Nervi's Design and Construction Methods for Two Thin-Shell Structures: The Leverone Field House and Thompson Arena by Momo T. Sun BASc., University of Toronto (2014) Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Master of Engineering in Civil Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2017 Momo T. Sun, MMXVII. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Eedacted Signature of Author ... SignatureSignature.re ...... Department of Civil and Environme al Engineeri May- 12,K Certified by...... Signature redacted /~ John A. Ochsendorf Class of 1942 Professor of Civil and Env. Engineering & Architecture 7Thvsi Supervisor Accepted by...... Signature redacted ...... y Jesse Kroll Associate Professor of Civil and Environmental Engineering MASS CH NSTITUTE Chair, Graduate Program Committee

JUN 14 2017

LIBRARIFS 2 Nervi's Design and Construction Methods for Two Thin-Shell Structures: The Leverone Field House and Thompson Arena by Momo T. Sun

Submitted to the Department of Civil and Environmental Engineering on May 12, 2017, in partial fulfillment of the requirements for the degree of Master of Engineering in Civil Engineering

Abstract This thesis studies two major thin-shell concrete structures by Pier Luigi Nervi (1891- 1979) - the Leverone Field House and Thompson Arena. These two similar parabolic vaults are two of the few international structures he has completed in the United States. Situated across the street from each other at Dartmouth College, these two thin-shell concrete structures designed only a few years apart and in a such mature stage of Nervi's engineering career deserve a closer look.

Access to Nervi's original calculations, specifications, and correspondences with Dartmouth College reveal a new level of refinement in his design methods and deci- sions. This study analyzes his structural design methods and compares them with approximated hand calculations assuming an asymmetric load on a 3-hinged parabolic arch. The maximum moment was calculated to be within 7% of Nervi's results. An arch was also explored by building a Finite Element (FE) model in SAP2000, how- ever, the results proved the model to be an unreliable representation of the behavior of the funicular concrete arch.

Furthermore, never before published construction photos give clues to the con- struction of the first structure built with the "Nervi System" in the United States. Slight changes were made to the construction method from his previous structures with the Nervi System in Rome. The types of different precast panels were reduced to increase repetition and refinement was made to the multi-step formwork system to reduce the amount of wooden formwork while keeping a high level of accuracy for the shape of the precast panels.

Thesis Supervisor: John A. Ochsendorf Title: Class of 1942 Professor of Civil and Env. Engineering & Architecture

3 4 Acknowledgments

I'd like to thank my adviser John for his direction for this thesis and Gordana for her continued support throughout the year. The MEng class has become my friends and family, I couldn't have done this without them all. In addition, I'd like to thank my parents for moving to three continents to allow me the best possible upbringing, exposing me to different opportunities at each place. It's been a long journey but well worth it. I'd also like to thank Tullia Iori and her students at University of Rome Tor Vergata, MAXXI archives, Tom Leslie, and Dartmouth College, for their guidance and resources.

5 6 Contents

1 Introduction 13 1.1 Pier Luigi Nervi and His Structures ...... 13 1.2 Literature Review ...... 14 1.3 Problem Statement ...... 15

2 Case Studies 17 2.1 Leverone Field House ...... 18 2.2 Thompson Arena ...... 22 2.3 D iscussion ...... 26 2.3.1 R oof ...... 27 2.3.2 B uttress ...... 27 2.3.3 Exterior and Front Face ...... 28 2.3.4 Antonio Nervi's Learning Experience ...... 28 2.4 Summary of Case Studies ...... 29

3 Structural Analysis 31 3.1 Nervi's Method ...... 31 3.2 Approximation Using Hand Calculations ...... 38 3.3 Finite Element (FE) Model ...... 39 3.4 Summary of Structural Analysis ...... 41

4 Construction Methods 43 4.1 Precast Panels ...... 44

7 4.2 In-Situ Concrete ...... 45

4.3 Construction Sequence Photos ...... 46 4.4 Summary of Construction Methods ...... 50

5 Conclusion 51

A Nervi's Original Calculations and Documents 55

B Moment of Three-Hinged Arch Derivation 101

8 List of Figures

2-1 Leverone Field House's front face [Photo by author, 2017] ...... 20 2-2 Leverone Field House's interior view [Photo by author, 2017] .... . 20

2-3 Leverone Field House's A-shaped columns [Photo by author, 2017] . 21

2-4 Leverone Field House's exterior side view [Photo by author, 2017] . 21

2-5 Thompson Arena's front and side view [Photo by author, 2017] . .. . 24

2-6 Thompson Arena's interior view [Photo by author, 2017] ...... 24

2-7 Thompson Arena's Y-shaped buttress [Photo by author, 2017] ... . 25

2-8 Thompson Arena's exterior side view of buttresses [Photo by author, 20 17] ...... 2 5 2-9 Roof plan for Thompson Arena showing precast panels and Y-buttresses [Dartmouth College, 1962b] ...... 27

3-1 Elementary arch broken into segments for analysis, from Calcoli Statici, 05-05-1961 [StudioNervi, 1961] ...... 33

3-2 Leverone Field House, T-section for design of reinforcement, from Cal- coli Statici, 05-05-1961 [StudioNervi, 1961] ...... 33

3-3 Leverone Field House, gravity loading conditions from Calcoli Statici, 05-05-1961 [StudioNervi, 1961] ...... 34 3-4 Leverone Field House, gravity loading schemes for structural analysis from Calcoli Statici, 05-05-1961 [StudioNervi, 1961] ...... 35

3-5 Thompson Arena, scaled resin model under seismic loading in the elas- tic range, 1:50 [Cassinello et al., 2010] ...... 37

9 3-6 Thompson Arena, Model displayed at Polytechnic University of the Marches, 1971 [Cassinello et al., 2010] ...... 37 3-7 Leverone Field House, section of rib for end arch reinforcement, draw- ing from 01-19-1962 [Dartmouth College, 1962a] ...... 38 3-8 Leverone Field House, arch contour truss spacing detail, drawing from 03-11-1961 [Dartmouth College, 1962a] ...... 38 3-9 Leverone Field House, exaggerated deflection under uniform symmetric loading of DL+LL, model from SAP2000 ...... 41 3-10 Leverone Field House, exaggerated deflection under asymmetric load- ing on half span of LL, model from SAP2000 ...... 41

4-1 Brick formwork on curved scaffolding surface [Dartmouth College, 1962b] 46 4-2 Concrete negative mold [Dartmouth College, 1962a] ...... 46 4-3 Forming reinforcement around negative mold for precast panel [Dartmouth College, 1962a] ...... 47 4-4 Precast panel enclosed in wooden formwork ready to be poured [Dartmouth College, 1962a] ...... 47 4-5 Complete formwork with precast panel [Dartmouth College, 1962a] . 48 4-6 Removing side panels of wooden formwork [Dartmouth College, 1962a] 48 4-7 Precast panel hoisted and removed from formwork, moving to storage [Dartmouth College, 1962a] ...... 49 4-8 Scaffolding on center rails, continuing to pour in-situ concrete forming ribs between laid out precast panels, some ribs are already filled from previous pours [Dartmouth College, 1962a] ...... 49

B-i Asymmetric load on a three-hinged parabolic arch ...... 101

10 List of Tables

1.1 Nervi's projects in the United States ...... 14

2.1 Basic information on geometry and cost ...... 18

3.1 Leverone Field House, Scheme 2 results comparison under asymmetric loading on half span ...... 40

11 12 Chapter 1

Introduction

1.1 Pier Luigi Nervi and His Structures

Pier Luigi Nervi (1891-1979) was a famous Italian structural engineer known for his economic and pragmatic innovation in reinforced concrete shell structures. He was one of the few designers of his time to bridge the gap between art and technology, in essence, architecture and structural engineering [Nervi, 1965]. He began his ca- reer by designing and building in Italy then developed his own innovative precast concrete system named the "Nervi system", which was a very effective way of build- ing thin-shell barrel vaults. He first used the system to build hangars which created long-spanned structures economically. The Nervi System used precast diamond or triangular-shaped panels as formwork which were connected by casting a thin layer of in-situ concrete, creating ribs between the precast panels. He continued to de- velop this system and gained international recognition for his elegant design and building techniques especially after the construction of one of his most iconic struc- tures, Palazzetto dello Sport, a thin-shell dome used for the 1960 Olympics in Rome [Bologna and Neri, 2013]. While Nervi's projects were widely celebrated among engineers, architects, and the general public for their daring and elegant designs, the two specific structures to be discussed in this thesis have received very little publicity. These were two of the few structures Nervi has ever built using his own system outside of Italy and arguably

13 the most involved international projects illustrated by his personal licensure. In the archives at Dartmouth College there were six professional engineering (PE) licenses for New Hampshire, one for each year from 1970 to 1976 for the design and construction duration of Thompson Arena, the same were likely obtained for earlier years for the Leverone Field House [Dartmouth College, 1962b]. This was the only licensure Nervi obtained outside of Italy [Leslie, 2018]. Table 1.1 shows the list of projects Nervi completed in the United States. The Leverone Field House was designed and constructed from 1960 to 1962, and Thompson Arena from 1967 to 1976. With the two projects at Dartmouth College only five years apart, the study of this pair of structures shows the refinement of his design methods in a mature stage of his career [Bologna and Neri, 2013].

Table 1.1: Nervi's projects in the United States

Project Year Completed Location Leverone Field House 1962 Hanover George Washington Bridge Bus Terminal 1963 New York Norfolk Scope Arena 1970 Norfolk Saint Mary's Cathedral 1971 San Francisco Thompson Arena 1975 Hanover

1.2 Literature Review

There is an abundance of literature and published works on Nervi and his major projects in Italy, however, his projects in the United States, namely Leverone Field House and Thompson Arena, are surprisingly understudied. The two structures at Dartmouth College are mostly neglected, sometimes mentioned in passing and occa- sionally the Field House received a short feature in books showcasing Nervi's projects but the Thompson Arena is largely missed. From the Database for Civil and Struc- tural Engineering, included are all of Nervi's projects worldwide and links to relevant publishing for each structure [Structurae, 2016]. Not surprisingly, there are 12 rele- vant articles or books for Palazzo dello Sport [Structurae, 2016]. To a lesser scale, the

14 Leverone Field House has three relevant articles [Structurae, 2016]. Lastly, Thomp- son Arena has zero listed relevant articles or books [Structurae, 2016]. Of course this is not an exhaustive list for all published works on each project but it shows a stark contrast between the detailed analyses that have gone into other similar Nervi projects and the two thin-shells at Dartmouth College. There have been no in-depth analyses on the design methodology for the Leverone Field House and Thompson Arena or on the construction process. This warrants fur- ther study because Nervi was a great designer who paid equal attention to architec- ture, structural expression, construction, and economics and yet these two important projects have not been studied. Nervi passed away in 1979, and arguably since the 1975 Thompson Arena is one of Nervi's last projects, it has some of the most im- proved and refined elements of design, making this project even more of a target to study [Britannica, 2016].

1.3 Problem Statement

Visits to Dartmouth College uncovered first hand materials found in the archives and seeing the structures themselves gave new insight into the otherwise poorly published structures. This thesis uses Nervi's original calculations and documents to analyze his design methods and decisions. With the addition of these new resources, this thesis seeks to answer the following questions:

1. What is the historical significance of the two structures?

" What materials are available in the Dartmouth Archives and the MAXXI Archives?

" What generalizations can be made from the material?

2. What was the structural design process for the two thin-shell concrete struc- tures?

* How did Nervi determine the shape of the structures?

15 * What methods did Nervi use to analyze the structures? What assumptions did he make?

3. What was the construction method of the two structures?

* How did Nervi adjust his design and methodology for building in the US?

16 Chapter 2

Case Studies

Dartmouth's Business Manager Richard W. Olmsted (Dartmouth '32) attended the 1960 Olympic Games at Palazzetto dello Sport. After seeing the elegant crystalline geometric units that made up the stadium, he decided to hire Nervi to design a similar sports facility for the college [Meacham, 2008]. From 1960 to 1962, Nervi designed and constructed the Leverone Field House, then he was rehired in 1967 to design the Thompson Arena. Both are parabolic vaults constructed with precast concrete panels from Nervi's system. The design philosophies were the same for both structures. A visit to the facilities showed how strikingly similar they were both on the exterior and interior. Located across the street and with the front of the stadiums facing each other, it could have created a unique mirroring effect with the near identical structures. However, this effect was never achieved because Thompson Arena hides behind a row of preserved early-twentieth-century houses and can barely be seen from the street. Records and sketches from college planning at Dartmouth showed that there were several attempts to join the two structures to make one large sports plaza, however, this was never realized due to the preservation of the historic houses [Dartmo., 2012]. Table 2.1 show details about geometry and cost for each of the two structures. No- tably, the Thompson Arena costs significantly more per square area than the Leverone Field House. Although Thompson Arena is smaller in terms of area, it is a more com- plex facility in terms of functionality and encloses a significantly larger volume for

17 more capacity. The ice surface and seating are completely below grade which resulted in increased costs from excavation. The hockey rink, cooling facilities, and seating itself also added largely to the total cost compared to the simple open field, on grade design for the Leverone Field House. A crude estimation of the excavation costs with the approximate volume of soil excavated and an assumed cost of $300/m2 yields $2.8 million USD in 2017 [Homewyse, 2017]. With this excavation cost and other additional costs for the facility, the cost for the two structures are fairly comparable.

Table 2.1: Basic information on geometry and cost

Leverone Field House Thompson Arena Span (m) 66.75 54.254 Height (m) 13.335 9.576 Length (m) 109 97.5 2 Area (M ) 7,276 (91,800ft 2) 5,290 (57,000ft 2 ) Number of Precast Units 1,240 1,024 Total Cost at Time of Construction (USD) 1.5mil 4.4mil Total Cost Adjusted* (2017 USD) 14.3mil 34.8mil Cost per Unit Area (2017 USD/m 2 ) 1,970 6,580 Seats N/A 3,620

*[RSMeans, 2017]

2.1 Leverone Field House

The Leverone Field House, shown in Figure 2-1, is the better known of the two facilities. Nervi had already gained international recognition by this point and had worked on several projects abroad, however, none were entirely in his iconic style of curved thin-shell concrete structures with interlacing ribs. Though he was an acclaimed and sought after designer, it was unexpectedly difficult for Nervi to convince owners to design and build purely using his precast system. Richard W. Olmsted being an admirer of Nervi's unique style, was the first to take the risk and approve the never-before used precast design to be constructed in the US at Dartmouth College. The vaulted long-span barrel roof created a large uninterrupted surface for indoor

18 track facilities, and an indoor practice ground for football and other field sports as shown in Figure 2-2. The field of the structure is on grade with buttresses propping up the barrel roof hidden along the length of the building creating offices and storage areas. Typical of Nervi's iconic thin-shell concrete designs, the roof is made up of a variation of four different types of precast panels, diamond-shaped near the crown and triangular panels closer to the landing edge beams. The roof consists of a slab with uniform thickness and ribs with tapering thickness from 0.8m at the spandrel to 0.5m at the crown [StudioNervi, 1973]. The edge beams are connected to sets of vertical and angled, A-shaped buttresses as shown in Figure 2-3. These buttresses have a rectangular section that gradually narrows as it approaches the ground. Figure 2-4 shows the exterior side view which is an inexpressive plain brick wall covering offices and storage areas. This facility is still in use today for track and field and other varsity team practices [Dartmouth Big Green, 2016b].

