPL >
N
X
Pj dPj
ð20Þ
j¼1
For tapered domical dome, the weight PL of the lantern not only has to satisfy (20) over the mechanism with fixed springing (Fig. 2), but it also has to exceed the right member as much as necessary so that Eq. (10) provides a k greater than the
value associated with the design live loads.
The virtual work done by the shell for the mechanism where the springing parallel and the drum translate laterally and
open is (Fig. 3):
Lv s ¼
N
X
Pj dPj > 0
ð21Þ
j¼1
According to (18-b) and (21), both the lantern and the shell do positive virtual work over the mechanism of Fig. (3). Thus,
the mechanism with the springing parallel and drum that translate laterally is resisted only by the drum, as for the hemispherical dome. However, all other parameters being equal, the virtual work done by the domical shell and lantern is less
than that of the hemispherical shell and lantern. This result can be immediately derived combining the shapes of the domes
and the slopes of the thrust lines that lie within the thickness of each geometry. Thus, the drum of a domical dome may not
be as strong as the drum of the hemispherical dome.
Ultimately, in the domical dome the lantern plays two dual roles—namely, it belongs to the resisting system in the mechanism with fixed springing but it belongs to the load system in the mechanism with the springing parallel and drum that
open. In the tapered domical dome, the lantern has to be heavy enough to prevent the formation of the mechanism with
fixed springing but lightweight enough to prevent the formation of the mechanism with the springing parallel and drum that
open.
5.3. The lantern of real historic domes
Architects and master builders (master masons) knew the effects of the lantern on a masonry dome [2,6–9,
11–14,16,29,32,35,42,44].
One example is the dome of the temple of Villa Barbaro (Fig. 8), in Maser, near Treviso (Italy), which was designed by
Andrea Palladio and was built between 1554 and 1580 (in 1560 for many historians).
This dome is a hemispherical masonry shell. Palladio’s plan for the temple included a large and massive lantern at the
crown. However, Palladio had realized that a lightweight lantern was mandatory for the equilibrium of the dome. Thus,
he rejected the idea of a lantern made of marble. In fact, the great weight of a large marble lantern either would have
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Fig. 8. Dome of the temple of Villa Barbaro in Maser, a little town near Treviso (Italy). The temple, including the dome, was designed by Andrea Palladio. The
figure shows the hemispherical masonry dome over Palladio’s temple of Villa Barbaro. The large lantern of the dome is made of timber; thus, it is not as
heavy as its massive form would suggest.
Fig. 9. Dome of the Santa Maria del Fiore cathedral, in Florence (Italy), designed by Filippo Brunelleschi. The masonry dome over the Florence cathedral is
composed of ribs and webs: octagonal ribbed domical dome. The lantern, which is a belvedere about 20 m high, is made of marble. Thus, Brunelleschi’s
lantern is heavy.
327
triggered the mechanism with fixed springing section (Fig. 1) or would have requested a thick shell that would have triggered the mechanisms with the springing section that opens (involving the drum). Palladio put his idea into effect by using
a timber lantern that imitated a typical marble lantern (Fig. 8).
This research has confirmed Palladio’s supposition. With a marble lantern, Eq. (10) applied to the mechanism with fixed
springing (Fig. 1) gives a negative value of k. Conversely, with a timber lantern, Eq. (10) gives:
k ¼
4
PN
N
X
dPj PL dPL
¼ 0:13 Pj
PN
4 i¼1 F i dFi
j¼1
j¼1 P j
ð22Þ
The forces Fi are the resultant forces that derive from a uniformly distributed load applied to the dome. Eq. (22) considers
the actual cracking pattern of the dome, which exhibits four uniformly spaced meridian cracks that pass through the entire
thickness. The four cracks initiate at the four openings of the windows located at the springing of the dome; each crack
extends vertically from the top of the opening up to the bottom of the crown, and from the bottom of the opening to the
base of the drum. Hence, the four meridian cracks cut the entire thickness of the drum, springing, and haunches; only the
crown is not cracked. Thus, Eq. (22) was obtained by applying Eq. (10) to the four arches of identical geometry that the dome
is split into. Accordingly, in this dome, the two summations of (10) add four identical terms; i.e., M = 4 in Eq. (10).
Another example is the dome of Santa Maria del Fiore cathedral in Florence (Italy), which was built by Filippo Brunelleschi
between 1420 and 1436 (Fig. 9). This is an octagonal ribbed domical masonry dome.
