Stress Analysis Along Tree Branches

1 Copyright © 2014 by ASME

STRESS ANALYSIS ALONG TREE BRANCHES

Allison Kaminski Manhattan College, Mechanical Engineering

Riverdale, NY, United States

Simon Mysliwiec Manhattan College, Mechanical Engineering


Zahra Shahbazi Manhattan College, Mechanical Engineering


Lance Evans Manhattan College, Biology Department


ABSTRACT Efforts have been made to develop various models to calculate

the stress due to weight throughout tree branches. Most studies

assumed a uniform modulus of elasticity throughout the branch

as well as analyzing the branch as a tapered cantilever beam

orientated horizontally or at an angle. However, previous

studies show that branches located lower on the tree have a

greater variance of modulus of elasticity values in the radial

direction and that branches located lower on a tree are more

curved. Also, different tree species have different morphologies,

some with curvy branches. In this work we have developed a

model which considers the curved shape and varying modulus

of elasticity values in order to determine stress across the tree

branches more accurately. To do this the cross sectional area

was divided into rings and each ring was assigned a different

modulus of elasticity. Next, the area of the rings was

transformed according to their modulus of elasticity. We then

considered the curved shape of the branch by generating a best

fit line for the diameter of the tree branch in terms of distance

from the end of the branch. The generated diameter equation

was used in the stress calculations to provide more realistic

results. Based on equations developed in this work, we have

created a Graphical User Interface (GUI) in Matlab, which can

be used as a tool to calculate stress within tree branches by

biologists without getting involved with the mathematical and

mechanical calculations. We also created a Finite Element

Model (FEM) in Abaqus and compared results.

INTRODUCTION A tool used to accurately calculate the stress on tree

branches from easily measured dimensions and properties can

have multiple applications. This tool will be beneficial to

biologists researching the relationships between tree

morphology and stress because it is hypothesized that tree

branches grow in a specific way which creates a uniform stress

throughout the branch [1].

Previous studies calculated the stresses on tree branches by

examining them as tapered cantilever beams of either an

elliptical or circular cross sections. These stress calculations

assumed the tree branch was of a uniform material with a

uniform Modulus of Elasticity (MOE) value [1]. The MOE is a

measure of the stiffness of a material. Materials with a small

MOE bend more easily while materials with a large MOE are

stiffer. This study accounts for the varying MOE values within

tree branches. Multiple studies have shown the MOE depends

on the age of the wood and its location on the tree. As a tree

grows, new layers of the tree branch grow on the outermost

parts of the branch. This growth pattern makes the outer most

part of the branch the youngest while the innermost part of the

branch is the oldest [2]. The innermost wood has a small MOE

value, and has little mechanical significance to resist bending.

The outer wood resists a majority of the bending in a tree

branch [3]. Therefore at a given cross section the MOE of the

branch varies in the radial direction - as the radius increases the

MOE value increases.

This study considers radial variances in MOE when

calculating stress. Branches located lower on the tree have a

greater variance of MOE in the radial direction than branches

located closer to the top of the tree [2]. For this reason it was

hypothesized that using the proposed tree branch model will be

more necessary for branches located lower on the tree.

As mentioned above, in addition to modeling tree branches

as being composed of a uniform material, previous studies

modeled tree branches as tapered cantilevered beams [1, 2].

Branches in nature do not resemble straight lines, they have

curves. In this study a tool to calculate stresses in a curved tree

branch is developed.

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-37726


θ mg

θ

V

Fa

x

To summarize, in this study a tool is developed to analyze

the stress in different tree branch models. Two new cases are

proposed in this study, the first considering the varying MOE of

branches in the radial direction and the second considering the

curviness of the branch. Four additional models are used for

branches with less complicated geometries and for comparison

purposes. These models include a cantilever beam of a fixed

circular and elliptical cross section, as well as a tapered

cantilever beam of a circular and elliptical cross section. In

addition, a Finite Element simulation is developed to verify

results.

