Near-Surface Mounted Reinforcement of Sawn Timber Beams-FEM Approach - MDPI
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materials Article Near-Surface Mounted Reinforcement of Sawn Timber Beams-FEM Approach Izabela Burawska-Kupniewska * and Piotr Beer Institute of Wood Sciences and Furniture, Warsaw University of Life Sciences-SGGW, Nowoursynowska 159, 02-776 Warsaw, Poland; piotr_beer@sggw.edu.pl * Correspondence: izabela_burawska@sggw.edu.pl Abstract: The demand for timber has increased significantly in recent years. Therefore, reliable tools are needed to predict the mechanical properties of sawn timber, especially for structural applications. Very complex models require a lot of input data for analysis, which cannot always be guaranteed, especially in industrial practice. Thus, a simplified model for material description was developed and assessed with experiments (static bending tests carried out in accordance with the guidelines suggested in the European standard EN 408) and an analytical approach (gamma method according to the guidelines given in the European standard EN 1995). The effective stiffness was calculated as a major parameter, which has an influence on the elements’ behavior. The model included a near- surface mounted (NSM) local reinforcement technique, with CFRP strips of Scots pine timber beams being subjected to bending stresses. It is anticipated that the developed model can be a starting point for the repair engineering field, contributing to decision-making regarding conservation technique selection and range. Next, improvements of the model will provide more and more realistic results for numerical analysis in terms of the obtained failure mechanisms for sawn timber elements. Keywords: strengthening; NSM; CFRP strip; Scots pine Citation: Burawska-Kupniewska, I.; Beer, P. Near-Surface Mounted Reinforcement of Sawn Timber Beams-FEM Approach. Materials 1. Introduction 2021, 14, 2780. https://doi.org/ 10.3390/ma14112780 In terms of the deficit of raw wood material, especially of high-quality, and the general increase in prices, it is justified to take up the issue of using inferior quality Academic Editor: Tomasz Sadowski wood in construction and using it for structural elements. A material of low quality class, loaded with structural defects, can be strengthened, which will significantly increase its Received: 25 March 2021 original strength parameters. An important issue, however, is the cost of the reinforcement Accepted: 20 May 2021 technique used. The reinforced wooden element must be economically competitive with a Published: 24 May 2021 full-value element of comparable strength properties, available in the required quantity on the market. Publisher’s Note: MDPI stays neutral The most important determinants of timber strength parameters is the presence of with regard to jurisdictional claims in structural defects. The basic and the most common wood defect is knots [1]. Therefore, published maps and institutional affil- the presence of knots and resulting fiber deviations is the basic criterion for the quality iations. classification of wood [2–6]. Single knots with a diameter of less than 5mm may be omitted when making the qualitative classification [7]. On the other hand, knots with a particularly unfavorable location, large size, and occurring in clusters may become the reason for the elimination of Copyright: © 2021 by the authors. sawn timber from construction applications. Therefore, in some aspects, it is more reliable Licensee MDPI, Basel, Switzerland. to analyze composite materials, e.g., CLT, where the influence of knots on the mechanical This article is an open access article parameters can be neglected. In the model developed for 5-layer CLT panels, the perfect distributed under the terms and connection between each layer (without taking into account the adhesive joint) and the conditions of the Creative Commons constancy of the mechanical parameters over the lamella thickness was assumed [8]. A Attribution (CC BY) license (https:// similar assumption was made in the analysis of notched connections in a composite system creativecommons.org/licenses/by/ of two materials: timber–concrete (TC system) [9]. Solid sawn timber was not analyzed, but 4.0/). Materials 2021, 14, 2780. https://doi.org/10.3390/ma14112780 https://www.mdpi.com/journal/materials
