Relationships between stiffness of material, lamellas and CLT elements with respect to out of plane bending and rolling shear

The use of cross laminated timber (CLT) for construction has increased greatly in recent years and the large volumes of wood used for CLT means that it is important to optimize the use of the material. This requires relevant grading of lamellas and knowledge of relationships between lamella and CLT properties. In the present study, the relationship between dynamic axial modulus of elasticity (MoE) of lamellas and the quasi-static out of plane bending stiffness of CLT is investigated. By means of four-point bending test, it is shown that the effective quasi-static MoE of lamellas in CLT is only 2–6% lower than the average axial dynamic MoE of the individual lamellas. With this knowledge, producers of CLT can easily predict and control the important out of plane bending stiffness of the produced CLT. Moreover, it is shown that effective rolling shear stiffness of layers in CLT can be accurately determined by means of digital image correlation performed in connection to four-point bending of CLT, even for long test spans. For layers of lamellas of Scots pine of size 40 × 190 mm2 the average apparent or effective rolling shear modulus was determined to be 159 MPa. The average rolling shear modulus of the same material was determined to be 56 MPa.


Background
Timber from sustainable managed forests provides material for construction that gives lower carbon footprint compared to alternative construction materials (Gustavsson et al. 2017;Dodoo et al. 2021) and during the last fifteen years, the interest in cross laminated timber (CLT) has increased enormously. In 2020 more than 1000,000 m 3 of CLT was produced only in central Europe and it is estimated that the production in this region will be well over 2000,000 m 3 already in 2023 (Jauk 2021). This means that it is becoming increasingly important to optimize the use of sawn timber in CLT, which in turn requires knowledge of all relevant material properties, as well as principles for grading of sawn timber for CLT lamellas that promotes optimal resource utilization.
The common strength classes used for softwood in Europe are C classes (bending strength classes) and T classes (tension strength classes). Grade determining properties (GDPs) of C/T classes are bending/tensile strength, bending/tensile modulus of elasticity (MoE) and clear wood density. In machine strength grading it is usually the requirement on strength, i.e. the lower fifth percentile of strength of all pieces graded to a class, rather than requirements on MoE or density, that determine the yield in a strength class (Ridley-Ellis et al. 2016;Olsson and Oscarsson 2017). This is mainly because the relationships between indicating properties (IPs) used in machine strength grading and the grade determining strength are rather low, even for the best grading machines on the market.
The statistical distribution of board strength of different batches of for example grade C24 or T14 may differ greatly, even if the compared batches are graded using the same machine. If grading is done using settings for 'single grade', a batch of C24 lamellas will contain many boards with bending strength far higher than 24 MPa, and of course some that are considerably weaker, while if settings for simultaneous grading of for example C14/C24/C35 is used, the C24 batch will contain boards with lower coefficient of variation (CoV) of strength. Due to the homogenization effect in CLT (combined lamination and system effects), stronger CLT plates are produced using lamellas with high CoV of bending strength compared to lamellas with low CoV of strength (Jöbstl et al. 2006). Therefore, Unterwieser and Schickhofer (2013) and Brandner et al. (2016) suggested definition of CLT strength classes taking CoV of lamella strength into account. Unfortunately, however, no information of CoV of strength accompany batches of C or T graded sawn timber and the CLT standard EN 16351 (2021) does not require the declaration of CoV of lamella strength but only that lamellas are graded according to EN 14081-1 (2016) + A1 (2019). Therefore, the preparation work for implementing CLT in Eurocode 5 led to a proposal to use a conservative assumption of CoV equal to 0.25 for strength of lamellas when calculating the out of plane bending strength of CLT plates from lamellas graded in C or T classes . Of course, this is a practical and reasonable proposal in the short perspective, but it does not promote optimized utilization of wood material. Thus, refined definitions of C and T classes, including information of CoV of strength, or another set of classes taking relevant measures of the statistical distributions of the GDPs into account should be considered for CLT. In the draft for the new Eurocode 5, which in contrast to the current version will include CLT, only a single strength class, CL24, is defined. Producers may, however, based on test results obtained following the European assessment document EOTA (2015) declare better properties than those of the proposed CL24 class. To produce CLT with higher stiffness and strength, the producer may choose to grade lamellas in higher C or T strength classes than C24/ T14, which are the common classes for base material both in production and in scientific studies (Brandner et al. 2016), or to grade lamellas according to other suitable criteria.
Regarding mechanical properties of CLT, the out of plane bending stiffness is usually a more crucial design parameter than what bending strength is. For example, in floor constructions it is usually requirements related to springiness and vibrations (i.e. stiffness and in particular dynamic stiffness) that determine the floor design (Jarnerö et al. 2014). Thus, it may be appropriate and expedient to take greater account of stiffness in the grading of lamellas, and it is worth noting that calculations of effective bending stiffness of CLT elements based on class specific MoEs of the base material give conservative values (Li et al. 2020).
The objectives of the present paper concern relationships between material, lamella and CLT stiffness properties, and between quasi-static stiffness properties and dynamic stiffness properties, as developed in the following two sub-sections. The scope of the study is limited to threelayer CLT, but the results presented should be essentially as valid for CLT with five or more layers.

