Structure and History of Guastavino Vaulting at the
Metropolitan Museum of Art
by
Jonathan Calman Ellowitz
Bachelor of Arts in English, Skidmore College, 2007
Post-baccalaureate certificate in Civil and Environmental Engineering, Tufts University, 2013
Submitted to the Department of Civil and Environmental Engineering in Partial Fulfillment of
the Requirements for the Degree of
Master of Engineering in Civil and Environmental Engineering
at the
Massachusetts Institute of Technology
June 2014
© 2014 Jonathan Calman Ellowitz. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any medium now known or
hereafter created.
Signature of Author:_____________________________________________________________
Department of Civil and Environmental Engineering
May 9, 2014
Certified by:___________________________________________________________________
John A. Ochsendorf
Professor of Architecture and Civil and Environmental Engineering
Thesis supervisor
Accepted by:___________________________________________________________________
Heidi M. Nepf
Chair, Departmental Committee for Graduate Students
Structure and History of Guastavino Vaulting at the
Metropolitan Museum of Art
by
Jonathan Calman Ellowitz
Submitted to the Department of Civil and Environmental Engineering
on May 9, 2014, in Partial Fulfillment of the Requirements for the Degree of
Master of Engineering in Civil and Environmental Engineering
Abstract
The R. Guastavino Company constructed structural masonry vaults for wings E and H of the
Metropolitan Museum of Art, New York (the Museum) between 1910 and 1912. In the early
1960s the Museum relocated the Egyptian and Near- and Far-Eastern galleries to these wings,
which in combination with growing numbers of visitors doubled the design live load for the
vaults. To accommodate this change, the Museum demolished the Guastavino vaults and
replaced them with steel beams even though consulting engineers had not performed a thorough
structural assessment prior to demolition. The vaults were a part of a landmark McKim, Mead
and White building and warranted appropriate analysis to determine their capacity under
increased loading demand.
This thesis investigates both history and structural analysis. Primary sources reveal that the
consulting engineers hired by the Museum were unfamiliar with the structural analysis of
unreinforced masonry vaults, leading to the decision to demolish them. These historical events
contextualize the quantitative focus of the thesis, which is to provide engineers with accessible
techniques to structurally assess unreinforced masonry systems. This enables decisions based on
evidence rather than a lack of comprehension of structural behavior.
Three important assumptions about masonry behavior are adopted: Masonry has no tensile
strength, it has unlimited compression capacity, and it will not fail from sliding between blocks
or segments. With these assumptions, masonry analysis is primarily a problem of stability rather
than elasticity. Analytical equilibrium and graphical techniques are used to determine vault
stability under dead and live loading. Two vaults are investigated: A scaled Guastavino vault
built for a recent exhibit, and a representative cross-vault from wing H of the Museum. Two
methods of modeling structural behavior are used for each analysis technique: The triangle-arch
and sliced parallel arches methods. Analysis shows that the Museum vaults had the capacity to
resist the increased live load.
Thesis Supervisor: John A. Ochsendorf
Title: Professor of Architecture and Civil and Environmental Engineering
Acknowledgements
It is amazing to study a subject for many months, and one day to pause and realize how many
people contributed to its development. A number of excellent minds made this thesis possible.
First, I owe mighty gratitude to Professor John Ochsendorf, who introduced me to the fabulous
achievements of the R. Guastavino Co. and to the particularly intriguing topic of the demolition
of the Guastavino vaults at the Metropolitan Museum of Art. John is a true “Guastafarian,” and
spreads his gospel well. One only has to see the growing number of engineers, architects, and
historians drawn to Guastavino structures to understand John’s ability to communicate his own
fascination with the topic. Thank you John for pushing me to really comprehend how these
structures work. I will be a better engineer for it.
I am equally thankful for the help and camaraderie of two graduate students in MIT’s Building
Technology program: William Plunkett and Caitlin Mueller. Despite his busy schedule, William
was always available to check my work or to hear out my ideas about the challenging structural
analysis of these vaults. He offered expert suggestions that have surely improved the quality of
this thesis. Despite her busy schedule finishing both a PhD in structural optimization and a
Masters in design computation, Caitlin, a true teacher, took great interest in my study and was
always on hand to field ideas and offer suggestions that improved the scope of my inquiry. I
extend my thanks also to the entire Structural Design Lab (SDL), a round-table research group at
MIT led by John Ochsendorf. Students at the SDL thoughtfully and acutely offered ideas and
support throughout the process of my work. I hope my input in their studies was as valuable as
their contribution to my progress. I thank Kristian Fennessy, a student of architecture at MIT,
who during his senior year explored the issue of the demolition of the Metropolitan Museum
vaults in wings E and H for a class paper. Kristian initially exposed the importance of this topic,
and I thank him for laying the foundation for future study of it.
This thesis would not have been possible without the assistance of two dedicated archivists: Janet
Parks of Columbia University’s Avery Drawings and Archives Collection, and James Moske,
Managing Archivist of the Metropolitan Museum of Art. Janet aided me in my search for
relevant drawings and primary sources. The results of our collaboration were instrumental in
moving the thesis forward. James patiently facilitated my research at the Museum, helping me
sort through hundreds of documents and drawings in search of useful sources. Thank you Janet
and thank you James.
I am fortunate to have the family support for my structural engineering aspirations. My parents
Jeralyn and David, my brothers Jake and Daniel, and my grandparents Lenore and Jack, and Sam
and Grace (z”l), have unflinchingly promoted my course of study and have even taken personal
interest in the elegant structures designed and built by the Guastavinos. Thanks to this fantastic
family for its loyalty.
And I offer enormous thanks to my beautiful wife Miriam, to whom this thesis is dedicated with
love. Miriam understands why these vault structures have had such intellectual intrigue for me.
For her devotion to my passions, I make it my goal to take equal interest in hers forever.
Contents
1
Introduction ......................................................................................................................... 10
1.1
Motivation ................................................................................................................................. 10
1.1.1 Prevalence of Beaux-Arts buildings in New York City ........................................................ 10
1.1.2 Impetus for study: Demolition at the Metropolitan Museum of Art ..................................... 10
1.1.3 The role of the structural engineer ........................................................................................ 10
1.2
History of the Guastavinos ........................................................................................................ 11
1.2.1 Rafael Guastavino Sr............................................................................................................. 11
1.2.2 Rafael Guastavino Jr. ............................................................................................................ 12
1.3
Thesis context and mission ........................................................................................................ 12
1.3.1 The future of historic masonry architecture .......................................................................... 12
1.3.2 Strategies for analyzing unreinforced masonry vaults .......................................................... 12
2
1.4
Problem statement ..................................................................................................................... 12
1.5
General thesis outline going forward......................................................................................... 13
Literature review ................................................................................................................ 15
2.1
Chapter objectives ..................................................................................................................... 15
2.2
Principles of masonry equilibrium ............................................................................................ 15
2.3
Guastavino’s masonry theory .................................................................................................... 18
2.3.1 His Essay at MIT, 1892 ......................................................................................................... 18
2.3.2 Disputing some of Guastavino Sr.’s assertions ..................................................................... 20
2.4
Masonry analysis ....................................................................................................................... 22
2.4.1 Graphical analysis ................................................................................................................. 22
2.4.2 Analytical equilibrium analysis ............................................................................................. 25
2.4.3 Membrane analysis................................................................................................................ 26
2.4.4 Thrust Network Analysis (TNA)........................................................................................... 29
2.4.5 Considering finite element analysis ...................................................................................... 29
2.5
3
Claim of thesis authenticity ....................................................................................................... 31
The Metropolitan Museum of Art additions, 1910-1912 ................................................. 33
3.1
Chapter objectives ..................................................................................................................... 33
3.2
Building wings E and H............................................................................................................. 33
3.3
Renovation program at the Met ................................................................................................. 34
3.4
Proof of Guastavino vaults ........................................................................................................ 35
3.4.1 Early drawings of wing E ...................................................................................................... 35
3.4.2 The Wills & Marvin sub-contract for wing H ....................................................................... 37
3.5
History of the vaults’ demolition ............................................................................................... 37
3.5.1 Early stages of renovating wings E and H ............................................................................ 37
3.5.2 The decision to demolish the vaults ...................................................................................... 39
7
3.5.3
3.6
4
Renovations are underway .................................................................................................... 40
Conclusions ............................................................................................................................... 42
Analysis of scaled Guastavino replica ............................................................................... 44
4.1
Chapter objectives ..................................................................................................................... 44
4.2
Replica design and geometry ..................................................................................................... 44
4.3
Modeling force flow for triangle-arches and sliced parallel arches........................................... 47
4.4
Structural analysis ..................................................................................................................... 48
4.4.1 Static equilibrium analysis .................................................................................................... 48
4.4.2 Results discussion – static equilibrium ................................................................................. 53
4.4.3 Graphical analysis ................................................................................................................. 55
4.4.4 Results discussion—graphical analysis ................................................................................. 69
4.5
5
Conclusions ............................................................................................................................... 70
Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H ................. 72
5.1
Chapter objectives ..................................................................................................................... 72
5.2
Authoritative drawings .............................................................................................................. 72
5.3
Loading assumptions ................................................................................................................. 73
5.4
Cross-vault geometry................................................................................................................. 73
5.5
Structural analysis: Cross-vault ................................................................................................. 75
5.5.1 Equilibrium analysis: Distributed loading............................................................................. 77
5.5.2 Results discussion—static equilibrium ................................................................................. 80
5.5.3 Graphical analysis: Distributed loading ................................................................................ 81
5.5.4 Graphical analysis: Distributed loading with point load ....................................................... 94
5.5.5 Results discussion—graphical analysis ............................................................................... 104
5.6
6
Conclusions ............................................................................................................................. 104
Concluding discussion ...................................................................................................... 106
6.1
Results summary ..................................................................................................................... 106
6.2
Suggestions for future work .................................................................................................... 107
7
References .......................................................................................................................... 109
8
Appendix ............................................................................................................................ 113
8.1
Derivation of arch equilibrium equations ................................................................................ 113
8.2
Authoritative drawings of the Metropolitan Museum vaults ................................................... 115
8.3
Selected letters and documents ................................................................................................ 127
8
9
1 Introduction
1 Introduction
1.1 Motivation
1.1.1 Prevalence of Beaux-Arts buildings in New York City
In New York City the Beaux-Arts style of architecture is widespread and a hallmark of the built
environment. Architects like McKim, Mead and White and Richard Morris Hunt were well
known in the late nineteenth and early twentieth centuries. A lesser known partner in these
monumental construction projects is the R. Guastavino Company. Rafael Guastavino, Junior and
Senior, were structural engineer-architect-builders whose ingenuity helped construct hundreds of
Beaux-Arts buildings in America. The Guastavinos built structural tile vaults, arches, and domes
in hundreds of American buildings from 1889 to 1962. In many applications, preserving the
Guastavinos’ structural systems means preserving an entire Beaux-Arts building. This thesis
argues quantitatively that this vaulting has significant strength and loading capacity. When
structural engineers preserve and reinforce these vaults they preserve history, but they also
maintain valuable structural components. Efforts must be made to understand the structural
mechanics behind Guastavino vaulting because many of these structures are at risk of
demolition. This thesis investigates a case when fundamental misunderstanding of a Guastavino
structural vault led to its demolition as part of the expansion of the Metropolitan Museum of Art
(the Museum) in New York.
1.1.2 Impetus for study: Demolition at the Metropolitan Museum of Art
In 1962, Guastavino vaults in the north wings (E and H) of the Museum were demolished in
order to prepare the wings for greater loading from the relocated Egyptian and Far- and NearEastern exhibits. It is evident from the correspondences of New York City engineering officials
and consulting engineers that a lack of understanding of masonry vault structural behavior led to
the decision to demolish the vaults. It is the goal of this study to show how the vaults can be
understood structurally and to assess whether the vaults were safe for increased loading. Two
different methods of structural analysis are used in this objective.
1.1.3 The role of the structural engineer
Structural engineers can play a pivotal role in the reinforcement, refurbishment, and continued
use of Guastavino systems, but their general lack of confidence with unreinforced masonry is a
barrier to this objective. Structural engineering curricula focus on steel and reinforced concrete.
While those topics are of great importance to the field, unreinforced masonry has been proven by
history to be robust, long-lasting, and viable structurally, particularly in locations of low to
moderate seismicity. Of the thousands of vaults built by the company, there is no record today of
any structural failure of a Guastavino system which led to collapse (Reese 2008). This fact alone
is reason for engineers to study how to analyze Guastavino unreinforced masonry systems.
Through the topics of structural mechanics, statics, and dynamics, an engineer can learn about
10
1 Introduction
these systems and scientifically apply methods of analysis to reinforce them, preserve them, and
maintain them.
1.2
History of the Guastavinos
1.2.1 Rafael Guastavino Sr.
Rafael Guastavino Sr. began his career in his native Spain. Born in 1842 in Valencia, Guastavino
Sr. showed an early interest in building design. By 1861, Guastavino Sr. had enrolled in the
Escola Especial de Mestres d’Obres—the Special School for Masters of Works—in Barcelona.
In Spain, the education was organized to produce builders foremost. The Masters of Works
education was practical and included courses in mechanics, geometry, and construction. Part of
what Guastavino Sr. was learning was a 500-year-old technology of tile vaulting known as
“boveda tabicada” or “boveda catalana” in Spain. In addition to succeeding in the American
building market by serving as architect, structural engineer, and builder, Guastavino Sr. can be
credited with making widespread the use of a traditional Mediterranean vaulting system
(Ochsendorf 2010).
Before his immigration with his nine-year-old son (Rafael Guastavino Jr.) to the United States in
1881, Guastavino Sr. had designed and constructed large-scale works such as the Battlo’ Factory
in Barcelona and the La Massa theater in Vilasar de Dalt to the north. With Battlo’, the twentyfour-year-old Guastavino designed and built a sprawling industrial facility using a mix of tile
vaulting and iron (rods and columns). With La Massa, Guastavino designed and built a tile dome
spanning 56 feet for a new theater. These projects began to earn him renown. But in February
1881, he, his son, and a housekeeper sailed for New York City, where presumably Guastavino
imagined his skills could fulfill the huge demand for buildings in America’s booming cities
(Ochsendorf 2010).
In America, after failing to establish himself as an architect, Guastavino Sr. found a niche for
himself in fireproof construction of tile vaults and domes. American cities had experienced
cataclysmic fires that required municipalities, building companies, architects, and politicians to
rethink building design. The Chicago fire, which destroyed over 19,000 buildings in 1871, is an
example. Cities needed to rebuild, and they wanted their buildings to be fireproof. In 1889,
Guastavino Sr. convinced Charles McKim of McKim, Mead and White, to hire him to build
structural tile vaults for the new Boston Public Library. This commission would launch
Guastavino Sr.’s career and his company’s success, including vaulting for the same architects’
1910-1912 additions to the Metropolitan Museum of Art, the subject of this study. Though
Guastavino Sr. passed away in 1908, he had built a profitable construction company and his
structural engineering was often instrumental to Beaux-Arts architectural design. Guastavino
Sr.’s son, also Rafael, would continue the family business (Ochsendorf 2010).
11
1 Introduction
1.2.2 Rafael Guastavino Jr.
Guastavino Sr.’s son, who would receive four US patents before he was twenty years old, would
continue to grow and innovate the R. Guastavino Co. after his father’s passing. Among
Guastavino Jr.’s many engineering achievements are his foray into acoustics. With a noted
Harvard professor he developed acoustical masonry tiles that could absorb sound. One of his
greatest achievements was the dome of the cathedral of St. John the Divine near Columbia
University in upper Manhattan. It is a spherical dome with a 66-foot radius; at the base it has a
diameter of 98 feet; the top of the dome is 161 feet above the ground; the four pendentives
connecting the dome to the four side-arches are a foot thick; but the dome itself starts at 7.5
inches thick at the pendentives and narrows to 4 inches thick at the crown (Dugum 2013). This
impressive project was built in 1909 in the short span of fifteen weeks. That means that only a
year after his father’s death, Guastavino Jr.’s assumption of company leadership was marked by
one of the most significant structural engineering feats of the company’s nearly eighty-year
history. Guastavino Jr. would expand the company so that by 1910, just a year after the
completion of St. John the Divine, the company was constructing vaults at Grand Central and
Pennsylvania Station in New York City (Ochsendorf 2010).
1.3
Thesis context and mission
1.3.1 The future of historic masonry architecture
This thesis presents a case where structural engineers who did not understand the structural
behavior of an unreinforced masonry system promoted its demolition and its replacement with
steel beams. A lack of formal education in the engineering of unreinforced masonry may lead to
the unnecessary demolition of architectural monuments, many of which may be sound structural
systems. It is the intent of this thesis to help engineers make educated decisions by presenting
valid methods to gauge the structural integrity of unreinforced masonry vaults.
1.3.2 Strategies for analyzing unreinforced masonry vaults
This thesis uses analytical static equilibrium and graphical analysis to estimate magnitudes and
components of thrust, lines of pressure, and masonry capacity of a) a scaled shallow Guastavino
vault supported on four end arches, and b) a now-demolished cross-vault from wing H of the
Metropolitan Museum of Art, New York. The intention of these studies is to investigate the
strength of the two systems. These methods and results will demonstrate analysis techniques for
engineers to implement.
1.4
Problem statement
These questions drive the inquiry of this thesis:
12
1 Introduction





1.5
To what extent did engineers’ lack of comprehension of unreinforced masonry behavior
lead to the demolition of Guastavino cross-vaults at the Museum in the early 1960s?
Can analytical and graphical analysis using two different force-flow models show valid
solutions for the shallow scaled vault at the Museum of the City of New York (MCNY)?
The methods are valid if they show stability because the vault is known to stand under its
own weight.
Can the same methods of analysis give valid solutions for a representative cross-vault
from wing H of the Museum?
Can the unreinforced masonry vault structural analysis techniques be presented in a way
that is accessible and easy to follow for engineers who are novices with the topic?
Do the valid solutions for the Museum cross-vaults reveal that engineers and
administrators acted erroneously in demolishing these vaults?
General thesis outline going forward
In chapter 2 Literature review, Guastavino Sr.’s assertions about his construction techniques are
presented, as are masonry analysis techniques and their applicability to the goals of this thesis.
This chapter establishes the novelty of this thesis, as the questions in the problem statement have
never before been answered in full.
In chapter 3 The Metropolitan Museum of Art additions, 1910-1912, the history of the
construction of wings E and H at the Museum is assembled from contracts and primary sources.
Then the decisions leading to the demolition of the vaults are presented and investigated.
In chapter 4 Analysis of scaled Guastavino replica, force-flow modeling techniques for crossvaults are presented using this shallow vault as an example. Then, using the same shallow vault,
analytical equilibrium and graphical methods of structural analysis are applied and their results
investigated.
In chapter 5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H, the same
analysis methods from the previous chapter are applied to a representative cross-vault from wing
H of the Museum and the results are investigated.
Chapter 6 Concluding discussion then reviews the main results of the analysis and suggests steps
for future work.
13
14
2 Literature review
2 Literature review
2.1 Chapter objectives
This chapter presents Guastavino Sr.’s assertions about his construction methods and provides an
overview of key techniques for masonry analysis. These techniques are judged for their
applicability to the goals of the thesis. The assertion is made that these techniques have never
before been used to answer the problem statement questions presented in section 1.4 Problem
statement.
2.2
Principles of masonry equilibrium
The problem of analyzing historical masonry structures is primarily a problem of stability, not
elasticity (Block, DeJong, and Ochsendorf 2006). Medieval cathedral builders understood well
that their designs had to be based on the assumption that the materials could not take tension
(pulling forces), yet could take very high levels of compression (pushing forces). Stability of the
great gothic cathedrals was made possible by the individual stones compacting against each other
through gravity forces (Heyman 1999). In the arches of cathedrals, for example, voussoirs of
stone push against each other to stay in equilibrium, and the friction between the blocks was
sufficient also to keep them in their orientation. To understand the enormous capacity of
masonry in compression, consider this example: Medium sandstone has a weight density of 20
kN/m3, and it will crush under a compressive stress of 40 MN/m2, which is its compressive
strength. Dividing the strength by the density will give the equivalent height of a column of
sandstone needed to crush itself:
40,000,000 𝑁/𝑚2
= 2,000 𝑚
20,000 𝑁/𝑚3
This means it would take two kilometers of sandstone piled upon itself for it to fail, or to reach
its compression limit and splinter through brittle crushing (Heyman 1999).
