Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs

materials

Article Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs

Natalia Staszak 1, Tomasz Garbowski 2 and Anna Szymczak-Graczyk 3,*

1 Research and Development Department, FEMat Sp. z o. o., Romana Maya 1, 61-371 Poznań,Poland; [email protected] 2 Department of Biosystems Engineering, Poznan University of Life Sciences, Wojska Polskiego 50, 60-627 Poznań,Poland; [email protected] 3 Department of Construction and Geoengineering, Poznan University of Life Sciences, Pi ˛atkowska94 E, 60-649 Poznań,Poland * Correspondence: [email protected]

Abstract: The need for quick and easy deflection calculations of various prefabricated slabs causes simplified procedures and numerical tools to be used more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Based on the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified Citation: Staszak, N.; Garbowski, T.; Szymczak-Graczyk, A. Solid Truss to homogenized plate FE model. The method allows for the correct homogenization of any complex Shell Numerical Homogenization of composite structures made of both solid and structural elements, without the need to perform Prefabricated Composite Slabs. advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a Materials 2021, 14, 4120. https:// representative volume element (RVE) and appropriate formulation of the transformation between doi.org/10.3390/ma14154120 kinematic constrains on the RVE boundary and generalized strains.

Academic Editors: Panagiotis Keywords: numerical homogenization; prefabricated ﬂoor slab; concrete; composite structure; strain G. Asteris and Luisa F. Cabeza energy equivalence

Received: 22 May 2021 Accepted: 22 July 2021 Published: 23 July 2021 1. Introduction Prefabricated concrete structures such as floor slabs, girders, and columns have nu- Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in merous advantages, including (a) quality standard, (b) saving of formwork, (c) short con- published maps and institutional affil- struction time, (d) durability of the structure, and (e) very low energy consumption [1–3]. iations. That is why prefabricated concrete elements, and in particular composite structures, have been widely used in industrial and residential buildings all over the world in recent decades [2]. Floor slabs and girders are one of the most frequently used prefabricated structural elements among all prefabricated structures [4]. One of the most popular prefabricated composite floor-slab is the so-called “Filigree” Copyright: © 2021 by the authors. ceiling, which is used all over Europe. It is a typical example of a semi-prefabricated ceiling Licensee MDPI, Basel, Switzerland. This article is an open access article used in industrial, residential, rural, and general construction, and it allows architects to distributed under the terms and follow their creativity, which this type of ceiling does not limit. In Poland, this ceiling first conditions of the Creative Commons appeared in 1996, while in Europe these structures could be found already in 1964–1965. Attribution (CC BY) license (https:// The most important feature in this type of floor is that the strength of floor slabs is adjusted creativecommons.org/licenses/by/ to a specific load, in accordance with conditions prevailing during the use of the floor. 4.0/).

Materials 2021, 14, 4120. https://doi.org/10.3390/ma14154120 https://www.mdpi.com/journal/materials Materials 2021, 14, 4120 2 of 16

Over the past years, various numerical studies have been developed to better under- stand the mechanical behaviour of precast composite structures. To present the global behaviour of prefabricated structures, various models have been developed. From the macro level, where fibre elements are used to simulate prefabricated beams, slabs, columns [5–7], and sandwich layered shells are used to simulate prefabricated slabs. Tzaros et al. [8] studied the bending behaviour of composite slabs. Modelling of structural elements at the macro level, however, presents some obstacles to, for example, an accurate simulation of the contact between a precast element and concrete poured in place. In the work published by Abdullah and Easterling [9], a force balance method was developed to calculate the shear-slip relationship in FEM modelling of composite slabs. Ren et al. [10] proposed a full 3D FE model of precast concrete bridge panels to model its nonlinear behaviour using a concrete damage plasticity constitutive model. The complete procedure for concrete damage plasticity model calibration is given in the work by Jankowiak and Łodygowski [11]. Other material models for concrete can be used for numerical analysis provided their constitutive parameters are properly identified [12–14]. A three-dimensional (3D) solid FE model was also used in the work of Gholamho- seini et al. [15], where interface elements were introduced to present the bonding between steel and concrete. Nonlinear 3D FEM models of composite panels were also used to present slip mechanics in “pull-out” tests [16]. On the other hand, the effective FEM model used to recreate the longitudinal shear behaviour of composite slabs with a profiled sheet was presented in the work by Ríos et al. [17]. In general, with the development of computational speed, 3D finite element modelling has become a promising tool for the numerical analysis of prefabricated concrete structures, primarily due to its ability to describe complex relationship between composite elements. If complex analysis is not particularly required, time consuming modelling of full 3D FE models is not economical. In such cases, the complex multi-layer composite cross- section can be replaced by a single-layer model with equivalent properties, which ensures that its behaviour is very close to the reference model. This kind of simplification is known as homogenization. Numerous studies have dealt with the homogenization of complex sections for dozens of years. In 2003, Hohe presented homogenization methods based on strain energy [18]. The author applied a procedure in the context of homogenization of sandwich panels with a honeycomb core. It uses strain energy with the assumption of mechanical equivalence between a simplified model and the representative volume element (RVE) of a sample. Another type of homogenization is a periodic homogenization method proposed by Buanic et al. [19]. It allows for calculating both the equivalent shearing and bending characteristics of the periodic plates and membrane. Biancolini used the strain energy equivalence between a single layer equivalent model and the numerical model of representative volume element (RVE) to compute the bending properties of corrugated core panels and membrane [20]. This method was later updated by Garbowski and Gajewski [21]. In the work by Marek and Garbowski [22], the compilation and comparison of various methods to homogenization of composite plates with corrugated and sandwich core were presented. A slightly different approach, based on inverse analysing can be found in the study by Garbowski and Marek [23]. An interesting approach to periodic composites based on the numerical implantation of asymptotic homogenization method is presented in [24–26]. The textile composite, on the other hand, can be homogenized by a multiscale approach based on mechanics of a structure genome [27–29]. The discussion about the differences between the static and dynamic homogenization can be found in the wark by Tallarico et al. [30] This paper presents an extension of the homogenization method presented by Biancol- ini [20] and later by Garbowski and Gajewski [21]. Originally, this method was developed for homogenization of shell structures. In this paper, the extension of the method is presented so that it can be applied to structures made of both solid and structural components and allows the transverse shear to be taken into account during homogenization. The transverse shear significantly affects the mechanical behaviour of, e.g., sandwich struc- Materials 2021, 14, 4120 3 of 16

tures; therefore, this issue has been addressed by many researchers, e.g., [31,32]. A similar effect is expected in the case of a structure consisting of a concrete slab with spatial truss reinforcement. The proposed approach uses the principles of strain energy equivalence between the simplified single layer shell model and the full 3D finite element RVE model of the prefabricated composite slab. The results obtained by the proposed homogenization method were compared with the ones obtained through the full numerical analysis of 3D modelling. The calculated displacements using the full 3D FEM model and the simplified shell (equivalent) model are in very good agreement.

