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Distribution of Concentrated Loads in Timber-Concrete Composite Floors: Simplified Approach

Distribution of Concentrated Loads in Timber-Concrete Composite Floors: Simplified Approach buildings Article Distribution of Concentrated Loads in Timber-Concrete Composite Floors: Simplified Approach 1 , 1 2 Sandra Monteiro * , Alfredo Dias and Sérgio Lopes The Institute for Sustainability and Innovation in Structural Engineering, Civil Engineering Department, University of Coimbra, Rua Luís Reis Santos—Pólo II, 3030-788 Coimbra, Portugal; alfgdias@dec.uc.pt Centre for Mechanical Engineering, Materials and Processes, Civil Engineering Department, University of Coimbra, Rua Luís Reis Santos—Pólo II, 3030-788 Coimbra, Portugal; sergio@dec.uc.pt * Correspondence: sandra@dec.uc.pt Received: 30 December 2019; Accepted: 13 February 2020; Published: 18 February 2020 Abstract: Timber-concrete composite (TCC) solutions are not a novelty. They were scientifically referred to at the beginning of the 20th century and they have proven their value in recent decades. Regarding a TCC floor at the design stage, there are some assumptions, at the standard level, concerning the action of concentrated loads which may be far from reality, specifically those associating the entire load to the beam over which it is applied. This naturally oversizes the beam and a ects how the load is distributed transversally, a ecting the TCC solution economically and mechanically. E orts have been made to clarify how concentrated loads are distributed, in the transverse direction, on TCC floors. Real-scale floor specimens were produced and tested subjected to concentrated (point and line) loads. Moreover, a Finite Element (FE)-based model was developed and validated and the results were collected. These results show that the “loaded beam” can receive less than 50% of the concentrated point load (when concerning the inner beams of a medium-span floor, 4.00 m). Aiming at reproducing these findings on the design of these floors, a simplified equation to predict the percentage of load received by each beam as a function of the floor span, the transversal position of the beam, and the thickness of the concrete layer was suggested. Keywords: TCC floors; transverse distribution of load; concentrated loads; simplified approach 1. Introduction Since its first scientific reference in the early decades of the last century [1], the use of a composite solution gathering timber beams with a thin concrete layer through an ecient connection has been spreading either for new or rehabilitation applications, on building floors or on bridge decks [2]. Timber-concrete composite (TCC) solutions may be as versatile as needed, by using di erent materials: di erent concrete strengths or densities, di erent timber species or engineered products, and di erent connection systems; or di erent sections (thicknesses and shape) [3–8]. They were initially developed with the aim of rehabilitating or strengthening timber floors [9]. However, in some cases of heritage cultural value buildings, their use may be overlooked by their insucient reversibility [10] or for being a non-dry technique. In fact, there are cases where TCC solutions were preferred relatively to other rehabilitation techniques and were also recognized as prize-worthy [11–13]. The rehabilitation of a building floor may be a consequence of physical or biological damages, lack of strength for the associated use (actual or new one), among others. Regardless of the motivation, there are common types of loading, such as furniture (point loads) or partition walls (line loads) that must be considered at the design stage. Concerning a timber-concrete solution, beside a document that is being prepared [14], there are no current standardization or code rules for the design for such composites. Annex B of the Eurocode 5 [15] is commonly used to perform the design computation. Buildings 2020, 10, 32; doi:10.3390/buildings10020032 www.mdpi.com/journal/buildings Buildings 2020, 10, 32 2 of 12 This computation considers the association of the entire load, point or line load aligned with the length of the timber beam, with the beam under consideration, but it can be far from the real behavior. In recent years, a few studies [16–21] aiming to understand how the load is distributed in the transverse direction were performed in this field. Parameters that might a ect that distribution were investigated. The work developed by the authors [16–18] proves that the share of load received by the loaded beam could be, in some cases, less than half. It is easy to understand the economic implications that an overestimated cross-section may have, associated with the unnecessary waste of material. Furthermore, there are also consequences at the mechanical behavior level. The thicker the concrete slab (using the same timber cross-section), the higher the transverse distribution of load. The opposite occurs with the increase of the timber beam height (keeping the concrete thickness unchanged), but with less expression [22], hence the importance of such studies. This paper aimed to present a simplified approach to be applied at a design stage in order to help to obtain an optimized TCC floor solution in terms of mechanical behavior and expenses. Therefore, an experimental set of results obtained from real-scale TCC floors tested under concentrated loads, together with the results of a parametric study using a Finite Element Method (FEM) model developed and validated by the authors was proposed. 2. Parametric Study To achieve the set goal, a comprehensive parametric study was developed. Aiming at studying the mechanical behavior of medium span TCC floors (4.00 m), a Base Simulation (BS) was established. The BS composite slab has a square plan and is composed of a 0.07 m-thick concrete layer, seven timber beams 0.60 m apart from each other and a 0.02 m-thick timber interlayer. Its material and geometric characteristics can be found in Monteiro et al. [23]. To perform such a study, several parameters (Table 1) were chosen and their e ect on the load distribution of TCC floors was analyzed. Only the loading of four beams, B1 (end beam) to B4 (central beam) was considered due to symmetry (for detailed information, see Monteiro et al. [23]). The analysis was accomplished by evaluating the percentage of support reaction (sr) received by each beam for the various loading cases: each beam loaded at 1 1 a time, with a point load at /2 span or /4 span, or line load. Moreover, the distribution of vertical displacements (vd) at mid-span and the distribution of longitudinal bending moment at mid-span (bm) were analyzed. Figure 1 summarizes the percentages of load received by the end beam (B1 or B7) and the central beam when loaded, at a time, associated with BS, in terms of the analyzed quantities. Regardless of the loading case or location, the loaded beam does not receive the entire load but a share of it, as well as the unloaded beams, which emphasizes the existence of load distribution. The share of load received by the loaded beam will be higher or lower the farther or nearer that beam is from the center of the slab, respectively. Beam B1 is the one associated with the highest share of load when loaded (more than 80% for sr). In addition, when considering the extreme load locations, B1 vs. B4, the maximum deviation of received share is associated with this beam location (di erence between the percentage of load associated with B1 when the load is applied at B1 and the percentage of load associated with B1 when the load is applied at B4 is about 90% for sr, for the three loading cases under consideration). On the contrary, intermediate beams show smaller deviations, with B3 presenting the minimum deviation (di erence between the percentage of load associated with B3 when the load is applied at B1 and the percentage of load associated with B3 when the load is applied at B4 is about 20% for vd and bm, 1 1 for /2 Pt and Ln, reaching less than 5% for /4 Pt). Concerning the studied parameters, it was found that they a ect in di erent amounts the quantities analyzed. The ones with a greater e ect (more than 10% of deviation) are displayed in Tables 2–4. The deviation was computed between two modeling tasks associated with a parameter. The Details column specifies among which “parameter values” was the maximum deviation obtained (e.g., a maximum deviation of 75% was found among BS with two di erent support conditions (Ss vs. Sae) when loaded with a point load at mid-span of B1). Buildings 2020, 10, 32 3 of 12 Table 1. Parameters studied. Type Parameter Symbol Span L Slab Width b slab Beam spacing s Geometrical Concrete h Thickness Interlayer h Cross-section Timber h Shape Beam , I, # Strength class According to EC2 [24] Concrete Normal-weight NWAC Aggregates Light-weigh LWAC Material Strength class According to EN 338 [25] Solid Timber Product Wood-engineered GL, LVL, OSB + LVL, CLT Low lK Connection sti ness Medium mK High hK Mechanical behavior Linear Elastic LE Material Non-linear Elastic Perfectly-plastic EPP Simply supported Ss Beams’ ends Support conditions, Sc Fixed Fx All ends Simply supported Sae Mid-span /2 Pt Point load Type Quarter-span /4 Pt Loading Linear load Ln Location On each beam at a time B1, B2, : : : , B7 Domestic and residential activities A According to EC1 [26] Floor use Areas where people congregate, C4 with possible physical activities. Undersized Un Degree of oversizing, DO Timber cross-section EC5 “tight-fit” According to EC5 [15] EC5 Oversized Ov With: GL—Glued laminated timber; LVL—Laminated veneer lumber; OSB—Oriented strand board; CLT—Cross-laminated timber. Table 2. Parameters with the greatest e ect on the load distribution referring to BS. Maximum Load Beam Quantity Parameter Details Deviation [%] Type 75 B1 /2 Pt sr Simply supported on beams’ ends, Support conditions, Sc Ss vs. Simply supported on all ends, Sae 36 B1 Ln bm B1 Ln vd B2 /2 Pt sr Concrete thickness, h 0.02 m vs. 0.07 m (BS) 31 c B1 Ln bm 28 B2 Ln sr Span, L 4.00 m (BS) vs. 16.00 m * 27 B1 /2 Pt sr Beams vs. deck BS vs. CLT deck 26 B1 /2 Pt vd Span, L 4.00 m (BS) vs. 16.00 m * 24 B1 /2 Pt bm 17 B1 Ln vd Concrete strength LC16/18 vs. C25/30 (BS) 16 B1 Ln bm B4 /2 Pt bm Beams vs. deck BS