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Proceedings of the International Association for

Shell and Spatial Structures (IASS) Symposium 2013

„BEYOND THE LIMITS OF MAN”

23-27 September, Wroclaw University of Technology, Poland

J.B. Obrębski and R. Tarczewski (eds.)

1

The Ribbed Floor Slab Systems of Pier Luigi Nervi

Allison B. Halpern1, David P. Billington2, Sigrid Adriaenssens3

1PhD Candidate, Civil and Environmental Engineering, Princeton University, Princeton, USA, abhalper@princeton.edu

2Professor Emeritus, Civil and Environmental Engineering, Princeton University, Princeton, USA, billington@princeton.edu

3Assistant Professor, Civil and Environmental Engineering, Princeton University, Princeton, USA, sadriaen@princeton.edu

Summary: This paper presents an historical and analytical evaluation of the ribbed floor slab systems developed by the Italian structural artist Pier

Luigi Nervi (1891-1979). The historical discussion includes the evolution of the floor slab from a non-structural element to an inspired structura l

system, culminating in Nervi’s patented ribbed floor systems. While the isostatic inspiration for the rib patterns of Nervi’s floor systems is well-

documented, the method used to generate these patterns is considerably unknown by comparison. The methods of experimental stress analysis

identified by Nervi and the mathematical theories available prior to the 1949 isostatic floor system patent are discussed to clarify the means used to

generate the isostatics. An Isostatic Line Tool developed for this paper is used to evaluate the correlation between the isostatics and Nervi’s

arrangement of ribs for the Gatti Wool Factory, Palace of Labor, and Large Sports Palace floor systems. This tool can be used by designers in the

conceptual design phase to develop ribbed floor systems inspired by Nervi.

Keywords: Nervi, ribbed slabs, isostatics, floor systems, computational tool

1. INTRODUCTION

Evolution of Concrete Floor Systems 1.1.

Prior to the 20th century, the prevalent materials used for floor systems

were timber, masonry, brick, and tile. Sa fety concerns arising from

several 19th century building fires and aspirations to construct taller

buildings provided impetus for engineers to develop stronger,

noncombustible floor systems. The absence of a universal system

prompted the ra pid filing of patents to protect the proprietary nature of

these new systems [1].

Despite the development of reinforced concrete in the latter half of the

19th century, concrete was first used for its fireproofing rather than its

structural qualities. The 1844 Fox and Barrett floor system, patented in

the UK by Fox, was the first to use concrete as a fireproof covering over

timber planks and cast-iron joists [2]. Though the first floor system to

use concrete in a structural capacity was patented in 1854 by Wilkinson

in the UK, it was Monier’s 1873 French patent that stimulated the spread

of structural concrete floor systems. Monier experimented with the

layout of iron reinforcement within concrete floor s. The widespread use

of the Monier system was due to the financial sponsorship of the

German firm of Wayss and Freytag, who in 1885 obtained the rights to

the Monier patent [1].

The first reinforced concrete framing and floor system was patented by

Hennebique i n 1892. The success of this system in France provided his

firm with the financial impetus to further develop and promote the

system internationally. Although the Monier and Hennebique systems

provided the fire resistance and structural capacity required for taller

construction, the orthogonal arrangement of ribs essential to these

systems produced an imitative timber joist and beam aesthetic [2].

Cement-based Floor Developments in Italy 1.2.

In 1883, Monier filed a series of Italian patents, which included

applications for floor systems. H owever, the development of reinforced

concrete floor systems in Italy was soon hindered by a nearly decade

long economic crisis. In 1892, Hennebique filed an I talian patent

application for his reinforced concrete floor and framing system,

followed five years later by an updated version with improvements for

beam reinforcement. While the first major projects using this system

could not begin until the start of t he economic recovery in 1898, the

spread of this system throughout Italy was largely due to Hennebique’s

marketing expertise. Not only did Hennebique promote the new system

as immune to fire a nd both lighter and cheaper than a comparable iron

system, he also appointed local engineers as agents of the patented

system authorized to promote its use in new construction projects [3].

