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Строительная механика инженерных конструкций и сооружений Structural Mechanics of Engineering Constructions and Buildings
2024. 20(5). 391-403 ISSN 1815-5235 (Print), 2587-8700 (Online)
http://journals.rudn.ru/structural-mechanics
АНАЛИТИЧЕСКИЕ И ЧИСЛЕННЫЕ МЕТОДЫ РАСЧЕТА КОНСТРУКЦИЙ ANALYTICAL AND NUMERICAL METHODS OF ANALYSIS OF STRUCTURES
DOI: 10.22363/1815-5235-2024-20-5-391-403 UDC 624.012 EDN: ZRBRFL
Research article / Научная статья
Strength Model for Concrete in Near-Reinforcement Region
Vladimir I. Kolchunov12 , Natalia V. Fedorova1 , Tatiana A. Iliushchenko3 и
1 Moscow State University of Civil Engineering (National Research University), Moscow, Russia
2 Research Institute of Construction Physics of the Russian Academy of Architecture and Construction Sciences, Moscow, Russia
3 Kursk State University, Kursk, Russia И [email protected]
Received: June 12, 2024 Accepted: October 1, 2024
Abstract. The relevant problem of concrete strength in the near-reinforcement zone is solved as a problem of volumetric stress-strain state with the "closure" of output integral parameters of this zone on the framework of the whole reinforced concrete element, synthesizing hypotheses and dependencies of various disciplines of solid mechanics, including fracture mechanics. The model of reinforced concrete element takes into account Vl.I. Kolchunov's effect of reinforced concrete, which describes the mechanism of formation and development of transverse and longitudinal cracks. In this respect, generalized hypotheses of linear and shear strains for warping and gradients of relative mutual displacements of reinforcement and concrete are adopted. New functionals of reinforced concrete are constructed, which are consistent with the physical interpretations of the strength of cross-sections of bar elements in near-reinforcement zones. Constitutive equations for the concrete matrix, which models zones between transverse cracks, are written. The displacement components for the near-reinforcement zone in relation to the crack opening width at the "concrete-reinforcement" contact interface in transverse, longitudinal and radial cracks, respectively, are found. The use of the adopted assumptions and a multi-level calculation approach for the near-reinforcement region brings the model significantly closer to a real evaluation of the physical phenomena.
Keywords: volumetric stress state, near-reinforcement zone, displacement, cylindrical coordinates, effect of reinforced concrete, linear and shear strains, generalized hypothesis
Conflicts of interest. The authors declare that there is no conflict of interest.
Vladimir I. Kolchunov, Corresponding Member of the Russian Academy of Architecture and Construction Sciences, Doctor of Technical Sciences, Professor, Professor of the Department of Engineering Graphics and Computer Modeling, Moscow State University of Civil Engineering (NIU MGSU), Moscow, Russia; eLIBRARY SPIN-code: 3990-0345, ORCID: 0000-0001-5075-1134; e-mail: [email protected]
Natalia V. Fedorova, Advisor of the Russian Academy of Architecture and Construction Sciences, Doctor of Technical Sciences, Head of the Department of Industrial and Civil Engineering, Moscow State University of Civil Engineering (NIU MGSU), Moscow, Russia; eLIBRARY SPIN-code: 3365-8320, ORCID: 0000-0002-5392-9150; e-mail: [email protected]
Tatiana A. Iliushchenko, Candidate of Technical Science, Senior Lecturer of the Department of Industrial and Civil Engineering Construction, Kursk State University (KSU), Kursk, Russia, eLIBRARY SPIN-code: 6913-5863, ORCID: 0000-0001-6885-588X; e-mail: [email protected]
© Kolchunov Vl.I.. Fedorova N.V.. Iliushchenko T.A.. 2024
ifä ^ ^ i This work is licensed under a Creative Commons Attribution 4.0 International License https://creativecommons.Org/licenses/by-nc/4.0/legalcode
Authors' contribution. Kolchunov Vl.I. — scientific supervision; conceptualization, methodology, data gathering and processing, writing of the source text, scientific editing of the text. Fedorova N.V. — participation in the development of the material; revision of the text; scientific editing of the text. Iliushchenko T.A. — participation in the development of the material, processing and editing the text.