Heyman organizes this and other features of masonry’s mechanical qualities in the following
three key assumptions: “(i) masonry has no tensile strength; (ii) stresses are so low that masonry
has effectively an unlimited compressive strength, and (iii) sliding failure does not occur”
(Heyman 1999). These powerful assumptions are interpreted:
1) Masonry has no strength in tension. This means that the structural engineer considering
masonry systems assumes they will have no capacity to resist tensile stresses. This is a
conservative assumption since masonry has some small tensile capacity.
2) Compared to compressive strength, the stresses developed in masonry buildings are so
low that there is essentially unlimited strength in compression. This means that even
massive arches, flying buttresses, and huge domes (i.e. St. Peter’s in Rome) can bear
15
2 Literature review
large magnitudes of weight and compressive forces without any risk of brittle failure
through crushing. Stress levels can be checked in critical regions of high compression.
3) Failure through sliding of masonry building components does not occur. This means that
the friction forces developed between masonry components—like the segments of a brick
arch—are fully capable of maintaining the orientation of the components. The brick or
stone segments of an arch will not slip past each other and fall out. As long as they are in
contact, masonry components exert enough pressure against each other for the system to
remain stable (Heyman 1999).
Most masonry systems are statically indeterminate—for any given loading condition there are
many ways for the structure to carry the loads (Heyman 1999). The load path within the masonry
is known alternately as the line of pressure or the thrust line. For the structure to be stable, the
line of pressure must pass within the geometry of the masonry (Reese 2008), (Block, DeJong,
and Ochsendorf 2006). For the arch to remain stable when the line of pressure leaves the
geometry, it must have tensile capacity, and it is assumed that masonry has no tensile capacity.
Instead, masonry cracks in response to support movements over time (Heyman 1999).
“Masonry,” writes Heyman, “is supposed to crack.” Cracks do not necessarily warn of
instability. Rather, they reveal a history of the building moving, of foundations settling. It is
common to see masonry geometry that has altered itself over centuries to contain changing lines
of pressure, as in Figure 2.1:
Figure 2.1: Deformed arches in Selby Abbey, England (Block, Ciblac, and Ochsendorf 2006), (photo by Jacques
Heyman)
16
2 Literature review
Cracks accommodate movement; cracks mean the masonry is behaving well. It would take
very large displacements to cause instability that leads to collapse in most historical masonry
buildings (Block, DeJong, and Ochsendorf 2006).
Both Rafael Guastavino Sr. and Kidder and Nolan write that for an arch to be stable or
“safe”, the thrust line must pass within the middle-third of the geometry (Guastavino 1893),
(Kidder and Nolan 1921). This assertion, called the theory of the middle-third, was a
common assumption for masonry in the late nineteenth century. Heyman explains that
masonry arches can still be stable even if the line of pressure passes outside the middle-third.
So long as the line of pressure remains within the masonry geometry, the system is stable.
Cracking is masonry’s way of accommodating the shifting line of pressure. For example, if
supports shift so that the arch is pulled outward, the masonry develops cracks so that the
changing line of pressure can be accommodated and equilibrium is achieved. In this scenario,
the thrust line passes through the arch apex at the extrados. This corresponds to a minimum
horizontal thrust reaction at the spring line. If supports move inward, cracking occurs to
accommodate the thrust line as it lowers to the intrados. This corresponds to a maximum
horizontal thrust reaction at the spring line (Heyman 1999). See Figure 2.2 below:
Figure 2.2: Minimum and maximum thrust states (Heyman 1999)
To further illustrate the nature of flow of forces in an arch—or in any two-dimensional masonry
structure working in compression—Robert Hooke asserted in 1675: “As hangs the flexible line,
so but inverted will stand the rigid arch” (Heyman 1999). In other words, if one takes a chain
such as a necklace, fixes its ends level with each other via pins to the wall, and lets the chain
hang under its own self-weight, a catenary shape is formed. If that same shape were flipped
upside down, the result would be the shape of the ideal arch in compression under its own
weight. Pure tension forces and pure compression forces are the mirror image of each other
(Figure 2.3). Heyman gives a similar example with alterations to reveal the same phenomenon:
Imagine the same “flexible line” hanging but with a small point load a bit off center. If the
resulting shape were flipped upside down, the result would be the thrust line of an arch with a
17
2 Literature review
concentrated load on it (Heyman 1999). Stability requires this line of pressure to remain within
the geometry of the masonry.
Figure 2.3: The catenary shape of a hanging line (top) and the inverted tension forces in pure compression
(bottom) (Huerta 2008)
Masonry structures are statically indeterminate to a very high degree, but as far as structural
engineers need to be concerned, any proof of a thrust line within the geometry is proof of global
stability. This is the central assertion of the master safe theorem (the safe theorem). Equilibrium
can be expressed by Hooke’s analogy of the hanging chain. The assumption that masonry has no
tensile capacity requires the line of pressure to lie within the geometry. The safe theorem states
that if any single position for the thrust line can be determined to lie within the geometry, then
the structure is proven to be stable, and collapse will not occur under the given loading
conditions (Heyman 1999). The implication of the safe theorem is that structural analysis has
only to prove one line of pressure under a given static loading to prove stability, and therefore
safety. This implication is both powerful and useful.
2.3
Guastavino’s masonry theory
2.3.1 His Essay at MIT, 1892
In his “Essay on the Theory and History of Cohesive Construction,” Guastavino Sr. presents how
he believes his systems function. He places himself within the long history of Spanish tile
vaulting which he called “timbrel vaulting.” Jumping from the construction methods of ancient
Egypt, Assyria, Greece and Rome, the European Middle Ages, and to the Renaissance,
Guastavino Sr. discusses how his take on the “cohesive system” was the next development of an
age-old engineering discipline. Much of what he says is salesmanship: He compares his
“cohesive construction” to the carved-out gigantic arches and vaulted spaces of nature fashioned
over millennia by flowing water (Guastavino 1893). He distinguishes his construction technique
from virtually all others by claiming that his is cohesive, meaning the entire finished structure
18
2 Literature review
acts as a single homogeneous mass, whereas the “mechanical construction” relies on gravity for
keeping all the pieces—i.e. voussoirs—in place (Guastavino 1893).
The genius of the cohesive system lies in its use of Portland cement, and in layering structural
tiles flat on each other by breaking the joints (Guastavino 1893). Figure 2.4 shows a typical twocourse shallow arch, with the 6-in. side of the tiles (1 in. thick) on the arch face, and with joints
broken rather than contiguous as found in a brick arch:
Figure 2.4: Two-course Guastavino tile arch (Guastavino 1893)
Guastavino Sr. gives a brief description of how such an arch functions. By breaking the vertical
joints, the pieces do not work as brick voussoirs, but instead as a single cohesive mass. Mortar
laid on a course of tiles bonds with it entirely, giving it a resistance to shear (or perpendicular
force) of 17.82 ksf (Guastavino 1893). The resistance and strength figures Guastavino Sr.
presents come from material tests he performed between 1877 and 1887 (Guastavino 1893). The
claim of cohesive bonding has three important consequences: In a quarter of an hour a twocourse tile arch (three inches thick) can be built to span 20 feet, only hours after construction the
arch has loading capacity, and the arch has reached its final settlement right after completion.
Guastavino Sr. says that these facts make his system perfect for architects of the day (Guastavino
1893), who wanted true masonry building systems that were quick to implement.
From material tests, Guastavino Sr. claims his cohesive system had strength in tension and
bending, not just in compression. The results follow in Table 2.1:
Stress
Days cured
Strength (psi)
5
360
n/a
n/a
n/a
n/a
2,060
3,290
90
287
124
34
Compression
Bending
Tension
Shear in Portland cement
Shear in plaster-of-Paris
Table 2.1: Guastavino’s tile material strength values (Guastavino 1893)
Note that the compressive strength is about ten times the tensile strength. This confirms
Heyman’s assumptions about the mechanics of unreinforced masonry.
These strength values are used to determine required arch thickness at the center:
𝑇𝐶 =
19
𝐿𝑆
8𝑟
Equation 2.1
2 Literature review
where C is the strength: C for compression, C’ for tension, C” for bending. T is the crosssectional area in “superficial inches”, i.e. (12 in.)*(the thickness T); r is the rise, in feet; S is the
span, in feet; L is the load in lbs including material self-weight. Similar to T, L is evaluated as
(12 in.)*(the span)*(the load in lbs per “superficial foot”). TC can be replaced with resistance, R.
The above equation is therefore equivalently expressed:
𝐿
∗ 12" ∗ 𝑆 ∗ 𝑆
′2
𝑅 ∗ 12" = 𝑇 ∗ 12" ∗ 𝐶 = 1
8∗𝑟
T will emerge as a thickness in inches; L will emerge as a load in pounds (Guastavino 1893).
Guastavino Sr.’s equation is the same as that for the horizontal thrust of an arch:
𝐹𝐻 =
𝑞𝐿2
8𝐷
Where q is the uniform gravity load per foot of length; L is the span; D is the rise of the arch
from the level of springing to the crown, or center.
In all his calculations, Guastavino Sr. sees the arches not like the usual ones with voussoirs, but
as a single homogeneous body that acts like a solid monolith of stone and which has some tensile
capacity, as if iron reinforcement were taking up the tension developed in the masonry
(Guastavino 1893). This point is important because it is the one Guastavino Sr. is selling: His
cohesive system is made of masonry, but it behaves in a unique way where it has tensile
capacity, which no other masonry possesses.
2.3.2 Disputing some of Guastavino Sr.’s assertions
Guastavino Sr. maintained that because of the “cohesive” nature of his construction method, his
tile-and-cement systems had tensile capacity, thus bending capacity, and therefore behaved in a
manner completely apart from the old voussoir systems. If that were truly the case, Guastavino
Sr. would not have included buttressing walls, tension ties, or dome hoop steel, but he did.
Whether it was intuition or hard-earned understanding, Guastavino Sr. knew masonry possessed
poor tensile capacity. He designed a barrel vault with a 54-ft. span for the Palace of Fine Arts in
St. Louis in 1904 (now the St. Louis Art Museum), which included three large lunettes on either
side of the arch. Lunettes help to support the barrel vault against tension forces. Guastavino Sr.
also accounted for the tendency of the barrel vault to kick out at its base (horizontal thrust) with
masonry buttressing walls. These walls were designed specifically because the barrel vault does
not have the tensile capacity to resist the horizontal thrust (Reese 2008). At the Boston Public
Library (the commission with McKim, Mead, and White which propelled him to success)
Guastavino Sr. used a composite tile vault-beam-girder-tension rod system (Figure 2.5). And as
20
2 Literature review
far back as Spain, Guastavino Sr. included iron rods to contain the tensile thrust of barrel arches
at the Battlo’ Factory (Figure 2.6). These measures reveal that Guastavino Sr.’s designs assumed
masonry had little tensile capacity.
Figure 2.5: Guastavino composite system, Boston Public Library (photo by the author)
Figure 2.6: Tension ties at Battlo’ Factory barrel vaults (Ochsendorf 2010), (text and arrow added by the author)
It is appropriate to conclude that Guastavino Sr.’s structural systems behave like conventional
unreinforced masonry systems, and their analysis can be aided by using the three assumptions
presented above. Guastavino Sr.’s work is nonetheless remarkable in its impressive economy and
malleability of form. As described in Chapter 1, this tile construction could facilitate a 98-ft.
21
2 Literature review
diameter dome only four inches thick at its crown. The R. Guastavino Co. was innovating
impressively on a building tradition that reached back centuries.
2.4
Masonry analysis
The equations of structural analysis used for masonry are those relating to static equilibrium,
compatibility of geometry, and stresses in the material. Only the first two kinds are applicable to
the subject of this study because, as shown in section 2.2, historic masonry structures do not
develop nearly the compressive stresses needed to cause brittle failure (Block, DeJong, and
Ochsendorf 2006). As explained in section 2.2, the structural adequacy of masonry depends on
its stability. This is because it cannot take tensile stresses and is assumed never to experience
crushing stress from compression. Therefore, masonry analysis becomes a process of
determining whether the geometry is adequate. The goal is to determine the nature of the
system’s equilibrium. Graphical analysis methods use scaled polygons which represent force
equilibrium; membrane analysis uses mathematical equilibrium to show stability; and finite
element analysis considers material stress. As will be seen, the first two methods are useful,
while the latter does not accurately reflect masonry behavior.
2.4.1 Graphical analysis
The stability of masonry structures relies on their geometry. Graphical analysis is a powerful tool
to determine stability because it uses vectors to construct the line of pressure. Early twentieth
century authors Kidder and Nolan explain the motivation behind and execution of graphical
analysis. The goal is to show that the line of pressure is within the geometry. The safe theorem
tells us that under a given loading, if a single thrust line scenario can be shown within the
masonry, then the structure is in equilibrium. Kidder and Nolan explain how this line of thrust is
visually constructed. The thrust at the crown of an arch needs to be determined. That thrust
magnitude is visually exerted on an arch segment (representing a standard mass of the arch-ring),
and combined with its own self weight and load to determine the resultant thrust on it from the
segment below. That resultant thrust is then combined similarly with the neighboring arch
segment to get a resultant thrust on it from the segment below. The process is repeated for each
arch segment until the segment resting at the spring-line (Kidder and Nolan 1921). Figure 2.7
shows a diagram of a segmental arch, and Figure 2.8 shows the end result of graphical analysis
on a half-arch.
Figure 2.7: Diagram of segmental arch (Kidder and Nolan 1921)
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2 Literature review
In Figure 2.7, R is the springing stone or skewback, and S is the springer—the last arch segment
in the graphical analysis. K is the keystone, line ai is the rise, and V represents a voussoir
(Kidder and Nolan 1921).
Figure 2.8: Line of pressure in unloaded semi-circular arch-ring (Kidder and Nolan 1916)
The points where the resultant thrusts intersect with the segment joint lines (dividing the
voussoirs) are the “centers of pressure.” Connecting them forms the line of pressure (Kidder and
Nolan 1921). Both the terms “line of pressure” and “thrust line” are used interchangeably in this
thesis.
Zalewski and Allen (1998) describe a similar process. A distributed loading—for example, selfweight—is divided equally among divided masonry segments, which are themselves divided
equally. Each same-size segment gets its equally divided loading as a point load applied to its
center of gravity. To determine the point loads:
𝑃 = 𝑓𝑙𝑤
P is the point load; f is material pressure; l is block length; w is block or arch depth (out of
plane). In this example, f = 3.06 kN/m2, l = 1.24 m, w = 1 m. P = (3.06 kN/m2)(1.24 m)(1 m) =
3.8 kN. This load is applied at each block’s center of gravity, and each load is laid one under the
other to form a vertical line (right side of Figure 2.9) depicting the total vertical load on the arch.
The arch is known to be a parabola, and lines oa and om are first drawn. Each of these lines starts
at the spring-line and passes straight to a point o on the centerline. The corresponding rays oa
and om on the “force polygon” on the right are drawn parallel to these, respectively, thereby
finding the pole o. With the scale in place, the resultant thrust and its horizontal component are
found (Zalewski and Allen 1998).
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2 Literature review
Figure 2.9: Graphical analysis example (Zalewski and Allen 1998)
This same approach can be used to evaluate the equilibrium of cross-vaults and rib vaults, which
are the focus of this thesis. In cross-vaults, compressive forces flow through the arch-like
quadrants to the ribs, or diagonal sections, and through the ribs to the four vertical piers at the
corners. The phenomenon of the “force flow” can be visualized in an infinite variety of patterns
and the analyst must choose a possible “slicing” technique to divide the vault into a series of
arches (Huerta 2008). To simplify analysis, vault quadrants, or webs, are typically divided into
arches that intersect on the diagonal, and the diagonals (ribs) themselves are also modeled as
arches (Figure 2.10). The ribs bring forces from the web arches to the vertical supports (Allen,
Zalewski, and Iano 2010). Web arches are secondary horizontal members, ribs are primary
horizontal members, and piers are the columns. The horizontal thrusts from any adjacent web
arches form the components of a vector which is the resultant horizontal thrust in the rib at that
location. Figure 2.10 shows how forces are equilibrated at each node analytically: HCD and HAB
are horizontal thrusts from adjacent web arches; HDA is the horizontal thrust from the rib-arch
segment above; HBC is the resulting horizontal thrust for the subsequent rib-arch segment.
The vertical force in each of the four piers (at the corners) could be estimated to be
approximately a quarter of the vault’s total weight:
𝑅𝑉 = 0.25𝑊
24
Equation 2.2
2 Literature review
Rv is the vertical reaction and therefore the vertical component of rib thrust; W is the total vault
weight. For one particular example, the horizontal thrust at each corner of a square cross-vault
has been shown to be between 21 and 32 percent of the vault’s weight W (Allen, Zalewski, and
Iano 2010). This corresponds to maximum (0.32W) and minimum (0.21W) horizontal reactions
in the rib-arches; maximum means the line of pressure approaches the intrados at the crown;
minimum means the line of pressure approaches the extrados at the crown.
This strategy of dividing the vault into primary (ribs) and secondary (web) arches will be used
for analysis of Guastavino structures in chapters 4 and 5. The analysis combines graphical
methods and analytical equations of equilibrium, such as Equation 2.2 and Equation 2.7 (below):
The thrusts and reactions are found analytically, and superimposed on the diagonal ribs. Then,
with those forces applied and scaled, the ribs are analyzed graphically.
Figure 2.10: Vault analysis (Allen et al. 2010)
2.4.2 Analytical equilibrium analysis
As mentioned in section 2.4.1, equations of equilibrium are directly related to graphical analysis.
This is because graphical analysis is a visual expression of force equilibrium. Just as resultant
horizontal thrusts from graphical methods can be summed for the rib-arches, these same thrust
resultants, obtained through analytical equilibrium equations, can be summed. If a vault is
divided into arches, and each of those arches are loaded with their own self weight, Equation 2.4,
Equation 2.5, Equation 2.6, and Equation 2.7 describe the horizontal thrust at each arch support
25
2 Literature review
(or springing). With self-weight per linear foot of arch q, the vertical reactions at the support are
found through vertical equilibrium:
𝑅𝑉 =
𝑞𝐿
2
Equation 2.3
L is the span. Maximum thrust corresponds to rise to intrados D, and minimum thrust to rise to
extrados, which is the rise to the intrados plus the arch thickness, t. This gives
𝐹𝐻,𝒎𝒂𝒙 =
𝐹𝐻,𝒎𝒊𝒏
𝑞𝐿2
8𝐷
𝑞𝐿2
=
8(𝐷 + 𝑡)
Equation 2.4
Equation 2.5
Therefore a range of horizontal thrust values exists between lower and upper bounds:
𝑞𝐿2
𝑞𝐿2
≤ 𝐹𝐻 ≤
8(𝐷 + 𝑡)
8𝐷
Equation 2.6
The average horizontal thrust is calculated using the rise to the center of the arch at the crown:
𝐹𝐻,𝒂𝒗𝒈 =
𝑞𝐿2
𝑡
8 (𝐷 + 2)
Equation 2.7
For analysis it is assumed that all thrust lines pass through the boundary points of x = 0 and x =
L, x being the horizontal axis and L being the entire span of the arch.
Derivation of the above equations is found in section 8.1 of the appendix. The three equations for
maximum, minimum, and average horizontal thrust, and the equation for the vertical reaction
form the basis of the iterative parametric procedure that will be used to determine the stability of
two Guastavino vault systems.
