Non-linear analysis of Eduardo Torroja’s Frontón de Recoletos’ roof using a discrete reinforcement approach

Engineering Structures 80 (2014) 406–417

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Non-linear analysis of Eduardo Torroja’s Frontón de Recoletos’ roof using a discrete reinforcement approach Adrián Cabello, Ignacio Paya-Zaforteza ⇑, Jose M. Adam ICITECH, Departamento de Ingeniería de la Construcción, Universitat Politècnica de València, Camino de Vera s/n, 46071 Valencia, Spain

a r t i c l e

i n f o

Article history: Received 24 April 2014 Revised 18 August 2014 Accepted 26 August 2014

Keywords: Eduardo Torroja Thin shell structures Finite element Non-linear analysis Concrete

a b s t r a c t Eduardo Torroja was a Spanish engineer and a famous figure in structural design. His vast production of technically demanding work includes thin reinforced concrete shells, daring structures that meet efficiency requirements through their thinness. One of his most remarkable works is the Frontón Recoletos, which is described in an extensive bibliography, enabling us to approach this milestone of structural engineering again with modern computational tools. Although thin shells have mainly been designed under the elasticity hypothesis, nonlinear analysis also deserves to be considered, since current computer processing speeds make this possible. The present work carries out a complete non-linear analysis of the roof of Torroja’s Fronton de Recoletos and can be considered as the continuation of the linear-elastic analysis previously carried out by Lozano-Galant and Paya-Zaforteza (2011). The key aspects of the investigation presented in this paper are the use of discrete reinforcement instead of the more common smeared approximation to modeling the roof reinforcement and the accurate definition of concrete through a robust constitutive equation, the Multi-crack model proposed by Jefferson (1999) for which FE software was used. The results have been checked against both the historical documentation in the Torroja Archive and the recent studies using linear models by Lozano-Galant and Paya-Zaforteza (2011). As it turns out, the relevance of geometric nonlinearities, material nonlinearities and stability behavior should not be neglected. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction A thin shell structure carries loads mainly by the membrane effect, i.e. it develops only negligible transversal stresses thanks to its shape, boundary conditions and the patterns of the applied loads. The era of modern concrete shells starts with the Zeiss Planetarium [4] (built in 1925), in which Dischinger and Bauersfeld successfully combined concrete and steel, relying on the shell theory. From this moment on, these structures were generally conceived under the following assumptions [4,5]: – Euler–Bernoulli hypothesis, i.e. plane cross sections remain plane after bending. – The hypothesis of thinness, i.e. the distance z of a point from the middle surface of the shell remains unaltered by deformations and the stress component normal to the middle surface is neglected.

⇑ Corresponding author. Tel.: +34 963877562; fax: +34 963877568. E-mail addresses: [email protected] (A. (I. Paya-Zaforteza), [email protected] (J.M. Adam). http://dx.doi.org/10.1016/j.engstruct.2014.08.044 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

Cabello),

[email protected]

– Geometrical linearity, i.e. geometrical imperfections and second order displacements are neglected. – Material linearity. These assumptions proved to be accurate and safe enough when based on a good understanding of structural behavior and enabled the extraordinary advancement of shell structures in the second third of the 20th century [6] when the famous architecture critic, Sigfried Giedion, proclaimed ‘‘Shell construction appears ever more strongly to be the starting point for the solution of the vaulting problem for our period’’ [7]. Eduardo Torroja (1899–1961), Félix Candela (1910–1997) and Heinz Isler (1926–2009) were some of the designers that took shell construction to its limits and justified Giedion’s statement. These designers combined the above-mentioned assumptions with outstanding structural intuition and the monitoring of both scale models and completed works to create works of Structural Art, as explained e.g. by Lozano-Galant and Paya-Zaforteza for Torroja [1], Garlock and Billington for Candela [8], and Billington for Isler [9]. In this paper, a complete non-linear analysis is carried out of the Fronton Recoletos’ roof designed by E. Torroja in 1935. By doing so, the influence of some of the simplifications commonly used in shell

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design as well as the use of advanced analysis models to compute and predict the response of complex concrete structures can be discussed. Previous works have performed linear or simplified non-linear analysis of some of Torroja’s structures [1,10] but, to the authors’ knowledge, this is the first complete non-linear analysis of one of Torroja’s major works. In addition, this study may be regarded as a useful guide for engineers involved in non-linear design or assessment of shells.

