Threads & Bolted Joints III: Bolted Joint Preload

追加: 2026-05-07

[00:00:00]today's preparation is about bolted joint preload so on the left we have a bolt going through a material we call it a bolt because it's going through a nut screws are what thread into a part just a matter of how you use it this portion here is called the shank and here's the threaded portion there are no threads up here this bolt is finger tight we have dropped the bolt through and turned the nut until it is quite finger tight we haven't used a wrench and really put a lot of torque on it so what's going to happen when we do take that wrench and torque it well the nut is going to advance up the threads but this material is in the way right but you know from practice that even once it touches the material you can get a few turns in so what happens when when you do that two things are going to happen the material itself the material inside the joint between the underside the head of the bolt and the top side of the nut here everything that's in between there this sandwich we call it the joint or we call it the members the members are going to compress you can see from the shape here that the it's exaggerated but this is kind of what it would look like there's been some material compression it's represented by the symbol here delta m it started out at this level and it's compressed to this level so the material the members have compressed and the second thing that's happened is because the material has some stiffness when it compresses it pushes back on the bolt and the bolt stretches so here is the stretching of the bolt you can see the originally the end of the bolt was here now the end of the bolt is down here we're representing that with a delta b for bolt so the member is compressed the bolt has stretched and when you stretch the bolt you develop a preload f subscript i think f i is for initial so that is some amount of tension that's inside the bolt some stress that's over the cross section even before you apply an external load to this joint normally we're going to apply some sort of load that's going to tend to separate the joint that's why we have put a bolt through it because we don't want these two parts to move apart from each other but even before you start trying to pull them apart there is some initial tension in the bolt called the preload and we give it the symbol f subscript i so preload is the tension of the bolt you get when you tighten it there are some advantages that it helps resist untightening of the bolt right when you apply that preload then there's this force that's pulling the nut up against the material that causes there to be friction between the nut and the material or the causes between the head of the bolt and the material for there to be some amount of friction so the more preload there is the more friction there's going to be that resists untightening it also resists the separation of the joined components so we've joined these two components and if this were just finger tight then you could pull on it and when you pull on these material and you try to pull it apart well the bolt is going to stretch a little bit because it's a spring and these two pieces are going to gap if the bolt is just finger tight if you've pre-loaded it then you've got a lot of pre-tension that is holding the joint together so even if you pull on this thing it's going to stretch the bolt a little bit more but it's not going to gap yet because the material is compressed it's squished together you have to pull on it pretty hard in order to separate the joint it also increases friction between the two components that you've joined so there's going to be more friction at this interface now why is that a good thing well consider if you had some sort of shear load right if you've got a force pulling this to the left and some force pulling this one over to the right well yes the shank of the bolt can handle that you don't want to put the threaded portion of the bolt in shear this has less cross-sectional area down here and there's also stress concentrations so it's better to use the smooth unthreaded portion of the bolt for that but look if you've got this friction between these two parts maybe you never even put the bolt in here maybe it's just friction between the members that can resist the shear load and finally it reduces amplitude stress from time varying loads so because the bolt is pre-loaded and the material is squished together now whenever i apply an external load a tensile load here it's going to stretch the bolt a little bit more but it's going to tend to decompress the material when the material decompresses that equal and opposite force that it applies in the bolt gets reduced and that causes effectively the total tension in the bolt is not just the sum of the preload plus an external load it is the preload plus a fraction of the external load that you apply and that fraction can be like 0.1 or 0.2 and it depends on the relative stiffness of the bolt compared to the members to this sandwich here now why is that good it turns out it has implications for fatigue and we'll study those later but the short of it is when materials see time varying loads they see fluctuation between a high load and a lower load the greater that change in load is the sooner that the part will fail due to fatigue that fracture that occurs the same sort of thing that when you're bending a paper clip back and forth so we'd prefer for fatigue purposes it'll be less likely to fail due to fatigue if you keep a constant comp tension on the bolt with very little fluctuation rather than having a little bit of tension in the bolt and large fluctuations in the load having