sustainability

Article Wooden Truss Analysis, Preservation Strategies, and Digital Documentation through Parametric 3D Modeling and HBIM Workflow

Angelo Massafra , Davide Prati * , Giorgia Predari and Riccardo Gulli

Department of Architecture, University of Bologna, 40136 Bologna, Italy; [email protected] (A.M.); [email protected] (G.P.); [email protected] (R.G.) * Correspondence: [email protected]; Tel.: +39-347-410-8978



Received: 2 May 2020; Accepted: 11 June 2020; Published: 18 June 2020


Abstract: The main focus of this paper is the most recent phase of a large research project that has studied several wooden roof structures in the area of Bologna, belonging to a set of important historical buildings, all dating back to the 16th and 18th centuries. In particular, the behavior of the wooden trusses that support pitched roofs is analyzed, according to a methodological approach, based on generative algorithms that can help researchers and technicians to improve the comprehension of wooden structures’ behavior during their entire lifespan. While all the previous case studies concerned churches, this latest step extends the survey to the roofing system of the Municipal Theater of Bologna, which has a span of approximately 25 m. The core of the process concerns the automatic transformation of the point cloud into 3D models using parametric modeling tools, such as Grasshopper generative algorithms. Following this workflow, it is possible to speed up the creation of different truss models by changing only a few input parameters. This updating of the research protocol automatically creates a Building Information Modeling (BIM) model and a calculation model for the wooden trusses to perform a structural stress analysis by linking Grasshopper tools with Dynamo-Revit features. The procedure that has been developed from previous studies is still evolving and aims to speed up the modeling procedure and introduce new tools and methods for interpreting the functioning of these structural elements when surveyed through terrestrial laser scanning (TLS) devices.

Keywords: heritage at risk; municipal theater of Bologna; terrestrial laser scanning; generative algorithms; structural systems reverse engineering; cloud to 3D model comparison; heritage building information modeling; digital documentation; cultural management

1. Introduction In the history of construction, wooden trusses are the main structures used to support timber roofs, commonly used to cover large halls and the naves of churches. These types of structures are conceived both as planar and three-dimensional systems, and they are made with solid wood beams having different cross-sections, depending on the span of the roof. The capability to redistribute the vertical load without producing a horizontal thrust on the side walls ensured the success of these structural configurations since the beginning of the 4th century. Studies on large span roofs, particularly wooden trusses, are usually based on relevant handbooks from the 16th century [1,2]. At the turn of the 18th and 19th centuries, the theme was always present, and was at first addressed during the encyclopedic cataloging of traditional techniques [3–5]; then, it focused on aspects related to the calculation and verification of static schemes [6–8], following the development of the disciplines of Building Science and Civil Engineering. Nowadays, the analysis of

Sustainability 2020, 12, 4975; doi:10.3390/su12124975 www.mdpi.com/journal/sustainability Sustainability 2020, 12, 4975 2 of 23 timber roof structures is widely present in the scientific literature, in several contributions. On the one hand, the topic is systematically analyzed by building science, architectural technology, and restoration theory [9–11]. On the other hand, only a few studies have systematically addressed the real specificity of these building systems relying on practical experience and on-site diagnoses [12]. Academics usually simplify the construction design and static scheme of these artifacts. If they are still the original ones, they show signs of a long life; or, if they underwent structural problems, their configuration has probably been modified over time, if they have not been completely replaced. Their behavior is determined by precise construction specificities, such as the different ways of assembling linear elements to form tie-rods or beams with larger cross-sections, the more or less accentuated stiffness of the joints, which is determined by the timber notching, the presence or absence of metal brackets, and other details [13,14]. In other words, the topic seems to concern the art of construction and not the theory of construction. The theme is therefore reduced to a matter of minor importance for restoration activity. The roof structures are hidden and difficult to access and do not attract the attention of technicians and researchers unless there are severe problems of deterioration or damage due to poor maintenance or exceptional events. In addition, various factors highlight the lack of systematic and complete information: the wooden objects are no longer original or difficult to date, and their behavior eludes scientific and analytical interpretations, or is determined by means of great approximations. However, the roof is an area of the building that bears many loads, which vary due to atmospheric phenomena, such as snow, wind, or earthquakes, and is particularly subject to material deterioration and fires. Moreover, the roofing systems are heavily affected by potential structural problems that may arise from the underlying structures. For these reasons, it is possible to obtain information on the “state of health” of the entire building by an accurate analysis of the wooden roof structures. An innovative form of investigation has been developed using innovative tools and following a structured methodological approach, after observing the interweaving of the historical, technological, and architectural aspects of the design of wooden trusses and considering the excessive simplification of the behavior of these building systems in the traditional literature. The study of wooden trusses in the area of Bologna is of great importance, since the city has many significant ecclesiastical buildings with wooden roofs. After having been the papal seat in the 15th century, for a brief time, the city continued to have a strong bond with the Vatican. In 1575 Pope Gregory XIII (1502–1585), the Bolognese Ugo Boncompagni, commissioned a faithful representation of his city to be painted in the Vatican palaces. The result would be the so-called Bologna Room [15], where the pope wanted all the religious buildings to perfectly depicted, with a golden roof [16]. The case studies that have so far been studied have formed the basis for setting up and refining the research protocol by providing various static schemes, which were used to cover naves of different spans with construction solutions that were among the most advanced in the European context at the time [17,18]. While the previous studies examined only churches (St. Peter’s Cathedral, the church of San Giovanni in Monte, the Basilica of San Petronio, the Basilica of San Domenico, and the church of San Salvatore Maggiore), this case study extends the analysis to the roofing system of the Municipal Theater of Bologna. Its roof is a remarkable example in terms of size and structural design (about 25 m span, 6 m high, and almost 8 tons in weight). The theater, built in the year 1763, is one of the most important opera houses in Italy.

2. State of the Art Nowadays, the awareness that the modeling of wooden trusses does not fit most simplified hypotheses has shown that increasing the knowledge behind these original static schemes and joints is a non-trivial procedure. The construction practice is often ignored in favor of handbook descriptions or regulatory verification. The choice to use digital techniques to better describe the behavior of wooden trusses is also due to the awareness that these techniques have recently reached high relevance and maturity. In particular, there are numerous studies and applications to the geometric survey of Sustainability 2020, 12, 4975 3 of 23 archaeological and architectural sites for the modeling of complex shapes, surfaces, and spaces [19–21] and educational dissemination [22–25]. If digital acquisition is an established research field, the virtual photorealistic reconstructions for the qualification, evaluation, and control of building characteristics and pathologies seem less developed. In recent years, as-built 3D models have been developed for the evaluation of building quality by using Terrestrial Laser Scanning (TLS) and photogrammetric techniques [26]. Still-high costs, the operative expertise need, and time-consuming process dependence strongly limited their widespread use [27]. If the use of TLS for the surveying of complex architectural surfaces is not new and is linked to many types of research in the field of restoration [28], its application to the study of hidden spatial structures, such as wooden roofs, is rarer or even absent. An example can be found in the analysis of past interventions on the wooden roof of the “Castello del Valentino” [29], where a TLS point cloud was manually rendered to obtain a finite element model useful for the numerical assessment of the roof. Working without a 3D model could be the solution, and much scientific literature deals with cloud-to-cloud and mesh-to-cloud comparison [30,31]. Many modeling strategies allow the generation of 3D models in various formats, such as meshes that interpolate the acquired point cloud, NURBS-based reconstructions based on manual, semi-automated, or automated procedures and parametric objects for Building Information Modeling. For example, a recent method attempts to reconstruct the visible surface of a vault using neural networks [32]. Even if the use of neural networks seems promising, treating wooden trusses with this approach would become quite complicated due to the three-dimensional and irregular shape of these structures. Therefore, this research aims to demonstrate that digital sensing technologies lead to considerations that would be almost impossible by following the traditional investigation methods, usually based on direct observation and simplified architectural surveys. Furthermore, the intention is to develop a scientifically defined, coherent, repeatable, and standardizable monitoring protocol to preserve this type of building heritage by cyclically monitoring the state of each truss. This paper describes the main steps of the protocol to achieve the transformation of the point cloud into 3D and BIM models using parametric modeling tools such as the Grasshopper® and Dynamo® generative algorithms. Once created for a single truss, these parametric algorithms allow for the automatic generation of 3D models of all the trusses, by modifying only the TLS input point cloud. Different generated 3D models can be used to reconstruct the roof, using computer-aided design (CAD) or BIM softwares, or to make a comparison with the point cloud itself to highlight deformations and displacements. The acquired data can then be used as an interpretative basis to evaluate the static safety of the analyzed elements from a holistic point of view, not limited to the compliance with specific regulations that these artifacts were not required to comply with at the time of their construction.

