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

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 1.4 Problem statement ...... 12 1.5 General thesis outline going forward ...... 13 2 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 Claim of thesis authenticity ...... 31 3 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

3.5.3 Renovations are underway ...... 40 3.6 Conclusions ...... 42 4 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 Conclusions ...... 70 5 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 Conclusions ...... 104 6 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

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 Near- Eastern 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

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 twenty- four-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).

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:

1 Introduction

 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?

1.5 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 cross- vaults 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.

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 푁/푚2 = 2,000 푚 20,000 푁/푚3

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

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)

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

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

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 two- course 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 two- course 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 2,060 Compression 360 3,290 Bending n/a 90 Tension n/a 287 Shear in Portland cement n/a 124 Shear in plaster-of-Paris n/a 34

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:

퐿푆 푇퐶 = Equation 2.1 8푟

2 Literature review

where C is the strength: C for compression, C’ for tension, C” for bending. T is the cross- sectional 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: 퐿 ′2 ∗ 12" ∗ 푆 ∗ 푆 푅 ∗ 12" = 푇 ∗ 12" ∗ 퐶 = 1 8 ∗ 푟 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:

푞퐿2 퐹 = 퐻 8퐷

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

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.

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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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, self- weight—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: 푃 = 푓푙푤 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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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:

Equation 2.2 푅푉 = 0.25푊

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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

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(or springing). With self-weight per linear foot of arch q, the vertical reactions at the support are found through vertical equilibrium: 푞퐿 푅 = Equation 2.3 푉 2

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 푞퐿2 퐹 = Equation 2.4 퐻,풎풂풙 8퐷

푞퐿2 퐹 = Equation 2.5 퐻,풎풊풏 8(퐷 + 푡)

Therefore a range of horizontal thrust values exists between lower and upper bounds: 푞퐿2 푞퐿2 ≤ 퐹 ≤ Equation 2.6 8(퐷 + 푡) 퐻 8퐷

The average horizontal thrust is calculated using the rise to the center of the arch at the crown: 푞퐿2 퐹 = 퐻,풂풗품 푡 Equation 2.7 8 (퐷 + 2)

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

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 휎 = 푅γ 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: 20푘푁 200푘푁 푅훾 = 10푚 ∗ = = 0.2푁/푚푚2 푚3 푚2 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:

Equation 2.8 푁휃 = −푞푎 cos 휃

With q the self-weight per unit area and a the radius (Heyman 1999).

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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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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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(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,

2 Literature review compressive stresses in masonry will rarely if ever reach the magnitude of its strength. Elasticity- based 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.

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 Beaux- Arts 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 main- floor 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 Far- and 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.

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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

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:

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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.

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):

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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 two- course 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.

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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:

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).

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.

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.

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 ½- scale 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.

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:

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 Q pcf ratio pcf ratio ft 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.

L Dintrados Dextrados Dcenter ft. ft. ft. ft. Long side 9.1 0.33 0.92 0.63 Short side 6.7 0.33 0.92 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

푊푇표푡푎푙 = 2(63 푝푠푓)[(9.1 푓푡)(1 푓푡) + (6.7 푓푡)(1 푓푡)] + (36 푝푠푓)(9.1 푓푡)(6.7 푓푡) ≈ 4,211 푙푏 ≈ 4,210 푙푏

Figure 4.2 is a photo of the vault as construction nears completion. In the foreground, the five- course 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).

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)

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:

4 Analysis of scaled Guastavino replica

Wweb Q web Warches Q arch lb psf lb 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 푞 = 푄푤

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:

퐹 = √푅2 + 퐹2 Equation 4.2 푡ℎ푟푢푠푡 푉 퐻

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.

w q FH,max FH,min FH,avg RV Fthrust,max Fthrust,min Fthrust,avg ft lb/ft lb lb lb lb lb lb lb 1.7 61 1,910 694 1,020 279 1,930 748 1,060 Long 1.0 63 1,950 710 1,040 285 1,970 765 1,080 side 124 3,860 1,400 2,060 564 3,900 1,510 2,140 2.3 83 1,400 510 749 279 1,430 582 799 Short 1.0 63 1,060 384 563 210 1,080 437 600 side 146 2,460 894 1,310 489 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: 푑푒푚푎푛푑 푢푛𝑖푡푦 푐ℎ푒푐푘 = < 1 Equation 4.3 푐푎푝푎푐𝑖푡푦

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.

