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Page 1The History and Future of Domes

The History and Future of Domes

Course Description

This course reviews the essential principles in designing a dome. It uses the Romans’ Pantheon Dome to in-troduce the structural elements of domes: meridians, parallels and materials such as concrete. It outlines how modern domes are engineered and what loads must be considered in the design, emphasizing how to calculate the loads on a dome from wind using the ASCE Standard, and how to size I-beams for a modern dome. It also shares the stories behind the building of Istanbul’s Hagia Sophia – a cathedral built 14 centuries ago that can still astonish a modern-day engineer – as well as one of the most revolutionary domes in the history of building: The Duomo (Dome) of the Basilica di Santa Maria del Fiore in Florence. The double-masonry dome and unique structure allowed its designer to span a greater distance than the Pantheon.

Chapters

• Chapter One: History of Dome Engineering• Chapter Two: Calculation of Wind Load• Chapter Three: Sizing an I-Beam Used in a Dome

Learning Objectives

Upon completion of this course, the participant will be able to:• Explain the basic principles of dome engineering• Identify the structural elements of domes• Describe the considerations for determining dome load, such as wind and snow• Use the ASCE Code to calculate the load from wind on a dome• Understand the technical aspect of Brunelleschi’s Dome in Florence,

including the unique structure that allows it to span a great distance• Explain how to size I-beams used in a modern triangulated dome


Chapter One: History of Dome Engineering

Overview

• Introduction • The Pantheon

» Brief History » Structural Elements of a Dome

• Why a Dome “Works”• The Pantheon Dome• Concrete

» How the Romans Poured Concrete to Make the Pantheon

• Modern Domes• Snow Loading

Learning Objectives

• Explain the basic principles of dome engineering• Identify the structural elements of domes• Explain how a dome structure works• Describe considerations for determining dome

load, such as wind and snow

Introduction

In this course, the huge impact engineers have had in creating spaces where the human spirit can soar will be examined, where it can celebrate its relationship to the mysteries of the universe, experience the adrenaline of a sporting event, or even just feel majesty. This struc-ture is something that is taken for granted: the dome! Think about it for a moment: Domes cap the world’s greatest churches in the world and crown sporting are-nas around the globe, places where twenty or forty or even eighty thousand people commune to rejoice, to carouse, to consecrate, to jubilate... to engage in all the activities that make us human.

In this course you will review how to estimate the loads from wind on a dome, and how to size the I-beams for a modern dome made of triangular aluminum plate. These calculations are necessary to construct a safe building. That role for engineers – great protector of the human population – is often underrated by the pub-lic because it is done so well. Just imagine the havoc if sporting arenas or cathedrals routinely collapsed.

Take this journey back through time and also around the world to examine three marvelous achievements of engineers: The Romans’ great Pantheon; the sixth century’s magnificent Hagia Sophia; and Florence’s amazing masterpiece, Bruncelleschi’s Duomo, the Dome of the Cathedral of Florence. The architect Ma-rio Salvadori astutely said, “Perhaps the dome is the nearest materialization of heaven, the only man-made representation of the sky, and this is why a dome seems to protect us like the sky of a clear night, embracing us and our smallness and solitude.” Learn about the mysteries of how these magnificent engineers built structures that have remained standing for centuries.

The Pantheon

The Romans built the Pantheon as a temple to all the gods of Ancient Rome, and then rebuilt it in 126 AD. Its exact use is still debated, and no one knows who designed it, but there is no debate that it is capped with the most majestic and oldest large-scale dome in Rome. It has been in continuous use throughout its history; in fact, one of the reasons it is so well preserved is that it became a Roman Catholic Church in the 7th Century.


Michelangelo found it so beautiful he called it “an-gelic” and declared it “not of human design.” In the Middle Ages most people believed the Pantheon’s dome stayed put because of sinister forces of demons because it did not have the dense row of arches as did conventional cathedrals. But, of course, it is human-built, a unique structure created by engineers.

The Pantheon’s dome provides us with a beautiful way to study how building materials must be used to create an elegant, but safe, structure. Here are some details to see why it stays standing.

Brief HistoryThe current building dates from about 126 AD, during the reign of the Emperor Hadrian. You can actually see “date-stamps” on its bricks. Hadrian was a cos-mopolitan emperor who traveled widely in the East, where Greek culture strong influenced him. It is tell-ing that he used a Greek name for the structure. Pan-theon comes from the Greek pan for “all” and theon for “gods”) rather than a Latin name more typical of Rome. With its hemispherical dome and orderly divi-sion of the interior walls into different levels, the Pan-theon represents an architectural embodiment of the Greek idea of cosmos.

Structural Elements of a DomeThe building is circular with a portico of three ranks of huge granite Corinthian columns (eight in the first rank and two groups of four behind) under a pediment opening into the rotunda, which is capped with a cof-fered, concrete dome, itself topped by a central open-ing – the oculus or Great Eye – open to the sky. The columns of the portico work as “foils” to break up the huge expanse of the rotunda. Thus when the doors are opened, the vast, round, apparently unsupported space

startles a first time viewer. The height to the oculus and the diameter of the interior circle are the same: 142 feet. Since the “Great Eye” at the dome’s apex is the source of all light in the interior, it draws the viewer’s eye toward it like a moth toward a flame. The interior is seamless: no straight lines, axial vistas, flat walls, or angles to give an observer a solid focal point. The dome simply dominates. Yet, many of the features serve more than an aesthetic function. The oculus, for example, also serves as a cooling and ventilation method. During storms, a drainage system below the floor handles the rain that falls through the oculus.


Why a Dome “Works”

Think for the moment of the dome as a perfect half-sphere, the thickness of which is very small relative to its span. It isn’t exactly, but it is close enough. This dome must carry its own weight and the weight of any load on its exterior surface; for example, wind, rain or snow. The dome is made up of meridians, curved ver-tical lines that remind one of an arch. The meridians meet at the top and have a common keystone.

Source: Why Buildings Stand Up: The Strength of Archi-tecture by Mario Salvadori (Norton 1980)

The loads on these meridians accumulate from top to bottom; the members become more and more com-pressed as they approach the dome’s support at its base. If you are familiar with cathedrals, you will

recall that their arches have what are called “flying buttresses.” These exist because the arch thrust out-ward in addition to downward. By contrast, a dome is a monolithic structure where all the “arches” are tied together, thus eliminating the need for buttresses. The linked meridians create a structure that works like the hoops that hold the staves in a barrel.