For citation: Kolchunov Vl.I., Fedorova N.V., Iliushchenko T.A. Strength model for concrete in near-reinforcement region.
Structural Mechanics of Engineering Constructions and Buildings. 2024;20(5):391-403. http://doi.org/10.22363/1815-5235-2024-20-5-391-403
Расчетная модель сопротивления железобетона в околоарматурной области Вл.И. Колчунов1'2 , Н.В. Федорова1 , Т.А. Ильющенко3 и
1 Национальный исследовательский Московский государственный строительный университет, Москва, Россия
2 Научно-исследовательский институт строительной физики РААСН, Москва, Россия
3 Курский государственный университет, Курск, Россия И [email protected]
Поступила в редакцию: 12 июня 2024 г. Принята к публикации: 1 октября 2024 г.
Аннотация. Решена актуальная задача сопротивления околоарматурной зоны бетона как задача объемного напряженно-деформированного состояния с «замыканием» выходных интегральных параметров этой зоны на стержневую схему всего железобетонного элемента, синтезирующую в себе гипотезы и зависимости механики железобетона и механики разрушения. В расчетной модели железобетонного элемента учтен эффект железобетона проф. Вл.И. Колчунова описывающий механизм образования и развития поперечных и продольных трещин. При этом приняты обобщенные гипотезы линейных и угловых деформаций для депланаций и градиентов относительных взаимных смещений арматуры и бетона. Построены новые функционалы железобетона, которые согласуются с физическими представлениями о сопротивлении поперечных сечений стержневых элементов в околоарматурных зонах. Записаны физические уравнения для бетонной матрицы, моделирующей зоны между поперечными трещинами. Найдены составляющие перемещений для околоарматурной области применительно к ширине раскрытия трещин на границе контакта «бетон-арматура» в поперечных, продольных и радиальных трещинах соответственно. Использование принятых предпосылок и многоуровневой расчетной схемы для околоарматурной области заметно приближает расчетную модель к реальной оценке физических явлений.
Ключевые слова: объемное напряженное состояние, околоарматурная зона, перемещение, цилиндрические координаты, эффект железобетона, линейные и угловые деформации, обобщенная гипотеза
Заявление о конфликте интересов. Авторы заявляют об отсутствии конфликта интересов.
Вклад авторов. Колчунов Вл.И. — научное руководство; концепция исследования; развитие методологии; написание исходного текста; итоговые выводы. Федорова Н.В. — участие в разработке материала; доработка текста; итоговые выводы. Ильющенко Т.А. — участие в разработке материала, обработка и редактирование материала.
