Structural failure investigations through probabilistic nonlinear finite element analysis: Methodology and application

Accepted Manuscript Structural failure investigations through probabilistic nonlinear finite element analysis: Methodology and application

Fulvio Parisi, Nicola Augenti PII: DOI: Reference:

S1350-6307(17)30325-4 doi: 10.1016/j.engfailanal.2017.07.004 EFA 3222

To appear in:

Engineering Failure Analysis

Received date: Revised date: Accepted date:

10 March 2017 26 June 2017 5 July 2017

Please cite this article as: Fulvio Parisi, Nicola Augenti , Structural failure investigations through probabilistic nonlinear finite element analysis: Methodology and application, Engineering Failure Analysis (2017), doi: 10.1016/j.engfailanal.2017.07.004

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Structural failure investigations through probabilistic nonlinear finite element analysis: Methodology and application

Fulvio Parisi*, Nicola Augenti

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Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio

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21, 80125 Naples, Italy

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Abstract

Structural failure investigations can be strongly influenced by high levels of uncertainty in modelling

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parameters, particularly in the case of historical constructions. This suggests forensic analysts to perform probabilistic simulations, allowing a risk-informed diagnosis and prognosis of structural failures. In this

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study, a failure investigation methodology including uncertainty characterisation, modelling and propagation is presented and applied to a historic piperno stone balcony, the collapse of which caused four casualties. High uncertainty in physical and mechanical properties of piperno stone, which has been

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widely used for a long time in the architectural heritage of Naples and Southern Italy, motivated

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stochastic finite element (SFE) simulations to account for spatial variability of material properties throughout the balcony. Based on field inspections, laboratory surveys and experimental testing, a three-

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dimensional finite element (FE) model with four alternative restraint conditions was developed and material properties were statistically characterised. Experimental data were found to be in agreement with

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those available in the literature for similar piperno stones. Deterministic nonlinear FE simulations with mean material properties showed a major influence of restraint conditions, providing an initial identification of the most realistic model that was able to reproduce the observed damage. Then, SFE simulations were performed on structural models having random fields of material properties. It is shown that the selected SFE model of the balcony had a mean load capacity very close to the total load expected at the time of collapse, allowing the lowest uncertainty level in the output of forensic analysis.

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Corresponding author. Tel.: +39-081-7683659; fax: +39-081-7685921. E-mail address: [email protected] (F. Parisi).

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Keywords: Historical constructions; Structural failures; Forensic analysis; Uncertainty; Stochastic nonlinear finite element analysis.

1. Introduction Failure investigations are a classical type of forensic engineering services that aim at

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identifying the causes and potential responsibilities of a failure, namely, a mismatch

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between actual and expected performance of a material, component or system. The

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importance of studying and disseminating real failure cases has been a matter of discussion by many authors who highlighted the social role of forensic engineering in

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preventing reoccurrence of accidents through the improvement of design, construction and management (e.g. [1]–[4]). Besides, modern forensic engineering services include

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quantitative risk assessment studies to support stakeholders and insurance companies in claims cases [5]. In the context of structural forensic engineering, various authors have

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proposed failure investigation procedures, evidencing the conceptual difference between

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design and forensic analysis (e.g. [6]–[9]). In fact, the design process aims at providing a solution that meets client’s requirements under some code-based assumptions on

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materials, loads, structural configuration and mechanical behaviour. By contrast, failure analysis relies on observational and metrological evidence so causation is determined

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according to realistic assumptions and physical-mathematical models. In the last decades, forensic engineering has benefited from major advances in science and technology that have allowed on one hand three-dimensional site (3D) surveys and sophisticated experimental testing at different scales [10][11], and on the other nonlinear numerical simulations of real engineering failures [12]–[15]. Previous studies have shown the successful use of finite element analysis also in structural damage assessment of historical constructions and their components. Forensic analyses based on

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nonlinear numerical simulations were carried out by, amongst others, Martini [16] and del Coz Díaz et al. [17] who investigated damage to the archaeological site of Pompeii, Italy, and to a masonry wall of an industrial building, respectively. A major issue in structural assessment and forensic analysis of ancient constructions is the high level of

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uncertainty associated with material properties, geometric imperfections, boundary conditions, and capacity models. Even when numerical techniques such as the finite

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element method (FEM) are used, uncertainties have a considerable impact on the output

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of structural response analysis. This fact has strongly stimulated the scientific community to recognize the importance of probabilistic approaches to design,

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assessment and failure analysis. Treating key system properties as random variables

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(RVs), processes or fields allows the classical FEM to be effectively applied to structural components and systems with some degree of uncertainty. In that way,

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uncertainties are modelled and propagated through structural analysis according to a

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rational and transparent method that provides a basis for risk-informed assessment and protection of structures. These objectives cannot be achieved by means of traditional, deterministic FEM approaches in which single values of system variables (e.g.,

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minimum, maximum, mean) or, less frequently, a few prescribed combinations of those

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values are assumed in structural modelling. Based on such considerations, the classical FEM was extended to structural problems with uncertain material, geometric and load variables, delineating the stochastic finite element method (SFEM) that is also named random finite element method or probabilistic finite element method [18]. Dealing with stone masonry structures, Parisi et al. [19] used this probabilistic variant of FEM to carry out a back-analysis of experimental tests and to develop a nonlinear micromechanical model of tuff stone masonry. Those numerical simulations allowed the

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performance limit states at both microscopic and macroscopic scales to be statistically characterised, hence providing uncertainty in the analysis output as a propagation of the uncertainty in the input variables. Zhu et al. [20] validated another micromechanical model for hollow concrete block masonry, highlighting the need for stochastic FE

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modelling that allows the spatial variability of material properties to be taken into account. One of the main challenges is thus the validation of SFEM in failure

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investigations, starting from relatively simple and emblematic case studies.

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In this study, a forensic analysis procedure including the experimental characterisation of material uncertainties and their propagation through SFEM is delineated and applied

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to a historical piperno stone balcony of a masonry building located close to Naples,

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Italy. That balcony collapsed in the presence of five people who were watching down a local parade in the main street running alongside the building. Four people died and a

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person remained hanging on the balcony railing. The importance of this failure case

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study is twofold: (i) the structural element is a typical balcony of monumental masonry buildings in Southern Italy that are protected by the Italian Ministry of Cultural Heritage, and (ii) the application of SFEM to a single structural element rather than a

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whole building strongly emphasizes the influence of material uncertainties on the

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identification of the structural model that provides the best reproduction of observed failure, allowing a risk-informed diagnosis and prognosis.

