Evolution+of+the+Roman+Dome

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 * Overview **

This article highlights the evolution of the dome through engineering advancements by the ancient Roman civilization and summarizes their progress through several case studies. The influence the perfection of this structure had on the design of St. Peter's Basilica Dome in Vatican City is then investigated.

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=Roman Dome Development=

As one of the most advanced ancient civilizations, the engineering accomplishments of the ancient Romans are typically recognized as extremely practical and purpose based. The engineers were always searching for structural solutions that would provide more open and usable spaces or more successful means of accomplishing their vision for a design. The concrete dome in particular is one of the most important structures that the Romans perfected through logical engineering and advancements and has had a profound impact on many large scale western civilization structures since.

Practicality of the Dome
The reason that the Romans began constructing domes is because they recognized the benefit of large spaces uninterrupted by columns, walls or any other roof supporting structure. However, with their technology of unreinforced concrete, masonry, flat roofs and columns, this vision was not possible because the unreinforced concrete or masonry roof could not span large distances without cracking and failure. Masonry and concrete both perform incredibly well in compression but poorly in tension. The difficulty in spanning long distances with flat forms of such materials is that the form will undergo bending which creates compression stresses on the top and tensile stresses on the bottom of the form as seen below in Figure 1. These bending stresses cause cracking on the tension side and the masonry or concrete form will collapse if the crack progresses through the cross section.



Perhaps the Romans had attempted such designs and seen that they needed to develop a more suitable method of obtaining large open rooms. For a civilization that had extensive knowledge of arches, the dome seems like a logical progression as it is merely an arch spun 360 degrees to create a hemispherical three dimensional form. The engineers of the Roman time had already realized the potential of the concrete or masonry arch in spanning large distances under heavy loads by compression forces, so it is likely they realized that a well-engineered dome would be similarly effective for large spans when applied to a three dimensional space.

Influence from Other Civilizations
When considering the beginning of dome usage in Rome, it is reasonable to think the engineers developed the idea of a dome on their own. However, evidence of dome structures dated prior to the Roman Empire have been found throughout the middle east and surrounding regions, though many of these are corbelled domes and not true domes. A corbel dome is unlike a true dome in that it does not rely purely on the compressive forces between the masonry or concrete components to maintain it's shape and structural integrity and therefore has a very limited span (Chant & Goodman, 1999). Some archaeological investigations have revealed other ancient civilizations that were building true domes and several have been uncovered in the ruins of the Sumerian city state of Ur (Chant et al., 1999). Chant and Goodman (1999) report that true domes and evidence of their wooden centering have been found in the royal #|cemeteries of ancient Ur and dated to 2500 BC. This would place knowledge of the masonry true dome long before the rise of the Roman Empire. Also, although not true domes, evidence of timber domes has been found in the ancient Etruscan area of Italy that date back to the beginning of #|the rise of Rome (Keinbauer, 1971).  Since Rome began with heavy Etruscan influence it is easy to see how the timber dome would naturally become a part of the early engineering ideas in the city and it is likely many were constructed that we have no evidence of. As the city of Rome grew in power and began to expand, it is unknown whether or not masonry domes were encountered in nearby regions because no conclusive evidence for such structures has been found in these areas. It seems equally as likely that the domes of ancient Ur were known to the Romans through expeditions or that the Romans developed the technology on their own from progression of the arch. Either way, the dome was not a Roman invention but they were the first civilization to overcome the challenges associated with it and perfect the form.

Formwork Development
One of the greatest difficulties associated with building a large self supporting curved shape out of masonry or concrete is the formwork necessary to support it during construction. This difficulty is increased by the double curvature of a dome because it requires more support and smaller pieces of wood in order to approximate the rounded shape. For arches and domes this formwork is termed "centering" and enabled the Romans to construct the desired shape without danger of the unfinished structure collapsing in on itself before compressive forces are established by inserting the keystone and achieving equilibrium. Early in their history, the Romans developed a method of centering for arches that was accurate, stable and provided a firm support to build the arch on as seen in Figure 2 below.