History of architecture explains that the gothic style of this dome is due to the fact that the competition to complete the
church of Santa Maria del Fiore, held in 1418, was not for the design of a new dome. In fact, the competition won by Brunelleschi was for the construction of a dome according to the preliminary design that had already been accomplished in the
previous century by Giovanni di Lapo Ghini. Moreover, history of architecture explains that Brunelleschi did not refine the
preliminary design toward the contemporary style also because the dome had to match the gothic style of the church and to
rest upon an octagonal drum (the church had been built in the XIV century, and the drum had been built together the
church). Although these style requirements, however, the rise of the dome could have been reduced and/or less emphasis
could have been put on the intersections between the webs (ribs).
Contrary to what history of architecture tells, Brunelleschi maintained the preliminary design because it was the solution
to all the problems that he had to face and overcome. In particular, Brunelleschi knew that a lower rise and weaker ribs at the
corners would have not provided the dome with adequate strength against the mechanism where the springing parallel
opens. This inadequacy would have been emphasized by the heavy lantern that Florentine wanted to crown the dome
(not designed yet).
The Florentine dome tapers in thickness over the span, from the springing to the crown. This research found that the virtual work done by the shell (i.e., without the lantern) for the mechanism with fixed springing parallel is negative, although
the crown is thinner than the springing section. This behavior did not happen by chance, but it was the result of some precise
decisions made by Brunelleschi. On one hand, the less the crown thickness the less the springing thrust; on the other hand,
however, a domical dome with a very thin crown cannot bear its own weight, unless it is topped by a heavy lantern. Brunelleschi knew the empirical aspects of these rules; consequently, he saved all the weight that was not strictly required by the
dome to prevent the failure mode of Fig. 2 from triggering when the dome did not have the lantern yet. To this end, the
degree of taper was not pronounced at the crown. In so doing, the dome resisted the failure mode of Fig. 2; moreover,
the lower thrust necessary to prevent the failure mode of Fig. 3 from triggering was reduced to the minimum.
Brunelleschi knew that his dome would have guaranteed the equilibrium against the mechanism with fixed springing
(Fig. 2) not only without the lantern, but also without the crown, because he thought that the dome would not have split
into vertical one-dimensional elements during the construction work. It was on the basis of this conviction that he proposed
to build the dome without temporary falsework (without centering). In particular, Brunelleschi decided to lay the bricks
along the parallels, rather than along the meridians. This construction technique implied to build the dome in subsequent
rings from the springing to the crown, rather than in subsequent arches starting from the ribs and ending with the webs.
The construction work of the shells was completed in 1436. The lantern was positioned in 1471, after the death of
Brunelleschi.
The preliminary design of the dome developed in the XIV century did not include the lantern, but only the idea of crowning the dome with a belvedere. Another competition was held for the lantern, which again was won by Brunelleschi. When
Brunelleschi designed the lantern he knew that stability did not require that weight; but above all, he knew that the heavier
the lantern the greater the lateral springing thrust. On one hand, he designed the lantern that the Florentine expected, which
allowed him to win the competition. On the other hand, however, he had realized that he had to solve the problem of the
transmission of the lateral thrust between dome and drum through the springing, although the dome was domical, since the
lantern that he was about to design would have been heavy and the drum was relatively thin and had eight large eyes.
6. Drum and buttresses
The structural system formed by the masonry dome and drum (tambour) that the dome rests upon is stable only if the
maximum springing thrust that the drum can bear is greater than the lower thrust of the dome. If conversely the lateral
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Fig. 10. Not only does this thrust line satisfies inequalities (1) and (2), since the thrust line is entirely contained within the arch, but it also satisfies (3),
since it is also entirely contained within the abutments. Therefore, the structure is stable under the applied load.
Fig. 11. In this masonry system, every thrust line that lies within the arch does not lie within the abutments, and vice versa. Since it is impossible to place a
thrust line entirely within the masonry, this structure is unstable under this load. Failure occurs with the mode of Fig. 3, i.e. the abutments and springing
sections open and the crown moves downwards.
thrust-carrying capacity of the drum does not exceed the lower thrust, the system cannot prevent the formation of the mechanism with the springing parallel and drum that open (Fig. 3).
The comparison between the hemispherical and domical shapes shows that, at the springing, the slope of the thrust line
of the former (Fig. 10) is less than that of the latter (Fig. 11), all other things being equal. Thus, the hemispherical dome has a
lower thrust greater than the domical dome. Hence, the drum of the hemispherical dome has to be thicker than that of the
domical dome, and may require buttresses or belts (Figs. 10 and 11).
