Nonlinear investigation of Gol’denveizer’s problem of a circular and elliptic elastic torus

The Gol’denveizer problem of a torus can be simply described as follows: the inner and outer equator of the torus shell is loaded by the vertical opposite balanced forces. Gol’denveizer proposed that the membrane theory of shells does not yield a valid solution to this problem. Following this, Audoly and Pomeau used a membrane theory of a toroidal shell and nonlinear boundary-layer to study the linear Gol’denveizer problem and gave a simple solution relating resultant force F and vertical displacement. Sun used the bending theory of shells to solve the linear Gol’denveizer problem, and demonstrated that the solution by Audoly and Pomeau for the linear Gol’denveizer problem is extremely accurate. However, to the best of our knowledge, no one has examined the nonlinear Gol’denveizer problem. In this study, the finite element method is used to observe the nonlinear mechanical property and buckling of the Gol’denveizer problem of a circular and elliptic torus. Investigations reveal that the radius ratios a / R and a / b have great influences on the deformation and force of a circular torus and elliptical torus, respectively, and that the negative Gaussian part of a torus has a stronger bearing than the positive Gaussian part. Additionally, when the vertical force is large enough, shear buckling may occur in the circular and elliptic tori. We propose the buckling failure modes of circular and elliptical torus. As a / b increases, the collapse load of an elliptic torus of the Gol’denveizer problem is enhanced gradually. • Studied the nonlinear Gol’denveizer problem of torus shells by using FEM. • Identified the negative Gaussian part of circular torus has a stronger bearing. • Force and displacement of circular tori are not equal to zero at crown line. • Studied the buckling behaviors of Gol’denveizer’s problem of half-circular tori. • Found a new buckling phenomenon that called a “skirt” shape.