Official Luthiers Forum!

Since it's popular topic in a few recent threads, I dug into a prior FEA model of a guitar body to explore pinned vs pinless bridge stresses. Most details on the model are here (and prior pages):
http://www.mimf.com/phpbb/viewtopic.php ... &start=140
Briefly, it is a symmetric 3D model of a guitar body and bridge, with orthotropic material properties (in correct orientations) for top, bridge, bridge plate, and braces. X-braces are forward shifted with typical scallops. Since it is symmetric (to save computation time), the LFBs are purely lateral (like a Larrivee). This model isn't fully vetted yet, but I think it's good enough to get some basic insight.


Here is a 2D view of the simulated loading on a standard pinned bridge (side view). I call this case "J1". There are a bridge pin holes, but no bridge pins (so, loads are like in a slotted bridge).


Here is a perspective view of the J1 loads. Light strings are assumed, using an average of individual treble and bass loads: E & E' = 24.2 lb, A & B = 25.9 lb, D & G = 29.5 lb. The reactions at the ramp are calculated.


Here is case J1's pressure distribution on the soundboard. Red is tensile stress; blue is compressive. Scale is psi. Later I'll show a graph of the stress along the red dotted line. Note that I haven't looked at shear stress yet; it is also important.


Below is the loading to simulate a simplified pinless bridge, case J2. The geometry is identical to J1; only the loads are different.


J2's pressure distribution -- note higher "red" stress at tail, but hold off on interpreting that:


I wondered if J2's higher tail stress was caused by the nearby ball-end loads, especially since they have a vertical component (directed towards the saddle). So I moved that anchor point to the string ramp as a test/exploration case (J4).


Sure enough, tail stress is similar to J1:


The next post will show a comparison graph.

_________________
David Malicky

This graph shows the pressure along the dotted red lines in the pics above:


Theoretically, the pressure goes to infinity at both edges of the bridge. In reality, the wood or glue will experience micro-yielding to resolve the infinite stress. But J2 goes to infinity "faster", so J1 and J4 are better. I.e., if pinless, avoid designs that put the ball ends forces right next to those peak tail stresses, and try to direct the ball end forces purely horizontally, no vertical component. Other comments are on the graph.

_________________
David Malicky

David thank You for this - it's excellent and a lot to take in which is what I will be doing.

Excellent post, one of the best I've seen on the topic.

Great info.

One item remains in my mind and that is the tendency of the top loaded bridge To want to be ripped off the soundboard surface versus the forced coupling of the bridge structure (bridge, top & plate) by the balls on a pinned bridge. Any thought here on the difference in shear forces developed at the bridge/top glue joint?

_________________
Brian

You never know what you are capable of until you actually try.

https://www.howardguitarsdelaware.com/

This is really great! What would be interesting is how the Elliot bridge compares. From a first look, it would have more vertical pull component because of the shallower break angle, but that doesn't take into account the fact that the ball end is captured by the post and isn't really pulling up on the rear of the bridge in the same way as if the ball ends, as in the first pinless example, pulled directly from the back of the bridge rather than from the exit point of the ramps.

Really interesting, David. Thanks for sharing your work.

_________________
George :-)

I think the j4 case for pinless on a steel string is closer than the j2 case. In every bridge I've seen either the balls end up closer to the saddle or there is a sharp angle up to the saddle from a location closer to the saddle. Basically almost no upwards pressure on the tail of the saddle.

_________________
http://www.Harvestmoonguitars.com

I'm not sure I'd agree. It's the break angle that's the issue, not the distance of the ball ends (for example, if the ball ends of a pinned bridge ended up down in the body, it wouldn't have an affect.) In the case of a pinless bridge it wouldn't matter if the back of the bridge were further because that force is parallel to the force of string tension. There is effectively zero torque from the back of the bridge as long as the string exits the ramps within a traditional distance from the saddle -- that's where the torque would be generated as that's where the moment arm is and consequently the upward force vector.