Для цитирования: Колчунов Вл.И., Федорова Н.В., Ильющенко Т.А. Расчетная модель сопротивления железобетона в околоарматурной области // Строительная механика инженерных конструкций и сооружений. 2024. Т. 20. № 5. С. 391-403. http://doi.org/10.22363/1815-5235-2024-20-5-391-403
Колчунов Владимир Иванович, член-корреспондент РААСН, доктор технических наук, профессор кафедры инженерной графики и компьютерного моделирования, Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ), Москва, Россия; главный научный сотрудник, Научно-исследовательский институт строительной физики, Российская академия архитектуры и строительных наук, Москва, Россия; eLIBRARY SPIN-код: 3990-0345, ORCID: 0000-0001-5075-1134; e-mail: [email protected]
Федорова Наталия Витальевна, советник РААСН, доктор технических наук, профессор, заведующий кафедрой промышленного и гражданского строительства, Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ), Москва, Россия; eLIBRARY SPIN-код: 3365-8320, ORCID: 0000-0002-5392-9150; e-mail: [email protected]
Ильющенко Татьяна Александровна, кандидат технических наук, старший преподаватель кафедры промышленного и гражданского строительства, Курский государственный университет (КГУ), Курск, Россия; eLIBRARY SPIN-код: 6913-5863, ORCID: 0000-0001-6885-588X; e-mail: [email protected]
1. Introduction
Mechanics of reinforced concrete is fundamental for ensuring mechanical safety of buildings and structures in the conditions of new challenges of man-made, natural and terrorist nature. One of the key and extremely controversial problems of the modern theory of reinforced concrete is the problem of crack opening. For solving this problem, in the last two or three decades, considerable amount of information on the mechanics of deformation and cracking in reinforced concrete has been accumulated worldwide [1-9], including such regulatory documents introduced into the design practice as ACI Committee 318-14, EN 1992-1-2: 2004, SP 5.03.01-2020, SP 63.13330.20181, etc. A lot of models, associated with a large number of theoretical and experimental investigations, have been developed over this period, among which the studies of Russian [10-14] and foreign [15-21] scientists can be mentioned. In the last two decades, in the framework of such concept and on a common methodological basis, studies on this problem are conducted also under Vl.I. Kolchunov's supervision [22-27]. In this regard, this paper discusses modelling of the bond between reinforcement and concrete, taking into account physical nonlinearity and the presence of transverse cracks, and using a two-level approach: at the first level, the entire reinforced concrete element is analyzed as a bar, while at the second level, the volumetric stress-strain state of the near-reinforcement zone is considered using a number of parameters obtained from the first-level model.
2. Methodology
Combined action of concrete and reinforcement in a reinforced concrete element is ensured by the near-reinforcement zone. This is a local zone of concrete directly adjacent to the lateral surface of the reinforcing bar and ending (according to the Saint-Venant principle) at some radial distance tb (Figire 1). Therefore, the problem will be solved using cylindrical coordinates. The positive directions of these coordinates are given in Figure 1.
Diagram of section I-I between cracks e and section II-II in the stress diagram;
sections I *-I* between cracks in the average relative deformation diagram;
1 — physical neutral axis of reinforced concrete,
2 — average neutral axis of reinforced concrete
Figure 1. Regarding the analysis of experimental and numerical studies in solving the problem of determining the stress-strain state of the near-reinforcement zone: a — characteristic sections; b — in the zone of adhesion of concrete to reinforcement S o u r c e: made by Vl.I. Kolchunov
1 See: ACI Committee 318-14. Building Code Requirements for Structural Concrete and Commentary. Farmington Hills, Mich: American Concrete Institute, 2014; 519 p.; EN 1992-1-2: 2004. Eurocode 2: Design of concrete structures — Part 1-1: General rules and rules for buildings, 2004. 225 p. SP 5.03.01-2020. Concrete and Reinforced Concrete Structures. Minsk; 2020. 236 p.; (In Russ.) SP 63.13330.2018. Concrete and Reinforced Concrete Structures. General Provisions. Moscow: Minstroy; 2018.152 p. (In Russ.)
Theoretical solution to the problem under consideration was preceded by a number of experimental and numerical studies [10-27]. As a result of the experimental studies (using continuous chains of strain gauges for measuring deformations), a qualitative picture of concrete strain along the Z-axis has been obtained (Figure 1).
The results of such studies allowed not only to find the qualitative nature of the strain distribution in concrete, reinforcement and their mutual displacements in the near-reinforcement zone along the Z-axis, but also to obtain a theoretical solution for determining a number of strength parameters of a reinforced concrete bar element, taking into account physical nonlinearity and the presence of transverse cracks.
Thus, the first aspect of the following solution of the volumetric stress-strain state in the near reinforcement zone is that a multilevel analysis approach is used here: at the first level, the entire reinforced concrete element is analysed as a bar; at the second level, the volumetric stress-strain state of the near reinforcement zone is considered using a number of parameters obtained from the first level model. In turn, the output integral parameters of the second-level model influence the parameters of the first-level model at the next iteration stage. Such an approach is possible, for example, when using parameter traditional for reinforced concrete, which allows to update the average value of the parameters of the first-level model without changing the model.
