- 1 - FINITE ELEMENT ANALYSIS OF A GUITAR NECK Matt Hayes CM2234 2/17/15 ME 422-01 Intro to Finite Element Fund. Professor S. Jones DEPARTMENT OF MECHANICAL ENGINEERING ROSE-HULMAN INSTITUTE OF TECHNOLOGY
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FINITE ELEMENT ANALYSIS
OF A GUITAR NECK
Matt Hayes
CM2234
2/17/15
ME 422-01
Intro to Finite Element Fund.
Professor S. Jones
DEPARTMENT OF MECHANICAL ENGINEERING
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
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Table of Contents Table of Contents ................................................................................................................................... - 2 -
List of Figures ......................................................................................................................................... - 3 -
List of Tables ........................................................................................................................................... - 3 -
Diagram of a Guitar and its Parts ........................................................................................................... - 4 -
Nomenclature ........................................................................................................................................ - 4 -
Introduction ........................................................................................................................................... - 5 -
Model ..................................................................................................................................................... - 6 -
Nut System .......................................................................................................................................... - 6 -
Headstock System ............................................................................................................................... - 8 -
Stress Analysis ..................................................................................................................................... - 9 -
Bending Stress............................................................................................................................... - 10 -
Axial Stress .................................................................................................................................... - 10 -
Total Stress ................................................................................................................................... - 10 -
Finite Element Analysis ........................................................................................................................ - 11 -
3D Solid Models ................................................................................................................................ - 11 -
Materials ........................................................................................................................................... - 13 -
Mesh ................................................................................................................................................. - 13 -
Loading & Constraints ....................................................................................................................... - 13 -
Results and Discussion ......................................................................................................................... - 15 -
Hand Calculation Results................................................................................................................... - 15 -
Finite Element Analysis Results ......................................................................................................... - 15 -
Disagreement .................................................................................................................................... - 16 -
Conclusion ............................................................................................................................................ - 18 -
Appendices ........................................................................................................................................... - 19 -
Appendix A: References ..................................................................................................................... - 19 -
Appendix B: MATLAB code ................................................................................................................ - 20 -
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List of Figures Figure 1: Neck model with machine heads, nut, and strings................................................................... - 6 -
Figure 2: Free body diagram of the nut and string system ..................................................................... - 7 -
Figure 3: Free body diagram of the headstock system ........................................................................... - 8 -
Figure 4: Free body diagram of equivalent neck system ......................................................................... - 9 -
Figure 5: Stresses on differential element at point H.............................................................................. - 9 -
Figure 6: Fully defined Fender Stratocaster neck model [2] ................................................................. - 11 -
Figure 7: Fully defined machine head model [3] ................................................................................... - 11 -
Figure 8: Simplifed Fender Stratocaster neck ....................................................................................... - 12 -
Figure 9: Simplified machine head model ............................................................................................. - 12 -
Figure 10: Final mesh used in finite element analysis ........................................................................... - 13 -
Figure 11: Loading model in Ansys ........................................................................................................ - 14 -
Figure 12: Loading on an actual Stratocaster [7] .................................................................................. - 14 -
Figure 13: Total deflection plot of guitar neck ...................................................................................... - 15 -
Figure 14: Normal stress in the X-direction plot displayed on break surface ........................................ - 16 -
Figure 15: Likely stress concentration at back of nut region ................................................................. - 17 -
List of Tables Table 1: String attributes used in the model ........................................................................................... - 7 -
Table 2: Nickel material properties [5].................................................................................................. - 13 -
Table 3: Sugar Maple (Acer Saccharum) material properties [6] .......................................................... - 13 -
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Diagram of a Guitar and its Parts
[1]
Nomenclature The first time any nomenclature is used in this document, the word will be bolded.
Headstock – The part of the guitar at the end of the neck where the strings wrap around the
machine heads.