2.4.3 Membrane analysis
Membrane analysis assumes that a shell-like structure carries load entirely through membrane
action—that is, the internal force flow is parallel to the structural shape. The assumption is that
in response to vertical or horizontal point loads or to distributed loads such as self-weight, all
internal stresses are axial tension or compression acting at the neutral axis, and no bending is
experienced (Reese 2008). This assumption allows the engineer to visualize the flow of forces in
masonry vaults as within the geometry of the masonry parallel to the intrados (underside) and
extrados (top surface) of the material. Graphical analysis works similarly in that the line of
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2 Literature review
pressure must remain within the masonry geometry to achieve stability. Any deviation of the
thrust line outside of the geometry means tension is engaged which implies the masonry can
bend. And like membrane analysis, graphical analysis neglects the masonry material properties,
treating the problem instead as one of pure equilibrium (Reese 2008). Guastavino structures are
suited to membrane analysis because their span-to-thickness ration, L/t, is very large: Grace
Universalist Church in Lowell, MA, has L/t = 200 (Ochsendorf 2010). To put that in perspective,
an eggshell has L/t = 100 (Heyman 1999). Because of their eggshell proportions, thin masonry
vaults and domes like those built by the R. Guastavino Co. do not have to be built with the
required thicknesses of the magnitude of two-dimensional arches. The restraint on thickness for
these membranes is only to avoid buckling in compression (Heyman 1999). In fact, thinner shells
are most efficient. Increasing thickness doesn’t necessarily increase loading capacity; it increases
material, which increases the self-weight and the area needed to support the monolith (Heyman
1999).
As with domes, the stresses in thin-shell cross-vaults are independent of vault thickness. Stresses
in a smoothly curving vault like the subject of chapter 4 are calculated as
𝜎 = 𝑅γ
R is the local radius of shell curvature; γ is the material unit weight; σ is stress. Typically, the
stresses developed in cross-vaults are orders of magnitude below the material’s compressive
strength. Consider a medium-sandstone vault of 15 m span, R = 10 m, γ = 20 kN/m3:
𝑅𝛾 = 10𝑚 ∗
20𝑘𝑁 200𝑘𝑁
=
= 0.2𝑁/𝑚𝑚2
𝑚3
𝑚2
The compressive strength of medium sandstone is 40 N/mm2 (Heyman 1999):
0.2
= 0.005 = 0.5%
40
The ratio above represents a “unity check,” with which demand is compared to capacity.
Under its own weight, a medium sandstone vault does not even develop 1% of the stresses
necessary to cause compressive failure. Large stresses do concentrate at abrupt changes in
geometry and at structural intersections, however. Medieval builders knew this and designed
vault ribs to carry the stresses to piers and abutments (Heyman 1999). Essentially, these ribs act
as diagonal reinforcing two-dimensional arches. This model of the ribs is adopted in chapters 4
and 5 of this thesis.
Membrane theory gives an expression for forces in the radial direction (θ) of a cross-vault:
𝑁𝜃 = −𝑞𝑎 cos 𝜃
With q the self-weight per unit area and a the radius (Heyman 1999).
27
Equation 2.8
2 Literature review
Figure 2.11: One bay of a cross-vault (left); half-bay of a cross-vault (right) (Heyman 1999)
According to Heyman, the stress resultant in the eighth of the vault ABD is expressed by
Equation 2.8, as is the stress resultant longitudinally along the crown ABC (Heyman 1999).
(Please note that Heyman uses w for self-weight per unit area.) And Heyman also proposes
slicing vault segments into parallel arches that can each be analyzed as two-dimensional systems,
a method adopted in this thesis. Furthermore, the thrust line of the rib acts within the geometry
toward the crown but deviates above the rib extrados and leaves the vault a distance z below the
level of the crown. This distance increases as the thrust moves away from the vault center:
Figure 2.12: Modeling cross-vault as sliced parallel arch segments (left and center); location z of thrust line exit
(right) (Heyman 1999)
The thrust line “exits” the plan of the vault a distance h = 0.534a below the level of the crown,
which is outside the geometry. To safely contain the resultant force from the ribs, fill, rubble, or
other masonry is used to fill in the recesses at the springing of the webs, thus containing the
thrust path:
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2 Literature review
Figure 2.13: Thrust line exiting rib geometry a distance z below the crown level (Heyman 1999)
To review, membrane analysis assumes membrane action in combination with the assumption
that only compressive forces act and they must act within the masonry. This leads to equilibrium
expressions that rely on geometry for stability independent of material properties.
2.4.4 Thrust Network Analysis (TNA)
While graphic statics is a powerful tool to determine equilibrium, it is suited to two-dimensional,
not three-dimensional models. Other than arches, masonry systems are three-dimensional, and it
is desirable to have a model that is also three-dimensional. This is the motivation behind Block’s
(2009) development of thrust network analysis (TNA)—a thrust-line equilibrium approach that
uses computer-generated reciprocal diagrams to produce a line of pressure in unreinforced
masonry. It preserves features of graphical analysis such as a) the intuitive visual representation
of forces within the structural system, and b) a way to interactively explore various thrust lines in
the indeterminate structure that are bounded by minimum and maximum thrust magnitudes
(Block 2009).
TNA operates on assumptions identical to those adopted by this thesis. In addition to using the
safe theorem (section 2.2), TNA adopts the same three assumptions about unreinforced masonry
as presented in section 2.2: Masonry has no strength in tension, there is no sliding between
structural elements, and the masonry has infinite compression capacity compared to its
compressive strength. In addition, the goal of TNA is to represent the vault’s structural action
represented by a network of “discrete” forces which respond to “discrete” applied loads (Block
2009). This is the same objective of graphical analysis. Block has developed an add-on to
Rhinoceros3D called “Rhino Vault” which implements TNA (Block, Lachauer, and Rippmann).
2.4.5 Considering finite element analysis
Finite element analysis is an elasticity theory-based approach that uses material properties such
as Young’s modulus and Poisson’s ratio to estimate stresses in a structure. The structural system
is first modeled geometrically, and assigned boundary conditions, and if applicable it is given
support settlements, too. The output shows stress distributions throughout the structural system
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2 Literature review
(Dugum 2013). Elastic solutions are very sensitive to even the smallest support movements, and
they can often show exacerbated stress magnitudes from small support shifts which undermines
their analytical worth (Reese 2008). Cracks in an unreinforced masonry system would manifest
as tensile stresses in the finite element software output. Therefore, finite element analysis,
showing tension stress in a material that can’t take tension, is an inadequate tool.
Block, Ciblac, and Ochsendorf (2006) compare finite element analysis (elastic) with thrust line
analysis (stability) for two different arch geometries (Figure 2.14). They find that in either case,
the finite element software output shows distributions of compressive and tensile stress in the
masonry. That is the limit of the information the program can tell the engineer. A simple thrust
line analysis reveals that for the same loading and thrust line, a thicker geometry ((b) of Figure
2.14) could maintain the line of pressure within the arch, whereas a thinner one could not ((a) of
Figure 2.14). The arch which cannot contain the line of pressure would fail, but the finite
element analysis cannot determine that stability on its own.
Figure 2.14: Comparison of finite element results to thrust-line analysis (Block, Ciblac, and Ochsendorf 2006)
In comparing finite element and thrust line analyses of a Guastavino barrel vault, Reese finds
that the stability-based thrust-line approach provides sensible physical answers that are easy to
interpret. The same is not true of the finite element results. Thrust line analysis shows the
necessity of diaphragm walls to contain the line of pressure which leaves the geometry of the
vault as it approaches the spring-line of the semi-circular geometry. The finite element results
merely show stress concentrations. But the unreinforced masonry stability is based on its
capacity to contain a thrust line (Reese 2008). As this literature review has established,
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2 Literature review
compressive stresses in masonry will rarely if ever reach the magnitude of its strength. Elasticitybased approaches look for stress concentrations for answers of structural adequacy, but at hand is
a problem of stability, not elasticity. Finite element analysis results for unreinforced masonry
structures have been shown to be difficult to interpret and inadequate for the current problem,
which pertains to stability. Therefore, only the methods of sections 2.4.1, 2.4.2, and 2.4.3 will be
considered in this study. They form the basis of the analytical static equilibrium and graphical
analysis methods presented in chapters 4 and 5.
2.5
Claim of thesis authenticity
The above techniques involving graphical, analytical, and membrane analysis have not been used
previously to answer the problem statement questions of section 1.4, particularly for the Museum
cross-vaults in wing H. Although the ½-scale Guastavino vault was designed by MIT’s Masonry
Research Group and has a record of analysis, the aim was not to perform analysis in a way that
engineers are meant to follow step-by-step. That is the purview of this thesis and one of its
contributions. Current literature addresses neither the reasons engineers promoted demolishing
the Museum’s vaults nor the stability of those vaults based on structural analysis. This thesis
achieves a number of goals for the first time:




The presentation of historical records and examination of engineer-client decisions
regarding the cross-vaults of the north wing of the Museum;
The investigation of a ½-scale Guastavino vault replica as an example of how to perform
structural analysis for unreinforced masonry systems;
The investigation of the stability of former cross-vaults in wing H of the Museum;
The use of a representative cross-vault from the Museum in a quantitative case study of
the structural analysis of unreinforced masonry systems.
31
32
3 The Metropolitan Museum of Art additions, 1910-1912
3 The Metropolitan Museum of Art additions, 1910-1912
3.1 Chapter objectives
This chapter provides a historical overview of the construction and eventual demolition of the
Guastavino vaults in wings E and H of the Museum using both secondary and primary sources.
Proof is given of the fact that the R. Guastavino Co. initially constructed the vaults.
Correspondence involving the Museum’s consulting engineers indicates a general unfamiliarity
with unreinforced masonry vaults.
3.2
Building wings E and H
In January 1904, the building committee of the Metropolitan Museum of Art awarded the BeauxArts architectural office of McKim, Mead and White the contract to design the north wing
additions to the Museum along Fifth Avenue. McKim took personal leadership over the project,
shifting the main north-south axis of the building east so that it would run through the grand
entrance designed by Richard Morris Hunt and Richard Howland Hunt (which opened in 1902).
With the shift in the main axis of the museum, McKim introduced a corridor running alongside
every new gallery in the new north wings. He also designed two-story courts with skylights
along these corridors. In 1908 Mead took over lead design of the project when McKim retired
from the firm (Heckscher 1995).
In May 1906, although construction on the first north wing addition, wing E, had begun,
Museum director Edward Robinson altered the plans so that McKim’s corridors in both wings E
and H would also serve as gallery space (Figure 3.1). Other alterations included raising the mainfloor ceiling to 25 feet in all new wings, including E and H (Heckscher 1995). Mead presented
the building committee with plans for wing H in October 1909. Main level designs featured a
central court accessible from anywhere in the surrounding cloister through arched openings. The
construction contract for wing H was signed in December 1909, and building of the foundation
started in March 1910. This exhibit space opened in June 1913. It was originally designed to
house the arms and armor display, and would later serve as the galleries for Egyptian and Farand Near-Eastern art starting in the 1960s (Heckscher 1995). Figure 3.2 shows the completed
two-story court of wing H before the arms and armor exhibit was installed. Plastered Guastavino
vaults are seen in the shadows in the galleries flanking the court. These are the subject of this
thesis.
33
3 The Metropolitan Museum of Art additions, 1910-1912
Figure 3.1: Undated design drawing showing additions E and H (“Metropolitan Museum of Art Archives”)
Figure 3.2: Wing H courtyard with Guastavino vaults in flanking corridors (Heckscher 1995)
3.3
Renovation program at the Met
In 1962, the Museum was undergoing sweeping changes, and they were expressed in large-scale
construction, renovation, and relocations of art collections from one wing to another. Part of the
new configuration affected wings E and H: It included moving the Egyptian exhibits to the main
floors, moving the Far- and Near-Eastern exhibits to the second floors, and renovating the
Costume Institute on the ground floors of these wings. The Museum claims in a document about
the updates that wings E and H, built between 1907 and 1912, were not structurally suited to take
the increased loading from a growing number of visitors and heavier exhibits (statues). For
34
3 The Metropolitan Museum of Art additions, 1910-1912
example, one Egyptian statue in the museum’s collection of Hatshepsut from the 18th Dynasty
weighs 8,000 lbs. The Museum states that renovations include removing the existing floor
structures and rebuilding the entire second floor with 142 tons of steel and 570 cubic yards of
concrete. This new floor system would be able to take a 300 psf live load (Metropolitan Museum
of Art, New York City 1962).
The Museum claims that the existing structural masonry in wings E and H could not support the
new required loading. This is information they derive from their hired consulting engineers.
Those consulting engineers, however, did not fully understand how the unreinforced Guastavino
vaults acted structurally, and consequently they strongly recommended demolishing them and
replacing them with a kind of system whose structural action they could predict: Steel beams and
girders.
3.4
Proof of Guastavino vaults
The R. Guastavino Co. often produced drawings that lacked the detail we expect from
consultants and constructors today. To further complicate efforts to determine with certainty that
the R. Guastavino Co. did in fact build the Met’s vaults in wings E and H, it was common
practice among firms like McKim, Mead and White to trust Guastavino with full responsibility
in planning and constructing the vaults (Ochsendorf 2010). This means it is rare to see more than
a mention of Guastavino by name on an architectural drawing. It is even possible that the
decision to use Guastavino vaults was made simply by telegram or over the phone. As this
chapter demonstrates, the R. Guastavino Co. did indeed build the vaults for wings E and H; that
is the structural system consulting engineers faced while charting the design of the Museum’s
renovations.
3.4.1 Early drawings of wing E
As early as 1906, McKim, Mead and White envisioned using the Guastavino system in their
designs for the Museum’s additions. Figure 3.3 shows a blueprint elevation of an early design for
wing E. On that elevation are hatch marks indicating Guastavino construction:
35
3 The Metropolitan Museum of Art additions, 1910-1912
Figure 3.3: 1906 elevation wing E (top); inclusion of Guastavino (bottom) (“Metropolitan Museum of Art
Archives”), (red outlines added)
This drawing shows that early in the design process, the architects already intended to include
the Guastavino vaulting systems in their large building commission.
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3 The Metropolitan Museum of Art additions, 1910-1912
3.4.2 The Wills & Marvin sub-contract for wing H
A list of construction and interior finish sub-contractors, vendors, and deadlines shows that the
construction company Wills & Marvin Co. was awarded the prime contract to build wing H on
December 9, 1909. The contract was completed on September 29, 1911 (The City of New York
Department of Parks 1913). In addition there is a sub-contract dated February 11, 1910, between
“Wills & Marvin Company, 1170 Broadway” and “R. Guastavino Co., Fuller Building” for
“Cohesive tile construction, Addition ‘H’ of the Metropolitan Museum of Art, New York City.”
The work is to conform to plans from McKim, Mead and White. Furthermore, the specification
included in the contract compels the R. Guastavino Co. to use its cohesive tile system. The tile is
to cover all steelwork, and be ready to receive a plaster finish. In addition, the completed vaults
must also be designed to support 6 in. thick concrete floors above (Wills & Marving Co. and R.
Guastavino Co. 1910). This contract strongly indicates the involvement of the R. Guastavino Co.
on the north-wing additions to the Museum.
3.5
History of the vaults’ demolition
As stated previously, it is evident that because the Museum’s consulting engineers did not fully
understand the structural behavior of the Guastavino vaults in wings E and H, they decided to
demolish them and replace them with a steel beam-girder system. This is unfortunate because
those vaults had architectural merit alone as original components to a McKim, Mead and White
Beaux-Arts architectural landmark. This thesis argues that without properly analyzing the vaults,
engineers could not know whether there was potential for increased loading capacity. This
chapter provides insight into how and why the decision was made to demolish the Museum’s
Guastavino vaults in wings E and H.
3.5.1 Early stages of renovating wings E and H
1948:
Correspondence about the Museum’s planned expansion in wings E and H dates back fourteen
years before construction, to 1948. In May of that year, the Museum writes the R. Guastavino
Co. looking for structural drawings and details about their floor systems in wing H. The Museum
reports 1910 contract specifications calling for a live load of 150 psf on the second floor. They
want to know if the floor will take a heavier load, but they have no structural details on file. To
show the R. Guastavino Co. which areas of the wings it is referencing, the Museum sends along
a small plan schematic with the areas of concern hatched in red (Tolmachoff 1948):
37
3 The Metropolitan Museum of Art additions, 1910-1912
Figure 3.4: Plans of wing H hatched in red where increased loading is anticipated (“Drawings & Archives, Avery
Library, Columbia University”)
A day later, on May 21, the president of the R. Guastavino Co., Malcolm Blodgett, responds. He
reports that his company has drawings for wing H dated 1911. He says that they indicate twocourse tile construction which was commonly used then, although for the twenty years leading
up to 1948 the company had been using at least three courses to increase the factor of safety. He
also states that a “large number” of two-course vaults constructed over 40 years before 1948
were “in perfect shape” still. Responsibly but very conservatively, Blodgett is careful to say that
he does not recommend loading the Museum’s vaults over their design capacity (150 psf) (R.
Guastavino Co. 1948).
Enter I.A. Berg, a consulting structural engineer often retained by the R. Guastavino Co. He
writes to Blodgett to update him on his two inspections of cracks in the floor above Guastavino
vaults at the Museum. Berg reports that in wings E and H there were cracks in the wood floor
and under the concrete fill that runs parallel to the vaults’ longitudinal axis. He also inspected all
the piers, walls, and soffits of the Guastavino vault bays. He even had the Museum’s
superintendent open a section of floor above the vault. Berg wrote that “there were no signs of
cracks in the Guastavino work.” Berg determined the vaults were structurally sound, and that any
cracking in the floor, concrete fill, or flatwork was from expansion and contraction from
temperature changes over the years. Berg says he told the superintendent of the Museum that the
vaults were structurally sound (Berg 1948).
1949:
On January 25, H.F. Seitz of R.B. O’Connor & Aymar Embury II, the architecture firm working
on the Museum expansion, writes Blodgett asking for Berg’s inspection results (Seitz 1949). The
R. Guastavino Co. responds simply repeating verbatim what Berg reported the past summer (in
August 1948) (Treasurer R. Guastavino Co. 1949). This means that in early 1949, the head
architects on the renovation of wings E and H had a professional engineer’s conclusions
regarding the safety of the Guastavino vaults: Despite superficial cracks, the vaults were
structurally stable.
38
3 The Metropolitan Museum of Art additions, 1910-1912
3.5.2 The decision to demolish the vaults
1950:
On June 7, Berg writes Blodgett updating him about a meeting he attended with the Museum’s
consulting engineer and the New York Building Department. He reports that neither the
consulting engineer nor the Building Department were familiar with the Guastavino system, so
they wanted the R. Guastavino Co. to furnish information showing that the loading on the second
floor is safe. Berg says he tried to show them with their current drawings that the vaults were
safe, but that the consulting engineer and Building Department wanted the same information and
conclusions on R. Guastavino Co. letterhead (Berg 1950).
A few weeks later, on June 29, a Mr. Harrison of the Museum sends the R. Guastavino Co. a
Western Union telegram with the Museum’s decision regarding the vaults. He says, “have
decided not to use arch for supporting ceiling will use steel instead am advising Mr. Berg
accordingly” (Harrison 1950). It appears that a lack of comprehension of the Guastavino
system’s structural action fueled suspicion of its safety. That suspicion compelled the Museum to
order the vaults’ demolition and replacement with structural steel.
