Toroidal shells under nonsymmetric loading

Geometry-Induced Rigidity in Elastic Torus from Circular to Oblique Elliptic Cross-Section

“…The torus has been studied for more than 110 years (Chang (Zhang) 1949 [4], Qian and Liang 1979 [5], Xia and Zhang 1986 [6], Zhang, Ren and Sun 1990 [7], Zhang and Zhang 1991 [8], 1994 [9], Audoly and Pomeau 2002 [10], 2010 [11] [35]). The most of publications have been focused on circular torus, and only small percentage on non-circular cross-section torus [25][26][27][28], in which, Zingoni, Enoma and Govender 2015 [26] considered a semi-elliptic toroid and tried to get an approximate bending solution that may be valid in regions adjacent to the horizontal equatorial plane.…”

Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

“…The existence of a turning point in a complete torus is one source of difficulty in finding a solution. Owing to the difficulty of solving differential equations with both hyperbolic and elliptic regions [17][18][19][20][21], various proposed asymptotic solutions have sigularity problem at the turning point of the Gauss curvature, where the Gauss curvature is zero [17,18,20,21]. The torus has been studied for more than 100 years [16,22], and various aspects have been extensively investigated.…”

“…In 1965, Steele investigated toroidal pressure vessels, and was able to make a concise comparison of the volume and weight properties of enough shapes to provide a convenient basis for their design. Qian and Liang [14], Xia and Qian [15], and Zhang and Zhang [17,18] conducted further study, sought an enhanced asymptotic solution valid for the full domain of θ ∈ [0, 2π]. Sun [20,21,23] was the first person to derive displacement-type equations of a torus and proposed a closed-form solution when the radius ratio tends to be null.…”

Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

“…The existence of a turning point in a complete torus is one source of difficulty in finding a solution. Owing to the difficulty of solving differential equations with both hyperbolic and elliptic regions (Chang (Zhang) 1949 [1], Qian and Liang 1979 [2], Xia and Zhang 1986 [3], Zhang, Ren and Sun 1990 [4], Zhang and Zhang 1991 [5], 1994 [6], Audoly and Pomeau 2010 [7], Sun 2010 [8] 2012 [9], Clark 1950 [10,11], Dahl 1953 [12], Novozhilov 1959 [13], Timoshenko and Woinowsky-Krieger 1959 [14], Flügge 1973 [15], Gol'denveizer 1961 [16] Sun 2013 [17]), various proposed asymptotic solutions have singularity problem at the turning point (or called as crowns) of the Gauss curvature (Gol'denveizer 1961 [16]), where the Gauss curvature is zero (Chang (Zhang) 1949 [1], Qian and Liang 1979 [2], Xia and Zhang 1986 [3], Zhang, Ren and Sun 1990 [4], Zhang and Zhang 1991 [5], 1994 [6], Audoly and Pomeau 2010 [7], Sun 2010 [8] 2012 [9], Clark 1950 [10,11], Dahl 1953 [12], Novozhilov 1959 [13], Wissler 1916 [20], Tölke 1938 [27], E. Reissner 1949 [28]).…”

“…The torus has been studied for more than 110 years ((Chang (Zhang) 1949 [1], Qian and Liang 1979 [2], Xia and Zhang 1986 [3], Zhang, Ren and Sun 1990 [4], Zhang and Zhang 1991 [5], 1994 [6], Audoly and Pomeau 2010 [7], Sun 2010 [8] 2012 [9], Clark 1950 [10,11], Dahl 1953 [12], Novozhilov 1959 [13], Föppl 1907 [18] Weihs 1911 [19], Wissler 1916 [20], H. Reissner 1912 [25], Meissner 1915 [26], Tölke 1938 [27], E. Reissner 1949 [28], Tao 1959 [29], Steele 1965 [30], Sun 2018 [31] ), and various aspects have been extensively investigated. In this paper, we will restrict ourselves to the small symmetrical deformation of elasic torus with a circular cross-section, and in particular with an emphasis on its theoretical formulation, associated analytical and numerical solutions.…”