CC BY
93
DOI: 10.17277/amt.2019.01.pp.073-081
Modeling Adhesive Strength of Corroded Rods in Reinforced Concrete Structural Elements Using a Genetic Algorithm
V.P. Yartsev*, A.N. Nikolyukin, T.M. Pluzhnikov
Tambov State Technical University; 112D, Michurinskaya St., Tambov, 392032, Russia * Corresponding author. Tel .: +7 4752 63 03 80. E-mail: kzis@nnn.tstu.ru
Abstract
We developed an analytical model to describe the behavior of the tangential stresses of the reinforcement adhesion xad in the anchoring zone, taking into account the influence of various parameters (concrete strength, rod corrosion, etc.). Under normal conditions, the development of such a model is a long process. The "Genetic Algorithm" (GA) of soft computing, being a powerful tool for evolutionary modeling of complex technical problems, was used to simplify and accelerate this process.
Keywords
Reinforced concrete; core reinforcement; adhesive strength; corrosion; experimental data; approximation; mathematical model; genetic algorithm.
© V.P. Yartsev, A.N. Nikolyukin, T.M. Pluzhnikov, 2019
Introduction
To speed up the construction of a building or structure, precast concrete elements have been widely used in construction practices. The formation of a unified structure occurs with the help of butt joints produced, as a rule, in the form of inserted (overlapped) rod bars, subsequently bonded in butt joints [1-5]. Under the influence of external loads or due to precipitation of the base, axial tensile and transverse shear forces occur in the butt joints of the
structure (Fig. 1); this leads to the formation of cracks or various defects in the bonded joint. As a result, rod reinforcement of the structure is subjected to corrosion while in operation [6-11].
Consequently, when designing precast concrete structures special attention should be paid to the development of rod joints. In this case, it is necessary to take into account such parameters as limiting the width of crack opening and determining the area of the reinforcement anchoring in concrete.
Fig. 1. Diagram of forces acting on a separate block of butt joint
For a long time, S.A. Mariupolsky was engaged in the study of this area. In his work, he described the main types of butt joints, which were done using reinforcing rods, subsequently bonded in the place of installation [11]. The author noted that one of the reasons for the premature destruction of structures is the insufficient strength of construction joints.
However, the destruction of the joints may occur due to the ductility of joints which is not taken into account during their construction, as well as the occurrence of transverse and longitudinal deformations of free ends of reinforcing rods in the concrete joints as a result of various factors. Therefore, the joints should not be considered as an absolutely stiff structure. In order to control displacements (cracks) in the joints, one should perform the selection of the number and diameter of the rods in the assembly, as well as specify the calculation of stresses and attenuation of the joint section caused by joint action of the structure elements [8].
Studying the issue of design modeling of butt joint destruction requires solving a number of important general problems related to the action of reinforcing elements in concrete under longitudinal and transverse loads, as well as setting physical parameters characterizing the mutual action of concrete and steel through different means of stress application [12].
Thus, in order to solve this problem, in this paper, we propose to use genetic algorithms (GA) in order to establish the relations between the input parameters and final shear stresses of the reinforcing rods, occurring in the joints while in operation.
Therefore, when building the model, six parameters were taken: compressive strength, concrete sample dimensions, reinforcing rod profile, reinforcing rod diameter, embedment depth, corrosion degree, maximum adhesive strength. On the basis of the obtained experimental data [13-30], the dependence was established between the input parameters and the final shear stress.
Experimental
To build the model, experimental data were used from [13-30]. Six critical parameters, which are presented in Table 1, were used.
These parameters were used to estimate the final adhesive strength between the corroded reinforcing rod and the concrete. In pulling tests, concrete cubes 150x150x150 mm in size were used with reinforcing rod of A400-A240class.
