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### Sometimes math is incredibly helpful, even when we like to pretend it isn't... ### I started writing this mostly for my good friend Jack on solar car — but as I was doing it, it became relevant enough to increase the generality a little bit and create a post out of it. Bolt pre-load and torque wrench settings are interesting and necessary topics in mechanical engineering. Considerations of preload are incredibly important to many high load mechanical designs (such as the suspension of a solar car). So what I'm going to try to do today is cover the basics of a few topics based on some work Jack himself did, and based on resources I found. Credit will be given where credit is due of course. You'd be surprised how much you can learn from a simple Wikipedia article, and the rest of the information I've pulled from the 31st edition of the Machinery's Handbook and a few other places. This post is more theoretical than practical, so we'll talk about resources for further learning on bolted joints later in the post. Ready? Set? Go! ### So what is bolt preload — essentially it's how hard you tighten down a bolt when using it to attach two things together. Here's another detailed post on this topic, maybe save that for later? Let's look @ the diagram of a bolted joint above. We have a bolt. Which is going through a hole in two parts what we wish to connect to each other with said bolt. And a nut on the other side. When we tighten this bolt down with a wrench, it exerts a force normal to the faces of the part (this is due to material properties, it essentially acts like a very stiff spring, the bolt actually elongates and sees some strain). This force the bolt feels (in the axial direction) is called the bolt pre-load.  You can imagine that because we've tightened this bolt down and there's a pre-load force on the bolt the part feels some force as well. The bolt transfers force to the face of the part where the head of the bolt and the nut touch the part (usually through a washer which I didn't draw purely out of laziness). And now you can also imagine the parts push against each other. The bolt essentially squeeeezzessss this whole assembly in the direction it's being tightened. And this squeezing action produces a pressure on the faces of the part that are touching each other, I'll call this the pre-load pressure. ### Now bolted joints can be loaded in two main ways, axially, or in shear. Each causes it's own set of problems. Shear is a little easier to visualize so I'll start with that. Assume the nut-side is connected to a rigid wall and cannot move.  In shear (and unlike in a pin joint) the bolt does not take the force of shear or stop the parts from sliding apart directly. Instead it's the friction between the faces of the parts that acts against any shear forces applied to the face. It's the goal of the bolted joint to provide enough normal force (or preload pressure times the area) such that the mu*Fn is greater than the force applied.  In compression, the force Fa pushes against the face of the part and the parts compress slightly. If the bolt preload is not high enough the bolt itself can "detach" from the face of the part, rattle around, cause parts to shift, and also decrease fastener fatigue life. This is a more complex loading condition and we won't go into great levels of detail about it here. But mainly all this is to say this is what preload is, and it's important. Very important. ### So those conditions above specify a minimum preload right? But the other thing to consider is the maximum. We can't keep tightening the bolt to infinity preload because either the bolt will break or the material we are bolting will yield or buckle. So we need to find this happy medium between the two. How you find the happy medium, greatly depends on the setup and the joint. Is the loading cyclic? What materials are you using? All of these can change your bolt preload requirements. ### This is a graph I first saw in the Machinery's Handbook — and he's a very nice graph (this specific one was stolen from here). But at first glance he might be somewhat meaningless. This graph shows the load the bolt itself takes in a bolted joint when you apply a force to the assembly.  The 1-to-1 line describes how if the bolt preload is zero, the bolt takes all the force. This makes sense because the friction between the parts is zero since the normal force is zero, and in the case of axial loading, any force applied must be taken by the bolt fully because that's the only thing holding the two parts together.  Now let's say we preload the joint and apply an axial force to the system. When we preload a joint the material inside the two parts we are joining compresses by a certain amount (because the material is elastic and the bolt preload is applying a force to them). Which means as an external compressive force is applied the materials just compress into each other more, friction goes up, the joint is fine (until the material itself yields). But in tension, the material UN-COMPRESSES until it's in the fully uncompressed (original) state, and at that point the bolt starts taking forces. That point is called the separation point because the normal force between the two parts has now gone to zero and all the force is being taken by the bolt alone. ### So how do we get preload — easy, tighten that boi wayyyy down with a wrench. If you know the number of rotations you've applied to the nut, and you know the thread pitch (how much elongation comes from one rotation), you can calculate the total bolt elongation and find the preload from the internal stress (for my 2.001 introduction to mechanics friends). But most times these days we use a torque wrench which allows you to apply a specific torque to the bolted joint which translates to a certain preload force. Either way, there has to be some rotational to axial translation that must happen to determine how to inflict a desired preload onto a joint. Note that this torque we apply has to "do" two things. It has to apply this elongation to the bolt itself that generates the preload force, and it has to overcome any friction between the head of the bolt and the face of the part. This is NOT insignificant and can be close to 40-50% of the preload torque if not more. So our math is slightly complicated by this fact.   ### All the equations above come from the Wikipedia article linked above. They were cross checked against a few sources before being placed in this post. Essentially we can boil down all of the complex physical interactions between the bolt, the parts, friction, the threads, down to these sets of simple formulas. So now let's talk about how to find these numbers, and write a MATLAB script for the calculation. I'll use metric bolts because that's what I use most often (sorry Jack, I do think it'll work for inches though). Note, we also referred to Shigley's Mechanical Engineering Design 8th Edition to fill in the blanks (page 422 relating bolt torque to bolt tension). Here's a free version from online. Here's some standard major and minor diameter data for metric screws, it includes thread angle data. Here's the bolt data itself from McMaster. We made a MATLAB script for torque-preload for bolts here as well. The MATLAB script does a good job explaining most of the values, and the links above provide resources for inputting the parameters. This is how you know how much torque to apply. ### This is true because the bolt preload generates the normal force, friction is proportional to normal force by the coefficient of friction, and we want friction to be resisting any shear forces. bottom of page