# Experimental Test of Non-Destructive Methods to Assess the Anchorage of Trees

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Data Analysis

## 3. Results

#### 3.1. Indicators of Anchorage Strength

#### 3.1.1. Tree Size

#### 3.1.2. Soil–Root Rotational Stiffness ${k}_{r}$

#### 3.2. Testing the Estimations of Anchorage Strength

#### 3.2.1. Linear Extrapolation of Rotational Stiffness (Equation (1))

#### 3.2.2. Non-Linear Extrapolation (Equation (3))

#### 3.2.3. Comparison between Predictors

#### 3.3. Species Effect on Anchorage Strength

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

M | Bending moment |

${M}_{a}$ | Anchorage strength |

${k}_{r}$ | Root–soil rotational stiffness |

$\varphi $ | Root plate rotation |

d | Stem diameter in m |

h | Tree height in m |

## Appendix A

**Table A1.**ANOVA of a linear model $ln{\left({M}_{a}\right)}_{i}={\beta}_{0}+{\beta}_{1}\times {\mathrm{Species}}_{i}+{\beta}_{2}\times ln{\left(\mathrm{Size}\right)}_{i}+{\beta}_{3}\times {\mathrm{Species}}_{i}\times ln{\left(\mathrm{Size}\right)}_{i}+{\u03f5}_{i}$.

DF | Sum Sq | Mean Sq | F-Value | Pr (>F) | |
---|---|---|---|---|---|

Species | 7 | 22.38 | 3.20 | 28.93 | 0.0000 |

Size | 1 | 100.90 | 100.90 | 913.03 | 0.0000 |

Species:Size | 7 | 1.14 | 0.16 | 1.47 | 0.1771 |

Residuals | 267 | 29.51 | 0.11 |

**Table A2.**ANOVA of a linear model $ln{\left({M}_{a}\right)}_{i}={\beta}_{0}+{\beta}_{1}\times {\mathrm{Species}}_{i}+{\beta}_{2}\times ln{\left({k}_{r}\right)}_{i}+{\beta}_{3}\times {\mathrm{Species}}_{i}\times ln{\left({k}_{r}\right)}_{i}+{\u03f5}_{i}$.

DF | Sum Sq | Mean Sq | F-Value | Pr (>F) | |
---|---|---|---|---|---|

Species | 7 | 22.4 | 3.2 | 59.9 | 0.00 |

$ln\left({k}_{r}\right)$ | 1 | 116.4 | 116.4 | 2178.4 | 0.00 |

Species:$ln\left({k}_{r}\right)$ | 7 | 0.9 | 0.1 | 2.5 | 0.02 |

Residuals | 267 | 14.3 | 0.1 |

**Table A3.**ANOVA of model $ln\left(\frac{2.5{k}_{r}}{{M}_{a}}\right)={\beta}_{0}+{\beta}_{1}\mathrm{Species}+{\beta}_{2}ln\left(\mathrm{Size}\right)+{\beta}_{3}\mathrm{Species}\times log\left(\mathrm{Size}\right)+{\u03f5}_{i}$.

DF | Sum Sq | Mean Sq | F-Value | Pr (>F) | |
---|---|---|---|---|---|

fSpecies | 7 | 5.36 | 0.77 | 9.08 | 0.000 |

logSize | 1 | 6.29 | 6.29 | 74.57 | 0.000 |

fSpecies:logSize | 7 | 1.08 | 0.15 | 1.83 | 0.081 |

Residuals | 267 | 22.53 | 0.08 |

**Table A4.**ANOVA of a linear mixed-effects model ${M}_{{a}_{ij}}={\beta}_{0}+{\beta}_{1}\times {k}_{{r}_{ij}}+{\beta}_{2}\times {\mathrm{Site}}_{ij}+{\gamma}_{0i}+{\gamma}_{1i}\times {\mathrm{Site}}_{ij}+{\u03f5}_{ij}$ with ${\u03f5}_{ij}$ modeled as a power function of ${k}_{r}$.