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6.1. The drum in real historic domes
Andrea Palladio wanted that the dome of the temple of Villa Barbaro had neither buttresses nor external circumferential
belts (Fig. 8). Although the dome was hemispherical, Palladio however was able to reach the target, since he had realized that
the less the weight of dome and lantern the less the lower thrust. Thus, Palladio designed the dome and lantern by using the
minimum weight that was possible.
There was hence another reason why the lantern was made as lightweight as possible, in addition to the reason explained
in Section 5.3—namely, also to provide the temple dome with adequate safety margin against the mechanism shown in Fig. 3.
The minimum thickness-to-span ratio for the hemispherical masonry dome without lantern is 1/32 [33,37]. The timber
lantern in imitation of a marble lantern (Section 5.3) increased that limit value only marginally. Moreover, the brick pattern
adopted by Palladio decreased that limit value (Section 7). Thus, Palladio could assign to the dome a thickness even less than
the minimum; the thickness-to-span ratio of this temple dome is 1/34 of the span. In so doing, Palladio made the dome as
lightweight as possible, which reduced the demand for springing thrust.
However, Palladio also wanted a thin drum. To this end, Palladio’s plan for the temple of Villa Barbaro included an internal
masonry annular beam at the top of the drum, encircling the springing. The masonry annular beam is a brickwork curved
beam whose brick pattern is designed so that the parallel tension forces are resisted by friction stresses exchanged between
the bricks and by tension stresses in the bricks. Therefore, the masonry annular beam (which seems a combination of contradictory terms) requires no help from the mortar, which can even be cracked.
The encircling action provided by the masonry annular beam allowed Palladio to further reduce the thickness of the
drum, as desired.
The competition won by Brunelleschi for the dome of the Santa Maria del Fiore Cathedral in Florence did not allow the new
design to modify the drum and lateral chapels, which had already been built in the previous century.
Brunelleschi knew that the dome he was designing was atypical, since it was very thick and would have carried a heavy
lantern; moreover, the drum was not particularly thick and had eight large eyes. Therefore, the lower thrust of the dome was
a huge problem to solve.
The first key design criterion was that the less the weight of the dome the less the value of the lower thrust. To this end,
Brunelleschi’s plan included an inner and an outer shell held together with a ring and rib system, where a significant fraction
of the volume was empty. Hence this dome is like a honeycomb sandwich structure. According to Brunelleschi’s plan, moreover, the two shells gradually tapers in thickness over the span, from the springing to the crown, which further reduced the
total weight. In so doing, the lower thrust was much less than that of the same dome composed of a unique shell.
In addition, Brunelleschi incorporated into the dome a series of wooden and stone chains, which run round the circumference, encircling the dome where the thrusting action was high. Unlike external belts, these circumferential rings were
invisible, since they were buried in the dome’s masonry. These construction techniques further reduced the lower thrust.
Although the drum was buttressed by the nave and chapels around the dome’s octagon, however the above solutions
were insufficient to guarantee the equilibrium of the dome. Fortunately, Brunelleschi had realized that the design still lacked
something.
7. The structural role of brick pattern
Meridian (i.e., vertical) cracks that pass through the entire thickness should be considered as intrinsic to a masonry dome,
since the own weight of the structure is sufficient to cause meridian cracking [2,3,6–9,13–19,31,32,35,40–47].
The circumferential stresses induced by the dead loads are substantially constant along each parallel and reach the maximum at the springing. Thus, a vertical crack initiates in the point of the springing where tension strength is minimum and
propagates vertically, cutting the entire masonry thickness, upwards toward the crown and downwards into the drum.
As aforementioned, the mortar may crack even under very low tension stresses [1,4,10,11,16,17,20,22,23,30,36,43,
45–47]. Nevertheless, the mechanism of interlocking and friction between the bricks (stones) allows masonry to transmit
significant tension stresses also when the mortar is cracked [34,48]. Therefore, the maximum value that the tension stress
can reach is dictated only by the brick (stone) pattern (i.e., the disposal of bricks or stones). This means that masonry tension
strength is dictated neither by the mortar, which may provide irrelevant tension stresses, nor by the brick (stone), which has
tension strength greater than the pattern.
Hence, the points along the springing where vertical cracks initiate depend on the brick (stone) pattern, which therefore
dictates the shape of the cross-section of the arches that the masonry dome splits into.