Table 1
Parameters
Parameters Designations in the formulas Designations in the expression tree
Compressive strength Rb d0
Concrete sample dimensions c d1
Reinforcing rod profile Sp d2
Rod diameter d d3
Embedment depth ^depth d4
Corrosion degree Km d5
Maximum adhesive strength Tad y
As the adhesion value, Tad was used to describe the average stress values throughout the sample anchoring zone
N„
Tad -'
(1)
where Nmax is ultimate load (N) in pull-out-test (kN), A is contact area (cm2).
To model adhesion properties, the database was arbitrarily divided into two parts, namely, training and testing. Among these experimental data, 10 % were used as a test base (26 sets), while the remaining 90 % were used to train the model (225 sets). The training database was required when developing a prediction model, while the test database monitored the repeatability and reliability of the proposed model.
Gene Expression Programming (GA)
Genetic programming (GA) proposed by J.R. Koza [31], essentially is the application of genetic algorithms for computer programs. Gene expression programming (GA) was introduced by C. Ferreira [32], and it can be viewed as a natural development of genetic algorithms and genetic programming. GA creates computer programs of various sizes and shapes encoded in linear chromosomes of a fixed length. The GA algorithm begins with the random generation of chromosomes of a fixed length for each individual for the initial population. The chromosomes are then expressed and the suitability of each person is evaluated based on the quality of the solution they represent. An important characteristic of GA is that it allows you to accurately derive the phenotype based
on the sequence of the gene and vice versa, which is called the Karva code.
GA parameters used for the derivation of mathematical models were listed earlier (Table 1),
Tad = a1 + a2 + a3 + a4 + a5 + a6 =
(2)
where Tad is final adhesive strength between the reinforcing rod and concrete; functions , ..., c6 .
In the present study, the GeneXproTools 5.0 application package was used to determine the functions O, ..., ^6 in equation (2). The GA parameters used for the derivation of mathematical models are listed in Table 2.
In this case, the genetic algorithm is designed to approximate experimental data and build dependency functions. The complexity of the function being optimized is determined both by the number of variables for which optimization is carried out and by the presence of local extremes. This, in turn, with a large area of research can lead to almost a huge number of solution options, so it is impossible to prove that the solution found with the help of GA is the best. However, such proof in this case is not required, it is only important that the installed function satisfies the required conditions for the convergence of the model.
The parameters presented in Table 2 were derived from a preliminary study to maximize correlation and minimize error.
Table 2
GA parameters used to build the model
Parameters GA
Values
Fundtions
Chromosome
Population size
Functions relation
Number of genes
Inversion
One-point and two-point recombination rate
Gene recombination and gene transfer rate
Number of genes
Number of generations
+, -, *, /, A, ln, exp, cos, sin, tg, ctg, log, xA1/3, xA2, xA1/4, xA1/5
30
8
Addition 0.045 0.1
0.3
0.1
6
1864590
To implement the exact model various mathematical operations were used. During training, it was found that the software can sometimes ignore some input variables due to a minor external impact on the overall model. In this case, the GA model presented in this study used each of the input parameters at least once. As a result, the following dependencies were obtained:
(
a, =
(G1C7 - Km )Rb - G1C0 -
1
A
1/3
cos K
1/3
G2C4
a =■
G2C0 - Rb + c
-yKm + G2C1
; (3)
(4)
a =
ln ((exp(Rb )1/4 + (Rb +G3C5) G3C1) - d) ; ln10
4 = cos(/depth - cos Sp - Km G4C0) - c ;
a5 =
tg(b + ^depth+G5C1 - Km + d)1/3
+ c ;
a6 = lnf d exp (sin(G6C5 +1(
depth
))) •
(5)
(6)
(7)
(8)
where the constants: G1C0 G1C7 G2C1 G2C4 G2C0 G3C1 G3C5 G4C0 G5C1 G6C5
= -35.6765470119042; = 2.6195898035416; 6.61345964007763; = 5.41190574148063; 7.07966697155402; = -8.92397791683817; = 4.84969634083071; = 0.563865309919699; = 6.18578923675139; = 7.1938173760206.