vs. CLT deck B2 Ln sr Concrete strength LC16/18 vs. C25/30 (BS) 10 B1 /2 Pt vd Support conditions, Sc Simply supported, Ss vs. Fixed, Fx *—underestimated timber section. Buildings 2020, 10, 32 4 of 12 Buildings 2020, 10, x FOR PEER REVIEW 4 of 12 0 B11234 B2 B3 B4 B5 567 B6 B7 8 -20% 0% 20% 100%/7=14% 40% vd B1 ½Pt vd B1 ¼Pt 60% vd B1 Ln vd B4 ½Pt 80% vd B4 ¼Pt vd B4 Ln 100% (a) 012 B1 B2 B3 B4 B5 B6 34567 B7 8 -20% 0% 20% 100%/7=14% 40% sr B1 ½Pt sr B1 ¼Pt 60% sr B1 Ln sr B4 ½Pt 80% sr B4 ¼Pt sr B4 Ln 100% (b) B1 B2 B3 B4 B5 B6 B7 012 3456 7 8 -20% 0% 100%/7=14% 20% 40% bm B1 ½Pt bm B1 ¼Pt 60% bm B1 Ln bm B4 ½Pt 80% bm B4 ¼Pt bm B4 Ln 100% (c) Figure 1. Percentage of load received by each beam when B1 or B4 is loaded, in terms of (a) vd; (b) sr; Figure 1. Percentage of load received by each beam when B1 or B4 is loaded, in terms of (a) vd; (b) sr; and (c) bm and. (c) bm. Although the DO has shown a great effect, both Ov and Un series, varying between 31% (sr) and 42% (vd) when considering spans of 4.00 m (the same as BS) and 16.00 m (reaching deviations of 66% and 58% when considering the extreme spans [2.00 m; 16.00 m]), the percentages found are only indicative of the trend. In contrast, for the EC5 series, the sections found were established based on an objective criterion: the design based on Annex B of EC5 [15], aiming at maximizing the section strength utilization ratio, for Un and Ov series that did not occur. Although a common procedure has been established, by changing only the timber height, no uniform percentage of over or under sizing was defined (for extra detail see Monteiro et al. [23]). Buildings 2020, 10, 32 5 of 12 Table 3. Parameters with the greatest e ect on the load distribution, referring to the boundaries of each parameter. Maximum Load Beam Quantity Parameter Details Deviation [%] Type 70 B1 /2 Pt sr CLT and Sc CLT + Ss vs. CLT + Sae 58 - /2 Pt sr Load location B1 vs. B4 for BS with lK connection 47 B2 Ln sr Span, L 2.00 m (BS) vs. 16.00 m* B1 /2 Pt vd 45 Concrete thickness, h 0.02 m vs. 0.20 m B1 Ln bm 44 B1 Ln vd Span, L 2.00 m (BS) vs. 16.00 m* 43 B2 Ln sr h 0.02 m vs. 0.20 m 41 B1 Ln bm Span, L 2.00 m (BS) vs. 16.00 m* 32 B1 /2 Pt sr CLT and s BS + juxtaposed beams vs. CLT deck Ln vd 29 Load location B1 vs. B4 for BS with LC16/18 concrete Ln bm 26 B1 Ln vd CLT and Sc CLT + Ss vs. CLT + Sae 23 B1 /2 Pt bm 18 B1 Ln vd Concrete strength 17 B1 Ln bm LC16/18 vs. C40/50 B2 Ln sr B1 Ln vd CLT and s BS + juxtaposed beams vs. CLT deck 11 B1 Ln bm CLT and s *—underestimated timber section. Table 4. Maximum deviation of load distribution associated with the DO. Maximum Load Beam Quantity Parameter Details Deviation [%] Type 60 - /2 Pt sr Load location B1 vs. B4 for Un with L = 16.00 m 43 B1 Ln vd EC5 42 B1 Ln vd Ov 41 B1 Ln bm EC5 39 B4 Ln sr Un 1 L = 4.00 m vs. L = 16.00 m B1 /2 Pt vd Un B1 Ln bm Ov 37 B1 Ln bm Un 36 B2 Ln sr EC5 31 B2 Ln sr Ov vd Ln 30 Load location B1 vs. B4 for EC5 with L = 4.00 m bm Ln 66 B1 Ln vd Ov 64 B1 Ln bm Ov 58 B1 Ln vd Un B1 /2 Pt vd EC5 B1 Ln bm Un L = 2.00 m vs. L = 16.00 m B2 Ln sr Un B1 Ln bm EC5 B2 /2 Pt sr Ov 53 B2 /2 Pt sr EC5 From this analysis, it becomes clear the significant e ect of the following parameters: 1. The support conditions, with a maximum deviation of 75% between BS (Ss) and BS with Sae); 2. The degree of oversizing, with a maximum deviation of 66% (Ov), 58% (Un) and 56% (EC5), considering the limit spans 2.00 m and 16.00 m; Buildings 2020, 10, 32 6 of 12 3. The loading position, with a maximum deviation of 58% considering the loading applied at B1 vs. applied at B4 when the modeling BS with lK connection is considered (reaching 60% when in association with DO—verified for the modeling task with the same cross-section as BS and L = 16.00 m (underestimated timber section)); 4. The span length, with a maximum deviation of 47% between spans of 2.00 m and 16.00 m; 5. The concrete thickness, with a maximum deviation of 45% between thicknesses of 0.02 m and 0.20 m; 6. The existence of a timber deck underneath the concrete layer, instead of timber beams and interlayer using juxtaposed beams or a CLT deck, with a maximum deviation of 27%; and 7. The concrete strength, with a maximum deviation of 18% between an LWAC LC16/18 and an NWAC C40/50. Although the DO has shown a great e ect, both Ov and Un series, varying between 31% (sr) and 42% (vd) when considering spans of 4.00 m (the same as BS) and 16.00 m (reaching deviations of 66% and 58% when considering the extreme spans [2.00 m; 16.00 m]), the percentages found are only indicative of the trend. In contrast, for the EC5 series, the sections found were established based on an objective criterion: the design based on Annex B of EC5 [15], aiming at maximizing the section strength utilization ratio, for Un and Ov series that did not occur. Although a common procedure has been established, by changing only the timber height, no uniform percentage of over or under sizing was defined (for extra detail see Monteiro et al. [23]). The analysis of the previous tables shows that most of the parameters that have the greatest e ect on the load distribution are associated with the end beam B1 (or B7). This was verified in 71% of the cases when considering the variation relatively to BS; 78% of the cases, when considering the variation of a parameter (except DO) among its extreme values; and in 67% of the cases, when considering the variation on the DO, as for L = [4.00 m; 16.00 m], as for L = [2.00 m; 16.00 m]. This is due to the fact that end beams tend to concentrate the load applied over it (and thus, a lower percentage of distributed load) when compared with the remaining ones, allowing a higher variation than the central beam (B4), for instance, where the opposite happens, and a smaller range of variation can occur. With regard to the loading, although most of the listed parameters were associated with a linear loading, the di erences found relatively to a point load at mid-span were, at most, 4% (disregarding the Sc modeling task, which, due to a di erent structural system, were associated with greater di erences). 3. Simplified Approach The design stage is a crucial stage where the designer must be able to come up with an economical structural solution, preferably in the shortest time possible. Knowing the percentage of load received by a specific beam, before the floor being built, without the need to numerically modeling it, will surely contribute to it. Thus, based on the findings of the parametric study and aiming at providing a practical tool capable to predict the sought percentage, a simplified equation was developed. As shown above, three quantities were used for evaluating the load distribution, vd, sr and bm, but only one was used on the simplified model: the longitudinal bending moment, given its importance in the design process. For that, three essential parameters were considered: the span length, the concrete thickness and the transversal location of the beam. For this last consideration, a dimensionless parameter designated “beam location”, bl, was defined in order to provide the transversal position of the beam in question, relatively to the longitudinal axis of the outermost beam (B1) (Figure 2). The results collected in the parametric study, specifically, the percentage of load received by the loaded beam for the three parameters listed above were treated and gathered. Four sets of “continuous” curves were obtained, based on the design considerations, BS, un, EC5, and ov, for the considered loadings, gathering the results for B1 to B4, for each loading by span. Various polynomial approaches to the BS curves were tried in the approximation process: from a first-degree polynomial simple Equation Buildings 2020, 10, x FOR PEER REVIEW 6 of 12 37 B1 Ln bm Un 36 B2 Ln sr EC5 31 B2 Ln sr Ov vd Ln 30 - Load location B1 vs. B4 for EC5 with L = 4.00 m bm Ln 66 B1 Ln vd Ov 64 B1 Ln bm Ov 58 B1 Ln vd Un B1 ½ Pt vd EC5 B1 Ln bm Un L = 2.00 m vs. L = 16.00 m 56 B2 Ln sr Un B1 Ln bm EC5 B2 ½ Pt sr Ov 53 B2 ½ Pt sr EC5 The analysis of the previous tables shows that most of the parameters that have the greatest effect Buildings on the load dist 2020, 10, 32 ribution are associated with the end beam B1 (or B7). This was verified in 71% o 7fof th 12 e cases when considering the variation relatively to BS; 78% of the cases, when considering the variation of a parameter (except DO) among its extreme values; and in 67% of the cases, when considering the (1) to a fourth-degree polynomial simple Equation (3), considering also first (2) and second-degree variation on the DO, as for L = [4.00 m; 16.00 m], as for L = [2.00 m; 16.00 m]. This is due to the fact that polynomial equations with crossed terms). end beams tend to concentrate the load applied over it (and thus, a lower percentage of distributed load) when compared with the remaining ones, allowing a higher variation than the central beam (B4), z = a + a  x + a  x + a  x , (1) 0 1 1 2 2 3 3 for instance, where the opposite happens, and a smaller range of variation can occur. With regard to the loading, although most of the listed parameters were associated with a linear loading, the z = a + a  x + a  x + a  x + a  x  x + a  x  x + a  x  x , (2) 0 1 1 2 2 3 3 4 1 2 5 1 3 6 2 3 differences found relatively to a point load at mid-span were, at most, 4% (disregarding the Sc modeling 2 2 2 3 3 3 z = a + a  x + a  x + a  x + a  x + a  x + a  x + a  x + a  x + a  x + 0 1 1 2 2 3 3 4 1 5 2 6 3 7 1 8 2 9 3 task, which, due to a different structural system, were associated with greater differences). (3) 4 4 4 a  x +a  x + a  x , 10 1 11 2 12 3 3. Simplified Approach where x —span length; x —beam location; x —concrete thickness; and a , with i = 0 to 1 2 3 i 12—polynomial coecients. The design stage is a crucial stage where the designer must be able to come up with an The attempt to obtain the best approximation with the various polynomial equation was made by economical structural solution, preferably in the shortest time possible. Knowing the percentage of obtaining a set of polynomial coecients, according to the polynomial under consideration through load received by a specific beam, before the floor being built, without the need to numerically the minimization of the sum of the squared di erences between the numerical and the polynomial