Calculation Concept for Wood‐Based Components in Mechanical Engineering

Herein, an approach is described for a calculation concept to close the gap between lightweight construction and necessary safety for structural components made of wood materials with use in mechanical engineering. The calculation of wood‐based materials represents a challenge due to the orthotropic material behavior and the many influencing parameters on the mechanical properties. The main objective of the considerations is to predict the mechanical component behavior as precisely as possible. In the next step, the designer can use the results of the calculations to make statements about the component safety. The analyses of this article are part of the joint project “Wood‐based materials in mechanical engineering (HoMaba)”.


Introduction
Within the development process of technical structures in mechanical engineering, calculations are carried out to predict the mechanical component behavior and to analyze the stresses and strains in the component. Based on the calculation results, safety verifications are carried out to guarantee the usability of the component over its entire service life. The main objective is to develop components that are as safe as necessary and as light as possible. Figure 1 shows some applications in wood composite construction that were constructed as part of the previous research and development work of the research group Application of Renewable Materials of the Professorship of Materials Handling and Material Flow Technology at Chemnitz University of Technology. [1,2] These applications have generally been dimensioned according to EUROCODE 5 in combination with component tests. [3] This includes either the development of new systems in wood construction or the substitution of metal by wood materials in existing components. On the one hand, this requires a large number of test components and testing effort with correspondingly high costs. [4] On the other hand, dimensioning according to Eurocode 5 is complex and generates oversized components by reducing the mechanical properties to characteristic 5% quantile values in combination with additional safety factors. [5][6][7][8] This may be appropriate in wood engineering but does not correspond to the sociopolitical demand for sustainability in mechanical engineering. [9][10][11] Due to its anisotropic structure and a number of influencing factors on the mechanical properties, such as the influence of wood moisture, the calculation of wood-based materials is a more demanding challenge than metallic construction materials with generally isotropic material behavior. [12][13][14][15] In addition to other influences, the orientation of the individual elements of a composite component has a significant influence on its composite properties and must be taken into account in the calculation. Using the example of selected demonstrators, this article is intended to show the possibilities and limits of the calculation of composite components made of plywood.
With a validated calculation concept, a more flexible safety concept for wood-based materials can be derived in the next step that makes lightweight construction possible. This gives the designer more safety when dealing with the material wood and its material-specific properties and opens up wood as a construction material better access to mechanical engineering, especially in the area of load-bearing applications in conveyor technology. Load-handling devices in various forms, workpiece carriers, crane systems, vertical conveyors, machine frames are just a few potential examples of use in wood construction.
The analyses and results described in this article were developed as part of the joint project "Wood-based materials in mechanical engineering (HoMaba) -calculation concepts, characteristic value requirements, characteristic value determination". For more detailed information, please refer to the final report. [16]

