Minimum Induced Drag for Tapered Wings Including Structural Constraints

For a wing in steady level flight, the lift distribution that minimizes induced drag depends on a tradeoff between wingspan and wing-structure weight. In 1933, Prandtl suggested that tapered wings have an advantage over rectangular wings due to this tradeoff. However, Prandtl’s solutions were obtained using assumptions that correspond to rectangular wings. Therefore, his claim was not analytically proven by his 1933 publication. Here, an approach similar to Prandtl’s is taken with more general approximations that apply to wings of arbitrary planform. This more general development is used to study Prandtl’s claim about tapered wings. Closed-form solutions for the optimum wingspan and corresponding induced drag are presented for wings having elliptic and linearly-tapered planforms with constraints of fixed wing loading and maximum stress. It is shown that induced drag is minimized with a triangular planform, which gives a reduction in induced drag of up to 24.44% over the rectangular planform and up to 11.71% over the elliptic planform. Numerical solutions for the lift distributions that minimize induced drag for each planform are also presented. It is shown that the optimum lift distribution produces up to 5.94% less induced drag than the elliptic lift distribution when the triangular planform is used