- 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

- 2 -

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 -

- 3 -

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 -

- 4 -

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

- 5 -

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.

- 6 -

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

- 7 -

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

- 8 -

∑ 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

- 9 -

∑ 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

- 10 -

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.

- 11 -

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

- 12 -

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

- 13 -

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

- 14 -

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]

- 15 -

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;