METHOD Six different tree branch models were created. Case 1 is a

fixed circular cross section, and Case 2 is a fixed elliptical cross

section. Case 3 is a tapered circular cross section, and Case 4 is

a tapered elliptical cross section. Case 5 is a fixed circular cross

section with a non-uniform material where the modulus of

elasticity varies in the radial direction. Case 6 is a curved

branch of uniform material and a circular cross section.

Equations to calculate stress were derived for each of the

six cases. These equations were used to write a code in Matlab

that allows measurable dimensions of the branch to be inputted

and then a stress analysis at a specified location would be

outputted. The Matlab code is able to calculate stress at the top,

bottom and sides of any desired cross section along the length

of the branch.

Only the stress due to the weight of the branch was

considered as no external loads were examined. The weight of

the branch was analyzed as a distributed load that acts vertically

downward. To perform the analysis the weight was broken into

two components, one parallel to the axis of the branch and one

perpendicular to the branch (Fig. 1).

Figure 1. Tree branch model showing the force due to weight

acting on the branch along with its components.

The component of the distributed weight that acts

perpendicular to the branch and parallel to the cross sectional

area is the shearing distributed load, ws(x). The component of

the distributed weight that acts parallel to the branch and

perpendicular to the cross sectional area is the axial distributed

load, wa(x). The shearing distributed load contributes to the

normal stress due to bending, while the axial distributed load

contributes to the normal compressive stress on the branch.

These two stresses need to both be considered in order to

calculate the total normal stress acting on a given cross section.

General equations were derived to calculate stress that can

be used for each of the six cases. All of the equations derived

assume a uniform modulus of elasticity (MOE) throughout the

branch. To calculate the axial compressive stress acting on a

cross section of a tree branch a general equation for the axial

distributed load needs to be determined, which is the load per

unit length (1).

x

xgxA

x

gxV

L

mgxwa

cos)(cos)(cos)(

Where m is the total mass of the branch, L is the total length of

the branch, g is the gravitational constant, x is the distance from

the tip of the branch where the stress analysis is desired, V(x) is

the volume of the branch up to the specified point x, A(x) is the

cross sectional area of the branch and θ is the angle of the tree

branch with respect to the vertical. For instance, if the branch is

perfectly horizontal then θ will be 90 degrees, making the axial

distributed load zero. Using Equation (1) the axial force acting

at any location from the tip of the branch, Fa(x), can be

determined from Equation (2).

dxxwxF aa )()(

The normal axial stress due to the axial component of the

weight is

)(

)()(

xA

xFx a

a

The normal stress due to bending is caused by the

component of weight acting perpendicular to the branch. The

shearing distributed load, ws(x), which acts perpendicular to the

length of branch can be calculated using

x

xgxA

x

gxV

L

mgxws

sin)(sin)(sin)(

The variables are the same as for the distributed axial load. The

component of weight that acts perpendicular to the length of the

branch and parallel to the cross sectional area is the shear force,

Vs(x), which can be determined from the shearing distributed

load.

dxxwxV ss )()(

The bending moment in terms of distance from the tip of the

branch (6) can be obtained by integrated the shear force, Vs(x),

with respect to distance from the tip of the branch, x.

dxxVxM )()(

The normal stress due to bending can be expressed as

)(

)()()(

xI

xcxMxb

Where c(x) is the radial distance from the center of the cross

section to the location in the radial direction where the stress

analysis is desired. For instance to obtain the bending stress at

the top of the cross section c(x) would be the radial distance

from the center of the cross section to the top of the cross

section. I(x) is the moment of inertia of the cross section. The

normal stress due to bending is compressive (negative) or

(1)

(2)

(3)

(4)

(5)

(6)

(7)


tensile (positive) depending on the location of the cross section

being examined. The top of the branch will be in tension and

the bottom will be in compression. The neutral axis is where the

stress passes from positive to negative and does not experience

the effects of bending. The left and right sides of the cross

section lie along the neutral axis and therefore have a bending

stress of zero.

The total normal stress acting on a cross section is the sum

of the stress due to bending and the axial stress. When adding

these two stresses together the sign convention of each stress

must be considered.