Section B-B 32.5 m Skylights

A

A

2. Eduardo Torroja and the Frontón Recoletos The Spanish engineer Eduardo Torroja (1899–1961) was one of the most important structural engineers of the 20th century [11]. For almost forty years he was intensely active as university professor, researcher, and consultant engineer [12–14]. He was especially outstanding in the design and construction of thin-shell concrete structures, a technical field in which his designs evoked admiration for their audacity, efficiency, and aesthetics [15]. The Algeciras Market Hall (1934), the Zarzuela Hippodrome Roof (1935), and the Fronton Recoletos (1935) are his three major concrete shell projects. Torroja explained his main works and structural philosophy in his two major books [16,17]. He also founded several associations, such as the International Association for Shell and Spatial Structures (IASS), which since 1959 has been promoting activities and gathering information related to the design, analysis and construction of shells and long-span roofs. The Fronton Recoletos (Fig. 1) was a sports facility built in 1935 for the playing of a traditional Spanish game called Pelota Vasca (Basque Ball). The architect of the project was Secundino Zuazo and the choice of the roof shape was governed by the need for a large open space for the ball court and stands and to make use of natural light. The final design is shown in Figs. 1–3 and was the result of a study of alternatives detailed by Torroja in [16]. The roof covered a surface of 55  32.5 m2 with a thin concrete shell typically 8 cm thick. The cross section of the roof is defined by two cylindrical horizontal and parallel lobes, of 12.2 m and 6.4 m in radius. The intersection of them is a line named by Torroja as a ‘‘seagull profile’’. Its 55 m of span length made Recoletos the longest span barrel vault shell when built. It was also the first time that two cylindrical sectors met without any expected support to define the cross section of the roof. In the words of Torroja [18]: ‘‘We believed it convenient to move away from the observer’s impression of the idea of two barrel vaults resting on a longitudinal beam, because it would have meant misunderstanding or even subverting the structural behavior of the thin shell. [. . .] If there is any beam, it is the shell itself’’. On the surfaces where skylights were needed the regular shell was replaced by a triangulated grid that worked as shell and as the frame for glass panes (Figs. 1 and 2), anticipating by fifty years the single-layer glass gridshells now in vogue. Throughout his career, Torroja perfected the idea of covering wide spaces by spatial beams supported on edge frames, combining curved shells or folded plates, as for example in his San Nicolas Church [10].

Frontal wall Lateral wall 55 m B

B

Forecourt Playing pitch Promenade

Rebound wall Z

Section A-A

X

Y

Fig. 2. Cross section (top) and plan view (bottom) of the Frontón Recoletos building. Source Torroja [16].

Fig. 3 details the main dimensions of the cross section of the roof. The thickness of the built shell is variable and is greatest, to provide sufficient stiffness, in the most highly stressed sections and in the areas where the shell meets the skylights. It was a courageous decision to choose a typical thickness of 8 cm, which resulted in a span to thickness ratio of around 690:1, more than twice the value of the first thin shell, the Zeiss Planetarium (290:1). The structure of the Fronton Recoletos was built in only 90 working days and was badly damaged during the Spanish Civil War (1936–1939). The roof collapsed on the night of the 15th August 1939 while it was being repaired. The Recoletos roof was a personal demonstration of Torroja’s technique, knowledge and generosity, as he considered the roof

Fig. 1. Interior view (a) and exterior view (b) of the Frontón Recoletos. Courtesy of Archivo Torroja – CEHOPU.

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Detail of the intersection of the two lobes 0.04

1.0

35

0.04

0.16

0 0. 3

0 0 .4

R6

0.25

.40

0.19 R

0.10

47 2.

0.7 5 1.2 0

0.08

3

0. 1

7

R 3.3

0.30

Y 1. 20

0.11 at the center 0.08 at the contact with the springings 1.2

0.75

0

C

1.

20

0.17

R3

.33

20 2. R1

0.08 0.16

0.30

X

1.80

A

0.20

27º 23' 16''

1.20

0.40

1.20

0 .4 R6

61º 16' 23''

13.91

B Fig. 3. Geometric definition of the directrix of the shell roof of the Frontón Recoletos. Courtesy of Archivo Torroja – CEHOPU.

to be a way of advancing science, and wrote many publications to share the knowledge he had gained with the scientific-technical community. In this regard, the report [18] written by Torroja on the occasion of his appointment as a member of the Real Academia de Ciencias Exactas, Físicas y Naturales (Royal Academy of the Exact, Physical and Natural Sciences) is of major importance. This text contains the details of the analysis, construction, monitoring, repair work and collapse of the Recoletos roof and provides detailed drawings of the structure. The structural analysis gave maximum deflections around 15 cm, compressive stresses between 0 and 5.7 MPa, tensile stresses between 0.6 and 7.8 MPa, transverse bending moments between 10 KNm/m and 10 KNm/m and transverse shear between 3.4 KN/m and 2.5 KN/m [18]. Torroja checked his theoretical results with a 1:10 scale model of the structure and with data from the monitoring of the finished structure as detailed in [18]. The Recoletos’ roof has recently been studied by Lozano-Galant and Paya-Zaforteza [1] using different finite element models of increasing complexity and precision. All the analyses were linearelastic and used a homogenized material for the reinforced concrete. Principal stresses, bending moments and displacements in the shell were obtained and compared to the results published by Torroja. The main conclusion of this work was to show the reliability of Torroja’s conceptual design, although his theoretical results seemed to have underestimated the internal forces and stresses in the shell. This study also pointed out the influence of the concrete elasticity modulus on the behavior of the structure and recommended that a non-linear analysis of the structure be carried out in the future to obtain more conclusive results. This is the starting point for the study presented in this paper.