The most recurring input parameter in the model was the corrosion degree and the concrete strength (it was repeated five times), while the size of the concrete sample was the least used (it was repeated only once). This may be due to the significance of the input parameters selected for modeling.
For example, frequent repetition of concrete strength and the corrosion degree of the reinforcing rod can be explained by the fact that these parameters have a great influence on the amount of adhesion, since the amount of adhesion of the cement gel between the reinforcing rod and the concrete mass increased with an increase in strength.
1
Shrinking Rust reduction
at
at
Fig. 2. Stress-strain state between reinforcing rod and concrete in pull-ou-test
With a low degree of corrosion, an increase in the adhesion strength occurred until the formation of cracks in the concrete mass as a result of a change in the surface properties of the reinforcing rod. In this case, an increase in the magnitude of the friction force between the surfaces was observed. At the same time, the filling of the layer between the reinforcing rod and
the concrete formed by rust led to a more dense combination of materials (Fig. 2). However, the occurrence of cracks and reduction in the cross section of the rod (at a higher degree of corrosion) resulted in a decrease in the amount of adhesion, which contributed to premature loss of bearing capacity.
The low importance of such a factor as the profile frequency was due to the insufficient number of reinforcement profiles in the presented sample to fully reflect the model.
As can be seen from equation (3), trigonometric functions are included in the proposed GA model. Preliminary studies showed that equations (3) - (8) with the parameters of the GA were the optimal characteristics for obtaining the best prediction model with the available data set.
Models developed by software in its native language can be automatically analyzed for visually attractive expression trees, thus allowing them to understand their mathematical logical complexity faster and more completely. Fig. 3 shows the expression tree for terms used in the formulation of the
GA model.
a)
b)
Fig. 3. Expression tree for the model:
a - Sub-ET1; b - Sub-ET2
(D0 is the compressive strength of concrete; D1 is the dimensions of the concrete sample; D2 is the reinforcing rod profile; D3 is the reinforcing rod diameter; D4 is the embedment depth; D5 is the corrosion degree; CO - C7 are constants)
a
y
a
x
a
c
a
0
T
T
Some mathematical operations from the expression tree were simplified in equation (8). The comparison of the experimental and predicted bond strength is graphically illustrated in Figs 4 and 5. The model was evaluated using the correlation coefficient R (equation (9)), which describes the correspondence between the curve of the output variables of the model and the actual variable of the test output curve. A higher R coefficient indicates a model with better ability
R =
(mi - mmean XPi - Pmean )
Vzc
mt - mn
i )2 - Pmean )2
(9)
where m0 and p0 are mean values of measured mi and predictedpi values, respectively.
As can be seen from Fig. 4 and Fig. 5, the trend of changing training data was better than that of testing. The calculated correlation coefficients (R) were 0.890 and 0.9 for the training and test databases, respectively. This can be viewed as the evidence that there was a satisfactory correlation between actual and predicted values. Moreover, close values of the correlation coefficients can be considered as an indicator of the reliability and appropriate suitability of the proposed GA model.
Analysis of the obtained correlation
To assess the possibility of predicting the proposed model, the normalized results were calculated, calculated by dividing the predicted results by actual ones (Fig. 6). In addition, for further comparison of the proposed models, Fig. 7 presents the estimation errors and the frequency of the corresponding data.
Considering the normalization of the predicted data, the ideal estimate was 1.0, which indicated that the predicted and observed values were exactly the same. As can be seen from Fig.6, the closest trend in the change of normalized values for this model was about 1.0. The normalized values of xad ranged from 0.48 to 7.12 for the model with test data, while for the training database, the range of values was from 0.22 to 4.72. However, observing the trend of distribution of normalized results together with the errors presented in Fig. 8, one can notice that the greatest fluctuations occurred for the actual adhesion strength of less than 5.0 MPa. Figs 6 and 7 show that most of the data exceeded this value and the estimation errors became relatively low as the adhesion strength values increased.