The ultimate behavior of a masonry dome depends only on the ultimate behavior of these arches, which, in turn, depends
on the thickness of their cross-section. The thickness of the cross-section is the distance between the rotation axis of the
pins at the extrados and intrados. Hence, the position of the meridian (vertical) cracks influences the ultimate behavior of
the dome.
A dome with circular plan splits into arches whose cross-section is an annular sector. The greater the tension strength of
the brick pattern the greater the distance between two consecutive vertical cracks, and, in turn, the greater the central angle
of the annular sector. The angle of the annular sector dictates the thickness of the arches that the dome splits into. Thus, the
stronger the brick pattern the greater t of (1). The empirical aspects of this relationship were known by architects and master
masons.
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passing-
passing-
through crack
through crack
corner crack
passing-
passing-
through crack
through crack
nave
Fig. 12. Crack pattern of the Florence dome (survey of Professor Andrea Chiarugi, who first provided a scientific interpretation of the behavior of the
Brunelleschi’s dome, including the construction technique). Every other web of the dome is cracked: Four cracks that cut the entire thickness of the two
shells in the middle of the web. The dome exhibits other cracks (e.g., in the vertexes, called corner cracks), but these cracks do not pass through the entire
thickness (they form pins). Conversely, the four cracks in the figure cut the dome from its inner to its outer surfaces, from the base of the drum to the crown.
Therefore, only the four cracks in the webs dictate how the dome splits into arches and the dome’s ultimate behavior, while the other cracks do not
influence structural safety.
In order to investigate the role of the brick pattern in domes with circular plan, Villa Barbaro’s temple dome (Fig. 8) was
analyzed by using (10); the virtual work done by the shell and lantern as they move through the displacements of the failure
mode shown in Fig. 3 provided the lower thrust. Eq. (10) was applied both to the actual arches that the dome is split into,
whose geometry had been obtained by a survey, and to the arches that the dome would have been split into according to the
no-tension assumption, whose geometry prevented the development of tension stresses. More specifically, Eq. (10) was
applied both to four identical arches whose cross-section was a circular annulus with central angle equal to 90 degrees,
and to arches whose cross-section was a circular annulus with infinitesimal central angle. In the former case, (10) used
the thickness t that had been provided by the survey of the dome; in so doing, (10) obtained the actual lower thrust, which
is the result of how the dome had cracked. Conversely, in the latter case, (10) used the thickness t of the shell; in so doing,
(10) obtained the lower thrust that the dome would have had if cracking had produced the weakest resisting system. The
difference between the former and latter lower thrust quantifies the role of the brick pattern in this dome.
The actual value of the lower thrust resulted to be 0.144 of the total weight of dome plus lantern. The lower thrust according to the no-tension assumption resulted to be 0.361 of the total weight of dome plus lantern. Contrary to the former value,
the latter value could not have been counterbalanced by the drum of the temple, which is thin and not buttressed.
A dome with polygonal plan (e.g., the ribbed domical dome) splits into arches whose cross-section has a shape that
depends on whether the vertical (meridian) cracks initiate at the ribs or in the webs (Figs. 12 and 13). If cracking initiates
and develops at the ribs (Fig. 14), the cross-section of each arch is a rectangle with the thickness of the web (which is the
thickness of the dome). Conversely, if cracking initiates and develops in the webs, the cross-section of each arch is a composite geometric figure with a thickness substantially greater than that of the dome (Fig. 13). Accordingly, both the thickness-to-span ratio and lower thrust of a dome with polygonal plan drastically depend on the position of the vertical cracks.
If the bricks (stones) are laid onto horizontal planes, the component of the meridian forces parallel to a mortar joint does
not lie within the friction angle (sliding failure). Thus, the bricks (stones) have to lie along the thickness of the dome, so that
the face of each masonry unit and each mortar joint are orthogonal to the meridian forces.
A web of a polygonal masonry dome is composed of bricks or stones that lie in continuous rows (courses of bricks or
stones); each course starts at a rib and ends at the consecutive rib. If a course lies both along the thickness of the dome
and along a parallel, this course lies on one plane. However, the courses that form a parallel of the dome, each one in its
web, lie on planes different from each other, i.e., a different plane per web. The only exception is when the tangent to the
rib is vertical (as it usually is at the springing section). Apart from this exception, hence, the intersection at a rib of two consecutive courses of the same parallel does not lie on a plane: The planes tangent to the course on the right and left of a rib are
different from one another.