3. Materials and Methods

3.1. The Case Study: The Municipal Theater of Bologna During the 18th century, the city of Bologna had a significant number of theatrical buildings, including public theaters and private villas in the urban surroundings. The Emilia Romagna Region, whose capital is Bologna, together with the Marche Region, currently offers one of the most formidable sets of theaters in Italy [33]. In this context, the Municipal Theater of Bologna was born with the task of representing the municipal authority, replacing the Malvezzi Theater, which was destroyed by a fire in 1745 [34] and was the favorite of citizens at the time. The original construction (Figure1) was designed by Antonio Galli Bibiena (1697–1774), who belonged to a famous family of theater architects and who had just returned to Italy after about thirty years spent as an architect in Vienna, at the imperial court [35]. However, Bibiena’s project was only partially completed in 1763 due to the great controversy raised by some Bolognese intellectuals and to serious financial difficulties, that led to the halving of the initial budget [36,37]. Sustainability 2020, 12, 4975 4 of 23 Sustainability 2020, 12, x 4 of 23

(a) (b)

FigureFigure 1. 1. TheThe New New Theater Theater of of the city of Bologna,Bologna, afterafter thethe changeschanges to to the the original original project project provided provided by byBibiena, Bibiena, engraved engraved by by Filippo Filippo Berti Berti and and Lorenzo Lorenzo Capponi Capponi in in 1773 1773 [38 [38]]. (a. )(a Section) Section (b )(b Floor) Floor plan. plan.

TheThe economic economic hardships hardships experienced experienced during during the the theater theater’s’s construction construction led led to to an an early early structure structure deteriorationdeterioration at at the the beginning beginning of of the the 19th 19th century century,, and and some some inadequacies inadequacies about about the the enjoyment enjoyment of of the the theatricaltheatrical performances performances were were shown. shown. Since Since then then,, the the building building has has undergone undergone various various transformations transformations toto solve solve these these critical critical issues issues.. Between Between 1818 1818 and and 1820 1820,, the the vault vault in in the the gr grandand theater theater hall hall was was rebuilt rebuilt,, asas it it appeared appeared to to be be in in a a dangerous dangerous condition condition [39] [39].. At At the the time time of of this this intervention, intervention, the the t theaterheater was was equippedequipped with with a a fascinating fascinating wooden wooden machine machine that that served served to to raise raise the the floor floor of of the the stalls, stalls, bringing bringing it it upup to to the the level level of of the the stage, stage, to to obtain obtain a a single single large large hall hall t too host host events events and and costume costume balls. balls. In In 1853 1853 the the architectarchitect Carlo ParmeggianiParmeggiani made made a restylinga restyling ofthe of wholethe whole building, building, and in and 1866 in the 1866 engineer the Coriolanoengineer CoriolanoMonti designed Monti thedesigned rear facade the rear of thefacade building of the in building a post-unification in a post-unification style [40]. style [40]. InIn 1931 1931 a a fire fire destroyed destroyed the the stage, stage, so so the the engineer engineer Armando Armando Villa Villa rebuilt rebuilt it it in in a a larger larger size size using using aa reinforced reinforced concrete concrete structure structure [41] [41].. After After the the new new stage stage was was built, built, in in 1935 1935,, the the architect architect Umberto Umberto RizziRizzi completed completed the the unsafe unsafe entrance entrance facade. facade. Finally, Finally, in in 1980 1980 the the restoration restoration of of the the entire entire building building was was carriedcarried out out,, including including the the strengthening strengthening of of all all the the wooden wooden structures structures,, the the verification verification and and reparation reparation ofof the the vault vault in in the the hall, hall, the the renovation renovation of of floors, floors, plasters plasters,, and and paintings, paintings, the the restoration restoration of of stuccoes stuccoes andand frescoes, frescoes, the the installation installation of of new new technological technological systems systems,, and and the the construction construction of of new new dressing dressing roomroomss [40,42] [40,42].. TheThe building building is is currently currently located located in in the the core core of of the the city, city, in in the the strategic strategic center center of of the the University University area,area, planned planned as as an an artistic artistic and and cultural cultural district district of of Bologna. Bologna. The The complex complex is is characterized characterized by by a a floor floor planplan with with a a depth depth of of about about 77 77 m m and and a a width width of of about about 54 54 m m.. Inside Inside the the theater theater,, the the great great hall hall used used for for thethe shows shows is is undoubted undoubtedlyly the the most most interesting interesting area area covered covered by by the the grand grand vault vault,, which which is is supported supported byby the the roof roof system system that that is is going going to to be be discussed discussed.. It It consists consists of of a atwo two-pitched-pitched roof roof with with a alow low slope, slope, arrangedarranged on on a a rectangular rectangular plan plan of of 29 29 × 2626 m m,, and and is is composed composed of of six six trusses trusses dating dating back back to to around around × 1760,1760, with with a a center center to to center center spacing spacing of of about about 4 4 m m,, covering covering a a span span of of about about 25 25 m m.. TheThe different different static static schemes that have beenbeen identifiedidentified amongamongthe the six six trusses trusses can can be be traced traced back back to tothe the primordial primordial scheme scheme of of the the “Palladian “Palladian truss,” truss, which” which includes includes principal principal rafters rafters (Figure (Figure2). 2). Furthermore, new struts have been added to the primordial scheme over the centuries, differently for each truss. In fact, at the same time as the building, the trusses underwent various interventions, at first during the 19th century and later in 1980, when a substantial strengthening was carried out on the wooden structures of the roof (Figure3). Following the interventions of 1980, the trusses now support the vault with the aid of metal tie-rods [42]. Sustainability 2020, 12, 4975 5 of 23

Sustainability 2020, 12, 4975 5 of 23 Sustainability 2020, 12, x 5 of 23

(a) (b)

Figure 2. The wooden trusses of the Municipal Theater (a) Plan of the roof. (b) Static schemes.

Furthermore, new struts have been added to the primordial scheme over the centuries, differently for each truss. In fact, at the same time as the building, the trusses underwent various interventions, at first during the 19th century and later in 1980, when a substantial strengthening was

carried out on the wooden(a structures) of the roof (Figure 3). Following (theb) interventions of 1980, the trusses now support the vault with the aid of metal tie-rods [42]. Figure 2. The wooden trusses of the Municipal Theater (a)) PlanPlan ofof thethe roof.roof. (b) Static schemes.

Furthermore, new struts have been added to the primordial scheme over the centuries, differently for each truss. In fact, at the same time as the building, the trusses underwent various interventions, at first during the 19th century and later in 1980, when a substantial strengthening was carried out on the wooden structures of the roof (Figure 3). Following the interventions of 1980, the trusses now support the vault with the aid of metal tie-rods [42].

(a) (b)

(a) (b) (c) (d)

FigureFigure 3. ConfigurationsConfigurations of of truss truss n.3 n.3 over over time. time. (a ()a Original) Original configuration. configuration. (b) ( bAddition) Addition of girders. of girders. (c) (Connectionc) Connection with with the the vault. vault. (d) (Installationd) Installation of new of new technological technological systems. systems.

TheThe woodenwooden vault vault covers covers a bell-shaped a bell-shaped plan with plan transverse with andtransverse longitudinal and spans,longitudinal respectively spans, of 18respectively m and 22m. of It18 consists m and of22 am. double-curved It consists of membranea double-curved made ofmembrane gypsum plaster,made of 8 cmgypsum thick [plaster,42], with 8 acm layer thick of [42], reeds with supporting a layer(c) of the reeds plaster supporting layer but the also plaster working layer as formworkbut also working during(d) theas formwork construction during [43]. Thethe construction membrane is [43]. stiffened The bymembrane a series of is woodenstiffened ribs by placeda series at of a distancewooden ofribs about placed 60cm at a and distance presents of aabout centralFigure 60 holecm 3. and Configurations where presents the chandelier a ofcentral truss n.3 hole of theover where great time. thehall(a) Originalchandelier is hung configuration. [44 of]. the great (b )hall Addition is hung of girders.[44]. (c) Connection with the vault. (d) Installation of new technological systems. 3.2.3.2. The InvestigationInvestigation Protocol Protocol Workflow Workflow The wooden vault covers a bell-shaped plan with transverse and longitudinal spans, The previous case study analyses allowed us to define a procedure to systematize the assessment respectivelyThe previous of 18 mcase and study 22 m analyses. It consists allowed of a doubleus to define-curved a procedure membrane to madesystematize of gypsum the assessment plaster, 8 of the wooden structures’ behavior in terms of displacement and deformations over time. This research cmof thethick wooden [42], with structures’ a layer of behaviorreeds supporting in terms the of plasterdisplacement layer but and also deformations working as formwork over time. during This protocol (Figure4) includes an on-site instrumental acquisition phase using TLS survey techniques, theresearch construction protocol [43] (Figure. The membrane4) includes isan stiffened on-site instrumentalby a series of acwoodenquisition ribs phase placed using at a TLSdistance survey of followed by the digital rendering of the acquired data. This last step provides a useful information about 60 cm and presents a central hole where the chandelier of the great hall is hung [44]. background (photos, drawings, diagrams, and models) that comprises the operational tools to correctly define3.2. The and Investigation interpret Protocol the behavior Workflow of roof systems [45]. Finally, the research protocol allows for a rapid and reliable digitization of wooden trusses by using parametric 3D and BIM modeling tools, The previous case study analyses allowed us to define a procedure to systematize the assessment of the wooden structures’ behavior in terms of displacement and deformations over time. This research protocol (Figure 4) includes an on-site instrumental acquisition phase using TLS survey Sustainability 2020, 12, x 6 of 23 techniques, followed by the digital rendering of the acquired data. This last step provides a useful informationSustainability 2020 background, 12, 4975 (photos, drawings, diagrams, and models) that comprises the operational6 of 23 tools to correctly define and interpret the behavior of roof systems [45]. Finally, the research protocol allows for a rapid and reliable digitization of wooden trusses by using parametric 3D and BIM while meeting the needs of monitoring, preservation, risk reduction, documentation, exploitation, modeling tools, while meeting the needs of monitoring, preservation, risk reduction, documentation, and dissemination. exploitation, and dissemination.

