6.3 Variation of Properties in a Tree
6.3.2 Influence of location in the tree
Simulations based on the experiments on clear wood specimens were carried out, using the distribution of the longitudinal modulus of elasticity and the density along the radius of the tree shown in Figures 3.24 and 3.28. From these experimental values for the longitudinal modulus of elasticity and the density, average microfibril angles were determined, using the procedure described in Appendix A. A structural wood model using irregular hexagonal cells was employed for determining the stiffness and shrinkage properties along a radius from the pith to the bark of a tree at increments of 10 mm. The mean measured values along the radius for the longitudinal modulus of elasticity and the density, together with the calculated average microfibril angles, are presented in Table 6.4.
a) 0 10 20 30 40 50 60 70 80
0.6 0.8 1 1.2 1.4 1.6 1.8x 10
Growth ring, no
Modulus of elasticity, MPa
b) 0 10 20 30 40 50 60 70 80
400 410 420 430 440 450 460 470 480 490 500
Growth ring, no
Density, kg/m3
c) 0 10 20 30 40 50 60 70 80
0 5 10 15 20 25 30 35 40 45
Growth ring no.
Microfibril angle. deg.
Figure 6.13: Input data for the model along the radius in a tree.
a) Mean of measured longitudinal moduli of elasticity.
b) Mean of measured densities.
c) Microfibril angles calculated by use of Eq.(2.40) and data in a) and b).
Table 6.4: Measured longitudinal moduli of elasticity and densities and the calcu-lated microfibril angles along the radius of a tree.
Ring Mod. of elast. Density, Microfibril No. EL, MPa ρr, kg/m3 angle, ϕ◦
0 6450 450 41.5
3 8700 424 32.4
7 11300 430 24.2
12 13200 436 17.9
18 14300 444 14.3
34 15600 459 10.1
45 16000 467 9.3
58 16300 470 8.0
69 16400 470 7.2
80 16500 470 6.4
In Figure 6.13 the same values are shown as curves along the radius, the curves near the pith being dotted so as to indicate that the simulation results close to the pith are uncertain, which can be due to various factors, one of which is that differences in chemical composition.
The stiffness and shrinkage properties were calculated using the values for the densities and microfibril angles shown in Table 6.4. Figures 6.14-6.17 show the stiffness and hygroexpansion properties obtained from the simulations. The modulus of elasticity in the longitudinal direction agrees well with the experimental data, see Figure 6.14. Both the radial and tangential moduli of elasticity were found to have high values close to the pith with first a steep decrease and then an increase towards the bark. The high radial and tangential moduli of elasticity in the pith are due to the large microfibril angles and the high density there.
The three moduli of shear also show high values near the pith, Figure 6.15. In the LR- and LT-planes the moduli of shear decrease towards the bark whereas in the RT-plane, outside the pith area, there is an increase in the modulus of shear.
The Poisson ratios in the LR- and LT-planes decrease from the pith towards the bark, whereas those in the RT-plane increase from the pith towards the bark, see Figure 6.16.
The hygroexpansion coefficients obtained from the simulations are shown in Fig-ure 6.17. The longitudinal hygroexpansion is large near the pith and first decreases rapidly and then increases again along the radius towards the bark. The radial and the tangential hygroexpansion are both low near the pith but increase towards the bark. The simulation results agree reasonably well with the experimental observa-tions obtained, cf. Figure 3.32.
0 10 20 30 40 50 60 70 80 0.4
0.6 0.8 1 1.2 1.4 1.6x 104
Growth Ring, No.
Longitudinal MOE, MPa
0 10 20 30 40 50 60 70 80
600 650 700 750 800 850
Growth Ring, No.
Radial MOE, MPa
0 10 20 30 40 50 60 70 80
1050 1100 1150 1200 1250 1300
Growth Ring, No.
Tangential MOE, MPa
Figure 6.14: Moduli of elasticity in the longitudinal, radial and tangential directions along the radius of a tree.
0 10 20 30 40 50 60 70 80 800
850 900 950 1000 1050 1100 1150 1200 1250
Growth Ring, No.
Modulus of shear LT, MPa
0 10 20 30 40 50 60 70 80
850 900 950 1000 1050 1100 1150 1200 1250 1300
Growth Ring, No.
Modulus of shear LR, MPa
0 10 20 30 40 50 60 70 80
10.5 11 11.5 12 12.5 13 13.5 14
Growth Ring, No.
Modulus of shear RT, MPa
Figure 6.15: Moduli of shear in the LT-, LR- and RT-planes along the radius of a tree.
0 10 20 30 40 50 60 70 80 0.02
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Growth Ring, No.
Poisson´s ratio RL
0 10 20 30 40 50 60 70 80
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Growth Ring, No.
Poisson´s ratio TL
0 10 20 30 40 50 60 70 80
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Growth Ring, No.
Poisson´s ratio TR
Figure 6.16: Poisson ratios νRL, νT L and νT R along the radius of a tree.
0 10 20 30 40 50 60 70 80 0
0.005 0.01 0.015 0.02 0.025 0.03 0.035
Growth Ring, No.
Longitudinal shrinkage coefficient
0 10 20 30 40 50 60 70 80
0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24
Growth Ring, No.
Radial shrinkage coefficient
0 10 20 30 40 50 60 70 80
0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41
Growth Ring, No.
Tangential shrinkage coefficient
Figure 6.17: Shrinkage coefficients αL, αR and αT along the radius of a tree.
6.4. Concluding Remarks
The results of the parametric studies presented show the parameters that governing the stiffness and hygroexpansion properties of wood here to be the microfibril angle of the S2-layer, the density and the properties of the chemical constituents. The choice of parameters to describe the cell structure geometry did not influence the stiffness and the hygroexpansion properties as much, except in a few cases. The stiffness in the radial direction was strongly influenced by irregularity and by ray cell stiffness. Some parameters were also affected by the ratio of the tangential cell widths that was chosen. To make an accurate prediction of the stiffness and hygroexpansion properties, it is essential that one have reliable material data on the chemical constituents and on the microfibril angle of the S2-layer. The analyses conducted of the variations in properties within the tree indicate the variations in the stiffness and hygroexpansion properties along the radius of the tree to be large.
These variations have a strong influence on shape stability and are important to take into account in analysing of the mechanical behaviour of wood.
7.1. Introduction
Wood is a cellular and porous material that exhibits a nonlinear behaviour at load-ing, especially when loaded in compression perpendicular to the grain and in shear in the radial-tangential plane. Above the limit of proportionality, wood behaves in a highly nonlinear way. The behaviour of wood is influenced by several factors, such as density, moisture content, temperature and duration of loading. For compression loading perpendicular to the grain, three basic failure patterns can be distinguished, depending on the orientation of the growth rings in relation to the direction of loading. For radial compression, local buckling of the cell walls in the earlywood zone occurs. Tangential compression results in a more macroscopic buckling failure where the latewood region buckles into the earlywood. Shear failure often occurs for loading at an angle to the growth rings.
In many applications wood is loaded beyond the elastic limit, where a compres-sional load perpendicular to the grain often leads to very large deformations. To improve the nonlinear constitutive models for wood, a better understanding of the basic behaviour of wood is needed. To achieve this, micromechanical models from which the nonlinear behaviour of wood can be simulated can be utilised. By use of the numerical models described in Chapters 4 and 5 extended and aided by the experimental work presented in Chapter 3, the behaviour of wood at higher loading was studied. The linear elastic material models for the cell wall layers need to be modified to elastic-plastic material models, since the strains can locally become very large in the cell wall.