Strength Model for Concrete in Near-Reinforcement Region

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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 63.13330.2018[1], 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 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 ψs, 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 σbi - εbi, μ(λ) - εbi, σbt - ω are used in the analysis. The projection of the σbi - εbi diagram onto axes τrz - γr 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 εq. 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 z1= 0,5lcrc) 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 nearreinforcement 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 = z1. 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 nearreinforcement 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 σbt and σ′bt (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 εb(z) in sections I-I and II-II are determined taking into account the strain incompatibility of concrete and reinforcement. They are denoted as εz,I and εz,II respectively. Using the σbi - εbi diagram for section I-I, stress σz,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 (εz,I and σbt, εz,II and σb) are specified, then the angle between the considered axes is needed only to determine the initial modulus of the σb,z - εb,z diagram: Eb z, =φEb, (1) where φ 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 σb,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: 2 ( )= 2 2 - r + ε - r σz,I r k1εz,ID 1 φEb z,ID 1 , (2) t1 t1 where k1 =σbt -φ2Ebεz z, ; D =-εz,I +Vεq; t1 = - -h0 xt a*. εz,I a c 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 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. ∂σθ Taking into account that the problem is axisymmetric, the values of and τ0z are equal to zero ∂θ, τθr and the equations of equilibrium in cylindrical coordinates take the form ∂τrz +∂σz +τzr = 0; (3) ∂r ∂z r ∂σr +∂τrz +σr -σθ = 0. (4) ∂r ∂z r The Couchy geometrical relations will also simplify, as displacement components υ and strains γrθ, γθz are also equal to zero due to symmetry. The rest of the strain components have the form εr =∂u; εθ =u ; εz =∂w; γ =rz ∂u+∂w. (5) ∂r r ∂z ∂z ∂r The constitutive equations for the concrete matrix, which is modelled as an elastoplastic isotropic body between the transverse cracks, are expressed as ε =r 1( ) σ μ λ σ σr - ( )( z + θ) ; (6) E λ γrz = , (7) where E(λ), ζ(λ) and μ(λ) 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 2 ∂τrz + A D1,I 2 1- r + A D2,I 1- +r τrz = 0. ∂r t1 t1 r The following notation can be introduced: ∂k1 ε +2 k1εz ∂εz = A1,I ; ϕEb ∂ε∂ z= A2,I.; t1 = - -h0 xt a*. ∂ z 2 ∂ (8) z z z z= 1 z z z= 1 The solution to differential equation (8) is adopted in the form of a product of two functions of r: τ =rz u r( ) ( )υ r . (9) The following is obtained after differentiation of both sides of equality (9) and their substitution into equation (8): ∂υ+υ +υ =-∂u A D2 2 u ∂r r ∂r 1,I 1- tr1 - A D2,I 1- tr1 . (10) Function υ 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: ∂υ=-∂r , (11) υ r and after integration: υ=r-1. By substituting the determined value of υ( )r into equation (10) and performing algebraic transformations and integration, the following is obtained: 2 2 2 3 2 4 2 3 u r( ) = A1,I - + - 2 + A2,I - + +C1. 2 3t1 4t1 2 3t1 After substitution of υ( )r and u r( ) into equation (9): (12) C τ =rz r A3 3,I +r A2 4,I +rA5,I + 1 . r Here (13) A = - A1,I ; A = - 2D2 + A D2,I ; A = A D2 D 3,I 2 4,I 5,I 1,I - + A2,I - . (14) D r 2D r D r Dr Dr 4t1 3t1 3t1 2 2 Integration constant С1 is determined from the condition that τrz = τr,I at r a= *, Here, τr,I is the shear stress in section І-І, which is known from the analysis of a reinforced concrete bar element assuming strain incompatibility of concrete and reinforcement3. Then = A a*2 C1 1,I ( - ) D a2 * - D a2 *2 + D a2 2*3 + A2,I ( a-*21) Da* - Dat1 *2 -a A*4 3,I -a A*3 4,I -a A*2 5,I. (15) a* 1 t1 3t1 r In the near-reinforcement zone adjacent to the transverse crack (section ІІ-ІІ), equation (13) will have a similar form. In this case ∂k2 ε2 + k2 2εz ∂εz II = A1,II; φEb ∂∂εzzz z= II = A2,II. (16) ∂z z ∂z z z= Parameters А3,ІІ -А5,ІІ and С2 differ in the fact that in relationships (14) and (15) tb needs to be substituted instead of t1. 3 Then, having stress components τrz 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 σθ = (σz -εzE( )λ )-σr. (17) Substituting expressions (17) and (13) into equation (4) results in a differential equation, which after algebraic transformations will take the following form: ∂σr +r B r B3 1 + 2 2 +rB10 + 1 B11 + 2σr =B12. (18) ∂r 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: σr =- 1 - 2 - 10 - 11 + 12 + 22 . 