By Kae Ransom
Photos by the author unless otherwise stated
Domes maintain an architectural precedent dating all the way back to the huts and tombs of the ancient Middle East. Though limited by small scale techniques like building solid mound domes — a far cry from the skyline-dominating wonders we have today — ancient cultures nevertheless made use of these structures where they could (Britannica, 2023). It was the ancient Romans who really began to test the limits of what could be achieved with domes, not just in size but in splendor. We will look closely at two hallmarks in dome history, respectively known for groundbreaking structural innovations that allowed them to build bigger than was thought possible — the Pantheon from Ancient Rome and Brunelleschi’s Duomo from the Renaissance period. We will examine the different techniques each employed to combat the structural difficulties of dome construction that led them to becoming precedent for dome building throughout history and into the modern day.
Engineering of a Dome
Domes are not only visually impressive, but structural beneficial as well. As ceilings, they enclose large interiors without the need for excessive support across the span, ultimately reducing material cost. Due to weight, a horizontal beam will deflect and experience a bending moment over long spans between supports as seen in Figure 1. The beam will experience tension along the bottom and compression at the top. Particularly in ancient times where brittle materials like stone, masonry, and unreinforced concrete were commonly used, material weight and tension failure posed huge issues for building large, long span roofs without significant reinforcement from columns along the span.
In this regard, domes are preferable because they behave similarly to arches. Arches distribute loads axially to supports and work purely in compression, making them ideal for the type of masonry building early civilizations were employing (Francis, 1980). This principle allows arches to span long distances while only requiring extra reinforcement at the end supports. The distributed load on the arch causes a tendency to bow out at the bottom called horizontal thrust, which stresses the supports (Francis, 1980). This is the primary concern when building arches and domes, as it is the point most likely to fail.
Domes as 3D Arches
Domes behave like arches rotated 360 degrees with a few key differences. Longitudinally, domes distribute loads down axially to supports like arches. However, as shown in figure 2, dome stresses act circumferentially as well as longitudinally, leading to uniform distribution across the shell (Francis, 1980). Under load the dome experiences only compressive stress longitudinally, however the circumferential hoop stress acts in compression at the top of the dome and tension around the bottom (Francis, 1980). This is a similar phenomenon to the horizontal thrust experienced by arches and why many brittle domes see vertical cracks appearing at their base but not top. Horizontal thrust is principally resisted by the surrounding building, but often this is not enough to fully relieve the stress accumulated – the material cracks when the pulling caused by horizontal thrust overcomes the material’s tensile capacity (Petroski, 2011).
Stress, Thickness, and the Importance of Radius
Assuming you can reinforce the dome perimeter to reduce the tensile stress, the next largest concern is the longitudinal compressive stress from the distributed loading. Stress is force divided by the area it is acting on; for a dome only supporting its own weight, the volume of the dome and area of the perimeter the force is uniformly acting on partially cancel such that the longitudinal compressive stress is purely governed by the radius and material density (Francis, 1980). This means that domes can be built thinner and with less material without sacrificing much structural integrity, making them highly economical. It also emphasizes the significance of the Pantheon and Duomo’s construction. Because of the large radii of the domes, Brunelleschi and the Romans had to get creative about how to reduce stress in other ways. Both domes found unique and revolutionary ways to tackle the two biggest considerations in dome integrity — tensile hoop stress and longitudinal compressive stress at the dome perimeter.
The Pantheon
The Pantheon standing today is a 2nd century reconstruction under Hadrian of Marcus Agrippa’s early Pantheon from 27-25 BC. Until Brunelleschi’s Duomo was completed in 15th century, it was the largest dome ever built, spanning a diameter of 43.4 meters across. It still remains today the largest made of unreinforced concrete (Archeoroma, 2024).