1958:
On May 8, John Zoldos of the consulting engineering firm Severud-Elstad-Krueger-Associates
(SEKA) writes R. Guastavino Co. vice president A.M. Bartlett. He wants to know if the second
floor of wing E is strong enough for the scaffolding to be used in renovations to the roof (Zoldos
1958). Curiously, Bartlett responds admitting unfamiliarity with these vaults. He writes on May
12, “we are unfamiliar with Wing ‘E’, of the above building, and find no reference to this wing
on the working drawings, which we have on file.” He gives a list of working drawings on file,
which include three that are related to the Museum:



One drawing dated June 17, 1911, for wing H for the first floor groin vaults and levelling
of the second floor adjacent to the stairs and elevator, based on McKim, Mead and White
plans;
One drawing dated June 23, 1911, for first floor groin vaults with brick arches and
levelling of the second floor, based on McKim, Mead and White plans;
One drawing dated July 5, 1911, (printed as 1958 but clearly a mistake) for wing H for
groin vaults and levelling adjacent to the stairs and elevator, based on McKim, Mead and
White plans (Bartlett 1958).
None of these drawings concerns wing E. The Museum’s consulting engineers relied on the R.
Guastavino Co. to furnish structural details of their past work. Unfortunately, they could not
fully honor this request. The lack of available relevant drawings may have precipitated the
decision to demolish the vaults. Nonetheless, it is also evident that the consulting engineers felt
unprepared to gauge the vaults’ strength without original drawings. Perhaps if they had had the
analytical facility to study the vaults’ structural capabilities, they would not have had to rely on
the R. Guastavino Co. for original drawings and structural strength information.
1960:
39
3 The Metropolitan Museum of Art additions, 1910-1912
On January 18, New York City engineer Alfred Engel writes the R. Guastavino Co. He says that
“recent investigation of structural conditions” of the vaults in wings E and H, “constructed
between 1908-12,” reveals that they may not be able to sustain 100 psf as required by the New
York City Building Code. Engel says that he had spoken with Berg, who suggested a program of
reinforcing the vaults if they were thought to be inadequate. First, however, Engel wanted more
information about design factors and construction details of the vaults. He requests “drawings
and data” and advice from the company about “reinforcing the Guastavino arches” (Engel 1960).
Barlett, vice president of the R. Guastavino Co., responds to Engel on March 22 in reference to a
phone call they had the day before. Bartlett says that he has searched all available files on the
project and could not find any safety factors; he assumes the files were destroyed “some time in
the past.” Bartlett writes, “there is no person now connected with this Company or any person
now living that is not connected with the Company, that we know of, who could recall the actual
construction of GUASTAVINO Vaulting in Wings ‘E’ and ‘H’ of this project.” Bartlett repeats
Engel’s own assertion that the project called for a live load of 150 psf in wing H and a live load
of 250 psf in wing E, and it can be assumed that construction was carried out to conform to those
design requirements. Bartlett also says that Berg had inspected the vaults many times since 1948,
each time determining that they were structurally adequate (Bartlett 1960).
It is true that the R. Guastavino Co. is unable to furnish the requested Museum vault drawings at
least twice—in 1958 and 1960—but the company was also on the verge of closing during these
requests. By the time the company closed in 1962, it had experienced many years of downsizing
as the market for its products diminished (Ochsendorf 2010). Consequently, it is understandable
that at the onset of the Museum’s renovations, the R. Guastavino Co. was less organized than it
had been at the height of its business success.
3.5.3 Renovations are underway
1963:
On April 20, the New York Times reports on the progress of immense renovations and
construction at the Museum. There is mention of 142 tons of steel beams being loaded into the
north wing (wings E and H) through windows facing Fifth Avenue (Figure 3.5). This indicates
that demolition of the Guastavino vaults in both wings is occurring around 1963. A caption in the
article reads, “Inside the building the beams are used to replace old arches under the flooring in
Near and Far Eastern Gallery. The old floor could not support the weight of exhibits.” And the
concrete that’s on the way could “pave a mile-and-a-half-long highway lane” (Knox 1963).
40
3 The Metropolitan Museum of Art additions, 1910-1912
Figure 3.5: Loading steel beams into the north wing of the Museum, 1963 (Knox 1963)
1966:
Work on the Museum’s north wing continues even after earlier renovations. On June 21, a New
York Times article reports that starting in January 1967, “most of the north wing’s three floors
will be closed for two years.” Work on the Costume Institute on the ground floor will finally
start. “At the same time the ancient Near East and Egyptian collection on the main and the
Chinese jade and pottery collection on the second floor, will be closed while the floors are
strengthened” (Shepard 1966). It seems that extensive reconstruction in 1963 might not have
been enough for the long-term loading requirements of wings E and H. One wonders if the
Guastavino vaults might have been retained initially, and reinforced in 1967.
41
3 The Metropolitan Museum of Art additions, 1910-1912
Figure 3.6: Demolition of the Museum’s Guastavino vaults underway, early 1960s (“Drawings & Archives,
Avery Library, Columbia University”)
3.6
Conclusions
This chapter has detailed the history of the construction and demolition of the Guastavino vaults
in wings E and H of the Museum. Throughout the process of renovations to the Museum, the R.
Guastavino Co. was unable to provide sufficient engineering data and drawings for its past
designs. At the same time, it is clear that the structural engineers involved in renovations had
little to no analytical facility with unreinforced masonry vaults.
The question remains whether the cross-vaults could have supported the increased loading.
42
43
4 Analysis of scaled Guastavino replica
4 Analysis of scaled Guastavino replica
4.1 Chapter objectives
This chapter presents methods of analytical static equilibrium and graphical analyses for a ½scale Guastavino vault built for the recent exhibit Palaces for the People (Museum of the City of
New York 2014). For both analytical static equilibrium and graphical analysis, two methods are
applied: The triangle-arches and sliced parallel arches methods. These two techniques are
distinguished by how they consider force flow. The analysis is organized as follows:
1) Analytical equilibrium
a. Triangle-arches method
b. Sliced parallel arches method
2) Graphical analysis
a. Triangle-arches method
b. Sliced parallel arches method
4.2
Replica design and geometry
The Palaces for the People exhibit started at the Boston Public Library in 2012 and opened in
late March 2014 at the Museum of the City of New York (MCNY). This chapter analyzes the
equilibrium of the ½-scale vault built for this exhibit, which was a ½-scale replica of a vault at
the Boston Public Library built in 1889. Dimensions and material properties are based on the
original 2012 design for the Boston Public Library (that design was recreated for MCNY), and
reproduced in Figure 4.1. The side-arches are of different lengths (see plan view), but the rise of
each arch is the same height, and the thickness of each arch is identical. The different lengths of
the sides means the thrust line in the two different arch types will be different, and so will their
horizontal thrusts.
44
4 Analysis of scaled Guastavino replica
Figure 4.1: ½-scale vault elevation (top) and plan (bottom) (MIT Masonry Research Group 2012)
The exhibition vault has a central opening, or oculus, to demonstrate the construction method. In
all following analysis procedures, however, the opening in the top will be assumed closed—that
is, filled with masonry, as though the design were built to completion. (Please note that the hole
remaining in the illustrations is from their origin as exhibit-planning documents.) This better
approximates the true action of a full vault. In other words, when each side-arch is analyzed as
though it is an entire quarter of the vault, the opening will be assumed closed; when the vault’s
quarters are sliced into series of parallel arches, the opening will also be assumed closed.
Understanding the vault’s thicknesses is key to a thorough analysis. The arches are five courses
thick. The vault itself is three courses thick. Each course of tile has a nominal thickness of 1 inch
(Reese 2008). It is assumed that a layer of mortar (Portland cement or plaster-of-Paris) is one
half-inch thick (Dugum 2013). That means the side-arches, each five courses thick, have two
inches of mortar, making those arches seven inches thick. The vault web, with three courses of
tile, has one inch total of mortar, making it four inches thick. The weight of mortar and plaster
together is half the weight of tiles (MIT Masonry Research Group 2012). For Guastavino
vaulting, therefore, the self-weight per unit area is approximated as follows:
45
4 Analysis of scaled Guastavino replica
From the AISC steel manual 14th edition, “common brick” is 125 pcf. A ratio of 4/5 is allocated
to the tiles and 1/5 to the mortar.
For the side-arches:
Tile density
Share of vault
Mortar & plaster density
Share of vault
Arch thickness
pcf
ratio
pcf
ratio
ft
Q
psf
125
0.71
62.5
0.29
0.58
63
Table 4.1: Determination of MCNY vault side-arch self-weight pressure
For the webs:
Tile density
Share of vault
Mortar & plaster density
Share of vault
Web thickness
Q
pcf
ratio
pcf
ratio
ft
psf
125
0.75
62.5
0.25
0.33
36
Table 4.2: Determination of MCNY vault web self-weight pressure
Table 4.3 gives key web dimensions, lengths L and rises D. The dimensions come from Figure
4.1. To review, the plan is rectangular, so there are two different lengths for the side-arches
(corresponding to two different lengths for the web), but their rises are the same.
Long side
Short side
L
ft.
Dintrados
ft.
Dextrados
ft.
Dcenter
ft.
9.1
6.7
0.33
0.33
0.92
0.92
0.63
0.63
Table 4.3: Dimensions of length and rise for ½-scale vault
In addition, each side-arch is 1 ft. wide in plan.
Total vault weight is then
𝑊𝑇𝑜𝑡𝑎𝑙 = 2(63 𝑝𝑠𝑓)[(9.1 𝑓𝑡)(1 𝑓𝑡) + (6.7 𝑓𝑡)(1 𝑓𝑡)] + (36 𝑝𝑠𝑓)(9.1 𝑓𝑡)(6.7 𝑓𝑡) ≈ 4,211 𝑙𝑏 ≈ 4,210 𝑙𝑏
Figure 4.2 is a photo of the vault as construction nears completion. In the foreground, the fivecourse side-arch is seen, with the vault web rising above and away from it toward the crown (left
open for the exhibit, but assumed closed for this analysis).
46
4 Analysis of scaled Guastavino replica
Figure 4.2: View of vault nearing completion (photo by the author)
4.3
Modeling force flow for triangle-arches and sliced parallel arches
Triangle-arches:
Figure 4.3 shows one simple way to visualize the structural action of the vault. The vault is cut
into triangular quarters, and each piece analyzed separately as a single arch. The loading from
the web is superimposed on the loading from the side-arch. In this model, the side-arches each
take a quarter of the distributed load from the vault and transfer it to the supports. This is a
method of obtaining a thrust line for the side-arches themselves. In this thesis, this technique is
known as the triangle-arches method.
Figure 4.3: Modeling quarter-vaults taking ¼ of entire load; arrows show force flow (MIT Masonry Research
Group 2012), (arrows added)
47
4 Analysis of scaled Guastavino replica
Sliced parallel arches:
Alternatively, the vault can be sliced into parallel arches to model force flow (Figure 4.4).
Through membrane action, the distributed self-weight of the masonry “flows” as compressive
forces toward the diagonal ribs that crisscross the rectangular plan. These diagonal ribs distribute
the forces in compression to the corner supports. The resultant force is a thrust with both a
horizontal and vertical component. The vertical component is taken in compression in the steel
column; the horizontal component is taken in tension in the horizontal steel angles (steel columns
and angles not shown in Figure 4.4).
Figure 4.4: Force flow in ½-scale vault (Lee 2010), (arrows added)
In this visualization of force flow, parallel strips from each quarter of the vault can be modeled
independently as arches, and their thrusts applied to the diagonal ribs, themselves modeled as
arches (Heyman 1999). In this way, a thrust line can be obtained for each parallel arch, but more
importantly, a thrust line can be determined for the diagonal ribs, and the resultant thrust at the
supports found. This technique is known as the sliced parallel arches method.
4.4
Structural analysis
4.4.1 Static equilibrium analysis
Triangle-arch method:
The weights of the web and the side-arches are given separately using the equation for total
weight above:
48
4 Analysis of scaled Guastavino replica
Wweb
lb
Q web
psf
Warches
lb
Q arch
psf
2,230
36
1,980
63
Table 4.4: Vault weight properties
Weights per unit length are now calculated in order to determine horizontal thrusts and vertical
reactions. Each side-arch has a tributary width of 1 ft. Each quarter-web needs a tributary width
to calculate a distributed uniform load per unit length. Because each arch takes a quarter of the
vault’s loading as shown in Figure 4.3, the resulting shape of area is triangular, not rectangular.
Therefore, the average tributary width, wavg, is used for each side. For the long side, the tributary
width is one-half times half the length of the short side (or a quarter of the short side); for the
short side, the tributary width is one-half times half of the long side (or a quarter of the long
side). Then the distributed load is determined as follows:
Equation 4.1
𝑞 = 𝑄𝑤
The uniform weight per unit length, q, is determined for the webs and for the arches separately,
and the two are summed for qtotal. Equation 2.3, Equation 2.4, Equation 2.5, and Equation 2.7 are
then used to assess the vertical reactions at supports and maximum, minimum, and average
horizontal thrusts, respectively. The resultant thrusts are determined according to the
Pythagorean Theorem:
𝐹𝑡ℎ𝑟𝑢𝑠𝑡 = √𝑅𝑉2 + 𝐹𝐻2
Equation 4.2
The following table shows the results from this equilibrium procedure. (For a visual
interpretation of what kind of line of pressure corresponds to maximum and minimum horizontal
thrust, see Figure 2.2). Values for q in order from top to bottom are qweb, qarch, qtotal, respectively.
Long
side
Short
side
w
ft
1.7
1.0
2.3
1.0
q
FH,max
lb/ft
lb
61 1,910
63 1,950
124 3,860
83 1,400
63 1,060
146 2,460
FH,min
lb
694
710
1,400
510
384
894
FH,avg
lb
1,020
1,040
2,060
749
563
1,310
RV
lb
279
285
564
279
210
489
Fthrust,max Fthrust,min Fthrust,avg
lb
lb
lb
1,930
748
1,060
1,970
765
1,080
3,900
1,510
2,140
1,430
582
799
1,080
437
600
2,510
1,020
1,400
Table 4.5: Determination of horizontal thrusts, vertical reactions, and resultant thrusts in quarter-vaults
Note that FH,min < FH,avg< FH,max, as expected. The total vertical reaction at each support is 564
lb+489 lb = 1,053 lb ≈ 1,050 lb. This is closely confirmed by dividing the total weight by the
number of vertical supports: (4,210 lb)/4 = 1,053 lb. The maximum resultant thrust is 3,900 lb.
From Figure 4.1, it is found that the face width of the arch springing is about one foot, and the
49
4 Analysis of scaled Guastavino replica
arch thickness is seven inches. The face area of the springing is therefore 7*12 = 84 in^2. The
pressure on the springer is then (3,900 lb)/(84 in^2) = 46 psi. From Guastavino Sr.’s material
tests (Table 2.1), the crushing strength of this tile brick is 2,060 psi, which is confirmed by
modern-day sources which put it at about 2,000 psi (The Brick Industry Association 2007). The
unity check mentioned in section 2.4.3 is re-introduced:
𝑢𝑛𝑖𝑡𝑦 𝑐ℎ𝑒𝑐𝑘 =
𝑑𝑒𝑚𝑎𝑛𝑑
<1
𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦
Equation 4.3
Demand is 46 psi, capacity is 2,060 psi. That gives a unity check of 46/2,060 = 0.022 ≈ 2%.
Furthermore, this gives the vault a factor of safety against crushing of 45. The masonry is not at
risk of compressive failure under its own weight. Masonry structural analysis is a stability
problem, not an elasticity problem.
Sliced parallel arches method:
Next, representative vault quarters (long and short) are sliced into series of parallel arches
(Figure 4.5). These follow the force flow model presented in Figure 4.4. Each of six arches is
analyzed separately for its vertical reactions and horizontal thrusts. The sums of the vertical
reactions of adjacent web arches become the vertical point loads on the diagonal rib in between
them (represented by the broken brown line in Figure 4.5). The resultant horizontal thrusts from
adjacent quarter-vaults become the horizontal thrust at that point on the rib in the diagonal
direction. The final resultant thrust, angled upward, compressing the face of the rib cross-section
is itself the resultant from those former resultants. In other words, the final resultant thrust is
found through vector addition of the horizontal and vertical forces. Figure 4.5 shows the sliced
parallel web arches for each side, numbered 1 to 5. Parallel sliced arches numbered 6 are simply
the side-arches under the uniform self-weight of Qarch. The web is sliced into arches numbered 1
to 5 and those strips are under the uniform self-weight Qweb. The numbered dot in the middle of
each arch width on the diagonal represents where the resultant forces are assumed to act.
50
4 Analysis of scaled Guastavino replica
Figure 4.5: Illustration of sliced parallel arches and locations of resultant forces on diagonal ribs
A key assumption made in the analysis is that the rises of the intrados and extrados are the same
for each parallel arch in the webs. This is appropriate because the vault has the characteristics of
a thin shell; its thickness is uniform throughout the web.
Refer to Table 4.1and Table 4.2 for the uniform self-weights per unit area for the side-arches and
for the web (the analysis accounts for the different thicknesses of the side-arches and the web).
Results for all the parallel arches including side-arches (arch number 6) for each of the long and
short directions are as follows:
51
4 Analysis of scaled Guastavino replica
Arch
Long
side
L
w
q
FH,max
FH,min
FH,avg
RV
Fthrust,max
Fthrust,min Fthrust,avg
ft
ft
lb/ft
lb
lb
lb
lb
lb
lb
lb
1
2
3
4
5
6
1.5
3.0
4.6
6.1
7.6
9.1
0.67
0.67
0.67
0.67
0.67
1.0
24
24
24
24
24
63
21
85
191
339
530
1,950
7.7
31
69
123
193
710
11
45
102
181
283
1,040
19
37
56
74
93
285
28
93
199
347
538
1,970
20
48
89
144
214
765
22
59
116
196
298
1,080
1.1
2.2
3.4
4.5
5.6
6.7
0.91
0.91
0.91
0.91
0.91
1.0
33
33
33
33
33
63
16
62
140
250
390
1,060
5.7
23
51
91
142
384
8.3
33
75
133
208
563
19
37
56
74
93
210
24
73
151
260
401
1,080
19
44
76
117
170
437
20
50
93
152
228
600
Short
side
1
2
3
4
5
6
Table 4.6: Equilibrium analysis for sliced parallel arches
Using the horizontal thrusts from sliced parallel arches, the resultant horizontal thrusts at each
numbered loading point 1 through 6 on the adjacent rib along its longitudinal axis can be
determined. The procedure is as outlined in section 2.4.1 with Figure 2.10, where the resultant
horizontal thrust at one point on the rib is the sum of the resultant thrust from the point i-1 above
plus the resultant thrust on the immediate point i:
2
2
𝐹𝐻𝑖 = 𝐹𝐻𝑖−1 + √𝐹𝐻𝑖,1
+ 𝐹𝐻𝑖,2
, 𝑖 = 1, … ,6
Equation 4.4
The vertical reaction at each loading point 1 through 6 on the rib-arch is the sum of the vertical
reactions from adjacent parallel arches plus the vertical reaction from the point above:
𝑅𝑉,𝑟𝑖𝑏,𝑖 = 𝑅𝑉,𝑙𝑜𝑛𝑔 𝑤𝑒𝑏,𝑖 + 𝑅𝑉,𝑠ℎ𝑜𝑟𝑡 𝑤𝑒𝑏,𝑖 + 𝑅𝑉,𝑟𝑖𝑏,𝑖−1
This procedure leads to the following results for the ribs:
52
Equation 4.5
4 Analysis of scaled Guastavino replica
Horizontal resultants
Vertical
Thrust resultants
Fthrust,rib|max Fthrust,rib|min Fthrust,rib|avg
FH,rib|max
FH,rib|min
FH,rib|avg
RV,rib
Load
point
lb
lb
lb
lb
lb
lb
lb
1
2
3
4
5
6
26
132
369
790
1,450
3,670
9.6
48
134
287
527
1,330
14
70
197
421
772
1,960
37
112
223
372
558
1,050
46
173
431
873
1,550
3,820
38
121
260
470
767
1,700
40
132
297
562
953
2,220
Table 4.7: Horizontal thrusts on a diagonal rib and resultant thrusts
4.4.2 Results discussion – static equilibrium
Results from the two prevailing methods are compared to gauge their accuracy relative to each
other.