2. Materials and Methods 2.1. Precasted Composite Slabs The work presents a solid truss to shell homogenization method using as an example a Filigree precast slab. Therefore, the subsection below provides some basic information about the ceiling. The Filigree plate is a prefabricated composite floor consisting of a reinforced concrete slab, spatial trusses, and concrete topping laid on the construction site. The thickness of prefabricated plates usually varies between 4.5 and 7 cm. The structural thickness of the ceiling is usually between 15 and 30 cm. Due to transport and optimal material consumption, filigree slabs are most often found in dimensions from 2.4 to 10 m (length) and 0.6 to 2.5 m (width). They can be formed in any shape from a trapezoid, rectangle, and triangle to a semicircle. Additionally, the location of the mounting holes can be taken into account already at the design stage. Slabs are made of concrete of at least C20/25 class, and their weight is on average 125–200 kg/m2. Reinforcement of prefabricated slabs is calculated in the same way as for monolithic ceilings. The main reinforcement of prefabricated slabs formwork is longitudinal bars, while transverse reinforcement only plays a role of a distributed reinforcement. In addition, slabs are joined on the construction site with the use of nets, which helps to avoid the keying effect found in various foundations. Moreover, prefabricated slabs make also a permanent formwork. Filigree slabs use the previously mentioned space trusses. They constitute an addi- tional reinforcement of the ceiling and its stiffening. The trusses are arranged parallel to the longer side of the slab, with a spacing not exceeding 75 cm. We can distinguish, among others, D, E, EK, EQ, SWE, and JD types of trusses. Recommended bar diameters, spacing of bottom bars, and diagonals as well as other necessary information can be found in the National Technical Assessment [33]. To demonstrate the homogenization technique proposed, two types of trusses were utilized. These were D and EQ types of spatial trusses. They consisted of an upper flange in the form of a single steel bar, a lower flange in the form of two bars, and steel diagonals arranged in two planes. The D-type truss consisted of a truss strip made of bars with a diameter of 12 mm and cross-braces made of bars with a diameter of 6 mm, see Figure1. Materials 2021, 14, 4120 4 of 16 Permissible bar diameters were as follows: belts—from 5 to 14 mm, crosses 5–7 mm. The truss height was assumed to be 18 cm, and the diagonal spacing was 20 cm.

(a) (b)

Figure 1. D-typeD-type spatial truss ( a)) X-X X-X view; view; ( b) Y-Y view.

(a) (b)

Figure 2. EQ-type spatial truss (a) X-X view; (b) Y-Y view.

Figure 3. Filigree ceiling on the construction site.

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The EQ-type spatial truss was made entirely of 6 mm diameter rods. The external spacing of(a) lower bars was 80 mm, and the( heightb) and spacing of diagonals were 140 mm and 200 mm, respectively. Figure2 shows the cross-section and dimensions of the EQ- Figuretype truss. 1. D-type spatial truss (a) X-X view; (b) Y-Y view.

(a) (b)

Figure 1. D-type spatial truss (a) X-X view; (b) Y-Y view.

(a) (b)

FigureFigure 2. 2. EQ-typeEQ-type spatial spatial truss ((aa)) X-XX-X view; view; ( b(b)) Y-Y Y-Y view. view. Filigree ceilings are used in industrial as well as residential, general, and rural construction [34], see Figure3 . The entire panel production process, from the delivery of the architectural(a design) to the delivery of panels to the(b construction) site, does not take long. In the case of a single-family house, the production time is approximately 3–4 days, and prefabricatedFigure 2. EQ-type panels spatial themselves truss (a are) X-X made view; in production(b) Y-Y view. plants on special heated tables.

Figure 3. Filigree ceiling on the construction site.

Prefabricated elements, before they fulfil their final task, go through many stages in which they are exposed to different support conditions and are subject to different loads. In the case of Filigree ceiling, transport and storage, as permanent formwork and as part of the structure phase, can be distinguished. During transport (assembly), a prefabricated slabFigureFigure is mainly 3. 3.Filigree Filigree affected ceiling ceiling onby theon its the construction own construction weight. site. The site. action of wind and snow can be neglected since the assembly of elements takes place in good weather conditions. In this situation, supportPrefabricatedPrefabricated is provided elements, by elements, the ones before beforeattached they they fulﬁl to spatialfulfil their ﬁnalth trusseseir task,final [35], go task, through see go Figure through many 4a. stages many in stages in whichwhichThe they theynext are arephase exposed exposed is storage. to differentto different Prefabricated support support conditions slabs conditions are andplaced are and subjecton are top subject toof differenteach to other, different loads. sepa- loads. In the case of Filigree ceiling, transport and storage, as permanent formwork and as part of ratedIn the by case wooden of Filigree beams ceiling,[35], as showntransport in Figureand storage, 4b. When as permanentthe slab has formwork been embedded and as part inthe the structure structure phase, and canconcreted, be distinguished. it transfers During the loads transport from (assembly),the weight aof prefabricated the ceiling and slabof the is mainlystructure affected phase, by can its own be distinguished. weight. The action During of wind transport and snow (assembly), can be neglected a prefabricated other layers placed on it. In addition, it must be able to withstand possible variable loads, sinceslab theis mainly assembly affected of elements by its takes own placeweight. in good The weatheraction of conditions. wind and In snow this situation,can be neglected for example, from people, furniture, and partitions. Filigree ceilings are able to transfer a supportsince the is providedassembly by of the elements ones attached takes toplace spatial in trussesgood weather [35], see Figureconditions.4a. In this situation, kN/m uniformlysupportThe next isdistributed provided phase is storage. loadby the of Prefabricatedones 10 attached slabsdepending to arespatial placed on trusses onthe top span [35], of each andsee other, Figurethickness separated 4a. of the board. by woodenThe next beams phase [35], is as storage. shown in Prefabricated Figure4b. When slabs the are slab placed has been on embedded top of each in theother, sepa- structurerated by and wooden concreted, beams it transfers [35], as the shown loads fromin Figure the weight 4b. When of the ceilingthe slab and has other been layers embedded placedin the onstructure it. In addition, and concreted, it must be ableit transfers to withstand the possibleloads from variable the loads,weight for of example, the ceiling and from people, furniture, and partitions. Filigree ceilings are able to transfer a uniformly other layers placed on it. In addition, it must be able to withstand possible variable loads, distributed load of 10 kN/m2 depending on the span and thickness of the board. for example, from people, furniture, and partitions. Filigree ceilings are able to transfer a uniformly distributed load of 10 kN/m depending on the span and thickness of the board.