modeling it, will surely contribute to it. Thus, based on the findings of the parametric study and predictions. To measure the “strength of the approximation”, the determination coecient, R , (4) was aiming at providing a practical tool capable to predict the sought percentage, a simplified equation used, for which the strongest approximation corresponds to R = 1 and the weakest approximation to was developed. R = 0. Detailed information about the coecients obtained for the various sets and loading cases, As shown above, three quantities were used for evaluating the load distribution, vd, sr and bm, together with the corresponding R , can be found in Monteiro [27]. but only one was used on the simplified model: the longitudinal bending moment, given its importance in the design process. For tha Xt, three essential par X ameters were considered: the span 2 2 R = z Z / Z Z , (4) i i length, the concrete thickness and the transversal location of the beam. For this last consideration, a dimensionless parameter designated “beam location”, bl, was defined in order to provide the where R—correlation coecient, z —value given by the polynomial fit for the i point, location, transversal position of the beam in question, relatively to the longitudinal axis of the outermost beam Z—average of the values to approximate, Z —value to approximate for the i point, location. (B1) (Figure 2). B1 B2 B3 B4 b =1/3 b =2/3 b =0.00 b =1.00 l l Figure 2. Beam location parameter. Figure 2. Beam location parameter. Since the goal was to obtain an equation to predict the behavior of TCC floors under concentrated loads, the polynomial coecients obtained for the various attempts were analyzed aiming at finding common tendencies among them, for di erent loadings. Given that the BS set was the only one for 1 1 1 which three loading cases, /2 Pt, /4 Pt, and Ln, were modeled (for the remaining sets only /2 Pt and Ln were modeled); this was the chosen set to perform that comparison. The polynomial coecients for all attempts for BS were compared with each other and among the various loading cases ( /2 Pt vs. Ln; 1 1 1 /4 Pt vs. /2 Pt; /4 Pt vs. Ln). This analysis evidenced similar coecients for comparable polynomial attempts and among those the one with the best approximation was identified: the second-degree polynomial simple equation. The polynomial coecients of the sought equation, designated Pr (since it intends to predict the percentage of load received by the loaded beam), were defined as the computed average polynomial coecients found for the three loading cases (5). Figure 3 shows its course as a function of the floor span. As the figure depicts, some di erences can be found between the predicted and the numerical percentages, with Pr approaching the Ln load case curve more than the other curves. In general, it tends to underestimate the percentage of load associated with /2 Pt, but it tends to overestimate the same quantities concerning /4 Pt and Ln. Although for both point loadings, the greatest deviation is about 20% (associated, essentially, with the thinner concrete layers), the mean Buildings 2020, 10, 32 8 of 12 1 1 di erences were rather lower: about 11% for /2 Pt and + 8% /4 Pt. For Ln the deviations are lower than the point loadings, ranging between 10% and +15%, with a mean di erence of +4%. 2 2 2 Pr = 0.90 0.05  x 0.472  x 4.696  x + 0.002  x + 0.299  x + 15.805  x , (5) 1 2 3 1 2 3 where x —span length; x —beam location; and x —concrete thickness. 1 2 3 Buildings 2020, 10, x FOR PEER REVIEW 8 of 12 100% 100% bm ½Pt bm ½Pt bm Ln bm Ln bm ¼Pt bm ¼Pt 80% 80% Pr Pr 60% 60% 40% 40% 20% 20% 0% 0% 2 444 444 445 8 12 16 244 444 4445 8 12 16 Span [m] Span [m] (a) (b) 100% 100% bm ½Pt bm ½Pt bm Ln bm Ln bm ¼Pt bm ¼Pt 80% 80% Pr Pr 60% 60% 40% 40% 20% 20% 0% 0% 244 444 444 58 12 16 2444 4444 458 12 16 Span [m] Span [m] (c) (d) Figure 3. Percentage of bm received by the loaded beam (a) B1; (b) B2; (c) B3, and (d) B4 of BS set for Figure 3. Percentage of bm received by the loaded beam (a) B1; (b) B2; (c) B3, and (d) B4 of BS set for the various loadings and xi. the various loadings and x . For evaluating its adequacy to predict the load distribution also in terms of vd and sr, percentages For evaluating its adequacy to predict the load distribution also in terms of vd and sr, percentages obtained with Pr were compared with the experimental results of five real-scale TCC floors’ obtained with Pr were compared with the experimental results of five real-scale TCC floors’ specimens, specimens, built and experimentally tested by the authors, subjected to point and line loads at built and experimentally tested by the authors, subjected to point and line loads at di erent locations different locations (S1, S2, S3, S4 and S5 differing between them essentially in terms of concrete (S1, S2, S3, S4 and S5 di ering between them essentially in terms of concrete strength and thickness, strength and thickness, and span) [16]. Experimental vertical displacements, at mid- and quarter- 1 1 and span) [16]. Experimental vertical displacements, at mid- and quarter-span (vd /2 L and vd /4 L, span (vd ½ L and vd ¼ L, respectively) and support reactions were recorded and worked in order to respectively) and support reactions were recorded and worked in order to obtain the corresponding obtain the corresponding percentage. By comparing the experimental and Pr percentage curves, a percentage. similar co By urse was comparing found; the however, d experimental ifferences were and Pr per relacentage tively higcurves, h in some a ca similar ses (ma course inly asswas ociated found; with sr—when computing the difference between the percentage obtained with Pr (independent of however, di erences were relatively high in some cases (mainly associated with sr—when computing the loading type) and the experimental percentage for a specific beam, loading type, and loading the di erence between the percentage obtained with Pr (independent of the loading type) and the location, the values range between −4% and −42%, with a maximum average partial difference −28%, experimental percentage for a specific beam, loading type, and loading location, the values range concerning a medium span floor (4.00 m) using NWAC, as has S1. In order to make simplified between 4% and 42%, with a maximum average partial di erence 28%, concerning a medium approach suitable to predict the percentage of load received by the loaded beam, regarding the three span floor (4.00 m) using NWAC, as has S1. In order to make simplified approach suitable to predict quantities, vd, sr and bm, an extra coefficient was defined, cfi with i = {vd, sr, bm} = {1.25, 1.60, 1.00}, for the percentage of load received by the loaded beam, regarding the three quantities, vd, sr and bm, which Pr was multiplied. Figure 4 presents a good agreement between experimental and Pr ∙ cfvd an extra coecient was defined, cf with i = {vd, sr, bm} = {1.25, 1.60, 1.00}, for which Pr was multiplied. curves. This is also proven by the decrease of the average partial differences computed between the Figure 4 presents a good agreement between experimental and Pr  cf curves. This is also proven by percentages obtained with Pr ∙ cfi, with i = {vd, sr, bm} and experimental one vd s (Table 5). The extreme the decr values v ease of arie the d from average −7% t partial o 8% for divd er ½ L ences , −9% and computed 6% for between vd ¼ L, and thefrom percentages −14% to 1obtained 9% for sr (wit with h Pr a maximum average partial difference of 11% for the S1 experimental specimen). Thus, concerning cf , with i = {vd, sr, bm} and experimental ones (Table 5). The extreme values varied from 7% to 8% the specimen with average span dimensions (4.00 m) and regular materials specifically concrete 1 1 for vd /2 L, 9% and 6% for vd /4 L, and from 14% to 19% for sr (with a maximum average partial (NWAC), S1, a slightly overestimated prediction was obtained with the simplified approach, with a di erence of 11% for the S1 experimental specimen). Thus, concerning the specimen with average span predicted percentage of load higher than that obtained experimentally. dimensions (4.00 m) and regular materials specifically concrete (NWAC), S1, a slightly overestimated Buildings 2020, 10, 32 9 of 12 prediction was obtained with the simplified approach, with a predicted percentage of load higher than that obtained experimentally. Buildings 2020, 10, x FOR PEER REVIEW 9 of 12 100% 80% 60% 40% vd ½L ½Pt vd ½L ¼Pt 20% vd ½L Ln cf cf vd .P .P rr vd 0% S1 (L=4.00m; S2 S3 S4 (L= 2.00m;S5 (L=6.00m; S1 (L = 4.00 S2 (L = 4.00 S3 (L = 4.00 S4 (L = 2.00 S5 (L = 6.00 NWAC; (L=4.00m; (L=4.00m; NWAC; NWAC; m; NWAC; m; LW AC ; m; NWAC; m ; NWAC; m ; NWAC; tc= 0.05m) LWAC; NWAC; tc= 0.05m) tc= 0.05m) t = 0.05 m) t = 0.03 m) t = 0.03 m ) t = 0.05 m) t = 0.05 m) c c c c c tc= 0.05m) tc= 0.03m) Figure 4. Percentage of vd ½ L received by the loaded beam for floors S1 to S5, for the various loadings. Figure 4. Percentage of vd /2 L received by the loaded beam for floors S1 to S5, for the various loadings. Table 5. Average partial di erences for the three loadings [%]. Table 5. Average partial differences for the three loadings [%]. 