Bionics in architecture and geometric modelling of thin shell surfaces

. In the scientific problem of design and calculation of thin elastic shells in the modern world, certain advances have already been made in mathematical and technical theory, based on hypotheses, experimental data, calculation equations and engineering calculations. Only such shells, which are designed based on calculation and used in building and technical constructions, can be referred to a small number of geometric surfaces. When designing thin shells, surfaces of rotation (sphere, torus, paraboloid, ellipsoid of rotation) and transfer surfaces (hyperbolic and elliptic paraboloid, circular transfer surface) are used. Trends in construction and engineering seek to apply complex mathematical models in harmony with environmental policy and the environment. This leads to the necessity of studying the influence of parameters when modeling an object on the parameters and properties of the created construction. Possessing a more complex shape the shells are realized as a result of experiment. As a result of active introduction of information technologies it became possible to introduce cardinally new methods in the application of geometric thin-walled spatial structures for the design of building and technical constructions, a number of machine-building parts. Modern analytical calculation programs and computer-aided design systems (Compass, Autocad, Archicad, etc.) make it possible to create a geometric projection model of a structure on the basis of primitives, to perform structural and static calculations of a project in an elementary manner. The solution of such layout problems is made possible with the support of computer geometry based on descriptive and analytical geometry, linear and vector algebra, mathematical analysis, and differential geometry. Modern bionics and environmental policy relies on the latest methods of mathematical modeling of architectural projects with a wide choice of computational and graphical software for calculation and 3d visualization.


Introduction
With the growth of technical progress, the need for additional scientific research in the field of body geometry and stress-strain state of technical products and building structures in the system of thin-walled spatial building and technical structures of complex geometric shape, but, at the same time, with a simple and efficient technology of their erection is increasing. Solving the problem of designing geometric surfaces is one of the main directions of engineering geometry. The issues of geometric modeling and their application in various fields of urban planning, mechanical engineering, aircraft engineering are considered in a number of research papers [1][2][3][4].
The preferred choice of surface geometry of spatial building structures is made taking into account the realization of all urban planning, functional and aesthetic requirements, as well as the necessary conditions for appropriate static operation, the conditions of detailing of surface parts into separate prefabricated elements used in the production and installation of the structure.
Relevance of the scientific problem is determined by the need to improve the effectiveness of computer and mathematical calculations of the geometric modeling system and the construction of thin elastic shells of surfaces, the construction of their sweep to the plane when solving architectural and construction problems.

Methods
It seems reasonable to classify the median surfaces of shells used in architecture and engineering with regard to their embodiment. Such thin-walled structures are considered in the following authors [1][2][3][4]. Since the questions of curvature, method of formation of linearity, number of constituent elements and other features of the surface are independent of each other, it is suggested to classify the median surfaces according to the following attributes: whether the shell surface contains straight formations or not and, depending on this, by deployability; whether it can be formed by parallel or circular transfer of a flat curve; by Gaussian curvature; whether the shell is a closed surface (its part) or an unclosed one; whether the shell consists of a single piece of curvature.
In research works, a number of authors consider the problem of the stress-strain state of thin geometric shells under a vertical load arising under given rigid displacements of a number of cross sections [5][6][7][8]. Scientific papers [9][10][11][12] are devoted to the problems of nonlinear deformation of the stress-strain state of thin-walled building structures of complex geometric shape.
In the present study, an attempt is made to analyze different predetermined requirements to the designed shells and their geometrical interpretation is given. For example, based on the calculation requirements, the dependence of the calculation complexity on the geometry of the shell is shown, and the static features of shells with parabolic, hyperbolic and elliptical points are given [10][11][12]. Studies of the shells delineated by the transfer surface have been carried out. Fig. 1 shows a hyperbolic paraboloid with an unshaped noncentral surface; the guide in this case is a branch of the hyperbola, the formant is a parabola. The surface is described by the canonical equation (1), where a and b are positive constants