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(a) (b)

Figure 4. Filigree ceiling:(a) (a) transport (assembly); (b) storage. (b) Figure 4. FiligreeFigure ceiling: 4. Filigree (a) transport ceiling: (assembly); (a) transport (b )(assembly); storage. (b) storage. All the described special features of Filigree ceilings, as well as the multitude of load cases Allto which the described the panelsAll the special may described be features subjected special of during Filigreefeatures transport of ceilings, Filigree and as ceilings, wellassembly as as the well make multitude as thethe needmultitude of of load forload quick cases calculations to whichcases to the whichof panelspanel the deflections panels may be may subjected highly be subjected justified. during during transport This transport can andbe achieved and assembly assembly in make two make the need ways:the need (i) using for quickfor a complexquick calculations calculations model of of panel theof panel full deflections 3D deflections structure highly highly of justified. the justified.slab This(reinforced canThis be can achievedslab be andachieved in two spatialin two truss) ways: orways: (i) (ii) using using(i) using a complexa simplified a complex model model model of the (single-layer of full the 3Dfull structure 3D shell structure structure) of the of slabthe with slab (reinforced effective (reinforced slab and parametersslab and spatial allowingspatial truss) truss)for or quick (ii) or using (ii) and using aprecise simplified a simplified calculations model model (single-layer of different(single-layer panel shell shell structure)variants. structure) Both with with effective approacheseffective parameters useparameters the finite allowing elemallowing forent quick (FE) for quickmethod. and precise and In precise the calculations first calculations one, of both different ofsolid different paneland truss variants.panel ele- variants. Both mentsBoth approaches and a fullapproaches model use the (with finiteuse several the element finite thousand elem (FE)ent method. FEs) (FE) of method. In the the complex first In the one, geometry first both one, solid ofboth the and solid entire truss and truss ele- slabelements need to and bements a applied, full modeland in a fullthe (with modellatter—a several (with small thousand several representative thousand FEs) of the FEs)model complex of the(RVE) complex geometry should geometry be of con- the of the entire structedentire slab for need properslab to need be homogenization applied, to be applied, in the latter—a soin thethat latter—a du smallring representative thesmall next representative step model a simple (RVE)model plate should (RVE) model beshould be con- (withconstructed several for dozenstructed proper FEs) for homogenization could proper be homogenizationused for so thatcalculation. during so that theIn the nextdu ringfollowing step the a simplenext section, step plate aboth simple model nu- plate model merical(with several approaches(with dozen areseveral FEs) described. could dozen be FEs) used could for be calculation. used for calculation. In the following In the following section, bothsection, both nu- numerical approachesmerical approaches are described. are described. 2.2. Full 3D Numerical Model 2.2. Full 3D Numerical Model In order to2.2. compute Full 3D Numericalthe reference Model deflectio ns (but also to validate the proposed ho- In order to compute the reference deflections (but also to validate the proposed homog- mogenization method),In order two toexamples compute were the referenceconsidered. deflectio In bothns cases, (but also FE commercial to validate thesoft- proposed ho- enization method), two examples were considered. In both cases, FE commercial software ware (ABAQUSmogenization FEA [36]) was method), used. twoThe examplesresults obtained were considered. from the fullIn both 3D modelscases, FE were commercial soft- (ABAQUS FEA [36]) was used. The results obtained from the full 3D models were further further comparedware with (ABAQUS the results FEA obtained [36]) was fr omused. the The simplified results obtainedshell model from (after the homog-full 3D models were compared with the results obtained from the simplified shell model (after homogenization). enization). In thefurther first compared model, a Filigreewith the ceiling results slab obtained with frD-typeom the trusses simplified was shellmodelled model as (after homog- In the first model, a Filigree ceiling slab with D-type trusses was modelled as a simply a simply supportedenization). plate. In The the slab first was model, modelle a Filigreed as 3D ceiling solid slab concrete with D-typestructure. trusses The wasslab modelled as supported plate. The slab was modelled as 3D solid concrete structure. The slab was was 2.10 m wide,a simply 7.20 msupported long, and plate. 0.06 Them thic slabk, wassee Figuremodelle 5b.d as The 3D reinforcement solid concrete ofstructure. the The slab 2.10 m wide, 7.20 m long, and 0.06 m thick, see Figure5b. The reinforcement of the slab was 2.10 m wide, 7.20 m long, and 0.06 m thick, see Figure 5b. The reinforcement of the slaband and the trussesthe trusses were were modelled modelled as steel as steel wire wi structures.re structures. In this In model,this model, D-type D-type trusses trusses with slab and the trusses were modelled as steel wire structures. In this model, D-type trusses withthe dimensions the dimensions and cross-sectionand cross-section shown shown in Figure in Figure1 were 1 used.were Theused. reinforcement The reinforcement mesh with the dimensions and cross-section shown in Figure 1 were used. The reinforcement meshwas modelled was modelled as the as main the main reinforcement reinforcemen consistingt consisting of bars of bars with with diameter diameter of φ of= 10𝜙 mm,= 10 mesh was modelled as the main reinforcement consisting of bars with diameter of 𝜙 = 10 mm,spaced spaced every every 100 mm, 100 andmm, the and separating the separating reinforcement reinforcement with bars withφ =bars 6 mm 𝜙 =every 6 mm 200 every mm mm, spaced every 100 mm, and the separating reinforcement with bars 𝜙 = 6 mm every 200(shorter mm (shorter bars), see bars), Figure see5a. Figure 5a. 200 mm (shorter bars), see Figure 5a.

(a) (b) (a) (b) Figure 5. Filigree ceiling slab with D-type trusses (a) rebars; (b) concrete slab.

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Materials 2021, 14, 4120 6 of 16 FigureFigure 5. Filigree 5. Filigree ceiling ceiling slab slabwith with D-type D-type trusses trusses (a) rebars; (a) rebars; (b) concrete(b) concrete slab. slab.

In theIn thenumerical numerical model model 2, a 2,Filigree a Filigree ceilin ceiling slabg slab with with EQ-type EQ-type trusses trusses was was modelled. modelled. TheThe dimension dimensionIn the numericalof EQ-type of EQ-type model trusses trusses 2, ais Filigree shown is shown ceilingin Figure in Figure slab 2. with The 2. The EQ-typetrusses trusses and trusses and reinforcement wasreinforcement modelled. werewereThe modelled dimension modelled as steel ofas EQ-typesteel wire wire structures, trusses structures, is shownwher whereas in easthe Figure theslab2 slab. was The was made trusses made of and concrete of reinforcementconcrete and and mod- mod- were elledelledmodelled as a as 3D a solid.3D as steelsolid. Reinforcem wire Reinforcem structures,ent entof the whereasof theslab slab had the had the slab thesame was same properties made properties of concrete as in as model in and model modelled1, i.e., 1, i.e., the themainas amain 3D reinforcement solid. reinforcement Reinforcement was was made made of of the bars of slabbars 𝜙 had =𝜙 10 = themm 10 samemm every every properties 100 100mm, mm, and as inand the model theseparating separating 1, i.e., the main reinforcement was made of bars φ = 10 mm every 100 mm, and the separating reinforcementreinforcement was was made made of bars of bars with with a diameter a diameter of 𝜙 of =𝜙 6 mm = 6 mm every every 200 200mm. mm. The The slab slab to to reinforcement was made of bars with a diameter of φ = 6 mm every 200 mm. The slab to analyseanalyse was was 1.50 1.50 m wide, m wide, 6.00 6.00 m long m long, and, and 0.05 0.05 m thick, m thick, see seeFigure Figure 6. 6. analyse was 1.50 m wide, 6.00 m long, and 0.05 m thick, see Figure6.