FigureFigure 4. 4. InvestigationInvestigation protocol protocol workflow. workflow.

3.3.3.3. The The Model Model Generation Generation Procedure Procedure TheThe assumption assumption behind behind the the algorithmic algorithmic generation generation procedure procedure of 3D of 3Dmodels models is to isexploit to exploit the vast the amountsvast amounts and the and accuracy the accuracy of spatial ofspatial information information provided provided by TLS bydevices, TLS devices, such as suchpoint asclouds. point Tclouds.hese data These allow data for allow an forextremely an extremely detailed detailed analysis analysis of wooden of wooden trusses trusses and and,, therefore therefore,, for for the the processprocessinging both ofof specificspecific and and comparative comparative information information regarding regarding their their static static behavior behavior and theirand their state stateof preservation. of preservation. TheThe available available geometrical geometrical data data are are very very interesting interesting.. O Onn the the one one hand, hand, they they allow allow for for an an accurate accurate 3D3D rendering rendering of of each each truss, truss, on on the the other other hand hand,, they they permit t thehe compar comparisonison of of static static schemes schemes and and constructionconstruction systems systems belonging belonging to to various various wooden wooden structures structures,, used used to to arrange arrange the the roofs roofs of of different different buildings.buildings. Besides Besides,, the the parametric parametric 3D3D models models of the of wooden the wooden trusses trusses should should become become the starting the starting point point for future monitoring cycles regarding the displacements and deformations of these structures over time. Sustainability 2020, 12, 4975 7 of 23

Another essential hypothesis is that the usual modeling methodologies, such as orthophoto vectorization or 3D modeling of “destructive geometries” using CAD or BIM softwares, strongly depend on the operator’s choices and execution modes and use workflows with a low degree of repeatability and accuracy. For these reasons, it was decided to transform the point cloud into 3D models using Grasshopper® generative algorithms. These tools, which are updated continuously, provide advantages when compared with manual modeling methods, both in terms of execution speed and elimination of inaccuracies. Once the case study had been identified in the Municipal Theater of Bologna, an accurate TLS survey was carried out. The entire TLS survey of the roof of the theater was conducted with a FARO CAM2 FOCUS 3D® laser scanner using a targetless approach. The survey campaign took one working day, and it was necessary to shoot more than 60 scans (Table1) with a resolution of 7.67 mm /10 m and a quality filter of 3 to get the whole roof system. In this case, many high-resolution scans were × necessary due to obstacles between the trusses, for example, air conditioning systems, electric cables, and walkways for maintenance. The alignment was performed with the FARO SCENE 2019® software using an interactive cloud to cloud registration with a subsampling average distance of 3 cm and a searching radius of 5 m. With a medium overlapping between the scans of 36% (Table2), despite the roof complexity and occlusions, it was possible to achieve an extremely accurate alignment, with an average standard deviation between corresponding points of 1.5 mm and a maximum deviation of 2.6 mm [46].

Table 1. Terrestrial laser scanning (TLS) survey campaign overall data.

Numbers of Scans (nr.) Project Point Cloud (pti) 64 1.014.651.419

Table 2. Accuracy parameters of the cloud-to-cloud alignment algorithm.

Maximum Standard Deviation (mm) Average Standard Deviation (mm) Overlapping Between Scans (%) Average 1.9 1.5 36.0 Min 1.1 1.0 21.1 Max 2.6 2.0 51.8

The primary purpose was to use parametric modeling for the generation of the BASIC 3D MODEL, which represents the current state of the trusses. Starting from this first model, the IDEAL 3D MODEL was created through a series of automated operations, to reconstruct the hypothetical condition of the structure when it was initially built. This model has been compared with the TLS point cloud to evaluate the displacements and deformation states [47]. The algorithmic generation procedure, used to obtain 3D models, has been divided into several subphases, to ensure better management and, above all, a more straightforward understanding of the adopted operating process (Figure5).

3.3.1. Automatic Generation of Cross-Sections While the previous version of the protocol [16,17] required some manual operations in Geomagic Studio® to generate cross-sections, the new updating transferred this phase to Grasshopper®, saving much time and solving several inaccuracies deriving from the operator’s choices. The cross-section generation was performed starting from the clipped point cloud of a single truss. Once the clipping boxes were created, the enclosed points were exported in .xyz format and then imported into Geomagic Studio®. In this environment, some quick editing operations were necessary to delete all the points not belonging to the beams (e.g., air systems, cables, etc.). Sustainability 2020, 12, 4975 8 of 23 Sustainability 2020, 12, x 8 of 23

By clipping ample portions of points near the joints and at the centerline of the beams, it was possible to isolate three cross-sections for each element. These groups of points were then exported in .dxf format, imported in Rhinoceros®, and associated with Grasshopper® plug-in algorithms (Figure6). By selecting each portion of the imported points, the “cross-section generation algorithm” recognized the angle of inclination of the rods in the vertical plane of the truss, identified the lying plane of the cross-sections, selected a 4-cm-wide parallel point slice (creating a sort of local clipping box), projected the enclosed points on the lying plane of the section, resorted them using a local polar coordinate system in the lying plane and interpolated all the resorted points through a third-degree curve. This “non-uniform rational basis-spline” (NURBS) curve was divided into 100 equal-length

Sustainabilitysegments, 20 and20, finally12, x a new third-degree NURBS curve was created as the interpolation of the8 of 100 23 points, obtaining the final cross-section curve (Figure7).

Figure 5. Parametric modeling procedure workflow.

3.3.1. Automatic Generation of Cross-Sections While the previous version of the protocol [16,17] required some manual operations in Geomagic Studio® to generate cross-sections, the new updating transferred this phase to Grasshopper®, saving much time and solving several inaccuracies deriving from the operator’s choices. The cross-section generation was performed starting from the clipped point cloud of a single truss. Once the clipping boxes were created, the enclosed points were exported in .xyz format and then imported into Geomagic Studio®. In this environment, some quick editing operations were necessary to delete all the points not belonging to the beams (e.g., air systems, cables, etc.). By clipping ample portions of points near the joints and at the centerline of the beams, it was possible to isolate three cross-sections for each element. These groups of points were then exported in .dxf format, imported in Rhinoceros®, and associated with Grasshopper ® plug-in algorithms (Figure 6). Figure 5. Parametric modeling procedure workflow. Figure 5. Parametric modeling procedure workflow.

3.3.1. Automatic Generation of Cross-Sections While the previous version of the protocol [16,17] required some manual operations in Geomagic Studio® to generate cross-sections, the new updating transferred this phase to Grasshopper®, saving much time and solving several inaccuracies deriving from the operator’s choices. The cross-section generation was performed starting from the clipped point cloud of a single truss. Once the clipping boxes were created, the enclosed points were exported in .xyz format and then imported into Geomagic Studio®. In this environment, some quick editing operations were necessary to delete all the points not belonging to the beams (e.g., air systems, cables, etc.). By clipping ample portions of points near the joints and at the centerline of the beams, it was possible to isolate three cross-sections for each element. These groups of points were then exported ® ® in .dxf format, imported in Rhinoceros , and associated with Grasshopper plug-in algorithms (Figure 6). (a) (b)

FigureFigure 6. 6TLS. TLS point point cloud cloud ( a()a) Clipped Clipped trusses trusses from from the the Municipal Municipal Theater’sTheater’s pointpoint cloud. ( (bb)) Groups Groups of pointsof point importeds imported in Rhinoceros in Rhinoceros®. ®.

By selecting each portion of the imported points, the “cross-section generation algorithm” recognized the angle of inclination of the rods in the vertical plane of the truss, identified the lying plane of the cross-sections, selected a 4-cm-wide parallel point slice (creating a sort of local clipping

(a) (b)

Figure 6. TLS point cloud (a) Clipped trusses from the Municipal Theater’s point cloud. (b) Groups of points imported in Rhinoceros®.

By selecting each portion of the imported points, the “cross-section generation algorithm” recognized the angle of inclination of the rods in the vertical plane of the truss, identified the lying plane of the cross-sections, selected a 4-cm-wide parallel point slice (creating a sort of local clipping Sustainability 2020, 12, x 9 of 23 box), projected the enclosed points on the lying plane of the section, resorted them using a local polar coordinate system in the lying plane and interpolated all the resorted points through a third-degree curve. This “non-uniform rational basis-spline” (NURBS) curve was divided into 100 equal-length segmentsSustainability, and2020 finally, 12, 4975 a new third-degree NURBS curve was created as the interpolation of the9 100 of 23 points, obtaining the final cross-section curve (Figure 7).

(a)

(b)

FigureFigure 7. 7. CrossCross-section-section generation generation algorithm algorithm.. (a ()a )Parametric Parametric 3D 3D modeling modeling operations operations in in Rhinoceros Rhinoceros®®. . (b(b) )G Grasshopperrasshopper®®’s ’sdiagram diagram:: automatic automatic calculation calculation of of the the lying lying planes planes of of the the cross cross-sections.-sections.