When the load of the dome presses down, it tends to expand at the bottom and flatten at the top.


This allows the dome to be made thinner than one might expect because, first, the compression makes the dome much stiffer; and, second, it keeps the bottom from opening up. In this drawing the degree is exaggerate to which this deformation takes place: A 100 foot rein-forced concrete dome that is two or three inches thick will deflect by less than one-tenth of an inch.


Before looking at the Pantheon in detail, a crucial dif-ference between an arch and a dome should be noted. An arch supports its load by compression in its ele-ments, but also by bending if necessary. A dome does not bend because it is monolithic; this means that a dome can carry any kind of load without changing shape and without developing bending stresses – un-til, of course, it fails catastrophically. This means that domes, even though thin, are stiffer and stronger than


any structure created by humankind and, as seen with the Pantheon, among the most beautiful.

The Pantheon Dome

The Pantheon holds the record for the largest unre-inforced concrete dome. The 5,000 ton weight of the concrete dome is concentrated on a 30-foot-diameter ring of voussoirs (the wedge-shaped pieces that form the vault) which forms the oculus, while the downward thrust of the dome is carried by eight barrel vaults in the 21-foot-thick drum wall; these are divided into eight piers. The thickness of the dome varies from 21 feet at the base of the dome to 4 feet around the oculus. The height to the oculus and the diameter of the inte-rior circle are both 142 ft, so the whole interior would fit exactly within a cube.

Concrete

The Romans build the Pantheon after they discovered how to make pozzolana concrete. This type of con-crete – also known as pozzolanic ash – is a fine, sandy volcanic ash, originally discovered and mined in Italy around the volcano Vesuvius. The origin of the word “concrete” gives away its ancientness. It is made by combining the Latin prefix con, meaning “together,” and crescere, meaning “to grow.” The name comes about because when the ingredients making up con-crete – water, gravel, sand, and a bit of cement – are mixed, they turn into a hard, rigid solid. The Romans discovered concrete by accident. A builder who was making some mortar happened to be working near Mount Vesuvius. He tossed in some volcanic ash and noticed that when his mixture dried it made a very hard substance. From this serendipitous beginning, the Romans fine-tuned the recipe for concrete. This was in contrast to simple lime mortar, which is made by mixing water with quicklime and sand; it sets when the water has been evaporated into the atmosphere or absorbed into surrounding masonry. Roman concrete sets by combining chemically with water; the cement doesn’t need to dry out like lime mortar. In fact, poz-zolana concrete is called “hydraulic” because it can even set when immersed in water.

A simple acid-base reaction between calcium hydrox-ide, also known as Portlandite, or (Ca(OH)2), and si-licic acid (H4SiO4, or Si(OH)4) forms a calcium sili-cate hydrate (CaH2SiO4 2 H2O). This reaction is:

Ca(OH)2 + H4SiO4 → Ca2+ + + H2SiO4

2-+ 2 H2O → → CaH 2 SiO · 2 H2O

In addition to this silicate, other substances can be add-ed to increase the strength of the concrete. For exam-ple, the Romans mixed horsehair to reduce shrinkage during hardening, and they also added blood, which made the concrete frost-resistant. Today plastics are used instead of horsehair and industrial chemicals in-stead of blood, but the same principles apply. With these innovations, Roman concrete reached a level of quality unmatched until the 20th century.

Today’s structural concrete improves on Roman con-crete for two main reasons: 1. Its mix consistency is more fluid and homoge-

neous, partly because of carefully prepared ag-gregate instead of the nearly random rubble used by the Romans. This means it can be poured into forms, thus requiring less hand layering; and

2. Reinforcing steel gives modern concrete great strength in tension.

Even though concrete is now high-tech – with smart concrete that can conduct electrical signals, for exam-ple – it is a testament to the ingenuity of engineers that, after 2,000 years, the concrete dome of the Pantheon is still standing. This concrete was both easier to work


with than lime mortar, and far superior in its compres-sive strength. Although there are no tensile test results are available for the Pantheon’s concrete, other Roman ruins of the same era give us a good guess. Roman ruins in Libya show a compressive strength of 2.8 ksi (20 MPa) and a tensile strength of 213 psi (1.47 MPa).

How the Romans Poured Concrete to Make the PantheonThe heroes in the pouring of the mighty concrete dome of the Pantheon are carpenters. Master carpen-ters skillfully build the wooden molds that shaped the concrete.

Carpenters built a complicated system of supporting timbers, propped on top of the lower walls and sturdy enough to hold the weight of the concrete. The inside of the dome was lined with coffers – sunken rectangu-lar sections – so intricate they had to be built by master carpenters. More than decorative, these coffers helped reduce the weight of the dome. They remove only five percent of the weight, but on a 5,000-ton dome, that’s a significant 250 tons. All work on the dome was dan-gerous, but perhaps the most dangerous was the job of those men who stood on the scaffolding and poured concrete layer by layer, as the dome grew higher and higher.

The dome is made of tapering courses or steps that are thickest at the base and thinnest at the oculus. The Ro-mans used the heaviest aggregate, mostly basalt, at the bottom and lighter materials such as pumice at the top. They embedded empty clay jugs into the dome’s upper courses to further lighten the structure and make the concrete cure. In fact, the Roman engineers overdid it and created too massive a base on the dome; later, they had to scrape out great chambers to reduce the dome’s weight.

At the very top they left an opening, the oculus, to avoid pouring concrete on a horizontal scaffold set high at the crown. They rimmed this opening with hard-burnt bricks, cemented by mortar, and then cir-cled it with a bronze cornice. The area encircled by the oculus is about four percent of the surface of the floor below. One visitor to the Pantheon notes, “There is no railing around either the oculus or the flashing. To lie upon the flashing and look down through the oculus to the paving forty-five meters (about 150 feet) below is

an experience not readily forgotten.” But the oculus is more than this; it functions as a compression ring or sort of a keystone of all the “arches” in the dome.