(a) (a) (b) (b) Figure 6. Filigree ceiling slab with EQ-type trusses (a) rebars; (b) concrete slab. FigureFigure 6. FiligreeFiligree ceiling ceiling slab slab with with EQ-type trusses ( a) rebars; ( b) concrete slab.

BoundaryBoundaryBoundary conditions conditions conditions in both in in both both 3D numerical3D 3D numerical numerical models models models of a of ofsimply a simply supported supported type type were were appliedappliedapplied at two at two opposite oppositeopposite shorter shortershorter sides. sides.sides. The TheThe Filigree Filigree Filigree ceiling ceiling ceiling slabs slabs slabs with with with Type-D Type-D Type-D and and and Type- Type-EQ Type- EQ EQtrusses trusses were were were loaded loaded loaded with with with a aload loada load evenly evenly evenly distributed distributed distributed over over over the the theentire entire entire surface surfacesurface of the ofof theplate plate (shown(shown(shown in Figure inin FigureFigure 7),7 which),7), which which amounted amounted amounted to to 150 150to 150kg/ kg/ mkg/mand2mandand 125 125 125kg/ kg/ mkg/mform2 for Type-Dfor Type-D Type-D and and and Type- Type-EQ Type- EQ EQtrusses,trusses, trusses, respectively. respectively. Additionally, Additionally, thethe reinforcementthereinforcement reinforcement mesh mesh mesh and and theand the bottom thebottom bottom horizontal horizontal horizontal bars barsbarsof of the the of truss thetruss truss were were were anchored anchored anchored in in the the in concrete theconcrete concrete slab slab usingslab using using the the embedded theembedded embedded techniques techniques techniques available avail- avail- in ableableAbaqus in Abaqus in Abaqus [36 ].[36]. [36].

FigureFigureFigure 7. Loading 7. 7. LoadingLoading and andboundary and boundary boundary conditions conditions conditions of Filigree of of Filigree Filigree ceiling ceiling slabs. slabs.

TwoTwoTwo materials materials were were used used to describe to describe the thechar char characteristicsacteristicsacteristics of individual of individual elements elements in the in the examplesexamplesexamples analysed analysedanalysed here: here:here: concrete concrete concrete and and and steel. steel. steel. The The The concrete concrete waswas was described described described by by the theby linear thelinear stress–linear stress–strainstress–strainstrain relations. relations. relations. For For steel, Forsteel, steel, an an elastic elastic an elastic perfectly-plastic perfectly-plastic perfectly-plastic modelmodel model was was used. used. The The The engineer- engineering engineering ingparameters parameters of of materials materialsof materials used used used in in the in the thetests tests tests are are shownare shown shown in Table inin TableTable 1, where1 1,, wherewhere 𝐸 is𝐸E the is is theYoung’s Young’s Young’s 0 modulus,modulus,modulus, 𝜈 is ν𝜈 the isis the thePoisson PoissonPoisson′s ratio,s′s ratio,ratio, and andand 𝜌 is ρ𝜌 isdensity. is density. density. In ModelIn Model 1, the 1, theconcrete concrete cross-section cross-section was was divided divided into into 3D 3Dsolid solid elements elements with with di- di- mensionsmensionsTable 1.ofMaterial 30 of mm30 parametersmm in the in thethickness ofthickness steel anddirection concretedirection of usedthe of the inslab the slab and tests. and 50 mm50 mm in the in theplane. plane. This This gives us 12,096 of 20-node brick elements with three degrees of freedom in each node gives usMaterial 12,096 of 20-node brickE(GPa) elements with threeν (degrees−) of freedomρ (kg/min each3) node concrete 30 0.2 2400 steel 210 0.3 7900

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In Model 1, the concrete cross-section was divided into 3D solid elements with dimensions of 30 mm in the thickness direction of the slab and 50 mm in the plane. This gives us 12,096 of 20-node brick elements with three degrees of freedom in each node (with reduced integration scheme, called C3D20R according to [36]). The steel components, i.e., truss and reinforcement mesh, were meshed using 2-node truss elements with three degrees of freedom (T3D2 according to [36]). Discretization gives a total of 963 and 1872 elements in the truss structure and reinforcement mesh, respectively. Model 2, similarly to Model 1, was also discretized using 3D solid elements (C3D20R according to [36]) for concrete slab and truss elements (T3D2 according to [36]) for steel elements. The slab was divided into 7200 brick elements with the following dimensions: 25 mm in the thickness direction (to have at least two elements through the thickness) and 50 mm towards the plane of the slab. The total number of truss elements in the spatial truss structure was 807, and in the reinforcements mesh—1140 elements. The total number of nodes in Model 1 equals 247,590 which gives in total 742,770 degrees of freedom (since all the elements have three degrees of freedom in each node). Model 2 consist of 147,894 nodes and thus has 443,682 degrees of freedom. These are rather sparse meshes, useful in this study, but if one would prefer to obtain more precise results, the number of nodes should be much higher, which in turn would cause much more demanding computations. In order to speed up computation but also to simplify the geometry for eventual variational study, a simpler model is required.

2.3. Representative Volume Element In order to perform numerical homogenization (details described further in the work), a proper definition of the stiffness matrix of a representative volume element (RVE) of a ceiling element is needed. RVE was created by extracting a periodic piece of structure from the entire 3D model. RVE for the Filigree ceiling slabs with D-type trusses was 0.70 m wide, 0.60 m long, and 0.06 m thick, see Figure8a. For the correct definition of RVE, model geometry containing only one repetitive period is usually used. However, based on the experience gained, it was finally decided to use RVE with three periods. This is dictated by previous observations (see [21]), which show that for RVE with base dimensions deviating from the square, the transverse stiffness for the shorter side is overestimated. The choice of RVE with three periods has no effect on the remaining stiffnesses in the ABD matrix determined by this method. The slab was modelled as a solid 3D structure made of concrete. The model used repeatable D-type trusses (three periods) and reinforcement shown in Figure8b,c. They were modelled as steel wire structures. In addition, the reinforcement mesh and the bottom bars of truss were embedded in the concrete slab. The slab was meshed with solid C3D20R elements, i.e., 20-node elements with three degrees of freedom, while the steel part was meshed with truss T3D2 elements (Figure8d). The RVE model for the Filigree ceiling slab with EQ-type trusses consisted of a plate with dimensions of 0.50 × 0.60 × 0.05 m, EQ-type truss, and slab reinforcement, see Figure9 . The steel trusses and bars were modelled as a wire structure and then meshed with T3D2 type elements. The concrete part of the model was a 3D solid and was divided into FE using C3D20R elements.