The second important aspect of the proposed solution is that after the formation of cracks (both transverse and longitudinal), fracture mechanics hypotheses are involved in the analysis.
3. Results and Discussion
The solution will be derived taking into account the physical nonlinearity and particular nature of concrete, being different resistance to tension and compression. Relationship diagrams oti—eti, ^(A) — eti, ott—ro are used in the analysis.
The projection of the oti — eti diagram onto axes Trz — jr allows to establish the relationship between shear stresses and shear strains or, when local shearing is introduced, between the relative mutual displacements of reinforcement and concrete eq. It is important to emphasize that, since the stresses in the concrete of the near-reinforcement zone adjacent to the crack are compressive in the cross-section of a reinforced concrete element (see Figure 1), the shear stresses are determined by the upper branch of the strain diagram. This will have a significant effect on the maximum value of shear stresses. Thus, the zones of maximum shear bond stresses are concentrated in proximity to transverse cracks. Therefore, on one hand, the greater the number of transverse cracks crossing the reinforcing bar, the better the bond between the reinforcing bar and concrete in the near-reinforcement zones (before yielding of the reinforcement or formation of radial cracks). On the other hand, as the load increases, the adhesion of concrete and reinforcement in the region between cracks (at zi= 0,5/crc) first increases, and then, due to the specificity of deformation of the concrete matrix (Figures 1, 2), begins to decrease.
Firstly, this is associated with the fact that in these regions, the stresses in the concrete of the near-reinforcement zone are tensile in the cross-section of the reinforced concrete element. This affects both the current value of the shear stress and its possible maximum.
Secondly, as transverse cracks appear, the area and magnitude of tensile stresses between cracks decrease. In view of the above, special attention is required for the highlighted sections (see Figures 1, 2) located at distances z = t and z = zi. An additional assumption concerning the near-reinforcement zones adjacent to sections I-I and II-II, where it is necessary to determine the stress-strain state of concrete, is taken. The strain diagrams of concrete in tension in section I-I and concrete in compression in the near-reinforcement zone in section II-II are assumed to be linear. This assumption is consistent with the ideas about the resistance of cross-sections of bar elements and is confirmed by numerous experiments. Stresses ott and o'tt (see Figure 1) in a reinforced concrete bar element are determined from the conditions of equilibrium of moments in sections I-I and II-II with respect to the point of application of the compressed concrete resultant force. The values of strain et(z) in sections I-I and II-II are determined taking into
account the strain incompatibility of concrete and reinforcement. They are denoted as sz,i and Sz,ii respectively. Using the Obi—Sbi diagram for section I—I, stress Oz,i(r) is found. Given that the strains and stresses along the z-axis are considered, the diagrams are projected onto this axis. Since it is known that the curvilinear segments of the diagram are described by a square parabola and the coordinates of the end points of these parabolas (ez,i and Obt, 8z,ii and Ob) are specified, then the angle between the considered axes is needed only to determine the initial modulus of the Ob,z—£b,z diagram:
Eb, z = Ф^
b
(1)
where 9 is the coefficient accounting for combined stress state and the presence of shear stresses along axis z2.
In this case, the value of the shear bond stress and normal stress Ob,z is determined at a load of 0.2 of the cracking moment Mcrc. After substituting the expression for determining the strain in section I-I into the expression for determining the stress in the same section, the following equation is obtained:
о z J (r ) = kfi2 J D2
1
. tx.
+Ф4 ez ,1D
f \ 1 -JL
V
t1
(2)
obt - фEb e z z
where k1 =-2-- ; D = -ez I +as ; t1 = h0 - xt - a*.
