Simplified analytical model and balanced design approach for light-weight wood-based structural panel in bending

http://dx.doi.org/10.1016/j.compstruct.2015.09.045 0263-8223/Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). ⇑ Corresponding authors. Tel.: +1 608 231 433; fax: +1 608 231 9582 (J.F. Hunt). Tel.: +1 608 231 9446; fax: +1 608 231 9582 (Z. Cai). E-mail addresses: jfhunt@fs.fed.us (J.F. Hunt), zcai@fs.fed.us (Z. Cai). Jinghao Li , John F. Hunt b,⇑, Shaoqin Gong , Zhiyong Cai b,⇑


Introduction
Sandwich composites provide light-weight and high-strength characteristics that are widely used for many applications, ranging from aerospace and marine, to less demanding applications in transportation and building systems [1][2][3][4]. The sandwich structures of two thin skins with a structural core are generally produced using metal or fiber-reinforced-polymer (FRP) composites. The typical structural core can be formed using various structural cores (i.e.: honeycomb cores, foam cores, corrugated cores, truss cores, and grid cores), each of these have unique characteristics for specific applications. In 1964, the concept of an isogrid structural core made from aluminum was first proposed in the field of aerospace. The structural pattern of triangular trusses was shown to be very efficient [5]. Thus subsequent researchers continued to study isogrid structures to determine its mechanical characteristics, manufacturing methods, and applications [5][6][7]. In recent years, FRP materials have been widely used for many types of isogrid structures, which have demonstrated excellent performance characteristics. Fan et al. demonstrated improved compression and bending performance for lattice grid panels made from carbon fiber [8]. Several manufacturing methods have been proposed to construct the isogrid structures [9][10][11]. An isogrid fabrication process using carbon fiber and epoxy composite prepreg tow and unidirectional tape was introduced [12], and another manufacturing method used interlocked composite grid arrangement that could be used to simplify the assembly by using prefabricated materials [13]. For the aerospace and marine applications, the development and application of isogrid structures were summarized [14,15]. These studies used metal or FRPs for higher-value applications. There is no known literature for a core or structural panel that uses the isogrid made from renewable materials such as wood-based materials. There is a unique challenge and opportunity to extend the field of engineering applications using the isogrid core with green materials to develop new structural composites from light-weight engineered materials.
Forest Products Laboratory (FPL) has developed an eco-friendly tri-gird or tri-axial isogrid core structural panel made using phenolic impregnated laminated paper composites from our renewable forestry [16,17]. This research was initially sponsored by U.S. Department of Defense to look for alternative sustainable materials to substitute the current aluminum and other expensive alloy http://dx.doi.org/10.1016/j.compstruct.2015.09.045 0263-8223/Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). materials for its air transportation pallets and tactical shelters. As required, high strength, light weight, and water resistance were the top criterions in selecting the alternative products. Thus, this developed renewable structural panel also could be used for building construction, transportation, packaging, container with benefits to protect our environment. To produce the cost effective tri-axial isogrid core structural sandwich panels for difference applications, we need to develop a predicted model to simulate the overall performance of the panel with the configuration of each constitutive member to ensure a proper design. In the previous research [16,17], the simple I beam equation was applied to evaluate the bending properties of these structural panels, but it only considered linear ribs in one direction and didn't consider any cross-rib influences for the core and was less accurate in predicting actual performance. Finite element analysis (FEA) with higher accuracy could be used to analyze the panels' complex behavior such as stress concentration and buckling. However, the FEA process requires more time to build an exact model that includes the complex nature of the structure and its multiple parameters, especially, during the preliminary design process and optimization. Thus, a new integrated equivalent model for these composite grid structures with or without skins is being proposed. The model used equivalent stiffness based on coordinate matrix transformation to evaluate the grid mechanical properties [9]. Other researchers have also used the equivalent modeling method to calculate the equivalent modulus for the material properties. Fang's model considered repeatable triangle elements for transforming the structural mechanical performance [18]. It provided an efficient method to evaluate the performance for preliminary design, but it was only designed for the equilateral triangle element structure in Fig. 1. In our study, a simplified orthogonal model was developed that included the ribs' intersection angle as a parameter ( Fig. 2(b)) and repeatable elements based on equivalent stiffness. These changes simplified the model by eliminating the coordinate transformation matrix as compared to Chen's model [9]. It also improved design flexibility to include non-equilateral triangles compared with Fang's equilateral model. This enhancement improved the design potential that allows for easier core parameter changes during the preliminary design process. Recently, higher order deformation plate theory and layerwise theory have been developed by Ferreira [19][20][21][22] to analyze sandwich structure plates. Both theories provided higher accuracy than classical plate theory and had better outcomes, particularly for shear analysis. Our goal is to simplify the analytical model for preliminary design while reducing complexity and computational effort. The simplified model with or without core shear effect was used to analyze and predict the bending behavior in this work.