Materials 2021, 14, 2780 2 of 11 rather wood-based composite with homogeneous material parameters (LVL—laminated veneer lumber). The negative effect of knots is manifested in the reduction of tensile strength along the fibers, bending, compression along the fibers, and the modulus of elasticity [10]. This impact is caused, among others, by stress discontinuity resulting from the different ori- entation of the wood fibers, which results from the different orientations of the branches in relation to the trunk. Additionally, the fibers in the trunk show a specific deviation around the knot. After the branches die back and the knot falls out, there is no longer any continuity between the tissues of the branch and the trunk. Therefore, the knot can be considered to be a hole [11]. Another issue is the deviation of the surrounding fibers in the knot itself in the plane tangent to the log, resulting from natural growth processes. The deflection of the fibers in the tangential plane generates shear stresses and stresses perpendicular to the fibers, even during axial loading [12]. In addition, the reduction in strength is also caused by the deflection of the knot fibers in the plane radial to the log [13]. Due to the complexity of the geometrical aspects of the fiber course, it becomes necessary to introduce some simplifications. The studies of wood on a macro scale show that analysis taking into account the simplification of knots to holes and the characteristics of the angle of fiber deviation only in the tangential plane is sufficiently accurate and does not cause significant errors in the estimation of strength parameters [14–16]. An analogous error in estimating the bearing capacity is obtained when performing analyses that do not take into account fiber deviation in any plane and assuming the analogy of the knot and the hole [17]. Other modeling methods using a material with a higher density, fully or partially adhering to the surrounding wood, and with a different orientation of the main material axes show a greater error in the estimation of strength properties [18,19]. Nevertheless, models including fiber deformation around knots are being developed. The assumptions for the analysis were taken from the theory of laminar flow around a still obstacle [10,20]. In one model, steeped elliptic obstacles reflected knots, and the fluid flow path represented fiber deflection [21–24]. Another model involving flow-grain analogy was suggested by Guindos [25]. There are many models developed to estimate the load capacity of a solid timber element, weakened by the presence of structural defects: usually knots and/or fiber deviation. There is a clear shortage of models representing composite materials, for example, timber reinforced with FRP (fiber reinforced polymers), and even more locally, not along the entire length of the reinforced element. The aim of this research was to develop a simplified model of NSM locally reinforced sawn timber elements. To achieve this aim, it was necessary to validate a model with the results obtained experimentally and analytically, with such comparisons being made at different levels. Validation was possible by verifying the displacement values, comparing the stress and strain distribution determined analytically on the basis of the data obtained from experimental and numerical investigation, and comparisons of the local bending stiffness related to sections weakened by the presence of knots calculated from such strain distribution [26]. 2. Experimental Details 2.1. Material The test materials were Scots pine (Pinus sylvestris L.) sawn timber beams. The nominal cross-section dimensions of the tested timber batch after drying and planing were 50 mm × 100 mm, the length was 2000 mm. A total number of 60 beams were subjected to the study. The reinforcement material was 1.4 mm thick CFRP strips (CFRP S&P Lamelle CFK 150.2000). The CFRP was selected as a reinforcement material because of its high strength parameters (Table 1), easy availability, and widespread use in the building industry.
Materials 2021, 14, 2780 3 of 11 Materials 2021, 14, x FOR PEER REVIEW 3 of 11 Materials 2021, 14, x FOR PEER REVIEW 3 of 11 Table 1. Properties of the CFRP strips (data provided by S&P Polska, Malbork, Poland). Table 1. Properties of the CFRP strips (data provided by S&P Polska, Malbork, Poland). Table 1. Properties of the CFRP strips (data provided by S&P Polska, Malbork, Poland). Property CFRP Strip Property CFRP Strip density Property (kg/m 3) CFRP 1500 Strip density (kg/m 3) 1500 Young’s modulus (kg/m3)2 ) 2 density (N/mm >1651500 Young’s modulus (N/mm 2) )2 >165 tensile strength Young’s (N/mm modulus (N/mm ) >2800 >165 tensile strength (N/mm2)2 >2800 tensile strength (N/mm ) >2800 2.2. Methods 2.2. Methods 2.2. Methods All beams, beams, before before being being reinforced, reinforced, were All were tested tested forfor elastic elastic range range by by four four point pointbending bend- (Figure All1)beams, in orderbefore to being reinforced, determine the wereproperties elastic tested for elasticof the range timber bybeams four point andtobend- touse use ing (Figure 1) in order to determine the elastic properties of the timber beams and ingbending the (Figure 1) in orderastoa determine stiffness the elastic characterization of properties their initial of the timber quality [27]. beams The and to use experimental the bending stiffness as a characterization of their initial quality [27]. The experimental the bending program stiffness assumed as a by testing characterization bending60 60solidof their solid wood initial qualityThen, samples. [27]. The experimental program assumed testing by bending wood samples. Then, allallofofthethesamples samples program assumed testing by bending 60 solid wood samples. Then, all of knot the