Experimental assessment of failure criteria for the interaction of normal stress perpendicular to the grain with rolling shear stress in Norway spruce clear wood

The mechanical behavior of Norway spruce (Picea abies) clear wood under stress perpendicular to the grain with rolling shear interaction was studied by means of an experimental setup. The material originates from the Experimental Forest and Research station in Asa, Sweden. The log of a Norway spruce tree was cut at 1.30 m and 2.70 m from the ground and sawn into pieces with cross-sectional dimensions of 75 × 75 mm², see Fig. 1. The wood pieces were positioned so that one of the edges follows the same annual ring at the outer side of the log (Fig. 1) and four pieces with similar annual ring patterns could be cut out. The objective was to obtain homogeneous material properties and similar annual ring structure in all specimens without any knots or defects. The wood pieces were then dried in a research oven at Luleå University of Technology with a targeted equilibrium moisture content of 12%. In the development of the test setup, geometry and setup effects on material behavior under compression, rolling shear and stress interaction in the R-T plane were studied. For this purpose, different shapes of specimens and different force imposing systems were investigated, as described in the following. Four different specimen geometries and shapes were cut out from the boards after drying, see Fig. 1,

All specimens had a length of 20 mm (in the longitudinal direction of wood), except for D-shaped specimens, which were 10 mm thick in the area of interest. Specimens were prepared with two orthogonal material orientations for testing with normal stresses in the radial (QR, VR, TR, DR) and the tangential (QT, VT, TT, DT) directions. Detailed notations and notch dimensions of the specimens are shown in Fig. 1. The prepared specimens were stored in a climate chamber at 20 °C room temperature and 65% relative humidity. Note that for the geometry and setup effect study, Q-type specimens were prepared from material at the east part of the stem, while, V- and T-type specimens were from the west side of the stem. The average density was 492 kg/m³ for Q-type and 481 kg/m³ for V- and T-type specimens. D-type specimens for loading in radial and tangential directions were prepared from material at the south and north side of the stem. The average density for the specimens of series DR was 490 kg/m³ and DT was 508 kg/m³. The average moisture content was 12.80%, which was measured by means of the oven drying method. A total number of 104 specimens were tested of which 72 tests were performed on D-shaped specimens to compare experimental findings with analytical failure criteria. A total of 38 uniaxial tests were performed for rolling shear, uniaxial compression and tension perpendicular to the grain, to study the geometry and force imposing setup effects.

The mechanical test setup was developed within the test frame 322 of manufacturer MTS, which was equipped with two servo-hydraulic actuators (MTS Model 661.20F) with a capacity of 100 kN for the vertical orientation and 50 kN for the horizontal orientation.

Two different test setups, with basically two different force imposing systems, were used, see Fig. 2. Spengler’s biaxial test setup (Spengler 1982) was considered as a basis to develop the setup, due to its very simple configuration that allows straight-forward testing, particularly when using it without adhesive bonding. The study on effects of the shape of specimens as described above was performed with this setup, consisting of two L-shaped steel blocks connected to the moveable sledges of the test frame, see Fig. 2a, d. The sledge below the test specimen is moveable in the horizontal direction, whereas the cross-head above the specimen is moveable in the vertical direction. Both are almost rigid, particularly when considering the weak stiffness of the test specimen. Q-, V-, and T-type specimens were then simply put into the test setup, which had 10 mm high steel plates on both sides of the bottom and the top of the L-shaped blocks to laterally support the specimens. There was a smooth surface on the side supports.