Results and discussion
Taking into account structural requirements, the geometry of shells [7,10,12,13] is considered in detail, depending on different ways of their support on: walls (supporting rings); columns (columns); directly on foundations (solid or separate); supports, which are an organic transition between the shell and the foundation; combined (mixed) support on walls and cables; console support; "floating" foundations. The influence of overlapped span size on the choice of shell shape was studied.
Technological requirements dictate the geometrical interpretation of the shells in monolithic and precast versions. Consideration is given to the existing practice of shells partitioning into prefabricated elements [5]. Of interest is the special geometry of shells made of reinforced concrete, wood, metal, plastics, etc.
Volume-planning requirements imply the dependence of the shape of the shell on the plan of the structure, which in turn depends on the functional features of the room and the place of action in it: the action occurs at the end, the action occurs in the center or the action occurs over the entire area. In the design, a detailed analysis of the plans of large implemented and projected structures. The influence of the shape of the envelope on the formation of the corresponding volume in the room is investigated.
Consideration is given to acoustic requirements in rooms of different purposes covered by envelopes and geometric factors affecting the acoustics: the volume of the room and its shape. Some space-planning considerations are taken into account to ensure the necessary acoustics.
An important factor is ensuring lighting requirements for simple and composite envelopes. Considerations on the form, size and location of light openings in the envelope.
Important significance is given to aesthetic and psychological requirements when erecting complex architectural structures. Of scientific interest are geometric spatial shells as a natural stage in the evolution of forms in architecture. Obligatorily must take into account the peculiarities of visual perception of the human shell in the interior and exterior. The study of such structural forms reveals the unity of the shells of constructive expediency and architectural expression. Thin spatial shells are one of the forms of implementation of bionics in architecture, based on natural forms, smooth lines, repeating the perfect contours of living nature. Co-modern bionics is based on the latest methods using mathematical modeling system with an extensive choice of software calculation and graphic software for the calculation and 3d-visualization.
The modeling and calculation of the dome cover structure, which is a convex lifted shell on a circular, elliptical plan, is interesting to study in this paper. An axisymmetric thin shell of rotation and a support ring are the structural elements of such a dome. The dome surface is formed by rotating a flat curve around the axis of rotation, which is vertically located in space.
When geometrically modeling domes (vaults) on a circular plan using surfaces of rotation, it is definitely preferable to refer to the parametric method when creating them. Applying the cylindrical coordinates φ and ρ, the calculation equation of the shell is shown in the following form where f is the function defining the forming geometry of the dome-shell surface (1    2 , 0    2).
We use as the shaping geometric surface of the ellipsoid of rotation, then, choosing the function f  to calculate the general dome model (3) and, applying the equation of the formant in the coordinate system Oz, we obtain:  In this case, the Oz axis is vertical and the O axis is at the base of the dome, a and b are the semi-axes of the ellipse forming the rotation surface, and H is the dome height. Taking the variable z from equality (4) and taking into account equality z  f , we obtain the following equation for the elliptic dome shape (at 0    R): The introduction of function f  described by equality (5)  Taking into account the functional series of peculiarities of the geometric surface of the dome [12,13], it is important to fulfill this relation b  H. С. Taking into account the design mathematical parameters of the surface, the following inequality corresponds to the above inequality: R  Hk.
For the mathematical calculation of the elliptic dome [14,15], the geometric model is also obtained by substituting the function f , defined by equality (5), into expression (3), in this case the value of parameter ρ is set in the following limits R1    R2. The height H and the parameters of the investigated calculation-mathematical model a and b find their expression by means of the structural-geometrical parameters of the system by substituting the following coordinates z  0,   R2 и z  H1,   R1 of the points of the dome into equation (5). Calculating this system of equations, we come to equality (7), where k  a/b. The design parameters in the calculation must be chosen to meet the ratio b  H, which corresponds to the following inequality   Based on the design features for the calculation, we consider a smooth shell dome. The thin shell of a smooth dome assumes a compression work under given stress conditions [5][6][7][8]. Under the influence of the acting load distributed over the surface, the area of bending moments influence is within the supporting ring, which under a certain vertically directed load acts in tension, with the lantern ring working in compression.
To produce certain conditions for torque-free operation of the smooth geometric shell of the dome, the support ring must move freely along both angular and radial lines under various temperature influences, in a wide range of fluctuations, the load values in the calculation are determined by the following formula Calculation of the annular longitudinal force N2 is calculated by this formula 2 = 2 ( − 1 1 ), (10) where F is the vertical resultant of the applied external load acting on the thin shell section located above the given horizontal section (solution angle 2) (Fig. 4); Fn is the normal component to the dome surface of the external load per unit surface area.
It should be taken into account that the meridian forces N1, despite the value of the angle , will be compressive. At the same time, the circular longitudinal force N2, having passed its zero value, in the pole region is converted from compressive to tensile. At zero values, the annular longitudinal force N2 will be called the transition joint, which corresponds to the angle 0 (at N2 = 0): As a result of calculations we obtain 0=51°49. It is assumed that when the central solution angle of the calculated hollow dome-shell is less than the value 20 = 103°38′, no tensile forces appear in this annular direction line as a result; whereas, at a larger value, annular tensile forces appear in the area of annular sections, well below the conditional transition joint,

Conclusion
The results of the study showed excellent characteristics of such hollow thin-walled shells as dome geometric surface, which, despite its apparent simplicity, can withstand decent loads and is able to combine power, reliability and structural elegance. The advantages of spatial structures of thin-walled shells is that they make it possible to cover large spans without intermediate supports, buildings and structures take unusual architectural expressiveness, combine bearing and enclosing functions, in the construction of thin geometric frameworks optimal use of building material, multilevel transfer of external loads disappears. With the advent of modern new construction materials it is possible to build a thin reinforced concrete frame, as well as walls made of glass. Accordingly, such structures have disadvantages, such as the high labor intensity of making the elements and their installation, as well as the difficulty of transportation. But these points are few and solvable.
Let's consider some economic considerations about the efficiency of wide implementation of thin-walled shells, the main factors influencing the cost of the shell. Thinwalled shells are one of the types of spatial geometric structures, which are used in urban planning in the construction of buildings and structures with a large area, such as public buildings (sports facilities, concert halls, exhibition halls, shopping centers) and industrial buildings (hangars, garages, markets, warehouses, reservoirs). The thin-walled shell used as a curved surface has a very considerable load-bearing capacity due to its geometric curvilinear shape and, taking into account its minimal thickness and therefore minimal weight and consumption of the material used, is economically efficient and profitable. By means of competent modeling when designing structures and parts with hollow sections, it is possible to save a significant amount of consumable material (up to 20-25%), which, nevertheless, does not affect the flexibility and strength of parts and structures. This is a definite advantage in today's quest to maintain functionality without threatening the environment. In addition to the unusual, original and spectacular design, modeling and implementation of thin-walled shells serves to save resources and energy, reduce costs with a clear increase in durability and strength of architectural structures, technical elements and parts. The use of thin shells in production and construction involves obtaining new yet unrealized opportunities for the erection of geometric structures of different scales.
It is advisable to replace the single-type buildings with constructions with modern bionic design, inspired by nature itself, lightweight, practical and economical building structures, and at the same time to solve the problem of environmental health and ecology, using the latest materials.