samples were weakened with a borehole of 20 mm in diameter, simulating a centric knot inin were weakened with a borehole of 20 mm in diameter, simulating a centric order order were weakened with a borehole ofof20knots mm in diameter, simulating a centric was knotplaced in order to investigate the effect of the size of knots in timber beams. The borehole was placed inin to investigate the effect of the size in timber beams. The borehole to investigate the middle of of the effect ofspan, the beam’s beam’s the size of knots inthe the tensilein zone. timberAgain, beams.samples The borehole was weretested testedplaced in the middle span, in tensile zone. Again, samples were bybyfourfour the middle point bending of bending to the beam’s to verify verifythe span, thechange in changein the tensile instiffness zone. stiffnessproperties. Again, properties.Then, samples Then,half were halfofofthe tested thesamples samples by four were point were point bendingbytoinserting strengthened verify thethe change in stiffnessmaterial reinforcement properties. insideThen, the half of the samples cross-section and were gluing strengthened by inserting the reinforcement material inside the cross-section and gluing strengthened with two-componentby insertingepoxy theresin. reinforcement shapematerial Theshape inside the cross-section and gluing with two-component epoxy resin. The ofofthe thereinforcement reinforcement reflectedthe reflected thecircular circular with two-component segment. The reinforcementepoxy was resin. The shapeinto introduced of the a reinforcement previously reflected prepared slot, the circular made with segment. The reinforcement was introduced into a previously prepared slot, made with aa segment. circular sawThe reinforcement with a diameter was of 250introduced mm (Figureinto2).a The previously thicknessprepared of the slot, made glueline, with a resulting circular saw with a diameter of 250 mm (Figure 2). The thickness of the glueline, resulting circular from the saw with a diameter differences in of 250 mmthe (Figureand 2). The reinforcement, thickness of the glueline, resulting from the differences in the the thickness thickness of of the slot slot and the the reinforcement,was was0.2 0.2mm. mm.Due Dueto from the the fact thatdifferences the in the thickness reinforcement method ofassumes the slotthe andstrengthening the reinforcement, of the was 0.2zone tensile mm.of Due the to the fact that the reinforcement method assumes the strengthening of the tensile zone of to thesection, fact thatthe theslot reinforcement method assumes the strengthening of the tensile zone of the cross section, the slot depth was 50 mm. Then, all samples were conditioned for about a cross depth was 50 mm. Then, all samples were conditioned for about the cross section, the slot depth was 50 mm. Then, all samples were conditioned for about amonth monthunder undernormalnormalclimate climateconditions, conditions,samples sampleswere weretestedtestedagain againby byfour fourpoint pointbending bend- a month under normal climate tests conditions, samples were tested again by four point bend-a ing until destruction. All bending tests were conducted using a displacement control with until destruction. All bending were conducted using a displacement control with ing until destruction. All bending tests were conducted using a displacement control with aspeed speedrate rateequal equaltoto3.0 3.0mm/min. mm/min. a speed rate equal to 3.0 mm/min. Figure 1. Loading and measuring system for four point pointbending. Figure1.1.Loading Figure Loading and and measuring measuring system system for for four four point bending. bending. Figure 2. Near-surface mounted (NSM) local reinforcement with a CFRP strip. Figure2.2.Near-surface Figure Near-surface mounted mounted (NSM) (NSM) local local reinforcement reinforcement with with aa CFRP CFRP strip. strip.
Materials 2021, 14, 2780 4 of 11 Materials 2021, 14, x FOR PEER REVIEW 4 of 11 2.3. 2.3.Results Results 3 3 , while the MC The Theaverage averagedensity densityofofall allsamples samplestested testedwas was563 kg/m3 ±±62.5 563kg/m 62.5kg/m kg/m 3, while the MC was 11.5± was11.5 1.8% (Table ± 1.8% (Table 2). 2). Table2.2.Laboratory Table Laboratorytest test results results (standard (standard deviation is given deviation is given in inparentheses). parentheses). All All Weakened Weakened Reinforced Reinforced Property Property Samples Samples Samples Samples Samples Samples 3 563 563 556 556 561 561 density density(kg/m (kg/m3)) (62.5) (62.5) (69.7) (69.7) (55.3) (55.3) 51,993 48,140 46,446 bending stiffness (Nm2 ) 51,993 48,140 46,446 bending stiffness (Nm2) (9557) (9688) (8641) (9557) (9688) 24.4 (8641) 41.9 bending strength (N/mm2 ) - 24.4 (7.5) 41.9 (7.7) bending strength (N/mm2) - (7.5) (7.7) The bending strength of the reinforced samples was significantly higher (by more The bending strength of the reinforced samples was significantly higher (by more than 70%, one-way ANOVA analysis) than the bending strength obtained for the samples than 70%, one-way ANOVA analysis) than the bending strength