Geometric model to predict twist in unrestrained boards

Abstract

A theory has been developed for calculating the twist that develops in boards during drying without restraint, as well as the deformation in cross-section that accompanies the development of twist. Calculations require a knowledge of only a limited number of parameters: width, thickness and length of the board, annual ring orientation, distance from the pith, radial, tangential and longitudinal shrinkage coefficients, and the variation of spiral grain angle (SGA) with distance from the pith. The theory is derived from geometrical and physical principles and shows that a complicated interaction between all the above parameters gives rise to twist. A novel coordinate system is used that is better adapted to the fact that spiral grain lies at an angle to the log axis rather than the usual Cartesian or cylindrical polar coordinates. Unlike the finite element models that have recently been developed this theory does not allow for the effect of stresses that develop in a board, although the theory in its present form can easily be extended to incorporate this effect. The advantage of this theory over the more exact finite element models lies in its educational value in that it clearly identifies the mechanisms that are responsible for twist. An associated MS Excel spreadsheet allows rapid analysis of different scenarios such as the effect on twist of changing the shrinkage coefficients, annual ring orientations and moisture content. The theory predicts that for radiata pine 100×50 mm boards maximum twist occurs near the pith, and that the direction of twist reverses when the distance from the pith is greater than about 120 mm. These predictions are shown to agree with experiment. The theory also predicts that if a radiata pine log is live-sawn (through-and-through sawn) there will be two regions in the mature wood where the quartersawn boards will have large negative twist values, but that this can be avoided by cant- or grade-sawing. In contrast, the theory also predicts that if the SGA is constant at 4° from pith to bark, board twist will decrease smoothly from pith to bark for all annual ring orientations without ever becoming negative.

Appendices

Appendix 1

Shrinkage factors to use after the isotropic shrinkage step

Assume a length H lying along the grain direction. In this analysis the first step is an isotropic shrinkage of the board by a fraction r. This reduces this length to H×(1−r). The real shrinkage fraction along the grain direction is L, so that the final dimension should be H×(1−L).

To achieve this, after the isotropic shrinkage step all lengths along the grain direction must be swelled by a factor of 1+(r−L)/(1−r), so that the expansion factor is (r−L)/(1−r). Similarly, after the isotropic shrinkage step all dimensions in the tangential direction must be reduced by a tangential shrinkage fraction of (t−r)/(1−r).

Appendix 2

Relative horizontal rotation of a point along the cylinder

A point F on the circumference of a board at height H rotates along a cylinder of diameter R_F (Fig. 2). The movement ∇ along the horizontal plane is equal to QU′−QN (Figs. 4 and 5). Then from Eq. 18 and Figs. 4 and 5:

$$ \begin{aligned} \nabla = & {\text{QS}^\prime} \times {\text{cos}}\,\phi + {\text{S}^\prime\text{F}^\prime} \times \sin \phi - {\text{QS}} \times {\text{cos}}\,\phi - {\text{SF}} \times {\text{sin}}\,\phi \\ = & ({\text{QS}^\prime} - {\text{QS}}) \times {\text{cos}}\,\phi + ({\text{S}^\prime\text{F}^\prime} - {\text{SF}}) \times {\text{sin}}\,\phi \\ \end{aligned} $$

Then substituting from Eqs. 4 and 5 for QS′ and S′F′ leads to:

$$ \nabla = - \frac{{(t - r)}} {{(1 - r)}} \times {\text{QS}} \times {\text{cos}}\,\phi + \frac{{(r - L)}} {{(1 - r)}} \times {\text{SF}} \times {\text{sin}}\,\phi $$

(23)

Substituting for QS and SF from Eqs. 11 and 15 into Eq. 23 leads to

$$ \nabla = - \frac{{(t - r)}} {{(1 - r)}} \times (R_{\text{F}} \times \theta _{\text{F}} - Z_{\text{F}} \times {\text{tan}}\,\phi ) \times {\text{cos}}^2 \phi + \frac{{® - L)}} {{(1 - r)}} \times (R_{\text{F}} \times \theta _{\text{F}} \times {\text{sin}}\,\phi + Z_{\text{F}} \times {\text{cos}}\,\phi ) \times {\text{sin}}\,\phi $$

After further trigonometric and algebraic manipulation this leads to:

$$ \nabla = \frac{1} {{(1 - r)}} \times [R_{\text{F}} \times \theta _{\text{F}} \times \{ ® - t) + (t - L) \times {\text{sin}}^2 \phi \} + Z_{\text{F}} \times {\text{sin}}\,\phi \times {\text{cos}}\,\phi \times (t - L)] $$

(24)

The term $\left( {{1 \mathord{\left/ {\vphantom {1 {(1 - r)}}} \right. \kern-\nulldelimiterspace} {(1 - r)}}} \right) \times R_{\text{F}} \times \theta _{\text{F}} \times (r - t)$ represents a small movement that is responsible for normal cupping (i.e. without a contribution from spiral grain). The term $\left( {{{(t - L)} \mathord{\left/ {\vphantom {{(t - L)} {(1 - r)}}} \right. \kern-\nulldelimiterspace} {(1 - r)}}} \right) \times R_{\text{F}} \times \theta _{\text{F}} \times {\text{sin}}^2 \phi$ is responsible for an additional cross-sectional distortion that is normally small. Both the first and the second term are very small compared to the third term. Moreover, these displacements are the same at the top and the bottom of the board, so that the relative distance of rotation # is:

$$ \begin{aligned} \# = & \frac{1} {{(1 - r)}} \times Z_{\text{F}} \times {\text{sin}}\,\phi \times {\text{cos}}\,\phi \times (t - L) \\ = & \frac{{0.5}} {{(1 - r)}} \times Z_{\text{F}} \times {\text{sin}}\,(2\phi ) \times (t - L) \\ \end{aligned} $$

In the range from 0° to 21° the relationship between sin(2φ) and φ is given by a linear equation with r²=0.998, where sin(2φ)=1.8358×φ, so that for SGAs up to 21°

$$ \# = \frac{{0.918}} {{(1 - r)}} \times \phi \times Z_{\text{F}} \times (t - L) $$

(25)

Hence the distance of rotation at a point on a board is proportional to the SGA in radians, the length of the board, and the difference between the tangential and longitudinal shrinkage fractions.