Modelling of Experimental Tests of Wooden Specimens from Scots Pine ( Pinus sylvestris ) with the Help of Anisotropic Plasticity Material Model

In order to describe the behaviour of wood when calculating wooden elements and structures with the use of the fi nite element method, orthotropic material model in combination with non-interactive (maximum stress criterion) or interactive failure criteria (Hoffman and Tsai-Wu criterion) is used. Another option is to use a general anisotropic plasticity material model complemented with a non-interactive failure criterion – maximum stress criterion, which allows to describe wood failure by brittle failure in tension. The presented general material model was used in combination with the idealization of annual rings by cylindrical surface for the modelling of wood specimen tests form Scots pine (Pinus sylvestris). The obtained results show good agreement between the results of numerical analysis and experimental testing of wood specimens. The use of the anisotropic material model can also be seen in cases when the level of the applied load is higher than the level when the failure of wooden material occurs.


Ključne riječi: obični bor (Pinus sylvestris), FEA, numeričko modeliranje, elastične konstante, materijalne konstante, anizotropna plastičnost
For the purpose of the numerical analysis of a prefabricated suspended staircase from Scots pine (Pěnčík, 2013), elastic and material constants of the Scots pine were selected with the use of results presented in the papers of Matovič (1993) and Požgaj et al. (1997;2004).The constants are shown in Table 1, where E is modulus of elasticity in material directions (longitudinal L -E L , radial R -E R and tangential T -E T ), G is shear modulus in material planes (G RT , G LT , G LR ),  Poisson's ratio ( RT ,  LT ,  LR ), f Lt , f Rt , f Tt are strengths in tension and f Lc , f Rc , f Tc are strengths in compression in material directions L, R and T, f LR , f LT , f RT are shear strengths in material planes.
The verifi cation of elastic and material constants (Table 1) was performed by a numerical modelling of experimental testing with the use of fi nite element calculation using software ANSYS (ANSYS, 2012a).

Eksperimentalno ispitivanje
Wooden specimens made of Scots pine (Pinus sylvestris) were prepared in compliance with ČSN EN 408.The dimensions of the cross section -width (b) and height (h) and length (L) of specimens were selected in compliance with the requirements of ČSN EN 408; b × h × L = 25 × 25 × 475 mm.Before the testing, the specimens were conditioned in a standard environment at the temperature of 20 ± 2 °C and relative

UVOD
Despite the high abundance of Scots pine (Pinus sylvestris) in mild and colder Eurasia climate (Úřadníček et al., 2001; 2012a), few literature sources contain information on elastic and material constants of this tree species, which could be used for an analysis with the use of anisotropic plasticity material model.Anisotropic plasticity material model defi ned by 9 elastic and 18 material constants is described by Moses and Prion (2004) as a general material model, which allows to defi ne different bilinear elasto-plastic behaviour with possible hardening of material in three perpendicular directions, including the defi nition of different behaviour in these directions in tension, compression, and shear.
Elastic constants for Scots pine (Pinus sylvestris), shown by Bodig and Goodman (1973) and Martin and Berger (2001), are based on a special report produced by R. F. S. Hearmon (Hearmon, 1948).Elastic constants for Scots pine used for the modelling of a section of a wooden string staircase and stair joints are also presented by Pousette (2006).Elastic and material constants of Scots pine are shown by Danielsson and Gustafsson (2013), who used these constants for the modelling of a double cantilever beam (DCB) and endnotched beams.According to Danielsson and Gustafsson (2013), the specifi ed constants are in compliance with the values for Norway spruce (Picea abies).The values of elastic and material constants of Scots pine are shown in the papers of Matovič (1993) and Požgaj et al. (1997;2004).humidity of 65 ± 2 % until reaching constant weight (Kuklík and Vídeňský, 2005).In total, 10 specimens were prepared.The specimens were simply supported during the experimental testing.The specimens were loaded by four-point bending (Fig. 1).The distance between supports was 450 mm (18•h).The testing set was complemented with a developed steel V-shape fi xture (Fig. 1), which prevented transversal displacement of the tested specimens, while allowing their bending.
For the purpose of the monitoring of vertical displacement w* (Fig. 1), testing specimens were fi tted on the right support at their upper surface, with an inductive standard displacement transducer HBM WA-T/50 mm (HBM, 2012), with the measuring range of 0 -50 mm and accuracy of 0.001 mm.The vertical displacement w at midspan was converted with the use of the measured vertical displacement w* (w = w*/2).The load was derived from the mechanical testing press FPZ100.The magnitude of the loading force was recorded by a calibrated resistance load cell.All sensors were connected to an 8-channel measuring data logger HBM Spider 8 (HBM, 2006).The data were recorded with the recording frequency of 2 Hz, i.e. in the time interval of 0.2 s.

