Relaxation of stress in elements of reinforced concrete structures

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Introduction In a global sense, the relaxation phenomenon represents the process of thermodynamic equilibrium establishing in a system consisting of a large number of particles. In structural mechanics, relaxation is understood as a reduction in stresses when the initial deformation is fixed by the bonds. From the point of view of physical chemistry, stress reduction occurs due to intermolecular displacements and reorientation of the intramolecular structure, and therefore the similarity of relaxation and creep phenomena is manifested. In this paper on the basis of the accepted concept of the strength structure of materials the modification of known in the linear creep theory L. Boltzmann’s superposition principle [1] is obtained allowing its applicability under nonlinear dependence of deformations on design stresses. Relaxation problems are associated with the phenomenon of creep - an increase in deformation -generated by stress when . Stresses decrease over time with constant deformation and this phenomenon is called stress relaxation. The phenomenon of relaxation is a consequence of the development of creep deformation in the material , because with a constant total deformation due to an increase in deformation , the portion of instantaneous deformation decreases and the stress decreases. This means that creep and relaxation phenomena take place simultaneously. Continuous redistribution of stresses between concrete and reinforcement is essential for their current stress-strain state. Stress relaxation in the concrete component of the structural element entails an increase in the stress in the reinforcement and its significant excess over the calculated one can lead to breaking of the reinforcement and to serious consequences in critical structures (in reinforced concrete reactor shells, water ducts). The standard method for solving the considered problems is the preliminary determination of the relaxation kernel by a given creep kernel. However, the solution of the corresponding integral equation is expressed in a very slowly converging series [2]. The application of the Laplace transform and the Poincare small parameter method is busy and time-consuming [3]. The solution of relaxation problems is greatly simplified when the corresponding integral equations of state are reduced to their differential forms. In this paper, the linear integral equation of state of uniaxially loaded concrete is reduced to a simple linear differential equation of the first order. This takes into account the presence of a single aging function that determines the evolution of the elastic modulus and the creep measure. Remark 1. The key point of this method is the linearity of the integral equation of state. In the nonlinear statement, the transformation to a linear differential equation becomes possible after the derivation of the linear relative to structural stress (based on a modification of L. Boltzmann's principle) statement equation of concrete. Rheological equations of mechanical state Creep deformation under simple loading , (1) Take the measure of creep of concrete in the form (2) where - the aging function; - the creep strain accumulation function; . Based on experimental data [4], the following structure of the aging function was established in [5]: . (3) Denoting = и = , according to (2) and (3) we have ; , (4) where - the stress level; - the strength of concrete at the age of 28 days. Corresponding to the increment of the stress level at the moment a partial increment of creep deformations at the moment t is . (5) Since each increment depends only on the magnitude of the level and , then the increment (6) responds to creep deformation increment . (7) By passing in (7) to the limit and integrating in parts we have (8) The increment of the total deformation + and, adding to the deformation, caused by stress we obtain a linear rheological equation of concrete under uniaxial loading (9) When the parameters are time-invariant for ageless concrete (10) Remark 1. Boltzmann's superposition principle is formulated for ageless concrete (ideal) material. When taking into account aging, the superposition of partial creep deformations is realized by partial increments of the reduced stress = , corresponding to increments of the level of stresses. The creep deformation accumulation function is selected in the form ; and thus [6]: . (11) Remark 2. When in (11) we have a summand and it is called short-term creep. Taking measure (11) with , some authors [7] assume the inertia-free nature of the creep phenomenon, while other authors [8] (since the creep measure determines precisely delayed deformations), assuming the inertia in time of this phenomenon, take the creep measure (11) with . (12) According to (9) and (12) we have . (13) The measure (12) is taken in this paper. Since with measure (12) we have , then according to (13) . (14) The value of (15) in building codes it is taken as the general compliant of instantaneous and delayed deformations and under simple loading . (16) The imposition of deformations is carried out with general compliance (17) and when loading , we assume (18) According to (16) and (18) , . (19) Equality (19) means, that with compliance the measure of creep deformations decreases by an amount . In the works [7-9], assuming ; the rheological equation is derived by applying partial increments of creep deformation , (20) equivalent according to (19) to equation (21) Remark 3. Equations (20) and (21) describe the same mechanical state of concrete, which was not noticed by the authors of [10]. This led to the statement [10] that the second component in (21) is unnecessary and to an incorrect representation of the superposition principle “as a fundamental error in the theory of creep” [11]. The force increasing by a normal cross-section entails the destruction of a part of the fractions, reducing the cross-sectional area to , formed by whole fractions at the moment The value associated with structural damage (22) is called the structural, and the value of (23) is called the calculated normal voltage in the structural element. According to (22) and (23) , (24) where the function describes the process of destruction of fractions, accompanied by a redistribution of loading on the area , because only fractions that are entire at this moment exert force resistance. In contrast to the linear formulation, which means the equal strength of all fractions of the element, the stepwise increment of stress does not correspond to mutually independent increments of creep deformations - the action at the moment = is enhanced by the action of where [12]. Note also that the area is determined by all increments ; . This circumstance leads to the need to modify the principle of L. Boltzmann's superposition. Since the cause of the dependence between is the equal strength of the fractions, we will (mentally) select in the concrete component V of the structural element its part Vt, consisting of entire fractions in the segment [t0, t]. Just for this part the rheological equation describes the stress-strain state (SSS) taking into account the rheology at [t0, t]. Since the (SSS) of the part Vt at the current moment[t0, t] coincides with the (SSS) in the part Vτ, consisting of entire fractions at the moment , the stress in Vt is a structural stress . Strength balance of the fractions Vt to entails mutual independence of increments and stress generates deformation (25) The relation (25) allows (by repeating the above constructions) the derivation of the rheological equation of the mechanical state in a nonlinear formulation [13; 14] , (26) . (27) Remark 4. Relation (25) represents the principle of superposition of creep deformations for an aging material (in the concept of its strength structure) in a nonlinear formulation. In [9], assuming the interdependence of partial increments, the equation is derived . (28) In (28) and - nonlinear functions of instantaneous deformations and creep deformations. Remark 5. In the physical aspect, both types of deformations are generated by a single force factor - structural stress - and therefore [ and . According to the unified stress function is represented as. The nonlinear function in applications is given by the equation [15] , (29) where and m are empirical parameters and for concrete m = 4 is usually assumed. Equation (9) does not take into account the nonlinearity of the diagram observed in experiments and, as A.A. Gvozdev first noted, is not suitable for the theory of reinforced concrete. Assuming the dependence of the instantaneous deformation on the linear one , and the creep deformation as the nonlinear one, he considered the surge in the initial section as a consequence of the rapidly flowing creep. According to A.A. Gvozdev [16], creep is two-component and deformation consists of the so-called partially reversible deformation of the 2nd kind and irreversible deformation of the 1st kind generated by force damage. Equation of two-component creep theory under uniaxial loading [3] , (30) where is a nonlinear stress function; - a function of the total duration of the stress to the moment t. Denoting as a part of the area , corresponding to the fractions destroyed at the moment ; ; ; ; . (31) The representation (31) of the structural stress corresponds to the equations ; , (32) meaning that instantaneous and delayed deformations are composed of their linear and nonlinear parts. According to (30) and (32), the last summand on the right side of the equality is the sum and is represented in [3] as a creep deformation of the 1st kind. Remark 6. The incorrectness of considering the nonlinear part of instantaneous deformation as creep deformation is also noted in [10]. In [17], the creep equation of concrete is given on the basis of nonlinear Eurocode diagrams , (33) where and represent the direct and inverse function of a nonlinear diagram . According to (33), in the equation of state, along with the nonlinear dependence of instantaneous deformations on , creep is represented as linearly dependent on . Relaxation problems Stress relaxation in concrete under uniaxial loading To determine the design and structural stresses for a given deformation according to equations (9) and (26), we have the following integral equations (34) . (35) Both equations (9) and (26) and equations (34) and (35) have the same structure with the same parametersand in linear and nonlinear formulations. Within the framework of the concept we have adopted the nonlinearity is determined by the structure of the material and force loading, and parameters that do not depend on these factors - and are determined by physico-chemical processes. The scheme of definition and is the same. Consider equation (34). Because and ; - elastic deformation, then according to (9) with measure (12), we obtain the equality ; (36) that represent a linear integral equation with respect to elastic deformation . Equation (36) is reduced to the differential form [18]: multiply both