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:

4 Analysis of scaled Guastavino replica

Arch L w q FH,max FH,min FH,avg RV Fthrust,max Fthrust,min Fthrust,avg Long side ft ft lb/ft lb lb lb lb lb lb lb 1 1.5 0.67 24 21 7.7 11 19 28 20 22 2 3.0 0.67 24 85 31 45 37 93 48 59 3 4.6 0.67 24 191 69 102 56 199 89 116 4 6.1 0.67 24 339 123 181 74 347 144 196 5 7.6 0.67 24 530 193 283 93 538 214 298 6 9.1 1.0 63 1,950 710 1,040 285 1,970 765 1,080 Short side 1 1.1 0.91 33 16 5.7 8.3 19 24 19 20 2 2.2 0.91 33 62 23 33 37 73 44 50 3 3.4 0.91 33 140 51 75 56 151 76 93 4 4.5 0.91 33 250 91 133 74 260 117 152 5 5.6 0.91 33 390 142 208 93 401 170 228 6 6.7 1.0 63 1,060 384 563 210 1,080 437 600 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:

퐹 = 퐹 + √퐹2 + 퐹2 , 𝑖 = 1, … ,6 Equation 4.4 퐻푖 퐻푖−1 퐻푖,1 퐻푖,2

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:

Equation 4.5 푅푉,푟푖푏,푖 = 푅푉,푙표푛푔 푤푒푏,푖 + 푅푉,푠ℎ표푟푡 푤푒푏,푖 + 푅푉,푟푖푏,푖−1

This procedure leads to the following results for the ribs:

4 Analysis of scaled Guastavino replica

Horizontal resultants Vertical Thrust resultants

FH,rib|max FH,rib|min FH,rib|avg RV,rib Fthrust,rib|max Fthrust,rib|min Fthrust,rib|avg Load point lb lb lb lb lb lb lb 1 26 9.6 14 37 46 38 40 2 132 48 70 112 173 121 132 3 369 134 197 223 431 260 297 4 790 287 421 372 873 470 562 5 1,450 527 772 558 1,550 767 953 6 3,670 1,330 1,960 1,050 3,820 1,700 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 2 퐹푡ℎ푟푢푠푡,푡푟푖푎푛푔푙푒 = (√퐹퐻,푙표푛푔 + 퐹퐻,푠ℎ표푟푡) + (푅푉,푙표푛푔 + 푅푉,푠ℎ표푟푡) Equation 4.6

The following table compares thrust resultants on the rib-arches for maximum, minimum, and average horizontal thrust magnitudes:

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 FH,rib|min FH,rib|avg FH,triangle|max FH,triangle|min FH,triangle|avg lb % W lb % W lb % W lb % W lb % W lb % W 3,670 87 1,330 32 1,960 46 4,580 109 1,660 40 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

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 1,053 1,050 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 side- arches 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 edge- arches. 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.

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: 퐿 퐿 6 − 𝑖 푃 = 푄 ( 푎 ) ( 푏) ( ) Equation 4.7 푤푒푏,푖 푤푒푏 10 2 5

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): 퐿 퐿 1 푃 = 푄 ( 푎 ) ( 푏) ( ) Equation 4.8 푤푒푏,6 푤푒푏 2 ∗ 10 2 10