When concrete dome construction began, the centering system used for arches was a good two dimensional starting point but the engineers needed to develop a new system for double curvature. The error and progression of their method can be seen in the earlier Roman domes. Two different ways were developed to construct the dome centering, radial and horizontal formwork as seen in Figure 3 below (Lancaster, 2005). With horizontal formwork, the planks had to be relatively short in order to approximate a smooth curvature. The ends of each plank typically would line up with a single meridional line running vertically so that they could all be supported with evenly spaced radial frames. In a radial formwork system, the planks could be longer and each plank would typically end on a common circumferential line running horizontally. This method allowed less radial frames to be used but it required the circumferential line framing to be supported in a curved shape which was difficult. It is difficult to tell which system the Romans used more often because many of the domes constructed did not leave evidence of formwork on the interior (Lancaster, 2005).



The way that the arches and domes of the Roman civilization were designed often provided ledges at the base of the curvature to place the centering in order to minimize the amount of ground support that was needed. For arches, often times long timber columns could be avoided through these designs and completely self supported centering was achieved (Lancaster, 2005). This avoided the problem of flimsy and unstable wood columns under heavy loads during construction that would deflect and result in a sagging arch. For domes, some columns were typically still required for support of the centering because the three dimensional shape and large spans made it difficult to achieve the same stability without added support from the ground. However, like arch centering, the Romans also had the technology to build self supporting centering for double curvature shapes but this system typically required support until all radial arches were in place and equilibrium was achieve (Lancaster, 2005). It took the engineers some time before designing a reliable method of centering for large domes that yielded a highly accurate completed shape but by the second century AD they had succeeded.

Dome Stiffening
Once the problem of formwork was solved, the stiffness of these large domes came into question. If a dome is too flexible, when the centering is removed the dome apex may sag down significantly and this deflection could result in extensive cracking at certain areas. This may have been discovered through observed deflections and failed designs or perhaps it was something that the Roman engineers intuitively knew would happen. They developed different methods of building domes with stiffening elements placed throughout the curved shape in order to avoid such large deflections. One method developed used ribs that were visible on the interior of the structure as those on the vault in Figure 4 below. The Romans used this same technique in many of the domes that they constructed. These ribs not only served to strengthen the structure but also provided a strong visual texture to the otherwise plain interior of the curved shape through indentations called coffers. The structure would not have been stable with a uniform thickness of only that at the deepest point of the coffer but it was also excessive to use a uniform thickness matching that of the ribs. By employing this ribbed technique, the dome was able to be stiffened and decorated through one method and, although increasing the difficulty of formwork, was used extensively. Another system of stiffening that is evident in domes is placed within the dome shell and is not visible from the interior or exterior. This system began being used during the fourth century AD and is characterized by a type of brick lattice that is then filled in with concrete to create a closed dome as in Figure 5 below (Lancaster, 2005). The vertical lines of bricks essentially form a series of arches that are connected at the top of the dome by a common keystone. These arches would transfer loads through pure axial forces down to the base of the dome and in many domes it can be seen that the Romans placed these brick lattices so that they landed on a main support (Lancaster, 2005). In this manner, they could avoid placing unnecessary stresses on openings and thinner areas of supporting walls.


 * = [[image:_MG_7085.JPG width="463" height="317" align="center"]] ||= [[image:DomeRibs.jpg width="429" height="320" align="center"]] ||
 * = Figure 4: Vault in Roman Forum with interior coffers forming stiffening ribs (photo by author) ||= Figure 5: Brick lattice ribbing at the Baths of Agrippa (Lancaster, 2005) ||

Additional purposes have been proposed for the use of the brick lattices in domes and they were likely also intended by the Roman engineers. During construction of a dome, the lattices likely served to divide the dome up into manageable sections of construction that could be completed in a day and also to aid in placing formwork for the general shape of the dome. Many domes have horizontal courses of bipedalis bricks at vertical intervals between the brick lattices and these could have been stood on by carpenters so they could place the formwork just ahead of the concrete pouring. In this way the formwork for the entire dome would not have to be built before any concrete placement began (Lancaster, 2005). The lattices would also keep the dome stiff while curing of the concrete took place (Lancaster, 2005). With the weight of the concrete, the wooden frames used to hold the formwork during curing would have slightly deflected and the brick lattices could help mitigate this problem and increase quality control.