Nut – The small piece of bone or corian which the strings slide over where the neck meets the
headstock
Luthier – A stringed instrument maker
Scale length – The length of the guitar neck from nut to bridge. The length of string free to
vibrate
Machine head – The machine on the headstock which can be turned to tune the string up or
down
Strings – The metal wires which are tensioned between the machine heads and bridge and
vibrate to create the sound of the guitar
Truss rod – A supportive metal rod installed in modern guitar necks to allow adjustment of the
relief (or bend) of the neck
Frets – The metal bumps on the neck which strings are held to in order to shorten the string
and raise the pitch of the vibration
Fingerboard/fretboard – The face of the guitar neck where the frets are located and which the
strings cover
Relief – The lengthwise forward curvature of the guitar neck
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Introduction Guitars, acoustic and electric, are beautiful looking and sounding instruments which have
become a staple in modern music. All too often, though, guitars suffer from broken necks. It is
generally regarded as the most common catastrophic failure experienced by guitars. Necks
have a tendency to break in the region where the neck meets the headstock, referred to as the
nut region. This fact is especially important to those who make guitars. Guitar manufacturers
and luthiers, both, would benefit from a neck design which is less likely to snap.
This tendency of guitars to break in the nut region is hypothesized to be caused by excessive
normal stress in the X-direction of the region. Although this stress can be calculated by hand,
finite element analysis lends itself well to this problem. Guitar necks have varying cross sections
which makes manual calculations of polar moment of inertia difficult. Additionally loads in an
actual neck distribute unevenly across the cross-section, but must be assumed uniform for the
sake of modeling simplicity. By performing hand calculations and finite element analysis for this
problem, normal stress in the nut region of a neck can be analyzed and possibly reduced.
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Model For this study, a standard scale length Fender Stratocaster style neck was selected. This iconic
design is one of the oldest and most popular style electric guitars. It has been produced and
reproduced in such large numbers that it is representative of a large cross section of guitars
used today. The neck to be modeled is shown in Figure 1, below, with machine heads installed
and strings drawn in.
Figure 1: Neck model with machine heads, nut, and strings
Although a 3D model was used extensively throughout the study, analysis was performed in
only the X and Y-directions. Forces in the Z-direction were considered negligible as they do not
contribute significantly to the normal stress in the X-direction.
Nut System For both hand calculations and software analysis, the reaction loading at the nut is needed. To
find this, a system was drawn around the nut and the parts of string that contact the nut. The
nut was assumed to be frictionless so that tensions on both sides of the nut will be equal. Nut
reactions in the X-direction were assumed negligible. It was also assumed that the tensions on
the playable side of the nut acted parallel to the neck and therefore contributed no component
in the Y-direction. Thus only the tensions on the headstock side of the nut will contribute to
loading in the Y-direction. Lastly, the reaction at the nut is assumed to be distributed evenly.
The free body diagram of the system reflecting these assumptions can be seen on the following
page in Figure 2.
X
Y
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Figure 2: Free body diagram of the nut and string system
The string tensions were found in a resource published by d’Addario, an instrument string
manufacturer. The guide shows readers how to calculate the tension according to
𝑇 =
𝑈𝑊 ∗ (2𝐿𝐹)2
386.4
(1)
where T is the tension in the string in pounds, UW is the unit weight of the string in pounds per
inch, L is the length of the string in inches, and F is the frequency in Hertz [1].
In addition to teaching readers how to calculate tension, the guide also provides charts to find
the tension in a string based on which d’Addario string is being used and what note the string is
tuned to. For this reason, d’Addario strings were assumed. Common string gauges were chosen.
String product numbers, notes, pitches, tensions and angles can be found in Table 1, below.
Angles were found using trigonometry and the measurement tool in SolidWorks.
Table 1: String attributes used in the model
String # Musical Note
D’Addario Product
Frequency (Hz) Tension (lbs) [1]
Angles (deg)
1 e’ PL010 329.6 16.2 2.54 2 b PL013 247.0 15.4 3.20 3 g PL017 196.0 16.6 4.02 4 d NW026 146.8 18.4 5.14 5 A NW036 110.0 19.5 6.88 6 E NW046 82.4 17.5 13.49
Knowing the string tensions, static analysis can begin. Conservation of linear momentum
dictates that for static equilibrium:
T1
T1
T2
T2
T3
T3
T2 T4
T2
T4
T2
T5
T2
T5
T2
T6
T2
T6
T2
Fretboard Side
Headstock Side
ϴ1
ϴ2
ϴ3
ϴ4
ϴ5
ϴ6
X
Y
Z
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∑ 𝐹 = 0 (2)
Summing forces in the y direction yields:
𝑃𝐴 − 𝑇1sin (𝛳1) − 𝑇2sin (𝛳2) − 𝑇3sin (𝛳3) − 𝑇4sin (𝛳4) − 𝑇5sin (𝛳5) − 𝑇6sin (𝛳6)= 0
(3)
Solving for P yields a pressure value of 42 psi. This value will be used in the FEA simulation and
the next step of hand calculations.