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 푃푎푟푐ℎ,푖 = 푄푎푟푐ℎ ( ) (1 푓푡) 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. lblock Lb/2 κ Pweb Parch Long side ft ft lb lb 1 0.91 3.4 1.0 112 57 2 0.91 3.4 0.8 89 57 3 0.91 3.4 0.6 67 57 4 0.91 3.4 0.4 45 57 5 0.91 3.4 0.2 22 57 6 0.46 3.4 0.1 5.6 29 Short side 1 0.67 4.6 1.0 112 42 2 0.67 4.6 0.8 89 42 3 0.67 4.6 0.6 67 42 4 0.67 4.6 0.4 45 42 5 0.67 4.6 0.2 22 42 6 0.34 4.6 0.1 5.6 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

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:

Equation 4.10 푃푖 [푙푏] ∗ 0.125 [𝑖푛. ] = 1 [푙푏] ∗ 푃푠푐푎푙푒푑 [𝑖푛. ]

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 Parch Ptotal in in in 1 14 7.1 21 2 11 7.1 18 3 8.4 7.1 15 4 5.6 7.1 13 5 2.8 7.1 10 6 0.70 3.6 4.3 Table 4.12: Scaled loads for graphical analysis

First, the scaled web loads alone are applied to the side-arch:

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).

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 side- arch, 1,040 lb = 130 in.

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

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

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] 𝑖푛. 12 = 2,236 푙푏 ≈ 2,240 푙푏 0.125 𝑖푛./푙푏

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: 퐹 (8퐷 ) (2,240 푙푏)(8 ∗ 0.6 푓푡) 푞 = 퐻 푐푒푛푡푒푟 = = 130 푙푏/푓푡 푒푞 퐿2 (9.1 푓푡)2

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

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:

4 Analysis of scaled Guastavino replica

Segment no. RV from all parallel arches Force P lb in 1 37 5 2 74 9 3 112 14 4 149 19 5 186 23 6 495 62 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):

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 = 2,072 푙푏 ≈ 2,070 푙푏 0.125

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:

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 rib- arches 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

4 Analysis of scaled Guastavino replica

Notice the new horizontal thrust magnitude of 8.25 (27 + ) ∗ 12 12 = 2,658 푙푏 ≈ 2,660 푙푏 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 self- weight the MCNY vault is stable. Throughout the assessment the vault is modeled as a two- dimensional 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 two- dimensional model to a side-arch for graphical analysis. Lumping material atop the arch still

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.

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).

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.

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

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-of- Paris 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 Qdead,vault pcf ratio pcf ratio ft psf 125 0.8 62.5 0.2 0.50 56 Table 5.1: Met vault properties Other pertinent dimensions and loading:

Long side Short side Area A Wdead,vault ft ft ft2 lb 26 12 326 18,400 Table 5.2: Lengths, area, and total vault weight

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:

푊 = (56 푝푠푓) ∗ (26.3 푓푡) ∗ (12.4 푓푡) ≈ 18,400 푙푏

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 over- approximation 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:

퐴푠푢푟푓푎푐푒,푣푎푢푙푡 = 휋푅퐿푎푑푗

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 Along Rshort Ashort Wdead,vault ft ft2 ft ft2 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 Thickness Long side Short side Area A Qdead,slab Wdead,slab pcf ft ft ft ft2 psf lb 145 0.5 26 12 326 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 Q dead,total Qdead,vault Qdead,slab Qlive,i Qlive,f lb psf psf psf psf psf 42,000 129 56 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

5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H

2 푊(퐷+퐿)|1 = 18,400 푙푏 + 23,700 푙푏 + (300 푝푠푓) ∗ (326 푓푡 ) ≈ 139,900 푙푏 = 140 푘𝑖푝

Whereas for comparison, total weight dead plus live, approximating the vault surface area as half a cylinder, is

2 푊(퐷+퐿)|2 = 28,800 푙푏 + 23,700 푙푏 + (300 푝푠푓) ∗ (326 푓푡 ) ≈ 150,300 푙푏 = 150 푘𝑖푝

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:

Dintrados Dextrados Dcenter ft ft ft Long side 7.3 7.8 7.6 Short side 7.3 7.8 7.6 Table 5.6: Vault rise dimensions In this assessment the total dead load plus the design live load is summed, giving

