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A LANA W OLFE Mechanical Properties of Tropical Trees Manhattan College – Class of 2009 Research Advisor – Dr. L. Evans

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TREES AND SHRUBS SHOW A VARIETY OF MORPHOLOGIES Some are medium height with short primary branches Some are tall and wide with a less dominant main stem and very long branches Inga vera Quercus bumelioides

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View of Tropic Forest – Panama

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With this view and additional views – we note typical branching patterns

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T O DATE, THERE HAS BEEN VERY LITTLE RESEARCH FOR A UNIFYING PRINCIPLE OF TREE AND SHRUB MORPHOLOGIES

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M ECHANICAL P ROPERTIES : B ENDING M OMENT (M) Beer and Johnston, 1981 Definition of bending moment

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Bending Moment (M) [low] Bending Moment (M) [intermediate] Bending Moment (M) [high] As branches enlarge-bending moment increases

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M ECHANICAL P ROPERTIES : S ECTION M ODULUS (S) S = I (Area 3 ) C I = (1)(b)(h 3 ) 12 Beer and Johnston, 1981 Definition of section modulus

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Calculation of section modulus

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Bending Stress = Bending Moment Section Modulus Definition of bending stress

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M ECHANICAL P ROPERTIES : B ENDING S TRESS Slope = Bending Stress Definition of bending stress (graphic)

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MATERIALS & METHODS: MEASUREMENTS Diameter of segment (vertical and horizontal dimension) Length of segment Weight of segment Weight of Side branches Volume of Side branches

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Diagram of method

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Trees of the tropics will have constant bending stresses. Hypothesis 1

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1. B ENDING STRESS IS CONSTANT FROM THE BASE TO THE TIP OF THE BRANCH : T ROPICAL [P ANAMA ] ( A VICENNIA GERMINANS ) One example – bending stress = 10.1 mPa

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T ABLE 1: P ROPERTIES OF TREE BRANCHES Species Bending Stress MPar2r2 Artocarpus altilis4.60.69 Avicennia germinans10.10.97 Bauhinia monandra4.70.99 Bursera simaruba9.20.90 Calycophyllum candidissimum6.70.93 Citrus (1)6.20.99 Diphysa americana7.40.96 Genipa americana8.30.99 Goethalsia meiantha8.60.96 Bending stresses of samples

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Species Bending Stress MPar2r2 Guarea rhopalocarpa5.90.85 Inga spectabilis8.70.94 Inga vera6.70.99 Laguncularia racemosa9.30.96 Myrcianthes fragrans4.70.96 Myrospermum frutescens14.40.98 Sideroxylon capiri3.00.98 Terminalia catappa7.10.92 Virola koschnyi5.50.74 Mean7.60.93 Standard Deviation2.50.086

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CONCLUSION FOR HYPOTHESIS 1 Bending stress is constant from tip to base for tropical trees

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HYPOTHESIS 2 Bending stress of tropical trees will be greater than bending stresses of temperate trees Rational: Tropical trees should have more side branches and thus more weight because few side branches die.

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2. Bending Stress is higher for Tropical species than for Temperate species. Section Modulus (m 3 x 10 -8 ) Bending Moment (N-m) 0 Temperate Tropical

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Tropical [Panama]: Bending Stresses

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1. B ENDING STRESS IS CONSTANT FROM THE BASE TO THE TIP OF THE BRANCH : T EMPERATE [N EW Y ORK ] (P INUS THUNBERGII ) One example

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Temperate [New York]: Bending Stresses

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2. Bending Stress is higher for Tropical species than for Temperate species. Section Modulus (m 3 x 10 -8 ) Bending Moment (N-m) 0 Temperate Tropical

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2. B ENDING S TRESSES OF T ROPICAL SPECIES ARE HIGHER THAN T EMPERATE SPECIES SIDE BRANCHESTropicalTemperate Mean7.6 MPa5.7 MPa Student’s T-Test Probability 0.021 Conclusion for Hypothesis 2 Branches of Tropical species have higher bending stresses than branches of Temperate species

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Growth of branches is a function of the addition of branches and enlargement and retention of existing branches

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HYPOTHESIS 3 Tropical trees will have larger volumes of side branches near their branch terminals than for temperate trees Rational: Tropical trees should not have death of small branches

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Temperate Tropical Growing tips produce more volume The branches of tropical trees should have more small branches near their terminals

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Branches of Various Sizes

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Proportional Length Proportional Volume of Side Branches 1.0 0 0 To compare a variety of branches each branch must be proportionalized.

