Bending and energy absorption performance of novel openwork wooden panels

Taking into account the shape and dimensional proportions of the structures designed by Murlak (2019) (Fig. 1), the tests were carried out on solid and openwork panels with a thickness and length of 18 and 1032 mm, respectively (Fig. 2). The solid panels were made of 265 mm wide solid timber (not glued) with fibers oriented along the longer side of the panel. The openwork panels were made by gluing together six curvilinear slats cut from 265 mm wide solid panels. According to Kollmann and Cote (1968), the bending strength of the curvilinear wood elements depends, among others, on the inclination of the wood fibers to the beam axis. This inclination should not exceed 10°. Therefore, in further considerations, it was assumed that the inclination of the side walls of the curvilinear slats to the longer side of the panel equals 9° and 11°. This way, the width of the openwork panels was increased to 328 and 355 mm, respectively. Thus, the width of the openwork panel compared to the width of the solid panels increased by 63 and 90 mm, respectively (Fig. 2). The authors decided to make the panels following the patent claim P.42481 (Murlak 2019). So, the side walls of the curvilinear slats to the longer side of the panel are considered for two cases only. The panels were made of beech (Fagus sylvatica L.), oak (Quercus robur L.), ash (Fraxinus excelsior L.), and walnut (Juglans regia L). The basic physical and mechanical properties of the selected wood species are presented in Table 1.

At first, the boards of 20 mm thickness were planed on both sides on a Format 4 planer (Felder Group, Żory, Poland) in order to obtain a constant thickness of 18.2 mm. Then, solid full panels and curved slats were made on the Infotec Professional milling plotter (InfoTEC CNC, Zasutowo, Poland), using a 3 mm end mill operating at a speed of 24,000 rpm (Fig. 3). All elements were sanded on a DMC MB 90 wide-belt sander (SCM Polska SP.Z.O.O., Suchy Las near Poznań, Poland) to a thickness of 18 mm. Curvilinear slats were glued with Chemolan B 4 M/D4 polyurethane adhesive (Interchemol S.A., Oborniki Sląskie, Poland). Based on previous experimental studies, an adhesive linear elasticity modulus of 820 MPa and shear strength of 16 MPa were assumed (Kamboj et al. 2020). The adhesive was applied on flat, rectangular surfaces with dimensions of 18 mm × 41 mm (Fig. 4). After applying a pressure of 0.3 MPa, the panels were left in the laboratory for 28 days for air conditioning. The relative air humidity and temperature in the laboratory room were 65 ± 1% and 21 ± 1 °C, respectively. Under those conditions, the wood reached the hygroscopic equilibrium without changing its absolute humidity. After seasoning, an average thickness of 0.1 mm of the adhesive layer was measured using a Kern OZM 922 microscope (KERN & Sohn GmbH, Balingen, Germany).

Table 2 presents the mass and volume of the designed panels, based on the density of wood presented in Table 1. The table shows that the volume of the openwork panel with the 9° and 11° inclination angle of the side walls is smaller by 10 and 8.2%, respectively, in relation to the solid panel.

Five identical solid and openwork panels with an inclination angle of the side wall of 11° were made from each wood species for the experimental tests. The panels were subjected to three-point bending on a Zwick 1445 universal testing machine (Zwick Roell GmbH & Co. KG, Ulm, Germany) (Fig. 5). For the bending tests, a support spacing of 800 mm was used. It is a representative dimension for shelves and top and bottom panels used in furniture design. The panel length of 1032 mm was due to technological limitations arising during formatting on a sawing machine. During the tests, the force was recorded with an accuracy of 0.01 N and the displacement in the direction of the force with an accuracy of 0.01 mm. The speed of increasing the displacement was set at 10 mm/min, according to EN 310 standard (EN 310:1993 1993). It was assumed that the structure produced is a type of wood composite. The tests were stopped when the panel deflection of 40 mm was obtained. The limitation resulted from preliminary tests, which revealed high susceptibility of the panels to bending without damage (even up to 60 mm). Moreover, for aesthetic reasons, deflections of furniture elements exceeding 4 mm/m are considered unacceptable (Smardzewski 2015).

Based on the direct results of experimental tests, the interdependencies between the load and deflection of the panels were determined. These dependencies were used to calibrate analogical numerical models. As a consequence of the satisfactory accordance between the results of experimental tests and numerical calculations for solid and openwork panels with an inclination angle of the side walls of 11°, an additional numerical model of the openwork panel with an inclination angle of the side walls of 9° was developed, which was smaller by 1° from the 10° value suggested in the literature, and smaller by 2° from the experimental model. Table 3 shows the markings of all test samples. Additionally, Table 4 presents the characteristics of the cross-sections of the bent panels.