The force imposing system with L-shaped steel blocks is very simple, but could obviously lead to stress concentration at the corners. The wooden specimens could have been adhesively bonded to the steel supports in order to get a more distributed force transmission. However, this would have been very cumbersome and difficult to realize in this biaxial setup. Thus, mechanical grips were developed to more homogeneously introduce loading to the test specimens. For this purpose, the side support plates were replaced with mechanical grips using spiked steel plates with a height of 15 mm, see Fig. 2c, e. The side support was fixed to the loading device while it was attached to the bolts and a fixed steel support on the other side of the test specimen. These mechanical grips allowed for the specimen to be grasped, while the pyramid texture of the steel plates gave a rough surface with high friction and good connection between the loading device and the wooden specimen. Dog-bone shaped specimens were tested in this device. The effects of the force imposing device on the stress and displacement state of the specimen could be studied by comparing the mechanical grips with the L-shaped steel block. Moreover, for this setup, an external multi-axial load cell of type GTM-00037 (Gassmann Thelss Messtechnik) with a capacity of 5 kN in the vertical and the horizontal directions, the two directions evaluated in this study, was used to assess the accuracy of the larger in-built load cells in the loading range of the tests. A maximum difference of up to 12% between the in-built load cells and the external load cell was found, while the average difference for all tests was 6.5%. The maximum difference was observed in the quasi-elastic part of the test at comparatively low forces, whereas the difference was less for higher forces. Only the results of the external load cell data are provided in the results section.

Experiments were carried out in displacement control mode along several displacement paths, as illustrated in Fig. 1e. Rolling shear combined with tension and compression stresses are denoted by ST and SC, followed by a number that is related to the angle with respect to the shear plane. A displacement rate of 2 or 1 mm/min was applied in case of compression and biaxial tests and 0.50 or 1 mm/min in rolling shear tests. No influence of these displacement rates on the overall force-displacement behavior was observed. In the case of combined loading, with a certain ratio of vertical to horizontal displacement, displacements were applied simultaneously with an overall displacement rate equal to 1 or 2 mm/min. In addition, two unloading sequences were applied, one in the quasi-elastic and one in the elasto-plastic domain. The first unloading cycle was performed at a force of 1 kN, while the second unloading cycle was applied at a displacement of 5 mm for the compression and biaxial tests on D-type specimens. At the beginning and the end of each unloading cycle, the force (for the first unloading cycle) or displacement (for the second unloading cycle) was kept constant for 5 s to reduce the influence of time-dependent effects on the unloading stiffness. The investigated displacement paths for the corresponding test specimens are illustrated in Fig. 1. Two or three specimens were tested in each loading path. Corresponding forces and displacements were measured by the internal actuator of the MTS test frame and the external load cell.

Note that the lateral boundary condition plays an important role, which is most obvious in uniaxial testing. Displacement controlled testing as outlined above means constraining the lateral displacement, which as a consequence of the Poisson effect leads to a biaxial stress state. In addition to the displacement path testing, shear tests and compression tests with unconstrained lateral boundary conditions were performed. For unconstrained testing, the vertical or horizontal force was limited to a maximum absolute value of 50 N, while the lateral displacements were not constrained. The latter test data was then compared to results from displacement-constrained loading to assess the effects on the determination of uniaxial mechanical properties.

A digital image correlation (DIC) system (Aramis, Gesellschaft für Optische Messtechnik mbH, Braunschweig, Germany) was used externally to measure the strain fields on the surface of the test specimens, as well as, through point markers, the displacements of the loading devices, during the experiments. This data allowed a comparison of the internal displacement measurement of the test frame with the optically determined displacement states. For DIC measurements, the specimens were sprayed with a very thin black speckle pattern on a very thin white base layer. The desired point size of the speckle pattern was 23 pixels (P). Two 12 MP cameras were used to continuously capture images at a rate of 1 Hz (one picture per second) in the elastic and the beginning of the elasto-plastic part, followed by a rate of 0.50 Hz (one picture every 2 s) during the remaining test period. The field of view for the DIC was chosen to approximately 190 × 140 mm². A facet size of 19 P together with a grid spacing of 15 P (parameters have been set based on recommendations of the supplier and a preliminary study) resulted in a distance of approximately 1.20 mm between the measurement points. A noise study was carried out before each experiment to check the suitability of combining the speckle pattern, illumination, and camera settings. Displacements measured by the control system of the test frame were compared to displacements of point markers measured by DIC. The maximum difference was less than 10%.

