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This is my article on the guitar as a structure. This was originally published by the Guild of American Luthiers in American Lutherie #100, in 2009. (Their graphics are better than mine!)
All content on this page: © Copyright 2009 James W. Blilie, 2009
The Guitar As A Structure (And Some Practical Information On Bracing)
I
am a structural engineer and have been working in the civil engineering,
aviation, and medical device industries for 25 years.
My engineering work has been, in essence, ensuring that structures are
strong (or stiff) enough. I am also
a guitar maker since 1998 (I’m building my 59th guitar) and a
fingerstyle guitar player. I have
found that my engineering experience has helped my guitar building in many ways,
such as reading plans, making jigs and tools, and understanding the stiffness
and strength of the guitar. This is
my $0.02 as an experienced structural engineer.
Figure 1 - The Guitar Idealized As a Simple Structure
What is a structure and why is the guitar a structure?
Basically every thing in the world that stands up on its own is a
structure. A house is a structure.
So is the
Measuring Structural Factors. To explain how the stiffness of the
braces (and the rest of the guitar) works, I need to introduce some engineering
measurements that are used to define the stiffness of structures.
These measurements are section properties and material properties.
Section properties are inherent to the shape of the structure.
Material properties are inherent to the material it is made of.
They act together to define strength and stiffness.
Because engineers understand section properties and material properties
(and properly apply them), buildings don’t fall down and airplane wings
don’t fall off.
Most
people have a tactile, “horse-sense” understanding of section properties:
If a wood plank is taller or wider, it will be stiffer (all other
things being equal – this caveat applies throughout; I’ll not repeat it.
In real life, things are never equal.)
A thick Section
properties are simply the exact way that engineers use to measure a “stiffer
than” or “stronger than” structural shape.
Section properties are determined exclusively by the shape and
size of the structure. Section
properties are not affected in any way by the material – this is the key
point. A two-inch by four-inch plank
will have the same section properties whether it is made of steel or plastic or
butter. Section properties are essentially
perfectly understood by engineers and have been tested and confirmed
millions of times. They are simple
to compute and formulas may be found in the suggestions for further reading
(SFR) at the end of this article. Please note that a number of topics that are
introduced in this article are discussed in much more detail in some of these
suggested readings. Separate
from section properties the other
important factor is material properties.
A two-inch diameter steel pipe is much stiffer than a two-inch diameter
plastic pipe (both pipes have identical inside diameter (ID) and outside
diameter (OD)). This is the material
effect. Because the ID and OD are
identical these two pipes have exactly
the same section properties; they are different in stiffness (and
strength) because they are made of different materials that have different
inherent material properties. These inherent material properties are also
precisely measured by engineers. The
primary material property of interest here is the modulus
of elasticity, abbreviated as E,
and generally referred to as “Young’s
Modulus.” The details on
Young’s Modulus and how it is determined and used can be found in the SFR. The
material properties of wood are quite variable and hard to predict.
The difference in stiffness and density from one piece of (one species
of) wood to another can be dramatic. Therefore,
no formula for sizing wood parts is practical where stiffness is critical.
Extensive testing of each individual piece of wood would be required. For
wood (and most other materials), the strength and stiffness vary in a linear way
with density (specific gravity). See
the table of density versus strength for various wood species below, and the
plot in fig. 2. The plot is a
straight line. The averaged data are
linear. In more homogeneous
materials such as metal, the data conform even better to a straight line.
Table 1 lists strength and stiffness material properties for common
lutherie woods . These values were
taken from an online reference from the U.S. Forestry Service, USFS
Wood Research Lab: http://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr113/fplgtr113.htm.
Note that the Young’s modulus listed is for the longitudinal direction (with
the grain) since this is the only direction of interest here.
This is because all the main loads under consideration are along the
length of the grain (with the minor exception of cross-grain deflections of the
top; but these are driven more by the longitudinal deflections than by stiffness
across the grain.) Another reference
work, Wood Handbook:
Wood As an Engineered Material:
http://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr113/fplgtr113.htm
contains an extensive table of values (pp. 4-4 through 4-24) and also has
nice discussions of wood strength behavior. Table
1 – Wood:
Material
Properties
Figure
2
– Wood:
Density versus Inherent
Stiffness for Various Species The fact
that the relationship between stiffness (Young’s Modulus) and density is
inherently linear shows that just changing wood species doesn’t affect the
stiffness to weight ratio very much. The
section properties of interest for us are the bending
modulus (abbreviated as I; also
sometimes called the “bending moment of
inertia” or “second moment of
inertia”, see SFR) and the area (abbreviated as A,
sometimes called “cross sectional
area”). The area is simple for
rectangular cross sections: A two by
four plank has area A = 2 inches X 4
inches = 8 inches2. Cross sectional area affects the strength and
stiffness of a structure when it is pulled (or pushed) along its length.