parts (36) by . (37) We differentiate all summands (37) by t, taking into account the known equality (38) Now, multiplying all the summands by , we obtain a linear differential equation of the first order with respect to ; . (39) The function represents the general solution of a homogeneous equation We look for the general solution of equation (39) in the form and, substituting it into (39), we get ; ; и . So, the general solution of equation (39) is ; (40) Since according to (37) we have and , then . Thus, (41) (42) In the nonlinear formulation, equation (26) is reduced to the form (39) using the above transformations and the structural stress is determined as (43) According to the stress found by this formula the required calculated stress is determined by the solution of the algebraic equation . (44) When the nonlinear function is taken as (29) for m = 4, according to (44) we have the equation (45) and the largest of the real roots is taken as an estimate of the calculated stress. In [18], model cases of forced deformations and ) are considered; v is the constant rate of deformation. In work [19-20] linear integral equations (34) and (35) are solved by iteration method. Remark 1. In (17), equation (33) is reduced to a first-order differential equation. In this case, the function according to the nonlinear Eurocode diagrams is taken as , where functions are selected empirically. As a result, in contrast to equation (39), a rather complex nonlinear equation is obtained [17], and the question of its solution arises. Stress relaxation in bent reinforced concrete elements Let's consider stress relaxation in a single reinforced concrete beam bent by a moment . At a distance from the neutral axis Ox . (46) Here , and are the areas of normal sections of the reinforcement and the concrete component of the beam; , where and are the moments of inertia of the concrete part of the beam and the reduced normal section relative to the Ox axis; is stress in the concrete layer in contact with the reinforcement. At the level according to the condition of compatibility of deformations, we obtain the equation , (47) where - instant elastic stress; ; ; - modulus of elastic deformations of reinforcement; - modulus of elastic deformations of concrete. For a given elastic stress the integral equation (47) is solved by reducing to a differential equation of the form (39) relatively to or by simple iterations with zero approximation . At a constant bending moment we have the equation (48) According to the solution of equation (48) with an initial condition for sufficiently large the estimate of is . (49) As a result of a prolonged redistribution of stresses from concrete to reinforcement its initial stress increases to value . (50) Structural damage leads to an intensification of this process and makes a significant contribution to the assessment and . Remark 1. In [3], a combination of the small parameter method by introducing a multiplier into the last summand of equation (30) and the Laplace transform is used to solve relaxation problems. The structure of equation (30), together with the methods used, makes finding stresses and quite a difficult task. Conclusion In the long-term forecast of the safety of reinforced concrete structures and buildings, estimates of the maximum stress values in the reinforcement are essential, because its rupture can lead to serious consequences. These estimates are obtained by solving the problem of stress relaxation in the concrete component of the structural element, entailing an increase of stress in the reinforcement. Based on the concept of statistical strength distribution of fractions which union forms an element of a reinforced concrete structure, an approach is proposed for solving relaxation problems by reducing integral equations of state to a differential form. At the same time, according to the linearity of the integral equation of state with respect to the so-called structural stresses and the generality of the aging function for the modulus of elasticity and the creep measure of concrete, a simple linear differential equation of the first order is obtained. The approach proposed in this paper, based on the strength structure of constructive materials (concrete, steel, wood, plastic), is significantly simpler than the known methods of stress assessment in the components of reinforced concrete structural elements.

About the authors

Evgeny A. Larionov

Peoples' Friendship University of Russia (RUDN University)

Email: evgenylarionov39@yandex.ru
ORCID iD: 0000-0002-4906-5919

Doctor of Technical Sciences, Professor of the Department of Construction, Academy of Engineering

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Vitaly G. Nazarenko

Research Center “Construction”, JSC

Email: kilativnazarenko@yandex.ru
ORCID iD: 0000-0002-9035-9214

Doctor of Technical Sciences, Professor

6 2-ya Institutskaya St, Moscow, 109428, Russian Federation

Marina I. Rynkovskaya

Peoples' Friendship University of Russia (RUDN University)

Email: rynkovskaya-mi@rudn.ru
ORCID iD: 0000-0003-2206-2563

PhD, Docent, Director of the Department of Civil Engineering, Academy of Engineering

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Elena A. Grinko

Peoples' Friendship University of Russia (RUDN University)

Author for correspondence.
Email: grinko-ea@rudn.ru
ORCID iD: 0000-0002-0459-8359

Head of the Materials Resistance Laboratory, Department of Mechanical Engineering Technologies, Academy of Engineering

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

References