Analytical model of deformation of reinforced concrete columns based on fracture mechanics

Full Text

Introduction When performing calculations of reinforced concrete buildings and structures, it is quite important to apply non-linear methods of analysis which allow to ensure economical and reliable structures and to reveal reserves of bearing capacity of the system. A widespread type of structural system used in seismic areas is the reinforced concrete frame, the feature of which is the perception of the horizontal component of seismic load due to the rigid joint between beams and columns. In the nonlinear stage of frame frames, local areas of elastoplastic deformations occur in the vicinity of the girder support nodes. In accordance with this, in the design diagrams of frame frames, the nonlinear properties are concentrated in separate areas, which are called plastic joints, while the columns and spanning sections of the beams work elastically [1]. The appearance of plastic joints in columns in ordinary cases is considered to be unacceptable. However, columns designed for elastic operation may suffer some damage during operation, e.g. caused by fire [2], reinforcement corrosion [3], mechanical damage, earthquake, etc. In such a case, due to the reduced mechanical characteristics, the behavior of the column in the elastic-plastic domain will have to be taken into account when carrying out verification calculations or justifying the reinforcement. Description of nonlinear behavior of columns and beams in the plastic hinge region is usually carried out with the help of hysteresis diagrams (Figure 1), which take into account degradation of strength and stiffness at low-cycle vibrations, loss of dissipative energy, change of stiffness at opening and closing of cracks (pinching effect) [4]. Hysteresis diagrams are usually plotted in the axes “bending moment-curvature” or “horizontal force-horizontal displacement”. The basic element of a hysteresis diagram is the monotonic loading curve, commonly referred to as skeleton curves. The monotonic curve limits the range of possible deformation under low-cycle loading (Figure 1). The monotonic curves should, wherever possible, take into account the greatest ductility of reinforced concrete structures and include areas of hardening and softening to establish the true nature of the redistribution of forces in the system. Many hysteresis models of varying degrees of accuracy have been developed by individual researchers. Let's consider these models in terms of monotonic curves used. In [5] a bilinear elastic-plastic diagram is proposed which has a linear-elastic first section with an equivalent stiffness Ke, after reaching the bearing capacity the stiffness becomes zero - a yield point occurs. The determination of the value of carrying capacity and equivalent stiffness for a particular structure is a rather complex task, for reinforced concrete columns the method proposed in [6] can be used. Despite its simplicity, the bilinear diagram [5] is quite popular for seismic calculations as it has clear computational advantages. Figure 1. Hysteresis diagram based on the model of Takeda et al. [7] In the bending of reinforced concrete columns and beams, the redistribution of stresses will cause the individual cross-sectional areas to engage gradually due to the non-linear behaviour of the reinforcement and the concrete. In this way, the bearing capacity of the elements will be realized and the possibility of absorbing a larger moment will be realized. In the deformation diagram, this can be accounted for by introducing a non-zero stiffness after the limit force is reached. This approach is implemented in the bilinear diagram proposed in [8]. The stiffnesses are usually determined by approximating real curvilinear diagrams obtained from experiments or numerical analysis. The bilinear diagram with strengthening combines computational simplicity and a more accurate account of strengthening effects, which justifies the choice of this model as the basis by other authors [9; 10]. A characteristic feature of reinforced concrete structures is the formation of normal and inclined cracks, which reduce the initial stiffness. In order to account for cracking, a three-line diagram of deformation has been proposed in [7] with a successive decrease in stiffness after the cracking force is reached and then when the yield strength in the reinforcement is reached. This approach makes it possible not only to take into account crack opening at the initial stages of deformation of the element, but also to provide a basis for describing the process of reopening and closing of cracks in subsequent cycles. In addition to piecewise linear diagrams, some researchers use curvilinear diagrams in their models [11]. This allows a more accurate approximation of the real deformation diagram of the element and takes into account the consistent reduction of stiffness. In practice, this approach is less popular, which is justified by the complexity of the calculations and analysis of the results. It should also be noted that curvilinear diagrams have a rather narrow field of application, since the dependencies describing eccentrically compressed and bendable elements will be different. Columns and beams adjacent to the frame nodes where the highest bending moments occur, according to the standards of most countries, must be reinforced by densely placed, closed cross clamps, which in addition to providing the strength of the sloping sections act as indirect reinforcement. The indirect reinforcement increases the load-bearing capacity of the elements and the plastic deformation capacity, which contributes to the redistribution of forces in the system and a fuller use of the load-bearing capacity reserves. In addition, after the limit forces are reached, when the stresses in the clamps reach the yield strength, there