19 Figure 2-1: Leverone Field House's front face [Photo by author, 2017]

-4

Figure 2-2: Leverone Field House's interior view [Photo by author, 2017]

20 Figure 2-3: Leverone Field House's A-shaped columns [Photo by author, 2017]

Figure 2-4: Leverone Field House's exterior side view [Photo by author, 2017]

21 2.2 Thompson Arena

The lesser known Thompson Arena started its lengthy design process in 1965 shortly after the completion of the Leverone Field House in 1962. Construction began in 1973 and the arena was open for its first game in 1975. When completed, the ice arena was the largest venue of its kind among US colleges to host varsity hockey games [Dart- mouth College, 1962b]. It gained some attention amongst hockey programs, however as one of the last structures Nervi ever designed, it received very little attention in the structural world. Nervi passed away in 1979, only four years after the comple- tion of this project, leaving this to be his final complete project in the US [Bologna, 2013]. His son Antonio Nervi, the apprentice designer on this project passed away later in the same year [Bologna, 2013]. Perhaps it is because the structure is hidden behind from the street view, or perhaps because it is so similar to the Leverone Field House, it went unnoticed. The Thompson Arena is rarely mentioned in publications on Nervi's works and sometimes even missed in lists of his completed works.

Figure 2-5 shows the Thompson Arena, it is a parabolic vault of 54m by 90m made up of 1,024 triangular precast concrete units, each weighing one ton [Dartmouth Big Green, 2016a]. At first glance it may seem that the Thompson Arena is virtually identical to its precedent across the street, however, upon closer examination, it is clear that Nervi has made many refinements to this second sports facility. Pier Luigi Nervi negli Stati Uniti in 2013 is the only publication with a dedicated passage for the Thompson Arena [Bologna, 2013].

This structure contains Nervi's two most distinctive styles: columns with vari- able cross-section which twists and tapers, and ribbed shell with precast geometric elements. It has the Y-shaped buttresses on the exterior, shown in Figures 2-7 and 2-8, transfer gravity and thrust loads to foundation, they are visually identical to the buttresses at the famous Palazzetto dello Sport. As shown in Figure 2-6, the arena is more grand than it appears from the outside, the building is excavated so that the ice surface and all the stadium seats sit below grade. This created a much larger space than what can be seen or expected from the square footage of the building. It

22 still remains one of the greatest venues for college hockey games, with 3,520 stadium seats and room for standees, and the largest crowd for a playoff game was recorded at 6,000 attendees [Dartmouth Big Green, 2016a].

23 1!

Figure 2-5: Thompson Arena's front and side view [Photo by author, 20171

IM~ --. A*N 4-

......

Figure 2-6: Thompson Arena's interior view [Photo by author, 2017]

24 -ww.....im.

Figure 2-7: Thompson Arena's Y-shaped buttress [Photo by author, 2017] 1

~: , .. -WI

Figure 2-8: Thompson Arena's exterior side view of buttresses [Photo by author, 2017] 25 2.3 Discussion

Throughout his career, Nervi had been developing his techniques for designing long- span concrete structures and using interlaced ribs as a way to decrease the thickness of the overall roof slab. This first began in 1935 when he designed aircraft hangars for WWII. Later he achieved his laudatory title "Poet in Concrete" for the sports palaces in Rome in the 1960 Olympic Games [Dartmouth Big Green, 2016b]. The elegant geometric roof with interlacing ribs were truly an astounding achievement of creating interesting architecture with structural design.

Nervi lectured at Harvard as a Charles Eliot Norton Professor for the year 1961- 1962 where he combined the ideas of technical and aesthetic aspects. Through his lectures, the goal was to advocate and support architecture and structural engineering as a synthesis of technology and art rather than technology as well as art [Nervi, 1965]. Since the lectures were concurrent with the construction of the Leverone Field House, it was the last project mentioned during his professorship. A few construction photos were included to illustrate the construction process of the precast panels working with in-situ concrete.

Both of Nervi's structures at Dartmouth College share many similarities with Palazzetto dello Sport in Rome in terms of the design features and principles, how- ever, there is a clear flow of design changes as he began his first building in the US. Starting with the Leverone Field House in 1962, Nervi wanted to keep the concept from Palazzetto dello Sport but scale back the complexity. First, because this facility is on a smaller scale since it is a college stadium as opposed to an Olympic stadium, and second, because he is working in a new environment where he has no previous experience with the construction materials or the level of skilled laborers available. For the Leverone Field House, many of the details from the Sports Palace were sim- plified or completely eliminated. Interestingly enough, for Thompson Arena, Nervi added back some details as he got a better sense of what could be achieved in the US and as a result, this second structure is more similar to Palazzetto dello Sport.

26 2.3.1 Roof

The roof geometry was simplified from a double curvature dome to a single curvature parabolic vault. This allowed a moving arch scaffolding on rails to construct the roof one strip at a time. Construction methods and further details will be explained in Chapter 4. The precast elements were also simplified from an array of various sized diamonds and other geometric shapes to only four different types of panels, these were either diamonds or triangles. For the Thompson Arena designed a few years later, all the basic concepts of a parabolic vault stayed the same except he further simplified the precast panels eliminating the diamonds and having only triangles. Only one single size precast element was used for the entire vault [StudioNervi, 1973.

Figure 2-9: Roof plan for Thompson Arena showing precast panels and Y-buttresses [Dartmouth College, 1962b]

2.3.2 Buttress

The buttresses supporting the roof of the Leverone Field House had the same struc- tural concept as Palazzetto dello Sport, transferring load from the roof directly to the ground on the perimeter leaving an unobstructed field in the interior. However, the details of the buttresses were largely simplified. The buttresses protrude through interior storage areas and have a rectangular section that linearly narrows as it ap- proaches the ground. These are hidden within an exterior wall. The shape of the first buttress can be seen through the front face of the building while the others along the length of the building are unceremoniously hidden within storage area surrounded by gym equipment. By the time Nervi got to designing the Thompson Arena, he brought back many of the details for the buttresses he had previously used at Palazzetto dello

27 Sport. The buttresses were expressed in a much more structurally artistic way, ex- posed and on the exterior, becoming one of the expressive features of Thompson

Arena. These were refined Y-buttresses with a complex curving taper on each side of the column, they now have a complex varying cross-section compared to the ones at Leverone Field House. These buttresses look identical to the ones at Palazzetto dello

Sport in Rome.

2.3.3 Exterior and Front Face

The front and back face of the Leverone Field House are made up of glass curtain walls. This consists of vertical steel tubes and glass allowing nature light to come in on either ends. This also created a distinct look that is different than the rest of the concrete building. The design of the front and back faces of the Thompson Arena are much simpler. There exist the same vertical lines as were with the Leverone Field House, however, the design of the structure is more uniform and has a purity for material usage. Concrete extruded verticals do not create the same effects as the extruded steel frames on tinted glass, Thompson Arena has a simpler design and the structure relies on the expressive buttresses as a focal point for structural and architectural detail. Since it is a facility that will house an ice rink, it was a wise decision to not replicate the previous curtain wall design with glass, the solid concrete wall eliminates the glare from the windows, the sunlight that would melt the ice, and also provides better insulation from the outside temperatures.

2.3.4 Antonio Nervi's Learning Experience

Since the Thompson Arena had such similar criteria and design goals as the Leverone

Field House, Pier Luigi Nervi saw it as an excellent learning opportunity for his son Antonio Nervi (1925-1979), in effort to pass on his studio to his sons. It took five years (1967-1972) for this project to be approved; this design process was longer than the Field House previously. Part of this was due to the learning curve for Antonio

Nervi and part was due to a slightly more complex structure [Bologna and Gargiani,

28 2006]. Although the intentions were for Antonio to complete most of the design work for this project, instead, he mainly handled correspondences with the client in the US. Nervi made all the design decisions. Therefore, in literature and in this paper, the name "Nervi" simply refers to Pier Luigi Nervi and not his sons.

2.4 Summary of Case Studies

In some aspects, the design of the Thompson Arena was simplified. For example, the types of different precast panels were reduced from diamond and triangular-shaped to only triangular-shaped, to streamline the construction process. There are other ways the Thompson Arena was more refined. Compared to the earlier Leverone Field House, the Thompson Arena is more expressive and is aesthet- ically superior. The multi-dimensionally tapered buttresses on the exterior greatly added to the structural expression and aesthetics of the structure. The concrete end walls with the accented extrusions is an improvement from the steel members and curtain wall design of the Leverone Field House in terms of purity in material usage. Lastly, the Thompson Arena is a better use of space, hosting a larger venue with more capacity in a smaller footprint.

29 30 Chapter 3

Structural Analysis

This research uncovered design documents at Dartmouth College archives with orig- inal calculations, specifications, and letters from Studio Nervi. There was one design package, "Calcoli Statici" of 77 pages for the Leverone Field House [StudioNervi, 1961]. The Thompson Arena has one main package of 72 pages which is clearly a copy from the design package from Leverone Field House. There were also additional packages B and C which were only a few pages each on the design of the buttresses [StudioNervi, 1973]. These design packages contain some diagrams and numerous detailed tables with precise values for each step of his calculation. How did he design the shape of the structures? What assumptions did he make to carry out his analyses? Deciphering the calculations showed Nervi's strategic design decisions. Not surprisingly, the de- sign packages revealed that he used the same methods for the two structures. For comparison, the following subsections will also show approximate hand calculations and FE models in SAP2000 (Computer and Structures, Inc., Version 15) to check against Nervi's analyses.

3.1 Nervi's Method

The structural analysis was carried out by assuming a transverse portion of the vault as a fix-fixed arch. Nervi establishes an "elementary arch" of just under 3m width

31 to analyze for each of the structures. Because the ribs of the roof are diagonally interlaced and complex, there are no constant transverse sections to simplify the analysis. He chose the elementary arch in such a way so that there is always at least one rib through the transverse section, see Figure 3-8. This way, he was able to design the roof in strips, approximating an elementary arch with a T-section. The design philosophy was to minimize moment with the shape of the arch. Nervi approximated the dead load and live load for the roof, then designed the curve of the arch to coincide with the curve of pressures from the Dead Load and half of the Live Load. This results in zero moment for the fix-fixed arch under a uniform load of DL+0.5LL. The elementary arch was subdivided into 20 segments with equal horizontal projection, working with 10 segments for the half arch when analyzing symmetric uniform loads. Instead of using an approximate parabolic shape, Nervi spent an immense amount of effort to determine the exact shape using an elastic equation from "Belluzzi Vol. II, page 217",

1 As AX = X" E A- (3.1) E y 2AW The original "Belluzzi" publication has not been identified as of now.

32 87 V

® 9

Figure 3-1: Elementary arch broken into segments for analysis, from Calcoli Statici, 05-05-1961 [StudioNervi, 1961]

Figure 3-2: Leverone Field House, T-section for design of reinforcement, from Calcoli Statici, 05-05-1961 [StudioNervi, 1961]

The calculation packages show many tables with geometric and section properties for each segment on the half arch, such as x- and y-coordinates, area, section modulus, moment of inertia, etc. These were used by Nervi to produce the moments, normal forces, and shear forces of each segment. Nervi defined two load conditions that were applied to an elementary arch with fixed-end conditions: 1. Symmetric Dead Load + Symmetric Live Load 2. Symmetric Dead Load + Asymmetric !(Live Load) The Dead Load is not a constant uniform distributed load because it largely

33 consists of self-weight, the loading increases closer to the ends of the arch as the projected self-weight increases, as seen in Figure 3-3. Instead of calculating the two complex loading conditions, two simpler loading schemes, as shown in Figure 3-4, were analyzed so they can be superimposed in ways to achieve the results of the loading conditions above.

9~;7o

Figure 3-3: Leverone Field House, gravity loading conditions from Calcoli Statici, 05-05-1961 [StudioNervi, 1961]

34 * dal 2* schema ehe sark studlate-appresso

Figure 3-4: Leverone Field House, gravity loading schemes for structural analysis from Calcoli Statici, 05-05-1961 [StudioNervi, 1961]

After defining these load schemes and conditions, he analyzed the fix-fixed arch as a pin-pinned arch with moments constraints counteracting at each reaction. This is what he used to calculate the reactions for the fixed-end arch for the loading schemes in Figure 3-4. Results from loading Schemes 1 and 2 were superimposed in various ways such as adding a reflection of a loading scheme or subtracting a loading scheme to arrive at the two loading conditions. The moments obtained from the asymmetric loading condition, condition #2, was governing for the moment resistance design for the buttresses.

Lateral resistance was analyzed in both transverse and longitudinal direction of the building. For the transverse direction, wind load governed over seismic. There was only one load case with a uniform lateral wind force applied to an elementary arch with fixed supports. Similar to the gravity calculations, the arch was divided into 20 segments and the moments, normal forces, and shear forces were found for each segment. In the longitudinal direction, seismic forces governed over wind. The

35 columns were designed to resist the lateral load in the longitudinal direction. Furthermore, Nervi was known to test scaled models to find the capacity and safety factor of his complex structures towards the end of his career. Figure 3-5 shows the scaled 1:50 resin model being tested under seismic loading in the elastic range. This was Nervi's last model tested at the Bergamo in 1970-1971 [Cassinello et al., 2010]. In addition to the load cases, Nervi checked the effects of horizontally displacing the columns by 25mm (1 in) to allow for construction tolerance [Long, 1967]. Since the buildings were located in New Hampshire where the weather is more frigid than the weather in Italy where he is used to working, he determined the effects of a 38*C(100*F) decrease in temperature [Long, 1967]. These two additional requirements were known to be evaluated for the Leverone Field House through correspondences, however, the calculations were not included in the package submitted to Dartmouth College. One obvious area missing from the calculations were serviceability checks. In the specifications at the beginning of each calculation package, there are precise instructions to make sure that construction would be carried out as precisely as possible. This is in line with the detailed calculations he for both the structures. There were no mention or calculations of deflection of the roof, it is likely that Nervi knew from the experience of his previous projects that deflection would be minimal. Deflection will be explored in Section 3.3 with a finite element model.

36 Figure 3-5: Thompson Arena, scaled resin model under seismic loading in the elastic range, 1:50 [Cassinello et al., 2010]

Figure 3-6: Thompson Arena, Model displayed at Polytechnic University of the Marches, 1971 [Cassinello et al., 2010]

37 Figure 3-7: Leverone Field House, section of rib for end arch reinforcement, drawing from 01-19-1962 [Dartmouth College, 1962a]

Figure 3-8: Leverone Field House, arch contour truss spacing detail, drawing from 03-11-1961 [Dartmouth College, 1962a]

3.2 Approximation Using Hand Calculations

The funicular shape of the arch for the roof is very close to a parabola. An ap- proximate hand calculation for moment for an asymmetric load is used by assuming a 3-hinge pin-pinned parabolic arch. See Appendix B for derivation of M1, for the maximum moment at .1 span.

38 M1 = (3.2) 64 For the Leverone Field house, the moment at the loaded side fixed-end was 209kNm per elementary arch. The moment obtained from the equation above was 194kNm per elementary arch, this is an excellent approximation compared with Nervi's detailed calculations with only a 7% difference. This provides a lower bound, equilibrium approximation to Nervi's elementary arch.

