{"type":"doc","content":[{"type":"paragraph","content":[{"text":"Purpose:","type":"text","marks":[{"type":"strong"}]},{"text":" The intent of this page is to give an intro to the necessary background to structural mechanics.","type":"text"}]},{"type":"heading","attrs":{"level":1},"content":[{"text":"Introduction to Fundamental Structures Concepts","type":"text"}]},{"type":"paragraph","content":[{"text":"Instead of attempting to summarize all the fundamentals necessary to understand structural mechanics, a good YouTube series by The Efficient Engineer gives a good a very informative and in depth summary.","type":"text"}]},{"type":"paragraph","content":[{"text":"Structural Mechanics Playlist:","type":"text","marks":[{"type":"strong"}]}]},{"type":"blockCard","attrs":{"url":"https://youtube.com/playlist?list=PLEYqyyrm-hQ3wtF34smyJSAOqUJqnf1ch&si=r_53Z3Wb7SCAfwYu"}},{"type":"paragraph","content":[{"text":" This playlist is < 1.5 hrs of content. I would recommend watching all of it and referring back to it when you encounter questions as you begin your structural analysis career. Feel free to skip the fatigue failure section of this playlist as it doesn’t affect our structures (this isn’t the case in industry, fatigue is one of the more common failures for vehicles with more than a few years of lifecycle).","type":"text"}]},{"type":"paragraph","content":[{"text":"Note:","type":"text","marks":[{"type":"underline"}]},{"text":" The playlist goes through a fair bit of derivations, which are great in understanding how the fundamental failure mode equations are developed. However, I want the key takeaway to be an understanding of what aspects of your design (variables) you can leverage to strengthen your structures.","type":"text"}]},{"type":"paragraph"},{"type":"paragraph","content":[{"text":"Material Properties:","type":"text","marks":[{"type":"strong"}]}]},{"type":"blockCard","attrs":{"url":"https://youtube.com/playlist?list=PLEYqyyrm-hQ1BC1Meqj7DMIoPSS7TTy16&si=cF2oHwSRniu_h43U"}},{"type":"paragraph","content":[{"text":" This playlist is < 0.5 hrs of content. I would strongly recommend watching all of it. Understanding material properties will help you understand what materials to select, how to select them, and why to select them. Again this series is here to help you establish the foundation. A further understanding is developed by applying your learnings and tuning into your MODS courses.","type":"text"}]},{"type":"rule"},{"type":"heading","attrs":{"level":1},"content":[{"text":"Cross Sectional Profiles","type":"text"}]},{"type":"paragraph","content":[{"text":"Understanding Area Moment of Inertia (a.k.a moment of inertia):","type":"text","marks":[{"type":"strong"}]}]},{"type":"paragraph","content":[{"text":"Goal","type":"text","marks":[{"type":"underline"}]},{"text":": This section is an attempt to build an intuitive understanding of the area moment of inertia of a beam’s cross section. ","type":"text"}]},{"type":"paragraph","content":[{"text":"The area moment of inertia of a beam dictates a beams ability to resist loading in a certain direction. The higher the moment of inertia, the stiffer the beam.","type":"text"}]},{"type":"paragraph","content":[{"text":"The more area a beam has away from the neutral axis, the higher the moment of inertia. ","type":"text"}]},{"type":"paragraph","content":[{"text":"Consider the beam with the following cross sections:","type":"text"}]},{"type":"mediaSingle","attrs":{"layout":"center","width":400,"widthType":"pixel"},"content":[{"type":"media","attrs":{"width":914,"alt":"image-20240830-230644.png","id":"f311fc73-f3ed-4f93-a6b7-f5ebe2f23ff8","collection":"contentId-3686432769","type":"file","height":541}},{"type":"caption","content":[{"text":"Figure 1: Beam in Pure Bending","type":"text","marks":[{"type":"em"}]}]}]},{"type":"table","attrs":{"layout":"default","width":760.0,"localId":"08822534-788d-483c-8d39-b750a76c92ba"},"content":[{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1},"content":[{"type":"mediaSingle","attrs":{"layout":"center","width":362,"widthType":"pixel"},"content":[{"type":"media","attrs":{"width":912,"alt":"image-20240830-231027.png","id":"4cb64bcb-583a-489d-b81f-43585dd4af63","collection":"contentId-3686432769","type":"file","height":940}},{"type":"caption","content":[{"text":"Section