2. Methodology for structural failure investigations In this section, a comprehensive methodology for the investigation of structural failures is discussed and summarised in Figure 1. When the forensic engineer is committed to investigate a structural failure, some key information provided by the client or judicial

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authority allows the accident to be identified in terms of facility location, structural type, early evidence on the failure event and its consequences (e.g. casualties, business interruption, emergency management). Field inspections are a second important step that actually begins soon after failure and proceeds forward with further on-site

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assessments during the forensic analysis process. Such activities should aim at delineating the post-damage configuration of the structure, which is used to confirm or

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reject computer simulations of structural performance under a given scenario. Field

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inspections may include visual examinations, terrestrial/aerial photographic surveys, terrestrial/aerial video recording, 3D laser scanning, photogrammetric surveys,

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topographic surveys, as well as sampling, taxonomy and tagging of structural and non-

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structural components. Those surveys allow the development of a virtual modelling of the damaged structure, which can be effectively used to compare the simulated and

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observed structural configurations each other. It is worth noting that field inspections in

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the aftermath of failure may be followed by the need for urgent remedial measures, in order to avoid damage propagation and hence additional consequences on the community. This is a typical situation of seismic emergency scenarios in Italy and other

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countries where earthquake clusters can last several months, producing major issues in

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the recovery process of the affected region, as shown by the 2009 L’Aquila earthquake sequence [21]. For instance, effective remedial measures installed on damaged buildings soon after the 2012 Emilia Romagna mainshocks allowed damage accumulation to be reduced over time [22]. Urgent remedial operations may include a temporary seizure of the site, a preliminary safety assessment of the residual structure, the identification of components prone to collapse, the design and execution of controlled demolition, remedial shoring, and structural health monitoring.

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A second group of failure investigation operations concerns data acquisition, which includes a documentation analysis, the collection of witness information, the identification of failure modes and alternative scenarios (especially in terms of loads) compatible with the observed damage, and experimental testing. The analysis of

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documentation regarding the facility under investigation is of paramount importance because it provides evidence on all main events prior to the failure event of interest. The

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forensic engineer should collect all documents related to the life of the facility, that is,

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project, construction and possible modification/retrofit interventions, and structural performance and damage observed during past events. Witness information plays a key

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role as it provides direct observational evidence on pre-damage and failure sequence of

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the structure (through photos and videos that often are shared on social networks and reported by media), but also on potential unauthorized modifications of the structure

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and misuse after construction (the latter influencing the actual loading conditions to

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include in computer simulations). All information collected by documentation and witnesses provides a sound basis for the failure event reconstruction, allowing one or multiple scenarios to be defined and used for structural performance assessment. In

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view of that fundamental stage of failure investigation, each scenario should be a

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deterministic combination of data related to the structural configuration (e.g. shape, size, boundary conditions) and loading at the time of failure. Dealing with loads, it should be noted that measurements of natural events (e.g. windstorms, earthquakes, slow-moving landslides) are frequently available and can be combined with structural health monitoring data, particularly in the case of strategic facilities. The fourth step of the data acquisition process is experimental testing, either on site or in laboratory. Nondestructive tests, such as endoscopy, georadar, tomography, ultrasonic test and thermo-

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vision, allows a qualitative even though useful characterisation of both typology and conservation state of materials, components and the system as a whole. In the case of monumental structures, non-destructive testing is the only one operation allowed by local offices for cultural heritage protection so material modelling can merely rely upon

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the identification of material types, which is followed by a literature-based assignment of statistics to their physical and mechanical properties. Otherwise, performing a

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number of minor destructive tests (e.g. single or double flat jacks) and destructive tests

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(e.g. compression, tensile, shear and shear-compression tests) can support a statistical characterisation of material properties to be directly used in a probabilistic performance

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assessment of the structure. In any case, experimental testing can provide useful

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information for characterisation of uncertainties especially in material properties. Sometimes, laboratory testing of a number of full- or reduced-scale replica of the

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structure is carried out to obtain also a statistical characterisation of structural response,

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although this is very expensive, time-consuming and very complex for historical constructions.

After that a scenario is selected, a probabilistic structural performance assessment can

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be carried out accounting for the uncertain nature of system variables. As most of

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variables can be measured in previous stages of the failure investigation process and a given scenario in terms of loads and boundary conditions has been chosen, a major fraction of uncertainty still remains in material properties, particularly in the case of historical constructions. Therefore, the selected scenario is deemed reasonable if the corresponding outcome of probabilistic simulations allows the observed damaged configuration of the structure to be reproduced with the minimum level of uncertainty. This requires several iterations over the assumed set of scenarios in order to get the

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solution of forensic analysis. Characterising uncertainty in simulation output means to perform a risk-informed diagnosis of structural failure, that is, the identification of the most probable failure mode along with its causes and potential responsibilities to be communicated to the client or the judicial authority in the framework of a criminal

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procedure. It is also emphasised that further simulations can be run on the residual structure in order to carry out a risk-informed prognosis, which means a probabilistic

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prediction of damage evolution under future hazardous events. If the residual structure

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needs to be preserved (as in the case of monumental constructions), a quantitative risk analysis can be performed to assess the probability of higher damage levels to be

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compared with the acceptable risk level. If the latter is exceeded, a risk mitigation

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strategy should be selected to reduce hazard, vulnerability or exposure of the residual structure.

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In the following section, the forensic analysis methodology discussed above is applied

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to a historical piperno stone balcony that suddenly fell down, producing four casualties. Such a case study, even though conceptually quite simple because related to a single structural component, is deemed methodologically appropriate to highlight the

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importance of probabilistic finite element simulations of structural failures when the

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engineering system under investigation is affected by high levels of uncertainty.

3. Description of case study and data acquisition 3.1. Identification of failure case study The balcony belonged to the main façade of a historical masonry building that was built in 1868 and is located in a village close to Naples, Italy (Fig. 2). Such a structural element falls in a wide class of ancient balconies that are made of a single piperno stone

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slab supported by the load-bearing masonry wall of the building façade. Each balcony may have either a constant or variable width in plan and is a cantilevered structural element. Dealing with the balcony material, piperno stone is a volcanic building material widely used in the architectural heritage of Naples, many other urban centres of

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Campania region and even in other regions of Southern Italy. According to Calcaterra et al. [23], piperno has been used for a long time since at least the Roman age until the

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first decades of the 20th century. Together with Neapolitan yellow tuff, piperno is a

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Late Quaternary product of Phlegraean Fields (Campi Flegrei in Italian) that is a large volcanic area situated to the west of Naples. Piperno has an eutaxitic texture composed

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of black collapsed scoriae called fiammae (with flattening ratio approximately equal to

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1:10) and a light grey ashy matrix. The macroscopic variability in textural features such as the scoriae-to-matrix ratio, scoriae dimension and welding degree produces high

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variability in physical-mechanical properties and weather-related degradation of piperno

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(e.g. alveolization of scoriae, disaggregation and salt crystallization of ashy matrix). This motivates an explicit modelling of uncertainties in structural properties of piperno stone elements, as presented in the following.