The direct outputs of the tests were the displacement of the loading device as well as the load cell data, which were then used for the calculation of nominal engineering strains and stresses. The tensile or compressive strains, \(\varepsilon _{RR}\) and \(\varepsilon _{TT}\), were calculated as the vertical displacement of the force imposing device divided by the unsupported height of the specimen, which was 50 mm for Q-type specimens, and 60 mm for V-, T-, and D-type specimens with L-profile, while for the gripped plate it was 20 mm for Q- and 30 mm for V-, T-, and D-type specimens. Rolling shear strain, \(\gamma _{RT}=2\varepsilon _{RT}\) was determined as the horizontal displacement divided by the unsupported height of the specimens given above. For comparison reasons, average strains in the center of the specimens were even determined as the average strains on the surface of the specimens measured with DIC.

The normal stresses, \(\sigma _{RR}\) and \(\sigma _{TT}\), were determined as the vertical force divided by the initial minimum cross-section; and the rolling shear stress, \(\tau _{RT}\), was calculated as the horizontal force divided by the initial minimum cross-section in the center of the specimens. Corresponding cross-sectional areas amounted to 1000 mm² for Q-shaped specimens, 800  mm² for V- and T-shaped specimens, and 300 mm² for D-shaped specimens.

The gradient from half of the first unloading path of the stress-strain curve, see Fig. 3, was considered when calculating the Young’s moduli of elasticity in the radial and tangential directions, \(E_R\) and \(E_T\), and the rolling shear modulus, \(G_{RT}\). The unloading path was chosen to determine the elastic material parameters, since elasticity is defined as the mechanically recoverable energy stored in the loaded sample (Bader et al. 2016).

To compare the test data with failure criteria, stress states at specific strain states were plotted in the \(\sigma _{RR}\)-\(\tau _{RT}\) and \(\sigma _{TT}\)-\(\tau _{RT}\) stress planes. Stress points were considered at 1% and 2% compressive strains. These stress points were chosen to study the onset and the development of plastic failure of the material. These stress points were then compared to previously suggested failure criteria discussed in Sect. 2. The predictions of failure criteria were calculated by using the uniaxial strength properties as determined in uniaxial tests. Uniaxial compressive strength was determined as the stress at 1% compressive strain, while the maximum stress was considered as strength in tensile and rolling shear tests. In the case of failure criteria that do not distinguish between tensile and compressive strength, the compressive strength was considered for calculations. To assess the suitability of phenomenological failure criteria, the prediction capability, R of these failure criteria were calculated instead of fitting the criteria. R-value equal to 1 means failure prediction is perfect. \(R>\)1 means the criterion underestimates and \(R<\)1 means the criterion overestimates the failure. R is calculated from the experimental stress and material strength values.

The value R, for example in the case of quadratic criteria, can be calculated aswhich for \(\sigma _{TT} = 0\) in the considered stress interaction gives

During the development of a suitable and efficient setup for biaxial testing in the R-T plane, test setup effects such as geometry and boundary or force imposing effects were studied. Loading was applied by means of two different devices, one with direct compression contact in L-shaped steel blocks and one with mechanical grips. Note that L-blocks restricted the horizontal (global x-direction) deformation of the test specimens in the corners, which led to shear stress even under pure compressive force. Moreover, compression tests were conducted with constrained horizontal displacement, which yielded a global shear force. The L-block setup developed a higher shear force than mechanical grips, which was particularly pronounced in biaxial loading. Mechanical grips, on the other side, allowed for deformation of the specimens in the horizontal direction, but led to stronger constraints in the load introduction area, where even deformation in the transverse (global z-direction) direction was constrained. Another important difference between the loading setup was that the mechanical grips reduced the unsupported height of the specimen.

