Copyright reserved © J.Mech.Cont.& Math. Sci., Special Issue-1, March (2019) pp 324-333 Tangential Developable Surfaces and Shells: New Results of Investigations *1Sergey N. Krivoshapko, 2Iraida A. Mamieva, 3Andrey D. Razin 1 Peoples' Friendship University of Russia (RUDN University), Moscow, Russia * Corresponding Author: [email protected] https://doi.org/10.26782/jmcms.2019.03.00031 Abstract After publication of a monograph Geometry of Ruled Surfaces with Cuspidal Edge and Linear Theory of Analysis of Tangential Developable Shells (Krivoshapko, 2009) with 386 references, new papers, devoted to geometry, application and strength analyses of thin shells with the middle developable surface were published. Some results of investigations have newness and definite scientific and practical in- terest but some works improve methods presented before or propose new variants of application of tangent developable surfaces. In a paper, new results derived past the last 10 years and connected with needs of engineer practice and architecture of man- ufactured articles, structures, and erections, are analyzed. The analyses of the whole spectrum of investigations of torse surfaces and shells presented in the publications till present time will help researchers concerned to plan further investigations and to economize their time not repeating a conclusion of theorems, equations, and proposi- tions the well-known already. Keywords : Tangential Developable, Design of A Torse, Geodesic Curve, Para Bending, Tangential Developable Shell Geometric Modeling, Stress-Strain State of the Shell I Introduction Shells in the form of developable surfaces are the cheapest structures among dif- ferent geometrical models of thin-walled shells because their fabrication is the sim- plest process due to their ability to develop on a plane without any lap fold or break (Krivoshapko, 2009). A developable surface is a ruled surface having Gaussian cur- vature K = 0 everywhere. The simplest classification of ruled surfaces is presented in Fig. 1. 324 Copyright reserved © J.Mech.Cont.& Math. Sci., Special Issue-1, March (2019) pp 324-333 Developable surfaces with cuspidal edge (Krivoshapko, 2009) Ruled surfaces (Krivoshapko, 2006) Conical surfaces (Mamieva and Razin, 2017) Surfaces of zero Gaussian curvature (K = 0) (Krivoshapko, 2009) Cylindrical surfaces (Krivoshapko and Ivanov, 2015) Surfaces of positive Gaussian curvature There are no ruled surfaces with K > 0 (K > 0) (Krivoshapko and Ivanov, 2015) Surfaces of negative Gaussian curvature (K < 0) (Krivoshapko and Ivanov, 2015) Fig. 1. The simplest classification of ruled surfaces General results in all areas of investigations of non-degenerated developable sur- faces and thin torse shells were stated in a monograph of Krivoshapko (2009) with 386 references. This monograph contains the references published before 2008. But researches, devoted to studying of geometry, application (Krivoshapko, 2018), and determination of stress-strain state of thin torse shells, did not cease after this date. Consider the most interesting scientific works published after 2006 and not being reckoned among the references of the monograph (Krivoshapko, 2009). In the first place, it is necessary to note a review paper of Sn. Lawrence (Law- rence, 2011) «Developable surfaces: their history and application», where initial pe- riod of studying of developable surfaces with the cuspidal edges is described and the works of L. Euler, G. Monge, and other scientists, published in XVIII – at the begin- ning of XX centuries, are quoted. A valuable series of works was published by researchers from RUDN on determi- nation of bending moments appeared in the process of parabolic bending of thin plate into a given torse shell. Conclusions of S.N. Krivoshapko, based on Love’s formulae (Krivoshapko, 2009), were assumed as a basis of the subsequent investigations which were continued by Мarina Rynkovskaya (Rynkovskaya, 2017). Geometry and design of tangential developable surfaces A developable surface is the envelope of a one-parameter family of planes (for ex- ample, a rectifying surface) and therefore is locally obtained by isometric deforming a piece of a plane. In engineering practice, at first, drawings of location and spacing of blanks of a plane sheet material are prepared and then slabs are cut out to scale of real structure and at last they are bent into project position without folds and breaks. S.N. Krivo- shapko (Krivoshapko, 2009) offered a method of design of surface developments with using of consistent calculation of lengths of the fragment of the contour lines and angles of the contour lines with rectilinear generatrixes. As an example, he con- structed a development of the torse with the contour lines in the form of two parabo- las of the second and forth orders. Later, the same development with application of the same method was designed with the help of computer (Fig. 2). 325 Copyright reserved © J.Mech.Cont.