Experimental Section
The dimensioning of components was intended to ensure that they remain functional throughout their entire service life. For this reason, dynamic loads or load combinations with the corresponding number of load changes were decisive for the DOI: 10.1002/adem.202300085 Herein, an approach is described for a calculation concept to close the gap between lightweight construction and necessary safety for structural components made of wood materials with use in mechanical engineering. The calculation of wood-based materials represents a challenge due to the orthotropic material behavior and the many influencing parameters on the mechanical properties. The main objective of the considerations is to predict the mechanical component behavior as precisely as possible. In the next step, the designer can use the results of the calculations to make statements about the component safety. The analyses of this article are part of the joint project "Wood-based materials in mechanical engineering (HoMaba)".
dimensioning. The basis for this was the dimensioning against static, uniaxial loads, which was the main subject of the project. Figure 2 shows the main structure of the HoMaba project. In the first step, a lot of material tests were carried out to determine the mechanical properties of the materials used in the project. That was a basic requirement for the calculation and simulation of the composite components to predict their mechanical behavior in form of a force-displacement curve. To be able to estimate the component safety, a semi-probabilistic safety concept adapted to mechanical engineering was developed in the next step. Therefore, the EUROCODE 5 was analyzed first and based on that relevant safety factors that are necessary for mechanical engineering were derived. At the end, the results of the calculations were validated by comparison with experimentally determined force-displacement curves of selected composite components made of plywood. On the one hand, hollow profiles in wood construction were used as part of a modular system for the frame construction of skid-conveyors or load carriers (Figure 1-1, 1-3, 1-4). Furthermore, load-handling devices as shown in Figure 1-2 were used for validation. This article refers to the developed calculation concept, that is, the analytical prediction of the force-displacement curve of composite components without simulation tools. To do this, the stiffness of the composite components and the yield point as the transition from linear to nonlinear behavior must be calculated using the material properties of the individual elements. The determination of characteristic values is only described to the extent necessary for the analyses in this article. Simulation and safety concepts are not part of this article. [16] The calculation concept focused the static three-point bending load case with static uniaxial loads. That was one of the most frequently occurring load cases for applications made of woodbased materials. For example, the side wall of the skid conveyor system in Figure 1-1 is classically subjected to three-point bending during the passage of a loaded skid. The load-handling device in Figure 1-2 is also classically subjected to three-point bending.
In the case of special load carriers, load-bearing individual components could be subjected to bending loads depending on the elements to be transported. It should be noted that depending on the application in reality, local combinations of different load Figure 1. Examples for applications made of wood materials for use in mechanical engineering (picture 1: skid conveyor system. Reproduced with permission. [42] Copyright 2014, Volkswagen AG WOB/Dr. F. Drechsler; picture 2: load-handling device [16] ; pictures 3 and 4: special load carrier. Reproduced with permission. [1] Copyright 2022, LiGenium GmbH).
www.advancedsciencenews.com www.aem-journal.com cases could act on a component, which must be taken into account when dimensioning. To take into account the combined bending and shear stress due to a classic three-point bending load case, the Timoshenko beam theory was used for the calculations. The components were considered as rigidly connected composite components made of panel-shaped wood veneer composite materials. Furthermore, only the linear component behavior was considered, failure analyzes were not carried out. The reason for this was that when the yield point was exceeded, plastic displacement was assumed and the component was classified as no longer usable. This was not necessarily the case with low load changes, but it became increasingly critical with higher load changes.

Material Properties
As part of the investigations of the HoMaba project, relevant material properties were determined in about 6500 material tests on veneers, solid wood, and plywood made from birch and beech. All samples were self-produced from selected trees, so that a consistent quality of the wood samples could be ensured. Depending on the existing measurement systems, the data were measured optically with extensometer or camera systems from the company Carl Zeiss GOM Metrology GmbH. All specimens were conditioned and tested at 20°C/50% relative humidity. This test climate was derived as the standard climate for mechanical engineering applications. The 20°C corresponds to a temperature according to the German workplace ordinance at which all activities from light to heavy work can be carried out. [16] The selected relative humidity of 50% is therefore a compromise. If the relative humidity is too low, problems with electrostatic discharge arise. [17][18][19] If the relative humidity is too high, this can lead to problems with corrosion. [20][21][22] The relevant material parameters include characteristic values for modulus of elasticity, shear modulus, as well as strength and strain values of the yield point and at failure for relevant directions and fiber load angles (depending on the wood material). These values can be obtained from the measured force-displacement curve as a result of a single material test, like Figure 3 shows. The modulus of elasticity can be determined from the slope of the curve in the linear range, which can be obtained by linear regression in the range between 10% and 40% of the maximum force (failure). To determine the yield point, the linear slope is compared to the slope of the tangents at each point on the curve. The first slope of the tangents that deviates by more than 5% of the linear slope in the curve corresponds to the yield point. The failure point is the point of maximum force. Depending on the type of material test, the force and displacement parameters as well as the modulus of elasticity can be converted into stress and distortion parameters using appropriate mechanical engineering formulas. [16] To determine the material properties of veneers, tensile and compression tests were carried out in the direction of the grain (fiber-load angle 00°) and perpendicular to the direction of the grain (fiber-load angle 90°). Shear tests were also carried out using the method described in ref. [23]. In the case of solid wood, the three main directions longitudinal (L), radial (R), and tangential (T) were tested in tensile und compression tests. Comparable shear tests as for veneers were also carried out in three relevant combinations of directions: LR, LT, and RT. In the three-point bending tests, only the bending in the longitudinal direction was tested, with a distinction being made between the radial and tangential direction when the force was introduced. In the case of plywood, tensile and compression tests were carried out in the direction of the midplane of the panel, with the grain direction of the cover layer being parallel to the direction of the force. This corresponds to a disk stress. Shear tests were also carried out according to ref. [23] in the middle plane of the plate. The bending properties were determined by applying force parallel to the center plane of the plate (disc) and perpendicular to the center plane of the plate (plate). Table 1 shows the tested configurations.
All data were collected centrally and consistently evaluated. As a result, the determined material properties were transferred to a material database. In addition to all relevant material parameters, the associated test parameters and statistical parameters were also entered. Test parameters include, among other things, all parameters that can influence the mechanical properties, such as the sample geometry, the test speed, the orientation of the samples, as well as the test and conditioning climate. Statistical parameters include,  www.advancedsciencenews.com www.aem-journal.com among other things, the standard deviation, the number of samples per test series, and various quantile values of the test series.
For the calculations described in this paper, only the bending and shear properties of birch and beech plywood, shown in Table 2, are relevant. These properties relate to the plywood bending test configuration shown in Table 1. To determine these properties, bending tests were carried out with different span-to-height ratios and evaluated using the Timoshenko beam theory.

