Predictive Models for Elastic Bending Behavior of a Wood Composite Sandwich Panel

Article Predictive Models for Elastic Bending Behavior of a Wood Composite Sandwich Panel

Mostafa Mohammadabadi 1, James Jarvis 2, Vikram Yadama 3,* and William Cofer 3

1 Material Science and Engineering Program and Composite Materials and Engineering Center, Washington State University, Pullman, WA 99164, USA; [email protected] 2 DCI Engineers, San Francisco, CA 94105, USA; [email protected] 3 Department of Civil and Environmental Engineering and Composite Materials and Engineering Center, Washington State University, Pullman, WA 99164, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-509-335-6261

Received: 24 April 2020; Accepted: 22 May 2020; Published: 1 June 2020

Abstract: Strands produced from small-diameter timbers of lodgepole and ponderosa pine were used to fabricate a composite sandwich structure as a replacement for traditional building envelope materials, such as roofing. It is beneficial to develop models that are verified to predict the behavior of these sandwich structures under typical service loads. When used for building envelopes, these structural panels are subjected to bending due to wind, snow, live, and dead loads during their service life. The objective of this study was to develop a theoretical and a finite element (FE) model to evaluate the elastic bending behavior of the wood-strand composite sandwich panel with a biaxial corrugated core. The effect of shear deformation was shown to be negligible by applying two theoretical models, the Euler–Bernoulli and Timoshenko beam theories. Tensile tests were conducted to obtain the material properties as inputs into the models. Predicted bending stiffness of the sandwich panels using Euler-Bernoulli, Timoshenko, and FE models differed from the experimental results by 3.6%, 5.2%, and 6.5%, respectively. Using FE and theoretical models, a sensitivity analysis was conducted to explore the effect of change in bending stiffness due to intrinsic variation in material properties of the wood composite material.

Keywords: small-diameter timber; wood-strand composite; sandwich panel; bending stiﬀness; ﬁnite element model; theoretical model; building envelope

1. Introduction Increasing world population and limited natural resources require us to rethink how we utilize our forests more productively to construct effective building products for houses. Forests play a critical role in sequestering carbon and building products and can further this cause by continuing to store the carbon for a prolonged period. As an order of magnitude, a tree on an average absorbs one ton of CO2 and produces 0.7 tons of O2 for every cubic meter of growth [1], and every cubic meter of wood as a building material can reduce CO2 emissions by an average of 2 tons compared to other building materials such steel and concrete [2]. A sustainable forest management plan to protect forests against fire, insects, and disease should consider thinning operations that result in improving forest health. Often, this requires the removal of small-diameter timber (SDT) at a cost and requires high-value markets or high-volume usage of these low-quality logs to recover the forest treatment costs. For example, the average cost for a forest service thinning (approximately $70/dry ton) is usually more than the market value of the SDT removed (energy and chip markets pay approximately $25 to $35/dry ton) [3]. Besides existing products such as medium-density fiberboard (MDF) and oriented strand board (OSB) produced from SDT, it is worth developing new products by converting these

Forests 2020, 11, 624; doi:10.3390/f11060624 www.mdpi.com/journal/forests Forests 2020, 11, 624 2 of 16 low-quality SDTs into more robust and versatile building products that would meet the structural performance as well as energy requirements. Sandwich structures are widely used in many areas like aerospace, automotive, civil, building construction, and marine industries because their clever construction reduces their weight while increasing their mechanical performance [4]. However, those with hollow cores have attracted interest from researchers because the hollow geometry of the core can be used to improve the thermal performance when they are filled with appropriate materials, such as closed-cell foam. This idea has also been used to develop wood-based sandwich panels with different 3D core geometries [5–10]. Some of the disadvantages of these panels is that they use a wet forming process for panel manufacturing, have a relatively low structural capacity, and have poor interfacial shear strength at the intersection between the core and face plies resulting in poor structural performance. Voth et al. [11] designed a 3D core with biaxial corrugated geometry and used wood strands produced from SDT to fabricate the panel shown in Figure1. The bending behavior of wood-strand-based sandwich panels with biaxial Forests 2020, 11, x FOR PEER REVIEW 3 of 16 corrugated core geometry was investigated experimentally, and the results showed the stiffness in both directions along the length and width is higher than that of oriented strand board (OSB) and then bonded with a two-part epoxy resin (Loctite Epoxy by Henkel) to the biaxial corrugated cores. 5-ply plywood. Additionally, sandwich panels with cavities filled with insulation foam decreased the A 400-grit sandpaper was used for a light sanding of the bonding area of both the flat and corrugated thermal conductivity of the panels by over 17% while improving the panel strength and stiffness by 34 panels. A thin layer of resin was only applied to both sides of the corrugated core. Flat panels were and 16 percent [12]. Creep [13] and impact [14] performances of the biaxial corrugated core sandwich placed on both sides of the corrugated core to form a sandwich panel as shown in Figure 1g, and the panelpanels were were also clamped investigated for 24 hours experimentally. before cutting and trimming.

Figure 1. Fabrication process process:: ( (aa)) disc disc-strander,-strander, ( (bb)) blender blender,, ( c) orientor orientor,, (d) preform,preform, (e) hot-pressinghot-pressing using mold,mold, (f) components ofof sandwichsandwich panel,panel, ((g)) clampedclamped panels,panels, andand ((hh)) sandwichsandwich panel.panel.

ForA unit application cell (UC), of the these simplest sandwich repeating panels element as a structural of the biaxial component corrugated in building core envelopesgeometry, (roofs,along floors,with its and dimensions walls) with, is longergiven in spans, Figure the 2a. bending Directions stiff nessalong and the the length bond and area width between of the the panel core andare thedefined faces as need longitudinal to be increased and transverse as they directions. typically limit Wood the strands load-carrying were oriented capacity. along Models the longitudinal to predict theirdirection behavior to make are the useful preform tools tofor design both the and corrugated engineer thecore geometry and the flat of corespanels. and sandwich panels as long as they are verified and validated. Additionally, it is critical to accurately determine the material properties that serve as inputs to these models. The finite element method is an effective way to simulate complicated geometries under different loading and boundary conditions to predict the behavior and explore distribution of stress and strain in the structures. It is also useful to develop theoretical models to derive closed-form solutions to understand the influence of material properties and geometrical features on the behavior of the structures. These models can then be used to analyze the behavior of structural components subjected to service loads. In this study, a finite element model and a theoretical model were developed to evaluate the bending behavior of a wood-strand composite sandwich panel with a biaxial corrugated core geometry designed by Voth [11]. Elastic constants of the

Figure 2. Biaxial corrugated core (a) highlighted unit cell (b) geometric parameters and their values in mm.

L1, L2, and h are the length, width and height of the UC. Core wall thickness was defined as t. Y21, Y22, and X1 are associated with the dimensions of the bonding area between the flat panels and the core to fabricate a sandwich panel. The angle of slanted areas is represented by θ1 and θ2 as shown in Figure 2b.