Modern Domes

Like the Romans, there are several domed structures built to celebrate our culture. Houston’s once mighty Astrodome showcased Muhammad Ali fights and El-vis Presley concerts. It played a part in our nation’s politics when it housed Hubert Humphrey’s 1968 presidential campaign rally and George H.W. Bush’s 1992 nominating convention, and it presented spec-tacles such as Evel Knievel’s motorcycle jumps and Billie Jean King’s defeat of Bobby Riggs in the 1973 tennis match billed as a “Battle of the Sexes.”

Unlike the Romans, there are other building options for creating domed structures. Perhaps the most amaz-ing of modern times is the Skydome – now renamed the Rogers Centre – in Toronto.

Opened in 1989, the Skydome houses the Toronto Blue Jays. While it is mainly a sporting arena, it also hosts conventions, trade fairs, concerts, fun fairs, and – my favorite – monster truck shows. The Skydome is the first stadium to have a fully retractable motorized roof.


This roof is an amazing engineering achievement. Al-though it covers eight acres and weighs 6,500 tons, it can be opened in twenty minutes, even in forty-mile-per-hour winds. But perhaps more amazing than even these statistics is its greatest achievement: It has done this safely thousands of times since its debut in 1989.

Today domes are not only for spectacular purpose, but also for such common uses as roofs of libraries, mosques, and synagogues, as well as water, sewage, and chemical tanks. Their safe use comes, of course, from the incredible body of knowledge about building materials developed over many millennia.

For example, in designing a modern dome, the engi-neer must consider loads from wind. In tall buildings, wind pressures require a structure that resists the wind load separate from the one that resists the gravity load. In some tall buildings, up to ten percent of the struc-tural weight, and thus ten percent of the cost, goes to wind bracing. The first question to be answered about a tall building’s structure is, “What would be the stron-gest wind ever at the building site?” For large projects such as the Toronto Skydome, an engineer would use a “100-year wind”, which is defined as a wind speed that has one change in one hundred years of occurring.

Engineers can use the ASCE Standard 7-05 to esti-mate the design pressures for wind loading of a dome. Engineering standards are one of the greatest triumphs of human ingenuity, saving countless lives. To design a dome safely, an engineer needs to find the maximum pressure exerted by the wind; that is, the wind in the worst possible conditions. We will learn how to do calculate loads in the next chapter. It isn’t always wind that can overload a dome; unlike the Romans, in North America we need to worry about snow.

Snow Loading

In the early morning hours of January 1978, the huge domed roof of the C.W. Post Center at Long Island University collapsed. “It looks like a giant cracked eggshell”, said a police officer as he viewed the wreck-age. Indeed, the 300-foot-diameter roof had shattered into thousands of fragments. Unlike the monolithic concrete dome of the Pantheon, this dome was reticu-lated: A space frame of hundreds of triangular trusses made of galvanized steel, held together by tension suppression rings, with a compression ring at the top like the Pantheon’s oculus, and a tension ring at the base of the dome.

A post-mortem on the building concluded it had been designed using a simplified theory that didn’t apply to reticulated domes. The principal error was in assum-ing that the dome would have uniform dead and live loads, or the weight of the dome itself and the load from wind or snow, respectively. Instead, the wind blew snow only on one side of the dome, thus stress-ing the structure unevenly. In fact, this uneven load-ing was only one-fourth of the load required by code if the load were applied evenly. ASCE Standard 7-05 now in effect requires engineers to estimate loads for wind and snow that are both balanced and unbalanced. Since the collapse occurred during the winter break at the university the campus was deserted, no one was even near the dome. Nevertheless, this highlights the importance of the ASCE Standards and their role in preserving human life.

The Roman engineers, without the benefit of comput-ing abilities and rigorous theory of structure, simply overbuilt the great dome of the Pantheon. So it is no surprise that the span of the Pantheon was unsurpassed


for 1,300 years, until the octagonal dome of Santa Ma-ria del Fiore (The Cathedral of Florence) exceeded it by three feet in maximum span. In the final chapter of the course, the amazing engineering achievement will be examined in detail.

Summary

We began this chapter by discussing the Pantheon Dome, which is the oldest large-scale dome in Rome, and looked at the structural elements of domes, which include meridians and parallels. We learned that the loads on meridians accumulate from top to bottom. In addition to meridians and parallels, concrete was used to hold everything together. Finally, we received a pre-view of how modern domes are engineered and what loads must be considered in the design.


Chapter Two: Calculation of Wind Load

Overview

• Hagia Sophia• Calculating the Loading from Wind on a Dome• General Method• Building Description• Velocity Pressure• Dome Roof Pressure

Learning Objectives

• Summarize how the dome of the Great Cathedral Hagia Sophia is supported

• Use the ASCE Code to calculate the load from wind on a dome

• Identify how wind affects three components of a building

Hagia Sophia

Hagia Sophia – from the Greek for “Holy Wisdom” – built 14 centuries ago, can still astonish a modern day engineer. No engineer today would fail to marvel at this magnificent structure, which was built in less than six years. And, unlike the Gothic Cathedrals of the Middle Ages, it did not result from a careful step-by-step evo-lution, but instead appeared as if from nowhere. Noth-ing comparable had been built for 200 years.

Justinian I, the fourth Christian Emperor of Rome, egged on by his wife, a former theater performer and courtesan, forcibly put down a tax revolt by the people of Constantinople. This “Nika Revolt” (citizens cried “Nika!” (“Conquer!”) as they took over the city) last-ed for a week in A.D. 532. Nearly half the city was burnt and 30,000 people were killed. The protesting populace had burned down the church Hagia Sophia. Justinian, outraged at this act, decided to rebuild the church on a scale never seen before.

The Emperor amassed material from all over the Ro-man Empire. As the English historian Edward Gibbon noted in his Decline and Fall of the Roman Empire:

“The memory of past calamities inspired Justin-ian with a wise resolution, that no wood, except for the doors, should be admitted into the new edi-fice; and the choice of the materials was applied to the strength, the lightness, or the splendor of the respective parts.”


Justinian brought in Hellenistic columns from the temple of Artemis; beautiful red-purple porphyry from Egypt; green marble from Thessaly in Greece, black stone from the Bosporus region; and yellow stone from Syria.