A","type":"text","marks":[{"type":"em"}]}]}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1},"content":[{"type":"mediaSingle","attrs":{"layout":"center","width":362,"widthType":"pixel"},"content":[{"type":"media","attrs":{"width":911,"alt":"image-20240830-231007.png","id":"8581e471-44eb-4f3f-8aa3-cd482f58cf7c","collection":"contentId-3686432769","type":"file","height":940}},{"type":"caption","content":[{"text":"Section B","type":"text","marks":[{"type":"em"}]}]}]}]}]}]},{"type":"paragraph","content":[{"text":"The neutral axis for both cross sections is the Y-axis (a result of symmetry). Note however, that the area for the two sections are different. Section B has the flats of the I-beam further away from the axis. This design leads to the following. Note that we are looking at the area moment of inertia about the Y-Axis.","type":"text"}]},{"type":"table","attrs":{"layout":"default","width":760.0,"localId":"dfb8999f-f197-46a8-b9e3-060cf6551657"},"content":[{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[252.0]},"content":[{"type":"paragraph","content":[{"text":"Area Moment of Inertia, I [m","type":"text"},{"text":"4","type":"text","marks":[{"type":"subsup","attrs":{"type":"sup"}}]},{"text":"]","type":"text"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[505.0]},"content":[{"type":"paragraph","marks":[{"type":"alignment","attrs":{"align":"center"}}],"content":[{"text":"I","type":"text"},{"text":"A","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" << I","type":"text"},{"text":"B","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[252.0]},"content":[{"type":"paragraph","content":[{"text":"Cross Sectional Area, A [m","type":"text"},{"text":"2","type":"text","marks":[{"type":"subsup","attrs":{"type":"sup"}}]},{"text":"]","type":"text"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[505.0]},"content":[{"type":"paragraph","marks":[{"type":"alignment","attrs":{"align":"center"}}],"content":[{"text":"A","type":"text"},{"text":"A","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" << A","type":"text"},{"text":"B","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[252.0]},"content":[{"type":"paragraph","content":[{"text":"Mass of Beam, m [kg]","type":"text"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[505.0]},"content":[{"type":"paragraph","marks":[{"type":"alignment","attrs":{"align":"center"}}],"content":[{"text":"M","type":"text"},{"text":"A","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" << M","type":"text"},{"text":"B","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]}]}]}]}]},{"type":"paragraph","content":[{"text":"One might think that this result means, more area == larger moment of inertia, ","type":"text"},{"text":"this characteristic is not always true","type":"text","marks":[{"type":"strong"}]},{"text":".","type":"text"}]},{"type":"paragraph"},{"type":"paragraph","content":[{"text":"What if we make the sections such that the area of A and B are the same? What happens to our moment of inertia?","type":"text"}]},{"type":"paragraph","content":[{"text":"Consider the following:","type":"text"}]},{"type":"table","attrs":{"layout":"default","width":760.0,"localId":"5dc27475-b1ca-4b35-a510-6466cbc8a851"},"content":[{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1},"content":[{"type":"paragraph"},{"type":"mediaSingle","attrs":{"layout":"center","width":362,"widthType":"pixel"},"content":[{"type":"media","attrs":{"width":912,"alt":"image-20240830-231027.png","id":"4cb64bcb-583a-489d-b81f-43585dd4af63","collection":"contentId-3686432769","type":"file","height":940}},{"type":"caption","content":[{"text":"Section A","type":"text","marks":[{"type":"em"}]}]}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1},"content":[{"type":"mediaSingle","attrs":{"layout":"center","width":362,"widthType":"pixel"},"content":[{"type":"media","attrs":{"width":791,"alt":"image-20240830-232340.png","id":"0ab788bb-6cc7-4906-864a-c49deb0a7ff4","collection":"contentId-3686432769","type":"file","height":937}},{"type":"caption","content":[{"text":"Section C","type":"text","marks":[{"type":"em"}]}]}]}]}]}]},{"type":"paragraph","content":[{"text":"assume that the beam cross sectional areas are the same. This change would result in a table as follows","type":"text"}]},{"type":"table","attrs":{"layout":"default","width":760.0,"localId":"7af4277e-3368-472f-a58b-0feeec95ab94"},"content":[{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[252.0]},"content":[{"type":"paragraph","content":[{"text":"Area Moment of Inertia, I [m","type":"text"},{"text":"4","type":"text","marks":[{"type":"subsup","attrs":{"type":"sup"}}]},{"text":"]","type":"text"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[505.0]},"content":[{"type":"paragraph","marks":[{"type":"alignment","attrs":{"align":"center"}}],"content":[{"text":"I","type":"text"},{"text":"A","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" < I","type":"text"},{"text":"C","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" ","type":"text"},{"text":"< I","type":"text","marks":[{"type":"textColor","attrs":{"color":"#ff5630"}}]},{"text":"B","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}},{"type":"textColor","attrs":{"color":"#ff5630"}}]}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[252.0]},"content":[{"type":"paragraph","content":[{"text":"Cross Sectional Area, A [m","type":"text"},{"text":"2","type":"text","marks":[{"type":"subsup","attrs":{"type":"sup"}}]},{"text":"]","type":"text"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[505.0]},"content":[{"type":"paragraph","marks":[{"type":"alignment","attrs":{"align":"center"}}],"content":[{"text":"A","type":"text"},{"text":"A","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" = A","type":"text"},{"text":"C","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" ","type":"text"},{"text":"< A","type":"text","marks":[{"type":"textColor","attrs":{"color":"#ff5630"}}]},{"text":"B","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}},{"type":"textColor","attrs":{"color":"#ff5630"}}]}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[252.0]},"content":[{"type":"paragraph","content":[{"text":"Mass of Beam, m [kg]","type":"text"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":[505.0]},"content":[{"type":"paragraph","marks":[{"type":"alignment","attrs":{"align":"center"}}],"content":[{"text":"M","type":"text"},{"text":"A","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" = M","type":"text"},{"text":"C","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}}]},{"text":" ","type":"text"},{"text":"< M","type":"text","marks":[{"type":"textColor","attrs":{"color":"#ff5630"}}]},{"text":"B","type":"text","marks":[{"type":"subsup","attrs":{"type":"sub"}},{"type":"textColor","attrs":{"color":"#ff5630"}}]}]}]}]}]},{"type":"paragraph","content":[{"text":"Notice how section C still had a larger moment of inertia than A (not as large as that of B, however), but was able to achieve the same amount of mass. ","type":"text"}]},{"type":"paragraph","content":[{"text":"Key Takeaway:","type":"text","marks":[{"type":"underline"},{"type":"em"}]}]},{"type":"paragraph","content":[{"text":"This example is intended to teach that ","type":"text"},{"text":"you don’t need to increase thickness to increase stiffness/moment of inertia","type":"text","marks":[{"type":"strong"}]},{"text":". Solely increasing thickness will increase your part mass. Increasing part thickness is the easiest way to increase moment of inertia. However, when given the opportunity to adjust your form/cross sectional area (i.e, no spatial constraint) look at leveraging that mode of design.","type":"text"},{"text":" ","type":"text","marks":[{"type":"strong"}]}]},{"type":"paragraph"},{"type":"paragraph","content":[{"text":"Tubing Common Cross Sections Used:","type":"text","marks":[{"type":"strong"}]}]},{"type":"paragraph","content":[{"text":"The 2 most commonly used tubing cross sections are the circular and the square cross sections. A good comparison chart is as follows.","type":"text"}]},{"type":"mediaSingle","attrs":{"layout":"center","width":760,"widthType":"pixel"},"content":[{"type":"media","attrs":{"width":1920,"alt":"image-20240830-235700.png","id":"076e714c-4cc8-4603-ab6b-25e7f9542d2b","collection":"contentId-3686432769","type":"file","height":1080}},{"type":"caption","content":[{"text":"Figure 2: Tubing Cross Section Comparison ","type":"text","marks":[{"type":"em"}]},{"text":"Link","type":"text","marks":[{"type":"em"},{"type":"link","attrs":{"href":"https://discuss.ardupilot.org/t/building-a-better-quad-frame/27703/61"}}]}]}]},{"type":"paragraph","content":[{"text":"Note that square tubing is much easier to use in terms of manufacturing in areas where a flat mounting section is needed. However, I (Mido) suggest that circular tubing is used almost everywhere else in the vehicle. Using circular tubing allows us to use commonly found tubing while leveraging the learnings earlier on this page about moment of inertia to increase stiffness while minimizing mass.","type":"text"}]}],"version":1}

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