On the construction of the diagram of calculation method of rod reinforced concrete structures under the action of low negative temperatures

As a result of generalization and analysis of the data in the field of experimental studies, the significant influence of low negative temperatures on the development of stresses and strains in reinforced concrete structures from the force effects was justified. It is shown that this influence largely depends on various structural and technological factors: (characteristics of the composition of concrete, its structures, water-cement ratio and of the natural moisture content of concrete). The necessity of taking into account these factors of influence in the calculation of reinforced concrete structures on the strength, deformability and crack resistance at the same time power and temperature effects has been justified. Diagrams of deformation of concrete in the frozen to minus 70 °C condition are developed. Taking into account these diagrams the development of the diagram method is given, which allows to calculate reinforced concrete core structures under simultaneous influence of power loads, and low climatic (to minus 70 °C) negative temperatures.


Introduction, purpose and objectives of the study
At present, in connection with the plans to expand oil and gas production in the Northern regions of Russia, including in the Arctic shelf, and the construction of appropriate floating terminals, as well as tanks for storage of liquefied natural gases and their transportation afloat in terminals, along with the problem of ensuring the durability requirements to reinforced concrete structures of these structures, the problem of developing new methods of calculation of such structures operated under the simultaneous impact of force loads and significant low negative temperatures becomes urgent.
It is known that the resistance of concrete under the influence of negative temperatures (frost resistance of concrete) is characterized by a certain number of cycles of alternate freezing and thawing, after which the strength of concrete is reduced to a regulated value (see GOST 10060-2012, the first basic method).
However, this characteristic is not enough for a real assessment of the durability of reinforced concrete structures. This requires an assessment of the influence of low negative temperatures, especially climatic (up to minus 70 °C), on the basic regulatory strength and deformation characteristics of concrete required for the calculation and design of concrete and reinforced concrete structures operated in such conditions. The results of the relevant experimental studies are presented in  [1][2][3]. Especially it is necessary to mention the fundamental monograph of Professor, doctor of technical Sciences V. M. Moskvin in coauthorship with his postgraduates [2], dedicated to the physical and technological basis of concrete and structures from it at the severe weathering.
The results of the above-stated experimental studies of the strength and deformation properties of concrete in frozen state to different temperatures (up to minus 70 °C) are summarized from the standpoint of solving the problem of durability of reinforced concrete structures operated in severe climatic conditions in [2,3].
In the work [4] it was shown that degradation of properties of such material (conglomerate type) as concrete, is reflected most fully in the diagram method of calculation and its development. The initial basis of this method was the deformation diagrams of concrete and reinforcement under normal conditions of positive temperatures.
In work [5] it is proposed the initial approach to the construction of the diagram method of calculation of reinforced concrete structures under the action of low negative temperatures. However, the results of experimental studies [1,2,6] showed that the deformation diagrams of concrete, in addition to low negative temperatures, are largely influenced by the structural-technological characteristics of concrete, and, first of all, such a technological factor as the initial moisture content of concrete at the time of its freezing. It's also necessary to take into account the effect of low temperatures on the coefficients of thermal strains of concrete and reinforcement, and the strain diagram of reinforcement. All these factors are taken into account in the equations of the calculation diagram method presented below.
Taking into account all the above in this paper, based on the analysis of available experimental investigation data [1][2][3]6], the following research tasks are set: 1) to perform mathematical processing of the available experimental data on strength and deformation characteristics of heavy concrete to determine the coefficients of change of these characteristics according to the appropriate formulas, depending on the initial moisture of the concrete and its freezing temperature; 2) to perform, using results of processing of experimental data (see task 1), correction of diagrams of deformation of concrete under load of axial compression in the conditions of action of low negative temperatures (to -70 °C) in comparison with the corresponding diagrams of deformation of concrete under the same load in the conditions of action of positive temperatures; 3) to determine the nature of the effect of such temperatures on the change in strength, the initial modulus of elasticity, the relative deformations of concrete in the tops of the diagrams constructed during the test under axial compression loads in the temperature range from + 20 °C to -70 °C; 4) to determine the influence on the characteristics of the constructed diagrams of such structuraltechnological parameters of concrete, as its water-cement ratio (W/C), and, consequently, the porous structure, as well as the initial moisture (mass ratio of moisture in %/%).

Diagrams of concrete deformation taking into account the influence of low negative temperatures and moisture of concrete on the change of its strength, the initial modulus of elasticity and limit deformations under axial compression
The relationships between the relative strains and stresses of concrete under axial compression in the frozen state are recorded as: where bT b b , ,    -correspondingly the relative strain, stress and modulus of elasticity of concrete in the frozen state Processing of data of the experimental researches [1][2][3] on concretes with various natural moisture content at W/C=0,4 and W/C=0,5 showed that on average at change of moisture in the range W=3,1%-5,1% it is possible to accept: The following dependencies are also defined for the range of moisture content of concrete from W=3,1% to 5,1% during storage to freezing and testing under load of 28 days in the chamber of normal-humidity curing (in terms of air temperature 20 ºC +/-2 ºC and its humidity 90% +/-5%).
Prismatic strength of concrete R b , which is equal to the tension T b  at the top of the frozen concrete diagram, is determined from the dependence:  -stresses at the top of the concrete diagram in the initial test conditions (at t = 20  C and corresponding moisture content W%), β R -the coefficient of influence of low temperatures on the increase of stress at the top of the diagram. The analysis of experimental data [1][2][3] showed that: where b  -relative deformations at the top of the concrete diagram at normal conditions. For heavy concrete: where R 0 = 20 MPa. Data processing of experimental studies [1][2][3] shows the following: for the ascending branch    According to experimental data [6] it can be taken The axial tension parameters