푄퐷+퐿 = 푄퐷,푣푎푢푙푡 + 푄퐷,푠푙푎푏 + 푄퐿,푓 = (56 + 73 + 300)푝푠푓 = 429 푝푠푓

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 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 Long side 26 3.1 1,330 15,800 14,700 15,200 17,500 23,500 22,900 23,200 Short side 12 6.6 2,820 7,410 6,940 7,170 17,500 19,000 18,800 18,900 Table 5.7: Determination of horizontal thrust, vertical reaction, and resultant thrust in quarter-vaults

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. 23,200 푙푏 퐹 = 2,060 푝푠𝑖 ≥ = 104 푝푠𝑖 the material is not in danger of crushing 푐 223 𝑖푛2

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:

5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H

Arch L w q FH,max FH,min FH,avg RV Fthrust,max Fthrust,min Fthrust,avg Long side ft ft lb/ft lb lb lb lb lb lb lb 1 4.4 0.52 221 73 68 71 486 491 491 491 2 8.8 1.0 443 583 546 564 1,940 2,030 2,020 2,020 3 13 1.0 443 1,310 1,230 1,270 2,920 3,200 3,160 3,180 4 18 1.0 443 2,330 2,180 2,260 3,890 4,530 4,460 4,500 5 22 1.0 443 3,650 3,410 3,530 4,860 6,080 5,940 6,000 6 26 1.0 443 5,250 4,910 5,080 5,830 7,850 7,630 7,730 Short side 1 2.1 1.1 470 34 32 33 486 487 487 487 2 4.1 2.2 941 275 257 266 1,940 1,960 1,960 1,960 3 6.2 2.2 941 618 578 597 2,920 2,980 2,970 2,980 4 8.3 2.2 941 1,100 1,030 1,060 3,890 4,040 4,020 4,030 5 10 2.2 941 1,720 1,610 1,660 4,860 5,150 5,120 5,140 6 12 2.2 941 2,470 2,310 2,390 5,830 6,330 6,270 6,300 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:

퐹 = 퐹 + √퐹2 + 퐹2 , 𝑖 = 1, … ,6 Equation 5.1 퐻푖 퐻푖−1 퐻푖,1 퐻푖,2

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 Vertical Thrust resultants

FH,rib|max FH,rib|min FH,rib|avg RV,rib Fthrust,rib|max Fthrust,rib|min Fthrust,rib|avg Load point lb lb lb lb lb lb lb 1 81 75 78 972 975 975 975 2 725 679 701 4,860 4,910 4,910 4,910 3 2,180 2,040 2,100 10,700 10,900 10,900 10,900 4 4,760 4,450 4,600 18,500 19,100 19,000 19,000 5 8,780 8,220 8,490 28,200 29,500 29,400 29,400 6 14,600 13,700 14,100 39,800 42,400 42,100 42,300 Table 5.9: Horizontal thrusts, vertical reactions, and resultant thrusts on a diagonal rib

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 푙푏 − 35,000 푙푏 ( ) ∗ 100 = 14% 35,000 푙푏

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:

2 √ 2 2 2 퐹푡ℎ푟푢푠푡,푡푟푖푎푛푔푙푒 = (√퐹퐻,푙표푛푔 + 퐹퐻,푠ℎ표푟푡) + (푅푉,푙표푛푔 + 푅푉,푠ℎ표푟푡) Equation 5.2

The following table compares the resulting thrusts for the sliced parallel arches and triangle- arches 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:

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 FH,rib|avg FH,triangle|max FH,triangle|min FH,triangle|avg lb % W lb % W lb % W lb % W lb % W lb % W 14,600 10 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 triangle- arches 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 35,000 39,800 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

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: 퐿 퐿 6 − 𝑖 푃 = 푄 ( 푎 ) ( 푏) ( ) Equation 5.3 푖 10 2 5