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The analysis of documentation related to the building did not indicate past maintenance

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nor structural interventions on the balcony under investigation. Based on witness information, the balcony fell down under the presence of five people which were rather uniformly distributed over the length. The collapse of the balcony occurred as a result of a single complete fracture in the central zone and fractures at the location of the interface between the balcony and the façade wall. The balcony split in two segments as shown in Figure 3. The consequences of that failure were four died and one injured.

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3.2. Field inspection and geometric survey The geometry and boundary conditions were investigated both on site and in laboratory by means of photographical surveys and measurements. The piperno stone balcony under investigation was supported by the tuff stone masonry wall of the building façade

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and was characterised by an irregular shape because of its variable thickness and the presence of large holes especially in the central zone, as shown by the on-site

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photographical survey of balcony-wall interface in Figure 4. There was no evidence of

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previous damage due to past accidental actions (e.g. impact) on the balcony and deterioration of external surface exposed to environmental actions. The state of

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conservation of the case-study balcony appeared to be the same of other balconies in the

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same building. As a consequence, the city mayor ordered all building owners to appoint practitioners for the safety assessment of other balconies.

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A simplified geometric representation of the balcony in plan is depicted in Figure 5,

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where a skewed fracture line is reported. The balcony was approximately 2.65 m-long and 0.63 m-wide. According to Figures 6 and 7, the balcony had a strongly irregular longitudinal cross-section with a central hole in the load-bearing wall and a composite

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marble-mortar pavement system. This resulted in a structural part of the balcony (made

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of piperno stone only) having a thickness between 30 mm and 155 mm (Fig. 7). The lowest thickness was basically associated with the marble-mortar system, the latter having a total length of approximately 1.65 m and being a part of the doorway. The presence of the marble-mortar pavement system induced a strongly different restraint condition to the piperno stone slab, if compared to thicker parts of the balcony. The net area of the longitudinal cross-section was roughly 2/3 of the ideal area inside the dashed line in Figure 7.

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3.3. Physical and mechanical properties of piperno stone After that the two segments of the balcony were carried in laboratory, eight cores with 94 mm diameter and height-to-diameter ratio approximately equal to 1 were drilled.

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Mass density was found to have a mean  = 1848 kg/m3 (corresponding to a unit weight

 = 18.13 kN/m3) and a coefficient of variation CoV = 4.70%.

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Given the importance of non-destructive testing for mechanical characterisation of

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historical constructions in compliance with worldwide conservation rules, the authors of

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this study decided to perform ultrasonic tests on piperno cores and to characterise the correlation between uniaxial compressive strength and ultrasonic pulse velocity. In that

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way, future structural assessments of historical constructions including piperno stone elements could benefit from a material modelling based on the estimation of

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compressive strength rather than its direct characterisation through destructive testing.

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Ultrasonic tests were carried out according to the European standard EN 12504-4 [24]. Ultrasonic pulse velocity was measured two times on each single piperno core, resulting

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in a mean velocity Vm = 2178 m/s with CoV = 7.75%. Ultrasonic tests also evidenced a mean bulk modulus K = 8815 MPa with CoV = 15.92%.

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The uniaxial compressive behaviour of piperno stone was experimentally characterised by means of destructive tests on cores, according to EN 1926 [25]. Figures 8a and 8b show a cylindrical specimen before and after a uniaxial compression test, respectively. All tests were carried out with displacement control in order to derive both the rising and post-peak falling branches of force–displacement curves. The vertical displacement was imposed by an MTS810 testing machine with displacement rate of 0.01 mm/s. That machine has a maximum load capacity of 500 kN, both in tension and compression,

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with maximum stroke equal to ±75 mm. Axial displacements were measured via three linear variable differential transformers (LVDTs). Displacement readings were significantly different from stroke readings of the testing machine and were considered to be more reliable measures of material deformation capacity. The maximum stroke of

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the LVDTs used for compression tests was 20 mm, which is approximately one-fifth of specimens height. The accuracy of those LVDTs was equal to 0.2 mm. The axial load

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was measured by a load cell in the testing machine. The sampling rate of measurements

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was set to 1 Hz. Figure 9 shows the stress–strain curves of compression tests. The peak compressive strength fc was found to have a mean value fcm = 20.21 MPa with CoV =

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8.55%. Young’s modulus of piperno stone was estimated at one-third and one-half of

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peak compressive strength, resulting in mean values E1/3 = 2260 MPa (CoV = 11.29%) and E1/2 = 2222 MPa (CoV = 11.94%), respectively. The axial strain at peak

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compressive strength was approximately equal to 1%, whereas maximum strain levels

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between 3% and 6% were attained to derive stress–strain curves up to one-half of peak strength on the falling branch. Based on E1/2 and K, the upper-bound estimate of the mean Poisson’s ratio of piperno stone was found to be  = 0.46 with CoV = 1.37%,

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according to the American standard ASTM C597-02 [26].

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The following linear correlation model (with a coefficient of determination R2 = 0.58) was fitted to experimental data (Fig. 10): fc  9.3 103V

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in which V is measured in m/s. Such a linear model allows the estimation of mean compressive strength given the ultrasonic pulse velocity. An alternative correlation model which allows the maximisation of R2 is the following quadratic function (R² = 0.63):

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fc  1105V 2  6.18 102V  57.23

(2)

To estimate tensile strength of piperno stone, two series of prismatic specimens were subjected to three-point bending tests. The specimens and experimental setup of series 1 were prepared according to EN 1015-11 [27], as shown in Figure 11a. In detail, six

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specimens with size 40×40×160 mm3 were simply supported with clear span equal to 100 mm and were tested with displacement control by means of the MTS810 machine.

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The displacement rate was set to 0.05 mm/s and vertical displacement of mid cross-

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section was measured by a single LVDT placed below the specimen. The LVDT had a maximum stroke of 10 mm and accuracy equal to 0.1 mm. The four specimens of series

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2 were 50×50×480 mm3 in size, resulting in a length-to-depth ratio significantly higher

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than that of series 1 (i.e. 8 rather than 2.5). The clear span of those specimens was equal to 400 mm (Fig. 11b). The experimental tests on slender specimens were aimed at

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eliminating potential arch resisting mechanisms, allowing a more correct use of the

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Navier’s equation for classical Euler-Bernoulli beam elements. Figure 12 outlines strongly different force–displacement curves for the two specimen series tested in three-

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point bending, evidencing the influence of the length-to-depth ratio. Series 1 led to estimate a mean tensile strength ftm = 3.91 MPa with CoV = 16.60%, whereas series 2

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provided ftm = 2.07 MPa with CoV = 12.08%. Uplift movements at support centrelines were not observed.

