Presentation on theme: "The Reiterative Nature of Tree Branches Christine Schneider."— Presentation transcript:
The Reiterative Nature of Tree Branches Christine Schneider
Trees and shrubs show a variety of morphologies. There are tall slender trees with predominant main stem with short primary branches.
Other trees have a less dominant main stem with very long branches.
There are many trees with other forms of morphologies.
To date, there has been very little research into a unifying principle of tree and shrub morphologies.
The next few slides show our concept of how branches grow and the series is the basis for our idea that bending stresses of all branches are constant based upon the reiterative process. The red branches are primary branches. The blue branches are secondary branches and green branches are terminal branches.
Hypotheses 1. Mechanical stress is constant from the base to the tip of a branch. 2. All terminal portions of branches will have similar mechanical stress. 3. The addition of secondary branches is a reiterative process in the mechanical structure of tree branches. 4. Mechanical stresses of primary branches are constant among tree species.
Mechanical Properties: Bending Moment (M) Wilson and Archer, 1977 The next series of slides gives the definition of bending moment. The next view shows how bending moment is related to size of the branch.
Bending Moment (M) [low] Bending Moment (M) [intermediate] Bending Moment (M) [high]
Mechanical Properties: Section Modulus (S) Beer and Johnston, 1992 The next few slides define section modulus and shows how when a board is vertical the section modulus is different when the board is horizontal.
Mechanical Properties: Stress Beer and Johnston, 1992 Bending stress is the slope of the line of bending moment versus section modulus. Our hypothesis is that bending stress is constant among primary, secondary and terminal branches.
Measurements Diameter of branch (segment) Length of branch (segment) Weight of branch (segment) Weight of side branches
Table 1: Properties of tree branches Species section modulus at tip (m^3 x 10^-8) section modulus at base (m^3 x 10^-8) Acer palmatum0.47236.8 Ailanthus altissima12.689.7 Aler carpinfolium0.04257.9 Arbutus unedo1.08746.5 Betula albosinesis0.30450.6 Betula pubescens0.11920.3 Broussonetia papyfera0.84897.4 Cedrus libani0.85477.5 Chranantagus retusus3.9641.7 Cornus alternifolia1.027.2 Cornus florida0.38216 Table 1 shows the section modulii of the primary branches of the study.
Species section modulus at tip (m^3 x 10^-8) section modulus at base (m^3 x 10^-8) Crataegus diffusa0.1814.2 Ligustrum lucidum0.536.3 Liquidamber styraciflua0.86121 Malus hypenhesis8.2185 Phillyrea latifolia2.186.7 Picea likingensis1.116.4 Picea stchensis0.135.7 Pinus strobus0.3316 Pinus thunbergii0.6670.8 Prunus serotina0.4227.4 Quercus palustrius1.8108 Rhododendron arboreum0.9417.1 Tilia tomentosa2.6726 Ulmus procera0.3427.4
1. Mechanical stress is constant from the base to the tip of a branch. Example 1: Ulmus procera The next two figures show the slope values for primary branches. The slope for Ulmus procera is 0.138.
1. Mechanical stress is constant from the base to the tip of a branch. Example 2: Pinus thunbergii The slope for Pinus thunbergii is 0.0246.
Speciesr2r2 Linear Equation Acer palmatum0.95y = 0.0551x + 0.689 Aler carpinfolium0.93y = 0.09.48x + 0.145 Ailanthus altissima0.99y = 0.0875x - 2.40 Arbutus unedo0.93y = 0.0227x – 0.366 Betula albosinesis0.93y = 0.0672x – 0.118 Betula pubescens0.85y = 0.105x - 0.827 Broussonetia papyrifera0.93y = 0.0555x - 1.68 Cedrus libani0.98y = 0.0338x + 1.05 Chranantagus retusus0.96y = 0.0335x + 1.63 Cornus alternifolia0.94y = 0.0531x + 1.3321 Cornus florida0.99y = 0.0709x + 0.8411 Crataegus diffus0.97y = 0.0447x – 0.0523 Ligustrum lucidum0.98y = 0.0604x – 0.0774 Here we see the equations for primary branches of the species of this study.
2. All terminal portions of branches will have similar mechanical stress. (SSPS Institute Inc, 2000) Species Branch Number Smallest Section Modulus Largest Section Modulus Stress Probability Value Acer palmatum 11.9918.9 20.4720.10.12 32.0212.9 40.9310.6 Aler carpinfolium 10.0424.19 20.0792.02 30.0793.66 40.0633.560.161 50.0630.72 60.1035.24 70.0421.16 80.0421.12 This table over two pages shows that terminal portions of branches had similar bending stresses as expressed by probability values.
Species Branch Number Smallest Section Modulus Largest Section Modulus Probability Value Betula pubescens 10.1966.2 0.645 20.1194.18 30.9657.07 40.4478.57 Crataegus diffusa 10.492.3 0.928 20.311.4 30.492.1 40.494.6 Ulmus procera 11.7127.4 0.601 21.322 31.2121.9 40.6817.3 50.3417.9
3. The addition of secondary branches is a reiterative process in the mechanical structure of tree branches.
SpeciesBranchSM at tipSM at base Stress Probability Acer palmatum A2.0297.0 0.33 B0.4781.7 Acer palmatum A0.4718.9 0.9 B2.0236.8 Ailanthus altissima A18.589.7 0.036 B5.1516.2 C12.747.3 Aler carpinfolium A0.04252.5 0.32 B0.04257.9 C0.04231.6 D0.07942.8 Arbutus unedo A1.0925.9 0.87 B1.0946.6 (SSPS Institute Inc, 2000) This table over three pages shows that secondary portions of branches had similar bending stresses as expressed by probability values.
4. Mechanical stresses of primary branches are constant among tree species. Mean = 5.64, Standard deviation = 1.89 This figure shows that bending stresses of primary branches were relatively constant among the species tested. Other data (not shown) have been procured to demonstrate that bending stresses of primary, secondary, and terminal branches were similar.
Summary 1. Mechanical stress is constant from the base to the tip of a branch. 2. All terminal portions of branches will have similar mechanical stress. 3. The addition of secondary branches is a reiterative process in the mechanical structure of tree branches. 4. Mechanical stresses of primary branches are constant among tree species.
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