is no sudden failure of the element, but a gradual reduction in the bearing capacity with increasing plastic deformations follows [12]. Strengthening can take place when such phenomena as loss of stability of compressed reinforcement, geometric nonlinearity, chipping of concrete cover layer are taken into account. The described processes can be taken into account in models which include a branch of unstrengthening [13]. The model allows to take into account the true nature of force redistribution in the framework more accurately and is particularly relevant when carrying out the analysis of bearing capacity reserves for elements with initial damage [14]. The unstrengthening branch allows taking into account an important aspect of low-cycle operation of reinforced concrete elements, such as within-cycle degradation of strength. This is especially important in loading programs with sharply varying amplitudes, which can lead to sudden collapse of the structure. The hysteresis diagrams were further developed in [15], which takes into account the presence of residual strength after unstrengthening, which is observed in tests of reinforced concrete structures under low-cycle loading. The presence of residual strength makes it possible to take into account the incomplete disconnection of an element from operation and an increase in the resilience of the structural system as a whole when performing calculations based on the criterion of no collapse. It should be noted that the description of the reference points of hysteresis diagrams can be made in differrent ways. The most common approach is the approximation of experimental diagrams or by using empirical dependencies, e.g. in [15]. This method can also be applied in combination with diagrams derived from numerical calculations [14]. This somewhat limits the scope of application of the model and does not allow direct consideration of processes related to concrete rebound, loss of reinforcement stability, etc. a b c Figure 2. Deformation diagrams of materials: a - reinforcement; b - unconfined concrete; c - confined concrete Another approach would be to plot the diagrams based on taking into account the dissipation energy on the oscillation cycle. In this case, the dissipation energy can have either a constant value [16] or it can decrease with time based on the experimental dependence [17]. This approach allows a more accurate account of the energy dissipated by the structure at each cycle, however, it has the same drawbacks as the first one. A more accurate way of setting the monotonic curve is the method based on the analysis of the stages of the stress-strain state (STS) of a reinforced concrete element. The procedure of the method is based on the identification of reference points in the diagram where the change in stiffness is observed. In this case, forces and displacements can be found both analytically and numerically. For bendable reinforced concrete elements such a diagram has been obtained in [18], which, however, does not take into account the branch of softening after reaching the limit force. In this paper, the method based on the stages of the stress-strain diagram for the monotonic deformation of reinforced concrete columns is used to construct a monotonic deformation diagram. This approach is a generalization of the method of ultimate forces, which is accepted in domestic and foreign design standards, taking into account the specific features of work of reinforced concrete element at the stage of failure: the value of axial load, the presence of indirect reinforcement in the form of clamps, concrete spalling of protective layer, loss of stability of compressed reinforcement, the presence of residual carrying capacity. Methods and materials As noted above, the basis for constructing a monotonic diagram will be to consider the actual deformation pattern of the reinforced concrete column and to identify the characteristic stages at which the stiffness will change and the transition to a new stage of the stress-strain state will be observed. The monotonic diagram will be plotted in the axis “bending moment M - curvature ρ”. For a given column, the longitudinal force N is assumed to be constant during all loading phases. Such a diagram can serve as a basis for the transition to the horizontal force-displacement relation, in which case not only the bending stiffness but also the shear stiffness must be considered, and the displacements caused by the slip of the reinforcement must also be taken into account [18]. The diagram is based on a number of general assumptions inherent to the limit force method: - flat section hypothesis - the cross-sections are flat before deformation and remain so afterwards; - the following state diagrams are adopted for the materials: bilinear for compressed concrete (Figure 2, a) and reinforcement (Figure 2, b); three-linear for concrete bounded by transverse collars (Figure 2, c) [19]; - geometric non-linearity caused by the longitudinal bending of the reinforced concrete column is taken into account by means of an appropriate coefficient h; - the work of the tensile concrete is taken into account only at stage 1 - before the formation of cracks; - stresses in concrete and reinforcement are found by composing and solving equations of equilibrium. Assumptions made at specific stages will be described in the course of the presentation. A general view of the deformation diagram of a reinforced concrete column is shown in (Figure 3). The diagram has 6 characteristic stages of deformation. Consider each stage separately and determine corresponding values of ultimate bending moment and curvature. Figure 3. General view of reinforced concrete column deformation model At the 1st stage (before cracking) the reinforcement and concrete in the tensile zone will be deformed together. The stress profile in the tensile concrete