3. Nonlinear finite element modeling 3.1. General approach Historically, engineers have been reluctant to use nonlinear analysis, due to its complex problem formulation and long solution time [13]. The usual approach has therefore been to design shells to work in the linear range. Non linear behavior has sometimes been considered by making use of international guidelines, such as those of the IASS [19], or the recommendations given by authors such as Tomás and Tovar [20]. In the study presented here, the authors use the Lusas Finite Element software [21] to analyze a masterpiece of shell construction, considering stiffness variation due to shape changes (geometrical nonlinearity) and material properties (material nonlinearity). In order to reach the design load level of the structure under a nonlinear analysis, an incremental step procedure is needed, tracking the path of the solutions. Each solution or load step must converge using an iterative loop and a preset allowable error. The Modified Newton–Raphson and Crisfield Arc-length formulations [22,23] were very helpful in this study. Considered the most significant parameter by the authors, convergence was tackled by the Euclidian Residual Norm cu, measuring the norm of the residual forces vector as a percentage of the norm of the external forces. This was kept below 0.6 in all cases (0.6 for the largest error), which means that convergence is achieved when the out of balance forces are less than 0.6% of the reactions. It also fulfills the recommendation [21] that establish a limit of 5% for problems in which material non-linearity predominates over geometrical nonlinearity.

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A. Cabello et al. / Engineering Structures 80 (2014) 406–417 Table 1 Main features of the FE models of Frontón de Recoletos’ roof.

Small lobe

E E

E = 29,400 E = 29,400

No Beams BMS3

Thickness (cm)

Convergence (%)

No Beams BMS3

No No

Constant = 8 Variable

– –

cu < 0.5 cu < 0.4 cu < 0.4

Ec = 29,400 Ec = 24,500 Ec = 16,500

No No No

No No No

No No No

Constant = 8 Constant = 8 Constant = 8

Bi

B1

B1-E B1-NLM B1-NLG

E NLM NLG

Ec = 29,400 Es = 200,000

No

No

Bars BRS3

Constant = 8

B2-E B2-NLM B2-NLG

E NLM NLG

Ec = 24,500 Es = 200,000

No

B3-E B3-NLM B3-NLG

E NLM NLG

Ec = 16,500 Es = 200,000

No

C1-E C1-NLM

E NLM

Ec = 29,400 Es = 200,000

Solid HX20

C2-E C2-NLM

E NLM

Ec = 24,500 Es = 200,000

Solid HX20

C3-E C3-NLM C3Le-E C3Le-NLM C3Le-NLG Torroja’s design (elastic) Reduced model [18]

E NLM E NLM NLG – –

Ec = 16,500 Es = 200,000 Ec = 16,500 Es = 200,000

Solid HX20

Ec = 29,400–24,500 E = 34,300–24,500

No Yes

C1

C3

e

Big lobe

L1c L2d

Reinforcement

NLM NLM NLM

C2

c

Skylight modelingb

A1-NLM A2-NLM A3-NLM

Ci

d

Elasticity modulus (MPa)

A1 A2 A3

B3

a

Analysis typea

Ai

B2

b

Model name

–

cu < 0.4 cu < 0.1 No

Bars BRS3

Constant = 8

–

cu < 0.6 cu < 0.1 No

Bars BRS3

Constant = 8

–

cu < 0.3 cu < 0.1 Shell QSL8

Bars BRS3

Variable

–

cu < 0.5 Shell QSL8

Bars BRS3

Variable

–

cu < 0.5 Shell QSL8

Bars BRS3

Variable

–

cu < 0.1 Solid HX20

Shell QSL8

Bars BRS3

Variable

–

cu < 0.6 cu < 0.1 No Yes

No Yes

Constant = 8 Constant = 0.8

– –

E: Elastic, NLG: nonlinear geometric, NLM: Nonlinear material. When the word ‘‘No’’ appears in this column it means that the roof was modeled in the skylight area as a shell of uniform thickness without any void for placing the glass panes of the skylights. In the Ci models the skylights are modeled using HX20 or QSL8 elements which define the beams of a 3D-trussed structure. Model L1 corresponds to Lozano-Galant and Payá Zaforteza’s FEM-1 [1]. Model L2 corresponds to Lozano-Galant and Payá Zaforteza’s FEM-3 [1]. Models C3L differ from models C3 in the fact that they have an extra dead load in the junction between the two lobes due to the high density of reinforcing bars existing in that area.

Table 1 lists and describes the numerical models analyzed in this paper. Note that L1 and L2 are two of Lozano-Galant and Paya-Zaforteza’s models [1] and are used as a reference and starting point for the studies described herein. A detailed description of the numerical modeling is given in the next subsections. 3.2. Geometrical and material nonlinear formulation The Total Lagrangian formulation was used to study geometric nonlinearity. This algorithm is suitable for dealing with conservative loads, large displacements and rotations but small strains [21]. This approach works with the deformation of the solid as a function of the original coordinates of each particle. It makes use of the Green–Lagrange strain tensor, which gives information on the change in the squared length of elements, and the second Piola–Kirchhoff stress tensor which expresses the stress relative to the reference configuration. The discrete reinforcement approach used in this paper to model the reinforcing bars demands the definition of two separate constitutive equations for steel and concrete. Steel responds according to an elasto-plastic bi-linear stress–strain diagram where the material yielding stress is determined by the Von Mises criterion. Concrete modeling is a delicate issue and a key aspect in this paper and comes together with the choice of crack modeling. A smeared crack approach is suitable because the global behavior of the structure is sought and not the assessment of a local element – where the more costly discrete crack model would be preferable. Within this philosophy, the robust Multi-crack concrete model

developed by Jefferson [2,3,24] and implemented in Lusas [21] was employed. Embedded damage-contact planes are integrated with a plasticity component by using a thermodynamically consistent plastic-damage framework. These planes are called planes of degradation (POD) and are able to record the damage state or return contact when necessary, among other features described in Jefferson [2,3,24]. In the Multi-crack model, the strains and stresses required for each element are checked at every iteration according to the constitutive curve of the material. The governing  ¼ D Me determines the local stresses in an element equation r from the strain, the elasticity matrix D and the damage matrix M. A POD is formed normal to the major principal axis when the principal stress reaches the fracture stress, this stress being the maximum tension or compression the material can bear. At that moment M is also modified accordingly. Thereafter, it is assumed that damage on the plane can occur with both shear and normal strains.