3.3 Finite Element (FE) Model

Another method to compare the results from Nervi's method is with elastic FE mod- els. A model of the elementary arch was made in SAP2000 for the Leverone Field House in order to compare the results from current FE models to the results Nervi obtained from the energy method by breaking up the arch using (SAP2000, Com- puter and Structures, Inc., Version 15). Set up in the same way as Nervi's method, the arch was broken down into 20 elements with the exact geometry, the input was determined from x and y coordinates in tables from Studio Nervi. Other parameters were also obtained from Nervi's calculation package to match the exact properties of the transverse arch from Nervi's calculation, such as slightly varying moment of inertia (I) and cross sectional area (A) values. The modulus of elasticity (E), stayed constant at 24,500MPa as defined in the specifications [StudioNervi, 1961]. The model was built in 2D space to restrict the forces in a 2D plane avoiding out of plane action. Nervi designed the shape of the arch to have zero moment under Dead Load and half of the Live Load, however, with this loading the FE model resulted in some bending moment throughout the length of the arch whereas this funicular shape should result in zero bending moment. The output of result were puzzling for this funicular arch, and resulted in about a 50% larger moment at the supports under the asymmetric loading scheme as summarized in Table 3.1. The largest bending moment from the FE model under the Dead Load and half of the uniform Live Load is 90kNm at the center of the crown and slightly less at the fixed ends. The FE model seemed to

39 be overly sensitive and therefore produced exaggerated bending moments compared to the energy method and the 3-hinged approximate hand calculations which were within reasonable tolerance of each other. In terms of analysis of funicular arches, FE modeling in SAP2000 was not a reliable method to predict the forces and reactions of an elastic arch, because it predicts bending moments in a no-bending arch geometry. With this in mind, this SAP2000 model was also used to investigate the deflection of the arch. The FE model proved to deliver more plausible results for serviceability. Under load condition 1 mentioned previously with a symmetric Dead Load and Live Load, the maximum predicted deflection is in the center with the crown deflecting 22mm downwards. This is well within the L/240 limit which allows 278mm. A test for the absolute worst condition with the full Live Load applied asymmetrically over the left half of the span gives a maximum deflection near the quarter point of the span. The deflection is 109mm which is closer but still within the L/360 limit which allows 185mm. After applying conservative load cases, the deflections predicted by FE are still well within current common deflection limits. It is reasonable that Nervi did not carry out detailed calculation for deflections as part of his submitted calculation package.

Table 3.1: Leverone Field House, Scheme 2 results comparison under asymmetric loading on half span

Nervi's Results FE Results % Diff. Hand Calc. % Diff. MA (kNm) 209 339 62% 194 7% Mc (kNm) 236 364 54% - -

40 Figure 3-9: Leverone Field House, exaggerated deflection under uniform symmetric loading of DL LL, model from SAP2000

Figure 3-10: Leverone Field House, exaggerated deflection under asymmetric loading on half span of LL, model from SAP2000

3.4 Summary of Structural Analysis

The structural design packages show that Nervi used an energy method to analyze an elementary arch. The shape of the arch was designed for zero bending moment under DL+0.5LL loading. A quick approximate hand calculations using a 3-hinged arch provided the closest resulting moments to Nervi's results. The FE model did not provide a good prediction for the arch because it produced moments for a zero- moment arch geometry. However, the deflections were reasonable estimates compared to the predicted deflected shape deflection limits.

41 42 Chapter 4

Construction Methods

The Nervi System is a construction method invented and tested by Pier Luigi Nervi through his projects beginning in the 1940s. It is an economical and rapid way to construct large concrete shell structures by use of prefabrication of concrete panels in addition to traditional in-situ concrete. He improved this method through his many projects in Italy but the Leverone Field House was the first time he brought his new construction method to the United States [Long, 1967]. At such a mature stage in Nervi's career, he continued to make changes to per- fect his construction system. As well, he made a few changes to adjust to the new environment in the US. Two major advantages of the system that minimized cost in Italy were lost in the US. First, this system significantly saved on cost from nearly eliminating the use of wooden formwork for the roof. Second, this system required more skilled laborers which were less expensive and more plentiful in Italy. Both benefits were lost in the US where timber was plentiful and inexpensive while skilled laborers were a struggle to find. Those reasons combined with not being able to su- pervise onsite to direct construction as he did with his previous projects prompted Nervi to make many changes to reduce the complexity of both Leverone Field House and Thompson Arena. Many advantages to the Nervi System still remained, working with precast panels and in situ concrete greatly reduced the length of the construction process. This allowed the construction site to be divided into two parts that could operate simul-

43 taneously. The prefabrication was done in a different area onsite with the precast panels laid out ready to be placed and assembled. At the same time, excavation and the erection of the perimeter columns were underway uninterrupted on the building's footprint [Iori, 2009].

4.1 Precast Panels

The precast panels were key to the rapid construction process of the Nervi System. Two thirds of the concrete for the roof were precast meaning they could be prepared while other concrete elements were cast to reduce wait time for curing [Long, 1967]. An intricate, multi-step process was developed for the precast panels which varied slightly from what was done previously in Palazzetto dello Sport. An arch scaffolding to full scale was erected to match the exact curvature for the intrados of the roof vault, the formwork was then built on this arch to achieve the proper curvature for the precast panels. Multiple sets of formwork were used to achieve the precast panels which can be described as inverted reinforced concrete pans. The construction sequence for the precast panels can be seen in Figures 4-1 to 4-8. Brick was used as the first set of formwork; they were laid out in either triangular or diamond shapes. The inside dimensions of the brick formwork were measured to match the interior faces of the final precast concrete panel. The brick formwork were then filled with concrete creating a solid triangular or diamond-shaped prism as shown in Figure 4-2, this shapes the interior surface of the desired precast element. These solid shapes were "negative molds" and served as a surface to guide the reinforcement for the precast element as seen in Figure 4-3. Once the reinforcement is assembled around the negative mold as shown in Figure 4-4, wooden panels were placed around the negative mold with an offset equal to the thickness of the final precast elements, this can be seen in Figure 4-5. Finally, concrete is poured into this final set of formwork. The bottom rim of the produced precast elements would have the exact curvature as the proposed intrados of the arch, forming the correct curvature for the ribs once assembled.

44 This elaborate way of casting the prefabricated elements with multi-step formwork was an efficient and material reducing technique. For the Leverone Field House, there were a total of 1,240 panels of varying shapes and sizes. Certain panels were repeated as many as 76 times, instead of making a large number of wooden formwork that could only be used to fabricate a few panels before they would need to be discarded and replaced, the solid concrete negative molds were durable enough to be used over and over again while maintaining their accuracy. The side portion of the final formwork is the only part that is made from wood, choosing this as the only part with conventional wooden formwork meant it was lightweight and easily removable which is shown in Figure 4-6. Finally, Figure 4-7 shows a finished precast panel being hoisted out of its formwork which would be stored for assembly later. Multiple of the same molds worked simultaneously to quickly produce these precast panels, they were then stripped from formwork after three days and moved to the side for storage until they reach their 28-day concrete strength.

4.2 In-Situ Concrete

In-situ concrete was used for the uniform slab and ribs of the roof, edge beams, and columns. Similar to previous projects with the Nervi System, the perimeter columns or buttresses were first to be erected followed by edge beams. The buttresses used traditional wooden formwork and were cast in place using in-situ concrete. To build the roof, a movable transverse arch section of scaffolding on rails was used to provide the curved surface to lay on the precast panels [Nervi, 1965]. The panels would then be hoisted and arranged one next to another, ready for the in- situ concrete to be casted filling in the ribs as shown in Figure 4-8. Once this was cured, the same section of arched scaffolding would move down the center rails and be ready to cast the next transverse section of the roof. Casting the roof in sections was a stable and efficient way to construct the vault while minimizing the amount of scaffolding. Joints between the elements were stuccoed and the concrete was painted to produce a smooth and uniform look, allowing no evidence of distinction between

45 the precast and in-situ concrete [Nervi, 1965].

4.3 Construction Sequence Photos

The following photos were taken on the Leverone Field House site and they clearly illustrate the construction sequence of the formwork for the precast panels. A similar approach was carried out for the Thompson Arena.

Figure 4-1: Brick formwork on curved scaffolding surface [Dartmouth College, 1962b]

Figure 4-2: Concrete negative mold [Dartmouth College, 1962a]

46 Figure 4-3: Forming reinforcement around negative mold for precast panel [Dartmouth College, 1962a]

Figure 4-4: Precast panel enclosed in wooden formwork ready to be poured [Dartmouth College, 1962a]

47 t,427 Figure 4-5: Complete formwork with precast panel [Dartmouth College, 1962a]

'bi

Figure 4-6: Removing side panels of wooden formwork [Dartmouth College, 1962a]

48 Figure 4-7: Precast panel hoisted and removed from formwork, moving to storage [Dartmouth College, 1962a] -442

7M_

Figure 4-8: Scaffolding on center rails, continuing to pour in-situ concrete forming ribs between laid out precast panels, some ribs are already filled from previous pours [Dartmouth College, 1962a]

49 4.4 Summary of Construction Methods

The Leverone Field House was the first use of the Nervi System in the United States, combining the use of precast panels with in-situ concrete [Bologna and Neri, 2013]. The Thompson Arena later uses the same construction method. Precast panels made up most of the roof while in-situ concrete filled in the ribs joining the panels. Cast- in-place concrete was used for the foundation, the buttresses, and the edge beams. A movable scaffolding on central rails was used to construct the roof in arch sections. The Nervi System was an economical method developed in Italy to reduce construc- tion time, however, not all the benefits successfully transferred to construction in the United States.

50 Chapter 5

Conclusion

The Leverone Field House and Thompson Arena are a unique pair of structures to be celebrated. These two projects are the last of their kind. The Thompson Arena completed in 1975 was the last legacy Pier Luigi Nervi left in the United States before he passed away in 1979. These two structures perfectly show the evolution of the design process of two similar parabolic vaults, two years apart. The Dartmouth Rauner Special Collections Reference Library contains valuable design calculation packages for both Leverone Field House and Thompson Arena [StudioNervi, 1973], [StudioNervi, 1961]. It also has various correspondences between Studio Nervi and Dartmouth College, some construction documents, and a collection of construction photos for the Leverone Field House [Dartmouth College, 1962b]. Similarly, the MAXXI (Museo nazionale delle arti del XXI secolo, or National Museum of the 21st Century Arts) Archives has a collection of construction photos for the Leverone Field House. In addition, it has some sketches and drawings for both of the projects [Dartmouth College, 1962a]. The shape of the roof in the transverse direction is close to a parabola. Nervi designed the funicular shape of the curve by minimizing the moment under a basic uniform dead and live load. He used an elastic method to analyze the structures, assuming an elementary arch of a certain unit width with fix-fixed supports. The arch was divided into 20 segments and detailed calculations were completed for each of the segments. A hand calculation approximation was made assuming an asym-

51 metric load on a 3-hinged parabolic arch. Using simplified equilibrium methods, the maximum moment was calculated to be within 7% of Nervi's results. A FE model of the elementary arch was also explored, however, the results proved the model to not be a reasonable representation of the behavior of the funicular arch. The construction method of these two projects used the Nervi System which used precast panels with in-situ concrete creating the ribs to join the precast panels. It was a process invented by Nervi and one that he used for many of his long span, thin shell concrete structures. Clues from the construction photos showed that slight changes were made to the construction method from his previous structures with the Nervi System in Rome. The philosophy of his construction method was to shorten the construction time and reduce cost with movable scaffolding on center rails and precast panels. The precast diamond and triangular-shaped panels were produced with a multi-step formwork system to reduce the amount of wooden formwork while keeping a high level of accuracy for the shape of the precast panels. The Leverone Field House was the first project Nervi completed in the United States and Thompson Arena was his last. These two thin-shell parabolic vaults combined some of his best characteristic design elements and construction system. A detailed study of the two structures not only showed his design methods but the refinement and an evolution in design between the two similar structures.

52 Bibliography

Bologna and Gargiani (2006). The Rhetoric of Pier Luigi Nervi. EPEL Press. Trans- lated from the Italian by Juliet Haydock. Bologna, A. (2013). Pier Luigi Nervi negli Stati Uniti : 1952-1979 : master builder of the modern age. Firenze University Press, Italy. Bologna, A. and Neri, G. (2013). Structures and Architecture: Concepts, Application and Challenges, chapter 235: Pier Luigi Nervi in the United States. The height and decline of a master builder. Taylor and Francis Group. Britannica (2016). Pier luigi nervi - italian engineer and architect. Retrieved from https://www.britannica.com/biography/Pier-Luigi-Nervi. Cassinello, P., Huerta, S., Miguel, J., and Lampreave, R. (2010). Geometry and proportion in structural design : essays in Ricardo Aroca's honour. Madrid. Essay: Reduced Scale Mechanical Models 20th Century Structural Architecture, the case study of Pier Luigi Nervi by Mario Chiorino. Chiorino, C. (2010). Eminent structural engineer: Pier luigi nervi. Structural Engineering International Journal, pages 107-109. Chiorino, M. A. (2012). Art and science of building in concrete: The work of pier luigi nervi. Concrete International. Dartmo. (2012). Piazza nervi. Retrieved from http://www.dartmo.com/archives/category/leverone-field-house. Dartmouth College (1962a). Construction photos and drawings. Museo nazionale delle arti del XXI secolo, or National Museum of the 21st Century Arts Archives. Dartmouth College (1962b). Construction photos, drawings, and correspondences. Rauner Special Collections Reference Library, Dartmouth College. Dartmouth Big Green (2016a). Dartmouth sports facilities, thompson arena. Re- trieved from http://www.dartmouthsports.com.

Dartmouth Big Green (20161)). Leverone field house. Retrieved from https://dartix.dartmouth.edu. Heimsath, C. B. (1960). Nervi's methodology. Architectural Forum.

53 Homewyse (2017). Cost to excavate land. Retrieved from https://www.homewyse.com/services/costtoexcavateand.html.

Iori, T. (2009). Casabella, chapter Pier Luigi Nervi, The Palazzetto dello Sport in Rome. Architettura e struttura.

J. IASS (2013). International Association for Shell and Spacial Structures, 54(177). Special Double Issue on Pier Luigi Nervi. Guest Editors: John F. Abel, Gorun Arun, Mario A. Chiorino.

Jacobus, J. (1976). Nervi's concrete aesthetic, rupert thompson arena. Dartmouth Alumni Magazine, pages 22-26.

Leslie, T. (2003). Form as diagram of forces, the equiangular spiral in the work of pier luigi nervi. Journal of Architectural Education, pages 45-54.

Leslie, T. (2018). Beauty's rigor. University of Illinois (unpublished manuscript, March 2017).

Long, C. F. (1967). The nathaniel leverone field house. From Dartmouth, with a special supplement for Thayer School Alumni Journal.

Meacham, S. (2008). Dartmouth College. Princeton Architectural Press.

Morrison, H. (1961). Nervi designs a field house. Dartmouth Alumni Magazine, pages 20-23.

Nervi, P. L. (1956). Structures. McGraw-Hill Book Company. Translated by Giuseppina and Mario Salvadori.

Nervi, P. L. (1965). Aesthetics and technology in building. Harvard University Press. Translated from the Italian by Robert Einaudi.

RSMeans (2017). RSMeans Cost Data. John Wiley and Sons.

Structurae (2016). International database and gallery of structures. Retrieved from https://structurae.net/index.cfm.

StudioNervi (1961). Leverone field house calcoli statici. Rauner Special Collections Reference Library, Dartmouth College.

StudioNervi (1973). Thompson arena calcoli statici. Rauner Special Collections Refer- ence Library, Dartmouth College.

