Introduction
Properties | Model | Literature | Notes | Limitations | ||
---|---|---|---|---|---|---|
\(T/w\) | \(l/r/t\) | \(lr/lt/rt\) | ||||
Elastic | Hook’s law | Keunecke et al. (2007) | Experimentally obtained | \(w\) | \(l/r/t\) | \(lr/lt/rt\) |
Ormarsson et al. (1998) | Expressions not all experimentally verified | \(T/w\) | \(l/r/t\) | \(lr/lt/rt\) | ||
Hygro-expansion | Moisture induced strain | Florisson et al. (2021) | Experimentally obtained, with values for each RH-cycle | – | \(t\) | – |
Bengtsson (2001) | Experimentally obtained, including variation between pith and bark | \(l/r/t\) | – | |||
Creep | Hereditary model (Dahlblom 1987) | Ormarsson (1999) | Only \(t\) experimentally verified and covers a time period of only \(70{\text{h}}\) | – | \(l/r/t\) | \(lr/lt/rt\) |
Mechano-sorption | Mechano-sorption limit model (Salin 1992) | Ormarsson et al. (1998) | Expressions not all experimentally verified | \(T\) | \(l/r/t\) | \(lr/lt/rt\) |
Svensson and Toratti (1997) | Experimentally verified | – | \(t\) | – |
Materials and methods
Theory
Moisture flow model
Stress model
Numerical model
Experimental program
Application 1
Application 2
Numerical simulations
Application 1
Material parameters
Elastic \(\left( {{\text{MPa}}/ - } \right)^{***}\) | Hygro-expansion \(\left( - \right)\) | Creep \(\left( - \right)^{{**}\left( *\right)}\) | Mechano-sorption \(({\text{MPa}}^{ - 1} /{\text{MPa}})^{***}\) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(E_{l}\) | \(\times\) | \(E_{l0}\) | \(\times\) | \(v_{lr}\) | 0.35 | \(\alpha_{l}\) | 0.0035 | \(\phi_{{\sigma_{l} }}^{1}\) | \(\times\) | \(\phi_{{\sigma_{l} }}^{2}\) | \(\times\) | \(m_{l}\) | \(\times\) | \(m_{l0}\) | \(\times\) |
\(E_{r}\) | 1168 | \(E_{r0}\) | 1000 | \(v_{lt}\) | 0.6 | \(\alpha_{r}\) | 0.17 | \(\phi_{{\sigma_{t} }}^{1}\) | 0.106* | \(\phi_{{\sigma_{t} }}^{2}\) | 0.162* | \(m_{t}\) | 0.28 | \(m_{t0}\) | 0.2 |
\(E_{t}\) | 378 | \(E_{t0}\) | 500 | \(v_{rt}\) | 0.55 | \(\alpha_{t}\) | 0.35–0.39 | \(\phi_{{\sigma_{lt} }}^{1}\) | 0.07* | \(\phi_{{\sigma_{lt} }}^{2}\) | 0.09* | \(m_{lt}\) | 0.011* | \(m_{lt0}\) | 0.008* |
\(G_{lr}\) | 417 | \(G_{lr0}\) | 460 | \(n_{l}\) | \(\times\) | ||||||||||
\(G_{lt}\) | 401 | \(G_{lt0}\) | 425 | \(n_{t}\) | 69 | ||||||||||
\(G_{rt}\) | 28 | \(G_{rt0}\) | 47 | \(n_{lt}\) | 500* |
Application 2
Material parameters
Elastic \(\left( {{\text{MPa}}/ - } \right)\)** | Hygro-expansion \(\left( - \right)\) | Creep \( \left( - \right)\)* | Mechano-sorption \(({\text{MPa}}^{ - 1} /{\text{MPa}})\)*** | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(E_{l}\) | 6736 | \(E_{l0}\) | 286 | \(v_{lr}\) | 0.35 | \(\alpha_{l}\) | 0.0071–0.038r | \(\phi_{{\sigma_{l} }}^{ }\) | 0.235 | \(m_{l}\) | 0.0004 | \(m_{l0}\) | 0.0004 | \(n_{l}\) | 12,000 |
\(E_{r}\) | 1525 | \(E_{r0}\) | 1000 | \(v_{lt}\) | 0.6 | \(\alpha_{r}\) | 0.12 | \(\phi_{{\sigma_{r} }}^{ }\) | 0.25 | \(m_{r}\) | 0.15 | \(m_{r0}\) | 0.15 | \(n_{r}\) | 52 |
\(E_{t}\) | 605 | \(E_{t0}\) | 500 | \(v_{rt}\) | 0.55 | \(\alpha_{t}\) | 0.35 | \(\phi_{{\sigma_{t} }}^{ }\) | 0.30 | \(m_{t}\) | 0.2 | \(m_{t0}\) | 0.2 | \(n_{t}\) | 69 |
\(G_{lr}\) | 610 | \(G_{lr0}\) | 460 | \(\phi_{lr}^{ }\) | 0.15 | \(m_{lr}\) | 0.008 | \(m_{lr0}\) | 0.008 | \(n_{lr}\) | 500 | ||||
\(G_{lt}\) | 575 | \(G_{lt0}\) | 425 | \(\phi_{{\sigma_{lt} }}^{ }\) | 0.15 | \(m_{lt}\) | 0.008 | \(m_{lt0}\) | 0.008 | \(n_{lt}\) | 500 | ||||
\(G_{rt}\) | 51 | \(G_{rt0}\) | 47 | \(\phi_{{\sigma_{rt} }}^{ }\) | 0.50 | \(m_{rt}\) | 0.8 | \(m_{rt0}\) | 0.8 | \(n_{rt}\) | 27.6 |
Results and discussion
Application 1
Deflection
Elastic*** | Creep*** | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
\(E_{l,\theta = 0} \left( {{\text{GPa}}} \right)\) | \(E_{l0,\theta = 0} \left( {{\text{GPa}}} \right)\) | \(E_{l} \left( {{\text{GPa}}} \right)\) | \(E_{l0} \left( {{\text{GPa}}} \right)\) | \({\text{RD}} ^{**} { }\left( \% \right)\) | \(\phi_{l,\theta = 0}^{k} \left( - \right)\) | \(\phi_{{\sigma_{l} }}^{k} \left( - \right)\) | \({\text{RD }}\left( \% \right)\) | |||
1*** | 2 | 1 | 2 | |||||||
EE | 13.6 | 17.9 | 15.8 | 22.6 | 16.2 | 0.103 | 0.126 | 0.105 | 0.135 | 1.94/7.14 |
NN | 15.2 | 21.4 | 18.2 | 27.6 | 19.7 | 0.148 | 0.103 | 0.158 | 0.110 | 6.76/6.80 |
