Sawing pattern influence as assessed by the continuum theory: The rolling shear modulus is, apart from one single configuration, not an intrinsic material property, but rather an apparent quantity depending decisively on the annual ring orientation within the board and the stiffness ratios in the radial-tangential growth plan. This can be demonstrated by means of a simplified orthotropic continuum mechanics approach. On the mesoscale, i.e. at edge lengths of a few millimeters, wood can be regarded as a rhombic anisotropic, and thereby termed orthotropic, material with three material coordinate axes orthogonal to each other. Irrespective of the spatial orientation of the three axes, there is always a unique 4^{th} order elasticity tensor *C*_{ijkl} relating strains and stresses in a precise manner by the relationship:

$${\sigma}_{ij}={C}_{ijkl}{\epsilon}_{ij}\text{\hspace{1em}}\mathrm{(}i,j=1,2,3\mathrm{)}\text{\hspace{1em}(1)}$$(1)

Several materials have defined, major axes orientations, termed on-axis material coordinates denoted by *x*, *y* and *z*. This is true for natural ideal rhombic orthotropic materials (specifically crystals) and scale dependent approximately orthotropic materials (such as wood) as well as man-made orthotropic composites. In the case of wood, the on-axis directions are related to the tangential *t*, radial *r* and longitudinal (fiber parallel) *l* growth directions; in the following, the association *x*=*t*, *y*=*r*, *z*=*l* shall hold. Concerning the regarded mechanical problem of (rolling) shear in the radial-tangential growth plane under plane stress conditions, the longitudinal growth direction has no impact and the constitutive equation (1) and thus the 3D problem is in reality a 2D-problem in the *i*,*j*=1,2 plane.

Let us consider a wooden board or stick cut from the periphery of a large diameter tree. Further, the board shall have a small side aspect ratio of about 1:2 to 1:4 and small absolute dimensions, for instance 20 mm thickness. In this case, the annual ring curvature within the board is engineering-wise negligible. Further, the annual ring angle *φ* between the off-axis direction 1, chosen deliberately to align with the wide board edge, and the on-axis material direction *x*=*t* is almost constant along the board width (Figure 1a). For such idealized board configurations the apparent in-plane (rolling) shear modulus can be easily calculated for any off-axis angle *φ* by transformation from the known constitutive on-axis properties. Without further derivations, which are given in detail in textbooks on composite theory (e.g. Tsai and Hahn 1980), the apparent rolling shear modulus in the *r-t* plane can be written as

Figure 1: Geometry and sawing pattern definitions as well as analysis of beech wood boards with respect to rolling shear modulus:

(a) definition of angle *φ* between on-axis radial(*r*)-tangential(*t*) plane axes and deliberate off-axis coordinates 1,2. (b) Relationship of normalized off-axis shear modulus depending on angle *φ* for typical ratios *E*_{r}/*G*_{rt} and *E*_{r}/*E*_{t} of beech and reference spruce wood. (c) View of a typical beech board including pith and the associated cracks. (d) Characteristic dimensions *d*, *e* of pith allocation and definition of angles *φ*_{i}. (e) Classification of sawing patterns of the rolling shear specimens based on unambiguous annual ring geometry parameters.

$${G}_{r}{\mathrm{(}\phi \mathrm{)}}^{\text{-}1}=4{m}^{2}{n}^{2}\mathrm{(}{S}_{xx}+{S}_{yy}\text{-}2{S}_{xy}\mathrm{)}+{\mathrm{(}{m}^{2}\text{-}{n}^{2}\mathrm{)}}^{2}{S}_{ss}\text{\hspace{1em}(2)}$$(2)

where *m*=*cosφ*; *n*=*sinφ*; $${S}_{xx}=\frac{1}{{E}_{t}};$$ $${S}_{yy}=\frac{1}{{E}_{r}};$$ $${S}_{ss}=\frac{1}{{G}_{rt}};$$ $${S}_{xy}=\text{-}\frac{{\nu}_{tr}}{{E}_{r}}=\text{-}\frac{{\nu}_{rt}}{{E}_{t}}.$$

(Note: definitions of Poisson indices here 1^{st} index strain, 2^{nd} index stress.)

With regard to on-axis elasticity properties of beech wood in the *rt*-plane, literature provides rather little original data (). For comparison, bench mark data on Norway spruce are also listed. Data in reveal that the on-axis (rolling) shear modulus *G*_{rt} of beech wood is in the range of about 400 to 500 N mm^{-2} and hence roughly ten times larger than the on-axis *G*_{rt} value of Norway spruce which is about 40 to 60 N mm^{-2}. The modulus of elasticity (MOE) ratio *E*_{r}/*E*_{t} of beech wood extends from about 1.3 to 2.0 and is hereby in the same range as spruce. In absolute terms, *E*_{r} of beech is considerably, roughly 1.6 times, higher, which reflects well the specified density difference of an average factor of 1.5.

