A numerical study of the shape stabilityof sawn timber subjected to moisture variationPart 2: Simulation of drying board

S. Ormarsson, O. Dahlblom, H. Petersson

Abstract A theory for analysing the shape stability of sawn timber wasimplemented in a nite element program. To illustrate the types of results thatcan be obtained, the behaviour of a board during drying was simulated. Thesimulation yields information about unfavourable deformations and stressesduring the drying process. To investigate factors that inuence dryingdeformations, a parameter study was performed in which the inuence ofdifferent constitutive models and different material parameters was studied.In addition, the inuence of the spiral grain angle was examined.

IntroductionThe characteristics of wood, including its material orientation, make the behaviourof timber subjected to mechanical loading and to moisture variation highlycomplex. It is difcult without the use of numerical simulations to predict thedeformations. Finite element simulations can be performed to investigate howinternal structure and material properties affect the shape stability in sawn timber.To perform accurate simulation, it is essential to have a sufciently detailed de-scription of the wood properties and their variation with position in the tree stem,and also how the bres are oriented in the stem. The theory described in (Or-marsson et al. 1998) is implemented here in the nite element program ABAQUS(Hibbitt et al. 1995) and is applied in the simulation of a board during drying.

The board studied is 3 m long and 100 50 mm in cross section. The geom-etry and pith location are shown in Fig. 1. The pith is assumed to be parallel to

Wood Science and Technology 33 (1999) 407423 Springer-Verlag 1999

Received 22 April 1997

S. Ormarsson, O. DahlblomDivision of Structural Mechanics,Lund Institute of Technology, Lund University,Box 118, SE-221 00 Lund, Sweden

H. Petersson (&)Department of Structural Mechanics,Chalmers University of Technology,SE-412 96 Goteborg, Sweden

The research presented in this paper is a part of the nationalresearch programme in Sweden for wood physics and drying. It hasbeen supported nancially by the Research Foundation of SwedishSawmills and the Swedish Council for Forestry and AgriculturalResearch.

407

the longitudinal direction of the board and the board to not be subjected to anyexternal support constraints. Displacements are only prescribed to avoid rigidbody motions. The element mesh of the board and its boundary conditions areshown on the right side of Fig. 1. The arrows denote the nodal displacements,which have been set to zero in the calculation. The board dries from a 27%moisture content to a mean moisture content of 10.75%. The following is a briefaccount of the assumptions made. For further details see Ormarsson (1995).

Material dataThe material parameters used and assumed to be representative for spruce (Piceaabies), are listed in Table 1. The parameters are dened in Ormarsson et al.(1998). Due to the limited experimental evidence available, some of the param-eters were estimated on the basis of data from other species. It has been exper-imentally observed that the longitudinal elastic modulus E10 and the longitudinalmoisture elongation coefcient a1 vary signicantly from pith to bark, seeWormuth (1993) and Dahlblom et al. (1996). The variation in a cross section isillustrated in Fig. 2. Based on these results, E10 and a1 are assumed to varylinearly from pith to bark. Due to the lack of experimental data, the remainingelastic strain parameters are assumed to be independent of distance from the pith.These parameters are mainly based on experimental data from Siimes (1967),Santaoja et al. (1991) and Hisada (1986). The parameters ar and at are set to

Fig. 1. Geometry, element mesh and prescribed displacements of the board studied

Table 1. Material parameters used in the simulation (distance from the pith r in m)

Elastic strain El0 9700 + 100000r MPa Er0 400 MPa Et0 220 MPaparameters Elw 21000 MPa Erw 2200 MPa Etw 1300 MPa

Glr0 400 MPa Glt0 250 MPa Grt0 25 MPaGlrw 1163 MPa Gltw 122 MPa Grtw 72 MPamlr 0.35 mlt 0.6 mrt 0.55

Moisture-inducedstrain parameters

a1 0:0071 0:038r ar 0:19 at 0:35

Mechano-sorption ml0 0.1 10)3 MPa)1 mr0 0.15 MPa)1 mr0 0.2 MPa)1strain parameters mlr0 0.008 MPa)1 mlt0 0.008 MPa)1 mrt0 0.8 MPa)1

llr 0:0 llt 0:0 lrt 1:0

408

constant values due to the lack of experimental data on radial distribution. Themechano-sorption parameters ml0, mr0 etc. are assumed to be constant in thetimber log and to be independent of the moisture content. The values are basedon results presented by Santaoja et al. (1991) and Martensson (1992).