Headstock System Once all the loads have been determined, the headstock can be analyzed to find the reactions
in the nut region, where guitar legend dictates that necks are most likely to fail. The system was
drawn around the headstock and through the nut region where it should break. The free body
diagram can be found in Figure 3 below.
Figure 3: Free body diagram of the headstock system
Again, conservation of linear momentum was used, this time to sum forces in both X-direction
and Y-direction. Equations 4 and 5 illustrate this.
𝑅𝑥 − 𝑇1cos (𝛳1) − 𝑇2cos (𝛳2) − 𝑇3cos (𝛳3) − 𝑇4cos (𝛳4) − 𝑇5cos (𝛳5)− 𝑇6cos (𝛳6) = 0
(4)
𝑅𝑦 + 𝑇1sin (𝛳1) + 𝑇2sin (𝛳2) + 𝑇3sin (𝛳3) + 𝑇4sin (𝛳4) + 𝑇5sin (𝛳5) + 𝑇6sin (𝛳6)
− 𝑃𝐴 = 0
(5)
Reactions Rx and Ry were subsequently be solved for and found to be approximately 103 psi and
0 psi, respectively.
Finally, conservation of angular momentum is used to find the bending moment acting at the
nut region. Conservation of linear momentum states that for static equilibrium:
T1 T2 T3 T4 T5 T6
P*A
Rx
Ry
Rx
Meq
ϴ6 ϴ5 ϴ4
ϴ3 ϴ2 ϴ1
xi
Ytuner\break
Xnut\break
Y
X
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∑ 𝑀𝑂 = 0 (6)
Summing moments about the break at the nut region yields
𝑀𝑒𝑞 − 𝑃𝐴𝑥𝑛𝑢𝑡\𝑏𝑟𝑒𝑎𝑘
+ 𝑦𝑡𝑢𝑛𝑒𝑟/𝑏𝑟𝑒𝑎𝑘[𝑇1 cos(𝛳1) + 𝑇2 cos(𝛳2) + 𝑇3 cos(𝛳3) + 𝑇4 cos(𝛳4) + 𝑇5 cos(𝛳5)
+ 𝑇6 cos(𝛳6)]− 𝑥1𝑇1sin (𝛳1)−𝑥2𝑇2sin (𝛳2)−𝑥3𝑇3sin (𝛳3)−𝑥4𝑇4sin (𝛳4)−𝑥5𝑇5sin (𝛳5)−𝑥6𝑇6sin (𝛳6)= 0
(7)
Which can be rewritten as
𝑀𝑒𝑞 − 𝑃𝐴𝑥𝑛𝑢𝑡\𝑏𝑟𝑒𝑎𝑘 + 𝑦𝑡𝑢𝑛𝑒𝑟/𝑏𝑟𝑒𝑎𝑘 [∑ 𝑇𝑖 cos(𝛳𝑖)
6
𝑖=1
] − ∑ 𝑥𝑖𝑇𝑖sin (𝛳𝑖)
6
𝑖=1
= 0 (8)
Solving for Meq yields the moment acting at the break to be around 40 inlbs counter clockwise .
This is the value that will be used to calculate the bending stress in the nut region.
Stress Analysis Once reactions at the break have been found, stress in the nut region can be analyzed. Because
fractures occur at cracks and because cracks cannot propagate under compression, we are
most likely to see crack formation on the back of the neck where it is in tension due to bending.
A differential element was placed at point H on the back of the neck in the nut region where
the neck has the smallest cross-sectional area. Point H can be seen on the free body diagram of
the entire system below.
Figure 4: Free body diagram of equivalent neck system
Normal stresses in the X-direction on this differential cube are comprised primarily of bending
stress and axial stress as shown in Figure 5 below. Shear stresses were considered negligible.