The first native Italian patent related to reinforced concrete was filed by

Carlo Poma in 1893, which improved upon the preexisting Monier

patent by providing a cheaper, more workable alternative through

modifications of the aggregate proportions [4]. The economic recession

not only slowed cement sales but also deterred the industrialization of

the construction site, which retained t he “artisanal” masonry methods of

construction. The rise of Hennebique’s reinforced concrete system

coupled with the artisanal construction site stimulated the integration of

masonry el ements into concrete floor systems. The combination of

hollow bricks (pignatta) and cement in concrete floors resulted in a

lighter structural system with improved material economy; the first

patent for a pignatta and concrete floor system was filed by Sigismondo

Ghilardi in 1902 [5].

Due to the destruction caused by the 1908 Messina and Reggio Calabria

earthquake, an international competition was established by the Società

Cooperativa Lombarda di Lavori Pubblici in 1909 to establish the best

material for withstanding earthquakes. Arturo Danusso (1880-1968), a

practicing Italian engineer and later a professor at the Polytechnic of

Milan in strict relation with Nervi his entire life, received highest

recognition in the competition. Danusso ascertained that reinforced

concrete structures would provide the most reliable resistance to

earthquakes, which established reinforced concrete as the preeminent

construction material for Italy’s seismically-active regions [6]. However,

when Italy declared war in 1915 construction plummeted due to the

ensuing supply and labor shortages. Postwar reconstruction saw

reinforced concrete restored as the main construction material of I taly

[7], which led to the development of new building regulations in 1927.

This first update in 20 years prompted the development of several

references with practical calculation methods for designing and

constructing reinforced concrete structures [3].

After graduating from the University of Bologna in 1913, Pier Luigi

Nervi worked for the Società Anonima per le Costruzioni Cementizie

(SACC), which was owned by Attilio Muggia, one of Nervi’s professors

and mentors at the University. As Muggia had obtained the rights to the

Hennebique patent for central Italy in 1895, Nervi was exposed to the

avant-garde of reinforced concrete construction early in his career [8].

Excluding a hiatus during WWI, Nervi worked at SACC from 1913 to

1923, after which he founded his own firm, Soc. Ing. Nervi & Nebbiosi,

with Rodolfo Nebbiosi. In 1932, Nervi transitioned Soc. Ing. Nervi &

Nebbiosi into a second company, Soc. Ing. Nervi & Bartoli, which

allowed Nervi to control both the design and construction of a project

[9]. Shortly thereafter, the Italian Fascist political atmosphere shifted

into a state of Autarchy, or self-sufficiency. As metal reinforcement was

primarily imported from foreign manufacturers, reinforced concrete

quickly became an Anti-Autarchic material [10]. This prohibitive

designation was a radical change from reinforced concrete construction

symbolizing the architecture of the regime [3].

When reinforcement was first partially and then fully banned in 1935

and 1939, respectively, engineers were forced to use either traditional

2

masonry arch systems or experiment with atypical reinforcement

materials manufactured within the Kingdom of Italy. As cement was a

material produced in mass quantities in Italy, this new research was

characterized by cement-based floor systems with nominal

reinforcement, no reinforcement, or reinforcement made with materials

other than iron or steel, e.g., bamboo, alu minum. Patents of floor

systems with a drastic reduction or complete elimination of

reinforcement were filed throughout Italy. One of the most promising

systems, the S.I.F. (Senza Impiego di Ferro, without the use of iron)

floor, was patented by Eugenio Miozzi in 1937. To compensate for the

absence of the required tensile capacity typically provided by

reinforcement, the S.I.F. floor was composed of several layers of

terracotta tile joined with high strength cement plaster and applied to the

extrados and intrados of a traditional pignatta and concrete rib floor.

While structurally sound, the S.I.F. floor was too time-consuming and

labor-intensive for widespread use [5].