Located in the heart of Rome, the Pantheon is nestled somewhat unassumingly amidst narrow alleys of shops and restaurants. In fact, my first day in Rome I stumbled upon it by accident — approaching from the back, the concrete and brick base layer of the rotunda appears unassuming on first glance. Continuing around to the plazza and portico in front unveils a different story, revealing a glimpse into the past of this impressive structure. The interior marbling and huge, single-piece granite columns call back to the grandeur of the Roman empire, when it was covered top to bottom in shining marble. Even though this marble exterior is lost, the pantheon still inspires a sense of awe to visitors and passerby’s alike.
The dome is a perfect hemisphere, but due to reinforcing on the exterior roofing this is not immediately apparent from the outside. Walking inside allows for true appreciation the dome’s curvature — from floor to oculus, the Pantheon structure encloses a perfect sphere capped by this incredible dome.
Materials and Weight
For a dome this large, minimizing weight and material density is of critical importance. The Pantheon dome addresses this in a few key ways related to its material choice. The dome is completely made of unreinforced concrete, but utilizes different aggregates throughout in order to reduce density toward the top of the dome. The lime and sand cement mixture includes bricks and tufa near the base, porous volcanic slag, pumice and light tufa in the middle, and small clay pots to make air pockets near the oculus (Archeoroma, 2024). Furthermore, the thickness of the dome is varied as it rises, ranging from 5.90 m at the base to 1.4 m at the oculus (Archeoroma, 2024). The 48 square cut outs around the perimeter, coffers, mainly serve as aesthetic value but were likely intended as a method of weight reduction as well. However, they only take out less than 5% of the dome’s total weight (Mark & Hutchinson, 1986). Figure 6 shows a close up of these coffer cut outs around the perimeter of the dome.
As previously mentioned, max compressive stress is dependent on radius and material density. The Pantheon varies in density from 1600 kg/m3 to 1350 kg/m3 at the oculus, and computer modeling of the density variation reveals the max compressive stress at the perimeter to be 2.8 kg/cm2 (Mark & Hutchinson, 1986). This is well within the the range of compressive stress this concrete can withstand, which is believed to be 5 MPa or about 51 kg/cm2 (Masi et. al, 2018). A computer model simulating stresses on the dome for a consistent heavy concrete aggregate of 2200 kg/m3 found stresses 80% higher than the gradational aggregate model that was actually employed (Mark & Hutchinson, 1986).
The Oculus
The circular opening at the top of the dome serves as the only natural light source into the building, illuminating the rotunda like a beacon. Such a feature is possible because of the stress distribution behavior of domes. Unlike arches where the middle piece is key in fixing the arch together and distributing the load, domes experience compressive hoop stress near their apex alongside longitudinal stress, allowing for the insertion of a compression ring without compromising structural integrity. The oculus acts like two semicircular arches, symmetrically allowing for equal force distribution about the center (Petroski, 2011). The oculus has an 8.8 m diameter and is reinforced by rings of long bricks at its edge (Archeoroma, 2024).
Reducing Tension with Cracks
Concrete’s low tensile strength (about 0.08 MPa for the concrete used in the Pantheon) is a huge concern in ensuring the dome’s structural integrity (Masi et. al, 2018). Ultimately, cracks can form a number of controllable and uncontrollable factors, especially since Roman concrete is particularly weak in tension. For example, shrinking of the curing mixture, rapid thermal changes like cold rain after a sunny day, and of course load bearing can all contribute to cracking on the structure. The Pantheon cracks almost exclusively climb vertically from the perimeter since the tensile hoop stress pulls on the concrete horizontally; these existing cracks span up to about a 57° latitude which is consequentially around the point where the hoop stress switches from tensile to compressive (Masi et. al, 2018). Figure 7 shows a map of the 14 main cracks around the perimeter of the dome.
There is some disagreement over the origin of the Pantheon cracks, with the predominant theory being that they formed from concrete shrinkage and gravitational force acting on the dome early on (Masi et. al, 2018). The good news is these cracks do not mean the Pantheon is at risk of falling anytime soon — in fact, the Romans may have found a way to manipulate them in a manor that strengthens the dome rather than breaks it.