2.4. Homogenization Algorithm The classical formulation of a displacement-based linear finite element method includes

Ke ue = Fe, (1)

where Ke is a statically condensed global stiffness matrix of the RVE; ue is a displacement vector of external nodes, and Fe is a vector of the nodal force applied to external nodes. The FE mesh and external nodes are visualized in Figure 10. Materials 2021, 14, 4120 7 of 16

(with reduced integration scheme, called C3D20R according to [36]). The steel components, i.e., truss and reinforcement mesh, were meshed using 2-node truss elements with three degrees of freedom (T3D2 according to [36]). Discretization gives a total of 963 and 1872 elements in the truss structure and reinforcement mesh, respectively. Model 2, similarly to Model 1, was also discretized using 3D solid elements (C3D20R according to [36]) for concrete slab and truss elements (T3D2 according to [36]) for steel elements. The slab was divided into 7200 brick elements with the following dimensions: 25 mm in the thickness direction (to have at least two elements through the thickness) and 50 mm towards the plane of the slab. The total number of truss elements in the spatial truss structure was 807, and in the reinforcements mesh—1140 elements.

Table 1. Material parameters of steel and concrete used in the tests.

Material 𝐄 𝐆𝐏𝐚 𝛎 − 𝛒 𝐤𝐠/𝐦𝟑 concrete 30 0.2 2400 steel 210 0.3 7900

The total number of nodes in Model 1 equals 247,590 which gives in total 742,770 degrees of freedom (since all the elements have three degrees of freedom in each node). Model 2 consist of 147,894 nodes and thus has 443,682 degrees of freedom. These are rather sparse meshes, useful in this study, but if one would prefer to obtain more precise results, the number of nodes should be much higher, which in turn would cause much more demanding computations. In order to speed up computation but also to simplify the geometry for eventual variational study, a simpler model is required.

2.3. Representative Volume Element In order to perform numerical homogenization (details described further in the work), a proper definition of the stiffness matrix of a representative volume element (RVE) of a ceiling element is needed. RVE was created by extracting a periodic piece of structure from the entire 3D model. RVE for the Filigree ceiling slabs with D-type trusses was 0.70 m wide, 0.60 m long, and 0.06 m thick, see Figure 8a. For the correct definition of RVE, model geometry containing only one repetitive period is usually used. However, based on the experience gained, it was finally decided to use RVE with three periods. This is dictated by previous observations (see [21]), which show that for RVE with base dimensions deviating from the square, the transverse stiffness for the shorter side is overestimated. The choice of RVE with three periods has no effect on the remaining stiffnesses in the ABD matrix determined by this method. The slab was modelled as a solid 3D structure made of concrete. The model used repeatable D-type trusses (three periods) and reinforcement shown in Figure 8b,c. They were modelled as steel wire structures. In addition, the reinforcement mesh and the bottom bars of truss were embedded in the concrete slab. The Materials 2021, 14, 4120 slab was meshed with solid C3D20R elements, i.e., 20-node elements with8 three of 16 degrees of freedom, while the steel part was meshed with truss T3D2 elements (Figure 8d).

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(a) (b)

Figure 8. RVE-1—Type-D (a) geometry; (b) rebars; (c); spatial truss structure (d) FE mesh.

The RVE model for the Filigree ceiling slab with EQ-type trusses consisted of a plate with dimensions of 0.50 × 0.60 × 0.05 m, EQ-type truss, and slab reinforcement, see Figure 9. The steel trusses and bars were modelled as a wire structure and then meshed with T3D2(c )type elements. The concrete part of the model (wasd) a 3D solid and was divided into FE using C3D20R elements. Figure 8.Figure RVE-1—Type-D 8. RVE-1—Type-D (a) (geometry;a) geometry; ( (bb) rebars;rebars; (c ();c spatial); spatial truss truss structure structure (d) FE mesh.(d) FE mesh.

(a) (b)

FigureFigure 9. RVE 9. RVE 2—Type-EQ 2—Type-EQ (a (a) )geometry; geometry; ((bb)) rebars;rebars; (c ()c spatial) spatial truss truss structure; structure; (d) mesh.(d) mesh.

2.4. Homogenization Algorithm The classical formulation of a displacement-based linear finite element method includes

𝐊 𝐮 =𝐅, (1)

where(c) 𝐊 is a statically condensed global stiffness (d)matrix of the RVE; 𝐮 is a displacement vector of external nodes, and 𝐅 is a vector of the nodal force applied to external Figure 9. RVEnodes. 2—Type-EQ The FE ( amesh) geometry; and external (b) rebars; nodes (c )are spatial visualized truss structure; in Figure ( d10.) mesh. Static condensation is the elimination of unknown degrees of freedom (DOF), also 2.4.called Homogenization secondary or Algorithm minor degrees of freedom, here in relation to internal nodes, and re- formulationThe classical of the formulation stiffness matrix of a with displaceme fewer unknownsnt-based reducedlinear finite to the element degrees ofmethod free- in- cludesdom in external nodes (referred to as the primary unknown or principal DOF).

𝐊 𝐮 =𝐅, (1)

where 𝐊 is a statically condensed global stiffness matrix of the RVE; 𝐮 is a displacement vector of external nodes, and 𝐅 is a vector of the nodal force applied to external nodes. The FE mesh and external nodes are visualized in Figure 10. Static condensation is the elimination of unknown degrees of freedom (DOF), also called secondary or minor degrees of freedom, here in relation to internal nodes, and re- formulation of the stiffness matrix with fewer unknowns reduced to the degrees of freedom in external nodes (referred to as the primary unknown or principal DOF).

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(a) (b)

FigureFigure 10. 10. RVE—externalRVE—external (in (in red red colour) colour) and and internal internal nodes nodes (a ()a )Type-D; Type-D; (b (b) )Type-EQ. Type-EQ.

