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International Journal for Computational Civil and Structural Engineering, 19(4) 147-154(2023)
DOI:10.22337/2587-9618-2023-19-4-147-154
DETERMINATION OF CONCRETE RHEOLOGICAL PARAMETERS USING NONLINEAR OPTIMIZATION METHODS
Anton S. Chepurnenko, Stepan V. Litvinov, Batyr M. Yazyev
Don State Technical University, Rostov-on-Don, RUSSIA
Abstract. The article proposes a method for processing concrete creep curves based on the nonlinear equation of V.M. Bondarenko. The experimental data ofA.V. Yashin is used. The problem offinding rheological parameters and the nonlinearity function is posed as a nonlinear optimization problem. The objective function represents the sum of the squared deviations of the experimental values of the creep strain from the theoretical values for all creep curves for one concrete at different stress levels. The minimum of the objective function is found using the interior point method, the surrogate optimization method, the pattern search method, the genetic algorithm, and the particle swarm method. It has been established that the first of these methods has the greatest efficiency. The proposed approach provides high quality approximation of experimental curves at all stress levels. It is shown that for concrete the nonlinearity of creep deformations is more pronounced than the nonlinearity of instantaneous deformations, and the same function cannot be used to describe these two types ofnonlinearity.
Keywords: concrete, creep, viscoelasticity, viscoplasticity, nonlinearity
ОПРЕДЕЛЕНИЕ РЕОЛОГИЧЕСКИХ ПАРАМЕТРОВ БЕТОНА ПРИ ПОМОЩИ МЕТОДОВ НЕЛИНЕЙНОЙ ОПТИМИЗАЦИИ
А.С. Чепурненко, С.В. Литвинов, Б.М. Языев
Донской государственный технический университет, г. Ростов-на-Дону, РОССИЯ
Аннотация: В статье предлагается методика обработки кривых ползучести бетона на основе нелинейного уравнения В.М. Бондаренко. Используются экспериментальные данные А.В. Яшина. Задача поиска реологических параметров и функции нелинейности ставится как задача нелинейной оптимизации. Целевая функция представляет сумму квадратов отклонений экспериментальных значений деформации ползучести от теоретических по всем кривым ползучести для одного бетона при различных уровнях напряжений. Минимум целевой функции отыскивается при помощи метода внутренней точки, метода суррогатной оптимизации, метода шаблонного поиска, генетического алгоритма и метода роя частиц. Установлено, что наибольшей эффективностью обладает первый из указанных методов. Предлагаемый подход обеспечивает высокое качество аппроксимации экспериментальных кривых при всех уровнях напряжений. Показано, что для бетона нелинейность деформаций ползучести более выраженная, чем нелинейность мгновенных деформаций, и для описания этих двух видов нелинейности нельзя использовать одну и ту же функцию.
Ключевые слова:бетон,ползучесть, вязкоупругость, вязкопластичность, нелинейность
INTRODUCTION
There are a large number of different theories of nonlinear concrete creep in the literature, which differ in the set of hypotheses used and approaches to their construction [1-10]. One of the first variants of the nonlinear theory of concrete creep, which generalized the linear Harutyunyan-Maslov equation, was the theory ofN.Kh. Harutyunyan and P.I. Vasiliev [11-13]:
"(t)
E (t) J
i
i:
E (r)
*(r)dr-
/
I
dC (t ,t)
(1)
dr
F [ct(-t)] dr,
where C(t,r) is the measure of creep, F is the nonlinearity function, r0 the
initial moment of time at which the load is applied.
Equation (1) takes into account the nonlinear component of creep deformations, but does not take into account the instantaneous nonlinearity of deformation.
Another version of the Harutyunyan-Maslov equation generalization, which allows taking into account both nonlinear creep and instantaneous deformation nonlinearity, was proposed by Yu. N. Rabotnov [14]:
The instantaneous nonlinearity of concrete deformation is well studied, and the definition of the function f does not cause great difficulties, which cannot be said about the function f2.
The purpose of this work is to develop a methodology for finding the function f2, as
well as other rheological parameters that determine the measure of creep, from concrete creep curves at various stress levels.
/ W' 4a
dC (' ,T)
ôt
E(') J dr
1
E M.
"(r)dr-
(2)
where f (<?) is the non-linear deformation function.
Close in essence to equation (2) is the viscoelastic-plastic model of hereditary aging of concrete, proposed by A.G. Tamrazyan [15]:
s(') =
_ f K')] _ f 8C('
E (' )
J"
ÔT
-f[a(r)]dr.
