Influence of long-term exposure to loads on the annular sections’ strength and rigidness

. A numerical experiment was carried out to study rigidity, critical force, and bearing capacity of reinforced concrete racks with annular section under short-term and long-term loading. The effect of long-term loading was estimated in the numerical experiment by changing the coefficient 𝜑𝜑 𝑙𝑙 from 1 to 1.8 times 0.2, as well as the modulus of deformations 𝐸𝐸 𝑏𝑏,𝜏𝜏 , taking into account the operating conditions (W = 40-75%). At the same time, the external load on the rack N was taken in the experiments as a multiple of the critical force N cr from 0.2 to 1.0, and the relative eccentricity 𝛿𝛿 𝑒𝑒 of the load application varied from 0.15 to 1.35, which made it possible to estimate the stress state of the struts in an extended range of possible loadings. The results obtained made it possible to identify qualitative and quantitative regularities of rigidity changes, critical force and bearing capacity of annular struts during short-term and long-term load application.

The research results of these authors were used as the basis for calculating the structures of the annular section according to the SNiP 2. 03.01-84 norms. The methodology for calculating such structures in the new standards BC 63.13330.2018 [7] has not undergone significant changes. However, we analyzed in detail some features of the calculation according to the new standards in [8].
It should be noted that most of the above-mentioned studies, including ours [8,10], were based on the results of the short-term loads' impact on structures. Insufficient attention has been paid to the study of the long-term loadings influence on the operation of structures with an annular section.

Main part
In this work, which is a continuation of the previous studies [8][9][10], the results of a numerical experiment on the study of the bearing capacity, bending rigidity and the critical force of annular struts under short-term and long-term loads are presented. A cylindrical support was adopted as a test sample PTL according to GOST 22687.2, the parameters of which are given in Table 1. The assessment of the stress state of the structure under short-term and long-term loading was carried out according to the norms BC 63.13330.2018 methodology [7].
The relative height of the annular elements' compressed zone of concrete ξcir: Where , = + ; Аsp , As are the areas of stress and non-stress reinforcement, respectively.
Bearing capacity of the annular section : Where zs=(0.2+1.3 )rs., by that 0.15 < ≤ 0.6 D-pillar flexural rigidity and critical force Ncrc were determined, respectively, by the formulas (3) and (4): Where Db is concrete section rigidity, Ds is reinforcement rigidity; Ib and Is are the moments of inertia, respectively, of the concrete section and reinforcement; = 0 is relative eccentricity of external load application N, H defines outside diameter of the strut beam; = 1 with short-term load. The effect of long-term impact of loads on bending rigidity and critical force was taken into account by changing the coefficient and concrete deformation modulus , by the formulas (5) and (6).
where β=1for heavy concrete; Ml and М are the moments from long-term and full loads, respectively.
where , is a creep coefficient of concrete, depending on the strength of concrete and ambient water percentage in accordance with [7].
In a numerical experiment to expand the field of possible loadings of the strut, the change in rigidity D and critical force Ncr was investigated for the short-term and long-term loading, depending on the relative eccentricity = 0,15-1,35 (multiples of 0.15). And when assessing the bearing capacity of the Mult compressive strut, depending on the relative height of the compressed zone the value of N was taken as a multiple of Ncr from 0.2 to 1.
Within the framework of this article, it is not possible to show the obtained research results in detail, however, the most characteristic and significant dependences are presented. Table 2 shows the numerical values of the rigidity Db concrete section of the compressive strut, and Table 3 shows rigidity D of a reinforced concrete section with shortterm and long-term loading for different values and operating conditions.  It should be noted that the concrete section rigidity of the compressive strut Db according to formula (3) varies proportionally and inversely proportional to the load duration factor . Therefore, in Tables 2, 3 and in the graphs in Fig. 1 the values Db are given under various for the initial modulus of elasticity (short-term load) and , (continuous load) at three values of operating conditions. Analysis of these graphs (refer with Fig. 1) shows that the rigidity of the compressive strut Db (all other things being equal), the main influence is exerted by the relative eccentricity of the load application . In this case, the functional dependence Db = f( ) and D = f( ) is nonlinear, which is obvious from the formula (3).
Operating conditions Wambient water percentage and associated values , have less impact on rigidity Db and D, and the form of the function Db = f( ) and D = f( ) remains unchanged, and the graphs converge with magnification .
So, for example, the rigidity Db ( by = 1.2 and W = more 75%) changed from 552.64·10 8 Table 2). Influence on eccentricity full rigidity D (refer with Table 3) affects several times less than Db, which is explained by the weight fraction of reinforcement rigidity Ds in the total rigidity of the compressive strut section.
At the same time, the change in the water percentage of the environment W with prolonged exposure to load on full rigidity D much less than Db.
For example, changing W=40-75% ( = 1.2, = 0.15) leads to a decrease in rigidity with 1359.42·10 8 kNmm 2 to 1159.85·10 8 kNmm 2 , i.e. by 17%. Moreover, in each row and each column of Table 3, this change will be different, which is associated with the influence of the rigidity of the reinforcement Ds on the total rigidity of the compressive strut D section. Table 4 shows the numerical values of the critical force of the compressive strut depending on the relative eccentricity and the coefficient , taking into account the duration of the load, and in fig. 2 -graphs of changes = f( ). It should be noted that the nature of the change in function = f( ) is similar to the function D = f( ), which is obvious from the formula (4).  These graphs analysis shows that the main influence on the critical strength of the compressive strut has a relative eccentricity of the load application . In this case, the influence of the load duration factor on the critical force of the compressive strut disproportionate to the change in value . This is due to the lack of a clear influence of the coefficient for rigidity Ds. Incomplete studies of the bearing capacity of the compressive struts Mult showed that the main influence on the bearing capacity is exerted by the value of the relative compressed zone of concrete and the relationship / с . Wherein Mult is increased with increasing / с , reaching the maximum value at a certain value / с (refer with Fig.3).