Hence, if the courses are laid along the parallels of the dome, the faces of the bricks (stones), respectively, on the right and
on the left of a rib form an angle different from the straight angle. In this case, the courses exhibit a discontinuity of tangent
at the ribs. This angular discontinuity of the brick pattern causes the tension strength at a rib to be very low; in particular, it
just reaches the tension strength of the mortar.
Ultimately, if the brick (stone) courses are laid along the parallels of the dome, the contiguous webs are not connected to
one another; so, the brick pattern provides the ribs with marginal capacity of bearing parallel (hoop) tension forces.
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Fig. 13. The four passing-through cracks in the middle of the webs split the dome into four arches with ‘‘C’’-shaped cross section (drawn in bright). The
actual thickness of each ‘‘C’’-section is the distance between the rotation axes of the potential hinges at the extrados and intrados of a section. Hence, the
thickness of the arches that the dome splits into when the passing-through cracks are in the webs is drastically greater than the thickness of the dome.
Fig. 14. Hypothetical splitting of the Florence dome if the passing-through cracks had formed at the ribs: Eight cracks that would have cut the entire
thickness of the ribs. With this cracking, the dome would have split into eight arches with rectangular cross-section (drawn in bright), whose thickness
would have been that of the dome.
The angular discontinuity can be eliminated only if the masonry units that are at a rib lie on the plane orthogonal to the
rib. It means that the faces of the masonry units at the beginning and the end of a course have to lie on the cross-section of
the rib they intersect. In so doing, the courses do not exhibit any angular discontinuity at the ribs. In order to have continuity
along the courses as well, the shape of the web has to be described by courses whose trajectory has double curvature. Conversely, if the courses have the trajectory of the parallels (i.e., single curvature), either there is an angular discontinuity at the
ribs or the courses do not describe the web.
The application of Eq. (10) showed that, if the Brunelleschi dome (Fig. 9) had split into webs (Fig. 14), the lower thrust
would have been 0.244 of the weight of the dome. This is a very great value for the lower thrust, which the drum would
not have withstood, although the nave and chapels around dome’s octagon buttressed the drum.
From the analysis of the construction technique of the Florence dome, it is clear that Brunelleschi knew that problem and
how to solve it [49]. The contemporary rules derived from experience (by trial and error) provided him with the information
that, if the dome had split into arches with ‘‘C’’-shaped cross-sections, the drum would have tolerated the lateral thrust.
Brunelleschi had also understood the relationship between the splitting of the dome into arches and the brick patterns.
In fact, the brick courses were laid so that the ribs were provided with greater tension strength than the webs [50].
Firstly, Brunelleschi adopted a herringbone pattern, where each brick laid on a plane directed towards what Brunelleschi
assumed to be the center of the dome. In so doing, the bricks could be laid without temporary falsework (without centering)
[51]. Then, Brunelleschi designed the brick pattern so that each course had a double curvature, with the bricks at the edges of
the course that lay on the cross-section of the ribs [49–51].
In so doing, each web of the Brunelleschi dome was strongly connected to the adjacent webs, along the entire rib.
Connecting the adjacent webs to one another, Brunelleschi succeeded in providing the ribs with greater masonry tension
strength than the webs. Consequently, the vertical cracks due to the parallel tension stresses initiated and developed in the
webs (Fig. 12), while the parallel tension stresses did not crack the ribs [51].
More specifically, few years after the completion of the Florence dome, the parallel tension stresses caused four out of the
eight webs to crack (Fig. 13); every other web cracked around the middle and each crack cut the entire thickness of the dome
(i.e., cut both the shells).
332
The octagonal drum rests upon four piles; thus, the drum is supported at every other span. Consequently, cracks occurred
in the four webs that were not directly supported by the piles. Each crack initiated at the springing and propagated towards
the crown and into the drum [49–51].
According to Brunelleschi’s plan for the dome, the brick pattern forced the dome to split into arches whose cross-section
was composed of a web and half of the two adjacent webs (Fig. 13). The thickness of the ‘‘C’’-shaped cross-section cut down
the lower thrust to a value that could be tolerated by the drum. Thus, the opening of the springing parallel and the drum
(Fig. 3) was prevented from occurring. Moreover, the dome was provided with a considerable safety factor. More specifically,
the multiplier of the uniformly distributed live load obtained by applying Eq. (10) to the arches of Fig. 13 over the mechanisms of Fig. 3, which is the mode that dictates the failure, results to be:
PL dPL þ
k ¼ PN
PN
i¼1 F i
j¼1 P j
dFi
dPj
¼ 0:18 PL þ
N
X
!
Pj
ð23Þ
j¼1
Eq. (23) shows that the Florence dome has a considerable safety factor.