The next stage of the research was to determine the energy absorbed during the bending of the panels. Energy absorption is key to testing the structural integrity in the event of failure. As shown in some articles (Jin et al. 2015; Yan et al. 2014; Schneider et al. 2016; Sun et al. 2017; Yazdani Sarvestani et al. 2018; Smardzewski 2019b; Ha and Lu 2020; Farrokhabadi et al. 2020), the honeycomb panels exhibit excellent energy absorption properties. The energy absorption \({E}_{a}\) (J) of the selected panels was calculated as follows:where \({F}_{x}\) (N) is the crushing force as a function of displacement \(x\) (m) during the crushing process, \({d}_{max}\) (m) is the effective deformation distance (deflection). The next parameter for analysis was specific energy absorption by panels per unit mass given by:where \({SE}_{a}\) (J) is specific energy absorption, \(m\) is the mass of the energy absorber. \({SE}_{a}\) is often used to compare the energy-absorbing ability of different materials and structures. The average crushing force \({F}_{c}\) (N) was defined as follows:where \(l\) (m) is the displacement corresponding to the maximum load, \({F}_{x}\) (N) is the instantaneous compression force.

Figure 6 presents models for numerical calculations of solid and openwork panels composed of 223,507 nodes and 174,804 elements (average), including quadrilateral elements of type R3D4, linear hexahedral elements of type C3D8R, and COH3D8 cohesive elements. Type R3D4 elements were used to model supports, type C3D8R elements for modeling wood, and COH3D8 for glue line. The numerical model of wooden elements was described with the use of the orthotropy plasticity material model (Kawashita et al. 2012; Qiu et al. 2014; Tran et al. 2015; Smardzewski 2019a). In the cited articles, the non-linear elastic properties of wood have been pointed out many times. Therefore, in this study, non-linear models were also used, both due to the orthotropy of the wood as well as the geometric nonlinearity of the model. The material constants are shown in Table 1, and the directions of the orthotropy are presented in Fig. 6, where L, R, T show anatomical directions of wood: longitudinal, radial, and tangential, respectively. A glue line joined the slats. The behavior of glue lines was modeled using the Cohesive Zone Model (CZM) through standard COH3D8 cohesive elements. The cohesive stiffness was estimated from the glue line isotropic mechanical properties (Smardzewski 2019a; Kamboj et al. 2020), assuming that the interfacial stiffness represents the elastic deformation of a glue-line layer of 0.1 mm in thickness. The panels were supported and loaded similarly, as shown in Fig. 5. The fixed supports and crosshead were assumed as perfectly rigid bodies. Between them and the panels, a surface-to-surface contact with a friction coefficient of 0.2 was modeled. The crosshead movement controlled the load value, causing the sample to be deflected by 40 mm at half its length.

Prior to starting numerical calculations, the models were first calibrated by selecting the appropriate mesh density and the number of iterations to obtain a satisfactory agreement between the load–deflection relationships determined numerically and in experimental tests. The calibration was finished successfully when the results were achieved, with no more than a 5% difference. The stresses at the wooden slats’ A–H points were then calculated (Fig. 6a). Stresses in the glue line were also calculated (Fig. 4), and then, using Eqs. 1–3, the absorbed energy, specific energy absorption, and crushing force were calculated. Figure 7 shows examples of deflections and stresses distribution in bending panels.

The commercial ABAQUS/Explicit v. 6.14 software (Dassault Systemes Simulia Corp., Waltham, MA, USA) was used for numerical calculations. All computations were performed at the Poznan Supercomputing and Networking Center (PSNC) using the Eagle computing cluster.