'z ,i
Figure 2. Diagram of the stress state: a — in the near-reinforcement zone, b, c — deformation of the concrete matrix in sections I-I and II-II respectively: 1, 2 — stress-strain state before and after crack formation, respectively; 3, 4 — direction of microcracks and macrocracks, respectively; 5 — deformation of concrete S o u r c e: made by Vl.I. Kolchunov
2 Veryuzhsky Yu.V., Kolchunov V.I. Methods of reinforced concrete mechanics: textbook. Kyiv: NAU Book Publ.; 2005. (In Russ.)
a
c
The fundamental equations for the near-reinforcement zone, with respect to the case when only transverse cracks are present and longitudinal and radial cracks have not been developed yet, are now considered.
doe
Taking into account that the problem is axisymmetric, the values of —- and T0z are equal to zero
d0,
'-Br-
and the equations of equilibrium in cylindrical coordinates take the form
dz„z do z t zr z + -^ + _ 0;
dr dz r do, — O0
rz I r
-+—^+ dr dz
= 0.
(3)
(4)
The Couchy geometrical relations will also simplify, as displacement components u and strains jrQ, jez are also equal to zero due to symmetry. The rest of the strain components have the form
= du = u =dw =du dw
Sr = ; S0 = ; Sz = ~> ; Yrz = ^ + .
dr r dz dz dr
(5)
The constitutive equations for the concrete matrix, which is modelled as an elastoplastic isotropic body between the transverse cracks, are expressed as
er=Ejxj r- ^z+;
(6)
Y =
i rz
w
(7)
where E(A), Z(A) and ^(A) are respectively the elastoplastic deformation moduli and the Poisson's ratio for concrete.
It should be emphasized that for expression (2) in the second parentheses, the differentiation with respect to z is first performed and then it is evaluated at z = z1.
Then for the near-reinforcement zone between transverse cracks, equation (3), considering expression (2), will have the form
dT
_rz
dr
+ Ali D2
'1 - rv
+4,i D
(
1 —
v -1J
V '1J
+-Z. = 0. r
(8)
The following notation can be introduced:
TT £2 + 2k1£ z % dz dz
= Au ; qEb
de z dz
= A2,i. ; t1 = h0 — xt — a*.
z=z1
The solution to differential equation (8) is adopted in the form of a product of two functions of r:
Trz = u (r )u(r ). (9)
396 Analytical and numerical methods of analysis of structures
t
rz
The following is obtained after differentiation of both sides of equality (9) and their substitution into equation (8):
u
f cv v J du
dr r
+ v—_- A1I D2 dr '
f J2 fi - ^' V t1J
A d
fi - üJ
V t1J
(10)
Function u is selected such that the expression in the parentheses of the left-hand side of equation (10) is equal to zero. Then, the following is obtain after separation of variables in this equation:
Эи _ dr v r
(11)
and after integration: u = r-1.
By substituting the determined value of u(r) into equation (10) and performing algebraic transformations and integration, the following is obtained:
u
(r)_ A
f D2r2 2D2r3 D2r4 J
2 3t 4t2
V 1 1 J
+ A.
f Dr2 Dr3 J
-2,1
3t
+C,
(12)
1 J
After substitution of v(r) and u(r) into equation (9):
Tz _ r3A3,I + r2A4,I + rA5,I + —
(13)
Here
_ A11 _ 2D2 A21D _ Ал _-4p ^ _-~3T+-33t~; _ Au
f D2 J
v " 2 J
4-D J
(14)
Integration constant Ci is determined from the condition that Trz = Tri at r = a*, Here, Tr,i is the shear
stress in section I-I, which is known from the analysis of a reinforced concrete bar element assuming strain incompatibility of concrete and reinforcement3. Then
C1 _ A1,I
a.
2
(a* -1)
D2a
D2 a2 D 2a3 J
3t,
+ A.
a
2
2,I
1 J
(r -1)
Da„
Da,
2
~ a* A31 a.3 A41 a* A51. (15)
l1 J
In the near-reinforcement zone adjacent to the transverse crack (section II-II), equation (13) will have a similar form. In this case
dk2 2 „ dsz s 2 + k22s
z 2 z
dz dz
_ A1,n; феъ
ds z
dz
_ A2,II.