Axial dynamic MoE of lamellas and quasi-static out of plane bending stiffness of CLT
Axial dynamic MoE of timber boards is determined based on the fundamental axial resonance frequency (f), average board density (ρ), and board length (L) as E x,dyn is widely used as indicating property to board strength (Hanhijärvi and Ranta-Maunus 2008) and to quasistatic board MoEs. However, according to several studies E x,dyn of Norway spruce and other softwood species is typically about 10% higher than quasi-static MoEs of boards determined by means of four-point bending tests (e.g. Larsson et al. 1998;Raymond et al. 2007) or even higher (Krajnc et al. 2019). A common explanation for this is related to the viscoelastic properties of the material, and some presented results of comparisons between static and dynamic MoE of clear wood specimens (Divos and Tanaka 2005) support this explanation. However, there is another explanation to consider as well. Local weakness due to a single large knot or a cluster of knots may have larger influence on the determined quasi-static MoE than on axial dynamic MoE. Furthermore, it is also prescribed in EN 384 (2016) that when evaluating local and global quasi-static MoEs, the anticipated weakest part of the board shall be placed inside the inner load points in a four-point bending test or between the jaws in a tension test. An investigation performed on the basis of data obtained from previous research projects indicates that the ratios between E x,dyn and quasi-static bending MoEs are larger for small dimension boards than what it is for large dimension boards. In Table 1, mean values of the ratio of E x,dyn and local static bending MoE, E m,loc,stat and between E x,dyn and static tensile MoE, E t,stat , determined according to EN 384, are presented for Norway spruce boards of different dimensions. For the relationship between E x,dyn and E m,loc,stat it is quite clear that the ratio is larger for small dimension boards than for large dimension boards. For the largest boards investigated, of size 70 × 220 mm 2 , the ratio is as low as 1.03. The same tendency is seen, although not as clearly, for the ratio between E x,dyn and E t,stat . More details of the boards included in Table 1 are given in Olsson and Oscarsson (2017) and in Briggert et al. (2020), respectively. However, the relationships and comparison for individual dimensions shown in Table 1 have not been presented before.
When boards are glued together in CLT, it is reasonable to assume that local weaknesses in the lamellas have limited impact on the stiffness of the CLT panels, just as local weakness due to knots has smaller influence on large dimension boards than on small dimension boards. Consequently, it is reasonable to expect the ratio between the average E x,dyn of the lamellas and the quasi-static bending stiffness of the CLT panels made out of the lamellas to be comparatively small. The relationship between the average E x,dyn of the lamellas and the bending MoE of the CLT element made of the lamellas should also be stronger than the corresponding relationship for lamellas (i.e. the values for R 2 in Table 1). Thus, knowledge of the ratio between average lamella E x,dyn and out of plane bending MoE of the CLT element should enable convenient and accurate prediction of the important out of plane bending stiffness of CLT elements.
One of two purposes of the present paper is to make a close investigation of the relationship between dynamic MoE of lamellas and quasi-static out of plane bending stiffness of the CLT panels that are manufactured from the lamellas.

Effective rolling shear stiffness of lamellas in CLT
In practice, out of plane bending of CLT plates comprises both bending and shear deformations, and the latter depend on the effective rolling shear modulus of lamellas in transverse layers of the CLT plate. Effective rolling shear modulus depends, in turn, on the wood material stiffness parameters, especially the rolling shear modulus G rt , and on the orientation of the annual rings of the lamellas (e.g. Aicher and Dill-Langer 2000). Several experimental studies on rolling shear modulus and rolling shear strength have been carried out but rather few on other species than Norway spruce. For example, Aicher et al. (2016) investigated properties of European beech and Li et al. (2021) properties of Radiata pine. Erhart and Brandner (2018) give a summary of results obtained in previous studies and of experimental methods used to evaluate rolling shear properties. Furthermore, they present new results of effective rolling shear modulus also for other species than Norway spruce, including Scots pine, which is foreseen to be a species more commonly used for CLT production in some countries in the future, for example in Sweden. For lamellas with width to thickness ratio equal to four, and a distance between pith and centre of the lamella of about 60 mm, Erhart and Brandner (2018) reported for lamellas of Scots pine an average effective rolling shear modulus of 158 MPa. For Norway spruce, tested under the same conditions they reported 100 MPa. This can be compared with the often-recommended value of 50 MPa (e.g. Gustafsson et al. 2019) which is also the value prescribed in the draft for the new Eurocode 5. The second purpose of the paper is to present a method to evaluate both the effective rolling shear modulus, and the material property G rt , from calculations and four-point bending tests of CLT elements that give reliable results even for long test spans, and to present results, complementary to those few available, for rolling shear moduli of Scots pine.

