Insolvent ways of development of the modern theory of reinforced concrete

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Introduction Eurocode is a system which includes scientific developments and experience of outstanding scientists from various countries, motivated formulation of the main Principles and Rules, the classical mechanics and general theory of computing of elastoplastic systems, detailed and numerous experimental data. Nonlinearity of deformational properties of reinforced concrete at short and long term loadings is the basis of standards of Eurocode 2 [1]. Dependence “strain - deformation” of concrete has a descending interval and limited extension creep deformations are nonlinear from the very low levels of strain. Violation of the Eurocode system, as a rule, leads to errors in the scientific and normative theories, additionally accompanied by a violation of the rules of mechanics and mathematics. The requirements for computational models of reinforced concrete to consider instant nonlinear properties are not met in current normative and technical documents of many countries, international institutes for standardization [2-5], Principles and Rules of Eurocodes, despite the prohibitions of methods by the Eurocode: · the theory of long-term resistance of reinforced concrete is built on an irrelevant computational model containing errors that cause fundamental defects in the theory; · short-term nonlinear properties of concrete are substituted by linear creep properties, causing gross errors in the evaluation of the bearing capacity of t εσ (t ) = σ(t0 ) J (t, t0 ) + ò J (t, t¢) dσ(t¢) , (1) t0 reinforced concrete, also leading to a violation of the principle of independence of the action of forces; where ε σ (t ) 1 is the complete strain from stress σ(t); φ(t, t ¢) - short-term nonlinear deformations of reinforced concrete are not taken into account in calculations of J (t, t ¢) = Ec c (t ¢) + E (t ¢) - compliance function; the bearing capacity. A jump occurring from the elastic stage of deformation to the missing plastic hinge, which is accompanied by the disappearance of the length of the structure is considered normal. These errors that we discovered are not a simplification of Ec (t ¢) is nonstationary modulus of elasticity; φ(t, t ¢) is nonstationary creep characteristic considering ageing. In scientific publications (1) is usually integrated by parts, thus obtaining standardization. For example, in [2] it is reported that σ(t ) t ¶ é 1 φ(t,t¢)ù the developed theory is “on an international scale εσ (t ) = - E (t ) òσ(t¢) ¶t¢ ê E (t¢) + E (t¢) ú dt¢ . (1′) the basis of a new advanced format for calculating creep”, Gordon Clark, director of Ramboll, president c t0 ë c c û FIB 2014 [13], warned designers about failure of the theory of creep in real design. We have identified The term φ(t, t ¢) Ec (t ¢) is a measure of the creep of and analyzed this inconsistency, the errors of theory and international standards, in particular [2-4]. This will be shown below in the materials of the article, at the end of which numerical errors of error are presented, which only amount to taking into account the instantaneous deformations in the creep theory of 300% [7]. Basic errors We investigate the fundamental errors of the normative theory of long-term resistance of reinforced concrete. The managers of its creation specify that this theory was coordinated and promoted by international standards institutes within the framework of the global harmonization scenario. It is implemented in the standards of a number of countries and now proposed for inclusion in Eurocode 2 [2]. Considering the ageing and the dependence of modulus of elasticity on time (non-stationary properties) of concrete are considered as the main achievements and distinctive features of