Stages of Resistance of Reinforced Concrete Frames in Accidental Design Situation

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1. Introduction Buildings and structures are exposed to physical impact of various nature and intensity during their service life, including accidental situations, i.e., situations not covered by the normal operating conditions of construction objects. Accidental impact occurs much less frequently than climatic or functional loads. However, the failure of one or more load-bearing elements as a result of an accidental impact can cause damage that is disproportionate to the initial impact, such as the collapse of part of the frame or the entire structure. One special aspect of designing reinforced concrete building frames taking into account the risk of an accidental situation (failure of a load-bearing element or connection) is the consideration of the arch and catenary action stages of slab structure deformation as allowable states [1-3]. The consideration of such stages of resistance allows to reduce the costs of ensuring mechanical safety, which can be generally limited to the installation of a system of additional restraints and local frame strengthening. In [4], experimental data on the deformation of fragments of reinforced concrete building structures in the form of plane and spatial frames are presented, demonstrating significant excess bearing capacity when resisting according to the catenary or membrane mechanisms. At the same time, [5-7] show that when transitioning to the catenary mechanism of resistance, the corner columns and columns around the perimeter of the building become vulnerable to failure. Studies [1; 5] have provided experimental data on the boundaries of the deformation stages in damaged frames of building structures. Based on experimentally proven assumptions, [8-11] provide analytical dependencies for evaluating the resistance of structures according to the arch and catenary mechanisms. However, it should be noted that such studies have a number of limitations. For example, experimental studies [9; 12] consider two-span beam structures over local collapse, restrained from horizontal displace-ments, which does not allow to take into account the influence of the bearing capacity and deformability of vertical elements (columns, pylons) on the mechanisms of secondary failure propagation in a structural system damaged as a result of an accidental impact. In order to overcome this limitation, studies [1; 5] consider single-storey fragments of reinforced concrete building frames. However, these results cannot be directly extrapolated to multi-storey frame structures, in which, in most cases, it is impossible to consider the behavior of isolated elements, since their load-bearing capacity and deformability depend on the topology of the structural system and kinematically possible failure mechanisms. Another limitation is the simplified application of concentrated loads and the approximation of the curvature diagram along the length of the two-span girder above a distant vertical element. Thus, despite the presence of a significant number of publications with the results of experimental and numerical studies on the issues of arch and catenary actions of reinforced concrete frame structures, there are no formalized force and deformation criteria for transitioning into these stages, taking into account the interaction of elements in the frames of multi-storey buildings, as well as the bearing capacity and deformability of vertical elements, the presence of a system of additional restraints, various scenarios of initial local collapse. In this regard, the purpose of this study was to formulate and substantiate the strength and deformation criteria for the stages of resistance of reinforced concrete multi-storey building frames in an accidental situation in case of failure of one of the elements of the structural system. 2. Models and Methods 2.1. Initial Assumptions To analyze the stages of resistance of reinforced concrete multi-storey building frames in an accidental situation, a zone of possible local collapse is identified [13] - an area bounded by one span in each direction from the element being removed, taking into account the superimposed restraints simplistically modeling the interaction with the rest of the structural system. Limiting the analyzed area allows to obtain an analytical solution to the problem, while it is consistent with the results of experimental and numerical studies, as well as the requirements of design standards for limiting secondary failure. Depending on the scenario (location) of the initial local failure, various mechanisms of secondary failure are possible in the structural system of a building damaged by accidental impact. Based on the analysis of the cases of collapse of buildings, structures and their parts as a result of failure of a bearing element, as well as data from experimental and numerical studies, the classification of secondary failure mechanisms was performed (Table 1). Table 1 Characteristic mechanisms of secondary failure in damaged reinforced concrete multi-storey building frames Structural elements of the building frame Scenarios of initial local failure Characteristics of the failure mechanism Removal of a corner column Removal of the column of the edge or middle row, in which the zone of possible local failure includes the contour of the building Removal of an edge or middle row column, in which the area of possible local failure does not include the contour of the building Structural elements of floors (beams, crossbars) Fracture in normal cross-sections during the bending stage of operation Fedorova N.V., Korenkov P.A. [14], Iliushchenko T.A. et al. [15], Adam J. et al. [4] No data No data Fracture along normal cross-sections at the stage of compressive arch action No data Fedorova N.V., Korenkov