Imagine a fishing line holding up a weight.
Think of 3 pound test versus 100 pound test monofilament fishing line.
The 100 pound test line has greater cross sectional area than the 3 pound
test line – approximately 33 times the area; but the material is the same.
The greater strength (and stiffness) of the 100-pound line is due to it
having larger cross sectional area, its section property is larger. The
bending modulus is a little more complex and has units of inches to the fourth
power (in4). This is
described in some detail in some of the SFR.
Think of bending a 2 by 8 plank flat-ways (floor board); then think of
bending it tall-ways (floor joists). It
takes much more force to bend it tall-ways, even though the area (and weight) is
the same in both cases (8 X 2 = 2 X 8 = 16 in2).
This difference in stiffness is the effect of the bending modulus.
An
example: If you take that 2 by 8
plank on edge and replace it with a 4 by 8, on edge, the area is twice as large.
And, the bending modulus is also twice as large – because the height
did not change. If you take that 2
by 8 plank and replace is with a 2 by 16, then, again, the area doubles.
But, this time, because the height doubled, the bending modulus goes up
by a factor of 8 (23 = 8). Note
that the weight of both the 4 by 8 and the 2 by 16 is the same (A
= 32); but the 2 by 16 is 4 times as stiff as the 4 by 8 when both are oriented
in the tall-ways direction. Weight
follows area (A) directly. The
original (2 X 8) beam: I1
= 1/12 (2)(8)3 = 85.3 in4 A1
= (2)(8) = 16 in2 The
4 X 8 beam: I2
= 1/12 (4)(8)3 = 171 in4 A2
= (4)(8) = 32 in2 The
2 X 16 beam: I3
= 1/12 (2)(16)3 = 683 in4 A3
= (2)(16) = 32 in2 This
shows how important height is for bending stiffness.
A guitar top ¼ inch thick is 19 times as stiff in bending as a top that
is 3/32 inches thick, while it is only about 3 times as
heavy (assuming the same wood – a big assumption in real life.)
Guitar top vibrations are essentially governed by bending stiffness and
weight of the top and braces. A little brace height or top thickness goes a long
way. Graphic Data On Brace Stiffness. In Figs. 3 and 4 you see section properties plots
for four brace shapes: Rectangular,
triangular, trapezoidal (top width = ½ bottom width), and “paraboloid”
which is the curved shape that most builders use, more or less.
What you see is that the area (and weight) of the brace changes
“directly” with the width of the brace (twice the width gives twice the area
and twice the weight). The bending
stiffness of the brace changes by the third power of the height (they fall on
slightly different curves because the different shapes mean different formulas,
although still proportional to h3.)
The thing to remember is that the effect of brace height on stiffness
is very strong and much bigger than the effect of height on weight
(area). If you want a light,
stiff brace, make it tall and narrow. If
you want it less stiff and maybe heavier, then make it wider and shorter.
Figure
3
– Plot of Brace Height Versus
Figure
4
– Plot of Brace Height versus Cross-sectional Area
What Do Changes In Section Properties Mean? How does this math and technical junk
apply to guitars? As shown above,
the thickness (section height) of the top is the dominant factor for stiffness.
The same thing is true of the braces and the neck and the fingerboard.
If you shave the height of a brace down from ½ inch to ¼ inch, then you
have reduced the bending stiffness of that brace by a factor of 8 – it’s now
1/8th as stiff as it was before and the weight
is ½ what it was before. If you
left the brace height alone and made the brace ½ as wide as it was before, then
the stiffness is ½ what it was before and the weight is also ½ what is was.