In the simulation, the material temperature T is assumed to be the same as thereference temperature T0 20 C, meaning that the temperature has no inuence

Fig. 2. Experimentally obtained variations in the longitudinal elastic modulus El and thelongitudinal moisture shrinkage/swelling coefcient a1 over the cross section of a board

409

on the material parameters. This results in a constant bre saturation pointof wf 0.3. For a more detailed description of the material parameters seeOrmarsson et al. (1998).

On the basis of experimental evidence, see e.g. Mishiro and Booker (1988),Harris (1989) and Thornqvist (1990), the radial distribution of the spiral grainangle is assumed to be a linear function of the distance from the pith. The spiralgrain angle is said to be positive if the bre deviation from the pith directionrepresents a positive rotation around the radial axis. This means that positivedeviation is to the left of the upper extremity of the longitudinal axis of thegrowing tree, as viewed by an observer from the ground. According to a com-monly used denition, see e.g. Harris (1989), this is called left-hand spiral grain.The expression used for spiral grain is h 4 40r, where r is the distance fromthe pith.

The conical angle is assumed to be constant for the board studied. The sign ofthe conical angle follows the sign of the bre rotation around the tangentialcoordinate axis. The direction of the longitudinal axis is dened as being positivefrom the bottom to the top of the tree, which means that the conical angle willnormally be negative since the diameter normally decreases with the distancefrom the bottom. The expression used for the conical angle is u 0:5.Moisture transportIn the present simulation, the board is assumed to be insulated on the ends andmoisture transport to be two-dimensional in the cross-section plane of theboards. The study focuses on the modelling of deformation and stress develop-ment. Moisture transport is assumed to be governed by the linear diffusionrelation

Dwo2woy2 o

2w

oz2

ow

ot 0 1

where Dw is diffusivity and w is moisture content. In the simulation the param-eters Dw, w0 and w1 are chosen as Dw 7 10)10 m2/s, w0 0.27 and w1 0.089.Parameter w0 represents the initial uniform moisture content in the board and w1the moisture content at the surface in contact with the environment. These valueswere selected in order to achieve approximate agreement with the experimentalvalues for initial moisture content, for moisture content in the surface layer, andfor moisture content in the centre after 6 days of drying, as obtained by (Rosen-kilde et al. 1996). The description of moisture distribution reects qualitativelythe conditions in a drying board. It should be noted, however, that in a detailedsimulation the nonlinearity and direction dependence of the moisture transportin wood needs to be considered, see Claesson and Arfvidsson (1992), Perre et al.(1993), Ranta-Maunus (1994).

In the nite element analysis performed, the moisture content history repre-sented the input data for the deformation simulations. The moisture content isprescribed in all the element nodes in accordance with Eq. (1). In Fig. 3 thedevelopment of the average moisture content during the drying process in theboards that were studied is shown.

The average moisture content is 27% at the beginning of the drying process,decreasing to 10.75% at the end of drying. Since the surfaces of the boards dryvery quickly and the moisture content values for the integration points of theelements are computed by linear interpolation based on the element node values,

410

the assumed average moisture content decreases very rapidly during the rst fewminutes of the drying process.

Computational resultsThe simulation yields information regarding how stress and deformation in theboard develop during the drying process. Deformation, stress and moisture contentof the board are shown in Fig. 4. Both undeformed and deformed element mesh areshown for the board there at four different times during the drying process. It shouldbe noted that the deformations displayed in this gure are shown at a scale factorof 5. A contour plot in colour on the deformed element mesh, showing stresses inthe tangential direction, is presented. On the right hand side of the gure, a two-dimensional contour plot of the moisture content of the board is shown in colour.Red denotes high moisture content and blue low moisture content.

The most signicant deformations are the twist and cup deformations. Sinceshrinkage is greater in the tangential than in the radial direction, considerable cupdeformation develops. This shrinkage difference, together with the spiral grainangle, results in considerable twist deformation developing as well. Other pa-rameters, such as the annual ring orientation, inuence the drying deformationscf. Perstorper et al. (1995). This has been investigated in Ormarsson et al. (1998).

