ALANA WOLFE

Mechanical Properties of Mechanical Properties of Tropical TreesTropical Trees

Manhattan College – Class of 2009Research Advisor – Dr. L. Evans

TREES AND SHRUBS SHOW A VARIETY OF TREES AND SHRUBS SHOW A VARIETY OF MORPHOLOGIESMORPHOLOGIES

Some are medium height with short primary branches

Some are tall and wide with a less dominant main stem

and very long branches

Inga veraQuercus bumelioides

View of Tropic Forest – Panama

View of Tropic Forest – Panama

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

TO DATE, THERE HAS BEEN VERY LITTLE TO DATE, THERE HAS BEEN VERY LITTLE RESEARCH FOR A UNIFYING PRINCIPLE OF TREE RESEARCH FOR A UNIFYING PRINCIPLE OF TREE

AND SHRUB MORPHOLOGIESAND SHRUB MORPHOLOGIES

MECHANICAL PROPERTIES: MECHANICAL PROPERTIES: BENDING MOMENT (M)BENDING MOMENT (M)

Beer and Johnston, 1981

Definition of bending moment

Bending Moment (M) [low]

Bending Moment (M) [intermediate]

Bending Moment (M) [high]

As branches enlarge-bending moment increases

MECHANICAL PROPERTIES: SECTION MODULUS (S)

S = I (Area3) C

I = (1)(b)(h3) 12

Beer and Johnston, 1981

Definition of section modulus

Calculation of section modulus

Bending Stress = Bending Moment Section Modulus

Definition of bending stress

MECHANICAL PROPERTIES: BENDING MECHANICAL PROPERTIES: BENDING STRESSSTRESS

Slope = Bending Stress

Definition of bending stress (graphic)

MATERIALS & METHODS: MEASUREMENTS

Diameter of segment (vertical and horizontal

dimension) Length of segment Weight of segment Weight of Side branches Volume of Side branches

Diagram of method

Trees of the tropics will have constant bending stresses.

Hypothesis 1Hypothesis 1

1. BENDING STRESS IS CONSTANT 1. BENDING STRESS IS CONSTANT FROM THE BASE TO THE TIP OF THE BRANCH: FROM THE BASE TO THE TIP OF THE BRANCH: TROPICAL [PANAMA] (TROPICAL [PANAMA] (AVICENNIA GERMINANSAVICENNIA GERMINANS ))

One example – bending stress = 10.1 mPa

TABLE 1: PROPERTIES OF TREE BRANCHESTABLE 1: PROPERTIES OF TREE BRANCHES

SpeciesBending Stress

MPa r2

Artocarpus altilis 4.6 0.69

Avicennia germinans 10.1 0.97

Bauhinia monandra 4.7 0.99

Bursera simaruba 9.2 0.90

Calycophyllum candidissimum 6.7 0.93

Citrus (1) 6.2 0.99

Diphysa americana 7.4 0.96

Genipa americana 8.3 0.99

Goethalsia meiantha 8.6 0.96Bending stresses of samples

SpeciesBending Stress

MPa r2

Guarea rhopalocarpa 5.9 0.85

Inga spectabilis 8.7 0.94

Inga vera 6.7 0.99

Laguncularia racemosa 9.3 0.96

Myrcianthes fragrans 4.7 0.96

Myrospermum frutescens 14.4 0.98

Sideroxylon capiri 3.0 0.98

Terminalia catappa 7.1 0.92

Virola koschnyi 5.5 0.74

Mean 7.6 0.93

Standard Deviation 2.5 0.086

CONCLUSION FOR HYPOTHESIS 1CONCLUSION FOR HYPOTHESIS 1

Bending stress is constant from tip to base for tropical trees

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.

2. Bending Stress is higher for 2. Bending Stress is higher for TropicalTropical species species than for than for TemperateTemperate species.species.

Section Modulus (m3 x 10-8 )

Ben

din

g M

om

ent

(N-m

)

0

Temperate

Tropical

Tropical [Panama]: Bending StressesTropical [Panama]: Bending Stresses

0

5

10

15

20

25

30

35

40

45

0 100 200 300 400 500 600

Section Modulus (m3 x 10-8)

Ben

din

g M

omen

t w

ith

sid

e b

ran

ches

(N-m

)

1. BENDING STRESS IS CONSTANT 1. BENDING STRESS IS CONSTANT FROM THE BASE TO THE TIP OF THE BRANCH: FROM THE BASE TO THE TIP OF THE BRANCH: TEMPERATE [NEW YORK] (TEMPERATE [NEW YORK] (PINUS THUNBERGII)PINUS THUNBERGII)

One example

-20

-10

0

10

20

30

40

50

60

70

0 200 400 600 800 1000 1200 1400 1600

Section modulus (m3x 10-8)

Ben

din

g m

omen

t w

ith

sid

e b

ran

ches

(N

-m)

Temperate [New York]: Bending StressesTemperate [New York]: Bending Stresses

2. Bending Stress is higher for 2. Bending Stress is higher for TropicalTropical species species than for than for TemperateTemperate species.species.

Section Modulus (m3 x 10-8 )

Ben

din

g M

om

ent

(N-m

)

0

Temperate

Tropical

2. B2. BENDING STRESSES OF ENDING STRESSES OF TROPICALTROPICAL SPECIES ARE HIGHER SPECIES ARE HIGHER THAN THAN TEMPERATETEMPERATE SPECIESSPECIES

SIDE BRANCHES Tropical Temperate

Mean 7.6 MPa 5.7 MPa

Student’s T-Test Probability 0.021

Conclusion for Hypothesis 2Conclusion for Hypothesis 2

Branches of Tropical species have higher bending stresses

than branches of Temperate species

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

HYPOTHESIS 3HYPOTHESIS 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

Temperate

Tropical

Growing tips produce more volume

The branches of tropical trees should have more small branches near their terminals

Branches of Various SizesBranches of Various Sizes

Proportional Length

Proportional Volume of Side Branches

1.0

1.0

0

0

To compare a variety of branches each branch must be proportionalized.

Proportional Volume

Proportional Length

CU

M P

ropo

rtio

nal V

olum

e of

Sid

e B

ranc

hes

1

CUM Proportional Length

1

1

Tropical

Temperate

Tip

Base

The above relationship should be true

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

LENGTHLENGTH

TEMPERATE [NEW YORK]:TEMPERATE [NEW YORK]: 3. PROPORTIONAL VOLUME VS. PROP. 3. PROPORTIONAL VOLUME VS. PROP.

LENGTHLENGTH

3. COMPARING PROPORTIONAL 3. COMPARING PROPORTIONAL VOLUMESVOLUMES

Tropical Slopes

TemperateSlopes

Tropical X-intercept

TemperateX-intercept

Mean 1.04 1.16 0.22 0.32

Standard Deviation 0.09 0.3 0.077 0.11

Student’s T-Test

Probability0.18 0.011

CU

M P

ropo

rtio

nal V

olum

e o

f Sid

e B

ranc

hes

1

CUM Proportional Length

1

1

Tropical

Temperate

0.22 0.32

Conclusion for Hypothesis 3Conclusion for Hypothesis 3

Tropical trees will have larger volumes of side branches near their branch terminals than for temperate trees

ConclusionsConclusions

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.

WORKS CITEDAlmeras, 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

ACKNOWLEDGEMENTSACKNOWLEDGEMENTS

Dr. Lance Evans

Patricia Evans

Christina Pereira

Elaina Petrone