these standards. However these major achievements are errors. The principle of superposition is the basis of both the modern scientific creep theory of concrete, which is called the “world harmonized format” by foreign scientists, and the developments “in recent decades of international standardization institutions... for recommendations, norms and technical guidance documents” [2-4]. These works also indicate that McHenry in USA (1943) “substantiated this trend by experimental studies of the creep of hermetic specimens using the principle of superposition which is characteristic for the theory of Volterra”. We give the basic law of creep of concrete in the original notation [2]: concrete C(t,t') used in publications in our country, which is preferable to application of the creep characteristics in the processing of experiments. We emphasize that ageing of concrete is taken into account in φ (t,t') and C (t,t'), and the modulus of elastic-instantaneous deformation Ec(t') essentially depends on the age of the concrete. Equations (1), (1′) are substantiated by two fundamental assumptions: the principle of linear connection between stresses and strains ε σ (t, t ¢) = σ(t ¢)J (t, t ¢) ; (1′′) the principle of superposition, verbally formulated in various versions in numerous well-known publications on the theory of creep of concrete, reference books, for example in [9]. Serious mistakes in (1) make the normative theory inconsistent with Eurocode, unreliable and uneconomical. Losses from such norms and calculations are significant as annual global volume of usage of concrete and reinforced concrete is 4 billion m3. Let us also recall the tragedy of the collapse of the Transvaal Park (Moscow, 2004), caused by creep problems in concrete. We note that the article has no relation to the “ongoing disputes, ...discrepancies and uncertainties” existing in this section of creep of reinforced concrete. Also, in this paper we do not discuss a different point of view. We, using the Eurocode system, identify and analyze the errors in that area of creep, where, as the leaders and developers of norms indicate, there is a “steady consensus” [2-4]. The main mathematical error in (1) lies in its basis - the principle of superposition, which appeared in the theory of reinforced concrete after the work of McHenry. This principle incorrectly builds the core of creep, incorrectly describes the processes of changing instantaneous deformations and creep strains. The errors in the principle of superposition can be determined in various ways: for example, by constructing and solving a differential equation corresponding to a linear connection (1′′); solving the inverse problem of classical mechanics; analysing the value of the total strain rate corresponding to (1′′). Applying the last method the following is obtained: v (t, t¢) = σ& (t¢)× J (t, t¢) + σ(t¢) ¶J (t, t¢) + σ(t¢) ¶J (t, t¢) . o ¶t ¶t¢ From this formula it is clearly seen that four terms, caused by the rate of change in the compliance factor, are lost in the main law (1): -σ(t¢) E&c (t¢) + σ(t¢) E2 (t¢) E 1 (t¢) ¶φ(t, t¢) + σ(t¢) ¶t E 1 (t¢) ¶φ(t, t¢) - ¶t¢ c c c -σ(t¢)φ(t, t¢) E&c (t¢), c E2 (t¢) (2) and the value of these terms is comparable with that of the remaining term. These losses cause considerable discrepancies between the theory and the experiments described in the scientific literature, e.g. [8]. Opposite mathematical actions, first differentiation and then integration, are performed (and without any need) over the known result (1′′) of the classical theory in the principle of superposition. Оne term for instantaneous deformations and several terms for creep deformations are lost in the process of differentiation. After integration, the losses rors [8; 10; 11], distorting the theory of creep of concrete: 1. incorrectly determines the values of short-term linear strains; 2. incorrectly finds the expression of a nucleus describing the process of changing linear creep strains; 3. erroneously classifies as instantaneous elastic deformations to creep strains. Let us consider them in more detail. 