is non-linear trapezoidal and the highest stresses reach the design tensile strength of concrete Rbt (Figure 4, a). Compressed concrete works elastically, stress diagrams have a triangular shape. The ultimate bending moment at stage 1 is given by the equation (1) where Wpl - elastic-plastic moment of resistance of the section; (2) ex - distance from the core point furthest from the tensile face to the force application point N; (3) The curvature corresponding to the moment M1 is determined by the equation (4) where D - bending stiffness of reinforced concrete section at stage 1; (5) In formulae (1)-(5) the geometric characteristics of the reduced section (area Ared, moment of inertia Ired, resistance torque Wred) are determined taking into account the entire cross-section of concrete and reinforcement. Stage 2 is characterised by the operation after the formation of cracks in the tensile zone. The ultimate force in this stage can be achieved in two cases: the stresses in the reinforcement reach the yield stress Rs (stage 2.1) or the stresses in the concrete throughout the compressed zone have reached their design resistance Rb (stage 2.2). As we know which case the ratio of the relative height of the compressed zone ξ to its boundary value determines ξR: ξ ≤ ξR - case of large eccentricities (stage 2.1); ξ > ξR - the case of small eccentricities (stage 2.2). Note here, however, that in the first case, unlike in the second, the element does not enter the fracture stage. Consider stage 2.1 in more detail (Figure 4, c). As noted, the stresses in the reinforcement at this stage reach the design resistance Rs. Stresses in the compressed concrete and in the compressed reinforcement do not exceed the corresponding design resistance Rb and Rsc. The compressive stresses in the concrete are assumed to be triangular. The values of the stresses in the concrete and the compressed reinforcement are then determined from the consideration of the deformations in the flat section (flat section hypothesis). If at stage 2.1 the deformations in the tensile reinforcement reach a value of εs = εso, then the required stresses are found from the expressions (6) (7) The symbols used in formulae (6) and (7) are given in Figure 4, c. Note that if the stresses in formula (6) exceed the design resistance, then the stress diagram should be corrected by taking it in trapezoidal form, whereby the boundary between the triangular and rectangular parts of the diagram will be the fiber where the condition is fulfilled εb = εb1. Composing the equilibrium conditions for the internal forces and the moments of these forces with respect to the center of gravity of the stretched reinforcement, we find the height of the concrete compressed zone and the ultimate bending moment (8) (9) where As and As' - the areas of tensile and compressed reinforcement respectively. The limiting bending moment with respect to the center of gravity will be found by taking into account the effects of longitudinal bending (10) where i - index denoting the stress train stage number; η - coefficient longitudinal bending (11) where - critical force at i stage; (12) where lo - design element length. image1 а b c d e f Figure 4. Stages of the stress-strain state of a reinforced concrete column: a - stage 1; b - stages 4 and 5; c - stage 2.1; d - stage 2.2; e - stage 3.1; f - stage 3.2 Stiffness in this and subsequent stages Di we find with the variable height of the compression zone xm and tensile concrete work between cracks, which is taken into account by the coefficient ψs. The relevant formulas are described in sufficient detail in the regulatory literature (SP63.13330.2018. Concrete and reinforced concrete structures) and, due to their cumbersome nature, are not given in the text of this article. Stage 2.2 will in turn correspond to the failure stage of the reinforced concrete section for the case of small eccentricities. The stresses in the compressed concrete are distributed according to a rectangular law and are equal to the design resistance Rb, in the compressed reinforcement, the stresses also reach the design resistance Rsc, and in the stretched one less than the value Rs (Figure 4, d). Composing and transforming the equilibrium equations we find (13) (14) The limiting bending moment and curvature will be obtained from formulae (4) and (10). Stage 3 will also be considered in two variants. In stage 3.1, for elements operating with large eccentricities, a subsequent increase in bending moment due to the yield strength of the tensile reinforcement will result in an increase in compressive stresses in the concrete to the value of Rb and stresses in the compressed reinforcement up to Rsc (Figure 4, e). The reinforced concrete section will enter the fracture stage. From the equilibrium conditions we have (15) (16) In turn, if the element failed at low eccentricities, a transition to stage 3.2 will follow (Figure 4, f). The deformation in this stage will take place until the yield point is reached in the stretched reinforcement. The compressed zone of the concrete will be divided into two parts: a protective layer and a concrete core bounded by transverse collars. If the bending moment increases, the concrete protective layer for the fibers will splinter off, where the relative deformations of the unconfined concrete reach the limit values εb = εb2. Then only the height of the compressed concrete protection layer will be taken into account in the calculation x'. At this stage the indirect reinforcement is activated, as a result of which the stresses in the concrete core will increase. The compressive stress profile is assumed to be trapezoidal with a minimum value at the neutral fibre equal to Rb and maximum value σb,tr. Applying the plane section hypothesis, determine the values of x' and σb,tr considering that the relative deformations in the reinforcement reach