Threads & Bolted Joints IV: Fastener and Member Stiffness Example

Ajouté : 2026-05-07

[00:00:00]today's lecture is about fastener and member stiffness in bolted joint connections so specifically learning how to calculate these things and we're going to do it through an example this is a water pipe at the wrf on campus and it's got 16 3 4 10 unc bolts we've got narrow washers on both sides and regular nuts on the sides with the threads now in reality it's actually threaded rod throughout and there's nuts on both sides but i'm going to pretend for the sake of the analysis that it's a regular old bolt the analysis with a threaded rod is actually easier it's just that the whole length inside the grip is going to be threaded as opposed to a bolt if it were a bolt then we're going to learn how to figure out what the unthreaded length was going to be and treat that separately from the threaded portion of the length because they're going to have different areas over which different stiffnesses as a result so it's going to make the stiffness calculations a little bit more complex and that's why we're going to try it that way the total thickness of the two the plate so we've got like a flange on the end of the pipe and we've got water pressure that's being held on by in this by this cap here the total thickness of the flange plus the cap is two inches let's say that the pressure of the water actually i was told by a student last year that they've seen it go up to 100 psi and they told me that the inside diameter of these pipes was 18 inches so this pressure is acting over that whole 18 inch circle so we've given that steel pipe it's got the operating pressure of 100 psi and the gasket has a diameter of 18 inches the combined thickness of the flange and the cap is 2 inches and the cap is secured by all those bolts what is the total tension in each of the bolts and is it less than the proof strength of the bolt does the bolt going to fail so our strategy here is going to be well we need to determine the tension in the bolt and see whether it's less than the proof strength right so our equation here is the tension in the bolt tension in the bolt less than or equal to the proof strength p for proof b for bolt so how do you calculate both of these well we've talked about in previous lectures that the proof strength is the product of the excuse me the proof load is the product of the proof strength which is a stress load being a force is the product of the proof strength and the tensile stress area of the bolt and the proof strength of the bolt you just look up in a table you can actually also look up for your bolts what a t is so we're going to go through that process but on the left hand side of this equation we've got the tension of the bolt that's the sum of f i the initial force in the bolt the preload of the bolt plus p sub b let's change that to a lowercase b which is the load in the bolt the tension the bolt that is caused by p the externally applied load i guess is the way to say it it's tension in the bulb that's caused by the externally applied load so we've got some preload that's due to the fact that you torqued the bolt and then pb the externally applied additional tension what is the preload in the bowl well that depends on whether you have a permanent or non-permanent connection or really how much you torque the bolt but the recommendation is that you should set a preload for non-permanent connections to 75 percent of the proof load of the bolt which we've already calculated now this additional tension in the bolt that's due to the applied load is given by c the joint stiffness constant or coefficient times p which is the externally applied load per bolt see we can calculate if we knew the stiffness of the bolts and the stiffness of the member which is the material in between the bolt in that sandwich there it's this ratio here and then p the externally applied load purple is the total applied load in this case the pressure times the area in this case divided by n the number of bolts we're going to assume that it's equally divided among all of them which is a reasonable assumption because of the symmetry in the problem now remember here the additional tension in the bolt that is due to the applied load is some fraction c that's less than one you can tell this is less than one if these are positive numbers the denominator is the sum of these two which is greater than the numerator so this is less than one it is a fraction less than one of the applied load per bolt and what we're after here is how to figure out for a given voltage joint that you might be designing what is c here and