During the partial reinforcement ban, Nervi started on a series of

hangars for the Italian Air Force, the first series of which could only be

made with ti mber formwork a nd concrete poured on-site. This

expensive, time-consuming construction process provided impetus for

Nervi’s 1939 patent for an efficient, economical construction procedure,

which used prefabricated concrete elements. Nervi also focused on

developing a new material to facilitate the consideration of fluid

structural forms. This material exploration resulted in the creation of

ferrocement, which is comprised of layers of flexible wire mesh encased

in sufficient cement to achieve complete coverage at a reduced thickness

[11]. In the years prior to his 1949 isostatic ribbed floor patent, Nervi

explored a means of using ferrocement in tandem with his 1939 patented

construction procedure [12]. The extension of the 1939 patent to include

mass-produced, reusable ferrocement forms on moveable scaffoldings

made constructing large floors supported by evenly spaced columns

fiscally possible.

Fig. 1. Tobacco Factory Floor System [13]

Nervi first tested this moveable formwork procedure (entitled the “Nervi

System”) with the construction of the Bologna Tobacco Factory in 1949

(Figure 1). While the formwork produced a pattern imitative of the

traditional beam and joist systems, the successful implementation of this

construction procedure inspired Nervi’s departur e from orthogonality to

the curved forms of the isostatic ribs. Although the isostatic rib patterns

are considered integral to the “Nervi System,” the concept of aligning

floor ribs along t he isostatic s of the principal bending moments is

attributed to Aldo Arcangeli, a practicing engineer who worked in the

office of Soc. Ing. Nervi & Bartoli [12]. Arcangeli is noted as the author

of the invention listed in the corresponding 1949 patent (No. 455,678),

filed by Soc. I ng. Nervi & Bartoli, which included the improvement in

the construction of a variety of structures, e.g., floors, domes, by the

arrangement of support ribs along the isostatics of bending moments or

normal stresses [14].

Arcangeli determined from classical plate theory that a 2D continuous

body subjected to normal forces produces two families of orthogonal

curves (isostatics), tangential to the principal bending moment

trajectories, along which torsional moments are equal to zero. If this

continuous body is replaced by ribs oriented along the isostatics, then

the rib structure and the continuous body would have identical structural

behavior under identical loading and support conditions [14]. Nervi

drew inspiration from the patterns formed by the isostatics; he stated: “it

was amazing to find that by thus limiting our task to t he interpretation of

a purely physical phenomenon, we were able to discover unexpected and

expressive new forms [12],” and asserted that “in harmony with such

inspiration, reinforced concrete beams lose the rigidity of wooden beams

or of metal shapes and ask to be molded according to the line of the

bending moments” [13].

2. NERVI’S METHOD FOR EVALUATING ISOSTATICS

While the structural behavior on which the isostatic rib patterns are

based has been highlighted by Nervi, the manner in which these patterns

have been generated has seen limited literary discussion. This section

explores the experimental and theoretical analysis methods emphasized

in Nervi’s writings and the application of these analysis techniques to

generating isostatics.

Strain Gauge Methods 2.1.

In Structures, Nervi i dentifies two domains of experimental stress

analysis: strain gauge methods and photoelasticity. Strain gauge

methods rely on devices capable of measuring strain via mechanical,

optical, electrical, acoustical, and pneumatic methods, to determine the

displacements and stresses at points on a small-scale model [15]. Nervi

first used mechanical strain gauges to determine the magnitude of the

stresses in the ribs of a scale model for the 1935 Orvieto hangars [13]. In

the late 1930s, Simmons and Ruge independently developed bonded-

wire electrical-resistance strain gau ges [15], which permitted the

calculation of principal stress directions.

To find the stress field on the surface of a flat slab, it is necessary to use

three-element strain gauge rosettes. These rosettes include three strain-

gauges, each oriented at a different angle relative to the two in-plane

axes (x and y), which provide three strain measurements corresponding

to the three orientation angles. Using strain-transformation equations,

the three Cartesian components of strain (, , and ) can be

calculated and can be used to find the principal strain direction at the

measurement location [15]. While calculating the principal stress

directions from the rosette readings is simple, the experimental

preparations and procedure are costly and time-consuming. Several

rosettes would be needed to obtain enough data to clearly represent the

full field of principal stress trajectories. As this method does not provide

a more a ccessible way to obtain isostatics than theoretical calculations,

strain gauge methods could not have been Nervi’s initial means of

finding isostatics.