Lightweight Concrete
Another construction technique the Roman engineers realized would enable larger and more stable domes is strategic variation of concrete weight in different areas of the dome by changing thickness and aggregate. Domes have complex stresses because of their double curvature and under a constant distributed gravity load over their entire surface they develop stress patterns as seen in Figures 6 and 7 below (Mehrotra et al, 2013). For our purposes the value of the numbers on the colored bar are not relevant but the sign and magnitude are. Negative numbers represent compressive stresses and positive numbers represent tensile stresses. It can be seen that the meridional stresses, which run vertically, are all compressive while the hoop stresses, which run horizontally, are a mix of compressive and tensile. What makes dome behavior difficult for engineering, especially with unreinforced concrete, is the band of tensile hoop stresses towards the bottom of the dome. In an unreinforced concrete dome these stresses are typically large enough to cause the concrete to crack into a series of arches connected by a common keystone and slump down as seen in Figure 8 below. The magnitude of the tensile hoop stresses is strongly dependent on the self weight of the dome and, therefore, it would be beneficial to decrease the weight of the dome where possible. The Romans recognized these characteristics of dome behavior and knew that towards the bottom of the dome the concrete needs to be thicker and use stronger aggregates in order to carry the weight of the top portion of the dome and to provide as much resistance as possible to the tensile hoop stresses. In contrast, towards the top of the dome there are much lower stresses and they are all compressive so the engineers knew the dome could be thinner and use lighter weight aggregates. The practice of varying the concrete weight began in the first century BC by using different lightweight aggregates such as tufo giallo della via Tiberina, Vesuvian scoria, and pumice which are all weaker than the heavier tuff and brick commonly used in Roman concrete (Lancaster, 2005). By the first century AD, this practice became standard in the larger domes that were being constructed and the thickness of the concrete also began to be varied according to the stress distribution (Lancaster, 2005).


 * = [[image:MeridionalStresses.png width="424" height="398" align="center"]] ||= [[image:HoopStresses.png width="427" height="404" align="center"]] ||
 * = Figure 6: Meridional stresses in a hemispherical dome due to constant distributed gravity loading (Mehrotra et al, 2013) ||= Figure 7: Hoop stresses in a hemispherical dome due to constant distributed gravity loading (Mehrotra et al, 2013) ||



Once the Romans had developed these three major advances, the technology to design and build large scale concrete domes was complete. However, these advances only resulted by constructing domes and making improvements over time and the earlier domes attest to this progression.

=Roman Dome Progression Case Studies=

The development of the Roman dome can be seen through unique aspects in their design of the structure as the civilization's history progressed. The Temple of Mercury at Baiae and Temple of Minerva Medica at Rome (see Figure 9 and 10 below for location) will be studied as each either resulted in or displayed major improvements that allowed Roman engineers to perfect their dome design.


 * = media type="custom" key="23831630" ||= media type="custom" key="23831588" ||
 * = Figure 9: Temple of Mercury at Baiae (Google Maps) ||= Figure 10: Temple of Minerva Medica at Rome (Google Maps) ||

Temple of Mercury at Baiae
The Temple of Mercury at the Roman resort of Baiae (seen in Figure 11 below) is the earliest surviving large scale concrete dome constructed by the Romans and is most likely one of the first. Archaeologists have dated the construction of the dome to the late Republic or early Imperial era and estimate that it must have been built before the first half of the first century AD because of the "packed tufa rubble" used in the concrete that is characteristic of the Augustan period (Adam, 1989/1994). The dome has a diameter of approximately 21.5 meters and was actually never a temple but was instead used for the city baths along with two other similarly sized concrete domes ("Baiae", n.d.).



