# Is Wood a Material? Taking the Size Effect Seriously

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## Abstract

**:**

## 1. Introduction

## 2. The Classical Understanding of Elasticity

“We might consider a wire as composed of a great number of minute threads, extending through its length, and closely connected together; if we twisted such a wire, the external threads would be extended, and in order to preserve the equilibrium, the internal ones would be contracted…”

## 3. Problems with the Application of Elasticity Theory to Wood

## 4. Problems in Discerning Trends in the Strength of Wood

## 5. Size Effect Theories

“Predicting yield of structural members under complex loading conditions is a difficult task for the engineer. Complex loading often results in the structural members being stressed biaxially or even triaxially, whereas yield strength data are usually only available for tests conducted in uniaxial (tensile or compressive) or torsional stress states. The test specimens are also typically much smaller than the actual structural members. The problem, therefore, is to predict structural member yield using only these uniaxial and/or torsional yield test results. The problem of relating the test results in simple stress states to full-scale members under much more complicated stress conditions is often solved using what is known as the maximum distortion energy theory.”

“One must now consider how the test results on small samples of material relate to the full-scale structural members. In many cases the yielded volume in a failed full-scale member will be orders of magnitude larger than the yielded volume in the average test specimen. It therefore seems reasonable that each test specimen’s distortion energy capacity can be considered a point measurement of the distortion energy capacity for large members. An engineer would therefore be interested in using the distribution of the mean distortion energy capacity of the material $\left(\theta \right)$ rather than the distribution of the test sample distortion energy capacities as a design guideline.”

## 6. Evidence of Size Effects in Wood

_{0}are experimentally determined constants.

_{N}, to the size of the specimen, D, scaled in terms of a parameter D

_{0}.

## 7. Modelling Wood

## 8. Conclusions and Matters for Further Study

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Electron micrographs of tracheids in the xylem of Radiata Pine (Pinus radiata). (

**a**) Transverse section. (

**b**) Longitudinal section. From [11].

**Figure 4.**(

**a**) Colour plate reproduction of a painting of the granular structure of granite observed in 1883 using polarized light microscopy. No scale bar or magnification was given. From [51]. (

**b**) Optical micrograph of the microstructure of acid-etched armour steel published by Bayles in 1883 [52]. This photograph was originally taken by Sorby and presented at a lecture he gave to the Sheffield Literary and Philosophical Society [53]. No magnification or scale bar was included.

**Figure 5.**Schematic student textbook plots showing load-deformation paths for (

**a**) the elastic (recoverable) and (

**b**) the inelastic (irrecoverable) deformation of wood. From [79].

**Figure 6.**Experimental compressive stress–strain plots of (

**a**) polyurethane foam tested at one strain rate and (

**b**) polyethylene foam tested at several (low) strain rates. From [80].

**Figure 7.**Schematic diagrams of rectangular wooden compression specimens cut at various angles to the grain of the wood. (

**a**) Parallel to the grain. (

**b**) Perpendicular to the grain radially. (

**c**) Perpendicular to the grain tangentially. (

**d**) at 45° to the grain. The ‘+’ signs indicate the position of strain gauges. From [84]. R means radial, L means length, T means tangential.

**Figure 8.**Experimental compressive stress–strain plots of a number of different woods measured using specimens (

**a**) cut parallel to the axis of the trunk of the tree and (

**b**) cut at a tangent to the trunk. From [4].

**Figure 9.**Strengths of Korean Pine as a function of grain angle in (

**a**) compression and (

**b**) tension. From [85].

**Figure 11.**Schematic diagram showing the asymmetry of the tensile and compressive response of defect-free (or ‘clear’) wood. From [86].

**Figure 12.**Photographs of four $25\times 25\times 206\text{}\mathrm{mm}$ Douglas Fir beam bending specimens. (

**a**) Clear wood; (

**b**) knot mid-height of the face; (

**c**) knot subjected to compression; (

**d**) knot subjected to tension. From [87].

**Figure 14.**(

**a**) Tensile and (

**b**) compression specimens for investigating the size effect for wood. From [88].

**Figure 15.**Schematic diagram showing how the strength of a wooden beam might vary along its length. From [100].

**Figure 16.**Schematic diagram showing how a wooden beam containing knots can fail at a different place from where it is loaded. From [102]. (

**a**) Failure occurs away from peak load due to a weak knot close by. (

**b**) Failure occurs at peak load as the peak load coincides with a knot, even though that knot is stronger than that shown in (

**a**).

**Figure 17.**Schematic comparisons of load-deflection curves in bending for defect-free (clear) wood, ‘strong’ and ‘weak’ lumber. From [86].

