Analytical Modeling of Reinforced Concrete Columns Under Lateral Impact with Shear Failure

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1. Introduction An important aspect of ensuring the safety of buildings and structures with reinforced concrete frames is checking the strength of columns and beam-columns against lateral impact. This issue has received considerable attention, which indicates the relevance of this research area. Factors such as the load-bearing capacity of reinforced concrete columns, taking into account the effect of confinement of concrete (confined concrete) [1], the deformability of composite reinforced concrete columns in carbon fiber reinforced plastic (CFRP) tubes [2], and the dynamic response and energy absorption capacity of structures with modified concrete, for example, with the addition of rubber crumb [3]. A large amount of experimental research, analytical and numerical modeling has been carried out, which shows interest in a comprehensive study of the behavior of compressed columns under impact loads. This includes corrosion damage [4], considering the longitudinal (vertical) coupler splicing of rebar along the length [5], and considering combinations of bending and compressive actions [6]. One of the most important aspects of research in column dynamics is its response and damage assessment due to horizontal impact [7]. The point of application of the impact load is important. For example, in [8], impact on a part of a column near its support node is considered. The authors of this study prove the effect of the type of supports on the load-bearing capacity of columns. A number of papers are devoted to the study of compressed structures under low-velocity impact, for example [9]. For such an impact, the damping properties of the column and its ability to absorb energy can be increased by strengthening it with CFRP [10]. Low-velocity impact can be modeled in calculations as both quasi-static [11] and impulse load [12], where it is noted that the limit state of columns can be reached either by fracture or buckling. Buckling usually prevails for slender reinforced concrete columns with a slenderness ratio of more than 50. The complexity of describing the deformation and modeling of columns under horizontal impact has necessitated the development of simplified analytical approaches to calculations and engineering methods. Various features of these methods are presented in the following papers. Article [13] compares simplified approaches to assessing the load-bearing capacity of conventional and CFRP-strengthened reinforced concrete columns, paper [14] describes the probabilistic nature of the loading effect and assesses the reliability of structures, and study [15] reveals the influence of the percentage of longitudinal reinforcement on the load-bearing capacity. Analytical models for determining the load-bearing capacity under dynamic impact are based on quasi-static equilibrium equations [16], which have been refined for cases of random corrosion damage [17] and various compressive and impact force ratios [18]. There are significant differences in calculation methods depending on the impact velocity, the stiffness of the impactor body, the stiffness of the column itself, and the penetration of the impactor tip into it. In [19], a significant reduction in the stiffness and load-bearing capacity of a compressed column as a result of corrosion is noted, but stiffness can also decrease due to cracking under combined loads, including static moments and shear forces [20]. A number of studies focus on the stiffness of the impactor. If the stiffness of the impactor body is high, then all the kinetic energy is transferred to the impacted structure (“hard” impact), and if the impactor itself can absorb energy during impact, such an impact is considered soft [21]. In addition to the stiffness of the impact, the shape of the cross-section has a significant effect on the dynamics of columns. Thus, when comparing the results of studies [22] (square cross-section) and [23] (round cross-section), it can be seen that round columns are more vulnerable to brittle shear failure than square ones. A number of considered studies [7; 8; 18; 22] involve the use of intensive computational procedures and require highly qualified researchers, which significantly hinders the practical application of the proposed developments. Therefore, the purpose of this article is to develop an engineering method for calculating compressed columns under impact causing shear failure. To achieve this goal, it is necessary to solve the problem of constructing a simplified method for evaluating the load-bearing capacity and to verify it on the basis of experiments and numerical modeling. The object of the study is a compressed reinforced concrete column subjected to horizontal impact at the support, and the subject of the study is the ultimate shear bearing capacity. 