obtained for the samples weakened with a borehole. Typically, the failure was caused by exceeding the shear weakened with a borehole. Typically, the failure was caused by exceeding the shear strength, followed by the crack propagation along the fibers (Figure 3). In view of the strength, followed by the crack propagation along the fibers (Figure 3). In view of the destruction occurring in wood within the elasic range, at a high load value (and not in destruction occurring in wood within the elasic range, at a high load value (and not in the the reinforcing material or in the joint), it can be assumed that the shape, location, and reinforcing material or in the joint), it can be assumed that the shape, location, and parameters of the reinforcing material were optimal and correctly selected. parameters of the reinforcing material were optimal and correctly selected. Figure 3. Failure mode in the reinforced samples. Figure 3. Failure mode in the reinforced samples. Localreinforcement Local reinforcement in in the the shape shape ofof aa segment segmentof ofaacircle, circle,using usingaamaterial materialofofadequate adequate strength, is a very effective method for reinforcement. It is related tothe strength, is a very effective method for reinforcement. It is related to theincrease increaseininthe the surfaceof surface ofthe the glue glue joint joint and and the the shape shape ofofthe thereinforcement reinforcementitself, itself,which whichresults resultsininthe thelack lack oflocal of localstress stressconcentrations concentrationsappearing appearingininplace placeofofthe thenotches. notches.Additionally, Additionally, thanks thanks to to the the possibility of hiding the reinforcement inside the cross-section, and thus possibility of hiding the reinforcement inside the cross-section, and thus the high aesthetics the high aes- thetics of of the method, the method, this typethis of type of reinforcement reinforcement can be can used beinused in historical historical buildings. buildings. 3.3.Analytical Analytical Analysis Analysis Thevalues The valuesofofstresses stressesand andstrains strainswere were calculated calculated according according to to thethe gamma gamma method method [28]. [28].stress The The stress and strain and strain distribution distribution was determined was determined in sections in five five sections of bending of the the bending el- element ement (Figures (Figures 4 and 4 and 5). The5). The effective effective bending bending stiffness stiffness can becan be determined determined usingusing Formula Formula (1): (1): ni ( EI )e f = ∑ Ei Ii + γi Ei Ai a2i (1) i =1 where:
( ) = ( + ) (1) Materials 2021, 14, 2780 5 of 11 where: ( )( EI−effective bending )e f —effective stiffness bending [Nmm stiffness [Nmm 2], 2 ], —number ni —number of different materials of different within materials within the section the section [-], [-], —modulus Ei —modulus of elasticity of the of elasticity of i-cross section the i-cross [N/mm section [N/mm2], 2 ], —moment Ii —moment of inertia of the of inertia of i-cross section the i-cross [mm[mm section 4 4 ], ], —area of the Ai —area ofi-cross-section the i-cross-section[mm[mm2], 2 ], —reduction γi —reduction factor [-], [-], factor —distance ai —distance fromfrom thethe middle middle plane plane of the of the whole whole cross-section cross-section to the to the middle middle plane plane of of the the analyzed analyzed section section [mm]. [mm]. Normal Normalstresses stresseswere werecalculated calculatedfrom fromEquations Equations(2) (2)and and(3):(3): γ ××Ei ××ai × ×M σi = = i (2) (2) ( EI( ) )e f 0,5××Ei × h×i × 0.5 ℎ M× ,== σm,i ( EI( ) )e f (3) (3) where: where: 2 ], σi —normal stresses (component 1) [N/mm −normal stresses (component 1) [N/mm2], 2 ], σm,i —normal stresses (component 2) [N/mm , −normal stresses (component 2) [N/mm2], hi —height of a given section [mm]. ℎ −height of a given section [mm]. Strain value can be calculated using Equation (4): Strain value can be calculated using Equation (4): M×z ε = ×1 (4) =( EI )e f (4) ( ) where: where: ε—strain [‰], —strain [‰], z1 —distance from the neutral axis of the section to the point for which the deforma- —distance from the neutral axis of the section to the point for which the deformations tions are calculated [mm]. are calculated [mm]. Figure4.4.Local Figure Localreinforcement reinforcementof ofaasawn sawntimber timberelement, element,A-A–E-E–sections A-A–E-E–sectionswere wereanalyzed analyzedwith withthe the use of the gamma method. use of the gamma method.
Materials 2021, 14, 2780 6 of 11 Materials 2021, 14, x FOR PEER REVIEW 6 of 11 Figure 5. Normal stress andFigure strain 5. distribution in various Normal stress sections and strain of the NSM distribution reinforced in various timber sections beam; of the whereas: NSM (a–e)timber reinforced section A-A–section E-E respectively, cross-section dimensions are in (mm), normal stress in (N/mm 2), and strain in (‰). beam; whereas: (a–e) section A-A–section E-E respectively, cross-section dimensions are in (mm), normal stress in (N/mm2 ), and strain in (‰).