General Procedure
The calculation concept for wood-based components under threepoint bending load case as well as for other load cases is a four-step process, as shown in Figure 4. First, the application needs to be analyzed with regard to existing stresses and geometric boundary L-longitudinal direction; R-radial direction; T-tangential direction; Symbol (⥊)-fiber direction (for plywood fiber direction of the cover layer).
www.advancedsciencenews.com www.aem-journal.com conditions. The component stiffness is then calculated from the stiffnesses of the individual elements. To do this, the Timoshenko beam theory is applied. [24] This determines the linear increase in the force-displacement curve. In step three, the yield point is determined using the Norris/McKinnon criterion. [25] This is the point at which linear component behavior changes to nonlinear behavior. The final step involves calculating the failure, also using the Norris/McKinnon criterion. This paper only refers to step two and three.

Calculation of the Component Stiffness
In the case of a three-point bending stress (as well as in the case of four-point bending), the beam experiences not only a bending moment but also a shearing stress as a result of the lateral force.
Due to the fact that wood, unlike steel, is a flexible material with low shear stiffness and strength, the shear influence should not be neglected. [26][27][28] The combination of bending and shear stress warps the cross section, causing it to no longer remain flat and perpendicular to its median plane during stress. The restrictions for the Euler-Bernoulli beam theory lose their validity and the Timoshenko beam theory needs to be applied, which is a common method for calculating wood constructions. [24,[29][30][31][32][33] According to the Timoshenko beam theory, the total displacement is the sum of bending and shear displacement [24] w total ¼ w bending þ w shear (1) w total -total displacement [mm] w shear -displacement from shear stress [mm] w bending -displacement from bending stress [mm] By inserting the differential equations of the bending and shear lines in Equation (1), an equation for calculating the component's modulus of elasticity as a function of the span-to-height ratio will be given, where the component's height is included in the cross section and the area moment of inertia [16,24,34]    www.advancedsciencenews.com www.aem-journal.com

A-cross-section area [mm 2 ] I-area moment of inertia [mm 4 ]
Equation (2) can be used to calculate the equivalent component elasticity modulus depending on the span L and the component height H. In addition to the geometry parameters, cross-sectional area A, moment of inertia I, and span L, a shear correction factor κ is required, which takes into account the cross-sectional warping caused by the combined bending and shear stress. [35][36][37] Furthermore, the bending modulus of elasticity E B and the shear modulus G are required for Equation (2).
The bending modulus of elasticity E B of a composite component results from the sum of the stiffnesses of the individual elements i (belts and webs) in relation to the total area moment of inertia of the component [3,34]  The shear modulus G of a composite component results from the sum of the shear stiffnesses of the individual elements in relation to the total cross-section area [3,34] G ¼