Test specimens were tested in two different orientations, where either the radial or the tangential direction was parallel to the global y-direction, see Fig. 1. In radial compression, cell layers are compressed and the void volume is continuously reduced until the cells are fully compressed and the cell layer material is fully densified (Mackenzie-Helnwein et al. 2003). In tangential compression, however, buckling of the latewood cell layers is the predominant failure mechanism (Bodig 1965; Tabarsa 1999), and thus, the height of the specimen is of importance for the overall force carrying capacity. Due to these differences in the material response in the two orthogonal directions, setup effects were more pronounced for testing in the tangential direction than in the radial direction. This is clearly visible in Fig. 4a, which shows stress–strain relationships as determined by compression tests on Q-type specimen, using L-shaped (represented by solid lines) and gripped loading setup (represented by dashed lines). Differences in the initial behavior due to further deformation in the contact case compared to the mechanical gripping, become obvious, while the overall shape of the curves and the stress levels are rather similar. For compression testing in the tangential direction, however, considerable differences in the stress levels have been observed with higher stresses in the case of L-profiles, which is a consequence of boundary effects. This was obvious when comparing Q- and D-type specimens under tangential compression, as discussed in the next subsection. All tests except ’L-QR’ tests were stopped at a strain level of about 0.30.

Rolling shear tests were performed by either constraining displacement or force in the global vertical direction. The vertical force was restricted up to 100 N for Q-, V-, and T-shaped specimens, and 50 N for D-shaped specimens, and thus limited to nominal compressive stresses of 0.10 N/mm² for Q-shaped specimens and 0.17 N/mm² for D-shaped specimens. Therefore, not only could the effects of the force imposing system be assessed, but also the loading protocol.

Figures 5 and 6 show the stress paths of shear tests in the \(\sigma _{RR}\)-\(\tau _{RT}\) and \(\sigma _{TT}\)-\(\tau _{TR}\) stress planes. This indicates the two possible material orientations of specimens in shear testing, which both lead to shear stresses in the R-T plane. Force and displacement-constrained testing is indicated by force and disp. The figures illustrate the effects of the loading protocol and highlight the development of compressive stresses of more than 1 N/mm² in cases of displacement constrained testing, which partly lead to higher shear strength. Q-, V-, and T-shaped specimens were tested with direct contact and mechanical grips, while D-shaped specimens were tested with mechanical grips only. The effects of the force imposing system on shear strength seem to be insignificant or overlaid by variation of the shear strength.

In the case of biaxial loading, direct contact through the L-blocks led to complex surface strain distributions, such as tensile stress in the two opposite sides of the L-plates with stress concentration near the notch, and thus cracks developed undesired failure modes. Moreover, differences in the stress paths were observed for the same displacement loading path. Mechanical grips yielded comparably lower shear stress, whereas force imposed by contact led to higher nominal shear stress but lower nominal normal stress. This effect was pronounced for compression in the radial direction. However, the post-elastic behavior of the material was similar for the two test setups. Hardening in biaxial testing with the radial compression led to strongly increased rolling shear stresses, while biaxial testing with tangential compression led to considerably increased compression stresses and rolling shear stress softening.

The assessment showed that mechanical grips led to a more homogeneous force transmission and avoided stress concentrations that occurred in the corners of the L-shaped contact device. The influences of the specimen’s shape combined with the force imposing system on the stress state in the specimens and on mechanical properties derived from testing are assessed next.

A balance in the shape and size of the specimens for biaxial testing is obviously required because shear testing would require flat specimens to reduce the eccentricity in the shear loading, whereas compression testing would require higher specimens to avoid any boundary effects. The influence of specimen shape on the shear behavior was mentioned in the previous section, where the unsupported height of the specimen was found to affect the mechanical response when testing in the tangential direction. The height of the specimens greatly influenced the development of higher nominal stress in compressive testing as well. This is illustrated in Fig. 4b. A reason for the difference is that the nominal compressive stress was calculated using the minimum cross-sectional area. For tangential testing, however, the material above and below the reduced cross-section contributes to the force distribution, as seen in the strain fields, which will be discussed later. Another possible reason is that the average density of the T-specimens is higher than the density of the R-specimens, which can yield higher compressive stress. No compression tests were carried out on V and T-shaped specimens. Therefore, no comparison is shown for these two types of specimens.

The effect of the specimen’s shape was also included in Figs. 5 and 6, where there were no considerable differences for rolling shear testing. The stress state under shear testing was more closely assessed by means of rolling shear strain distributions across the specimens, as measured by digital image correlation. As with stress plane figures, strain sections are evaluated for specimens tested in the two material orientations, RT and TR. The strain sections in Figs. 7 and 8 clearly show the influence of the annual ring structure, since a more homogeneous strain level was found for R-oriented specimens, while alternating shear strain levels were visible in T-oriented specimens. Figures 7 and 8 also show the differences between specimen shapes as well as some variation when testing similar specimens but with different loading protocols. Shear strains were taken at a nominal shear stress level of 1 N/mm². Note that all shear tests with force constrained loading were accomplished by using mechanical grips. D-shaped specimens showed the least strain concentrations in the area of interest and least edge concentrations, compared to other shapes.

Rolling shear strain–stress relationships for D-shaped specimens are shown in Fig. 9 for RT- (R) and TR- (T) orientations. When the shear plane is parallel to the radially stacked annual rings, RT-orientation led to an almost perfect brittle failure, whereas when the shear plane crosses several annual rings, TR-orientation testing led to a more progressive brittle failure through the development of several cracks. The latter failure mechanisms occurred at comparably higher stress levels. However, as a consequence of the homogeneous orthotropic material, no difference was considered between rolling shear strength in RT- and TR-orientation compared to failure criteria.

The strain field under biaxial testing is assessed next for SC-45 testing. Figure 10 shows the normal strain and shear strain fields for all investigated specimen shapes and testing in the two orthogonal directions R and T. Strain fields represent the state at a nominal compressive stress of 3 N/mm² and corresponding shear stresses between 0.11 and 1.48 N/mm². Because D-shaped specimens were tested with mechanical grips and all other shapes with L-shape steel plates, considerably lower shear stresses developed in the D-shaped specimens. The DIC assessment revealed comparatively uniform surface strain distributions for all geometries, except for some stress concentration at the edges. The reduced cross-section in the notched area in V-, T-, and D-shaped specimens helped to yield a concentrated strain distribution and initiate a crack in this part. However, this was not always achieved in Q-type specimens of biaxial testing with compression in the radial direction. Biaxial testing with compression in the tangential direction highlighted the sample shape effect on the tangential strain–stress response. The nominal tangential compressive stress increased with the increased notch size. Activation of the material above and below the notched area is well visible in the corresponding shear strain fields.

Finally, the material properties derived from nominal strain–stress relationships measured on different specimen shapes are compared. The Young’s moduli in the radial and the tangential directions, and the rolling shear moduli were calculated from unloading paths as described in Sect. 3.3, which yielded values shown in Table 1. For testing under radial compression, minor differences were observed for the different types of specimens and force imposing devices, as well as when comparing tensile and compressive stiffness. The corresponding strengths between 4 and 5 N/mm² were measured. A slightly higher variation was found for testing under tangential compression. The rolling shear stiffness and strength, however, showed higher variations for different shapes of specimens, and thus, different stressed volume. As a consequence, Q-type specimens yielded higher rolling shear stiffness and strength than V-, T-, and D-type specimens. The high influence of stressed volume on shear strength was mentioned by Steiger and Gehri (2011). The rolling shear modulus was between 50 and 60 N/mm² for almost all setups and specimen shapes with notches. This corresponds well with values given in previous scientific works (e.g. Dumail et al. 2000; Hassel et al. 2009 and material standard EN 338 2009). Dog-bone shaped specimens exhibited the lowest rolling shear strength value, which is expected as a consequence of the reduced cross-section and less effect of the curvature of the annual ring structure. The highest rolling shear stiffness and strength were found for Q-type specimens, which showed some curvature of the annual ring structure, even if the test specimens were cut from the outermost part of the tree. The influence of the sawing pattern on the rolling shear modulus is well-known in wood science, see for example Aicher and Dill-Langer (2000). Because dog-bone shaped specimens have been further used for biaxial testing, the following average material properties have been considered for the assessment of biaxial failure criteria.Uni-axial compression and rolling shear testing of dog-bone shaped specimens resulted in typical material behavior. The radial compression tests yielded linear elastic behavior, followed by a stress plateau with slightly increasing stress (Bodig 1963; Tabarsa 1999). However, tangential compression resulted in a non-linear behavior and a stress peak, a consequence of latewood layer buckling (Tabarsa 1999). Modulus of elasticity in the radial direction was higher than in the tangential direction, and agrees well with previous findings (e.g. Madsen et al. 1982; Hall 1980; Gehri 1997; Farruggia and Perré 2000; Hoffmeyer et al. 2000; Kristian 2009; Zhong et al. 2015). \(E_{R}\) was 1.50 times higher than \(E_{T}\), which is in agreement with Zhong et al. (2015), but comparatively lower than the findings of Gehri (1997), Farruggia and Perré (2000), Hoffmeyer et al. (2000) and Kristian (2009), who reported a factor of around 2.