퐿 퐿 1 푃 = 푄 ( 푎 ) ( 푏) ( ) Equation 5.4 6 2 ∗ 10 2 10

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

5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H

Segment no. lblock Lb/2 κ P P Long side ft ft lb kip 1 2.6 6.2 1.0 7,000 7.0 2 2.6 6.2 0.80 5,600 5.6 3 2.6 6.2 0.60 4,200 4.2 4 2.6 6.2 0.40 2,800 2.8 5 2.6 6.2 0.20 1,400 1.4 6 1.3 6.2 0.10 350 0.35 Short side 1 1.2 13 1.0 7,000 7.0 2 1.2 13 0.80 5,600 5.6 3 1.2 13 0.60 4,200 4.2 4 1.2 13 0.40 2,800 2.8 5 1.2 13 0.20 1,400 1.4 6 0.62 13 0.10 350 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-to- span 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 7.0 2 5.6 3 4.2 4 2.8 5 1.4 6 0.35 Table 5.14: Scaled loads, long side Figure 5.6 shows the loading and labeling scheme for the graphical analysis.

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:

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

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:

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.

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

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

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.

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. RV from web arches Force P lb kip ft 1 972 1.0 1.0 2 3,900 3.9 3.9 3 5,830 5.8 5.8 4 7,780 7.8 7.8 5 9,720 10 10 6 11,700 12 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.

5 Analysis of Guastavino cross-vault, Metropolitan Museum of Art, wing H

Figure 5.18: Force polygon for ribs Giving a line of pressure:

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

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.)

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:

Segment no. lblock Lb/2 κ P P Long side ft ft lb kip 1 2.63 6.2 1.0 6,550 6.5 2 2.63 6.2 0.80 3,640 3.6 3 2.63 6.2 0.60 2,730 2.7 4 2.63 6.2 0.40 1,820 1.8 5 2.63 6.2 0.20 910 0.91 6 1.3 6.2 0.10 227 0.23 Short side 1 1.24 13 1.0 6,550 6.5 2 1.24 13 0.80 3,640 3.6 3 1.24 13 0.60 2,730 2.7 4 1.24 13 0.40 1,820 1.8 5 1.24 13 0.20 910 0.91 6 0.62 13 0.10 227 0.23 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 6.5 2 3.6 3 2.7 4 1.8 5 0.91 6 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 quarter- vault. 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:

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): 푃퐿 푞퐿2 퐹 = + Equation 5.5 퐻 4퐷 8퐷

The new horizontal thrust values are given for both long and short sides:

FH,max FH,min FH,avg lb lb lb Long side 12,000 11,300 11,600 Short side 5,670 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.

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: (푃 + 푞퐿) 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

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.

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:

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. RV from web arches Force P lb kip ft 1 4,630 4.6 4.6 2 2,530 2.5 2.5 3 3,790 3.8 3.8 4 5,060 5.1 5.1 5 6,320 6.3 6.3 6 7,580 7.6 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).

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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

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.

6.2 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.

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References

7 References

Allen, Edward, Wacław Zalewski, and Joseph Iano. 2010. Form and Forces: Designing Efficient, Expressive Structures. Hoboken, N.J: John Wiley & Sons, Inc.

Bartlett, A.M. Letter to Alfred Engel. 1960. “RE: METROPOLITAN MUSEUM OF ART New York, N.Y.”, March 22. Drawings & Archives, Avery Library, Columbia University.

———. Letter to John Zoldos. 1958. “Re: #2089 Alterations to Metropolitan Museum of Art New York, N.Y.”, May 12. Drawings & Archives, Avery Library, Columbia University.

Berg, I.A. Letter to Malcolm Blodgett. 1948. “Re: Metropolitan Museum of Art New York”, August 3. Drawings & Archives, Avery Library, Columbia University.

———. Letter to Malcolm Blodgett. 1950. “Re: Metropolitan Museum of Art New York City”, June 7. Drawings & Archives, Avery Library, Columbia University.

Block, Philippe. 2009. “Thrust Network Analysis: Exploring Three-Dimensional Equilibrium”. Massachusetts Institute of Technology.