how do we calculate how do we figure out what the kb and km are in order to calculate what c is so that's what this module is really about is calculating those stiffnesses so to get started we really have to look at well in my book um this is version 9 page 426 table 8-7 but there's a similar table in the 10th and 11th editions and so you can find something that looks like that is the suggested procedure for finding the fastener stiffness so i'll zoom in a little bit here let's talk about the different uh these two different diagrams here we've got an a and a b and we've got a little bit different instructions depending on whether we're looking at something that looks more like figure a or something that looks more like figure b let's talk about these different dimensions d the major diameter the bolt right um the total length of the bolt capital l is from the bottom of the head the base of the head to the very tip that's not a super important parameter another problem or it's just kind of an intermediate thing what we're really going to be after is what is lowercase l sub d which is the distance between the head of the bolt to the beginning of the threads the run out is what it's called where the threads begin and that look this is a cylinder of a constant radius that's the major diameter of the bolt times two or divided by two i guess and so it's going to have this is just a cylinder of steel it's going to have a stiffness associated with it and then also we're going to have this threaded portion here which is going to have a length l sub t lowercase lt which is the part of the grip i guess the lowercase l is the grip the part that's in between the base of the head and the nut that's lowercase l it's the part of the grip that's threaded and this has a different area effectively it's it's a bit like a cylinder of a different area because it's got the threads on there it's going to have you're going to use the tensile stress area for calculating its stiffness instead notice something between uh the the lower case and the upper case lower case is related to the grip length it's the part of the fast of the fastener that is sandwiching material between it the upper case l is the whole length is what it's referring to so uppercase you can think of hole length the entire length and lowercase l is grip length and it's consistent we're talking about l subscript t if it's a capital l we're talking about the whole threaded length whereas a lowercase t excuse me lowercase l subscript t we're talking about the portion of the threaded length that's within the grip and similarly here lowercase l sub d is the portion of the unthreaded link that's within the grip but the unthreaded length is the same the the part of it that's in the grip and the whole of it it's always all of it is always in the grip so we could give this a symbol capital l subscripty as well we've got a situation like this we've got a bolt and we've got a washer and we've got a nut in the end actually we've got a washer on this side as well in our case we said there's washers on both sides we've got a bolt situation as opposed to a screw situation we call the same fastener a screw when it threads into a part and doesn't have a nut on the other end so a and b different procedures we've got procedure a that's relevant to us here so the first thing we're just going to follow this to the letter if we've got a fastener diameter d yep we've got it it's three quarters of an inch a pitch p what's the pitch in this case well we've got 10 threads per inch so let's find out what the pitch p is going to be p the pitch is 10 threads per inch so it's actually just one over the number of threads per inch is equal to 0.1 inches what's the washer thickness well we get it for a fastener with this diameter from table a32 or a33 well in my book i'm jumping over to page 1056 for this and it just says what the dimensions of standards are and we've got the fastener size well we've got a three quarter inch fastener now i said it was a narrow washer instead of a wide washer and for narrow w for wide so if it was a narrow washer from the problem description then the thickness here this column here for a three-quarter inch and when we're talking about narrow washer is 0.134 inches so t the washer thickness is 0.134 inches what's the next step in the procedure it says find the nut thickness h from table a31 okay so we go to table a31 and we find for the nominal size of the nut which is three quarter inches we are interested in the height h of the nut we have we've got a regular nut it said so in the problem statement and the regular nut has a height h of 41 64.