Work on the structure commenced on February 23, 532. Ten thousand workers were split into two rival groups: One worked on the southern half the building, the other on the northern. The competition was de-signed to spur them to make tremendously fast prog-ress. As Gibbon notes, Justinian took an avid interest in the construction:

“The emperor himself, clad in a linen tunic, sur-veyed each day their rapid progress, and encour-aged their diligence by his familiarity, his zeal, and his rewards.”

The completed cathedral, Hagia Sophia, was conse-crated by the Emperor in an amazing five years, eleven months, and ten days from the first foundation. Gib-bon poked fun at the final structure, but also praised precisely the aspect that still stuns a modern engineer:

“The dome, illuminated by four-and-twenty win-dows, is formed with so small a curve, that the depth is equal only to one sixth of its diameter; the measure of that diameter is one hundred and fifteen feet, and the lofty center, where a crescent has supplanted the cross, rises to the perpendicu-lar height of one hundred and eighty feet above the pavement. The circle which encompasses the dome lightly reposes on four strong arches, and their weight is firmly supported by four massy piles, whose strength is assisted, on the northern and southern sides, by four columns of Egyptian granite.”

Although Gibbon was not a structural engineer, his careful eyed zeroed in on the main structural elements.

At no time in history had a structure so completely broken with the past. Its composition of space, sur-face, and light were like no other structure before it. The building’s interior does not impart a feeling of structural mass, but instead one of an enfolding sur-face. This effect has a specific purpose, because any building with a social or societal function conveys a message. Pope Nicholas V outlined why the Roman Catholic Church built magnificent structures:

“To sustain the faith of the unlettered masses there must be something that appeals to the eye or it will wither away. Bu if authority were visibly displayed in magnificent building, imperishable monuments and everlasting witnesses seemingly built by God himself, the belief implanted by the doctrine would be confirmed and strengthened.”

Hagia Sophia still stunned Byzantine Emperor John VI Kantakouzenos 800 years after its construction: “In size the greatest of all Churches under the sun, sur-passing in beauty and magnitude all others whereso-ever they may be.” Indeed, until the Dome built by Brunelleschi, described in the next chapter, no dome even came close to the dome of Hagia Sophia.

The key engineering problem in building Hagia So-phia was how to support the dome. Four main arches make up the sides of a square.


The dome of Hagia Sophia sits only on the top of the arches that make up the four walls of the Church. This alone would not support the dome sufficiently; it would crack under its own weight. Source: http://videos.howstuffworks.com/hsw/13391-discover-magazine-engineering-secrets-video.htm

Each arch is a half-circle whose rise equals half its span. Hagia Sophia’s great round dome must sit on this square; in fact, it only touches at the very top of the rounded arches, essentially at a point on the top of each rounded arch. This would not support the dome well enough; it would crack under its own weight. Its designers used a most ingenious and original method to overcome this flaw. Between the arches, its builders placed pendentives.

To “fill in” the spaces between the arches, Hagia Sophia’s designers used pendentives, triangular segments of a sphere. Source: http://videos.how-stuffworks.com/hsw/13391-discover-magazine-engi-neering-secrets-video.htm

Pendentives are triangular segments of a sphere, which taper to points at the bottom and spread at the top. They fill in between the arches to create a continuous circular structure for the dome. In other words, they

receive the weight of the dome, concentrating it at the four corners where it can be received by the piers beneath. This was revolutionary. Prior to the penden-tive’s development, builders used corbeling – a piece of stone jutting out of the wall to carry weight – or col-umns inside the dome’s rotunda. Without pendentives, Hagia Sophia would not have its majestic interior that provides the awesomeness of an immense open space unobstructed by columns.

Calculating the Loading from Wind on a Dome

Let’s imagine for a moment that we are building a replica of Hagia Sophia in the United States. A key design consideration would be the load caused by wind and snow on the structure. Each dome has, of course, a dead load from its own material, but we need to consider the extra load created by wind and snow. In this example, I will focus on how to use the ASCE Standard 7-05 Minimum Design Loads for Buildings and Other Structures. While this was the most current standard when I performed the calculations below, be sure to check the ASCE Website to find the most up-to-date standards.

To estimate the wind load so that a structure can be safely designed, we need to assess the impact of wind on three aspects of the structure: 1. The first is the impact on the main wind force-

resisting system (MWFRS); 2. The second is that on the components and clad-

ding (C&C); and 3. The third is the loads directly on the walls.

The first is, of course, the overall structure receiving wind loading from more than one face. The second re-ceives wind loads directly and generally transfers the load to other components or to the main system. Gen-erally the C&C design pressures will be higher than the MWFRS because localized pressures act over small areas; also, the MWFRS receives pressure from sever-al surfaces. So, with spatial averaging and correlation, the pressures are likely to be smaller than for C&C. In general, an engineer calculates all these loads and then uses them to design the various components. For in-stance, in the next chapter I show an example of how to size the I-beams used to support a triangulated dome.


General Method

To calculate MWFRS and C&C, we need to know some details of the building beyond its dimensions: its location and use are key in applying the ASCE Stan-dard. For small structures there are simple ways to es-timate the wind load, but typically a domed structure is large enough that we need to use the analytical pro-cedure described in the Standard.

In brief, the procedure is: 1. Calculate the velocity pressure caused by the

wind; then2. Use the appropriate gust effect factors and force

coefficients to estimate the loads felt by the MW-FRS and then the C&C.

In this example, calculate only the wind load on the MWFRS holding up Sophia Hagia; the standard can be used to calculate C&C and the load on the walls in a similar way.

Building Description

In order to apply the standard, it is necessary to know its use and location in addition to building dimensions. So, let’s say you’re building a replica of Hagia Sophia in Marion County, Florida, but using modern tech-niques of steel construction.

Location: Marion County, Florida

Topography: Homogeneous

Terrain: Open

Dimensions: • 102 feet diameter for the dome• 156 feet height to eave (the spring line of the dome)• 186 feet distance from the ground to the top of the

dome

Framing: Steel frame dome roof, metal decking roofing

Exposure: The Standard lists three ground expo-sure categories, which depend on surface roughness; roughness can dramatically affect wind. Our Sophia Hagia replica will be in an area with scattered obstruc-tions having heights less than 30 feet; typically flat open country and grasslands. This would be, then, ex-posure category C.