3.3.2.3.3.2. Basic Basic Algorithm Algorithm TheThe b basicasic algorithmalgorithm providedprovided the the BASIC BASIC 3D model 3D model of the of truss the starting truss starting from previously from previously generated generatedcurves. It curves. consists It ofconsists applying of applying the “loft ”the command “loft” command to the sections to the sections belonging belonging to the same to the wooden same woodenbeam to beam perform to perform the creation the creation of a surface of a onlysurface through only through the extremity the extremity and the and centerline the centerline sections. sections.Once the Once “loft the” operation “loft” operation on each on truss each elementtruss element was completed,was completed, it was it was necessary necessary to completeto complete the themodeling modeling of the of joints.the joints Contrary. Contrary to the previousto the previous versions ofversions the protocol, of the that protocol adopted, that diff erentiatedadopted differentiatedprocedures according procedures to diaccordingfferent connections to different between connection theswooden between elements, the wooden the elements, latest update the latest uses a updategeometric-optimization uses a geometric-optimization algorithm to closealgorithm the beams to clos ine the correspondence beams in correspondence with the joints with (Figure the joints8). (FigureFirst 8). of all, the algorithm automatically identified the “projection planes”: the vertical plane of the truss, the axial planes of the girders, the vertical planes passing through the lateral bearings, and the horizontal plane passing through the ridge, which were all perpendicular to the vertical plane of the truss. Subsequently, the centroids of the end sections were projected on the planes along the respective axes of the beams. Then, the end curves were extruded in correspondence with the projection planes. The correct association between the end sections and the projection planes was obtained through the Galapagos® evolutionary solver using a parametric optimization principle (Figure9). In evolutionary models, a population of candidate solutions is maintained and new candidate solutions are randomly generated by mutating or recombining variants in the existing population. Periodically, the population is pruned by applying a selection criterion (a fitness function) that allows only the better candidates to survive into the next generation. Iterated over many generations, the average quality of the solutions in the candidate pool gradually increases [48]. (a) (b) After completing the automatic modeling of the rod parts at the joints, the beams were joined using the “bake” command on Rhinoceros®, and the so-called BASIC 3D model was obtained. A first comparison between the model and the detected point cloud was carried out using Geomagic Studio® to evaluate the degree of compliance of the previous parametric modeling operations. In particular, the software detects and reports in a chromatic scale the distances between the point cloud, considered as the object reference, and the BASIC 3D model, considered as the test object. The extracted summary reports derived by the comparison indicate the percentage of points within the nominal overlap threshold of the two compared objects. In this case, the highlighted row contains the points within the threshold of 5 mm, whose deviation is considered not significant, ± therefore treated as correctly overlapped points. The percentage of points within the threshold indicates Sustainability 2020, 12, x 9 of 23 box), projected the enclosed points on the lying plane of the section, resorted them using a local polar coordinate system in the lying plane and interpolated all the resorted points through a third-degree curve. This “non-uniform rational basis-spline” (NURBS) curve was divided into 100 equal-length segments, and finally a new third-degree NURBS curve was created as the interpolation of the 100 points, obtaining the final cross-section curve (Figure 7).

(a)

(b)

Figure 7. Cross-section generation algorithm. (a) Parametric 3D modeling operations in Rhinoceros®. (b) Grasshopper®’s diagram: automatic calculation of the lying planes of the cross-sections.

3.3.2. Basic Algorithm The basic algorithm provided the BASIC 3D model of the truss starting from previously generated curves. It consists of applying the “loft” command to the sections belonging to the same wooden beam to perform the creation of a surface only through the extremity and the centerline sections. Once the “loft” operation on each truss element was completed, it was necessary to complete Sustainabilitythe modeling2020 , 12of, 4975the joints. Contrary to the previous versions of the protocol, that adopted10 of 23 differentiated procedures according to different connections between the wooden elements, the latest thatupdate the uses total a overlap geometric degree-optimization between thealgorithm point cloud to clos ande the the beam 3D models in correspondence is 75% on average with forthe all joints the modeled(Figure 8) trusses. (Figure 10).

Sustainability 2020, 12, x 10 of 23 (a) (b)

(c)

(d)

FigureFigure 8.8.Parametric Parametric 3D 3D modeling. modeling. (a) ( Froma) From cross-sections cross-sections to the to creation the creation of elements of elements of the truss of the through truss thethrough “loft” command.the “loft” command. (b) Extrusion (b of) Extrusion end sections of in end correspondence sections in correspondence with the joints. (withc) Grasshopper the joints.® (’sc) diagram:Grasshopper loft® operation.’s diagram: (d loft) Grasshopper operation. ®(d’s) G diagram:rasshopper extrusion®’s diagram of end: extrusion sections. of end sections.

3.3.3.First Projection of all, Algorithmthe algorithm automatically identified the “projection planes”: the vertical plane of the truss,The following the axial planes steps aim of the to generategirders, the the vertical IDEAL planes 3D MODEL passing of through the wooden the lateral trusses, bearings which,was and createdthe horizontal using aplane series passing of algorithms through the reconstructing ridge, which the were possible all perpendicular configuration to the of thevertical truss plane at the of timethe truss of its. originalSubsequently construction,, the centroids tracing of the the deformations end sections and were displacements projected on that the theplanes structure along hasthe hypotheticallyrespective axes undergone of the beams. over time.Then, the end curves were extruded in correspondence with the projectionAfter runningplanes. The the “automaticcorrect association cross-sections between generation” the end sections algorithm and (Figure the projection 11), the planes first phase was consistedobtained ofthrough applying the the Galapagos “projection® evolutionary algorithm” tosolver the vectorized using a parametric curves. This optimization algorithm recovered principle the(Figure rotation 9). of the wooden rods outside the vertical plane of the truss. Once each cross-section centroid was created, the vertical plane passing through the centroids of the two external curves of the tie-beam was defined. This operation assumed that the lateral bearings of the tie-beam were still in their original position or resisted negligible movement throughout the life of the roof, particularly along the longitudinal axis of the building.

(a)

(b) Sustainability 2020, 12, x 10 of 23

(c)

(d)

Figure 8. Parametric 3D modeling. (a) From cross-sections to the creation of elements of the truss through the “loft” command. (b) Extrusion of end sections in correspondence with the joints. (c) Grasshopper®’s diagram: loft operation. (d) Grasshopper®’s diagram: extrusion of end sections.

First of all, the algorithm automatically identified the “projection planes”: the vertical plane of the truss, the axial planes of the girders, the vertical planes passing through the lateral bearings, and the horizontal plane passing through the ridge, which were all perpendicular to the vertical plane of the truss. Subsequently, the centroids of the end sections were projected on the planes along the respective axes of the beams. Then, the end curves were extruded in correspondence with the projection planes. The correct association between the end sections and the projection planes was Sustainabilityobtained2020, 12 through, 4975 the Galapagos® evolutionary solver using a parametric optimization principle 11 of 23 (Figure 9).

(a)

Sustainability 2020, 12, x (b) 11 of 23

(c)

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Figure 9.FigureProjection 9. Projection algorithm algorithm workflow workflow (a) Projection (a) Projection planes. planes. (b ) ( Modelingb) Modeling of of the the joints. joints. (c ) (c Galapagos) Galapagos algorithm: evolutionary optimization. (d) Galapagos algorithm: diagram and evolutionary algorithm: evolutionary optimization. (d) Galapagos algorithm: diagram and evolutionary function function in Grasshopper®. in Grasshopper®. In evolutionary models, a population of candidate solutions is maintained and new candidate solutions are randomly generated by mutating or recombining variants in the existing population. Periodically, the population is pruned by applying a selection criterion (a fitness function) that allows only the better candidates to survive into the next generation. Iterated over many generations, the average quality of the solutions in the candidate pool gradually increases [48]. After completing the automatic modeling of the rod parts at the joints, the beams were joined using the “bake” command on Rhinoceros® , and the so-called BASIC 3D model was obtained. A first comparison between the model and the detected point cloud was carried out using Geomagic Studio® to evaluate the degree of compliance of the previous parametric modeling operations. In particular, the software detects and reports in a chromatic scale the distances between Sustainability 2020, 12, x 12 of 23

the point cloud, considered as the object reference, and the BASIC 3D model, considered as the test object. The extracted summary reports derived by the comparison indicate the percentage of points within the nominal overlap threshold of the two compared objects. In this case, the highlighted row contains the points within the threshold of ± 5 mm, whose deviation is considered not significant, therefore treated as correctly overlapped points. The percentage of points within the threshold indicates that the total overlap degree between the point cloud and the 3D model is 75% on average for all the modeled trusses (Figure 10).

Sustainability 2020, 12, x 12 of 23 the point cloud, considered as the object reference, and the BASIC 3D model, considered as the test object. The extracted summary reports derived by the comparison indicate the percentage of points within the nominal overlap threshold of the two compared objects. In this case, the highlighted row contains the points within the threshold of ± 5 mm, whose deviation is considered not significant, therefore treated as correctly overlapped points. The percentage of points within the threshold indicates that the total overlap degree between the point cloud and the 3D model is 75% on average forSustainability all the modeled2020, 12, 4975 trusses (Figure 10). 12 of 23

Figure 10. Graph and table indicating the distribution of distances between the point cloud and the basic 3D model.

3.3.3. Projection Algorithm The following steps aim to generate the IDEAL 3D MODEL of the wooden trusses, which was created using a series of algorithms reconstructing the possible configuration of the truss at the time of its original construction, tracing the deformations and displacements that the structure has hypothetically undergone over time. After running the “automatic cross-sections generation” algorithm (Figure 11), the first phase consisted of applying the “projection algorithm” to the vectorized curves. This algorithm recovered the rotation of the wooden rods outside the vertical plane of the truss. FigureFigure 10. Graph and table indicating the distribution of of distances distances between between the the point point cloud cloud and and the basicbasic 3D model model..