& Math. Sci., Special Issue-1, March (2019) pp 324-333 Fig. 2. A development of the torse with the contour lines in the form of two parabolas of the second and forth orders Kinematics of bending of spatial curve placed on a developable surface in the process of developing of the surface on a plane was described in a paper of S.A. Be- restova et al (Berestova, Belyaeva, Misyura, Mityushov, and Roscheva, 2017). The authors of a paper (Berestova et al, 2017) hope to use a new algorithm offered by them for the solution of diverse problems of design of spatial structures in building and industry. Analogous method of construction of developments was illustrated on concrete examples in a work of Z.V. Belyaeva et al (Belyaeva, Berestova, Mityushov, 2017). Tangential developable surfaces with a circle and a parabola, with an ellipse and a parabola as the director curves were considered in this manuscript (Belyaeva et al, 2017). This method can be used if two director curves and the edge of regression of a given tangential developable are known. The development of the edge of regres- sion on a plane is built firstly and after that, two director curves are built. Investiga- tions about common bending of the fragments of intersected non-degenerated deve- lopable surfaces were not carried out from 1972 till present time. Properties of the focal curve Сγ of a spatial curve γ are studied in a work of P. Alegre et al (Alegre, Arslan, Carriazo, Murathan and Öztürk, 2010). The focal curve is defined as the centers of the osculating spheres of the γ curve. In this work several new theorems are proved for torses connected with the curves γ and Сγ. A paper of Zhao H. and Wang G. (Zhao H., Wang G., 2008) proposes a new method for design- ing a developable surface by constructing a surface pencil passing through a given curve. H. Zhao and G. Wang give necessary and sufficient conditions for this prob- lem. Sometimes a special condition is put for a given curve to be by a geodesic curve. Such problem is appeared due to an inquiry of boot and shoe industry. The same problems are studied by R.A. Al-Ghefari and R.A. Abdel-Baky (2013). It would be interesting to use materials of Hongyan Zhao and Guojin Wang (2008) for answer to the question: are the contour curves, presented in Fig. 2, geodesic curves? Körpinar T. and Turhan E. (2012) proved several new theorems connected with tangential developable surfaces and derived parametrical equations of several tangen- tial developable surfaces. 326 Copyright reserved © J.Mech.Cont.& Math. Sci., Special Issue-1, March (2019) pp 324-333 Approximation of complex surfaces by the torses Some geometricians consider that approximation of complex surfaces by torses with following approximation of the derived torses by folds is a major problem. The new concept of the design of conical meshes with planar quadrilateral faces is offered in a paper of Liu, Pottmann, Wallner, Yang, and Wang (2006). This method gives a possibility to design the perspective interiors and exteriors for freeform archi- tectural shapes and it was demonstrated in the paper. In a work of Obradović, Beljin, and Popkonstantinović (2014) with 22 references, the recommendations for the con- struction of compound model surface of transition with the plane curve in the form of a circle or an ellipse and a quadrilateral contour, placed in parallel or in intersected planes on the opposite sides, are given. This process is fulfilled with the help of four conical surfaces and four plane triangles. The authors (Obradović et al., 2014) use as much as possible the methods of computer graphics Substitution of complex surfaces by the fragments of tangential developable sur- faces in the form of the ribbons or strips is considered in a paper of Tang Ch., Pengbo Bo, Wallner J., and Pottmann (2016). An apparatus of geometric modeling is used. They (Tang et al., 2016) refer to a great number of investigation (44 references) that were performed by them or other geometricians earlier, but on the basis of these in- vestigations, they offered new approach to geometric modeling of tangential develop- able surfaces. This approach was illustrated by several examples. Approximation of complex surfaces by the fragments of tangential developable surfaces in the form of the ribbons was used also by Gonzalez-Quintial F., Barrallo Javier, Artiz-Elkarte A. (2015). Bruno Postle (2012) presented many examples of substitution of complex surfaces of positive and negative Gauss curvature by the fragments of tangential de- velopable sur-faces (cones, cylinders, and torses) in the form of the ribbons or plane triangular and quadrangular elements. He also built developments of the approx- imated surfaces and gave the examples of the real sculptures containing of the com- pound ribbon torses. Approximation of complex surfaces by the torses and substitu- tion of torses by the folds was applied in a work of Kilian et al (Kilian M., Flöry S., Chen Z., Mitra N.J., Sheffer A., Pottmann H., 2008).