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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 propositions 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 different geometrical models of thin-walled shells because their fabrication is the simplest 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 curvature K = 0 everywhere. The simplest classification of ruled surfaces is presented in Fig. 1.

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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 surfaces 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 determination 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 example, 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 parabolas 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).

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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 regression 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 developable 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 designing 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 problem. 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 tangential developable surfaces.

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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 architectural shapes and it was demonstrated in the paper. In a work of Obradović, Beljin, and Popkonstantinović (2014) with 22 references, the recommendations for the construction 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 surfaces 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 investigations, they offered new approach to geometric modeling of tangential developable 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 developable 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 compound ribbon torses. Approximation of complex surfaces by the torses and substitution 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). Both in a paper of Bruno Postle (2012) and in a paper of Kilian et al (Kilian et al., 2008), graphical methods, based on the well-known properties of rectilinear generatrixes of torses, were used.

Stress-strain state of torse shells Researches on the determination of the stress-strain state of shells with tangential developable middle surfaces are conducted in RUDN, in general, for developable open helicoids. It was shown by S.N. Krivoshapko (2017) that the equilibrium equations of A.L. Goldenveiser (1961) for an arbitrary system of curvilinear coordinates, represented in a monograph (Krivoshapko, 2009), can be transformed into the equilibrium equations, derived by S.N. Krivoshapko. The equilibrium equations of A.L. Goldenveiser (1961) contain the force factors shown in Fig. 3 with asterisks, but the equilibrium equations of S.N. Krivoshapko contain the force factors shown in Fig. 3 without asterisks. The formulae, connected

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Copyright reserved © J.Mech.Cont.& Math. Sci., Special Issue-1, March (2019) pp 324-333 the pseudo-forces of A.L. Goldenveiser with the formulae, used by engineers, have the following form (Krivoshapko, 2017):

Fig. 3. Pseudo-forces and forces used by engineers

  Nv  Nv sin , Nu  Nu sin ,     Sv  Sv  Nv cos, Su  Su  Nu cos,   Mvu  Mvu sin , M uv  M uv sin ,     M v  M v  M vu cos, M u  M u  M uv cos. Thus

Su  Sv and M uv  M vu ,

*  because    / 2, but the physical equations give Su  Sv even if    / 2. So, the both systems of governing equations are equivalent and they can be used for the determination of stress-strain state of torse shells, the middle surface of which is given in curvilinear non- orthogonal coordinates. Series of works of М.I. Rynkovskaya (Rynkovskaya, 2012; Rynkovskaya, 2015) are devoted to the subsequent improvement of analytical methods of analysis of a long developable helicoid subjected to dead loads. Moreover, in a work of M. Ryn- kovskaya (2012), advisability of application of Bernoulli’s numbers for the determination of dis-placements is shown if formulae contain the integrals not defining ana- lytically.

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An equal slope surface The A cone The B cone

а) b) c) Fig. 4. Developable surfaces

In the monograph (Krivoshapko, 2009), static analysis of the elliptical shell of equal slope (Fig. 4,а) and two right conical shells with the same director ellipse (Fig. 4,b,c) is given. All of these shells are subjected to the surface load in the direction of the z axis. Later, the same shells were calculated for general stability (Krivoshapko and Timoshin, 2012). Critical compressive linear load distributed along the upper edge of the shells in the direction of the z axis was determined and the first three forms of buckling were defined. A thin shell in the form of Monge’s ruled surface attracted the attention of J. Fili- pova (Filipova, 2016) (Fig. 5). This shell is an equal slope shell and that is why it is a tangential developable surface. She determined tangential forces in this shell using the equations of momentless theory of shells given in lines of principal curvatures and compared these results with the results derived with the help of FEM. The diver- gences in values of normal forces Nβ directing along rectilinear generatrixes are aver- age 2 per cent. Normal Nα and tangent forces, that are equal to zero in momentless theory, tend to zero in FEM i.e. they are much less of Nβ. Apparently V.I. Olevs’kyy (Olevs’kyy, 2011) and D.M. Annenko with M.S. Bab- kin (Annenko and Babkin, 2018) were the first who proposed to use nonlinear analysis for tangential developable shells be cause the practical application of them widens. As an example, V.I. Olevs’kyy ex- amined an elliptical shell of equal slope (Fig. 4, а). For study of its deformation and interaction between the form of shells and a complex of technological imperfection, he proposed the modified method of parameter continuation and used the generalized summation of solutions by two-dimensional approximation Pade. In a paper of An- nenko and Babkin (2018), a geometrically nonlinear theory of analysis was applied for the shell in the form of Monge’s ruled surface (Fig. 5).

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Fig. 5. Monge’s ruled surface with the circle cylindrical director surface

Experimental study A single work (Savićević et al., 2017) dealing with experimental study of helical shells was found. The method presented in it can be applied to developable open helicoidal shell. A manuscript of Solomon J., Vouga E., Wardetzky M., & Grinspun E. (2012) also can be designated as experimental work because the authors model diverse developable surfaces from a sheet material. They have proposed a ﬂexible discrete structure for representing developable surfaces. They suppose that this exploratory work sug- gests the potential for an assortment to follow up research. The work contains 68 references.

II. Conclusion Examining in conference, the presented paper and the monograph of S.N. Krivo- shapko (2009), one can have the whole spectrum of investigations of torse surfaces and shells presented in the publications till present time. It will help researchers concerned to plan further investigations and to economize their time not repeating a conclusion of theorems, equations, and propositions the well-known already. Additional information on the application of tangential developable shells can be taken in Krivoshapko (2018), Lawrence (2011), in Perez and Suarez (2007), in Soley Ersoy and Kemal Eren (2016), and in Yi-chao, Eliot Fried (2016). A number of de- signers have focused on free-form surface shell to realize a free architectural form that is different from the analytical curved surface. Developable surface is a special form of ruled surface generated by continuous movement of the straight line. It gives an opportunity to create continuum shells of interesting shape (Cui J., Ohsaki M., Na- kamura K., July 2017). In this study, several developable surfaces are combined to form a curved roof structure.

III. Acknowledgement The publication was prepared with the support of the “RUDN University Program 5-100”.

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Copyright reserved © J.Mech.Cont.& Math. Sci., Special Issue-1, March (2019) pp 324-333 XXVIII. Savićević S., Ivandić Ž., Jovanović J., Grubiša L., Stoić A., Vukčević M., Janjić M. (2017), The model for helical shells testing. Tehnički Vjesnik 24, 1: 167-175. XXIX. Soley Ersoy, Kemal Eren (2016). Timelike tangent developable surfaces and Bonnet suefaces. Abstract and Applied Analysis, Volume 2016, Article ID 6837543, 7 pages http://dx.doi.org/10.1155/2016/6837543 XXX. Solomon J., Vouga E., Wardetzky M., & Grinspun E. (2012). Flexible Deve- lopable Surfaces. Eurographics Symposium on Geometry Processing 2012. Eitan Grinspun and Niloy Mitra (Guest Editors), 31 (5): 1-10. XXXI. Tang Ch., Pengbo Bo, Wallner J., Pottmann H. (January 2016). Interactive design of developable surfaces. ACM Transactions on Graphics, 35(2), Ar- ticle 12:12 p. XXXII. Yi-chao, Eliot Fried (2016). Möbius bands, unstretchable material sheets and developable surfaces. Proc. of the Royal Society. A. Math., Phys. and Engi- neering Sciences. 472 (2192). XXXIII. Zhao Hongyan, Wang Gupjin (2008). A new method for designing a developable surface utilizing the surface pencil through a given curve. Progress in Natural Science, 18: 105-110.

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