Table 1 On-axis stiffness properties in the radial-tangential growth plane of defect free beech wood (*Fagus sylvatica*) and spruce (*Picea abies*) in small clear dimensions.

The ratio of *E*_{r}/*G*_{rt} of beech ranges roughly from 3 to 5 and in case of spruce 20–40. This stiffness ratio has a fundamental impact on the variation of the apparent (rolling) shear modulus in off-axis material orientations. This is highlighted by the graph in Figure 1b, which depicts the shear modulus ratio *G*_{r}(*φ*)/*G*_{rt} drawn on the ordinate axis as a function of off-axis angle *φ*. The curves reveal the extremely different situation for both species as depending parametrically on the ratios *E*_{r}/*G*_{rt} and *E*_{r}/*E*_{t}. Consistently, the extreme values are at *φ*=0°, 45° and 90° whereby *G*_{r}(0,90)/*G*_{rt} =1 throughout. The extreme value of apparent shear modulus *G*_{r} at 45° is fully independent of *G*_{rt} (Eq. 2) and extremely sensitive to the stiffness ratios. For most of the beech wood stiffness configurations given in , the ratio *G*_{r}(45)/*G*_{rt} is in the range of 1.0–1.1; some on-axis data (Niemz et al. 2015) deliver a ratio of slightly below 1. Thus, on the basis of the rough assessment of the shear stiffness behavior of a beech board as being dependent on annual ring angle *φ* based on orthotropic continuum assumptions, it can be concluded that *G*_{r}(*φ*) of beech wood, in significant contrast to spruce, should not vary in an extreme manner for different board configurations. A deeper theoretical insight would be gained on the basis of a polar anisotropic modeling of the board’s stiffness as done for spruce by Aicher and Dill-Langer (2000) and Jakobs (2005), which is however, outside the scope of this paper.

Considerations on two-plate compression-shear tests: Currently, no internationally recognized test procedure for direct determination of the rolling shear strength and apparent shear modulus of a single board exists, representing some of the most important properties for design and calculation of cross-laminated timber (CLT) plates. The European product standard on CLT (EN 16351 2015) does, however, contain detailed provisions on the determination of the integral rolling shear properties of a cross layer consisting of several adjacent boards in three or more layered plates by two somewhat different compression-shear test methods. One test method refers to the two-plate shear test in the European standard EN 789 (2004) which is intended for the determination of in-plane shear properties of panels. Alternatively, a modified version of the stated test method allows for a wider dimensional variability of the specimen, and avoiding the lateral steel plates by transfer of the loads via the lengthwise oriented CLT boards.

Two further possible two-plate compression-shear test principles are provided by i) the European standard EN 408 (2012) on determination of shear strength parallel to the fiber of solid wood and ii) by the ASTM standard D 2718 (2016), serving for the determination of planar shear, i.e. rolling shear of structural panels. Contrary to the European shear tests in EN 789 and EN 408, which prescribe the specimen dimensions very rigorously, ASTM D 2718 followed by Zhou et al. (2014) for determination of rolling shear properties of down-sized black spruce laminations, allows for significantly different specimen sizes by geometrically similar adjustment. In addition, the issue of angle *α* between the line of the applied shear force *V*, and the specimen axis or bondline is handled inconsistently in the different mentioned standards. In the case of EN 789, angles in the range of 8–25°, which are dependent upon specimen thickness (6–80 mm), are obtained, whereas the cited alternative EN 16351 procedure, like EN 408, prescribes a fixed load line angle *α* of 14°. ASTM D 2718 also prescribes a minimum width and height as a function of the thickness, giving a considerably shallower load line angle of about 3 to 5°.

The issue of angle *α* in the context of direct (two-plate) shear tests, which has a strong impact on the superpositioned normal stresses perpendicular to the rolling shear plane, has been investigated by Feldborg (1991) and Mestek (2011). There are both advantages and disadvantages for the different angles in the range of 3–14°, which is to be further investigated. For lack of any directly applicable standard, the test setup shown in Figure 2a, based on a realization of *α*≈14°, has been chosen. It should be further mentioned that apart from (quasi) two-plate shear tests, several further test methods for determination of rolling shear properties exist. A bending test with short spans is possible (EN 16351), enabling determination of rolling shear modulus and strength. An additional test approach with shear loaded notched, so-called Iosipescu specimens for on-axis *G*_{rt} determination, as employed by Dumail et al. (2000) is also possible. However, the latter test is poorly suited for the determination of rolling shear strength.