2. Methods When a horizontal force is dynamically applied to the column, most researchers, including [24], identified the following failure patterns: local crushing, which may be accompanied by chipping or punching (local failure), bending failure (along normal sections), and shear failure (along inclined sections) (Figure 1). Considering experimental and theoretical studies of column shear failure, as well as qualitative results of numerical modeling, it is noted that the load-bearing capacity is significantly affected by the level of compressive loading, as well as the magnitude and direction of the static shear force present in the frame structure. As a result of dynamic impact, the column is loaded by both shear force and bending moment. Therefore, similar to normal operation, two strength conditions must be formed: strength along the inclined section under shear force taking into account added stress from dynamic loading; strength along the inclined section under bending moment taking into account its change under dynamic loading: k Qd1 max ≤Qсd ult, , (1) k Md2 max ≤Mсd ult, , kd1, kd2 are the dynamic load coefficients of the system; Qmax , Mmax are the internal forces due to load for the considered inclined section; Qсd ult, , Mсd ult, are the ultimate force values determined by the resistance of concrete and reinforcement in case of shear failure. Figure 1. Failure modes under horizontal impact: a, b, c - local crushing at various impact velocities; d - diagonal shear failure of a compressed column; e - flexural failure: 1 - impactor; 2 - column; 3 - local crushing zone; 4 - concrete spall on the side opposite to the contact area between the impactor and the column S o u r c e: made by A.V. Alekseytsev. The method chosen for solving the system of inequalities (1) consists of compiling static equilibrium equations with respect to the lateral axis of the column and with respect to the point passing through the beginning of the inclined section. Cases of local failure (Figure 1, a-c) are not considered in this study. For numerical verification, the method of direct integration using the implicit scheme of differential equations of motion of the system, discretized using the finite element method (FEM), was adopted. For a reinforced concrete frame, this equation can be represented in the form of (2). = , (2) M y + С y K y F + τ where y y y , , are the acceleration, velocity and displacement vectors respectively, F is the vector of nodal forces, M is the mass matrix. Damping matrix С and global shear stiffness matrix Kτ in formula (2) are determined using formula (3): С=βKτ; Kτ = Kсo +Kro +Kso, (3) where β is the structural damping constant; Kсo,K Kro, so are the shear coefficient matrices for concrete, reinforcement and supports (in case when the column rests on a deformable base). 3. Results and Discussion 3.1. Calculation Procedure The first strength condition is obtained from expression (1). If there is static load present in the system before the dynamic impact, the following formula is proposed: ΔQd ± Qst ≤1, (4) k QNd dult QbN +QswN ΔQd is the added shear force caused by the horizontal impact; kNd is the coefficient accounting for the restraints and the column stress state under service load; Qdult is the ultimate shear force in the section taking into account dynamic strengthening of concrete and reinforcement and the confinement effect; Q Qst, bN , QswN are respectively: the shear force from the design static load, the shear force resisted by concrete, and the shear force resisted by transverse reinforcement during normal operation, taking into account the presence of longitudinal force Ne . Coefficient kNd is determined according to formula (5): d = 1.51-μ kе + NNultе PPNeultult -1, NNultе < 0.6, (5) kN PPNeultult +kе -1, NNultе ≥ 0.6, where PNulte is the static equivalent of the lateral impact load, at the value of the service longitudinal force equal to Ne and when bending failure of the column occurs; Pult is the same for longitudinal force Ne =0; μ is the coefficient for converting the design column length to the geometric length. When one end of the column is fixed and the other is pinned, μ= 0.7; when both ends are fixed, but compression remains possible, μ= 0.5. Evaluation of PNulte , Pult is performed taking into account the influence of bending moments and deflections caused by these forces. The value of Nult is determined for the case of small eccentricity (virtually axially compressed bar) using the formula from SP 63.13330[1]: Nult =φ(R A R Ab b + sc sc), (6) where φ is the buckling coefficient; R Rb, sc are the design compressive strengths of concrete and reinforcement respectively; Ab,Asc are the areas of concrete and reinforcement. The value of ke refers to the level of the confinement effect. For a square cross-section based on the basic recommendations of J. Mander’s model [25], it is determined based on the percentage of longitudinal reinforcement and the geometry of transverse reinforcement: 2 1 Sw -dw , (7) ke = -μsc 1- 2d 1 where μsc,dw are the percentage of longitudinal reinforcement taking into