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I never heard of this before seeing What was the actual Q-alpha limit for Saturn/Apollo launches, and what was the typical max Q-alpha? which refers back to this answer to What does “angle of attack 100%” mean in Apollo abort rules?

Q is maximum dynamic pressure, and I believe that using the small angle approximation the products Qα and Qβ are ways to gauge the effects of dynamic pressure perpendicular to a rockets axis when its angles of attack α and β deviate from zero.

What exactly are the mechanical risks when the limits are exceeded? I can think of several possible problems and there certainly may be more:

  1. distortion, crushing or buckling (see GIF in this question) of the rocket's cylindrical form due to non-axisymmetric pressure along the sides of the rocket's body
  2. flexure/bending, (especially for flying noodles) due to the variation of non-axisymmetric aerodynamic pressure
  3. flexure/bending caused by thrust vectoring trying to compensate some aerodynamic torque, thereby exceeding some mechanical limit
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