4 5 4 2 3 r Integration constant С2 is determined from the condition that σr = 0 at r = b*: (19) b B b B6 5 b B4 b B3 b B2 C2 = * 1 + * 2 + * 10 - * 12 + * 11 . (20) r B4 r B3 r B2 B rB C 6 5 4 3 2 Here, parameter b* is determined from the condition that at r = b* local stresses τrz in the zone adjacent to the reinforcement practically decay, i.e. their values approach zero. Then it follows from equation (13) that b A*3 3,I +b A*2 4,I +bA* 5,I + =C1 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 τrz. Numerical studies show that at b* = 3…4a, the values of τrz 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 σz, σr, σθ, τrz, the strain components are determined using formulas (6), (7). Then, the displacement components are found from the Cauchy relationship (5): u = εrdr+ f z1 ( ); (22) w= εzdz+ f2 ( )r ; (23) γrz = τrz = ∂u + ∂w. (24) G( )λ ∂z ∂r Here εr, εz are determined according to equation (6), and τrz is determined according to equation (13). Substituting the values of σz, σθ and σr 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 +C z3 +C5 ; r ( z,I )1 r6 2 r5 3 r4 w = D 1- ε dz z - z,I+ η + η + η + t1 6 5 4 (25) +η4 (r -a)4 +η5 r3 +η6 r2 -2C1 1+μ λ( ) lnr -C r4 +C6. (26) 4 3 2 Ε( )λ Here, the values of η1 -η8, С1-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 εz,II, A1,II, tb, τz,II, εq,II are substituted in these equations instead of εz,I, A1,I, t1, τz,I, εq,I respectively. In all of the above formulas, deformation modulus E(λ) and coefficient μ(λ) are determined from the σbi - εbi and μ diagrams. The strains are determined by the formula εbi = 2 (εz -εr )2 - -(εr εe )2 + -(εe εz )2 + 3 γrz2 . (27) Stress σbi is determined as a function of concrete strain εbi. Then E( )λ = σbi . εbi A more complex relationship is used for μ(λ): • at εbi < εcrc.c, the value of μ(λ) is equal to 0.2; • at εcrc.c < εbi < εcrc.v, the value of μ(λ) is calculated according to formula (28) μ λ( )= 0,2+0,3 εbi -εcrc,c , εcrc,v -εcrc,c • at εcrc.v < εbi < εv the value of μ(λ) is equal to 0.5; • at εv < εbi < εbu the value of μ(λ) is calculated according to formula (29) μ λ( )= 0,5-(0,5-μb ) εbi -εv . (30) 2 1 +μ λ( ) 2 εbu -εv Here, concrete parameters εbi, εbu are taken according to the tables, parameters εv, μb are determined according to diagrams σbi - εbi. Considering that the analytical solution for the near-reinforcement zone has been obtained, projecting the σbi - εbi diagram onto any axes, for example, onto the σz - εz axes, does not cause difficulties. 4 Longitudinal cracks (Figure 3, a) cancel the adhesion resistance at Δu ≤ ωu, where ωu is determined from the relationships of fracture mechanics. The longitudinal crack opening width is calculated by the formula acrc,c =Δ = -u u ub s, (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. а b Figure 3. Analysis of longitudinal cracks: а - 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 us =aεrsdr. (32) 0 Here ε ≈μ σsi ( )z . (33) rs s Es More precise values of εr,s and us 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 σrs and σθs on the value of εrs 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: us =μ σs si ( )z . (34) Es The integration constant is determined by satisfying the condition that us = 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 Δ ≥u ωu . 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 Δ <u ωu ), 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 σθ 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 acrc,r ≥ωu. 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 εi is calculated by formula (27), where the strain components are calculated at z z= ІІ 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 hypo- theses for cross-section warping, jumps from relative mutual displacements of reinforcement and concrete εq, 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 σθ 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 ≥ωu. 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 nearreinforcement 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.

About the authors

Vladimir I. Kolchunov

Moscow State University of Civil Engineering (National Research University); Research Institute of Construction Physics of the Russian Academy of Architecture and Construction Sciences

Email: vlik52@mail.ru
ORCID iD: 0000-0001-5075-1134
SPIN-code: 3990-0345

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, Russia

Natalia V. Fedorova

Moscow State University of Civil Engineering (National Research University)

Email: fedorovanv@mfmgsu.ru
ORCID iD: 0000-0002-5392-9150
SPIN-code: 3365-8320

Advisor of the Russian Academy of Architecture and Construction Sciences, Doctor of Technical Sciences, Head of the Department of Industrial and Civil Engineering

Moscow, Russia

Tatiana A. Iliushchenko

Kursk State University

Author for correspondence.
Email: tatkhalina93@yandex.ru
ORCID iD: 0000-0001-6885-588X
SPIN-code: 6913-5863

Candidate of Technical Science, Senior Lecturer of the Department of Industrial and Civil Engineering Construction

Kursk, Russia

References