What is the maximum mean stress in a bolted joint?

[Q_Goest]

There's a common belief that for any given bolted joint, the higher the preload, the longer the fatigue life. I’d like to challenge that preconception and suggest that increasing preload above that needed to keep the joint together and acting as a pair is generally a bad thing and results in shorter fatigue life for the bolt. Here's why.

Bolted joints are generally modeled as two linear springs acting against one another – the bolt and the joint. As long as the bolt doesn't come away from the joint, the spring rate of the assembly is constant. Because the spring rate is constant, the stretch (or change in length of the bolt) that occurs when the bolted joint undergoes a given change in load is independent on the initial and final load. The stretch of the bolted joint due to a given change in load dF, is given by dx = dF/k where dF is the change in load and k is the spring constant of the bolted joint. So the strain in the bolted joint is linearly dependant on the change in force imposed on the joint as long as the bolt and joint stay together and act as a pair. This proved to be the case when we modeled a joint using FEA and also modeled it using the canned formulas in texts on the subject.

This change in force therefore produces a given change in strain which produces a given change in stress. From this, we can find the limits of stress that the joint undergoes. The limits are:
- The stress produced under the preloaded condition.
- The stress produced under the loaded condition. For the sake of simplicity, let's assume the loaded condition only acts in one direction, and that force is parallel to, and along the centerline of, the axis of the bolt.

The median of these two stress levels is called the mean stress. The mean stress is equal distance from these two stress extremes, and that distance is called the alternating stress.

The question then is, "If we have a given alternating stress and are only able to change the mean stress, then what is the best mean stress to have in a bolted joint to maximize fatigue life given that the alternating stress remains the same regardless of mean stress?" Figure 8.16 (attached) shows various types of stress combinations, all of which correspond to an equal fatigue life. From left to right, what it shows is that as the mean stress increases, alternating stress must decrease in order to maintain the same fatigue life. This can be seen also on a Goodman diagram in which we plot the fatigue life against alternating and mean stress as shown in Figure 8.21 (attached). On this Goodman diagram, the parallel lines running horizontally and then converge on the Uts (150 ksi) represent the fatigue limit of the material for given cycles. The operating point is then placed on this chart to determine if it is inside or outside any of these lines. If the operating point is outside the line, fatigue failure is predicted. The operating point is simply a point on this graph which is determined from the mean stress and alternating stress. That point is plotted where x is the mean stress and y is the alternating stress. Since the y value of alternating stress stays constant, it is obviously best to minimize the x location (mean stress) because all the lines drop off to the right, and the farther you go out to the right (positive x direction) the less alternating stress it takes to exceed the fatigue limit of the material.

Unfortunately, calculating the fatigue life of a part is never easy given all the various factors involved, but fortunately, those factors don't change depending on stress level. The factors such as gradient factor, load factor, surface factor and stress concentration factor, all remain the same regardless of the mean and alternating stress. The conclusion I've come to therefore is that to maximize the strength of a bolted joint in pure tension, I want to have the minimum mean stress possible that allows the bolted joint to act as a pair under all conditions which means the best bolted joint will have the lowest possible preload (torque) on the joint that is sufficient to keep the bolted joint together and acting as a pair during all foreseeable operations.

Thoughts?

 

[FredGarvin]

Hey Q,

Bickford does quite a bit of talking about this issue. I get the impression that he is on the fence because he has data that supports both arguments, especially in terms of the level of mean stress. He does state that there are mitigating circumstances that will change this view because a standard analysis of a joint usually does not include nasty subjects like prying. I also get the impression that, as a guide, data seems to support the preloading to yield approach. He does go on to list the top factors that will increase fatigue life, specifically related to reducing the load excursions a bolt sees in operation. They are:

- Prevent prying
- Proper preload
- Controlling joint-bolt stiffness ratio
- Achieving correct preload

In my research, I tend to think that the joint-bolt stiffness ratio holds the key in many issues. Bickford does admit that this argument of stiffer or less stiffer bolts should be looked at for every individual case.

While I don't disagree with your analysis, I don't think I, personally, could follow it. It may be my nature, but because there are so many factors that are impossible to account for in an analysis, I stay away from trying to minimize a bolt preload. I just wouldn't trust it. If fatigue life does turn out to be an issue, the first things I do are look at using a radiused root thread form or adjusting the bolt loads. Again, that is just me. I don't run into this scenario enough to really have a large database built up to draw conclusions from.

 

[Q_Goest]

Yes, I have Bickford's book also. I didn't notice anywhere in there that he really addresses the issue.

I think most people think along the lines of a pipe flange or other joint that has sufficient room to put a very large number of fasteners that results in negligable cyclic stresses (ie: less than 5% overall change in stress).

In comparison, a reciprocating pump that puts the pump shaft in tension and goes to high pressure won't have room to put lots of extra bolts in. The subsequent joint has to be very carefully designed such that fatigue won't destroy any bolted connections internal to the machine. It's because of this I'm looking at the issue.