(16)
Parameters A3,ii -A5,ii and C2 differ in the fact that in relationships (14) and (15) tb needs to be substituted instead of t1.
3 Veryuzhsky Yu.V., Kolchunov V.I. Methods of reinforced concrete mechanics: textbook. Kyiv: NAU Book Publ.;
2005. (In Russ.)
Then, having stress components Trz calculated, the second differential equilibrium equation (4) can be considered. It is easier to solve the problem in terms of stresses. From the equation for determining the strains in section I-I it follows that
= - -8E))-• (17)
Substituting expressions (17) and (13) into equation (4) results in a differential equation, which after algebraic transformations will take the following form:
^ + r3 B + r2 B2 + rBio + -Bn + ^ = Bi2. (18)
dr r r
Here the values of parameters B1- B7 are determined according to the work of Y.V. Veryuzhsky, V.I. Kolchunov3.
The solution of differential equation (18) is also adopted in the form of a product of two functions, which after a number of similar transformations is reduced to the following expression:
r4 B1 r3 B2 r2 B10 B11 rB12 C2
o =--1--1--10--11 + —— + —r. (19)
r 4 5 4 2 3 r2
Integration constant C2 is determined from the condition that or = 0 at r = b*:
bB1 b* B2 b* B10 b*3 B12 b*2 B1 4 3
C _ + u*u2 + U*±J1Q — U*±J12 + (20)
Here, parameter b* is determined from the condition that at r = b* local stresses Trz in the zone adjacent to the reinforcement practically decay, i.e. their values approach zero. Then it follows from equation (13) that
b*3 A3I + b*2 A4I + b* A5I + C = 0. (21)
b*
Relationship (21) can be used to determine parameter b*. Taking into account that the rate of change of this function is quite substantial, even small changes of b* lead to significant changes of stress Trz. Numerical studies show that at b* = 3...4a, the values of Trz can be considered as approaching zero. Moreover, at b* > 4a, the outer radius of the near-reinforcement zone can be considered infinitely large (with an error of less than 6%). In this case, the solution is no longer related to the shape of the outer contour. Thus, formulas (13), (17), (19) characterize the stress distribution for the near-reinforcement zone with any shape of the outer contour of the cross-section of a reinforced concrete element.
Knowing stress components oz, or, oe, Trz, the strain components are determined using formulas (6), (7). Then, the displacement components are found from the Cauchy relationship (5):
u = J erdr+f (z); (22)
w = J s zdz+f (r); (23) Trz du dw
= gW = aU(24)
Here sr, sz are determined according to equation (6), and Trz is determined according to equation (13). Substituting the values of oz, oe and or from relationships (2), (17) and (19) into equation (6) and
performing algebraic transformations taking into account (22) and (23), the values of displacements u, w are obtained. After substituting the latter and the stresses from equation (19) into equation (24), considering the integration of the equations, the following is obtained:
u = D1 + C3 z + C5; (25)
w = D
f r V . r 6 r 5 r 4
v ti J
((e zidz )|z—z,I+n^+П2-5+Пз —+
(r-a)4 r3 r2 1 + ц (X) +П4—+ П5у + ln r-C4r + C6. (26)
Here, the values of n1 -n8, С -C6, D1, E1, H1 are determined according to the work of Y.V. Veryuzhsky, V.I. Kolchunov4.
In case of the near-reinforcement zone adjacent to the transverse crack (section II-II), equations (13)-(26) will have similar form. In this respect, equation (16) is used, and values ez,n, A1,ii, tb, Tz,ii, £q,n are substituted in these equations instead of ez,i, Ai,i, ti, Tz,i, eq,i respectively.