Observation Order
Points lto 196
a)
> 15
T3
tu £
Observation Order
b)
Fig. 4. Comparison of the obtained values using the model with the experimental:
a - training data set; b - tesingt data set
35 30 25
£ 20 u
=L
* »
M
♦ * » *
♦ 9 V? * ST*
« f* * *
*
10 15 20 25 30 35
Experimental data, MPa
a)
35
ata, d d te
O
3 20 d e
P
♦
* ♦
♦
* * ♦
4» *
Fig. 5. Regression analysis of the model:
a - training data set; b - testing data set
10 15 20 25 30 35 Experimental data, MPa
b)
ata d
■
■
■ 2
i r /
♦ A
♦ ■
10 15 20 2? 30 35 Maximum adhesion Tad, MPa
Fig. 6. Comparison of the effectiveness of the proposed model for training and testing data sets:
1 - training data set; 2 - testing data set
cN 90
r, o SO
rro
E 70
60
50
40
30
20
10
0
Tad< 5 5 < Tad < 10 10 < Tad < 15 15 < Tad < 20 Tad > 20
Intervals of maximum adhesion values xad, MPa
Fig. 7. Error analysis of the proposed GA model for training and testing data sets:
1 - training data set; 2 - testing data set
The fluctuation of normalized data usually tends to re-evaluate for the GA model in the training array. Sufficiently high errors occurred for the GA model with overestimated data. The number of confirmed or understated data obtained for the model in the test database is much less. Considering that most of the data, the values of Tad which exceeded 5.0 MPa, the average value of the absolute error was about 10 and 20 % for the proposed GA models, respectively.
The predicted values obtained for the GA model indicate an underestimate. For a more detailed test of the prediction capabilities of the proposed data set models, a statistical analysis was also conducted. The following statistical parameters were calculated using equations (10) - (12) and are presented in Table 3.
Mean Absolute Percentage Error:
1
MAPE = — Y" ,
N^J=1
m - Pi
m,
100% ;
Mean Square Error:
Y"-1(mi- Pi)2
MSE = ^^-;
n
Root Mean Square Error: 1
RMSE =
Y"-1(mi- Pi)2
(10)
(11)
(12)
where m and p are measured values (mi) and predicted values (pi), respectively.
Table 3
Statistical parameters of the proposed models
Model MSE RMSE MAPE RSE RRSE RAE R R2
Training 9.242286 3.040113 2.400503 0.19611 0.442844 0.429236 0 896753 0 804167
Testing 7.319179 2.705398 1.941344 0.176835 0.420518 0.367689 0 909323 0 826869
1
0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0
0 0 0
w" w" w"
S MS S
RM M
R
Fig. 8. Histogram of model regression analyses:
1 - training data set; 2 - testing data set
As can be seen from Table 3 and Fig. 8, the lowest errors were observed for the proposed model, especially for the training database. However, the statistical errors calculated for the testing database were also close to the training database. Error analysis, shown in Figs 6, 7 and Table 3, proves that the proposed model, in terms of accuracy and efficiency of prediction, works within an acceptable range.
However, for low experimental values of Tad up to 5 MPa, the average error between the predicted and experimental data was significant compared with other ranges. According to the observed correlations and determination, the reliability of the proposed models can be considered satisfactory and higher for values of xad more than 5 MPa. The correlation coefficients for the database of training and testing were 0.9 and 0.89, respectively. In addition, for testing databases, correlation coefficients of 0.89 were achieved for the first and 0.9 for the latter stages. Although the database for testing was not used for training, a high level of prediction was obtained for both the training data sets and for testing ones, which was caused by a low average absolute error rate and high correlation coefficients.
The statistical analysis based on the MAPE and MSE values also showed that the proposed wording of the testing data set had relatively lower errors than the training set. Observing the general trend of efficiency evaluation, it turned out that the GA-based model can be considered as the preferred prediction model.
