The aerodynamic theory of the lifting line was first solved by Multhopp1. This solution uses a spanwise distribution of the control points accordingly a sinus-function and is applicable to unswept wings. For the bell-shaped lift distribution a direct solution of the lifting line formula can be derived and is described here.

The integral for total lift can be written as (Truckenbrodt2):



(Tru. 5.42)

 For a symetrical lift distribution this formula can be transformed:


 with ;


(Equ. 00-01)

If the spanwise control points are aranged along a "central-angle" similar to the Multhopp procedure, the following terms can be written:


local lift load and spanwise station


(Mul. 07)

This leads to the final integration and its solution:




(Equ. 00-02)



(Equ. 00-03)

The total required geometrical twist is, accordingly to Truckenbrodt, the sum of effective and induced angle of attack . Angles are calculated in [rad.]:




(Tru 7.64)

The local effective angle of attack can be found with the term:



(Tru. 7.11 & 7.28)

 This term is now used in Equation (Equ. 00-01 b):



(Equ. 00-04)

 The solution for the induced angle of attack is more complex. Truckenbrodt gives the induced angle of attack for unswept wings:


 with the circulation


(Tru. 7.67 & 7.70)

 Using the following terms, we can transform this integrartion in the formulation as given in Multhopp:



(Equ. 00-05)



(Mul. 13)

  This formula now has to be solved for the bell shaped lift distribution:



(Equ. 00-06)



(Equ. 00-07)

For this the term is split up in several parts, which can be integrated seperately. The calculation is described in the Appendix A1 . The final solution is:




(Equ. 00-08)


 This gives the required calculation method for the geometrical twist , which can be transformed, using equation (Equ. 00-03):



(Equ. 00-09a)



(Equ. 00-09b)