P.A. [14], Iliushchenko T.A. et al. [15] Pham A.T. et al. [1; 5] Fracture from rebar breakage at the stage of catenary action No data Pham A.T. et al. [1; 5], Fedorova N.V. et al. [16; 17] Pham A.T. et al. [1; 5] Fracture along inclined and spatial sections during combined resistance of elements Kolchunov V.I., Moskovtseva V.S. [18] Kolchunov V.I., Moskovtseva V.S. [18] No data Vertical elements (columns, pylons) Fracture along normal cross-sections Savin S.Yu. et al. [19] Pham A.T. et al. [1; 5] Pham A.T. et al. [1; 5] Fracture along the supporting inclined sections Kolchunov V.I. et al. [6] Kolchunov V.I. et al. [6] No data Fracture along inclined sections in the anchoring zone of floor slab reinforcement in tension No data Choi H., Kim J. [20], Kolchunov V.I., Moskovtseva V.S. [18] No data S o u r c e: made by S.Yu. Savin Three characteristic scenarios of initial local failure are identified here, depending on the degree of influence on the possibility of achieving specific stages of resistance of damaged building frames (arch, catenary). Thus, when a corner column (pylon) is removed in case of accident, there is no thrust in the slab structure above the local failure due to the absence of horizontal restraints, which makes it impossible to develop compressive arch action and subsequently transition to resistance according to the catenary mechanism. When removing the column of the edge or middle row, in which the zone of possible local failure includes the contour of the building, the deformability of the vertical elements of the edge rows has a significant impact on the possibility and parameters of the arch and catenary resistance mechanisms. In addition, in this scenario of initial failure, vertical elements along the contour of the building are vulnerable to fracture along normal or inclined sections. The third scenario for removing the column of the edge or middle row, in which the zone of possible local failure does not include the contour of the building, almost always allows for the compressive arch action of the slabs, and with appropriate reinforcement and anchoring parameters, the transition to the catenary action after the fracture of concrete in the support zone and in the middle of the span. At the same time, there is a decrease in the risk of failure of the vertical elements, which in this case have excess bearing capacity according to the criteria of the accidental limit state. 2.2. Force and Deformation Criteria for Multi-Level Resistance A reinforced concrete frame structure in the area of possible local failure is considered (Figure 1). At this step, the cases of collapse associated with failure of elements under combined resistance, such as combined action of bending moment and torque, shear forces, which require independent detailed analysis, are not considered. a b Figure 1. Reinforced concrete frame structure in the zone of potential local collapse: а - fully braced; b - partially braced S o u r c e: made by S.Yu. Savin Taking into account possible mechanisms of secondary failure discussed in Table 1, multi-level resistance of the structural system of a building can be represented as a piecewise linear diagram of the relationship between the generalized load P and the displacement of the structural node above the local failure, as shown in Figure 2. The characteristic points of this diagram represent the force and deformation criteria for changing the stages of resistance and exhaustion of the bearing capacity of the analyzed area of possible local failure. Then, from the energy balance condition [21-23], the maximum static load that can be resisted by the load-bearing system in the arch and catenary action stages of resistance can be determined from the following expressions: ¡ for the compressive arch action stage: (1) ¡ for the catenary action stage: (2) Upon removal of a corner column P2 = P3 = P4 = 0. The characteristic points of the diagram in Figure 2 are discussed further using assumptions based on the results of experimental studies and kinematic analysis. Total vertical reaction P, kN Catenary action Compressive arch action Flexural action Vertical displacement z, m Horizontal reaction H, kN Figure 2. Schematic deformation diagram of the reinforced concrete frame in the zone of potential local collapse S o u r c e: made by S.Yu. Savin 2.3. Flexural Stage of Resistance of Slabs The ultimate load and deflection during the flexural stage of resistance are determined from the conditions: (3) Here, according to the design code approach[14], in the margin of stiffness, the curvature can be determined from a conditional elastic calculation: (4) where zs, Ab is the moment arm of the inner couple of forces and the area of compressed concrete, determined from the calculation of the ultimate forces for the flexural stage of resistance of the element. The ultimate bending moment in the normal section of the element during the transition to the compressive arch action stage of resistance in terms of the generalized load P is expressed as: (5) where (6) The following notation is adopted in (6): F is the concentrated force applied to the node of a two-span girder over local failure in the structural system of a building; q is the uniformly distributed load on the two-span girder. The relationship between loads F, q and the generalized load P: (7) In quasi-static analysis, simultaneous consideration of loads F and q, which is inherent in the pull-down and push-down approaches, respectively, allows to partially eliminate the disadvantages of these approaches, discussed in more detail in [24]. 