3. Experiments This section is divided into two subsections. The first introduces the experiments conducted to establish material properties employed in the FE and theoretical models. The second part describes

Forests 2020, 11, 624 3 of 16

Forestswood-strand 2020, 11, x compositeFOR PEER REVIEW material required to develop both the finite element (FE) and the theoretical3 of 16 models were evaluated, and the models were verified against the experimental results. Considering thenthe intrinsic bonded naturewith a oftwo wood,-part theepoxy models resin were (Loctite further Epoxy applied by Henkel in a sensitivity) to the biaxial analysis corrugated to understand cores. Athe 400 influence-grit sandpaper of variation was inused elastic for a constants light sanding on the of bendingthe bonding stiffness area ofof theboth sandwich the flat and panel. corrugated panels. A thin layer of resin was only applied to both sides of the corrugated core. Flat panels were 2. Materials placed on both sides of the corrugated core to form a sandwich panel as shown in Figure 1g, and the panelsUsing were a clamped disc-strander for 24 (manufactured hours before cutting by CAE) and operating trimming. at a rotational speed of 500 rpm (shown in Figure1a), thin wood strands with an average thickness of 0.36 mm were produced from ponderosa pine (pp) logs ranging in diameter from 191 to 311 mm. Wood strands were then dried to a target moisture content of 3%–5% and sprayed with an aerosolized liquid phenol formaldehyde (PF) resin in a drum blender (shown in Figure1b) to a target resin content of 8% of the oven-dry weight of the wood strands. Subsequently, wood strands sprayed with resin were oriented and hand-formed unidirectionally using a forming box (shown in Figure1c) to fabricate a wood strand mat, or preform, as shown in Figure1d. Unidirectional mats were consolidated to a thickness of 6.35 mm into flat panels for the outer plies or corrugated core panels (shown in Figure1f) with a matched-die mold (shown in Figure1e) in a hot press. For both flat and corrugated panels, the preform was hot-pressed for 6 min at an operating temperature of 160 ◦C to reach a target thickness of 6.35 mm. Applied pressure by the press was an uncontrolled variable and was dependent on the density and the target thickness of the panel. To fabricate the sandwich panels as shown in Figure1f–h, flat panels were then bonded with a two-part epoxy resin (Loctite Epoxy by Henkel) to the biaxial corrugated cores. A 400-grit sandpaper was used for a light sanding of the bonding area of both the flat and corrugated panels. A thin layer of resin was only applied to both sides of the corrugated core. Flat panels were placed on both sides of the corrugatedFigure 1. Fabrication core to form process a sandwich: (a) disc-strander panel as, (b shown) blender in,Figure (c) orientor1g, and, (d) thepreform panels, (e were) hot-pressing clamped for 24 husing before mold cutting, (f) components and trimming. of sandwich panel, (g) clamped panels, and (h) sandwich panel. A unit cell (UC), the simplest repeating element of the biaxial corrugated core geometry, along A unit cell (UC), the simplest repeating element of the biaxial corrugated core geometry, along with its dimensions, is given in Figure2a. Directions along the length and width of the panel are with its dimensions, is given in Figure 2a. Directions along the length and width of the panel are defined as longitudinal and transverse directions. Wood strands were oriented along the longitudinal defined as longitudinal and transverse directions. Wood strands were oriented along the longitudinal direction to make the preform for both the corrugated core and the flat panels. direction to make the preform for both the corrugated core and the flat panels.

FigureFigure 2. 2. BiaxialBiaxial corrugated corrugated core core ( (aa)) highlighted highlighted unit unit cell cell ( (bb)) ge geometricometric parameters parameters and and their their values values inin mm mm..

LL11, ,LL22, ,and and hh are the length,length, widthwidth and and height height of of the the UC. UC. Core Core wall wall thickness thickness was was deﬁned defined as t .asY 21t. , Y2122, ,Y and22, andX1 areX1 are associated associated with with the the dimensions dimensions of theof the bonding bonding area area between between the the ﬂat flat panels panels and and the thecore core to fabricate to fabricate a sandwich a sandwich panel. panel. The The angle angle of slantedof slanted areas areas is represented is represented by byθ1 andθ1 andθ2 θas2 as shown shown in inFigure Figure2b. 2b.

Forests 2020, 11, 624 4 of 16

3. Experiments Forests 2020, 11, x FOR PEER REVIEW 4 of 16 This section is divided into two subsections. The first introduces the experiments conducted to theestablish experimental material bending properties tests employed performed in the to FEverify and the theoretical sandwich models. panel bending The second stiffness part describes predictions the ofexperimental the FE and theoretical bending tests models. performed to verify the sandwich panel bending stiffness predictions of the FE and theoretical models. 3.1. Tensile Test 3.1. Tensile Test Wood is an orthotropic material and requires nine elastic constants to be fully defined [15]. Wood is an orthotropic material and requires nine elastic constants to be fully defined [15]. However, because the fiber orientation in individual strands varies in the width direction with However, because the fiber orientation in individual strands varies in the width direction with respect respect to an idealized orthotropic material axes and the preform is a conglomeration of wood strands to an idealized orthotropic material axes and the preform is a conglomeration of wood strands oriented oriented uniaxially, transverse isotropy was assumed to define the material properties of both flat uniaxially, transverse isotropy was assumed to define the material properties of both flat plies and plies and the core [16]. Therefore, five elastic constants, including E1, E2 = E3, G12 = G23, ν12 = ν13, and ν23, the core [16]. Therefore, five elastic constants, including E , E = E , G = G , ν = ν , and ν , were required to fully define the sandwich panel. Subscripts1 1,2 2, and3 312 refer23 to the12 material13 axes23 were required to fully define the sandwich panel. Subscripts 1, 2, and 3 refer to the material axes (local (local coordinate system) of the composite material (Figure 2b). The 2–3 plane is the plane of isotropy coordinate system) of the composite material (Figure2b). The 2–3 plane is the plane of isotropy for for this transversely isotropic material. To determine these representative material properties, tension this transversely isotropic material. To determine these representative material properties, tension specimens shown in Figure 3a were cut from flat panels at different angles (0°, 15°, and 90°) with specimens shown in Figure3a were cut from flat panels at di fferent angles (0 , 15 , and 90 ) with respect to the longitudinal axis (x axis in Figure 2) and tested (Figure 3b) as per◦ the standard◦ ◦testing respect to the longitudinal axis (x axis in Figure2) and tested (Figure3b) as per the standard testing guidelines in ASTM D1037 [17]. Testing was performed using an 8.8-kN Instron test frame Model guidelines in ASTM D1037 [17]. Testing was performed using an 8.8-kN Instron test frame Model 4466, 4466, change in gauge length (elongation) was recorded using a 25-mm Epsilon extensometer, and change in gauge length (elongation) was recorded using a 25-mm Epsilon extensometer, and then then strain measurements were obtained by dividing the elongation by the initial gauge length (25 strain measurements were obtained by dividing the elongation by the initial gauge length (25 mm). mm).

Figure 3. MechanicalMechanical testing testing:: ( (aa)) tension tension specimens specimens,, ( (bb)) tensile tensile testing testing,, ( (cc)) bending bending of of corrugated corrugated core core,, and ( d) bending of sandwich beam.