the small fluctuations with a large preload is preferable so disadvantages what's wrong with preload uh just that it complicates analysis really everything else mechanically is good and that's why we do it so we're gonna put up with the more complicated analysis the tension of the bolt is no longer just equal to the applied load it becomes a little bit more intricate how you size a bolt for a given joint you also have to calculate what the preload should be how much you should torque it you don't just look in a catalog and see oh this bolt can handle this much load and use it it's not quite that simple and so we're going to learn how to do that today so here's the process of analyzing a bolted joint first you need to determine what your external load per bolt is so i'm going to give an example today that is a pressurized cylinder and it's got an end plate on it that's held together by many bolts and so the pressure inside the cylinder is pushing on the end plate with some force but then we've got several bolts so we can assume that the load is divided up equally among the bolts and we get the external load per volt p that way now it's not always that simple loads are not always going to be evenly distributed among all your bolts so that's problem dependent and i'm calling that the first step to determine that external load per volt p we've got a lot of p's and f's that are going to be showing up as variables so that's why i'm emphasizing that we call the external load per volt p next step is to determine the tension the resultant bolt load in each bolt f sub b is the total tension in the bolt and it's the sum of f sub i which is the preload the preload which is proportional to the torque that's applied to the bolt and we're going to also talk about some recommended values for what that should be or you might need to choose it based on your circumstances it's a sum of that preload and pb the additional tension the bolt that's caused by p so p is that total external load per bolt but remember not all of it ends up inside the bolt the tension of the bolt is not just the sum of the preload plus the applied external load p no pb is in general less than much less than p and this should be surprising if you haven't done this sort of analysis before i was surprised anyways and we'll see why how that's the case i've got some videos that go through the theory of why that's the case so you can watch those separately today we're gonna just use the equations that you can derive by considering the relative stiffnesses of the different components so it depends on the joint stiffness the bolt stiffness and the members okay so once we've determined the tension in the bolt we are going to ensure that the tension the that force in the bolt doesn't exceed the limits and there are several limits we need to consider we don't want to exceed the proof load of the bolt you've already been exposed to proof load sometimes we're going to call it yielding the bolt that's the failure state even though there's a difference between proof strength and yield strength sometimes we'll say that we don't want the bolt to yield even though we mean we don't want it to exceed the proof load the joint can gap that's another thing right we said that if you just finger tighten the bolt and if you try to pull it the joint apart then as soon as you stretch the bolt even just a tiny little bit then there's going to be a gap in the joint let's say you're trying to hold a pressurized seal in that cylinder well air is going to start leaking you don't want that so even with preload you need to make sure that you have enough preload so that when you pull there's not going to be any gap and also you need to consider the fatigue strength of the bolt material we haven't talked too much about fatigue so we're not going to be doing this analysis here but we'll learn these later today we're going to talk about the preload how to figure out what the preload is and how to calculate the total tension of the bolt assuming you know the bolt stiffness and member stiffness and next time we'll look at how to calculate what those are so what i want you to get out of this module is that you should be able to determine the typical recommended preload for a bolt when you're designing something you might want to adjust this up or down but here's some general recommendations you'll also be able to calculate the torque that's required to produce that preload so you get out your wrench and you don't just crank it down until you feel snug get out your torque wrench and torque it until the torque wrench clicks or if you're using a dial sort of torque wrench then until it reaches the recommended preload and then given the stiffness of the bolt and the joint material we're not going to calculate those today but if you have those then you'll be able to compute the total tension in the bolt when you apply an external load not just the sum of the preload plus the external load it's the preload plus a fraction of the external load a fraction that depends on the stiffness of the bolt and that joint material the members so remember that proof load is the tension that a bolt can withstand without acquiring a measurable permanent set it's the smallest amount you can measure and how is that different from yield well it's the same concept as yield permanent set that's what yield means but when we divide the yield strength of material we use the 0.2 percent offset method the proof load of a volt material they measure much more carefully and they say all right what's the smallest amount of permanent deformation we can measure that's what we're going to call the proof load so the proof load for a bolt is the proof