Meq
Rx
Ry
Rx
Point H Y
X
Y
X
σaxial σaxial
σbend σbend
Figure 5: Stresses on differential element at point H
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Bending Stress The bending stress at point H was calculated using
𝜎𝑏𝑒𝑛𝑑 =
𝑀𝑒𝑞𝑦𝑚𝑎𝑥
𝐼=
(40.95𝑖𝑛𝑙𝑏𝑠)(0.388𝑖𝑛)
0.04 𝑖𝑛4= 397.2 𝑝𝑠𝑖
(9)
where Meq is the moment at the break, ymax is the furthest distance between the neutral axis
and the edge of the cross section, and I is the polar moment of inertia of the cross section
about the Z-axis.
Axial Stress The axial stress at point H was calculated using
𝜎𝑎𝑥𝑖𝑎𝑙 =
𝑅𝑥
𝐴𝑏𝑟𝑒𝑎𝑘=
−103.5𝑙𝑏𝑠
1.01 𝑖𝑛2= −101.6 𝑝𝑠𝑖
(10)
Where Rx is the reaction in the X-direction at the break and Abreak is the cross sectional area in
the YZ plane of the neck.
Total Stress The total normal stress in the X-direction of the neck at point H was found by summing the
bending and axial stresses according to
𝜎𝑥,𝑛𝑜𝑟𝑚𝑎𝑙 = 𝜎𝑏𝑒𝑛𝑑 + 𝜎𝑎𝑥𝑖𝑎𝑙 = 397.2 𝑝𝑠𝑖 − 101.6 𝑝𝑠𝑖 = 295.6 𝑝𝑠𝑖 (11)
taking care to account for the direction of the stresses.
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Finite Element Analysis 3D Solid Models Solid modeling files for a fully defined neck and fully defined machine heads were acquired
from grabcad.com [2] [3]. The original parts can be seen in figures 6 and 7, below. These models
could not directly be used in the finite element analysis because they include details which are
unnecessary or cause errors in Ansys. For the files to be used, they must first be simplified.
Figure 6: Fully defined Fender Stratocaster neck model [2]
Figure 7: Fully defined machine head model [3]
The simplified guitar neck can be seen in Figure 8, on the following page. The neck was
simplified by suppressing a number of features. One such feature is a large cavity located on
the back of the neck model. In a functional modern guitar, this space would house the truss
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rod. However, this isn’t a necessary feature of a guitar since the first truss rods didn’t appear
until the early 20th century and guitar-like instruments with long, fretted necks have existed
since the 12th century [4]. The fret slots on the fingerboard and the mounting holes on the back
of the neck were also removed because they are unnecessary intricacies with regards to the
normal stress in the neck.
Figure 8: Simplifed Fender Stratocaster neck
The machine heads were also simplified, because Ansys could not successfully import them.
Assuming this was due to the complexities in the functional machine head assembly, a
completely new file was created. The new model took the basic shape of the machine head and
was made to the same dimensions. The simplified model is shown below in Figure 9.
Figure 9: Simplified machine head model
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Materials The simplified solid neck assembly was imported to ANSYS. The neck and machine heads were
assumed to be sugar maple and nickel, respectively. The values used for these materials’
properties can be found in the table, below. Maple, being orthotropic, had many more
necessary properties than nickel, which is isotropic
Table 2: Nickel material properties [5]
Table 3: Sugar Maple (Acer Saccharum) material properties [6]
Ex Ey Ez νxy νyz νxz Gxy Gyz Gxz
Maple 12.6 GPa
1.66 GPa
819 MPa
0.424 0.774 0.476 1.40 GPa
793 MPa
793 MPa
Mesh The model was initially solved with the default mesh. The FEA results using this default mesh
seemed reasonable. To be sure the solution had converged, the mesh was repeatedly refined in
important areas such as the back and sides of the neck and in the nut region. Once the results
no longer changed significantly between refinements, the solution had converged and the mesh
didn’t need to be any finer. The final mesh can be seen in Figure 10, below.
Figure 10: Final mesh used in finite element analysis
Loading & Constraints A diagram of the neck’s loading can be found in Figure 11, on the next page. Guitar necks are usually
fixed to their bodies by screws or glue. In either case, this feature can be modeled in Ansys using a fixed
support on the part of the neck which normally contacts the body. String tensions which act on the
machine heads were modeled using point forces defined by X and Y components. The magnitude of the
Young’s Modulus E Poisson’s Ratio ν Nickel 207 GPa 0.31
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forces are equal to those listed in Table 1. A pressure due to the force of the strings over the nut is
applied to the appropriate region.