Photoelasticity 2.2.

Nervi’s fascination with stress visualization in photoelasticity

experiments (led by Danusso at the Polytechnic of Milan) suggests

Nervi’s possible use of photoelasticity to generate isostatics [12].

Photoelasticity is derived from the strain- and stress-optics laws

(Neumann 1841, Maxwell 1852) on the theory of artificial double

refraction (anisotropic birefringence) in a stressed isotropic, transparent

solid. In 1816, Brewster coined the t erm photoelasticity due to the color

pattern produced in clear glass when stressed and examined under

polarized light. When certain transparent materials undergo stress, the

material exhibits birefringence. As polarized light passes through the

material, the rays refract and separate into t wo perpendicular

components each parallel to the principal refractive indices of the

material. A condition of t he stress-optics laws states that these principal

indices correspond to the principal stress directions [16].

3

In 2D cases, small-scale models with plane stress conditions are placed

in a polariscope, which allows analysis of a model under polarized light.

There are two types of optical interference patterns, isoclinics and

isochromatics. Isoclinics designate the locus of all points where the

principal stress directions are parallel to the directions of the polari zing

axes, appearing as black bands. Isochromatics define the locus of all

points having equal difference between the two principal stresses

(constant maximum shear stress) appearing either as a field of dark

fringes or a continuous range of the visible spectrum, depending on the

light source (Figure 2) [17].

Fig. 2. Example Photoelasticity Pattern (outlet conduit with a

concentrated load at the top and uniformly supported at base) [18]

Isoclinics allow the user to determine the principal stress directions at all

points of the model, whereas the isochromatics indicate the differences

in the principal stresses in the model and the stress at the free

boundaries. As the isoclinics indicate the lines along which the

directions of the principal stresses within the stressed model are

constant, the isoclinics are not identical to the isostatics. Although

isostatics can be drawn from the isoclinic diagram, the manual process is

time-consuming to generate a full field of orthogonal isostatics [19].

While plane stress is an assumption of thin plate theory, thin plates

subjected to transverse bending cannot be observed solely using the

aforementioned 2D photoelasticity method. As thin plates subjected to

transverse loading experience tension on one side and compression of

equal absolute value on the opposite side, any phase difference resulting

from light passing through the first half of the plate thickness would be

cancelled out when travelling through the second half of the plate

thickness [20], thu s producing no photoelastic patterns. The first

successful photoelasticity experiment to capture t he principal stresses in

plates subjected to transverse loading was documented by Goodier and

Lee in 1941 [21], yet photographs of the optical interference patterns

were not published. Ba sed on the model discussed by Goodier and Lee,

Kuske developed a “time-efficient” method u sing traditional 2D

photoelasticity methods to view isoclinics occurring at the surface of the

plate [20]. However, this method was not published until 1953, the same

year as the completion of the Gatti Wool Fa ctory and four years after the

submittal of the 1949 isostatic ribbed floor patent.

Although Nervi referred to the photoelastic phenomenon as “the beauty

and poetry of this transmutation of stress i nto a play of light rays [13],”

photoelasticity experiments could not have been used to determine the

ribbed floor slab patterns. Nervi asserted that photoelasticity was “more

efficient in the study of local stresses in bodies of limited dimensions

(crane hooks, chain links, a nd parts of machines), than in t he analysis of

entire structures [12].” Given this mindset, the authors assume that he

would not have been inclined to use photoelasticity to evaluate the

stresses present in a large flat slab. Despite his dedication to

experimental methods, Nervi was cognizant of their time-consuming and

costly aspects – the preparation, measurement, and analysis phases – and

thus recommended that experimental modeling be reserved for structures

of “special technical and architectural importance” and the analysis of

structural problems unable to be solved theoretically [ 12]. The problem

of determining the stresses in a thin plate subjected to transverse loading

could be solved theoretically at the time of the patent submi ttal in 1949,

indicating that the use of models to evaluate the stresses was not

mandatory. This conclusion is substantiated by Iori [14] who asserted

that the shape of the isostatics relating to the ribs of the Gatti Wool

Factory must have been determined using calculations derived from thin

plate theory.