I think we are saying the same thing, at least as I intended. Closer to the saddle increases the break angle, I also call out the sharp angle to the saddle. I was thinking two different styles one where the ball end could move forward and one where the ball stays at the tail but the string breaks up to the saddle closer to the saddle. I thing the model for j4 covers both cases. That is it does not matter much where the ball terminates.

_________________
http://www.Harvestmoonguitars.com

Thank you, all, glad to hear it's helpful! Those are all great and helpful comments. I appreciate the feedback as I want the model to be relevant to luthier's needs. And I'm glad to answer any questions.

Brian, good question -- that is what originally motivated the model. I haven't looked at shear stress yet (and it needs a finer FEA mesh for smooth plots), so I don't know for sure, yet. Initially, though, for a pinned bridge, the ball-end's clamping forces are resolved by a "stress column" that mostly goes straight up to the string ramp (pic below), with no help at the tail. In other words, those ball-end forces appear to have just a local effect. If the glue fails and the bridge starts to peel, then the ball ends would help clamp and generate some shear friction in the process. But if the tail glue is sound, it probably doesn't benefit from that clamping. We'll see!


David, yes, the Elliot bridge has both of those effects, and so I'm not sure how it will compare. My guess is that moving the force application away from the tail will help a lot, but it's just a guess. With FEA, I've learned that I can predict some things, but I also get surprised and always learn more than I expect.

John, David, thanks. I think the J2 model may be similar to an old Taylor pinless, though its ball forces are a bit inset, and I can't tell if the string is redirected inside: http://i192.photobucket.com/albums/z94/ ... 203321.jpg

For J5, mainly because it's a very simple change, I'm going to simulate loading like this: http://img62.imageshack.us/img62/9591/t ... dge.th.jpg
This let's me use the same bridge geometry, with purely horizontal loads at the tail and new forces inside the pin holes. I'd like to keep 1 bridge design for a bit longer, as version control is more work after geometry diverges.

After that, I could add relief cuts at the tail, and then also on top of the bridge (both are easy)--the design seems to be common (?) and these will create interesting hinges: http://www.imagineguitars.com/archive/a ... bridge.jpg
And the Elliot bridge (more work): http://www.doolinguitars.com/articles/bridgejigs/

_________________
David Malicky

Thank you! Much to think about. The J4 case is probably pretty close to what the Elliot design does. It seems to me the thing that determines when the bridge will pull up is the ultimate loading at the back edge, and that does not seem to be significantly different for the J1 and J4 cases. If that's true then pinned and pinless bridges should not be different in that regard.

Results are in for case J5: Pinless loading, with the ball ends applying only horizontal loads at the tail.
Similar to this bridge design: http://i1224.photobucket.com/albums/ee3 ... inless.jpg

To simplify the CAD work (in Creo, nothing is easy), I also modified the basic bridge design at the tail (the profile is flatter in the top view), but this change had no effect on the line graphs -- I reran J1-J4 to verify. The FEA mesh is also refined now, so the shear plot is reasonably smooth.

3D geometry -- there are 3 sub-cases to study different hole depths:


2D side view of simulated string loads:


J5b pressure distribution (J5a and J5c are similar, although J5a has more tensile stress at the tail, and J5c has less).


Graph of pressure along the dotted red lines in the pics above. No surprises.


Zoom-in of the graph just above, focusing on the last 0.1" by the tail.
J5a is substantially better than J2, but still worse than the others. J5b is almost as good. J5c is ~equal to the others.


Graph of shear stress along the dotted red lines in the pics above -- all cases are pretty similar at the tail, but J2, J5a, and J5b are a bit worse (trends are like in the pressure graph, but not as dramatic):


So, for this bridge design, some findings:
1. If the ball-ends are at the tail, purely horizontal loading (J5a) is better than loading with a vertical component (J2).
2. If the ball-ends are set in by at least 0.2-0.3" from the tail, they will have negligible effect on the tail stress. (The Elliot bridge is much thinner at the tail, so it will probably behave differently -- I'm guessing it will need less than 0.2").