Block, Philippe, Thierry Ciblac, and John Ochsendorf. 2006. “Real-Time Limit Analysis of Vaulted Masonry Buildings.” Computers & Structures 84 (29-30): 1841–52. doi:10.1016/j.compstruc.2006.08.002.

Block, Philippe, Matt DeJong, and John Ochsendorf. 2006. “As Hangs the Flexible Line: Equilibrium of Masonry Arches.” Nexus Network Journal 8 (2): 13–24. doi:10.1007/s00004-006-0015-9.

Block, Philippe, Lorenz Lachauer, and Matthias Rippmann. RhinoVAULT.

“Drawings & Archives, Avery Library, Columbia University.”

Dugum, Hussam. 2013. “Structural Assessment of the Guastavino Masonry Dome of the Cathedral of Saint John the Divine”. Cambridge, MA: Massachusetts Institute of Technology.

Engel, Alfred. Letter to R. Guastavino Co. 1960, January 18. Drawings & Archives, Avery Library, Columbia University.

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References

Guastavino, Rafael. 1893. Essay on the Theory and History of Cohesive Construction Applied Especially to the Timbrel Vault. Second. Ticknor & Company.

Harrison. Western Union Telegram to R. Guastavino Co. 1950, June 29. Drawings & Archives, Avery Library, Columbia University.

Heckscher, Morrison. 1995. The Metropolitan Museum of Art: An Architectural History. Vol. LIII. The Metropolitan Museum of Art Bulletin. New York, New York: The Metropolitan Museum of Art.

Heyman, Jacques. 1999. The Stone Skeleton: Structural Engineering of Masonry Architecture. Cambridge, United Kingdom: Cambridge University Press.

Huerta, Santiago. 2008. “The Analysis of Masonry Architecture: A Historical Approach.” Architectural Science Review 51 (4): 297–328.

Kidder, Frank, and Thomas Nolan. 1916. The Architects’ and Builders’ Handbook: Data for Architects, Structural Engineers, Contractors, and Draghtsmen. Sixteenth. New York, New York: John Wiley & Son, Inc.

———. 1921. The Architects’ and Builders’ Handbook: Data for Architects, Structural Engineers, Contractors, and Draghtsmen. Seventeenth. New York, New York: John Wiley & Son, Inc.

Knox, Sanka. 1963. “Work Proceeding at Museum of Art.” The New York Times, April 20.

Lee, Suk. 2010. “Catalan Design”. AutoCAD drawing.

“Metropolitan Museum of Art Archives.” Metropolitan Museum of Art Archives.

Metropolitan Museum of Art, New York City, (The Museum). 1962. “Planning for the Future: The Building Program of the Metropolitan Museum of Art.”

MIT Masonry Research Group. 2012. “Palaces for the People: Half-Scale Guastavino Vault Installation.”

Museum of the City of New York. 2014. “Palaces for the People.”

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References

Ochsendorf, John. 2010. Guastavino Vaulting: The Art of Structural Tile. First. New York, New York: Princeton Architectural Press.

R. Guastavino Co., President. Letter to Helen Tolmachoff. 1948, May 21. Drawings & Archives, Avery Library, Columbia University.

Reese, Megan. 2008. “Structural Analysis and Assessment of Guastavino Vaulting”. Cambridge, MA: Massachusetts Institute of Technology.

Seitz, H.F. Letter to Malcolm Blodgett. 1949. “Re: Metropolitan Museum of Art”, January 25. Drawings & Archives, Avery Library, Columbia University.

Shepard, Richard F. 1966. “Metropolitan Museum Is Making Costly Changes.” The New York Times, June 21.

The Brick Industry Association. 2007. “Technical Notes on Brick Construction: Specifications for and Classification of Brick.” www.gobrick.com.

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.

