Physical and mechanical modeling of shaping pneumatic formwork by bicubic splines

The article deals with the hierarchical design structure, consisting of four stages of the stress state of pneumatic shells. The main tasks of designing and constructing pneumatic formworks during the construction of thin-walled architectural shells are considered. The introduction of pneumatic formwork into architectural and construction practice has a significant impact on the ability of the architect, builder and designer in the search for new and more advanced forms, as well as the builder in translating the creative ideas of the architect into steel and modern concrete. Thanks to the flexibility and elasticity of the material, it allows the construction of curvilinear spatial shells of a new elastic quality level. The proposed geometric model makes it possible to obtain a variety of software as well as the formation of a discrete-point frame of a finite shape by bicubic splines, which provides a predetermined accuracy for solving subsequent software design and construction problems.


Introduction
The introduction of pneumatic formwork (PF) into architectural and construction practice has a significant impact on the ability of the architect, builder and designer to search for new and more advanced forms, as well as the builder to translate the creative ideas of the architect builder into steel and modern concrete.Pneumatic formwork, due to the flexibility and elasticity of the material, allows the construction of curved spatial shells of a new elastic quality level.
There are the following methods of shaping and erection of reinforced concrete shells using pneumatic formwork: • bending method; • spray method; • complex method; The deflection method is carried out by bending the reinforced concrete layer from the planar position by lifting the pneumatic formwork to the design position and compacting the pneumatic formwork to the design position [1].Compaction is carried out by vibrating rails in the design position.With the spraying method, the pneumatic formwork raised to the design position, instead of with reinforcement, is sprayed with a quick-hardening concrete mixture.Both methods allow the mechanization of erection; exclude the re-application of the inner protective layer of reinforcement, but the spray method is preferred, which eliminates the compaction of the concrete mixture.With the spray method, it becomes possible to completely mechanize and automate the process of erecting a reinforced concrete shell (Figure 1).It is known [2], that mainly wooden and metal formwork is used in construction practice.Wooden formwork is 1.5 to 2 times more expensive than pneumatic formwork and 5 times more labor consuming to erect shells with it.There is an opinion that the secondary use of wooden formwork is limited.An integrated approach to the automated design of pneumatic formwork involves the use of a modern computer and peripherals in the search for optimal design solutions.The use of mathematical methods and modern computer technology makes it possible to increase the technical level and quality of the designed object, to move from the automation of individual engineering calculations to complex automation, i.e. to the creation and implementation of a computer-aided design system (CAD) of the object.Computer-aided design of pneumatic formwork is connected with the solution of a number of tasks, the main of which are the geometric modeling of the object and the design process and the degree of interaction between the designer and the computer in the design process.

Research materials and methods
Below we illustrate a mathematical model of the process of designing pneumatic formwork and a description of the physical state of the object at various stages of calculations and design [3,4].This process can be grouped into four stages, which approximately correspond to the four main states of pneumatic formwork (Figure 2).Based on the above, we can propose a general structure for formalizing the problem of designing pneumatic formwork (Figure 3).Now let's consider the geometric modeling of the process of shaping pneumatic formwork.A formal description of the process of shaping pneumatic formwork can be carried out by the associated analytical, differential geometry, also bearing in mind the differential characteristics of surfaces with the statics of soap films or highly elastic rubber in a loaded state [].
As a result of creative searches for complex architectural forms obtained with the help of pneumatic formwork, the designer creates a number of difficulties, including the ambiguous assignment of a functional dependence that is designed by him to the object, i.e., the shell.In such cases, it is expedient to specify the forms discretely.
Determining the cutting (initial) form, operating with discrete values of the models, leads to significant errors in the case of a sparse one, and requires a large computer memory for dense grids.
In this regard, in this work, the method of spline approximations is used to form the surface on a discrete frame of points.Indeed, in engineering geometry, spline approximation provides a high accuracy of approximation of functions and their derivatives, partial derivatives, when the values of the functions are given on a number of finite grid nodes (Figure 4).

Results
To form a discrete surface of pneumatic formwork and solve metric, positional problems associated with determining the cutting shape, the use of bicubic splines significantly minimizes the errors of approximation of functions to predetermined conditions.Interpolation by bicubic splines with their piecewise polynomial representation is expedient both in solving creative and theoretical issues, and in automatic computational terms, in particular, using Auto CAD, CAD [6].
(3) The first derivatives of the cubic polynomial are calculated by the following formulas.
The values of the function  =  + 2 are given in Table 1.

Conclusion
Thus, the following conclusions can be drawn: • a hierarchical design structure is proposed, consisting of four stages of the stress state of pneumatic formwork.The sequence of solution and the structure of formalization of the design problem are determined; • a certain method of automated shaping of pneumatic formwork based on the theory of soft (elastic) shells was proposed; • the developed geometric model allows to obtain a variety of forms of pneumatic formwork and by solving a linear problem; • formation of a discrete-point framework of a finite form by bicubic splines provides a predetermined accuracy (required) for solving subsequent problems of software design and construction.

Figure 1 .
Figure 1.Erection of reinforced concrete shells using pneumatic formwork.

Figure 2 .
Figure 2. Mathematical model of the design process and design of pneumatic formwork.

Figure 3 .
Figure 3. Shaping the task of designing pneumatic formwork.

Table 1 .
The values of the function  =  + 2 .