Material
The lamella used for production of the CLT elements investigated in this research were boards of Scots pine (Pinus sylvestris) and Norway spruce (Picea abies (L.) H. Karst) of size 45 × 195 × 4800 mm 3 from a sawmill in the south-west of Sweden. In total, 220 pieces of Scots pine graded C24, 34 pieces of Scots pine graded C16 and 220 pieces of Norway spruce graded C24 (single grade) were prepared. Settings for C24 single grade using a Dynagrade strength grading machine (grading based on axial resonance frequency, no assessment of density) were used throughout. Whether the pieces were finally graded C24 or C16 depended on fulfilment of visual override inspection requirements according to EN 14081-1. Before the boards were planed and glued to CLT elements, axial resonance frequencies and mass were determined manually for each piece. For every fifth board, moisture content (MC) and actual dimensions were also determined (MC determined using a resistive, pin-type instrument). Based on this, axial dynamic MoE and density, corrected to 12% MC in accordance with EN 384, were calculated and denoted E x,dyn,corr and ρ corr , respectively. Mean value (mean) and coefficient of variations (CoV) of these properties are presented in Table 2, along with determined MC. Mean and CoV of E x,dyn,corr are also presented separately for those 85 lamellas of Norway spruce and those 85 lamellas of Scots pine that were actually used in top and bottom layers of the produced CLT elements. Prior to manufacturing of CLT panels, the lamellas were planed to size 40 × 190 mm 2 and finger joints were applied, using a moisture curing one-component polyurethane adhesive (Loctite HB S049) following the prescriptions of EN 15497 (2014), to produce continuous lamella. This lamella was then cut in lengths corresponding to the lengths of the main direction of the respective layers of the CLT panel. Four three-layer CLT panels, size 120 × 3050 × 6050 mm 3 , glued (one-component PUR adhesive, Loctite HB S209) between the layers, were manufactured. Lamellas within the same layer were not glued to each other. Lamellas in the outer layers, i.e. layer one and three, were oriented along the longer panel direction. Lamellas in the mid layer, i.e. layer two, were oriented along the shorter direction. The subsets of boards used as lamellas in the different layers of the four panels are specified in Table 3. Thus it was known, not only that a specific layer in a specific panel was manufactured of boards of for example Scots pine C24 but of certain boards for which E x,dyn,corr and ρ corr are known. However, since boards were cut and finger jointed in the CLT manufacturing process, the exact positions of the boards within the panels/ layers are not known.
Each of the four panels was cut into pieces of size 120 × 377 × 4000 mm 3 and 120 × 377 × 2000 mm 3 , respectively. Each piece/specimen was two lamellas in width and the specimens were numbered as shown in Fig. 1 (for panel one). Panels two to four were cut in the same way and specimens were numbered 2:2-8; 2:11-17; 3:2-8; 3:11-17; 4:2-8; 4:11-17. Seven long and seven short specimens were manufactured from each panel. However, the short elements from panel three and four were partly used for other purposes, and therefore no short elements from these panels were included in the present study. Thus, in total 28 fourmeter specimens and 14 two-meter specimens were prepared for the present investigation.