G-component's shear modulus [MPa] A-component's cross section area [mm 2 ] G i -shear modulus of the individual elements [MPa] A i -cross-section area of the individual elements [mm 2 ]
With the calculated component's modulus of elasticity, the displacement of the component can be calculated as a function of the force according to the classic equation for calculating the displacement of a three-point bending load case. This displacement includes both the bending and shear displacement, when using the modulus of elasticity of the component. Equation (5) can also be used to calculate the displacement resulting from the bending stress by using the bending modulus of elasticity instead of the component's modulus of elasticity. [24] w component ¼

Yield Point
The yield point describes the transition point from the linear to the nonlinear behavior of a force-displacement curve and thus represents the limit of the calculated force-displacement curve. If a material sample is loaded with a central point force in a three-point bending test, the material experiences a combined bending and shear stress. Depending on the cross-sectional geometry, however, deviating stress combinations can arise in the component. The shear stress in particular depends to a large extent on the shear flow that occurs in the cross section. To take these combined stresses into account for a component, a separate consideration of the yield point for bending and shearing from the material test is not sufficient. In that case, the empirical failure criterion according to Norris-McKinnon was used to record the stress combination in the component's cross section [25,38,39] 1 Equation (6) describes a stress-based approach to model stress combinations. By applying Hooke's law, the stresses can be replaced by strains, allowing a strain-based calculation [34] 1 For uniaxial bending, ε y can be assumed to be zero. The normal strain ε x as a result of a three-point bending stress can be calculated as a function of the bending displacement w bending according to Equation (8) [34]  Similarly, shear strain γ xy can be represented as a function of the load F. First of all, the formula for calculating the shear stress is required [24] (9) and (11) into Equation (7) gives a formula whose solution delivers a yield point force when the allowable strains of the critical element (material properties) are used F, Q-load F and resulting lateral force Q [N] X, Z-allowable normal strain X and shear strain Z [-] Figure 5. Bending strain pattern of loaded hollow beams (left picture) and loaded load-handling devices (right picture). Reproduced with permission. [16] Copyright 2022, HoMaba project/B. Buchelt, University of Technology, Dresden, Institute of Natural Materials Technology, professorship in wood and fiber material technology.  As a final step, the calculated force value of the yield point must be substituted into Equation (5) to calculate the associated displacement of the yield point.

Validation
All component tests were measured optically with a GOM-camera system. This enabled a separate evaluation of all displacements resulting from the bending and shearing stresses. The left picture in Figure 5 shows an example of the bending strain pattern of a loaded hollow beam. It can be seen that a compressive stress zone forms in the upper area (blue-colored area) and a tensile stress zone forms in the lower area (red colored area). The right picture in Figure 5 shows an example of the bending strain pattern of a loaded load-handling device.  www.advancedsciencenews.com www.aem-journal.com There a compressive stress zone forms in the lower area (bluecolored area) and a tensile stress zone forms in the upper area (red-colored area).

