Estimation of the stress state of a cylindrical shell by analogy with a beam

. The analysis of the stability and strength criterion of reinforced concrete shells during deformation according to the harmonic law under the action of axial compression is carried out. A method for determining the criterion of durability of reinforced concrete according to the momentless theory is presented. Options for estimating the stress-strain state of shells at a constant and variable modulus of elasticity are considered. An analogy is given between the behavior of quantities in the semi-momentless and momentless theories of zero-curvature shells. A method for calculating reinforced concrete structures of coatings under conditions of rheological deformation is presented, reflecting their real work, as well as taking into account the random nature of the values taken in the calculation.


Introduction
The development and improvement of calculations of the strength and durability of reinforced concrete constructions of large-span shells of coverages is still an important modern task, since both during the construction and in the conditions of reconstruction of buildings and structures, when determining the method of restoring the bearing capacity or strengthening, it is necessary to assess the actual state of such a structure.
It is known that the nonlinear and non-equilibrium properties of materials lead to a redistribution of forces from more to less loaded sections and components of cross-sections of reinforced concrete constructions, as well as between concrete and reinforcement, which saves materials, but reduces reliability [1][2][3][4].
The forces and the law of redistribution depend on the adopted method of reinforcement, and thus, the process of regulating the stress-strain state is possible.In real conditions of exploitation of buildings and structures, the loads acting on reinforced concrete constructions are not constant, and the law of their change is random and obeys statistical distribution curves [5][6][7][8][9][10][11].

Method
An effective way to reduce the material consumption of coverage structures is associated with the use of spatially working shells characterized by high technical and economic indicators.The stress-strain state of these structures under load has not been fully studied, and the calculation issues need to be improved.
For the calculation, the torqueless theory of reinforced concrete cylindrical shells was originally proposed by Dischinger (Germany).According to this theory, to determine the forces in the shell, they proceed from the consideration of the equilibrium conditions of the element cut by transverse and longitudinal sections, considering that only three kinds of forces act on the element: normal forces in cross-sections per unit length; normal forces in longitudinal sections per unit length; tangential forces.This method of calculation gives the distribution of normal and chipping forces in the shell, in addition, in many cases, the influence of moments in the shell cannot be neglected.According to the torqueless theory, the entire cross-section of the shell is compressed, and the side elements are stretched, and the magnitudes of compressive forces and the distribution of stresses along the crosssection do not depend on the thickness of the shell, nor on the rigidity of the side elements adjacent to it.
In [12], an example of calculating a circular smooth cylindrical shell related to the system of curvilinear coordinates is considered ,   (fig. 1).
With this solution, w in the ABCD band will be outward, and in the DCEF band it will be inside the shell.In fig. 2 shows the cross-section of a cylindrical shell deformed according to the harmonic law.
The shell acquires such deflections under the action of a distributed load (which can be only one harmonic component of a more general case of load distribution), or, as follows from experimental and theoretical studies, with loss of stability under the action of axial compression.
In the process of deformation, the points K, M, N move to the position K1, M1, N1.
The KM arc increases its curvature, the MN arc decreases.

Results
Let the edges of the shell be rigidly pinched.Cut the ABCD strip out of the shell and load its rectilinear edges AB and CD with zero internal forces on them (Fig. 3).
The ABCD strip under the action of the transverse load Z will be deformed as a beam having a cross-section in the form of a sector of a thin-walled ring with geometric characteristics: Here: yс -ordinate of the center of gravity of the section, IZ -axial moment of inertia of the section, F is the cross-sectional area.
Using formula (3), taking into account the smallness of the α, we obtain the moment of inertia IZ relative to the neutral axis Z 1 The equation for the bending of the strip beam is written as follows: Methods for calculating reinforced concrete structures are based on the following initial assumptions: hypotheses about the smallness of deformations and straight normals are valid, the deformation of materials is subject to nonlinear phenomenological equations of the state of an elastic -creeping body, the physical and mechanical characteristics of materials and the modes of changes in external loads, taking into account environmental influences, are random.Then the resolving system of calculation equations will have the following form: here: Kb(z,t) -is a coefficient considering the degree of corrosion damage to concrete, changing over time of observation; ωs,x (t) -is a similar coefficient for taking into account corrosion damage to reinforcement; E e (v,t) is an integral modulus of deformations, taking into account the nonlinearity of deformation; E 0 (t) is an elastically instantaneous modulus of deformations; C(t,t 0 ) is a measure of creep; x is the height of the compressed concrete zone; h0 and g0 are, respectively, the distance from the center of gravity of the stretched reinforcement and the center of gravity of the adduced cross-section to the compressed zone; Es,Es' is the elastic modulus of the compressed and stretched reinforcement; As,As' is the areas of the -reinforcement; x is the height of the compressed zone of the reinforced concrete element;  0 is the distance from the center of gravity of the adduced section to the compressed zone; σ(t) -is the magnitude of the acting stresses; R(t) is the strength of the material n and m are the parameters of the nonlinearity of deformation determined from experimental data obtained on the basis of solving a system of two logarithmic equations composed for two points of the experimental curve (ε-σ); w and φ are, respectively, the vertical displacement and the tension function; Kx, Ky-curvature of the shell.Closed integration of system (9) meets insurmountable mathematical difficulties, therefore, to obtain numerical results, one should resort to linearization of the problem using the method of integral estimates.The essence of this technique consists in a step-by-step review of solutions within each time interval, into which the entire changing external load mode is divided, and creep processes and corrosion are considered fixed within the boundaries of each interval.
Due to the adoption of such assumptions, the system of integro -differential equations ( 9) turns into a nonlinear algebraic one, which allows a solution based on the finite difference method using sequential iterations.
For a grid with a square cell of width b, the discrete equations in finite differences will take the following form: ( 2 ) ( , The second group of equations of system (12) will look similar, it is enough to replace in expression the value of w by φ, and take qij(t) =0.
Considering the mechanical characteristics of materials as random variables obeying normal distribution curves, we ca n assume: Where is λ -a safety characteristic corresponding to the specified reliability.

Taking
note that equations ( 7) and ( 8) have the same structure and differ only in a coefficient of 1.25, reflecting the inaccuracy of the construction of the bending equation ( 7) and ( 8) coincides, and this allows us to conclude that there is an analogy between the behavior of quantities in the semi-momentless and momentless theories of zero curvature shells.The analysis of the stability and the criterion of strength of reinforced concrete shells during deformation according to the harmonic law under the action of axial compression and the method for determining the criterion of durability of reinforced concrete according to the momentless theory are carried out.A variant of estimating the stress-strain state of shells at a constant modulus of elasticity is considered.An analogy is given between the behavior of quantities in the semi-momentless and momentless theories of zero-curvature shells.
The possibility of taking into account the processes of long-term deformation of reinforced concrete from the standpoint of a probabilistic approach to the assignment of calculated reliability coefficients for load and material is considered.In real conditions of operation of buildings and structures, the loads acting on reinforced concrete shells are not constant, they change both during exposure and during operation as a result of environmental influences, and the law of their change is random and obeys statistical distribution curves.And to a greater extent, this applies to temporary loads, to a lesser extent to permanent ones, which can usually be considered unchanged.

Fig. 2 .
Fig. 2. Diagram of curvature and displacement of points K, M, N.

-Fig. 3 .
Fig. 3. Scheme of deformation of the ABCD strip under the action of the transverse load Z At R = 1, the expression (5) is reduced to the equation: