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Below is a GIF I prepared and used in an earlier question, and the answer seems quite reasonable. With a height to diameter ratio of about 70 m to 3.7 m (nearly 20:1) a weight-conscious design, focused primarily on withstanding axial stresses (e.g. thrust + drag), and allow for some flexing.

But if it bends this much in a breeze, what about in flight?

This answer references the detailed, knowledgable Flightclub simulation of a recent Falcon 9 launch. At one point when the speed of the rocket (wrt Earth's rotating frame presumably) is about 1000 m/s and the pressure must be roughly 0.06 or 0.07 bar at 22km altitude, the angle of attack is estimated to be 4.6 degrees. That would present a crushing-type force, but would it tend to bend the rocket since the fairing is so much wider than the body?

Just how much can tall skinny rockets bend? (roughly, safely)

enter image description here

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uhoh
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    Are you asking specifically about how much the rockets can safely bend without breaking (if, say, a giant flying gorilla were to grab and bend them in mid-flight), or about how far they would be expected to bend under typical flight stresses (i.e. assuming no giant flying gorillas)? – Ilmari Karonen Apr 05 '17 at 15:46
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    @IlmariKaronen I want to remain a little flexible (pun intended) so as not to prevent an informative answer from being posted. This particular question is asking for fairly specialized knowledge, and if I over-specify, someone may just choose not to post. I have a strong feeling that since rockets are such highly optimized designs that strive to minimize structural weight, the expected limit and the safe limit are not so far apart from each other. – uhoh Apr 05 '17 at 16:02
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    Hmmm - is it definitely bending, or is that movement inline with tolerances of the clamp mechanism? – Rory Alsop Apr 05 '17 at 18:00
  • I can not answer how much bend is possible. But it depends on the stress how many bend cycles are possible without failure. See https://en.wikipedia.org/wiki/Fatigue_(material)#The_S-N_curve
    Brittle aluminium will fail under a stress of 320 MPa in one cycle only, but under a stress of 150 MPa, 10000 cycles are possible. If a first stage is reused, it depends on construction if this stage may be reused 10 times or 100 times due to failure by bend cycles. Unfortunately the structural mass would increase when the stage is constructed for more bend cycles.     – Uwe Apr 05 '17 at 19:36
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    Related: shuttle stack twang: https://www.youtube.com/watch?v=ExfjSuJxOP8 – Organic Marble Apr 05 '17 at 20:33
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    Related: bendy Saturn V: https://m.youtube.com/watch?v=PnsFFhKqXXo – Russell Borogove Apr 11 '17 at 15:11
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    I've heard that a spacex vp has described F9 as a "wibbly wobbly noodle" (sic) in flight (due to the high aspect ratio) and that that's one of the biggest dynamics problems that spacex faces. – randomUsername Apr 18 '17 at 15:17
  • @uhoh unfortunately it was word of mouth about an in person remark, it happened to stick with me because I thought it was funny but unfortunately I don't have anything better than that. – randomUsername Apr 18 '17 at 15:27
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    @randomUsername ya I know how that goes, OK thanks! I'm never going to forget that description - a flying noodle! Hopefully at least al dente – uhoh Apr 18 '17 at 15:33
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    Related: Crushing a soda can with a hydraulic press: https://www.youtube.com/watch?v=YP_UBNwEoGs – David Hammen Mar 19 '21 at 08:48

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Nothing can be perfectly inflexible without causing undue stress on the hold downs. Similar to using a longer crowbar to gain leverage making any wind load transfer directly to the hold-down clamps, etc. The motion displayed in the image is likely a light breeze as the clamping mechanism lets go and wouldn't affect the launch vehicle itself. The Falcon 9 fairing is listed as 5.2m in diameter, the motion I estimate about a .1m in 10kt wind with a 30kt wind limit would put it at about 25% of the Acceptable loading giving an estimated flex of about .4m for the Stack. The maximum flexion would vary due to the construction materials and design.

Daryl Morning
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    An answer to this question should specifically address "How much..." during flight, not on the launch pad. "But if it bends this much in a breeze, what about in flight?" Can you address structural loading during flight? – uhoh Jan 02 '18 at 23:42
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    There's no way of accurately addressing in-flight dynamic loads unless you have the engineering designs and can get the computer models for the materials and designs. I certainly don't work in the aerospace industry and have no way of obtaining those. The best answer is to have the wind blowing hard enough they want to take the LV back to the barn and that is likely within 50-70% of the Maximum allowable flexion. Unless one of the design engineers can get on here, realistically there cannot be an answer. – Daryl Morning Jan 02 '18 at 23:54
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    @DarylMorning If you feel you can not answer the question as asked, then why do you still post one? You can ask clarification from the OP in comments, but answers should actually, you know, answer the question, and after reading yours I still don#t know more then before. – Polygnome Jan 03 '18 at 00:22
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    I was trying to give the OP an inferred answer based on the generic question of "Just how much can tall skinny rockets bend?" towards the Falcon 9 rocket and now that I looked at my answer I realized I deleted a sentence from my edit which would have made my answer a little more relevant. I'm going to re-edit my answer. As it stands my answer really says next to nothing of what I wanted to express. – Daryl Morning Jan 03 '18 at 00:32
  • Answer updated, Mea Culpa. – Daryl Morning Jan 03 '18 at 00:49
  • You've given numbers that apply to what can already be seen in the image, in which the rocket is still on the ground. I'll reiterate my first comment; " An answer to this question should specifically address "How much..." during flight, not on the launch pad. "But if it bends this much in a breeze, what about in flight?" Can you address structural loading during flight?" – uhoh Jan 03 '18 at 01:13
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    All bending counts towards failure, wind loads would get an approximation of the MAXIMUM bending possible. Just because the LV isn't going forward doesn't matter to the actual bending itself. The ratio of acceptable % of possible bending before failure is usually to a max of 150% the given thresholds. As I said, without the engineering models the BEST we can do is an inference or estimate. It is the BEST answer possible without hard numbers that SpaceX probably has secured with NDAs. My answer also gives the methodology I used and can be used for other "tall skinny rockets" as well. – Daryl Morning Jan 03 '18 at 01:31
  • @DarylMorning I've come back to this question again and I think this is the best possible answer without actually publishing detailed engineering data on real rocket designs, which is what you pointed out in the beginning. Thanks for the answer! – uhoh May 17 '18 at 06:23