Tullia Iori (2009). Pier Luigi Nervi. Milano : Motta architettura.

54 Appendix A

Nervi's Original Calculations and Documents

This appendix includes the calculations from Studio Nervi for the Leverone Field House. It shows the detailed methods Nervi used to design the structure, a very similar set of calculations are available for the Thompson Arena at Dartmouth's Rauner Special Collections Library.

55 STUDIO NERVI Prog. IIZO Peg. I

S P E C I FI C A T IO N

OF DARTMOUTH COLLEGE FIELD HOUSE

U.S.A. HANOVER, NEW HAMPSHIRE

1- CONCRETE - a) Controlled concrete conforming with paragraph

913,2 (c) page 135 of National Building code and paragraph

C 26 - 3640 of New York Building Laws (from page 203 on)

shall be used throughout the Building both for precast and

cast-in-situ concrete. Thereby allowing a working stress not

less than 1400 lbs/sq. in.

b) Proportions - Unless otherwise specified the cost-in-situ

concrete shall consist of the adequate mixture , which can al-

low the working stress mentioned in the above paragraph. The

precast concrete shall consist of a minimum of 850 lbs of cement

per cubic yard.

c) - Aggregate - 1) - Throughout the work the aggregate shall

not exceed 1 gauge . I1) - In the rib-beams,to be poured 4 between the precast units,the aggregate shall not exceed 3" 4 gauge. I1) - In th precast panels the aggregate shall not 1"5 exceed gauge.

Ratio of grain sizes must 0e proportioned to conform with the ap-

propriate American Building Regulations.

56 STUDIO NERVI Prog. Pag. 2

2 - REINFORCEMENT - The steel reinforcement must have an allowable

working stress of 20.000 lbs/sq.in.

Reinforcemsnt shall consist of round deformed steel bars. Hooks

shown on drawings indicate only ends of bars and are not neces-

sary, where deformed steel is used.

3) - FORMWORK - a) All the surface of the structure which are to be

cast-in-situ, and which will be visible shall be cast by wsing

forms mode up of timoer planks not wider than 3" - 4" with sur-

faces planed and sprayed with a liberal quantity of form oil (1)

to prevent adherence of concrete to formwork.

The plsochVgof, the pa* surface., whilol

the visible surfaces of the castings, shall be executed exactly as

shown in the drawings. Particular attention shall be given to the

corners of formwork.

b) - Spacers - All the reinforcing steel shall be kept back from

the formwork by the use of adequate spacers made of concrete

1" - 2" In width, as shown on the drawings, in order to ensure

proper coverage for the steel.

c) - Type of cement - Surfaces of the structure which are to remain

visible shall have a perfectly uniform colour. This shall be obtal-

ned by maintaining a strict contro) of the materials for uniformity

as well as maintoing a constant proportion for the mixes.

57 STUDIO NERVI Prog. Peg. 3 , I

Further the concrete mint be continuously controled to avoid any

irregularity of the surface finish. A light vibration must be given to the oon-

crete.

d) - Precast roof Units.

I) - As shown on the drawings the entire roof will be formed of 1240

precast elements The units will be repeated as follows :

Type 1, 4, 5, 7, 8, 10, 11, 13, 14, 16 76 TIMf S

Type 2, 3, 6, 9, 12, 15 72 "

2 % 2 R, 3 R, 6 R, 9 R, 12 t, 15 R,

Type 2 L. , 3 61 9 12 L, 15 , I LiI' 6 L I L I' L I I 2 - 2 R., 3 R., 6 R., 91 ., 12 R., 15R., I I I I I I

Ii) - For the making of these precast uniq it will be necessary to

construct 26 different formn from each of which shall be obtavei several units

as shfws abOve, eath of which shall be identical.

We believe that the nost economical system for oitaining the form

Isto buIld a skeleton of brick which shall be finished off with cement mortar

and gypsum plaster. This final delleate operation of smoothing shall be dons

oy skilled workmen in order to obtain a surfac, perfectly smooth and without

any irregularities. See DRG. 3 w. The forms shall be well oiled (1) in order

58 STUDIO NERVI Prog. P,,. 4 1

to prevent the adherence of the precast units to the formwork . When the Ole

ments have hardened sufficiently, depending on the quality of the cement and

the air temperature, they may be removed from the forms. Under normal con-

dltions (in Italy) a minimum of three day3 will be sufficient before the removal

of the elements. Additional concrete forms can be obtained as shown in DRG.

No. 3 w.

After the element ha been removed and before casting another unit,

the form shall be checked for accuracy and smoothed over again with gypsum

plaster, in order to repair any damage that may have been caused by the remo-

val of the previous element. The form shall then be covered with form oil and

the procedure as described above repeated.

Ill) - Storage of precast units.

The precast elements, as they are being removed from the forms,

mwt be handled with care and stored horizontally and in such a way as to pre- of vent any plastic deformation the concrete not completely hardened which

could have serious effects when they are placed in their final position.

(1) - Since we hove no experience of American proprietary products, we

cannot recommend any particular type of form oil. The choice of the form

oil is, however, very important.

59 STUDIO NERVI Prog. 1120 Pag. 1.'

STUDIO NERVI PROF OOTT MOiPIER LUIGI R ER vj STRUCTURAL ANALYSIS G0OT A"CH ANTO0 0 5 ANsV soIT .vU A.. . S 11CMIC A 801.5IA

DOTT VITT C O L A T A L. 0.044

CALCOLI STATICI The roof conslshof a ribbed cylindrical vault with slab of uniform thickness and ribs tapering from a thickness of 2' -0' (at the spandrel) to a thickness of 1' - 0 (at the crown). The longitudinal spOndreT beam has a considerable flexural rigidity. Thereby an arch behaviour is prevailing along the cram section. LAVO RO; The structural analysis has been made by assuming a tiinsverse por- DARTMOUTH COLLEGE tion of the vault as a fixed-end arch. The central curve of the arch has been deenelned re e vnF FIELD - HOUSE res due to the dead load and to half the live fead. U.S.A. HANOVER, NEW HAMPSHIRE It has Deen calculated also the very light flexural interference, which I asises In the fixed end arch Indicated above, because of the deformation work of the normal strain. Prog. N. 1120 Therefore the stresses corresponding to the various loading conditions Calcohl have been calculated separately as follows :

1) - Uniform load over half the vault (see scheme 2 page 18)

2) - Horiazontal force uniformly acting only In one direction (see action of wind pressure page 49)

Thus, by properly combining the above loading conditions with the live load conditions, the most critical stresses have been obtained for each section of the arch. These sections have been checked for stability accordingly.

Rome, 5 Magglo

L I STUDIO NERVI Prog. 1120 Peg. STUDIO NERVI Prog. 1120 Peg. I

.a, vso.t VOLTA DJ COPERTURA

As for as the seismica I forces are concerned their action in the troarver- -7An-lici dl carieht se direction is for inferior to the wind pressure action. On the contrary it is La poLezaione normele eU& selta.

necessary to take care of th* seismical action in the longitudinal direction. ) I a) - In chLave It

&Glett- 1- X2, lxit, 70x0, 10. - 0. 870 me. prefabbricato ~brdol1 4.5OxO. 0x0. 045 - 0. 326 "

1, 196 z2. 500*2. 980 K4 I

Liagonali ISxO, 456 NOTE : Untees otherwise specified the metric system is used throughout getto in operajtravettl 7,60z0. - 0, 18 rc. the calculation. aevereajj 1, 30x0. 24x0. 50 - 0, 156 2.500 xO, 674 z 1.680

6 6 0 K. A--. .C-pph-."- i. 4 . 2 84 Tale carico insate asu un' area A S7. 25 x 22 10. 30 mq,

U Leare (peso proprio) a mq. risulta * -. 4. 0110 ril mq. + 204 K PB 1080 .308 / mp.rweLu aaxion. - 470 Kg/mq. b) - All'imposta

soletta x 2,60x6, 70x0. 10 - 0. 870 me. prefabbricato IC436 bordo 4. SMx. 8C x0.045 0. 522 " b-., 2.500 x 1. 392 - 3.480 Kg

travetti d aganali S0x. I5x0, 755 0. 860 mc. getto in opera 7, * trasversall 1, 30x0, 24x0, 80 - 0, 250 " 2.500 a 1.110-2.780 6.260 Kg

'~-~- 3 STUDIO NERVI Prog. 1120 Pa.2 STUDIO NERVI Prog. 1120 Pe. /

4 ,4 #4 . A^e -f /~ r .s I... 2,84 -''.+s..+enr~ - 0 A.+ A-i -ws tA a.... Tale carico insiste mu un'area A a 7, 25 x 2 10, 30 mq. y - ordinata del punti d'asse rispetto alU'asse x passanto er Ia him e

x a actame y I. . y 'eq._ 1./_d, q,S 11 carico (peso proprie) a mq. rimulta: pB A c .richia mq. in chiave E all'imposta 61260 ''' Jr'-. Kg/mq. L - semiluce p'A 10.30 . 610 Kg/mq/ + 20 imperntab 630 In .p .a "; .-. m riata c, X - spinta in chilave relativa ad una striscia di 1, 00 m. In pro;esimne orisaoatam I cartehi' rieltano : E ('si) -- / ./J. 1-e wr4f, L'equazione (1) al pub anche scrivere p'B 470_ . Kg/mq. pB * * 470 in 2 3 Ces y . Ki x + 122

j. P A A 6 - 810 Kg/mq. (2) Con (K, B cooA 39' 0 776 2 X

(- .~-- .. Delia volta sono imposte 6 L X I,3~,,, f= 4'- x ..Lr.X.. 1/2 itce L 33,375 m ' 1'2= ks 11 valore della spinta X Si ricava dalla (1) per y = e x L -1- .~.I .Jr., .j..-. 5 (la (1) infattl vale per ogni punto dell'arco) : Sla frcea f 1s, 335 m. ., b=33~2 .. -- -~ 1 L2 PA -EB L I at ha X B 2 ( 6- x-- 7- -(p +2 pB. +0,. ".$,-1Jam -... #.hC -jf-I*.,t b' -. - --- #,A #.A,. La linea d'asse mar& ricavata :imponendo I& condizione che essa sia 2 B- A ilmases del carico permanente pio mett sovraccarico accidentale, L 1 332375 S , ( 2 x 570 + 910) 2 28.540 Kg/ml. me da schema

ft,ea. 1 ' Mk.' '00 ,7o . I 0M .. e. - (" a-'' K 2 570 - on9s s () i coefficenti (2) risultano 1 2 x 28.540

on - 340 -5 2 6 x33, 375M8. $40 ,4 .U

in considerazitne une Pac 4. / i4 ./ a. - Per U1 calcolo della volta a prenderemo striscia larga m 2, 84 (part &llalargbeaa di I tavellone) . Talee--- k here-9 m u! t ~a- Equa zone della Unea dasse utrieta;dlsb sark in seguito indicate con Ia denominazione di "arco ele- t pA B x 3 conci; di x 2 mentar ". Ogni arco elementare sarA poi Asuddivieo. in ugual .. d ,'-e ./ . . AA. C0 Y) X B 2 L .4 p.y..., h...z#..I p,..+,.. protezione ori montele e quisdi di sviluppo diverso.

144 _ =.A"' G) '-% .- '.j I.(l..

M M IIII STUDIO NERVI Pro g. 1120 p,. 4 STUDIO NERVI Prog. 1120 Peg 5 T ELLA 1 T--

1 2 3 2 Sezione x x- x I x 1K 2 .3 A IA~'~~I'LAL4.. 0 - B 0,001 00o 0,00 0,00" 0,00 0,0o 3, 337 0, t13 0,0338. 1- 57' 11,139 0,013. 11,152 3495 9.11.1. Q.00; o'.114 k33375 0,1350 0,1047 5- 59' 11,139 0422. 11,261 3. 3M -2a, 3.31L9131.8T 5D 44.SR 558 _.297-408U_3411 8 3371 0. 598 . 0.1792 10" 10' . 11. 139. 0,358 1LA97. 3.3908 -3 1-0-01M 1109 i . 003, ? kl -1,001 0, 060 .0_61 4 1-4371M40 , D.2&7l JA 27'. 1L 13-. 2.740 .l1l m1.AUI 1 3. 31101 __78. 2225 2.L 379.,2704 LiiQ QA4g JAjl 3, 337(l1A38. 044Q4 19* 49' ._1L139 1.290 . l429t 3Aa5.. I I -7 1784271 A-4L 8132 2.71 0.2?1 .. 0h7 rho 5, - _L_11 LARS 027213*2Q13 ' . 13. 01 . ali17 LS L_ 20, 4f L0MA. 03D.031f 4 , 3, 33_7 ,727 0,5174 2 22' _11,139 2,982 14,121 3,7572 7 _"_3 54, 8064 12. 751, 402 5.45. 675 t e, 209 (.41 Lu .61 1 A31-' 24' .l013U- C1 . 5,109-3,41M0 -L70OL 72L81PQ J03 1q30 7, LI9 _J13-a AM25 22 - 1 3.3375 2,371 C.7104 35- 24' .1LLU 5.52 121l 4.02 0 S ID 9 302 2)0 t-27, .0L 374 9,010 1,61 1, 62 .4 10 -. 13314713 0, 812 -I 0W. 1L139. L 310 11. ,49 ._4,3l1 10 A 33.375 1.113,8906 37.176,0988 11,123 2,21 13 335 Z1 38, 752

------~

7I

8I

I I ' STUDIO NERVI Prg. 1120 peg. 5 ! e STUDIO NERVI Prog. 1120 Peg. 6 TABELLA N. 2 7 it il. Z

Llarco elementare si immagina suddiviso in 20 conci (10 per ogni smiarco) di ugale proieneAuUor;.zontale. -0-l -- ]-oXL 1,oo 0,0001 c..l Acm S/w,./.P., -1 Ar,,,, -0 .- ,.. .,( e.-.4 I116873 0. on Ricerea dei valor angolari relativi ale sezioni di estremit dei sin - - 3. 3373 0.11 02 3375 0.M2 -., 716 41 28' .$ I .n : . a 3-n n At 35. 472 0. 1 1. "* 4

4 1~" 1 A7 1432

4 13.3500 1. 2i 3. 3375 1, 013 0. 3035 16-53' I __ 5 15. 0187: 2. 45j a 16. 6875 3. 0 3. 3375 1. 28a 0,1841 21_

20. 050 . 3.337 1. 57 0 471 3315 S21. 9937 5. 30d ? 2. 6256.203.3375, 1. 884 0. 36411 29'27'. 12-t. 1121 7. 1fig 8 _Z6, 11000 8, 25$ 3 ,3375i 8.,200 0, 6600 33-27' 929. 3687 9 3 31..46 11.7v 10wA 33,3750113,33 t3, 33751 2, 89 0, IGI1 40-53'1 ( STUDIO NERVI Prog. I1120 pag. 8! STUDIO NERVI Prog. 1130 Peg. 7

e Tabella N. 2 -r . M. 2 c..,'d Per l a condizione di carico considerata (carico permanente

+ met& sovraccarico accidentale) 1. linea d'asse dell'arco coincide con la funicol are dei carichi. 0-B 0 00 1.000 ,-,, . b" , &e4...// -f 7.sf L'arco si pub dunque calcolare aol metodo della caduta di "'eA. ( spinta. 2 La fo rmula che fornisce La caduta di spinkin un arco inca- 3 -8 's S C - - - - 6tat(ved . Belluzzi Vol. 11. pagg. 217 e segg.). 4 _1. 202, 0,978 S 0.3149 0. 933 A 0. 20 0. 957 s _L ,- 0, 359 0, 93$ g o. 4200. 904