3.4. Comparison with existing experimental data After that case-specific experimental data are collected, their comparison with those available in the literature for similar materials and systems plays an essential role in order to validate subsequent structural modelling and failure analysis.

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Based on the high relevance of piperno stone in the architectural heritage of Naples and Campania region, some researchers investigated the piperno stones extracted from quarries of two districts of Naples named Soccavo and Pianura ([23], [28]–[30]). Therein, the visible piperno stone deposit is approximately 20 m-deep. Maggiore [31]

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detected six piperno stone layers with different petro-physical properties. On the basis of that field investigation, Calcaterra et al. [28] derived a layered model where each

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layer is defined in terms of depth range and main petro-physical properties. In 2000,

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Calcaterra et al. [23] authored the first comprehensive paper on the main mineralogical, petrographic and engineering-geological properties of piperno stone. Those researchers

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performed a very high number of experimental tests, including 30 compression tests on

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cubic specimens from Soccavo district and 10 similar tests on specimens from Pianura district. The maximum length of fiammae ranged between 300 mm and 400 mm.

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Although the piperno stones of those quarries had approximately the same

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mineralogical and petrographic characteristics, the experimental tests showed significant discrepancies in statistics of physical and mechanical properties. Table 1 outlines the mean and CoVs of dry mass density d, unit weight , porosity , ultrasonic

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pulse velocity V, and uniaxial compressive strength fc. The variability of almost all

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properties of piperno stones from Pianura district was lower than in the case of Soccavo district. A strong difference between compressive strength values of the two series of piperno samples is observed. In detail, the mean compressive strength related to Soccavo district was found to be 4.11 times higher than that related to Pianura district, the latter having a significantly higher level of porosity. All experimental results on piperno stone properties were aggregated in a single data set by Calcaterra et al. [28], the statistics of which are summarised in Table 2 that shows

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higher dispersion particularly in dry mass density, porosity and uniaxial compressive strength. Table 2 provides also some statistics on Young’s modulus at one-half of peak compressive strength. According to data subsets in Table 1, most of dispersion in such material properties can be associated with piperno stones from Soccavo district. It

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should be noted that the flat inclusions in piperno stones (i.e. fiammae) are an important source of material orthotropy, increasing dispersion in physical and mechanical

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properties. For instance, non-volumetric parameters such as ultrasonic pulse velocity

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and uniaxial compressive strength show higher values in the direction parallel to fiammae.

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The piperno stone of the balcony under study showed a uniaxial compressive behaviour

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very close to that highlighted by Calcaterra et al. [23] in the case of 3/5H layers, confirming an axial strain at peak compressive strength approximately equal to 1% and

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a peak compressive strength roughly ranging from 14 MPa to 27 MPa. Furthermore,

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Table 2 shows an overall consistency between the case-study piperno stone and those investigated in previous works, especially in terms of mass density (1848 kg/m3 versus 1733 kg/m3, respectively), ultrasonic pulse velocity (2178 m/s versus 2695 m/s),

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uniaxial compressive strength (20.21 MPa versus 20.95 MPa) and Young’s modulus at

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one-half of peak compressive strength (2222 MPa versus 2542 MPa). By contrast, no experimental data on Poisson’s ratio and tensile strength of piperno stone were found in the literature.

4. Numerical modelling 4.1. Geometry and boundary conditions Data collected through surveys and experimental tests were used to develop a finite

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element (FE) model within ABAQUS/Standard software [32]. The model consisted of 5248 solid elements, each of them having six nodes (Fig. 13). The critical modelling aspects of the case-study structural element were the low tensile strength of piperno stone and restraint conditions. The latter were strongly influenced by two construction

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features: (i) the presence of the door opening, a significant thickness drop due to the marble slab and two mortar beds above and below it, two holes with steel screws, and a

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large hole in the central zone of the balcony; and (ii) a gradual thickness reduction on

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the left-hand side of the balcony due to the absence of a consecutive balcony segment (see e.g. Fig. 2). In that respect, it is noted that the right-hand side of the balcony

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element was not structurally connected to the adjacent balcony element. Based on such

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considerations, the balcony-wall interface section was divided in two edge zones (Fig. 14a) and a central zone (Fig. 14b) in order to assign them either equal or different

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considered as follows:

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restraint conditions. Four alternative assumptions on restraint conditions were

– RC1 (Fixed): Section totally fixed to the wall. – RC2 (Fixed-Hinged): Section with fixed nodes in the edge zones and hinged nodes

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in the central zone.

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– RC3 (Hinged): Section totally hinged to the wall. – RC4 (Hinged-Free): Section with hinged nodes in the edge zones and free nodes in the central zone. Each of those restraint conditions was considered as a prescribed scenario in which either deterministic or probabilistic FE analysis can be run.

4.2. Loads

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Mass density of piperno stone was considered as dead load. Based on witness information, five people were placed on the balcony at the time of collapse. Each person was considered as uniformly distributed live load over a squared area of 300×300 mm2 (Fig. 15). Nonlinear analysis consisted in the application of incremental live loads so

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that vertical displacement in a control point of the mid-section perpendicular to the balcony-wall interface was monotonically increased. The total vertical load on the

4.3. Selection and calibration of material model

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balcony at the moment of collapse was thus expected to be approximately 10.53 kN.

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The mechanical behaviour of piperno stone was simulated through the Concrete

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Damaged Plasticity (CDP) model [32]. The CDP model was originally proposed for concrete ([33]–[36]) and was successfully applied to several masonry assemblages and

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structures (see e.g. [37], [38]). The chance to use the CDP model both in the case of

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concrete and masonry relies upon their quasi-brittle behaviour and moderate orthotropy ratio under biaxial compressive stresses, namely the ratio between biaxial and uniaxial (initial) compressive strengths denoted by fb0 and fc0, respectively. More in detail, the

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CDP model is a continuum, plasticity-based, damage model that assumes tensile

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cracking and compressive crushing as main failure modes. The material is assumed to have the same elastic properties, but different strengths and damage properties, in tension and compression. The failure surface of the CDP model was developed by Lubliner et al. [35] on the basis of the classical Drucker-Prager model, with some modifications proposed by Lee and Fenves [36]. The evolution of the failure surface is controlled by two hardening variables, namely, the tensile and compressive equivalent plastic strains. Therefore, the CDP model consists of the combination of isotropic

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damaged elasticity with isotropic, non-associated, multi-hardening plasticity to represent the inelastic behavior. Furthermore, the material model is based on conventional stress–strain coaxiality, so the total strain rate is basically considered as the sum of an elastic component and a plastic component, without further

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decomposition of the plastic strain in its coaxial and non-coaxial parts. Accordingly, the plastic strain turns out to be perpendicular to the failure surface. Some considerations on

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potentialities of non-coaxial plasticity models for numerical analysis of slabs can be

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found, for instance, in [39].