#|Surveys of this one remaining dome have shown large variations in the building plan when compared with the ideal circular plan that a dome should have. There are six locations in which the footprint of the dome deviates drastically from the ideal perfectly circular plan and the maximum of these variations reaches a value of 22 centimeters (Lancaster, 2005). These variations do not occur at equal spacing around the perimeter of the plan and can be seen in Figure 12 below (Lancaster, 2005). Two individuals have expressed theories for the large variations in the dome footprint. F. Rakob has proposed that the centering was composed of eight equally spaced timber trusses that were supported with a central column and the discrepancies in footprint perimeter occurred because the centering was not well constructed or positioned (Lancaster, 2005). Alternatively, J. J. Rasch argues that the centering was composed of eight timber trusses that were accidentally unequally spaced and there was no center column support which resulted in the footprint error from sagging during construction (Lancaster, 2005). Both of the centering layouts proposed by Rakob and Rasch can be seen in Figure 12 below and the section view of Rakob's plan can be seen in Figure 13 below.


 * = [[image:MercuryBaiae.PNG width="456" height="410" align="center"]] ||= [[image:MercuryBaiae2.PNG width="541" height="411" align="center"]] ||
 * = Figure 12: Temple of Mercury plan variations and proposed Roman centering systems by Rakob and Rasch (Lancaster, 2005) ||= Figure 13: Rakob's proposed Roman centering for the Temple of Mercury at Baiae (Lancaster, 2005) ||

The difficulty in discovering the actual cause of the plan variation is the lack of formwork imprints left on the inside of the dome from construction and no written records of this structure's building technique (Lancaster, 2005). From the imprecision at the Temple of Mercury, which is uncharacteristic of Roman engineering, it can be seen that the stable centering and stability needed to construct a large scale concrete dome had not been mastered. With the precision evident in other Roman engineered shapes, we can imagine that the Roman's immediately began developing better methods to deal with this first challenge of large dome construction.

Temple of Minerva Medica at Rome
The second problem that the Romans encountered with large domes is the flexibility associated with such a small thickness to span ratio structure. This undoubtedly was an additional part of the problem with the Temple of Mercury at Baiae and is likely something the engineers realized after building that dome. The dome stiffening method of brick lattice ribs discussed in the Roman Dome Development section was employed in the fourth century AD dome for the Temple of Minerva Medica in patterns seen below in Figures 14 and 15 (Lancaster, 2005). In the three dimensional view of the Temple, it can be seen that the large lattice ribs land directly on portions of the supporting decagon that continue uninterrupted down to the ground. This seems logical for a load transferring system but there are other placements of lattice ribbing that don't appear to follow the same logic. The smaller lattice ribs in the plan view of the Temple actually land on portions of the supporting decagon that have openings below as seen in the three dimensional perspective. From a structural point of view for load resistance, this seems like the least logical place to support the weight of the dome. Instead, an alternative explanation is that the Romans did not use lattice ribbing only for carrying the loads of the dome but also to distribute loads more evenly throughout the dome and prevent stress concentrations (Lancaster, 2005). A basic rule for stress flow is that stresses tend to follow the stiffest path in a material. If the dome is stiffly supported only at certain areas around its perimeter, the stresses will flow away from the areas that are less supported (those supported by walls containing holes in the Temple of Minerva Medica) and be directed towards the stiffer areas which results in large stress variations throughout the dome. These variations could cause excessive cracking and the Roman engineers were likely preventing this by stiffening the dome at locations that would otherwise be flexible and, therefore, encouraging the stress distribution to be more uniform.


 * = [[image:MinervaMedicaPlan.jpg width="490" height="464" align="center"]] ||= [[image:MinervaMedica3D.jpg width="455" height="456" align="center"]] ||
 * = Figure 14: Plan of the Temple of Minerva Medica showing lattice ribbing (Lancaster, 2005) ||= Figure 15: Three dimensional perspective of the Temple of Minerva Medica showing lattice ribbing (Lancaster, 2005) ||

A second advancement that the Temple of Minerva Medica exhibits is the use of lighter weight concrete in the upper portion of the dome. In Figure 16 below, the gray area of the dome ruins represents the portion made from concrete with lightweight pumice aggregate (Lancaster, 2005). Any part of the dome above this would also have been constructed from the same material which, in theory, would decrease the outward thrusting forces on the decagonal walls and tensile hoop stresses in the base of the dome. Unfortunately, one of the drawbacks to using lightweight aggregates is that the concrete becomes more brittle and there is a strong possibility this contributed to the collapse of the dome over time despite the fact that a larger dome from the Roman era has survived.