3.3.3. Projection Algorithm The following steps aim to generate the IDEAL 3D MODEL of the wooden trusses, which was created using a series of algorithms reconstructing the possible configuration of the truss at the time of its original construction, tracing the deformations and displacements that the structure has hypothetically undergone over time. ® Figure 11. Projection algorithm: Grasshopper ®’s diagram. After running the “automaticFigure 11. crossProjection-sections algorithm generation: Grasshopper” algorithm’s diagram (Figure. 11), the first phase consistedThe barycenterof applying of the all “ otherprojection cross-sections algorithm was” to projected the vectorized on this curves. vertical This plane algorithm using the recover “project”ed Once each cross-section centroid was created, the vertical plane passing through the centroids thecommand. rotation Theof the connecting wooden rods lines outside between the the vertical original plane and newof the centroids truss. were used as reference lines of the two external curves of the tie-beam was defined. This operation assumed that the lateral toSustainability execute the 20 “move”20, 12, x command. This geometrical operation simply moved the original curves13 of from23 bearings of the tie-beam were still in their original position or resisted negligible movement the old barycenter to the projected one. In other words, this algorithm managed the projection of the throughoutthe old barycenter the life toof thethe projectedroof, particularly one. In other along words, the longitudinal this algorithm axis manage of thed building. the projection of the curves by moving their centroids to the hypothetical vertical plane of the truss. curvesThe by barycenter moving their of all centroids other cross to the-sections hypothetical was projected vertical planeon this of verticalthe truss. plane using the “project” 3.3.4.command. Rectification The connecting Algorithm lines between the original and new centroids were used as reference lines 3.3.4. Rectification Algorithm to execute the “move” command. This geometrical operation simply moved the original curves from The last step for the creation of the IDEAL 3D model of the trusses was mostly focused on the The last step for the creation of the IDEAL 3D model of the trusses was mostly focused on the reconstructionreconstruction of of the the main main non-deformable non-deformable outer outer triangle, triangle, consisting consisting of of tie-beams tie-beam ands and rafters. rafters. This This last phaselast wasphase obtained was obtained by scriptingFigure by 11. the scriptingProjection “rectification algorithmthe “rectification algorithm”,: Grasshopper algorithm which®’s diagram eliminated”, . which theeliminate displacementsd the ofdisplacements the joints and of the the flexural joints and deformations the flexural deformations of the wooden of beamsthe wooden in the beams vertical in the plane vertical of the plane truss (Figureof Oncethe 12 truss ).each (Figure cross -12section). centroid was created, the vertical plane passing through the centroids of the two external curves of the tie-beam was defined. This operation assumed that the lateral bearings of the tie-beam were still in their original position or resisted negligible movement throughout the life of the roof, particularly along the longitudinal axis of the building. The barycenter of all other cross-sections was projected on this vertical plane using the “project” command. The connecting lines between the original and new centroids were used as reference lines to execute the “move” command. This geometrical operation simply moved the original curves from

FigureFigure 12. 12.Elimination Elimination of offlexural flexural deformations:deformations: G Grasshopperrasshopper®’®s ’sdiagram diagram..

TheThe main main hypotheses hypotheses to to recover recover thethe theoreticaltheoretical displacements that that the the truss truss has has undergone undergone in inits its verticalvertical plane plane over over time time were: were:

• LateralLateral bearings bearings of of the the tie-beam tie-beam are are stillstill inin theirtheir original position. position. • • The bending deformation of the tie-beams is due to their weight and creep phenomena. • Inward translation and lowering of the joints between rafters and posts are due to roof loads. • Rotation of bottom rafters hinged to the lateral bearings. • Rotation of the queen posts happens toward the symmetry axis of the truss. • Rigid connection of the joint between the rafters and tie-beam is due to the presence of metal brackets. • Axial deformations of beams are disregarded. The kinematic deformation system was based on precise geometrical assumptions, so a preparatory algorithm was initially needed. The “nodes individuation algorithm” allowed us to quickly recognize the point-like nodes and to apply the simplified planar hypothesis to a three- dimensional environment to evaluate the theoretical translations of the nodes. Besides, this algorithm, working as a first machine learning algorithm, should be fundamental for the generation of the BIM model that is going to be described in the next paragraphs. The procedure was therefore developed by recovering to the following kinds of displacements (Figure 13): • Displacements of the main joints: all cross-sections connected with the posts and the rafters were moved upwards by the same distance for each node. • Straightening of rafters: the top and bottom rafters were rotated to ensure that the centroids of their cross-sections lay on the same plane passing through lateral bearing and ridge nodes, orthogonal to the vertical plane of the truss. • Rectification of the posts: rotation of the posts in the direction of the symmetry axis of the truss. • Bending deformations: projection of the centroid of the centerline section on the straight line that joins the centroids of the two end sections of each element. Sustainability 2020, 12, 4975 13 of 23

The bending deformation of the tie-beams is due to their weight and creep phenomena. • Inward translation and lowering of the joints between rafters and posts are due to roof loads. • Rotation of bottom rafters hinged to the lateral bearings. • Rotation of the queen posts happens toward the symmetry axis of the truss. • Rigid connection of the joint between the rafters and tie-beam is due to the presence of • metal brackets. Axial deformations of beams are disregarded. • The kinematic deformation system was based on precise geometrical assumptions, so a preparatory algorithm was initially needed. The “nodes individuation algorithm” allowed us to quickly recognize the point-like nodes and to apply the simplified planar hypothesis to a three-dimensional environment to evaluate the theoretical translations of the nodes. Besides, this algorithm, working as a first machine learning algorithm, should be fundamental for the generation of the BIM model that is going to be described in the next paragraphs. The procedure was therefore developed by recovering to the following kinds of displacements (Figure 13):

Displacements of the main joints: all cross-sections connected with the posts and the rafters were • moved upwards by the same distance for each node. Straightening of rafters: the top and bottom rafters were rotated to ensure that the centroids • of their cross-sections lay on the same plane passing through lateral bearing and ridge nodes, orthogonal to the vertical plane of the truss. Rectification of the posts: rotation of the posts in the direction of the symmetry axis of the truss. • Bending deformations: projection of the centroid of the centerline section on the straight line that • Sustainabilityjoins the 2020 centroids, 12, x of the two end sections of each element. 14 of 23

FigureFigure 13.13. Hypothetical truss displacements.displacements.

Once all thethe describeddescribed operations were carried out,out, and the curves had been moved to the new position,position, all thethe woodenwooden beamsbeams cross-sectionscross-sections of the trusstruss reachedreached theirtheir finalfinal position,position, thethe so-calledso-called “rectified”“rectified”position. position. After performingperforming all the proceduresprocedures forfor thethe “automatic“automatic generationgeneration ofof thethe cross-sections”cross-sections” andand subsequentlysubsequently running thethe “projection”“projection” andand “rectification”“rectification” algorithms,algorithms, allall woodenwooden beamsbeams ofof thethe trusstruss couldcould bebe modeledmodeledusing using the the “loft” “loft” operation operation again again and and completing completing the the rod rod parts parts at theat the joints, joints as, in as the in ® “basicthe “basic algorithm.” algorithm In the.” In end, the using end, theusing “bake” the “ commandbake” command on Rhinoceros on Rhinoceros, it was® possible, it was topossible generate to thegenerate so-called the so IDEAL-called 3D IDEAL model 3D (Figure model 14 ().Figure 14).

.

Figure 14. Ideal 3D model and Basic 3D model.

3.4. Truss Analysis through 3D Models and Point Cloud Comparison Once the IDEAL 3D model definition was reached, a comparative analysis was performed using Geomagic Control® software to evaluate the deviations between the outer faces of the robust geometric model and the point cloud. The IDEAL 3D model of each truss could be compared with the corresponding segmented portion of the original point cloud to highlight the differences. The minimum and maximum reference deviation threshold were manually set, in a chromatic scale, to have a better indication of the extent of the movements that occurred between the surveyed configuration and the IDEAL 3D model for all the trusses. The graphic representation through the most appropriate chromatic scale clarifies the entities of the displacements. It is possible to select any position on each element of the truss to highlight the numerical value of the local deviations using the graphical representation. This representation provides both qualitative and quantitative information and facilitates an accurate and widespread understanding of all the displacement and deformations that each truss has undergone over time. In addition, the lowering, rotations, and deformations of each truss can be analyzed and locally highlighted. Comparisons can be made in a 3D space (Figure 15) and on 2D projection planes selected by the user according to the wooden element and the type of deviations that need to be highlighted. The software also allows for the precise quantification of deviations using control points. With the “annotation” tool, it is possible to report the exact value of the distance measured between the Sustainability 2020, 12, x 14 of 23

Figure 13. Hypothetical truss displacements.

Once all the described operations were carried out, and the curves had been moved to the new position, all the wooden beams cross-sections of the truss reached their final position, the so-called “rectified” position. After performing all the procedures for the “automatic generation of the cross-sections” and subsequently running the “projection” and “rectification” algorithms, all wooden beams of the truss could be modeled using the “loft” operation again and completing the rod parts at the joints, as in ® Sustainabilitythe “basic 2020algorithm, 12, 4975.” In the end, using the “bake” command on Rhinoceros , it was possible14 ofto 23 generate the so-called IDEAL 3D model (Figure 14).