Longitudinal andand transversetransverse Young Young modulus modulus (E 1(Eand1 andE2 E) were2) were obtained obtained from from test test results results of 0 ◦ofand 0° and90◦ specimens,90° specimens, respectively. respectively. These These material material properties, properties along, with along the with test resultsthe test of results coupons of preparedcoupons preparedat 15◦ angles, at 15° were angles used, towere determine used to thedetermine shear modulus, the shearG12 modulus,, using the G transformation12, using the transformation equation [15]. equation [15]. 1  1 cos4θ sin4θ   − 2υ12 1Eθ − E1 − E2  G12 =  +  (1) 2E (sin2θ)(4 cos2θ) 4 −1 = 1 + (1) ( 𝑐𝑐𝑐𝑐𝑐𝑐)(𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠) 𝜃𝜃 12 𝜃𝜃 − 1 − 2 12 𝜐𝜐 𝐸𝐸 𝐸𝐸 𝐸𝐸 where E1, E2, Eθ, υ12, and θ are modulus𝐺𝐺 � of elasticity in2 the longitudinal,2 � transverse, and θ-directions, 𝐸𝐸1 𝑠𝑠𝑠𝑠𝑛𝑛 𝜃𝜃 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 wherePoisson’s E1, ratio,E2, Eθ, and υ12, assumedand θ are angle modulus in this of studyelasticity (15 ◦in), respectively.the longitudinal, The transverse, value for Poisson’s and θ-direction ratio, υ12s,, Poisson’s ratio, and assumed angle in this study (15°), respectively. The value for Poisson’s ratio, υ12, was assumed to be 0.358, based on a previous study [18] which evaluated the mechanical properties of a wood strand composite material with similar structure and manufacturing process to the composite material used in this study. Among commercially produced wood-strand-based materials,

Forests 2020, 11, 624 5 of 16 was assumed to be 0.358, based on a previous study [18] which evaluated the mechanical properties of a wood strand composite material with similar structure and manufacturing process to the composite material used in this study. Among commercially produced wood-strand-based materials, structural composite lumber (SCL) [e.g., parallel strand lumber (PSL), oriented strand lumber (OSL), and laminated strand lumber (LSL)] are similar to the material in this study. They are manufactured using long veneer strips or wood strands that are also aligned in a longitudinal direction. Therefore, Poisson’s ratio, υ23, of PSL as determined by Clouston [19] was assumed for the material of interest in this study. Considering the transverse isotropy assumption resulting in material isotropy in plane 2–3, G23 was calculated as: E2 G23 = (2) 2(1 + υ23) Since wood is a natural material influenced by variations in growing conditions, its properties are influenced by naturally occurring growth characteristics, such as knots, variation in cell wall thickness between the annual rings, and grain deviations [20]. This inherent variability can significantly influence the properties of wood strands produced from logs or lumber, and this variation in turn can affect the properties of the composite material produced using these strands. Furthermore, variations in the mat architecture can also be expected due to variation in the orientation of the wood strands with respect to the longitudinal axis as the mat is formed. In addition, mat architecture would also influence the heat and mass transfer during the hot-pressing process due to variations in moisture and heat distribution within a mat [21]. Effort was made to reduce these variations by tightly controlling the processing variables and the press schedule during the production process. Yet, variation in density within a panel and between the panels after consolidation is inevitable and should be expected. As mentioned, this variation in density can affect the material properties obtained from tensile test. To explore the influence of density on the longitudinal modulus, panels with density varying between 600 and 950 kg/m3 were fabricated and submitted to tensile test.

3.2. Flexural Test To measureForests 20 the20, 1 bending1, x FOR PEER stiff REVIEWness, a four-point bending test was conducted as per ASTM D72496 [ of22 16] on corrugated coreForests and2020, 1 sandwich1, x FOR PEER panel REVIEW specimens, as shown in Figure3c,d, respectively. The bending6 of 16 specimens were cut in the longitudinal direction of the panel with the width of one UC. As per the standards, five specimens of each panel were tested. The bending stiffness (EI) was determined using Equation (3) and load-deflection results obtained from the bending test.

Pa a= L , m= P 23mL3 EI = 3L2 4a2 3 ∆ EI = (3) 48∆ − → 1296 where P, ∆, L, and a are bending load, deflection at mid-span, span length, and distance of loading point from the support, respectively. The configuration of the bending test is shown in Figure4. Table1 summarizes the dimensions of all flexural test specimens. Note that the specimen length in Table1 indicates the span length and does not include a 25.5-mm overhang at each end of the flexural specimens (i.e., total length of these specimens was 51 mm longer than span length). Deflection was measured using a 25-mm linear variable differential transformer (LVDT), located at the mid-span of ± the test specimens.Figure 4. Finite element (FE) model of 3-D core and sandwich panel in the longitudinal direction. Figure 4. Finite element (FE) model of 3-D core and sandwich panel in the longitudinal direction. Table 1. DimensionsTable 1. Dimensions of flexural of flexural specimens specimens used used in the in bending the bending test. test. Table 1. Dimensions of flexural specimens used in the bending test. Length LengthWidth WidthThickness Thickness HeightHeight SpecimenSpecimen Length Width Thickness Height SpecimenL (mm) L (mm)b (mm) b (mm) t (mm)t (mm) hh (mm) (mm) L (mm) b (mm) t (mm) h (mm) Corrugated Core 572 108 6 25 Corrugated CoreCorrugated 572 Core 572 108 108 66 25 25 SandwichSandwich Panel Panel 559 559 108108 —--- 3838 Sandwich Panel 559 108 --- 38

4. Finite4. Finite Element Element Model Model To evaluateTo evaluate the thebending bending behavior behavior of ofboth both the the sandwich sandwich panel panel and and thethe corrugatedcorrugated corecore withinwithin the elasticthe elastic region, region, FE modelsFE models were were developed developed using using Abaqus Abaqus finite finite element element softwaresoftware (Figure 4).4). For For both boththe flatthe outerflat outer layers layers and and the the core, core, shell shell elements elements (S4R) (S4R) with with hourglass hourglass contrcontrol and aa reducedreduced integrationintegration rule rule were were employed. employed. To Toeasily easily assign assign material material orientation orientation toto thethe corecore withwith complexcomplex geometry,geometry, a shell a shell element element was was chosen. chosen. Since Since this this type type of of elementelement considerablyconsiderably decreases thethe simulationsimulation and andrunning running time time of theof the FE FE model, model, a finera finer mesh mesh was was adopted adopted toto increaseincrease the accuracy.accuracy. In In addition,addition, all elements all elements were were given given the the capability capability to to undergo undergo finite finite strains strains and and rotations.rotations. ConsideringConsidering the thefabrication fabrication process, process, both both the the corrug corrugatedated core core andand thethe flatflat outer pliesplies werewere assumedassumed to be to transversely be transversely isotropic. isotropic. To To simulate simulate a perfecta perfect bond bond between between thethe faceface-sheets andand thethe 3 3-- D core, rigid links were modeled between the nodes of these two components using a tie constraint. D core, rigid links were modeled between the nodes of these two components using a tie constraint. As for the boundary conditions, the nodes in the contact area between the specimens and the supports As for the boundary conditions, the nodes in the contact area between the specimens and the supports were constrained in the z direction. Since the supports were free to rotate about the y axis to be were constrained in the z direction. Since the supports were free to rotate about the y axis to be consistent with ASTM D7249 [22], no other boundary condition was applied. However, to avoid consistent with ASTM D7249 [22], no other boundary condition was applied. However, to avoid instability in the structure, the centerline of the sandwich panel exactly at the mid-span was fixed to instabilityavoid anyin the translation structure, in the the centerlinex direction. of Loading the sandwich was applied panel as exactly prescribed at the downward mid-span deflection was fixed of to avoidthe any loading translation heads. in the x direction. Loading was applied as prescribed downward deflection of the loadingTo heads.determine an acceptable element size, a mesh convergence analysis was performed on the flexuralTo determine 3-D core an model. acceptable In Figure element 5, the size, results a mesh of this convergence convergence analysisstudy are was displayed, performed comparing on the flexuralthe 3bending-D core loadmodel. that In corresponds Figure 5, the to resultsa deflection of this of convergence 25.4 mm in the study center are ofdisplayed, the specimen comparing to the the bendingnumber ofload elements. that corresponds As the number to a of deflection elements increasedof 25.4 mm from in 4263 the centerto 59,814 of (correspondingthe specimen toto anthe numberelement of elements. size of 5 mmAs the to 1.3 number mm), thereof elements was a 1.73% increased change from in the 4263 resulting to 59,814 bending (corresponding load. Because to ofan elementthis sizenegligible of 5 mm change to 1.3 in mm), the therebending was load a 1.73% and changenoticeable in thesavings resulting in the bending computation load. Becausetime, the of this smallestnegligible number change of inelements the bending considered, load and4263 noticeable(element size savings of 5 mm),in the was computation chosen to meshtime, thethe smallestspecimens. number of elements considered, 4263 (element size of 5 mm), was chosen to mesh the specimens.