Four-point bending tests with application of digital image correlation
Quasi-static stiffness properties of the CLT elements were evaluated using a setup for four-point bending tests. This is illustrated in Fig. 2 showing (a) a drawing with measures and definitions of local and global deflections, w loc,exp and w glob,exp , respectively, and (b) the electro mechanical testing machine of mark Alwetron used to conduct the tests. Prior to testing, the 2000 mm long elements were cut to 1600 mm. The setup was arranged, and the test performed with the aim to fulfil the test prescriptions of EN 16351 and EN 408 (2010) + A1 (2012). However, some practical limitations of the equipment and the material tested led to deviations from the standards, namely the following (none of which constitute any significant limitation given the purposes of this research): -Due to limitations of the width of the testing machine the tested elements were only two lamellas, or 377 mm, wide (not four lamellas as prescribed in EN 16358), -The local deflection, w loc,exp , was determined on the top of the element rather than at the neutral layer as prescribed in the standard, -The global deflection, w loc,exp , was determined on the bottom of the element rather than at the neutral layer, -The width of the steel plates used to apply point loads and supports were 160 mm wide, rather than half the depth of the test piece which would be only 60 mm, -The distance between the supports and the end sections of the elements were longer than what is prescribed in EN 16358. For the 4000 mm and 1600 mm elements, respectively, l 2 (see Fig. 2a) was 200 mm and 80 mm, respectively. According to EN 16358 l 2 should be only half the depth of the test piece, i.e. 60 mm.
Regarding the global deflection, w glob,exp , this was adjusted to compensate for settings in the supports of the testing machines since no gauges were mounted on the tested CLT elements at the supports. The stiffness of each of the two supports was, in a separate evaluation, determined to be 12.5 kN/mm.
Digital image correlation (DIC) was performed to determine strains as a function of the applied load over parts of CLT element edges, between the left support and the left point load. For each of seven 4000 mm long elements, namely element 1:4, 2:2, 2:3, 2:5, 2:6, 2:7 and 2:8, two DIC measurement projects were performed, each covering 400 mm of the respective CLT element in longitudinal direction and the entire element depth of 120 mm. The areas covered by the DIC projects are marked by rectangles in Figs. 2a,b. In Fig. 2c, a surface on an element edge, which is covered by a DIC measurement, is shown. The current shear stress state is indicated by arrows. As can be seen, a speckle pattern was applied on the surface. This was done to enable tracing of positions on the surface using two cameras, which are parts of the DIC system. The two cameras were directed towards the surface in different angles, taking pictures of the surface as the applied load and the deformations increase. For more details about DIC and applications to wood, see for example Serrano and Enquist (2005).
Regarding the practical implementation of the test, the CLT element was first loaded to a level of 24 kN, corresponding to a maximum normal stress of about 20 MPa, while the cameras of the DIC system were directed towards the first of the two studied areas and used to collect data of displacements as a function of the applied load. Then the element was unloaded, the cameras moved to the second position and the CLT element was loaded again while the second DIC project was performed.
In Fig. 2d, an example of an engineering shear strain (i.e. shear angle) field, γ yz,exp (y,z), calculated based on determined displacements from DIC is visualised. (Indexes y and z refer to the local coordinate system of  Fig. 2c). The fact that lamellas in the same layer were not glued to each other shows by the seemingly high strains at the borders between lamellas in the mid layer. Regarding accuracy of strains, this depends on the resolution. For the high resolution displayed in Fig. 2d, the accuracy is rather poor, which is shown by the noise in the image representing the strain field. However, for the average engineering shear strain in the mid layer over an area of 40 × 190 mm 2 (the size of the cross section of a lamella) γ exp,40,190 , or over an area of 40 × 760 mm 2 (the size of the cross sections of four adjacent lamellas) γ exp,40,760 , the errors become small, which is shown below.
The effective rolling shear modulus of the mid layer, or a part of the mid layer, is assumed to be proportional to the applied load step ΔF and the corresponding shear strain. For a part of the mid layer of length 190 mm (one lamella), this can be expressed as and for a part of length of 760 mm (four lamellas) either as or as Equations (3a, 3b) give similar but not identical results, which is shown in the results section below.
Regarding accuracy, the lowest coefficient of variation between F and yz, ,40,190 obtained for any of the 7 × 4 evaluated areas of size 40 × 190 mm, was as high as 0.998 (when considering 100 load steps over an interval ΔF ≈ 20 kN). This can be compared with the minimum acceptable coefficient of determination of 0.99 between load and displacement in four-point bending tests given in EN 408.
13 of the 28 long elements were tested to failure. Results of ultimate loads for these elements are not presented herein but are available in Meulmann and Latifi (2021).

Determination of location of pith in lamellas in middle layers
The location of the pith in relation to the cross sections of a lamella has significance for the effective rolling shear modulus of the lamella. Therefore, the location of pith was determined for all mid layer lamellas within the parts of the CLT elements investigated with DIC. The position of the pith was determined by fitting the annual rings visible on the cross ions of the lamella to concentric circles drawn on a transparent plastic sheet, on which also an orthogonal coordinate system and a grid were drawn. Figure 3 shows (a) the plastic sheet used and (b) lamellas 4-7 of element 1:4 together with the horizontal (y-) and vertical (z-) coordinates of the pith in relation to coordinate system with origin placed at the lower left corner of each lamella cross ion.