the limit values εs = εso (17) (18) Composing and solving the equilibrium equations we obtain (19) (20) where ηs = 0-1 - a coefficient which takes into account the reduced contribution to the load-bearing capacity of the part of reinforcement bars which have lost stability due to ineffective retention by transverse reinforcement in the free-bending section of the clamp. It is important to note that in stage 3.2 it is possible to increase the bending moment limit as well as to decrease it. This depends on the fraction of the resistance that the cross-section loses when the concrete protection layer rebounds and part of the compressed reinforcement becomes unstable. In Figure 3 the possible directions of unstrengthening are shown by the dotted arrow lines. At stage 4, the load-bearing capacity of the section will be exhausted. As the bending moment increases, the stresses in the concrete core will reach their design resistance Rb,tr, which will be accompanied by the transverse clamps flowing (Figure 4, b). The strength of confined concrete Rb,tr depends on the strength of unconfined concrete Rb and the effective lateral pressure Re which results from the resistance of the clamps to the transverse deformations of concrete. According to [12] the strength of confined concrete can be determined as (21) The effective lateral pressure Re in the case of a square cross-section is (22) where Rsw - yield strength of transverse reinforcement; ke - retention factor, which takes into account the uneven compression of concrete in cross-sections other than circular; ρs - transverse reinforcement coefficient by volume. In view of the considerable deformations in the cross-section and the consequent low height of the compressed zone x', the component related to the resistance of the unconfined concrete at this stage will be neglected. Composing and solving the equilibrium equations we obtain (23) (24) At stage 5, the load-bearing capacity of the reinforced concrete cross-section will be reduced, which is reflected in the diagram by the presence of a softening branch. Stresses in the concrete core will decrease to the value of krRb,tr, where kr - is the residual strength factor of the confined concrete (Figure 4, b). Otherwise, the design dependencies will be similar to the corresponding ones in stage 4. The curvature in stage 6 will increase with a constant value of bending moment until the longitudinal or transverse reinforcement reaches the limit of relative strain εs2, which will be accompanied by a rupture of the reinforcement and complete exhaustion of the load-bearing capacity. It is worth noting that the latter criterion must be monitored at all stress-strain stages. Results and discussion The dependencies obtained will be considered on the example of a reinforced concrete column of a frame structure. We will carry out the calculation in two variants - with the coefficient of longitudinal force ν = 0,3 and ν = 0,6. (25) a b Figure 5. Cross-section of a reinforced concrete column (a) and “moment - curvature” diagrams based on the results of calculations based on the proposed model (b) The cross-section of the column is square 300×300 mm, the geometric dimensions are given in Figure 5, а. Longitudinal reinforcement of 4 bars Æ25А400, As = As' = 982 mm2. Cross reinforcement from Æ8А400 with pitch sw = 100 mm, ρs = 0,005. The design length of the column is assumed to be lo = 3 m. Consider all reinforcement effectively secured against loss of stability ηs = 1. The residual strength coefficient is assumed to be kr = 0,25 [20]. Concrete class B20. The calculation results are shown in Figure 5, b. It can be seen from the graphs that the ultimate bearing capacity for the column with a higher longitudinal force coefficient ν higher, although this column shows less load-bearing capacity prior to the failure of the protective layer than with ν = 0,3. It should be noted that due to the inclusion of the compressed zone of concrete in the work, the more loaded column has greater stiffness in all stress-strain stages. While the less loaded column shows greater capacity for plastic deformation, especially at the stage after the inclusion of indirect reinforcement. Residual load-bearing capacity for column at ν = 0,6 is slightly higher. Failure in both cases is due to clamp rupture when the relative strain limits are reached. Conclusion The analytical model for construction of monotone diagram “moment - curvature” for reinforced concrete columns at different level of axial load, taking into account indirect reinforcement by transverse collars, loss of stability of compressed reinforcement, residual strength of concrete is obtained. The model takes into account all stages of the static deformation of eccentrically compressed reinforced concrete elements, including the non-critical phases of operation. The model can also be used for calculation of frame beams. The authors consider that the main purpose of constructing such a monotonic diagram is to use it as a basis for a hysteresis diagram which describes the behaviour of reinforced concrete elements under low cycle seismic loads. It is worth considering that bringing reinforced concrete elements to supercritical stages, when there is destruction of concrete protective layer and loss of stability of compressed rods, is not always justified in terms of efficiency of repair and further operation of structure. But when designing buildings based on the concept of non-destruction, the proposed model will allow to reveal reserves of bearing capacity of the system. The developed model is suitable for solving by hand calculation, however in case of more complex deformation, e.g. oblique eccentric compression, section damage due to fire or corrosion, sections other than rectangular shape it is possible to apply for solving equilibrium equations at each stage a non-linear deformation model.