1 1 Quantity vd /2 L vd /4 L sr Quantity 1 1 vd ½ L 1 1 vd ¼ L 1 1 sr Pr  cf vs. /2 L /4 L Ln /2 L /4 L Ln /2 L /4 L Ln S1 (L = 4.00 m; NWAC; t = 0.05 m) 4 7 2 4 3 1 11 6 2 Pr ∙ cfi vs. ½ L ¼ L Ln ½ L ¼ L Ln ½ L ¼ L Ln S2 (L = 4.00 m; LWAC; t = 0.05 m) 7 5 3 5 9 4 3 6 14 Experimental specimen S1 (L = 4.00 m; NWAC; tc = 0.05 m) 4 7 2 4 3 1 11 6 −2 S3 (L = 4.00 m; NWAC; t = 0.03 m) 3 5 8 6 1 6 19 9 1 S4 (L = 2.00 m; NWAC; t = 0.05 m) 6 5 3 3 4 2 3 8 9 S2 (L = 4.00 m; LWAC; tc = 0.05 m) −7 −5 −3 −5 −9 −4 3 −6 −14 S5 (L = 6.00 m; NWAC; t = 0.05 m) 1 2 1 0 -3 1 4 0 5 Experimental specimen S3 (L = 4.00 m; NWAC; tc = 0.03 m) 3 5 8 6 1 6 19 9 −1 S4 (L = 2.00 m; NWAC; tc = 0.05 m) −6 −5 −3 −3 −4 −2 −3 −8 −9 4. Conclusions S5 (L = 6.00 m; NWAC; tc = 0.05 m) −1 2 1 0 -3 −1 4 0 −5 Concentrated loads are common loads in building floors, a consequence of heavy furniture or partition walls. The usual design of TCC floors considers the entire load associated with the loaded 4. Conclusions beam. However, this may be far from reality, more so if the loaded beam is nearer to the floor center (mid-width). That assumption may lead to overestimated sections, which will be consequently Concentrated loads are common loads in building floors, a consequence of heavy furniture or uneconomic and, at the same time, detrimental concerning the load distribution. An extensive partition walls. The usual design of TCC floors considers the entire load associated with the loaded parametric study developed using Finite Element (FE) numerical models was performed and the beam. However, this may be far from reality, more so if the loaded beam is nearer to the floor center parameters that most a ect the distribution of concentrated loads in the transverse direction were identified. The floor ’s support conditions, the degree of oversizing, the loaded beam, the span length, (mid-width). That assumption may lead to overestimated sections, which will be consequently the concrete thickness, the structural system (deck vs. timber beams underneath the concrete layer) uneconomic and, at the same time, detrimental concerning the load distribution. An extensive and the concrete strength were the parameters that showed the highest e ect. The goal of this study parametric study developed using Finite Element (FE) numerical models was performed and the was to obtain a simplified approach capable of predicting the behavior of TCC floors subjected to parameters that most affect the distribution of concentrated loads in the transverse direction were a concentrated load, which could be applied at the design stage. Thus, a polynomial equation that identified. The floor’s support conditions, the degree of oversizing, the loaded beam, the span length, can predict the percentage of load received by the loaded beam based on the floor span, the concrete thickness, and beam location, in terms of vertical displacement, support reactions and longitudinal the concrete thickness, the structural system (deck vs. timber beams underneath the concrete layer) bending moment was devised. Compared with the results of real-scale floor specimens, the simplified and the concrete strength were the parameters that showed the highest effect. The goal of this study approach leads to di erences usually small (<10%) and “safe”, as the prediction tends to be higher was to obtain a simplified approach capable of predicting the behavior of TCC floors subjected to a than the experimental value. Nevertheless, this equation is not yet in its simplest form as the authors concentrated load, which could be applied at the design stage. Thus, a polynomial equation that can predict the percentage of load received by the loaded beam based on the floor span, the concrete thickness, and beam location, in terms of vertical displacement, support reactions and longitudinal bending moment was devised. Compared with the results of real-scale floor specimens, the simplified approach leads to differences usually small (<10%) and “safe”, as the prediction tends to be higher than the experimental value. Nevertheless, this equation is not yet in its simplest form as the authors would like. Therefore, further studies are ongoing to deepen the subject with the hope that in the near future, designers may easily use the simplified approach. Author Contributions: A.D., S.M. and S.L have read and agree to the published version of the manuscript. A.D. and S.M. conceived the studied issue; S.M. performed the experimental tests, developed the numerical model, performed the modelling tasks, analyzed the data and wrote the paper; A.D. and S.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 Buildings 2020, 10, 32 10 of 12 would like. Therefore, further studies are ongoing to deepen the subject with the hope that in the near future, designers may easily use the simplified approach. Author Contributions: A.D., S.M. and S.L have read and agree to the published version of the manuscript. A.D. and S.M. conceived the studied issue; S.M. performed the experimental tests, developed the numerical model, performed the modelling tasks, analyzed the data and wrote the paper; A.D. and S.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Operational Program Competitiveness and Internationalization R&D Projects Companies in Co-promotion, Portugal 2020, within the scope of the project OptimizedWood–POCI-01-0247-FEDER-017867. Acknowledgments: The authors would like to thank the ISISE-Institute for Sustainability and Innovation in Structural Engineering. Conflicts of Interest: The authors declare no conflict of interest. Abbreviation bm is the longitudinal bending moment at mid-span BS is the Base Simulation b is the width of the slab slab CLT is the Cross Laminated Timber cf is the extra coecient to obtain a better approximation with i = {vd, sr, bm} EPP is the elastic-perfectly plastic behavior Fx is the fixed support condition GL is the Glued Laminated Timber, Glulam h is the concrete thickness h is the interlayer thickness hK is the high sti ness h is the height of the timber beam I is the I-shape cross-section L is the span LE is the linear elastic behavior lK is the low sti ness Ln is the line load LVL is the Laminated Veneer Lumber LWAC is the lightweight aggregate concrete mK is the medium sti ness NWAC is the normal strength concrete Ov is the overestimated sizing section Pr designation of the simplified approach Pt is the point load R is the correlation coecient Sae is the simply supported condition in all ends sb is the beam spacing Sc is the support condition sr is the support reaction Ss is the simply supported condition sw is the self-weight Un is the underestimated sizing section vd is the vertical displacement at mid-span Z is the average of the values to approximate z is the value obtained by the polynomial fit for the i point, location, Z is the value to approximate for the i point, location. is the rectangular shape cross-section # is the round shape cross-section Buildings 2020, 10, 32 11 of 12 References 1. Emperger, F. Der Holzbeton. Dinglers Polytech. J. 1920, 335, 109–112. 2. Dias, A.; Skinner, J.; Crews, K.; Tannert, T. Timber-concrete-composites increasing the use of timber in construction. Eur. J. Wood Wood Prod. 2016, 74, 443–451. [CrossRef] 3. Van Der Linden, M. Timber Concrete Composite Floor Systems. Ph.D. Thesis, University of Delft, Delft, The Netherlands, December 1999. Available online: https://repository.tudelft.nl/islandora/object/uuid: 6b2807c2-258b-45b0-bb83-31c7c3d8b6cd?collection=research (accessed on 12 June 2019). 4. Jorge, L.; Lopes, S.; Cruz, H. Interlayer Influence on Timber-LWAC Composite Structures with Screw Connections. J. Struct. Eng. 2011, 137, 618–624. [CrossRef] 5. Yeoh, D. Behaviour and Design of Timber-Concrete Composite Floor System. Ph.D. Thesis, University of Canterbury, Christchurch, New Zealand, 2010. Available online: https://ir.canterbury.ac.nz/handle/10092/4428 (accessed on 12 June 2019). 6. Lukaszewska, E. Development of Prefabricated Timber-Concrete Composite Floors; Luleå University of Technology: Luleå, Sweden, 2009. 7. Wacker, J.; Dias, A.; Hosteng, T. Investigation of Early Timber-Concrete-Composite Bridges in the USA. In Proceedings of the 3rd International Conference on Timber Bridges, Skellefteå, Sweden, 26–29 June 2017. 8. Zhu, W.; Yang, H.; Liu, W.; Shi, B.; Ling, Z.; Tao, H. Experimental investigation on innovative connections for timber–concrete composite systems. Constr. Build. Mater. 2019, 207, 345–356. [CrossRef] 9. Yeoh, D.; Fragiacomo, M.; De Franceschi, M.; Heng Boon, K. State of the Art on Timber-Concrete Composite Structures: Literature Review. J. Struct. Eng. 2011, 137, 1085–1095. [CrossRef] 10. Branco, J.M.; Descamps, T.; Tsakanika, E. Repair and Strengthening of Traditional Timber Roof and Floor Structures. In Strengthening and Retrofitting of Existing Structures; Costa, A., Arêde, A., Varum, H., Eds.; Springer: Singapore, 2018; pp. 113–138. 11. Croatto, G.; Turrini, U. Restoration of historical timber structures—Criteria, innovative solutions and case studies. In Proceedings of the Intervir em Construções Existentes de Madeira, Guimarães, Portugal, 5 June 2014; Lourenço, B.P., Sousa, J.B.M.H.S., Eds.; Universidade do Minho: Braga, Portugal, 2014; pp. 119–136. 12. Projetos, Miguel Guedes Arquitetos, Lda. Available online: http://www.miguelguedes.pt/pt/projetos/ver/ palacio-do-raio-centro-interpretativo-de-memorias-da-misericordia/ (accessed on 12 June 2019). 13. Vencedores, 2016, Reabilitação do Palácio do Raio, in Prémio Nacional de Reabilitação Urbana. Available online: https://premio.vidaimobiliaria.com/ (accessed on 12 June 2019). 14. Dias, A.; Fragiacomo, M.; Gramatikov, K.; Kreis, B.; Kupferle, F.; Monteiro, S.; Sandanus, J.; Schänzlin, J.; Schober, K.; Sebastian, W.; et al. Design of Timber-Concrete Composite Structures. A State-of-the-Art Report by COST Action FP1402/WG 4; Dias, A., Schänzlin, J., Dietsch, P., Eds.; Verlag: Aachen, Germany, 2018. [CrossRef] 15. EU. Eurocode 5: Design of Timber Structures—Part. 1–1: General—Common Rules and Rules for Buildings; EN1995-1-1; European Committee for Standardization: Brussels, Belgium, 2004. 16. Monteiro, S.; Dias, A.; Lopes, S. Transverse distribution of internal forces in timber–concrete floors under external point and line loads. Constr. Build. Mater. 2016, 102, 1049–1059. [CrossRef] 17. Monteiro, S.; Dias, A.; Lopes, S. New guidelines for design of timber-concrete systems for point and line loads. In Proceedings of the WCTE 2016—World Conference on Timber Engineering, Vienna, Austria, 22–25 August 2016. 18. Dias, A.; Monteiro, S.; Martins, C. Reinforcement of timber floors—Transversal load distribution on timber-concrete systems. Adv. Mater. Res. 2013, 778, 657–664. [CrossRef] 19. Kieslich, H.; Holschemacher, K. Investigations on load sharing e ects in timber-concrete composite constructions. In Proceedings of the 9th International Conference on Structural Analysis of Historical Constructions (SAHC 2014), Mexico City, Mexico, 14–17 October 2014; Peña, F., Chávez, M., Eds.; 2014. Available online: https://www.semanticscholar.org/paper/INVESTIGATIONS-ON-LOAD-SHARING- EFFECTS-IN-COMPOSITE-KieslichHolschemacher/ e798847d3b910f0f9266072092c4a7b3b7752a (accessed on 12 June 2019). 20. Kieslick, H.; Holschemacher, K. Transversal load sharing in timber-concrete floors—Experimental and numerical investigations. In Proceedings of the 14 World Conference on Timber Engineering, Vienna, Austria, 22–25 August 2016. Buildings 2020, 10, 32 12 of 12 21. Mudie, J.; Sebastian, W.M.; Norman, J.; Bond, I.P. Experimental study of moment sharing in multi-joist timber-concrete composite floors from zero load up to failure. Constr. Build. Mater. 2019, 225, 956–971. [CrossRef] 22. Antunes, S. Distribuição Transversal de Cargas em Lajes Mistas Madeira-Betão, Estudo da Influência da Secção Transversal. Master ’s Thesis, University of Coimbra, Coimbra, Portugal, 2019. 23. Monteiro, S.; Dias, A.; Lopes, S. Transverse distribution of concentrated loads in timber-concrete floors—Parametric study. Proc. Inst. Civil Eng. Struct. Build. in press. [CrossRef] 24. EU. Eurocode 2: Design of Concrete Structures—Part. 1–1: General—Common Rules and Rules for Buildings; EN1992-1-1; CEN: Brussels, Belgium, 2004. 25. EU. Structural Timber; EN338; European Committee for Standardization: Brussels, Belgium, 2003. 26. EU. Eurocode 1: Actions on Structures—Part. 1–1: General Actions—Densities, Self-Weight, Imposed Loads for Buildings; EN1991-1-1; European Committee for Standardization: Brussels, Belgium, 2001. 27. Monteiro, S. Load Distribution on Timber-Concrete Composite Floors. Ph.D. Thesis, University of Coimbra, Coimbra, Portugal, 2015. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Buildings Multidisciplinary Digital Publishing Institute

