This is a picture of an assembly that is part of my full scale (human size) quad-copter rotor blade.
For background, the FlexElement is the structural member that will connect the rotor blade to the rotor hub or head. (shaft). The leading edge rod is an internal structure to give the leading edge of the rotor blade airfoil durability for small impacts. The composite beam is a composite layup which holds it all together, but mainly its function is to transfer the centrifugal forces of the rotor blade (which extends to the left off the page) to the FlexElement.
Originally I did a simple calculation to determine how much surface area I needed for the glue joint between the FlexElement and the composite beam. This was based on the shear strength of the epoxy resin I was using.
A = F / U where U is the maximum stress I want the glue to experience.
In my case F=1600 Lb
U = 5000/4 psi (Lb/in^2) (divide by safety factor of 4:1)
A (minimum) = 1600 / 1250 = 1.28 in^2
My FlexElement rod has an OD of 5/16" or 0.3125", so the circumference is pi * .3125 = 0.982"
So every 1" of bond length along the rod, will give 0.982 in^2 of bond area, so my bond length needs to be at least:
1.28 / 0.982 = 1.30" long. That's how much the rod must insert into the composite beam to provide enough bonding area.
In actual fact, I made the joint length about 4" because of other design factors, so I theoretically had 12 x more area than I needed (safety factor of 4 * (4"/1.3") ~ 12)
I was VERY surprised when this joint failed in simple tension at well below 1600 Lbs of tension.
Lets look why.
My simple bond joint area calculation made one very innocent, but deadly assumption:
That the force or stress would be evenly distributed across the full area of the joint.
WRONG WRONG WRONG WRONG!!
Lets look at a simpler geometry to help understand.
Here is a cut-away of a simple rod glued into a larger cylindrical rod.
The problem with this design is that it produces a concentration of stresses in the glue joint at the point where the rod first enters the cylinder. This is because of the huge, and sudden difference in cross sectional area between the rod, and the rod/cylinder combination at that point.
All materials have elasticity and deform (stretch or compress) when a force is applied to them.
The rod is being pulled in tension, so it is stretched longer than it's normal relaxed length.
Since the glue joint between these two parts is very very thin, we assume that the rod and cylinder dont move relative to each other. But that is not the case here, because the rod is a small diameter, so it is under much greater stress (force per unit cross sectional area) than the green cylinder. So... the rod will be stretching more, so it MUST deform more than the cylinder which is under less stress because the tensile force is spread across a much larger cross sectional area.
When you pull the rod in tension, you quickly cause a stress concentration in the glue joint at the spot indicated.
When the rod pulls to the right, all of that force is concentrated to a small portion of the glue joint. That section of the glue joint fails, and the rod stretches a minute amount. But now, the portion of the glue joint just to the left of the stress concentration area becomes the new stress concentration area, and so on until the joint failure/ stress concentration has moved all the way along the glue joint, and the rod simply pulls out of the cylinder.
This is very hard to describe and visualize because it happens essentially instantaneously, but seeing the results (failure of the joint WAY below what was expected) make it impossible to deny.
THE SOLUTION:
The solution to the problem is to design the two parts (or for simplicity in this case - the cylinder) so that there is a very gradual change in cross sectional area, so that there is a gradual change in the stress, and hence a gradual change in the stretch (strain).
Care must be taken to make this transition gradually so that at no point along the glue joint, does the stress excess the shear strength of the cured glue material.
So back to the stress concentration area, the glue joint only needs to transfer enough force to the "cylinder" (now a cone) to make that tiny, thin green area on the far right to stretch along with the rod which is under full stress.
This force transfer keeps happening along the length of the glue joint until eventually all of the force is tranferred to the green cylinder.
Thanks for taking the time to "bond" with me. :)
Comments
Here is the assembly before testing with the "sucker rod end" epoxyed to the fiberglass rod.
Below is a close-up of the part after fracture.
You can clearly see the epoxy joint strength exceeded the rod itself.
You can see that the left-most section of epoxy did de-bond from the fiberglass rod, but that may have been caused by the taper on the end of the rod to facilitate insertion. Also, I was reluctant to put large gouges in the rod surface so close to the end (to help give some mechanical bonding) because of the risk of tear-out, so close to the end of the rod. I've tested enough of these to be confident that I can consistently build a strong joint. I can also tensile test each rod up to 2x working stress to "verify" before building into rotor.
One of the fabrication "tricks" is to line the inside diam of the threaded end with a mold release (parting wax, or PVA or both) to ensure the epoxy slides on the steel part under stress, and stays connected to the rod. As the rod stretches under load, the epoxy gets "wedged" in the tapered ID which helps the epoxy stay bonded to the rod.
Yes to all of the above. I will plan to taper the smaller rod at it's end otherwise I could get a similar failure in the joint at the opposite end. I was thinking this wasn't necessary because the stress is low at the full combined cross section, but the amount of force supported by the smaller rod is still enough to potentially fracture the glue joint because of the "stress-riser" effect. In woodworking I've seen manufactured joints to bond two boards end-to-end with a multi-interlocking taper with a very steep taper angle which is essentially the same as what I'm trying to do, except I don't think that tool would work on composites.