Figure 11: Loading model in Ansys
The actual loading of a functional Stratocaster can be seen in Figure 12, below. It can be seen
that the model’s loading is consistent with the way strings are typically wrapped around machine
heads.
Figure 12: Loading on an actual Stratocaster [7]
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Results and Discussion Hand Calculation Results Using hand calculations, normal stress in the guitar neck at point H on the back of the nut
region was found to be 295.6 psi tension. This total normal stress in the X-direction is made up
of a bending stress of 397.2 psi tension and an axial stress of 101.6 psi compression.
Finite Element Analysis Results An easy first check of a finite element solution is total deformation. If the system turns out
more or less deformed than expected, it may be a quick first indicator that something is
incorrect. The total deformation plot of the guitar neck can be found in Figure 13, below. The
deflection at the nut is 4.2 thousands of an inch. This seems reasonable as the right relief for an
electric guitar is 7 thousands of an inch measured at the seventh fret [8]. Because the distance
between the strings and the neck is greatest at the seventh fret, it should be a little larger than
the deflection at the nut.
Figure 13: Total deflection plot of guitar neck
The solution to the original problem is not deflection, however. It is normal stress in the X-
direction and at point H specifically. To see this, normal stress in the X-direction was plotted on
a mid-section surface located at the break. Figure 14, on the next page, shows the normal stress
plot. It is apparent that bending stress is occurring because stress on the back is tensile and
stress on the fretboard side is compressive. The stress at point H is near-maximum at
approximately 530 psi tension. The maximum compressive stress occurs near the fretboard on
the E-string side of the neck. This is likely due to the E-string have the steepest angle of descent
of all the strings.
Δynut = 0.0042in
Y
X
- 16 -
Figure 14: Normal stress in the X-direction plot displayed on break surface
Disagreement The hand calculations and finite element results are fairly different. The hand calculation of
normal stress, at 295.6 psi tension, is 44% smaller than the finite element result of 531.5 psi
tension. There are many reasons this could be. The two most likely ones are 1) that axial stress
varies over the cross section and cannot accurately be modeled as uniform and 2) Stress
concentrations are too close to the break for the simple bending model to apply.
Because the neck has an inconsistent cross section in the nut region and because the strings act
so far off the front face of the guitar, the axial loading of this cross section may not be uniform
as it’s assumed in the hand calculations. This means the axial stress near the fretboard side of
the cross section may be higher than near the back side of the cross section. Because the axial
and bending stresses at point H work against each other, less axial compressive stress on point
H means more tensile bending stress.
Again because the neck has an odd shape in the nut region, it may be experiencing a stress
concentration there. If this is the case, the already high stress in the area is being amplified due
to the geometry of the neck. Additionally, stress concentrations tend to double or triple the
stress, which is what is seen between the FEA and hand calculation results. Figure 15, located
on the next page, is a plot of the normal stress in the X-direction all along the neck. A red patch
of high stress can be seen in the nut region with a maximum of around 760 psi. This is a further
indication that a stress concentration is increasing stress in the region.
- 17 -
Figure 15: Likely stress concentration at back of nut region
- 18 -
Conclusion According to general musician consensus, neck breaks are widely considered the most common
catastrophic failure of guitars. The tendency is for breaks to occur where the neck meets the
headstock in the nut region. People who make guitars like luthiers and manufacturers could
benefit from a neck design which is less susceptible to breakage. It was hypothesized that this
tendency to break in the neck region was due to excessive normal stress in the X-direction.
Hand calculations of stress yielded a normal stress of 295.6 psi tension, comprised of a 397.2 psi
tensile bending stress and a 101.6 psi compressive axial stress. FEA results found the total
normal stress in the X-direction to be 44% higher at approximately 530 psi. Due to the shape of
the guitar neck and the nature of its loading, it’s most likely that there is a stress concentration
acting at the nut region which is making the normal stress much higher than predicted by the
hand calculations.