Distribution of Concentrated Loads in Timber-Concrete Composite Floors: Simplified Approach

Monteiro, Sandra; Dias, Alfredo; Lopes, Sérgio
Buildings , Volume 10 (2) – Feb 18, 2020
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buildings Article Distribution of Concentrated Loads in Timber-Concrete Composite Floors: Simplified Approach 1 , 1 2 Sandra Monteiro * , Alfredo Dias and Sérgio Lopes The Institute for Sustainability and Innovation in Structural Engineering, Civil Engineering Department, University of Coimbra, Rua Luís Reis Santos—Pólo II, 3030-788 Coimbra, Portugal; alfgdias@dec.uc.pt Centre for Mechanical Engineering, Materials and Processes, Civil Engineering Department, University of Coimbra, Rua Luís Reis Santos—Pólo II, 3030-788 Coimbra, Portugal; sergio@dec.uc.pt * Correspondence: sandra@dec.uc.pt Received: 30 December 2019; Accepted: 13 February 2020; Published: 18 February 2020 Abstract: Timber-concrete composite (TCC) solutions are not a novelty. They were scientifically referred to at the beginning of the 20th century and they have proven their value in recent decades. Regarding a TCC floor at the design stage, there are some assumptions, at the standard level, concerning the action of concentrated loads which may be far from reality, specifically those associating the entire load to the beam over which it is applied. This naturally oversizes the beam and a ects how the load is distributed transversally, a ecting the TCC solution economically and mechanically. E orts have been made to clarify how concentrated loads are distributed, in the transverse direction, on TCC floors. Real-scale floor specimens were produced and tested subjected to concentrated (point and line) loads. Moreover, a Finite Element (FE)-based model was developed and validated and the results were collected. These results show that the “loaded beam” can receive less than 50% of the concentrated point load (when concerning the inner beams of a medium-span floor, 4.00 m). Aiming at reproducing these findings on the design of these floors, a simplified equation to predict the percentage of load received by each beam as a function of the floor span, the transversal position of the beam, and the thickness of the concrete layer was suggested. Keywords: TCC floors; transverse distribution of load; concentrated loads; simplified approach 1. Introduction Since its first scientific reference in the early decades of the last century [1], the use of a composite solution gathering timber beams with a thin concrete layer through an ecient connection has been spreading either for new or rehabilitation applications, on building floors or on bridge decks [2]. Timber-concrete composite (TCC) solutions may be as versatile as needed, by using di erent materials: di erent concrete strengths or densities, di erent timber species or engineered products, and di erent connection systems; or di erent sections (thicknesses and shape) [3–8]. They were initially developed with the aim of rehabilitating or strengthening timber floors [9]. However, in some cases of heritage cultural value buildings, their use may be overlooked by their insucient reversibility [10] or for being a non-dry technique. In fact, there are cases where TCC solutions were preferred relatively to other rehabilitation techniques and were also recognized as prize-worthy [11–13]. The rehabilitation of a building floor may be a consequence of physical or biological damages, lack of strength for the associated use (actual or new one), among others. Regardless of the motivation, there are common types of loading, such as furniture (point loads) or partition walls (line loads) that must be considered at the design stage. Concerning a timber-concrete solution, beside a document that is being prepared [14], there are no current standardization or code rules for the design for such composites. Annex B of the Eurocode 5 [15] is commonly used to perform the design computation. Buildings 2020, 10, 32; doi:10.3390/buildings10020032 www.mdpi.com/journal/buildings Buildings 2020, 10, 32 2 of 12 This computation considers the association of the entire load, point or line load aligned with the length of the timber beam, with the beam under consideration, but it can be far from the real behavior. In recent years, a few studies [16–21] aiming to understand how the load is distributed in the transverse direction were performed in this field. Parameters that might a ect that distribution were investigated. The work developed by the authors [16–18] proves that the share of load received by the loaded beam could be, in some cases, less than half. It is easy to understand the economic implications that an overestimated cross-section may have, associated with the unnecessary waste of material. Furthermore, there are also consequences at the mechanical behavior level. The thicker the concrete slab (using the same timber cross-section), the higher the transverse distribution of load. The opposite occurs with the increase of the timber beam height (keeping the concrete thickness unchanged), but with less expression [22], hence the importance of such studies. This paper aimed to present a simplified approach to be applied at a design stage in order to help to obtain an optimized TCC floor solution in terms of mechanical behavior and expenses. Therefore, an experimental set of results obtained from real-scale TCC floors tested under concentrated loads, together with the results of a parametric study using a Finite Element Method (FEM) model developed and validated by the authors was proposed. 2. Parametric Study To achieve the set goal, a comprehensive parametric study was developed. Aiming at studying the mechanical behavior of medium span TCC floors (4.00 m), a Base Simulation (BS) was established. The BS composite slab has a square plan and is composed of a 0.07 m-thick concrete layer, seven timber beams 0.60 m apart from each other and a 0.02 m-thick timber interlayer. Its material and geometric characteristics can be found in Monteiro et al. [23]. To perform such a study, several parameters (Table 1) were chosen and their e ect on the load distribution of TCC floors was analyzed. Only the loading of four beams, B1 (end beam) to B4 (central beam) was considered due to symmetry (for detailed information, see Monteiro et al. [23]). The analysis was accomplished by evaluating the percentage of support reaction (sr) received by each beam for the various loading cases: each beam loaded at 1 1 a time, with a point load at /2 span or /4 span, or line load. Moreover, the distribution of vertical displacements (vd) at mid-span and the distribution of longitudinal bending moment at mid-span (bm) were analyzed. Figure 1 summarizes the percentages of load received by the end beam (B1 or B7) and the central beam when loaded, at a time, associated with BS, in terms of the analyzed quantities. Regardless of the loading case or location, the loaded beam does not receive the entire load but a share of it, as well as the unloaded beams, which emphasizes the existence of load distribution. The share of load received by the loaded beam will be higher or lower the farther or nearer that beam is from the center of the slab, respectively. Beam B1 is the one associated with the highest share of load when loaded (more than 80% for sr). In addition, when considering the extreme load locations, B1 vs. B4, the maximum deviation of received share is associated with this beam location (di erence between the percentage of load associated with B1 when the load is applied at B1 and the percentage of load associated with B1 when the load is applied at B4 is about 90% for sr, for the three loading cases under consideration). On the contrary, intermediate beams show smaller deviations, with B3 presenting the minimum deviation (di erence between the percentage of load associated with B3 when the load is applied at B1 and the percentage of load associated with B3 when the load is applied at B4 is about 20% for vd and bm, 1 1 for /2 Pt and Ln, reaching less than 5% for /4 Pt). Concerning the studied parameters, it was found that they a ect in di erent amounts the quantities analyzed. The ones with a greater e ect (more than 10% of deviation) are displayed in Tables 2–4. The deviation was computed between two modeling tasks associated with a parameter. The Details column specifies among which “parameter values” was the maximum deviation obtained (e.g., a maximum deviation of 75% was found among BS with two di erent support conditions (Ss vs. Sae) when loaded with a point load at mid-span of B1). Buildings 2020, 10, 32 3 of 12 Table 1. Parameters studied. Type Parameter Symbol Span L Slab Width b slab Beam spacing s Geometrical Concrete h Thickness Interlayer h Cross-section Timber h Shape Beam , I, # Strength class According to EC2 [24] Concrete Normal-weight NWAC Aggregates Light-weigh LWAC Material Strength class According to EN 338 [25] Solid Timber Product Wood-engineered GL, LVL, OSB + LVL, CLT Low