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Variables, functions and mappings
Published in Alan Jeffrey, Mathematics, 2004
In fact the lemniscate considered here with b = a is a special case of a family of curves called the Cassini ovals. When a <b the curves form single loops enclosing the two foci, and the loop does not pass through the origin, though the oval shaped curves may be slightly pinched in along the y-axis as in Fig. 2.12(c), but when a > b the curves separate into two distinct loops, each enclosing one of the foci as in Fig. 2.12(d). ■
Effect of thickness on the buckling strength of egg-shaped pressure hulls
Published in Ships and Offshore Structures, 2018
Jian Zhang, Minglu Wang, Weicheng Cui, Fang Wang, Zhengdao Hua, Wenxian Tang
To resolve these disadvantages, researchers have advocated non-spherical pressure hulls, shells of revolution with positive Gaussian curvature, to replace the spherical one (Wong 2012; Blachut 2014). For instance, Jasion and Magnucki presented a numerical and analytical study into the elastic buckling of barrelled shells resembling Cassini oval (Jasion and Magnucki 2015a), clothoidal-spherical (Jasion and Magnucki 2015b) and circular arc (Jasion and Magnucki 2007; Magnucki and Jasion 2013) shapes. Moreover, Blachut proposed a family of shells of revolution in the shape of circular arc (Blachut 2002) and generalised ellipse (Blachut 2003). The elastic–plastic buckling of these shells was numerically and experimentally explored. However, these studies fall into a tentative stage, in which these non-spherical shells are not in match with the application maturity as spherical ones, and their applications into deep pressure hulls are not evaluated. Also, little attention has been paid on the effect of wall thickness on the buckling.
Buckling of an egg-shaped shell with varying wall thickness under uniform external pressure
Published in Ships and Offshore Structures, 2019
Jian Zhang, Zhengdao Hua, Fang Wang, Wenxian Tang
Many strengthening methods have been used to improve the buckling resistance of such shells, including changing shapes (Blachut and Wang 2001; Blachut 2002; Blachut 2003; Pan et al. 2012; Jasion and Magnucki 2015a, 2015b; Zhou et al. 2017; Zhang, Wang, Cui, et al. 2017; Zhang, Wang, Wang and Tang 2017; Zhang, Wang, Wang, Tang, et al. 2017; Zhang, Zhu, et al. 2017; Zhang et al. 2018), adding stiffeners (Ghanbari et al. 2014; Foryś 2015), constructing sandwich walls (Malinowski et al. 2015), and incorporating corrugated walls into the design (Ghanbari, Dizaji et al. 2015; Ghanbari, Jiao, et al. 2015). Among these methods, changing shapes into shells with a positive Gaussian curvature is effective. Spherical shells are typical examples of such shells, and they are widely used and studied owing to the constant stress and strain distributions in the material. For example, the buckling of spherical shells under uniform external pressure is experimentally and numerically studied to ensure their application in deep manned submersible (Pan et al. 2012) and neutrino detector. (Zhou et al. 2017). However, it is very difficult to fabricate the spherical configuration owing to its extremely high imperfection-sensitive properties, and to house the equipment because of its high radius curvature (Zhang, Zhang, Tang et al. 2017). Therefore, the buckling of atypical shells of revolution with a positive Gaussian curvature has been widely investigated. For example, Jasion and Magnucki analytically and numerically studied the elastic buckling of Cassini oval (Jasion and Magnucki 2015a) and clothoidal-spherical (Jasion and Magnucki 2015a) shells. Moreover, Blachut et al. numerically and experimentally studied the elastic-plastic buckling of circular arc (Blachut and Wang 2001; Blachut 2002) and generally ellipsoidal (Blachut 2003) shells. Recently, Zhang et al. proposed an egg-shaped configuration from a bionic viewpoint. They have analytically, numerically, and experimentally investigated the buckling properties of egg-shaped pressure hulls and the effect of the shape index and wall thickness size on buckling (Zhang, Wang, Cui, et al. 2017; Zhang, Wang, Wang and Tang 2017; Zhang, Wang, Wang, Tang et al. 2017; Zhang, Zhu, et al. 2017). Although various atypical shells of revolution with a positive Gaussian curvature have been evaluated and demonstrated to show high buckling resistance, most have uniform wall thickness. Few studies have focused on atypical shells with varying wall thickness.