Figure 5 – Relationship of Brace Height to Brace Stiffness and Weight
Figure
5 shows the change in bending stiffness for change in brace height for braces of
a constant area, and therefore constant mass. Stiffness and Vibration Response. The big question is: What does changing the stiffness of the top, sides, back, neck, and braces do to the sound of the guitar? A good friend of mine is an engineer and an expert on vibrations. She gave me a simple way to think about vibration and stiffness. If you take nothing else from this article, I hope you find this useful: If you want something to vibrate at a
higher frequency,
then you should make it stiffer or lighter, or both. If you want to make something vibrate at a
lower frequency,
then you should make it less stiff or heavier, or both. Please
note: As you make something stiffer,
you tend to make it heavier (adding material to add stiffness) – this works
against high frequencies (lighter would be better).
And, as you make something less stiff, you tend to make it lighter
(removing material to reduce stiffness) – this works against low frequencies
(heavier would be better, all things being equal). Therefore, the stiffness and
weight relationships to height and width in Fig. 5 are important.
In general, if you want to make a plate (guitar top) vibrate at higher
frequency, make it thicker and use tall, thin braces, which are lighter for the
same bending stiffness. If you want
to make a plate vibrate at a lower frequency, make it thinner and use wider,
flatter braces, which are heavier for the same bending stiffness. How Stiff/Strong Should I Make the Brace? In his article entitled “On
the Selection and Treatment of Bracewoods”, American
Lutherie #60, Ervin Somogyi wrote, “… I’d been careful only to
use vertical-grained material [for guitar braces].
That was, after all, what everyone had told me to do, since the very
beginning. But I’d not done
anything to determine whether any selections of vertically-grained woods really
were the stiffest. I did know, from
previous experience of overbuilding, that you could make things too stiff.
And, likewise, too loose. So,
where to draw the line?” Exactly;
there’s the eternal question: How
stiff? After hearing the information presented here, luthiers always ask,
“so what brace size/shape/height/width/material should I use?”
I always answer with a question, “What do you want the brace to do?”
The braces at the neck end of the top are there for strength.
Upper bout transverse bars, usually the heaviest braces in the guitar,
keep the upper bout area from collapsing under the string forces.
The braces around the sound hole reinforce this big weakness in the top,
which is right in the line of the force of the strings.
(You would have a hard time picking a worse place to put that hole, from
a strength standpoint.) The other
braces, in my opinion, are there to control the vibrations of the top
and, to a much lesser extent, to react the twist of the bridge (for strength.)
You could make a guitar with no top braces.
It would have to have a very thick top to not collapse.
It probably wouldn’t sound very good.
Clearly, some sharing of the load by the braces is needed in order to
free other parts of the top to move more. The
top (and back) is a stiffened-shell type structure, like most structures that we
use (airplane fuselage, airplane wing, house walls, floors and roofs, car
bodies, boat hulls, etc.), for a very good reason.
A stiffened shell is the most weight-efficient way to make these
structures. I
have no formula for a good sounding top/guitar.
I use my ears and tap parts all through the stages of construction so I
can hear how they are reacting to vibration.
I think my guitars succeed quite well:
I am pleased with the volume, tone, and overall effect.
But I have no formulas. I
flex and tap, flex and tap – and listen. What Shape Brace to Use? The choice of brace shape (rectangular, trapezoidal,
triangular, “paraboloid”, whatever) is purely a matter of personal choice
and working methods. None of them
would be noticeably superior for sound quality.
Observe all the different methods that have been used to brace very
successful guitars: X-brace,
fan-brace, ladder-brace, lattice bracing, double-tops with foam or honeycomb
cores, Kasha-style bracing, etc. There
are many ways to skin this cat; there is no silver bullet. A
“paraboloid” (more or less a parabola or half-oval in cross section) brace
shape looks nice, is easy to form, does not induce stress concentrations, and is
resistant to dent damage (no sharp corners); but these imply nothing about its
acoustic performance. I use
paraboloid shaped braces most of the time. But
for some designs, I use rectangular or triangular brace shapes.
You see every possible brace shape in successful existing guitars.
A rectangular brace or brace of other shape, as long as it has the same
stiffness and weight and has the same “footprint” (attachment glue area) on
the top should give an identical effect.
Special
Case: I-Beams. One special case
of stiffness is the I-beam, which you have seen in steel construction.
The I-beam places most of the working material far from the center of the
cross-section. The
I-beam section generally gives the best strength and stiffness for a given
weight of material, given practical considerations like fastening it into
the structure. That’s why it is so
common in steel construction: The
big cost driver in steel construction is weight of steel. Most
of the time, lighter is better, all things being equal.