lK Connection sti ness Medium mK High hK Mechanical behavior Linear Elastic LE Material Non-linear Elastic Perfectly-plastic EPP Simply supported Ss Beams’ ends Support conditions, Sc Fixed Fx All ends Simply supported Sae Mid-span /2 Pt Point load Type Quarter-span /4 Pt Loading Linear load Ln Location On each beam at a time B1, B2, : : : , B7 Domestic and residential activities A According to EC1 [26] Floor use Areas where people congregate, C4 with possible physical activities. Undersized Un Degree of oversizing, DO Timber cross-section EC5 “tight-fit” According to EC5 [15] EC5 Oversized Ov With: GL—Glued laminated timber; LVL—Laminated veneer lumber; OSB—Oriented strand board; CLT—Cross-laminated timber. Table 2. Parameters with the greatest e ect on the load distribution referring to BS. Maximum Load Beam Quantity Parameter Details Deviation [%] Type 75 B1 /2 Pt sr Simply supported on beams’ ends, Support conditions, Sc Ss vs. Simply supported on all ends, Sae 36 B1 Ln bm B1 Ln vd B2 /2 Pt sr Concrete thickness, h 0.02 m vs. 0.07 m (BS) 31 c B1 Ln bm 28 B2 Ln sr Span, L 4.00 m (BS) vs. 16.00 m * 27 B1 /2 Pt sr Beams vs. deck BS vs. CLT deck 26 B1 /2 Pt vd Span, L 4.00 m (BS) vs. 16.00 m * 24 B1 /2 Pt bm 17 B1 Ln vd Concrete strength LC16/18 vs. C25/30 (BS) 16 B1 Ln bm B4 /2 Pt bm Beams vs. deck BS vs. CLT deck B2 Ln sr Concrete strength LC16/18 vs. C25/30 (BS) 10 B1 /2 Pt vd Support conditions, Sc Simply supported, Ss vs. Fixed, Fx *—underestimated timber section. Buildings 2020, 10, 32 4 of 12 Buildings 2020, 10, x FOR PEER REVIEW 4 of 12 0 B11234 B2 B3 B4 B5 567 B6 B7 8 -20% 0% 20% 100%/7=14% 40% vd B1 ½Pt vd B1 ¼Pt 60% vd B1 Ln vd B4 ½Pt 80% vd B4 ¼Pt vd B4 Ln 100% (a) 012 B1 B2 B3 B4 B5 B6 34567 B7 8 -20% 0% 20% 100%/7=14% 40% sr B1 ½Pt sr B1 ¼Pt 60% sr B1 Ln sr B4 ½Pt 80% sr B4 ¼Pt sr B4 Ln 100% (b) B1 B2 B3 B4 B5 B6 B7 012 3456 7 8 -20% 0% 100%/7=14% 20% 40% bm B1 ½Pt bm B1 ¼Pt 60% bm B1 Ln bm B4 ½Pt 80% bm B4 ¼Pt bm B4 Ln 100% (c) Figure 1. Percentage of load received by each beam when B1 or B4 is loaded, in terms of (a) vd; (b) sr; Figure 1. Percentage of load received by each beam when B1 or B4 is loaded, in terms of (a) vd; (b) sr; and (c) bm and. (c) bm. Although the DO has shown a great effect, both Ov and Un series, varying between 31% (sr) and 42% (vd) when considering spans of 4.00 m (the same as BS) and 16.00 m (reaching deviations of 66% and 58% when considering the extreme spans [2.00 m; 16.00 m]), the percentages found are only indicative of the trend. In contrast, for the EC5 series, the sections found were established based on an objective criterion: the design based on Annex B of EC5 [15], aiming at maximizing the section strength utilization ratio, for Un and Ov series that did not occur. Although a common procedure has been established, by changing only the timber height, no uniform percentage of over or under sizing was defined (for extra detail see Monteiro et al. [23]). Buildings 2020, 10, 32 5 of 12 Table 3. Parameters with the greatest e ect on the load distribution, referring to the boundaries of each parameter. Maximum Load Beam Quantity Parameter Details Deviation [%] Type 70 B1 /2 Pt sr CLT and Sc CLT + Ss vs. CLT + Sae 58 - /2 Pt sr Load location B1 vs. B4 for BS with lK connection 47 B2 Ln sr Span, L 2.00 m (BS) vs. 16.00 m* B1 /2 Pt vd 45 Concrete thickness, h 0.02 m vs. 0.20 m B1 Ln bm 44 B1 Ln vd Span, L 2.00 m (BS) vs. 16.00 m* 43 B2 Ln sr h 0.02 m vs. 0.20 m 41 B1 Ln bm Span, L 2.00 m (BS) vs. 16.00 m* 32 B1 /2 Pt sr CLT and s BS + juxtaposed beams vs. CLT deck Ln vd 29 Load location B1 vs. B4 for BS with LC16/18 concrete Ln bm 26 B1 Ln vd CLT and Sc CLT + Ss vs. CLT + Sae 23 B1 /2 Pt bm 18 B1 Ln vd Concrete strength 17 B1 Ln bm LC16/18 vs. C40/50 B2 Ln sr B1 Ln vd CLT and s BS + juxtaposed beams vs. CLT deck 11 B1 Ln bm CLT and s *—underestimated timber section. Table 4. Maximum deviation of load distribution associated with the DO. Maximum Load Beam Quantity Parameter Details Deviation [%] Type 60 - /2 Pt sr Load location B1 vs. B4 for Un with L = 16.00 m 43 B1 Ln vd EC5 42 B1 Ln vd Ov 41 B1 Ln bm EC5 39 B4 Ln sr Un 1 L = 4.00 m vs. L = 16.00 m B1 /2 Pt vd Un B1 Ln bm Ov 37 B1 Ln bm Un 36 B2 Ln sr EC5 31 B2 Ln sr Ov vd Ln 30 Load location B1 vs. B4 for EC5 with L = 4.00 m bm Ln 66 B1 Ln vd Ov 64 B1 Ln bm Ov 58 B1 Ln vd Un B1 /2 Pt vd EC5 B1 Ln bm Un L = 2.00 m vs. L = 16.00 m B2 Ln sr Un B1 Ln bm EC5 B2 /2 Pt sr Ov 53 B2 /2 Pt sr EC5 From this analysis, it becomes clear the significant e ect of the following parameters: 1. The support conditions, with a maximum deviation of 75% between BS (Ss) and BS with Sae); 2. The degree of oversizing, with a maximum deviation of 66% (Ov), 58% (Un) and 56% (EC5), considering the limit spans 2.00 m and 16.00 m; Buildings 2020, 10, 32 6 of 12 3. The loading position, with a maximum deviation of 58% considering the loading applied at B1 vs. applied at B4 when the modeling BS with lK connection is considered (reaching 60% when in association with DO—verified for the modeling task with the same cross-section as BS and L = 16.00 m (underestimated timber section)); 4. The span length, with a maximum deviation of 47% between spans of 2.00 m and 16.00 m; 5. The concrete thickness, with a maximum deviation of 45% between thicknesses of 0.02 m and 0.20 m; 6. The existence of a timber deck underneath the concrete layer, instead of timber beams and interlayer using juxtaposed beams or a CLT deck, with a maximum deviation of 27%; and 7. The concrete strength, with a maximum deviation of 18% between an LWAC LC16/18 and an NWAC C40/50. Although the DO has shown a great e ect, both Ov and Un series, varying between 31% (sr) and 42% (vd) when considering spans of 4.00 m (the same as BS) and 16.00 m (reaching deviations of 66% and 58% when considering the extreme spans [2.00 m; 16.00 m]), the percentages found are only indicative of the trend. In contrast, for the EC5 series, the sections found were established based on an objective criterion: the design based on Annex B of EC5 [15], aiming at maximizing the section strength utilization ratio, for Un and Ov series that did not occur. Although a common procedure has been established, by changing only the timber height, no uniform percentage of over or under sizing was defined (for extra detail see Monteiro et al. [23]). The analysis of the previous tables shows that most of the parameters that have the greatest e ect on the load distribution are associated with the end beam B1 (or B7). This was verified in 71% of the cases when considering the variation relatively to BS; 78% of the cases, when considering the variation of a parameter (except DO) among its extreme values; and in 67% of the cases, when considering the variation on the DO, as for L = [4.00 m; 16.00 m], as for L = [2.00 m; 16.00 m]. This is due to the fact that end beams tend to concentrate the load applied over it (and thus, a lower percentage of distributed load) when compared with the remaining ones, allowing a higher variation than the central beam (B4), for instance, where the opposite happens, and a smaller range of variation can occur. With regard to the loading, although most of the listed parameters were associated with a linear loading, the di erences found relatively to a point load at mid-span were, at most, 4% (disregarding the Sc modeling task, which, due to a di erent structural system, were associated with greater di erences). 3. Simplified Approach The design stage is a crucial stage where the designer must be able to come up with an economical structural solution, preferably in the shortest time possible. Knowing the percentage of load received by a specific beam, before the floor being built, without the need to numerically modeling it, will surely contribute to it. Thus, based on the findings of the parametric study and aiming at providing a practical tool capable to predict the sought percentage, a simplified equation was developed. As shown above, three quantities were used for evaluating the load distribution, vd, sr and bm, but only one was used on the simplified model: the longitudinal bending moment, given its importance in the design process. For that, three essential parameters were considered: the span length, the concrete thickness and the transversal location of the beam. For this last consideration, a dimensionless parameter designated “beam location”, bl, was defined in order to provide the transversal position of the beam in question, relatively to the longitudinal axis of the outermost beam (B1) (Figure 2). The results collected in the parametric study, specifically, the percentage of load received by the loaded beam for the three parameters listed above were treated and gathered. Four sets of “continuous” curves were obtained, based on the design considerations, BS, un, EC5, and ov, for the considered loadings, gathering the results for B1 to B4, for each loading by span. Various polynomial approaches to the BS curves were tried in the approximation process: from a first-degree polynomial simple Equation Buildings 2020, 10, x FOR PEER REVIEW 6 of 12 37 B1 Ln bm Un 36 B2 Ln sr EC5 31 B2 Ln sr Ov vd Ln 30 - Load location B1 vs. B4 for EC5 with L = 4.00 m bm Ln 66 B1 Ln vd Ov 64 B1 Ln bm Ov 58 B1 Ln vd Un B1 ½ Pt vd EC5 B1 Ln bm Un L = 2.00 m vs. L = 16.00 m 56 B2 Ln sr Un B1 Ln bm EC5 B2 ½ Pt sr Ov 53 B2 ½ Pt sr EC5 The analysis of the previous tables shows that most of the parameters that have the greatest effect Buildings on the load dist 2020, 10, 32 ribution are associated with the end beam B1 (or B7). This was verified in 71% o 7fof th 12 e cases when considering the variation relatively to BS; 78% of the cases, when considering the variation of a parameter (except DO) among its extreme values; and in 67% of the cases, when considering the (1) to a fourth-degree polynomial simple Equation (3), considering also first (2) and second-degree variation on the DO, as for L = [4.00 m; 16.00 m], as for L = [2.00 m; 16.00 m]. This is due to the fact that polynomial equations with crossed terms). end beams tend to concentrate the load applied over it (and thus, a lower percentage of distributed load) when compared with the remaining ones, allowing a higher variation than the central beam (B4), z = a + a  x + a  x + a  x , (1) 0 1 1 2 2 3 3 for instance, where the opposite