Along side the thick rotunda walls supporting the dome, the Romans also added a series of 7 concentric steps bracing the lower section of the dome where tensile hoop stress is highest. The extra weight of these rings compresses the structure at the perimeter, working to reduce hoop stress in this critical area (Archeoroma, 2024). Figure 8 shows these stepped rings as they appear on the roof exterior — they significantly hide the hemispherical shape of the dome interior.
Interestingly, it is the cracking that makes these rings most effective. The presence of the longitudinal cracks breaks up the uniform load distribution property of the dome and instead allows it to function as a series of wedge-shaped arches (Masi et. al, 2018). This does cause bending forces in the dome to double, increasing the likelihood of tensile failure (Mark & Hutchinson, 1986). But in modelling the stress profile of the cracked dome with and without the stepped rings, it was found that the rings localized much of the tension in predominantly compressive areas, meaning any tension cracks that formed would not propagate as far (Mark & Hutchinson, 1986). Figure 9 shows the differences in stress profiles for the Pantheon with and without the step rings and the cracked versus uncracked dome. The maximum hoop stress was found to be about ~27.1% less in the ringed dome than the unringed dome. Overall, the rings offer extra support and stability to the weakest part of the dome, reinforcing preexisting cracks to prevent further spreading as well as reducing hoop stress around the perimeter.
The Duomo
In the 13 centuries after the Pantheon’s construction, no dome came close to toppling its precedent as the largest dome ever built — until 1436. Spanning a 45 meter diameter interior and beginning 55 meters off the ground, the duomo of the Basilica of Santa Maria del Fiore in Florence set out to overcome the bar set by the Pantheon (Jones et. al, 2008).
Today the cathedral and dome dominate the Florentine skyline from every angle. The cathedral itself is grand, but it is the dome that really draws the eye. From its elegant cupula and lantern to the peculiar octagonal shape, it is not hard to appreciate its achievement as a perfect embodiment of the sought-after skill and ideals driving the Renaissance.
The Basilica began construction in 1296 without a solid idea of how the dome would be completed. In 1418, a competition was held to obtain the contract for the dome’s design, won by Filippo Brunelleschi (Mueller, 2014). He was tasked to not only build bigger than anyone before him, but under unique constraints like the dome’s prebuilt octagonal base and the extreme height of the construction site. His innovative solutions are what mark the Duomo as the historical and technical marvel it stands as today.
Self Supporting Masonry
The 1418 competition tasked architects to build off an earlier concept by Neri di Fiorivante who proposed a double shelled dome bounded by internal rings for support rather than external buttresses. The primary concern with this model was the shear amount of scaffolding it was theorized to require. Masonry arches are most often built with wooden framework for centering, which, for a dome of this size, would have required an estimated 700 trees and crowded the building space too much for efficient work (King, 2000). Laying masonry on a spherical dome provides horizontal and vertical support that reduces the need for scaffolding, but the octagonal shape of the Duomo prevented this principle from being applicable. The dome is more structurally similar to an array of arches than a true dome, which was why centering formwork seemed the inevitable option. The economy of Brunelleschi’s design, which instead proposed self-supporting construction from start to finish, won him the contract. Brunelleschi was incredibly secretive so the specific methods of construction employed are unknown. However, analysis of the existing dome and a modeled replication of theorized techniques give a good basis for how and why he constructed the dome the way he did.
The double dome design involves a heavier interior and lighter exterior that also helps protect the decorations inside. The two are connected via vertical ribbing in the space between, as well as four horizontal stone chains dispersed at 35′ intervals up the length of the dome (King, 2000). These were also supplemented with iron and wood chains near the base of the dome, as iron and wood have far higher tensile strength than stone (King, 2000). Using these kinds of chains near the bottom helps to reinforce against horizontal thrust. Between these ribs are a pattern of bricks laid in a compressive arch pattern that allows for self-support throughout the process (Petroski, 2011). Brunelleschi built in progressive horizontal layers around symmetrically opposite sides such that the masonry remained in compression from start to finish, using plumb lines and center measurements to exact the curvature for each rib and side (Jones et. al, 2008).