About the authors

Ashot G. Tamrazyan

National Research Moscow State University of Civil Engineering

Author for correspondence.
Email: tamrazian@mail.ru
ORCID iD: 0000-0003-0569-4788

Doctor of Technical Sciences, Professor, Head of the Department of Reinforced Concrete and Stone Structures

26 Yaroslavskoye Shosse, Moscow, 129337, Russian Federation

Vladimir I. Chernik

National Research Moscow State University of Civil Engineering

Email: chernik_vi@mail.ru
ORCID iD: 0000-0001-6240-9993

postgraduate, lecturer, Department of Reinforced Concrete and Stone Structures

26 Yaroslavskoye Shosse, Moscow, 129337, Russian Federation

Tatiana A. Matseevich

National Research Moscow State University of Civil Engineering

Email: MatseevichTA@mgsu.ru
ORCID iD: 0000-0001-6292-0759

Doctor of Physical and Mathematical Sciences, Associate Professor, Head of the Department of Applied Mathematics

26 Yaroslavskoye Shosse, Moscow, 129337, Russian Federation

Ivan K. Manaenkov

National Research Moscow State University of Civil Engineering

Email: ivanadekvatniy@mail.ru
ORCID iD: 0000-0002-5260-8793

Candidate of Technical Sciences, Associate Professor of the Department of Reinforced Concrete and Stone Structures

26 Yaroslavskoye Shosse, Moscow, 129337, Russian Federation

References