Forests 2020, 11, 624 6 of 16 Forests 2020, 11, x FOR PEER REVIEW 6 of 16

Figure 4.FigureFinite 4. elementFinite element (FE) ( modelFE) model of 3-Dof 3-D core core and and sandwichsandwich panel panel in inthe the longitudinal longitudinal direction. direction. 4. Finite Element Model Table 1. Dimensions of flexural specimens used in the bending test. Length Width Thickness Height To evaluateSpecimen the bending behavior of both the sandwich panel and the corrugated core within the elastic region, FE models wereL developed(mm) b (mm) using t Abaqus (mm) finiteh (mm) element software (Figure 4). For both the flat outer layersCorrugated and Core the core, 572 shell elements108 (S4R)6 with hourglass25 control and a reduced integration rule were employed.Sandwich To Panel easily assign559 material108 orientation--- to38 the core with complex geometry, a shell element was chosen. Since this type of element considerably decreases the simulation and running time of the4. Finite FE model, Element a finerModel mesh was adopted to increase the accuracy. In addition, all elements were given the capabilityTo evaluate to undergothe bending finite behavior strains of both and rotations.the sandwich panel and the corrugated core within Consideringthe elastic theregion, fabrication FE models process, were developed both the using corrugated Abaqus core finite and element the flat software outer plies(Figure were 4). For assumed to be transverselyboth the flat isotropic. outer layers To and simulate the core, a perfect shell elements bond between (S4R) with the hourglass face-sheets contr andol and the a 3-Dreduced core, rigid integration rule were employed. To easily assign material orientation to the core with complex links weregeometry, modeled a shell between element the was nodes chosen. of these Since twothis componentstype of element using considerably a tie constraint. decreases Asthe for the boundarysimulation conditions, and running the nodes time inof the FE contact model, areaa finer between mesh was the adopted specimens to increase and the the accuracy. supports In were constrainedaddition, in the allz elementsdirection. were Since given the the supportscapability to were undergo free finite to rotate strains about and rotations. the y axis to be consistent with ASTM D7249Considering [22], nothe otherfabrication boundary process, condition both the wascorrug applied.ated core However, and the flat to avoidouter instabilityplies were in the structure,assumed the centerline to be transversely of the sandwich isotropic. panel To simulate exactly a at perfect the mid-span bond between was the fixed face to-sheets avoid and any the translation 3- D core, rigid links were modeled between the nodes of these two components using a tie constraint. in the x direction. Loading was applied as prescribed downward deflection of the loading heads. As for the boundary conditions, the nodes in the contact area between the specimens and the supports To determinewere constrained an acceptable in the z direction. element Since size, the a supports mesh convergence were free to analysisrotate about was the performed y axis to be on the flexural 3-Dconsistent core model. with ASTM In Figure D72495 ,[2 the2], resultsno other of boundary this convergence condition studywas applied. are displayed, However, comparingto avoid the bendinginstability load that in corresponds the structure, to the a deflectioncenterline of of the 25.4 sandwich mm in panel the center exactly of at the the specimenmid-span was tothe fixed number to of elements.avoid As theany numbertranslation of in elements the x direction. increased Loading from was 4263 applied to 59,814 as prescribed (corresponding downward todeflection an element of size of 5 mmthe to 1.3loading mm), heads. there was a 1.73% change in the resulting bending load. Because of this negligible To determine an acceptable element size, a mesh convergence analysis was performed on the change in the bending load and noticeable savings in the computation time, the smallest number of flexural 3-D core model. In Figure 5, the results of this convergence study are displayed, comparing elements considered, 4263 (element size of 5 mm), was chosen to mesh the specimens. Foreststhe 20 bending20, 11, x FOR load PEER that REVIEW corresponds to a deflection of 25.4 mm in the center of the specimen to7 ofthe 16 number of elements. As the number of elements increased from 4263 to 59,814 (corresponding to an element size of 5 mm to 1.3 mm), there was a 1.73% change in the resulting bending load. Because of 2290 this negligible change in the bending load and noticeable savings in the computation time, the smallest number of elements2280 considered, 4263 (element size of 5 mm), was chosen to mesh the specimens. 2270

Load (N) 2260

2250

2240 0 10000 20000 30000 40000 50000 60000 Number of elements FigureFigure 5. Mesh 5. Mesh convergence convergence results results of ofthe the ﬂexuralflexural specimen specimen in inthe the longitudinal longitudinal direction. direction.

5. Theoretical Model Due to the complex geometry of a biaxially corrugated core with cavities caused by three- dimensional geometry, it is not easy to develop a theoretical model to capture all deformations under different types of loading. One method to overcome this difficulty is homogenization. In this section, the inhomogeneous sandwich panel with corrugated core geometry is replaced with an equivalent homogenous and continuous layer. The effective properties for this homogenous and continuous layer consist of effective extensional moduli in the x and y directions, effective Poisson’s ratio, and effective shear modulus in the x–y plane. Because it is a common practice to compute these effective properties with only consideration of in-plane loading [23–25], they were calculated based on the laminate extensional stiffness matrix, generally referred to as the [A] matrix. Besides the assumption of neglecting resultant moments to calculate effective properties, in developing the theoretical model, all stresses through the thickness were also neglected. Additionally, the [A] matrix components were computed by integrating over the UC volume instead of UC height. The constitutive equation for a laminate in terms of resultant forces [N], resultant moments [M], mid-plane strains {ε0}, and mid-plane curvatures {κ} can be written as

= (4) 0 𝑁𝑁 𝐴𝐴 𝐵𝐵 𝜀𝜀 where A, B, and D are the extensional stiffness,� � the� bending� � � –extension coupling, and the bending 𝑀𝑀 𝐵𝐵 𝐷𝐷 𝜅𝜅 stiffness matrices, respectively, and their components are defined as

= , = , = (5) ℎ⁄2 ℎ⁄2 ℎ⁄2 2 The symmetric and𝐴𝐴𝑖𝑖𝑖𝑖 balanced∫−ℎ⁄2 𝑄𝑄 𝑖𝑖𝑖𝑖configuration𝑑𝑑𝑑𝑑 𝐵𝐵𝑖𝑖𝑖𝑖 ∫− ℎof⁄2 the𝑄𝑄𝑖𝑖𝑖𝑖 𝑧𝑧𝑧𝑧𝑧𝑧sandwich𝐷𝐷𝑖𝑖𝑖𝑖 panel∫−ℎ⁄ 2about𝑄𝑄𝑖𝑖𝑖𝑖𝑧𝑧 its𝑑𝑑𝑑𝑑 mid-plane results in a zero bending–extension coupling matrix, [B] (see Appendix). In addition, because the sandwich panel was assumed to be a simplified 2-D structure and evaluated like a plate, all stresses through the thickness were neglected. Therefore, the constitutive equation for the sandwich panel is defined [23] as 0 = 0 0 (6) 𝑥𝑥 11 12 𝑥𝑥 𝑁𝑁 𝐴𝐴0 𝐴𝐴0 𝜀𝜀0 𝑁𝑁𝑦𝑦 12 22 𝜀𝜀𝑦𝑦 � � �𝐴𝐴 𝐴𝐴 � � 0 � 𝑥𝑥𝑥𝑥 66 and the components of the extensional𝑁𝑁 stiffness matrix ([𝐴𝐴A]) given𝛾𝛾𝑥𝑥𝑥𝑥 in Equation (2) expanded over three layers can be rewritten as