Excessive normal stress in the X-direction is most probably what makes guitar necks more likely
to break at the nut region than anywhere else. Fracture begins with crack propagation, and
crack propagation forms under excessive tensile stress. From Figure 15, the normal stress in the
nut region on the surface could be as high as 760 psi. According to the MatWeb material
property database, the ultimate tensile strength of Sugar Maple (Acer Saccharum) is 770 psi [9].
So with only an additional 10 psi of tension, a crack could form in the neck causing it to fail in a
brittle and instantaneous manner. This observation explains why it seems that all it takes to
snap a guitar neck is a solid fall forward out of a guitar stand.
- 19 -
Appendices Appendix A: References [1] partsofaguitar.com. “Parts of a Guitar.” [Online]. Available: http://www.partsofaguitar.com/
[2] J D’addario & Co, Inc. “A complete technical reference for fretted instrument string
tensions.” [Online]. Available: http://daddario.com/upload/tension_chart_13934.pdf
[3] Mitchell, Jason. Stratocaster Style Guitar Neck. (2011). [Online]. Available:
https://grabcad.com/library/stratocaster-style-guitar-neck
[4] Golphin, Christopher. Guitar Machine Head. (2014). [Online]. Available:
https://grabcad.com/library/guitar-machine-head-2
[5] Wikipedia.org. “Guitar.” [Online]. Available: http://en.wikipedia.org/wiki/Guitar
[6] MatWeb.com. “Nickel, Ni.” [Online]. Available:
http://www.matweb.com/search/DataSheet.aspx?MatGUID=e6eb83327e534850a062dbca3bc
758dc
[7] Kretschmann, David. “Mechanical Properties of Wood.” [Online]. Available:
http://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr190/chapter_05.pdf
[8] strat-talk.com. “2 String Trees vs. 1.” (2010). [Online]. Available: http://www.strat-
talk.com/forum/stratocaster-discussion-forum/47677-2-string-trees-vs-1-a.html
[9] Guitarrepairbench.com. “Truss Rod Adjustment.” [Online]. Available:
http://www.guitarrepairbench.com/electric-guitar-repairs/adjust_truss_rod.html
[10] MatWeb.com. “American Maple, Rock (Sugar Maple).” [Online]. Available:
http://www.matweb.com/search/DataSheet.aspx?MatGUID=e30c1ad86e814c359e61b4c34490
09bb&ckck=1
- 20 -
Appendix B: MATLAB code %MSH 2/18/15
%FEA FINAL PROJECT
clc
clear all
close all
%Parameters
w = 0.14;
l = 1.68;
h = 0.13;
Axz = l*w;
Ayz = h*l;
xNutBreak = 0.523;
yMid = 0.097;
xdist = [6.76 5.83 4.91 3.96 3.16 2.10];
%strings denoted e' b g d A E
T = [16.2 15.4 16.6 18.4 19.5 17.5];
%STRUNG LOW
%Nut/Machine Head Measurements
xshort = [6.16199 5.22959 4.29719 3.36479 2.43240 1.50000];
yshort = [0.56312 0.58248 0.59236 0.59275 0.58365 0.56506];
%Tension Angles
thetaShort = atand(yshort./xshort);
%Tension Components
TxShort = T.*cosd(thetaShort);
TyShort = T.*sind(thetaShort);
%Nut Pressure
PShort = sum(TyShort)/Axz;
%STRUNG MID
%Nut/Machine Head Measurements
xmid = [6.16199 5.22959 4.29719 3.36479 2.43240 1.50000];
ymid = [0.27312 0.29248 0.30236 0.30275 0.29365 0.27506];
%Tension Angles
thetaMid = atand(ymid./xmid);
%Tension Components
TxMid = T.*cosd(thetaMid);
TyMid = T.*sind(thetaMid);
%Nut Pressure
PyMid = sum(TyMid)/Axz;
- 21 -
%Equivalement Moment at Break
Mmid = PyMid*Axz*xNutBreak - yMid*(sum(TxMid)) - sum(TyMid.*xdist)
%STRUNG HIGH
%Tension Angles
thetaHigh = [0.94 1.33 1.76 2.27 2.96 4.18];
%Tension Components
TxHigh = T.*cosd(thetaHigh);
TyHigh = T.*sind(thetaHigh);
%Nut Pressure
PHigh = sum(TyHigh)/Axz;