strength of that material multiplied by the tensile stress area and what's that tensile stress area it's the area of a circle that has a diameter that's equal to the mean of the pitch diameter and the minor diameter so you just use the average of the pitch diameter dp plus the minor diameter dr r for root take the average of those two diameters so it's a little less than the pitch diameter and you calculate the area of a circle with that diameter that's your tensile stress area now fortunately we don't have to actually run through this calculation that's just conceptually what it is these are tabulated for bolts and so is the proof strength for different bolt materials for different sae grades we know what the proof strength of it is we can look up for the bolt size and number of threads per inch we can look up the tensile stress area and we can calculate the proof load of the bolt so what's the recommended preload this is what it says in shigley for non-permanent connections if you plan on reusing your fasteners you're not supposed to reuse your bolts you're supposed to toss those after you torque them once but you can reuse the fasteners if you torque it up to 75 percent of the proof load of the bolt if you have a permanent connection you can go up to 90 percent of the proof load of the bolt so you are almost yielding the bolt and when i say yield i mean exceeding the proof strength you are almost yielding the bolt when you torque the bolt up to 90 of the proof strength but don't worry when you apply an external load that doesn't mean you've got 10 percent of the capacity the bolt left because of what we're saying with the relative stiffness of the members and the bolt you take that into account you only add a small fraction of that external load so it turns out that this is still going to allow you to handle a lot of applied load even if you preload it very high it also defines what we said in the last slide that the proof load of the bolt is the product of the tensile stress area and the proof strength of the material the bolt's made out of and it also mentions that the proof strength is around 85 percent of the yield strength so that's a good estimate so let's do an example what's the recommended preload for a permanently installed sae grade 5 quarter 20 bolt so it says permanently installed so according to the last slide we're going to want to make go up to 90 percent of the proof load of the bolt so we need to calculate the proof load of the bolt according to this formula here so we are going to first look up the proof strength of the bolt material it says it's saa grade 5 and if you look in the appropriate table the book then you'll find for a quarter 20 volt the proof strength is 85 ksi and then we can also look up a quarter 20 bolt with its tensile stress area is so we look that up for a quarter 20 volt if you had a fine threads you'd look over here but for a quarter 20 the tensile stress area is 0.0318 inches squared so the proof load is just going to be the product of this and that the proof load is the product of those two 2 700 pounds and so what is the preload it's 90 percent of that for a permanently installed connection preload is 2432 pounds so when you apply torque to a nut or the head of the bolt that's what causes the preload because the nut advances up the threads squishes the material causes the bolt to stretch and that causes the preload in the bolt and here's the general equation right for the torque relationship between torque and force we learned this in the context of power screws but guess what we don't have to use it it simplifies down for standard fasteners because they follow a certain relationship between the diameter and the lead and the collar in this case is going to be the friction between the head of the bolt or usually the nut and a washer turns out that it simplifies to something like this the torque is just the preload we need multiplied by the major diameter of the fastener and some constant k all of this stuff simplifies down to a constant k so it's just an approximation but it's a pretty good one and jiggly gives table 8 15 might vary in your version of the book for different bolt conditions we've got different constants k so if you've just got a black finish bolt hasn't been plated there's no special lubrication or anything you might get a k that's as high as 0.3 it's going to take more torque to get it given preload if it's zinc plated then it takes less torque if you lubricate it even less now in some cases figuring out what the preload is going to be or how much you need to torque it to get a given preload is not good enough sometimes you want to measure the preload directly inside the bolt and there are some technologies for doing that but they tend to be kind of expensive and so this is often what we end up doing so let's do an example calculation what torque is required if you want to produce the recommended preload for a permanently installed galvanized sae grade 5 quarter 20 bolts the same bolts as before we already calculated the proof load on that but now we know that it is galvanized and galvanized means zinc plated so we're going to use the k factor 0.2 that's in that table and here's the calculation the torque is that k times the preload which we calculated before times the diameter of one quarter inch so 0.2 coming from this the 2 400 pounds that we calculated before is the preload times the major diameter of a quarter inch and that's 122 inch bounce force so get out a wrench and torque it up to that value in order to get the recommended preload which was 90 of its proof strength so we said that once you calculate the preload once you know the preload and the torque required