This study analyzed the bending behavior and developed a failure criterion for the wood-based structural panels made from phenolic-impregnated laminated paper with or without carbon fiber fabric bonded to the surfaces. The panels were made having an isogrid core. Four types of wood-based structural panels were fabricated each having a different configuration, and three replicates were tested for each type of wood-based structural panels. The panels were tested using four point bending test to determine bending load and deflection. Eight normal strain gages and four shear strain gages were attached to the panels to measure the strains at critical locations. The test results were compared with the analytical model and FEA model and then a preliminary failure criterion for bending was determined using failure modes and strain distributions plotted on a 3-Dimensional diagram. The 3-D diagram was further proposed as a possible method for balanced design for a structural panel with a fixed isogrid core.

Analytical modeling approach
2.1. The elemental transformation of tri-axial core structure In our previous study, we transformed the tri-axial isogrid core into an equivalent solid core using repeatable element properties to achieve an equivalent stiffness approach. The tri-axial pattern of repeatable elements, Fig. 2(a), can be described as consisting of five ribs in different directions, Fig. 2(b). An equivalent rib section dimensions in the orthogonal directions were transformed based on angle h under equivalent lengths, m and n, and core height, h c shown in Fig. 2(a) or (b). The parameters of an equivalent solid core were obtained by transforming the elastic moduli, Poisson's ratios, and the shear moduli of rib material. The plane stress assumption was used in this model so that the relative properties along the 3-axis (Fig. 3) were ignored. The equivalent properties of E X , E Y , l XY , l YX , and G XY for the tri-axial core can be obtained from the materials properties of E 1 , E 2 , l 12 , l 21 and G 12 in the repeatable element through transforming the effective sections of the ribs to orthogonal equivalent sections, and then to an equivalent solid core. The progression of transformation is shown in Fig. 2   The equivalent properties can be obtained from the transformation equations below [23][24][25][26]: where h is angle between the longitudinal rib and cross rib; a is rib thickness in the core; n is width for the equivalent solid element that perpendicular to X-axis; m is equivalent solid element width that perpendicular to Y-axis; E 1 is the elastic modulus in the machine direction for the relative coordinate system 1-axis for the laminated paper composites material (Fig. 3); E X and E Y are the elastic moduli on the X-axis and Y-axis of the equivalent solid element of the panel (Fig. 2); I X , I Y are the moment of inertias for orthogonal equivalent ribs on YZ-plane and XZ-plane; I 0 X , I 0 Y are the moment of inertia for solid plate element on the YZ-plane and XZplane; l 12 , is the Poisson's ratio for the rib material and l XY , and l YX are the Poisson's ratios for the equivalent solid plate element in both directions; and G XY is the shear modulus of the equivalent solid plate element in the XY-plane.