From the triangle-arch analysis, the total vertical reaction at a support is ΣRV = 1,053 lb ≈ 1,050
lb using three significant figures. This number represents the vertical reaction at point 6 of the
rib-vault at the vault corner (Table 4.7). This value has been found by summing all the vertical
reactions for the long and short web arches, and is the resulting vertical force pushing up on the
end (springing) of the rib. The percent error between the two methods is 0%. Evidently the
analysis must distinguish between the different thicknesses of the vault web and side-arches to
obtain accurate results, and must not approximate the entire density with an average thickness.
Similarly, the resulting thrusts of the triangle-arch method and the sliced parallel arches method
can be compared. For the triangle-arch method, the final resultant is determined in the following
manner: 1) take the resultant of the horizontal thrusts from the two adjacent triangle-arches; 2)
sum the vertical reactions from the two triangle-arches; and 3) take the total thrust resultant using
the resultant horizontal thrust and the summed vertical reactions (Equation 4.6). This is in order
to get the thrust resultant acting between the two triangle-arches—in other words, the diagonal
resultant thrust which acts on the ribs between two triangular arch-areas. (This resultant has a
horizontal and vertical component.) Then this value can be compared to the results from the
sliced parallel arches method.
2
2
2
𝐹𝑡ℎ𝑟𝑢𝑠𝑡,𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 = √(√𝐹𝐻,𝑙𝑜𝑛𝑔
+ 𝐹𝐻,𝑠ℎ𝑜𝑟𝑡
) + (𝑅𝑉,𝑙𝑜𝑛𝑔 + 𝑅𝑉,𝑠ℎ𝑜𝑟𝑡 )
2
Equation 4.6
The following table compares thrust resultants on the rib-arches for maximum, minimum, and
average horizontal thrust magnitudes:
53
4 Analysis of scaled Guastavino replica
Thrust resultants in diagonal direction
Rib-arch
Triangle-arch
Fthrust,rib|max Fthrust,rib|min Fthrust,rib|avg Fthrust,max Fthrust,min Fthrust,rib|avg
lb
lb
lb
lb
lb
lb
3,820
1,700
2,220
4,700
1,970
2,660
Table 4.8: Comparing diagonal thrust resultants for two methods
Relative difference between maximum states: 23%.
Relative difference between minimum states: 16%.
Relative difference between average states: 20%.
There is no difference between the two analysis methods in the vertical component of thrust, and
less than 25% relative differences between the two methods for total thrust. This thesis is most
concerned with the average horizontal thrust state, for which there is 20% relative difference.
These results argue for some incompatibility between the triangle-arches and sliced parallel
arches methods of modeling force-flow.
The horizontal thrusts along the rib’s diagonal axis are compared to the vault’s total weight as
percentage after the practice of Allen et al. (see section 2.4.1):
Horizontal diagonal resultants % total vault weight
From sliced parallel arches method
From triangle-arches method
FH,rib|max
lb
%W
3,670
87
FH,rib|min
lb
%W
1,330
32
FH,rib|avg
lb
%W
1,960
46
FH,triangle|max
lb
%W
4,580
109
FH,triangle|min
lb
%W
1,660
40
FH,triangle|avg
lb
%W
2,440
58
Table 4.9: Horizontal thrusts as % total vault weight for both analysis methods
For horizontal thrust this gives an upper bound (maximum) of 0.87W and a lower bound
(minimum) of 0.32W for the sliced parallel arches method and between 1.1W and 0.4W for the
triangle-arches method. Allen et al. (see section 2.4.1) performs this procedure for a steeper vault
geometry (a groin vault) and generates a range of horizontal thrusts with an upper bound of
0.32W and a lower bound of 0.21W. The MCNY vault is a very shallow almost dome-like vault
with a smooth geometry in the web, in contrast to a groin vault which resembles intersecting
pointed barrel vaults with ribs much thicker than surrounding webs. Therefore, it is expected that
the values for horizontal thrust as a percentage of total vault weight may not conform precisely to
the predictions of Allen et al. Most likely, the high values of horizontal thrust as percentage of
total weight are a function of the low rise of vault—D is less than a foot for maximum,
minimum, and average horizontal thrusts.
The analysis methods are quite different considering their load path assumptions. In the sliced
parallel arches method, the vault is subdivided and each section analyzed separately (locally),
then the results are combined to obtain a global picture of structural behavior. Whereas in the
triangle-arch method, the vault is divided into large quadrants, and a side-arch alone bears the
54
4 Analysis of scaled Guastavino replica
entire quadrant weight plus its own weight and transmits this load into a single thrust magnitude.
Both these methods account for the differences between side-arch and web density. The sliced
parallel arches method accounts in more detail for the force flow from webs and arches to
diagonal ribs, and therefore this method is recommended for analysis if one of the two methods
is to be chosen.
The following table collates the key analytical results:
Triangle-arches
Sliced parallel arches
Long side
Short side
On diagonal
Rib
FH,max
lb
3,860
2,460
4,580
3,670
FH,min
lb
1,400
894
1,660
1,330
FH,avg
lb
2,060
1,310
2,440
1,960
RV
lb
564
489
1,053
1,050
Fthrust,max
lb
3,900
2,510
4,700
3,820
Fthrust,min
lb
1,510
1,020
1,970
1,700
Fthrust,avg
lb
2,140
1,400
2,660
2,220
ΣRV
lb
1,053
1,050
1,053
Table 4.10: Collated analytical equilibrium results
4.4.3 Graphical analysis
Graphical analysis according to the methods described in section 2.4.1 is carried out for the sidearches via the triangle-arches method and for diagonal ribs via the sliced parallel arches method.
For the long side-arch the triangle-arch method is used, where a quarter of the vault is assumed
to be carried by its corresponding edge-arch. A thrust line is drawn for the long-sided edgearches. Next, using force values from the sliced parallel arches method in the preceding analysis,
the thrust line for the diagonal ribs is constructed. Please note: Throughout the graphical
analysis, the values for average horizontal thrust, FH,avg, are used, and lines of pressure indicate
possible load paths that fit within upper and lower bounds defined by maximum and minimum
horizontal thrusts, respectively. The analytical method of determining horizontal thrust is
approximate since it assumes a uniform distributed load per unit length. A graphically
determined horizontal thrust can be found and compared to the analytical equivalent if the
analytical horizontal thrust leads to a line of pressure outside vault geometry.
55
4 Analysis of scaled Guastavino replica
Figure 4.6: Division of areas for graphical analysis for triangle-arches method
Triangle-arches method:
Reference is made to Table 4.4 for the uniform self-weight per unit area for the web Qweb and
Qvault for the vault. Q is multiplied by a rectangular tributary area taken from each triangular
quadrant such that Q times the area AT gives a load Pi (in lb) on divided segments of the arch
(numbered 1 to 6 in Figure 4.6). Then these loads are scaled to represent distances instead of
forces. This follows from the approach of Zalewski and Allen, except that not all loads Pi will be
identical because not all their tributary areas are identical. The resulting point load acts at the
center of gravity of each subdivided segment. The rectangular tributary area for the web
segments corresponds to an average depth of the triangle that passes through the location of the
point load. Figure 4.6 shows the geometry of this method for both web segments and side-arch
segments. The widths lblock of each segment are an equal fraction of the arch length, except for
the ends (denoted number 6) which are each lblock/2. The depths of the rectangular areas for the
web are as deep as the location on the diagonal corresponding to the midpoint of the segment
width lblock (locations shown as a green dots in Figure 4.6).
An equation is developed to determine the point loads Pi on web segments 1 through 5:
𝑃𝑤𝑒𝑏,𝑖 = 𝑄𝑤𝑒𝑏 (
𝐿𝑎 𝐿𝑏 6 − 𝑖
)( )(
)
10 2
5
56
Equation 4.7
4 Analysis of scaled Guastavino replica
La = the primary arch length which is being divided into segment widths lblock for analysis; Lb =
the adjacent arch length which divided in two is the maximum depth of the rectangular area; i =
the number of the segment; the term on the right controlled by i is the “depth scale factor” κ. The
equation is adjusted for the smallest segments (segment number 6):
𝑃𝑤𝑒𝑏,6 = 𝑄𝑤𝑒𝑏 (
𝐿𝑎
𝐿𝑏 1
)( )( )
2 ∗ 10 2 10
Equation 4.8
Equation 4.8 uses the same labeling as Equation 4.7, except instead of using i, the fraction of half
the adjacent length (for depth) is 0.1—this is the “depth scale factor” κ. This is because it is half
the depth as any 1/5 the same length. Similarly, the segment width is lblock/2—half that used in
Equation 4.7.
For the side-arches, the equation is
La
𝑃𝑎𝑟𝑐ℎ,𝑖 = 𝑄𝑎𝑟𝑐ℎ ( ) (1 𝑓𝑡)
10
Equation 4.9
Equation 4.9 changes by a half for segment 6, where lblock is simply (1/2)*(La/10)= La/20. The
point loads Pweb and Parch acting at the center of gravity of each arch segment for both long and
short sides are calculated for both the web and side-arch segments:
Segment no.
Long side
lblock
ft
Lb/2
ft
κ
Pweb
lb
Parch
lb
1
2
3
4
5
6
Short side
1
2
3
4
5
6
0.91
0.91
0.91
0.91
0.91
0.46
3.4
3.4
3.4
3.4
3.4
3.4
1.0
0.8
0.6
0.4
0.2
0.1
112
89
67
45
22
5.6
57
57
57
57
57
29
0.67
0.67
0.67
0.67
0.67
0.34
4.6
4.6
4.6
4.6
4.6
4.6
1.0
0.8
0.6
0.4
0.2
0.1
112
89
67
45
22
5.6
42
42
42
42
42
21
Table 4.11: Deriving loads on arch segments
The table above shows that despite their different total lengths, each triangle-arch web quadrant
of the vault is loaded the same way. The side-arches are not loaded in exactly the same way, but
57
4 Analysis of scaled Guastavino replica
the long side experiences a greater magnitude of load along its side-arch. Graphical analysis will
be performed three times for the triangle-arch method on the long arch only. The break-down of
graphical analysis on the long side-arch is:
1) Apply loads from web segments
2) Apply loads from side-arch segments
3) Apply superimposed loading from web and side-arch segments
The loads are scaled here by a factor of 1/8 in.—that is, the scale is 1 lb = 1/8 in. as follows:
𝑃𝑖 [𝑙𝑏] ∗ 0.125 [𝑖𝑛. ] = 1 [𝑙𝑏] ∗ 𝑃𝑠𝑐𝑎𝑙𝑒𝑑 [𝑖𝑛. ]
Equation 4.10
Pscaled is in inches. Figure 4.7 shows the set-up of the graphical analysis. Scaled loads act in the
middle of arch segments denoted AB, BC, CD,…, KL. The segments between loads are denoted
oa, ob, oc,…, ol, with o being the intersection at the vault centerline of the two equal resultant
thrusts on the springings. The right side of the force polygon (Figure 4.8) is made by
superimposing the applied vertical loads, in inches, atop each other in order from left to right
along the arch. The orientations of the thrust resultants are drawn by connecting o to a, and o to l
to close the force polygon. Horizontal thrust is the horizontal distance between o and the
midpoint of the superimposed vertical loads; the thrust resultant is the scaled magnitude of rays
oa and ol. The objective here is to use the values from the static equilibrium triangle-arch
method and see whether a resulting line of pressure acts within the arch geometry, which would
indicate vault stability.
The scaled loads are as follows:
Segment no.
Pweb
in
Parch
in
Ptotal
in
1
2
3
4
5
6
14
11
8.4
5.6
2.8
0.70
7.1
7.1
7.1
7.1
7.1
3.6
21
18
15
13
10
4.3
Table 4.12: Scaled loads for graphical analysis
First, the scaled web loads alone are applied to the side-arch:
58
4 Analysis of scaled Guastavino replica
Figure 4.7: Application of web point loads to arch-segment centers of gravity
The applied scaled loads (red downward arrows) are stacked atop each other in order from AB to
KL. Then the average horizontal thrust for the web loading from Table 4.5 for the long side is
scaled: 1,020 lb ≈ 127 in. This value is added to the force polygon (blue horizontal ray through
center). Then rays oa through ol are drawn (Figure 4.8).
Figure 4.8: Force polygon for web loads
The distance from point o to the midpoint of FG gives the scaled horizontal thrust; half the
vertical distance from A to L gives the vertical reaction at either support.
The rays are then kept in their orientation and applied to the center of gravity of each arch
segment AB to KL. With all the thrust rays for each arch segment superimposed upon the arch, a
common path shown in yellow is traced by connecting the intersections of the rays at segment
centers of gravity to show a possible line of pressure within the arch (Figure 4.9).
59
4 Analysis of scaled Guastavino replica
Figure 4.9: Line of pressure for long side-arch (yellow line), web loads
At first glance the line of pressure in yellow shown in Figure 4.9 might seem alarming—it does
not appear to adhere to the safe theorem. It shows that the line of pressure for the web leaves the
geometry toward the supports. A line of pressure may be shifted to fit within the geometry,
however. This is equivalent to simply moving a force vector along its axis of action—a common
practice in the study of vector physics. The line of pressure is shifted to fit within the geometry:
Figure 4.10: Adjusted line of pressure for long side web loads
In addition, remember that a generous assumption is made in the triangle-arch method that the
entire triangular quadrant of the vault acts as one two-dimensional arch having the geometry of
the side-arch. Web material actually extends behind the side-arch and rises in height until the
actual vault crown, above the altitude of the arch shown in Figure 4.9. This will be explored
further in section 4.4.4, but is mentioned to illustrate the complex three-dimensional nature of the
structural system. Notwithstanding that complexity, the line of pressure is made to fit within the
geometry, indicating stability under web loading.
Now the arch is subjected to the scaled loads from the long side-arch alone.
Figure 4.11: Application of side-arch point loads to arch-segment centers of gravity
The force polygon is constructed using the value of average horizontal thrust for the long sidearch, 1,040 lb = 130 in.
60
4 Analysis of scaled Guastavino replica
Figure 4.12: Force polygon for arch loads
Giving the following line of pressure:
Figure 4.13: Line of pressure for long side-arch (yellow line), arch loads
The thrust line falls within the middle-third of arch geometry for the state of average horizontal
thrust. This is predictable because the scaled loads on each arch segment are essentially the same
and do not grow in magnitude due to any portioning of triangular surface area into rectangular
tributary areas.
Now the arch is subjected to the combined scaled loads for the webs and the side-arch:
Figure 4.14: Application of web and side-arch point loads to arch-segment centers of gravity
61
4 Analysis of scaled Guastavino replica
The force polygon is generated using the average horizontal thrust from both web and side-arch
loads together, 2,060 lb ≈ 257 in.
Figure 4.15: Force polygon for combined web and arch loads
Giving the following line of pressure:
Figure 4.16: Line of pressure for long side-arch (yellow line), combined web and arch loads
The line of pressure acts within the geometry of the side-arch, indicating overall stability in this
model under superimposed web and side-arch weight. The variable loads from the web model
tend to push the line of pressure down outside of the middle-third, but still within the overall
geometry. The line of pressure could be shifted slightly upward to fit within the middle-third.
At a glance, it is evident that a slightly greater magnitude of horizontal thrust might help fit the
line of pressure better within the middle-third of the side-arch geometry. As previously stated,
the analytically calculated horizontal thrust is only an approximation. It derives from a uniform
distributed loading, which is also only an approximation, especially of vault weight. A more
accurate horizontal thrust can be found graphically. The force polygon can be constructed using
the same scaled vertical loads found analytically, but with a horizontal thrust found graphically.
The process is now amended: a) use the scaled vertical loads from Table 4.12; b) draw rays oa
and ol tangent to the curvature of their arch segments (Figure 4.17); c) maintain the orientation
of those two lines and connect the left then the right lines to points A and B respectively at the
62
4 Analysis of scaled Guastavino replica
right to start building the force polygon (Figure 4.18); d) extend these thrust rays until they
intersect—this is the new location of the pole o (Figure 4.18); e) continue constructing the thrust
line as before.
Figure 4.17: Drawing rays oa and ol starting on load diagram
Figure 4.18: Graphical interpretation of horizontal thrust
Notice the new scaled horizontal thrust of 23 ft.-3 ½ in. This is
3.5
[(23 + 12 ) ∗ 12] 𝑖𝑛.
= 2,236 𝑙𝑏 ≈ 2,240 𝑙𝑏
0.125 𝑖𝑛./𝑙𝑏
63
4 Analysis of scaled Guastavino replica
This slightly overestimates the 2,060-lb average horizontal thrust found analytically. Using only
graphical methods for the rays gives (2,240/2060)*100% = 109%, or only 9% greater magnitude.
This comparison confirms the validity of the methods adopted earlier.
The equivalent uniform distributed load per unit length from self-weight, qeq, is calculated using
the graphically determined horizontal thrust as representing the average state:
𝑞𝑒𝑞 =
𝐹𝐻 (8𝐷𝑐𝑒𝑛𝑡𝑒𝑟 ) (2,240 𝑙𝑏)(8 ∗ 0.6 𝑓𝑡)
=
= 130 𝑙𝑏/𝑓𝑡
(9.1 𝑓𝑡)2
𝐿2
The total distributed load shown in Table 4.5 for the long side is 124 lb/ft. The calculation above
gives 105% of that distributed load. This result stems from an agreement between the actual
physical mechanics of the shallow vault and the modeling strategies employed here. The
associated force polygon for this technique follows:
Figure 4.19: Force polygon from purely graphical horizontal thrust interpretation
Giving a line of pressure that also fits within the geometry:
Figure 4.20: Line of pressure drawn from graphical interpretation
Notice that this line of pressure acts neatly through the center of the arch. It even appears to act
within the middle-third. The purpose of the preceding process is to show the compatibility of
loading the system with scaled point loads yet generating the resultant thrusts (green rays)
graphically only. Finding horizontal thrust either analytically or graphically is valid. If one
64
4 Analysis of scaled Guastavino replica
method does not show the line of pressure within the geometry, the other may be used as a check
and for an alternative solution. The goal is to find a valid equilibrium solution in compression
which lies within the masonry. It is suggested to begin by following the practice of the first
iteration, however, so that external loading is found analytically. Calculate the externally applied
loads—vertical and horizontal—and then graphically evaluate their influence on the structure’s
geometry. If the line of pressure leaves the geometry, as in Figure 4.9, try shifting the line of
pressure to fit (Figure 4.10). If it still does not fit, determine the horizontal thrust graphically and
observe the adjusted line of pressure (Figure 4.17 and Figure 4.20).
Note: If either of the above two methods does not yield a valid solution for stability according to
the safe theorem, then an iterative process can be undertaken involving different trials of
graphically found horizontal thrust along with the current vertical loading to find a stable line of
pressure. The process is as follows:
a) With the original vertical loads superimposed on each other such as on the right in Figure
4.19, attempt a trial horizontal thrust extending from the center of the vertical loads to the
pole o that will try to keep the line of pressure within the geometry;
b) Construct the force polygon;
c) Superimpose the rays onto the arch geometry as performed elsewhere in this thesis;
d) Connect the lines at their intersections along the vertical axis passing through the
segment centers of gravity (as before);
e) Observe the line of pressure. If the line of pressure fits within the geometry, then a valid
solution for stability has been found. If it does not fit, then iterate the horizontal thrust
again (part a)) with the same set of vertical loads. Repeat to find a valid solution.
This procedure is adopted in chapter 5, sections 5.5.3 and 5.5.4.
Sliced parallel arches method:
In this analysis, the vertical reactions for loading-points 1 through 6 from analytical static
equilibrium for the parallel arches (Table 4.6) are added at each loading point, scaled in inches,
and applied to corresponding nodes on the diagonal rib-arch. The scaling is the same 1 lb = 0.125
in. Then with the calculated average horizontal thrust from Table 4.7, resultant scaled forces will
be superimposed on the rib-arch geometry to see whether they form a line of pressure acting
within the diagonal rib-arch. Refer to Figure 4.5 for illustration and labeling of the web arches
interacting with diagonal ribs.