The elastic behaviour was characterised in terms of secant Young’s modulus E1/2 and

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Poisson’s ratio , the latter ranging between 0.2 and 0.46. The upper bound to  was

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derived from ultrasonic tests (see Section 3.2), whereas the lower bound was collected from the literature [40]. The initial compressive strength was assumed to be fc0 = fc/3,

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whereas the initial tensile strength was directly set equal to the peak strength, that is, ft0

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= ft. The plastic behaviour in compression was simulated by a uniaxial stress–strain model with parabolic hardening up to peak compressive strength and nonlinear softening up to an ultimate compressive strain of 3%. By contrast, the material was

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assumed to have a quasi-brittle behaviour in uniaxial tension with an ultimate

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displacement (corresponding to vanishing tensile strength) set to 0.2 mm according to experimental tests on specimen series 1. A linear softening rule was adopted to simulate tensile strength degradation up to failure, and hence a macroscopic description of the gradual formation of micro-cracks in the material. The dilation (or dilatancy) angle at high confining pressures was assumed to be  = 30°, whereas the eccentricity parameter was set to e = 0.1 and influences the change in dilation angle under varying confining pressure. The ratio between biaxial and uniaxial compressive strengths was

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set to 1.16 and the Kc coefficient affecting the shape of failure surface in the deviatory plane was set to 1. Preliminary elastic analyses of the balcony showed that structural response was rather insensitive to fc while highlighting a major influence of restraint conditions and tensile

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strength. As a matter of fact, fc was not achieved in any case whereas ft strongly influenced the structural capacity. This motivated the calibration of ft which was carried

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out so that each three-point bending test on piperno stone specimens was reproduced.

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FE simulations were performed for both specimen series and 972 eight-node cubic finite elements with edge equal to 7 mm were assumed. In detail, the force–displacement

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curve of each single specimen was iteratively simulated until the numerical peak force

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attained the experimental resistance. For instance, Figure 16 shows the experimentalnumerical comparison for specimen series 1. That procedure led to numerical estimates

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of tensile strength between 1.40 MPa and 2.40 MPa (i.e. 7%–12% of mean compressive

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strength), with mean ftm = 2.09 MPa and CoV = 17.67%. It is interesting to note that this mean value is approximately equal to that directly estimated by means of three-point bending tests on slender specimens (series 2).

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Given that three-point bending tests are typically used to estimate ft through the

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Navier’s equation according to EN 1015-11 standard [27], it was interesting to characterise a correction factor  = ft,exp/ft,FEM denoting by ft,exp the experimental estimate of ft according to [27] and ft,FEM the numerical estimate of ft derived by model calibration. In the case of specimen series 1,  was found to have mean m = 1.88 and CoV = 3.38%. The calibration of tensile strength was repeated on the basis of experimental results related to specimen series 2 through a FE model composed of 6144 eight-node cubic elements. The numerical estimates of tensile strength were between

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1.10 MPa and 1.70 MPa (i.e. 5%–8% of mean compressive strength), with mean ftm = 1.33 MPa and CoV = 19.85%. The correction factor  was found to have m = 1.58 and CoV = 5.27%. For both specimen series, it should be noted that a nonlinear distribution of axial strains over the depth of specimen mid-section was found in FE analysis. Thus,

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the Euler-Bernoulli assumption of plane section after flexural deformation in the Navier’s equation was strongly inaccurate, resulting in an overestimation of tensile

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strength according to mechanical nonlinearity of piperno stone [41]. This is reflected by

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the fact that tensile strength was overestimated even in the case of slender specimens (series 2), namely when the arch resisting mechanism was negligible. In addition, the

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analysis of sectional axial strains in both specimen series evidenced that neutral axis

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was located approximately at one-half of sectional depth. It is worth noting that several building codes and standards take into account the difference between tensile and

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flexural strength of materials. For instance, if standard prismatic specimens with the

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same size of series 1 are subjected to flexural tests, the ratio between tensile and flexural strengths is assumed to be 0.44, 0.64 and 0.83 by the 2010 fib Model Code [42], Eurocode 2 [43] and Italian Building Code [44], respectively. The three-point bending

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tests on piperno stone specimens allowed the estimation of the mean of such a ratio, i.e.

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1/m, which was equal to 0.53 and 0.63 for specimen series 1 and 2, respectively.

5. Discussion of analysis results 5.1. Deterministic simulations Nonlinear static analysis of the balcony with mean values of material properties and different restraint conditions (from RC1 to RC4) was carried out. Figure 17 shows that restraint conditions significantly influenced the peak load capacity Fmax of the balcony,

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which was equal to 180.00 kN for RC1, 119.50 kN for RC2, 82.80 kN for RC3, and 13.70 kN for RC4. The restraint condition RC4 was considered to be more realistic than others because of the poor construction features of the central zone and the presence of the door opening that definitely did not provide a fixed or hinged restraint condition

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(see Section 4.1). Therefore, the analysis case RC4 was further investigated under varying tensile strength of piperno stone. The latter was assumed to range between 1.4

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MPa and 2.4 MPa with step equal to 0.2 MPa, according to the model calibration based

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on specimen series 1 (see Section 4.3). As ft ranged from the lower to the upper bound, Fmax oscillated from 9.89 kN to 15.05 kN. This means that an increment of 1 MPa in

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tensile strength determined an increase of 5.15 kN in peak load capacity. Principal

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stresses in the FE model were monitored during nonlinear static analysis. Figures 18a and 18b show the distribution of principal stresses (above and below the balcony,

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respectively) corresponding to the attainment of peak load capacity in the balcony with

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restraint condition RC4. According to those different 3D views of the FE model, the peak load-bearing capacity was reached when the edge zones of the balcony-wall section and the transverse mid-section failed in tension, confirming the failure mode

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described by witnesses. According to preliminary elastic analyses, the maximum

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compressive stress did not reach fc in any finite element and the maximum stress-tostrength ratio was equal to 0.18. After that the nonlinear FE model was found to provide a good reproduction of observed damage (see Fig. 1), probabilistic simulations were run to assess the uncertainty in the analysis output.

