I apologize for the long delay in producing this, classes and all that, but here is my introductory to mechanics of deformable bodies explanation and analysis of VEX robotics shafts. Note, this is a huge topic, and I will only cover the small portion of it that is directly applying to this analysis.
Disclaimer: This is college level material, I will explain it in a general case, but it is a complicated but useful subject.
The three categories that are important to this calculation are loading, material data, and dimensions. Loading includes the type of load (tension, compression, shear, and torsion), magnitude of the loads, and cycle information (not covered). Material data is what characterizes the physical properties of each material, there is a large number of these covering thermal properties, optical properties, and physical properties. The most common is probably density. Note that mass is not included as it is dependent on the amount of material there is. The big values used in this analysis are yield strength, ultimate strength, and elasticity. Dimensions refer to the shape of the body being loaded and the relative distances between relevant parts of the system.
The easiest way to visualize the force per area required to elongate a particular material is a stress-strain curve. Stress (lowercase sigma) is a measurement of pressure, force per unit area, while strain is a unit-less quantity describing the amount of stretch per original length. It is sometimes denoted as length per length, in/in, or cm/cm. The formula is the change in length divided by the original length. The stress-strain curve is different for each type of material.
The stress-strain curve can be broken down into four different sections, the first section being the elastic region, where the material behaves like a spring. In both the yielding (very small) and the strain hardening sections, the material undergoes plastic deformation where it does not return to its original length. In necking, the cross-section of the part begins to shrink as the strain increases. Advanced studies in this subject will explain more of the interesting phenomenon that occur in this situation.
There are two classifications for materials, ductile and brittle. VEX materials are almost entirely ductile, meaning they experience significant yield, while brittle materials will break after stretching elastically.
The four types of loading that I will discuss in relation to shafts are: normal, shear, torsion, and bending.
VEX shafts are made of AISI 1018 Steel, with a Rockwell hardness C of 20, which makes it a bit stronger than the materials in my textbook. Which makes this an excellent time to introduce the concept of the factor of safety. It is defined by the maximum failure stress divide by the allowable stress. It is an important concept in engineering which allows for a margin of error for variables that were not accounted for, assumptions and simplifications that were made, and unexpected loadings. It is good practice to include them, but competition robots may ride very close to a safety factor of one. I am using 64ksi for ultimate tensile strength and 54ksi for yield strength. I got my values from Shigley’s Mechanical Engineering Design. The last relevant piece of information is that the cross section of VEX standard shafts are 1/8” squares and high strength shafts are 1/4” squares.
Normal stress occurs when you load an axel lengthwise in a pull or push configuration. I will not cover axels under compression as it is much more complicated. Under tension the formula is simply: stress = F/A. Using this the maximum load for yielding is 843.75lbf for standard shafts and 3375lbf for high strength shafts. The load at the ultimate tensile strength is 1000lbf for standard shafts and 4000lbf for high strength shafts.
Shear stress is when you load a shaft perpendicular to its length but without bending it. It is similar to if you were trying to cut it with scissors. It occurs where ever there is a pin or shaft in drivelines and in lifts. The hitch is that the yield and ultimate stresses are for tension, not shear. Fortunately, there is a conversion. Skipping the derivation, the conversion rate is about 50%. The formula for shear stress is visually identical to normal stress, just conceptually slightly different. As we need the area of the face that is being sheared, which is perpendicular to the force. Using this data, the maximum load for yielding is 421.9lbf for standard shafts and 1687.5lbf for high strength shafts. The load at the ultimate tensile strength is 500lbf for standard shafts and 2000lbf for high strength shafts. Note if the axel is supported at two points, the area doubles, doubling the allowable load.
Torsion is any twisting of the shafts. Exceeding the yield strength is what is responsible for twisted shafts. It is actually a little more complicated to derive the formula for since it is a square shaft, but there is an established formula for square shafts. Shear stress is equal to 4.81 times the torque divided by the cube of the side length. This produces a maximum load for yielding is 10.96lbf-in for standard shafts and 87.71lbf-in for high strength shafts. The load at the ultimate shear strength is 12.99lbf-in for standard shafts and 103.95lbf-in for high strength shafts. Note the 393 motor in standard gearing produces 14.78lbf-in at stall.
Bending is very simple to understand on a broad level, but comparatively difficult to calculate. First off, bending produces a tensile stress, linearly increasing from its centroid (center of area). VEX shafts are nice squares, so it is right in the middle. The maximum stress is on the outside edge. The formula for bending stress is the moment times the distance from the centroid divided by the area moment of inertia. Some derivation will get you: stress equals six times the moment divided by the cube of the side length. The maximum load for yielding is 17.58lbf-in for standard shafts and 140.625lbf-in for high strength shafts. The load at the ultimate tensile strength is 20.83lbf-in for standard shafts and 166.67lbf-in for high strength shafts. But this is only if the shaft is orthogonal to the force, what if it was rotated 45 degrees so the force was on a corner? The distance from the centroid increases, so the formula is stress equals the moment times 6 times root two, divided by the cube of the side length. This makes the new maximum load for yielding is 12.43lbf-in for standard shafts and 99.44lbf-in for high strength shafts. The load at the ultimate tensile strength is 14.73lbf-in for standard shafts and 117.85lbf-in for high strength shafts.
This was a huge amount of information. Don’t worry if some of it was confusing, it is a sophomore level engineering course, so feel free to ask me any questions and I will do my best to answer them.
Reference: Mechanics of Materials 9th ed. by Hibbeler
Nice write up!