Figure 2: Employed test setup: (a) technical scheme with dimensions (in mm); (b) view of the realized test setup with fixed LVDT slip measurement.

The beech wood (*Fagus sylvatica*) originated from a forest in Southern Germany, about 50 km north of Stuttgart. The logs from which the boards were cut had bottom and top diameters of 0.7 to 1 m and 0.6 to 0.8 m, respectively. The logs were sawn flatwise as is usual in hardwood processing to slabs/planks of 50 mm thickness. The planks were then air-dried for one year prior to kiln drying to a target moisture content (MC) of 12% within a period of one week. After drying, the planks were sawn to 140 mm wide and 38 mm thick boards, comprising the whole spectrum of possible annual ring configurations. Further, boards were frequently obtained where the annual ring pattern changes within the board, for instance from quarter- to semi-quarter-sawn. A more rigorous definition is given later on. Regarding the growth characteristics, such as, e.g. knots and fissures, the boards cut from the outer stem periphery were almost defect free and had no cracks whereas the boards containing pith or which were located close to the pith contained almost throughout cracks as shown exemplary in Figure 1c. A batch of 200 boards with lengths in the range of 2.5 to 4 m were ordered for various testing purposes, one of which being the here described rolling shear tests. Prior to cutting of the specimens, the boards were planed to a final thickness and width of *t*=33 mm and *l*=135 mm, respectively. A total of 50 specimens were then cut with a length (parallel to fiber) of *b*=100 mm, comprising all of the below discussed annual ring patterns.

In order to assess the effect of the annual ring orientation within a board, termed sawing pattern, this specimen property was measured in a quasi-unambiguous manner and then classified in close alignment with the established annual ring classifications. The unambiguous classification of the sawing pattern was based on a best possible allocation of the pith of the stem from which the board was cut vs. a reference point located at mid-width of the wide cross-sectional board’s edge oriented towards the pith. This edge and the related board’s face is generally termed “right” board edge or face. The characteristic dimensions for the pith allocation are (Figure 1d): *e*=eccentricity of the pith vs. mid-width of the board and *d*=distance of the pith vs. the “right” board edge. In case the pith is located within the thickness *t* of the board, *d* is negative and further defined by |*d*|≤*t*/2.

The actual determination of *e* and *d* was performed by means of both respective end grain face scans of the specimen slabs, which were then processed by a graphical algorithm searching for the best matching pith position with regard to the annual ring contours of the board. The bias related to the specified “exact” origins has not been investigated and is dependent on various factors, such as for instance the stem contour geometry, here throughout assumed to be circular.

As the mathematical description of the annual ring orientation of the board does not enable an immediate grading classification of the board with regard to the established sawing pattern classes, e.g. type quarter-sawn, all specimens were classified with regard to these easily visually perceivable groupings as well. Nevertheless, todays established definitions on sawing patterns based on an average annual ring angle *φ* for the whole board were found to deliver a too imprecise classification of, most notably, the boards with a pronouncedly changing annual ring angle *φ*.

To address this issue in a more transparent manner, the determination of the annual ring angle was based on the value *φ*_{mean}= (*φ*_{1}+*φ*_{2}+*φ*_{3})/3 where angles *φ*_{2}, *φ*_{1}, and *φ*_{3} represent the means of both end grain faces measured along a line at half the board’s thickness at mid-width *x*=0 of the board and at the ends *x*=∓*b*/2, respectively (Figure 1d). Based on the *φ*_{i}, *φ*_{mean} and *d* definitions, the established sawing patterns a, b, c (see below) and a further pattern (d) are specified more precisely as:

$$60\xb0\le {\phi}_{\text{mean}}\le 90\xb0\text{\hspace{0.17em}quarter-sawn\hspace{0.17em}(a)}$$

$$30\xb0\le {\phi}_{\text{mean}}<60\xb0;\text{\hspace{0.17em}}{\phi}_{1}>0\xb0\text{\hspace{0.17em}semi-quarter-sawn\hspace{0.17em}(b)}$$

$$0\xb0\le {\phi}_{\text{mean}}<30\xb0;\text{\hspace{0.17em}}{\phi}_{1}\le 0\xb0;\text{\hspace{0.17em}}d\text{>5\hspace{0.17em}mm\hspace{0.17em}flat-sawn\hspace{0.17em}(c)}$$

$${\phi}_{\text{1}}<0\xb0;\text{\hspace{0.17em}}d\le \text{5\hspace{0.17em}mm\hspace{0.17em}including\hspace{0.17em}pith\hspace{0.17em}(}d\text{)}$$

Figure 1e reveals for all specimens the relations between the mathematically unambiguous annual ring definition of *e* and *d*, and the conventional, here given definitions of sawing patterns (see also ). The geometry parameters *e* and *d* enable a cylindrical or polar anisotropic finite element continuum analysis of the board’s apparent in-plane stiffness properties, so for instance of shear modulus *G*_{r}, not presented in this paper.