FE model of CLT elements
A two-dimensional (2D) finite element (FE) model of CLT elements was created and used to simulate four-point bending tests. The purpose was to compare results from simulations with results from laboratory tests, and to determine crucial stiffness properties of the layers of the CLT elements based on such comparisons.
In the model, illustrated in Fig. 4, linear elasticity and orthotropic material properties were assumed, and these material properties were related to the directions of the lamellas. The x-direction is the longitudinal direction, the y-direction is the perpendicular direction within the plane of the CLT-element and the z-direction is the direction perpendicular to the plane of the CLT-element. Thus, the actual material direction, i.e. the longitudinal (l), radial (r) and tangential (t) directions are not directly considered in this model. For the MoE in the longitudinal direction of the lamella, E x , and for the effective rolling shear modulus, G yz , different values were considered, as explained below. For other stiffness properties, values given for sawn timber of class C24, EN 338 (2016), were used, i.e. E y = E z = 370 MPa and G xy = G xz = 690 MPa. For Poison's ratios v xy = v xz = v yz = 0.40 were used.
The finite element used was a 2D eight node isoparametric element. Alternative assumptions of plain stress and plain strain, respectively, were evaluated. The three-layer CLT element was divided in m elements in longitudinal, horizontal direction of the CLT-element and n 1 + n 2 + n 3 elements in the perpendicular, vertical direction. Convergence studies showed that for the elements of length 4000 mm, m = 300 and n 1 + n 2 + n 3 = 3 + 3 + 3 were sufficient for all calculations made in the present research. At supports and load application points, constraints were used to model the 160 mm wide steel plates as rigid elements. The same distance between supports and CLT-element ends as those used in the laboratory tests were used in the simulations. Local and global deflections, w loc,sim and w glob,sim , corresponding to the measures used for the laboratory tests (w loc,exp and w glob,exp , see Fig. 2a) were extracted from simulation results. In the illustration of the 2D FE model in Fig. 4, dashed lines are used to visualize the actual 3D structure of the CLT element. A global coordinate system is defined for the 2D model as well as local coordinate systems for each of the three layers of the CLT element.
In this model, the mid layer was modelled as a continuous solid and thus the fact that lamellas in the same layer were not actually glued to each other was ignored. Consequently, Fig. 3 a Plastic sheet with a coordinate system and grid drawn upon it and b photograph showing end cross sections of lamellas 4-7 in the mid layer of CLT element 1:4. Given pith coordinates (y;z) are determined using the plastic sheet and relate to the coordinate systems indicated at the lower left of the lamella cross sections the effective rolling shear modulus, that is determined herein by comparing results from experiments and simulation of for point bending, represents the shear stiffness of the mid layer including gaps between lamellas, rather than the shear stiffness of the material itself.

FE model of single lamellas
A 2D, FE model was established for the radial-tangential (rt) plane of a single lamella. This model was used to evaluate how the location of pith in relation to the cross section of the lamella affects the relationship between the material parameters, G rt , E r , E t , and the effective shear modulus. Values of material parameters for clear wood of Scots pine were adopted according to Dinwoodie (2000), see Table 4, and used in this model as 'nominal values'. The element type was a four node isoparametric element and a mesh with 100 elements in y-direction (i.e. over 190 mm) and 20 elements in z-direction (i.e. over 40 mm) was used. A convergence study was performed to verify that this mesh was sufficient for the purpose of the analysis.
The model was used to simulate a state of shear for lamellas with different pith locations. In Fig. 5, an example of deformed element mesh is shown along with illustrations of applied boundary conditions. In the simulation, all nodes on the lower horizontal edge were locked in horizontal as well as vertical direction and all nodes on the upper horizontal edge were displaced by a distance u in horizontal direction. No rotation of the upper edge was allowed. The horizontal displacement, u, corresponds to a horizontal force, T, per unit area in the xy-plane, see Fig. 5, and the relationship between u and T, determined using the FE model, forms the basis to define the effective rolling shear modulus for the lamella as G yz,eff,nom,40,190 is thus a function of the location of pith, the geometrical relationship between b = 40 mm and h = 190 mm, the boundary conditions for the lamella (free/ unglued vertical edges of the lamella) and the material parameters with nominal values, see Table 4.
Regarding the boundary conditions of the vertical edges, free/unglued edges correspond to the case when lamellas in the mid layer are not glued to each other, and this is the main case in the present study. However, the present FE model of a single lamella can easily be modified to represent the case with glued vertical edges. This can be done by applying cyclic boundary conditions such that any displacement of a node at the right vertical edge of the lamella is forced to be the same as the displacement of the node with the same z-coordinate (coordinate system shown in Fig. 5) at the left vertical edge.

Determination of effective rolling shear modulus of mid layers
For each of the CLT elements that were tested in four-point bending with DIC, the effective shear moduli, G yz,eff,exp, 40,190 and G yz,eff,exp,40,760 were derived as follows.
-The average engineering shear strain, within four adjacent areas, each 40 × 190 mm 2 in size, were determined, using DIC, as a function of the applied load F and denoted γ yz,exp,40,190 (F G yz,eff,exp,40,190 which corresponds to Eq. (3b).

Determination of wood material properties of mid layer lamellas
The definition of G yz,eff,nom,40,190 according to Eq. (4), derived using the FE model of a single lamella, implies that the mid layer lamellas are 'coupled in parallel' and, consequently, the effective rolling shear modulus for a layer, or for a part of a layer of the CLT element, should simply be calculated as the average of the values of G yz,lam,eff,nom,40.190 for the lamellas within the considered part of the mid layer. Thus,G yz,eff,nom,40,760 ,was calculated as Both G yz,eff,nom,40,190 and G yz,eff,exp,40,190 were determined for four adjacent parts (lamellas) of seven CLT elements (namely elements 1:4, 2:2, 2:3, 2:5, 2:6, 2:7 and 2:8) and this formed the basis to calculate a relationship between the two shear moduli as The ratio k was then used to estimate actual material properties of the wood used in the mid layer lamellas as (9) G rt = kG rt,nom ; E r = kE r,nom ; E t = kE t,nom