Materials 2021, 14, 2780 7 of 11 4. FEM Analysis Numerical tests were carried out in the Abaqus 6.13 environment (Dassault Systemes, Waltham, MA, USA) using the following assumptions and input data: E1 : E2 : E3 ≈ 20 : 1.6 : 1 (5) G12 : G13 : G23 ≈ 10 : 9.4 : 1 (6) E1 : G12 ≈ 14 : 1 (7) where: E1 —modulus of elasticity in the longitudinal direction (x axis) (N/mm2 ), E2 —modulus of elasticity in the radial direction (y axis) (N/mm2 ), E3 —modulus of elasticity in the tangential direction (z axis) (N/mm2 ), G12 , G13 , G23 —Kirchhoff modules in the xy, xz, and yz directions, respectively (N/mm2 ). The values of the Poisson coefficients and the yield limits of Scots pine wood were adopted on the basis of literature reports [29,30]. The created FEM model refers to the internal reinforcement (CFRP strip) in the shape of a segment of a circle. The FEM model was developed on the basis of three representative real sawn timber elements, weakened with a borehole simulating a knot and internally reinforced with CFRP. The thickness of the applied adhesive joint was 0.2 mm, while the thickness of the reinforcing tape was 1.4 mm. Necessary parameters, such as the beam’s dimensions, bending stiffness, and bending strength, were adopted from laboratory tests; the other remaining parameters necessary for the analysis were taken from literature or from calculations from the available data (Tables 3–6). Table 3. The yield limits of Scots pine wood [30]. Parameter σ11 σ22 σ33 σ12 σ13 σ23 σ0 N/mm2 15.5 3.6 3.6 6.0 6.0 3.0 15.5 Table 4. Material parameters of the sawn timber beams. E1 E2 E3 µ12 µ13 µ23 G12 G13 G23 Beam/Parameter N/mm2 N/mm2 N/mm2 - - - N/mm2 N/mm2 N/mm2 Beam 1 13,500 1080 675 0.42 0.37 0.47 964 906 96 46 × 100 × 1995 mm3 Beam 2 13,050 1044 653 0.42 0.37 0.47 932 876 93 47 × 100 × 2002 mm3 Beam 3 10,663 853 533 0.42 0.37 0.47 762 716 76 47 × 100 × 2002 mm3 Table 5. Material parameters of the epoxy resin [data provided by Havel Composites]. E µ Parameter N/mm2 - 3000 0.3 Table 6. Material parameters of the CFRP strips [data provided by S&P Polska]. E1 E2 E3 µ12 µ13 µ23 G12 G13 G23 Parameter N/mm2 N/mm2 N/mm2 - - - N/mm2 N/mm2 N/mm2 165,000 10,000 10,000 0.3 0.3 0.03 5000 5000 500
Materials 2021, 14, 2780 8 of 11 Materials 2021, 14, x FOR PEER REVIEW 8 of 11 Materials 2021, 14, x FOR PEER REVIEW 8 of 11 Scots pine sawn timber was represented by square tetrahedral, 10-node spatial el- ements (C3D10). The types of the of FEM elements were automatically selected and coating elements optimized by the (S4). The FEM software. Themodel of beam adhesive joint 1and consisted of 58,548 the CFRP elements strip were and 114,908 modeled as four- coating nodes node elements (Figure coating 6), (S4). whileThe elements theFEM (S4). Themodel models of of the FEM beam 1 consisted remaining model of beambeamsof had 58,548 elements of 58,548and a comparable 1 consisted 114,908and number elements of nodes (Figure elements 114,908 6), and nodes nodes while thewhile to beam (Figure 6), models 1. the of models the remaining of the beams had abeams remaining comparable had anumber of comparable elements number ofand nodes and elements to beam 1. to beam 1. nodes Figure 6. FEM 6. mesh applied FEM applied to the analyzed mesh applied element. to the analyzed Figure Figure 6. FEM mesh to the analyzed element.element. Due Dueto Due tothe to thefact the factthat fact thatinin that inbending bendingtests, bending tests,the tests, thecrack the crackpropagation crack propagationtypically propagation occurs typically typically occurs occurs within within within the elastic the range, the elastic elastic range,it was range, it decided was decided it was not decided not to model not to to model the modelthe plastic theplasticbehavior plasticbehavior of the behaviorofofthe pine thepinewood. pinewood. Based Basedon wood.Based failure on modes on failure failure modes modesanalysis (Figure analysis analysis (Figure (Figure3),3), it was 3), ititwas wasassumed assumed assumed that the that that the the maximum maximumdeformation maximum leading deformationlead- deformation lead- to ingcrack to initiation crack did initiation not did exceed not the exceed maximum the maximum deformation transferred deformation ing to crack initiation did not exceed the maximum deformation transferred by the CFRP by transferred the by CFRP the strip. CFRP strip. strip.As a failure criterion, the following principle was applied: As As aa failure failure criterion, criterion, thethe following following principle principlewas wasapplied: applied: ε FRP ≥ ε max (8) ≥≥ (8) (8) where: where: where:ε max —maximum deformation at which crack initiation occurs [‰]. −maximum −maximum deformation at which which crackcrackinitiation initiationoccurs occurs[‰]. [‰]. Figures 7 anddeformation at 8 show the individual stress components obtained based on the numeri- Figures Figures 7 and 8 show the individual stress components the individual stress components obtained based obtained basedon onthe thenu- nu- cal tests of beam 1. In order to illustrate the results as clearly as possible, the figures show merical merical tests of beam 1. In order to illustrate the results as clearly as possible, order to illustrate the results as clearly as possible, the figuresthe figures views in various planes. show show views views in various planes. planes. Figure 7. Figure 7. Normal Normal stress stress distribution distribution in in the thereinforced reinforcedelement. element. Figure 7. Normal stress distribution in the reinforced element.