, 0.379 05 0 0, 831 Nella quale Cfl jo . 0 0 79 10 x . 28. 540 Kg. = spinta in chiave - 0.?751 10Ai0.631 ,,, /,, P4 -C rd-d-fM Jegs-- A5.viiuppo singoli conci

1. 44 ..... 4.Awes 4C A 0 sezi one baricentrica dei vingoll conci Nella tab*J1a precedents sono atal trovatl g~l elementi geornetrici

delle aezionl di eatremitk d aL rgol conci. . S distanza dei baricentri del conci dall'orisaontale per U centro elastico. -y.A sos-+

speso elasti dei igli co.ci

.. ;s e..- s~aione conc to generico

4- - 4-, CLT

CL ------Mgm-, - ~

Prog. 1120 p,. 9 / STUDIO NERVI STUDIO NERVI Prog. 1120 Pag.10 'r 61 ". 9 e e n.n . fl e.-." - = j TABELLA f. 3 4 ca del peo elasti

Ricerca del A%* -

' 4.s ( se -T..be a) L sem M As noti (ved. Tab. 2) 1 333 95 Sh3 2S3L85 $5.946 11. 70 77,646 2.Jf 4 2 .0# 7- E s2 5 . 105 Kg/cmq. - cost. 2.340 I 3SLI7f54.9 192.5114 R1 . 1.704 84.43 2.704 . 044_ J(etone a ") (50 a + 284 b -234 c3 L jiIn MA 56.5 3231-R3 30. 79i 11. 7 92- 4L2. RAi 2. 340 5.111 S r-r L; A 4 344,46 4 59,64 3W56, 93 84. 923 11.70 100. 823 2.3982 2.340 5.322 2 h 2 10 + 234 x 4e,-Mt _7.U9 09.244 3.1244234 -- DA4 Sr-r 20 2 6 3903.17, _L362. 91 CL4 4277. 16 1L0L U*- I .L0I 18. MPI 3.7 2.4 LU

A * 50 . h . + 234 x 10 7 375,78 68, 42 4681,30 117.032i 11.70 123.732 3.421 2,340 5.761

c e b - 10 -. B-"1.26 71 5 119.40 _12_.98S 11.7 139.6 .53.77 2. 340 5.915

9 40940 74.8215598,03 139.951 11.704151.6513.741. 2.3401 6.081 a h - b 7 11 A I9,1 78, 624 0 1. 1 153,0381 1.70164. 14 3912,3.3401 6.252

I

A

I. I I4

r08L~~~~~~6T~~00686 isC9~T181~'1B'g 0 '0- ~

Zee'~B Tel's LOLa i 00~ i 0O9 x SL66') Sq 01- SO 'ol LOm g u - n r -- g -wt

tT61 :OILSmO61016 bsg'L t89 P L - U9

!01- Fl p-I 0T WHOUA~'t 08'199f, Iow IGOV-090- OOgs-LS t O IZOLi' b~L ' 16-9 L~L 6,91 1

a _ u _ ___ -I!- -S ru-Vq -VL u- T 16f ~

- C9KI:9i -1. ol Q Li~'i 0_____ f cIlf'TOIxeg a -fnN f

6 6 11 @d OZIT - -d IAJJN 6o i~.s STUDIO NERVI Pr. 1120 Pag.13/ STUDIO NERVI Prog. 1120 Pa. 14

TABELLA N. 4 - T-m- ^/- 4 Segue Tabella N. 4 Determnsnaxlono del centro elastico e dell& caduta dI apinta.

SA 5 . .A ii

10 - O2, 004 _ _0 -- x 1__ 320. 9 103.fl1 0 10413 25, 267,699 x 110 295,36 87. 23 2 908.686 16jj3 50441335, 57 0. 060553' 1a b0 I&1149S- D&._. n01L4.& A07x1 7_9 S_3. 96 x 1610 14LJ.A 61.52 4 53.I49 x 1~0 532444,j_ 0, Og$- S 1 145. 1 1158. 55. x 10 177.26 -10 00 51-76 x 10- A464 P52,0 06q452 _ 7,2061 x 10 245, 1 1766,215 x 10 75, 96 5.770 -1o -10 a -7 51 x - 10 91 006469L _6 6.04J xD .b0-t 73.3 2428. 279 x 110 - 52,2a 4 2.2 251. 72z 100761 0,0623 7 5, 9187 x 10 .30, 6 3140. 462 x 110 -209,54 43.90 75.78 -10 -10 8 5413 x10 718,9 3899 026 x 10 397. t_ 158.A" 0.1L. 431 x 10 M1 1U.. 0- 0 L

- t- n 10 081 09. 40 0. 0&732. 9 4,9975 x 1-010 939,5 4695, 151 x 110 61.8, 44 302.4641 1035 -10 10 3.518.132 x 110 6252 430, 10 .0.O6879 10 5x 10 193, 2 5518,908 x 10 -872,144 760.628 0 --- Z' 4--9.572. 276x10 A -0,66146 ( 4 Edw.73,3174 x10 ZAe'-235139,465 1 10x 0

. _ w. 23.539, 45 -321,06 cm. y Z 73,3174

I orw-, Prog. 1120 Pa..1 IS TU D IO N E RVI P,,g. 1120 Peg. IS [ iTi , N

TABELLA N. 5 - CALCOLO DELLA SPINTA ADDIZIONALE. Determinazione di N e T a ml. di arco.

AS- I 0 66146 10 x 28.540 W 2, 5. 105 9. 572.276 8 0 Kg/mi.

Per effetto della spinta addizionale mi gen.~ano neUe sezioni dell'ar- sez. 4M C.i co del momenti flettenti. 11 massimo Yalore di tali momwnti at ha Ie in "haevLt.: 0, 655.0,755 416.176 16.644 16.641 L 893 3, 3375 2.960 5 ,796 14.491' 19..417.218 Mc . 4aX. f - 60 x 13, 335 - 1. 066 Kgm. 0 886 3 33751 2 667 n-rh ,C....i ' .sm //e_','A _ 11 ,. d SI ritengono percib trascurabili quei momenti, data aloro mode- to a ,337 2,730 5500_8634 33.3 20.591 1:.731 7 825_.7Q1.f 31, I371 3.. 7 4 4 sta entit&. 2 .410A7Q-lka6 21 35 .' A 0) 791 3,33752. 640 6 0, 426 0,904 10.520 22. 325. 12. 1241 Q0 6 757 ____ 252 CALCOLO DEGLI SFORZI NORMAL! E DEOL! SFORZI DI TAGLIO 5 0,359 0,933 8.866, 23.041 10.217 2.163024.622,1.277907O0761 4723 3,3375 2.413 NELLE VARIE SEZIONI DELL-ARCO DOVUTI9AL PRIMO SCHEMA S MC-.1 00 . + , L..,c. 1- J 4 _ :0,290 0,957 7.162 23.634, 8.253 DI CARICO ( carico permanente + 1/2 sovraccarico). 4 089 3 3375 2.209 3 0,208 0,978 5.137 24.153 5.2 -rhe ~...Au f,. .. e Acs.k...-.k- .. d4 n-_.. /.,.s Le formule che danno 1li sforzi di taglio e glf eforsi normali 3 655 3, 3375 2.166 &.A. . fez j-CG+1.C jy : La.. .4_c.w_. - 4&.. 44P 3.84 mono rispettivamente: fa 3,31.20 . Tvcos -X eon.- CO84 Pr I.- 0 0 .000 1,000 0 24.696 0 () 2 F24.696 Kg/ml. . son (x, + X coo a-N P

y .a hJLre&C4-,.

I STUDIO NERVI Prog. 1120 Pag., 18. STUDIO NERVI Prog. 1120 Po,.17 - Segue Tabella N. 5 CARICHI DISSIMMETRICI. ) Larco in *same ear& ancora utudiato per le seguenti condillo- ni di cart6& come Indicate negli eami seguenti

21.487 . 0,0000 0, ONO .f7 8o4 r"r .. o e e 2, 604 2. 980 1. 803 2,3 4 . 5,79168

23. 736 5.847 8.316 4.876 10 6? 7 24 .o 600 42A3 . S$ 32.0433t .1 6 25. 728'11. 240 4 788 10.161 31. 46"0i -. 5 4LIU31i,766 A 942 12. 844 30.477 20 Del 1' acheme giA etudieto 4 27 6 12.17Q 4. 92 "A- _24m 1O7 02 '...... ,...... ,, Cv1sp 8 1 43 1I.01L 29.A1 62

28 7 20.q64 2 410 - 49-A0 -. - Z8-j75247j7 1-198 .22.669 28.524210 28.46O24.696 ,00_0 24.e 6 28.46 Dal 11IE~allili!~ scem sgIAr stue d djrto

e del 2* echema che sarA studiato appresno

2* Schema ( 9 STUDIO NERVI Prog. 1120 peg. 1 STUDIO NERVI Prog. 1120 pag. 20

b 4, A. /.d. 1 per sovragposiziane degli effetti si possono ricavare la I e Il con- Le 3 Incognite iperstatiche assunte (MA, M , XA C diaione di carico come segue ;aranno determinate Lm;;.endo le 3 condlU;Pni seguenti

La I condizione di carico si ottiene per savrapposiione degli effet- fSeh- / __a s-hr a + At . l<. .0t eh'-a 1) - A 0 ti del 1 schema + 2' schema + simmetrico del 2- schema. 2) - A . 0 La IX condisione di carico si ottiene r .ovrapposisione degli effet-

ti del I^ schema + emisimnvtrico del 2- .chema. 3) - . - 0

.-. io..1/A tA~ W ., -d- .SA.,, -d- STUDIO DELL'ARCO CON IL 2^ SCHEMA DI CA CHI. Per le equazioni della statica *I ha -q ile r/...-of -j-5 /. -1 0 A-f /-- ( ... _ ^ CI Sark ancora studiato un arco elementare di larghezza pari a 1 2 in- x A = c X .. orJ'A <-4a -411 -,// &e 'f.-d"drd -1 betde -ne. .' terasse (m. 2. 84) e questo sari suddiviso. come precedentemente ~.S n~- -f ,- /- h..14 S/.-) 0 1 h -' -Q d. e - *-$ - Mc -M A ;dto, in 20 conci (10 per oemiarco) dl ug;&l pro;eione ortz.zoae, Ycy y c + L

L'arco In esame 6 per tiamente incastrato alle estremIth. -- M A - Me -K! y A A L I,

dove

L * luce dell'arco - 66, 75 m.

* ale -- S''k k.. r - YA' YC Reazioni di trave appoggiata

C Y A 7.109Kg.

Y = 2. 369,50 Kg.

XA ~XC

xA x - -~-~- - I:

22 STUDIO NERVI Prog. I1130 peg. 21 1 STUDIO NERVI prog. 1120 Pog. /

Da t 3 condizioni di vizlcoUi acatariacono 1. oquationt riaolutive

(1) -- -M amM X" 0 4 2 2' 4 47 7, (2) -7-M aM (MA)' ws' o x 4, i 30 9

(3 :ma M(Mc) A

uieU. quai

M - M A L Me L--x)-X S ~ M-L 2 L 2 5-~- M () a

3 (4) IM -, I L1 a''* j di 1 (MA) (+ X9 i- easandoM I nomenti 3) 6,- .- 4 - - - trav. appoggiata nelle va- ri sejn debarco. Le eapresalont 1) .2), 3) sotto sommatoria, aotituendo a MI ( L ~2 2 1 L 14 M ) M C) "i 1oro4 4sori fh-nlti dall (4), al traaformana

in Leaprostone dt M 8 rioulta 1) m MM'AW -f Mi + y M +-LM +~ MA- M- - - P (u -K - x ) - a L a a a 2 A L a A 2 a A 2 B '0' r r a A '2 a

-Zt pr Ur + x P rdalla sezione 10alaaezione 0 A L 'a , A 2 c L a c 2

L L +xay8m +f 2X-2 IYa X+Y a .XX V '. a". c L c A l2 x.) s 2: r (ur - xe) -YA (T~ - x 3 0 p..., ge0. . f.. 's ,. - .1 5 P- U +x P dalla sezione 0 alla sezione 10' r a (MA) ~2 a 4 A 2 L A 4

a 2L c 2 2 L a 2L 2 2 .M + x MA +~L Ma m--y-M + -a X A A 2 L c L c L

- L '-7 0

I M ------IN - - 23 STUDIO NERVI ,rog. 1120 P.g. 24 STUDIO NERVI prog. 1120 pg.

-r- l . ,Y . 3)- I' M jw+--M -m + *-M + TABELLA N. 6. 2,30 3,375, 947,85 (MC) -2 a 4 A 2 L A 4

- m m + x - X + - m2 SesloX VS - t - - M --- M + 2 a f X + L2 A L 2 L2 L _0 .. 33.37150 13.333 .00 .lfL.-31,_7A= 11-.2,12 247, Al a u + L -9-- 30. 0375 10. 922 . 2179 2R. SR27 O sAL a&- -- -A 267000 A,25.1 2 _ 650 - 2025.. 312 ' _ __ 23,3M2 IL209 1A.Am25

. 20.U021 4.AU I3.5an 11.32562 -2.. U3 14.5. - 1 6AL _051 75 IR M27 2 AR "i as A. . 1. 3500L 01 .. 0250.~ L 6212 1 31 942.853.