SS | 26.6 | 45.0 | 37.6 | 68.0 | 41.4 | 0.032 | 0.108 | 0.010 | 0.095 | 68.8/12.0 |
Mechano-sorption**** | ||||||||
---|---|---|---|---|---|---|---|---|
\(m_{l,\theta = 0} \left( {{\text{MPa}}^{ - 1} } \right)\) | \(m_{l0,\theta = 0} \left( {{\text{MPa}}^{ - 1} } \right)\) | \(m_{l} \left( {{\text{MPa}}^{ - 1} } \right)\) | \(m_{l0} \left( {{\text{MPa}}^{ - 1} } \right)\) | \({\text{RD }}\left( \% \right)\) | \(n_{l,\theta = 0} { }\left( {{\text{MPa}}} \right)\) | \(n_{l} { }\left( {{\text{MPa}}} \right)\) | \({\text{RD }}\left( \% \right)\) | |
EE | 32.2* | 23.0* | 22.4* | 16.0* | 30.4 | 11,000 | 12,000 | 8.3 |
NN | 28.0* | 20.0* | 22.4* | 16.0* | 20.0 | 14,000 | 18,000 | 22.2 |
SS | 25.2* | 18.0* | 14.0* | 10.0* | 44.4 | 34,000 | 70,000 | 51.4 |
The effects of spiral grain on material parameters
Stress distribution
Application 2
Moisture data
Deflection
Strain terms
\(\varepsilon_{l}\) \({\text{(\%) }}\) | \(\varepsilon_{e} ^{*}\) \({\text{(\%) }}\) | \(\varepsilon_{h}\) \({\text{(\%) }}\) | \(\varepsilon_{c}\) \({\text{(\%) }}\) | \(\varepsilon_{ms}\) \({\text{(\%) }}\) | |
---|---|---|---|---|---|
\(A\,2\) | 0.257 | 0.176 | − 0.025 | 0.0387 | 0.067 |
\(c\,\left( \% \right)\) | 100 | 68.5 | 9.7 | 15.1 | 26.1 |
\(A\,3\) | 0.225 | 0.152 | − 0.049 | 0.0391 | 0.083 |
\(c\, \left( \% \right)\) | 100 | 67.6 | 21.8 | 17.4 | 36.8 |
Stress distribution
Conclusion
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The small clear-wood beams analysed as part of the second application show the strong effect that spiral grain and climate have on deflection, calibrated material parameters and normative stress states.
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The results from the third application show that the larger beam exposed to mechanical load and natural climate experience a slower change in moisture, smaller moisture gradients, more seasonal fluctuation in longitudinal stress, tangential stress and longitudinal-tangential shear stress, and higher drying stress in tension compared to the smaller beam.
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Analysis of the material orientation contributes to an understanding of why bending tests are not a particularly suitable method of determining how reliable various material parameters are here. The results obtained showed spiral grain in particular to have a strong effect on the stress distribution and deflection that is characteristic of wooden beams generally and is difficult to be eliminated from slender beams.
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Climate had the strongest effect on the stress states concerning the \(t\)-direction of the material orientation. For each beam configuration and each type of climatic condition that was involved, stress change between tension and compression was found to occur, this leading to an alternating of the stress states involved between the surface and the inner sections of the beams. This created an environment in which both surface and inner checks could develop and could result in splits or cracks, weakening the shear capacity of the beams.
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Calibration of the numerical model benefitted from the experimental methodology that was adopted, whereas it provided a framework in which not only moisture content but also elastic, creep, hygro-expansive and mechano-sorptive behaviour could be fitted to the experimental data. Although the experiment to validate the model was sufficient, it would have benefitted from constant recordings of changes in mass.
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A major challenge in regard to three-dimensional models is that of obtaining a sufficient amount of reliable experimental data for gaining an adequate understanding of the many material parameters needed to describe the moisture flow, and the hygro-mechanical and viscoelastic behaviour of Norway spruce, or of whatever type of wood is of interest.
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The model generally employed for describing creep trends towards an asymptote, making it difficult to predict creep outside of the time frame in which the model is calibrated.