Diagrams of deformation of reinforcement in conditions of low negative temperatures
Deformation diagrams of the reinforcement are constructed in accordance with the dependencies given in [4] with additional consideration of the influence of low negative temperatures. Low negative temperatures affect the yield strength σ 0,2 and the elastic modules of the reinforcement E s, which in conditions of low temperatures are designated σ 0,2T and E sT. According to the data presented in [6], it is possible to accept ; 210 ; 210 The relationship ratio between the stresses and deformations of the reinforcement, as in (1), is assumed to be:

Changes in the coefficients of thermal strains of concrete
According to experimental studies [1,2], the coefficient of thermal deformation (  bt  ) also largely depends on the initial moisture content of concrete under the influence of negative temperatures. So, for concrete with natural moisture content after aging in the mode of normal-humidity hardening in the temperature range from + 20 °C to 0 °C , the average value  bt  0,910 -5 (C) -1 , when the temperature changes from 0 C to -40 C, then  bt  1,1310 -5 (C) -1 . When the temperature changes from -40 °C to -70 C, then  bt  0,910 -5 (C) -1 . The coefficient of temperature deformation of the value in the temperature range from +20 C to -70 C according to [1,2] remains constant and equal  st0  110 -5 (C) -1 .

Construction of physical relations of the diagram model taking into account the influence of low negative temperatures
The above relationships between stresses and deformations of concrete and reinforcement (diagrams of their deformation), as well as the coefficients of thermal deformations, serve as the basis for the construction of physical relations to the calculation of reinforced concrete structures operated under conditions of low negative temperatures [4,5].
Let's consider the construction of such relations in relation to the calculation of reinforced concrete structures.
In Figure 3, the calculation scheme of the normal section of the reinforced concrete element exposed to the action of two moments is presented: M x (in the plane ZOY) and M y (in the plane ZOX) and the normal force N acting along the axis Z. The position of the point 0 at the origin x, y, z can in principle be arbitrary, but it is rational to place it in the center of gravity of the section, which is determined in the elastic stage of deformation of the element.
In the derivation of physical relations tensile stresses and forces are taken for positive and compressive for negative. Freezing temperatures of concrete and reinforcement are also taken as negative. In the part of the section with cracks, the concrete is turned off from work and all efforts are transferred to the reinforcement. In this area the free rebar diagram σ s − ε x is replaced by a diagram σ s − ε sm , where σ s -the stress of the rebar in the crack, ε sm is the average relative strain on the participation between the cracks. Average deformations ε sm are determined by the method of V. I. Murashev. The change in the relative deformations of concrete and reinforcement ε e (e=b, bt, s, sm) along the height of the section of the reinforced concrete element follows the hypothesis of flat sections [4].
Numerical integration is used to derive physical relations. The cross section of the concrete element ( Figure 3) is divided into i elementary sections of concrete with areas A bi and coordinates of their (25) D ijstiffness of the element, which is calculated by the formulas: Physical relations (25) for complex constructive systems are rationally record in finite increments, which leads to economical weakly iterative and non-iterative methods for solving physically nonlinear tasks.
It should be noted that the results of the studies carried out in this work on the construction of diagrams of deformation of heavy concrete corresponds: -with presented in the works of S. N. Leonovich [7], and Yu. V. Zaitsev [8] physical models of the phase transition of water into ice in the pores and capillaries cement stone of concrete; -with the theories and hypotheses of frost destruction of concrete according the characteristics of its structure, in particular its differential porosity, which has a significant impact on the development of hydraulic pressure in the gel pores and pores-capillaries of the cement stone concrete; the latter, provided that the original moisture content of concrete W exceeds the limit of W (W>W cr , where W cr corresponds to the degree of saturation of cement stone of the concrete is the critical value for more than 90%) [1,2]. This can lead first of all to the development of irreversible microcracks in the walls of the above pores and capillaries, and later -to the formation and development of the so-called "magistral (main) crack" [8] and then -to the progressive destruction of concrete; -with the nature of the diagrams "axial compression stresses in frozen concrete under different temperatures -longitudinal deformations of concrete" given in [7,8], as well as in the works of foreign researchers [3,[9][10][11][12].

Conclusion
The general physical relations connecting moments M x , M y and normal force N with generalized deformations (curvatures 1/r x , 1/r y and ε 0 -relative deformations at the level of the chosen z axis) taking into account influence of low negative temperatures are established. Physical relations allow to calculate reinforced concrete structures taking into account the influence of temperature factors by modern computational methods, for example, the finite element method. The initial basis of the physical relations are diagrams linking the stresses and strains of concrete and reinforcement under the action of low temperatures. Based on the generalization of the available experimental data, the correction of concrete diagrams was performed. This takes into account the effect of low temperatures on increasing the strength of concrete, its initial modulus of elasticity and relative deformation at the tops of the diagrams to a temperature of approximately minus 70 °C. It is shown that the increase in strength, the initial modulus of elasticity and the limiting relative deformations at the tops of the diagrams largely depends on the moisture of the concrete at the time of its freezing. At temperatures lower than minus 70 °C, most experiments indicate that the increase in the strength and deformation