1. The rate of elastic deformation equals are included into the values of deformations, and then into the theory of design calculations. The principle of superposition distorts the classi- ε& (t ¢) = σ& (t ¢) 1 у Ec (t ¢) + σ(t ¢) ¶ 1 . ¶t ¢ Ec (t ¢) cal linear connection (1′′), causing three types of er- Integrating, we obtain t 1 t ¶ 1 ò ε у (t ) - ε у (t0 ) = ò E dσ (t¢) + σ (t¢) (t¢) ¶t¢ E dt¢. (t¢) t0 c Integrating the first term by parts, we find t0 c ε ( ) ε ( ) σ(t ) σ (t0 ) σ ( ) ¶ 1 σ ( ) ¶ 1 . t t у t t = - t¢ dt¢ + ¢ ¢ t¢ dt¢ ¢ ¢ у 0 E (t ) E (t ) ò ¶t E (t ) ò ¶t E (t ) c c Hence the short-term deformation equals 0 t0 у e (t ) = c t0 c s(t ) . Ec (t ) It is also clear that the first term under the integral sign (1′) is superfluous, and the use of the overlapping principle in (1) and (1′) ε (t ) · (t0 ) 1 dσ (t ) · (t ) o (t ) ¶ 1 dt , t t = - ¢ = - ¢ ¢ (4) у ò ò (t0 ) E (t ) E (t ) ¶t E (t ) E ¢ ¢ ¢ is strongly erroneous. c t0 c c t0 c The principle of overlapping erroneously recont e t = òs t¢ 1 e-g( - ¢)dt¢, structs the actual, real elastic linear model of concrete with the Ec(t) module; the prinicple attaches to it a non-existent and unreal model of a linear viscous t t 1 f ( ) ( ) h (t¢) t0 1f 2 h ¢ E (t¢) = с (6) fluid with a viscosity coefficient K (t¢) = Eс (t¢) , 1 f (t ) ; &j (t¢) 1 thus forming Maxwell's scheme. E& c (t¢) ¥ t 1 Let us consider an example, putting e2 f (t ) = ò s(t¢) h dt¢ , (t¢) (t ) = s0 = const σ0 in (3), (4), we will receive σ0 t0 2 f E2 t¢ εe (t ) = Ec (t ) and εe (t0 ) = Ec (t0 = const. Com- ) h2 f (t¢) = с ( ) E& c (t¢) j¥ 1 ; (t¢) (7) parison of these deformations is shown in figure 1. t 1 ¢ -g(t -t¢) ¢ , 0 e3 f (t ) = òs(t ) ( ) e dt t h3f t¢ h3 f (t¢) E2 (t¢) 1 = - с ; c ¥ E& (t¢) j (t¢) (8) Figure 1. Comparison of εу(t0) and εу(t) t h e4 f (t ) = òs(t¢) t0 4f 1 dt¢ , (t¢) h (t¢) = - Eс (t¢) , (9) Curve 2 in figure 1 corresponds to the VNIIG ¥ 4 f &j (t¢) data on the changing of modulus of elasticity with time. Errors in the value of elastic deformation are about 300% at t = 360 days. 1. In the region of creep deformations, the number of additional (fictitious) bodies arising due to an incorrect scheme for constructing the creep kernel (hereditary function of type I) increases substantially. It depends on the form of the function φ(t,t') describing the nonstationary creep characteristic in where η1ф, ... , η4ф are the viscosity coefficients or the coefficients of internal resistance of the fictitious bodies; moreover, the bodies (8) of Voigt and (9) of the viscous element expand under compression. The creep deformations (6) - (9), caused by the effect of the superposition principle on the classical bond (1′′), are a fiction; they are also summed up with a short-term fictitious deformation t the main law (1). We write this function in a wellknown, widely used in the scientific literature form e5ф (t ) = -òs(t¢) t0 ¶ ¶t¢ Eс 1 (t¢) dt¢: (10) φ(t, t¢) Eс (t¢) φ (t¢) é1- e-γ(t -t¢) ù = ¥ ë û , (5) Ec (t¢) εσф 5 (t )= åεiф i=1 (t ), where φ¥ (t¢) is a function considering the ageing of concrete. In the famous monograph of I.E. Prokopovich the creep behavior φ(t,t') used by foreign scientists has the designation C (t, τ) , these are identical quantities. In case (5) the fundamental law (1) forms four extra (fictitious) bodies: two Foigt type bodies and and introduce large errors in the value of the total deformation εσ(t) determined by the creep law (1′). For example (Recommendations, 1988), at constant stresses, the error from applying the superposition principle for creep strains reaches 100%: t ε (t ) òΩ(τ) f (t - τ) dτ cσ mistakes t0 two viscous elements connected in series with each other. Deformations of these bodies are equal εcσ (t ) principle = 1- Ω(t 0 ) f (t - t0 , ) dτ where Ω(τ) is “the function of the effect of ageing on the measure of creep”; f(t-τ) is - “a function that takes into account the increase in time creep measure”. 