2.4. Compressive Arch Action Stage of Resistance of Slabs The maximum load carried at the compressive arch action stage in a two-span girder with equal spans and the corresponding deflection are determined taking into account the physical constraints from the following conditions: (8) where l is the length of one span (according to the initial model) in the two-span girder; Mult = Mult(z) is the ultimate bending moment resisted by the support section of the girder at the compressive arch action stage, calculated as for an eccentrically loaded element, taking into account the acting thrust: (9) Here, С1, С2 are the reactions due to unit horizontal displacement of the left and right support sections of the arbitrary arch. At the same time, the following conditions must be satisfied to ensure the bearing capacity of vertical elements at the flexural and compressive arch action stages of the slab structures: (10) The following notation is adopted in (10): Nb is the thrust force as a result of girder deformation, determined by formula (9); Mc is the bending moment in the upper support section of the lower storey column; Qcn is the shear force in the upper support section of the lower storey column; Qcv is the shear force in the lower support section of the upper storey column. The angle of deflection of the corresponding node of the girder to column connection θi is determined by the following formula for a fully braced frame (see Figure 1, а): (11) For a partially braced frame (see Figure 1, b): (12) At the moment of transition to the catenary action stage, by assuming the first-order approximation z3 = (h0 - a’), the following is obtained: (13) Here, σs1, σs2 can be determined according to[15]: (14) Where τb,max is the maximum bond stress, corresponding to elastic resistance of the reinforcement; ds is the diameter of longitudinal rebar; ΔLCAA is the maximum elongation of the girder edge upon transitioning from the compressive arch action to the catenary action, determined by the formula: (15) For longitudinal rebars located at distance zsi from the center of gravity of the compressed longitudinal reinforcement in the section, ΔLsi is obtained as: (16) By substituting ΔLsi instead of ΔLCAA into formula (14), stress σsi can be determined in all longitudinal rebars in the section. If σs > Rs as a result of evaluating formula (14), the stress in the reinforcement in tension is determined by formula , (17) where τbт,pl = 0.27τb,max is the average bond stress in the plastic region of the rebar; εsu is the ultimate strain of the longitudinal rebar. The adjusted value of deflection upon the transition from the compressive arch action stage to the catenary action is determined from expression (18) 2.5. Catenary Action Stage of Resistance of Slabs If the condition of σs < Rsu is satisfied for the deformed structure in the transition state (P3, z3), the maximum deflection z4 at the moment of rupture of the most stretched reinforcing bars in one of the support sections at the catenary action stage of resistance can be found by assuming σs = Rsu: (19) The utlimate load P4 is calculated by formulas (13), substituting z4 instead of (h0 - a’). At the same time, for the normal sections of the edge columns at the point of connection with the girder in partially braced frame structures, the following condition must be satisfied: (20) Here, by following the approach considered in ACI 318[16], the coefficient of buckling upon relative horizontal displacement of the slabs of one floor is determined taking into account the action of all the columns of the floor in question on one side of the local failure in the structural system, where ∑P, ∑Pcrit are the sums of the effective longitudinal forces and critical forces in these columns. In the case when the analyzed zone of possible local failure includes the columns of the edge row or the corner column, the signs of the sum in (20) disappear, and the formula is modified to the known form for the analysis of individual elements. 3. Results and Discussion 3.1. Comparison with Results of Studies by A.T. Pham and K.H. Tan To verify the proposed strength and deformation criteria for the stages of resistance of reinforced concrete frame structures of buildings, a comparison was made with the results of experimental studies by A.T. Pham and K.H. Tan [1], who tested several substructures of reinforced concrete frames at different levels of gravitational load applied to the node at the midspan and the rate of its application. The general view of the frame structure, its dimensions and loading scheme are shown in Figure 3, a. Mechanical characteristics of the structural materials: yield strength of longitudinal reinforcement is at least 535 MPa, temporary rupture strength is at least 615 MPa; the average cylinder strength of concrete under uniaxial compression is 35 MPa. The calculation results in the form of a P - z diagram are shown in Figure 3, b. а b Figure 3. Comparison with experiments by Pham and Tan [1]: a - dimensions and reinforcement layout of the RC frames; b - calculated and experimental Load (vertical reaction) vs. Midspan Joint Displacement curves S o u r c e: a - made by A.T. Pham, K.H. Tan [1]; b - made by S.Yu. Savin As seen from the graphs, the curves constructed according to the relationships proposed in this study quite well reproduce the experimental diagrams at the characteristic points of transition of the resistance stages. However, due to the absence of intermediate points in the diagram at the catenary action stage, there is a more significant gap between the experimental curves and the analytical line passing through characteristic points (P3, z3), (P4, z4). The result of such a discrepancy may be an overestimation of the ultimate static load according to the energy balance method, which is resisted at the catenary action stage of the slabs above local failure. In this regard, an approximation by a tangent function can be used to refine the region of the diagram corresponding to the catenary mechanism of deformation. 