Use: The Standard defines four categories that range from low hazard to human life to essential facilities such as hospitals and fire stations. This sets the value of I – the “importance factor” – as described below. Since this building will be used as a church with a capacity of more than 300 people, it is a Category III building as per Table 1-1 of the Standard.

Wind Speed: You can use the maps in the Standard to determine the basic wind speed, defined as 3-second gust speed at 33 feet above the group for a particular exposure category. Associated is a typical occurrence of once every 50 years. From the map shown below, we can estimate V=120 miles per hour for Marion County, Florida.

Source of map: American Society of Civil Engineers. 2006. Standard 7-05. In Standard Minimum Design Loads for Buildings and Other Structures. Reston, VA. ASCE.

Velocity Pressure

The wind loading calculation begins with an estimate of the velocity pressure felt by the structure. Again, we estimate this and then use the appropriate gust coeffi-cients to apply this to the MWFRS, C&C, and the load on the walls. In this example, we start with:

The velocity pressure, , is given by equation 6-15 in the Standard:• is a coefficient based on the exposure category• corrects for topography; in this case 1.0, since

it is homogeneous• is a wind directionality factor of 0.95 as per

Table 6-4 of the Standard• V = 120 mph• I = 1.15 for Category III as in Table 6-1 of the Stan-

dard


With the data above, we can calculate the velocity pressure. This is the key calculation for all the neces-sary wind loads for design: MWFRS roof and wall and the C&C. The engineer simply uses different pressure coefficients to calculate the forces and loads on the walls and roof, and on the cladding and components. Here it is calculating only the roof pressures, so we need this value at the top of the dome.

From Table 6-3 of the Standard, the value of for calculating MWFRS is for a height of 180 feet (the top of the dome) and exposure category C, we find

= 1.43 – the difference between 180 and 200 feet is so small that interpolation isn’t needed. This is the final value we need in order to calculate the velocity pressure:

Section of Table 6-3 Velocity Pressure Exposure Coeffi-cients

Height above ground

(ft)

Exposure (Note 1)

B C D

Case 1 Case 2 Cases 1 &2

Cases 1 & 2

180 1.17 1.17 1.43 1.58

200 1.20 1.20 1.46 1.61

Notes for Table 6-3:Case 1: a: All components & cladding; b: Main wind force resisting system in low-rise building designed using Figure 6-10 of the Standard.

Case 2: a. All main wind force resisting systems in build-ings except those in low-rise building designed using Fig-ure 6-10 of the Standard; b. All main wind force resisting systems in other structures.

Domed Roof Pressure

We next use this velocity pressure to calculate the domed roof pressures, or the loads caused by wind. This is the net pressure on any surface which arises from a difference of the external and internal pres-sures. To calculate this, we use equation 6-17 from the Standard:

G is the gust effect factor, which is always 0.85 for rigid buildings as noted in 6.5.8.1 of the Standard.

From Figure 6-5 of the Standard, shown below, you learn that for enclosed buildings = +/- 0.18.

Source: American Society of Civil Engineers. 2006. Stan-dard 7-05. In Standard Minimum Design Loads for Build-ings and Other Structures. Reston, VA. ASCE.

For an enclosed building, is always the velocity pressure at the dome, and so = 57.59 psi. Thus the pressures are

To use this equation, you must estimate the values of the pressure coefficients. These vary with height,

and according to the Standard you need to calculate them in two ways using figure 02-103:


Figure #02-103 [width of cylinder is 102 ft, height of cylin-der is 156 feet, base of cylinder to dome top-point B is 186 feet – should be drawn to scale]

Case A: Use a linear interpolation to calculate values of from Point A to Point B

Case B: Use the pressure coefficient at A for the entire front area of the dome up to an angle of = 25 de-grees (as measured from the spring line to the top of the dome), and then interpolate the values for the rest of the dome as in case A.

To calculate the coefficients, we use Figure 6.7 and find the values at points A, B, and C using the ratio of the height at spring line (hD) to the dome diameter (D) and the ratio of the dome height above the spring line (f) to the diameter of the dome (D).

For our replica of Hagia Sophia, these values are:

Reading Figure 6.7 of the Standard, we find

In Table 02-101 below, I’ve tabulated the values for Case A for the windward side. A typical line is calcu-lated as follows:

Step 1: Estimate Cp by interpolation

The Standard recommends using linear interpolation from the windward point A to the top of the Dome

point B. For a Distance of 20 feet from the spring line, this yields:

Cp = slope*distance from edge of dome + intercept

intercept = -1.52 [We start from the edge of the dome where the distance from along the spring line is zero]

Cp (@ 20 feet along spring line) = -0.0075*20 - 1.52 = = -1.37

Step 2: Estimate Pressures

Table 02-101:

Case A: Pressures along Dome Case from Wind-ward Point A to Top of Dome Point B

Distance along spring line (ft)

Cp(pressure coefficient)

Pressure-minus(psi)

Pressure-plus(psi)

External pressure

(psi)

0 -1.52 -84.8 -64.0 -74.4

5 -1.48 -83.0 -62.2 -72.6

10 -1.45 -81.1 -60.4 -70.8

15 -1.41 -79.3 -58.6 -68.9

20 -1.37 -77.5 -56.7 -67.1

25 -1.33 -75.7 -54.9 -65.3

30 -1.30 -73.8 -53.1 -63.5

35 -1.26 -72 -51.3 -61.6

40 -1.22 -70.2 -49.5 -59.8

45 -1.18 -68.4 -47.6 -58.0

50 -1.15 -66.5 -45.8 -56.2


Table 02-102:

Case B: Pressures along Dome Case on Leeward Side from Top of Dome Point B to Point C

Distance valong spring line (ft)


Pressure-minus(psi)

Pressure-plus(psi)

External pressure

(psi)

51 -1.14 -66.2 -45.4 -55.8

60 -1.03 -60.6 -39.9 -50.3

70 -0.9 -54.5 -33.8 -44.1

80 -0.78 -48.4 -27.6 -38.0

90 -0.65 -42.2 -21.5 -31.8

100 -0.53 -36.1 -15.3 -25.7

Case C: Using Edge Coefficients Up to = 25o

The Standard requires that we calculate the pressures on the Dome in a second way: Determine the point of the dome at which = 25o; use the values at the lead-ing edge (Point A) until that point. From there, the cal-culation follows that of the Case A. For the replica of Hagia Sophia, = 25o when we have traveled 14.38 feet along the spring line – from the outer edge to the center.