(3)
METHODS
Let us consider the procedure for processing creep curves using the example of the curves presented in the work of A.V. Yashin [17]. In this paper, concrete with the compressive strength Rb =30 MPa was tested at the age of
r0 = 28 days, with an initial modulus of
elasticity E0 = 4 -104 MPa at stress levels a I Rb
from 0.4 to 0.8 with a step of0.1.
Thus, the total number of experimental curves
for one class of concrete was 5.
As a measure of creep, the expression proposed
by A.G. Tamrazyan was used [15]:
Equations (2) and (3) postulate the same nature of nonlinearity for instantaneous and long-term deformations, which does not correspond to the real rheological behavior of concrete. More general is the equation of V.M. Bondarenko [16], having the form:
-i
!
Í
E (' )
dC (',T)
d_
8T
'jJ E M
f [s(r)]dT-
(4)
dr
f2 [S M]
where s (t} = G(t) IR (t) , R (t) is the instant concrete strength, fx and f2 are nonlinear
functions corresponding to instantaneous and long-term deformation.
a' az
C (' ,T) = c^ + B (e---e ).
(5)
This expression contains 4 rheological parameters C, B, a, y to be determined. The advantage of the creep measure (5) is that its exponential form allows to pass from the integral creep law to the differential one and to calculate creep strains using the Euler or Runge-Kutta method. This transition is shown in[18]. Since the experimental data of A.V. Yashin do not contain information about the aging of concrete (change in time of its modulus of elasticity), then we will neglect this effect, i.e. we omit in equation (4) the term containing
_8_
ÔT
1
E M,
Then the dependence of the creep strain on time at a constant stress level s = a / R takes the form:
Table 1. Rheologicalparameters ranges
£cr (t ) - /2(s)
f at ax,
ce -e
at
e -
— + B (e^ - e^ )
.(5)
s . (t-) = k-G-
cr ,i\J / i i
c
at- ax,
e 1 - e
ate 1 -
I-+B(
e ^ - e~rt]
(6)
where <ji is the stress in i -th experiment.
Next, the objective function Z calculates the sum of the squared deviations of the theoretical values of creep strains from the experimental values, multiplied by 1000, over all experimental curves:
5 n
Z = -hhheor (tj ),htest (tj )]-1000)2.(7)
i=1 j=1
parameter a, 1/day Y, 1/day c, MPa1 B, MPa"1
min 10"4 10-4 10"5 10"5
max 0.1 0.1 10-4 10"4
We pose the problem of determining the parameters C, B, a, y, as well as the function /2(s), as a problem of nonlinear optimization.
The objective function Z is built, the input parameters of which are C, B, a, y, as well as 5
coefficients k according to the number of
experimental curves, taking into account the nonlinear creep component depending on the stress level.
For the time points tj at which the strains were
measured, the objective function calculates the theoretical values of the creep strains using the formula:
The solution to the problem of creep curves processing was implemented in the MATLAB environment using the Optimization Toolbox and Global Optimization Toolbox packages. The following optimization methods were used to find the minimum of the objective function:
1. Interior point method [19]
2. Surrogate optimization method [20]
3. Pattern search method [21]
4. Genetic algorithm [22]
5. Particle swarm method [23-24].
The first of these methods allows to find a local minimum in the vicinity of the starting point of the search, the remaining methods are searching for a global minimum.
At the starting point of the search, the coefficients k were taken equal to 1. The values
recommended in [15] were taken as the initial values for C, B, a, y. They are given in Table. 2.
Table 2. Values o/rheologicalparameters at the startingpoint ofthe search
a, 1/day y, 1/day C, MPa"1 B, MPa"1
0.032 0.062 3.77-10"5 5.68-10"5
Note that the quality of the A.V. Yashin's experimental curves approximation with the indicated in the Table 2 values is unsatisfactory. They are probably defined for concrete of a different class or composition.
The objective function Z must reach a minimum.
The coefficients kt must be greater than or equal
to 1. We accept the range from 1 to 20 for them. The ranges for the parameters C, B, a, y are presented in Table 1.
RESULTS AND DISCUSSION
Table 3 shows the values of the objective function obtained as a result of solving the problem of nonlinear optimization by the five methods indicated above.
