# Probability of Failure - Seven expansion joints

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#### daledun

Hello,

My company made seven expansion joints that were placed into a closed loop pipe system. Two joints failed prematurely and were replaced. Shortly after replacement, the new joints, which were placed into exactly the same locations as the originals, failed. So a first set of joints, then a second, located in the same place in the system failed. The other five located in different locations in the system have functioned as expected.

There is no evidence that manufacturing deficiencies were to blame for the failures. What statistical/probability data can I present to the customer to reinforce my belief that it is a problem with their system, not my product? I do not have MTBF info.

#### Tim Folkerts

Super Moderator
Re: Probability of Failure

Here are two different ways to look at it:

(This argues your side). Suppose the parts follow an exponential sort of failure mode - so that previous history is not an indication of future performance (which is common for electronics). If you run the system long enough, you would expect eventually that two joints in each would fail. Suppose position x and y fail first. When you replace them, the system is like new (due to the nature of the exponential failure mode). The second time around, there is a 2/7 chance that x or y position will fail first and a 1/6 chance that the remaining x or y will fail next = 2/42 = 1/21 = 4.8%.

(this argues their side). Suppose that, unknown to you, the joints are bad 1/3 of the time. In the system, you would expect about 2 bad joints. If you replace these with two other random joints, there is a 1/3 * 1/3 = 1/9 =11% chance that you would randomly select 2 more bad joints as the replacements and that they would again fail prematurely. Not very likely, but certainly possible, so it is hard to rule out bad parts as the cause

My guess would be that vibrations/stress/heat/etc at those two spots is generating the problems. More info on the MTTF and the failure distribution would obviously help nail down the answer.

Tim

#### Steve Prevette

##### Deming Disciple
Staff member
Super Moderator
I'd suggest doing this more as a binomial.

Let us assume that each of the 7 joints at any point in time are equally likely to be the "next to fail". Even if some joints were replaced more recently than others, if we do assume exponential (memoryless) failure, then the assumption is valid that each equally likely to be the next to fail.

Now - what is the probability that the first four failures will ONLY involve two of the joints? If we pick two specific joints, then the probability of the first failure includes one of the two specific joints is 2/7. Same for the second, the third, and the fourth. Thus we end upon with (2/7)^4 = 0.006664

However, we don't know the specific two joints. We must multiply by the numbers of ways we can take two joints out of seven. That is 7! / (2! x 5!) or 7*6/2 = 21.

So the probability that the first four failures involve only two of the joints is 21 * 0.006664 = 0.14

#### bobdoering

Trusted Information Resource
The problem may be you do not have all of the system variables to determine the root cause. If you have any data that compares the print requirements of the parts that are passing to the parts that are failing to show they are in print (thickness, tensile and/or compression testing, etc.) that is a good starting point. You do not likely have key system data, such as: 1) system pressure in passing areas vs failing areas 2) flange condition (flatness, surface finish) 3) flange bolt torque 4) flange spacing after installation 5) perpendicularity of flanges (is there a cantilevered load cocking the flanges in the failing installation?). There may be sufficient difference in any of these factors that could cause a failure in one area, but not another - however, acquiring that data may be difficult..

If the root cause is in the system and not product variation, they are independent occurrences, and that would influence the probability of failure - they may never fail in the other installations, and they may always fail in the suspect installations. They are trying to determine product MTTB (blame) - and the raw failure data may not accurately represent that. It may represent installation MTTB (shared blame).

Last edited:

#### Tim Folkerts

Super Moderator
I'd suggest doing this more as a binomial.

....
So the probability that the first four failures involve only two of the joints is 21 * 0.006664 = 0.14
(For the sake of argument, let's assume that joint 1 & joint 2 are the ones that failed)

My concern with your analysis, Steve, is that you are allowing for some possibilities that could not easily have occurred in the actual setting. For example, you include the possibility that joint 1 failed all four times. In other words, you are replacing the failed joint as soon as it fails. But in the actual setting, they had a pair of failures replaced together, so there are fewer possibilities. I think my estimate of 4.8% is more appropriate than 14%.

Perhaps more importantly for the original post, all of this assumes the process is "memoryless", which I would not expect for mechanical parts. Steve & I were both being mathematicians and not engineers.

If you drive your car until 1 tire goes flat (and replaced it with a new tire), you would be highly surprised if that new tire was the next to go flat. And it it did go flat again then I would look for a problem with the car -- basically what Bob was discussing. If you take into account this real life wear & tear & stress & strain, then the odds that either steve or I calculated would be WAY too high.

Tim

#### Miner

##### Forum Moderator
Staff member
While this is a good discussion, can you not perform a failure analysis of the failed joints? You did not say whether these were metal joints, so this may be out in left field. A good metallurgical analysis can tell many things about the type of stress that caused a failure. You can tell the difference between an overstress, a fatigue failure, material or heat treat issues.

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