=Pantheon - The Pinnacle=

Even though Roman dome engineering progressed during the third and fourth centuries AD, such as with the Temple of Minerva Medica discussed above, their perfect dome had already been constructed. The iconic Pantheon built in the second century AD is still the largest unreinforced concrete dome in the world at an approximate diameter of 44 meters and it was larger than any the Romans had previously constructed (Moore, 1995). Today it remains one of the most monumental feats of engineering in the world and analysis of the dome is still being done as the complexities of it are not fully understood. Dome construction advances from previous works were employed in the Pantheon and together they resulted in what is now viewed as the iconic Roman dome.

Pantheon Formwork
The massive size of the Pantheon has fueled discussion regarding the centering system used to construct it. Did it use a ground supported centering system or a completely self supporting system with all radial trusses framing into a central compression ring at the top? In order for the dome to be constructed without many column supports upholding individual formwork pieces, radial trusses would have to span a distance from the base of the dome curvature to the edge of the oculus. This angular distance is approximately 26 meters as seen in Figure 17 below and coincides with the centering span required to build the Basilica Ulpia which was also in Rome and constructed at roughly the same time as the Pantheon (Lancaster, 2005). The Romans strategically designed both buildings so that the same centering could be used which would save time and materials (Lancaster, 2005). The wood construction knowledge at the time would have allowed radial trusses of this dimension to be built so it would have been unnecessary to have many columns supporting small pieces of formwork from the ground (Lancaster, 2005). The next question is whether or not the centering had a large central tower or not. According to Vitruvius, towers reaching 53 meters high and having a base of 10.4 meters wide had been designed and built by Diades, an engineer for Alexander the Great, and these would have been more than adequate to support the Pantheon centering (Lancaster, 2005). So the technology to build a central column supported centering system was available to the Romans at the time of the Pantheon's construction (Lancaster, 2005). The alternative method of a completely self supporting centering system would also have already been developed and would have required a large central compression ring against which the radial trusses would rest as seen in Figure 18 below.



Discounting the system of many columns supporting formwork, of the centering systems proposed for the Pantheon, it is likely that the Romans used the easier and less complicated method, which would suggest the large central tower system. The use of a self supporting system would have been overly complex and required extra supports or cranes to hold the system in place until all the radial frames were positioned and equilibrium was achieved. Any of the systems discussed could have been used but with the construction of a large central tower also came the ability to easily reach the top of the centering by ladders or elevators for construction and material transport (Lancaster, 2005). It seems logical that for a dome as large as the Pantheon the Romans would have chosen the more stable and safe method that was still economical.

Concrete Weight
Because the dome of the Pantheon was twice as large as any earlier dome the Romans constructed, they were concerned with the outward thrusting forces it would exert on the rotunda walls supporting it and the hoop stresses in the lower portion of the dome (Lancaster, 2005). As discussed earlier in the Roman Dome Development section and the Temple of Minerva Medica case study, using lighter weight concrete towards the top of the dome is one of the best ways to address both of these concerns. For the Pantheon, the Romans chose to vary the concrete weight by decreasing the thickness of the dome as the height increases and to change the aggregate weight within the concrete three different times. The dome starts at a thickness of 5.9 meters thick at the bottom and decreases to a thickness of 1.5 meters at the top (Moore, 1995). The aggregate in the concrete changes from broken bricks at the base of the dome to bricks and tuff in a small midsection and ends with tufo giallo and scoria for the top portion as seen in Figure 19 below (Lancaster, 2005). It is likely that the Romans also chose the placement of the aggregates based on the presence of tensile hoop stresses. Towards the base of the dome where the tensile stresses are higher the use of tufo giallo and scoria would have been a poor choice because of the brittle nature of these materials and their higher susceptibility to cracking.