TheStatic stiffness condensation matrix condensed is the elimination to external of unknownnodes (principal degrees DOFs) of freedom can be (DOF),calculated also usingcalled the secondary following orequation: minor degrees of freedom, here in relation to internal nodes, and re-formulation of the stiffness matrix with fewer unknowns reduced to the degrees of 𝐊 = 𝐊 − 𝐊 𝐊𝐊 (2) freedom in external nodes (referred to as the primary unknown or principal DOF). whereThe the stiffnessglobal stiffness matrix condensedmatrix is partitioned to external nodesinto external (principal (subscript DOFs) can𝑒) and be calculated internal (subscriptusing the 𝑖 following) nodes on equation: four subarrays as follows:

𝐊 𝐊 𝐮 −1 𝐅 K e = Kee −Kei =Kii Kie (3)(2) 𝐊 𝐊 𝐮 𝟎 whereThe thetotal global elastic stiffness strain energy matrix stored is partitioned in the system into after external static(subscript condensatione) and is reduced internal to(subscript the work iof) nodes external on forces four subarrays on the corresponding as follows: displacements and amounts to 1 Kee Kei ue Fe 𝐸= 𝐮 𝐅 = (4)(3) Kie Kii 2ui 0 The energy equivalence between the simplified shell model and a full 3D finite element modelThe total can elastic be presented strain energy by a proper stored defin in theition system of displacements after static condensation at external isnodes reduced of to the work of external forces on the corresponding displacements and amounts to RVE to trigger both membrane and bending behaviour. The generalized displacements at each node on the RVE edge surfaces are rela1ted to generalized strains. Originally pro- E = uT F (4) posed by Biancolini [20], later extended by Garbowski2 e e and Gajewski [21], the homogenization method was modified here to include only the translational degrees of freedom in both typesThe energy of finite equivalence elements: betweentruss and the continuum. simplified shellTherefore, model the and relationship a full 3D finite between element generalizedmodel can beconstant presented strains by aand proper the position definition of ofexternal displacements nodes on at the external RVE nodesboundary of RVE is expressedto trigger by both the membrane following transformation: and bending behaviour. The generalized displacements at each node on the RVE edge surfaces are related to generalized strains. Originally proposed by Biancolini [20], later extended by Garbowski𝐮 =𝐀 𝛜 and, Gajewski [21], the homogenization(5) method was modified here to include only the translational degrees of freedom in both where 𝐮 is a displacement vector, 𝛜 is a strain vector. Here, for a single node (𝑥 =𝑥, types of finite elements: truss and continuum. Therefore, the relationship between general- 𝑦 =𝑦, 𝑧 =𝑧), the 𝐀 matrix, adopted for a solid truss RVE model (after [20,21]), can be derived:ized constant strains and the position of external nodes on the RVE boundary is expressed by the following transformation: 𝜀 ui = Ai i,⎡ ⎤ (5) 𝜀 ⎢ ⎥ u 𝛾x = x y = y where is a𝑢 displacement𝑥0𝑦2 vector,⁄ 𝑧2⁄is a strain0𝑥𝑧0 vector. Here, for a single⁄ 𝑦𝑧2 node ( i , i , ⎢𝛾 ⎥ zi = z), the A𝑢 i matrix, adopted for a solid truss RVE model (after [20,21]),⎢ can ⎥ be derived: =0𝑦𝑥2⁄ 0𝑧2⁄ 0𝑦𝑧𝑥𝑧2⁄ 𝛾 . (6) ⎢ ⎥ 𝑢 00 0 𝑥2⁄ 𝑦2⁄ −𝑥 ⁄2 −𝑦 ⁄2 −𝑥𝑦⁄ 2 ⎢𝜅ε⎥x  𝜅 ⎢ ε⎥y  ⎣𝜅 ⎦      γxy  ux x 0 y/2 z/2 0 xz 0 yz/2    γxz  The aboveuy  matrix=  0 𝐀y defines x/2 the 0 relationshipz/2 0 betweenyz displacements xz/2  and effective . (6)  γ  strains, whichu are applied0 0 to the 0 nodesx/2 ony/2 the boundary−x2/2 − conditionsy2/2 −xy (red/2 points yz in Figure z i  κ  10) to which the stiffness of the full model is condensed.  x   κ  Recalling the definition of the elastic strain energy for a discrete model: y κxy i

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The above matrix Ai deﬁnes the relationship between displacements and effective strains, which are applied to the nodes on the boundary conditions (red points in Figure 10) to which the stiffness of the full model is condensed. Recalling the deﬁnition of the elastic strain energy for a discrete model:

1 1 E = uT K u = T AT KA (7) 2 e e 2 e e e e and considering that for a shell (or plate), ﬁnite element subjected to bending, tension, and transverse shear, the elastic internal energy is

1 E = T A {area} (8) 2 e k e thus, the stiffness matrix for a homogenized composite can be easily extracted from a discrete matrix as AT KA A = e e . (9) k area The presented homogenization method adopted to truss-and-solid to shell transformation is based on the equivalence between the full 3D FE model and the simplified flat shell model by appropriate calculation of its effective stiffness. This allows for a significant acceleration of numerical analyses while maintaining a very high accuracy of the effective model. In general, the Ak matrix (which is the ABDR matrix) can be directly used in calculations by, e.g., general shell stiffness definition. However, if it is required to determine the effective material constants describing the orthotropic linear elasticity, first the effective thickness must be computed with the following equation: s trace(D∗) t = 12 , (10) trace(A)

where D∗ = D − BA−1B, (11) Equation (11) can be easily suppressed by choosing the position of the neutral axis that minimizes the value of the B matrix. In such a case, D∗ = D The corresponding respective submatrices used in Equation (11) are       D11 D12 0 B11 B12 0 A11 A12 0 D =  D12 D22 0 ; B =  B12 B22 0 ; A =  A12 A22 0 . (12) 0 0 D33 0 0 B33 0 0 A33

In order to compute the effective parameters in the orthotropic model (represented by the four in plane material constants), the following decomposition of the D matrix (which describes bending stiffness) can be used:

12 12 12 D ED = ; ED = ; GD = ; νD = 12 . (13) 11 3 −1 22 3 −1 12 3 −1 12 D t D11 t D22 t D33 22 The same can be done with the A matrix (which describes the membrane stiffness):

12 12 12 A EA = ; EA = ; GA = ; νA = 12 . (14) 11 −1 22 −1 12 −1 12 A tA11 tA22 tA33 22 Materials 2021, 14, 4120 11 of 16 Materials 2021, 14, 4120 11 of 16

However, the most accurate results can be obtained by using average values (this is particularly important if the analysed model𝑅 is expected𝑅 to work in a complex membrane 𝐺 = ; 𝐺 = , (16) and bending state), which is calculated as𝑡 follows: 𝑡

where 𝑅 and 𝑅 are the transverse shear stiffnesses from the ABD matrix: ED + EA ED + EA GD + GA νD + νA = 11 11 = 22 2 = 12 12 = 12 12 Materials 2021, 14, 4120 E11 ; E22 ; 𝑅G12 0 ; ν12 .11 of 16 (15) 2 2 𝐑= . 2 2 (17) 0𝑅 The effective transverse shear can be computed directly from the R matrix using the 3.following Results equations: R𝑅 𝑅R G𝐺 == 11 ;; 𝐺G == 22, ,(16) (16) Below are the results of the numerical13 t𝑡 analysis23 presented𝑡t for full 3D numerical models, Type-D (see Figure 11) and Type-EQ (see Figure 12). Figure 11 shows the side view of wherewhere 𝑅R andand 𝑅R are are thethe transversetransverse shear stiffnesses from the the ABD ABD matrix: matrix: the slab loaded11 and22 displaced. The maximum displacement read from the numerical model equals 36.63 mm. 𝑅 0 𝐑=R11 0 . (17) R = 0𝑅 . (17) 0 R22

3.3. Results Results BelowBelow are are the the results results of of the the numerical numerical anal analysisysis presented presented for for full full 3D 3D numerical numerical mod- models, els,Type-D Type-D (see (see Figure Figure 11 )11) and and Type-EQ Type-EQ (see (see Figure Figure 12 12).). Figure Figure 11 11 shows shows the the side side view view of of the theslab slab loaded loaded and and displaced. displaced. The The maximum maximum displacement displacement read read from from the numericalthe numerical model modelequals equals 36.63 mm.36.63 mm.

