# How can one calculate the maximum instantaneous bank angle a plane can do in a certain time? [duplicate]

I know that there is a limitation to do an instantaneous banking angle a plane can do. These limitations are due to the airplane (infrastructure - aerodynamics) and limitation on the pilot (due to his ability to withstand a certain maneuvering calculated in g), so how can one calculate an accurate or approximate value for the maximum instantaneous bank angle taking into account the (military and civilian) planes infrastructure and also taking into account the pilot (with or without) a g-suit.

Also i want to know how long can a pilot withstand that instantaneous bank angle?

• I assume you're asking about the maximum bank angle in a level turn; is that right? Outside of a level turn, many planes can easily bank through 360 degrees. – Terran Swett Jul 28 '18 at 20:03

This question has been asked and answered in over 20 various forms, a few are....

How much g-force is experienced in a 45° turn?

What is the relation between airspeed and rate of turn?

Best procedure to Turn around in Canyon, Turn radius as a function of velocity

How to calculate angular velocity and radius of a turn?

• you did not answer the last question "Also i want to know how long can a pilot withstand that instantaneous bank angle?" could you answer it ? – AAEM Sep 10 '18 at 11:18

If by instantaneous you mean what is commonly called instationary, the rules of the turn allow to trade altitude for turn rate such that the maximum load factor can be achieved even if the drag involved in doing so far exceeds the thrust available from the engines.

Contrast this with a stationary turn were no more drag may be created than can be handled by the engines such that altitude is maintained.

Then the answer is easy: The maximum bank angle $\varphi$ for a coordinated, instationary turn can be obtained from the maximum g load factor $n_z$ the plane can sustain. For small vertical speeds, use the formula for level flight: $$\varphi = \arctan\left(\sqrt{n_z^2-1}\right)$$ If the vertical component of the flight path angle $\gamma$ should not be neglected, the load factor can become slightly higher since gravity has a forward component which does not add to the loads in the z-direction: $$\varphi = \arctan \left (\sqrt{n_z^2-\cos\gamma}\right)$$

The answers to this question discuss the issue in more detail.

As to the time the pilot can sustain the bank angle: This is really about the load factor, and for that we have Eiband diagrams. Essentially, much depends on pilot position and the techniques used to avoid blackout. A few trained pilots when sitting can suffer through 9g for a few seconds, but a more normal limit would be 12g (equivalent to $\varphi = 85^\circ$) for 0.04 seconds, decreasing to 5 g (equivalent to $\varphi = 78^\circ$) when durations in excess of 0.2 seconds are involved.