**Coffers** With a dome as large as the Pantheon the need for stiffening is imperative to keep the shell in place during curing and to provide a strong system to carry the load down to the supporting rotunda walls. The use of brick lattice stiffening was not developed by the time the Pantheon was constructed so the engineers used a system of internal coffered ribs as seen in Figure 20 below (Lancaster, 2005). These coffers also provided interior visual texture as described in the Dome Stiffening portion of the Roman Dome Development section above. An additional benefit of coffers that can be particularly valuable in large domes is the weight saved by removing unnecessary concrete. The collective concrete removed from the 140 coffers inside the Pantheon is quite substantial and undoubtedly had a significant reduction on the overall weight of the dome. Near the top the Romans did not remove any concrete to create ribs which was likely a precaution against the thinner concrete and more brittle lightweight aggregates crushing. The use of coffers in any structure, and particularly one with 140, increases the difficulty of formwork construction. The formwork for the coffers cannot be connected to the general formwork for the dome because the walls of the indentations are perpendicular to the face of the dome. As the dome curvature changes, the angle of the coffer walls also change and if they were built into the formwork the entire system would be locked into the concrete and unable to be removed because removal of each coffer's formwork must occur at a different angle (Lancaster, 2005). For this reason, the engineers must have constructed individual formwork pieces for the coffers and they must have been temporarily connected to the Pantheon's formwork for construction. Despite this difficulty, the inclusion of strengthening ribs, visual texture, and weight reduction were deemed a priority and the Romans constructed the Pantheon with the interior that we see today.



Thrusting and Tensile Hoop Stress Mitigation
The thrusting forces and tensile hoop stresses were perhaps the greatest concern for the engineers of the Pantheon and rightly so. It is the primary reason they used lighter weight concrete and decreased the thickness of the dome towards the top and a strong reason they formed the 140 coffers on the interior. Despite these efforts, the engineers chose to employ one last method of reducing these forces and stresses in the form of the exterior step rings near the base of the Pantheon dome as seen in Figure 21 below. There are seven of these step rings and they are not of uniform dimensions or spacing (Moore, 1995). The first of the rings is by far the largest while the next six are smaller and approximately one size (Moore, 1995). The rings have been the subject of much engineering debate and there have even been proposals that there are iron bands inside each ring that act to reduce the thrusting forces of the dome and prevent spreading. However, no evidence of such material has been discovered and, indeed, cracks in the dome from tensile hoop stresses would suggest that the rings are purely concrete. This does not negate the design of these steps though. The Romans knew that a dome as large as the Pantheon would have massive thrusting forces even with the decreased dome weight. It is for this reason that they built the exterior of the supporting rotunda walls up above the base of the dome curvature as seen in Figure 19 above. By constructing the building like this, they were able to create a platform on which the step rings could rest. The Romans also placed the step rings at the ideal location on the dome to resist tensile hoop stresses as the stress changes from compression to tension at approximately 51 degrees from the apex as in Figure 22 below (Design of Spherical Shells). The weight of the step rings was designed to resist the thrusting forces of the massive dome by pushing down on the haunch of the dome and acting against the dome spreading. The Pantheon has developed cracks in the lower portion of the dome but ultimately the step rings have achieved their design intention and prevented the dome from pushing out the rotunda walls.


 * [[image:PantheonStepRings.PNG align="center"]] || [[image:DomeStressProfile.PNG width="437" height="349" align="center"]] ||
 * Figure 21: Section of the Pantheon dome showing step rings (Achwal et al., 2006) || Figure 22: Hemispherical dome hoop stress profile due to self weight ("Design of Spherical Shells," n.d.) ||

Buttressing
Not all of the cracking in the Pantheon walls can be attributed to the thrusting of the dome though. The location of the Pantheon as seen in Figure 23 below is on top of soft clay that the Tiber river deposited when it would frequently overflow into the city of ancient Rome (Jones, n.d.). This clay had differential settling when a building as large and heavy as the Pantheon was placed on it and this resulted in cracking throughout the structure. The Romans chose to build the large buttressing structure on the back of the Pantheon as seen in Figure 24 below to restrain the rotunda and prevent complete collapse from further soil settlement (Jones, n.d.). Studies by M.W. Jones have concluded that construction on this buttress actually began before the rotunda had even been finished. Towards the bottom of the rotunda, the two structures are completely separate with a cold construction joint between them, but towards the top of the Pantheon there are several places where the buttress is bonded into the rotunda (Jones, n.d.). This leads to the conclusion that the rotunda must have started cracking very soon after construction began, sparking the construction of the buttress which then caught up and the two were bonded together in the upper third of the rotunda to create one unit (Jones, n.d.). Undoubtedly, the massive buttress does help resist the thrusting forces from the dome but it appears that was not its primary purpose.