/ / / = = / + / + / (7) ℎ⁄2 ℎ𝑓𝑓 2 ℎ𝑐𝑐 2 ℎ𝑓𝑓 2 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑐𝑐 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝐴𝐴 ∫−ℎ⁄2 𝑄𝑄 𝑑𝑑𝑑𝑑 �∫−ℎ𝑓𝑓 2 𝑄𝑄 𝑑𝑑𝑑𝑑� �∫−ℎ 2 𝑄𝑄 𝑑𝑑𝑑𝑑�𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �∫−ℎ𝑓𝑓 2 𝑄𝑄 𝑑𝑑𝑑𝑑� where Qij are components of the stiffness matrix𝑇𝑇𝑇𝑇𝑇𝑇 in the global coordinate system (𝐵𝐵x𝐵𝐵–𝐵𝐵𝐵𝐵y𝐵𝐵–𝐵𝐵z) [26]. Additionally, hf is the thickness of the outer layers and hc is the height of the corrugated core. For

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5. Theoretical Model Due to the complex geometry of a biaxially corrugated core with cavities caused by three-dimensional geometry, it is not easy to develop a theoretical model to capture all deformations under different types of loading. One method to overcome this difficulty is homogenization. In this section, the inhomogeneous sandwich panel with corrugated core geometry is replaced with an equivalent homogenous and continuous layer. The effective properties for this homogenous and continuous layer consist of effective extensional moduli in the x and y directions, effective Poisson’s ratio, and effective shear modulus in the x–y plane. Because it is a common practice to compute these effective properties with only consideration of in-plane loading [23–25], they were calculated based on the laminate extensional stiffness matrix, generally referred to as the [A] matrix. Besides the assumption of neglecting resultant moments to calculate effective properties, in developing the theoretical model, all stresses through the thickness were also neglected. Additionally, the [A] matrix components were computed by integrating over the UC volume instead of UC height. The constitutive equation for a laminate in terms of resultant forces [N], resultant moments [M], mid-plane strains {ε0}, and mid-plane curvatures {κ} can be written as ( ) " #( ) N AB ε0 = (4) M BD κ where A, B, and D are the extensional stiffness, the bending–extension coupling, and the bending stiffness matrices, respectively, and their components are defined as

Z h/2 Z h/2 Z h/2 2 Aij = Qijdz, Bij = Qijzdz, Dij = Qijz dz (5) h/2 h/2 h/2 − − − The symmetric and balanced configuration of the sandwich panel about its mid-plane results in a zero bending–extension coupling matrix, [B] (see AppendixA). In addition, because the sandwich panel was assumed to be a simplified 2-D structure and evaluated like a plate, all stresses through the thickness were neglected. Therefore, the constitutive equation for the sandwich panel is defined [23] as

    0   Nx   A11 A12 0  εx        N  =  A A 0  ε0  (6)  y   12 22  y      0  Nxy 0 0 A66  γxy  and the components of the extensional stiffness matrix ([A]) given in Equation (2) expanded over three layers can be rewritten as Z Z  Z  Z  h/2  h f /2   hc/2   h f /2  A = Q dz =  Q dz +  Q dz +  Q dz ij ij  ij   ij   ij  (7) h/2 h /2 hc/2 h /2 − − f Top − Core − f Bottom where Qij are components of the stiffness matrix in the global coordinate system (x–y–z)[26]. Additionally, hf is the thickness of the outer layers and hc is the height of the corrugated core. For uniaxial corrugated cores, such as those with sinusoidal or trapezoidal configurations where the corrugated geometries can be easily specified with a known function, computing these integrals over the height to calculate the extensional stiffness matrix ([A]) is straightforward [27–29]. However, since the geometry of the core analyzed in this study is biaxially corrugated and varies along both the x and y axes, the second term in Equation (7) representing the core layer cannot be easily computed. Unlike other methods which use a known function describing the core geometry to compute Equations (5) and (7), a discretization technique was used to simplify the integration in this study. Therefore, one quarter of the Forests 2020, 11, 624 8 of 16

UC (Figure2b) was broken down into seven simpliﬁed domains as shown in Figure2b. Subsequently, integration was carried out over the simpliﬁed domains and averaged as

L L Z h /2 Z 1 Z 2 Z h /2 7 Z ! c 4 2 2 c 4 X Qijdz = Qijdzdydx = QijdV (8) hc/2 L1L2 0 0 hc/2 L1L2 V − − k=1 k where L1 and L2 are dimensions of the unit cell, and Vk is volume of each domain shown in Figure2b. It should be noted that the components of the global stiffness matrix (Qij) of the core layer given in Equations (7) and (8) are obtained by transforming the stiffness matrix in the local coordinate system (1–2–3) using the transformation matrix, [T]. Considering the global coordinate system (x–y–z) and local coordinate system (1–2–3) that varies from domain to domain, the transformation matrix for each domain is expressed as   1 0 0 0 0 0    2 2   0 m n 0 2mn 0    (  0 n2 m2 0 2mn 0  m = cos(0) T =   For domain #1, 3, 5, 7 :  2 2 − ,  0 0 0 m n 0 0  n = sin(0)  −   0 mn mn 0 m n   − −   0 0 0 0 n m    m2 0 n2 0 2mn 0      0 1 0 0 0 0    (  n2 0 m2 0 2mn 0  m = cos(30) T =   For domain #4 :  − , (9)  0 0 0 m 0 n  n = sin(30)  −   mn 0 mn 0 m2 n2 0   − −   0 0 0 n 0 m    m2 n2 0 0 0 2mn    2 2   n m 0 0 0 2mn   −  (  0 0 1 0 0 0  m = cos(56) T =   For #2 and 6 :  ,  0 0 0 m n 0  n = sin(56)  −   0 0 0 n m 0     mn mn 0 0 0 m2 n2  − − and inverting the extensional stiffness matrix ([A]) in Equation (6) results in

 0    1      εx   A11 A12 0 −  Nx   a11 a12 0  Nx            ε0  =  A A 0   N  =  a a 0  N . (10)  y   12 22   y   12 22  y   0          γxy  0 0 A66 Nxy 0 0 a66 Nxy

The stress-strain relation for a homogenous and continuous lamina with the thickness of h is expressed as

         ε0  1/E υ /E 0  σ  1/E υ /E 0  N   x   x xy x  x   x xy x  x   0   −   1  −   ε =  υyx/Ey 1/Ey 0  σy =  υyx/Ey 1/Ey 0  Ny (11)  y   −   h  −    0         γxy  0 0 1/Gxy  τxy  0 0 1/Gxy  Nxy  and comparing Equation (10) for the biaxial corrugated core sandwich panel with Equation (11) for a lamina gives the eﬀective material properties of a lamina that is equivalent to the biaxial corrugated core sandwich panel. These material properties are summarized [23] as