Therefore, soft computing methods can be handy tools for processing empirical data and determining the final adhesion strength of specimens susceptible to corrosion. All predicted values were valid and comparable with the observed ones. Therefore, we can conclude that both proposed models can be considered as useful systems with satisfactory prediction ability.
This study focuses on predicting the adhesive strength between concrete and reinforcing rod, subject to varying degrees of corrosion. From the point of view of practice, the use of models on structural elements is quite relevant. For example, the ultimate strength of adhesion is necessary to calculate the length of the anchoring [5]. In this case, the rod perceives the force acting upon pulling out if the value of ld is sufficient:
ld=—d , (13)
' 4lad
where R is design yield strength of reinforcing rod, Tadis ultimate adhesive strength and d is a rod diameter. In the future, the considered value of R and the estimated value of xad, the proposed empirical models, as well as the depth of embedding of structural elements can be calculated in more detail.
Conclusion
Adhesion is the ability of concrete to resist slipping of reinforcing rods under load in concrete products. "Reinforced concrete" is considered as a single composite material until such time as there is adhesion along the contact surface between the reinforcing rod and concrete. In operation, the metal reinforcing rod is subject to structural degradation of the surface (corrosion). Therefore, special attention should be paid to the process of breaking the interaction of reinforcing rod with concrete due to corrosion. For evaluations of this interaction, a soft computing method based on the genetic algorithm programming was used.
The application of the GA model is convenient for the user, since it receives data without normalization. However, if predicting accuracy is more important with new parameters and in a different range of specified parameters, the use of the existing models becomes incorrect. GA computing on the basis new empirical data will result in retraining the model or correcting the existing one.
References
1. Semchenko A.S., et al. Osobennosti scepleniya s betonom sterzhnevoj armatury razlichnyh profilej [Adhesion of concrete and reinforcing rods of various profiles]. BST
EHkspertiza. 2008, Issue 8, pp. 58-62.
2. Trost H. Verbundfestigkeit von Spannglieren und ihre Bedentung für Rissbildung und Rissbreiten beschränkung. Dt. Aussch. Für Stahlbeton. 1980. H.310, pp. 1-151
3. Tevelev Yu.A., Dmitriev A.N. Zhelezobetonnye konstrukcii v vodohozyajstvennom stroitel'stve: proektirovanie i izgotovlenie. M-vo sel'skogo hoz-va Rossijskoj Federacii. Moskva: Rosinformagrotekh [Reinforced concrete structures in water construction: design and manufacture. Ministry of the Russian Federation. Moscow: Rosinformagrotech], 2006, Moscow: Rosinformagrotekh. p. 21.
4. Kashevarova G.G., Martirosyan A.S., Travush V.I. Raschetno-ehksperimental'noe issledovanie processa razrusheniya svyazej scepleniya pri vdavlivanii sterzhnya
zhestkoj armatury v beton [Computational and experimental study of the process of destruction of adhesion bonds when pressing hard reinforcing rods into concrete]. Vestnik Permskogo nacional'nogo issledovatel'skogo politekhniches-kogo universiteta. Mekhanika. 2016, issue 3, pp. 62-75.
5. Benin A.V. Deformirovanie i razrushenie zhelezo-betona: analiticheskie, chislennye i ehksperimental'nye issledovaniya: monogr. [Deformation and destruction of reinforced concrete: analytical, numerical and experimental studies: monograph]. PGUPS, SPb, 2006, 27 p.
6. Migunov V.N. , Ovchinnikov I.I., Ovchinnikov I.G. Ehksperimental'no-teoreticheskoe modelirovanie armiro-vannyh konstrukcij v usloviyah korrozii. Monograf. [Experimental and theoretical modeling of reinforced structures in corrosion conditions. Monograph]. Penza, PGUAS, 2014, 362 p.