The following table shows the sum of two vertical reactions from long and short web arches
applied as scaled vertical loads (in inches) to the rib-arch:
65
4 Analysis of scaled Guastavino replica
Segment no.
1
2
3
4
5
6
lb
Force P
in
37
74
112
149
186
495
5
9
14
19
23
62
RV from all parallel arches
Table 4.13: Determination of scaled rib thrusts and angles
The graphical representation of loads on rib-arch segments is created according to the process
outlined in the previous section:
Figure 4.21: Application of point loads to rib-arch segments’ centers of gravity
Now the force polygon is generated for the state of average horizontal thrust. From Table 4.7, the
average horizontal thrust on the ribs is 1,960 lb, which scaled is (1,960 lb)*0.125 = 245 in ≈ 244
in. This is applied to the force polygon, and rays oa,…,ol are drawn (Figure 4.22):
66
4 Analysis of scaled Guastavino replica
Figure 4.22: Ribs force polygon
The vertical dimension of 21 ft.-7 in. corresponds to the vertical reactions at two supports of the
rib-arch:
7
(21 + 12) ∗ 12
0.125
= 2,072 𝑙𝑏 ≈ 2,070 𝑙𝑏
Dividing the vertical reaction by two gives the reaction at each end support: 2,070/2 = 1,035 lb.
From Table 4.7, the calculated vertical reaction at an end support from static equilibrium is 1,050
lb. The % error in transferring the loads from analytical to graphical analysis is negligible for this
shallow arch:
(
1,050 − 1,035
) ∗ 100 = 1.4%
1,035
The force polygon above generates the following line of pressure (in yellow) for the rib-arch
geometry:
67
4 Analysis of scaled Guastavino replica
Figure 4.23: Line of pressure, in yellow, in the diagonal rib-arch
The line of pressure leaves the geometry between points C and D, and I and J, seeming to prove
instability. The arch is stable, however, which can be confirmed by the structure standing in the
exhibit. Shifting the line of pressure down shows stability:
Figure 4.24: Line of pressure in diagonal rib-arch adjusted to fit within geometry
In this case, using the analytical average horizontal thrust leads to a valid solution. The ribarches are stable under the loading from self-weight.
As in the triangle-arches method above, the horizontal thrust and line of pressure can also be
found graphically. The force polygon is drawn using the same vertical loading scheme as in
Figure 4.21 and using resultant thrusts found graphically as in Figure 4.17:
Figure 4.25: Force polygon for rib-arches with graphically found horizontal thrust
68
4 Analysis of scaled Guastavino replica
Notice the new horizontal thrust magnitude of
8.25
(27 + 12 ) ∗ 12
= 2,658 𝑙𝑏 ≈ 2,660 𝑙𝑏
0.125
This exceeds the calculated average horizontal thrust of 1,960 lb by ((2,660-1,960)/1,960)*100%
= 36%. The high percentage difference may be due to the fact that the horizontal thrust on the
rib-arch comes from superimposing the analytical horizontal thrusts from the webs, whereas the
above graphically evaluated horizontal thrust ignores the effect of the web-arches on the rib-arch.
This force polygon is an example of a trial where building the horizontal thrust according to the
method of Figure 4.17 and Figure 4.18 does not lead to a valid equilibrium solution. This is
evident in the associated line of pressure:
Figure 4.26: Line of pressure for rib-arches with horizontal thrust found graphically
Note that even when the line of pressure is shifted along the vertical axis, it does not fit within
the rib-arch geometry. This is an invalid solution, which contrasts with the stability shown by the
valid solution of Figure 4.24 which used the average horizontal thrust determined analytically.
4.4.4 Results discussion—graphical analysis
For the MCNY vault, Section 4.4.3 has shown the lines of pressure to act within the side-arch for
superimposed web and arch loads and within the rib-arch. This shows that under its own selfweight the MCNY vault is stable. Throughout the assessment the vault is modeled as a twodimensional arch, but the vault is, of course, a three-dimensional structure. Lines of pressure that
result from loading may not act exactly as modeled, and the physical force-flow modeling
adopted here does not entirely portray the true structural action of this very shallow vault.
Truthfully, the thrust line may act at an angle, into the page with forces “flowing” from near the
crown and out through the geometry parallel to the extrados and intrados to the supports. If this
three-dimensional understanding of the force-flow is true, then the analysis presented in section
4.4.3 represents a conservative assessment, and perhaps the vault has an even higher factor of
safety.
It can be extrapolated from Figure 4.1 that if the vault were built to completion (without the hole)
as understood in this assessment, then the vault’s crown rises approximately 6.75 in. above the
triangle-arch crown. This is essentially 6.75 in. more material that can be added in the twodimensional model to a side-arch for graphical analysis. Lumping material atop the arch still
69
4 Analysis of scaled Guastavino replica
considers only two dimensions, however. In reality the vault rises to the crown in three
dimensions. Perhaps the line of pressure could be shown to fit within the three-dimensional
geometry. The thrust line of Figure 4.9, representing only web loads applied to the side-arch,
may in fact act into the page with its base at the side-arch springings and its crown near the
actual crown of the vault. Nonetheless, the same line of pressure can be shifted to fit within the
two-dimensional geometry of the model (Figure 4.10).
4.5
Conclusions
Analytical static equilibrium shows that the vault has a factor of safety of more than 45 under its
own material self-weight. Methods of analytical equilibrium analysis have been presented in
such a way that the reader may repeat the process for other vaults. An important aspect of the
analysis is distinguishing between material densities of the side-arches and webs if they are
different.
Graphical analysis shows that the side-arch and rib-arch are stable under the vault’s self-weight,
but that the line of pressure leaves the rib-arch at the extrados about 1/8-span from the supports
when the horizontal thrust is found graphically. This agrees with the discussion by Heyman
regarding cross-vault rib action, but calls into question the accuracy of applying the force-flow
modeling assumptions presented in section 4.3 to shallow vaults where the rise-to-span ratio is
1/3 and 1/4 as here. An alternative thrust line strategy is postulated in section 4.4.4 which
accounts for the three-dimensional nature of the vault, but executing the strategy is outside the
scope of this thesis. Unlike ribbed Gothic vaults and cross-vaults, the shallow vault here has very
smooth transitions from one quadrant to another. This resembles the geometry of a dome more
than a cross-vault. Like a dome, the shallow vault may have both meridional and hoop stresses.
Yet in this assessment, only arch-like compressive action is considered.
The graphical methods presented here are generally compatible with the shallow vault geometry
with the exception of the rib-arch when horizontal thrust is found graphically. The underlying
objective is to present these methods step by step so the engineer may apply them to proper
cross-vaults. Analyzing cross-vaults is the main theme of this thesis and the subject of the next
chapter.
Important points this chapter contributes and an idea for future development:
1) Both the triangle-arches and sliced parallel arches methods can be used to approximate
thrust lines and find a safe compression solution.
2) For the shallow vault, the horizontal thrust is between 0.32W and 0.87W and the uniform
loading assumption is valid.
3) Stresses are very low in the shallow vault.
4) Thrust network analysis (TNA) could be used to better model the three-dimensional
behavior of the vault geometry and obtain more accurate solutions.
70
71
5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
5.1 Chapter objectives
This chapter applies the same methods of analytical static equilibrium and graphical analysis
presented in Chapter 4 to the former cross-vaults in wing H of the Museum. Authoritative
drawings of the now-demolished vaults help recreate the cross-vault geometry for analysis. Then,
as in Chapter 4, both the triangle-arches and sliced parallel arches methods are applied for the
analytical static equilibrium and graphical analysis techniques. The analysis follows this outline:
1) Analytical equilibrium for distributed loading
a. Triangle-arches method
i. Long side
ii. Short side
b. Sliced parallel arches method
2) Graphical analysis for distributed loading
a. Triangle-arches method
i. Long side
ii. Short side
b. Sliced parallel arches method
3) Graphical analysis for superimposed distributed loading and point load
a. Triangle-arches method
i. Long side
ii. Short side
b. Sliced parallel arches method
5.2
Authoritative drawings
A representative cross-vault from wing H which supported the second level is analyzed. As
correspondence reveals, the R. Guastavino Co. had working drawings from wing H but none
from wing E extant. Those drawings of wing H are now at the Avery Drawings and Archives
Collection at Columbia University (hereafter referred to as Avery Archives). This is thanks to
Columbia Professor George Collins, who saved the documents when the Company closed its
doors in 1962 (Ochsendorf 2010). The drawings used here are dated June 23, 1911, and July 5,
1911 (see Figure 8.1 to Figure 8.7 in the appendix).
The City of New York Department of Parks also produced drawings of existing structural
conditions in wings E and H. These drawings, dated February 28, 1962, are also used here to recreate the geometry of the Guastavino vaults in wing H. On these drawings, SEKA (Severud) is
listed as the structural engineer for the project (Figure 8.8 and Figure 8.9).
McKim, Mead and White architectural detail elevations dated October 9, 1911, are used. These
show vault heights and widths, and confirm the roughly sketched details and dimensions of the
Guastavino drawings themselves (Figure 8.10 to Figure 8.12).
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
June 9, 1943, tracings by the Museum of second floor framing plans dated December 22, 1909,
by McKim, Mead and White are given in the appendix (Figure 8.13 and Figure 8.14). They
include a column schedule and beam sizes for the steel that was part of the composite Guastavino
masonry-plus-steel beam construction. Although the scope of this thesis is to investigate the
structural capacity of the unreinforced masonry systems alone, it is instructive to note the use of
steel for vertical support elements. Furthermore, this knowledge could be useful in future
assessments of the early structural loading capacity of the Museum’s wings E and H.
These drawings, all of which are found in the appendix, are sufficient to recreate the geometry of
the Guastavino vaulting in wing H of the Museum.
5.3
Loading assumptions
From 1910 construction specifications (see Chapter 3), it is determined that second floor
concrete fill was 6 in. thick and the slab was unreinforced. Concrete unit weight is taken as 145
pcf. Also from correspondence, it is determined that the original design live load capacity of
wing H on the second floor was 150 psf. The Museum’s 1962 renovation program publication
presents the new design capacity as 300 psf, which presumably included point loads from
monolithic pieces such as Egyptian statues weighing 8,000 lbs (4 tons). The density of
Guastavino tile-and-Portland-cement systems can be taken as 112 pcf (Reese 2008), but as in
Chapter 4 a separate determination gives 125 pcf, which is used here.
These loading assumptions drive analysis methods in determining the loading capacity and safety
of the Guastavino cross-vaults in wing H of the Museum. The key questions are a) could the
vaults support the increased 300 psf live load, and b) could the vaults alternately support an
8,000-lb point load superimposed on the design 150 psf live load?
5.4
Cross-vault geometry
From the drawings referenced in section 5.2, a three-dimensional recreation is generated of the
wing H cross-vaults which formerly surrounded the central courtyard of that wing. In the
following images two consecutive cross-vaults are shown. In the analysis, a single cross-vault
comprised of a short side and a long side—or two intersecting barrel vaults—is assessed. Also
note the presence of the piers in some of the drawings. These are to indicate the location of steel
columns which supported the vaults. These steel columns were cloaked in a masonry veneer,
presumed aesthetic rather than structural. The steel columns are not considered in this
assessment, but attention is given to them in the conclusion, where future prescriptions for
analysis are recommended.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.1: Rhinoceros 3D recreation of vaults in wing H near stairwell and elevator; left: with floor slab, and right:
without floor slab
Figure 5.2: Long side of the vault with dimensions, including floor slab
Figure 5.3: Short side of the vault with dimensions
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.4: Plan view of the vault with dimensions
5.5
Structural analysis: Cross-vault
R. Guastavino Co. drawings dated 1911 and drawings of existing conditions dated 1962 reveal
that these vaults were four courses thick. With tile and mortar that is about six inches of total
thickness. Using the same assumptions in Chapter 4 about tile unit weight, mortar and plaster-ofParis unit weight, and the ratio of the weight of tiles and mortar or plaster to the total weight of
the vault, the following table is generated for the Museum vault’s properties:
Tile density
Share of vault
Mortar & plaster density
Share of vault
Vault thickness
pcf
ratio
pcf
ratio
ft
Qdead,vault
psf
125
0.8
62.5
0.2
0.50
56
Table 5.1: Met vault properties
Other pertinent dimensions and loading:
Long side
ft
Short side
ft
Area A
ft2
Wdead,vault
lb
26
12
326
18,400
Table 5.2: Lengths, area, and total vault weight
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
To review, total vault dead weight is calculated by multiplying stress by area:
𝑊 = (56 𝑝𝑠𝑓) ∗ (26.3 𝑓𝑡) ∗ (12.4 𝑓𝑡) ≈ 18,400 𝑙𝑏
In Table 5.2, the surface area is assumed rectangular and flat, giving a uniform distributed load
per unit area. Alternatively, the surface area can be approximated as half of a cylinder with the
radius being half the span of either the long side or the short side. The surface area is the same
using either the long side or short side for the radius (Table 5.3). In this approach, the surface
area of one of the intersecting barrel arches only is used, because it would be an overapproximation to add the two total half-cylindrical surface areas of the intersecting vaults (there
is cylinder material on the underside subtracted where the vaults intersect). The surface area is
half the surface area of a cylinder with that radius:
𝐴𝑠𝑢𝑟𝑓𝑎𝑐𝑒,𝑣𝑎𝑢𝑙𝑡 = 𝜋𝑅𝐿𝑎𝑑𝑗
Where Ladj is simply the length of the equivalent half-cylinder and R is the radius. Then total
weight is found as the product of self-weight in psf and the surface area in square feet:
Rlong
ft
Along
ft2
Rshort
ft
Ashort
ft2
Wdead,vault
lb
13
513
6.2
513
28,800
Table 5.3: Weight found with cylindrical surface area
The concrete floor slab properties are shown and its dead weight is calculated:
Slab density
pcf
Thickness
ft
145
0.5
Long side
ft
Short side
ft
Area A
ft2
26
12
326
Qdead,slab Wdead,slab
psf
lb
73
23,700
Table 5.4: Slab properties
The slab and vault dead weights are added together for total dead weight. Table 5.5 shows this
dead weight plus other applicable stresses for the assessment. The subscripts i and f denote
“initial” and “final,” respectively—initial for the design live load from Guastavino’s era, and
final for the design live load for the Museum’s renovations.
Wdead,total
lb
42,000
Q dead,total Qdead,vault
psf
psf
129
56
Qdead,slab
psf
Qlive,i
psf
Qlive,f
psf
73
150
300
Table 5.5: Including slab loads, and presenting design live loads
Before proceeding, total weights modeling the cross-vault surface area as flat versus cylindrical
are compared. Total weight dead plus live, approximating the vault surface area as flat, is
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
𝑊(𝐷+𝐿)|1 = 18,400 𝑙𝑏 + 23,700 𝑙𝑏 + (300 𝑝𝑠𝑓) ∗ (326 𝑓𝑡 2 ) ≈ 139,900 𝑙𝑏 = 140 𝑘𝑖𝑝
Whereas for comparison, total weight dead plus live, approximating the vault surface area as half
a cylinder, is
𝑊(𝐷+𝐿)|2 = 28,800 𝑙𝑏 + 23,700 𝑙𝑏 + (300 𝑝𝑠𝑓) ∗ (326 𝑓𝑡 2 ) ≈ 150,300 𝑙𝑏 = 150 𝑘𝑖𝑝
The relative difference between the two is
150 − 140
(
) ∗ 100% = 7%
140
The difference is less than 10%. The vault surface area is assumed rectangular and flat for the
structural assessment. The entire weight of the vault and slab, dead plus live, is 140 kip.
Vault rise dimensions, D:
Long side
Short side
Dintrados
ft
Dextrados
ft
Dcenter
ft
7.3
7.3
7.8
7.8
7.6
7.6
Table 5.6: Vault rise dimensions
In this assessment the total dead load plus the design live load is summed, giving
𝑄𝐷+𝐿 = 𝑄𝐷,𝑣𝑎𝑢𝑙𝑡 + 𝑄𝐷,𝑠𝑙𝑎𝑏 + 𝑄𝐿,𝑓 = (56 + 73 + 300)𝑝𝑠𝑓 = 429 𝑝𝑠𝑓
5.5.1 Equilibrium analysis: Distributed loading
Triangle-arch method:
Reference is made to Chapter 4 sections 4.3 and 4.4.1 regarding the analysis procedure.
Reference is made especially to Equation 4.1 and Equation 4.2. Using here the entire pressure
from dead and live, QD+L, the following results are generated:
L
ft
w
ft
q
lb/ft
FH,max
lb
FH,min
lb
FH,avg
lb
RV
lb
Fthrust,max Fthrust,min Fthrust,avg
lb
lb
lb
Long
26 3.1 1,330 15,800 14,700 15,200 17,500 23,500 22,900 23,200
side
Short
12 6.6 2,820 7,410
6,940 7,170 17,500 19,000 18,800 18,900
side
Table 5.7: Determination of horizontal thrust, vertical reaction, and resultant thrust in quarter-vaults
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
The total reaction at each of four supports is 2*RV = 2*17,500 lb = 35,000 lb = 35 kip. This is
confirmed by dividing the entire weight by the number of supports (Equation 2.2): 140 kip / 4 =
35 kip.
As in chapter 4, the average horizontal thrust is emphasized in this chapter for all analysis. The
highest average resultant thrust is 23.2 kip. The area the thrust acts on is the vault thickness
multiplied by the average width, or 3.1 ft * 12 in/ft * 6 in = 223 in^2. The compressive strength
of the masonry is 2,060 psi. It can be determined whether the vault is safe by requiring the
compressive strength (stress) to be greater than the demand stress, a common procedure in
structural design.
𝐹𝑐 = 2,060 𝑝𝑠𝑖 ≥
23,200 𝑙𝑏
= 104 𝑝𝑠𝑖
223 𝑖𝑛2
the material is not in danger of crushing
This gives a factor of safety FS of 2,060/104 ≈ 20. This means that even when the design live
load was doubled from 150 psf to 300 psf for wing H at the Museum, the Guastavino cross-vault
would have remained safe; it would not have come close to failing due to masonry reaching its
crushing strength. This result agrees with the unity check in section 4.4.1 which shows that
demand can be as low as 2% of the crushing strength for structural Guastavino masonry vaults.
Sliced parallel arches method:
Now a series of parallel arches are analyzed as in Chapter 4, section 4.4.1. Reference is made to
the way in which the vault from Chapter 4 was “sliced,” or divided, into six parallel web arches
for each of two representative triangular sections of the vault (Figure 4.5). As in Chapter 4, it is
necessary to execute the sliced parallel arches method for two portions of the vault—one
associated with the long side and one associated with the short side. (If the vault were perfectly
square, the process could be done once.) As before, vertical reactions from each parallel arch are
applied to the diagonal rib-arch; all those vertical reactions are summed to produce the resultant
vertical reaction at the corner support. And as before, the resultant horizontal thrusts from two
adjacent sliced arches become the single horizontal thrust on the diagonal-running rib-arch at
that location. The resultant diagonal thrust is then the resultant of the diagonal horizontal thrust
and summed vertical reactions at that node. See Equation 4.4 and Equation 4.5.
The two representative quarters of the cross-vaults of wing H are each divided into six parallel
arches, similar to the analysis of the MCNY replica vault in Chapter 4. However, because this
cross-vault is assumed to have a uniform thickness, the pressure QD+L is maintained for every
web arch, whereas different self-weight pressures are calculated for the side-arches and webs in
Chapter 4 because their densities differ.