account the area of the confinement contour (рис. 2) and the transverse rebar diameter respectively; d is the transverse rebar length along the side, which is parallel to the plane of impact P within the bounds of the centers of gravity of the transverse rebars perpendicular to P. Ultimate force Qdult can be determined based on the expression that is valid for shear failure with a projection length of the inclined sectionс = 2h0 : , (8) Rbt is the design tensile strength of concrete; kb is the coefficient of dynamic strengthening of concrete; b h, 0 are the width and the design height of the cross-section; qsdw is the intensity of the load resisted by the transverse reinforcement, taking into account dynamic strengthening of steel and the stress from confined concrete core (Figure 2). a b Figure 2. Determination of the confinement level: a - column cross-section; b - section s-s: 1 - confinement boundary; 2 - core zone of the confinement; 3 - longitudinal rebars; 4 - transverse rebars S o u r c e: made by A.V. Alekseytsev. The value of qsdw is determined as qswd =ks R Asw sw 1-(0.2 +ke ) Ne , (9) Sw Nult where Rsw, Asw are the design strength and the area of transverse reinforcement; ks is its coefficient of dynamic strengthening. The value of the shear force resisted by concrete, taking into account compression by the longitudinal force, is calculated as follows, based on the methodology described in SP 63.13330 for the design of prestressed reinforced concrete structures: 2 QbN = 1.5φn btR bh0 , 0.5φn btR bh0 ≤ QbN ≤ 2.5R bhbt 0 , (10) c where c is the projection of the inclined section onto the vertical axis, and the value of φn accounts for the presence of normal stress caused by the compressive force. It is determined as follows: 1.25, 0.25Rb ≤ <σb 0.5Rb φn = 2.5- - 1 σRbb , 0.5Rb ≤ ≤σb Rb , σb =Ab +α εENRb beb 0 Asc , (11) where in the case of small eccentricity (all of the section is non-uniformly compressed) Asc is the area of the longitudinal reinforcement; Ab is the area of concrete; α= Es / Eb is the ratio of the elastic moduli of concrete and reinforcement, determined taking into account the stress state of the column; Rb is the design compressive strength of concrete; εb0 =0.002 is the concrete strain for short-term load. The value of c can be determined as: с=, с≤ 2h0 , (12) where the intensity qscw of the load resisted by the transverse reinforcement is determined according to (5) at ks =1. The second strength condition from expression (1) has the form given in SP 63.13330 and, in general, if the required development length of longitudinal reinforcement is achieved, it is satisfied kd2Mmax ≤ Mсd ult, = Ms + Msw , (13) where Ms,Msw are the bending moments resisted by the longitudinal and transverse reinforcement of the column, respectively. 3.2. Analytical Calculation A column with the following parameters is considered: cross-section of 400×400 mm, length of 4.0 m, В25 grade concrete (Rb =11.5 MPa, Rbt =0.9 MPa), А500 grade longitudinal reinforcement with Rs = 435 MPa, transverse reinforcement of the same grade, but with Rsw =300 MPa, coefficients of dynamic strengthening taking into account [26] k1= 1.1 for concrete, and k2= 1.2 for reinforcement, fixed at the base, pinned connection to the upper floor slab, μ= 0.7, the distance from the exterior face of concrete (horizontal and vertical) to the center of gravity of the longitudinal reinforcement а =5 cm (Figure 3). Figure 3. Calculation example for a column under horizontal impact: a - initial structure; b - cross-section S; c - design model with an inclined section (i.s.); d - bending moment diagram from a unit impact load; e - principal tensile strains in concrete at the peak dynamic force under service load Ne S o u r c e: made by A.V. Alekseytsev. The column is reinforced with 4d28 longitudinal bars, Аs4 28сd0 = 24.63 cm2 , symmetrically at the corners. The transverse reinforcement represents a closed frame made of 4d8, and this frame is arranged at a constant spacing of 250 mm along the height of the column, rebar area Аsdw8 = 0.503 cm2 . The column is assumed to be virtually axially compressed under service load Ne = 2000 kN, shear force Qst =0 kN. The column is subjected to dynamic action in the form of a horizontal impact from car collision. This impact is simulated by a mechanical force applied at a distance of 0.8 m from the base support. It is necessary to determine the maximum value of this dynamic force, assuming that the shape of the impact impulse is rectangular. In the calculation, it is assumed that the failure occurs along an inclined section with a projection length of с = 2h0 , and the strength condition for the inclined section under the action of bending moment is satisfied (the value of the moment is small compared to the beam elements). Condition (4) takes the form ΔQd ≤ k QNd dult . For determining kNd , the following values are calculated: μsc =24.63/30 30⋅ = 2.73%, where d = 30 cm is the size of the square confinement region (Figure 3, b, c). Then the