Figure 14. Ideal 3D model and Basic 3D model.model. 3.4. Truss Analysis through 3D Models and Point Cloud Comparison 3.4. Truss Analysis through 3D Models and Point Cloud Comparison Once the IDEAL 3D model definition was reached, a comparative analysis was performed Once the IDEAL 3D model definition was reached, a comparative analysis was performed using using Geomagic Control® software to evaluate the deviations between the outer faces of the robust Geomagic Control® software to evaluate the deviations between the outer faces of the robust geometric model and the point cloud. The IDEAL 3D model of each truss could be compared with the geometric model and the point cloud. The IDEAL 3D model of each truss could be compared with corresponding segmented portion of the original point cloud to highlight the differences. The minimum the corresponding segmented portion of the original point cloud to highlight the differences. The and maximum reference deviation threshold were manually set, in a chromatic scale, to have a better minimum and maximum reference deviation threshold were manually set, in a chromatic scale, to indication of the extent of the movements that occurred between the surveyed configuration and the have a better indication of the extent of the movements that occurred between the surveyed IDEAL 3D model for all the trusses. configuration and the IDEAL 3D model for all the trusses. The graphic representation through the most appropriate chromatic scale clarifies the entities of the The graphic representation through the most appropriate chromatic scale clarifies the entities of displacements. It is possible to select any position on each element of the truss to highlight the numerical the displacements. It is possible to select any position on each element of the truss to highlight the value of the local deviations using the graphical representation. This representation provides both numerical value of the local deviations using the graphical representation. This representation qualitative and quantitative information and facilitates an accurate and widespread understanding provides both qualitative and quantitative information and facilitates an accurate and widespread of all the displacement and deformations that each truss has undergone over time. In addition, understanding of all the displacement and deformations that each truss has undergone over time. In the lowering, rotations, and deformations of each truss can be analyzed and locally highlighted. addition, the lowering, rotations, and deformations of each truss can be analyzed and locally Comparisons can be made in a 3D space (Figure 15) and on 2D projection planes selected by the user highlighted. according to the wooden element and the type of deviations that need to be highlighted. The software Comparisons can be made in a 3D space (Figure 15) and on 2D projection planes selected by the also allows for the precise quantification of deviations using control points. With the “annotation” tool, userSustainability according 2020, to12, thex wooden element and the type of deviations that need to be highlighted.15 ofThe 23 it is possible to report the exact value of the distance measured between the corresponding points of software also allows for the precise quantification of deviations using control points. With the thecorresponding two compared points objects of the and two its subdivision compared alongobjects the and x, y,its and subdivision z axes (Figure along 16 the). x, y, and z axes “annotation” tool, it is possible to report the exact value of the distance measured between the (Figure 16).

FigureFigure 15.15. Three-dimensionalThree-dimensional representationrepresentation ofof thethe displacementdisplacement observedobserved byby overlappingoverlapping thethe idealideal 3D3D modelmodel andand thethe pointpoint cloudcloud ofof aa singlesingle truss.truss.

The analysis allows for the highlighting of the typical displacements that the trusses manifest over time by choosing the most appropriate control points. Thanks to the IDEAL 3D model, it is possible to quantify with sufficient precision the rotations of the posts, the lowering of the rafters, and the deformation of the tie-beam and the straining beam inside and outside the truss plane. The collected numerical data can be organized in specific tables to obtain an overview of all the recorded values through the described analysis. Data collection, therefore, becomes a crucial step to guarantee the traceability of the acquired information; once it has been correctly systematized, it becomes the tool to

Figure 16. Bidimensional representation of the displacement observed by overlapping the ideal 3D model and the point cloud of a single truss. Annotation for rafters, tie-beams, and posts control points.

The analysis allows for the highlighting of the typical displacements that the trusses manifest over time by choosing the most appropriate control points. Thanks to the IDEAL 3D model, it is possible to quantify with sufficient precision the rotations of the posts, the lowering of the rafters, and the deformation of the tie-beam and the straining beam inside and outside the truss plane. The collected numerical data can be organized in specific tables to obtain an overview of all the recorded values through the described analysis. Data collection, therefore, becomes a crucial step to guarantee the traceability of the acquired information; once it has been correctly systematized, it becomes the tool to correctly interpret the behavior of these structural systems from a quantitative and not only qualitative point of view.

3.5. The BIM Model and Calculation Model Generation Procedure After the displacement analysis, based on the acquired data and the followed generative algorithm approach, it is possible to deepen the understanding of wooden trusses by automatically creating informative models. These models can maintain the gathered information already acquired and could be useful for further structural analyses. The “axes generation algorithm” was scripted in Grasshopper® to obtain “line” elements corresponding to the axes of the rods. This algorithm could be applied to the previously modeled axes in the BASIC 3D model or the IDEAL 3D model to generate the BIM model in Revit® and the structural model in Robot Structural Analysis®. First, the “nodes individuation algorithm,” formerly adopted for the creation of the IDEAL 3D model, was run to solve some inaccuracies due to the previous modeling procedures. Then, the “axes generation algorithm” was applied. It consists of Sustainability 2020, 12, x 15 of 23 corresponding points of the two compared objects and its subdivision along the x, y, and z axes (Figure 16).

Sustainability 2020, 12, 4975 15 of 23 correctlyFigure interpret 15. Three the-dimensional behavior of representation these structural of the systems displacement from a observed quantitative by overlapping and not only the qualitative ideal point3D of model view. and the point cloud of a single truss.

Figure 16. Bidimensional representation of the displacement observed by overlapping the ideal 3D model and the point cloudcloud of a single truss. Annotation Annotation for rafters, rafters, tie tie-beams,-beams, and posts control points. 3.5. The BIM Model and Calculation Model Generation Procedure The analysis allows for the highlighting of the typical displacements that the trusses manifest over Aftertime theby displacementchoosing the analysis,most appropriate based on thecontrol acquired points. data Thanks and the to followed the IDEAL generative 3D model, algorithm it is approach,possible to itquantify is possible with to sufficient deepen the precision understanding the rotations of wooden of the posts trusses, the by lowering automatically of the creating rafters, informativeand the deformation models. Theseof the models tie-beam can and maintain the straining the gathered beam informationinside and outside already the acquired truss plane and could. The becollected useful numerical for further data structural can be analyses. organized in specific tables to obtain an overview of all the recorded ® valuesThe through “axes the generation described algorithm” analysis. Data was collection, scripted intherefore, Grasshopper becomesto a crucial obtain step “line” to guarantee elements correspondingthe traceability to of the the axes acquired of the information; rods. This algorithm once it has could been be appliedcorrectly to systematized, the previously it modeled becomes axes the ® intool the to BASIC correctly 3D modelinterpret or thethe IDEALbehavior 3D of model these to structural generate thesystems BIM modelfrom a in quantitative Revit and theand structural not only ® modelqualitative in Robot point Structural of view. Analysis . First, the “nodes individuation algorithm,” formerly adopted for the creation of the IDEAL 3D model, was run to solve some inaccuracies due to the previous modeling procedures.3.5. The BIM Model Then, and the Calculation “axes generation Model Generation algorithm” Procedure was applied. It consists of joining, rod by rod, the barycenter of the two end sections of the rod (coinciding with the truss joints) through the line After the displacement analysis, based on the acquired data and the followed generative component. The result is a set of lines that accurately represents the static scheme of the truss. Another algorithm approach, it is possible to deepen the understanding of wooden trusses by automatically algorithm was run to recognize the dimensions of the cross-sections automatically. creating informative models. These models can maintain the gathered information already acquired The Grasshopper’s algorithm was linked with the Dynamo®’s “BIM model generation algorithm” and could be useful for further structural analyses. through the Speckle® plug-in. In Dynamo®, some lists were imported, containing: The “axes generation algorithm” was scripted in Grasshopper® to obtain “line” elements 1.corresponding Axes of the to beams: the axes line of elements the rods. in This the algorithm same 3D coordinate could be applied system ofto thethe original previously point modeled cloud. 2.axes inAverage the BASIC numbers 3D model of base or and the height IDEAL of 3D the model typological to generate cross-section the BIM of model each rod. in Revit® and the structural model in Robot Structural Analysis®. First, the “nodes individuation algorithm,” formerly ® adoptedA new for “structuralthe creation framing” of the IDEAL family 3D was model, created was in run Revit to solveto model some theinaccuracies girders. Thedue familyto the consistedprevious modeling of wooden procedures. beams with Then, rectangular the “axes sections generation to add algorithm information” was about applied. the materialIt consists and of physical characteristics (e.g., density, Young’s modulus, Poisson’s coefficient, resistance, and thermal properties). Using the list of bases and heights of the cross-sections, which were provided by Grasshopper®, different family types were automatically created for each truss element (posts, tie-beams, struts, rafters, etc.) using the “Family Types” node in Dynamo®. Then, the linear axes and the different family types were given as input to the “StructuralFraming.BeamByCurves” node, which automatically generates “Beam” elements in Revit® (Figure 17). These “Beam” elements were not simple 3D objects (Figure 18). They also contained some vital information for the structural analysis, forming, along with joints and connections between elements, the so-called analytical model. These elements could be used in external analyses, to determine how the beam conditions contribute to the static and lateral structural behavior. Sustainability 2020, 12, x 16 of 23