Not only was the pivotal role of Brunelleschi’s brick pattern proven theoretically in this research, but it is also field-proven
by the dome of the sanctuary Basilica of Madonna dell’Umiltà, in Pistoia (Italy) (Figs. 15 and 16).
The construction work of this sanctuary date back to 1495 and lasted more than seventy years. The church was designed
by Giuliano da Sangallo. The first level of the drum was designed by Ventura Vitoni, who continued the construction job after
the death of Giuliano da Sangallo. The upper level of the drum and the dome were designed by Giorgio Vasari, from 1561 to
1567, in imitation of the Brunelleschi dome in Florence (Fig. 17). The design of this dome reproduced the main key features of
the Florence dome, including the double shells, materials and outer decorative details (Figs. 15 and 16). But the design
shifted towards the contemporary style; in particular, the domical shape shifted towards the hemispherical shape. Moreover,
Vasari used a brick pattern different from that of Brunelleschi’s dome, since he had not understood its pivotal role.
Hence, Vasari designed a quasi-hemispherical dome although the lantern was heavy (Fig. 18); he laid the brick courses
along the parallels, disregarding or ignoring the double curved herringbone courses of the Florence dome; and he connected
the inner shell to the outer shell with weak joints, disregarding or ignoring the strong ring and rib system that held together
the inner and an outer shell of the Florence dome.
Fig. 15. Dome of the Santuario della Umiltà, in Pistoia (Italy).
333
Fig. 16. Octagonal ribbed domical dome of the Umiltà sanctuary, which was designed by Giorgio Vasari in imitation of Brunelleschi’s dome.
Fig. 17. The dome designed by Vasari was about to collapse, and required to be strengthened. Circumferential metal belt were installed just after the
construction, from the springing to the haunches of the dome.
The brick courses along the parallels caused the tension strength at the ribs to be dramatically lower than in the webs,
due to the angular discontinuity at the ribs. Thus, the parallel tension stresses reached masonry tension strength at the ribs,
instead of in the webs. Consequently, the dome cracked at the ribs.
The meridian (vertical) cracks occurred already during the construction of the dome, before installing of the lantern. The
parallel tension stresses due to the dead load of the dome caused some ribs to crack; the cracks initiated at the springing and
propagated towards the crown, cutting the entire thickness of the two shells at the ribs, and into the masonry drum, cutting
334
Actual
shape
Outer
Hemispherical
shape
shell
Inner
shell
Fig. 18. Vertical cross-section of the dome of the Santuario della Umiltà, in Pistoia (Italy). The section cuts the dome and the drum (Survey G. Tucci). The
graphical construction of the figure shows the deviations of this dome from the hemispherical shape. This graphical construction was performed to prove
that this dome is closer to the hemispherical dome than to the domical dome.
its entire thickness. More specifically, now the dome exhibits eight meridian cracks at the eight ribs; but it is impossible to
know whether all the eight cracks were caused by the own weight of the dome, and therefore all of them occurred during the
construction, or some of them were caused by the own weight of the dome and the other cracks by the weight of the lantern,
and therefore some cracks occurred after its completion.
The cracks at the ribs split the dome into arches with rectangular cross-section, i.e. the section of the web, as in Fig. 14.
Hence, the dome splits into arches whose thickness was equal to the thickness of the web (i.e., of the dome), which was very
small in comparison to the span of the dome.
The consequence of having ignored Brunelleschi’s design was fatal for Vasari’s design (fortunately not for the dome). The
virtual work theorem proved that the cracked dome and drum could not prevent the formation of the mechanism shown in
Fig. 3 even without the lantern. In fact, on the assumption that all the eight cracks at the ribs were caused by the own weight
of the sole dome, and hence that all the eight cracks had already formed before the lantern was installed:
PN
j¼1 P j
dPj
i¼1 F i
dFi
k ¼ PN
¼ 0:12 N
X
Pj
ð24Þ
j¼1
The negative value of k of (24) confirms that that the dome could not bear even the own weight, since the rectangular
cross-section of the arches that the dome had split into implied a lower thrust greater than the maximum value that the
drum could provide the springing with.
Fig. 19. Metal tie-rods installed by Bartolomeo Ammannati, four years after the completion of the dome. The tie-rods eliminated the demand of springing
lateral thrust of the dome.
335
The negative value of k provided by Eq. (24) is the result of the quasi-hemispherical shape, the weak connections between
the two shells, brick courses laid along the parallels, and a drum neither thick nor buttressed.