[00:10:13]okay so that's the first few parts of our procedure here i just wanted to show you that these are reasonable things and what it looks like when you're actually choosing or doing some design work so mcmastercar.com we've talked about using that before it's a good source of parts and let's take a look at what the height of nuts are that work for a three-quarter inch bolts so let's say we've got hex nut medium strength just whatever the first thing gives us is and we're talking about for a thread size three quarter ten and look the height of all these nuts is 41 64. so the table in jiggly is giving us reasonable information now there might be nuts that are a little bit longer than that etc but what shiglee is giving us for kind of common ones is pretty reasonable same sort of thing for washers i'm just going to pick a washer general purpose nothing too special for a screw size three quarters and we see that there's actually a wide variety of thicknesses of washers and inner and outer diameters and it's all over the place right but what was our thickness we said 0.134 and that's in the range that we've got here some of these are a little bit thinner this one it's within this thickness range it's within this thickness range so it's a reasonable number i would you know if you're doing this carefully and doing for a real design we'll pick out your washers and then you know know what the thickness of those are um the recommendations in shigley these suggested numbers are for doing back of the envelope calculations when you haven't actually chosen your parts just getting kind of approximations okay so the next step is to find what the grip length l is and we're talking about a situation like in figure a and it says the grip length is the thickness of all material squeezed between the face of the bolt and the face of the nut now we've got a washer on this end that's squeezed between the face the bolt and the face the nut we've got another washer on this end and then we've got the sandwich of material in between so from our original figure you know i said that there was a washer here there was a washer on the other side and there's a two inch sandwich in between so it's the two inches plus two times the thickness of the washers is our grip length and the grip length was 2 inches plus twice the washer thickness this was 0.134 inches and that number is 2.268 inches what's the next step from that we can find the fastener length we need a fastener with total length l that's greater than the grip length plus capital h this is just from geometry we need a fastener that is at least as long l as the grip length plus the height of the nut i mean you can see there needs to be threads sticking out when we install this it has to go all the way through the nut it can be a little longer that's fine but we need to choose some sort of fastener that is greater than the sum of those two things so let's go ahead and calculate what this is what this l plus h is we need the total length of the fastener to be greater than l plus capital h which is that is equal to 2.268 inches plus 41 64.

[00:13:58]that means it's equal to 2.91 inches and l has to be bigger than that the total fastener length has to bigger than that we're going to choose a fastener from that table a17 table a17 says that the next largest we've got 2.9 i want to say 2.91 the next largest is 3 and these list preferred sizes so 3 inch fastener is what we're going to pick l is equal to 3 inches and let's also see that on mcmaster so if we go to bolts hex head screws we had um let's use grade five that's kind of medium strength three quarter ten looks like some are fully threaded you can get bolts that are fully threaded and you can get ones that are partially threaded and shigley is telling you that is giving you how to estimate what those minimum threaded lengths are going to be and we're going to see that in a second but look we've got kind of preferred sizes here two two and a quarter two and a half two and three quarters three inches there's not one that is two point nine one inches long we have to pick from one of these preferred sizes and schickley's fraction of inch table does a pretty good job of telling you what those are without you having to look up a particular part number okay so assuming it's a partially threaded fastener the size that we're going to need the length we're going to need is three inches so let's move on the next step is to figure out what the total threaded length l subscript t is if it's an inch series bolt and its total length is less than six inches long which it is it's only three inches then the threaded length lt is two times the diameter plus one quarter inch so lt is two times the diameter plus one quarter inch that would be 1.75 inches what does mcmaster say about that well yep for a three quarter inch bolt the minimum threaded length for any of the partially threaded fasteners is one and three quarter inches when we start getting above six inches when it's greater than that we've got a two inch minimum threaded length let's see what shigley has to say about that if jiggly agrees if it's greater than six inches then we add our quarter an inch to it so yep exactly what shigley said so shigley again just a handy way to give you an estimate of what you're going to find in a catalog of real parts okay what is the length of the unthreaded portion that's in the group and this is just geometry the unthreaded portion lowercase ld is equal to the total length of the fastener minus the threaded length of the fastener you can see here this part l sub d is equal to the total length of the fastener l minus capital l sub t right this threaded portion remembering that you know maybe we should write the equation capital l sub d is equal to capital l minus capital l t maybe it makes more sense if you think about it like l is equal to lt the threaded portion plus ld that makes it really obvious right well ld then is equal to l minus lt and ld is really itself equal to lowercase ld because the portion of the fastener that is unthreaded is equal to the grip portion the portion within the grip that's unthreaded so actually what i'm going to do is i'm just going to replace this symbol with lowercase l sub d and we're going to calculate this it is 3 inches minus 1.75 inches is equal to 1.25 inches what's the next step find out the length of the threaded portion that's in the grip that is the grip length minus l sub d the portion that is unthreaded within the grip so lowercase l sub d is equal to l minus l sub t the total portion that's in the grip is the unthreaded portion plus the threaded portion we could say the total portion that's in the grip is equal to the threaded part of that plus the unthreaded part of that and that's going to tell you how to find l sub t l sub t is equal to l minus ld looking at our numbers 2.268 minus l sub d which was from above 1.25 we get 1.018 and it says find the area of the unthreaded portion find the area of the threaded portion so a sub d the area of the unthreaded portion is pi major diameter squared divided by 4.