In all of the above formulas, deformation modulus E(A,) and coefficient |(A,) are determined from the Obi—£bi and | diagrams. The strains are determined by the formula
ebi = (£ z -6 r)2-(£ r- e-)2- e z)2+2Y 2 • (27)
Stress obi is determined as a function of concrete strain ebi. Then
E (X ) = . (28)
ebi
A more complex relationship is used for |(X):
• at ebi < ecrc.c, the value of |(X) is equal to 0.2;
• at ecrc.c < ebi < ecrc.v, the value of |(X) is calculated according to formula
ц(X) = 0,2 + 0,3 4l 6crc,c , (29)
e — e
crc,v crc,c
at ecrc.v < Ebi < Ev the value of p.(A) is equal to 0.5;
at Ev < Ebi < Ebu the value of p.(A) is calculated according to formula
R (X ) = 0,5 -(0,5 - r b). (30)
£bu e v
Here, concrete parameters Ebi, Ebu are taken according to the tables, parameters Ev, p.b are determined according to diagrams Obi — Ebi.
Considering that the analytical solution for the near-reinforcement zone has been obtained, projecting the Obi — Ebi diagram onto any axes, for example, onto the Oz — Ez axes, does not cause difficulties.
4 Veryuzhsky Yu.V., Kolchunov V.I. Methods of reinforced concrete mechanics: textbook. Kyiv: NAU Book Publ.; 2005. (In Russ.)
Longitudinal cracks (Figure 3, a) cancel the adhesion resistance at Au < ro«, where ro« is determined from the relationships of fracture mechanics. The longitudinal crack opening width is calculated by the formula
accc =AU = «b , (31)
where ub and us are the radial displacements of concrete and reinforcement respectively at r = a.
In this case, it is possible to take into account the aspects associated with continuity violation at the concrete — reinforcement interface through the boundary conditions.
Figure 3. Analysis of longitudinal cracks: a — in the vicinity of section I—I; b — fracture pattern as a result of radial cracks in the vicinity of sections II—II
S o u r c e: made by Vl.I. Kolchunov
Longitudinal cracks along the contact surface of concrete and reinforcement are the most important in the sense of ensuring adhesion of these materials. Displacement ub is determined by formula (25) at r = a. Displacement us is determined by formula
|0£rA (32)
Here
(z) (33)
s« ^. (33)
Es
More precise values of £r,s and u. can be obtained by using the aspects of the proposed solution for the near-reinforcement zone with respect to the reinforcing bar. However, numerical analysis shows that the influence of stress components Ors and Oe. on the value of Er. is less than 3%. Therefore, it is logical to neglect this influence when solving the problem under consideration in order to simplify the calculations.
Then from (5), the following is obtained after integration:
u. = . (34)
Es
The integration constant is determined by satisfying the condition that u. = 0 at r = 0. Then C = 0.
The value of us is calculated at r = a according to formula (34).
Longitudinal cracks appear, generally, in the near-reinforcement zones between transverse cracks due to the difference in radial displacements of concrete and reinforcement at the point of their contact (see Fig. 3, a). The longitudinal crack profile is close to a triangle with its maximum opening between the transverse cracks (section I-I) and zero opening at a distance of 2t* from the transverse cracks. In this case, it is assumed that there is no adhesion between concrete and reinforcement in the region of the main longitudinal crack at Au > . However, if the stiffness and strength of the structure is ensured by the bond
between concrete and reinforcement in other regions adjacent to transverse cracks (where Au < ), then,
taking into account that the presence of longitudinal cracks at the reinforcement surface does not damage the concrete cover, the operation of the structure can be continued.
The proposed analysis approach allows not only to detect transverse cracks, but also longitudinal and radial cracks. Considering that the loading scheme from Oe does not cause crack retardation at their apex, the emergence of radial cracks effectively cancels adhesion resistance in this zone, i.e. results in its failure at acrcr > qh . The cover layer is destroyed, the reinforcement is bare, so further operation of the structure
should be prohibited, even if its resistance due to other regions is not exhausted and it meets the criteria set by the standards for strength and stiffness. Failure from radial cracks (Figure 3, b) is characteristic for the near-reinforcement zones adjacent to the transverse crack. Here (depending on the design features), fracture by concrete crushing at the near-reinforcement zone is also possible. In this case, the value of Ei is calculated by formula (27), where the strain components are calculated at z = zn and r = a.
