Pseudospheric shells in the construction

The architects working with the shell use well-established geometry forms, which make up about 5-10 % of the number of known surfaces, in their projects. However, there is such a well-known surface of rotation, which from the 19th century to the present is very popular among mathematicians-geometers, but it is practically unknown to architects and designers, there are no examples of its use in the construction industry. This is a pseudosphere surface. For a pseudospherical surface with a pseudosphere rib radius, the Gaussian curvature at all points equals the constant negative number. The pseudosphere, or the surface of the Beltram, is generated by the rotation of the tracersis, evolvent of the chain line. The article provides an overview of known methods of calculation of pseudospherical shells and explores the strain-stress state of thin shells of revolution with close geometry parameters to identify optimal forms. As noted earlier, no examples of the use of the surface of the pseudosphere in the construction industry have been found in the scientific and technical literature. Only Kenneth Becher presented examples of pseudospheres implemented in nature: a gypsum model of the pseudosphere made by V. Martin Schilling at the end of the 19th century.


Introduction
The most famous modern architects can be divided into three groups. One group seeks to build [1] high -rise facilities while saving expensive land, especially in urban areas. These architects believe that big -flying structures are necessary only if they are functionally needed. In some cases, large -flying structures have even been demolished to make room for high -rise buildings (Moscow, Minsk, Belarus) or for a more profitable building (The King Dome, Seattle, USA). Another group believes that it is more comfortable for a person to be closer to the ground [2]. They join the former President of the International Shell Association (1966), Prof. A.M. Haas: "The people who build the shells are advanced people; They are united by the desire for new forms, new ways of solving problems" [3] and, by applying scientific approaches to analyse and design large -span structures, they have achieved outstanding success [4]. The third group believes that using traditional constructions (walls, columns, flat overlays) and rectangular shapes, it is also possible to get good results. They focus on low -cost model designs, shapes and high -end materials.
The architects working with the shell use well established geometry forms, which make up about 5-10 % of the number of known surfaces, in their projects. These are paraboloids of revolution [5], umbrella -shaped, apple -shaped, cyclic, propeller, cylindrical shell [6], conic shell [7], mid -surface shell in the form of conoid and cylindroid [8], elliptical paraboloid [9], single -striped hyperboloid [10] and some others [11]. In each particular case, architects and design engineers chose the most optimal shape of the shell based on functional necessity, strength, aesthetics, etc.
However, there is such a well -known surface of revolution, which from the 19th century to the present is the focus of mathematicians -geometers [12-18], but it is almost unknown to architects and designers, there are no examples of its application in the construction industry. This is a pseudo -sphere surface.

Characteristics and methods of setting the pseudosphere
For pseudosphere surface ( Figure 1, Figure 2) radius a Gaussian curvature К = k1k2 at all points equals a constant negative number K = -1 / а 2 .
Pseudosphere, or the surface of the Beltram, is formed by rotation of tracersis, chain line involutes r = ach(z/a), relative to the z-axis.
The equation of the tractix is where u -involute of a catenary line, the angle between the y-axis and tangent to tractices. The tractix equation can also be written as where the upper signs refer to the positive branch z > 0, lower to negative z < 0 ( Figure 2). The length of the segment tangent to the tractor from the point of contact to the point of intersection with the z-axis -constant and equal to a > 0. The section line of the pseudosphere by the хОу plane (edge of the pseudosphere) -is a circle of radius a, for all other parallels r < а. The volume of one floor of the pseudosphere: V = πa 3 /3. Three forms of pseudosphere definition are known using parametric equations [19]. For further application, let us use the following parametric form of pseudosphere surface setting: ( ) [ ] x x r r y y r r z z r a a a r r a r where r is the distance from the rotation axis to the corresponding pseudosphere point (r < а), circle r = r max = аis the pseudosphere edge. Area between parallels r = a and r = ro: S = 2πa(a -r o ). In this case, the coefficients of the basic quadratic forms of the surface and its main curvatures are: a a r k a r ar a a r k a r ar a a r k a r ar

Overview of pseudosphere shell analyses
The membrane theory of analysing pseudospherical shells was realized by V.G. Rekach [20] for the case of a homogeneous problem. The solution was made in the form of a trigonometric series. He also determined the tangential forces in the pseudospherical shell of constant thickness from its self-weight, supported hinged-movably in the normal direction in a parallel circle r = 0,5a. The bending theory of analysis of pseudospherical shells in a linear formulation subject to surface symmetric and inversely symmetric loads was considered in the work of D. Werner [21]. The solution was made in analytical form. A numerical example of the analysis is presented in tabular form. To simplify the analysis, the Poisson's ratio was assumed to be zero.
The first attempt to make an overview of all the works devoted to pseudospherical surfaces and shells was made in 1998 in the article [22]. In it, in addition to the investigations of E. Beltrami, V.G. Rekach [20], D. Werner [21], the results obtained by B. Bhattacharya