Figure 5 shows the development of twist, cup and bow deformations during thedrying process as a function of drying time. The results indicate each of thedeformation types to increase during drying, the rate of increase being greatest inthe early stages of the process. Since the average moisture content decreases veryquickly during the rst few minutes of drying, see Fig. 3, drying deformationslikewise develop very rapidly. Crook deformation is not shown in the gure sinceit is very small, due to the vertical symmetry section of the board. Equations (2)(5) have been used to calculate the different deformation types examined in thesimulation. The nodal displacements used to calculate the deformation are shownin Fig. 6.

Twist: utwist arctan at1 at2=Ly arctan at3 at4=Ly 2Cup: acup ac1 ac3=2 ac2 3Bow: abow ab1 ab3=2 ab2 4Crook: acrook acr1 acr3=2 acr2 5

Fig. 3. Development of theweighted average of themoisture content during thedrying process in the boardsstudied

411

The sign of the cup, bow and crook deformation is positive if the mean value ofthe end displacements is greater than the midpoint displacement (see Fig. 6).

The colour plot of the tangential stress component shown in Fig. 4 indicateslarge tensile stresses to occur at the surface at the beginning of the drying process,large compression stresses developing at the end of the drying process. Thus therisk of crack initiation at the surface is greatest at the beginning of the dryingprocess. To illustrate in greater detail how stresses develop, stresses in the tan-gential direction at two different positions are presented as a function of dryingtime in Fig. 7. The large tensile stress at the surface at the beginning of the dryingprocess is due to the large moisture gradient (see Fig. 4). After about a day, thesurface stress has changed to one of compression. The latter increases as thedrying process progresses. Due to the large shrinkage at the surface, the stress rt2inside the board develops to a state of compression early in the process. Afterabout two days, when the board starts to dry in the centre, the stress rt2 changesto tensile stress, which increases during the drying process.

Fig. 4. Development of deformation, tangential stresses and moisture content in the boardduring drying

412

Influence of the constitutive modelThe simulation just described was based on the material model for wood de-scribed in Ormarsson et al. (1998). In the following, the inuence which theassumed modications of the material model exert on deformation and stress

Fig. 5. Development of thetwist, cup and bowdeformation during thedrying process

413

development during drying is examined. The total strain rate _e is assumed to bethe sum of the elastic strain rate _ee, the moisture strain rate _ew and the mechano-sorptive strain rate _ewr, i.e. _e _ee _ew _ewr. Four alternative assumptions formodelling the strain rate components are presented in Table 2.

The rst material model is the reference model employed in the precedingsimulation example. In terms of this model, the elastic strains present are pro-portional to the stresses. Accordingly, the elastic strain rate is dependent onchanges in the material stiffness. In the second model, the elastic strain rate isassumed to be proportional to the stress rate. In models 3 and 4 the elastic strainrate is described just as in models 1 and 2, but the mechano-sorption effect is notconsidered. In Figs. 89, the drying deformations and the tangential stress for theboard are plotted for each of these four models as a function of drying time.

The results indicate that simulation, with and without mechano-sorption yieldquite different results in terms of drying deformations. The choice of formulationfor the elastic strain rate has very little inuence on the drying deformations. Theresults shown in Fig. 9 indicate that mechano-sorptive strain has a strong in-uence on tangential stress development. The mechano-sorption effect reducesthe stresses during the drying process, these thus being lower than those devel-oping in the elastic formulation. The choice of formulation for the elastic strainrate has little inuence on the results for stress when the mechano-sorption effectis considered, but is found to have a strong inuence on stress when mechano-sorptive strain is not taken into consideration.

Influence of material parametersTo investigate the inuence of different material parameters on simulated dryingdeformations, calculations using different values for the material parameters were

Fig. 6. The nodal displacements used to calculate the different deformation types

414

performed and were compared with the reference calculation board. In eachcalculation, one parameter was reduced to 50% of its reference value, the otherparameters being the same as in the reference case. For each calculation, twist,cup and bow deformations were studied. The results are presented in three guresthat show the inuence on the twist, cup and bow deformations. Each gurepresents the quantity in question as a function of drying time, both for thereference calculation and for the results of calculations in which the value of theone parameter was reduced to 50% of its reference value.