Note that the proposed solution in terms of stresses has an advantage over a similar solution in terms of displacements. The latter, even within the adopted assumptions, leads to a non-homogeneous second-order differential equation with a large number of particular solutions, which clearly complicates the calculation. An attempt to abandon the use of a multi-level calculation approach complicates the solution to the problem of the volumetric stress-strain state in the zone under consideration so much that it becomes analytically indeterminable. As a result, the solution is possible only by variational methods. Comparing the considered approach with variational methods of solution, which allow to obtain an approximate solution of differential equations with sufficient accuracy for engineering calculations, it can be noted that the proposed solution is first of all simpler. Nevertheless, with respect to the considered problem for reinforced concrete bar elements, this solution is on par in accuracy with the variational methods, when a sufficiently large number of independent functions and terms of the corresponding series are specified in the latter. At the same time, even with such a refined approach, the known methods do not allow to take into account the aspects of continuity violation of the concrete matrix when cracks emerge in it. Hypotheses and methods of fracture mechanics have not yet been properly applied here. The same can be said for the consideration of the strain incompatibility of concrete and reinforcement.
Thus, the differentiated approach to the analysis of the near-reinforcement zone allows to introduce more reasonable criteria for the operation of reinforced concrete structures taking into account not only transverse, but also longitudinal and radial cracks. With that, considering the specific aspects of concrete strength in the near-reinforcement zone (including the strain incompatibility of concrete and reinforcement) brings the analysis substantially closer to the real assessment of the physical phenomena occurring here. At the same time, the proposed methodology preserves the relative simplicity of the calculation and its physical essence, and, consequently, its engineering observability.
4. Conclusion
1. The relevant problem of concrete strength in the near-reinforcement zone has been solved as a problem of the volumetric stress-strain state with "closure" of the output integral parameters of this zone on the framework of the whole reinforced concrete element. In spite of the complexity of such a problem, taking into account the strain compatibility of concrete and reinforcement and the violation of concrete continuity, the solution of the differential equations written in this case is obtained in a closed analytical form.
2. An important aspect of the proposed solution of the problem is the hypotheses of fracture mechanics and the deformation effect of reinforced concrete established by Vl.I. Kolchunov, as well as the consideration of both transverse and longitudinal cracks. At the same time, generalized longitudinal and shear strain hypotheses for cross-section warping, jumps from relative mutual displacements of reinforcement and concrete Sq, are developed.
3. Physical equations are written for the concrete matrix modeled as an elastoplastic isotropic body between transverse cracks. The solution is obtained considering the physical aspect of concrete, being its different resistance to tension and compression.
4. The developed differentiated approach to the strength analysis of the near-reinforcement region allows, with respect to the reinforcing bar and concrete matrix, to introduce more reasonable criteria for the operation of reinforced concrete structures, taking into account not only transverse, but also longitudinal and radial cracks. Here, stresses oe do not cause crack retardation at their apex, and the emergence of
radial cracks effectively cancels the cohesion resistance in this zone and leads to its fracture at acrc r > Qu. The
cover layer is destroyed, the reinforcement is bare, so further operation of the structure should be prohibited, even if its resistance due to other regions is not exhausted and it meets the criteria set by the standards for strength, stiffness and crack resistance for the characteristic failure due to radial cracks.
5. Comparing the proposed method with the variational methods of solution, which allow to obtain only an approximate solution of differential equations, when there is a sufficiently large number of independent functions and terms of the corresponding series, the proposed solution using a multilevel calculation approach does not complicate the volumetric stress-strain state in the considered zone. In such a differentiated approach, hypotheses and methods of fracture mechanics allow to introduce more reasonable criteria for the operation of reinforced concrete structures taking into account not only transverse, but also longitudinal and radial cracks. With that, considering the noted aspects of concrete resistance in the near-reinforcement zone brings the analysis significantly closer to the real assessment of the physical phenomena occurring here. At the same time, the proposed methodology retains the relative simplicity of calculation and engineering observability.
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