So, use the pressures calculated from Case A at the point A until 14.38, from there we would interpolate between the value at 14.38 and the top of the Dome, Point B.

In Table 02-103 below, you’ll see the tabulated the values for Case B for the windward side. A typical line after = 25o is calculated as follows:

Step 1: Estimate Cp by interpolation

The Standard recommends using linear interpolation from the windward point A to the top of the Dome, point B. For a Distance of 20 feet from the spring line, this yields:




Table 02-103:

Case D: Pressures along Dome Case from Wind-ward Point A to Top of Dome Point B

Distance along spring line (ft)


Pressure-minus(psi)

Pressure-plus(psi)

External pressure

(psi)

0 -1.52 -84.8 -64.0 -74.4

14.38 -1.52 -84.8 -64.0 -74.4

20 -1.46 -81.9 -61.2 -71.6

25 -1.41 -79.4 -58.6 -69.0

30 -1.36 -76.8 -56.1 -66.5

35 -1.25 -74.3 -53.6 -63.9

40 -1.25 -71.8 -51.0 -61.4

45 -1.2 -69.2 -48.5 -58.9

50 -1.15 -66.7 -45.9 -56.3

The pressures shown on the dome in the figure below for cases A and B.

Summary

In this chapter, we focused on how to calculate wind loads on a dome. We were introduced to one of the most magnificent domes in the world: Hagia Sophia, which still astonishes modern-day engineers. We de-tailed the calculations necessary to estimate loading from wind, noting that to safely assess the impact of wind the engineer needs to consider its effect on three aspects of the structure: a) the main force-resisting system, b) the components and cladding, and c) the loads directly on the walls. To illustrate the calcula-tion, we looked at how to build a full-sized steel rep-lica of Hagia Sophia in Marion County, Florida.


Chapter Three: Sizing an I-Beam Used in a Dome

Overview

• Brunelleschi’s Dome• Sizing I-Beams in a Modern Dome

Learning Objectives

• Understand the technical aspect of Brunelleschi’s Dome in Florence, including the unique structure that allows it to span a great distance

• Explain how to size I-beams used in modern triangulated domes

• Analyze the charts of the Aluminum Association Standard to select an I-beam

Brunelleschi’s Dome

The dome of the Basilica di Santa Maria del Fiore dominates the skyline of Florence, Italy. Finished in 1436, its “peaked” dome – a shape called “a quinto acuto” – distinguishes it immediately from the Pan-theon, Hagia Sophia, and other ancient domes.

A contemporary observer remarked: “It was incredible in our own time that it could be done; it was not con-ceived or known by the ancients.” Indeed, this was a step beyond the Pantheon: Instead of that dome’s thick walls, the Duomo, amazingly, uses only two compara-tively thin domes nested together. It was the first to exceed the Pantheon’s diameter (142 vs. 145). From its completion in the 1st century A.D. no dome came close to the Pantheon. The mighty Hagia Sophia had a diameter only two-thirds that of the Pantheon. The Duomo is still the largest brick dome ever built.

In 1417, the town leaders of Florence decided to com-mission a dome for their unfinished cathedral. Its first stone had been laid in 1296 and its construction con-tinued for 170 years, interrupted by the death of its ar-chitect and the Black Plague of 1348. Work continued fitfully until 1418, when only the dome remained in-complete. The semi-finished cathedral had an octago-nal drum on which the dome was to be built.

This drum stood 177 feet high and presented the tower leaders with a difficult problem: With such a tall drum, how could enough resources – timber and other mate-rials – be amassed, both logistically and financially, to build scaffolding to support a dome as massive as San-ta Maria del Fiore deserved? Even though the Opera


del Duomo (the committee in charge of building the cathedral) owned several forests, it would still require a huge expenditure: At the time timber was nearly as difficult to harvest and transport as marble was. Today, power saws and sophisticated trucking make it look simple compared to mining stone. But in 1417, even if they could muster the resources to build the center-ing, removing it would be another huge task. Typically this type of demolition was one of the most hazardous parts of construction in the Middle Ages. One member of the Opera proposed (perhaps in jest) that they fill the cathedral with coin all the way to the top of the drum so the workers would have a “platform” to work on. The virtue of this, in the eyes of its proposer, was there would be no expense to having them removed: Once done, they could simply open the doors and let the greedy townspeople carry the coins away!

The leaders held a contest in 1417 to find the right person and the right design for the dome:

“Whoever desired to make any model or design for the vaulting of the main Dome of the Cathe-dral under construction by the Opera del Duomo – for armature, scaffold or other thing, or any lifting device pertaining to the construction and perfection for said cupola or vault – shall do so before the end of the month of September. If the model be used he shall be entitled to a payment of 200 gold Florins.”

This gave interested parties six weeks to build models or draw a design. The winner was from a well-known local goldsmith.

Filippo Brunelleschi, known as “Pippo”, submitted a model that called for a unique double-dome construc-tion – two domes nested together – and required, he claimed, no need for “centering”, the scaffolding tra-ditionally used to hold the dome as it was built. The committee awarded the prize, and a contract, to Pip-po. But because they were unsure of his abilities as a builder, they required him to partner with his main rival, a fellow goldsmith named Lorenzo Ghiberti.

The committee did not realize that Brunelleschi was a mechanical genius. His father had intended him to become, like his father, a civil servant, but as a young boy he showed no interest for this. Instead he dis-played an uncanny talent for solving mechanical prob-lems. Although disappointed his son would not follow in his footsteps, Brunelleschi’s father set him up as an apprentice to a goldsmith, where Young Pippo thrived on learning to mount gems with precision, engrave on silver, and also embossing. This led him to study the engineering of motion: how to use wheels, weights, and gear to move things. He is said to have used his mechanical knowledge to build the world’s first alarm clock.