**Figure 18.**Plot of the tensile strength of a wooden beam against the number of weak sections that it contains. From [101].

**Figure 19.**Variability in the modulus of rupture (MOR) of unseasoned clear-wood Sitka Spruce (Picea sitchensis). From [112]. White circles are data points.

**Figure 20.**Data obtained from 1348 tests for the modulus of rupture (MOR) and the modulus of elasticity (MOE). The mean regression line and the 5th percentile exclusion line are plotted. The (over-precise) equation that Dinwoodie gave for the mean regression line was MOR = 0.002065(MOE)

^{1.0573}with a correlation coefficient of 0.702. From [89]. ‘+’ signs are data points.

**Figure 21.**Plot of the longitudinal modulus of elasticity against specific gravity for more than 200 species of tree tested in green and dry states. From [92].

**Figure 22.**Effect of knot area ratio on the strength of Douglas Fir boards of two different widths. From [89]. The black circles and ‘x’ are data points.

**Figure 23.**Plot of the maximum compression strength against specific gravity for wood taken from more than 200 species of tree, both green and air-dried. From [89].

**Figure 24.**Variation in ultimate bending strength with duration of stress (time between initial application of load and failure) for unseasoned clear-wood Sitka Spruce (Picea sitchensis). From [112].

**Figure 26.**Plot showing the variation of bending size parameter S

_{Rb}with modulus of rupture. From [154].

**Figure 27.**Plots of the measured tensile strengths for two-inch (5 cm) dimension lumber of four different nominal widths and seven different grades cut from two types of tree: (

**a**) Douglas Fir and (

**b**) Hem-Fir. From [155].

**Figure 29.**Plot of bending strength, f

_{m}, against specimen depth, h, of for 349 specimens. Note the high degree of scatter relative to variation in mean value between values of h. From [130]. The different symbols distinguish between specimens of different sizes.

**Figure 30.**Cylindrically symmetric compression specimens used to investigate size effects in compression testing of Norway Spruce. Dimensions given in mm. From [125].

**Figure 31.**Plot of the effect of specimen size on compression strength. Comparison is made for three different theories. WLT means ‘weakest link theory’; SEL means ‘size effect law’, MFSL means ‘multifractal scaling law’. From [125].

**Figure 33.**Plots of data obtained for modulus of rupture (MOR) for specimens of structural size against MOR for small specimens of mixed hardwoods. The lines in each plot represent various functions of structural size. (

**a**) Exponential; (

**b**) logarithmic; (

**c**) power; (

**d**) polynomial. From [160].

**Figure 34.**(

**a**) Assessment of power law assumption for mean strength of European White Oak (Quercus robur and Quercus petraea) boards simulated by four different models. (

**b**) Simulated length effect of tensile strengths obtained using four fitted models showing the variation for all grades studied. From [138].

**Figure 35.**Plot of the equation discussed above. The axis labelled ‘c.v.’ plots the values of the coefficients of variation of strength within each board. This is related to ${k}_{1}$. From [86].

**Figure 37.**Effect of grain angle on the tensile, bending and compression strengths of timber. From [89].

**Figure 39.**(

**a**) Photograph of failed wood specimen that had been subjected to a bending load. (

**b**) The load–displacement graph for the specimen shown in (

**a**). The graph shows that no plastic deformation occurred before failure, i.e., the failure was brittle. From [100].

**Figure 40.**(

**a**) Photograph of failed wood specimen that had been subjected to a bending load. (

**b**) The load–displacement graph for the specimen shown in (

**a**). The graph shows that plastic deformation occurred before failure, i.e., the failure was ductile. From [100].

**Figure 41.**Total of 27 different specimen sizes for investigating length, depth, thickness size effects for spruce, pine, and fir. From [104].

**Figure 42.**Plot showing the effect of length on the strength of wooden beams. From [86].

**Figure 43.**Fifth and fiftieth percentile strength data obtained from three-point loading of Hem-Fir wooden beams of the same span-to-depth ratio, but with different depths. These data were obtained in 1976. From [86].

**Figure 44.**Plots of log(strength) vs. log(volume) and log(aspect ratio) for defect-free Douglas Fir. From [103].

**Figure 46.**Photographs of groups of five test specimens of Poplar and Pine wood that had been subjected to an impact bending strength test. The labels give the span lengths of the original specimens in each group. From [166].

**Figure 47.**Plots of the impact bending strengths of Poplar and Pine specimens as a function of the span length of the original specimens. From [166].