5.2. Stochastic finite element simulations Based on the statistical characterisation of physical and mechanical properties of

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piperno stone through laboratory testing and calibration of numerical model, stochastic finite element (SFE) simulations were carried out for each of the four alternative restraint conditions of the balcony. SFE simulations allowed the spatial variability of material properties (i.e. inhomogeneity of piperno stone) to be taken into account in the

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forensic analysis. Regardless of the restraint condition under study, the following material properties were considered as RVs: mass density, secant Young’s modulus at

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one-half of peak compressive strength, Poisson’s ratio, peak compressive strength, and

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peak tensile strength. Poisson’s ratio was assumed to be uniformly distributed over the range [0.2,0.45], whereas the other RVs were modelled by means of truncated normal

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distributions. One thousand realizations of material properties were randomly generated

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by sampling RVs according to Monte Carlo simulation technique. Each realization vector of RVs was randomly assigned to the finite elements of the structural model.

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Figure 19 shows one of the random realizations of the FE model.

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Nonlinear static analysis was carried out on each SFE model and for each restraint condition assigned to the balcony. SFE simulations confirmed that the structural capacity of the balcony was strongly influenced by the assumption on restraint

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conditions (Figs. 20a–d). Table 3 outlines the statistics of Fmax and the corresponding

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vertical displacement Fmax for each restraint condition. Since SFE simulations allow material uncertainties to be propagated, Table 3 shows that the level of uncertainty in the analysis output (herein measured by CoV) strongly reduced from restraint condition RC1 to RC4. The latter analysis case produced a mean value of peak load capacity equal to 10.76 kN, namely, very close to that expected at the time of collapse. As the lowest level of uncertainty was that associated with the restraint condition RC4, that analysis case was assumed to be the most realistic representation of the balcony under

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investigation.

6. Conclusions A failure investigation methodology based on characterisation, modelling and

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propagation of uncertainties has been presented and applied to the forensic analysis of a historic piperno stone balcony, the collapse of which induced four casualties. Such a

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case study was selected because of the wide use of piperno stone in the architectural

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heritage of Naples and Southern Italy, as well as the high level of uncertainty in material properties due to piperno stone inhomogeneity and orthotropy. The latter feature

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highlights the need for a risk-informed failure diagnosis and prognosis especially in the

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case of structural damage to historical constructions. This motivated the use of stochastic finite element simulations in which the spatial variability of material

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properties throughout the volume of the balcony was explicitly considered.

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After that on-site inspections and laboratory surveys were carried out to define geometric and construction characteristics of the balcony, physical and mechanical properties of piperno stone were statistically characterised and were found to be in good

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agreement with data sets available in the literature. A 3D finite element model was then

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developed, considering the strong geometric irregularity of the balcony. Four alternative restraint conditions of the balcony were assumed as prescribed scenarios for probabilistic simulations. A preliminary sensitivity analysis showed the major influence of restraint conditions and tensile strength. Therefore, the tensile strength of piperno stone was calibrated so that force–displacement curves of prismatic stone specimens subjected to three-point bending tests were numerically reproduced. Both deterministic and probabilistic variants of nonlinear finite element simulations

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were performed under different restraint conditions. The first group of simulations was carried out on FE models with mean material properties, evidencing the importance of restraint conditions in the assessment of peak load capacity. It was found that the deterministic FE model with the lowest level of restraint produced a peak load capacity

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quite close to the expected total load at the time of collapse. In addition, the distribution of principal stresses in the same FE model was consistent with the actual failure mode

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observed by witnesses.

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In the probabilistic simulations, a Monte Carlo sampling technique was used to generate random fields of material properties and one thousand simulations were run for each

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restraint condition. Those simulations confirmed that modelling assumptions on

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restraint conditions strongly influenced the structural capacity of the balcony, causing significantly different predictions of mean load-carrying capacity and the corresponding

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uncertainty level. Stochastic simulations allowed the more realistic model to be

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determined by comparing the mean of peak load capacity of the balcony to the total load expected at the time of collapse. The best estimate of peak load capacity was also associated with the lowest uncertainty level.

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Given that on one hand the case-study balcony is a typical structural element of

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historical constructions in Naples and Southern Italy, and on the other, piperno stone has been also used for other types of structural elements (e.g. columns, arches, building corners), this study provides experimental data and a probabilistic analysis procedure that can be effectively used to assess the safety level of piperno stone elements and to design strengthening systems for prevention of future accidents. Practitioners and researchers can use not only statistics of physical and mechanical properties of piperno stone reported in this paper, but also the regression models derived from ultrasonic and

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compression tests to estimate mean compressive strength through non-destructive ultrasonic tests. Besides, the CDP model calibrated in ABAQUS was proven to provide rather accurate response predictions of structural piperno stone elements. Nonetheless, the major contribution of this paper is the methodology for probability-based diagnosis

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and prognosis of structural failures of historical constructions.

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References

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[1] Carper KL, ed. Forensic engineering, 2nd ed. Boca Raton: CRC Press; 2000.

Society of Civil Engineers; 2003.

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[2] Lewis GL, ed. Guidelines for forensic engineering practice. Reston: American

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[3] Delatte N. Failure literacy in structural engineering. Eng Struct 2010; 32(7): 1952– 4.

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McGraw-Hill; 2010.

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[4] Ratay RT, ed. Forensic structural engineering handbook, 2nd ed. New York:

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[6] Augenti N, Chiaia B, ed. Ingegneria forense. Palermo: Dario Flaccovio Editore;

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[7] Bontempi F. Ingegneria forense strutturale: Basi del progetto e ricostruzione dei collassi. In: Augenti N, Bontempi F, eds. Ingegneria Forense, Crolli, Affidabilità Strutturale e Consolidamento ― Proc IF CRASC ’15 conference, Rome; 2015. p. 3–20 [in Italian]. [8] Brady SP. Role of the forensic process in investigating structural failure. J Perform Construct Facil 2012; 26(1): 2–6.

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[9] Brady SP. Effectively responding to structural engineering failure: Expertise and cognitive entrenchment. J Perform Construct Facil 2014; 28(5): 04014020. [10] Beasley KJ. Laboratory testing for building failure investigations. J Perform Construct Facil 2004; 18(1): 2–3.

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[11] Drerup MJ, Carper KL. Forensic investigations using nondestructive techniques. J Perform Construct Facil 2005; 19(1): 2.

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[12] Caglayan O, Yuksel E. Experimental and finite element investigations on the

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collapse of a Mero space truss roof structure –– A case study. Eng Fail Anal 2008; 15(5): 458–70.

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[13] Calderón PA, Adam JM, Payá-Zaforteza I. Failure analysis and remedial measures

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applied to a RC water tank. Eng Fail Anal 2009; 16(5): 1674–85. [14] del Coz Díaz JJ, Lozano Martinez-Luengas A, Adam JM, Martin Rodriguez A.