Relationship between dynamic axial MoE of lamellas and quasi-static bending MoE of CLT elements
The MoE in the longitudinal directions of lamellas in the two outer layers of a CLT element is decisive for the bending stiffness of the CLT elements. As explained in Sect. 2, the dynamic MoE in longitudinal lamella direction, E x,dyn,corr , was determined for the lamellas used in the different layers of the CLT panels. Values of E x,dyn,corr of individual lamellas were also used in FE simulations of four-point bending, using the FE model presented in Sect. 3.2.1. From the simulations, local and global deflections, w loc,sim and w glob,sim , were determined and compared with the corresponding deflections from real four-point bending tests, w loc,exp and w glob,exp . However, since it was not known which lamellas were placed in which position in the CLT panels it was not known exactly which lamellas were parts of which CLT element. Therefore, Monte Carlo simulations were performed in the evaluation, which was carried out as follows.
-Four-point bending tests were simulated for two types of 4000 mm long elements (elements with Scots pine in outer layers and elements with Norway spruce in outer layers, respectively) and one type of 1600 mm (originally 2000 mm) long elements (elements with Scots pine in outer layers) as described in Sect. , respectively, were derived (load steps between F ≈ 5 kN and F ≈ 20 kN considered in linear regression). -Finally, relationships between stiffnesses derived from Monte Carlo simulations and from laboratory experiments, respectively, were calculated as and The obtained values of k loc and k glob represent values of the ratio between average lamella E x,dyn and out of plane bending MoE of the CLT elements, as discussed above in Sect. 1.2.
Regarding the performed Monte Carlo simulations, these were needed since each CLT element only contained two lamellas in each of the outer layers. If mean values of E x,dyn,corr of the whole set of lamellas used in the outer layers are used instead in a single simulation of four-point bending, the stiffness obtained for the element becomes 1.7% higher compared to the average stiffness obtained by means of Monte Carlos simulation.

Effective rolling shear modulus of mid layers
The effective rolling shear modulus was determined for seven 4 × 190 = 760 mm long parts of CLT elements as described in Sect. 3.3.1. In Fig. 6, shear strains of the examined parts, determined using DIC and valid for the load P = 20 kN (corresponding to a maximum normal stress in outer layers of approximately 17 MPa) in four-point bending tests, are shown. The areas of the strain fields used to estimate the effective shear modulus, G yz,eff,exp,40,760 , by comparing strains from DIC with strains from FE simulation, are marked by thin white lines drawn on top of the images of strain fields in Fig. 6. The value of G yz,eff,exp,40,760 , calculated according to Eq. (6a) are also given in Fig. 6. The values given in parenthesis in Fig. 6 are G yz,eff,exp,40,760 calculated according to Eq. (6b). Mean values of G yz,eff,exp,40,760 for the seven elements were calculated to 159 MPa (156 MPa using Eq. 6b) with a CoV of 0.16. Thus, the difference in results of using Eq. (6a). compared to using Eq. (6b) G yz,eff,exp,40,760 was small.
It is possible to see, in the photographs of the CLT elements shown in Fig. 6, the annual ring pattern of the mid layer lamellas and to estimate approximate pith locations. It can also be seen that shear strains tend to be large in areas where the annual ring orientation is close to horizontal or Fig. 6 Photographs of parts of CLT elements and engineering shear strains determined by DIC and four-point bending tests. Displayed parts of elements were placed in the zone subjected to shear in the bending tests. Estimated effective rolling shear modulus is displayed for each of the seven parts of CLT elements vertical and smaller where annual rings are oriented in an angle to the horizontal plane of about 45 degrees. Borders between lamellas in the mid layer (lamellas in the same layer are not glued to each other) as well as superficial cracks in the wood material caused by drying (the latter are typically oriented in radial direction of the wood material) appear as large strains. Note that six of the seven CLT elements shown in Fig. 6 were cut from the same CLT panel, namely CLT panel 2, and the investigated 760 mm long parts of CLT element 2.3, 2.5 and 2.7 were from the same longitudinal position along the major direction of the CLT panel, see Fig. 1. Thus, the midlayer lamellas of these elements are often parts of the same timber board. For example, it can be seen in Fig. 6 that the first lamella from the left in elements 2.3, 2.5 and 2.7 originate from the same physical timber board. Correspondingly, the investigated parts of CLT elements 2.2, 2.6 and 2.8 are placed at, among themselves, the same longitudinal position along the CLT panel, which is different from the longitudinal position of elements 2.3, 2.5 and 2.7 (a similar distance from the opposite end of the CLT panel). The selection of CLT elements/parts of CLT elements was a compromise, which aimed at a diversity of investigated lamella cross-sections but taking practical aspects of the work in laboratory into account. The mutual order of the CLT elements in Fig. 6 is arranged to facilitate relevant comparisons between the CLT elements.