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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: 푞퐿 Σ퐹 = 0 = −푞퐿 + 2푅 ∴ 푅 = 푦 푉 푉 2 Cut the arch at an arbitrary distance x from the left springer and take moments about that point: 푞푥2 푞퐿푥 Σ푀 = 0 = − + 퐹 푦(푥) 2 2 퐻 Solve for y(x): 1 푞퐿푥 푞푥2 푦(푥) = ( − ) 퐹퐻 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: 퐿 푞퐿2 푦 ( ) = 2 8퐹퐻 Now take y(L/2) = D, the rise of the arch, and solve for the horizontal thrust by rearranging: 푞퐿2 퐹 = 퐻 8퐷 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: 푞퐿2 푃푥 푞퐿 Σ푀 = 0 = −푃푥 − + 푅 퐿 ⇒ 푅 = + 퐴 2 푉퐵 푉퐵 퐿 2

Now sum forces in the vertical direction to solve for RVA: 푃푥 푞퐿 푃푥 푞퐿 Σ퐹 = 0 = −푞퐿 − 푃 + ( + ) + 푅 ⇒ 푅 = 푃 − + 푦 퐿 2 푉퐴 푉퐴 퐿 2

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Check the result. For the case when P is at x= L/2 expect RVA = RVB: 퐿 (푃 + 푞퐿) 퐿 (푃 + 푞퐿) 푅 ( ) = = 푅 ( ) = 푉퐴 2 2 푉퐵 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): 푞푥2 푃푥 푞퐿 Σ푀 = 0 = − (푃 − + ) 푥 + 퐹 푦(푥) 2 퐿 2 퐻 This gives an expression for y(x): 1 푞퐿푥 푞푥2 푦(푥) = (푃푥 − 푃푥2 + − ) 퐹퐻 2 2

Again find the value for y(x= L/2), which is ymax, the crown of the arch: 퐿 1 푃퐿 푞퐿2 푦 ( ) = ( + ) 2 퐹퐻 4 8 Replace y(L/2) with the variable D, the vertical rise of the arch. Now solve for the horizontal thrust by rearranging: 푃퐿 푞퐿2 퐿 푞퐿 퐹 = + = (푃 + ) 퐻 4퐷 8퐷 4퐷 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.

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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”)

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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”)

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Figure 8.4: Guastavino Co., wing H, June 23, 1911 (“Drawings & Archives, Avery Library, Columbia University”)

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Figure 8.5: Guastavino Co., wing H, July 5, 1911 (“Drawings & Archives, Avery Library, Columbia University”)

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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”)

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Figure 8.8: Wing H from City of New York Department of Parks, February 28, 1962 (“Drawings & Archives, Avery Library, Columbia University”)

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Figure 8.9: Representative cross-section of long side from existing conditions, 1962 (“Drawings & Archives, Avery Library, Columbia University”)

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Figure 8.10: McKim et al. drawing of wing H courtyard; steel cols. with masonry cover, October 9, 1911 (“Metropolitan Museum of Art Archives”)

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Figure 8.11: Drawing showing general dimensions of cross-vaults around wing H courtyard, October 9, 1911 (“Metropolitan Museum of Art Archives”)

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Figure 8.12: View of short sides of cross-vaults; entrances to stairs and elevator; Oct. 9, 1911 (“Metropolitan Museum of Art Archives”)

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Figure 8.13: Original second floor steel framing plans, wing H, December 22, 1909 (“Metropolitan Museum of Art Archives”)

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Figure 8.14: Tracing of existing steel framing plan, June 9, 1943 (“Metropolitan Museum of Art Archives”)

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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”)

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Figure 8.16: Letter from Tolmachoff to R. Guastavino Co. (“Drawings & Archives, Avery Library, Columbia University”)

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Figure 8.17: Letter from Berg explaining that consulting engineers are unfamiliar with Guastavino systems (“Drawings & Archives, Avery Library, Columbia University”)

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Figure 8.18: Telegram showing 1950 decision to replace vaults with steel (“Drawings & Archives, Avery Library, Columbia University”)

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Figure 8.19: Letter from R. Guastavino Co. to New York City engineer (“Drawings & Archives, Avery Library, Columbia University”)

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