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Non-linear hygrothermal failure analysis of an external clay brick wall by FEM ––

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A case study. Construct Build Mater 2011; 25(12): 4454–64. [15] Augenti N, Parisi F. Buckling analysis of a long-span roof structure collapsed during construction. J Perform Construct Facil 2013; 27(1): 77–88.

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[16] Martini K. Ancient structures and modern analysis: Investigating damage and

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reconstruction at Pompeii. Autom Construct 1998; 8: 125–37. [17] del Coz Díaz JJ, Adam JM, Lozano Martínez-Luengas A, Pedro Alvarez Rabanal F. Collapse of a masonry wall in an industrial building: Diagnosis by numerical modeling. J Perform Construct Facil 2013; 27(1): 65–76. [18] Haldar A, Mahadevan S. Reliability assessment using stochastic finite element analysis. New York: Wiley; 2000. [19] Parisi F, Balestrieri C, Asprone D. Nonlinear micromechanical model for tuff stone

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masonry: Experimental validation and performance limit states. Construct Build Mater 2016; 105: 165–75. [20] Zhu F, Zhou Q, Wanga F, Yang X. Spatial variability and sensitivity analysis on the compressive strength of hollow concrete block masonry wallettes. Construct

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Build Mater 2017; 140: 129–38. [21] Augenti N, Parisi F. Learning from construction failures due to the 2009 L’Aquila,

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Italy, earthquake. J Perform Construct Facil 2010; 24(6): 536–55.

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[22] Parisi F, Augenti N. Earthquake damages to cultural heritage constructions and simplified assessment of artworks. Eng Fail Anal 2013; 34(6): 735–60.

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[23] Calcaterra D, Cappelletti P, Langella A, Morra V, Colella A, de Gennaro R. The

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building stones of the ancient centre of Naples (Italy): Piperno from Campi Flegrei. A contribution to the knowledge of a long-time-used stone. J Cultural Herit 2000; 1: 415–27.

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[24] EN 12504-4. Testing concrete – Part 4: Determination of ultrasonic pulse velocity. Comité Européen de Normalisation, Brussels; 2004. [25] EN 1926. Natural stone test methods – Determination of compressive strength.

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Comité Européen de Normalisation, Brussels; 1999.

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[26] ASTM C597-02. Standard test method for pulse velocity through concrete. ASTM International, West Conshohocken; 2002. [27] EN 1015-11. Methods of test for mortar for masonry – Determination of flexural and compressive strength of hardened mortar. Comité Européen de Normalisation, Brussels; 1999. [28] Calcaterra D, Langella A, de Gennaro R, de’ Gennaro M, Cappelletti P. Piperno from Campi Flegrei: A relevant stone in the historical and monumental heritage of

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Naples (Italy). Environ Geol 2005; 47: 341–52. [29] Calcaterra D, Cappelletti P, de’ Gennaro M, de Gennaro R, de Sanctis F, Flora A, Langella A. The rediscovery of an ancient exploitation site of Piperno, a valuable historical stone from the Phlegraean Fields (Italy). In: Prykril R, Smith BJ, eds.

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Building Stone Decay: From Diagnosis to Conservation. London: Geological Society, Special Publications 2007; 271: 23–31.

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[30] de’ Gennaro M, Calcaterra D, Cappelletti P, Langella A, Morra V. Building stone

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and related weathering in the architecture of the ancient city of Naples. J Cultural Herit 2000; 1: 399–414.

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[31] Maggiore L. Notizie sui materiali vulcanici della Campania utilizzati nelle

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costruzioni. Estratto Relazione Servizio Minerario Statistiche Industria Estrattiva 1934; 45(60): 57–61 [in Italian].

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[32] ABAQUS 6.10-1. ABAQUS Documentation. Rhode Island: Dassault Systèmes;

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2010.

[33] Ottosen NS. A failure criterion for concrete. J Eng Mech 1977; 103(4): 527–35. [34] Oñate E, Oller S, Oliver J, Lubliner J. A constitutive model for cracking of concrete

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based on the incremental theory of plasticity. Eng Comput 1988; 5(4): 309–19.

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[35] Lubliner J, Oliver J, Oller S, Oñate E. A plastic-damage model for concrete. Int J Solids Struct 1989; 25(3): 299–326. [36] Lee J, Fenves GL. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 1998; 124(8): 892–900. [37] Illampas R, Charmpis DC, Ioannou I. Laboratory testing and finite element simulation of the structural response of an adobe masonry building under horizontal loading. Eng Struct 2014; 80: 362‒ 76.

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[38] Yekrangnia M, Mobarake AA. Restoration of historical Al-Askari Shrine – II: Vulnerability assessment by numerical simulation. J Perform Construct Facil 2016; 30(3): 04015031. [39] Brunesi E, Nascimbene R. Numerical web-shear strength assessment of precast

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prestressed hollow core slab units. Eng Struct 2015; 102: 13–30. [40] Augenti N. Il calcolo sismico degli edifici in muratura. Turin: UTET; 2004 [in

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Italian].

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[41] Farmer IW. Engineering properties of rocks. London: E&FN Spon; 1968. [42] Fédération internationale du béton (fib). fib Model Code for concrete structures

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2010. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA; 2013.

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[43] EN 1992-1-1:2005. Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings. Comité Européen de Normalisation, Brussels; 2005.

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[44] DM 14.01.2008: Norme tecniche per le costruzioni. Italian Ministry of

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Infrastructure and Transportation, Rome; 2008 [in Italian].

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List of Figures Fig. 1. Flowchart of failure investigation methodology. Fig. 2. Piperno stone balcony under investigation. Fig. 3. Segments of the balcony in laboratory.

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Fig. 4. Photographical survey of the balcony-wall interface. Fig. 5. Plan view of the balcony with skewed fracture line in the central zone

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(dimensions in metres).

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Fig. 6. Longitudinal cross-section of the balcony.

Fig. 7. Thickness (vertically oriented figures) versus progressive distance (horizontally

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oriented figures) from left-hand side edge of the balcony (dimensions in millimetres).

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Fig. 8. Piperno stone specimen (a) before and (b) after uniaxial compression test. Fig. 9. Stress–strain curves of piperno stone under uniaxial compression.

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velocity of piperno stone.

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Fig. 10. Correlation between uniaxial compressive strength and ultrasonic pulse

Fig. 11. Specimens subjected to three-point bending test: (a) series 1; (b) series 2. Fig. 12. Force–displacement curves of piperno stone under three-point bending.

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Fig. 13. Mesh of the 3D finite element model of the piperno stone balcony.

zone.

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Fig. 14. Boundary conditions of the 3D finite element model: (a) edge zone; (b) central

Fig. 15. Loading surfaces corresponding to five people on the piperno stone balcony. Fig. 16. Experimental versus numerical force–displacement curves of piperno stone specimens (series 1) under three-point bending. Fig. 17. Deterministic force–displacement curves of the balcony with mean material properties under varying restraint conditions.