The Ribs
Both shells are connected via 24 stone ribs arching up to the compression ring upon which the lantern sits. The ribs start after a thick layer of stone at the base of the duomo where the interior and exterior domes first separate (Jones et. al, 2008). During construction a series of light iron chains where attached across the diagonals of the dome to further compress the ribs until the dome could fully support itself (Jones et. al, 2008). The 8 major ribs supporting the corners of the octagon extend beyond the outer dome and are visible from the exterior. These ribs act as arches to distribute the weight of the lantern down to the reinforced base of duomo. They arch around 30 degrees near the base of the dome before sloping to 60 degrees by the oculus — far past the angle thought to be achievable without centering support frame work (King, 2000). Figure 12 shows the layout of the 24 vertical ribbings between the two domes as well as the placement of the stone and wood rings laid horizontally.
The Spiral Herringbone Brickwork
The brickwork in between the ribs of the domes hides another inventive technique implemented for ease of construction as well as dome stabilization. The bricks were laid horizontally in a curved, arch-like pattern between the ribs to maintain compression, however each layer had one brick placed vertically and offset from the previous. This array forms a spiraling herringbone pattern that is continuous between ribs and rises all the way to the top of the dome, shown in Figure 13 below.
In a model study attempting to build a replica of the dome true to the construction techniques and materials available to Brunelleschi, the researchers noted both structural and construction benefits. They confirmed the theory that the vertical bricks helped bridge one layer to the next, strengthening the self-supporting structure during construction, particularly on inclined planes. They also noted however that the vertical pieces helped them keep track of the layers when working on different sides of the dome, which allowed for different segments of the dome to be worked on independently of each other (Jones et. al, 2008). This would have made considerable impact on speed of construction.
Reducing Horizontal thrust
Like the Pantheon, Brunelleschi employed several tactics to reduce the weight of the dome in order to reduce stress. Besides the reinforced base, most of the dome is made of lightweight bricks and tufa. The interior dome thickness decreases from 7′ to 5′ while the exterior decreases from 2′ to 1′ at the oculus (King, 2000). The 8 exterior ribs also taper toward the top of the dome, further reducing the weight at the apex. The ribs also work to resist horizontal thrust by attaching the two domes together.
The first 15 meters of the dome are reinforced with rings of sandstone beams held together with iron clamps to serve as compression rings where the tension is highest on the dome (King, 2000). These sandstone rings are embedded in the masonry around the perimeter of the dome and notched to slot into smaller transverse beams perpendicular to the sides of the octagon; the corners are connected at 45 degrees with specially-cast iron clamps coated to be rust proof (King, 2000). Rusting could have caused cracking at the vulnerable corners of the dome.
Innovations in Construction
Though he did not use scaffolding, Brunelleschi did build a heavy platform at the base of the dome. The platform was solidly built and extremely precise — 48 square put-holes were cast 2.3 m deep around the stone base of the octagon, allowing for huge wooden beams to be inserted and cantilevered for the platform’s support. The precision of the placement of these put-holes — all on the same horizontal axis — indicates the working platform also doubled as a reference plane for precise measurements of the dome curvature (Jones et. al, 2008). Brunelleschi also designed two key inventions that allowed construction to overcome the difficultly of manipulating heavy stone and marble 55 meters off the ground — an reversible lift and a crane called a castello.
