Post on 08-Dec-2021
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
Riverdale, NY, United States
Zahra Shahbazi Manhattan College, Mechanical Engineering
Riverdale, NY, United States
Lance Evans Manhattan College, Biology Department
Riverdale, NY, United States
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
2 Copyright © 2014 by ASME
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)
3 Copyright © 2014 by ASME
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)
4 Copyright © 2014 by ASME
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)
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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)
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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)
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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
versus maximum stress.
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
versus maximum stress.
8 Copyright © 2014 by ASME
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.
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