3.3. Description of the model: constitutive equations, boundary conditions and loads applied The model will be explained in the same ascending order as it was built up, from its constituent parts (finite elements) to their links, overall geometry and attributes. The first step was to choose suitable finite elements whose specifications [21] met the nonlinear formulation requirements. Steel bars were modeled using the Lusas BRS3 element, which is a 3D straight or curved bar isoparametric element with three Gaussian nodes and three translational

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A. Cabello et al. / Engineering Structures 80 (2014) 406–417

degrees of freedom per node. Concrete was modeled using a semiloof quadrilateral curved thin shell element (Lusas QSL8) which can accommodate generally curved geometry of varying thickness. The formulation of the QSL8 element takes into account both membrane and flexural deformations and excludes transverse shear deformations. The element has nine Gaussian integration points and uses the 5-point Newton Cotes rule for through-thickness integration. The concrete beams of the skylights of the large roof lobe are made of 3D solid hexahedral continuum elements (Lusas HX20 element). This is a higher order isoparametric element with 20 nodes (3 per edge) and three degrees of freedom per node. The element has 3  3  3 = 27 Gauss points, which is a rough indication of the processing speed it demands. The concrete beams of the skylight in the small lobe are made of QSL8 shell elements. Further information on the finite element modeling can be found in Section 3.4 and in Table 1. Element interconnection is explained next. The most usual way to create the system formed by the matrix and the reinforcement is by smearing the last in the shell elements, according to a widely accepted simplification. The reasons for using a discrete reinforcement approach are: (a) this approach is closer to reality and more reliable than some others, since each physical structural component is included in the structural model; (b) this approach provides data on the exact stress distribution between the different materials; and (c) this approach made it unnecessary to define a specific anisotropy angle for each finite element – due to the highly irregular reinforcement directions, as seen in Fig. 4a. Perfect bond was assumed between steel and concrete [25,26].

trol point to compare the results given by all the numerical models studied in this paper. This control point was also used by Torroja [18] and Lozano-Galant and Paya-Zaforteza [1]. 3.3.3. Loads applied to the model Torroja considered the worst load combination to be the sum of dead load, wind pressure, and snow. Safety factors were not applied to the loads; it was the designer who decided his own safety margins to work with. He also decided that wind suction was negligible, due to the proximity of other buildings. In the FE model, the loads were distributed and applied on the surface elements, according to the laws defined by Torroja and shown in Fig. 6. Note that the sum of dead load, wind, and snow is called ‘‘load level’’ in this paper and that the goal of the incremental loading procedure implemented in Lusas was to reach this load level in order to check the safety of the structure. In this procedure the magnitudes of wind and snow loads have been expressed as a percentage of the magnitude of the dead load, thus a single load factor could be used to climb up the solution path. This load factor is a dimensionless number that increases at each loading step i, and is defined in reference to the dead load of the structure (which is equal to 2452 N/m2) as follows:

Load factori ¼

Load applied at step i  2452 Dead load þ Wind load þ Snow Load

Therefore, the load factor takes values from 0 (no loads on the structure) to 2452 (when all the loads considered by Torroja are applied on the structure).

3.3.1. Constitutive equations The structural materials of the roof were steel and concrete. A thorough search of the bibliography available in the Torroja Archive in Madrid, the data given by Torroja in [18] and of the first Spanish code for reinforced concrete [27] – dating from 1939 – provided the following mechanical properties: Steel was assumed to have a yield stress of 240 MPa, a Poisson’s coefficient of 0.285 and a Young’s modulus Es equal to 200,000 MPa. The authors could not find any compressive strength value for the concrete (fc) used to build the roof, so fc equal to 15 MPa was assumed, based on the cement content (300 kg/m3) specified in [18]. Concrete tensile strength (fc,t) was obtained from [27], which provided a value of fc,t equal to 3.04 MPa. Another important parameter is the elastic modulus Ec whose remarkable influence has already been pointed out by Lozano and Paya-Zaforteza [1]. As Ec is not a pre-known input, three moduli were considered, leading to the FE Models labeled with the indexes 1, 2 and 3 in Table 1. The first value (Ec1 = 30,000 MPa) came from Torroja’s design and is used in FE models A1, B1 and C1. The second value (Ec2 = 24,500 MPa) was Torroja’s correction after observing building displacements and is used in FE models A2, B2 and C2. The third, (Ec3 = 16,500 MPa), was the result of placing the Frontón Recoletos in its historical context and corresponds to the value obtained by calculating fc,t according to the 1939 Spanish code for reinforced concrete [27] with a value of fc equal to 15 MPa. Ec3 is used in models A3, B3, C3 and C3L. The concrete Poisson’s coefficient was taken as 0.15 according to [27] in all the analyses.