to get it you also would be interested in the total tension in the bolt f subscript b that is the sum of the preload and the additional tension in the bolt that's due to the external load and that pb is not equal to the external load it's a fraction of it so let's do an example the end cap of a six inch diameter 2000 psi pressure vessel is secured with eight volts so the first calculation i'm not going to do on paper i'll do it in this slide because it's pretty simple we've got a six inch diameter seal we'll assume and 2000 psi within that so we need to calculate what is 2000 psi multiplied by the area of a circle that's a six inch diameter calculate that area of that six inch diameter circle and we calculate the total tensile load in the joint due to the pressure that's the pressure times the area that's 56 and a half kips so what's the external tensile load per bolt p it's not the additional tension in the bolt due to p it's just the tensile load per bolt so we take the total and we divide it by the number of bolts assuming they get it equally so 56.5 kips divided by eight bolts is seven thousand pounds what if we wanted to know what the additional tension in each bolt is due to the external load pb it's not just 7000 pounds it's going to be a fraction of it and we do not have enough information from that we need some more we need to know the relative stiffnesses of the bolt and the members so here's how we determine pb we need to know c it is called the joint stiffness coefficient p subscript b the additional tension due to the external load is c the stiffness constant times the external load but what c well c depends on the stiffnesses it's given by the ratio of the stiffness of the bolt relative to the combined stiffness of the bolt and the member the sum of those two okay so the same slide as last time except now we are given the stiffness of the bolt and the stiffness of the member stiffness of the bolt is 3000 kips per inch it would take 3000 kips in order to make the bolt stretch a full inch and it would take 12 000 kips to make the members compress by an inch and we want to know what's the additional tension in each bolt that's due to the external load what is pb and we also want to know assuming the bolts are unplated unlubricated and are pre-loaded for a non-permanent connection what torque is required to achieve the proper preload and what is the total bolt tension when the load due to the 2000 psi pressure is applied what is the total tension in the bolts so let's do that first part so we saw on the slides that pb the additional tension the bolt due to the external load is c the joint stiffness coefficient times p the load per volt the external load per volt what is c c we saw in the slides is kb the stiffness the bolt over kb plus k sub m stiffness of the members and we've got those numbers right here so 3000 over three thousand plus twelve thousand that's fifteen thousand on the bottom this is just one fifth zero point two and i chose these numbers to be nice but that is a very reasonable stiffness coefficient these are approximately correct for a bolt if you make certain assumptions about how long it is we'll learn how to calculate those soon okay so let's calculate this out 0.2 times 70 70.

[00:20:20]14 14. okay and that's our answer that is the additional tension in the bolt due to the external tensile load it's not 7000 it's a small fraction of that 1400 so we're going to assume that the bolts are unplated unlubricated and are pre-loaded for a non-permanent connection if they're unplated that means that our k factor that relates torque and load so i guess let's remember that t is equal to k times f i the desire to preload times the major diameter of the bolt and this k here is 0.3 in our case because it's unplayed and unlubricated so we've got 0.3 times what's fi what's the recommended preload it says there's a non-permanent connection so in that table it says 75 of the proof strength is the appropriate preload what's the proof strength proof strength is excuse me i should say proof load what is the proof load it is the proof strength sp times the tensile stress area 80 and where do we get these these are in the book so let's look up the tensile stress area for a 5 8 18 bolt so 5 8 is the major diameter 18 is the fine thread 0.256 inches squared is the tensile stress area and what's the proof strength of this material it's sae grade 2 sae grade 2 up to 3 quarter inches major diameter minimum proof strength is 55 ksi all right let's multiply those out 14 000 pounds is the proof strength 75 of that recommended preload is 10 000 pounds and plugging that into our formula for the torque as a function of the desired preload is 1980 inch pounds and what's the total bolt tension when this bolt is under load well the total bolt tension is the force in the bolt is equal to the preload f sub i plus c times the external load f sub i is 10560 c is 0.2 as we determined above based on the stiffnesses and p was 70 70. oh but we actually already calculated that product 0.2 times 70 70 this here is just 1414.

[00:23:07]and so the sum of those two total tension the bolt is eleven thousand nine hundred and seventy four and notice that that is even though we're applying seventy thousand seven thousand pounds and we've got a prelude of 10 000 pounds we only go up to a total tension in the bolt of 11 974 pounds so we are less than the proof strength of the bolt despite the fact that 10 000 plus 7000 is well over the proof strength of the bolt so preload's good for a lot of reasons and it doesn't hurt your capacity to handle external load it takes a little bit more effort to do calculations with bolts but it's worth the effort