Eq. (10) was also applied to the same dome, but split into arches with cross-section composed of a web plus half of the
two adjacent webs, as in Fig. 13. In this case, Eq. (10) provided a positive load multiplier, although small. Thus, the failure of
Vasari’s design was mainly due to the brick pattern and the weak connection between the external and internal masonry
shells. If Basilica of Madonna dell’Umiltà dome in Pistoia had been cracked in the same way as the Santa Maria del Fiore dome
in Florence, at least it would not have failed under the own weight. Moreover, the application of Eq. (10) to the dome with its
lantern proved that the dome split into "C"-shaped arches would also have carried the lantern, although heavy.
Also the quasi-hemispherical shape in lieu of the domical shape had a certain influence on the failure (Fig. 18), but much
less than the other two issues. Moreover, also the fact that the lateral force-carrying capacity of the drum was not high had
certain effects but not decisive on failure.
Observed the gaps at the ribs during the construction of the dome, the initial design was modified with the insertion of
some tie-rods buried into the masonry. In so doing, the construction of the dome could be completed.
Then, the lantern was installed at the crown. The lantern, which was designed by Vasari and, again, imitated that of the
Brunelleschi dome, was particularly heavy. Consequently, the lantern further increased the lower thrust. As aforementioned,
the weight of the lantern either cracked the ribs that had not yet cracked or increased the gaps of the eight cracks that had
already formed under the own weight of the dome.
In 1571, i.e. four years after the end of the construction work, the dome exhibited other symptoms of structural malfunctioning. In 1576, the crack at the ribs had produced considerable gaps between the webs, and the opening of the springing
had worsened. For this reason, Bartolomeo Ammannati was entrusted with the rehabilitation of the dome. Ammannati
installed several layers of circumferential (annular) metal tie-rods (belts), which have become part of the dome since then
on (Fig. 19). These belts reduced the lower thrust to zero; thus, the drum was not obliged any longer to provide the dome
with a certain lateral thrust. In so doing, the dome became a stable and safe structure.
The Umiltà church was inaugurated in 1582 only. The structural work lasted until 1584.
8. Conclusions
This paper has shown that the resisting system of a masonry dome is comprised of arches; these arches are the result of
meridian (vertical) cracking caused by the parallel (circumferential or hoop) tension stresses. Ultimately, a dome splits into
arches, and therefore the masonry dome is a one-dimensional trusting structure and not a two-dimensional shell.
Due to the lateral thrust transmitted through the springing sections, the resisting system also has to include the structure
that the dome rests upon, i.e. the masonry drum (tambour). Moreover, since the parallel tension stresses cause cracking of
the drum too, the resisting system has to include the structures that may buttress the drum (e.g., in a church, the nave and
lateral chapels).
The shape of the resisting structure depends on how the dome splits into arches. The splitting of the dome is dictated by
the tension strength of masonry, which does not depend on the mechanical characteristics of the materials that masonry is
made of, but instead on the brick (stone) pattern. In particular, the brick (stone) pattern dictates the thickness of the arches
that the dome splits into, and therefore it dictates the thickness of the resisting system.
Depending on the shape of the shell (hemispherical, domical), the resisting system may also include the lantern, while in
the other cases the lantern must belong to the load system. Usually, in the former case, the lantern is heavy.
This paper has provided the governing formulas, which are expressions that can be used to predict the configuration of
the resisting system, to interpret the splitting of the dome into arches, to identify the failure mode that dictates the ultimate
behavior, and to assess the load-carrying capacity of the masonry dome.
The paper has applied those formulas to well-known masonry domes taken from architectural Italian cultural heritage—
namely, the domes of the temple of Villa Barbaro in Maser (Treviso) designed by Palladio, of the Santa Maria del Fiore cathedral in Florence, designed by Brunelleschi, and of the L’Umiltà sanctuary in Pistoia, designed by Vasari.
The difference between the way in which architects and master masons operated, and the way in which this research has
been developed is that past builders used the contemporary rules derived from experience, which were field-proven by trial
and error. Conversely, this paper has derived and used theoretical expressions that allow the designers to operate without
reproducing prior work. The reason for which masonry arches and domes do not require mechanics is that the ultimate
behavior and safety of these structures depend on the geometric proportions only. Hence, geometry is for masonry structures
what mechanics is for reinforced concrete and steel structures. Note that the safety of masonry structures is dictated by the
ultimate limit state only, while the serviceability limit state is always satisfied in masonry buildings. In fact, masonry does
not exhibit any structural degradation; moreover, masonry structures carry the loads by means of in-plane behavior, whose
stiffness is ample.