3.3.4. Reinforcement modeling Torroja provided a reinforcement drawing in [18] which included a complete description of the diameters and spacing of the rebars used in the roof. Fig. 4a shows the roof reinforcement pattern and Fig. 4b gives two 3D views of the reinforcement as first introduced in Lusas. It should be noticed that Torroja’s reinforcement pattern was based on the flow of stresses and pursued structural efficiency. The following procedure (see Fig. 4b) was used to model the steel bars. First, a 3D model of the reinforcement was created using CAD software and every intersection point of two rebars was obtained. Second, all the intersection points were transferred from the CAD software to the FE software and then linked with BRS3 bars. Third, the gaps between the reinforcing bars were filled by assembled shell elements. This approach forced the QSL8 elements to adopt an excessive aspect ratio and a very small size (see Fig. 4c), thus reducing the reliability of the results and increasing calculation times. The authors therefore decided to bundle the bars together (see Fig. 4d) to get a more flexible meshing and a better performance of the quadrilateral elements. Note that Torroja heavily reinforced the junction between the two lobes of the roof as he placed there 16 solid 50  50 mm2 reinforcement square-bars. This resulted in a dead load higher than the original typical 2.45 KN/m2 load (see Fig. 6). Including this extra dead load forced the authors to develop the C3L model whose main features are given in Table 1.

3.3.2. Boundary conditions Fig. 5 shows the roof boundary conditions based on Torroja’s own data [18]. Supports have been placed to restrain the displacements of the roof perimeter in the X and Y directions. Following Torroja’s design, points A and B in Fig. 5 also have their Z displacement restrained. For reasons of symmetry and the use of appropriate boundary conditions, only half of the structure is modeled. Fig. 5 also shows point C at the intersection of the two lobes in the central directrix (Section B-B in Fig. 1), which is used as a con-

Some tests were undertaken to validate and improve the numerical models. First of all, the authors checked that the transverse shear force was low in order to avoid the need to use ‘‘thick shell’’ elements. A further test was carried out to accept the geometric imperfections caused by the generation of multiple surfaces, as the shape of the structure after filling the gaps between the rebars with shell elements differed from the idealized cylindrical shape of Recoletos roof. The last test focused on the proper representation of the skylights. A preliminary reinforced and loaded

3.4. Finite element model validation tests

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A. Cabello et al. / Engineering Structures 80 (2014) 406–417

(a)

(b)

Front view

Back view

(c)

Before rebars bundling

After rebars bundling

(d)

Fig. 4. (a) Plan view of the reinforcement pattern designed by Torroja [18]. (b) Front and back view of the total reinforcement projected on the roof by 3D-CAD software. (c) Detailed view of the part of the roof marked in red in (a). It represents the FE meshing before and after grouping reinforcement bars in bundles. Shell elements exceeding the maximum allowed aspect ratio are colored in blue and extremely small shell elements are circled. (d) Bundled reinforcing bars for the FE model. Only half of the roof is shown in (a, b and d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

model (B1-NLM in Table 1) described a strong cracking pattern across the skylights of the large lobe and on top of the small lobe (see Fig. 7a), whereas those in the small lobe remained working in the elastic range. It was therefore decided to use the trustworthy HX20 solid elements (Fig. 7b) for the large lobe skylights while keeping the QSL8 shell element to model the concrete in the area of the small lobe skylights. Reinforcement bars went through the central axis of the theoretical position of the skylight beams, connected to the surrounding solid or shell elements, depending on the case. It must be noticed that the final mesh was also the result of a convergence study that enabled to obtain the proper finite element size as a compromise between calculation times and results accuracy. This study showed that a reduction in the order of 50% of the finite element size finally chosen more than doubled the calculation times, but resulted in differences of the vertical displacement of the control point smaller than one mm.

4. Results Figs. 8 and 9, containing the main results of this study, plot the vertical displacements of the control point defined in Fig. 5 versus the load factor. The load factor is a dimensionless number defined in Section 3.3.3 that affects the sum of the loads applied to the model and was chosen here to rise parallel to the dead load of the structure. Note that the solution path depends on the variables that control the Newton–Raphson and Arc-Length algorithms and also on the maximum error accepted in the incremental procedure used to perform the non-linear analysis. Each load test therefore finally converged on a different load factor and the curves in Figs. 8 and 9 plot the average values obtained in five numerical tests. A thick blue line has been added in Figs. 8 and 9 to mark the first load factor in which cracks were found.

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A. Cabello et al. / Engineering Structures 80 (2014) 406–417

Fig. 5. Boundary conditions (restraints) of the finite element model in global coordinates. The red dot marks the control point. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Dead Load

2.45 (kN/m 2)

Wind

Snow

0.98 sinϕ (kN/m2)

0.64 cosϕ (kN/m2)

Fig. 6. Loads considered by E. Torroja as function of the angle u.