1. The fact of appearance of a single short-term the properties of deformations of the hereditary type εe,1(t,t′). The error is corrected by making new mistakes. Concrete has essentially non-linear properties at shortterm and long-term loading. The short-term load diastrain 1 Eс (t¢) in the nucleus of creep of the integral gram has a falling section and a limited extent, see figure 2. In the main law (1), (1′) only linear deformation equation (1′): εl (t) = εe (t) is taken into account, and the nonlinear deformation εn(t) is ignore, see figure 2. S.V. Aleksandrovsky indicates the reason for this circumstance: ¶ éε (t¢) + C (t, t¢)ù = ¶ éεe (t¢) + C (t, t¢)ù , “It is very difficult to take into account the dependence ¶t¢ ë e,1 û ¶t¢ ê σ (t¢) ú ë û led to the temptation of erroneous substitution of the properties of short-term deformation εe,1(t′) by of the modulus of elasticity on stresses and age of concrete simultaneously. Therefore, the modern theory of creep of concrete takes into account only a change in the modulus in time...” Figure 2. Distortion of the σ-ε diagram of concrete Let us consider two types of such substitution. The first substitution. A representative forum poses the erroneous task of “taking into account the influence of the pre-history of deformation on the modulus of elastic-instantaneous deformations”. The basic equation of the creep theory takes the form (in the original notation): t An “experimentally valid” expression appears for the modulus of elastic deformation of concrete An “experimentally valid” expression appears for the modulus of elastic deformation of concrete Et,τ = Et + an,τφt Eτ , where φt is characteristic of creep of concrete. ε( ) σ(t) σ(τ) ¶ é 1 ù C(t, τ) dτ And other erroneous forms of the main creep t = -ò ê + ú . (11) c (t,t ) t ¶τ ë Ec (t, τ) û law appear E ¢ 0 ε (t ) = σ ( ) + ò σ (τ) ¶ χ (t, τ) dτ - ò σ(τ) ¶ C* (t, τ) dτ, (12) t t t E (t ) τ ¶τ τ ¶τ ¶ * ¶ é 1 + 1 1 ù tion properties of the hereditary type εn(t,t′), the er- ¶ ( ) where C (t, τ) = ê ¶τ τ ë E τ C(t, τ)ú ; χ(t,τ) has û roneous overlapping principle is used, and, instead the name “reducing correction... to the current specific elastic-instantaneous deformations”. of the simple algebraic formula ( ) ( ) 2 ( ) εn t = B2 t σ t The second substitution. The nonlinear short-term strain εn(t) is erroneously attributed to the deforma- (B2 is a known coefficient), the integral following is contrived: ε ( ) σ ( ) ¶ εn (t, t¢) σ ( ) ¶ ( , ) t t n t = ò t0 t¢ ¶t¢ σ(t¢) dt¢ = ò t0 t¢ C ¶t¢ n t t¢ dt¢ , (13) where Cn(t,t′) is called the measrue of fast-flowing creep. 1 ë n û C (t, t¢) + Cn (t, t¢) = Eс éφ (t, t¢) + φ (t, t¢)ù , (t¢) (14) taken into account in (1′). The gross errors in the theory from such a substitution of the short-term nonlinearity of concrete we considered in [10] and [8]. J (t - t¢) = F (t - t) = a - be-b(t -t) , Famous foreign scientists renamed “fast-flowing creep” into “minute creep”, and the erroneous idea of the Second substitution is presented as their imwhere 1 0 a = C0 + E0 ; E C0 E0 C0 is an elastic modulus; portant achievement. The principle of superposition in the theory of b = ; η = E0 β , η is a stationary coefficient of creep of concrete is a mathematical error committed in the exptensive interpretation of the principle of the linear superposition of Boltzmann. In international norms of reinforced concrete, it is estimated incorrectly: it is