happens, and a smaller range of variation can occur. With regard to the loading, although most of the listed parameters were associated with a linear loading, the z = a + a  x + a  x + a  x + a  x  x + a  x  x + a  x  x , (2) 0 1 1 2 2 3 3 4 1 2 5 1 3 6 2 3 differences found relatively to a point load at mid-span were, at most, 4% (disregarding the Sc modeling 2 2 2 3 3 3 z = a + a  x + a  x + a  x + a  x + a  x + a  x + a  x + a  x + a  x + 0 1 1 2 2 3 3 4 1 5 2 6 3 7 1 8 2 9 3 task, which, due to a different structural system, were associated with greater differences). (3) 4 4 4 a  x +a  x + a  x , 10 1 11 2 12 3 3. Simplified Approach where x —span length; x —beam location; x —concrete thickness; and a , with i = 0 to 1 2 3 i 12—polynomial coecients. The design stage is a crucial stage where the designer must be able to come up with an The attempt to obtain the best approximation with the various polynomial equation was made by economical structural solution, preferably in the shortest time possible. Knowing the percentage of obtaining a set of polynomial coecients, according to the polynomial under consideration through load received by a specific beam, before the floor being built, without the need to numerically the minimization of the sum of the squared di erences between the numerical and the polynomial modeling it, will surely contribute to it. Thus, based on the findings of the parametric study and predictions. To measure the “strength of the approximation”, the determination coecient, R , (4) was aiming at providing a practical tool capable to predict the sought percentage, a simplified equation used, for which the strongest approximation corresponds to R = 1 and the weakest approximation to was developed. R = 0. Detailed information about the coecients obtained for the various sets and loading cases, As shown above, three quantities were used for evaluating the load distribution, vd, sr and bm, together with the corresponding R , can be found in Monteiro [27]. but only one was used on the simplified model: the longitudinal bending moment, given its importance in the design process. For tha Xt, three essential par X ameters were considered: the span 2 2 R = z Z / Z Z , (4) i i length, the concrete thickness and the transversal location of the beam. For this last consideration, a dimensionless parameter designated “beam location”, bl, was defined in order to provide the where R—correlation coecient, z —value given by the polynomial fit for the i point, location, transversal position of the beam in question, relatively to the longitudinal axis of the outermost beam Z—average of the values to approximate, Z —value to approximate for the i point, location. (B1) (Figure 2). B1 B2 B3 B4 b =1/3 b =2/3 b =0.00 b =1.00 l l Figure 2. Beam location parameter. Figure 2. Beam location parameter. Since the goal was to obtain an equation to predict the behavior of TCC floors under concentrated loads, the polynomial coecients obtained for the various attempts were analyzed aiming at finding common tendencies among them, for di erent loadings. Given that the BS set was the only one for 1 1 1 which three loading cases, /2 Pt, /4 Pt, and Ln, were modeled (for the remaining sets only /2 Pt and Ln were modeled); this was the chosen set to perform that comparison. The polynomial coecients for all attempts for BS were compared with each other and among the various loading cases ( /2 Pt vs. Ln; 1 1 1 /4 Pt vs. /2 Pt; /4 Pt vs. Ln). This analysis evidenced similar coecients for comparable polynomial attempts and among those the one with the best approximation was identified: the second-degree polynomial simple equation. The polynomial coecients of the sought equation, designated Pr (since it intends to predict the percentage of load received by the loaded beam), were defined as the computed average polynomial coecients found for the three loading cases (5). Figure 3 shows its course as a function of the floor span. As the figure depicts, some di erences can be found between the predicted and the numerical percentages, with Pr approaching the Ln load case curve more than the other curves. In general, it tends to underestimate the percentage of load associated with /2 Pt, but it tends to overestimate the same quantities concerning /4 Pt and Ln. Although for both point loadings, the greatest deviation is about 20% (associated, essentially, with the thinner concrete layers), the mean Buildings 2020, 10, 32 8 of 12 1 1 di erences were rather lower: about 11% for /2 Pt and + 8% /4 Pt. For Ln the deviations are lower than the point loadings, ranging between 10% and +15%, with a mean di erence of +4%. 2 2 2 Pr = 0.90 0.05  x 0.472  x 4.696  x + 0.002  x + 0.299  x + 15.805  x , (5) 1 2 3 1 2 3 where x —span length; x —beam location; and x —concrete thickness. 1 2 3 Buildings 2020, 10, x FOR PEER REVIEW 8 of 12 100% 100% bm ½Pt bm ½Pt bm Ln bm Ln bm ¼Pt bm ¼Pt 80% 80% Pr Pr 60% 60% 40% 40% 20% 20% 0% 0% 2 444 444 445 8 12 16 244 444 4445 8 12 16 Span [m] Span [m] (a) (b) 100% 100% bm ½Pt bm ½Pt bm Ln bm Ln bm ¼Pt bm ¼Pt 80% 80% Pr Pr 60% 60% 40% 40% 20% 20% 0% 0% 244 444 444 58 12 16 2444 4444 458 12 16 Span [m] Span [m] (c) (d) Figure 3. Percentage of bm received by the loaded beam (a) B1; (b) B2; (c) B3, and (d) B4 of BS set for Figure 3. Percentage of bm received by the loaded beam (a) B1; (b) B2; (c) B3, and (d) B4 of BS set for the various loadings and xi. the various loadings and x . For evaluating its adequacy to predict the load distribution also in terms of vd and sr, percentages For evaluating its adequacy to predict the load distribution also in terms of vd and sr, percentages obtained with Pr were compared with the experimental results of five real-scale TCC floors’ obtained with Pr were compared with the experimental results of five real-scale TCC floors’ specimens, specimens, built and experimentally tested by the authors, subjected to point and line loads at built and experimentally tested by the authors, subjected to point and line loads at di erent locations different locations (S1, S2, S3, S4 and S5 differing between them essentially in terms of concrete (S1, S2, S3, S4 and S5 di ering between them essentially in terms of concrete strength and thickness, strength and thickness, and span) [16]. Experimental vertical displacements, at mid- and quarter- 1 1 and span) [16]. Experimental vertical displacements, at mid- and quarter-span (vd /2 L and vd /4 L, span (vd ½ L and vd ¼ L, respectively) and support reactions were recorded and worked in order to respectively) and support reactions were recorded and worked in order to obtain the corresponding obtain the corresponding percentage. By comparing the experimental and Pr percentage curves, a percentage. similar co By urse was comparing found; the however, d experimental ifferences were and Pr per relacentage tively higcurves, h in some a ca similar ses (ma course inly asswas ociated found; with sr—when computing the difference between the percentage obtained with Pr (independent of however, di erences were relatively high in some cases (mainly associated with sr—when computing the loading type) and the experimental percentage for a specific beam, loading type, and loading the di erence between the percentage obtained with Pr (independent of the loading type) and the location, the values range between −4% and −42%, with a maximum average partial difference −28%, experimental percentage for a specific beam, loading type, and loading location, the values range concerning a medium span floor (4.00 m) using NWAC, as has S1. In order to make simplified between 4% and 42%, with a maximum average partial di erence 28%, concerning a medium approach suitable to predict the percentage of load received by the loaded beam, regarding the three span floor (4.00 m) using NWAC, as has S1. In order to make simplified approach suitable to predict quantities, vd, sr and bm, an extra coefficient was defined, cfi with i = {vd, sr, bm} = {1.25, 1.60, 1.00}, for the percentage of load received by the loaded beam, regarding the three quantities, vd, sr and bm, which Pr was multiplied. Figure 4 presents a good agreement between experimental and Pr ∙ cfvd an extra coecient was defined, cf with i = {vd, sr, bm} = {1.25, 1.60, 1.00}, for which Pr was multiplied. curves. This is also proven by the decrease of the average partial differences computed between the Figure 4 presents a good agreement between experimental and Pr  cf curves. This is also proven by percentages obtained with Pr ∙ cfi, with i = {vd, sr, bm} and experimental one vd s (Table 5). The extreme the decr values v ease of arie the d from average −7% t partial o 8% for divd er ½ L ences , −9% and computed 6% for between vd ¼ L, and thefrom percentages −14% to 1obtained 9% for sr (wit with h Pr a maximum average partial difference of 11% for the S1 experimental specimen). Thus, concerning cf , with i = {vd, sr, bm} and experimental ones (Table 5). The extreme values varied from 7% to 8% the specimen with average span dimensions (4.00 m) and regular materials specifically concrete 1 1 for vd /2 L, 9% and 6% for vd /4 L, and from 14% to 19% for sr (with a maximum average partial (NWAC), S1, a slightly overestimated prediction was obtained with the simplified approach, with a di erence of 11% for the S1 experimental specimen). Thus, concerning the specimen with average span predicted percentage of load higher than that obtained experimentally. dimensions (4.00 m) and regular materials specifically concrete (NWAC), S1, a slightly overestimated Buildings 2020, 10, 32 9 of 12 prediction was obtained with the simplified approach, with a predicted percentage of load higher than that obtained experimentally. Buildings 2020, 10, x FOR PEER REVIEW 9 of 12 100% 80% 60% 40% vd ½L ½Pt vd ½L ¼Pt 20% vd ½L Ln cf cf vd .P .P rr vd 0% S1 (L=4.00m; S2 S3 S4 (L= 2.00m;S5 (L=6.00m; S1 (L = 4.00 S2 (L = 4.00 S3 (L = 4.00 S4 (L = 2.00 S5 (L = 6.00 NWAC; (L=4.00m; (L=4.00m; NWAC; NWAC; m; NWAC; m; LW AC ; m; NWAC; m ; NWAC; m ; NWAC; tc= 0.05m) LWAC; NWAC; tc= 0.05m) tc= 0.05m) t = 0.05 m) t = 0.03 m) t = 0.03 m ) t = 0.05 m) t = 0.05 m) c c c c c tc= 0.05m) tc= 0.03m) Figure 4. Percentage of vd ½ L received by the loaded beam for floors S1 to S5, for the various loadings. Figure 4. Percentage of vd /2 L received by the loaded beam for floors S1 to S5, for the various loadings. Table 5. Average partial di erences for the three loadings [%]. Table 5. Average partial differences for the three loadings [%]. 