Fig. 7. Development oftangential stress during thedrying process at twodifferent positions

Table 2. Four alternative constitutive models of wood during drying

Material assumption Elastic Moisture Mechano-sorptivestrain rate strain rate strain rate

Model 1 _ee C _r _Cr _ew a _w _ewr mrj _wjModel 2 _ee C _r _ew a _w _ewr mrj _wjModel 3 _ee C _r _Cr _ew a _w ewr 0Model 4 _ee C _r _ew a _w ewr 0

415

The inuence of the reduction in the elastic moduli and shear moduli to 50% ofthe reference value is shown in Figs. 1011. The results indicate that the variousstiffness parameters differ considerably in their inuence on the drying defor-mations. The most signicant inuence on the twist deformation is obtained for

Fig. 8. Development of dryingdeformations of a board duringthe drying process according tofour different constitutiveassumptions

416

the elastic moduli Er and Et and for the shear moduli Glr and Glt. The otherstiffness parameters, El and Grt, affect the results to only a very slight degree. Thestiffness parameters having the strongest inuence on cup deformation are Er, Etand Grt. One should note that the elastic moduli Er and Et are very similar in theirinuence on twist and cup deformation, both twist and cup deformations in-creasing with a decrease in Et and decreasing with a decrease in Er. The results forbow deformations show each of the stiffness parameters to have a signicantinuence on bow deformation. The parameters Er, Et and Grt have the strongesteffect on bow results. The inuence of the parameters Er, Et on bow deformationis the reverse of their inuence on twist and cup deformations.

The inuence of a reduction in the moisture shrinkage parameters to 50% ofthe reference value is shown in Fig. 12. The results indicate that the moistureshrinkage parameters at and ar strongly inuence twist, cup and bow deforma-tion. The longitudinal shrinkage parameter a1 has only a slight inuence on twistand cup deformations, but a very strong inuence on bow deformation. A largevalue of the ratio at=ar yields large cup deformation.

The inuence of a reduction in the mechano-sorption parameters to 50% of thereference value is shown in Fig. 13. The computations show the drying defor-mations twist, cup, and bow to be strongly inuenced by the mechano-sorptionparameters. The strongest inuence is that obtained for the tangential componentmt. The parameter mr also has a signicant inuence on drying deformations. Thenormal component in the longitudinal direction, ml, has very little inuence ondrying deformation.

Influence of spiral grain angleIn this section, the inuence of the spiral grain angle on drying deformations isconsidered. As noted in Ormarsson et al. (1998), the spiral grain angle is denedas the angle between the pith and the bre direction in the 1-t plane. This angleoften shows a radial variation within the timber log, see e.g. Harris (1989). In thereference board, use was made of the spiral grain function h 4 40r, where r isthe radial distance from the pith in m. To investigate the inuence of the spiralgrain angle, four calculations using different spiral grain functions were per-formed.

Figure 14 shows the twist, cup, and bow deformations for these calculations asa function of drying time. The results indicate that differences in the spiral grain

Fig. 9. Development of thetangential stress component rt1during the drying processaccording to four differentconstitutive assumptions

417

have a very large inuence on twist deformation. Twist deformation increaseswith increasing spiral grain angle. In addition, it can be observed that no twistdeformations develop when the spiral grain angle is zero. Thus, the spiral grain

Fig. 10. Inuence of theelastic moduli on twist, cupand bow deformation

418

angle is one of the most important parameters in simulations of twist deforma-tion. It also inuences bow deformation, but has almost no inuence on cupdeformation.