confinement level ke = -1μsc 1- Sw2-ddw 2 = 1- 0.02731 1- 252 30-⋅ 2.8 2 = 0.408, 1 and Nult =φ(R A R Ab b + sc sc0) =0.9(1.15 40 40⋅ ⋅ +43.5 24.64)⋅ = 2620 kN . Ratio Ne / Nult = 2000/ 2620=0.763≥0.6, the equilibrium equation at the beginning of the inclined section passing through section S (Figure 3, c), is used, and the conditional ultimate horizontal forces at Ne = 0 kN, Ne = 2000 kN, which would cause bending failure, are calculated. The maximum moment from the action of these forces will be at the fixed support (Figure 3, d), an equilibrium equation is formed for this section. The deflection from the horizontal force at the fixed support is zero, so the equation will take the form: М, the maximum moment is determined using the displacement method under the condition that it is caused by force PNulte , eccentricity ef = ((35-5) / 2) =15 cm. Constant αR , associated with ensuring plastic failure mode, is determined, for which the boundary value of the relative height of compressed concrete under impact is calculated: ξR = 0.8÷(1+ (435÷ 2 10⋅ 5) ÷ 0.0035) = 0.493 , then αR = 0.493(1- 0.493/ 2) = 0.37 . Substituting all the obtained values into the equilibrium equation yields: 0.144 400 2000 15 1.1 1.15 40 35 0.37 1.2 43.5 12.32 35 5 , 57.6PNulte = 22934 +19293-30000, PNulte = 212.74 kN. When Ne = 0 kN, Pult = 733.1kN. Then kNd = (0.403+ 212.74 / 733.1)-1 =1.4427 . The intensity of the impact load resisted by two d8 transverse reinforcement bars taking into account the confinement effect level and the presence of service compressive force (5) is calculated: qswd kN/cm. 25 2620 The adopted diameter of the transverse reinforcement, taking into account the presence of longitudinal force and the adopted reinforcement spacing, must be checked for developing its full strength in concrete. Elasticity coefficient: νb = Rb ÷ Eb b0ε =11.5÷(27.5 0.002 10⋅ ⋅ 3) = 0.209. Average stress is calculated as σ = N / Ared = 2000 / 40 40( ⋅ + 0.209-1 ⋅(2 / 2.75) 10 24.64⋅ ⋅ ) =1.017 kN/cm2, coefficient φn = 2.5 1 σ -1.017 = 0.289 , - = 2.5 1 Rв 1.15 qswd ≥ qsw,min = 0.25φn btR b = 0.25 0.289 0.09 40⋅ ⋅ ⋅ = 0.2601. The condition is satisfied. Then, according to (4): Qdult = 3 1.1 0.09⋅ ⋅ ⋅(1+ 0.403)(1+ 2/2.62) ⋅40 35⋅ 2 ⋅0.7849 =146.8kN. The value of kN. That is, with an impact time of 1 second, the column can withstand a mechanical force of 211.8 kN. 3.3. Numerical Verification of Calculation Results Due to the complexity of setting up and conducting a full-scale experiment that reproduces the calculated situation, the problem is verified using a three-dimensional finite element model. The Drucker - Prager plasticity model [27] was used for concrete, with the possibility of material softening under shear stress, which simulates shear failure. The reinforcement was modeled using a bilinear diagram with a limit on the rupture strain. The parameters of the concrete and reinforcement deformation models, as well as the hyperparameters of the calculation algorithm, are given in Table 1. Verification of the finite element method (FEM) model with respect to bending strain with concrete deformation model parameters was carried out in [28]. For compressive strain, a design comparison at the ultimate force Nult = 2620 kN is performed. Numerical simulation of this process, taking into account slow (close to static) loading, yielded a value of 2778 kN. Numerical and analytical calculations with other material parameters also yielded similar results for the ultimate compressive force. However, there are the following differences in the strength parameters obtained on the basis of the numerical and analytical models. The compressive stress in concrete obtained in the finite element model corresponds to the value of Rb , but the stress in the longitudinal reinforcement at the limit state is at the level of σ=(0.4÷0.7)Rsc , and the main criterion for stopping the load growth is the stress in the transverse reinforcement reaching its limit σ= Rsw . It has been established that the growth of strain in the transverse reinforcement leads to concrete failure. It is evident that analytical models need to take into account the concrete confinement effect and the formation of strength criteria in terms of strain in the presence of transverse reinforcement, as well as the need to consider the aspects of concrete deformation in the area surrounding the reinforcement, which has been done in [29]. Using the verified finite element model, calculations of the considered column were made at various values of Ne , and these were compared with the results obtained using the proposed analytical method (Table 2). Table 1. FEM-analysis constants Material / Hyperparameters Parameters Cohesion stress Internal friction angle Dilation angle Yield strength Ultimate strain В25 concrete 3.3 MPa 38 deg. 28 deg. 0.9 MPa (tension) 11.5 MPa (compression) 0.0001 (tension) 0.0035 (compression) A500 longitudinal rebars - - - 435 0.025 А500 transverse rebars - - - 300 0.025 General damping ratio 5% Convergence