Table 3. Minimal objective/unction values when using various optimization methods
It can be seen from Table 3 that the most effective of the methods used was the interior point method. The values of the constants C, B, a, y, as well as the correction factors kr.. k5, obtained by the internal point method, are given in Table 4.
Table 4. Rheologicalparameters ofconcrete obtained by the internalpoint method
Fig. 1 shows the theoretical creep curves constructed using formula (6) and the parameters presented in Table. 4. Experimental points are marked with round markers.
1 1
о • -(j/I^ = 0.4
-сг/1^ = 0.5
-<t/R. " 0.6
f 5 5 ° -Kh = 0.7 -ff/R. =0.8
^-лт- и и -- t
\iVе
p Л-" в-e-в в "
0 100 200 300 400 600 600 700 800 900 t, days
Figure 1. Comparison of theoretical creep curves with experimental results
The reliability of the approximation R2 for each stress level is given in Table 5.
Anton S. Chepurnenko, Stepan V. Litvinov, Batyr M. Yazyev Table 5. Reliability of experimental curves
approximation at different stress levels
a 0.4 0.5 0.6 0.7 0.8
R2 0.918 0.990 0.994 0.982 0.991
Fig. 2 shows the graph of the stress function /2 (s), built on the basis of the calculated values of the coefficients k1... k5. For comparison, the graph of the function /1(s) is also shown, which
corresponds to the Sargin formula [25] used in the Eurocodes to describe the instantaneous nonlinearity of concrete deformation:
/ (s ) = ^ (k-(k - 2 )• s -
2_ (8)
-^[(k-2)• s-k]2 -4• s),
where Eo is the concrete initial modulus of elasticity (at c = 0), sR is the deformation at the
top of the diagram a-e, k = E0sR /Rb is the
coefficient characterizing the curvature of the diagram o — s.
Thedeformation sR duringplottingthe /1(s) graph was determined by the empirical formula [26]:
f ^.5
sR = 0.058 I^M . (9)
1QO 90 80 70 60 50
Ц
40 30 20 10 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
■s = a./R, b
Figure 2. Comparison ofthefunctions f (s) and f2(s)
Method Z
Interior point 0.183
Surrogate optimization 2.26
Pattern search 2.25
Genetic algorithm 2.06
Particle swarm 1.42
a, 1/day y, 1/day С, MPa1 В, MPa"1
0.0014 0.0061 1.02-10"5 1.78-10"5
h h2 кз к4 к5
1.03 1.51 2.57 3.61 3.74
sR = 0.058
f \ 0.5
Rl
V Eo
It can be seen from Fig. 2 that for concrete, the nonlinearity of creep deformations is much more pronounced than the instantaneous nonlinearity of deformation, and these two types of nonlinearity cannot be described by one function, as is done in equations (2) and (3). The dependence for /2 (s) obtained by us is well approximated by the polynomial /2(s) = s • (a ■ s2 + b ■ s + c) at
a = 203.6 MPa, b = -51.25 MPa, c = 30.19MPa. This approximation is shown in Fig. 2 by dashed line. Reliability of approximation is R2 = 0.986.
CONCLUSIONS
A technique for processing concrete creep curves based on the nonlinear theory of V.M. Bondarenko using nonlinear optimization methods is proposed. To solve the problem of nonlinear optimization, the interior point method, the surrogate optimization method, the pattern search method, the genetic algorithm, and the particle swarm method were applied. It has been established that the most effective method for the considered problem is the interior point method. On the basis of experimental concrete creep curves obtained by A.V. Yashin, its rheological parameters are determined. The use of the interior point method made it possible to achieve a high quality of approximation of the experimental curves. It is shown that the nonlinearity of creep strains as a function of stress is more pronounced than the instantaneous nonlinearity of deformation, and these two types of nonlinearity cannot be described by a single function. On the basis of experimental data, the nonlinearity function for creep deformations was selected.
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СПИСОК ЛИТЕРАТУРЫ
1. Zhang С., Zhu Z., Zhu S., He Z., Zhu D., Liu J., Meng S. Nonlinear creep damage constitutive model of concrete based on fractional calculus theory //Materials. -2019. - Vol. 12. - No. 9. - Article 1505.
2. Li Y. et al. Verification of concrete nonlinear creep mechanism based on meso-damage mechanics // Construction and Building Materials. - 2021. - Vol. 276. -Article 122205.
3. Han В., Xie H.B., Zhu L., Jiang, P. Nonlinear model for early age creep of concrete under compression strains
//Construction and Building Materials. -2017.-Vol. 147. - Pp. 203-211.