Ex = 1/ha , Ey = 1/ha , Gxy = 1/ha , νxy = a /a , νyx = a /a (12) 11 22 66 − 12 11 − 12 22 Forests 2020, 11, 624 9 of 16

Considering these effective material properties, two different beam models, classical beam theory (Euler–Bernoulli) and first-order shear deformation beam theory (Timoshenko), were employed to investigate the bending behavior of the equivalent structure. Using displacement fields of these beam models, the principle of minimum potential energy, and the variational method, the governing equations for this sandwich beam [30,31] were derived as

4 Euler–Bernoulli : Q I ∂ w = q(x) 11 ∂x4  2  ∂ φ ∂w  Q11I 2 ksQ55A φx + = 0 (13)  ∂x − ∂x Timoshenko :  2 ∂φ  k Q A ∂ w + + q(x) = 0  s 55 ∂x2 ∂x where beam deﬂection and rotation of the cross section about the y axis with respect to the thickness direction are shown with w and φ respectively. Additionally, Q11 and Q55, ks, A, and I, which are components of the stiﬀness matrix of the homogenized beam, shear correction factor, cross section area, and moment of inertia, respectively, are given as Q = Ex/ 1 νxyνyx and Q = G 11 − 55 13 1 3 (14) I = 12 bh , A = bh, ks = 5 + 5νxy / 6 + 5νxy

As shown in Table1, b and h are the width and the height of the sandwich panel test specimen and the material properties are given in Equation (12). It should be noted that eﬀective out-of-plane shear modulus (G13) is assumed equal to that of in-plane shear modulus (G12)[23]. Since Fourier series expansions satisfy the boundary conditions [32], they were used to solve the governing equation(s). Based on Euler–Bernoulli and Timoshenko beam theories, the closed form solution for the deﬂection of this sandwich beam with biaxial corrugated geometry under a four-point bending test, as shown in Figure4, is obtained as:

n a nπ(x a) pL3 sin( π )+sin − P L L nπx Euler–Bernoulli : w(x) = n∞=1 4 sin L " Q11I(nπ) # (15) ( )2+ 2 nπ + 2nπ P (Q11I nπ ksQ55AL )PL(sin( 3 ) sin( 3 )) nπx Timoshenko : w(x) = n∞=1 4 sin L ksQ55Q11IA(nπ)

6. Results and Discussion Results of tensile testing were used to obtain the elastic constants of the wood composite material. Using these properties, FE and theoretical models to investigate the bending behavior were applied and their results were compared with experimental results.

6.1. Elastic Constants To establish the properties of the wood composite material, 19 coupons were tested at each angle (0◦, 15◦, and 90◦). A typical stress–strain curve, along with the failure mode, of a tensile coupon is presented in Figure6. Since the stress–strain curve is almost linear, as proven by the regression model (red dotted line) and the coefficient of determination (R2), material nonlinearity for this wood composite sandwich panel can be ignored. Density plays an important role in affecting stress–strain curve and load-carrying capacity of this composite material. Therefore, test results of those coupons with a density of around 640 (kg/m3) were chosen to obtain the material properties. This avoided any variation caused by the density. All panels were manufactured to a target density of 640 kg/m3 in this study, since this density has been used for similar wood-strand products [11–14,18] and also is a typical average density for commercially available OSB [20]. These material properties are summarized in Table2 and coe fficients of variation in percent are presented in parentheses. Forests 2020, 11, 624 10 of 16 Forests 2020, 11, x FOR PEER REVIEW 10 of 16 Forests 2020, 11, x FOR PEER REVIEW 10 of 16

FigureFigure 6. 6.Typical Typical results result ofs tensileof tensile testing: testing (a:) ( load–straina) load–strain graph graph (blue (blue line) line) along along with with regression regression model Figure 6. Typical results of tensile testing: (a) load–strain graph (blue line) along with regression (red dotted line), (b) failure mode.model (red dotted line), (b) failure mode. model (red dotted line), (b) failure mode. Table 2. Table 2. ElasticElastic constants constants of of wood wood strand strand composite composite material. material. Values Values in parenthesis in parenthesis are COVs are in COVs percent. in Table 2. Elastic constants of wood strand composite material. Values in parenthesis are COVs in percent. 3 percent. E1 (GPa) E2 (GPa) G12 (GPa) ρ (kg/m ) E1 (GPa)9.80 (9.4%)E 1.712 (GPa) (13.4%) 2.56G (37.8%)12 (GPa) 640 (kg/m3) E1 (GPa) E2 (GPa) G12 (GPa) (kg/m3) 9.80 (9.4%) 1.71 (13.4%) 2.56 (37.8%) 640 9.80 (9.4%) 1.71 (13.4%) 2.56 (37.8%) 𝝆𝝆 640 Variation in material properties is to be expected expected in in all all heterogeneous heterogeneous and and anisotropic anisotropic materials, materials, Variation in material properties is to be expected in all heterogeneous𝝆𝝆 and anisotropic materials, as can be seen in coecoefficientfficient of variation values in Table2 2.. AsAs explained,explained, thisthis isis especiallyespecially truetrue forfor as can be seen in coefficient of variation values in Table 2. As explained, this is especially true for wood. For For wood wood-based-based products, products, variation in density c canan affect affect the material properties. The The influence influence wood. For wood-based products, variation in density can affect the material properties. The influence of the density on longitudinal Young modulus of the wood strand plies tested in this study is shown of the density on longitudinal Young modulus of the wood strand plies tested in this study is shown inin Figure 77.. Results indicate that aa 46%46% changechange inin densitydensity causedcaused aa 107%107% changechange inin longitudinallongitudinal in Figure 7. Results indicate that a 46% change in density caused a 107% change in longitudinal Young’s modulus.modulus. Young’s modulus.

Figure 7. Effect of density on longitudinal Young modulus. Figure 7. EffectEﬀect of density on longitudinal Young modulus.modulus. Considering the material properties obtained from tensile testing and the assumption of Considering the material properties obtained from tensile testing and the assumption of transverseConsidering isotrop they, the material material properties properties obtained of the from wood tensile strand testing composite and the are assumption summarized of transverse in Table transverse isotropy, the material properties of the wood strand composite are summarized in Table 3.isotropy, These elastic the material constants properties served of as the inputs wood for strand the compositeFE model areas well summarized as for the in theoretical Table3. These model elastic to 3. These elastic constants served as inputs for the FE model as well as for the theoretical model to computeconstants the served stiffness as inputs matrix for in the Equation FE model (7 as). well as for the theoretical model to compute the stiﬀness computematrix in the Equation stiffness (7). matrix in Equation (7). Table 3. The material properties of specimens. Table 3. The material properties of specimens. E1 E2 E3 υ υ G12 G13 G23 E1 E2 E3 12 υ13 23 G12 G13 G23 (GPa) (GPa) (GPa) υ12 [18] υ υ23 [19] (GPa) (GPa) (GPa) (GPa) (GPa) (GPa) [18] 13 [19] (GPa) (GPa) (GPa)

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Table 3. The material properties of specimens.

