 In the previous videos we looked at the different failure modes under static loading for a mechanical fastened joint, net section tension, bearing, shear tearout, fastener shear, and fastener pullout. In this video we are going to develop simple analytical models for predicting the onset of these failure modes. Now before we do that we need to cover a few basic assumptions for our predictions and the first very critical assumption is that we will only consider the failure of ductile materials both in the fastener as well as the sheets. Now the reason for this can be illustrated with this picture of a sheet with a fastener hole in it under a tensile load and on the right we will consider what would happen in a brittle material and on the left we will consider what would happen in a ductile material. In a brittle material well actually in both materials as we apply a load as we've seen earlier there will be a stress concentration associated with that fastener hole. Now if we continue to increase the load the stress at the edge of the hole will increase because of that stress concentration and if that material is brittle that will continue until the stress at that stress concentration reaches the ultimate strength of the material and then we will have failure propagate along the plate. In a ductile material things are a little bit different. It will not suddenly fail like this rather as we increase the load the material will start to yield or plastically deform which will cause a redistribution of stress along the width of the plate. And in fact it's a very easy and simple approximation that at final failure you will have a uniform stress along the width of the plate equal to the ultimate strength of the material. This is convenient for developing our models for failure because we can ignore the stress concentration of the hole and just utilize the fact that plasticity will redistribute loads and result in a uniform, approximately uniform stress equal to the ultimate strength at final failure. Related to this plasticity assumption is that all fasteners carry the same load at static failure. So here we have 12 rivets connecting these two plates and below static failure you will actually have a different amount of load transferred by each row. But just like plasticity redistributes the stress along the width of the plate the highly loaded rows will start to plastically deform which will redistribute load to the lower loaded rows. So at final failure we can assume that the total load is distributed amongst all of the fasteners that transfer load from one plate to another. Now that load transferred from one plate to another is a very important thing to keep in mind and that's related to the load path. So if we consider these two joints here one with two rows of bolts and this one has four rows of bolts you have to be mindful of how many rows actually transmit the load. So in this upper case the load is transferred from one plate to another by two rows. In the lower case you actually have two rows transferring it from the upper to the lower plate and then two rows transferring it back from the lower plate to the next upper plate. So in both of these cases the load F is transmitted from one plate to another by only two rows of bolts. So you would look at the failure in relation to those two rows of bolts. Okay so now let's jump into developing our simplified models for failure modes. We will start with net section tension failure which we saw as a failure that will occur when the stress in the net section so that is the smallest cross section of the plate reaches the ultimate strength of the sheet material. So here we're using that plasticity assumption. The stress is uniform and equal to the ultimate strength. So we can define that force as the net section area times the tensile ultimate strength of the material. If we look at this net section cross section which is this hatched area we would have this width minus the bolt, the regions where the bolt are. It basically cuts out area. So we would get width minus the number of fasteners times the diameter of the fastener and then multiplied by the thickness. That will define our net section area and we multiply that by the ultimate tensile strength of the material to get our net section tension failure for the plate. Next if we look at bearing failure this failure will occur when the stress is acting on the projected area of the sheet. On the bolt exceeds the bearing stress allowable for the sheet material. So this is a material property. So the bearing failure force is our bearing area times that bearing stress allowable. Now to define the bearing area we need to look at that projected area and when there's multiple plates you can have more than one bearing area. So here we have the upper sheet is resisting in one direction, the lower sheet in the other direction so we have a purple and green area. Now if both of the sheets are the same thickness and same material all of this will come out the same but you can join dissimilar materials or different thickness materials so you can get a difference in the bearing failure force for those different plates. In this simple case our bearing area is the diameter times the thickness and then we would multiply that by our bearing strength of the sheet material. Now it's important to realize that this equation gives you the force acting on the sheet by the bolt at failure and not the applied force to the entire joint. It's only looking at a single bolt so you will have to then look at the load path and figure out how many bolts transmit the load from one plate to another and multiply by this force by that appropriate number to get the failure force for the entire joint and we'll look at this a little bit later when we do an example problem. Next we have shear tear out failure which is a failure that will occur when the shear stress is acting on the area between the bolt and the joint edge exceeds the shear strength of the sheet material. So the shear tear out force is the area for shear tear out times the shear ultimate strength of the material. If we look at the image of the bolt our shear tear out area is defined by this cross hatch area in the same area on the other side because we have two areas resisting that failure mode. So that gives us two times the edge distance B times the thickness T. And again this equation gives you the force acting on the sheet by the bolt at failure and not the force applied to the entire joint. Next we have bolt shear failure which will occur when the shear stress acting in the area of the bolt exceeds the shear strength of the bolt material. So our bolt shear failure force is the area resisting that shear times the shear strength of the bolt. Okay not the plate this is a failure mode of the bolt material. And so that will be the cross-sectional area of a bolt which is pi over 4d squared times the shear strength of the bolt. And this N refers to how many planes actually resist the load transfer. So I can illustrate that with this sub figure where if we have two plates joined there's one plane resisting the load transfer. So N would be equal to 1. If we have three plates we have two planes of shear resisting and when we have four plates we actually have three planes of shear resisting the load transfer. So you have to be mindful of how many planes within the bolt resist the shear force. And again this equation is for a single fastener so you have to look at how many fasteners actually transmit the load when looking at the failure force for a entire joint. The final failure mode is fastener pullout failure. However this failure mode is much more complex it's a little bit more difficult to develop an analytical model because it's highly dependent on the bolt head geometry. So you can imagine the shape of the bolt head will have a huge influence on whether or not it gets pulled through the sheet. So it's a little bit beyond what we will cover within this course so just be aware that it would be something that you would typically look at in terms of test data and you may have to do things like adding washers or using a different fastener head to prevent this failure mode if it was at risk of occurring.