# a) If a bolt is over-tightened, which will fail first the bolt, or the plastic?

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1 Problem Set 3: Solutions to ME problems Fall 2013 Jacob Bayless Problem 1: Bolted joint a) If a bolt is over-tightened, which will fail first the bolt, or the plastic? The bolt is made of grade 5.8 steel (yield strength 420 MPa) and the flanges are made of polycarbonate (yield strength 60 MPa). When the bolt is tightened, it puts the threads in tension and the flange in compression. Given a root diameter of 4.8 mm, the cross-sectional area of the bolt is: So for a clamping force, the tensile stress in the bolt is The area of the polycarbonate that is in compression follows a stress cone ; it spreads into the thickness of the plastic. The stress will be highest where the cone is narrowest, which is right under the head of the screw. Assuming a clearance bore diameter of, the area directly under the head of the screw will be: The flange and bolt both carry of equal magnitude (although one is tensile and the other is compressive). To determine which will yield first, the stresses are compared with the yield strengths: This ratio is less than one, so the polycarbonate will yield first. (Polycarbonate is quite ductile so it won t break, but when the bolt is unscrewed, there will be a depressed area where the plastic was deformed). b) The bolts are tightened until a tensile stress of 300 MPa is developed in the bolts. What is the stiffness of the joint? As long as the bolt is tightened, the stiffness can be determined without needing to know the exact amount of stress in the bolt. The relevant equations are found on page 20 of chapter 9 of FUNdaMENTALs:

2 The Poisson s ratio of steel is: And of polycarbonate: and Young s modulus and Young s modulus We can assume a strain angle of. Some additional bolt dimensions are required. For these we can look up a typical M6 bolt on McMaster- Carr to determine the measurements:

3 And for the nut: From these drawings,,. Note that much of the relevant information (such as the dimensions of an M6 screw, the diameter of the clearance hole, and the strain cone angle) were deliberately omitted from the question. The reason is that in a real design situation, engineers are not usually handed problems to solve with all of the numbers already given, and so it s important to know how to make educated guesses and research appropriate numbers. In some settings, Before a design is finalized, Now we have enough information to find the stiffness. Just plug in the numbers, and: But there are four bolts, and these are the values from just one of them. The stiffness of the joint is the stiffness of all four bolts in parallel (and in parallel with the stiffness of the flange also times four). So,

5 This is way higher than the yield strength of the bolts, so we can reasonably expect the joint to break well before it would otherwise lose preload. Both the flange and bolts will yield well below such a high applied load. Yielding of the flange will not necessarily cause the joint to fail, but the bolts breaking definitely would. So if there s a chance that the joint will be subjected to very high loads, it would be worth re-calculating the maximum force that the joint can bear before the bolts break: This is a more reasonable load limit. Problem 2. Cantilever Beam A) At each location x, what is the maximum tensile stress along the top surface of the beam? We will let at the wall. (You could also use at the end of the beam if you wanted). The moment in the beam is the product of the force and its distance from, so. The tensile stress in a beam is given by: Where is the distance from the neutral axis (ie, the midplane) of the beam. In this case then,. is the second moment of area (also called the area moment of inertia), which for a rectangular profile is.

6 Therefore, b) How can we remove material to change the profile of the beam so that it is stressed equally along its length when is applied? Clearly, and cannot be constants with in our modified beam. Since it is done only by removing material from the existing beam, we can say that at, we will retain the original beam profile. With both and being free functions of x, there is an unlimited number of valid solutions. One solution would be to taper, so that the thickness is kept constant and Then The width decreases linearly along the length, creating a triangular-tapered beam. Does this seem familiar? An example was shown in FUNdaMENTALs: From FUNdaMENTALs of Design,Chapter 9: Structures, by Alex Slocum Another option would be to allow to vary, holding constant. In that case, This creates a beam which tapers with a parabolic thickness profile.

7 For bearing the point-force load, both of these tapered beams could be considered more structurally efficient than the conventional, uniform-profile beam, especially if a lightweight structure were needed, or if the materials were expensive. Sometimes it s a good idea to go beyond the simple textbook designs! c) If the force is changed to a moment, which design of beam is more efficient and why? With an applied moment, the maximum tensile stress in the beam is now In other words, the stress no longer varies with ; thanks to the moment loading, the stress is already uniform along the beam when and are constant. In that case, the uniform-profile beam is the most efficient structure. Note: Structural efficiency typically refers to a structure s ability to support a load with the least amount of material being wasted. The efficiency is generally highest when all of the materials are stressed to the limit. If there are parts of the beam that aren t highly stressed while other areas are highly stressed, it suggests that the material in the un-stressed areas is being wasted, and that same material would serve its purpose better if it were relocated to reinforce the highly-stressed areas. But as shown above, it depends strongly on the type of loading being considered.

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