7. Alekseev S.N., Ivanov F.M., Modry S., SHissl' P. Dolgovechnost' zhelezobetona v agressivnyh sredah [The durability of reinforced concrete in aggressive environments]. Sovm. Izd SSSR - CHSSR - FRG. Moscow: Strojizdat, 1990, 320 p.
8. Frajfel'd S.E., Pal'chinskij O.V. Prakticheskij metod rascheta zhelezobetonnyh konstrukcij s uchetom reologicheskih svojstv materialov [A practical method for calculating reinforced concrete structures with the rheological properties of materials]. Stroitel'nye konstrukcii: Sb. tr. YUzhNII. issue 3, Har'kov, 1959, pp. 17-22.
9. Bajkov V.N., Sigalov EH.E. Betonnye i zhelezobetonnye konstrukcii: Ob-shchij kurs [Concrete and reinforced concrete structures: General course]. Moscow: Strojizdat, 1991,768 p.
10. Garai T. Issledovanie ankerovki armatury v betone. Issledovanie prochnosti ehlementov zhelezobetonnyh konstrukcij [The study of reinforcement anchoring in concrete. Study of the strength of elements of reinforced concrete structures]. Collection of papers of NIIZHB. Moscow: Gos- strojizdat, 1959, Issue5, pp.78-109.
11. Holmyanskij M.M. Kontakt armatury s betonom [Contact of reinforcement with concrete]. Moscow: Strojizdat, 1981, 184 p.
12. Kol'ner V.M., Aliev SH.A., Gol'dfajn B.S. Sceplenie s betonom i proch-nost' zadelki sterzhnevoj armatury periodicheskogo profilya [Adhesion to concrete and the sealing strength of the reinforcing rod of periodic profile]. Beton i zhelezobeton. 1965, issue 11, pp. 25-27.
13. Tevelev Yu.A. Zadelka armatury v betone pri peremennom sceplenii po dline zony ankerovki [Seal reinforcement in concrete with variable adhesion along the length of the anchoring zone]. Sceplenie armatury s betonom. M.,1971, pp. 14-21.
14. Almussallam A.A., Al-Gahtani A.S., Aziz A.A., Rasheeduzzafar Effect of reinforcement corrosion on bond strength. Constr Build Mater 1996, vol. 10, issue 2, pp. 123-129
15. Auyeung Y., Balaguru P., Chung L. Bond behaviour of corroded reinforcement bars. ACI Mater J. 2000, vol. 97, issue 2, pp. 214-220
16. Shima H. Local bond stress-slip relationship of corroded steel bars embedded in concrete. In: Proceeding of
the third international symposium on bond in concrete. Materials and Structures, Budapest, Nov 2002, pp. 153-158.
17. Zhao Y., Jin W. Test study on bond behavior of corroded steel bars and concrete. J Zhejiang Univ (Engineering Science Edition). 2002, vol. 36, issue 4, pp. 352-356 (in Chinese).
18. Fang C., Lundgren K., Chen L., Zhu C. Corrosion influence on bond in reinforced concrete. Cem Concr Res., 2004, vol. 34, issue 11, pp. 2159-2167.
19. Horrigmoe G., Sœther I., Antonsen R., Arntsen B. (2007) Laboratory investigations of steel bar corrosion in concrete. Background document SB3.10. Sustainable bridges: assessment for future traffic demands and longer lives. A project co-funded by the European Commission within the Sixth Framework Programme 2007.
20. Chung L., Kim Jh.J, Yi ST. Bond strength prediction for reinforced concrete members with highly corroded reinforcing bars. Cem Concr Compos., 2008, vol. 30, issue 7, pp. 603-611.
21. Yalciner H., Eren O., Serhan S. An experimental study on the bond strength between reinforcement bars and concrete as a function of concrete cover, strength and corrosion level. Cem Concr Res., 2012, vol. 42, issue 5, pp. 643-655.