Mathematical Theory 2.3.

As no efficient experimental approach existed at the time, Arcangeli

theoretically studied the concept of placing ribs along the isostatics of

principal moments in proposing the idea to Nervi [14]. The two most

commonly used plate theories are the Kirchoff -Love and Reisner-

Mindlin plate theories. The Kirchoff-Love theory, applicable to thin

plates, was developed by Love in 1888 using Kirchhoff’s 1850 boundary

condition assumptions [22]. Reissner-Mindlin plate theory, an extension

of Kirchoff-Love plate theory and applicable to thick plates, takes into

account shear deformations through the thickness of a plate and was

proposed by Reissner in 1945, but not fully developed by Mindlin until

1951. Given this timeline, Arcangeli’s theoretical calculations for the

principal bending moment directions must have been based on Kirchoff-

Love thin plate theory. Although thin plate theory involves high-order

partial differential equations, numerous analytical (Navier, Lévy,

Timoshenko [23]), approximate (Ritz), and design solutions

(Westergaard and Slater) for thin plate -bending theory were already

well-established and in widespread use when the patent for isostatic rib

floors was filed in 1949. Additional resources were developed in Italy,

including the analytical solutions of Botasso [22] and the design

solutions of Santarella, who wrote and edited a plethora of practical

manuals and theoretical t exts on reinforced concrete produced as a result

of the 1927 updates to the building regulations [3].

3. ISOSTATIC LINE TOOL

While Arcangeli and Nervi relied on theoretical calculations to

determine the isostatics of principal moments for flat slabs,

contemporary engineers can employ two manual methods. The first

method includes theoretically calculating the principal bending moment

directions at a selection of nodes, hand drawing lines at set lengths in the

respective directions, recalculating the directions at the next nodes, and

repeating the process until reaching a boundary [16]. This process is

described in detail below and illustrated in Figure 3:

1) Select a start node (e.g., Node 0)

2) Calculate the maximum principal bending moment direction

at that node

3) Draw a straight line of a set length in the calculated

maximum principal bending moment direction (e.g., Segment

a)

4) Recalculate the principal bending moment direction at the

new node (e.g., Node 1)

5) Iterate through this method until crossing a boundary

6) Perform this method for a selection of start nodes to obtain a

field of primary isostatics

To draw the secondary isostatics, one can simply use the minimum

principal bending moments.

4

Fig. 3. Isostatic Path Determination

Alternatively, one can print out the principal bending moment

trajectories displayed by commercially available Finite Element

Software and manually draw the isostatics tangent to these trajectories or

import a picture of the plot into a CAD or graphics editing program to

draw the lines electronically. T he main disadvantages of these methods

are: 1) the plots produced by commercial FEM software are typically of

low resolution, which complicates drawing lines at the accurate angles;

2) the principal bending moment directions are averaged to the center of

the finite element, which leads t o biased isostatics; and 3) the principal

bending moment trajectories are displayed a s arrows, rather than

continuous lines, which forces the drawer to interpolate between

elements.

As an alternative to these manual drawing options, an a utomatic,

computational method of drawing the isostatics was developed using the

FE Software SAP 2000v15 and MATLAB. First, a flat slab FE model is

generated with the appropriate geometry, external loa ding, boundary

conditions, and material properties. After running the analysis, the

primary and secondary principal bending moment angles are obtained.

The MATLAB algorithm follows the six-step procedure previously

outlined. Termination of the algorithm occurs when the generated

isostatic line has reached a slab boundary, resulting in a plot of the

isostatic line (Figure 3).