Equations (1) through (7) are the general equations used to

perform the stress analysis for each of the cases. For some of

the cases additional analyses are needed. For Case 1 the generic

procedure described above can be followed exactly.

Case 2 is a fixed elliptical cross section. The equations

described above are used by substituting in the cross sectional

area of an ellipse, HV RRA , and the moment of inertia for an

ellipse, 3

4VH RRI

.

Cases 3 and 4 are tapered; therefore additional calculations

must be considered. Case 3 is tapered and has a circular cross

sectional area (Fig. 2).

Figure 2. Case 3 tree branch model, tapered with a circular cross

section. Where α is the angle of taper.

The angle of taper can be calculated using

L

R

x

xR o)(

tan

Where Ro is the radius at the base of the branch. Rearranging for

the radius of the branch in terms of distance from the tip of the

branch, R(x), gives Equation (9).

L

xRxR o)(

The cross sectional area in terms of distance from the tip of the

branch then becomes 2

2)()(

L

xRxRxA o

The moment of inertia also varies with x and must be calculated

using R(x). By making these adjustments to the generic

equations the stress analysis can be performed for Case 3.

Case 4 models the branch as a tapered elliptical beam. In

this case there are two angles of taper, one in the x-y plane and

the other in the x-z plane (Fig. 3).

(a). Taper in x-z Plane

(b). Taper in x-y Plane

Figure 3. Case 4 taper in the x-y planes and x-z planes. RHo and RVo are

the radii in the horizontal and vertical directions respectively at the

base of the tree branch. ϕ is the angle of taper in the horizontal

direction. β is the angle of taper in the vertical direction.

The angle of taper in the x-z plane is determined from the

horizontal radius of the ellipse and can be calculated using

Equation (11).

L

R

x

xR HoH )(

tan

Where ϕ is the angle of taper in the x-z plane, RHo is the

horizontal radius at the base of the branch, and RH(x) is the

horizontal radius with respect to distance from the tip of the

branch. Equation (12) provides the horizontal radius at any

location, x, along the branch.

L

xRxR Ho

H )(

The angle of taper in the x-y plane can be calculated using

L

R

x

xR VoV )(

tan

Where β is the angle of taper in the x-y plane, RVo is the vertical

radius at the base of the branch, and RV(x) is the vertical radius

with respect to distance from the tip of the branch.

x

y

x

RHo

ϕ

RHo

L

ϕ

L x

x

β

RVo

RVo

L

β

L

(8)

(9)

(10)

(11)

(12)

(13)


L

xRxR Vo

V )(

The radiuses of the branch in the horizontal and vertical

directions vary along the length of the branch; therefore, the

cross sectional area of the branch needs to be expressed in

terms of x, the distance from the tip of the branch.

2

2)()()( x

L

RR

L

xR

L

xRxRxRxA HoVoHoVo

HV

The moment of inertia equation for the branch becomes

3)()(

4xRxRI VH

Substituting these equations into the generic equations the stress

analysis can be performed for Case 4.

Case 5 is a fixed circular cross section with a non-uniform

material. The MOE varies in the radial direction (Fig. 4).

Figure 4. Case 5, a tree branch model of a fixed circular cross

section and varying modulus of elasticity in the radial direction.

The cross section of the branch is broken into concentric rings

of equal width each having a different MOE value. The derived

equations assume the branch is made of a uniform material. In

order to use these equations the moment of inertia of each ring

must be transformed based on their MOE values. After doing

this the branch can be analyzed as being composed of a uniform

material and the generic equations for the stress analysis can be

used.

The Matlab program for Case 5 was written so that the

branch can be divided into any number of rings. Therefore

based on the outer diameter of the branch and the desired

number of rings inputted, the radius of each ring is calculated.

The outer radius of each concentric ring is

m

Rhrh

Where m represents the total number of rings that the cross

section is divided into and h represents the number assigned to

each ring. The inner most ring is h=1, while for the outer most

ring h=m. R represents the outer radius of the branch. Using the

radius of each ring the area of each ring can also be calculated.