Rolling shear stiffness of the wood material
The finite element model of the rt-plane of a single lamella, see Sect. 3.2.2,was employed to evaluate G yz,eff,nom,40,190 for lamellas with different location of pith. In Fig. 7a, calcu lated values of G yz,eff,nom,40,190 for different relative position between pith and centre of the lamella are displayed. In the image, the pith location is placed at the origin and marked by a white, filled circle. Four examples of lamellas with different position in relation to the pith are marked by rectangles drawn with white dashed lines. The centre of each of these is marked by a small square drawn with solid white lines. The colours seen within the squares represent the effective shear modulus G yz,eff,nom,40,190 for the lamellas (with reference to the colour scale bar at the bottom of Fig. 7). Clearly, the position of the lamella in relation to the pith has a great impact on the effective shear modulus. If, for example, pith is located in the middle of the lamella it has a rather low effective shear modulus, even if it is considerably higher than G yz,nom . If instead pith is located at say a 50 mm distance from the centre of the lamella in the vertical direction, the effective shear modulus becomes about twice as high. This is because annual rings within such a lamella would be oriented in a close to 45˚ angle in a large part of the lamella cross section. At very large vertical distance between the pith and the lamella (pith centred in horizonal direction), the annual rings would be close to horizontal and, consequently, the effective shear modulus would again be low.
The pith location in relation to each of the lamellas within the parts of the elements that were investigated with DIC (i.e. parts of element 1:4, 2:2, 2:3, 2:5, 2:6; 2:7 and 2:8) was determined manually, as described in Sect. 3.1.2. Figure 7b shows photographs of the investigated parts of these CLT elements (same mutual order as in Fig. 6) with mid layer lamellas marked with white lines. The coloured rectangles drawn below the CLT elements represent the calculated effective shear modulus, G yz,eff,nom,40,190, of the respective lamellas, with reference to the colour scale bar shown at the bottom of Fig. 7a. The four lamella cross sections marked in Fig. 7a represent the locations of the four mid-layer lamellas of CLT element 1:4. In Fig. 7b, the value of G yz,eff,nom,40,760 , calculated according to Eq. (7), is also given for each of the seven elements. Table 5 shows values of G yz,eff,exp,40,760 and G yz,eff,nom,40,760 for each of the CLT elements and the average values of all the seven elements. The ratios between G yz,eff,exp,40,760 and G yz,eff,nom,40,760 are also given (forth column). In Eq. (8), k was defined as the ratio between the average values of G yz,eff,exp,40,760 and G yz,eff,nom,40,760 and from the results it is calculated that k = 0.85. Thus, the adjusted values of the rolling shear modulus were determined, using Eq. (9), to G rt = k G rt,nom = 0.85·66 = 56 MPa. Adjusted values of effective mid layer shear modulus are calculated as kG yz,eff,nom,40,760 and are shown in the fifth column of Table 5.
The clear wood density of Scots pine according to Dinwoodie (2000), ρ cw,nom = 550 kg/m 3 is about 3% higher than the density of the Scots pine timber used in the investigated elements, see Table 2, and since timber including knots has about 5% higher density than clear wood (EN 384), it can be estimated that the difference in clear wood density of Dinwoodie (2000) and the current wood material is about 8%, to compare with the difference of G rt , which is 15% (G rt = 56 MPa vs G rt,nom = 65 MPa). In previous research on Norway spruce, Dumail et al. (2000) reported G rt = 58 MPa for a clear wood density of 447 kg/m 3 ; Aicher et al. (2001) reported G rt = 50 MPa for clear wood density of 460 kg/m 3 ; and Neuhaus (1981) reported (according to Aicher et al. 2000) G rt = 42 MPa for a clear wood density of 460. Thus, the ratios of G rt /clear wood density for these studies were 0.13, 0.11 and 0.09 [m 3 MPa/kg], respectively. For pine, it is according to Dinwoodie (2000) 0.12 and according to the results obtained in the present study 0.11.
It can be noted that G rt = 56 MPa is close to the oftenrecommended value of 50 MPa to use for CLT. However, the current results show that the average effective rolling shear modulus of 159 MPa is almost three times higher, and practically the same as the 158 MPa obtained for similar Scots pine boards in Ehrhart and .
Finally, it should be noted that G rt is hardly constant, as assumed herein. Of course, there is a difference between different lamellas, and it is likely that G rt increases with increasing distance to pith, just like density and E l tend to do (e.g. Ormarsson 1999;Wagner et al. 2013). This explains the differences between the experimentally obtained values of G yz,eff,exp,40,760 (second column in Table 5) and the adjusted calculated values kG yz,eff,nom,40,760 (fifth column in Table 5).
The results presented and discussed this far are representative for the case when lamellas in the middle layer are not glued to each other. However, using cyclic boundary conditions and the 2D FE model of a lamella (Sect. 3.2), also the case with glued edges between lamellas was modelled. Of course, for this case the effective shear modulus increases. How much it increases depends on the orientation of the annual rings at the edges of the lamellas and varies between 3 and 16%. For pith locations in the region shown in Fig. 7a, the average increase of G yz,eff,nom,40,190 was calculated as 8%. This implies that the effective shear