4. Sanjarovskiy R., Ter-Emmanuilyan Т., Manchenko M. Creep of concrete and its instant nonlinear deformation in the calculation of structures //CONCREEP 10. - 2015. - Pp. 238-247.
5. Sanjarovsky R., Manchenko M. Creep of concrete and its instantaneous nonlinearity of deformation in the structural calculations //Scientific Israel-Technological Advantages. - 2015. - Vol. 17. - No. 1-2. -Pp. 180-187.
6. Yu P., Li R.Q., Bie D.P., Yao X.M., Liu X. C., Duan Y.H. A coupled creep and damage model of concrete considering rate effect // Journal of Building Engineering. -2022. - Vol. 45. - Pp. 103621.
7. Su L., Wang Y.F., Mei S.Q., Li P.F. Experimental investigation on the fundamental behavior of concrete creep //Construction and Building Materials. -2017. - Vol. 152. - Pp. 250-258.
8. Bu P., Li Y., Wen L., Wang J., Zhang X. Creep damage coupling model of concrete based on the statistical damage theory //Journal of Building Engineering. - 2023. -Vol. 63. - Article 105437.
9. Mirzaahmedov A.T. et al. Accounting For Non-Linear Work Of Reinforced Concrete In The Algorithms Of Calculation And Design Of Structures //The American Journal of Engineering and Technology. -2020. - Vol. 2. - No. 11.- Pp. 54-66.
10. Wang Q., Le J.L., Ren X. Numerical modeling of delayed damage and failure of concrete structures under sustained loading //Engineering Structures. - 2022. - Vol. 252. - Article 113568.
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19. Byrd R.H., Hribar M.E., Nocedal J. An
interior point algorithm for large-scale nonlinear programming // SIAM Journal on Optimization. - 1999. - Vol. 9. - No. 4. -Pp. 877-900.
20. Gutmann H.-M. A radial basis function method for global optimization // Journal of Global Optimization. - 2001. - No. 19. Pp. 201-227.
21. Kolda T.G., Lewis R. M., Torczon V. A
generating set direct search augmented Lagrangian algorithm for optimization with a combination of general and linear constraints // Technical Report SAND2006-5315. SandiaNational Laboratories, 2006.
22. Mirjalili, S. Genetic algorithm // Evolutionary algorithms and neural networks. - Springer, Cham, 2019. - Pp. 43-55.
23. Mezura-Montes E., Coello Coello C.A. Constraint-handling in nature-inspired
numerical optimization: Past, present and future. // Swarm and Evolutionary Computation, 2011.-Pp. 173-194.
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гражданских инженеров. - 2020. - №. 5. -С 108-116. 26. Ishchenko A.V., Pogodin D.A. Calculation of reinforced concrete arches on stability when creeping // IOP Conference Series: Materials Science and Engineering. - IOP Publishing, 2019. -T. 698. -№. 2. -C. 022086.
Chepurnenko Anton Sergeevich, DSc, Professor of the Department of Strength of Materials, Don State Technical University, 344000, Russia, Rostov-on-Don, Gagarin sq., 1, tel. +7 (863) 201-91-36, e-mail: anton_chepurnenk@mail.ru
Litvinov Stepan Victorovich, PhD, Head of the Department of Strength of Materials, Don State Technical University, 344000, Russia, Rostov-on-Don, Gagarin sq., 1, tel. +7 (863) 201-91-02, e-mail: litvstep@yandex.ru
Yazyev Batyr Meretovich, DSc, Professor of the Department of Strength of Materials, Don State Technical University, 344000, Russia, Rostov-on-Don, Gagarin sq., 1, tel. +7 (863) 201-91-36, e-mail: ps62@yandex.ru
Чепурненко Антон Сергеевич, доктор технических наук, профессор кафедры «Сопротивление материалов» Донского государственного
технического университета, 344000, г. Ростов-на-Дону, пл. Гагарина, 1, тел. +7 (863) 201-91-36 , e-mail: anton_chepurnenk@mail.ru
Литвинов Степан Викторович,кшдидал технических наук, заведующий кафедрой «Сопротивление
материалов» Донского государственного технического
-1, тел. +7 (863) 201-91-02 , e-mail: litvstep@yandex.ru
Языев Батыр Меретович, доктор технических наук, профессор кафедры «Сопротивление материалов»
Донского государственного технического университета, -201-91-36, e-mail: ps62@yandex.ru