Stress-strain state of shells of revolution with close geometric parameters
There are a number of works, for example [29], where some criteria are put forward for assessing the optimality of the selected design solution. V.V. Novozhilov [29] suggested using the results of their analysis according to the membrane theory for an approximate estimate of the optimality of the chosen form of the thin-walled shell of revolution. Shells with similar geometric parameters (boom lifting and diameter of the shell at the base) were chosen for the analysis.
Let's follow his example. Figure 3 shows five types of shells of revolution. The pseudospherical (Figure 3, a), conical shell (Figure 3, b), a shell with a median surface of revolution of the hyperbola around the z-axis (Figure 3, c), a shell with the median surface of revolution of the asteroid (Figure 3, d) and in the form of a onesheeted hyperboloid of revolution (Figure 3, e).
where parameter b can take any value, but b < r2 . All unspecified geometric parameters are shown in Figure 3. In the same figure, the meridians of the pseudosphere are shown by a solid line, and the meridians of the remaining surfaces of revolution are shown by a thin line with dots.
Shell thickness h = 0.05 cm, self-weight type surface load q = 100 kg / m 2 , the radius of the base is r 1 = 4 m, the radius of the hole in the apex is r 2 = 2 m, the boom of lifting H = 5.15 m is the same for all shells, Under axisymmetric loading of the shells of revolution, the surface distributed load in the direction of the curvilinear coordinate β is zero (Y = 0), normal forces (N r , N β ), shearing forces (Q r ), bending moments (M r , Mβ), deformations (ε r , ε β , κ r , κ β ) and displacements                        B. Bhattacharya [23] proposed to use a pseudospherical shell as the foundations of reinforced concrete chimneys. D. Werner [21] proposed to use a pseudospherical thin-walled shell as the base of the tower of the television station.
Basing his research on parametric equations of the pseudosphere, Zh. Kaydasov [31] introduced a new type of surfaces-multi-tube "pseudospheres" with external cycloidal and sinusoidal corrugations. In his opinion, these surfaces can attract the attention of architects. We take the name of the surface -"pseudosphere" -in quotes, because the proposed surface does not have a constant negative Gaussian curvature.
Additional information on the application of pseudospherical shells in building industry can be taken in a manuscript [33].

Conclusion
The areas of fragments of the considered surfaces of revolution with close geometric parameters are naturally compared with the minimum surface of revolution -the catenoid -the area of the fragment of which is determined by the formula: where a ≤ r2 -is the radius of the throat circumference lying in the xOy, plane, which is determined from the expression: (1) The above two formulas must be used if the throat circumference of the catenoid is outside the considered fragment of the catenoid.
If the throat circumference is within the considered surface, as in our case, then these two formulas must be written in the following form: where a ≤ r2 -the radius of the throat circle lying in the xOy, plane, which is defined from the expression: (2) You can take the optimality factor over the area in the form: Naturally, for the catenoid in this case ρpow = 1. However, for the case in question, r1 = 4 m, r2 = 2 m, H = 5.15 m the formulas (1), (2) have no solution, so compare the shells in question (Fig. 3) with a shell having a median minimum surface with the same geometric parameters, it is impossible. You can select the inverse task. First determine by the formula (2) Hmax = 3.65m at a = 1.448 m for r1 = 4 m, r2 = 2 m, and then pick up the other shells of revolution.
We introduce the optimality factor over the area: The analyses showed that of the five shells presented in Fig. 3, the smallest area of the median surface has a shell with a hyperbola z = b/x as a Meridian (ρ = 98.25/100.5 = 0.98). It is followed by a pseudosphere (ρ = 1), a shell with an asteroid (ρ = 102.86/100.5 = 1.02) as a Meridian, a unicellular hyperboloid of revolution (ρ = 103.1/100.5 = 1.03) and a cone (ρ = 104.1/100.5 = 1.04). The difference in surface areas shown in Fig. 3 is very small.
Comparing the isofields of efforts and displacements presented in Fig. 4-28, we can conclude that the pseudospherical shell has no particular advantages in this indicator either.
V.V. Novozhilov [29] wrote about the lack of demand for the catenoid, i. e. the only minimum surface rotation in construction in the near future. It will probably also be with a pseudospherical shell.