Fig. 7. (a) Cracking pattern of the FE model named B1-NLM used to decide the types of finite elements to be used (b) HX20 Solid elements mesh defining a unitary part of skylight, employed only in the big cylinder. Embedded in red are the BRS3 bars representing the steel. Only half of the roof is drawn in (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

When there is a kink point, this means there is an obstacle, the iterative algorithm fails to converge and is unable to pass it, or if it does it might be forced to stop growing soon after this point. A slack tolerance for convergence cu was adopted to address this setback. When it was set at cu < 0.6 it showed as ±0.2 mm or ±5 MPa in the graph. 4.1. Nonlinear analysis of the structure with constant thickness and no skylight modeling The first set of models (A1, A2 and A3 in Table 1) simulate the entire shell as a structure of uniform thickness equal to 8 cm. A material non-linear analysis was carried out following the procedure detailed in Sections 3.1 and 3.2. The main goal of these anal-

yses was to evaluate the importance of the reinforcement in the behavior of the structure. The results of the analyses (Fig. 8) show that the stiffness of the structure increases as the Elasticity Modulus E rises. The collapse of this structure took place for a load factor around 340, or 13.8% of its design load, and failed because the seagull profile defined by the intersection of the two roof lobes was overstressed in the region close to the shell mid-span (areas with Z coordinate close to 27.5 m). The Bi set of models is obtained when the concrete is reinforced. In models B1-NLM, B2-NLM and B3-NLM the load carrying capacity and stiffness increased considerably. The structure collapsed when the load factor was circa 1200, or 49% of its design load. The failure started in the upper section of the small lobe and was caused by overstress due to the lack of steel or insufficient thickness

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Load factor

A. Cabello et al. / Engineering Structures 80 (2014) 406–417

2500

stretch of the load path of the models B1-NLG, B2-NLG and B3NLG is plotted in Fig. 8.

2000

4.2. Nonlinear analysis of the structure with variable thickness and skylight modeling

1500 B2-E

B3-E

Model B1-NLG Model B2-NLG Model B3-NLG Load level Model L1-E Model B1-E Model B2-E Model B3-E Model B1-NLM Model B2-NLM Model B3-NLM Model A1-NLM Model A2-NLM Model A3-NLM Cracking outset

B1-E

1000

500

0.17

0.16

0.15

0.14

0.13

0.12

0.1

0.11

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0

0.01

0

Displacement δY [m] Fig. 8. Load factor – Control Point vertical displacement curves for the family of models Ai and Bi. Note that for clarity reasons, only the last part of the curves of the models B1-NLG, B2-NLG and B3-NLG has been plotted.

2500 C3L-E

C2-E C1-E

2000

Load factor

C3-E

1500 Load level Model L2-E Model C1-E Model C2-E

1000

Model C3-E Model C3L-E Model C1-NLM Model C2-NLM

500

Model C3-NLM Model C3L-NLM Cracking outset

0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

0

Desplacement δY [m] Fig. 9. Load factor – Control Point vertical displacement curves for the set of models Ci.

(Fig. 7a). The reinforcement did not yield at any time. The maximum vertical displacement of the control point was double that reached in the Ai models. The concrete started cracking, as expected, at the same deformation state as in the previous models. The influence of material nonlinearity is remarkable. The observed kink points occurred when a new crack front appeared and the stresses were redistributed. To reach the load level it would have been absolutely necessary to implement thickness variation; the constant-thick shell assumption is not enough. Fig. 8 also plots the result of the Bi models for geometric nonlinearity and shows that as concrete Elasticity Modulus Ec decreased, geometric nonlinearity increased. Note that for the sake of clarity, only the last

Passing the load level required including the reinforcement in the FE model as well as accounting for the increment of stiffness coming from thickness variations in the roof – profile shown in Fig. 3. The resulting FE models are identified as Ci in Table 1 and can be regarded as the closest approach to the behavior of the real structure, especially the C3L models due to their enhanced selfweight definition. Fig. 9 and Table 2 show the results of these analyses. If the nonlinear curves are compared to their corresponding elastic curves at the load level, then it can be stated that material non-linearity increases the vertical displacements of the control point in the models C1-NLM, C2-NLM, C3-NLM and C3L-NLM by 25.6%, 21.3% and 13.8% and 17.2% respectively. When Ec is reduced the influence of material nonlinearity is also reduced, except for the singular model C3L-NLM. Convergence turned out to be very difficult to achieve beyond the load level, so general failure could not have been much further away. Model C3L-NLG, which includes geometric nonlinearity, is not plotted in Fig. 9 since its influence was negligible for control point displacements due to the increased stiffness associated with the Ci models. Nevertheless, its importance is visible at other points of the structure, as will be explained at the end of this section. Fig. 10 plots, on the deformed Recoletos’ roof, the planes of degradation of the structure (areas where concrete is damaged) and the isobars of the principal tensile stresses in the lower shell fiber at some key load steps for Model C3L-NLM. It must be noted that the development of a plane of degradation does not necessarily imply a clear macroscopic crack and that positive stress values indicate tension stress. The complete loading sequence can be seen in the appended video. A tension limit value of 2.9 MPa is considered instead of a value of fct = 3.04 MPa because it is considered a stress state close enough to fct, where quite a lot of concrete degradation has already occurred. If a tension limit of 3.04 MPa had been considered in Fig. 10, only a few points would have shown material degradation. The most stressed sections were the first three tensioned diagonals of the large skylight, which are closer to the edge support and the top of the small lobe at midspan, where a visible ring of overstressed material left behind a vast zone of damaged concrete. This damage, not tracked by an elastic analysis, seems to have happened in the real structure, because the monitored deflections and the numerical model agreed on the presence of a subsidence in the shape of the structure at this point. Fig. 10a shows the evolution of the deformations and crack patterns. For a load factor close to 600, C3L-NLM left the elastic state and the seagull profile started cracking. The second crack pattern appeared in the overtensioned diagonals of the skylight. There was a third crack front