The equilibrium analysis is carried out for each sliced parallel arch with the following results:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Arch
L
w
q
FH,max
FH,min
FH,avg
RV
Long
side
ft
ft
lb/ft
lb
lb
lb
lb
lb
lb
lb
4.4 0.52
8.8 1.0
13 1.0
18 1.0
22 1.0
26 1.0
221
443
443
443
443
443
73
583
1,310
2,330
3,650
5,250
68
546
1,230
2,180
3,410
4,910
71
564
1,270
2,260
3,530
5,080
486
1,940
2,920
3,890
4,860
5,830
491
2,030
3,200
4,530
6,080
7,850
491
2,020
3,160
4,460
5,940
7,630
491
2,020
3,180
4,500
6,000
7,730
2.1
4.1
6.2
8.3
10
12
470
941
941
941
941
941
34
275
618
1,100
1,720
2,470
32
257
578
1,030
1,610
2,310
33
266
597
1,060
1,660
2,390
486
1,940
2,920
3,890
4,860
5,830
487
1,960
2,980
4,040
5,150
6,330
487
1,960
2,970
4,020
5,120
6,270
487
1,960
2,980
4,030
5,140
6,300
1
2
3
4
5
6
Fthrust,max Fthrust,min Fthrust,avg
Short
side
1
2
3
4
5
6
1.1
2.2
2.2
2.2
2.2
2.2
Table 5.8: Equilibrium analysis for sliced parallel arches
Now using Equation 4.4, the resulting horizontal thrust on the rib along its diagonal axis is
determined for each loading point. This equation is repeated:
2
2
𝐹𝐻𝑖 = 𝐹𝐻𝑖−1 + √𝐹𝐻𝑖,1
+ 𝐹𝐻𝑖,2
, 𝑖 = 1, … ,6
Equation 5.1
Then the thrust resultant on the rib at the load points 1 through 6 is determined with Equation
4.2. This process results in the following:
Horizontal resultants
FH,rib|max
FH,rib|min FH,rib|avg
Vertical
Thrust resultants
RV,rib
Fthrust,rib|max Fthrust,rib|min Fthrust,rib|avg
Load
point
lb
lb
lb
lb
lb
lb
lb
1
2
3
4
5
6
81
725
2,180
4,760
8,780
14,600
75
679
2,040
4,450
8,220
13,700
78
701
2,100
4,600
8,490
14,100
972
4,860
10,700
18,500
28,200
39,800
975
4,910
10,900
19,100
29,500
42,400
975
4,910
10,900
19,000
29,400
42,100
975
4,910
10,900
19,000
29,400
42,300
Table 5.9: Horizontal thrusts, vertical reactions, and resultant thrusts on a diagonal rib
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
5.5.2 Results discussion—static equilibrium
From the triangle-arches analysis method, the total vertical reaction at a support is 35,000 lb.
From the sliced parallel arches method, the total vertical reaction at a support (load point 6) is
39,800 lb. The relative difference between the two methods gives
39,800 𝑙𝑏 − 35,000 𝑙𝑏
(
) ∗ 100 = 14%
35,000 𝑙𝑏
The total weight of the vault with dead and live load is 140,000 lb. Summing the vertical
reactions from the triangle-arches method gives 4*35,000 lb = 140,000 lb, equal to the total
weight. Summing the vertical reactions from the sliced parallel arches method for four supports
gives 4*39,800 lb ≈ 159,000 lb. This indicates that the sliced parallel arches method
overestimates the structural action of the vault by 14%. Depending on the uncertainties with
which engineers are or are not comfortable, this method, giving more conservative results, may
be preferable to the triangle-arches method.
Using Equation 4.6, the resultant diagonal thrust for the triangle arches is determined so it can be
compared to the thrust acting on the ribs. First, the horizontal thrust resultant of the two
perpendicular horizontal thrusts is calculated. Then the final resultant thrust is found by using
that value along with the sum of the vertical reactions:
𝐹𝑡ℎ𝑟𝑢𝑠𝑡,𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 =
2
√(√𝐹𝐻,𝑙𝑜𝑛𝑔
+
2
2
𝐹𝐻,𝑠ℎ𝑜𝑟𝑡 )
+ (𝑅𝑉,𝑙𝑜𝑛𝑔 + 𝑅𝑉,𝑠ℎ𝑜𝑟𝑡 )
2
Equation 5.2
The following table compares the resulting thrusts for the sliced parallel arches and trianglearches methods for maximum, minimum, and average horizontal thrust states:
Thrust resultants in diagonal direction
Rib-arch
Triangle-arch
Fthrust,rib|max Fthrust,rib|min Fthrust,rib|min Fthrust,max Fthrust,min Fthrust,avg
lb
lb
lb
lb
lb
lb
42,400
42,100
42,300
39,100
38,600
38,800
Table 5.10: Comparing diagonal resultants for the two primary methods
Relative difference between maximum states: 8 %.
Relative difference between minimum states: 9 %.
Relative difference between average states: 9%.
These are favorable results. They show that there is little overall discrepancy between the two
prevailing methods of vault structural analysis presented above.
The horizontal thrusts along the rib’s diagonal axis are compared to the vault’s total weight:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Horizontal resultants % total vault weight
From sliced parallel arches method
From triangle-arches method
FH,rib|max
FH,rib|min
lb
%W
14,600
10
FH,rib|avg
FH,triangle|max
FH,triangle|min
FH,triangle|avg
lb
%W
lb
%W
lb
%W
lb
%W
lb
%W
13,700
10
14,100
10
17,400
12
16,300
12
16,800
12
Table 5.11: Horizontal thrusts as % total vault weight for both analysis methods
For the sliced parallel arches method, the horizontal thrusts are all 0.1W, and for the trianglearches method, the horizontal thrusts are all 0.12W. The bounds are merely a single value
because the horizontal thrust magnitudes for maximum, minimum, and average states are
approximately identical compared to total system weight. It is first suspected that perhaps this is
due to the analysis accounting for the slab dead weight and the 300 psf live load. However, when
excluding those loads from the analysis entirely and only including the self-weight of the vault
material, the bounds remain 0.1W for the sliced parallel arches method and 0.12W for the
triangle-arches method. This means that these values are a function of the cross-vault geometry.
In terms of the structural system’s safety, the analytical results prove that the vaults would not
come close to reaching their crushing strength under the increased 300 psf live load. Even under
the doubled live load the factor of safety is 20. This means that in order to fail in brittle crushing
of the material the cross-vaults would need to experience 20 times the stress levels they would
experience under the doubled live load.
The following table collates the key analytical results:
Triangle-arches
Sliced parallel arches
Long side
Short side
On diagonal
Rib
FH,max
lb
15,800
7,410
17,400
14,600
FH,min
lb
14,700
6,940
16,300
13,700
FH,avg
lb
15,200
7,170
16,800
14,100
RV
lb
17,500
17,500
35,000
39,800
Fthrust,max
lb
23,500
19,000
39,100
42,400
Fthrust,min
lb
22,900
18,800
38,600
42,100
Fthrust,avg
lb
23,200
18,900
38,800
42,300
ΣRV
lb
35,000
39,800
35,000
Table 5.12: Collated analytical equilibrium results
5.5.3 Graphical analysis: Distributed loading
Graphical analysis is briefly reviewed: Scaled vertical loads are applied to the centers of gravity
of arch segments. These vertical loads are lined up one atop the next in order from AB to KL. At
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
the midpoint of FG (the load on the arch center), the scaled average horizontal thrust is added,
connecting the midpoint of FG to the leftmost point o. The corresponding rays oa, ob,…,ol are
drawn from o to points A through L. These rays represent the resultant forces on the centers of
gravity of the arch segments. A line of pressure is interpolated by connecting the rays where they
intersect at arch segment centers of gravity. For example, the line of pressure passes through the
point where the ray oe intersects the ray od along the vertical axis representing the center of
gravity of arch segment DE.
Triangle-arch method:
Refer to Chapter 4, section 4.4.3 (Graphical analysis) for an explanation of how to perform this
analysis procedure. As in that section, the Museum cross-vault is subdivided into 11 rectangular
segments for each of the long and short sides. Refer also to Equation 4.7 and Equation 4.8 to
determine the equivalent point loads using rectangular segments that serve as tributary areas. For
convenience, the relevant equations are reproduced here:
𝐿𝑎 𝐿𝑏 6 − 𝑖
)( )(
)
10 2
5
Equation 5.3
𝐿𝑎
𝐿𝑏 1
)( )( )
2 ∗ 10 2 10
Equation 5.4
𝑃𝑖 = 𝑄 (
𝑃6 = 𝑄 (
The triangle-arches are divided into tributary areas to determine equivalent point loads (Figure
5.5), then the point loads for each loading point are determined with the above two equations
(Table 5.13):
Figure 5.5: Division of areas for graphical analysis based on triangle-arches
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Segment no.
Long side
lblock
ft
Lb/2
ft
κ
P
lb
P
kip
1
2
3
4
5
6
Short side
1
2
3
4
5
6
2.6
2.6
2.6
2.6
2.6
1.3
6.2
6.2
6.2
6.2
6.2
6.2
1.0
0.80
0.60
0.40
0.20
0.10
7,000
5,600
4,200
2,800
1,400
350
7.0
5.6
4.2
2.8
1.4
0.35
1.2
1.2
1.2
1.2
1.2
0.62
13
13
13
13
13
13
1.0
0.80
0.60
0.40
0.20
0.10
7,000
5,600
4,200
2,800
1,400
350
7.0
5.6
4.2
2.8
1.4
0.35
Table 5.13: Deriving loads on arch segments
Table 5.13 shows that despite their different overall lengths, each triangle-arch quadrant of the
vault is loaded the same way. However, the long and short arches have much different rise-tospan ratios such that the long arch is shallower than the short arch compared to its length. The
ratios are 0.29 for the long side and 0.63 for the short side. The line of pressure may fit
differently within the long arch versus short arch geometry. For this reason both arches are
graphically evaluated – first the long side and then the short side.
Long side:
The forces Pi are scaled as in Chapter 4 and applied to the long arch. Here, however, the scale is
1 kip = 1 foot.
Segment no.
Force P
ft
1
2
3
4
5
6
7.0
5.6
4.2
2.8
1.4
0.35
Table 5.14: Scaled loads, long side
Figure 5.6 shows the loading and labeling scheme for the graphical analysis.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.6: Application of point loads to long arch-segment centers of gravity
For the scenario of average horizontal thrust, (15.2 kip)*(1 ft/1 kip) = 15.2 ft. This horizontal
thrust is applied to the center of the superimposed vertical loads and the force polygon is drawn.
Figure 5.7: Force polygon, long side
Leading to a line of pressure:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.8: Line of pressure for long side-arch (yellow line)
The thrust line leaves the intrados in the centers of blocks EF and GH, very near the crown. As
discussed in chapter 4, the analytically calculated horizontal thrust values are approximate. Using
the same vertical loading as in Figure 5.7, a new force polygon and thrust line are drawn
corresponding to a graphically determined resultant thrust found according to the methods of
chapter 4 (see Figure 4.17 to Figure 4.19):
Figure 5.9: Force polygon for long side with horizontal thrust found graphically
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Note the horizontal thrust of 15 ft-5.5 in. = (15+5.5/12) kip = 15.5 kip = 15,500 lb. Contrast this
with the average horizontal thrust used before:
15.5 − 15.2
(
) ∗ 100% ≈ 2%
15.2
The associated line of pressure:
Figure 5.10: Line of pressure for long side-arch using graphically found horizontal thrust
Figure 5.10 like Figure 5.8 shows the thrust line leaving the geometry. A valid solution where
the thrust line acts inside the geometry is still sought.
Remember that unreinforced masonry vaults are statically indeterminate. There are essentially
infinite possible lines of pressure representing valid and invalid solutions. Although the
preceding graphical solutions show the line of pressure leaving the geometry, valid solutions for
stability can still be found by determining a horizontal thrust through a graphical iterative
procedure.
A valid solution is found for a horizontal thrust of 20 ft. = 20 kip = 20,000 lb:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.11: Valid force polygon for the long side-arch
There is a 32% relative difference between the new horizontal thrust and the analytically
determined average horizontal thrust, but the solution is valid for the geometry and the vertical
loading. The line of pressure follows:
Figure 5.12: Valid thrust line for long side-arch showing stability
The thrust line fits within the arch geometry, indicating stability. The geometry requires a higher
horizontal thrust for stability, but that horizontal thrust is valid because it leads to a valid
solution.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
The graphical analysis procedure is amended going forward. First the analytically calculated
average horizontal thrust is used. If a valid solution is not found, then the same iterative
procedure for horizontal thrust is used as in Figure 5.11 and Figure 5.12.
Short side:
Now for the short side the analysis proceeds as above. The scaled loads are identical to Table
5.14. Figure 5.13 shows the loading scheme, Figure 5.14 shows the force polygon, and Figure
5.15 shows the thrust line. The scaled average horizontal thrust is 7.2 ft.
Figure 5.13: Application of point loads to short arch-segments centers of gravity
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.14: Force polygon, short side
Figure 5.15: Line of pressure for short side-arch (yellow line)
The line of pressure falls outside the geometry in the centers of segments EF and GH, very close
to the crown. Even if the line of pressure were shifted along the vertical axis toward the crown it
89
5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
would still exit the intrados. As with the long side, a new line of pressure is found by graphically
iterating a horizontal thrust that will cause the line of pressure to fit within the geometry. The
new force polygon is shown alongside the new thrust line:
Figure 5.16: Line of pressure (left) and force polygon (right) for the short
side-arch using graphically iterated horizontal thrust
With a horizontal thrust of 10 ft. = 10 kip = 10,000 lb, the line of pressure fits within the
geometry under the same vertical loads. This shows a valid solution for vault stability.
Comparing this to the analytically determined average horizontal thrust gives a relative
difference of 39%. This means that the analytical results are quite a bit lower than the valid
solution given through graphical analysis. This is due to the load assumption inherent in the
analytically determined average horizontal thrust. That assumption is of a uniform distributed
load per unit length along the arch segment. By turning to the iterative graphical procedure, a
valid solution has been shown.
So far it has been proven through graphical analysis that both long and short side-arch models of
the Museum cross-vault are stable under this loading scenario.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Sliced parallel arches method:
The procedure outlined in Chapter 4 section 4.4.3 for the parallel arches is followed to generate a
force polygon and line of pressure for the ribs. Table 5.15 shows the summed vertical forces
from adjacent sliced arches at each node 1 through 6, along with the scaled load P that is applied
in the graphical analysis. Here as above, the scale is 1 kip = 1 foot.
Segment
no.
1
2
3
4
5
6
RV from web arches
lb
kip
972
3,900
5,830
7,780
9,720
11,700
1.0
3.9
5.8
7.8
10
12
Force P
ft
1.0
3.9
5.8
7.8
10
12
Table 5.15: Scaled vertical loads on rib-arches
Figure 5.17 shows the scaled loads in red arrows applied to the centers of gravity of the
subdivided rib-arch segments:
Figure 5.17: Application of point loads to rib-arch segments’ centers of gravity
For Figure 5.18, the average horizontal thrust on the rib is 14.1 kip = 14.1 ft.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.18: Force polygon for ribs
Giving a line of pressure:
92
5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.19: Line of pressure, in yellow, for the diagonal rib-arch
The line of pressure exits the geometry. As the above analyses for the long and short side-arches,
a new line of pressure is drawn using a horizontal thrust found graphically by an iterative
procedure. Note that if the line of pressure leaves the extrados the vault could still be stable
because side walls and fill were likely a part of the structural system, providing material through
which the thrust could pass above the thin vault. In other words, one cylindrical barrel section of
the cross-vault is buttressed by the cylindrical barrel section perpendicular to it. The iterative
procedure gives the following:
Figure 5.20: Line of pressure (left) and force polygon (right) for rib-arches
using graphically determined horizontal thrust
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Finding the horizontal thrust graphically gives 23 ft. = 23 kip = 23,000 lb. This is nearly 10,000
lb more than the analytically calculated average horizontal thrust used in Figure 5.18 and Figure
5.19. The line of pressure leaves the extrados, closely follows the arch contour, then reenters the
geometry. The shape of the thrust line resembles the rib-arch geometry. Although the thrust line
exits the rib-arch extrados, bear in mind that fill and side-walls helped support this system. There
was most likely material through which the thrust line could pass if it left the geometry so long
as it only exited at the extrados. The thrust line combined with this historical likelihood indicate
that under the prescribed loading the rib-arches are generally stable. The thrust line of Figure
5.20 could be contained if the vault corners had fill (the weight of the fill would also tend to push
the thrust line down).
The Museum vault must resist its own dead weight, the dead weight of a concrete slab, and a
very high 300 psf live load. This results in the slender profile of the force polygon of Figure
5.18. Figure 5.19 shows the thrust line first leaving the extrados near points E and H, then
dipping back into the geometry. Iterating the horizontal thrust graphically ultimately leads to a
valid solution for the thrust line (Figure 5.20).
5.5.4 Graphical analysis: Distributed loading with point load
When the Museum was planning to relocate the Egyptian and Far- and Near-Eastern exhibits to
wings E and H, the concern was the increased loading on the existing Guastavino vault systems.
The doubled live load from 150 psf to 300 psf assumes a uniform stress loading on an entire
tributary floor area, but in actuality the character of loading can differ significantly from that
kind of model. For example, recall that the pamphlet the Museum published detailing its
renovations mentions the mammoth Egyptian statues that would be moved to the north wings.
One of these weighs 8,000 lb. This four-ton point load influences the line of pressure in the vault
masonry differently than the distributed loading. It is instructive to investigate the effect of a
point load on the vault’s structural response. The point can be placed anywhere, but this thesis
investigates the effect of a point load at the center of the vault.
The procedure is as follows: The Guastavino-specified distributed live load of 150 psf is used
instead of the Museum’s 300 psf. Then the four-ton point load is superimposed at the crown of
the vault. This is a reasonable analysis technique. A live load of 300 psf is very high. The design
code of the American Society of Civil Engineers specifies 100 psf for lobbies and other inside
spaces that experience high foot traffic.
In order to graphically evaluate the effect of the 150 psf live load and an 8,000-lb point load, the
steps from analytical static equilibrium are repeated with modifications to account for the point
load. The development of the equations is found in section 8.1 of the appendix. (The procedure
of analytical static equilibrium is not shown here, as it follows the same steps outlined in
previous sections.)
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Triangle-arch method:
Refer to Figure 5.5 for the division of the quarter-vault into tributary areas and associated
loading points 1 through 6. The point load P is divided by four because each triangle-arch
represents a quarter of the vault, and the load is applied directly to the vault’s center in plan. This
gives the following loads:
Lb/2
ft
κ
Long side
lblock
ft
P
lb
P
kip
1
2
3
4
5
6
2.63
2.63
2.63
2.63
2.63
1.3
6.2
6.2
6.2
6.2
6.2
6.2
1.0
0.80
0.60
0.40
0.20
0.10
6,550
3,640
2,730
1,820
910
227
6.5
3.6
2.7
1.8
0.91
0.23
1.24
1.24
1.24
1.24
1.24
0.62
13
13
13
13
13
13
1.0
0.80
0.60
0.40
0.20
0.10
6,550
3,640
2,730
1,820
910
227
6.5
3.6
2.7
1.8
0.91
0.23
Segment no.
Short side
1
2
3
4
5
6
Table 5.16: Deriving loads on arch segments including point load at center
The scaled loads for both the long and short sides are then
Segment no.
Force P
ft
1
2
3
4
5
6
6.5
3.6
2.7
1.8
0.91
0.23
Table 5.17: Scaled loads
Long side:
A 8,000 lb / 4 = 2,000 lb point load is superimposed at the center arch segment of the quartervault. It is P/4 because each triangle-arch represents a quarter of the vault, and the load is applied
directly to the center of the vault in plan. The loading diagram is created:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.21: Triangle-method long-side with point load at center
The new horizontal thrust is calculated with Equation 5.5 (see appendix section 8.1 for
derivation):
𝐹𝐻 =
𝑃𝐿 𝑞𝐿2
+
4𝐷 8𝐷
Equation 5.5
The new horizontal thrust values are given for both long and short sides:
Long side
Short side
FH,max
lb
FH,min
lb
FH,avg
lb
12,000
5,670
11,300 11,600
5,310 5,480
Table 5.18: New horizontal thrusts with imposed
load P
For the long side, this gives a scaled average horizontal thrust of 11.6 kip = 11.6 ft. The
distributed load per unit length q is 864 lb/ft. Figure 5.22 shows the resulting force polygon.
