Modelling of the deformation of wood

The shrinking and swelling of wood and wood based materials due to a change of moisture content leads to a corruption of the geometry. This effect must also be simulated in the virtual reality with algorithms to predict the deformation and get correct process data.

There are mainly two kinds of methods to simulate the material behaviour as the numerical and the analytical approaches. The numerical methods as the FEA requires a high number of input data and a software for perform the simulation [13] [14]. Furthermore, all the finite element studies do not consider the influences of different shapes in the cross sectional deformation. In contrast, the analytical methods have an immediate application in the solution of problems because they are simple and they require few input data. The intrinsic problem in the analytical methods is that they consider certain assumptions in the calculation that lead to errors or inaccurate results because most of them consider the annual rings as straight lines and the profile shape is always considered rectangular [10][14][15]. There is no presence in literature of any calculation tool, forecasting the behaviour of the cross sectional wood with respect to the profile shape and it is necessary to develop an analytical calculation system.

Development of the new algorithm

The standard analytical approaches consider the deformation of a rectangular cross section considered as a transformation of coordinates or a transformation of external points but they do not consider the internal influences causing the deformation. The innovation in this approach is the consideration of the deformation as a mechanism happening due to linear expansion or contraction of latewood. The input data needed for calculate the deformations of a wood cross sectional shape with the graphical method are the tangential and radial differential shrinkage and swelling parameter, the difference between the initial moisture content and the end moisture content in the cross sectional piece and a picture of the profile cross section.

The series of steps needed to perform the calculation of the deformation start with a picture of the profile cross section.

1. The picture has to be transferred in a program for analyse images, for example a CAD system.

2. The perimeter around the shape of the profile has to be traced with some lines or arcs of circle.

3. With the perimeter of the profile in a vector image, the centre of area has to be calculated and identified in the figure.

4. After that the outline shape is traced it is important to analyse carefully the image searching the biggest latewood arcs of circle present in the image, because they represent the annual rings subjected to more deformation in the wood profile and are the major cause of deformation of the overall profile shape. Also the position of the latewood in each side of the contour of the wood profile has to be considered, because for simplify the process of graphical calculation it is crucial to determine at least two annual rings connecting one side of the board and assuming further that between two points only one segment can pass through. A selection of more latewood annual rings leads to a longer process for calculate the shape deformation.

5. The selected latewood annual rings have to be traced as done for the profile shape contour. Subsequently the nearest latewood annual ring passing through the centre of area has to be traced and this is then considered as the annual ring used to define the pith position of the fitted trunk in respect to the cross section of the board.

6. From the centre of the pith a radius connecting the centre of area of the profile has to be drawn as represented in Fig 3. The segment intersecting the centre of area of the board from the pith of the fitted trunk is considered as the equilibrium segment for the profile cross section where the tangential stresses are acting opposite each other in respect to this segment.

7. For calculate the tangential shrinkage and swelling of the profile it is necessary to draw a radius from the centre of the pith to the profile board sides intersected by the latewood annual rings. The radius and the profile shape, the radius and the angle with the equilibrium segment as represented in Fig. 3 have to be reported. The tangential displacement is computed with the subsequent formulation 1:

u r

l

  

 

  

Where is the radius from the latewood intersection whit the board sides to the pith; is the angle between the radius and the

equilibrium segment, converted in radians; is the tangential differential shrinkage and swelling factor and is the difference between the initial moisture content and the final moisture content state.

Figure 3: Procedure for calculate deformation of a board

For calculate the deformation in the radial direction the process is similar:

9. For every intersection between the selected latewood annual rings and the profile shape, it is necessary to draw and quote segments, connecting the intersections to the centre of area of the board cross section as represented in Fig 3. Subsequently the angle between the radius and the segment connecting the centre of area of the board as represented in Fig. 3 has to be determined.

10. The remaining length after the deformation of the segment connecting the latewood intersections to the centre of area of the profile is calculated by the subsequent formulation:





l

l_new _centrearea 





Where is the length of the subsidiary line connecting the centre of area, is the angle measured in step 9, is the

differential shrinkage parameter in the radial direction, is the difference in the moisture content states, is the new calculated distance between the segment connecting the centre of area to the latewood intersection in the shape of the profile. The segment is computed from the centre of area.

For compute the results originated in the calculation the direction of the deformation movements is oriented with some rules.

The first deformations applied are the deformations due to the radial strains and all the vectors are calculated with the direction against the centre of area as represented in Fig. 3, because the components of the radial stresses seek to be in balance in respect to the centre of area. The calculated length after the deformation has to be computed from the centre of area along the segment. Subsequently, the reported deformations in the radial direction, it is necessary to report the deformation due to a tangential deformation and apply the deformed vector in an arc of circle parallel to the arc of circle representing the latewood, with the origin in the cusp of the vector of the radial movements and with the direction towards the equilibrium segment as represented in Fig. 3. In Fig. 4 is represented an example of application of the algorithm in a profile shape.

Figure 4: Example of application in the algorithm

The results of this algorithm were investigated with experiments showing a good agreement with the reality.