Conclusion
1. Rigidity D of the annular struts with increasing eccentricity of load application decreases with both short-term and long-term exposure to loads. In this case, the functional dependence D = f( ) is non-linear. 2. Prolonged exposure to loads leads to a decrease in the rigidity of the concrete section Db proportional to the increase in the coefficient and inversely proportional to the change in the deformation modulus Eb,τ. (refer with Fig. 1). For any combination and change in environmental water percentage W from 75% till 40% leads to a decrease in rigidity Db for 1,565 times. A change in the relative eccentricity of the load application from 0.15 to 1.35 leads to a decrease in rigidity Db in 3,66 times for any values and W.

The parameters
and Eb, τ influence to the full rigidity of the compressive strut section D affects less than the concrete section Db rigidity (refer with Table 3). W change from 75% till 40% for each from 1,2 till 1,8 leads to a decrease in total rigidity D by = 0,15 from 17% to 13%, and at =1,35 from 6% to 4,1%. This is a decrease in the influence of parameters and W the total rigidity is related to the weight fraction of the reinforcement rigidity Ds in the overall rigidity of the compressive strut D. 4. Critical force of the annular strut (ceteris paribus) is decreased with increasing eccentricity of load application both for short-term and long-term loading. Functional dependence с = f ( ) is nonlinear (refer with Fig. 2). With increase there is a decrease in critical force out of proportion to the value . 5. Preliminary study of the bearing capacity of the compressive strut Mult (according to formula 2) revealed that Mult is increased with increasing ratio / , reaching a maximum within the acceptable values / =0.1÷1 (refer with Fig.3).
The authors plan to continue researching the bearing capacity of the compressive struts Mult depending on different combinations , , Eb, τ, / с for the purpose of building dependency monograms Mult from / с and percentage of compressive strut reinforcement.