Some people use I-beam shaped braces in guitar construction.
If you want really light and really stiff, this is a good way to go. You
can achieve the same effect as an I-beam by various techniques.
You can build (by gluing strips) or carve a true I-beam shape.
You can also laminate a brace. If
you laminate a much stiffer material (carbon fiber, hardwood, fiberglass, etc.)
to the top and bottom of basic
softer brace material, you have essentially the same thing as an I-beam.
(The top wood itself can act as an I-beam flange; but adding a stiffer
laminate works better.) It looks
like a rectangular section, but because the top and bottom laminates are so much
stiffer, you can think of them as being wider (not taller) than the rest
of the brace. It’s important to
laminate the stiff material to both the top and the bottom of the brace.
Otherwise, you will add little to the stiffness and will just shift the
stiffness center of gravity of the section – for which there is no
“luthierly” reason. Strength
and Structural Failure. Strength follows stiffness very closely.
People generally think of strength and stiffness as the same thing.
The important aspect of strength is:
How strong is the
structure compared to the load
it must carry? The usual ways that
guitars fail are well known. The
headstock breaks off – the crack usually starts right under the nut, the
bridge pulls off, the top cracks parallel with the grain, the neck block slips
raising the action. The guitar has
plenty of strength to carry the loads of the strings and general handling –
the loads for which it is designed. I
have run the analysis on the normal loads on a guitar and the applied stresses
are very low, far below the capacity of wood and glue. The margin of safety is
large. Guitar failures occur when
loads or conditions are applied that the guitar was not designed for. In my experience, guitar structural
failures are caused by two things: Accidents
and weather. Any of the thin plates
on a guitar are obviously susceptible to impact damage.
If they are pressed in a direction they weren’t designed for, they will
break. The
headstock breaks off because the guitar gets knocked off its stand and gets
whacked in a direction it was never intended to take a load (accident).
It breaks at the nut because that is usually the smallest cross-section
of the neck – it has the least strength at that location.
There’s also often a glued joint at the nut area.
Joints are often structural failure points.
They are natural failure points because of changes in section, changes in
materials, and stress concentrations. The
top cracks because it was taken into too low a humidity environment (weather).
The wood tries to shrink with the dropping humidity but is prevented from
doing so. It cracks to relieve the
stresses induced by the shrinkage. Cross-grain
(transverse) strength is much lower than strength in the grain direction
(longitudinal.) The
bridge pulls off because the guitar was left in the car (“just a few
minutes,” “I came right back,” “it wasn’t that hot out” –
this is weather.) The glue was
heated to the point where its strength fell below the strength required to react
the string tension. Essentially all
materials get weaker as you heat them. I
never leave a guitar in a car – it will either be stolen or damaged by
heat/cold. A good rule of thumb:
If the temperature and humidity are comfortable for you (want to sit in
that trunk for 3 hours?) then they are OK for your guitar.
If not, don’t do that to your guitar. Slow
permanent deformation of structure under load is called creep.
A neck slip might be caused by creep; but this is very unlikely in my
opinion because glues used in guitars are not very susceptible to creep and the
typical applied stresses are very low. Instead,
this is almost certainly heat and humidity cycling working on the glue, or a
one-time high temperature event (“guitar in the car” again). Choose
good quality wood. Use fresh, good
quality glue. Make your joints clean
and tight and well clamped. Keep
your plate thicknesses, neck section, and brace sections within the normal
proven ranges. You will never have
strength problems. Loads
and Stresses on the Guitar. We are going to look at the internal loads and
stresses on some significant locations on the guitar.
Before tackling these, we need to identify the internal loads applied to
the guitar. The applied load on the
guitar is the tension of the strings. This
load is distributed (carried) through the guitar as internal loads:
Axial loads, shear loads, and bending loads
(also called moments), which are
defined briefly below and are discussed in more detail in the SFR.
Note
that all objects deflect (compress, stretch, or bow/bend) when loaded.
The way they carry the load is by deflecting:
like a spring. These
deflections are mostly so small (Young’s modulus is large compared to the
load) that you don’t notice them. Deflection
of the guitar (the top, mostly) is what produces the sound.
No deflection, no sound. But
excessive deflection will change the load paths, often resulting in collapse. Internal
loads are the loads inside the structure, not applied from outside. The internal
loads react the loads coming from outside the structure.