Table 2 Compilation of densities and of geometry parameters for sawing pattern classification as well as rolling shear modulus and strength results depending on sawing pattern of the investigated specimens.

Apart from the sawing pattern, the specimens were further classified or measured with regard to density and eventual cross-sectional cracks. gives a statistical evaluation of the densities and the geometrical parameters separately for the respective sawing patterns and for the whole sample of specimens. Density *ρ*_{12} of the specimens ranged from 600 to 740 kg m^{-3}. The mean densities of the four different sawing pattern groups lay closely together, ranging from 651 to 685 kg m^{-3}. Further, the scatter of the densities in each group was comparably small with coefficients of variation of about 5%. Furthermore, the extreme values of the densities of the respective sawing pattern groups did not show very pronounced differences. The specified density values classify the material as typical for beech wood of medium density. According to literature (Kollmann 1982), the density range *ρ*_{12} of the species is 530…700…890 kg m^{-3}.

In contrast to the density, the geometrical quantities *d*, *e*, *φ*_{mean} and *φ*_{1}, describing the sawing pattern, differed definition-wise pronouncedly for the respective annual ring groups. Hereby the sample means of the mean annual ring angles of 15, 47, and 78° obtained for the flat, semi-quarter- and quarter-sawn group conformed very well with the central values 15, 45 and 75° of the conventionally assumed *φ*_{mean} ranges specified above for the respective sawing patterns.

With regard to macroscopic cracks visible at the end grain faces and on the wide sides, the different sawing pattern groups showed in some instances pronounced/extreme differences. In the case of the quarter- and semi-quarter-sawn specimens, no cracks at all were existent. Three of the 15 flat-sawn board specimens contained cracks. In contrast to the other groups, all 13 specimens containing pith had at times very expressed cracks; Figure 1c shows a typical end-grain crack appearance.

Figure 2a specifies the test scheme and dimensions and Figure 2b shows a view of the realized test setup. The test specimens were glued at room temperature between the steel plates with a special two-component epoxy resin (WEVO EP32 S with hardener WEVO B 22 TS), with a technical approval for glued-in steel rods in timber (DIBt 2014). In order to increase the strength of the bond between steel and beech, the surface of the steel plates was profiled by shallow (1.5 mm), rectangular grooves with a width of 10 mm along the width of the plates. The tests were performed in a screw-driven test machine at a constant rate of cross-head movement of 0.75 mm min^{-1}, whereby failure was achieved within 300±120 s. The relative slip between the two steel plates *u*, giving the average shear angle of the beech specimen tan*γ*=*u*/*t*, was measured by an LVDT, which was fixed on one steel plate by measuring the movement of a steel bracket which was fixed opposite to the other steel plate at mid-length of the specimen. The tests were performed at 20±1°C in a non-climatized test chamber. The load and displacement were recorded continuously. The MC of the beech specimen during testing was in the range of 9–10.5% (matched specimens, oven-dry method).

The rolling shear modulus *G*_{r} as well as the rolling shear strength *f*_{v,r}, were determined according to the relationships given in EN 789 (2004), with a correction for the load-application angle *α* as specified in EN 408 (2012) as

$${G}_{r}=\frac{\mathrm{(}{F}_{2}\text{-}{F}_{1}\mathrm{)}\cdot \mathrm{cos}\alpha \cdot t}{\mathrm{(}{u}_{2}\text{-}{u}_{1}\mathrm{)}\cdot l\cdot b}\text{\hspace{1em}(3)}$$(3)

and

$${f}_{v,r}=\frac{{F}_{\mathrm{max}}\cdot \mathrm{cos}\alpha}{l\cdot b}\text{\hspace{1em}(4)}$$(4)

where *F*_{2}>*F*_{1} are load levels within the elastic material range, and *u*_{2}, *u*_{1} are the corresponding slip values; for dimensions *t*, *l* and *b*, see Figure 2a.

The significance of the different sawing patterns on rolling shear modulus and strength values was checked by ANOVA with a further two-sided t-test with unequal variances and confidence level of *α*=0.05.

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