The inner most ring has the smallest MOE value and

therefore its moment of inertia does not need to be transformed.

All the other rings have a MOE value that is greater than the

inner most ring. To account for the larger MOE values, the ratio

of each ring’s MOE value to the MOE value of the inner most

ring will be used to transform the moment of inertia of each

section. For example, if a ring has a MOE that is two times

greater than that of the inner most ring, then the moment of

inertia of the outer ring needs to be two times greater as well.

This represents how a ring of a larger MOE is able to resist

bending more easily and therefore can be examined as having

the same MOE as the inner most ring if a larger moment of

inertia is used. After transforming the moment of inertia values

the branch can be analyzed as a having uniform material. The

ratio of the MOE of an outer ring to the MOE of the inner most

section is calculated using Equation (18).

1E

En h

h

Note for h=1 the ratio becomes 1. The transformed moment of

inertia becomes

4

1

4

4 hhhh rrnI

The new moment of inertia of the entire cross section is the sum

of the transformed moments of inertia for each ring.

mnew IIII ...21

The stress due to the bending moment on each ring can be

calculated using the new moment of inertia and the MOE ratio.

new

hh

bI

rxMnx

h

)()(

The axial stress on each ring was calculated by considering

the different MOE values of each ring. First the distributed axial

load, wa(x), and the total axial force, Fa(x), were calculated

using the total area of the cross section, Atotal(x). These variables

are independent of the MOE of the branch.

Each ring was of a different area and a different modulus of

elasticity value therefore each ring had a different amount of the

axial force acting on its section. The sum of the axial forces

acting on each ring is equivalent to the total axial force (22).

)()(...)()( 21 xFxFxFxF am

Where Fa(x) is the total axial force due to the weight of the

branch. Forces F1(x) through Fm(x) are the compressive forces

acting on each ring. The forces on each ring cause the branch to

deform along the axis of the branch, making the branch shorter.

This deformation can be calculated as

hh

h

hEA

LF

Where δh is the deformation of the ring, Fh is the axial force

acting on the ring, Ah is the area of the ring, and Eh is the

modulus of elasticity of the ring. Since all of the rings are

connected to one another they will have the same deformation.

m ...21

mm

m

EA

LF

EA

LF

EA

LF ...

22

2

11

1

θ m

g θ

V

F

a

x

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)


Using Equation (24) the axial force on each ring can be solved

for in terms of the axial force acting on any of the other rings.

The example to follow is all done in terms of ring 1, however

the same procedure can be followed to determine the axial force

for any of the rings. The axial force on any of the rings in terms

of the axial force on ring 1 is

1

11

FEA

EAF hh

h

The total axial force on the branch in terms of the force on ring

1 becomes

1

11

1

11

22

1 ... FEA

EAF

EA

EAFF mm

a

Equation (26) can be rearranged to solve for the axial force

acting on ring 1 (27).

1111

22

1

...1EA

EA

EA

EA

FF

mm

a

mm

a

EAEAEAEA

FF

...1

2211

11

1

This procedure was repeated to determine the axial forces

acting on each of the rings. Using the axial force acting on each

ring the axial stress on each ring can be calculated as

h

h

haA

xFx

)()(

The total stress for Case 5 is the sum of the axial and bending

stresses.

Case 6 accounts for the curves of a branch in the x-y plane,

seen from the side view. The curviness of the branch is

determined by measuring the height of the branch above and

below the axis of the branch at incremental distances along its

length. The axis line is the line that connects the center of the

cross section at the base of the branch to the center of the cross

section at the tip of the branch (Fig 5.).

Figure 5. Case 6: curvy tree branch model. Where yt is the height of

the branch on top of the axis, yb is the height below the axis.

The distance from the axis to the top of the branch is yt. The

distance from the axis to the bottom of the branch is yb. The

diameter of the branch is the sum of yt and yb. The branch is

assumed to have a circular cross section in this model because

the varying heights of the branch were only considered in the x-

y plane (side view), not the x-z plane (top view).