Figure 8a shows the average values for the experimental dependence between the load and deflection of the bent panels. This figure shows that for the same deflection value of 40 mm, the maximum forces are greater for solid panels and smaller for openwork panels. For solid panels made of beech, walnut, ash, and oak wood, the maximum bending forces are equal to 2983 N, 2935 N, 2842 N, and 2455 N, respectively. The decreasing values of bending forces result from decreasing values of linear elasticity modulus for selected wood species, 13,969 MPa, 13,760 MPa, 13,200 MPa, and 12,530 MPa, respectively (Table 1). For openwork panels with an 11° inclination angle of side walls made of beech, walnut, ash, and oak wood, the maximum bending forces are respectively lower by 12.3%, 3.4%, 22.5% and 13.0%. The obvious reason for the lower values of bending forces of openwork panels is their 20.6% lower moment of inertia of the \({I}_{x}\) (m⁴) cross-section compared to the moment of inertia of the solid panels’ cross-section (Table 4). However, what is interesting is that in the case of walnut wood, the stiffness of the solid panels was only 3.4% higher compared to the openwork panels, and in the case of ash wood, this difference equaled 22.5%. This may give preference to walnut wood and limit the use of ash wood as a raw material for making openwork panels. Figure 8b shows the results of numerical calculations for analogically loaded solid and openwork panels with an 11° angle of inclination of the side walls. This figure shows that for a deflection of 40 mm, the maximum forces for solid panels made of beech, walnut, ash, and oak wood equal to 2861 N, 2858 N, 2787 N, and 2508 N, respectively, whereas for openwork panels with an 11° inclination angle of the walls, the maximum bending forces are respectively lower by 7.7%, 5.6%, 17.1% and 13.9%. Thus, Fig. 8 shows that the results of numerical calculations are accordant with the results of the experimental studies. The differences between the values of experimental bending force for solid panels equal 4.1%, 2.6%, 1.9%, 2.2% for beech, walnut, ash, and oak, respectively. However, the differences for openwork panels equal 0.1%, 6.3%, 3.4%, 0.2% for beech, walnut, ash and oak wood, respectively.

To fully illustrate the satisfactory consistency of the numerical calculations with the results of laboratory tests, not only in relation to the values of maximum forces but also in terms of the load–deflection relationship, Fig. 9 compares the curves determined experimentally and numerically. This figure clearly shows that the curves determined numerically correspond satisfactorily with the experimental curves. Therefore, it was concluded that the calibration of the numerical model was carried out correctly, and the results of the numerical calculations could be used in the further argument. On this basis, numerical calculations were carried out for openwork panels with a 9° inclination angle of the side walls. Figure 10a summarizes the load–deflection relationships for openwork panels, while Fig. 10b shows the values of the maximum forces causing a deflection of 40 mm for solid and openwork panels.

Figure 10a shows that the load–deflection curves for openwork panels with 9° and 11° inclination angles of the side walls coincide. In contrast, Fig. 10b shows that the values of maximum forces causing deflection of these panels are minor and do not exceed 37 N. Of course, there is a significant difference between the values of the maximum forces causing bending of solid and openwork panels. In the case of panels with an 11° inclination angle of the side walls, the maximum forces are lower by 8.5%, 6.9%, 18.3%, and 15%, respectively, for beech, walnut, ash, and oak, and for panels with an inclination angle of 9° equal to 7.7%, 5.6%, 17%, and 14%.

The presented results show that making openwork panels out of solid panels with an inclination angle of the side walls 9° and 11° reduces their stiffness from 7.7 to 18.3%. At the same time, a minor decrease in stiffness was recorded for openwork panels made out of walnut wood (from 5.6 to 6.9%). Therefore, it is recommended to produce openwork panels from walnut, and then beech and oak wood, preferring the 11° angle of inclination of the side walls, which ensures the greatest width of the panel.

Table 1 shows that the difference between the greatest modulus of elasticity \({E}_{L}\) of beech and the walnut, ash, oak is equal to 2%, 6%, and 11%, respectively. For \({E}_{R}\) and \({E}_{T}\) modules it is 66%, 38%, 4% and 62%, 10%, 45%, respectively. The difference of the maximal forces causing the beech sample (9° and 11°) to deflect by 40 mm compared to the other samples is equal to 2%, 15%, and 23%, respectively. This regularity illustrates that the panel stiffness increases with an increase in the modulus of linear elasticity of the wood. At the same time, the differences between linear modules of wood elasticity and the differences between the bending forces of panels are adequate to each other. The orthotropic nature of wood and differences between the modules of wood elasticity across the grain also significantly influence the measured forces.

The differences in the strength of the designed and manufactured panels are interesting too. Figure 11a, b shows the variation of normal stresses \({\sigma }_{z}\) (MPa) at points A-H on the cross-section of the bent panel. This figure shows that the stresses in openwork panels have a non-linear course. Their value increases from the outer edges (points A and H) towards the panel axis, reaching a maximum at points B and G, which is on the inside of the first outer curvilinear slats. Then the stresses \({\sigma }_{z}\) (MPa) decrease and reach a minimum at points D and E, near the middle slats’ outer surfaces. The difference between the highest and the lowest normal stress does not exceed 10 MPa. At the same time, the maximum stresses (Fig. 11c) are significantly lower than the bending strength of the selected wood species (Table 1). In this case, the safety factor for wooden elements of openwork panels varies from 1.89 to 2.09. For solid panels (Fig. 11b), the \({\sigma }_{z}\) stresses (MPa) run almost linearly in the direction of the panel axis. As in the case of openwork panels, the maximum value of these stresses (Fig. 11c) does not exceed the bending strength of wood, which in this case is also manifested by a high safety factor varying from 1.72 to 1.83. This indicates that each proposed type of wood and each angle of inclination of the side walls in the openwork panels ensure good bending strength of the wooden slats.

