Abstract: Several improvements are made for existing asymptotic expansions for the axisymmetric toroidal shells. The new expansions are numerically satisfactory and satisfy the accuracy of the theory of thin shells. All of them are expressed in terms of generalized Airy functions, instead of Bessel or Airy function for the homogeneous and Lommel function for the particular solutions, respectively, as in the existing work. In this paper, three particular solutions are given, one of which is just the solution obtaine d by Tumarkin (1959) and Clark(1963).

Key words: toroidal shell, asymptotic, generalize d Airy function