Results of experimental testing 2.2. Rezultati eksperimentalnog ispitivanja
The relation of applied load F on vertical load w (load -displacement diagram F -w) for the whole inter-val of loading and all specimens is shown in Fig. 2. The load displacement diagram for individual specimens is shown in Fig. 5.The average value of limit load F max is 2.693 kN (Table 2).The regression analysis and the determination of modulus of elasticity in bending used the part of graph in the interval 0.1•F max to 0.4•F max , i.e. the part with linear relationship between load F and displacement w (Fig. 2) with the value of reliability R 2 higher than 0.99 (Kuklík and Vídeňský, 2005).Modulus of elasticity in bending E b (MPa) (Table 2) was determined with the use of relation (3) according to ČSN EN 408, where l -free sample length, m, b -width of cross-section, m and h -height of crosssection, m, and a -distance of load from support (6•h), m (Fig. 1).The difference (F 2 -F 1 ) and (w 2 -w 1 ) shows the growth of loading and vertical displacement on the linear part of the load -displacement diagram.The values F 1 , F 2 , w 1 and w 2 were determined with the use of load-displacement diagram in Fig. 2; F 1 = 0.269 kN and F 2 = 1.077 kN.The measured values are arranged in Table 2, where m -weight, g, F max -limit load, kN, w* -measured vertical displacement, mm, w -converted vertical displacement at midspan, mm and R 2value of reliability.Bending strength R b (MPa) (Table 2) was determined with the use of the relation (4) according to ČSN EN 408, where b -width of cross-section, m, and hheight of cross-section, m, a -distance of load from support (6•h), m F max -limit load of specimen.

Numerical analysis of experimental testing 2.5. Numerička analiza eksperimentalnog ispitivanja
When modelling wood by the fi nite element method (Bathe, 2006;Tankut et al., 2014), several approaches to wood modelling can be used.The approaches are based on simplifi cations of the real natural material.The modelling idealizes the real trunk of generally cone shape as a cylindrical object (Bodig and Jayne, 1993), idealizes annual rings as constant shape, thickness and curvature, local failures (knots, cracks), variable structure of material, and differences between earlywood and latewood.
The numerical modelling of experimental testing of specimens idealized annual rings of individual specimens as cylindrical surfaces (Fig. 4) and the behaviour of wood was described with the use of anisotropic plasticity material model (Moses and Prion, 2004).The model was described with elastic and material constants shown in Table 1.The material constants were adjusted in order to meet the conditions of plasticity incompressibility and closed surface plasticity of elliptical shape mentioned by Moses and Prion (2004) and ANSYS (2012a).The material model was complemented with non-interactive failure criterion -maximum stress criterion (Vinson and Sierakowski, 2002), in order to allow the description of wood failure by brittle failure in tension.Neither the hardening of material, nor the limitation concerning the plasticity reserve was considered for compression.
Ten 3D numerical models of specimens were made in the calculation system ANSYS version 14.0 (ANSYS, 2012a).The models were created with the In order to reduce the time demanding nature of calculations, symmetric boundary conditions were considered (Cook et al., 2001), i.e. specimen halves were modelled of a size of 25 × 25 × 237.5 mm (Fig. 3).In the transversal direction, the line load (line A) was distributed on specimens with the use of a distribution board (Fig. 3).
The cross sections of numerical models considered the characteristics of annual rings.The modelling of the characteristics of annual rings was based on one of two front views of specimens.General cone shape of the trunk, as well as the spiral arrangement of grains along the whole length of the trunk, was neglected.The characteristics of annual rings in the front were replaced with cylindrical surfaces, and the specimens were expected to have constant curvature along the length.The cross sections of specimens are shown in Fig. 4.
The calculations were made in steps, taking into account material non-linearity.In addition, the effect of large displacements and rotations of fi nite elements, i.e. geometric non-linearity, was taken into account.