supposedly “a tendency to study creep... linear viscosity. In the theory of creep, the fundamental solution of the corresponding differential equation is known to have the form t according to the principle of superposition peculiar σ ε (t ) = σ(t ) - ò σ(t ¢) 1 ¶φ(t - t¢) dt ¢ , (15) to Volterra's theory”. Let us consider this in more detail. We investigate the essence and the secondary nature of the Boltzmann scheme for the theory of creep of concrete on the example of concrete considered in E0 t0 where φ(t - t ¢) = E 1 0 C0 of creep. E0 ¶t¢ [1 - e -β(t -t¢) ] is characteristic the well-known paper of G.N. Maslov No. 4. Here the concrete has stationary properties corresponding to the classical theory. In the notation of G.N. Maslov the compliance function has the form The Boltzmann case is obtained from the solution of (15) by means of a number of its transformations mathematically valid only under the conditions of stationary properties é 1 1 ù t é 1 1 ù εσ (t ) = σ0 ê + φ (t - t0 )ú + ò ê + φ (t - t¢)ú dσ (t¢) . (15′) ë E0 E0 û t0 ë E0 E0 û Unlike (15), the compliance function is used in the transformation (15'), which attracted the attention of scientists. However, the transformation (15′) is possible only with substantial and very strong restrictions. In the exptensive interpretation of compliance, these restrictions were not taken into account, and the theory of creep of concrete proved to be deeply erroneous. Here, firstly, the property of the process that creates the temptation to expand the theory and transforms into the above-mentioned gross error for nonstationary E(t′) accompanying the normative linear creep theory of concrete is imposed on instantaneous deformation with an extremely simple physical meaning for an arbitrary t. In scientific literature there is even an authoritative statement that “elastic-instantaneous deformations strictly obey... the principle of superposition”. Secondly, it is necessary to integrate (15) by parts, that in the exptensive interpretation of the compliance function under the conditions of ageing of concrete (1) creates another temptation, traditionally leading to another gross error in finding the core of the integral equation. As it is known, for non-stationary properties of concrete, the creep strain is obtained from another solution of differential equation, a solution written in a more complex form · it satisfies the requirements of classical mechanics; · it does not satisfy the conditions of the Boltzmann principle. The Boltzmann principle distorts the essence of the nonstationary Maslov model. It replaces one classical body of creep of concrete with a chain model of successively connected bodies with a set of erroneous properties. In the theory of creep of concrete, there is a case when extensive interpretation of the compliance function is unacceptable even with a difference kernel. For example, the nucleus of creep in a number of known works is represented in the form (the second case) -β(t-t¢) K (t - t ¢) = Ae . (t - t¢)α-1 é t 1 ù ε cc (t )= e-F (t )êε c0 + ò σ(t ) ( ) e F (t )dtú , ë û ê t0 η t ú t F (t ) = òβ(t )dt , t0 where the parameters η(t) and β(t) in (15) are functions of time. In the concrete of G.N. Maslov the rate of deformation degenerates due to the difference kernel. In the case of an extensive interpretation of the compliance factor, the application of the Boltzmann principle usually becomes incorrect. The nonstationary model of Maslov concrete with a coefficient of vis- Certain forces correspond to this kinematic equation of motion in connection with the solution of the inverse problem of mechanics. The analysis of the differential creep equation reveals that in this nucleus there is a resistance force with a coefficient of viscosity of the linear model equal to η(t, t¢) = 1 (t - t¢)α-1 , which