DF | F-Value | Pr (>F) | |
---|---|---|---|

(Intercept) | 68 | 297.73 | 0.00 |

${k}_{r}$ | 68 | 157.16 | 0.00 |

Site | 7 | 0.34 | 0.58 |

**Table A5.**ANOVA of model $log\left(\frac{{M}_{\mathrm{est}}}{{M}_{a}}\right)=\mathrm{Species}\times log\left(\mathrm{Size}\right)$.

DF | Sum Sq | Mean Sq | F-Value | Pr (>F) | |
---|---|---|---|---|---|

Species | 7 | 5.36 | 0.77 | 9.08 | 0.000 |

log(Size) | 1 | 6.29 | 6.29 | 74.57 | 0.000 |

Species:log(Size) | 7 | 1.08 | 0.15 | 1.83 | 0.081 |

Residuals | 267 | 22.53 | 0.08 |

**Table A6.**ANOVA of model $log\left({\varphi}_{a}\right)=\mathrm{Species}\times log\left(\mathrm{Size}\right)$.

DF | F-Value | Pr (>F) | |
---|---|---|---|

(Intercept) | 1 | 2108.16 | 0.00 |

Species | 5 | 2.40 | 0.04 |

log(Size) | 1 | 46.95 | 0.00 |

Species:log(Size) | 5 | 3.19 | 0.01 |

DF | F-Value | Pr (>F) | |
---|---|---|---|

(Intercept) | 1 | 3482.60 | 0.00 |

Species | 8 | 21.81 | 0.00 |

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**Figure 1.**Species effect on anchoring strength (fitted lines: ${M}_{a}={y}_{0}(1-{e}^{-a\mathrm{Size}})$, ANOVA: Table A1).

**Figure 2.**Robust correlation of bending moments at a root plate inclination of 0.25° ${k}_{r}$ and at failure ${M}_{a}$. The blue line illustrates Equation (1).

**Figure 4.**Depending on the degree of non-linearity in the relationship between rotational stiffness and anchorage strength, the ratio between predicted (Equation (1)) and measured anchorage strength approaches 1 (red line) for larger trees. Fitted lines: $\frac{2.5{k}_{r}}{{M}_{a}}={y}_{0}{e}^{-a\mathrm{Size}}+b$, ANOVA: Table A3.

**Figure 5.**Estimated (Equation (3)) and measured anchorage strength (with four outliers beyond 1000 kNm removed). The blue line illustrates 1:1.

**Figure 8.**Species effect on anchoring strength. ANOVA: Table A7.

Species | n | d in cm | h in m |
---|---|---|---|

Betula pendula | 78 | 33 | 24 |

Picea abies | 65 | 32 | 25 |

Fraxinus excelsior | 48 | 34 | 25 |

Pinus sylvestris | 48 | 31 | 25 |

Fagus sylvatica | 21 | 33 | 26 |

Populus sp. | 18 | 44 | 30 |

Platanus acerifolia | 12 | 28 | 18 |

Acer saccharinum | 7 | 28 | 17 |

Acer pseudoplatanus | 5 | 22 | 23 |

Prunus sp. | 3 | 33 | 13 |

Quercus robur | 3 | 54 | 19 |

Tilia sp. | 3 | 32 | 14 |

Robinia pseudoaccacia | 2 | 48 | 19 |

Alnus glutinosa | 1 | 72 | 21 |

Acer platanoides | 1 | 34 | 10 |

Pseudotsuga menziesii | 1 | 31 | 22 |

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Detter, A.; Rust, S.; Krišāns, O. Experimental Test of Non-Destructive Methods to Assess the Anchorage of Trees. *Forests* **2023**, *14*, 533.
https://doi.org/10.3390/f14030533

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Detter A, Rust S, Krišāns O. Experimental Test of Non-Destructive Methods to Assess the Anchorage of Trees. *Forests*. 2023; 14(3):533.
https://doi.org/10.3390/f14030533

**Chicago/Turabian Style**

Detter, Andreas, Steffen Rust, and Oskars Krišāns. 2023. "Experimental Test of Non-Destructive Methods to Assess the Anchorage of Trees" *Forests* 14, no. 3: 533.
https://doi.org/10.3390/f14030533