To perform a stress analysis for Case 6 calculations were

done to determine the stresses at the base of the branch, where

x=L. At this location the axis of the branch coincides with the

center of the cross section of the branch, because the axis was

defined earlier as the line that connects the centers of the cross

sections at the base of the branch and at the tip of the branch.

Performing the stress analysis at the base provides the

maximum stresses experienced by the branch.

To account for the curviness of the branch a best fit

polynomial equation was determined for yt and yb in terms of x.

Because the branch is curved, the centroid of the branch may

not lie on the branch axis. Therefore the axial component of

weight applied at the centroid must be moved to the axis to

perform the stress analysis. To account for moving the forces an

additional bending moment must be added.

The centroid of the area in the x-y plane was determined by

first calculating the centroid of the areas above the axis and

below the axis separately. Then the two centroids were

combined to obtain the centroid of the entire branch. To do this

the area of the top and bottom parts must be calculated

separately. This was done by integrating the yt and yb equations

to find the area under the curve. Using the areas the x-

coordinate of the centroid for the top (29) and bottom (30)

portions of the branch were calculated using

dxxxfA

xA

t

t

t 1

dxxxfA

xA

b

b

b 1

Where yt and yb are both functions of x, yi=fi(x). To determine

the y-coordinate of the centroid, two additional best fit

polynomial equations were determined for x in terms of yt and

yb, giving xi=fi(y). The y-coordinates of the centroids above

(31) and below (32) to the axis line are

dyyyfA

yA

t

t

t 1

dyyyf

Ay

A

b

b

b 1

Next the x and y coordinates of centroid, relative to the branch

axis, were calculated for the entire branch (33) and (34).

bt

bbtt

i

i

i

i

i

AA

xAxA

A

xA

x

bt

bbtt

i

i

i

i

i

AA

yAyA

A

yA

y

To perform the stress analysis, the axial force, Fa, must be

moved from the centroid to the center of the branch’s cross

section. The perpendicular distance that Fa, must be moved

x

y

θ

x

yt

yb

(25)

(26)

(27)

(28) (29)

(30)

(31)

(32)

(33)

(34)


is y . Therefore the additional bending moment that must be

added to account for moving the axial force is

yFM aaddtional

The bending moment that is due to the shear force (36) is the

product of the shear force and the perpendicular distance

between the x-coordinate for the centroid and the base of the

branch.

)( xLVM sshear

Recall that x is measured from the tip of the branch, so it must

be subtracted by the length to get the distance between the base

and the centroid.

The total bending moment is the sum of these two bending

moments, when adding them together the signs must be

considered. After the total bending moment has been calculated,

the stress due to bending can be determined. The axial stress is

calculated using Equation (3) after the force has been moved to

the center of the cross section. The total normal stress acting on

the cross section at the base is the sum of normal stress due to

bending and the axial stress. Once again, when adding stresses,

the sign convention must be considered. The equations derived for each of the cases were used to

write codes in Matlab that can perform a stress analysis on any

branch after measureable dimensions are inputted. The Matlab

code provides an exact solution for the stress analysis. To make

the code more user friendly a Graphical User Interface (GUI)

was generated to allow anyone to easily determine the stresses

on a tree branch (Fig. 6).

Figure 6. Image of the Graphical User interface for Case 1.

FINITE ELEMENT ANALYSIS The results were compared to the results obtained from

Finite Element Analysis. The Finite Element Analysis was

performed using Abaqus CAE. Two and three dimensional tree

branch models were created. Material properties were assigned

to the models based on literature. Boundary conditions were

applied to the base of the branch to keep it fixed. No load was

applied to the tree branch other than gravity, which the program

applies by using the material’s density and shape. Gravity is

distributed evenly throughout the model. Once the model is

completed, Abaqus performs a stress analysis and provides a

contour image displaying stress, strain or displacement

throughout the entire branch (figures 7 and 8).

Figure 7. 2D “Simple Beam” branch model

Figure 8. 2D Model containing secondary branches

RESULTS AND DISCUSSION Results from 2D finite element model and our calculations

explained above match exactly (with less than 0.06 % error).