4. ANALYSES OF THREE FLOOR SLABS

Three of Nervi’s floor slabs are evaluated u sing the Isostatic Line Tool:

1) the Gatti Wool Factory warehouse floor (1953, Rome), 2) the Palace

of Labor mezzanine floor (1961, Turin), and 3) the Large Sports Palace

gallery floor (1960, Rome). As all of these floors were designed to

support uniformly-distributed loading conditions, and as scaling the load

does not alter the directions of the principal bending moments, each

system is evaluated under equivalent uniformly distributed loads. This

consistency in the applied loading patterns leaves the principal bending

moments unbiased so as to highlight how alterations of the dimensions

and boundary conditions influence the resulting isostatic rib patterns.

Gatti Wool Factory 4.1.

The use of isostatics to inform the arrangement pattern for a ribbed floor

system was first realized for the warehouse floor of the Gatti Wool

Factory. Nervi and Arcangeli collaborated with architect Carlo Cestelli

Guidi for the factory design, which required a wide-spanning floor

system capable of supporting heavy wool-spinning machinery [14]. The

curved isostatic pattern of the Gatti Wool Factory floors (Figure 4) fully

exploited the moldable flexibility of Nervi’s ferrocement forms.

Fig. 4. Gatti Wool Factory Floor System [13]

Each 5m x 5m slab of the Gatti Wool Factory is supported by a central

column. All slabs are monolithically joined along the perimeter edges

[14]. A quarter of the slab is evaluated u sing the Isostatic Line Tool. T o

satisfy the one-eighth symmetry in each quarter slab, a sufficiently fine

step size must be used. A relation between element length and step size

was prescribed to evaluate symmetry convergence, a s illustrated in

Figure 5, with the chosen step size of one-sixty-fourth the element

length yielding a symmetry error of only 0.31%.

Fig. 5. Stepping Procedure Symmetry Convergence

The analysis of an upper-right quarter of a 5m x 5m Gatti Wool Factory

slab, using the Isostatic Line Tool, is shown in Figure 6. Symmetry

conditions (axial rotation and orthogonal translation restrained) are

applied to the l eft and bottom boundaries and monolithic perimeter

conditions (full r otational fixity a nd in-plane tra nslation restraints) are

modeled on the top and right boundaries.

5

Fig. 6. Gatti Wool Factory Quarter Slab Analysis

The red lines in Figure 6 represent the primary isostatics, corresponding

to the maximum principal bending moments, and the blue lines represent

the secondary isostatics, corresponding to the minimum principal

bending moments. Both pairs of lines were generated from start nodes

set at increments equal to half of the finite element lengths and placed

along all four boundaries. The concentrations of lines indicate

convergence of the principal bending moment directions, highlighted by

the prescribed method of starting node generation. The grey and white

outline shows an approximate plan of the Gatti Wool Factory floor to

illustrate the correlation between the theoretical isostatics and the as-

built rib pa ttern. The concentric secondary isostatics produced around

the column support at the bottom left of the quarter slab reflects the

reasoning for placing concentric curves of reinforcement around the

column head and verifies Nervi’s placement of the rib encircling the

column head. The ribs emanating from the central column also show

close correlation to the generated primary isostatics.

Fig. 7. Palace of Labor Floor System [13]

Palace of Labor (Palazzo del Lavoro) 4.2.

In 1961, a celebration was held in Turin for the centennial of Italy’s

unification, for which Nervi designed and constructed the Palace of

Labor [24]. The interior border along the building perimeter is formed

by the mezzanine, which has repeated reinforced concrete slabs with

isostatic ribs (Figure 7). Each 10m x 10m slab is supported by columns

at the four corners and the isostatic patterns follow one-eighth symmetry

[13]. The analysis of an upper -right quarter of a 10m x 10m Palace of

Labor slab is shown in Figure 8, with symmetry conditions at the left

and bottom boundaries, monolithic perimeter conditions at the top and

right boundaries, and a column support at the top right corner. The

presented results show strong correlation exists between the theoretical

primary and secondary isostatics and the as-built plan.

Fig. 8. Palace of Labor Quarter Slab Analysis

Large Sports Palace (Palazzo dello Sport) 4.3.