Figure 11. Vertical displacements of Model-1 (Type-D).

Figure 12 shows the vertical displacements of the numerical full 3D model of the slab (Type-EQ). The maximal deflection is in the centre of the plate and equals 38.15 mm. Both deflections are rather safe for the assembly of ceilings (in terms of failure); however, the development of models and their computation took significant time. If it is necessary to model a more complicated shape of the slab, the reconstruction of the full 3D geometry after each change seems unreasonable and it is certainly too time-consuming. FigureFigure 11. 11. VerticalVertical displacements displacements of of Model-1 Model-1 (Type-D). (Type-D).

FigureFigure 12. 12. VerticalVertical displacements displacements of of Model-2 Model-2 (Type-EQ). (Type-EQ).

InFigure contrast, 12 shows the simplified the vertical single-layered, displacements homogenized of the numerical shell model full 3D of modelthe slab of is the very slab easy(Type-EQ). to rebuild. The In maximal this model, deflection the most is in thedem centreanding of work the plate is done and prior equals to 38.15 geometrical mm. Both modelling,deflections namely, are rather during safe forthe thehomogenization assembly of ceilingsprocess, (in which, terms however, of failure); is however,only done the oncedevelopment for all. The of stiffness models andmatrix their ABD, computation taking only took the significantsymmetrical time. part If into it is account, necessary is to model a more complicated shape of the slab, the reconstruction of the full 3D geometry presented in Table 2. The table summarizes all three stiffness submatrices, namely, A, B , after each change seems unreasonable and it is certainly too time-consuming. Figure 12. Vertical displacements of Model-2 (Type-EQ).

In contrast, the simplified single-layered, homogenized shell model of the slab is very easy to rebuild. In this model, the most demanding work is done prior to geometrical modelling, namely, during the homogenization process, which, however, is only done once for all. The stiffness matrix ABD, taking only the symmetrical part into account, is presented in Table 2. The table summarizes all three stiffness submatrices, namely, A, B,

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In contrast, the simpliﬁed single-layered, homogenized shell model of the slab is very easy to rebuild. In this model, the most demanding work is done prior to geometrical modelling, namely, during the homogenization process, which, however, is only done once for all. The stiffness matrix ABD, taking only the symmetrical part into account, is presented in Table2. The table summarizes all three stiffness submatrices, namely, A, B, and D (shown in Equation (12)), as well as the transverse shear stiffness matrix R presented in Equation (17) for both RVE models. The model is referred to as the ABD Model hereafter in the results.

Table 2. ABD stiffness matrix for RVE-1 and RVE-2 models.

Stiffness RVE-1 Model ABD RVE-2 Model ABD A11 (105 MPa mm) 19.11 15.99 A12 (105 MPa mm) 3.83 3.19 A22 (105 MPa mm) 21.46 17.73 A33 (105 MPa mm) 7.50 6.25 B11 (105 MPa mm2) 0 0 B12 (105 MPa mm2) 0 0 B22 (105 MPa mm2) 56.93 20.07 B33 (105 MPa mm2) 0.04 0.04 D11 (105 MPa mm3) 5658.9 3281.1 D12 (105 MPa mm3) 1162.2 676.7 D22 (105 MPa mm3) 15,319.5 6828.7 D33 (105 MPa mm3) 2293.9 1349.7 R44 (105 MPa mm) 0.67 0.53 R55 (105 MPa mm) 0.69 0.59

The effective thickness and all effective constitutive orthotropic constants computed from Equations (10)–(17) are presented in Table3 for the RVE-1 model (Type-D), while the effective thickness and all effective material constants computed from Equations (10)–(17) for RVE-2 model (Type-EQ) are summarized in Table4.

Table 3. Material properties computed after homogenization of the RVE-1 model. Three different sets of parameters using Equations (13), (14) or (15).

Material Par. Shell Model A Shell Model D Shell Model AD t (mm) 75.97 75.97 75.97 E11 (MPa) 24,259 15,245 19,752 E12 (MPa) 27,234 40,849 34,042 ν12 (−) 0.18 0.08 0.13 G12 (MPa) 9869 6278 8074 G13 (MPa) 883 883 883 G23 (MPa) 907 907 907

Table 4. Material properties computed after homogenization of the RVE-2 model. Three different sets of parameters using Equations (13), (14) or (15).

Material Par. Shell Model A Shell Model D Shell Model AD t (mm) 58.57 58.57 58.57 E11 (MPa) 26,315 19,175 22,745 E12 (MPa) 29,190 39,793 34,492 ν12 (−) 0.18 0.10 0.139 G12 (MPa) 10,678 8060 9369 G13 (MPa) 895 895 895 G23 (MPa) 1003 1003 1003 Materials 2021, 14, 4120 13 of 16

the mid-span vertical deflection of the full 3D FE model and the corresponding simplified shell model (RVE-1, Type-D) is shown in Table 5. Figure 14 shows a qualitative comparison of the deflected shape of the full 3D FE model of the Filigree slab (Type-EQ) and its simplified version (RVE-2). Table 6 summarizes the vertical displacement at the mid-span of the slab, calculated using the full 3D numerical model and the simplified model with three different cross-section definitions and material models (columns 3–5). Materials 2021, 14, 4120 13 of 16 Table 5. The deflection at the mid-span computed using the full 3D model (Type-D) and three simplified shell models (based on RVE-1 homogenization FE model).

Full 3D FE Shell Model Shell Model Shell Model A qualitative comparison of the deflected shape of the full 3D FE model and its Model ABD 1 AD D simplified version (RVE-1, Type-D) is shown in Figure 13, while a quantitative comparison Deflection at the centre (mm) 36.63 35.70 42.97 36.13 of the mid-span vertical deflection of the full 3D FE model and the corresponding simplified 1 shell model (RVE-1, This Type-D)result were is obtained shown inby Tabledirect 5use. of the ABD matrix.

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FigureFigure 13. 13. ComparisonComparison ofTable vertical of vertical5. The deflections: deflection deﬂections: at ( athe) RVE-1 mid-span (a) RVE-1shell computed model shell usin ABD; modelg the (b full) ABD; full 3D 3D model (b FE) full (Type-D)model 3D FE(Type-D). and model three simplified shell models (based on RVE-1 homogenization FE model). (Type-D). Full 3D FE Shell Model Shell Model Shell Model

Model ABD 1 AD D Table 5. The deﬂectionDeflection at the mid-span at the centre computed (mm) using 36.63 the full 3D 35.70 model (Type-D) 42.97 and 36.13 three 1 simpliﬁed shell models This (based result on were RVE-1 obtained homogenization by direct use of the FE ABD model). matrix.