The out-of-plane shear properties of the tri-axial structural core were analyzed under bending. For a structural panel with a large span-to-thickness ratio, shear deformation is small in the core and has little effect on core-skin bending deflection. However, core shear failure was one of the typical failure modes observed in the testing. It is important to determine the shear characteristic of the core. Based on the previous simplified orthogonal equivalent modeling, the out-of-plane shear modulus of the equivalent solid element in the span direction can be obtained from sectional transformation of the orthogonal equivalent element ( Fig. 2(c)) based on the equivalent shear stiffness given by: For core analysis in the orthogonal directions, the equivalent shear moduli can be given by: where G 12 is shear modulus of the rib; G XZ and G YZ are the shear moduli of the equivalent solid core in XZ-plane and YZ-plane; A X and A Y are sections of equivalent orthogonal element with respect to X-axis and Y-axis; A 0 X and A 0 Y are equivalent sections of solid element with respect to X-axis and Y-axis.
According to the geometric deformation, the shear strain relationship between the longitudinal rib and cross rib in the exact tri-axial structural core can be calculated based on trigonometric function: where c l is shear strain in the longitudinal rib, which is in 1-2 plane of rib material or XZ-plane of panel, and c c is shear strain in the cross rib in 1-2 plane of rib material.
If testing were in the Y-direction of the panel, the shear strain in the longitudinal rib becomes zero, and the shear strain in the Y-direction can be estimated within the cross ribs, which can be transformed by the equation: where c t is the transverse shear for the core orientation.
For bending test, the normal strain in the cross ribs between the loading points were ignored because the cross ribs were not continuous along the beam directions.

Laminated plate theory
According to the laminate plate theory, the wood-based structural panel composed of two skins sandwiched a structural core can be assumed to be three layers, thus the stress resultants N and stress couples M in Fig. 4 are given by: where W is width of panel; N X and N Y are normal force in X-axis and Y-axis; N XY is in-plane shear stress in XY-plane; M X , M Y and M XY are moments with respect to N X , N Y and N XY ; k is the layer number of the sandwich panel; h is thickness of each layer. e X 0 and e Y 0 are strain respect to the neutral axis of X and Y directions; e X 0 Y 0 is shear strain in XY-plane; j X and j Y are curvature in X-axis and Y-axis; j XY is the curvature respect to XY combination; z is location of plate boundary; and ½Q k is the stiffness matrix for the particular layer. The stiffness matrix of skin and core are given by: where ½Q s is stiffness matrix of skin and ½Q c is stiffness matrix of core.
We assumed that out-of-plane shear force in the two skins had no noticeable effect on the bending deflection because the core thickness was more than 6 times thicker than the skin thickness. Therefore, the shear force in the skins can be ignored in the calculation. The shear force was supported primarily by the structural core during bending, so the out-of-plane shear force of the equivalent solid core can be expressed as: where h c is the height of the core; Q XZ , c XZ , and Q YZ , c YZ are shear forces and shear strains in the XZ-plane and YZ-plane, respectively. Based on the equivalent transformed Eqs. (6) and (7) combined with shear force Eq. (14), the shear forces in the equivalent solid core also can be calculated by the shear modulus of ribs in the core, Eq. (15): In this analytical model, both normal and shear forces can be calculated using the corresponding strains and material properties according to the load and boundary conditions. Similarly, the inverse matrices (10), (11), (14) and (15) can be used to calculate the strain and bending characteristics based on bending load.

Reduced beam formulas based on plate theory for four point bending
Based on laminated plate theory, the relationship between the bending moment and geometry for bending analysis of a symmetric beam can be given by: where the matrix ½D Ã is the inverse of the bending-twisting stiffness matrix ½D. For a simply supported beam, the assumption was given by: Then the bending strain of our panel in the mid-span section under third point loading configuration can be given by: where d is the distance from the neutral axis to strain position on the beam transverse section; P is the total bending load; Relative D Ã is obtained from matrix ½D Ã ; R is the distance from the load point to the neighboring support; and W is the width of the panel.