**Figure 48.**Schematic plot showing how the size effect parameter ‘g’ is calculated. From [86].

**Figure 49.**Proportional limit stresses for five different sizes of specimens of Fagus sylvatica cut in different orientations with respect to the trunk of the tree. From [134].

**Figure 50.**Plot of the ratio of elastic moduli measured two different ways for Maritime Pine specimens against specimen length for three different cross-sections. The two methods used were (i) optical (digital image correlation, DIC) and (ii) mechanical (a displacement transducer). From [167].

**Figure 52.**Plots of (

**a**) Modulus of Elasticity (MOE) and (

**b**) Modulus of Rupture (MOR) for logs of various grades (labelled C40 through to C18) of Scots Pine taken from various parts of the tree trunks (B, butt; M, middle; T, top). Data obtained using four-point bend experiments. From [169].

**Figure 53.**Simple representative volume element for a porous hardwood. From [13].

**Figure 54.**Main types of knots according to their position on the piece. Listing from left to right and top to bottom: edge and face knots, inner through knot, outer through knot, arris knot, splay knot. From [175].

**Figure 55.**Schematic diagram showing the types of knots used in the parametric study: (

**a**) cylindrical, (

**b**) truncated conical, (

**c**) shallow conical, (

**d**) edge, and (

**e**) inclined. From [178].

**Table 1.**Early 19th century data (quoted to an absurd level of precision) on the failure loads of wrought iron, steel and timber pillars of various lengths and diameters. From [48].

Length. | Pillars with Both Ends Rounded. | Pillars with One End Flat, and the Other Rounded. | Pillars with Both Ends Flat. | ||||
---|---|---|---|---|---|---|---|

Diameter. | Breaking Weight. | Diameter. | Breaking Weight. | Diameter. | Breaking Weight. | ||

Wrought iron. | inches. | inch. | lbs. | inch. | lbs. | inch. | lbs. |

$90\frac{3}{4}$ | 1·017 | 1808 | 1·02 | 3355 | 1·02 | 5280 | |

$60\frac{1}{2}$ | 1·015 | 3938 | 1·03 | 8137 | 1·02 | 12,990 | |

$30\frac{1}{4}$ | 1·015 | 15,480 | 1·015 | 21,335 | 1·015 | 23,371 | |

$30\frac{1}{4}$ | 1·015 | 15,480 | 1·015 | 21,187 disc. | 1·015 | 25,387 disc. | |

$15\frac{1}{8}$ | 1·005 | 23,535 | 1·015 | 26,227 | 1·005 | 27,099 | |

Steel. | 29·95 | ·87 | 10,516 | ·87 | 20,135 | ·87 | 26,059 |

Timber. | $60\frac{1}{2}$ | Side of square. 1·75 | 3197 | Side of square. 1·75 | 6109 | Side of square. 1·75 | 9625 |

Specimen Label | Dimensions | ||
---|---|---|---|

Cross-Sectional Area/mm^{2} | |||

Width/mm | Length/mm | Height/mm | |

1 | 10 | 10 | 10 |

2 | 10 | 10 | 20 |

3 | 10 | 10 | 30 |

4 | 20 | 20 | 30 |

5 | 30 | 30 | 30 |

**Table 3.**Average values and standard deviations of the longitudinal modulus of elasticity for various lengths and cross-sections of specimens of Pinus pinaster. From [167].

Cross-Section/mm^{2} | Height/mm | ||
---|---|---|---|

30 | 60 | 120 | |

$20\text{}\times $ 20 | 15.7 ± 2.7 GPa | 15.9 ± 3.1 GPa | 14.5 ± 2.0 GPa |

$30\text{}\times $ 30 | 16.9 ± 2.9 GPa | 15.1 ± 3.0 GPa | 15.1 ± 2.9 GPa |

$40\text{}\times $ 40 | 18.1 ± 1.7 GPa | 16.1 ± 2.7 GPa | 15.8 ± 2.3 GPa |

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**MDPI and ACS Style**

Walley, S.M.; Rogers, S.J.
Is Wood a Material? Taking the Size Effect Seriously. *Materials* **2022**, *15*, 5403.
https://doi.org/10.3390/ma15155403

**AMA Style**

Walley SM, Rogers SJ.
Is Wood a Material? Taking the Size Effect Seriously. *Materials*. 2022; 15(15):5403.
https://doi.org/10.3390/ma15155403

**Chicago/Turabian Style**

Walley, Stephen M., and Samuel J. Rogers.
2022. "Is Wood a Material? Taking the Size Effect Seriously" *Materials* 15, no. 15: 5403.
https://doi.org/10.3390/ma15155403