The ribbed floor slab systems of pier luigi nervi

Abstract

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 structural 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.

Full text

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.
7. REFERENCES
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Materials, and Technology, 2nd Ed, W.W. Norton & Co (2010).
[2] Addis, B., Building: 3000 Years of Design, Engineering and
Construction, Phaidon Press (2007).
[3] Iori, T., 150 anni di storia del cement in Italia: Le opera, gli
uomini, le imprese, Gangemi Editore (2011).
[4] Gulli, R., Struttura e costruzione – Structure and construction ,
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