A contact problem in the theory of leaf spring bending

The weak joint bending (unbonded contact without friction) of the stack of slim non-uniform curved beams (leaves) with rectangular cross-sections is considered. Each leaf has one end clamped and the other free. The leaves have the same widths and different lengths (the lengths decrease upwards). The given loading is applied (upwards) to the lower leaf. This structure is the model of a leaf spring. The basic problem is to find the shapes of the leaves under bending. This problem is reduced to the problem of finding the densities of the forces of interaction between the leaves. The accurate formulation of the latter problem is propounded. The uniqueness of the solution of the problem is proved. The analytical solution is constructed in the special case of two uniform straight leaves.