Materials 2021, 14, 2780 9 of 11 Materials 2021, 14, x FOR PEER REVIEW 9 of 11 Materials 2021, 14, x FOR PEER REVIEW 9 of 11 Figure 8. Shear stress distribution in the reinforced element. Figure 8. Shear stress distribution in the reinforced element. Figure 8. Shear stress distribution in the reinforced element. Strain distributions obtained on the basis of FEM analysis and the analytical model Strain distributions obtained on the basis of FEM analysis and the analytical model (gamma method) Strain showed obtained distributions a very goodon thecorrelation, although basis of FEM the strain analysis and the distribution analytical ob- model (gamma method) showed a very good correlation, although the strain distribution obtained tained (gamma by the gamma method) method showed was typically a very rectilinearalthough good correlation, (Figure 9). theThe FEMdistribution strain model was ob- by the gamma method was typically rectilinear (Figure 9). The FEM model was validated validated tained byinthe terms of the method gamma deflection values was at destructive typically force rectilinear (Table9). (Figure 7).The There were FEM slightwas model in terms of the deflection values at destructive force (Table 7). There were slight differences differences validated between in terms deflection of the values, deflection not exceeding values at 15% force destructive of the(Table actual7). deflection There value were slight between deflection values, not exceeding 15% of the actual deflection value obtained during obtained during differences laboratory between tests, thus deflection it can values, notbeexceeding assumed that 15% the of FEM the model shows actual a high deflection value laboratory tests, thus it can be assumed that the FEM model shows a high compliance with compliance obtained with the actual during laboratory behavior of tests, thus the reinforced it can under element be assumed that under loading conditions. the FEM model shows a high the actual behavior of the reinforced element loading conditions. compliance with the actual behavior of the reinforced element under loading conditions. Figure 9. Comparison of the strain obtained on the basis of the FEM and analytical models. Figure Table 7. 9. Comparison Comparison of the of the strain obtained deflection on the basis values obtained ofbasis on the the FEM and of the analytical laboratory andmodels. numeri- Figure 9. Comparison of the strain obtained on the basis of the FEM and analytical models. cal tests. Table 7. Comparison of the deflection values obtained on the basis of the laboratory and numeri- Table 7. Comparison of the deflection values obtained on the basis calBeam/Parameter tests. Deflection (mm) and numeri- of the laboratory Destructive Force (N) cal tests. Experimental FEM Beam 1 Beam/Parameter 11,030 Force (N) Destructive 19.1 Deflection (mm) 17.2 Deflection (mm) Experimental Beam 2 Beam/Parameter 11,369 Destructive Force (N) 23.6 20.3FEM Beam Beam 3 1 11,030 10,214 19.1 Experimental 27.2 24.617.2 FEM Beam Beam 1 2 11,369 11,030 19.1 23.6 17.220.3 Beam 2 3 Beam 5. Conclusions 11,369 10,214 23.6 27.2 20.324.6 Beam 3 10,214 27.2 24.6 Local reinforcement in the shape of a segment of a circle using a material of adequate 5. Conclusions strength is an effective method for reinforcement. It is related to the increase in the surface 5. ofConclusions the Local glue joint and the shape reinforcement of shape in the the reinforcement of a segmentitself, whichusing of a circle results in a lackof a material ofadequate local stress concentrations Local strength reinforcement is an effectiveappearing method in forthe in the shape place of of the notches. a segment reinforcement. of a circle It is Additionally, relatedusing to the due to a material increase inthe of thepos- adequate surface sibility of the of strength ishiding glue anjointthe effective andreinforcement method the shape inside the cross-section forofreinforcement. the reinforcement It is relatedand itself, tothus whichthe the highin increase results aesthetics ina the ofoflocal lacksurface the of method, the stressglue this and joint typethe concentrations ofappearing reinforcement shape of the in the can be used reinforcement place of theinitself, historic notches. buildings. which results in adue Additionally, lacktoofthelocal pos- stress The sibility ofvalidation concentrations hiding the ofreinforcement the developed appearing in the model place inside showed the of the anotches. very good cross-section agreement andAdditionally, thus the high indue terms to of aestheticstheof the thecomparison possibility method, of type of hiding this the thedeflection values, reinforcement of reinforcement stress inside beand can the strain cross-section used distribution, in historic and thusand the bending buildings. stiff- high aesthetics ness of theprofiles. method, However, this type the of modeling reinforcement parameter can be was used inset to three historic The