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How To Optimize the Cantilever Beam Cross Section!
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👶 Meme Of The Week
🙋♂️ Interview Practice Question of the Week
Company: Meta and Tesla
Which massless, fixed end beam will minimize deflection for the given point load? (solve for F1 and F2)
✅ The Answer
This is one of our favorite questions as there’s a spicy twist at the end. Furthermore, we’ve received this question at both Meta and Tesla, so it’s definitely worth understanding.
Let’s start with Force 1 (F1):
Classically, the tensile stress is equal to force / area. The stress is related to strain in the elastic region by a factor of E (Young’s Modulus). Strain is equal to change in length divided by original length (dL / Lo). Written out, we have:
stress = force / area = E * strain = E * dL/Lo
As E and Lo are fixed for a given beam and material, an increase in force, divided by the cross sectional area, will result in an increase in length. Because of this relationship, we choose option C as the best beam for force F1 given it maximizes cross sectional area, minimizing the change in length dL.
Force 2 (F2).
When you think of resisting bending, there should be one shape you immediately think of: the I-beam! This is a trap. They are testing your ability to think critically, and understand what makes an I-beam so effective at resisting bending. It is not always the right answer when it comes to bending.
FYI: A fun bit of Hardware is Hard lore is that the H’s in our logo were designed to look like sideways I-beams!
Generally speaking, as we know, bending stress = (M*y)/I. When moment of inertia increases, bending stress decreases. Given that we just established a direct relationship between stress and deflection, it can be said that a higher stress in a given beam results in greater deflection.
You could calculate out the inertia for each shape, but honestly this is unnecessary. All it does is prove your ability to do use the equation for inertia of a rectangle ((1/12)B*H^3) and the parallel axis theorem (Ix’ = Ix + Ad^2), neither of which is that impressive (though you should be able to do it if asked). Instead, as when confronted with most interview questions (or any question for that matter), I recommend pondering for a moment. What is inertia? What gives an object high cross sectional inertia?
It’s not just about area, though that is a primary consideration. It’s about distance from the neutral axis (the plane of the beam which experiences no change in length (and no stresses). Imagine the forces/stresses inside the beam. As it bends, the top of the beam becomes longer than the bottom of the beam. The top is therefore in tension and the bottom is in compression. The middle of the beam correspondingly, has no stress and no deformation. If I can maximize the area of the beam where the forces are greatest (far away from the neutral axis), I can most effectively decrease the stress inside the beam and its resultant deflection.
That is why the I-beam is typically considered the best shape for resisting bending. The top and bottom of the “I” are far away from the center, and are very effective at opposing bending in “up” or “down” directions. However, an I-beam is MUCH less effective at opposing lateral forces because it concentrates the mass/area where the stress is lowest. It is not an efficient use of mass in such a case.
Back to our original problem, optimizing for F2! Though the I-beam (solution B) is the most efficient solution to oppose the proposed force, note that its moment of inertia is not actually higher than solution C. If the question had mentioned efficiency or optimization across mass/cost, then the I-beam takes the cake. But since the problem asked solely to minimize deflection, solution C is again, the correct answer. Note that the interviewer will usually ask you to explain why, and it is not sufficient merely to provide the answer. What a great opportunity to practice your first principle reasoning!
Note that instead of forming the relationship between stress and deflection, we could also have used the formula for deflection of a massless cantilever beam with a point load at the end:
Deflection = (FL^3) / 3EI
Force, length, and material properties are assumed to be fixed, turning this back into another inertial manipulation problem. Since this equation also optimizes for inertia, the ultimate answer is the same.
Thus far, this problem has really been about careful listening and avoiding red herrings. As promised, the exciting part comes from the add-ons!
What happens to the bending stress for load F2 when beam height is doubled?
Design the optimal cross section for a beam simultaneously experiencing F1 and F2?
Note that despite answering correctly the initial optimization, my initial answer was wrong in both technical interviews that I received these add-ons. Though I ultimately came to the right answer, these two are trickier than they first appear.