The maximum shear strain in the rib of the core can be calculated by the shear stress and shear modulus. The shear stress can be obtained by Eq. (20): where V is total shear force, which equals 1/2P of total load for bending; J is the static moment; and I z is moment of inertia of the entire cross sectional area. The sandwich panel can be considered as a composite beam to evaluate the bending behavior and then the core with different mechanical properties, apart from the skins, can be simplified using the equivalent I-beam approach.
Therefore, the shear strain in the core can be written by Eq. (21): where / is shear coefficient that depends on the geometry and / was assumed 1 for this analysis. The bending displacement equation without considering shear effects can be given by Eq. (22): The bending deflection combined with shear deflection in the core for total deflection of our panel was compared with the result of Eq. (22), thus the mid-span deflection for bending of the panel could be calculated including bending displacement of the panel plus vertical displacement due to core shear strain. The shear contribution to bending deflection for third point loading configuration can be calculated by Eq. (23) in terms of geometrical relationship: The total mid-span defection is given by: where x z;t is mid-span displacement; relative D Ã is obtained from ½D Ã .

Materials properties
The mechanical properties used for this study were obtained by testing according to ASTM test methods D638 and D695 (ASTM 2010; ASTM 2010) [29,30]. Phenolic impregnated laminated paper 2.36 mm thick with a density of 1387 kg/m 3 , NP610 (Norplex-Micarta Inc., Postville, IA) was used for the core and skins. Carbon fiber fabric, a tri-axial woven material, QISO, A&P Technology, (San Jose, California, USA) was bonded to both outside skins of panels using U.S. Composites (West Palm Beach, Florida, USA) epoxy resin no. 635 [16].The materials properties were shown in Table 1.
The epoxy resin no. 635 (US Composites Inc.) was formed into a dog-bone test coupon and tensile tested. The epoxy resin was assumed to be isotropic. The shear modulus of epoxy resin was estimated by the formula for isotropic materials: Fig. 4. The stress resultants and stress couples for a plate.
The epoxy resin strength was determined according lap shear test method ASTM D5868 [16,32], the average epoxy shear strength was 17.9 MPa between the laminated paper.

The experimental design for bending test
Four types of experimental panels were made were fabricated for testing. The panels were fabricated with tri-axial core configuration using laminated paper ribs. The nominal height of rib was 33.0 mm. The slots in the core pieces were cut slightly oversized to account for the 60°angular orientation between the ribs when assembled. The distance between slots for all pieces was 117.3 mm, thus creating an equilateral triangle with a core density of 65 kg/m 3 . Three main ribs were aligned with either the MD or CD of skins for different panels, see Fig. 2. The panels were fabricated so that the ribs were centrally positioned and width determined to include three complete longitudinal ribs. Panel 1 was made with 1 layer laminated paper skins and MD core only; panel 2 was made with two layers of laminated paper skins and MD fiber direction core, panel 3 was made with two layers laminated paper skins and CD fiber direction core; and panel 4 was made with laminated paper composited carbon fiber fabric skins with MD fiber direction core, see Table 2. When additional carbon fiber fabric or laminated paper sheet were added the surfaces were sanded before epoxy was applied. The primary alignment for the carbon-fiber fabric or additional laminated paper sheet aligned with the MD of the skins. The fabrication process of tri-axial structural panel is shown in Fig. 5.