Full 3D FE Shell Model Shell Model Shell Model Model ABD 1 AD D Deﬂection at the centre (mm) 36.63 35.70 42.97 36.13 1 This result were obtained by direct use of the ABD matrix.

Figure 14 shows a qualitative comparison of the deflected shape of the full 3D FE model of the Filigree slab (Type-EQ) and its simplified version (RVE-2). Table6 summarizes (a) (b) the vertical displacement at the mid-span of the slab, calculated using the full 3D numerical Figuremodel 14. Comparison and the simplified of vertical(a) model deflections: with ( threea) RVE-2 different shell model cross-section ABD;(b) (b) full definitions 3D FE model and (Type-EQ). material modelsFigure (columns 13. Comparison 3–5). of vertical deflections: (a) RVE-1 shell model ABD; (b) full 3D FE model (Type-D). Table 6. The deflection at the mid-span computed using the full 3D (Type-EQ) model and three simplified shell models (based on RVE-2 homogenization FE model).

Full 3D FE Shell Model Shell Model Shell Model

Model ABD 1 AD D Deflection at the centre (mm) 38.15 38.32 44.24 38.37 1 This result were obtained by direct use of ABD matrix.

4. Discussion The results presented in the previous section concern two models of Filigree type floor concrete prefabricated plates, namely, Type-D and Type-EQ, which differ in the (a) (b)

FigureFigure 14. Comparison 14. Comparison ofof vertical vertical deflections: deﬂections: (a) RVE-2 (a) shell RVE-2 model shell ABD; model (b) full ABD;3D FE model (b) full (Type-EQ). 3D FE model (Type-EQ). Table 6. The deflection at the mid-span computed using the full 3D (Type-EQ) model and three simplified shell models (based on RVE-2 homogenization FE model).

Table 6. The deﬂection at the mid-span computed usingFull 3D the FE full Shell 3D (Type-EQ)Model Shell model Model and Shell three Model

simpliﬁed shell models (based on RVE-2 homogenizationModel FE model).ABD 1 AD D Deflection at the centre (mm) 38.15 38.32 44.24 38.37 1 This result wereFull obtained 3D FE by directShell use ofModel ABD matrix. Shell Model Shell Model Model ABD 1 AD D Deﬂection at the centre4. (mm)Discussion 38.15 38.32 44.24 38.37 1 This result were obtained byThe direct results use of presented ABD matrix. in the previous section concern two models of Filigree type floor concrete prefabricated plates, namely, Type-D and Type-EQ, which differ in the

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4. Discussion The results presented in the previous section concern two models of Filigree type floor concrete prefabricated plates, namely, Type-D and Type-EQ, which differ in the thickness of the concrete slab and the geometry of the truss and the diameters of some bars. Both cases were analysed as simply supported plates with their own weight. In both cases, the maximum deflection did not exceed 40 mm. The results obtained from the homogenization of relatively small periodic (repeatable) fragments of the plate, the so-called representative volumetric elements (RVE), are presented in Table2. In both cases (RVE-1 and RVE-2), the membrane and bending stiffness in direction 2, which is parallel to the plate length, are much greater than in direction 1 (perpendicular to the L direction in plane). The ratio between the stiffnesses in the 2 and 1 direction for RVE-1 exceeded 3 and for RVE-2 was approximately 2.5. This is due to the peculiarities of the arrangement of spatial trusses that are oriented in the second direction. RVE-1 generally had much higher stiffness values than RVE-2, due to the slightly thicker concrete slab (60 mm vs. 50 mm), the lower truss structure, and greater main bar diameters (φ = 12 mm for RVE-1 and φ = 6 mm for RVE-2). It is worth noting that due to the coordinate system adopted in the homogenization process, the centre of which coincides with the centre of gravity of the concrete slab (in both analysed cases), the matrix B was practically zero. The calculated material constants (summarized in Tables3 and4) present an orthotropic elastic material. Orthotropy is the result of the homogenization process, which takes into account both the stiffness of the concrete slab (isotropic material) with the reinforcement in the form of a steel mesh and spatial steel trusses (again, isotropic material). The specific arrangement of the trusses and the number of rebars naturally generate direction- dependent behaviour for both compression/tension and bending/torsion stiffness, yet the orthotropy in both RVE-1 and RVE-2 was more apparent when the material parameters were calculated using the bending stiffness D than when calculating the membrane stiffness A. The results (in Figures 13 and 14 as well as in Tables5 and6) show that virtually all three simplified models provided a precise displacement solution. This means that using the ABD stiffness directly (without defining the plate thickness or a constitutive model) in the numerical model as well as using models with the calculated material constants and effective thickness can be successfully used for quick engineering calculations. The main limitation of the proposed method, particularly when calculations use the ABD matrix directly, lies in the possibility of correct calculation restricted to displacements only. Models using the determined effective material parameters and effective thickness allow the analysis of stresses and strains as well; however, for an effective model, these will be effective values. In further study, an extension to inelastic analysis of Filigree slabs is planned based on the general nonlinear constitutive law [37], already developed for 2D steel frames [38], composite beams [39], and corrugated sheets [40]. In this approach, plastic deformation can be captured by progressive degradation of stiffness.

5. Conclusions The paper presents the numerical solid truss to shell homogenization method, based on the principle of strain equivalence. As a working example, an analysis of prefabricated floor slabs was carried out. The selected two types of filigree floor slabs differ in the spatial type of the truss. In order to verify the effectiveness of the proposed homogenization method, the full 3D models of the analysed slabs were built. Homogenized simplified models, both qualitatively and quantitatively, fully reflect the elastic behaviour of their full 3D FE models. The calculated displacements are more than in satisfactory agreement with the displacements calculated by solid truss FE numerical models. Thus, the method can be effectively used for quick and engineering calculations of composite plates with spatial Materials 2021, 14, 4120 15 of 16

trusses using simpliﬁed models, assuming that the correct homogenization of the selected plate was performed earlier.

Author Contributions: Conceptualization, T.G.; methodology, T.G.; software, T.G. and N.S.; vali- dation, T.G. and N.S.; formal analysis, N.S.; investigation, N.S. and A.S.-G.; resources, N.S.; data curation, N.S.; writing—original draft preparation, T.G. and N.S.; writing—review and editing, A.S.-G.; visualization, N.S.; supervision, T.G.; project administration, A.S.-G.; funding acquisition, A.S.-G. All authors have read and agreed to the published version of the manuscript. Funding: The APC was funded within the framework of Ministry of Science and Higher Education program as ‘’Regional Initiative Excellence” in years 2019–2022, Project No. 005/RID/2018/19. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors thank Femat Sp. z o. o. for providing the commercial software. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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