The compressive strength in the radial direction that was determined higher than in the tangential direction is in agreement with Gehri (1997), Hall (1980) and Hoffmeyer et al. (2000). However, when considering the stress peak in the tangential direction, \(f_{c,T}\) was found to be higher than \(f_{c,R}\), see Table 1. Since the value in the tangential direction was dependent on specimen shape and height, it may not reflect the real material strength. It is emphasized that Table 1 only summarizes uniaxial tests for compression and tension perpendicular to the grain as well as rolling shear, where a total of 38 tests were performed.

The tensile strength in the tangential direction was higher than in the radial direction, with values of \(f_{t,T}\) = 3.28 N/mm² and \(f_{t,R}\) = 2.75 N/mm², which is in contradiction to Kristian (2009), who reported \(f_{t,R}\) = 4.90 N/mm² and \(f_{t,T}\) = 2.80 N/mm². By considering the brittle behavior under tensile loading, tests on a higher number of specimens could give better insight into tensile strength.

The determined rolling shear modulus of \(G_{RT}\) = 55 N/mm² is in good agreement with findings by Dumail et al. (2000) and Hassel et al. (2009), while the rolling shear strength, \(f_{v,RT}\) = 1.54 N/mm², is slightly lower than the value of 1.60 N/mm² reported by Kristian (2009), Dumail et al. (2000) and Hassel et al. (2009).

Biaxial testing of dog-bone shaped specimens was performed along 12 displacement paths, including combinations of rolling shear with compression and tension perpendicular to the grain. The corresponding stress paths are shown for normal stresses in the radial (Fig. 11) and tangential (Fig. 12) material directions. Note that the datasets shown in Figs. 11 and 12 consist of 36 tests each for DR and DT specimens, i.e., a total of 72 tests, performed on DR and DT specimens. Brittle failure was observed for the combination of tensile stresses with rolling shear, while ductile behavior was observed for the combination of compressive stresses with rolling shear. A transition zone from brittle failure under pure rolling shear to a ductile behavior in combination with compressive stresses was found for displacement paths SC-10 and SC-20. By comparing stress interaction paths in the radial and tangential directions, it is interesting that comparably higher shear stresses were observed for tangential compression than for radial compression, which means a stiffer shear behavior for this combination. This is a consequence of the displacement controlled testing and the ratio between Young’s modulus and rolling shear stiffness that is lower in the tangential direction; cf. stiffness properties given in Sect. 4.2.

Another interesting observation was the different post-elastic behavior of clear wood in the two orthogonal orientations. When combined with radial compressive stresses, rolling shear stresses increased more than compressive stresses after the elastic limit was reached, see SC-45, SC-35, SC-30, SC-20 in Fig. 11. However, when combined with tangential compressive stresses, rolling shear stresses decreased after the elastic limit, see SC-45, SC-35, SC-30, SC-20 in Fig. 12.

Material strength properties were derived to assess the suitability of failure criteria. The strength was simply set equal to the maximum stress in case of brittle material failure under tension, shear and tension-shear stress combinations. For ductile failure in compression and shear-compression interaction, stress points at 1%, and 2% compressive strains were evaluated instead. This is a typical procedure that is also specified in the material testing standard (EN 408 2010), which prescribes certain levels of permanent strain, though absolute strain was used here.