In the performed tests, there is no detected failure in the slats. The main reason for this is the character of the test performed. In this case, bending deflection equal to 40 mm stopped the test. For this reason, the maximum stresses (Fig. 11c) are significantly lower than the bending strength of the selected wood species (Table 1).

Figure 12a shows the distribution of reduced stresses in the selected glue line, characteristic for all calculations (see Fig. 4). This illustration shows that the greatest stresses are concentrated in the corners of the glue line, and the smallest stresses close to zero value occur in the center of the surface of the glue line. This distribution is characteristic for adhesive and shape-adhesive corner joints subjected to bending (Smardzewski 1996, 2002, 2008; Imirzi et al. 2015). The scope of the cited works and the results of numerical calculations (Smardzewski 2019a, b) show that the adhesive joint’s crack initiation may take place where the most significant stresses occur, i.e. in the corners of the adhesive joint. However, the average shear stress of the adhesive joint is at a much lower level. The value of the maximum reduced stresses at the corners of the glue line for analyzed openwork panels is shown in Fig. 12b. In the case of panels with a side wall inclination angle of 9°, the presented values exceed the shear strength of the adhesive used (16 MPa). During the tests, however, no damage was observed in the glue lines of the bent panels. This is due to the fact that the average reduced stress in the adhesive joint for all analyzed openwork panel structures did not exceed 3 MPa, thus it was significantly lower than the strength of the adhesive joint. At the same time, the values of shear stresses in the adhesive joints were significantly lower, as shown in Fig. 13.

The numerical calculations confirmed by experimental tests show that apart from sufficient stiffness, the openwork panels are characterized with very high strength of both wooden slats and glue lines. This means that the tested structures can be safe and durable.

Energy absorption is an important indicator for assessing the amortization effectiveness of the engineering structure under study. Comparing the energy absorption capacity of the openwork structure with the solid panels allows choosing the best solution. Figure 14a, b shows the dependence of the absorbed energy on a deflection for openwork and solid panels, while Fig. 14c compares the values of the maximum energy absorbed by individual panels. This figure shows that the energy absorption curves are progressive, and the amount of energy absorbed disproportionately rises with increasing deflection of the panels. During bending, most of the energy is absorbed by solid panels made of ash wood (69.2 J). The remaining solid panels made of beech, walnut, and oak wood absorb energy in the amount of 61.6 J, 61.4 J, and 51.1 J, respectively. In the case of openwork panels with a 9° inclination angle of the side walls, the amount of energy absorbed depends on the type of wood. Compared to solid panels, openwork panels made of ash and beech absorb respectively 19.3 and 7.0% less energy, while openwork panels of walnut and oak consume 2.5 and 2.9% more energy, respectively. The increase in the inclination angle of the side walls of the curved slats from 9° to 11° results in a slight reduction in the amount of absorbed energy. Compared to solid panels, openwork panels of ash and beech (11°) absorb 20.6 and 7.6% less energy, respectively, while openwork panels of walnut and oak consume respectively 3.7 and 3.9% more energy.

The change of specific energy absorption depending on the deflection of solid and openwork wooden panels is shown in Fig. 15a, b. Figure 15c shows the maximum values of the specific absorption energy. These figures show that walnut wood panels absorb the most energy per unit of mass. Openwork panels with 9° and 11° inclination angles of the side walls absorb respectively 24.4 and 23.7 J/kg, while solid panels absorb 21.1 J/kg. Openwork panels of ash with side walls inclination angles of 9° and 11° absorb respectively 19.4 and 18.7 J/kg, and solid panels—21.0 J/kg. Thus, an increase in the specific energy absorption was observed for solid ash wood panels. This is directly related to the large amount of energy absorbed by these panels during bending (Fig. 14). For panels of beech and oak, these trends are similar to those made of walnut. Openwork panels with side walls inclination angles of 9° and 11° absorb 18.6 J/kg, 16.1 J/kg, 18.1 J/kg, and 15.7 J/kg, respectively, while solid panels absorb 17.6 and 15.7 J/kg, respectively.

The presented analysis shows that the most favorable are openwork structures made of walnut wood, followed by beech and oak wood.

Figure 16 shows a histogram of the maximum bending force of the panels per unit of deflection. According to the histogram (Fig. 14) for a constant deflection value (40 mm), the maximum bending force of 1.68 kN is for a solid ash panel. The lowest bending forces of 1.33 and 1.31 kN fall on oak openwork panels made of curvilinear slats with the inclination angle of the side walls of 9° and 11°, respectively.