The Large Sports Palace was designed by Nervi, with ar chitect Marcello

Piacentini, as one of three structures constructed for the 1960 Rome

Summer O lympic Games. The intradoses of the perimeter gallery floor

include isostatic rib patterns (Figure 9) [25]. In contrast to the Gatti

Wool Factory a nd Palace of Labor slabs, the Large Sports Palace slabs

are rectangular. W hile these slabs are supported by columns at the four

corners, the shift from a square to rectangular boundary shape induces

different isostatic patterns.

Fig. 9. Large Sports Palace Floor System [25]

6

The analysis of the Large Sports Palace floor required further evaluation

as applying identical boundary conditions as the Palace of Labor case

study yielded isostatic s satisfying the assumed bounds of only t wo of the

four outlined floor ribs (Figure 10). Additional boundary conditions

were evaluated to try to shift the isostatics towards the two disparate

ribs, yet no combination yielded exact results for all ri bs, as seen in the

Gatti Wool Factory and Palace of Labor analyses. The analysis shown in

Figure 11 removed the fixity of the top boundary and the Figure 12

analysis placed simply-supported conditions at t he top and right

boundaries. While the fully simply-supported conditions show close

correlation between the topmost transverse rib and t he neighboring

secondary isostatics, the other three ribs lack correlation with the

isostatics. The rib closest to the left boundary captures the strictly

vertical portions of select primary isostatics, whose starting nodes start

along the left axis of symmetry, yet they soon diverge towards the top-

right corner support.

Despite this divergence, the two disparate ribs from Figure 10 appear to

be offsets of the two curves closely correlated to the isostatics. This

departure from pure conformity to the isostatics supports Nervi’s

statement of prescribing the design to an interpretation of t he true static

behavior. Naturally, the addition of the ribs dictates the structural

behavior of the floor system, reorienting the isostatics along these

predefined paths.

5. CONCLUSION

The successes of Nervi’s isostatic floor systems and the discussed

historical precedents highlight the catalysts for t he widespread

recognition of a reinforced concrete floor system: 1) the protection of

patents; 2) the personal ownership or financial sponsorship of an

established engineering firm; and 3) the connection to academic research

and experimentation. I n addition to these influential factors, the artisanal

construction sites endemic to Italy and the political atmosphere at the

time helped foster the construction and material innovations developed

by Nervi related to floor system design. The elegant design of these

floor systems i s expressed by the rib patterns, which allude to the

isostatics of the principal moments produced in a corresponding flat slab

system. The experimental stress analysis methods (strain gauge and

photoelasticity) referenced by Nervi and the mathematical analysis

methods available at the time (pertaining to transversely loaded flat

slabs) were discussed to evaluate the potential for these methods to ha ve

been used by Nervi to generate the isostatics. Although isostatics can be

generated via these experimental and theoretical methods, it was

determined that a theoretical approach was used by Nervi and Arcangeli.

The theoretical approach to generating isostatics has been implemented

in a computational I sostatic Line T ool, the efficacy of which was proven

in analyses of three of Nervi’s floor systems. A modern designer can

explore the efficiency of the Isostatic Line Tool to develop conceptual

designs for ribbed floors according to new geometries, boundary

conditions, and loading patterns. This tool provides designers with a

visual method of automatically determining the primary and secondary

isostatics in flat slabs, which provides an efficient alternative to the

potentially biased, time-consuming FE-based drawing methods

composed of centrally averaged principal bending moment trajectories.

The close correlation between the theoretical isostatics and the as-built

rib patterns for the evaluated floor slabs, as well as the artistic

interpretation of the static theory, highlights Nervi’s desire to construct

correctly while expressing elegance.

6. ACKNOWLEDGEMENTS

The authors are grateful for the financial support of the Sherrerd Fund

and Princeton University . The authors would like to thank Marco Nervi

(Pier Luigi Nervi Project Association) for permitting the use of the

images of Nervi’s floor systems. The authors would also like to thank

Serguei Bagrianski (Princeton University) for his help in developing the

Isostatic Line Tool and Ted Segal (Princeton University) for his research

recommendations.

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Figs. 10-12. Large Sports Palace Quarter Sla b Analysis – Full Fixity, Partial Fixity, Simply -Supported

7

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