[00:18:53]that makes sense we've got just a basically this portion is just a cylinder with radius d or diameter d so let's go ahead and calculate that 0.442 inches squared it says the area of the threaded portion is the tensile stress area from table 8-1 or 8-2 depends on whether you're using metric series or not 8-2 looks like it is for unc threads so we've got a three-quarter it is the 10 threads per inch and it has a tensile stress area tensile stress area for the coarse version is 0.334 inches squared we know that the deflection of a cylinder a cylindrical rod is equal to the force like pl over ea right we've seen that before so the stiffness which is basically the force divided by the deflection force per inch of deflection is equal to ea over l and what we're saying is e well it's the elastic modulus of steel which is uh 30 times 10 to the sixth psi and a well if we're talking about the unthreaded portion it's going to be this if it's the threaded portion it's a sub t and then length well if we're talking about the unthreaded portion it's l sub d if we're talking about the threaded portion it's l sub t so we can kind of just apply this in order to find the individual stiffnesses so the two parts so let's try that k of the unthreaded portion is equal to e times a d over ld is 30 times 10 to the sixth 0.442 1.25 inches we've got 10 million six hundred and two thousand eight hundred and seventy-five pounds per inch very stiff kt use the threaded area the tensile stress area over the threaded portion that's within the grip so that's 0.334 inches third portion that's within the grip is 1.018 inches 9 million forty two thousand eight hundred and twenty nine pounds per inch right and these are springs that are in series so we kind of remember that we can add them and find the total stiffness one over the total stiffness is equal to one over k so the stiffness of the bolt one over k of the bolt is equal to one over kd plus one over kt you could rearrange this you could figure out that kb is kd kt over kd plus kt we're just substituting in these two numbers here five million one hundred four 104 362.

[00:22:15]i guess this is really a three pounds per inch which is one of the things that we're after the stiffness of each of the bolts now there's a formula in the book that is basically just substituting all this together you could just use this formula and all it's done is exactly what we have you see these same sorts of equations that we've got they've just substituted in ae the areas and the elastic modulus and all that into our equation from before so if you prefer you can instead of calculating the stiffness of the threaded portion and unthreaded portion separately and then adding them as springs in series you could just use this equation here i just wanted to show conceptually where that was all coming from okay the next thing we might want to find is the stiffness of the material that is within the grip the member stiffness and there's two ways that we can do that and they're going to give us approximately the same answer so i'm going to show you the equations in the book that are relevant here so all this is derivation it's boils down to this equation being really useful if you've just got um a consistent material if it's both both sides are the same material of your sandwich or everything in your sandwich is the same material like in this case both sides are the same sort of steel if we had one that was aluminum one that's steel this equation isn't going to be valid because we're going to have different elastic modulus but if all of it's steel it comes down to you get a stiffness the member that is given by this all right so let's take a look at what these things are here we've got e which is 30 times 10 to the sixth we've got d which is three quarter again we've got d is three quarter and same thing here and these l's are the grip lengths 2.268 inches so if we were to plug all that in we get 20 million 986 874 pounds per inch and then that's formula is kind of long so they have an approximate formula that's based on a research paper stiffness of the member divided by the elastic modulus the material divided by the diameter is equal to you know some exponential function and so if we want to find the stiffness of the member then we are going to just multiply both sides by e d so this equation is telling us that km is equal to e d elastic modulus major diameter bolt of some constant a this is a constant we're going to look it up in a table exp that's the exponential function e to the power of b some other constant times the major diameter of the bolt divided by the grip length so if the entire joint is made up of the same material their constants for steel a is 0.787 and for b 0.628 so if you had a joint that was entirely made up of aluminum then you would use this a and this b instead one of the main things though to keep in mind is a in this case is not the area it's not any area it's just a constant these are two constants that are coming from the table in the book this a is given by the table and this b is given by the table right we've got d which is three quarters we've got l 2.268 as before got another example of d and then we've got elastic modulus 200 well 30 times 10 to the 6.