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Proportional Volume

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Proportional Length

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CUM Proportional Volume of Side Branches 1 CUM Proportional Length 1 1 Tropical Temperate Tip Base The above relationship should be true

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TROPICAL [PANAMA]: 3. PROPORTIONAL VOLUME VS. PROP. LENGTH

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T EMPERATE [N EW Y ORK ]: 3. P ROPORTIONAL V OLUME VS. P ROP. L ENGTH

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3. C OMPARING P ROPORTIONAL V OLUMES Tropical Slopes Temperate Slopes Tropical X-intercept Temperate X-intercept Mean 1.041.160.220.32 Standard Deviation 0.090.30.0770.11 Student’s T-Test Probability 0.180.011

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CUM Proportional Volume of Side Branches 1 CUM Proportional Length 1 1 Tropical Temperate 0.220.32 Conclusion for Hypothesis 3 Tropical trees will have larger volumes of side branches near their branch terminals than for temperate trees

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Conclusions 1. Bending stress is constant from tip to base for tropical trees. 2. Branches of Tropical species have higher bending stresses than branches of Temperate species. 3. Tropical trees will have larger volumes of side branches near their branch terminals than for temperate trees.

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WORKS CITED Almeras, T, Gril, J, and Costes E, 2002. Bending of apricot-tree branches under the weight of axillary productions: confrontation of a mechanical model to experimental data. Trees 16: 5-15. Beer, F.P., and Johnston, E.R. 1992. Mechanics of Materials. McGraw-Hill and Co., New York. Cannell, M., Morgan, J., and Murray, M. 1988. Diameters and dry weights of tree shoots: effects of Young’s modulus, taper, deflection and angle. Tree Physiol. 4: 219-231. Castera P, and Mortier, V. 1991. Growth patterns and bending mechanics of branches.Trees 5: 232- 238. Dean, T., Roberts, S., Gilmore, D., Maguire, D., Long J., O’Hara K, and Seymour, R. 2002. An evaluation of the uniform stress hypothesis based on stem geometry in select North American conifers. Trees 16: 559-568 Evans, L.S., Kahn-Jetter, Z., Torres, J., Martinez, M., and Tarsia, P. 2008. Mechanical stresses of primary branches: A survey of 40 woody tree and shrub species. Trees 22: 283-289. King DA (1986) Tree form, height growth and susceptibility to wind damage in Acer saccharum. Ecology 67: 980-990. Mattheck C, Bethge K, Schafer JJ (1993) Safety factors in trees. Theor. Biol. 165: 185-189. McMahon TA (1973) Size and shape in biology. Science 179: 1201-1204. Milne R, Blackburn P (1989) The elasticity and vertical distribution of stress within stems of Picea sitchensis. Tree Physiol. 5: 195-205. Morgan J, Cannell M (1987) Structural analysis of tree trunks and branches: tapered cantilever beams subject to large deflections under complex loading. Tree Physiol. 3: 365-371. Morgan J, Cannell M (1994) Shape of tree stems- a re-examination of the uniform stress hypothesis. Tree Physiol 14, 49 (1994) Niklas KJ, Spatz H-C (2000) Wind-induced stresses in cherry trees: evidence against the hypothesis of constant stress levels. Trees 14: 230-23

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ACKNOWLEDGEMENTS Dr. Lance Evans Patricia Evans Christina Pereira Elaina Petrone

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