Another approach to the modelling could be to include the mass of the root plate in the total tree structure weight (p).
If only the geometry and unit weight of the tree are available, it is informative to predict the failure values of the tree. From the Winkler model [Figure 2-33], uplift failure bending moment is Mu= pb/3,
Where, p=total weight of the structure and b=radius of the foundation= 1.63 m.
To estimate the total weight of the tree including the root-soil plate (p), the following calculations are conducted:
Tree mass above ground = 1613 kg [Section 2.4.1 & 2.4.2];
Root-soil plate volume measured using ImageJ software (a hemispherical shape is assumed for further calculations with 0.9 m depth) = 1.16 m3 [Section 2.4.3];
1/4th of the stem volume is taken as the root volume and the rest of the root-soil plate volume as the soil volume [Danjon et al. 2005];
Stem volume = 1.34 m3; root density= 1000 kg/m3 (green weight) [Dupuy et al 2005]; and soil density=20 kN/m3 [Section 2.5.3];
Therefore, the above ground tree weight+ root-soil plate weight =32 kN; With p=32kN and b=1.63 [Section 2.4.3], therefore Mu= pb/3=17.4 kNm.
It is clear from this analysis that the ultimate bending moment from the Winkler foundation calculations (Mu= pb/3) is now still very low compared to the tangent intersection method
(385 kNm) [Figure 2-31] and the previous Winkler foundation stiffness estimates (490 to 500 kNm) [Table 2-4]. It can be postulated that for a tree root structure that the equivalent stiffness (keq=kw) of the Winkler foundation is the combination of tree-root-soil weight and
the anchorage strength of the root soil plate. Coutts [1983 & 1986] and Blackwell et al. [1990] identified the components of tree anchorage as windward root resistance, soil root plate weight, leeward hinge resistance to bending, soil resistance and the weight of stem and crown. So some of the additional ‘missing’ capacity should be due to the rotational bending stiffness of the roots at one end of the root plate (kr) and the tensile (or pullout)
strength along the root plate and the total weight of the entire tree and root-soil plate are the remaining components of anchorage.
2.10.2.1 Components of tree-root anchorage:
Further calculations are presented below to investigate this hypothesis and to include the aspects of tree anchorage shown in Figure 2-35. It is assumed five windward and five leeward roots of an average diameter of 0.044 m and 1.0 m length contribute to tensile and compressive strength respectively. The major anchorage components are i) tree weight (wt)
and root plate weight (wr), ii) tensile resistance of the windward root, iii) soil shear strength
along the base of the root plate and iv) bending strength of the leeward roots, and are explored further as shown below:
1) Anchorage resistance from tree and root plate weight:
The total tree and root-soil plate weight = 32 kN;
Anchorage resistance component from the structural weight=Pw = 32 kN;
The corresponding resisting moment of the weight component, Mw=Pwb/3=17.4 kNm.
2) Tensile (or pull-out resistance) of the windward roots:
(a) Bischetti et al. [2005] suggested the mean tensile strength of Norway spruce tree roots was 38.94 MPa.
In this study, the Norway spruce tree had around 10 structural roots evenly distributed with an average diameter of 0.044 m.
Cross sectional area of 5 roots =5X 𝜋𝑑
2
4 = 0.0076 m
2
Giving an overall tensile strength= Pt =38.94(0.0076) = 0.296 MN =296 kN
total resisting moment from root tensile strength component = Mt =160.83 kNm;
(OR)
(b) Abe & Ziemer [1991] provided an empirical equation of tree root pull-out resistance as:
Pull-out resistance (in pounds of force) =278.7X (root diameter in inches) ^1.03 As almost all the roots were subdivided each with an average diameter of 0.044 m, so 5 roots were taken as the roots resisting the pull-out
total resisting moment from root pull-out resistance component = Mr = 5.8 kNm.
(c) Soil shear strength:
Shear strength of the root reinforced soil from vane shear field test = 120 kPa [Figure 2-10]. As 70% of the total root volume contributes to tree stability [Lundstrom et al. 2008], hence 70% of the root soil plate surface area is considered to contribute to the shear strength [Figure 2-35]
=0.7[ 𝜋𝑟√(𝑟2+ ℎ2)] = 6.68 m2
Overall shear force estimated from the vane shear tests =Ps=120X6.68= 800.4 kN
Resisting moment component from soil shear strength = Ms =435 kNm.
(d) Root compressive bending strength:
Stokes and Mattheck [1996] reported the root compression strength of Norway spruce varied from 29.3(±1.6) to 24.5(±1.9) MPa, (± standard error).
Average compressive strength of root = 27 MPa (Stokes and Mattheck 1996), is taken to estimate the root compressive strength.
Compressive strength=Pc= 27(0.0076) = 0.21 MN =210 kN
total resisting moment from compressive strength component = Mc= 114.1 kNm.
Therefore, the total anchorage strength/capacity is approximately 730 kNm. Root-soil weight appears to account for only 2 to 3% of the capacity and the compressive and tensile strength of roots appears to account for 38% of the capacity. The soil shear strength component seems to provide a much higher resistance for the tree root-plate to failure along the surface area of the root-soil plate base. However, the later component will reduce as the soil fractures and the plate uplifts, losing contact with the soil below.
A new predictive method can to be generated along the lines of the Yim and Chopra [1984] model for tree stability analysis, and some of the possible ways of estimating the equivalent stiffness and moment and rotational response are explained in this section. With the availability of large numbers of tree response data sets, one of these simple methods could be quite useful to estimate tree root anchorage strength.