As mentioned, there are three types of internal load. Axial load is
simple: It is pull or push (tension
or compression) acting along the length of an object.
This is tension in a rope, compression in a column.
Shear is also simple, at least in its simplest form. It’s the load
acting perpendicular to the length of an object.
Shear is a little harder to explain because most people don’t have an
intuitive feel for it. Shear loading is what makes your fence gate sag out of
true and become a rhombus shape (not 90-degree corners), as in Fig. 7.
Figure
7 – Shear Loading and Deflection
Figure
8 – Cantilever Beam:
Illustrates Stresses
are computed using the internal loads (PAXIAL, PSHEAR, M)
and the section properties A and I. Axial
stress is computed simply as the load (P, pounds) divided by the area (A, in2):
P/A = stress, units pounds per square inch (psi).
Shear stress is also computed as
P/A, units psi (symbol t, refer to Figure 7). Figures
9 through 14 give detailed internal loads and stresses for important locations
on the guitar. Note that all
stresses are rounded to 2 significant figures.
Peak compression stress (at top
edge of fretboard) = ~640 psi (bending
+ compression) Peak tension stress (at bottom of
neck) = ~490 psi (bending +
compression) Assumes entire neck section is
effective and no support from the truss rod. Stresses similar to those quoted
for Figure 9. Figure
11
– Loads On the Neck At the Heel FAXIAL = Compressive
stress in the guitar top at neck joint: ~820
psi Assumes 2-inch width of the top is
effective: A = 2.0*0.10 = 0.20 in2
P/A = 163/0.20 = 820 psi FSHEAR = Shear stress
in the glue at the heel: ~30 psi
(assumes 1 in2 effective) Top stress similar to Figure 11.
Back and sides stresses: Very
low. FSHEAR = Shear stress
in the sides = ~ 163#*sin(2degrees)/((0.07 inch)(3.5 inch))/2(two sides) = 12
psi
Figure
13
– Loads on the Guitar Top at the Sound Hole (top view) Open hole stress concentration
factor = ~3.0 FAXIAL = Compressive
stress in the guitar top near the soundhole = ~3.0 * 820 = 1600 psi
Figure 14
– Loads on the Body at the Bridge (top, top brace, and bridge only shown) Shear
stress in the bridge glue: ~33psi Peak
tension stress in the bridge glue: ~98psi Assumes
1 inch long by 5 inch wide bridge contact; string
height above top at saddle = 0.5 inches FSHEAR
= 163#/5.0 in2 = 33 psi FBEND
= Mc/I = 82 in-#(0.5 in)/(5/12 in4) = 98 psi Note
that I do not calculate the bending stresses in the top and top braces.
These stresses are highly dependent on the design of the guitar and will
therefore vary a great deal. (They
are also too complicated for the scope of this article.)
However, I have run some numbers and the stresses are low compared to the
strength of the wood and glue. Grain
Orientation. I have often seen the admonition that the grain direction in
the braces must be vertical – 90 degrees to the surface of the top or back
plate. There is no structural
justification for this (except that gluing a plate to the free edges of the
brace grain could conceivably inhibit splitting in the brace.)
Ervin Somogyi’s test data mentioned earlier support this conclusion.
Note that neither of the U.S. Forestry Service documents referenced
earlier lists separate strength values for tangential and radial grain
directions: Just the transverse
direction (perpendicular to the grain.) Very
few braces ever get very far from square proportions (height and width equal):
The height is rarely much more than twice the width.
Therefore, there is no structural reason to choose one orientation or
another. Unless
the grain spacing becomes close to the smallest dimension – what I’ll call
the critical dimension, of the part (brace, top plate, back plate,
whatever) the part can’t tell the difference between one orientation or the
other. I think that the vertical direction is favored for braces because
vertical grain is easier to plane, trim, scallop, and taper and perhaps for
aesthetic reasons. Grain direction
could become significant for very small braces.
Most braces are taller than they are wide.
Therefore a horizontal grain direction, rather than vertical grain
direction, would be more homogeneous. In
contrast with braces, vertical grain becomes very important for tops, sides, and
backs. This is because the grain
spacing becomes close to the critical
dimension (the thickness) of the plate.
If the thin plates of the instrument had flat-sawn grain, the wood would
no longer behave like a homogeneous material.