Table 2 Control point vertical displacements. Model

dY (cm)

Accuracy achieved (%)

L2-E C1-NLM C2-NLM C3-NLM C3L-E C3L-NLM C3L-NLM + C3L-NLG C3L-NLM + C3L-NLG + Shrinkage Built structure

9.11 6.47 6.94 8.2 7.82 9.45 9.45 11.95 14.3

64 45 46 57 55 66 66 84 Reference value

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(a) Planes of degradation

(b) Isobars of tensile stresses σ1 in the lower shell fiber

σ1 (MPa) -4.0 -3.6 -3.3 -2.9 -2.5 -2.2 -1.8 -1.4 -1.0 -732 E-3 -368 E-3 0 358 E-3 721 E-3 1.1 1.4 1.8 2.2 2.5 2.9

Fig. 10. (a) Planes of degradation. Results are plotted on the deformed shape of the structure. (b) Isobars of tensile stresses r1 in the lower shell fiber. Positive values indicate tension and negative values indicate compression. Only half of the roof is shown.

at the top of the small lobe. The kink point of the curve around load factor 1800 shows the moment when the crack fronts from the seagull profile and the top of the small lobe merge. There is a second kink point at 2000, representing the unloading of the overstressed diagonals. Indeed, the survey referred to in Torroja’s report [18] found considerable cracking in this area. In the last load increment of Model C3L-NLM, the concrete also reached the compressive limit (15 MPa) in the lower fiber of the shell near the small skylight springings. Another noteworthy compressed area (10 MPa) was the top fiber of the small lobe in areas with the Z coordinate close to 27.5 m. Torroja estimated a 4.9 MPa working stress for the concrete. This value is slightly exceeded on the mid surface of the shell cross

section for all the Ci models but never reached fc = 15 MPa. It should be noted that the reinforcement bars did not yield in any of the models analyzed, although the working load of the steel bars assumed by Torroja (107.8 MPa for the general reinforcement and 88.2 MPa for the seagull reinforcement) was exceeded in some of the skylight rebars. Fig. 11 shows a map of the steel stresses in Model C3L-NLM. The skylight zones shown in red in Fig. 11 are in excess of the prescribed working load. The maximum (absolute) values of the compressive and tensile stresses in the reinforcement bars are 92.2 MPa and 149.6 MPa, respectively. Fig. 12 compares the transverse bending moments along the central directrix obtained in this research work using C3L models with those obtained by Torroja [18] and Lozano-Galant and

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Fig. 11. Layout of gathered reinforcement bars. Model C3L-NLM; at the load level: in color blue are the less stressed bars. They turn toward red as they participate more (either in tension or compression). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Transverse Bending Moments (kNm/m)

Big lobe

Small lobe

20 Model L2-E

15

Model C3L-E Model C3L-NLM Model C3L-NLG

10

Torroja's design

5 0

0

5

10

15

20

25

30

35

40

45

-5 -10 Skylights

Skylights

-15

Distance along the central developed directrix (m) Fig. 12. Transverse bending moments along the central developed directrix for different models. Positive bending moments produce tension in the top face of the shell.

Fig. 13. Comparison of vertical displacements of the central directrix including different effects studied. Results scaled by 200:1.

Paya-Zaforteza [1]. Note that, for the sake of clarity, only C3L models have been plotted because all the Ci models gave similar results. The introduction of the reinforcement reduced the absolute value of the maximum transverse moments on the shell. The effect of material nonlinearity is visible in the small lobe, in agreement with

the crack pattern. The influence of geometric nonlinearity covered the large lobe, moving slightly toward the first buckling mode. Indeed, a geometric nonlinear analysis of Model C3L was performed beyond the design load and found that the upper critical load is 3.72 times the design load. Based on an eigenvalue

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extraction analysis, the bifurcation point was found to be 8.92 times the load level. This value does not meet the IASS design guidelines [19] that account for imperfections, nonlinearities and rheological behavior. Fig. 13 shows the deformed central directrix of the Recoletos’ roof for the most representative models analyzed, for Torroja’s reduced model and for the built structure. Table 2 details the vertical displacements of the control point. A new model that includes the effects of shrinkage has been included. Shrinkage was considered as a uniform shortening es equal to 54105 obtained according to the Spanish Code EHE-08 [28], considering the concrete to be 20 days old, since this was the lapse of time between pouring and monitoring of the displacements [18]. Autogenous shrinkage was 0.7105 and drying shrinkage 53105. Shrinkage increases vertical displacement by 2.49 cm at the control point. Fig. 13 and Table 2 show that numerical models provide a very good approximation to the deformed shape of the structure. The following specific conclusions can be drawn:  Model C3L-E is slightly stiffer than L2-E because it includes reinforcement.  Model C3L-NLM predicted the subsidence in the small lobe that the linear analyses were not able to capture.  If displacements due to geometrical nonlinearity are added to Model C3L-NLM, the resulting deformed shape (C3L-NLM + C3L-NLG) almost overlaps the previous one along the small lobe and the control point, but rises and folds the big lobe, which tends to match the data from the built structure.  If the effects of shrinkage and geometrical and material nonlinearity are superimposed, the deformation given by the numerical models closely matches the values provided by Torroja (e.g. control point vertical displacement difference of the order of 16%). It must be noted that the displacements in Model C3L-NLM + C3L-NLG + Shrinkage were obtained as a progressive sum of the different effects studied, as the authors consider that this is the best way to visualize the scope of the analysis, although the physical inconsistency of splitting these phenomena is acknowledged.