Material strength properties for strain levels of 1% and 2% compressive strain were then compared to failure envelopes predicted by Hill, Hoffman and the SIA 265 design equation, see Figs. 13 and 14. Rolling shear strength and tensile strength were kept constant for the two strain states, whereas compressive strength levels were changed to follow the evolution of failure surfaces. In Figs. 13 and 14, Hill-1, Hoffman-1, SIA 265-1 curves denote the failure surfaces by Hill, Hoffman and SIA 265 design equation by considering compressive strength at 1% compressive strain level, whereas Hill-2, Hoffman-2, SIA 265-2 curves denote the failure surfaces by Hill, Hoffman, and SIA 265 design equation by considering compressive strength at 2% compressive strain level. However, because compressive strength changes, the failure envelope changes slightly in the tension-shear interaction state. For the Hill failure criterion, the compressive strength was used as the strength perpendicular to the grain. A stronger difference in the compression-shear interaction area was observed, due to increased stresses at higher strain levels. The suitability of the failure criteria is assessed by the prediction capability value R, given in Table 2.

From the mean R-values in Table 2, none of the failure criteria yielded good prediction in both tension and compression interaction states in both material orientations. Overall, Hoffman’s criterion shows better prediction in DR-specimens than in DT-specimens. When considering both stress states, the average R-value for Hoffman’s criterion was found to be 0.95, in comparison to Hill’s criterion, which yielded a R-value of 0.85 for 1% compressive strain. Yet, in the DT-series, Hill’s criterion yielded better prediction than Hoffman when considering both stress states (compressive and tensile), while Hoffman was found better in compressive stress states. However, both criteria showed a small difference in compressive stress states, since the shape of the failure envelope is very similar. The difference in tensile stress states is high, which is a reason to set the tensile strength equal to the compressive strength for non-consideration of material tensile strength by Hill’s criterion. The latter was chosen here to assess the prediction quality of Hill’s criterion in compressive stress states. It could then possibly be combined with a brittle failure criterion for tensile stress states.

The experimental data given in Figs. 11 and 14 shows that the combination of rolling shear with compressive forces perpendicular to the grain influences the strength positively (Steiger and Gehri 2011; SIA 265 2012) in failure up to a certain compressive stress level. The positive effect of compressive stress on rolling shear strength was confirmed by Mestek (2011) as well. Regarding the assessment of failure criteria, both Hill and Hoffman’s criteria underestimate the positive effect of shear loading. Comparatively, Hoffman’s criterion yielded a closer limit curve than Hill’s criterion. This is a consequence of the possibility to account for different tensile and compressive strengths and the corresponding shift of the elliptical curve. The SIA 265 design equation gives, however, a good prediction for this transition zone. It would require a combination with a failure criterion for the compressive failure of wood in the radial or tangential direction, and thus, a multi-surface failure criterion.

The failure patterns observed in the experiments are shown in Fig. 15. Radial compressive stress yielded ductile behavior with strength hardening, where wooden cells were progressively compressed and the material densified, leading to an increase in stress. Rolling shear, tensile, and combined tensile force with rolling shear in specimens tested in the R-orientation led to brittle failure modes. A cascading type of brittle failure occurred under pure shear in this case, since cracking neither produced a smooth failure surface, nor followed that same annual ring. However, the radial tensile force and combined tensile force with rolling shear led to pure brittle failure with straight and smooth surfaces. Under combined compressive force with rolling shear, mixed failure modes, i.e., in between ductile and brittle failure modes, occurred, depending on the stress path (displacement ratios of normal stress to shear stress).

Tangential compressive stress led to buckling of the cells, causing damage to earlywood cells. A similar behavior was observed under biaxial stresses of combined compressive stress with rolling shear. Tensile, rolling shear, and combined tensile with rolling shear stress in this testing orientation yielded brittle failure. An almost straight crack was noticed for tangential tensile and combined tensile with rolling shear loading. Contrarily, failure started close to the notched area but extended at 45° for rolling shear for testing with tangential orientation.