E E E G G G 1 2 3 υ [18] υ υ [19] 12 13 23 Forests 20202020,, 1111(GPa),, xx FORFOR PEERPEER(GPa) REVIEWREVIEW(GPa) 12 13 23 (GPa) (GPa) (GPa) 1111 of 1616 9.80 1.71 1.71 0.358 0.358 0.2 2.56 2.56 0.71 9.80 1.71 1.71 0.358 0.358 0.2 2.56 2.56 0.71

6.2.6.2. Bending Behavior When subjectedsubjected to a four four-pointfour--point bending load, the corrugated core specimensspecimens inin the longitudinal directiondirection failedfailed duedue toto tension,tension,, asasas shownshown inin FigureFigure8 a.8a. ComparisonComparison betweenbetween thethe averageaverage ofof the thethe experimentalexperimental resultsresults andand FEFE resultsresults forfor deflectiondeflection inin thethe middle of the specimens subje subjectedcted to bending isis shown shown inin Figure 8 8b.b. SinceSince thethe FEFE simulationsimulation waswas developeddeveloped toto capturecapture thethe bendingbending behaviorbehavior withinwithin thethe linearlinear region,region, there there is is aa closeclose agreementagreement between between these these two two models models inin thethe linearlinear elasticelastic region.region. TheThe didifferencefference betweenbetween experimentalexperimental andand FEFE bendingbending stistiffness,ffness, calculatedcalculated usingusing EquationEquation (3),((3)),, withinwithin thethe linearlinear elasticelastic regionregion isisis 1.4%.1.4%.1.4%.

Figure 8. 8. FourFour-point--point bending bending test test of the of 3 the--D core 3-D layer core in layer the longitudinal in the longitudinal (a)) failurefailure (a )modemode failure ((b)) modeloadload– (deflectionb) load–deﬂection curve. curve.

Sandwich specimens were also subjectedsubjected toto four-pointfourfour--point bending,bending, andand aa typicaltypical load–deflectionload–deflection curvecurve andand failure failure mode mode are are presented presented in Figurein Figure9. Unlike 9. Unlike the corrugated the corrugated core samples, core samples, the initial the failure initial modefailurefailure of modemode longitudinal ofof longitudinallongitudinal sandwich sandwichsandwich panels was panelspanels debonding waswas debondingdebonding between the betweenbetween core and thethe face-sheets corecore andand (Figurefaceface--sheets9b) due((Figure to interfacial 9b) due to shear.interfacial Bending shear. load Bending versus load deflection versus deflection at the midpoint at the midpoint of these of sandwich these sandwich panels forpanels theoretical for theoretical and FE modelsand FE weremodels compared were compared against the against average the experimental average experimental results in Figure results 10 ina. BasedFigure on 10a. the Based linear on region the linear of these region load–deflection of these load– curvesdeflection and curves Equation and (3), Equation the bending (3),(3), thethe sti ffbendingbendingness of sandwichstiffness of beams sandwich for di beamsfferent for models different was calculatedmodels was and calculated compared and in Figurecompared 10b. in For Figure experimental 10b. For results,experimental the bending results, sti theffness bending was calculated stiffness was based calculated on the results based between on the results 20%–50% between of maximum 20%–50% load of (i.e.,maximum the portion load (i.e. of the,, thethe curve portioportio betweenn of the about curve 1000 between N and about 2200 1000N N in Figure and 2200N 10a). in Figure 10a).

Figure 9. Four-point bending test of the sandwich specimens in the longitudinal direction: (a) typical Figure 9.9. Four-pointFour-point bending test of the sandwich specimens inin thethe longitudinallongitudinal direction:direction: (aa)) typicaltypical load–deflection curve, (b) failure mode, delamination due to interfacial shear between the outer plies load–deﬂectionload–deflection curve,curve, ((bb)) failurefailure mode,mode, delaminationdelamination duedue toto interfacialinterfacial shearshear betweenbetween thethe outerouter plies and the core. andand thethe core.core.

The difference between experimental and theoretical bending stiffness within the linear elastic regionregion are 3.6% and 5.2% forfor sandwichsandwich beambeams based on Euler–Bernoulli and Timoshenko models, respectively.respectively. Assumptions in the theoretical models could be the sources of error when compared against the experimental results. These assumptions include considering thethe composite panel as a beam instead of a plate to develop the theoretical model;; thisthis resultsresults inin neglectingneglecting anyany deformationdeformation inin thethe y direction,, andand ignoringignoring both shear and normal stresses through the thickness by assuming

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the sandwich panel as a simplified 2-D structure to compute the effective properties. The difference between the Timoshenko beam model, which considers shear deformation, and the Euler–Bernoulli beam model, which does not consider this deformation, is not noticeable within the elastic region. Neglecting shear and normal stresses to compute effective properties as explained can be an explanation for this small difference. However, this difference can be noticeable if these two theories Figureare used 10.FigureComparison to 10.model Comparison non of- (lineara) of load–deflection ( abehavior.) load–deflection As curvethe curve theoretical and and (b )(b bending ) resultsbending stiobtained stiffnessffness ofoffrom sandwichsandwich both models beams are under bendinggenerallyunder load conservative bending as shown load in ascompared Figure shown3 ind Figure basedto the 3d onexperimental based di fferent on different models. ones models, the th. eoretical model can be a good predictive model to use in design of new geometry; however, it is limited to predicting the elastic Bending stiffness predicted by the FE model differs by 6.5% compared to the experimental Thebehav diffiorerence. between experimental and theoretical bending stiffness within the linear elastic results. This slightly larger difference compared to the theoretical bending stiffness could be due to region arethe rigid 3.6% link and used 5.2% for modeling for sandwich the bonding beams area based between on the Euler–Bernoulli core and outer layers. and Failure Timoshenko initiation models, respectively.for sandwich Assumptions specimens instarted the theoreticalat the bonding models area, which could results be thein the sources failure mode of error of delamination when compared againstin the that experimental area, as shown results.in Figure These9b. However, assumptions this bonding include area was considering not modeled the in the composite FE simulation panel as a beam insteadand the ofrigid a platelink, which to develop does not the undergo theoretical any deformation model; this, was resultsused to bond in neglecting the core to face any-sheets. deformation in the yTherefore,direction, the and FE ignoringmodel experiences both shear less deformation and normal compared stresses to through the actual the one, thickness and results by in assuming a higher bending stiffness. The deflected shapes of the FE simulation of the corrugated core and the sandwich panel as a simplified 2-D structure to compute the effective properties. The difference sandwich panel subjected to a four-point bending test, which were used to obtain the results betweenpresented the Timoshenko in Figures 8b beam and 10 model,, are shown which in Figure considers 11. shear deformation, and the Euler–Bernoulli beam model, which does not consider this deformation, is not noticeable within the elastic region. Neglecting shear and normal stresses to compute effective properties as explained can be an explanation for this small difference. However, this difference can be noticeable if these two theories are used to model non-linear behavior. As the theoretical results obtained from both models are generally conservative compared to the experimental ones, the theoretical model can be a good predictive model to use in designFigure of 10. new Comparison geometry; of (a) however, load–deflection it is curve limited and to(b) predicting bending stiffness the elasticof sandwich behavior. beams Bendingunder stiff bendingness predicted load as shown by in the Figure FE model3d based di onff differenters by 6.5%models compared. to the experimental results. This slightly larger difference compared to the theoretical bending stiffness could be due to the rigid Bending stiffness predicted by the FE model differs by 6.5% compared to the experimental link used for modeling the bonding area between the core and outer layers. Failure initiation for results. This slightly larger difference compared to the theoretical bending stiffness could be due to sandwichthe specimensrigid link used started for modeling at the bondingthe bonding area, area which between results the core in and the outer failure layers. mode Failure of delaminationinitiation in that area,for assandwich shown specimens in Figure started9b. However, at the bonding this area bonding, which arearesults was in the not failure modeled mode inof delamination the FE simulation and thein rigid that area link,, as which shown does in Figure not undergo9b. However, any this deformation, bonding area waswas not used modeled to bond in the the FE core simulation to face-sheets. Figure 11. Deformed shape of the FE model of (a) 3D core and (b) sandwich panel. Therefore,and the rigid FE modellink, which experiences does not undergo less deformation any deformation compared, was used to to bond the the actual core to one, face and-sheets. results in Therefore, the FE model experiences less deformation compared to the actual one, and results in a a higher bending stiffness. The deflected shapes of the FE simulation of the corrugated core and higher bending stiffness. The deflected shapes of the FE simulation of the corrugated core and sandwichsandwich panel subjectedpanel subjected to a four-point to a four-point bending bending test, whichtest, which were were used used to obtain to obtain the the results results presented in Figurespresented8b and in 10 Figure, ares shown 8b and in10, Figureare shown 11. in Figure 11.