In addition to all the good stuff such shot peened threads, eliminating the shank (machine to the root), maximizing the stiffness of the joint, having large radiuses, etc... I think it will be helpful to minimize the preload as long as the subsequent joint can be guaranteed to remain intact under all conditions.

Side note: One interesting thing about Bickford is that he puts his stress concentration factor into his calculation of fatigue life. See page 595 & 596. All other texts I'm aware of apply the stress concentration factor to the calculated stress at the thread root. In other words, Bickford suggests applying a factor of 0.3 (rolled threads) to the fatigue life while others apply a stress concentration factor of 3 to the calculated stress at the thread root.

Bickford, "An Introduction to the Design and Behavior of Bolted Joints" 3'rd edition.

 

[FredGarvin]

I have seen the use of derating factors for the base material fatigue limit, but I didn't see anything with the actual stress concentration.

Those are pretty tight places on pumps. I can understand the concern.

 

[CFDFEAGURU]

I want to have the minimum mean stress possible that allows the bolted joint to act as a pair under all conditions which means the best bolted joint will have the lowest possible preload (torque) on the joint that is sufficient to keep the bolted joint together and acting as a pair during all foreseeable operations.

Depending on how you are defining "keep the bolted joint together".

One potential problem with using the lowest preload possible could be this;

If you have a flanged joint, say a RFWN (Raised Face Weld Neck) pair and it requires a Flexitallic style spiral wound gasket you might not be crushing the gasket if you use the minimum preload. Therefore, more initial torque is going to be needed in order to avoid a leak.

The conclusion I've come to therefore is that to maximize the strength of a bolted joint in pure tension ...

Pure tension might be an oversimplification if you have metal to metal contact outside of the bolt circle. If there is metal to metal contact outside of the bolt circle the additional bending loads will be introduced into the bolt in addition to the tension load. The ASME Section VIII, Div. 1 pressure vessel code has a non-mandatory appendix to design these types of joints.

Those are my thoughts.

Thanks
Matt

 

[Q_Goest]

Thanks for the feedback Matt. I agree with what you say and that's consistent with other feedback I've gotten at work here. Most folks think "flanged joint", but I intended this to be a bit more generic. In the case of a flanged joint we must be concerned with gasket loading as you say. I guess the argument still holds though. Once we torque a bolt to a given value that consistently guarantees a dependable joint, additional torque reduces fatigue strength.

What brought this up is an issue regarding a pump connection that was found to fail due to fatigue. The connection is not a gasketed joint, and the load is purely axial. I guess I just wanted to verify by peer review that I wasn't missing something.

 

[FredGarvin]

Putting a gasket in the joint changes a lot of things. It's not an apples to apples comparison. While I think that most times joints are not pure tension, I would think Q's application in this case is indeed one of those times. However, it doesn't take much to induce bending or prying under the head.

 

[CFDFEAGURU]

However, it doesn't take much to induce bending or prying under the head.

Fred,

I was very suprised to how large of a moment can be developed with a moment arm of only at most a couple of inches.

Thanks
Matt

 

[CFDFEAGURU]

Here is some more information.

From "Theory and Design of Pressure Vessels" by John F. Harvey 2nd Edition chapter 6.

Thread and Nut Design

"Whitworth threads have shown increased fatigue life over American Standard threads of 16 percent because of their rounded roots."

"Using very high nuts so that, even though the tendency prevails to load highly on the first few threads of engagement, the actual stress on these threads is reduced over that existing for a standard height nut because of the greater total number of threads involved. Fatigue life has been doubled by this means. Another method of countering this effect is to increase the flexibility of either the nut by recessing it, or the bolt threads by cutting a taper on the first few threads of the nut so that the thread bearing takes place farther out toward its peak. Either means introduces more cantilever bending action in these first few engaged threads with the result that more load is tranfered to the upper ones. Reducing the net bolt area in the region of the upper threads, or a slight negative lead on the nut threads, will also tend to reduce this effect. Nut and bolt design modifications of types have shoen fatigue life increase of 30 percent of more."

Thanks
Matt

 

[DTM]

Gaskets aside, for a generic bolted joint, I think Q-Goest is absolutly correct. You want just enogh preload to not get any joint seperation. Anymore increases your mean stress and does nothing to prevent alternating stress.

I used to design nuclear reactors with many experts in bolted joints, etc. A number of the bolts had surprizing low preloads for this very reason. Also, many of the bolts had the root radius control on the threads.

 

[Q_Goest]

Gaskets aside, for a generic bolted joint, I think Q-Goest is absolutly correct. You want just enogh preload to not get any joint seperation. Anymore increases your mean stress and does nothing to prevent alternating stress.