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Fig. 18. Distributions of principal stresses (in Pa) at the attainment of peak load capacity: (a) top view; (b) bottom view. Fig. 19. Stochastic finite element model of the balcony. Fig. 20. Probabilistic force–displacement curves of the balcony under varying restraint

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condition: (a) RC1; (b) RC2; (c) RC3; (d) RC4.

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List of Tables Table 1. Statistics of piperno stone properties according to Calcaterra et al. [23]. Table 2. Statistics of piperno stone properties according to Calcaterra et al. [28]. Table 3. Statistics of peak load capacity and corresponding displacement of the

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balcony.

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Table 1 Statistics of piperno stone properties according to Calcaterra et al. [23]. Soccavo district

Pianura district

Property Max

Mean

CoV

Min

Max

Mean

CoV

d [kg/m3]

1566

2257

1860

11.61%

1295

1684

1456

7.35%

 [kN/m ]

25.18

26.26

25.74

0.78%

24.97

26.07

25.68

0.97%

 [%]

12.03

39.64

27.80

30.94%

34.09

49.90

43.30

10.07%

V [m/s]

2229

3239

2707

10.55%

2249

3105

2628

8.86%

fc [MPa]

13.46

67.47

25.91

50.17%

4.75

8.50

6.30

19.52%

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3

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Min

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Property

Min

Max

Mean

CoV

d [kg/m ]

1295

2259

1733

15.00%

 [kN/m3]

25.18

26.12

25.71

0.78%

 [%]

12.03

49.90

32.81

31.88%

V [m/s]

2229

3239

2695

9.94%

fc [MPa]

4.75

67.47

20.95

66.06%

E1/2 [MPa]

865

6495

2542

57.97%

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3

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Table 2 Overall statistics of piperno stone properties according to Calcaterra et al. [28].

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Table 3 Statistics of peak load capacity and corresponding vertical displacement of the balcony. Restraint condition RC2

Fmax [mm]

Fmax [kN]

170.39

3.05

14.95%

27.21%

Mean CoV

Fmax [mm]

Fmax [kN]

115.34

3.10

11.51%

23.23%

RC4

Fmax

Fmax

[mm]

Fmax [kN]

[mm]

81.79

2.90

10.76

0.54

7.68%

21.38%

2.04%

7.41%

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Fmax [kN]

RC3

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RC1

Statistic

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Fig. 1. Flowchart of failure investigation methodology.

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Identification of failure case study

Field inspections

Data acquisition

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Documentation analysis

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Witness information

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Identification of failure modes and alternative scenarios

Characterization of uncertainties in structural properties

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Experimental testing

Selection of scenario

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Urgent remedial measures

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Probabilistic structural performance assessment for a given scenario

Reproduction of observed damage

Selection of alternative scenario

NO

YES Risk-informed diagnosis

Risk-informed prognosis

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Fig. 2. Piperno stone balcony under investigation.

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Fig. 3. Segments of the balcony in laboratory.

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Fig. 4. Photographical survey of the balcony-wall interface.

Fig. 5. Plan view of the balcony with skewed fracture line in the central zone

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(dimensions in metres).

1.25

2.65

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1.40

1.32

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0.63

1.33

Fig. 6. Longitudinal cross-section of the balcony. piperno

screw

marble

mortar beds

screw

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Fig. 7. Thickness (vertically oriented figures) versus progressive distance (horizontally

155

155

180

190

200

210

220

230

240

250

260 265

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SC

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Fig. 8. Piperno stone specimen (a) before and (b) after uniaxial compression test.

(b)

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(a)

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Fig. 9. Stress–strain curves of piperno stone under uniaxial compression. 25

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s [MPa]

20

C1 C2 C3 C4 C5 C6 C7 C8

15 10 5 0

0

1

2

3

e [%]

4

5

6

7

40

155

155

170

155

160

130 / 25

150

90

140

90

130

90

120

85

110

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85

100

85

80

90

40 / 35

80

50

10 / 62

70

35

60

80

115

50

30

40

145

145

30

15 / 30

20

145

145

10

50

0

145

30

oriented figures) from left-hand side edge of the balcony (dimensions in millimetres).

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Fig. 10. Correlation between uniaxial compressive strength and ultrasonic pulse velocity of piperno stone. 25

fc = 0.0093V R² = 0.58

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15

5

Experimental data Best-fit model

0 1700

1900

2100

2300

2500

2700

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V [m/s]

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10

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fc [MPa]

20

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(a)

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Fig. 11. Specimens subjected to three-point bending test: (a) series 1; (b) series 2.

(b)

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Fig. 12. Force–displacement curves of piperno stone under three-point bending. 2

Series 1 Series 2

1.8 1.6

F [kN]

1.4

1.2 1 0.8

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0.6 0.4 0

0.2

0.3

0.4 d [mm]

0.5

0.6

0.7

0.8

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0.1

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0.2

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Fig. 13. Mesh of the 3D finite element model of the piperno stone balcony.

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Fig. 14. Boundary conditions of the 3D finite element model: (a) edge zone; (b) central

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zone.

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(a)

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(b)

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Fig. 15. Loading surfaces corresponding to five people on the piperno stone balcony.

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Fig. 16. Experimental versus numerical force–displacement curves of piperno stone specimens (series 1) under three-point bending. 2.5

1 0.5 0 0

0.2

0.4

0.6

0.8

1

1.2

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d [mm]

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1.5

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F [kN]

2

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S1-1 S2-1 S4-1 S6-1 S7-1 S8-1 ft = 1.40 MPa ft = 1.60 MPa ft = 1.80 MPa ft = 2.00 MPa ft = 2.23 MPa ft = 2.40 MPa

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Fig. 17. Deterministic force–displacement curves of the balcony with mean material properties under varying restraint conditions. 200

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160

120 100 80

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60 40 20

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F [kN]

140

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RC1 RC2 RC3 RC4

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Fig. 18. Distributions of principal stresses (in Pa) at the attainment of peak load

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capacity: (a) top view; (b) bottom view.

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Fig. 19. Stochastic finite element model of the balcony.

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Fig. 20. Probabilistic force–displacement curves of the balcony under varying restraint

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RESEARCH HIGHLIGHTS

 Uncertainty in structural failure investigations is discussed.  A methodology for risk-informed diagnosis and prognosis of failures is presented.  The collapse of a historical piperno stone balcony is considered as case study.

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 Uncertainty in material properties is considered via stochastic finite element analysis.

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 Observed damage is reproduced and uncertainty in failure analysis is evaluated.

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