The difference (in density and strength) between the late wood and early
wood is often very large and would become apparent in uneven stiffness in
different areas of the plate. Surface
Quality. I have regularly seen claims that the smoothness of the internal
parts of the guitar (how smoothly sanded the braces are, etc.) causes
measurable/perceptible differences in sound quality.
Don’t believe it. Fine
sanding of braces is done for reasons of visual or tactile aesthetics or for
traditional concepts of workmanship. There’s
nothing wrong with that. Just
don’t expect any acoustic changes. Whether
you sand to 150-grit, 320-grit, or 600-grit will not affect the stiffness of the
braces and therefore has no way to affect the generation of sound.
The sound frequencies of interest to the human brain have wavelengths on
the order of 1 to 50 inches (frequencies of a couple hundred Hz up to around
10,000Hz). There is no physical
mechanism by which sanding scratches of much less than one one-hundredth of an
inch width/depth could have any perceptible effect on the sound waves.
The sound wave cannot “feel” (be affected by) features this small.
Conceivably, if the surface were left so rough that the bracing was
absolutely fuzzy with wood fibers, it might damp the vibration.
But short of major fuzzy (think fake fur glued on), there will be no
effect on the sound. Tapering,
transitions, and joints. Be very careful with all joints.
They are a natural failure point. Bad
joint design, bad gluing, old glue, bad surface condition, big changes in
section, and poor clamping can all cause a joint to fail.
Very sudden changes in section lead to failure.
This is because of an effect called stress
concentration (see SFR). Just
one example: A 2 X 6 with a 2-inch
hole in its center will fail in tension (pulling straight along its length)
before two 2 X 2 boards of the same material, pulled lengthwise together.
The area is the same; but the hole causes the stress to be higher at the
edge of the hole (imagine water flowing around a rock in a stream) and therefore
the piece with the hole fails at a lower load. Sudden, big changes in section should be avoided.
Smoothly taper joints if possible. Use
long scarf joints. Use a scarf joint
instead of a butt joint. Taper the
ends of braces. Round off the
corners of components. All these
tend to reduce stress concentrations and make a more reliable structure. If you end a brace in the middle of a plate and don’t taper the
end, a lot of that brace mass is carrying no load (useless mass) and the end
creates a stress concentration (sudden change in section) that will likely be
the location of the first crack in the plate. Personal
opinion. I have found this to be true for all structures and for guitars as
well: Lighter really is better, all
things being equal. My experience is
all with acoustic guitars; mass may have some intrinsic benefit in electric
guitars. It seems to me that lighter
acoustic guitars generally sound better than heavier ones.
I have found that the most important factor in achieving a loud,
good-sounding guitar is using top wood that is very stiff, particularly in the
cross-grain direction (others will disagree.)
A couple of my older guitars had significantly heavier tops due to low
stiffness in the top wood. The
resulting guitars did not sound as good as my guitars with stiffer tops. Summary:
Material
properties and section properties are the two components of stiffness and
strength. Changing
materials will change stiffness. Generally,
stiffer materials are also more dense (heavier.) Tall,
thin braces are stiffer relative to their weight than wide, flat braces. If
you want something to vibrate at a higher frequency, make it lighter or stiffer. If
you want something to vibrate at a lower frequency, make it heavier or softer. Grain
orientation isn’t structurally significant until grain spacing approaches the
part’s smallest dimension. Surface
smoothness has no effect on stiffness or strength (within reasonable limits.) Guitars
generally have plenty of strength
to carry their loads. Just don’t
expect them to carry extraordinary loads: From
accidents and weather (excessive heat, cold, moisture, or dryness.)
I think I have indicated how important stiffness
is. Proper tailoring of stiffness
produces a satisfying sound. Suggestions
for Further Structures:
Or Why Things Don’t Fall Down,
J. E. Gordon (there’s a recent printing from 2003 by Da Capo) Engineering
the Guitar: Theory and Practice, Richard Mark French, Springer “Forces
On Archtop Guitars”, Franz Elferink, American
Lutherie Number 74, pp. 30, 31 For
more on bending moment of inertia (I): http://en.wikipedia.org/wiki/Area_moment_of_inertia For
more on bending: http://en.wikipedia.org/wiki/ For
more on stress concentration: http://en.wikipedia.org/wiki/Stress_concentration For
more on shear stress: http://en.wikipedia.org/wiki/Shear_stress For
more on tensile stress: http://en.wikipedia.org/wiki/Tensile_strength |