22. Migunov V.N., Ovchinnikov I.G. Dlitel'nye ehksperimental'nye issledovaniya vliyaniya prodol'nyh treshchin v zashchitnom sloe betona na izmenenie dolgovechnosti, kratkovremennoj zhyostkosti i prochnosti vnecentrenno szhatyh s malym ehkscentrisitetom stroitel'nyh obychnyh zhelezobetonnyh ehlementov [Long-term experimental studies of the effect of longitudinal cracks in the protective layer of concrete on the change in durability, short-term stiffness and strength of non-centrally compressed reinforced concrete elements with low eccentricity]. Izv. vuzov. Stroitel'stvo, 2010, issue 2, pp. 125-130.
23. Mulin N.M. Ehksperimental'nye dannye o sceplenii armatury s betonom [Experimental data on the adhesion of reinforcement to concrete]. Beton i zhelezobeton. 1968, issue 12.
24. Oatul A.A. Predlozheniya k postroeniyu teorii scepleniya armatury s betonom [Proposals to build a theory of reinforcement adhesion to concrete]. Beton i zhelezobeton. 1968, issue 12, pp. 8-10.
25. Cekhmistrov V.M. Raschet napryazhenij i deformacij pri vydergivanii armatury iz betonnoj prizmy, opertoj torcom (obrazec na vydergivanie) [Calculation of stresses and deformations when pulling out the armature
from the concrete prism supported by the end face (the holding sample)]. Issledovanie po betonu i zhelezobetonu. Collection of papers CHPI - Chelyabinsk, 1967, issue 46, pp. 27-43.
26. Oatul A. A., Ivashenko Yu.A. Ehksperimental'noe issledovanie scepleniya armatury s betonom na rastyanutyh obrazcah pri kratkovremennom, povtornom i dlitel'nom dejstvii nagruzki [Experimental study of the adhesion of reinforcement to concrete on stretched specimens with a short-term, repeated and long-term load]. Issledovaniya po betonu i zhelezobetonu. Collection of papers CHPI -Chelyabinsk, 1967, issue 46, pp. 44-71.
27. Ovchinnikova I.G. Napryazhenno deformirovannoe sostoyanie betona v oblasti silovogo vozdejstviya s armaturoj [Stress-strain state of concrete in the field of force action with reinforcement]. In Sceplenie armaturoj s betonom. CHelyabinsk. CPNTO Strojindustriya, MISI, CHPI, NIIZHB, VNIIZHelezobeton, 1971.
28. Gvozdev A.A., Shubin AB., ZHumagulov E.Sh. Ob uchete nakopleniya povrezhdenij struktury betona pri vychislenii deformacii polzuchesti, vklyuchaya psevdoplasticheskie [On concrete structure damage accumulation in the calculation of creep deformation, including pseudoplastic one]. Voprosy tekhnologii i konstruirovaniya zhelezobetona. Moscow: NIIZHB, 1981, pp. 32-39.
29. Karpenko N.I., Sudakov G.N. O zadache scepleniya armaturnogo sterzhnya s cilindricheskim obrazcom [On the problem of adhesion of a reinforcing rod with a cylindrical sample]. Sceplenie armatury s betonom. M., NIIZHB, 1971, pp. 22-30.
30. Madatyan S.A., Tuleev T.D., Surikov I.N. i dr. Vliyanie geometricheskih razmerov periodicheskogo profilya sterzhnevoj armatury na ee mekhanicheskie svojstva [Influence of the geometric dimensions of the periodic profile of reinforcing rod on its mechanical properties]. Izv. VUZov. Stroitel'stvo i arhitektura. 1991, pp. 132-136.
31. Koza J.R. Genetic programming: On the programming of computers by means of natural selection. 1991, MIT Press, Cambridge.
32. Ferreira C. Gene expression programming: a new adaptive algorithm for solving problems. Complex Syst., 2001, vol. 13, issue 2, pp. 87-129.
33. Ngo D., Scordelis A. Finite element analysis or reinforced concrete beams. Journal of the ACL 1967, issue 8, pp. 152-163.