Results of experimental testing 3.1. Rezultati eksperimentalnog ispitivanja
On average, the failure of tested specimens occurred under the force of 2.693 kN (Table 2).The min-imum force of 2.171 kN at the failure was measured when testing specimen No. 7. In contrast, the maximum force of 3.062 kN at the failure was measured when testing specimen No. 6.The experimentally measured variation strength coeffi cient at failure is 9.83 %.
The average value of the modulus of elasticity in bending E b , determined by experimental testing of 10 specimens, is 12935.4MPa.The measured values of the modulus of elasticity in bending ranged within the interval of 10624.0MPa (min; specimen No. 1) to 15738.0MPa (max; specimen No. 3).The experimentally measured variation coeffi cient of the modulus of elasticity in bending is 14.0 %, which is a lower value than that of 22 % mentioned by Bergman et al. (2010) and Jírů (1970).
The average value of bending strength R b is 75.9MPa.Minimum and maximum values of bending strength do not correspond to the same samples used for measuring modulus of elasticity in bending.The measured values of bending strength ranged within the interval of 61.2 MPa (min; specimen No. 7) to 86.3 MPa (max; specimen No. 6).The experimentally measured variation coeffi cient of bending strength is 9.8 %, which is again a lower value than that of 16 % mentioned by Bergman et al. (2010) and Jírů (1970).
The value of modulus of elasticity in bending E b and bending strength R b depends on the orientation and density of annual rings (Table 2 and Fig. 4).Higher values E b and R b were shown by samples with higher density of annual rings and specimens, whose annual rings were convex when looking at the specimen front.The load curves F -U Y at point B (Fig. 3) are shown in Fig. 5. Very good conformity between experimental tests and numerical analysis was reached with the specimen No. 4, 5, 6, 9 and 10.Regarding the specimen No. 1, 2 and 7, the hardness values of the numerical and material model, respectively, were higher than those of wood specimens.Load curves of specimen No. 3 and 8 are located under the experimentally determined load curves, indicating lower rigidity of the numerical and material model, respectively.
The average strength at the failure limit determined by experimental testing (2.693 kN) differs from the average strength at the failure limit determined by numerical modelling (2.650 kN) by -1.62 % (Table 3).The difference of the average measured vertical displacement at midspan (15.32 mm) and numerically determined values (14.93 mm) is -2.61 %.

ZAKLJUČAK
The determined value of average modulus of elasticity in bending E b 12935.4MPa by experimental testing differs from the values stated in literature for Scots pine (Pinus sylvestris) 12800 MPa (Table 1) by 1.05 %.The average bending strength R b 75.9 MPa determined with the use of the same group of specimens differs by 11.74 % from the published value of 86.0 MPa.The determined average values E b and R b confi rm the values of these constants mentioned in the paper of Matovič (1993) and Požgaj et al. (1997Požgaj et al. ( , 2004)).The partial results of experimental testing of specimens show that constants E b and R b are dependent on the orientation and density of annual rings.
The use of anisotropic plasticity material model (Moses and Prion, 2004) for wood modelling with elastic and material constants according to Table 1, while taking into account non-interactive failure criterion and the idealisation of the annual rings characteristics, showed good agreement between the results of numerical analysis and experimental testing of wood specimens.The difference between experimentally and numerically determined results is on average up to 2.7 %. Numerically determined load curves follow the load curves determined by experimental testing.The use of anisotropic plasticity material model can be seen on examples of load test modelling, numerical determination of load bearing capacity, i.e. in cases when the level of the applied loading is higher than the load when the material failure occurs.Otherwise, the above described approach can be replaced by a simpler, orthotropic material model with elastic constants according to Table 1 and interactive failure criteria -Hoffman (Hoffman, 1967) and Tsai-Wu criterion (Tsai and Wu, 1971).

Figure 3 Figure 4 F
Figure 3 Geometry and 3D numeric model of wooden specimens in the calculation system ANSYS Slika 3. Geometrijski i numerički 3D model drvnih uzoraka u računalnom sustavu ANSYS

Table 3
Results of experimental measurement of strength at the failure limit F (kN) and vertical displacement U Y at point B at midspan with numerically determined values Tablica 3. Rezultati eksperimentalnog mjerenja čvrstoće pri maksimalnom opterećenju F (kN) i vertikalnom pomaku U Y u točki B u sredini raspona s numerički utvrđenim vrijednostima