validation of the developed model showed a very good agreement in terms of sawn buildings. timber beams. Therefore, theThe furtherof validation comparison model of the the validation developed deflection is needed, values,model including showed stress and strain a testing very program good for andsubsequent agreement distribution, in terms bending stiff- sawn ofness timber the comparisonelements. Among the of the deflection profiles. However, the modelingreasons values, for the need stress and parameter for wasstrain the future development set todistribution, three sawn and timber and bending beams. Therefore, further model validation is needed, including a testing program for subsequent sawn timber elements. Among the reasons for the need for the future development and
Materials 2021, 14, 2780 10 of 11 stiffness profiles. However, the modeling parameter was set to three sawn timber beams. Therefore, further model validation is needed, including a testing program for subsequent sawn timber elements. Among the reasons for the need for the future development and improvement of the model is the remarkable heterogeneity of the wood structure, which causes a significant variation in the values of the physical and mechanical parameters. When it comes to antique wood, the topic is even more important. The heterogeneity of wood may be additionally increased due to the presence of biological corrosion, cracks, or locally increased wood moisture. Due to the large possible variability of the data, it is difficult to develop a universal model for predicting the properties of wood. It must be noted that under both models (analytical and numerical), the mechanical properties were obtained for each individual beam with laboratory conditions. This approach is unactable in the case of practical and commercial applications. It must also be emphasized that the developed model should not be applied to predict the load capacity when assuming real timber structures, due to the more complex loading conditions, such as cyclic loads. As a consequence of the variety of loading possibilities, more research must be performed in the future to develop more consistent and universal models of internally reinforced timber elements. Many possibilities become real once a verified model is developed. One application is repair engineering caused by the need for the reinforcement of local cross-section disconti- nuity. FEM analysis allows us to study timber elements much more easily and thoroughly compared to laboratory experiments. By numerically differentiating the configuration of the critical, weakened area and the crucial range of the local reinforcement, it is possible to successfully study the simultaneous influence of many parameters, which would be unattainable or very expensive when testing in the laboratory. Author Contributions: Conceptualization, I.B.-K.; methodology, I.B.-K.; software, I.B.-K.; validation, I.B.-K.; formal analysis, I.B.-K.; investigation, I.B.-K.; data curation, I.B.-K.; writing—original draft preparation, I.B.-K.; writing—review and editing, I.B.-K. and P.B.; visualization, I.B.-K.; supervision, P.B.; project administration, I.B.-K.; funding acquisition, I.B.-K. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. 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. Conflicts of Interest: The authors declare no conflict of interest. References 1. Kretschmann, D.E. Mechanical Properties of Wood—Wood Handbook; Forest Products Laboratory, United States Department of Agricultural Forest Service: Madison, WI, USA, 2010. 2. Johansson, C.J.; Boström, L.; Bräuner, L.; Hoffmeyer, P.; Holmquist, C.; Solli, K.H. Laminations for Glued Laminated Timber— Establishment of Strength Classes for Visual Strength Grades and Machine Settings for Glulam Laminations of Nordic Origin; A Nordic Wood Project, Sp. Report 38; SP Swedish National Testing and Research Institute: Borås, Sweden, 1998. 3. Teischinger, A.; Patzelt, M. XXL-Wood. Materialkenngrößen als Grundlage für Innovative Verarbeitungstechnologien und Produkte zur Wirtschaftlichen Nachhaltigen Nutzung der Österreichischen Nadelstarkholzreserven. In Berichte aus Energie und Umweltforschung; 27/2006; Universität für Bodenkultur Wien: Wien, Austria, 2006. 4. Mettem, C.J. The Principles Involved in Stress Grading with Special Reference to Its Application in Developing Countries; UNIDO—United Nations Industrial Development Organization: Vienna, Austria, 1981. 5. Ridley-Ellis, D.; Stapel, P.; Baño, V. Strength grading of sawn timber in Europe: An explanation for engineers and researchers. Eur. J. Wood Wood Prod. 2016, 74, 291–306. [CrossRef] 6. Johansson, C.J. Grading of Timber with Respect to Mechanical Properties. In Timber Engineering; Thelandersson, S., Larsen, H.J., Eds.; Wiley & Sons: London, UK, 2003; pp. 23–43. 7. PN-D-94021. Tarcica Konstrukcyjna Iglasta Sortowana Metodami Wytrzymałościowymi; Polski Komitet Normalizacyjny: Warsaw, Poland, 2013.