The four point bending test for third point loading configuration, ASTM C393 (ASTM 2006) [31], was used to test the panels. To verify the strain distributions in the panels during the bending test, strain gages were placed inside one panel from each configuration. Twelve strain gages including 8 normal strain gages (CEA-06-250UN-350) and 4 shear strain gages (CEA-06-187UV-350) (Micro-Measurements Inc) were bonded to the surface of the ribs or faces for measuring the strain of the critical locations on the panels [27]. The strain gage locations are shown in Fig. 6. Five LVDTs were placed beneath the panels at 1/6, 1/3, 1/2, 2/3 and 5/6 the span to measure bending deflection. The test set-up dimension and testing device positions are shown in Fig. 7. This was our first attempt to investigate both shear and tension strain distributions within the sandwich construction panel. There were a considerable complexity and high cost of applying strain gages within the panels. Since all the panels were fabricated from impregnated laminated paper composite material which was very uniform and consistent in their mechanical performance with less than 5% in coefficient of variance [27]. After consulting with a statistician, we constructed one panel for each configuration and believed that the strain distribution results were representative for the other panels tested with the same configuration. Table 3 shows the experimental, finite element analysis (FEA) and analytical modeling results for the four bending test panels. The top row within each group shows the results for the single panel test with strain gages. The second row within each group   shows the average results for all panels including the results from the panel with strain gages. For all the panels, the maximum normal strain was on the compression side which was larger than the tensile strain. We believe this occurred because localized compression buckling caused the additional strain. Panel 1 had the thinnest skins (Table 2), thus compression/buckling in the face occurred before the core shear failure occurred. In contrast, panels 2-4 had either twice the skin thickness (panels 2 and 3) or a layer of carbon fabric (panel 4). These thicker and stiffer skins were less susceptible to buckling and forced the failure to the core rather than failing in the skins. The stiffness ratio of faces-to-core significantly affected the location of the failure so that failure occurred on the lower stiffness component. Failure due to face compression buckling on the thin skinned panel (Panel 1) and core shear failure on the stiffer skinned panels (Panels 2-4) was as expected, Fig. 8  (a) and (b). The critical shear strain at failure in the core (Table 3) was measured from the strain gage located at the neutral axis of the rib. The measured maximum shear core strain in our panel was slightly higher than what the model predicted. This was expected because   the actual shear across the rib was distributed nonlinearly [28]. However, the analytical model with linear assumption predicted strain slightly less than experimental results, because it used a constant shear coefficient factor /¼ 1. The top skin compression buckling also had a small effect on the core that could have increased the measured shear strain on the rib. Thus the shear strain from the analytical model, while less than the experimental strain, was reasonable and acceptable estimate for preliminary design purposes. The results of FEA with each exact model of four types of panels had slightly higher accuracy than analytical model, but it required more time and computational effort to simulate the bending performance. Deflections with and without core shear were evaluated. The analytical model with shear effect or without it fit within 10% error of the experimental data. The model predicted deflections for all panels with core shear effect that compared well with the experimental data. The average difference was approximately 2.5% and maximum difference was 3.5%. Without core shear effects, the model predicted deflections that had a higher error than when core shear effects were included. This indicates that shear strain deformation, while relatively small, still contributes to provide a better model prediction. The three panels with thicker skins configurations core shear failure were expected. The reason was that compared with core stiffness, the skin stiffness was significant larger in bending, thus the causing the core components to fail at the material's yield strength, while the strength of the skin components were still below its ultimate strength. We estimate that damage first occurred in the core-face interface and then propagated quickly to ribs causing core failure. The epoxy resin's mechanical properties are significantly lower than the laminated paper's properties and failure may have initiated at the interface. , the core rib orientation had the paper laminate MD orientation turned 90°or perpendicular to the panel direction. This rotation provides higher shear strength because of the fiber orientation that provides better shear resistance within the ribs. The result of this rotation was an increase in the bending load for panel 3 that was higher than that of the other panels. With the two skin layers, the normal strain still did not reach the maximum failure strain before the panel failed. Panel 4 had even stiffer skins with thin carbon fiber fabric bonded to the surfaces of the panel. The result shows the maximum shear strain on the ribs was around 4827 le while the maximum normal strain on the skins was only around 5290 le at failure.

Three-dimension analysis diagram
Usually sandwich panels are dominated by either failures of compression or tensile failure on skins or shear failure on the core. The failure criterion can be determined using the maximum normal strain on the skins and maximum shear strain on the ribs. In Fig. 8, the blue triangle represents the simultaneous failure of the skins and the core when making full use of the material's strength properties under bending. To achieve the light-weight and high-performance characteristics, the balanced parameters for configurations for wood-based structural panel need to be analyzed to achieve a balanced design.