96
5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.22: Force polygon
Note that the force polygon is consistent with analytical results. We know from equilibrium
analysis that with the point load at the center the full vertical reaction must be:
(𝑃 + 𝑞𝐿)
2[
]
2
Graphical analysis gives 24+11.5/12 = 25 ft = 25 kip = 25,000 lb. This agrees approximately
with 2,000 lb + (864 lb/ft)*(26 ft) = 24,464 lb ≈ 24,500 lb from analytical equilibrium.
The line of pressure follows:
Figure 5.23: Thrust line (in yellow) for long triangle-arch with point load at center
97
5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Assuming the vault is stable under its original 150 psf design live load, superimposing the 2,000
lb point load at the center appears to cause the thrust line to leave the geometry. A valid solution
adhering to the safe theorem is still sought. As in section 5.5.3, horizontal thrust is found
graphically, and a new force polygon and line of pressure generated for a valid solution for
stability:
Figure 5.24: Force polygon (above) and line of pressure (below) for long side-arch, including point load, using
graphically determined horizontal thrust
The new horizontal thrust found graphically is 15 ft. = 15,000 lb. The resulting line of pressure
acts within the geometry in contrast to Figure 5.23. This indicates that the vault is stable. The
horizontal thrust found graphically differs from the average horizontal thrust found analytically
by about 29%. This is similar to the relative difference of 32% observed before for the valid
solution of the long side-arch under the 300 psf live load (section 5.5.3). This demonstrates the
consistency of this method across varied loading.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Short side:
Again with the live load of 150 psf and the 8,000-lb point load, the same process is performed
for the short side of the vault using the triangle-arches method.
Figure 5.25 shows the loading scheme and force polygon using a scaled average horizontal thrust
of 5.48 kip = 5.48 ft.:
Figure 5.25: Left) short arch loading scheme including point load;
right) force polygon
Leading to the following thrust line:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.26: Thrust line (in yellow) for short triangle-arch with point load at center
As in the case for the long side, the thrust line does not fit within the arch geometry. Even if the
thrust line were shifted upward it would still exit the intrados. As with the long side-arch, the
thrust line for the short side is evaluated using a horizontal thrust found graphically:
Figure 5.27: Line of pressure (left) and force polygon (right) for short side-arch
including point load and using graphically determined horizontal thrust
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
The new horizontal thrust is 7 ft. = 7,000 lb. The line of pressure acts within the arch geometry,
indicating stability. The % error between the analytically and graphically determined horizontal
thrusts is about 28%.
A valid solution of stability has been found for both the long and short side-arch models of the
cross-vault.
Sliced parallel arches method:
Next is an investigation of the effect on the diagonal rib-arches of an 8-kip point load applied to
the center of the vault. The point load is divided in half, assuming each of the two ribs takes one
half of this load if it is applied in the center. The scaled loads on segments 1 through 6 are
Segment no.
1
2
3
4
5
6
RV from web arches
lb
kip
4,630
2,530
3,790
5,060
6,320
7,580
4.6
2.5
3.8
5.1
6.3
7.6
Force P
ft
4.6
2.5
3.8
5.1
6.3
7.6
Table 5.19: Scaled vertical loads on rib-arch, incl. point load
These loads are used to construct a loading diagram:
Figure 5.28: Setup for graphical analysis for rib with point load at center
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
The force polygon is first generated with the scaled average horizontal thrust. The average
horizontal thrust on the rib-arch is determined with Equation 5.5. The length of the rib is L =
(26^2+12^2)^(1/2) ≈ 29 ft. The addition to the existing average horizontal thrust is calculated
using the rise to the arch center: PL/4D = (4 kip)*(29 ft)/(4*7.6 ft) ≈ 3.8 kip. This is added to the
9.2 kip average horizontal thrust on load point 6 found analytically: 9.2 + 3.8 = 13 kip = 13 ft.
The force polygon is drawn:
Figure 5.29: Force polygon for rib
with point load at center
The application of these rays to the vault geometry gives the following thrust line:
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
Figure 5.30: Thrust line for the rib-arch with point load at center
The thrust line leaves the masonry geometry near points D and I. As in the preceding sections, a
better fit is sought using a horizontal thrust found graphically.
Figure 5.31: Line of pressure (left) and force polygon (right) for rib-arch
including point load and using graphically determined horizontal thrust
The new horizontal thrust is 18 ft. = 18,000 lb, a 38% relative difference compared to the 13,000
lb average horizontal thrust calculated. Unlike in Figure 5.30, the thrust line acts within the
geometry. The shape of the thrust line resembles the case of Figure 5.20 compared with Figure
5.19 in section 5.5.3.
The rib-arch has been proven to be stable under a 150 psf live load and an 8,000-lb point load.
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5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H
5.5.5 Results discussion—graphical analysis
Graphical analysis in section 5.5.3 shows that under a doubled live load of 300 psf, the Museum
cross-vault is stable when using a horizontal thrust found through an iterative graphical
procedure. This is in contrast to results found with analytically determined average horizontal
thrusts which show the vault is unstable. It is recognized that the cross-vault may have had fill in
the corners. This would provide material in which the thrust line could pass even if it exited the
extrados.
In section 5.5.4, decreasing the live load to the original 150 psf and applying a point load of
8,000 lb at the cross-vault center causes the vault to be unstable if the vault is evaluated using the
analytically determined average horizontal thrust. However, as in section 5.5.3, using the
graphically determined horizontal thrust causes the thrust line to act within the geometry. This
gives a valid solution and shows overall stability of the masonry system.
Additionally, the geometry presented here is approximate and is rebuilt from century-old
architectural plans. Associated architectural elevation drawings show that the short side-arches of
the cross-vault may have been thicker in profile (Figure 8.12).
It is evident from this assessment that the Museum could have had reason to trust the safety of
the Guastavino cross-vaults under the increased loading from moving the Egyptian and Far- and
Near-Eastern art exhibits to the north wings. This conclusion is reached through analysis
procedures based on the structural mechanics of unreinforced masonry vaults. This is in contrast
to the Museum’s consulting engineers who promoted the demolition of the vaults without first
attempting analytical assessments. Knowledge of the methods presented here would have helped
engineers decide on ways to continue using the cross-vaults for their full loading capacity.
5.6
Conclusions
The important question this thesis asks is whether the Museum might have maintained its
Guastavino cross-vaults in wing H instead of demolishing them. The analytical static equilibrium
methods demonstrate a cross-vault that is well within safe compressive stress levels even when
the Museum doubles the design live load. Unreinforced masonry structures require a trajectory of
compressive force to act within the geometry of the material. The graphical analysis sections
show that this trajectory does indeed lie within the material with the exception of the rib-arch
under the 300 psf live load. However, it is recognized that the rib-arch would have been
buttressed by fill in the cross-vault corners, providing material through which the thrust could
pass above the vault. This indicates that the cross-vaults would experience no tension due to
bending and would act only in compression. In the case of the rib-arch mentioned above, fill in
the vault corners could take the line of pressure as it exits the extrados. From the graphical
analysis for the cross-vaults it is concluded that the line of pressure, situated well within the
geometry, could not cause a collapse mechanism. The analysis reveals that the cross-vaults could
safely resist heavy statues (four tons) or doubled live load (300 psf).
104
105
6 Concluding discussion
6 Concluding discussion
This chapter summarizes the most significant findings from the thesis, provides ideas for further
work, and identifies some areas of future research.
6.1
Results summary











Chapter 3 examines primary sources to determine the origin of the decision to demolish
the cross-vaults at the Museum. Spanning more than a decade, letters and telegraphs
reveal consulting engineers' unfamiliarity with the analysis of unreinforced masonry
structures. It is determined that their lack of technical expertise in this field directly led to
the decision to demolish the vaults.
Chapter 4 presents force-flow modeling techniques and associated analytical and
graphical analysis methods step by step applied to the ½-scale MCNY Boston Public
Library vault replica. These techniques are designed for application to appropriate
unreinforced masonry vault systems in engineering practice.
Analytical equilibrium shows a factor of safety of 45 for the MCNY vault against
crushing under its own weight.
For the MCNY vault, the triangle-arches and sliced parallel arches methods agree with
each other for the vertical reactions but less for total thrust calculations. There is a 0%
difference between vertical reactions but a range of 16% to 23% relative difference
between the resultant thrusts from the two methods.
Horizontal thrusts found analytically show bounds of between 0.87W and 0.32W for the
sliced parallel arches method and between 1.1W and 0.4W for the triangle-arches method.
The triangle-arches graphical analysis confirms vault stability when web and arch loads
are superimposed on the side-arch geometry in two dimensions. These analysis results are
confirmed by using graphically determined horizontal thrust. The line of pressure found
this way lies within the middle-third of side-arch geometry. It only requires 9% greater
horizontal thrust and 5% greater distributed loading per unit length to force the line of
pressure into the middle-third.
The sliced parallel arches graphical analysis gives a valid solution for a thrust line that
fits within the geometry.
Chapter 5 applies the methods presented in Chapter 4 to a representative cross-vault from
wing H of the Museum, thereby providing engineers with a valuable, detailed example of
how to analyze unreinforced masonry cross-vaults.
Analytical equilibrium finds the Museum cross-vault has a factor of safety of 20 against
crushing failure under the doubled live load of 300 psf.
The triangle-arches and sliced parallel arches methods closely agree with each other for
the Museum vault analytical analysis in Chapter 5. There is a 14% relative difference
between vertical reactions, and a 9% relative difference between resultant thrusts from
the two methods.
Horizontal thrusts found analytically for the Museum vaults are 0.1W for the sliced
parallel arches method and 0.12W for the triangle-arches method. For each method, those
106
6 Concluding discussion

6.2
percentages are true for the entire range of horizontal thrust values—between maximum
and minimum states.
Graphical analysis for both principal methods shows that alternately under the doubled
live load (300 psf) and under the original live load with a point load from monolithic
statues (150 psf + 8,000 lb) the Museum cross-vaults are stable. Analysis attempts using
the analytically determined average horizontal thrust show thrust lines outside the
geometry. However, analysis using graphically determined horizontal thrust shows thrust
lines acting within the geometry. By the safe theorem, only one valid solution for a line
of pressure within the geometry needs to be found to prove stability. Therefore, the
Museum cross-vaults would have been safe under the increased loading.
Suggestions for future work




For the MCNY vault of chapter 4, it is suggested that future work develop different or
altered analytical methods of analysis which reflect the uniqueness of this very shallow
vault geometry. The MCNY vault in many ways resembles a shallow dome resting on
four edge arches. An alternative method based on dome analysis might account for hoop
stresses in the material. The results from this method could then be compared to the same
results found in this thesis.
The Museum cross-vault geometry was recreated from drawings and documents that are
in many cases over a century old. The Guastavino drawings consulted for this purpose are
very roughly sketched. McKim, Mead and White plans and elevations are used to
confirm the Guastavino drawing dimensions. Nonetheless, the resulting geometry created
for the analysis is approximate. For example, it is possible that the short-side cross-vault
geometry was much thicker and may have been able to contain the thrust line. Future
work could hone the geometry and repeat the same graphical analysis to confirm the
conclusions of this thesis.
As stated at the beginning of chapter 5, many extant drawings show that steel beams were
part of the original construction of wings E and H. Future work could incorporate the
steel sections and experiment with portioning different fractions of total load to each
system and observing the effects, especially on the line of pressure in the vaults.
The demolition of the cohesive tile masonry system was certainly costly, disruptive, and
required a great amount of energy in the process. Even if the vaults were deemed unsafe
for the new loads, it may have been preferable to retain them aesthetically and rebuild the
floor system above with new steel. Future study could perform life cycle analysis of the
demolition and compare the economic and environmental costs for alternative
renovations that retain the vaulting.
107
108
References
7
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The City of New York Department of Parks. Letter to Conrad Hewett. 1913, June 11.
Metropolitan Museum of Art Archives.
Tolmachoff, Helen. Letter to R. Guastavino Co. 1948, May 20. Drawings & Archives, Avery
Library, Columbia University.
Treasurer R. Guastavino Co. Letter to H.F. Seitz. 1949. “RE: METROPOLITAN MUSEUM OF
ART New York, N.Y.”, January 26. Drawings & Archives, Avery Library, Columbia
University.
Wills & Marving Co., and R. Guastavino Co. 1910. “Contract.” Drawings & Archives, Avery
Library, Columbia University.
Zalewski, Waclaw, and Edward Allen. 1998. Shaping Structures: Statics. New York: Wiley.
Zoldos, John. Letter to A.M. Bartlett. 1958. “Re: #2089 Alterations to Metropolitan Museum of
Art New York, N.Y.”, May 8. Drawings & Archives, Avery Library, Columbia
University.
111
112
Appendix
8 Appendix
8.1 Derivation of arch equilibrium equations
Two derivations follow: For the horizontal thrust FH under q, uniform distributed load per unit
length, only, and for FH when a point load P is superimposed on an arch that experiences the
same distributed load per unit length, q.
First the case of uniform loading only:
Sum forces in the vertical direction. From symmetry the two vertical reactions are equal:
Σ𝐹𝑦 = 0 = −𝑞𝐿 + 2𝑅𝑉
∴ 𝑅𝑉 =
𝑞𝐿
2
Cut the arch at an arbitrary distance x from the left springer and take moments about that point:
𝑞𝑥 2 𝑞𝐿𝑥
Σ𝑀 = 0 =
−
+ 𝐹𝐻 𝑦(𝑥)
2
2
Solve for y(x):
𝑦(𝑥) =
1 𝑞𝐿𝑥 𝑞𝑥 2
(
−
)
𝐹𝐻 2
2
If the arch were cut at its center, the horizontal force there must be equal to the horizontal thrust
at the support. Calculate the rise D which is equivalent to finding y at x = L/2:
𝐿
𝑞𝐿2
𝑦( ) =
2
8𝐹𝐻
Now take y(L/2) = D, the rise of the arch, and solve for the horizontal thrust by rearranging:
𝐹𝐻 =
𝑞𝐿2
8𝐷
Next the general case when an additional load P is some distance x from the springer. In this
derivation, the leftmost extent (at the springer) is used as the origin for x (x = 0), and is denoted
point A whereas the rightmost point (the right springer at x = L) is denoted B.
Begin with P located a distance 0 < x < L/2. First solve for the vertical reactions RVA and RVB.
Take moments about A:
Σ𝑀𝐴 = 0 = −𝑃𝑥 −
𝑞𝐿2
𝑃𝑥 𝑞𝐿
+ 𝑅𝑉𝐵 𝐿 ⇒ 𝑅𝑉𝐵 =
+
2
𝐿
2
Now sum forces in the vertical direction to solve for RVA:
Σ𝐹𝑦 = 0 = −𝑞𝐿 − 𝑃 + (
𝑃𝑥 𝑞𝐿
𝑃𝑥 𝑞𝐿
+ ) + 𝑅𝑉𝐴 ⇒ 𝑅𝑉𝐴 = 𝑃 −
+
𝐿
2
𝐿
2
113
Appendix
Check the result. For the case when P is at x= L/2 expect RVA = RVB:
(𝑃 + 𝑞𝐿)
𝐿
(𝑃 + 𝑞𝐿)
𝐿
𝑅𝑉𝐴 ( ) =
= 𝑅𝑉𝐵 ( ) =
2
2
2
2
Cut a section at the arbitrary point x where P is applied and take moments about that point to find
an expression for the arch height, y(x):
Σ𝑀 = 0 =
𝑞𝑥 2
𝑃𝑥 𝑞𝐿
− (𝑃 −
+ ) 𝑥 + 𝐹𝐻 𝑦(𝑥)
2
𝐿
2
This gives an expression for y(x):
𝑦(𝑥) =
1
𝑞𝐿𝑥 𝑞𝑥 2
(𝑃𝑥 − 𝑃𝑥 2 +
−
)
𝐹𝐻
2
2
Again find the value for y(x= L/2), which is ymax, the crown of the arch:
𝐿
1 𝑃𝐿 𝑞𝐿2
𝑦( ) =
( +
)
2
𝐹𝐻 4
8
Replace y(L/2) with the variable D, the vertical rise of the arch. Now solve for the horizontal
thrust by rearranging:
𝐹𝐻 =
𝑃𝐿 𝑞𝐿2
𝐿
𝑞𝐿
+
=
(𝑃 + )
4𝐷 8𝐷 4𝐷
2
Note that cutting a section on the rightmost side of the arch and taking moments at that section
would give the same result.
These are the equations for the horizontal thrust. The maximum value is taken when D is the
vertical distance from the spring line to the intrados, and the minimum value is taken when D is
the vertical distance from the spring line to the extrados.
114
Appendix
8.2
Authoritative drawings of the Metropolitan Museum vaults
Figure 8.1: Guastavino drawing of wing H vaults, June 23, 1911 (“Drawings & Archives, Avery Library,
Columbia University”)
115
Appendix
Figure 8.2: Guastavino Co., wing H, June 23, 1911 (“Drawings & Archives, Avery Library, Columbia University”)
Figure 8.3: Guastavino Co., wing H, June 23, 1911 (“Drawings & Archives, Avery Library, Columbia
University”)
116
Appendix
Figure 8.4: Guastavino Co., wing H, June 23, 1911 (“Drawings & Archives, Avery Library, Columbia
University”)
117
Appendix
Figure 8.5: Guastavino Co., wing H, July 5, 1911 (“Drawings & Archives, Avery Library, Columbia University”)
118
Appendix
Figure 8.6: Guastavino Co., wing H, July 5, 1911 (“Drawings & Archives, Avery Library, Columbia University”)
Figure 8.7: Guastavino Co., wing H, July 5, 1911 (“Drawings & Archives, Avery Library, Columbia University”)
119
Appendix
Figure 8.8: Wing H from City of New York Department of Parks, February 28, 1962 (“Drawings & Archives, Avery
Library, Columbia University”)
120
Appendix
Figure 8.9: Representative cross-section of long side from existing conditions, 1962 (“Drawings & Archives,
Avery Library, Columbia University”)
121
Appendix
Figure 8.10: McKim et al. drawing of wing H courtyard; steel cols. with masonry cover, October 9, 1911
(“Metropolitan Museum of Art Archives”)
122
Appendix
Figure 8.11: Drawing showing general dimensions of cross-vaults around wing H courtyard, October 9, 1911
(“Metropolitan Museum of Art Archives”)
123
Appendix
Figure 8.12: View of short sides of cross-vaults; entrances to stairs and elevator; Oct. 9, 1911 (“Metropolitan
Museum of Art Archives”)
124
Appendix
Figure 8.13: Original second floor steel framing plans, wing H, December 22, 1909 (“Metropolitan Museum of
Art Archives”)
125
Appendix
Figure 8.14: Tracing of existing steel framing plan, June 9, 1943 (“Metropolitan Museum of Art Archives”)
126
Appendix
8.3
Selected letters and documents
Figure 8.15: Specification from 1910 contract between Wills & Marvin Company and R. Guastavino Co.
(“Drawings & Archives, Avery Library, Columbia University”)
127
Appendix
Figure 8.16: Letter from Tolmachoff to R. Guastavino Co. (“Drawings & Archives, Avery Library, Columbia
University”)
128
Appendix
Figure 8.17: Letter from Berg explaining that consulting engineers are unfamiliar with Guastavino systems
(“Drawings & Archives, Avery Library, Columbia University”)
129
Appendix
Figure 8.18: Telegram showing 1950 decision to replace vaults with steel (“Drawings & Archives, Avery Library,
Columbia University”)
130
Appendix
Figure 8.19: Letter from R. Guastavino Co. to New York City engineer (“Drawings & Archives, Avery Library,
Columbia University”)
131