I used to design nuclear reactors with many experts in bolted joints, etc. A number of the bolts had surprizing low preloads for this very reason. Also, many of the bolts had the root radius control on the threads.

Thanks for the comments.

 

[raghu.mech]

There's a common belief that for any given bolted joint, the higher the preload, the longer the fatigue life. I’d like to challenge that preconception and suggest that increasing preload above that needed to keep the joint together and acting as a pair is generally a bad thing and results in shorter fatigue life for the bolt. Here's why.

Bolted joints are generally modeled as two linear springs acting against one another – the bolt and the joint. As long as the bolt doesn't come away from the joint, the spring rate of the assembly is constant. Because the spring rate is constant, the stretch (or change in length of the bolt) that occurs when the bolted joint undergoes a given change in load is independent on the initial and final load. The stretch of the bolted joint due to a given change in load dF, is given by dx = dF/k where dF is the change in load and k is the spring constant of the bolted joint. So the strain in the bolted joint is linearly dependant on the change in force imposed on the joint as long as the bolt and joint stay together and act as a pair. This proved to be the case when we modeled a joint using FEA and also modeled it using the canned formulas in texts on the subject.

This change in force therefore produces a given change in strain which produces a given change in stress. From this, we can find the limits of stress that the joint undergoes. The limits are:
- The stress produced under the preloaded condition.
- The stress produced under the loaded condition. For the sake of simplicity, let's assume the loaded condition only acts in one direction, and that force is parallel to, and along the centerline of, the axis of the bolt.

The median of these two stress levels is called the mean stress. The mean stress is equal distance from these two stress extremes, and that distance is called the alternating stress.

The question then is, "If we have a given alternating stress and are only able to change the mean stress, then what is the best mean stress to have in a bolted joint to maximize fatigue life given that the alternating stress remains the same regardless of mean stress?" Figure 8.16 (attached) shows various types of stress combinations, all of which correspond to an equal fatigue life. From left to right, what it shows is that as the mean stress increases, alternating stress must decrease in order to maintain the same fatigue life. This can be seen also on a Goodman diagram in which we plot the fatigue life against alternating and mean stress as shown in Figure 8.21 (attached). On this Goodman diagram, the parallel lines running horizontally and then converge on the Uts (150 ksi) represent the fatigue limit of the material for given cycles. The operating point is then placed on this chart to determine if it is inside or outside any of these lines. If the operating point is outside the line, fatigue failure is predicted. The operating point is simply a point on this graph which is determined from the mean stress and alternating stress. That point is plotted where x is the mean stress and y is the alternating stress. Since the y value of alternating stress stays constant, it is obviously best to minimize the x location (mean stress) because all the lines drop off to the right, and the farther you go out to the right (positive x direction) the less alternating stress it takes to exceed the fatigue limit of the material.

Unfortunately, calculating the fatigue life of a part is never easy given all the various factors involved, but fortunately, those factors don't change depending on stress level. The factors such as gradient factor, load factor, surface factor and stress concentration factor, all remain the same regardless of the mean and alternating stress. The conclusion I've come to therefore is that to maximize the strength of a bolted joint in pure tension, I want to have the minimum mean stress possible that allows the bolted joint to act as a pair under all conditions which means the best bolted joint will have the lowest possible preload (torque) on the joint that is sufficient to keep the bolted joint together and acting as a pair during all foreseeable operations.

Thoughts?

Dear Goest,

This makes an interesting reading. For the time being I will take it with a pinch of salt.

In your post the crucial line, IMO, is

"... increasing preload above that needed to keep the joint together..."

How do you define that?

Thanks and cheers.. r.m

 

[Q_Goest]

Hi raghu, welcome to the board. The analysis of bolted joints is done by modeling the bolt as one "spring" and the joint as the second "spring". Yes, they are both very stiff springs, but they are 2 springs nevertheless. The stiffness of each spring is a function of cross sectional area under stress, modulus and other geometric factors of each of the two parts. The total spring constant (k_total) of the system is then equal to 2 springs in parallel, one is the bolt, the other is the part/parts being held together by the bolt. So k_total = k_joint + k_bolt

To answer your question then, in order to keep the joint together, the two springs must act as a pair (in parallel) and not come away from each other. For example, imagine a bolt holding two blocks together and there is a load tending to separate the two blocks. At first, the compression of the blocks is reduced, while the bolt stretches some very tiny amount. These two "springs" are acting in parallel at this point. The amount of load, F, and the overall spring load of the joint, k_total, results in a small movement of the two blocks away from each other, dx. The blocks don't simply separate as soon as you put load on them.

Now as you continue to apply more force on these two blocks, they eventually come away from each other such that the blocks no longer make contact, and only the bolt is providing any spring load. At this point, the spring load makes an abrupt change from some value that is the sum of the joint and bolt spring loads, to a spring load that is only equal to the bolt spring load. Thus, the load has increased above that needed to keep the two together and acting as a pair.