3.2. Comparison with Results of Experimental Studies of Multi-Storey Frames To verify the initial assumptions about the strength parameters of the elements of a reinforced concrete structural system in the zone of possible local collapse as a result of an accident at different stages, taking into account the design features, initial stress-strain state due to static operational load and specific resistance mechanisms, the results of experimental studies of reinforced concrete frame structures subjected to two-step static-dynamic loading from an accident event are analyzed. A semi-precast reinforced concrete frame is considered, the test results of which are given in [6]. The frame structure is shown in Figure 4, a. The height of the precast section made of grade B35 concrete is h2 = 70 (mm); the height of the cast-in-situ section made of grade B50 concrete is h1 = 30 (mm); the distance between the longitudinal axes of the two components is ω = 50 mm. The longitudinal reinforcement of compound girders is made of grade B500 reinforcement with a diameter of 4 (mm). The transverse reinforcement of the girders is made of wire reinforcement with a diameter of 2 mm and characteristics similar to grade A240 reinforcement. The experimental frames were loaded with a static load transmitted through a system of arms. The load was applied in the form of concentrated forces in the thirds of the girder spans. The values of the concentrated loads applied to the frame girders were: P1 = 3.2 kN, P2 = 2.64 kN, P3 = 2.04 kN. Here, the highest load value corresponded to the upper floor, and the lowest to the lower floor of the frame. а b Figure 4. Comparison with experiments by [6]: a - dimensions and reinforcement layout of the RC frames; b - calculated and experimental Load (vertical reaction) vs. Midspan Joint Displacement curves S o u r c e: made by S.Yu. Savin According to the methodology, the physical model of the reinforced concrete frame is tested in two steps. At the first step, gravitational load is applied incrementally (in 10 increments) to the nodes of the experimental model of the structure using a system of arms. At the second step, the intermediate support, imitating the column of the middle row, is instantly removed. When modeling the accidental impact on the physical model, the column turns into an instantaneously variable system, which is equivalent to the instantaneous removal of connection in the structural system. To simplify the procedure of physical modeling, the boundary conditions of the node of the frame connection to the intermediate support represent a pin support. Figure 4, b shows the calculated load-displacement diagrams of an equivalent model of a two-span girder of the experimental reinforced concrete frame, obtained using the relationships proposed in the study. Horizontal lines Pes, Ps,u,CAA, Ps,u,CA represent the equivalent static load on the frame, the ultimate static load in the arch and catenary action stages of the structure, respectively. It should be noted that when constructing the diagrams, the stiffness of the supports to horizontal displacement and rotation was taken into account, however, the load-bearing capacity of the columns was not checked. Therefore, the point corresponding to the fracture of the frame columns along inclined and normal sections occupies an intermediate position between the extreme points of the flexural and compressive arch action stages of resistance in the theoretical P - z diagrams. This confirms the need to check the load-bearing capacity according to the criteria of the accidental limit state not only of the slab structures above local failure, but also of the vertical elements (columns, pylons) in the area of possible local collapse, primarily for corner columns and columns of the extreme rows. The general view of the deformed state of the frame as a result of an accidental impact is shown in Figure 5. а b Figure 5. Frame of the second type after accident impact tests: a - general view; b - the crack and fracture pattern S o u r c e: made by V.I. Kolchunov et al. [6] As a result of an accident related to the failure of the middle row column (simulated by a removable support), the load was redistributed along alternative paths. The formation of normal cracks in the support and span sections of the two-span girders above the local failure on all floors of the frame structure was noted. Longitudinal cracks formed along the contact of the cast-in-situ and precast parts of the girders. The columns of the edge rows on almost all floors fractured along inclined sections. The column of the first floor along axis C fractured along the normal section with the formation of a plastic hinge in the upper support section. 4. Conclusion 1. The force and deformation criteria of the stages of resistance of reinforced concrete frame structures of buildings in the zone of possible local collapse due to sudden failure of a vertical element (column, pylon) are formulated. 2. Using the obtained criteria and the energy balance conditions, simplified relationships were constructed to estimate the ultimate static load for the arch and catenary action stages of resistance of the slab structures. 3. The comparison of the calculation results based on the relationships obtained in the study for the force and deformation criteria of the resistance stages with the data of experimental studies confirmed their reliability, and also demonstrated the need to take into account the joint deformation of slab structures and vertical load-bearing elements for correct evaluation of the deformed state of building frames in case of an accident.

About the authors

Sergei Yu. Savin

Moscow State University of Civil Engineering (National Research University)

Author for correspondence.
Email: suwin@yandex.ru
ORCID iD: 0000-0002-6697-3388
SPIN-code: 1301-4838

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

26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation

References

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