Observing Strain Cones and Predicting Member Stiffness in Bolted Connections

(Theoretical, FEA and Experimental Verification)

For this weekend's seek and geek, we will look at one of the most commonly used structural connection methods: Bolted Joints. These joints are widely used because of ease of assembly/disassembly, ability to get very high clamping forces within a small footprint and ease of design and manufacture. If we look at the spring model of a bolted connection, it looks like this:

As we can see from the spring model, the member stiffness is in parallel to the bolt stiffness. This means that when an external force is applied onto a bolted connection, the bolt and the clamped members will share the applied load according to their stiffness ratios. Higher the member to bolt stiffness ratio, higher will be the percentage of load shared by the clamped members. Also, the stiffness of the members play a key role towards the prevention of joint separation.

However, the predicting the stiffness of bolted members is not easy. This was a key research question during 1970s - 1990s. These studies mainly looked at how strain zones below the bolt head are formed, what kind of geometric shapes best approximate these zones and how to calculate the stiffness based on these strain zone geometries. It was an exciting problem!

For this week, I decided to look at stress field formed in case of a preloaded bolted connection via photo-elastic viewing techniques. Using this stress field geometry, we will then calculate the axial stiffness of the members and compare our theoretical calculations with experimental results and FEA.

Short Note on Stress Birefringence

We are are able to see the stress field in these acrylic member plates because of a property cause stress birefringence. So, how does this work? Basically, when we apply stress onto certain clear amorphous solids, the spacing between the molecules in solid change accordingly. This causes light which which is polarized in one direction to pass faster/slower compared to light that is polarized in the other direction. (Why? - because light in particular direction can travel more freely if the molecules in that direction are spaced farther apart and viceversa). So, in a sense we introduce two different refractive indices into the material and make it birefringent (double-refracting) with the application of stress.

To observe stress birefringence, we can place the material in between two crossed polarizers. If we do not have the material in the light path, then no light should pass through the second polarizer. But, introducing a birefringent material in between the two polarizers causes light to undergo a phase shift that is dependent of the difference between the two refractive indices and the thickness of material. Phase shift alters the polarization of light which then causes some light starts passing through the second polarizer. Now, considering that we pass white light through this stress induced birefringent material, the different individual wavelengths (VIBGYOR) in the white light will have different indices of refraction. This means that the phase shifts for these individual wavelengths will also be different.

If the phase shift for a particular wavelength is 2pi, that would essentially be equivalent to that wavelength not passing through that material at all. That wavelength will be blocked by second polarizer and would not be seen. Similarly, if the phase shift is pi, that wavelength will be able to pass through the second polarizer without obstruction and will be seen by eye. So, what the eye sees as different colours of stress field are essentially different wavelengths of visible light that are expressed based on the amount of phase shifts while passing through the birefringent material.

Someday, it will be a fun to do a prediction on stress level in a component by looking at the fringe patterns. It will be a great seek and geek exercise.

Back to Bolted Connections

Experiment:

For closing the loop on member stiffness of the bolted connections, we will start with the experimental data. Doing the experiment for member stiffness is not easy. So, I relied on past research papers to get the experimental data. Maruyama[1] used a simple experimental setup to determine axial stiffness of the members. The setup consisted of a circular plate with a hole and two pressure pieces (simulating bolt head and nut) to apply the axial load. Figure below shows the setup used by Maruyama. The diameter of the hole was 25 mm and the diameter of circular plate was 100 mm. The material of the circular plate is S45C (this is Japanese steel grading system).Its american equivalent is AISI 1045 (E=200 GPa)

Experimental Setup (Maruyama[1])

Note that here that single circular plate is used for as a replacement to 2 clamped members found in a bolted connection. The experimental values for stiffness of this member was reported to be 565 kgf/um or 5540 N/um.

FEA:

I then used the same geometry of the circular plate and pressure plates used by Maruyama and simulated its deformation in an FEA package. I subdivided the faces of the circular plate at the interface locations of the pressure plate.The bottom pressure plate interface was fixed and the top pressure plate interface was given a fixed constraint in X-Y while the Z was kept free. The X-Y constraint on the top is important because the assumption in the model is that there is no slip between the circular plate and the pressure plates. If we forget the X-Y constraint on the top, the stiffness will be come out to be artificially lower - so, one has to be careful here. This is one of the things about FEA - it will always give you an answer irrespective of whether the model (meshing, loads and boundary conditions) is right or wrong. It is very tempting to believe that whatever the FEA predicts is true.

Based on the FEA, the stiffness of the clamped plate is 1000N/0.18um = 5555 N/um, indeed very close to Maruyama's experimental result of 5540 N/um :) I just have the below figure of strain energy in the clamped member to show that majority of the strain energy is stored either just under the bolt head or just above the nut. This is what we see via photo-elastic experiments as well - lot of 'colorful fringes' occurring specifically at those locations.

Simulation of the deformation of Clamped Member

Simulation of Strain Energy in Clamped Member

Theory:

Below are the theoretical calculations for axial stiffness of bolted members. The theory predicts stiffness to be 6440 N/um.

Comparison

Experimental: 5540 N/um | FEA: 5555 N/um | Theory: 6640 N/um

From the comparison between the experimental, FEA and theoretical values, it seems clear that current analytical model overpredicts the axial stiffness of the clamped members by about 16%. The experimental and FEA values are almost a spot on match, which is indeed very encouraging. The reason why theory over-predicts stiffness most likely has to do with the approximation in the shape of the strain zone.. I assumed the shape to be a frustum with 30 degree cone angle which is what is generally recommended by text books. However, in reality, the shape of strain zone as can be seen from photo-elastic experiments is a bit curved at the inflection points. Also, as a side note, one needs to keep in mind that the theory does not take Saint Venant's principle into account. The impact of a local force field on an object extends only upto 3-5 times the characteristic dimension (here, it may be the bolt head diameter). Beyond that limit the material should not feel any impact.

References:

[1] Kazuo Maruyama, Isamu Yoshimoto, Yoshiyuki Nakano"On Spring Constant of Connected Parts in Bolted Joints", Bulletin of JSME, Vol. 18, No. 126, December 1975.