. 3437 .129M - 14 L k S_ $.750 0. 443j2.7, 2M2 n.M 25'.I

4.-W, 0,001 94z,85 0.0000 0.-000 3),47 . S . 1,6685 0, 001 1' - 3315 0, 113 3S0 125 2 - 5,0062 0,251 -21 6,6750 . 0 - 4. 0500 -_S. 3437 U1.3 -100125 1,061 43,375 4r'-11. 6812 1, 438 A 1 -1.3500 J.321 46 7250 ,51. -13.0187 2.451 5' 16,6875 3.057 50.045 -18,3562 3.733 -20 0250 4,482 53 4000 - 21. 6037 5 3Q06 7'.4 - ~2.3L 610 56,75 8 - 25,03L318 8' 267000 8 251 .60 0750

9' - 30.0375 1 0.622, 63. 4125 375 13,33 11,932 667500 STUDIO NERVI Prog. li0 peg. 25 STUDIO NERVI Prog. 1120 Peg. 26/

Segue Tabella N. 6 Segue Tabella N. 6

se -. r Xc P

I-a in a n m,00 .nan ,G 517B I . sA SA , A87% 10 30. 052 F22a4L AiO2- .230 . 22. Q90_ 2-..-0 .. L_ - - Q~I1 47.6 1 3 76 L1 SAIL 56.941, 1595.70 . 1f 4. 2 41.128 548. # 339.$31 5.3340 _.tl lL 7 2432 4.- 7 66,2nJ945 5j4 A M51j _ " 4,9672 L.M

65AA.O3 316.4. 4.0005 2. 2410

_11, 26 47324 7-2. 2 10.221L81.1425 Th2 4 14.112.31 2AL, 7901 1054,674 3 26 3. 3;37 158 4 4 85.419-11139.062 _164.090 2._ 47 0x. $kS_ 2 71 1518.2 47l32.2Lak5 7m .a a s It I 3 15.25K 1 3 88.583 ! 1 24 93.986 2,0002 05305 17. 08 I 0 I ".584 1181.297 41.014 13336 0.2315 1 .566 '530S5 . 8. ~ S 85.422 JLJH-102- "02 607966 41 102.A8I61, 01L4fl 0_.002123-109,442 281 1 - 79.096 1054.745 0,0000 9_ 0 _ _.0000

6. -- 51 158. 187.197124714,% J lL 50 . SS210. 9".4. i 18 n4 71,1S8e 949, 292 6.044 S156. 167 !47.5L ... n , .39a.2.L2 i. 12 v152 14.47 L 21 63,2791 643.825 -Ac N L 00335 0.2315 W- AL. JILIS824AIL50- .2.453 - .903 55.3721 732.386 58.750' - 2,0002 0,5305

A 474l 1a12 61.19 17l 6610 5 39 555 827.466 120. 920'

158.167,9478,50 35.6 1 S31.447 4--2 qL3 ;2410_ -4,6702 441 V 3.739 316.560 147.395, 1.1045 6 a, F 1Al.4 .167]47825 1$-7 19478,50 47.7O61 A 111. Q93 10.613 -5.3340 'I .A8. 1!7 9476, 5O 45O.79 - 204, 7.10 91 7. 922 105.640. 84. 147 - 6,0007 5,31101

10, LSJetI1947A0s 141.11 - 31.34S I 10, 0, 0000 10,0000 0.0000 -6, 6675 49.675 14. 028. 185 2602.777 140,0175 0,0000 49. 514 STUDIO NERVI Prog. 1120 peg. 27, STUDIO NERVI Prog. 1120 p.g. 28

-r.be Ta.. e.-l Segue TabeUa N, 6 Segue Tabe*a N. 8

Box. s 4 ..2

.242.._35L4A4 1 -10- A 0.A1OQ 99 9 . A- 999AL-1.1=ahI 0.2100-, -0.-000. 283.2687 11!1!269 1102 PMU~oAAMIL - 202.151 0. 202 A 4- a-. 220,30170 3.30012 220. 0542 68.0790 o 0- loon IS1.20.7M9fl1. JAMII .110 11 - 142 ~1 145.05776 2.17296 165, 5940 36, ss 17 - H 7 28. 472 0. 17498 0,34997 545.806 0,1225 13929 0 9 89.75s205 ,44 " 119. 5349 20,0883 H 6 34.79 014 99 0, 2999I 49.0AQ0 A P3jQ 3OU.7. 81. 5302 9, 3452 __ a - _0,76418 .L S~a..i4 1240s_ ,24Afl .-. 22s1. .0,0423. .2*7 -4 51.2331 3, 6902 cTq 42,709 O -99S9 9.993 ._ . 16222 . N.0400 1 I 14 55 Da4L " .. 10.62326 0.15914 28, 2969 1,1257 I 3 44 291 -097499 _0,91 .1.as0...0oaa 3283 *12,5462 0;144-- 44.202 -... ,214 Q09999-.09 -- A. - 0.o a011L.. A. 3.01.37 0.0128 -- 1 4I 711 -. 7J 0, 0210 . 0491 11.131 -0.0=25 -427 M Q000 [2- S.4B. %4.. .00 0,lo 00000 0.000 010000 - 01

. 37714 - 0, 00565 13. 0137 0,0128 I 3. 5.594 - 0.02199 -,.04999 1.1.133 0.00n5 35518,

2' J 09052 0, 04630 12.,3482 0,2144 2 . 31 639 -0,04909 -0,09999 44,5 . Q O00 :MITI 27 r3, _ -10,62326 - 0,15914 28, 2969 1,.1257 3' 27.686 -0,07499 - 0, 14996 100.250 0.0225 L 4, AU 51. 233 1 3,96902 4 -2.64535 - 0,38417 4s 23.731 -0,09393 - 0,19998 178,222. 0.0400 -8411.1

-51. 01369 - 0,76418 81, 5302 963452 19.777 -0, 12495 -0.2497 278,472 0,0625 0 9137. 05

6 -9,75205 - 1,34449 119, 5349 20, 0883 -16' .. A.9 Oas . MI:9 OmASI .401.000.. 00, 0 .ann 8423,5 -145,05776 - 2,17296 165, 5940 " 38. 5517, 7v 11.869 -0,17498 0,34917 . 545,006 . -. 122. -. 1024

IL 220 170 - - 3,30012 " 220, 0542. 68, 0790. at 7.915 Q., 19998 199- 7 6 712,6!90 01600 -A343.

' -319, 0532 - 4,77949 263,2867 112, 8269 of 3.961 10,22498 -9,44996 902,251 0,202 , 10 I 7-L0;_2 j 10' 0.0000 O-, 22496 -0 ,40996 1. 113. 991 0.2500 0.0000 19' -M0A 0000 3734,"2662 2641.0766 e T1T-Lgt ILSS'oft-C98 0000 0 0000.0 r :zo'ieo 39Lt'9T'-91'LLI'-L1- - cote I'Iit .01 1 00010 ssw i99 961't SI 00Vi0 0000 ' x 9Tt'g .01 910'99 '1 1 1'IV 909665'- .6 LL L1'9Z- 0090'99 19909 U*L9Og H 0 19 to

lIT' I 9 OizL 'M 90'991 g 9398 )991 8'LI- F - 6C3999- 9 19'LC 6I9 19 1e6''T 80119! 'L9 It I'Li it ro 9999s 00'IW aise't - a3U9'666 ~ ~ -ttI~~ ~1XOfl - -Z'16 ~1flL '91 son lift I89 *91 1I L110'9 I TOM 07 4 Me"II'Tt S60 '98 l9X'19' z

Tog IT Ofor'fs tote 4'of tou'Itt'; t9Ott'9 -tX 3FOT9O9U rff t L 69T?'VZU OW'0tU' t 60P 9M11 be

Tel c 11T MVio' 9LT'ss39UT tiU- _10"O it ft 0-Vi 60I'l gO-0999 lot 9-- fXfg11 ---

- 0C1'98't 6LC90'1 +

C1-09 OmttlO M 1999'01 - *!,r6.'1- Ol19V0 -- 1~ 11;001 90 = 0T0'9I5 9d9'LWO 'l - Txln T

955 00at 'O - 1fl9'966'! L6999'0 - 1 T Lwt' I l ut r9 14090 t 1C''O i fT fi T T T

0000' 0 0909'I9 TE 000'0 0 - 10'- EWU 10D 0 L990L U1 01 f9T0~'T iT'-66, 1os6eo0 1SS9L~t6~-L90 - bOI 01 969C'IT -004 t [9!6'9EVI O t-gYso 1.091'Cl 9990 '99 EBE'009 099O' 0111:91 i t

1918 vLC'T VSLIf,6 9 I-

906OSE tZEWOLSE'l QLIOYZ ve (t g - 9'93'I 9x983'9 't T , _is~F (( 0 - o -9 wu To0go-c I I-otz ' - - -,iw rar -06LV W io~ -019'1.99 T- SV -M-Z

tool 9u1se 6 1 , Uw 'U L

' 910so it 18199i 'i 5 'A1 17it VW WT O~u~l~991009991909' 9113LC Vi 0'L , 0 ~ 9 0 ! x W1to 'V 6 L7 Ts L s' E rotit : iiiif t 006 0- 909 9 vs ' LCLU93 t ' U 4T -- [6,ILlrtIw TOu T l mn'

6 ., t a OIW. OOS , . ' o.,, . j v-n9 7 13C -l N Ln IA LN VI V .L L *N vjleq' et~aS 10 L -/ t V4 id ' 0.11 f. IA -N 010 fLS / oc-6-d 0:11 -0M IA 3N olanis - -- _

STUDIO NERVI Pog. 1120 Peg. 81 3 2 STUDIO NERVI Prog. 1120 peg.

I"gb: Tbe, 7 . 44 Begue Tabells N. 7 Sostituendo nell. tre precedenti pr*eionl i valori trovati nol- la tabella (7) ii ottisne Il Mste :

1) - - 117. 177. 158 + 16.275.788 + (191, 0225 - 239, 06316) Al,

+ (991, 0225 - 239, 06316) MC + (26. 430, 5671 - 12. 721, 82263 10 0U7612 1,15625 0,57816 0,57810 I L0000'O -419M + 3512 7842) X - 0 24 22.A 421 h 1.2n. 47.942 2 5

3~ ~ .040 ~ I ,aeL.im1 I , 20244 115.797 4M41 2) - - 4. 393. 619 + 37, 18868 MA + 37,15868 M0 + __tJ ,99234: 1,984_7_3_ 0. 99472Q1 41779 . 1A13. .2 11_854U + (991, 0225 - 239, 00316) S0. 03171; 1, 863 35 .0.0Ait0h6 15. 95 129. 687 X - 426. 326 + 9. 38438 M -

-- 0QA85689 1.71367 , 42847 . 13.87_21 L - 9. 38498 M a 0

4 0.70306t "281.3_ 0,30528- 1i.fi-8 ign6 4 3 9 3 3 7 7 3)- - . .619 + , 158 M A + 3 , 15868 MC + " U_43 1.38717 0.19310 2.14557 114.02 4. F5291

2 0, 48801 0,9761k 0,09762 2. 44052 M. 4-8 2. 2591J + (991. 0225 - 239. 06316) X + 426. 396 - 9, 38498 MA +

1 080 0,56180 0,02810- 2A0959 7, 9j + 9, 38498 Me a 0 -A-000 -!9m--_ 0-0 - 111- 0000 0,63 9 K ciob I' -O, 28085 0, 56180 + 0, 02810 2,80359 - 39, 994 0. Q-00 751.95934 MA + 751,95934 MC + 17. 221, 4785 X - 100.901. 370 2' 0.48801-0,+97611 +0, q9762 2,44052 - 61. 767

1L64.3D8 -1.21717 + 0.1 31D. 1.14557 71.273 46,54366 MA + 27, 77370 M + 751,95934 X - 4.820.015

A!40.713D6 1.52 13 0.30526. 1.20785 - 72.415 7.zlaae 27.77370 MA + 46. 54366 MC + 751.95934 X - 3.967.223

-t 5 09 - 1.71367 . 142647 1,71387 -M6L43 .10,42*4 F. Slatema e risolto fornisce i segu;nA valori delle incogloe 6 -0,931 _- 1,86335 0, 55906 1, 55295 - 58,099 X - 5.980 Kg. MA * 21.319 Kgm. Mc * - 24.115 Kgm. 7' -0,99234 1.98473 0.69473 1,41779 - 47.116 *17, 8654 Tks - - g (r/ &- fI- Agy 1 0 4 2 0 2 Tali valori delle reasioni sono relativi ad una wtriscLa larga in. 2,84 *'r - 2,09401 0.1319 10264 -3L110 21. 496.1 (un arco elementare). 9 61, 08247 -2,l164 0,97431 1,20285 17. 101

10' -0, 57812 1. 15623 0, 57816 0,57616 0, 000 15, 4195% 0, 000 0 .0000 9,38498 37. 15868 426. 396 1239, 0631

-V STUDIO NERVI Pr,. 1120 P,. 3S i STUDIO NERVI Prog. 1120 Peg. 34 /

RICURCA DpOLI SFORZI NORMAl. D5GLI SFORZI DI TAGLJO E TABELLA N. 8. DELL'ARCO. 1 FLETTUNTI ML3 IWOLE 3ZIONI Voarl ai elmotldgli sfonl iormlt s degl sforzi di tello in I aio" ev ., -f# e dowaj provocaii l _^ r c---- dl boricw valwr Ir) Le formale riS" sme

MA L M M.~e M,= ~M iL 2 ~ ) - -L f(2 y

4 10-A2.75&L 13,3350 M.,7.0 ,1 0 0,n00m 41 1 7i~*+ .S+ No X con + YAen - son 2t 9,7 1030220 h3.A12A . 1s. ,21M0 20,253

S&.~ To 'A con (-XWsn o( -cOS&ZIc A 10 7 23,3625 6,2090 56,7373 j9 ,0125 7 _120 18,121

2,2 4,42 AR -MAQtM 12,24M AAn 17,055 -KI X a 5.980 Kg. 'XA -x 00 2 MA MC 21. 19 + 24.115 7.790 K4 4 13,35__ AL0 14.923 A L L 66.75 66,75 -- QA125 -1,061Al3 5 A 23.3"5 12.270 13,&57 ye 2.39,50 + c 1.688,50 Kg. 3 - S,50 0.4A30 40.O 2A,7000 i 12.B2 1C,791

1 ,375L AQlQ---L 625_ -03Q7= M2220- UM,22 K LI ., - A1 3Q 30.0375 .367125 13.20 9J94 2 -~ 36.75 3273 -1 . 4430, .J0QQ 40,0500 12.270 _4. 1 M 32.A3,3Z7- 12,-27AD 7.4 34 1,9210 20.=2W. 4",7250 11,4140- it 6' -- 767 3A0 11-4M7- &Qj 10.27Mo &3M

7.2a 23, 3W25 10A125 L6J7 L12hW L1U2

6,2510 6,A5M 0.'5O 5,040 2-.22 -l' Z-700D 1Q,6L20 4 33WS M,41M . 10' 304m J0-C - 33,375 13,3350 9,000 rWJ 5Z 0,0000 0,0000 1 > 40

to t ;4

117

4, 0 I 0I .

w

Ikt

C',j

79 STUDIO NERVI Prog. Ino Peg. 37/ - STUDIO NERVI prog. 1120 Peg. 38/

See". Tdeile N. S Avendo coiiderato p I1y I momentl che tendono I* fibr. di Intradoo, g91 sfori normall quando sono di compressione e gl forazi di tagllo quando provo- CanO ucotrimento vO. I'alto della pelsone di sinls rispetto alla ponuicne di ------.

10- 9,617 964

9 Z5 a sn 1 8" ,. ECAd.,.

1 a SONDI IONE Dl CAlICO. '1 .45 .. 1 .

.CAOs&r o --

00 A489- -.- 114 3 32

*9,450 .6.090 .229 *2! -- 2- La I1 condizione d carocast ub otteners, some procedenteamnte detto p., so- Acrne +#e eJ.c~ -4 xek-e i +. Se-;- ~4 + " A esi-,4- 3 9.270 A.199 - 407 vropposizione &gli effetti del I^ schema + 2' schema + slaufefrico del 2 Schaeme. 41 Jshe-~ 1 -Frh P,-4 w,.. F.- -. , elf-e,,wy me.4 C Ir * Le reazioni vincolori relative ad un arco elementor. (1 tavellen -m.2,84) ' a 10150 I _ 7' . 8. .12 .43 572 1^ schema g M - M -0 IL '8.501 6.126 -1,020 ... I. c-fJ A.. (permanent. + 1/2 sovroccarico In tutto)

- 60) x 2,84 - 10.126 8 7.905 Z L916 1-81 XA - X - (26.540

37 _5 I.2,IV yA - 70.1 Kg. Y - 24.696 x 2,84 - 70.137 Kg. 15Z--"LA21- 3--

STUDIO NERVI Prog. 1120 Peg. 39/ STUDIO NERVI Prog. 1120 Pag. 40

26 schema i M -- 21.319Kgm. L* Formuls rl: , ilvo rissutano I cllaytro) (2 "Wroccarico su metb 9 Me - + 24.115 Kgm. XA - Xc . 5. 0 Kg.

X ro i. )+ - - Y - 7.109 Ka. Y- 2.309 Kv.