is impossible by the A same reasons as in the above-mentioned case of applying the hereditary properties of the elastic modulus E(t,t′). The third case corresponds to the extensive interpretation of the compliance function in the “chain model”. This case is present in theoretical rheology, cosity η(t ) = C0 (t ) β and a time-dependent module and as a repetition - in the norms of reinforced E0(t) demonstrates this: - it satisfies experiments with simple loading at low levels σ » 0,1Rпр ; concrete. We preliminarily write the Boltzmann scheme for the Maxwell body in the form é 1 1 ù t é 1 1 ù εσ (t ) = σ0 ê + (t - t0 )ú + ò ê + (t - t¢)ú dσ (t¢) , (16) ë E0 η û t0 ë E0 η û where η is a stationary coefficient of viscosity. E0 , In the “chain model”, by successively connecting bodies (15) and (16), we have an extension With a variable viscosity coefficient η(t ) = φ& (t ) we obtain the theory of ageing of concrete record of the compliance function (Dischinger, Whitney); φ(t ) = φ¥ (1 - e-bt ), which J (t - t¢) = + φ(t - t¢) + (t - t¢) . (17) 1 1 1 by series expansion gives the function of Freudenthal E0 E0 η φ(t ) = φ¥t 1 b + t , substantiated by the experiments of A pair of integral equations corresponding to the expansion hypothesis (17), and solved either with Davis and Glanville. respect to deformations εσ(t), or relative to the stres- ses σ(t), in theoretical rheology are called “Boltzmann - Volterra equations”. It is also indicated that this pair “represents a complete mathematical formulation of the principle of linear superposition”. However, such a chain model, with its extensive interpretation of the compliance coefficient, is essentially erroneous. This is evidenced by its reduction to a differential form: &ε&σ (t ) σ η + ε& (t )η = &σ& (t ) η æ + σ& (t )ç η + 1 + η ö ÷ + σ(t ) . (17′) β E0β è E0 β C0 ø It can be seen from (17') that there is a resistance lished the unacceptability of such forces in both force &ε&σ (t )η β proportional to the acceleration, which problems of mechanics and in applications [6]. Unfortunately, in the scientific literature on is incompatible with classical mechanics, and, in connection with Art. 5.1.1(3)P Eurocode 0, the chain model is an inappropriate design model. The components of the force of the computaconcrete, in international norms, there are a number of errors analogous to those described, and consisting in an extensive interpretation of the compliance function in the form of a chain model [2], tional model can be a function of position ε σ (t ) , including for taking into account the rapidly flowing creep. speed ε& σ (t ) , time and other quantities. If there is Thus, in the case of consistent merging of (among others) a force proportional to acceleration &ε&σ (t ) , then the fundamental principle of mechanics about the independence of the action of forces is violated. The well-known scientist L. Pare has estab- Maslov's theory and the theory of ageing of concrete (D. McHenry, A.V. Yashin, T. Hansen, I.E. Prokopovich and I.I. Ulitsky), the creep equation has the form &ε&(t ) + βε& (t ) = &σ& (t ) 1 + σ& (t )æ φ& t + β φ β ö æ & · + σ(t ) t + φ& t ö . ç ÷ ç ÷ E0 è E0 E0 C0 ø è E0 E0 ø If another viscous element (with viscosity η(t )= Δe -α1t ) is added to this chain in order to take into account the rapidly flowing creep, that was previously assumed by the Eurocode developers before its approval, then we get another erroneous version of the theory (written without averaging) 1 æ φ& t β 1 ö æ φ&t βφ& t β η& (t ) ö &ε&(t ) + βε& (t ) = &σ& (t ) E + σ& (t )çç E + + ÷ + σ (t )ç + E η(t ) ÷ ç E E 2 ÷ + - ÷. η(t ) η (t ) (*) 0 è 0 0 ø è 0 0 ø When Eurocode 2 was adopted, the theory of It is obtained by decomposing e -β(t -t0 ) in a seageing and the viscous element were removed from this model, the error was annulled. In the