joining, rod by rod, the barycenter of the two end sections of the rod (coinciding with the truss joints) through the line component. The result is a set of lines that accurately represents the static scheme of the truss. Another algorithm was run to recognize the dimensions of the cross-sections automatically. Sustainability 2020, 12, x 16 of 23 The Grasshopper’s algorithm was linked with the Dynamo®’s “BIM model generation ® ® joining,algorithm rod” by through rod, the the barycenter Speckle plug of the-in. two In Dynamo end sections, some of lists the rodwere (coinciding imported, containing: with the truss joints) through1. Axes the of line the component. beams: line elementsThe result in is the a setsame of 3Dlines coordinate that accurate systemly represents of the original the pointstatic cloud.scheme of the2. truss.Average Another numbers algorithm of base was and run height to recognize of the typological the dimensions cross-section of the crossof each-sections rod. automatically. The Grasshopper’s algorithm was linked with the Dynamo®’s “BIM model generation A new “structural framing” family was created in Revit® to model the girders. The family ® ® algorithmconsisted” ofthrough wooden the beams Speckle with plug rectangular-in. In Dynamo sections, tosome add lists information were imported about ,the containing: material and 1. physicalAxes characteristicsof the beams: (e.g.line ,elements density, Young in the’ ssame modulus, 3D coordinate Poisson’s coefficient,system of theresistance, original and point thermal cloud. 2. properties).Average Usingnumbers the of list base of andbases height and ofheights the typological of the cross cross-sections,-section which of each were rod. provided by Grasshopper®, different family types were automatically created for each truss element (posts, tie- ® beams,A new struts, “structural rafters, etc.) framing using ”the family “Family was Types created” node in in Revit Dynamo to® .model Then, thethe linear girders. axes The and familythe consistdifferented offamily wooden types beams were given with asrectangular input to the sections “StructuralFraming.BeamByCurves to add information about the” node material, which and physicalautomatically characteristics generate s(e.g. “Beam, density,” elements Young in ’Revits modulus,® (Figure Poisson 17). ’s coefficient, resistance, and thermal properties). Using the list of bases and heights of the cross-sections, which were provided by Grasshopper®, different family types were automatically created for each truss element (posts, tie- beams, struts, rafters, etc.) using the “Family Types” node in Dynamo®. Then, the linear axes and the different family types were given as input to the “StructuralFraming.BeamByCurves” node, which Sustainability 2020, 12, 4975 16 of 23 automatically generates “Beam” elements in Revit® (Figure 17).

Figure 17. BIM model generation algorithm in Dynamo®.

These “Beam” elements were not simple 3D objects (Figure 18). They also contained some vital information for the structural analysis, forming, along with joints and connections between elements, the so-called analytical model. These elements could be used in external analyses , to determine how the beam conditions contribute to the static and lateral structural behavior. ® FigureFigure 17. 17.BIM BIM modelmodel generation generation algorithm algorithm in in Dynamo Dynamo®..

These “Beam” elements were not simple 3D objects (Figure 18). They also contained some vital information for the structural analysis, forming, along with joints and connections between elements, the so-called analytical model. These elements could be used in external analyses , to determine how the beam conditions contribute to the static and lateral structural behavior.

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SustainabilityFigureFigure 20 2018,.18. 1BIM2, BIMx model model of the of thetrusses trusses in Revit in Revit®. (a)® Architectural.(a) Architectural modelmodel (b) Analytical (b) Analytical model. model. 17 of 23

TheThe analyticalanalytical modelmodel waswas completedcompleted withwith thethe missingmissing information:information: external constraints, internalinternal releases,releases, andand structuralstructural loadsloads (Figure(Figure 19 19)).. At this point, the model was sentsent toto RobotRobot StructuralStructural AnalysisAnalysis®®,, a software performing advanced structural analysis usingusing BIM-integratedBIM-integrated workflowsworkflows toto exchangeexchange datadata withwith RevitRevit®® continuouslycontinuously.. (a) (b)

Figure 18. BIM model of the trusses in Revit®. (a) Architectural model (b) Analytical model.

Figure 19. Calculation model of a truss. Figure 19. Calculation model of a truss. Summarizing the whole process, once the TLS survey is performed, the protocol makes available, trussSummarizing by truss, the TLSthe pointwhole cloud, process, the BASIConce the 3D TLS model, survey the IDEAL is performed, 3D model, the the protocol displacement makes analysisavailable, through truss by the truss, described the TLS method, point the cloud, BIM model,the BASIC and the3D calculationmodel, the modelIDEAL through 3D model, a single, the fast,displacement and integrated analysis workflow. through the described method, the BIM model, and the calculation model through a single, fast, and integrated workflow. 4. Results 4. Results From the synthesis and integration of all the data deriving from the 3D models and the point cloudFrom comparison, the synthesis it was and possible integration to provide of all some the data interesting deriving interpretations from the 3D models regarding and the the overall point cloud comparison, it was possible to provide some interesting interpretations regarding the overall displacement and behavior of the entire roof system of the Municipal Theater of Bologna. Each truss displacement proved to be related to the other ones both inside and outside the trusses’ vertical planes (Figure 20).

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(b) Sustainability 2020, 12, x 17 of 23

The analytical model was completed with the missing information: external constraints, internal releases, and structural loads (Figure 19). At this point, the model was sent to Robot Structural Analysis®, a software performing advanced structural analysis using BIM-integrated workflows to exchange data with Revit® continuously.

Figure 19. Calculation model of a truss.

Summarizing the whole process, once the TLS survey is performed, the protocol makes available, truss by truss, the TLS point cloud, the BASIC 3D model, the IDEAL 3D model, the displacement analysis through the described method, the BIM model, and the calculation model through a single, fast, and integrated workflow.

4. Results Sustainability 2020, 12, 4975 17 of 23 From the synthesis and integration of all the data deriving from the 3D models and the point cloud comparison, it was possible to provide some interesting interpretations regarding the overall displacement and behavior of the entire roof system of the Municipal Theat Theaterer of Bologna. Each truss displacement provedproved toto be be related related to to the the other other ones ones both both inside inside and outsideand outside the trusses’ the trusses vertical’ vertical planes planes(Figure ( Figure20). 20).

(a)

Sustainability 2020, 12, x 18 of 23 (b)

(c)

Figure 20. Graphs represent representinging the displacements of the trusses in their vertical planes (different (different scale values).values). (a)) Struts. ( b) Tie Tie-beams.-beams. ( c) Rotation of the posts.

4.1.4.1. Displacement Analysis in the Vertical Plane of the TrussTruss All thethe externalexternal trianglestriangles ofof thethe analyzedanalyzed trussestrusses showedshowed aa similarlysimilarly asymmetricalasymmetrical behaviorbehavior inin theirtheir vertical plane, plane, unbalanced unbalanced toward toward the the east east side side of of the the building. building. A careful A careful visual visual analysis analysis of the of theorthophotos orthophotos and and some some measurements measurements deriving deriving from from the the point point cloud cloud highlighted highlighted that that the easterneastern bearingsbearings of the tie tie-beams-beams were were about about 10 10 cm cm lower lower than than those those on on the the western western side. side. Since Since this this was was not notcompatible compatible with with the initial the initial hypothesis hypothesis,, which which assumed assumed that the that lateral the lateral bearings bearings of the of tie the-beams tie-beams were werestill in still their in theiroriginal original position, position, it was it wasnecessary necessary to adjust to adjust the hypothesis the hypothesis of the of static the static behavior behavior of the of thetrusses. trusses. InIn general, thethe realreal staticstatic behaviorbehavior reportedreported byby eacheach studiedstudied trusstruss providedprovided (Figure(Figure 21 21)):: • InflectionInflection ofof thethe tie-beamstie-beams unbalancedunbalanced towardtoward thethe easteast sideside ofof thethe buildingbuilding • • CounterclockwiseCounterclockwise rotation ofof nodenode BB (bottom(bottom rafters-westernrafters-western queenqueen post-strainingpost-straining beam)beam) andand • consequentconsequent rotationrotation ofof thethe westernwestern queen-postqueen-post inin thethe samesame direction.direction. • ClockwiseClockwise rotation ofof nodenode CC (ridge(ridge node),node), lowerlower thanthan thethe rotationrotation ofof nodenode B,B, andand consequentconsequent • rotationrotation ofof thethe kingking postsposts inin thethe samesame direction.direction. • Clockwise rotation of node B’ (straining beam-eastern queen post-bottom rafter), quite more extensive than the rotation of node B, and consequent rotation of the eastern queen-post in the same direction. • Almost zero lowering of the B nodes. • Almost zero lowering of the C nodes. • Lowering of the B’ nodes with a value between 1–4 cm. • Lowering of node A’ (eastern lateral bearing), as anticipated. • Clockwise rotation of the straining beam, congruent with the movements of nodes B and B’.

Figure 21. General static behavior of the Municipal Theater’s trusses. Joints displacements and rotations. A - Bottom rafter/tie-beam joint. B - Straining beam-queen post joint. C Top rafter-king post joint (ridge)

These results have been attributed to two different causes. On the one hand, the difference in altitude between the eastern and western bearings seemed a small value if compared to the vast span Sustainability 2020, 12, x 18 of 23

(c)

Figure 20. Graphs representing the displacements of the trusses in their vertical planes (different scale values). (a) Struts. (b) Tie-beams. (c) Rotation of the posts.