Next I'll look at the effect of relief cuts at the tail, since those are common and likely to reduce the tail stress concentration.

_________________
David Malicky

David:
Please consider publishing these results in the on-line Savart Journal when you get it all done.

Alan, Yes, thanks, definitely planning to publish it. Savart looks like the best (only?) option for science of stringed instruments now?

Case J6 is Pinless with a relief slot along the rear face, leaving a more flexible ledge of wood at the tail edge. Shown here is case J6c2:


There are 4 sub-cases for variable ledge dimensions:


J6c2 pressure distribution.
-The thin yellow line at the tail indicates relatively low stress there. It still goes to 'infinity' (theoretically), but only very very close to the edge.
- There is a line of higher stress (with 3 'hot spots') about 0.2" in front of the tail, where the bridge is full-thickness and stiff. But these levels are very modest (<200 psi), and cannot go to infinity, so no risk of failure here. Basically we are 'trading' very high stress at the rear corner for moderate stress over a bigger area. A good trade.
- Once the flexible ledge ends, the stress increases back to typical values.
- J6b2 and J6c1 are similar but higher stress at tail edge; J6b1 is higher yet.

If the rear slot is not fully across, but instead comes to an end like this:
http://www.guitarrepairshop.co.uk/wp-co ... 4x5951.jpg
Then the stresses would be worse at those transitions, probably worse than no slot at all.


Case J7 is a beveled edge, as another way to make that tail edge more flexible.
- Compared to a ledge, a bevel can naturally sweep all the way across the rear profile.
- I tried 30 (modest) and 45 degree bevels.


J7-45 pressure distribution.
- The fully swept bevel makes the entire rear edge flexible, and lowers stress all along the rear profile.
- Compared to J1, J2, J4, etc, there is a larger area of yellow and yellow-green just in front of the tail edge. This is good because that (small) tension relieves the peak stresses at the very tail. Another good trade.
- J7-30 was similar but higher stress at tail


This graph summarizes pressures of J6 and J7 along the dotted red line, focusing just on the last 0.020" inch since that is the zone of highest risk of peeling:
- Compare J5b to J6b1: The ledge for J6b1 was 0.1" long, 0.07" thick: this was almost flexible enough to bring the stresses back to J1 Pinned levels.
- Compare J5b to J6b2: its somewhat thinner ledge (.05") is much more flexible (cube rule), and actually gives lower stress than J1 Pinned (with a square rear edge).
- Both J6c cases also give lower peak tail stresses than J1 Pinned.
- For J7, even the modest 30 degree bevel gave lower stresses than a square-corner Pinned bridge, especially in the last 0.005".
- In the last 0.002", the J7-45 deg case has the lowest peel stress of all. (The graph shows it peaking at 300 psi, but theoretically it would go to infinity, too -- my mesh was just not fine enough to resolve that).



My interpretations:
- The peak stress at the tail is much more correlated to the shape of the bridge there, than whether the bridge is pinned or pinless. I.e., a pinless bridge can be better or worse than pinned, depending on how flexible the tail edge is.
- The thin ledge and beveled face both make the rear corner of the bridge more flexible, and thus lower the stress concentration at the tail edge. Of course, these are similar to how we taper the ends of top and back braces, both in design and effect. We are trading very high stress on a very small area for modest stress on a big area.
- If a ledge, it needs to be quite thin to be effective at reducing tail stresses. Remember the cube rule. 0.09" thickness would likely not help at all.
- Any bevel would need to come to a 'knife edge' to realize the benefits.


The take-home message: everything you know about tapering the ends of braces to keep them from peeling also applies to bridges.

_________________
David Malicky

I don't follow your conclusions. In J6, the stresses at the back of the ledge are minimal. The problem comes where the ledge runs out. Surely the stress distribution of the ledge depends on the length as well as the thickness. Ideally it would fair to zero thickness, but even a .020" ledge would help if it was carried back far enough (or cut in deep enough). I'm curious about how a classical bridge would look in your analysis. A few years back, I strung up a classical guitar with light gauge steel strings to see what would happen. Two years later, it was still doing fine. As far as I know, it still is. (This was a mid-price import with an over-built top, and a truss-rod.) The bridge on that guitar had the faired-in trailing edge that was introduced by Torres in the 1800's.