[00:26:14]so if you plug all that in you'd get 21 million 800 and 3900 pounds force per inch look these are two different methods for calculating the stiffness of the member we don't have just one formula because there is not one that perfectly describes reality we're going to have to make approximations in either case this one was based on some theory that said that the stresses are going to distribute themselves over a cone and we're going to kind of figure out how squishy is that material that's it's where the stress is experienced versus this one which is kind of fea driven they did some curve fits based on fea experimental data i mean computer experiments in this case and look we get a similar result so we could pick something in the middle of these two we could pick one of the two it's our choice as engineers we see that they're similar and they're not going to affect the outcome of our problem probably they might affect the exact factor of safety but probably not whether the joint is safe or not so let's go ahead and for the rest of this we're going to use that the stiffness of the member inside the joint is 21 million so again looking at our strategy one thing that we needed these kb and km for was in order to find capital c the joint stiffness coefficient let's go ahead and write that let's actually take a screenshot of this this will help guide the rest of the analysis all we've done so far by the way is pick let's figure out what kb and km are but the rest is easy right you can see all the equations here so we're just going to plug and chug really for the rest it's all stuff that we've done before our kb is this number our km we said we're going to use this number and so our c if you plug those things in we're going to get a c of 0.1897 so in order to find the applied load per bolt p we have to find p total our total applied force that's equal to the pressure which was 100 psi on the cap times the area of the gasket so that's 100 psi times well it's pi times some diameter the diameter of the gasket that was 18 squared divided by 4.

[00:28:51]so the total force acting on that cap is trying to pull it off is 25 440 pounds and the total number of bolts is 16 and so our applied load per bolt that's trying to pull each bolt away is that 25 000 divided by 16 is 1590.43 pounds okay so let's fill in from these numbers here 0.1897 is our c 1590.43 well c times p that's 301.7 pounds force in each bolt is the additional tension that's caused by that external load even though we've got 25 000 pounds spread over 16 bolts which would be an average of 1590 pounds per bolt the total additional tension in each bolt due to that load is only 301.7 pounds so not very much now what's the preload let's see if in order to find out whether we're going to be the total tension is less than the proof load we need to find out what the preload is so 0.75 times the proof load well what's the proof load all right let's look these things up for our bolt at our tensile stress area we already calculated that actually we already have that uh we found that from the table 0.334 inches squared and the proof strength of our material let's say we're using grade 5 bolts all right grade five the material for bolts between one quarter and one inch in diameter our proof strength is 85 ksi okay so our proof load is 28.39 kips so our preload for a non-permanent connection is just 75 of that so this here is going to be 21.29 kips all right then the total tension in the bolt is the sum of these two things we've got fb is equal to 21 293 actually pounds plus just a measly 301.7 pounds that's due to the externally applied load so almost all of it is preload and we get a grand total of 21.50 9 kips all right so is 21.59 kips less than or equal to 28.39 kips yes it is so our bolts are fine we've only got an additional tension of 301.7 pounds for each of these bolts compared to the 21 000 pounds that we pre-loaded it to so it's almost inconsequential what the applied load is we could increase the apply load greatly and we still would not exceed the proof load of those bolts