Eurocode rules, only classic concrete G.N. Maslov is left; from its creep characteristics, a normative coefficient of creep development is obtained 0,3 ries using two terms. The exponent 0.3 of the power function takes into account on average the ageing of the concrete. In the case of nonlinear creep and short-term non-linearity in Eurocodes, the use of the Boltzmann β (t, t ) = é t - t0 ù , scheme is also erroneous. For nonlinear creep of c 0 where β n = 1 β . · t - t ê ú ëβn 0 û concrete of G.N. Maslov (the fourth case) within the framework of generally accepted hypotheses, the rate of deformation is v t, t¢, F éëμ t¢ , t¢ùû = σ& t¢ × F éëμ 1 t¢ , t¢ùû φ t - t¢ + σ { ( ) } ( ) ( ) ( ) E0 ë û ¶F éμ (t¢), t¢ù +σ(t¢)× μ& (t¢) ¶μ ë û 1 ¶F éμ (t¢), t¢ù φ(t - t¢) + σ(t¢)× E0 ¶t¢ 1 φ(t - t¢) + E0 +σ(t¢)× F éëμ (t¢), t¢ùû × 1 é¶φ(t - t¢) ¶φ(t - t¢) ù ê + ú , E0 ë ¶t ¶t¢ û which is not taken into account in the traditional nics show that such an assumption is a very supertheory. Here F [μ(t ¢), t ¢] is a non-linearity function, ficial assumption. We will devote a separate article in which the voltage μ(t ¢) = σ(t ¢) is usually taken (after the work of Leaderman) as a nonlinearity parameter, which is incorrect: the methods of classical mechato this problem. For example, under this assumption, a series of multiple Volterra - Frechet integrals t t t εσ (t ) = ò J1 (t - t¢) dσ(t¢) + ò ò J2 (t - t¢, t - t ¢) dσ(t¢) dσ(t ¢) +K -¥ -¥ -¥ ε (t ) = J (t )σ + J (t, t )σ2 + J (t, t, t )σ3 +K σ 1 2 3 is a nonintegral form [12]. Recently, some papers have appeared that develop “a modification of the principle of superposition of deformations for nonlinear creep” in the form t é 1 ù t ë û ε (t, t0 ) = ε (t0 ) + ò ê E (τ) + C (t, τ)ú dσс (τ), 0 where sс (t) = S éës (t)ùû is the known stress function σ[τ]. The error of this formulation is similar to that used in (1). The total strain rate here is (18) & é 1 ù d 1 ¶ ¶ (18') vσ (t, τ) = S [σ (τ)]ê + C (t, τ)ú + S [σ (τ)] + S [σ (τ)] C (t, τ) + S [σ (τ)] C (t, τ). ë E (τ) û dτ E (τ) ¶τ ¶t From this it is clear that the last three terms in (18') are lost in (18). The significance of these terms is identical to the significance that we described in items 1-3 above. We must additionally pay attention to the fact that the identity of the nonlinear function S[σ(τ)] for short-term and long-term deformations is also incorrect. But even if another function Sg[σ(τ)] is used for creep strains, then, as it is noted above, this assumption is a very superficial assumption that does not correspond to the real nonlinear creep theory of concrete, which will be published later. This theory has nothing to do with the principle of superposition. Conclusions In conclusion, we will estimate the errors of the considered models of standards. From formulas (1′), (2), for example, it follows that the superposition principle complicates and distorts the classical elastic model of concrete; adding to it an unreal model of a viscous fluid with a viscosity coefficient Curve 2 in figure 1 corresponds to well-known data (RRIHE) on the modulus of elasticity in time. The error in the value of elastic deformation reaches at t = 360 days, ≈ 300%.

About the authors

Rudolf S Sanjarovsky

L.N. Gumilyov Eurasian National University

Author for correspondence.
Email: salsa87@bk.ru

Dr Sci. (Eng.), Professor, Principal Researcher

2 Satpaev St., Astana, 010000, Republic of Kazakhstan

Tatyana N Ter-Emmanuilyan

Russian University of Transport (MIIT)

Email: tanya_ter@mail.ru

Dr Sci. (Eng.), Professor, Department of Theoretical Mechanics

9 Obrazcova St., bldg. 9, Moscow, 127994, Russian Federation

Maxim M Manchenko

Krylov State Research Centre

Email: salsa87@bk.ru

Cand. Sci. (Eng.), Senior Researcher

44 Moskovskoe Shosse, St. Petersburg, 196158, Russian Federation

References