This, though, was not enough to convince the Opera that he could build a huge stone dome alone. But when the time came to start building, Brunelleschi feigned illness, letting his rival head up the effort. Soon it be-came apparent that Ghiberti was not up the task and Brunelleschi, alone, directed the building of the dome for the next sixteen years. (In fairness to Ghiberti, Brunelleschi had omitted key elements from his mod-el so that no one but he could build it.)

Brunelleschi was a most exacting builder. He chose the clay and specified the exact dimensions of the bricks. He gave exact instructions about which quar-ries could provide stone, wanting only the highest quality. He managed the project with an iron hand, go-ing so far as to construct kitchens inside the cathedral so his workers would not waste time looking for food. He refused to pay dues to the stonemasons and wood-workers guilds, and even trained workers to replace them when they struck.


The most important aspect of his design was its dou-ble-masonry dome. It had a thick inner octagonal shell connected by meridianal arched ribs connected to a thinner outer shell. In fact, one can still walk between these inner and outer domes. The main structural ele-ment was six horizontal rings of sandstone reinforced on their outer surface with iron chain. These chains prevented the dome from bursting. This is in contrast to the Pantheon, where there are extremely thick con-crete walls to keep the dome from collapsing. In the Duomo, the stone and iron hoops allowed Brunelleschi to make two thin domes instead. Probably Brunelles-chi chose this method in part because he had likely studied the Pantheon during an extended stay there in his youth and understood the problem with cracking; and partly because the secrets of Roman concrete has long ago been lost. So, how did Brunelleschi build his dome without using centering? (Recall that, for the Pantheon, building the wooden scaffolding used for center was one of the most exacting and dangerous jobs in that dome’s construction.)

He used two clever methods, one of which is reflected in the herringbone pattern for the bricks. Between the ribs he constructed horizontal “arches”, creating nine concentric circles from the bottom of the dome to the top. Thus his Duomo works, structurally, just like a circular dome, which is stable at all times during its construction because the uppermost complete ring acts as a keystone for its meridianal arches and keeps them for falling inward.

But to accomplish this, Brunelleschi needed to use one more trick. During construction, the uppermost ring can act as a keystone only when it is complete, because only then can it resist compression. How did Brunelleschi solve this problem? He connected the uppermost incomplete ring to the completed ring be-low it. For the bricks and masonry to function like the concrete in a dome, the bricks could not be laid on the horizontal, but instead with an inward inclination so that every brick increases the height of the dome. This yields the herringbone spirals characteristic of the fa-mous Duomo.

Perhaps most amazing of all is that the great engineer-ing genius Brunelleschi had only an intuitive feel for the stresses in the dome. He would not have thought in terms of “compression” and “tension” as we do to-day, but would have understood the dome’s structure and stability as manifestations of crushing and pulling apart or tearing. He also would have known rough-ly in what direction these catastrophic events would happen. Today, of course, we have very sophisticated ways to estimate the structural properties of a roof. In the remainder of this chapter, we’ll look at how we can use engineering standards to size the components of a modern-day dome roof.

Sizing I-Beams in a Modern Dome

The most common type of dome roof is the triangulated dome popularized by Buckminster Fuller. Thousands of examples exist. Perhaps the most famous is Space-ship Earth at EPCOT Center in Orlando, Florida. This iconic structure dominates the skyline at Disney World.


In addition to being the main focal point of Epcot, the 18-story geodesic sphere takes guests on a “time ma-chine tour” using a conveyor belt: The 13-minute ride celebrates the achievements of engineers in advancing human communication by taking the viewer from pre-historic times to the 21st century. Ray Bradbury, the great science fiction writer, helped to write the story told inside the structure, although its exterior was inspired by the work of Buckminster Fuller, shown below.

Over forty thousand workers took 26 months to build Spaceship Earth. The “sphere” sits on a massive founda-tion formed by deep trusses and six legs, none of which are directly under the sphere in order to give the illusion of the ball floating in air. Quadrupod structures support the smaller beams on which the skin is built. Interest-ingly, a small service car at the top of Spaceship Earth carries an engineer down the sides to make any neces-sary repairs. Geometrically, Spaceship Earth is a penta-kis dodecahedron; that is, it consists of 60 isosceles tri-angle faces divided into 16 smaller equilateral triangles, which are then divided further. As in most such roofs, each individual triangle is flat, but the lattice approxi-mates a sphere when connected on a spherical surface. In analyzing such a roof, we can treat it like a dome.

There are many calculations one could make in design-ing such a dome, but here I was to use the Aluminum Association’s Specification for Aluminum Structures to illustrate how to size the members in a triangulated dome. The type of dome we’ll look at is a single-layer triangular lattice. To make this calculation concrete, here’s the dome we’ll analyze:• Diameter, D, at base of dome is 150 feet • Height of dome, f, above spring line is 30 feet

As above, although each member is flat, they tile the structure in such a way that the resulting structure can be treated as a dome. The main roof structure is a se-ries of I-beams which compose the triangular lattice. Clamped to these beams are panels made from light gauge aluminum. Batten bars hold the panels in place by clamping the edges to the top flanges of the I-beams. In addition to I-beams, panels, and batten bars, there must be a way to hold the “I-beam triangle” together. Typically this is done with a dished gusset plate that sandwiches the joint. The question asked is: What size standard I-beam should be used to build this dome?

While one could use extremely sophisticated com-puter programs to calculate all the properties of the dome, the goal will be to make a “quick and dirty” calculation that sizes the I-beams. To do this, we make use of research that shows single-layer lattice domes to be similar to thin solid shells. On the latter there has been a lot of analysis. This means we can use a thin solid shell to model the lattice dome; bearing in mind, though, that the validity of this model relies on the assumption that the joints in the lattice frame are rigid enough for the frame to act like a continuum – a not unreasonable assumption for the method typi-cally used to connect the I-beam triangles. Also, the method assumes no significant imperfections from its fabrication, nor errors in tolerance when the structure is erected.

An equation developed by engineers Douglas Wright and Kenneth Buchert relates the structural properties of a dome:

• Where pcr = critical pressure applied uniformly and normally to the shell


• A = cross section area of the individual members in the lattice

• E = modulus of elasticity • r = radius of gyration • L = typical length of members • R = spherical radius of dome

To estimate the proper size of the members (I-beams) in the dome’s lattice, we’ll use this equation to find the typical value of the product Ar and then use the Aluminum Association’s Standard I-Beam Tables to pick out the correct size I-beam. Thus, we’ll rearrange our equation to be:

Let’s consider the value of each term.