 * = media type="custom" key="23831656" ||
 * = Figure 23: Location of the Pantheon on Tiber flood plain (Google Maps) ||



Thrust Line Analysis
One of the most well known methods of analyzing arches without software or mathematical methods is known as thrust line analysis. It involves a scaled graphical drawing from which the magnitude and line of action for the forces may be determined. If this scaled line of action fits inside an equal scaled structural system envelope by a certain criteria then the structure is considered stable. Lancaster (2005) used this method to test the efficacy of the Pantheon dome by performing several thrust line analyses for different configurations of the structure. First, she tested the actual configuration (P1), then lightweight concrete used at the dome haunch (P3), then no lightweight concrete used at the dome crown (P4), and finally the removal of the step rings (P5) (Lancaster, 2005). The results of each configuration was compared using a criteria know as the Rankine Factor. The Rankine Factor uses the geometry of the thrust line and abutment to calculate a ratio as seen in Figure 25 below to determine if the structural system is adequate (Lancaster, 2005). If the factor is less than or equal to one then the structure is predicted to collapse and if it is greater than or equal to three then it is considered stable and safe (Lancaster, 2005). Any resulting factor between one and three characterizes a structure that is unconservatively designed. A summary of the thrust lines by Lancaster's (2005) analysis for each configuration of the Pantheon is shown in Figure 26 below. The analysis configuration consisted of a forty five degree wedge of the dome as seen in Figure 26 below because there are eight piers around the perimeter of the Pantheon each supporting such a portion and Lancaster (2005) assumed each pier had the capacity to carry such a load. Tensile hoop stresses were ignored because this accurately replicates the effect of the cracks in the lower portion of the Pantheon dome (Lancaster, 2005). From the analysis, the actual dome configuration (P1) returned a Rankine factor of 4.23 indicating that the Romans indeed conservatively designed a structure that was stable (Lancaster, 2005). For the configuration with lightweight concrete at the dome haunch (P3) a Rankine factor of 4.10 resulted while without lightweight concrete at the crown (P4) a Rankine factor of 3.60 resulted (Lancaster, 2005). Both of these results have interesting implications but by far the greatest effect was seen by removing the step rings around the base of the dome (P5) which resulted in a Rankine Factor of 2.60 (Lancaster, 2005). These results imply that the Romans really were engineering experts. By placing the lightweight concrete at the level they did, they found the balance between too much and too little as seen by the increased thrust of the P3 and P4 analyses. Their inclusion of the step rings is the ultimate verification that they understood the complexities of dome design though because without such a configuration the dome thrusts would have increased by nearly forty percent and resulted in an unsafe design.





=Roman Dome Influence and Impact=

Prior to the Romans no civilization had mastered the concepts of engineering necessary to construct a concrete dome spanning nearly 44 meters that would last for over 2000 years. The design was so advanced that the Pantheon became the inspiration for many large domes of the Renaissance over a millennium later, such as the dome of Santa Maria del Fiore in Florence and St. Peter's Basilica in Vatican City (Parsons, 1939). Interestingly, although both of these Renaissance domes have a greater rise, neither of them spans as far as the Pantheon which is a testament to the masterful engineering of the structure (Parsons, 1939).

St. Peter's Basilica Dome Case Study
Near the Pantheon, St. Peter's Basilica rises above the rest of the skyline and can be seen from nearly everywhere in Rome. This monumental structure constructed during the Renaissance took over 150 years to build due to changes in designers and setbacks. The prominent dome as seen in Figures 27 and 28 below is perhaps the most important part of the building and one which drew inspiration from the nearby Roman masterpiece just to the East (see Figure 29 below for location).