Fig. 11. Inuence of theshear moduli on twist, cupand bow deformation

419

Conclusions and discussionIn the present paper, a three-dimensional theory for shape stability of sawntimber has been employed in the simulation of a drying board. A parameter study

Fig. 12. Inuence of themoisture shrinkageparameters on twist, cup andbow deformation

420

was performed to investigate the inuence of different parameters on dryingdeformations. The simulations provide valuable information concerning theimportance of an accurate description of the material properties and an accurate

Fig. 13. Inuence of themechano-sorption parameterson twist, cup and bowdeformation

421

description of the material orientation. The drying deformations depend to a highdegree on the values for moisture shrinkage and of mechano-sorption parame-ters. It can be concluded, therefore, that having a relevant theoretical description

Fig. 14. Inuence of the spiralgrain angle on twist, cup andbow deformation

422

of moisture shrinkage and mechano-sorptive effects, as well as accurate values forthe corresponding material parameters, is highly important. Variation in thematerial parameters with respect to the distance from the pith also have aninuence on drying deformations. The spiral grain angle is an important para-meter that has a considerable inuence on such drying deformations as twist andbow. Radial variation in the spiral grain angle likewise has an inuence on thedevelopment of deformation.

ReferencesClaesson J, Arfvidsson J (1992) A new method using Kirchhoff potentials to calculatemoisture ow in wood, Contribution to international conference on wood drying, Un-derstanding the wood drying process: A Synthesis of Theory and Practice, ViennaDahlblom O, Ormarsson S, Petersson H (1996) Simulation of wood deformation processesin drying and other types of environmental loading. Annales des Sciences Forestieres 53(4):857866Harris JM (1989) Spiral Grain and Wave Phenomena in Wood Formation, Springer-Verlag,Berlin HeidelbergHibbitt; Karlsson & Sorenson, Inc. (1995) ABAQUS, Version 5.5. Pawtucket, RI.Hisada T (1986) Creep and set behaviour of wood related to kiln drying. Bull. For. & For.Prod. Res. Inst. No. 335, pp. 31130Martensson A (1992) Mechanical behaviour of wood exposed to humidity variations, Re-port TVBK-1006, Lund Institute of Technology, Department of Structural Engineering,LundMishiro A, Booker R (1988) Warping of new crop radiata pine 100 50 mm (2 by 4)boards, Bull Tokyo Univ. For., No. 80, pp. 3768Ormarsson S (1995) A nite element study of the shape stability of sawn timber subjectedto moisture variations, Report TVSM-3017, Lund Institute of Technology, Division ofStructural Mechanics, LundOrmarsson S, Dahlblom O, Petersson H (1998) Numerical study of shape stability ofsawn timber subjected to moisture variations, Part 1: Theory. Wood Sci. Technol. 32: 325334Ormarsson S, Dahlblom O, Petersson H (1999) Numerical study of shape stability of sawntimber subjected to moisture variations. Part 3: Inuence of Annual Ring Variations. WoodSci. Technol. 33 (in press)Perre P, Moser M, Martin M (1993) Advances in transport phenomena during convectivedrying with superheated steam and moist air. Int. J. of Heat and Mass Transfer 36(11):27252746Perstorper M, Pellicane PJ, Kliger IR, Johansson G (1995) Quality of timber products fromNorway spruce, Part 2: Inuence of spatial position and growth characteristics on warp.Wood Sci. Technol. 29: 339352Ranta-Maunus A (1994) Computation of moisture transport and drying stresses by a 2-DFE-programme. In: 4th IUFRO international wood drying conference: Improving wooddrying technology, RotoruaRosenkilde A, Arfvidsson J (1997) Measurement and evaluation of moisture transportcoefcients during drying of wood. Accepted for publication in HolzforschungSantaoja K, Leino T, Ranta-Maunus A, Hanhijarvi A (1991) Mechano-sorptive structuralanalysis of wood by the ABAQUS nite element program. Technical Research Centre ofFinland, Research notes 1276, EspooSiimes FE (1967) The effect of specic gravity, moisture content, temperature and heatingtime on the tension and compression strength and elasticity properties perpendicular to thegrain of Finnish pine, spruce and birch wood and the signicance of these factors on thechecking of timber at kiln drying, VTT Publication 84, HelsinkiThornqvist T (1990) Juvenile Wood in Coniferous Trees, Report No. 10, Swedish Universityof Agricultural Sciences, Department of Forest-Industry-Market Studies, UppsalaWormuth E-W (1993) Study of the relation between atwise and edgewise modulus ofelasticity of sawn timber for the purpose of improving mechanical stress methods. Diplomawork, University of Hamburg, Department of Wood Technology, Hamburg

423