Results of the Hollow Beam
As already mentioned, the hollow beams are part of a modular system for rack systems or frame constructions and consists of two flanges and webs glued together. Figure 6 shows the geometry of the considered hollow beams. In the case of a three-point bending stress, the direction of the force is perpendicular to the plate center plane of the flanges (plate stress) and parallel to the plate center plane of the webs (disc stress). For this reason, the material parameters for the plate configuration of birch plywood in Table 2 has to be used for the flanges, while the material properties of the disc configuration should be used for the webs. The material used was symmetrically constructed wood veneer composite in the form of birch plywood (21 layers, 0°/90°alternating) with a panel thickness of approx. 20 mm, with the fibers of the top layers running parallel to the span direction for all elements. The cross section had a height and a width of 100 mm. Four different spans were tested (750, 1000, 1250, 1500 mm) which results in span-to-height ratios of 7.5, 10, 12.5, and 15, respectively. For the calculations shown in Table 3, the material properties of birch shown in Table 2 were used. The parameters for the calculation determined from the cross-section geometry are shown in the first line of Table 3. Next, the bending and shear moduli of the hollow beam are calculated using Equations (3) and (4), and the bending and shear moduli of the flanges and webs (two flanges and two webs). With these two parameters, it is possible to calculate the component's modulus of elasticity according to Equation (2) for the existing span-to-height ratio. With the help of the component elasticity modulus, the displacement of the component can be calculated as a function of the force according to Equation (5). This allows the linear component curve to be mapped. Finally, Equation (12) is used to calculate the yield point force. For this purpose, the allowable bending and shear strains of the lower flange (plate) are used, since this is the critical element with the highest normal stresses and is responsible for the failure. Knowing the force of the yield point, Equation (5) is used to calculate the associated displacement.
The results in Table 3 allow the calculated force-displacement curve for the different spans to be plotted in a force-displacement diagram. The addition of experimentally determined test curves allows a comparison with the calculated curves, as shown in Figure 7. The comparison shows that there is good agreement with regard to the stiffness (increase in the linear curve area)  for all spans. The deviation of the calculated moduli of elasticity from the mean values of the experimentally determined values is between 1% and 10%. With a small span of 750 mm (large shear influence), the calculated curve is in the lower range of the experimentally determined curves, which is advantageous from a safety-relevant point of view. As the span-to-height ratio increases, the calculated curves approach the mean value of the experimental curves more and more. The same applies to the yield point. The deviation from the experimental values is greatest for the smallest span with the highest shear influence and approaches the experimental values more and more as the span increases, respectively, the shear influence decreases. As a conclusion, it can be seen that the shear becomes more influential on the calculation models with a decreasing spanto-height ratio, which must be taken into account in the safety analysis in the next step.

Results of the Load-Handling Device
The load-handling device represents the interface to the hoist and is used to attach loads. It was made of beech plywood and has a more complex geometry which can be seen in Figure 8. It also consists of two flanges and webs which are connected via loadbearing elements. A rigid connection of the individual elements is assumed. Five load-handling devices were built with a span of 1500 mm (span-to-height ratio approximately 10 in the middle).
Due to the small number of components, two of them were tested up to failure and three were tested to a defined force of 20 000 N. To take the bending moment curve into account, the component height increases toward the middle, which leads to a change in the cross-section area and the area moment of inertia over the length. This was taken into account in the calculation by using the finite difference method. [40,41] For this purpose, the load-handling device was divided into 1500 areas with a width of 1 mm across its span. Then, the local area moments of inertia and the cross-section areas of the whole cross section and the single elements were calculated in the middle of each area and used in the formulas of the finite difference method. The resulting system of linear equations was solved using a calculation software to determine the displacement of the component. Otherwise, the calculation was carried out in the same way as for the calculation of the hollow beams, only with material parameters for beech (see Table 2). To calculate the different moduli, the mean value of the local area moments of inertia and the crosssection areas were used. The results of the calculations are shown in Table 4. Figure 9 shows the experimental curves (black curves) and the calculated curve (red curve) of the load-handling device. Three load-handling devices were tested in the linear range up to a defined force of 20 000 N, and two were tested up to failure. As with the hollow profiles, good agreement between calculation and experiment could also be achieved here.

Conclusion
A calculation concept for rigidly connected composite components made of wood-based materials was presented. This makes it possible to predict the mechanical behavior of the components up to the yield point on the basis of the material properties of the individual elements, as the validations have shown. As a result, time-consuming and cost-intensive component tests to determine component behavior can be minimized. In addition, the designer can obtain reliable statements about the load-bearing capacity of statically loaded components, which means that the property potential of wood as a construction material can be better exploited. The use of greatly reduced characteristic material properties according to EUROCODE 5, among other things, to take model uncertainties into account is therefore not necessary. In contrast, wood-specific influencing factors such as the influence of wood moisture on the mechanical properties must be taken into account. So, future work will deal, among other things, with a semi-probabilistic safety concept to supplement the calculation concept with partial safety factors. Furthermore, the work on the calculation concept will be continued with the dynamic load case, so that a complete dimensioning of a component can be carried out and its functional maintenance can be guaranteed over its entire service life.