Balanced design analysis
The initial panel configuration and analysis stages are important for a balanced design of any structural panel. An approximate method to help achieve this balanced design was developed based on optimization of the face normal strain and core shear strain. Fig. 9 shows the normal vs core shear strain test plots as a function of bending load for panels 1 and 2. The only difference between the two panels was that panel 1 had a single skin layer while panel 2 had a double thickness skin layer, Table 2. A ''balanced" failure area was developed based on the area described by Eq. (27).
where D is the failure area calculated from strain and bending load; is the apparent maximum normal strain on the skins; c is the apparent maximum shear strain in the ribs; and P is the bending load. In Fig. 9, the estimated failure triangle area was determined from the results for panels 1 and 2. At point 1 (Panel 1), it illustrates that the skin thickness was too thin and failure would occur at the apparent maximum normal strain failure in the skins. At Point 1, the core shear for panel 1 was low and the core shear strain was not sufficient to cause failure. If the skin layer was doubled, point 3 (Panel 2), then the bending load could be increased to cause failure to shift to core shear. Having too thin or too thick of skins is still not balanced for the current panel core design. If the skin thickness were increased slightly from point 1 then the bending load increases as the skin apparent maximum normal strain was maintained. The effect on core shear strain was that it increased to maximum point 2 of the balanced design. If the skin thickness continued to increase, the load stayed the same but core shear would be the dominant failure criteria as shown by point 3. This process is useful to find the excepted or balanced design by determining simultaneously the skin normal strain and core shear strain failure. The balanced design case for panels 1 and 2 assumed the material properties were elastic and the relationship between the thickness of skin and failure triangle area had linear behavior for simplification. The apparent failure normal strain on both skins was assumed to be 7520 le and the failure apparent shear strain on the ribs was assumed to be 4701 le. Therefore, the balanced thickness of the skin using linear interpolation can be initially evaluated using Eq. (27) based from Fig. 9: where T Panel1 was the thickness of skins of panel 1, T Panel2 was the thickness of skins of panel 2, D ob is the balanced failure triangle area, D sf is the skin failure triangle area of panel 1, D cf is the core failure triangle area of panel 2. For this study, the balanced thickness for the skins was determined using FEA and compared with the diagram analysis and the analytical model. All these models used the apparent failure normal strain for the skin and apparent failure shear strain for the rib, Table 4. The diagram analysis and FEA analysis provided similar results for the optimum skin thickness. The skin thickness calculated using analytical model was approximately 20% less than the diagram analysis and FEA analysis. The skin thickness was less because the analytical model used linear material property assumption that provides stiffer results. Even though the analytical model predicted a thinner balanced skin thickness it was still thicker than the skin thickness for panel 1. This result would have been acceptable for a first approximation and useful for preliminary design purposes.

Conclusions
The bending behavior of the wood-based structural panel can be evaluated and predicted using the analytical model. The prediction of the bending deflection by the model with core shear effects  Table 4 Comparison between actual and optimal thickness of skins, normal strain, and shear strain for the actual structural panels and calculated by ANSYS, diagram analysis, and analytical modeling. had better accuracy than the model without core shear effects, especially on the core shear failure panels. The apparent normal strain for the skin and apparent shear strain for the rib were also analyzed using the analytical model approach. The model results fit well with the experimental results. Core shear strength affected the bending performance and failure mode. The failure criterion was defined using either the apparent maximum normal strain on the skins or the apparent maximum shear strain on the ribs to determine the failure load for the structural panels. The 3-D diagram approach with axes of normal strain, shear strain and bending load was creatively produced to show the typical failure map based on the experimental results. The balanced design of the analytical model, 3-D diagram statistical analyses and FEA model were discussed based on panels 1 and 2 to predict the balanced thickness skins for the panel with the specific core. The skin thickness obtained using the diagram analysis and the analytical model were within +4% and À17% error which was checked using the FEA model, respectively. These balanced methods can be used to maximize the full use of all components in the structural panel for bending.