Materials 2021, 14, 2780 11 of 11 8. Martínez-Martínez, J.E.; Alonso-Martínez, M.; Rabanal, F.P.Á.; Díaz, J.J.d.C. Finite Element Analysis of Composite Laminated Timber (CLT). Proceedings 2018, 2, 1454. [CrossRef] 9. Bedon, C.; Fragiacomo, M. Three-Dimensional Modelling of Notched Connections for Timber—Concrete Composite Beams. Struct. Eng. Int. 2017, 27, 184–196. [CrossRef] 10. Phillips, G.E.; Bodig, J.; Goodman, J.R. Flow grain analogy. Wood Sci. 1981, 14, 55–64. 11. Baňo, V. Numerical Analysis of Wood Strength Loss Due to Presence of Knots. Ph.D. Thesis, Polytechnic University of Madrid, Madrid, Spain, 2009. 12. Guindos, P.; Guaita, M. The analytical influence of all types of knots on bending. Wood Sci. Technol. 2014, 48, 533–552. [CrossRef] 13. Foley, C. A three-dimensional paradigm of fiber orientation in timber. Wood Sci. Technol. 2001, 35, 453–465. [CrossRef] 14. Guindos, P.; Guaita, M. A three-dimensional wood material model to simulate the behavior of wood with any type of knot at the macro-scale. Wood Sci. Technol. 2013, 47, 585–599. [CrossRef] 15. Lukacevic, M.; Füssl, J.; Lampert, R. Failure mechanisms of clear wood identified at wood cell level by an approach based on the extended finite element method. Eng. Fract. Mech. 2015, 144, 158–175. [CrossRef] 16. Lukacevic, M.; Lederer, W.; Füssl, J. A microstructure-based multisurface failure criterion for the description of brittle and ductile failure mechanisms of clearwood. Eng. Fract. Mech. 2017, 176, 83–99. [CrossRef] 17. Oscarsson, J.; Olsson, A.; Enquist, B. Strain fields around knots in Norway spruce specimens exposed to tensile forces. Wood Sci. Technol. 2012, 46, 593–610. [CrossRef] 18. Baňo, V.; Arriaga, F.; Guaita, M. Determination of the influence of size and position of knots on load capacity and stress distribution in timber beams of Pinus sylvestris using finite element model. Biosyst. Eng. 2013, 114, 214–222. [CrossRef] 19. Baňo, V.; Arriaga, F.; Soilan, A.; Guaita, M. Prediction of bending load capacity of timber beams using a finite element method simulation of knots and grain deviation. Biosyst. Eng. 2011, 109, 241–249. [CrossRef] 20. Goodman, J.R.; Bodig, J. Mathematical Model of the Tension Behaviour of Wood with Knots and Cross Grain. In Proceedings of the First International Conference on Wood Fracture, Banff, AB, Canada, 14–16 August 1978. 21. Foley, C. Modelling the Effects of Knots in Structural Timber. Ph.D. Thesis, Lund University, Lund, Sweden, 2003. 22. Konopka, D.; Gebhardt, C.; Kaliske, M. Numerical modelling of wooden structures. J. Cult. Herit. 2016, 27, 93–102. [CrossRef] 23. Jockwer, R.; Serrano, E.; Gustafsson, P.-J.; Steiger, R. Impact of knots on the fracture propagating along grain in timber beams. Int. Wood Prod. J. 2017, 8, 1–6. [CrossRef] 24. Lang, R.; Kaliske, M. Description of inhomogeneities in wooden structures: Modelling of branches. Wood Sci. Technol. 2013, 47, 1051–1070. [CrossRef] 25. Guindos, P. Method for the integral calculation of the fiber orientation and the fundamental material properties of softwood logs and lumber. Holzforschung 2016, 70, 981–991. [CrossRef] 26. Hu, M.; Olsson, A.; Johansson, M.; Oscarsson, J. Modelling local bending stiffness based on fibre orientation in sawn timber. Eur. J. Wood Wood Prod. 2018, 76, 1605–1621. [CrossRef] 27. EN 408 +A1. Timber Structures. Structural Timber and Glued Laminated Timber. Determination of Some Physical and Mechanical Properties; European Committee for Standardization (CEN): Brussels, Belgium, 2012. 28. EN 1995-1-1. Eurocode 5: Design of Timber Structures—Part 1-1: General—Common Rules and Rules for Buildings; European Committee for Standardization (CEN): Brussels, Belgium, 2010. 29. Kollmann, F.F.P.; Côté, A.C.J. Principles of Wood Science and Technology: Solid Wood; Springer: Berlin, Germany, 1968. 30. Nowak, T. Analiza Pracy Statycznej Zginanych Belek Drewnianych Wzmacnianych Przy Użyciu CFRP. Ph.D. Thesis, Wroclaw University of Science and Technology, Wrocław, Poland, 2007.
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