John-interesting view of tight epoxy bonds. I am planning to add "macro" pores by placing minute notches in the smaller rod at the right side of the bond area, giving way to larger notches. In thinking about it, the notches ensure that the two parts strain at equal rates. Now to apply some math to determine if the taper of the outside cylinder needs to be linear, or parabolic or what. Or, just make it twice as long as I think and call it good.
Regarding bending again, Yes Paul, and I can even design the FlexElement / Hub joint to match the expected coning angle (or a compromise between 0 deg and fully loaded cone angle) to reduce bending stresses in the FlexElement. The coning angle for this design is about 1.6 degrees. It could be larger, but I've added rotor tip weights to give more inertia to the rotors to buy me some time to transition to auto-rotation in the event of a power
failure. The rotor tip weights represent 4% of the gross design weight which is significant, so I may re-think this, but only after I've got some experience with the control system and its capability.
Now I've got one of these rotor assemblies completed (with airfoil all glassed) so the bond joint is concealed within the rotor. I think I can make a strategic slice(s) into the fiberglass skin, peel it back and modify the composite beam to minimize stress risers in that rotor. I really don't want to have to start over from scratch on that rotor, and it's only a test rotor for aerodynamic performance testing anyway, so the sliced skin won't be a big deal. There is plenty of safety factor in the skin to hold the centrifugal forces.
Gentlemen, thanks for the comments.
@John, that's actually a very difficult thing to do when you're talking about gluing two cylindrical objects together. The tolerance of the hole and rod need to be very small in order to have a design gap that is suitable for the glue. It would be interesting to try adding a slight taper to the holes (not positive engagement, but something like a 1 degree draft). That draft angle would allow the hole and rod to be pressed together, create a very small gap for the glue to reside, and create the appropriate amount of clamp in the joint (since the axial clamp force would be multiplied within the joint).
In my experience when it comes to epoxy, less is more. Meaning you should use just the right amount of epoxy so that the contact areas of the parts fully touch, and the epoxy only fill the micro-pores between them.
Done correctly this will make a incredibly strong bond. For example in a test using a stainless steel plate with a circular hole and a thick hardened glass view port bonded to it. The shear test split the glass in two, leaving a layer of broken glass on the metal plate instead of breaking the epoxy bond.
Oh in this type of joint I would also cut the end of the smaller tube off at a sharp angle ( around 7:1 or so) and extend it to "taper it's strength"as otherwise you have another stress point where the thinner tube stops.
Nice to see a scientific approach to design of glue joints. Makes me think I should try it on some of my model planes :) . I'm not sure where I picked it up, but I remember an old saying "always taper the strength" when making this type of joint . Not very scientific but I think it is trying to say the same thing in non-scientific language.
Now where is that slide-rule :)
regards
Andy
Hi Randy - "Stress-risers", a very common term in machined metal parts design. Probably less so in glued assemblies, but may be even more important due to differences in material properties.
Just one suggestion - you could also try machining the flex element rod to change its elastic qualities, either by drilling out the center (in various small steps) or tapering it on the outside, especially over the part of the joint length which will be more highly stressed.
James - true, in a rigidly-mounted and rigid propeller the bending stresses are high. And of course almost all small drones use propellers, but Randy's machine is very different, it is in certain respects more of a helicopter.
In a helicopter you most often have some coning to reduce the bending forces, or if the blades can flap then they reach a cone angle where there is pure tension in the blades themselves, and when fed into the hub those forces finally exhibit their vertical component.
P
James - You are right about bending loads. In my case the max stress from bending is about half the pure tensile stress. The main point here is to make a gradual change in cross-section when relying on a glue joint so that the total force is evenly distributed across the joint area.
Regarding your comment about using adhesive to attach a blade - yes, I know it is unconventional. I have limited machining/casting capability and being able to use a stock rod is attractive to me, so I'm trying to make it work, although I am considering incorporating notches or other "reverse relief profiles" which will provide a form of positive engagement between the two parts. The trick is to avoid any notches in the rod in the high stress areas as this will only reduce the cross-section and increase stresses.
I have played with positive engagement methods (see other block posts) for different joint geometries but for this one, I'm hoping to keep it simple. Also, if I can perfect the design concepts of a glued joint, it will be incredibly useful to me for other areas such as rotor support arms and air-frame design.
I always knew that "sharp" corners were stress concentrators, but until this specific problem, I never understood exactly what was going on. Having this understanding now will allow me to move forward with a more scientific approach to these problems in the future. Thanks for commenting. I always like hearing others thoughts, even if folks think I am barking up the wrong tree. I have a pretty sound background in engineering, and as long as I keep doing the incremental tests to prove out these concepts as I go, I think I could end up with something quite unique, functional and ..... cool!
R