Z' schema I m. + 24.115 M - - 21.319 P't Slmmatrico del C

XA - Xc - 5.960 YA - 2.369 Y -7.109

Complessivament. s -- 4-

MA - + 2.796 Kgm. M - + 2.79 X - 92.786 c XA - c

YA - 79.615 Y-79.615 per la I condizione di carico. 00 I prUCedOntI wierT del momentI at Intendano positivi quando tendono l.%hw M YA 2 4 -) A r r a A (I- di Intodosso, negatlvi 1uand t .ndana I. flbre di estQ;&;. I valorI del

, sforal normCill ti ntencldc positivi quando son di acmpreasIone, fIgatIvI quan- -XA ) + tr O + x (f MA r Pr do sana 4i trazione. Ivalori degli sforzi dl taglo ml i nndono ilstlvl allaech della sealone rispetto determnal uno scarrimento In alto dilia parts a Cinia N a- Y A in 0(2 + x An N- A s AOCSIfco '_ - sins n 1, 2 P, all, parts destra.

T - Y Cos s - XAsIn N -cos k 5 Pr RIC!ACA DEl MOMENTI FLtTENTI DEGLI SPORZI NORMALI E DEGLI A A r - -F ---* * F*677 SFORZI TAGLIANTI NELLE VARIE SEZIONI DELL'ARCO, per un interasa .E -. 4 - di m. 2,64.

I PINION.- 77"R o6

F lit

[H

lws-

C,,4

L8 C61

0

II

0 0

I, a

uft

i83 STUDIO NERVI 4 Prog. 1120 pog. 5 4 STUDIO NERVI Prog. 12O Peg. 6 L..d- -. C-,-L go CONDIZIONE D1 CARICO.

A. ar F "'- , 'A. - '. a ,M .- , "'a V- - Y k- -1 0- - woe scorilmento veno lalto dello parte a slnistra delic sezione rispetto alla La 2 cardizion. di corico si puo attenere come precedentemen edetto, parts jestre. per svrappasionsdegli effetti del 1' schema + esmiimmstgico del 2 schema. 1z I.. I- .sopRACCARICO__ RICERCA DEI MOMENT PLEWTENTI. DEGLI SFORZI NORMALI E DEGLI "ZCO P~eMlfAN(N7e SFORZ DI TAGUO NELLE VANIE SEZION1 DELLARCO.

1^ Schema (Iorloo prmnte 1 savroccarcoo su tutto I'arco).

Kg. MA - 0,00 Kgm. M -0,00 Kgm. XA - 80.826

00 Xc - 00.B26 Kg. Y - 70.130 Kg. Ye 70.130 . Kg. 0 A

E'Phwc 1-1 C - ** s..I..eG .she.r'e [ khi. jt Elisumaseeriao del r scheom (meth sovroccarloo negative su metb arco) Per quota c4n;z; di corico s i rice se 2 moment; Pl*entl, A I% .sea 4' .ce S ! - 1 de #er-.-ea r4 -~ 5 gli sforzi nosmalI a gli sforzi di togilo per savrsjszle .(ii *FettI M A-U.115Kgm. M -+21.319 Kgm. X A - .960Kg. A c A mu 1^ o dell'emsimestrico dell. schesa 2-, somarodo aigebrl'oeente I valori d1 dtte sol ctozioni Azione p.. sezione. Xc - 5.90 Kg. YA -- 1.8 Kg. Y c 7.790 Kg. Da tener preoente che it primo schema di crI fst- 'e. s j n o +& dc dc tenti nelie soxilas solo in virf delta caduta lspt Complessivam. MA 24.115 Kgm. Mc - + 21.319 XA - 74.846 Kg.

6 8 4 XC - 74.6 K9. Y A- . 2 K9. Y c 62.340 Kg.

I precedenti volori del momenti si intendono poitivi quando tendono Ia fibre di Intradosso, negativi quando tendono Is Fibre di estradoss . I valori degll sforzi normall si Intondono positivi quando son 9 di com- pressione, negativi quando san di trazlone. I valori degil sfarzi di togilo si meendonAo positivi allorchb determinano 0L J

~ ~ I II I I I mEl

3 7.7 9 T rsm mC1 Xff r -. 2 V171 -- l W1rfj

I1A

111

iti

"IN

2 C4C4c1 cz

I - z 1z- .4jAA, kA

85 STUDIO NERVI p . 1 . STUDIO NERVI Prog. 1120 Pg. 501

EF0T mE VEAO ;-eo . AC JAAS f.-" - , Doll. .q-zwlm dell. setloe ni h m pol

fX, 2F. - XA

A - M S'A A A L

-: -M MA I 2' t~ y y + cA

'Dov YA eye Sa * reezient verticull e@e prinedpole hlms'* . L'rd In 4a.s 6 perfestament inamtrato al I tweat. It aiste pr Incipa- is A L. rL rL.- -760 Ke.

76 Y A - 0 Kg. 00 "- A,0

dells 3 oeadlz I i v -n=mw sc mu wIs I. eqeinuiin -oleffe

1) - I m M,) '- 0 2)- -2 . =* 0-

3)- 0

Ingnite Ipustatich. Ountg S , ill pomona determinare nells quail -. Ae39. 1/ .sVN , I..,4. ,,A ( ,j,i,-, Ipuon.nd. 1. 3 ;.XdlzIA l (Wti 1tizlant di viRColo) S rN - AL N6 M M LA~ 2(j +x), - T-T (j-2 ax,) +XA(UA ( - y) 1)- . 0

2)- 'A -o 3)- 'I -0 (Mml A2 L 2 a 2I L N -- (-x)---+

I C jiuj-~.-- STUDIO NERVI Frog. 1U0 P49.51I STUDIO NERVI prog. 1120 pag. 52/ ( x ) Ii -O wkere .. 0is 4C ... h ~ . nn wA - J( r p.t - A . .1A oendo M I momentl nelle varie slaina dell'arco net sistma principale T le L TABILLA N. 11 F, 20Dx 2,84 x Y54 Ay. luostatico. F~~~ - x,4 y-~ _____y L'usrms one dI M risud : Sez. .-- - 10 4 . - L fro le sex. 10 . 0 M -YA s F 10- A .750 13,335 $4 L -11.7082 ii43mzJu. 1.$40 18,375 2I* A I x r 9 1300375 1.2 fr Isn . "0'm V, , 3W1 9 -39 ,%371 1.347 12.k5 _1 S2 72000A,5A 5,127,187 L20 4AW JA.33 S23,3625 6,9 .-1 2 4 2JMV A5.3 1M2 980 5.200u YA F V + y Z F 20,0250 4,482, lI r -- (,-x)-Aa A 2 a .. 18,32 3,73 425 MI! 3.0 U La esprouulne 1. 2. 3. sotto sommatorisastituendo a M' ,). M ' MA U "13,3500 1,92L 00 -AAL 40. 7 -KI M) Ivalori forniti dalle 4,sI trasformano In i 3 10.0123 4J 0.2~O-J-d L llL A l 1. Q,8&l.U . 488 7(2 c fi 3 0.8343 O7 9 0,59 339 247 M' M - 7 M( - -- M -- M + -2 .475-- Q.463 E W (x) L .2 A L= A 2 c yy x y. + M f X- 2 X-y M+ -*- M + - -- M + L c A Ys A- , 2 A L A .11 -. A A 75- 0J .L + yl M 2 X A ) -0 v- 3375 . 0,113 2 c L c 2 - 6.90 .463. 6,7Q ,43 - 8,37 0,729 I' 1 N 3' 0125 1,061 2)- .M M' ,+-M +M - S(MA) 2 4 A 2 A 4 4c 4 - 11,6812 1,4*_ 4' -13,3500 . I921, f 2 5 , - 15,0187 2,451 x2K -T 2Ax + 2L.x AL- + F MAm - L 2mc - Lx A - 18,352 3,733 20, 2MM 4.482 + -" x )A-0 '-4 L A. 7' - 21,9 5,306 B' 26.3005,251

4 3)-E N'1 - 1 + - - 3) - 2 M M4A- (c I + ~ ,IM-_ ' 3 , 575 t k I Kc L- K2 4 A c2 1 A I A fi A+ A2 2 - MA + 's + X 10. -- 2,A, ~ A A T L2 L 2 L- A

I I I C,11

IIuIv[ir ir II'ruwar IL I I' V1~ I 1Y'S Luv*~t~III~I v~~ 4

-IRS

e u to cx z ,

> ~. III

88 K ~~1 [c~ ~SA

QA4- 1 Ii V77E#V. Ma ~ - 74

~5A- K -4 ~IA- -z w L1-- -! 4-is [A I II II cn -,.. I ~ -~ v~I.ai...' -~ 'I) K il-I KY .1 Em2~a~ I

P~T 1 -r I 4- zr ; 14 ~i4~4IN

4 : ~~WCl4*1

1I 14~ K-AI- w

TZ~Z ~ L

Cf,

U

89 STUDIO NERVI Prog. Peg. 57/ his STUDIO NERVI Pg. 112 p.,. V

Sostituendo nollo tre precodenti espresmioni i valori trovatin;Z4 o- tabalallsi .3tiene i1 iatema: Sor&6 Tb.Ila N. 12. 2 2 1) - 751, 95934 MA - 751,95934 Mc + 17. 1.4785 XA - $4.149. 810

+ 27,77370 Me - 751,95934 XA W-3.762.147 Ax' 2) 46.54366 MA

3) 27, 77370 MA + 46, 54366 Mc - 751. 95934 XA -4.334.667

Siatema che risol fornice isguntI valori doll, incognito iperstaticho: 7. MA 20.783 Kgm.

M - - 8.66$ gm. e A XA -5.996 Kg.

Dalle equazioni della statica o1 ricavno poi. 3 5. X .2-F - X - 7569 - 5996 - 1.573 Kg. 2 SA

A A MAL- Us . 760 - 20.783 + 8.668 .760-440 320 K A A- L 86.75 Oukb L7 - "_ .760-440 320 ye C M L+M . 760 + - 8.6886620713

- A I momenti flttenti. gli storsi normali e gli forzi di taglio nelle

vare sezioni dell'orco sono orniti dall. foruwl.!

-- -1 0 - r-M M A x S M MMc + xaMa XAS-LA2Lc- 2

N - Y en - X co + cosn P

T Acos N + XA sen s - ien.5 P

tO ~ 12'.721.8228 3.512,7342 0,0000 9,28498 37,1U4S

I rii iF

0. I', 10 I.g :8 g Sz 'C>

N N N 9 z WI' 4:-4D 0.,

--V N, Nz C;

w a 0.* z a 0 o 0 0 C> 0 R 0 0 0 o a 0 FE 0 C; 111111 M. -A. i 1-4 1

I I . -

~ ~ ~~~~~~.rn ~ ~ ~ --~ Tf 1:1 ',XX ~'~:~ i~jr~v!r~Ti i IVi.W Ii

I ri

i, N C7 2j R ;

c-4- .4 40 - - C; c t;a

t' A I I

,% aIcol' -- - '.1

91 STUDIO NERVI Prog. 1120 Peg. 60/ STUDIO NERVI Prog. 1120 Peg. 61 /.

-1b.Ta "i. ,3 C..4 S Me~.f- -cr -%"~nr 41-1-+' -j,,.f.. e-j.S ;.be-, .enej Segue T.tle N. 13. I momenti sI intendono positivi qrcfdo le f'. di ntvMso, gil sforzi normall quando sono di aompresione, gli sforzi di ogIlIo quor provecono cotrrlment vwo l pat dell d p di sinistra risetto a qllada destr. K YA C"' X,~AA~ T, VUItIMCA SEZIONI. -S.bo.. I* L.. .d ' 0 Sezion. 10 - 20 condizione dl c.rico. 4 4.2lI 2.440 I-. N .101.370 LM.9 M - 24.115 Kgm. K. j0.1 - 3.J-0- 1Aa M -24.115 .37 -0,238 .- 23,S ca.

S 2.214 .4 .03 124 "44 co 4. . 2.2 5, W4 2 -241

S 1.247 L64 885,- ND 839 5.936 1.25 501

S se 1.470 - 4331 0-SI 00 5.99 sL5? - 3__ S4P I)

I - 239 _S.9. J.A02> Q

247 4 If centro di presione code entro 11 nocclolo del intera sezlane omogenelaa- 31 . 5.864 1.404 14 se'- '. to. La sollecitozione a d 0 C mi iulteno c ax c mln ' 2. 153 ._4 L3L N + Ney1 oflembo lnferere 2.354 5.420 1.5% 381 ~c mx A + 4, I

.9d4 5.216 1.526 49 N No y 2 -4A .a mfn + loo Seriors 8, 3.296 5.000 1.488 596 c f t 3.427 4.773 1.446 697 10 3.927 4.527 1.397 -89j STUD IO NE RVI ,,og. 1120 peg. 62;, ST UDIO NE RVI gP.g. 1120 peg. 63 /

l 40,o" do.

2 64 x W2 2 + 69,8 x 39 x 45,1 Y2- 2 -24,50 cm. 284 x 10,2 + 69,80 i 39

y- 80,00 - 24,50 - 55,50 am. 2"29

+ 2--4 AjJ1ou~ - ( 39 .4 - 245 i + 8 x 11,40 x p E~A f (d -u) +A' fn(d', ) 4 x 21,5 8 x 11,40 x j2,s53 - 3.84.500 + 292.000 - 4.156.500 cm -W-- (11,40 x 60,o0 - 11,40 x 13,20 - 39 -

Ai - A + n A, - 294 x 10 20 + 39 x 69,60 + 8 x 2 x 11,40 - 5.802cmq. - 1,23 (693 - 150 - 640) - - 119 1. sollctaziln al lembo Inferior. a superior. ru , r 2+ A', (d' - u)2 - 101.370 101.370 x 23.8 x 55,50 Omax 5.802 4.156.500 49,4Kg/mq. 48 iF--21,40 - -T (11,40x,802 . 11,40 . 133+ .1203

c - 101.370 101,370 x 23.8 x 24,50 -3,30 Kg/mq. min 5.802 4.156.500 -1,23 ( 42.141 + 1.96 + 6.9) - 42.773

IulL'quazione cubica In y risulta

y + p y - q - 0 4-

3 - 119 y - 62.773

4q940 rinulta y 40,70 cm.

a qul: AMcli tonondo conto dolla sola sexin rettangolare 39 x 80 sl ho y I Y + u - 56,90 cm.

Momento statico 1 12 S - b y - nA (d-y)+nA (y-d')"

I . 39. "9, + 8 x 11,40 x 20, 10 + 6 x 11,40 x $3,90 6 6.215 GM 3

-21.. I - - ii STUDIO NERVI Prog. 1120 Peg. 64/ STUDIO NERVI Prog. 1120 Peg. 65

, . ,,,,,. s,.,- + A..we I centro di prelone code entro 1i nocciolo d'ierzla della cezione. Le sollealtonziont sultano qoundi t Le sllecitazioni max min rsultano quindi :

- . - 15 . 56,90 - 87 Ko/cmq. , . N + Ne c Max A + c n A f ~lS'~'~~" .215 1-4K'o 0 N W,)-8 $3 x 20,10 -245 Kg/cxq. S N N y 2 cain A+nA; J

VERIFICA SEZIONE DI CHIAVE. Dove :

1 oandizion di carico. 4 A .760 oq.