criterion Load convergence tolerance 0.1% Nonlinear iteration 25 Newton - Raphson iterations per step, stiffness matrix updates each 5 iterations Integration step and time Δ =t 0.05 s , t =1.5 s S o u r c e: made by A.V. Alekseytsev. Table 2. Comparison of the ΔQd values Method ΔQd , with axial compression force Ne equal to 2500 kN 2000 kN 1500 kN 1000 kN 500 kN FEA 109 195 256 287 309 Proposed method (M) 257.28 211.8 272.4 275.3 298.2 Tolerance δ =(FEA-М)/М -91 % -0.07 -0.048 0.026 0.022 S o u r c e: made by A.V. Alekseytsev. The FEA results in this problem agree well with the analytical ones. As the table shows, the applicability of the proposed methodology should be limited by the level Ne / Nult <0.8, since the error of the method as Ne → Nult becomes very large. Calculations have shown that in the presence of service transverse force Qst condition (1) yields results, which correspond with numerical modeling quite satisfactorily. 3.4. Discussion and Perspectives The presented method is based on one of the possible scenarios of column failure, when the dynamic force increases at a relatively low rate, so that the stress in the concrete does not exceed the crushing stress, and its strain rate does not exceed the critical values leading to the formation of crack fans. If the load application rate is high, then during contact interaction, a failure pattern in the form of punching or chipping may occur. The calculation models for compressed elements given in a number of regulatory documents do not currently take into account the effects of concrete dilation, confinement effect, and the emergence of stress in the transverse reinforcement due to this confinement, but numerical models and experiments confirm these effects. Numerical models show that the load-bearing capacity of a compressed column along an inclined cross-section, taking into account lateral impact, significantly depends on the stress state of the concrete. The proposed model indirectly takes into account the effects of the emergence of stress in the transverse reinforcement when calculating parameter qsw (5), however, microcracking and, as a result, concrete dilation are not taken into account, nor is the resistance of concrete on the descending branch. Therefore, it is assumed that a significant error arises as Ne → Nult . The developed method can be used as an additional tool for solving problems related to assessing the survivability of buildings and structures [30; 31] in the event of man-induced accidents of mechanical nature. Prospects for improving this method include refining it for calculating elongated rectangular crosssections (pylons) and slender reinforced concrete columns (λ ≥50 ). It is also interesting to introduce various types of initial or acquired damage into the model, as well as to adapt the method to the calculation of reinforced concrete columns strengthened with steel or carbon fiber. 4. Conclusion 1. A method for analytical calculation of the lateral impact load capacity for reinforced concrete beamcolumns causing shear failure has been developed. Verification showed satisfactory agreement with the obtained results based on the calculation of the solid FEM model in the range of compressive force values corresponding to characteristic loading of columns in civil buildings. The difference in results at Ne <0.8Nult is no more than 5%. 2. The analytical model takes into account the confinement effect of concrete under compression, considering different spacing, diameter, and grade of transverse reinforcement. Limitations in the application of the method have been identified in terms of the compressive force, which must be less than 80% of the ultimate value. 3. A possibility has been established to quickly, compared to a solid FEM model, assess the safety of reinforced concrete columns in accidental situations such as collisions of technological or other transport with columns, man-induced mechanical impacts on columns in support areas, when failure occurs due to the action of a lateral force with a projection length of the inclined section 2h0 ≤с ≤3h0 . 4. The proposed relationships are recommended for use in the design of preventive measures aimed at improving the mechanical safety of buildings and structures, including their protection against progressive collapse.

About the authors

Anatoly V. Alekseytsev

Moscow State University of Civil Engineering (National Research University)

Author for correspondence.
Email: aalexw@mail.ru
ORCID iD: 0000-0002-4765-5819
SPIN-code: 3035-5571

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

26 Yaroslavskoye Shosse, Moscow, Russian Federation

Konstantin V. Yurusov

Moscow State University of Civil Engineering (National Research University)

Email: walrk@mail.ru
ORCID iD: 0009-0004-1970-3491
SPIN-code: 8084-3827

Postgraduate student of the Department of Reinforced Concrete and Masonry Structures

26 Yaroslavskoye Shosse, Moscow, Russian Federation

References

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