EDIT: I see now that you address the issue of the ratio of ledge length to thickness in your discussion of J6. How is the stress distribution affected by the materials properties of the bridge wood? Ebony is common for steel-strings, but very rare for classicals. These days, many builders are choosing low-density (flexible) woods such as walnut. I would guess that these would be slightly more forgiving of stress risers.

My thanks to you for taking this on and sharing the results. I've always wondered about how these stresses actually played out.

These models seem to suggest that a rabbet, cut to cover a ledge of finish, not only reduces glue surface in the most stressed area, but also concentrates stress though increases stiffness of the back edge. Essentially a reverse bevel. No? hmmm.

Good questions and info, Eric and David, thanks! Lots of helpful input and it will advance the project.

Eric, Yes, there is quite a range of effectiveness for the J6 cases, depending on the ledge thickness and length. A 0.020" ledge could indeed be very short and still be effective -- I had stopped at 0.050" only because I thought that was pretty thin (machining without chip-out, care in handling, special gluing cauls, etc). That's interesting that the classical bridge held up to steel strings. Thanks for the info on the Torres style bridge. The narrower classical profile would raise stresses, but it's probably only a linear effect (1" is probably ~50% worse than 1.5"), whereas the trailing/beveled edge would reduce stresses more dramatically.

I haven't studied the material props yet, but I agree the softer woods should be better. Currently the bridge has orthotropic rosewood properties in a rift-sawn orientation. So, grain angle is yet another variable in the mix.

David, Interesting, thanks, it will definitely go on the to-do list. I'm guessing the rabbet does raise the edge stress a little, but not as dramatically as a ledge or bevel helps. My thinking is that a 90 deg edge is nearly "fully" stiff. The bevel works because the last ~0.015" of material is quite thin, and so it makes a tiny flexible "beam" at the edge. You can see this in the last graph: the dark blue lines have similar stress as J1 at 1.48", but they remain flatter as they approach 1.5", indicating this is where their flexibility helps. Still, with a rabbet, the overhang of extra material will help reinforce that edge and stiffen it, so stresses should go up -- I just don't know how much--yet!

Thanks again!

_________________
David Malicky

Some years ago I read a book called 'The New Science of Strong Materials; or why you don't fall through the floor". He talked about the shearing stress in a glue line. From what I gathered the stress in the center of the line was a function of the nature of the glue line itself and/of the materials being joined. It tended to be at a fairly low and constant level. It rose at either end to, as you say, theoretical infinity. The area under the curve was the total stress. By making the glue line longer in the direction of the pull the center part took more of the total load, which reduced the maximum stress at the edges. I've always understood that this was why the 'belly' bridge, with little or no more gluing area than Classical bridge, could withstand the larger force of the steel strings without coming off to frequently. Of course, there are also major differences in the overall structure that probably tend to concentrate stress in the middle of the Classical bridge, but it's still interesting. At any rate, in might be worthwhile to see if anything like that shows up in your model. I may, of course, have totally misunderstood him; he didn't include any math.

Thanks, Alan, good points and suggestions. Yes, your description of the glue line stress distribution is the same as I understand, and consistent with the model. It sounds like the book was looking at pure shear loading, while the guitar bridge has both shear and tension/compression, the T/C due to the string torque. A hard question is whether it is shear or tension on the tail edge that initiates the glue failure. Probably some combination, as that is typical for most failure models. So far at least, the model is conveniently showing that the tensile pressure and shear on the tail edge are highly correlated from one geometry or loading change to the next. So I've focused on the pressure distribution since that data is a little clearer.

Yes, the longer glue line in the belly bridge helps shear, and also tension/compression due to better leverage to resist the string torque. I will look at changes for that dimension, too, but it will probably be pretty predictable (linear effect).

_________________
David Malicky