Safety Factor SF: Note that we’ve added in a safety factor against buckling, SF. We’ll use a value of 1.65; this is 65% greater than where we estimate buckling would occur. Typically this value would be listed in local codes, or even some kind of office standard.

Critical Pressure pcr: In order to do this calculation, we need to determine the loads on the dome based on local conditions. In the previous chapter we learned, for example, how to calculate the loading from wind. For this calculation, let’s assume that the dead load is about 10 psf and the snow/wind load is 70 psf. This means the total load is 80 psf. We’ll use this as the critical pressure in our equation.

Typical member size L: The value of L, the typical length of an I-beam, is determined by the materials available. For an aluminum roof this limit would be the width of the available coil material. For this dome we’ll use 3003-H16 alloy, for which the greatest width manufactured is 108 inches. Since the roof will be covered with equilateral triangles, the side lengths will be about 125 inches, using 108 inches as the altitude of the triangle. Some members might be a little longer and others a little shorter, but L = 130 inches will give a good enough estimate.

Spherical Radius R: A dome is, of course, a section of a sphere; although the spring line doesn’t neces-sary contain a radius. Typically the spring line of a dome is a chord, rather than a radius. The radius of the

sphere that forms the basis of the dome is a function of the height of the dome above the spring line and the dome’s radius (length of the spring line). Using a little solid geometry one can show that:

For the dome under consideration:

Elastic Modulus, E: We can easily look up the elastic modulus for this type of aluminum alloy. Its value is 10,100,000 psi.

Using these values in our main equation, we find a value for Ar of:

Recall that we are using the product of the area (A) and the major axis of gyration (r). We can use the Alumi-num Association Standard I-beams to pick out a beam. Table 03-101 from the Aluminum Association shows many properties of I-beams. Of greatest interest to us is the Area and the radius of gyration along the X-X axis. If we extract these two columns, we can make a table (Table 03-102) that contains the product of the two. Scanning down this list shows that I-beam “I 4 x 2.79” has an Ar value of 4.00; I-beam “I 5 x 3.70” has a value of Ar equal to 6.65. The value we calculated is between these two, so we’ll choose the larger I-beam.


Table 03-101: Aluminum Association Standard I-beams (excerpt)

Designation (Dept x width)

Depth d (in.)

Width bf

(in.)

Flange Thickness

tf (in.)

Web Thickness

tw (in.)

Fillet Radius R (in.)

Area A

(in.2)Axis X-X Axis Y-Y

Warping ConstantCw (in6)

Torsion Constant

J (in4)

Ix (in4)

Sx (in4)

rx (in)Iy

(in4)Sy

(in4) ry (in)

I 4 x 2.31 4 3.00 0.23 0.15 0.25 1.96 5.62 2.81 1.69 1.04 0.691 0.727 3.68 0.033

I 4 x 2.79 4 3.00 0.29 0.17 0.25 2.38 6.71 3.36 1.68 1.31 0.872 0.742 4.5 0.061

I 5x 3.70 5 3.50 0.32 0.19 0.30 3.15 13.9 5.58 2.11 2.29 1.31 0.853 12.5 0.098

I 6 x 4.03 6 4.00 0.29 0.19 0.30 3.43 22.0 7.33 2.53 3.10 1.55 0.951 25.3 0.089

Table 03-102: Ar for selected I-beams

Designation (Depth x width) Area A (in.2) rx

(in) Ar

I 4 x 2.31 1.96 1.69 3.31

I 4 x 2.79 2.38 1.68 4.00

I 5x 3.70 3.15 2.11 6.65

I 6 x 4.03 3.43 2.53 8.68

Summary

We opened this chapter by describing one of the most revolutionary domes in the histo-ry of building: The Duomo of the Basilica di Santa Maria del Fiore. We revealed how to size the I-beams in a modern dome. The most common type of dome is the triangulated dome popularized by Buckminister Fuller. Using a method developed by two engineers in the 1960s, which assumes that the joints in the dome lattice frame are rigid enough for the frame to act like a continuum, we showed how to estimate the area and radius of gyration of a beam for particular loading conditions. The product of these quantities can then be used to select proper sized I-beam using tables from the Aluminum Association Standards.


1. A dome is, in a way, a set of arches linked monolithically together. What are these “arches” called in a dome?a. Parallels b. Meridians c. Medians d. Voussoirs

2. One of the main differences between how an arch and a dome supports a load is:a. The arch not only compresses, but bends; a

dome does not bend, but only compresses throughout

b. An arch does not bend, but compresses throughout; a dome not only compresses, but bends slightly

c. A dome expands as a load is placed on it, while an arch contracts

d. There are no differences in how a dome and an arch support a load

3. The key ingredient in Roman concrete is:a. Potassiumb. Nickelc. Sandd. Volcanic ash

4. The Rogers Centre in Toronto, Canada, has this feature that surpasses the Pantheon:a. Doorsb. A motorized roof c. Better concreted. All of the above

5. The key engineering issue in building Hagia Sophia was:a. Find the proper type of stone to useb. Mixing the concretec. Supporting a round dome on a square base d. Choosing the site

6. A pendentive is:a. A five-sided regular polygonb. A type of concretec. The shape of Hagia Sophia’s domed. A triangular segment of a sphere

7. The Standard lists “exposure” as a category which influences windspeed. Exposure depends primarily on:a. The location of the buildingb. The type of material used in the buildingc. The use of the buildingd. The roughness of the ground

8. The spring line of a dome is typically the ___ of the sphere associated with the dome’s dimensions.a. Radius b. Chord c. Diameterd. Circumference

9. Bruncelleschi could make his dome thinner than the Pantheon because:a. He used iron chain to reinforce the dome and

keep it from bursting b. He had better quality concretec. He had less span to cover than in the Pantheond. He had more workers

10. In sizing an I-beam for a reticulated dome, the structure may be treated as:a. A space frameb. A thin-shelled structure c. A solid blockd. A membrane

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