M - + 3.504 Kgm. n Af - 8 x 2 x 11,40 - 162

N + 92.756 Kg. A -A + n A - 4.760 + 102 - 4.942 cmq. I e I

NMi -9 92.786 0,0378m -3,78 co. J- . 47 x9,8 + 284 x14,90 -237x4$ +8 x 11,40xx 90A +

- 84 + 8 x 11,40 . 52102 - -L( 2.950.000 + 940.00 - 24.400) + 13.00 + 3

+ 94.000 - 3.865.600 + 13.000 + 94.000 - 3.972.600

Solleclazioni max a mln i A40 92.78 + 92.78 x 3,78 14, 18,80 + 1,32 - 20,12 Kg.ca c nax 4.942 3.972.600 I.

on 9'8 In 4.942 - .f3.9W2. x 307 8x35r 18,80 - 3,10 - 15,70 Ko/emq.

La distanza dol baricentro dal Iembo svpeior* vale . -2 237 x "-+ 47 x 02 yq* _ 2 --- 12.300 4 58.600 70.900 4 0 1 '90 cm. 237 x 1U,2 + 50 x 47 2 _ 2.420 + 2.340 4 . 7 6 Prog. 1120 STUDIO NERVI Prog. 1120 Peg. 66/ STUDIO NERVI Peg. 67

DARTMOUTH COLLEGE Ammettendo aicora il cartco uniformemento ripartito, lo sfopse V..,,4 'edi er /3..e, di acorA mlnto risulta: TRAVE DI IMPOSTA DELLA VOLTA S 2 x 145. Kg. If-,,-.- 13... -,# -. q , sr-s - 1 212, 80 x 80 2 000 Trave continua con campate da ml. 5, 69 - ir" & .- b-.' '. f-t Pa,*,ee/- ' -Je e A 12 ferri 0 1" piegati assorbono ano stol purl a: La travddi imposta I sollecitata , nel piano taente all'arco nella sez.Ione d . ad una azione N asuliae. it cui valore massimo ai ha S = 12x4,90x*F2 x 1440 - 116.000 Kg. nella 2- condizione di carico eal. Tht., -, .- ;- f4 Lo sfo*o rim - * N 122. 201 Kg. lmente, part a 145.000 116.000 29. 000 Kg. vie V*.. -/ -- -'* a -,' -""'''' ""'(' -" 1-0 t4 ne largamente assorbito dalle staffe. quosto valore di N 6 relattvo ad una fascia di arco larga m. 2,84 (pari -.. I d IF+. - eaw.tn Z er- ) -r-h.. e alia Iargbesta di un tammlone) . Il corrits ointe carico equivalente a v -,,.,.i. 46.. VERIFICA A TOAMONE metro 1jneare risult : . U valore massimo del momento in corrispondenza deUa sezione S 1222 1 43. 000 Kg/ml. 2.84 d'imposta rimulta:

M - 24.115 Kgm. Componendo tale carco con ilpeso a metro lineare della suddetdtra- Ve di imposta part a P - 3.600 Kg/ml. e con il carico dovuto alla pen 11 valore retivo a 1 ml. vale C.- C-I silina laterale di valore P 2. 280 Kg/ml. Tk #.#.I I .. .. ,, 21.._s,_,a_ *_*6.,-3 M' . -24.1152.8 . 8.* 500Olg/l Km/ml. 11 totale del carico a metro lineare viene a riaultare: 2,h 4 it '- * 4. tI di S - 47. 000 Kg/ml. U momente torcente Ama.simo auglf appoggf b conseguenxa pa a:

In corrispondenaa agli appoggi e sempre nel piano inclinato corAenente S, Mt *5.500 x 2.84 - 24.115 Km. * 2.411.500 Kglem. of avrk: max

. hidamento suun retanmgoiare 60x150 at M x 47. 000 5 2 =127. Kgm. acendo szione cm. 1 12 x 000 ha per quewta sezione:

d 0, 362 * (f 150 cm.) 63/140C a )127.000 0.80 145 ' - 3 + 2.6 3 + 2, 4A. 4,12 45 +h 0,45 +150 A 55 x 80 x 145 - 69 cmq. 0, r 100 - .b s0 II o quindi: L~a verifica &l taglio lomporta:; Mt 2. 411,500 9.900.000 max 4, 12 x - x 15, 00 T' x 47.000 x 5,6 134. 000 Kg. xh 302 x 150 960.000 2 134. 000 ., 4004 a 12. 80 Kg/cmq. 0. 9x80X145

1 co,1111 8 6 9 STUDIO NERVI P,o. 1120 Peg. 6 / STUDIO NERVI Prog.1 80 peg. i

Analisi del carichi : Area del ferir lagttudinali: P A- A r 2,411.50X4.20 36, S0cmq. (40 1"+903/4 Soletta . 400 x 9. 800 peso proprie 0. 17 x 2.500 425 Kg/mq. J f0T 2 2 J,-i I...L& - uovraccarieo 200 Area compleusnva di staffe per unit di lunghezza deUa trave: . imperrleabillzmaione 40 665 Kg/mq. a 2Mi A 3460 . 0,087 eaq/cm. * 8,7 cmq/ml.

In effetti Bono state diapoSte 8 staffe a ml. a 4 braccia da 3/8, cio*, . Soletta 0, 22 x 2.500 550 Kg/mq. considerando reagenti a toraione 210 le racia aderenti alla superfiele .1.y AtC It M8L. . e ae- , 4. ., * sovraccarico 200 eaterna, un are effic;e3icc parta r.. r, imper abilizzazione 40 790 Kg/mq. aat - 8 x 2, 00 - I $"Acmq/m. &ff PC e . 1 5-1.de M 2 x 665 .l451xO. Kgm/ml. F.let 5, !eIc rt r p 3 ndante di co p! u a Trscura~do 1'effetto di questo momento sulla trave C - B. at ha:

CO 1 B 0 B 120 (7k +8) whe c. -J*l dove 4/72~ K a 665 'F B

-2 Schema: M 790X 5.4 (5.88+8,00) -:2.320 I) Kgm/mI. B 120

7 4 t'-J d - 0. 416 6. 320 20 ( -22) 53/1400

A 0 .20x14 * 9,20 cmq/ml.

Rt ir As-+, _.. _ ,- It ud, * .u ,ar . In messeria il moment* vale , considerando un carico medio part a: 66 + 790 727 Kg/mq. b'*'' ~T'rJ!n[~ 2

I- ad 5.04e 2320 STUDIO NERVI Prog. W2O STUDIO NERVI Prog. 1120 Pag. 70 Peg. 71

Schema in szIe della colonna di sostegno: ( 4"x4"xS/16") M M7 . 5;R - 2 1.140 Kgm/mI. 1 2

Verifies a taillo Ammittendo in prima approssimazione un carico uniforme part a: --- o 20 - 47msn - (NIS0I lt CE sob +790 ,) . tITIM T2RI 2 727 Kg/mq. rista: . 9. e6 TS 727 x50 + 2. 320 + 460 - 2.280 Kg/mI. Momento d'lnerzia barice-itrico della sezione Ideale: 2 5,04 12 -4 .,264 + 8 - 1-4- (10 - 44) 0. x100x20 ,. 26 Kg/cmq. 12 1922u-~~-~~ 1~)

9.26-- 4 8 -z 4 T 727 ( + 0, 66) - 2 2. 300 - 460 1. 840 Kg/MIl. 12 , 6 +g =21-B 0 i4 .41282 7.216 -2.927 c ,A-a - J. , a.c.t , Calcolo Putrelle di sostogno Area ideale

-1 Carico: A, 9. 26x9,26 + 4 x 9, 73 x 0. 47 z 8 p T e1.840 Kg/mI. ' 85. 74 + 146, 33 - 232. 07 cm2 S -2 M I x 1. 840 x2.4 1. 490 Kgm. 2.927 Y 3 W - 140 000 * 106 cm 1.400 x Snellezza 6112,232,07A3,55-Jcm.- Si adottano due profilati affianesti, ciascuno avente un W' pari a A s0 3 3 W' . - 53 cm 3,40 in.

dove L a - 0. 7L - 0, 7 x4, 090 *2, 86 m.

Analial del carichi sul pilastri diuoutegno - 4" x 4" - 2,6. 80,56 3.55

Analiul del carichi Risulta W a 1 59

P a 1.840 x 2, 84 . 5. 200 Kg. a 11.400 pounds quindi

C*N ,200 x 1.9 So 35. 50 Kg/cmq. 232-

I I !V9www=!1M - 1 1 2 0 3 STUDIO NERVI Prog.11ZO Peg7. STUDIO NERVI Prog. Peg.7 /

AZXON SiSMICUE-- l(SENSO &&;nVDMNLX F. l'azione orisontale agent* al1'estremith superior. del pilastro 4.. '& "-'-- ' 1- ej M . 11 momento Incognito alla steasa estremita Schema in senso treaversale: 4Ia. in - I . fl momento d'Lneraia della bezione Ivi

h n La lungbezca del pilastro

Ia condixione = a ml traduce nella

D 6 fMM', dw a o schiLnlongituenale: dove

M 1M-F

Jr U K 'iv U I =v.1 ado , 5A f 8.40 11 carico'permanente risulta pari a 736 Kg/mq. n fattore moltiplica- MM' M-F Uva del carico permanente per ottenre trze jismcht part a: 21, . - t5 io _tear~ , 'I a , 1., 1e,.. , ~-i4.p e., e quindi C= 2x 015 -- ., = 0.0666 00 4, 5 4,5 d0 JM1 a EZ In corrispondensa di ogni pilastro la forca erissotale ; mense lon-

*tudnle) cbe at genera per aione sismiche risulta pertento: dove, arnettedo 11 momenta d'inerzla variable linewrment* de valo- . . / +. a r. .. # , - h re 1, In A al valore 2 10 in E, si ha F - 0. 0668 x 736 x 80. 41 x 5.6 11.213 Kg. 2 I -I (1 + -- ) Tr;;c;raidl p"tstro verticale A - C, data la sue scarsa Ajde.- flea I* , nl dell'axione sismics considerata vi prende in con e quindi 1-- A 4i a il b .,s, . -r ' -. I.,., .. e,...ee eiderazione ii solo pilastro A - E inclinato. Questo pilastro sfpub consi- f... . #&., -S. d-., +. ,";s am . -. +K +#-f . . , _#s a do M-x M If dx F d, derare incastrato sia all* base in corritpondenta del sue atcco con le J M sET uJm( x ) El] r E 0 1+-h X, 0 1+- + - fondationi, *La in sommith In cOrrlapondenza del suo attacco con 1 trave 0 , h di bordo decopertura di rigidzsa molto superiore a quella del pilastro ,,, . //,.r ,- 4. 1.a _2 1. t e ,,... . M " log 2 P b x4+h)dx - e quindi ripetto a questo pratieamente indeforffabile. Ne deriva un mo- E 1 0 E I, h+ x

mento b i 'estrelnitA supexbre che i calcola impone nul a Ia rota- 00 2 Mh.! -Ph Ph Zione ivi ndicendo aatti con: So 0

e quindi

M l6g2 a *h (1-log2) I

STUDIO NERVI Prog.1120 Peg. 74, STUDIO NERVI Prog. 1120 Pag.75

- = 0.44 Fh M uFh ( 10g2-1- - 1) v Fh (14426 1) N _.___.__ maxIe 5 ,5x101+8x4,9x85II+84 x + 3.787.0003.600.000 x 50, 5 ed ese do

F - 11. 213 Kg. 192. 000 + 0, 95.50, 5 - 97, 4 Kg/en-A. n a ,0.1m. S49, 4 - 48. 00 -" 0

M - 0. 44 x 11. 213 x 5, 70 - 28. 000 Kgz1. I...,-k A. .. 1 -o- i-r Verifica sezione di sommitA Momento In sommith a pilastro. Immeadtamente mA ricava i valore del momento M all'extremiti tnfrii re del pilamtro par a',8 Ph: 9016f C". m M - Fh a 28. 000 - 11. 213 x 5. 70 *36. 00 Kgm.

D'altra parte it carico assial. provnlent della copertura per effetto del carico permanente, relativau nte ad un interasse, vale -4 Momento d'inerzia: E 183.200Kg. 3 Decornponendo secodo Z verticale a x 91 x i1 + 2 x 20 x 8 x ii, 52 _ 1. #27. 000cm secondo lasse del pilastro. si ha in defi- nitiva lungo quasto M * 2.800.000 Kg/cm. S = 192.000Kg. N * 192.000 Kg.

Verifica sezione di base ,. , -, Eccentricitu Momen to d'inerzia: 441 e a 2.800.000 14.6cm. -- 3 -2 192.000 9t12 35,5x 101 +2xx2Ox48 4 cm .000 2 . - .1 406 0, 239 Lsh 61 Eccentricjtk:

M. 36.000 N 192. 000 * 0,187m.

La rsuwltark cade ratteanmente al Umit. del nocciolo d'lnerzia delLa

sesione . Pertanto le L scitazioni max e min. risultano:

AIMI 6 I STUDIO NERVI Prog. 1120 P.g.7 / STUDIO NERVI Prog.1120 P4g.77 0

Verifica a preesoflessione Azioni sismiehe in senso trasvereale

Schema in aenso trauversale dell'arco: P73" 4A

p 4 20 x 43,1 + 20 x 13, 9 - 1 61 16

Agli ltetti della dethrminasione delle fors. orizzontali, i1 carico per- 0,59 ( 860.2 + 278 - 1285) =-81 manente viene molpliesto per U coefficente

2 2 3 q 0,59 [20 x 41 + 20 x 13i + - 24 .9 0.15 0,30 c =2 x 4 00666

- 0.59 [37.152+3. W4+13.50 a 32.164 T-), dc" - Alas 73L .T m--~- 2l carico permanente vale 731 Kg/mq. Di conseguenza Ilso is I- $,AeX -- _. 4.L 1 #.,,4es sontale su met& arco e per metro di 1w4aaa. vale: Y1+ py -1 * 0

F = 0, 0666x73. 6x33, 375 . 1. 636 Kg/ml. yj - 81 y a 32.164

y = 32. 6 cm. Rimilta quindi:

y * y +u a 32.6 + 15,9 a 48.5 p 1. 636 130 Kg/mq. < 200 Kg/mq.

*-- 1x65-8x20x10, 1."s ~. less 4<.. - 0 tr .- 11f ns~eed E, n 5 + Wz2046, 3 - 95.000-1680+7450 - Tale carico rioulta inferiore a quello precedentemente considerate per lazione del vento. 3 w 100.770 cm

6c . dy ,19000 x 48, 5 - 92, 50 Kg/cmq. 3 1-.7 Appendix B

Moment of Three-Hinged Arch Derivation

This appendix shows the derivation of the maximuni moment along with other reac- tions of three-hinged arch under asymmetric loading.

A FBD C, H, H, 4 RA L -R,

M,

BMD

LA 4

Figure B-1: Asymmetric load on a three-hinged parabolic arch

101 Vertical Force (RA)

MC = 0

W )(R(L) = 0

3wL 2 RAL=0 8 3wL -R 8 3wL RA= 8 Horizontal Force (H)

MB = 0

w( ) (L/4) + H(f) - RA(L/2) =0 2

wL2 3wL -(8 )(8)=0 8 +HAf 8

HA = = -Hc 16d Maximum Moment at -1 Span

- HA( ) +RA( ) M = -w( 4)( 8 ) 4 4

wL 2 3wL 2 3wL 2 32 64 32

64

102