Relationship between dynamic axial MoE of lamellas and quasi-static bending MoE of CLT elements
In Table 6, the ratios k loc and k glob , defined in Eqs. (10, 11), between stiffnesses obtained from experiments and FE simulations are presented. In the simulations, E x,dyn,corr was used for longitudinal MoE of the lamellas in the layers, as described in Sect. 3.3.3, and plane strain was applied. (If plain stress is used instead of plane strain, global and local displacements increase with 0.5%). Coefficients of variation of the inverses of the local and global deflections, which are proportional to the respective bending stiffnesses, are also given in Table 6. Regarding the ratios, k loc varies for the different types of elements between 1.01 and 1.03 and k glob varies between 1.06 and 1.12. Regarding reliability of the determined ratios, the value of k glob = 1.12 for the short elements is regarded the least reliable of those presented, since for short elements, k glob is sensitive both to the effective rolling shear modulus employed in the model (159 MPa was used for all mid layers) and to the limited stiffness of the supports (a stiffness of 12.5 kN/mm for each of the two supports was used, as explained in Sect. 3.1.2). For example, a 10% change of the effective rolling shear modulus gives for the short elements a change of k glob of 2%, and a 10% change of the stiffness of the supports gives for the short elements a 4% change of k glob . For the longer elements, k glob is less sensitive to changes in effective rolling shear modulus and the stiffness of the supports. The values obtained for k loc , 1.01-1.03, imply that only a very small correction factor of about 1.02 is needed for accurate calculation of the quasi-static bending stiffness of CLT element when the calculation is based on values of dynamic axial MoEs of the lamellas. This is in line with expectations as discussed in Sect. 1.2. The values obtained for k glob of 1.06-1.07 (disregarding the uncertain value of k glob for the short elements) indicate a slightly higher viscoelastic effect than what the values of k loc do. In total, considering results of both k loc and k glob , the results obtained herein imply that the axial dynamic MoEs of lamellas should be reduced by 2-6% to give basis for accurate calculation of quasi-static bending stiffness of the CLT elements.
Regarding the CoVs of stiffnesses presented in Table 6, these can be compared with the CoVs of the E x,dyn,corr of the lamellas presented in Table 2, which are about twice as high. Of course, the lower CoVs of the CLT elements are due to the system effect.
Finally, it should be noted that out of the 85 lamellas of Scots pine used in top and bottom layers of CLT elements, 12.5% of the material was removed due to trimming, as indicated in Fig. 1. Likewise, out of the 85 lamellas of Norway spruce used in top and bottom layers of CLT elements, as much as 42% was removed due to trimming and to the fact that only the 4000 mm long elements (no 2000 mm elements) were included in the investigation. Of course, this gives some uncertainty regarding the ratios presented in Table 6.

Conclusion
Three-layer cross laminated timber with Scots pine in mid-layers and either Scots pine or Norway spruce in the outer layers was manufactured and evaluated. Axial dynamic MoE, for each lamella used, was determined prior to manufacturing of the CLT elements. Local and global quasi-static bending stiffness of the CLT elements were evaluated in four-point bending tests. Shear strains were determined as a function of the applied load by means of DIC. Simulations corresponding to the four-point bending tests were performed using an FE model. Another FE model was employed to calculate the influence on the location of pith for the relationship between rolling shear modulus and effective rolling shear modulus. The following was concluded: -DIC in combination with an employed FE model forms the basis to determine effective rolling shear modulus of transversal layers in CLT subjected to four-point bending, even for long spans. (Of course, rolling shear strength cannot be evaluated using long spans since long spans will not give rolling shear failure.) -The average quasi-static effective rolling shear modulus for layers of Scots pine with lamellas of size 40 × 190 mm 2 and varying location of pith (no gluing between lamellas in the same layer) was determined to be 159 MPa. -The corresponding average rolling shear modulus of the wood material in the same Scots pine lamellas was calculated as 56 MPa. -Based on calculations, it was estimated that gluing the lamellas in the mid layer to each other would give an increase of the effective shear modulus of about 8%, compared to the case with no glue between the lamellas. -Knowledge of axial dynamic MoE of lamellas forms the basis for calculation of local and global out of plane bending stiffness of CLT elements manufactured of the lamellas. Comparisons of local and global deflections from quasi-static bending tests and from corresponding simulations using an FE model, show that the longitudinal MoE used in the FE model for the lamellas in outer layers of CLT elements should be reduced by 2-6%, compared to the axial dynamic MoEs of the lamellas used in the outer layers, to give the same average stiffness according to simulations and experiments. Thus, producers of CLT may use this result to control the important out of plane bending stiffness of the CLT produced.
Furthermore, results that were derived from data of previous research, and presented in Sect. 1.2 of the introduction, showed that: -The ratio of dynamic axial MoE and local quasi-static bending MoE of sawn timber is considerably higher for small dimension boards than what it is for large dimension boards.