4.1. Displacement Analysis in the Vertical Plane of the Truss All the external triangles of the analyzed trusses showed a similarly asymmetrical behavior in their vertical plane, unbalanced toward the east side of the building. A careful visual analysis of the orthophotos and some measurements deriving from the point cloud highlighted that the eastern bearings of the tie-beams were about 10 cm lower than those on the western side. Since this was not compatible with the initial hypothesis, which assumed that the lateral bearings of the tie-beams were still in their original position, it was necessary to adjust the hypothesis of the static behavior of the trusses. In general, the real static behavior reported by each studied truss provided (Figure 21): • Inflection of the tie-beams unbalanced toward the east side of the building

Sustainability• Counterclockwise2020, 12, 4975 rotation of node B (bottom rafters-western queen post-straining beam18) ofand 23 consequent rotation of the western queen-post in the same direction. • Clockwise rotation of node C (ridge node), lower than the rotation of node B, and consequent Clockwiserotation of rotationthe king ofposts node in B’the (straining same direction. beam-eastern queen post-bottom rafter), quite more • • extensiveClockwise than rotation the rotation of node of B node’ (straining B, and beamconsequent-eastern rotation queen ofpost the-bottom eastern rafter queen-post), quite in more the sameextensive direction. than the rotation of node B, and consequent rotation of the eastern queen-post in the Almostsame direction. zero lowering of the B nodes. • • AlmostAlmost zerozerolowering loweringof ofthe the C B nodes.nodes. • • LoweringAlmost zero of thelowering B’ nodes of withthe C anodes. value between 1–4 cm. • • LoweringLowering ofof nodethe B’ A’ nodes (eastern with lateral a value bearing), between as 1 anticipated.–4 cm. • • ClockwiseLowering rotationof node A of’ the(eastern straining lateral beam, bearing), congruent as anticipated. with themovements of nodes B and B’. •• Clockwise rotation of the straining beam, congruent with the movements of nodes B and B’.

Figure 21. General static behavior of the Municipal Theater’s trusses. Joints displacements and Figure 21. General static behavior of the Municipal Theater’s trusses. Joints displacements and rotations. A - Bottom rafter/tie-beam joint. B - Straining beam-queen post joint. C Top rafter-king post rotations. A - Bottom rafter/tie-beam joint. B - Straining beam-queen post joint. C Top rafter-king joint (ridge). post joint (ridge) These results have been attributed to two different causes. On the one hand, the difference in altitudeThese between results the have eastern been and attributed western bearingsto two different seemed acauses. small value On the if compared one hand, to the the difference vast span ofin thealtitude trusses. between So, it wasthe eastern probably and caused western by a bearings tolerance seemed error in a thesmall 18th-century value if compared construction to the site, vast which span certainly could not benefit from the precision of modern instruments. On the other hand, the data could be related to the overall static behavior of the original structures of the entire building. Probably, there was a profound subsidence phenomenon along the eastern side of the historic building. The historical research provided information about an ancient water channel close to this area that might have involved ground movements. It would be necessary to analyze in detail the crack pattern of the walls of the theater to confirm this hypothesis.

4.2. Displacement Analysis Outside the Vertical Plane of the Truss All data concerning the displacements outside the trusses’ vertical plane also proved to be compatible with the overall movement of the roof hypothesis. All the trusses rotated around their lateral bearings, in the theater proscenium direction (on the North). It was noted that, except for the second truss, while the western rafters have all undergone a rotation toward the theater proscenium (maximum displacement of 5 cm), the eastern ones have undergone a rotation in the opposite direction (Figure 22). In particular, the trusses’ displacement toward the proscenium was limited on the eastern side of the central trusses, probably due to the metal walkway located near this area. This platform, bound to the central tie-beams and the floor, worked as a brace element, allowing the central trusses to locally oppose the overall movement. As we interpret these effects, the importance of integrating the historical research and the data from the new technologies used by the protocol emerges. If the interventions suffered by the building during its life cycle are known, it becomes easier to define the pathologies that currently threaten the health of the historic building. Some of the consulted archival documents reported that the proscenium area had always shown structural problems at the roof level since the construction of the theater. This area suffered the thrusts from the wooden vault, which has been connected to the trusses with metal tie-rods since 1980. The vault suffered displacements toward the proscenium area, which has Sustainability 2020, 12, 4975 19 of 23 Sustainability 2020, 12, x 20 of 24

always been a structural weakness of the theater. Therefore, the metal tie-rods dragged the joints of the trusses in the same direction.

(a)

North

South

(b)

FigureFigure 22. 22. MovementsMovements of ofthe the rafters rafters outside outside their their vertical vertical plane. plane. (a) ( aSection) Section representing representing the the western western rafters.rafters. (b) ( bPlan) Plan of ofthe the roof roof extracted extracted from from the the point point cloud. cloud.

5. ConclusionsIn particular, the trusses’ displacement toward the proscenium was limited on the eastern side of the centralThe parametric trusses, probably 3D modeling due protocol,to the metal aimed walkway at evaluating located and near interpreting this area. This the platform, deformation bound states toof the the central 18th-century tie-beams wooden and the trusses floor, in worked Bibiena’s as Theater,a brace haselement, proven allowing to be an the innovative central trusses and useful to locallytool to oppose deal with the overall similar movement structures.. On the one hand, the application of TLS systems and parametric softwareAs we tointerpret this kind these of historicaleffects, the structures importance has of revealed integrating a fascinating the historical combination, research and which the allowsdata fromthe the research new technologies activity to focus used onby the protocol field of buildingemerges. restoration.If the interventions On the suffered other hand, by the the building research duringprotocol its life helps cycle to acquireare known information, it becomes that easier is usually to define diffi thecult pathologies to find through that currently the classical threaten methods the healthof investigation. of the historic building. Some of the consulted archival documents reported that the proscenium area had always shown structural problems at the roof level since the construction of the theater. This area suffered the thrusts from the wooden vault, which has been connected to the trusses Sustainability 2020, 12, 4975 20 of 23

The displacement analysis, carried out through the protocol, has proven a valid interpretation method for the overall kinematics of the roof system. Although the detected movements do not threaten the current state of the building, they highlighted the need for multi-year monitoring cycles. By simply updating the point cloud given as input for the generative algorithms, it is possible to understand if the currently identified movements are evolving and to assess the risk affecting the wooden trusses of the Municipal Theater of Bologna. Regarding the structure of the algorithms, although the main goal was to automate as many manual operations as possible, it was not possible to mechanize all the processes. However, by combining the algorithms with artificial intelligence and machine learning tools, it is possible to streamline all the procedures and standardize the method. The possibility of permanently checking the data on the trusses in the point cloud and the aim of scripting algorithms to create digital models that are compatible with the cloud have implicitly required a continuous comparison with the historic trusses of the theater. This has led to a complete knowledge of the studied wooden elements. Besides, the study has also revealed the extraordinary historical and construction values of the Municipal Theater of Bologna. The vast repertoire of technological and construction features, belonging to different historical periods, has highlighted the strong sense of identity of the building and the need to investigate all its parts in detail. Therefore, the archival research that has been conducted consists of a valid general framework for any further investigation. After understanding the building’s construction peculiarities through proper historical research, conducting on-site inspections, digitally surveying the most exciting spaces of the theater, analyzing the trusses (their materials, their static schemes, their joints, and their transformations over the centuries), scripting three-dimensional modeling algorithms to digitize the wooden rods, and analyzing the displacements through the explained methodology, the protocol could be considered a valid tool for monitoring the wooden structures, for evaluating the need of restoration interventions, and for supporting the designers’ activity. Finally, for the first time, the protocol has been integrated with BIM tools. In this case, the generation of a structural analysis model has been chosen among the countless applications that BIM methodologies could offer, to complete the structural analysis of the wooden trusses. However, BIM applications could further add significant value in terms of heritage digitization. For example, it is possible to improve the dissemination of built heritage by integrating BIM models with VR technologies. Another application could contribute to the generation of a “digital twin” of the historical building to manage the heritage at risk. A digital twin is a digital model reproducing the behavior of a real system as a response to external physical data of various kinds. It might be interpreted as an opportunity to enhance asset performance, help in anticipating adverse environmental effects during the life cycle stages, forecast maintenance activities based on sensor data, and bring further benefits to built heritage. Currently, BIM technologies seem the most appropriate tool to promote this digital process. In the future, through the technological advancement of sensor technologies, the data deriving from the protocol might be added to the remote sensing measurements almost in real-time, to investigate other fundamental factors during the building life-cycle (e.g., humidity conditions, atmospheric conditions, solar radiation, structural vibrations, displacements, quantification of consumption, etc.). Nowadays, a similar process appears to have been set for new buildings. However, it also seems essential for monitoring the “state of health” of heritage at risk and for preserving its historical, architectural, artistic, social, and economic value both in terms of efficient conservation and of the enhancement and dissemination of historical heritage for its transmission to future generations.

Author Contributions: Conceptualization, D.P. and A.M.; methodology, D.P.; software, A.M.; validation, D.P.; formal analysis, A.M.; investigation, A.M.; resources, D.P. and G.P.; data curation, A.M.; writing—original draft preparation, A.M.; writing—review and editing, D.P. and G.P.; supervision, G.P. and R.G. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Sustainability 2020, 12, 4975 21 of 23

Acknowledgments: We would like to thank the management and the whole staff of the Municipal Theater of Bologna, who guaranteed access to the building, providing an indispensable cultural and operational support for the activities that have been carried out. This research is part of a series of activities jointly carried out between the research group of the Department of Architecture and the administration of the Theater, which is currently underway. Conflicts of Interest: The authors declare no conflict of interest.

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