Unfortunately, masonry codes, including those of the new generation [24–26], do not wholly recognize the resisting system obtained in this paper. In particular, they do not recognize the role of the brick (stone) pattern in any way.
For masonry domes, consequently, codes are overly-conservative, and sometimes even unrealistic. For instance, safety
assessment in compliance with the Italian masonry code of the Santa Maria del Fiore dome and of the L’Umiltà dome provide
almost the same results. However, the former has a significant safety margin, which has been demonstrated theoretically in
336
this paper, but above all is field-proven by its long lifespan without any treatment. On the contrary, the latter had a negative
safety margin, which has been demonstrated theoretically in this paper, but above all was field-proven by the installation of
external metal belts immediately after its construction. For the Italian masonry codes, the Brunelleschi dome would require
metal circumferential belts at the springing and drum, like those installed around the Vasari dome by Ammannati. The only
reason why this treatment has been avoided for the Florence dome is that Italian authorities have not prescribed structural
assessment for this dome.
In total opposition was the destiny of the dome of Villa Barbaro’s temple, designed by Palladio. In order to celebrate five
centuries on from the birth of Palladio, the dome was subjected to restoration, which included a safety assessment. The point
is not whether safety assessment considered the mechanisms, according to the new codes, or the maximum stresses, which
is the approach of the previous Italian code. The point is that also if safety verification had accomplished a mechanism analysis, the result would not have justified the equilibrium of this dome. In fact, the lower thrust, which was here estimated
according to the no-tension assumption, resulted in being greater than the maximum thrust that the drum could provide
the dome with. In compliance with the present Italian code (or with the past code), hence, a circumferential pre-stressed
steel belt was installed onto the drum just under the springing of the dome. From then on, Villa Barbaro’s dome was no
longer authentic; it was something different from that which Palladio designed; and ironically the cause was an activity
started to celebrate the anniversary of his birth. Moreover, the same increment of strength could have been obtained by
using Fiber-Reinforced-Polymer composites instead of steel, for the belt. In so doing, the belt would have not provided
the dome with any encircling action until the structure is stable, preserving authenticity.
The domes of Brunelleschi, Palladio, and Vasari prove that the only way to avoid invasion of space of safeguarding to conservation is to avoid safety assessment. Until adequate balance is found between safeguarding and conservation, structural
interventions on cultural buildings should be executed only if the building suffers from serious structural damage or has to
be remodeled. Structural interventions that only aim at guaranteeing safeguarding should be executed only when the verbs
to conserve and to assess are combined into a unique verb—to restore.
The reason for which some contributions to structural capacity are not recognized is that a code incorporates only the
contributions that are general, while it ignores the contributions that are specific to a building. However, almost every historical building was constructed using specific techniques to obtain particular results. The purpose of this paper is to generalize some of the contributions with regards to the masonry dome, so that in future they can be included in masonry codes.
To this end, this paper proposes that the resisting system of a masonry dome should be defined by considering the tension
strength of masonry due to the brick (stone) pattern, as well as taking into consideration the construction techniques. In particular, the brick pattern together with some construction techniques provides tension strength due to interlocking and friction. This proposal does not require the value of masonry tension strength but only a comparative analysis that determines
where masonry tension strength reaches the minimum values, so as to predict or justify the positions of the meridian cracks.
According to this proposal, masonry tension strength has no role in assessing the load-carrying capacity of the resisting
system, which is calculated under the no-tension assumption, as prescribed by codes.
Ultimately, the resisting system according to this proposal is the actual structure that bears the ultimate load, while the
codes consider the resisting system that results from the weakest splitting possible.
This proposal does not include seismic assessment. While the cracks produced by the static loadings are localized, the
cracks produced during a severe earthquake are diffused throughout the entire dome. Thus, while under static loads the
masonry dome splits into an arch system, during a severe earthquake a masonry dome turns into an assemblage of dry
masonry units.
In conclusion, the mechanical behavior of the masonry dome during a severe earthquake has to be predicted under the
no-tension friction assumption. Accordingly, the resisting system that bears the seismic forces derives from the masses only.
Therefore, the resisting system that bears the seismic forces has a lower capacity than the resisting system that bears the
gravitational (and wind) loads. Nevertheless, the brick (stone) pattern provides the dissipation system with a capacity
greater than that recognized by the codes.
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Resisting system and failure modes of masonry domes

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