 * [[image:_MG_6547-20130830-DigitalPicasa.jpg width="513" height="354" align="center"]] || [[image:_MG_6807.JPG width="518" height="349" align="center"]] ||
 * Figure 27: The dome of St. Peter's Basilica at Vatican City through the Aventine peephole (photo by author) || Figure 28: The dome of St. Peter's Basilica at Vatican City (photo by author) ||


 * = media type="custom" key="23834102" ||
 * = Figure 29: Location of St. Peter's Basilica at Vatican City (Google Maps) ||

Pantheon's Influence
The original architect for the construction of St. Peter's Basilica in the fifteenth century was a man named Rossellino but only the foundations for the new church were placed before construction halted due to the Pope's death (Parsons, 1939). The next architect appointed to the project by Pope Julius II was Bramante in 1503 who developed the Greek cross plan for the church along with a magnificent dome (Parsons, 1939). This design was where the Pantheon influence was first seen through the hemispherical shape and single shell design as seen in Figure 30 below. Further, Bramante included seven step rings around the base of the dome that resulted in a nearly identical structure as the Roman design. Unfortunately, the architect died before the project could be carried out and the only progress made over the next few decades was purely design. During this phase, the architect Sangallo was appointed by Pope Leo X to the project and he made several significant changes to the church. The new design he proposed changed the shape to a more pointed profile and eliminated the step rings while adjusting the layout of the columns supporting the dome to achieve better buttressing against the thrusting forces but the single shell influence was retained (Parsons, 1939). Pope Leo X died before construction could start and again the project was put on hold. Finally, during the papacy of Pope Paulus III, and without any official design having been adopted, Michelangelo was appointed the architect for St. Peter's Basilica in 1546 with complete control over every aspect of the plan and construction (Parson, 1939). With him, much of the Pantheon's obvious influence was removed in that he chose to use an ovoid profile, dual skinned dome with a cupola at the peak (Lees-Milne, 1967). The Pantheon's main influence on the Michelangelo design is more in the changes that he made to the dome design of the Romans. Michelangelo recognized the thrust of a hemispherical dome is larger than that of an ovoid shaped dome and also that the single skin technique required much more material to achieve a dome as stiff as one using two skins. This latter realization came from studying Brunelleschi's dome at Santa Maria del Fiore.



**Double Skin** The design of a two skin dome relies on an inner dome connected to an outer dome through ribs. By using such a system, the stiffness of the dome is greatly increased in comparison to a single skin dome and allows for each of the two skins to be relatively thin in comparison to what would be needed for a single skin (Parsons, 1939). This system was so successful in Brunelleschi's dome at Santa Maria del Fiore in Florence that Michelangelo chose to use the same at the dome of St. Peter's Basilica. Michelangelo was originally from Florence and as such he had seen Brunelleschi's dome so when designing St. Peter's Basilica he sent many letters inquiring for dimensions of the two skin system (Lees-Milne, 1967). The visually pleasing aspects of a hemispherical dome were undeniable though and, as a result, Michelangelo chose to make the inner dome at St. Peter's a hemisphere and connecting it to the ovoid shaped outer dome as seen in Figure 31 below (Lees-Milne, 1967). This system still greatly reduced the thrusting forces in comparison to a completely single skin hemispherical dome which would have been much heavier.



Tension Rings
Despite the efforts of Michelangelo to resist thrusting forces by using a two skin system on a circular, two column thick buttress support (see Figure 31 above) the dome at St. Peter's Basilica began to show cracks in the eighteenth century and had to be reinforced with iron chains (Norwich, 1975). The use of these chains essentially act as tension rings to resist the spreading of the dome in much the same way that modern tension rings are used on domes. The chains are wrapped around the lower portion of the dome where the cracking occurs and, in this manner, the tensile stresses are taken off the masonry of the dome once the chains are engaged by a small amount of further spreading. This technique has prevented further cracking of the dome and ensured its stability.

=Conclusion=

Over approximately a century, the Romans took the dome from poor construction and quality control as seen in the Temple of Mercury to a visually refined and brilliantly engineered system as seen in the Pantheon. Over this time period the development of accurate centering and formwork, stiffening ribs and lattices, and concrete weight variation all reached a level that enabled the dome to be built on a large scale at numerous baths, halls and temples. All of this was accomplished without many of the modern technological advances that are necessary for building design and construction today. The engineers used intuition and judgement to advance their techniques with each dome that was constructed and by building on previous knowledge. The results provided the basis for constructing a dome that has lasted nearly two millennia and influenced all dome design since.

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