The lift was a supersized version of previous crack-style lifts, normally pulled by workers or slaves and used to raise materials on a much smaller scale than needed for the Duomo. Brunelleschi’s model used oxen rather than workers to turn the wheel that raised the necessary material. 600 ft and 1000 lbs of rope were used, at a thickness of 2.5″ across (King, 2000). Most revolutionary was the helix screw at the base of the contraption. This screw could turn in either direction and adjust the main rotor height to engage one of two pinions, thus allowing for the lift to reverse directions without needing to adjust the oxens’ rotation (King, 2000). Figure 14 shows a drawing of this lift mechanism using a horse rather than an ox. By only having the oxen move in one direction, construction could move much quicker because it wasn’t necessary to re-yoke the animals for every piece lifted.
The castello was another lifting mechanism set up on the massive platform at the base of the dome. This device was a precise wooden crane with a pivotable horizontal beam for moving and placing stone carried up by the lift. Using a series of screws and a slideable counterweight, loads could be lifted and kept in equilibrium as they were moved to their proper place. A turnbuckle allowed for precise placement of stones along the domes frame. An additional arm was implemented to press on the rope and prevent swinging in the winds at such great heights (King, 2000). Figure 15 shows a drawing by Leonardo Da Vinci of Brunelleschi’s contraption.
These two inventions were key to realizing the size and economy of Brunelleschi’s design. They weathered over a decade of heavy use, winter storms and rainfall and inspired inventors like DaVinci in further work.
Modern Advancements
It is inspiring to analyze these buildings from a modern day, particularly from an engineering perspective. Considering the Romans built the Pantheon long before Isaac Newton proposed his gravitational theory, it is fascinating to study how they realized their massive structures. They didn’t have a formalized understanding of dead load and the relationship between stress and density, yet they knew to heavily reduce the weight of the Pantheon near the oculus while reinforcing the base.
Brunelleschi’s story is also impactful to me as a studying engineer. He was only a goldsmith, yet took on one of the biggest, most puzzling engineering projects of his time. The construction of the Duomo embodies the core of engineering — innovation and thinking outside the box even when its deemed impossible. The creative problem solving exhibited by Brunelleschi delivered one of the most iconic engineering feats of the renaissance.
Though the Pantheon still holds its title of largest reinforced concrete dome in the world, the diameters of both the Pantheon and Duomo have been far surpassed by modern domes. However, their relevancy lies in the historical precedent they set. Brunelleschi drew from the Pantheon for inspiration on minimizing hoop stress and dead load on large constructions. In turn, the Duomo set a new standard from which domed buildings like St. Peters Basilica and the U.S. Capital building took inspiration from. Despite nearly 2000 years of history between us, we are still able to marvel and learn from the techniques and skills required to build beautiful, massive structures with half the knowledge we have now.
References
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Britannica, T. Editors of Encyclopaedia (2023). Dome. Encyclopedia Britannica. https://www.britannica.com/technology/dome-architecture
Francis, A. J. (1980). Introducing structures : A textbook for students of civil and structural engineering, building and architecture. Elsevier Science & Technology.
Jones, B., Sereni, A., & Ricci, M. (2008). Building Brunelleschi’s Dome: A practical methodology verified by experiment. Construction History, 23, 3–31. http://www.jstor.org/stable/41613926
King, R. (2000). Brunelleschi’s Dome: The Story of the Great Cathedral in Florence. Pimlico: London, England. (King, 2000)
Mark, R., & Hutchinson, P. (1986). On the Structure of the Roman Pantheon. The Art Bulletin, 68(1), 24–34. https://doi.org/10.2307/3050861
Masi, F., Stefanou, I., & Vannucci, P. (2018). On the origin of the cracks in the dome of the Pantheon in Rome. Engineering Failure Analysis, 92, 587-596. https://doi.org/10.1016/j.engfailanal.2018.06.013.
Mueller, T. (2014). Brunelleschi’s Dome. National Geographic. https://www.nationalgeographic.com/magazine/article/Il-Duomo
Petroski, H. (2011). Engineering: Arches and Domes. American Scientist, 99(2), 111–115. http://www.jstor.org/stable/23019241
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