FigureFigure 11. Deformed 11. Deformed shape shape of of the the FE FE modelmodel of of (a ()a 3D) 3D core core and and (b) sandwich (b) sandwich panel. panel.

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Another explanation for the diﬀerence between model results and the experimental results could be the variation in material properties. Material properties shown in Tables2 and3, which were used to develop both the theoretical and FE models, were established considering the tensile test results of coupons with a density of 640 kg/m3. However, since there is always a variation between the target density that a panel is pressed to and the actual density, the bending behavior from panel to panel and even from specimen to specimen within a same panel could vary. Therefore, predictions of bending stiﬀness by these two models could be expected to vary from the experimental results, especially in the case of wood-based materials.

7. Sensitivity Analysis Intrinsic variation in material properties of wood strands, inconsistencies in the fabrication process, and variations in density may influence the material properties of the wood strand composite panels, as shown in the results so far. Even with extreme care during the fabrication process and the testing, there still will be noticeable variation in the material properties determined (reflected by the coefficient of variations in Table2). Such a variation leads to variation in the bending sti ffness of the sandwich panel and its components. In this section, a sensitivity analysis is presented to clarify the extent of variation in bending stiffness that could be expected due to specified variation in material properties. To this end, using the FE simulation and theoretical model, one material property was changed while others were held constant, and then change in the bending stiffness of the sandwich specimen due to change in the material property was obtained. The coefficient of variation (COV) for each property listed in Table2 was used as a guide to vary a material property to perform the sensitivity analysis. Results of the sensitivity analysis are given in Table4. Based on COVs listed in Table2, e.g., the longitudinal Young’s modulus (E ) was varied by 9.4%, and the bending stiffness was computed using FE and theoretical 1 ± models. A 9.4% increase in E1 indicates an increase of 8.5%, 9.3%, and 9.2% in FE, Euler–Bernoulli, and Timoshenko bending stiffness, respectively. Meanwhile, a 9.4% reduction in E1 shows a decrease of 8.7%, 9.2%, and 9.1% in FE, Euler–Bernoulli, and Timoshenko bending stiffness. However, a higher change in E , 13.4%, creates less than 1% change in both FE and the theoretical bending stiffness. 2 ± ± Theoretical bending stiffness is not as sensitive as that of FE to change in shear modules because the effects of shear stresses and deformation through the thickness of the panel to calculate effective properties, which were used to develop the theoretical model, were neglected.

Table 4. Percent variation in the longitudinal bending stiﬀness of the sandwich panel caused by change in material properties.

% Change in Bending Stiﬀness Property % Change in Material Properties FE Euler Timoshenko 9.4% increase: E = 10.72 GPa 8.8 9.3 9.2 E1 1 9.4% decrease: E = 8.88 GPa 8.7 9.2 9.1 1 − − − 13.4% increase: E = 1.94 GPa 0.41 0.89 0.34 E2 = E3 2 13.4% decrease: E = 1.48 GPa 0.5 0.85 0.25 2 − − − 37.8% increase: G = 3.53 GPa 3.3 0.57 0.97 G12 = G13 12 37.8% decrease: G = 1.59 GPa 5.9 0.47 1.5 12 − − −

8. Conclusions Theoretical and ﬁnite element models were developed and applied to predict the linear ﬂexural behavior of the sandwich panel with a complex three-dimensional core geometry manufactured from small-diameter timber under a four-point bending load. The material properties were measured to input into the models. The bending stiffness of just the corrugated core specimens obtained by the FE model showed a 1.4% difference from the average of the experimental results. In case of the sandwich panel, the average experimental bending stiffness differed from Euler-Bernoulli, Timoshenko, and FE predictions by 3.6%, Forests 2020, 11, 624 14 of 16

5.2%, and 6.5%, respectively. The results indicated that a 46% increase in material density increases the longitudinal Young’s modulus by as much as 107%. A sensitivity analysis to understand the effect of variations in material properties on composite sandwich bending stiffness revealed that the longitudinal Young’s modulus is the most important property that influences the bending stiffness. Theoretical and FE models developed for the wood strand sandwich panel were utilized to conduct a parametric study [33] to understand the influence of the core geometry on the bending stiffness of the sandwich panel. Based on the results, new core geometry with higher performance was designed to meet the structural performance requirements of building envelope components, such as wall, floor, and roof. Theoretical models for the sandwich structure with new core geometry were developed for beam [34] and plate [35] configurations.

Author Contributions: Conceptualization, V.Y.; methodology, M.M., J.J. and V.Y.; software, M.M. and J.J.; validation, M.M. and J.J.; formal analysis, M.M., J.J. and V.Y.; investigation, M.M., J.J. and V.Y.; resources, V.Y.; data curation, V.Y.; writing—original draft preparation, M.M. and J.J.; writing—review and editing, V.Y. and W.C.; visualization, M.M. and J.J.; supervision, V.Y.; project administration, V.Y.; funding acquisition, V.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation, grant number CMMI-1150316. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A Components of bending–extension coupling matrix ([B]) given in Equation (5) can be written as Z Z  Z  Z  h/2  h f /2   hc/2   h f /2  B = Q zdz =  Q z dz +  Q z dz +  Q z dz ij ij  ij   ij   ij  (A1) h/2 h /2 hc/2 h /2 − − f Top − Core − f Bottom where h, hf, and hc are thickness of the sandwich panel, face-sheets, and core, respectively. Top and bottom subscripts refer to the top and bottom face-sheets. The ﬁrst and third terms in Equation (A1) (top and bottom) are zero because of the symmetric and balanced conﬁguration of the face-sheets and can be written as Z h f /2 2 ! z z = h f /2 Qijz dz = Qij = 0 (A2) h /2 2 z = h f /2 − f − Based on Figure2, the second term, representing the corrugated core, can be written as

L1 L2 h R hc/2 4 R 2 R 2 R hc/2 4 RRR Qijzdz = Qijz dzdydx = Qijz dzdydx + hc/2 L1L2 0 0 hc/2 L1L2 1 − RRR RRR− RRR Qijz dzdydx + Qijz dzdydx + Qijz dzdydx + (A3) RRR 2 RRR 3 RRR 4 i Q z dzdydx + Q z dzdydx + Q z dzdydx ij 5 ij 6 ij 7

Subscripts 1 to 7 represent the segmented parts of the corrugated core as shown in Figure2b, which are used as the domain for integration. Because of the symmetric and balanced conﬁguration of UC, parts 1 and 7, parts 2 and 6, and parts 3 and 5 cancel out each other. Part 4 is zero on its own because of its symmetric geometry about the mid plane. Therefore, the bending–extension coupling matrix (matrix [B]) for this sandwich panel is zero.

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