Next, the developed model (6 cases) was used to examine

trends that result from changing a single input. For Cases 1

through 5 all of the input variables were kept constant except

for one in order to examine how changing that single variable

affects the results. The variables considered include branch

length, diameter, density and aspect (or angle) with respect to

the vertical. For Case 5 changes in MOE were also considered.

For each of the scenarios tested, all of the cases produced

the same trends. As the length of the branch increased, so did

the mass, because the density remained fixed. However the ratio

of length to mass remained fixed because they both grow at the

same rate. Therefore as the length of the branch increased so

did the maximum stress (Fig. 9).

(35)

(36)


Figure 9. Length of branch versus maximum stress.

As the length increases the maximum stress appears to begin to

grow more rapidly, this would be due to the fact that an

increased length correlates to an increased bending arm. As for

the tree branch models, Cases 1, 2, and 5 all produced nearly

identical results. These were the fixed cross sectional area cases

that were not tapered. According to the stress analysis the

benefits of having an elliptical over a circular cross section

appear to be limited. The tapered Cases 4 and 5 had smaller

stress values than the non-tapered models. As the length

increased the difference between the tapered and non-tapered

cases became more apparent. This makes sense because a non-

tapered branch is more massive, creating greater stress.

When the diameter of the branch was varied, the mass also

varied because the density was kept constant. However, the

diameter and the mass did not increase at the same rates. When

examining the changes in diameter solely, an increase in

diameter appears to produce a decrease in stress (Fig. 10).

However, an increase in diameter per unit mass leads to an

increase in stress (Fig. 11).

As the diameter of the branch increases the stress decreases.

This shows that when comparing two tree branches from the

same tree a thicker branch can resists more stress. As the

diameter per unit mass ratio increases the stress increases. This

shows that when mass is not considered a branch with a smaller

diameter will have less stress. Therefore over a random

sampling of tree branches, an increase in diameter results in an

increase of stress. The relationships between the different tree

branch models remained the same; the tapered models have less

stress than the fixed models, and the shape of the cross section

had a negligible effect on the stress.

For each of the cases the density and the aspect of the

branch with respect to the vertical were also analyzed. The

density was varied by keeping the dimensions of the branch

constant and varying mass. As the density of the branch

increased the stress increased (Fig. 12) and as the aspect of the

branch from the vertical increased (became more horizontal) the

stress increased (Fig. 13).

Figure 12. Mass versus maximum

stress.

Figure 13. Aspect from vertical

versus maximum stress.

Here there is a more noticeable difference between the different

cases. Cases 1 and 5, the two cases with fixed circular cross

sections produced results that were much greater than any of the

other cases. Cases 1 and 5 also produced very similar results.

Here Case 2 has significantly lower stress values than Case 1,

this shows that when considering branch aspect and mass the

elliptical shape of the branch is beneficial for reducing the

affects of stress. This is further supported by the significantly

lower values produced by Case 4, the tapered elliptical branch,

than Case 3, the tapered circular branch. Again the tapered

branches produce less stress than the non-tapered branches.

For Case 5 the effects of varying MOE values were

analyzed. Case 5 was divided into three rings for this analysis,

and MOE values were selected for each ring based on results

from previous research. The percent increase in MOE values

between the first and second rings was 113%, while the percent

increase between the second and third rings was 25%. These

percent increases were increased or decreases by 5% until there

was either an additional positive or negative 20% from the

reference percents (Fig. 14).

Figure 14. Percent change in MOE from reference percent


As the percent change in MOE increased between concentric

rings the stress increased. This means that a branch composed

of a material that has a more uniform MOE throughout will

Figure 10. Diameter versus

stress.

Figure 11. Diameter to mass ratio



experience less stress than a branch that has drastic changes of

MOE in the radial direction.

Case 6 accounted for the curviness of the branch. Different

dimensions were entered into the program to create branches

with varying degrees of curves. For consistency all of the

branches had the same base diameter and the same tip diameter.