5. Conclusions This paper describes a series of non-linear analyses of one of Eduardo Torroja’s masterpieces. The numerical models developed use the discrete reinforcement approach and the Multi-crack constitutive model for concrete. By doing so, an insight is provided into the nonlinear behavior of complex shells, and an analysis tool for designers for the assessment of historical structures, more likely to be affected by nonlinear factors, is verified. The results confirm the unpredictability of nonlinear performance, i.e. the real flow of stresses, and therefore the structural behavior, may be widely different from that given by the undeformed and uncracked case, since the evolution of the loss of stiffness in the material will not follow a predictable linear pattern throughout the structure. As far as the authors are aware, these analyses are the first nonlinear analyses carried out on one of Torroja’s major shells and, in agreement with Lozano and Payá-Zaforteza [1], confirm the engineer’s outstanding, daring and accurate designs, showing his mastery of the technique together with his vast experience. However, the FE nonlinear analysis does reveal that he could perhaps have underestimated the resultant stresses, not to the extent that would have led to the collapse of the structure, but the safety factor would probably have been lower than expected. The analyses performed also enable us to establish: (a) the stress state of concrete

and steel, the crack pattern and the stress distribution throughout the shell, and (b) to bear witness to the relative importance of both geometric and material nonlinear analysis with respect to the linear-elastic assumption. Concerning the models, C3L, with its low Elasticity Modulus, was the most reliable. On the other hand, the safety factor preventing buckling does not meet the conservative IASS recommendations [19]. Future research should focus on reproducing the shell failure mode as described by Torroja [18]. To reach this goal, it would be interesting to perform a wider stability analysis and to reproduce the severe bomb damage suffered by the building, which would involve a study of the response of the shell to expansive waves with numerical models, after Zhao et al. [29] or as described by López Cela [30]. This work could also include effects such as the formwork removing sequence and creep. Acknowledgements Funding for this research was provided by the Spanish Ministry for Science and Innovation (Research Project BIA 2011-27,104). The authors are grateful to J. A. Lozano Galant, whose work is the starting point of the present research. The authors also want to thank the Archivo Torroja and their librarians María Isabel Sanchez de Rojas and Virtudes Azorin Albinana, for their kind cooperation and for the drawings provided. Finally, the authors want to thank Prof. Dr. Ester Giménez Carbó for her help on the definition of concrete properties. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.engstruct.2014. 08.044. References [1] Lozano-Galant JA, Payá-Zaforteza I. Structural analysis of Eduardo Torroja’s Frontón Recoletos’ roof. Eng Struct 2011;33(3):843–54. [2] Jefferson AD. Craft, a plastic-damage-contact model for concrete, I. Model theory and thermodynamics. Int J Solids Struct 2003;40(22):5973–99. [3] Jefferson AD. Craft, a plastic-damage-contact model for concrete, II. Model implementation with implicit return-mapping algorithm and consistent tangent matrix. Int J Solids Struct 2003;40(22):5973–99. [4] Hoefakker JH, Blaauwendraad J. Theory of shells. Delft (The Netherlands): Delft University of Technology; 2003. [5] Peerdeman B. Analysis of thin concrete shells revisited opportunities due to innovations in materials and analysis methods. Delft (The Netherlands): Delft University of Technology; 2008. [6] Joedicke J. Shell architecture. New York, USA: Reinhold Publishing Corporation; 1963. [7] Giedion S. Space, time and architecture: the growth of a new tradition. 1st ed. Cambridge, MA, (USA): Harvard University Press, 1967; 1941. [8] Garlock MEM, Billington DP. Félix Candela: engineer, builder, structural artist. New Haven, USA: Yale University Press; 2008. [9] Billington DP. A swiss legacy the art of structural design: a swiss legacy. Princeton, USA: Princeton University Press; 2003. [10] Nuñez-Collado G, Garzon-Roca J, Paya-Zaforteza I, Adam JM. The San Nicolas Church in Gandia (Spain) or how Eduardo Torroja devised a new, innovative and sustainable structural system for long-span roofs. Eng Struct 2013;56:1893–904. [11] Billington DP. The tower and the bridge: the new art of structural engineering. Princeton (USA): Princeton University Press; 1985. [12] Jordá C, editor. Eduardo Torroja, la vigencia de un legado. Eduardo Torroja, the validity of a legacy. Valencia, Spain: Universidad Politécnica de Valencia; 2002 [In Spanish]. [13] Fernández-Ordóñez JA, Navarro-Vera JR. Eduardo Torroja Ingeniero – Engineer. Pronaos, Madrid, Spain: Pronaos; 1999 [Bilingual edition Spanish – English]. [14] Monje-Vergés G, editor. Eduardo Torroja: su obra científica. Eduardo Torroja: his scientific work. Madrid, Spain: Ministerio de Fomento; 1999 [In Spanish]. [15] Jones C. Architecture today and tomorrow. New York, USA: McGraw-Hill; 1961. [16] Torroja E. The structures of Eduardo Torroja; an autobiography of engineering accomplishment. New York (USA): Dodge Corporation; 1958.

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