1 1 Quantity vd /2 L vd /4 L sr Quantity 1 1 vd ½ L 1 1 vd ¼ L 1 1 sr Pr  cf vs. /2 L /4 L Ln /2 L /4 L Ln /2 L /4 L Ln S1 (L = 4.00 m; NWAC; t = 0.05 m) 4 7 2 4 3 1 11 6 2 Pr ∙ cfi vs. ½ L ¼ L Ln ½ L ¼ L Ln ½ L ¼ L Ln S2 (L = 4.00 m; LWAC; t = 0.05 m) 7 5 3 5 9 4 3 6 14 Experimental specimen S1 (L = 4.00 m; NWAC; tc = 0.05 m) 4 7 2 4 3 1 11 6 −2 S3 (L = 4.00 m; NWAC; t = 0.03 m) 3 5 8 6 1 6 19 9 1 S4 (L = 2.00 m; NWAC; t = 0.05 m) 6 5 3 3 4 2 3 8 9 S2 (L = 4.00 m; LWAC; tc = 0.05 m) −7 −5 −3 −5 −9 −4 3 −6 −14 S5 (L = 6.00 m; NWAC; t = 0.05 m) 1 2 1 0 -3 1 4 0 5 Experimental specimen S3 (L = 4.00 m; NWAC; tc = 0.03 m) 3 5 8 6 1 6 19 9 −1 S4 (L = 2.00 m; NWAC; tc = 0.05 m) −6 −5 −3 −3 −4 −2 −3 −8 −9 4. Conclusions S5 (L = 6.00 m; NWAC; tc = 0.05 m) −1 2 1 0 -3 −1 4 0 −5 Concentrated loads are common loads in building floors, a consequence of heavy furniture or partition walls. The usual design of TCC floors considers the entire load associated with the loaded 4. Conclusions beam. However, this may be far from reality, more so if the loaded beam is nearer to the floor center (mid-width). That assumption may lead to overestimated sections, which will be consequently Concentrated loads are common loads in building floors, a consequence of heavy furniture or uneconomic and, at the same time, detrimental concerning the load distribution. An extensive partition walls. The usual design of TCC floors considers the entire load associated with the loaded parametric study developed using Finite Element (FE) numerical models was performed and the beam. However, this may be far from reality, more so if the loaded beam is nearer to the floor center parameters that most a ect the distribution of concentrated loads in the transverse direction were identified. The floor ’s support conditions, the degree of oversizing, the loaded beam, the span length, (mid-width). That assumption may lead to overestimated sections, which will be consequently the concrete thickness, the structural system (deck vs. timber beams underneath the concrete layer) uneconomic and, at the same time, detrimental concerning the load distribution. An extensive and the concrete strength were the parameters that showed the highest e ect. The goal of this study parametric study developed using Finite Element (FE) numerical models was performed and the was to obtain a simplified approach capable of predicting the behavior of TCC floors subjected to parameters that most affect the distribution of concentrated loads in the transverse direction were a concentrated load, which could be applied at the design stage. Thus, a polynomial equation that identified. The floor’s support conditions, the degree of oversizing, the loaded beam, the span length, can predict the percentage of load received by the loaded beam based on the floor span, the concrete thickness, and beam location, in terms of vertical displacement, support reactions and longitudinal the concrete thickness, the structural system (deck vs. timber beams underneath the concrete layer) bending moment was devised. Compared with the results of real-scale floor specimens, the simplified and the concrete strength were the parameters that showed the highest effect. The goal of this study approach leads to di erences usually small (<10%) and “safe”, as the prediction tends to be higher was to obtain a simplified approach capable of predicting the behavior of TCC floors subjected to a than the experimental value. Nevertheless, this equation is not yet in its simplest form as the authors concentrated load, which could be applied at the design stage. Thus, a polynomial equation that can predict the percentage of load received by the loaded beam based on the floor span, the concrete thickness, and beam location, in terms of vertical displacement, support reactions and longitudinal bending moment was devised. Compared with the results of real-scale floor specimens, the simplified approach leads to differences usually small (<10%) and “safe”, as the prediction tends to be higher than the experimental value. Nevertheless, this equation is not yet in its simplest form as the authors would like. Therefore, further studies are ongoing to deepen the subject with the hope that in the near future, designers may easily use the simplified approach. Author Contributions: A.D., S.M. and S.L have read and agree to the published version of the manuscript. A.D. and S.M. conceived the studied issue; S.M. performed the experimental tests, developed the numerical model, performed the modelling tasks, analyzed the data and wrote the paper; A.D. and S.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 B1 B1 B2 B2 B3 B3 B4 B4 B5 B5 Buildings 2020, 10, 32 10 of 12 would like. Therefore, further studies are ongoing to deepen the subject with the hope that in the near future, designers may easily use the simplified approach. Author Contributions: A.D., S.M. and S.L have read and agree to the published version of the manuscript. A.D. and S.M. conceived the studied issue; S.M. performed the experimental tests, developed the numerical model, performed the modelling tasks, analyzed the data and wrote the paper; A.D. and S.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Operational Program Competitiveness and Internationalization R&D Projects Companies in Co-promotion, Portugal 2020, within the scope of the project OptimizedWood–POCI-01-0247-FEDER-017867. Acknowledgments: The authors would like to thank the ISISE-Institute for Sustainability and Innovation in Structural Engineering. Conflicts of Interest: The authors declare no conflict of interest. Abbreviation bm is the longitudinal bending moment at mid-span BS is the Base Simulation b is the width of the slab slab CLT is the Cross Laminated Timber cf is the extra coecient to obtain a better approximation with i = {vd, sr, bm} EPP is the elastic-perfectly plastic behavior Fx is the fixed support condition GL is the Glued Laminated Timber, Glulam h is the concrete thickness h is the interlayer thickness hK is the high sti ness h is the height of the timber beam I is the I-shape cross-section L is the span LE is the linear elastic behavior lK is the low sti ness Ln is the line load LVL is the Laminated Veneer Lumber LWAC is the lightweight aggregate concrete mK is the medium sti ness NWAC is the normal strength concrete Ov is the overestimated sizing section Pr designation of the simplified approach Pt is the point load R is the correlation coecient Sae is the simply supported condition in all ends sb is the beam spacing Sc is the support condition sr is the support reaction Ss is the simply supported condition sw is the self-weight Un is the underestimated sizing section vd is the vertical displacement at mid-span Z is the average of the values to approximate z is the value obtained by the polynomial fit for the i point, location, Z is the value to approximate for the i point, location. is the rectangular shape cross-section # is the round shape cross-section Buildings 2020, 10, 32 11 of 12 References 1. 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A State-of-the-Art Report by COST Action FP1402/WG 4; Dias, A., Schänzlin, J., Dietsch, P., Eds.; Verlag: Aachen, Germany, 2018. [CrossRef] 15. EU. Eurocode 5: Design of Timber Structures—Part. 1–1: General—Common Rules and Rules for Buildings; EN1995-1-1; European Committee for Standardization: Brussels, Belgium, 2004. 16. Monteiro, S.; Dias, A.; Lopes, S. Transverse distribution of internal forces in timber–concrete floors under external point and line loads. Constr. Build. Mater. 2016, 102, 1049–1059. [CrossRef] 17. Monteiro, S.; Dias, A.; Lopes, S. New guidelines for design of timber-concrete systems for point and line loads. In Proceedings of the WCTE 2016—World Conference on Timber Engineering, Vienna, Austria, 22–25 August 2016. 18. Dias, A.; Monteiro, S.; Martins, C. Reinforcement of timber floors—Transversal load distribution on timber-concrete systems. Adv. Mater. Res. 2013, 778, 657–664. [CrossRef] 19. Kieslich, H.; Holschemacher, K. Investigations on load sharing e ects in timber-concrete composite constructions. In Proceedings of the 9th International Conference on Structural Analysis of Historical Constructions (SAHC 2014), Mexico City, Mexico, 14–17 October 2014; Peña, F., Chávez, M., Eds.; 2014. Available online: https://www.semanticscholar.org/paper/INVESTIGATIONS-ON-LOAD-SHARING- EFFECTS-IN-COMPOSITE-KieslichHolschemacher/ e798847d3b910f0f9266072092c4a7b3b7752a (accessed on 12 June 2019). 20. Kieslick, H.; Holschemacher, K. Transversal load sharing in timber-concrete floors—Experimental and numerical investigations. In Proceedings of the 14 World Conference on Timber Engineering, Vienna, Austria, 22–25 August 2016. Buildings 2020, 10, 32 12 of 12 21. Mudie, J.; Sebastian, W.M.; Norman, J.; Bond, I.P. Experimental study of moment sharing in multi-joist timber-concrete composite floors from zero load up to failure. Constr. Build. Mater. 2019, 225, 956–971. [CrossRef] 22. Antunes, S. Distribuição Transversal de Cargas em Lajes Mistas Madeira-Betão, Estudo da Influência da Secção Transversal. Master ’s Thesis, University of Coimbra, Coimbra, Portugal, 2019. 23. Monteiro, S.; Dias, A.; Lopes, S. Transverse distribution of concentrated loads in timber-concrete floors—Parametric study. Proc. Inst. Civil Eng. Struct. Build. in press. [CrossRef] 24. EU. Eurocode 2: Design of Concrete Structures—Part. 1–1: General—Common Rules and Rules for Buildings; EN1992-1-1; CEN: Brussels, Belgium, 2004. 25. EU. Structural Timber; EN338; European Committee for Standardization: Brussels, Belgium, 2003. 26. EU. Eurocode 1: Actions on Structures—Part. 1–1: General Actions—Densities, Self-Weight, Imposed Loads for Buildings; EN1991-1-1; European Committee for Standardization: Brussels, Belgium, 2001. 27. Monteiro, S. Load Distribution on Timber-Concrete Composite Floors. Ph.D. Thesis, University of Coimbra, Coimbra, Portugal, 2015. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Journal

Buildings – Multidisciplinary Digital Publishing Institute

Published: Feb 18, 2020

References

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