First the stress on a branch with no curves was compared to

Case 3. They should produce similar results because they both

had the same base diameter and get very narrow at the tip of the

branch. However, they had noticeably different results (Table

1).

Table 1. Comparison of the stresses in a non-curved branch as

calculated by cases 3 and 6. The percent error calculation is based on

the assumption that case 3 is more accurate.

Case 3 Stress (Pa) Case 6 Stress (Pa) Percent Error 5

1073.2 51063.3 32.97%

The substantial difference in the stress values could be due to

the fact that Case 6 generates a best fit polynomial equation

even when the branch may be more accurately represented by a

straight line. Based on this Case 6 should only be used when the

branch is curved, or when there is an abnormality in the branch

which would shift the centroid of the branch away from the

branch axis.

Figure 15 shows the shapes of two branches developed by

the best fit polynomial equations and their corresponding

maximum stress values.

(a). Max Stress=5

1089.2 Pa (b). Max Stress= 5

1029.3 Pa

Figure 15. Branch Shapes generated by polynomial equations and their

corresponding stress values.

The tree branch generated in Figure 13a has very slight curves.

The stress value calculated for this branch more closely aligns

to the stress calculated using Case 3 displayed in Table 2. This

could be due to the fact that Fig. 13a has minimal curves so it

resembles Case 3 and yet curves are present so a polynomial

equation will fit the shape of the branch better than a straight

line. Fig. 13b has more prominent curves, and produces a larger

stress value. The code was written so that the polynomial fit

equation will pass through all of the input points. Therefore

whenever a branch has a drastic change in height it should be

entered. In addition, input points should be added at locations

between the drastic changes to guide the path of the line.

CONCLUSION

In this study we performed exact stress analysis and a finite

element simulation on tree branches for 6 different cases. Next,

using the developed model, we analyzed the effect of several

different variables on the stress within branches.

The cases that had fixed cross sections produced stresses

that were greater than those with tapered cross sections. This is

due to the fact that non-tapered branches are more massive.

When the mass and angle of the branch were varied, having an

elliptical cross section proved to be advantageous in resisting

stress. Case 5 considered variances of MOE in the radial

direction. This produced results very similar to Case 1. Case 5

proved to be most beneficial when there are drastic changes in

MOE values between two concentric rings. Case 6 is most

beneficial for branches that have curves or abnormalities that

shift the center of mass of the branch away from the branch

axis.

In the future, with further collaboration with the biology

department, the Matlab code will be utilized to examine stress

patterns that form in actual trees as they grow. Additional cases

will be considered. Secondary branches will be added to the

main branches to determine how they affect stress. Also, Case 5

will be modified to account for compression wood and tension

wood. Compression wood is able to resist greater compressive

stresses, while tension wood is able to resist greater tensile

stresses. To resist different types of stresses, the biological

center of the wood does not coincide with its geometric center.

This causes the tree to develop rings that are off center;

therefore, variances in the MOE will be off center [4]. To

account for these wood properties additional programs will be

written for compression and tensile wood, where instead of

having concentric rings of different MOE values, the rings will

be off center. In addition, we will expand our FE model to study

3D and more geometrically complicated branches.

REFERENCES

1. Evans, L., Z. Kahn-Jetter, J. Torres, M. Martinez and P.

Tarsia. 2008. Mechanical stresses of primary branches: a

survey of 40 woody tree and shrub species. Trees 22:283-289.

2. Sone, K., K. Noguchi and I. Terashima. 2006. Mechanical

and ecophysiological significance of the form of a young acer

rufinerve tree: vertical gradient in branch mechanical

properties. Tree Physiology 26:1549-1558.

3. Mencuccini, M., J. Grace and M. Fioravanti. 1997.

Biomechanical and hydraulic determinants of tree structures in

Scots pine: anatomical characteristics. Tree Physiol 17:105-113.

4. Almeras, T., A. Thibaut, J. Gril. 2005. Effect of

circumferential heterogeneity of wood maturation strain,

modulus of elasticity and radial growth on the regulation of

stem orientation in trees. Trees 19:457-467.

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Stress Analysis Along Tree Branches