The Bearing Capacity of Compressed Corrosion-Damaged Reinforced Concrete Elements under Lateral Pulse Loading

Abstract

This article addresses the relevant problem of the stress–strain behavior of compressed reinforced concrete columns under lateral pulse loading. A simplified engineering method of analyzing the limit value of lateral pulse loading P, depending on longitudinal force N acting on the column, is developed. The proposed method involves the construction of the P-N curve that has three portions. Portion 1 describes the plastic deformation of concrete and rebars of that part of the structure that is mostly in bending. Portion 2 describes the state of the column that can trigger the brittle failure of the concrete along the normal section, and Portion 3 describes the high compression of the column that predominantly triggers its shear failure. For Portions 1 and 2, analytical relationships are obtained using equilibrium equations. Corrosive damage is taken into account in the analytical model as a reduction in the strength and deformability characteristics of the material. A conventional local corrosion spot can be considered; it can be located both in and outside of the area of action of lateral pulse loading. The results obtained using the proposed model were compared with the results of numerical studies and a full-scale experiment. As a result of testing the developed engineering technique, it was found that it provides a safety margin for corrosion-damaged elements of 0.20–0.8 of the ultimate value of horizontal impulse at operational values of compressive force. The spot corrosion damage considered in the paper leads to a 10–60% strength reduction in compressed columns, depending on their location.

1. Introduction

The study of the bearing capacity and safety of reinforced concrete structures subjected to combined actions is a relevant problem of science and technology. This problem deals with the deterioration of mechanical characteristics of materials under the action of adverse forces and medium-induced factors, as well as emergency situations. In particular, corrosion-damaged structures under emergency force actions are considered in the article. Corrosion processes must be experimentally and theoretically substantiated to evaluate the stress–strain state of such structures. This area of research is addressed in several works. For example, in [1], modeling of corrosion-damaged slabs was considered in case of changes in the effective area of the rebars and several points of concrete damage. These slabs were subsequently subjected to a low-velocity impact. A verification experiment was conducted. The experiment involved accelerated electrochemical corrosion induced by direct current. In [2], the effect of the bond between rebars and concrete on the load-bearing capacity of beam structures is studied if rebars are corrosion-damaged. It is emphasized that the anchorage of rebars in concrete is the determining factor in case of corrosion damage of reinforced concrete beams. Corrosion-damaged reinforced concrete slabs, subjected to multiple similar impacts, were tested [3], and CFRP (carbon fiber reinforced plastic) rebars were applied to improve the resistance of slabs. An important issue investigated by the authors is the reliability and durability of corrosion-damaged structures. For example, in [4], an aspect of this problem is considered under cyclic loading that causes fatigue failure. It is found that corrosion is the cause of acceleration and development of fatigue cracks. As a consequence, it triggers a substantial reduction in the durability of beam structures. This process is typical for bridge spans. Another aspect affecting the durability of corrosion-damaged beam systems is the non-uniform corrosion of rebars; it affects the bearing capacity of structures [5]. In this case, the main factor is the failure of the bond between concrete and rebars; non-uniform degradation of the bond is more dangerous than its uniform degradation. The study of corrosion is closely related to the identification of the ultimate bearing capacity of damaged systems and the improvement of their bearing capacity. In this case, the problem of optimal design of reinforcement systems arises. This problem was solved in [6]. In this work, reinforcement efficiency is considered for several options of pre-stressed reinforced concrete frames. A number of works address the modeling of corrosion propagation, in particular, corrosion propagation deep into a reinforced concrete structure. In the case of accelerated corrosion induced by direct current, scenarios of corrosion propagation deep into reinforcement bars may be diverse and arbitrary. The theory of distribution of random variables and sequences of random events, such as the Markov chains, is used for this purpose [7].

Of great interest is the study of reinforced concrete columns damaged by special effects, including fire [8], corrosion damage, and various types of lateral impact loads. Such structures include piers subjected to cyclic wave impact loading [9]. In the case of lateral impacts, many different aspects are experimentally and theoretically investigated, including the cross-sectional (square, circle) shape of the column, the reinforcement ratio, the eccentricity of the longitudinal force, and the impact velocity [10,11,12], etc.

The lateral impact on corrosion-damaged columns is very complex; it often requires labor-intensive finite element models. However, they are not rational at the preliminary design stages. Hence, simplified analysis techniques are proposed, for example, those addressed in [13]. However, they are based on the detailed study of mechanisms of resistance to lateral impacts on columns [14].

Several studies of resistance of corrosion-free and corrosion-damaged reinforced concrete columns to impacts focus on finite element models. Here, a 3D scheme is often applied, and a method of direct integration is used to simulate low-velocity impacts characteristic of the loading of these structures. Towards this end, the LS-DYNA software package [15,16] is effectively used to analyze accidental collisions between a structure and a ship and to add composite columns to raise their impact strength. Some works address the study of lateral impacts from accelerating bodies. In particular, as far as free-falling bodies are concerned, research efforts focus on reinforcement schemes, the impactor mass, and the impact time frame [17,18]. The tools for numerical studies include software packages such as Abaqus and Ansys Autodyn. The ability of the impactor to transfer the energy of impact to the column is investigated. As far as this property is concerned, impacts can be soft and rigid. For example, in [19], a soft lateral impact is comprehensively investigated. In general, the finite element method (FEM) is applied in various fields [20,21], such as mechanical engineering, environment, energy, etc.

It is very important for the results to take into account the actual mechanical characteristics of concrete. As the authors show, for example, in [22], the strength of structures depends significantly on the ratio of cement to sand in the concrete, as well as on the size fraction of the aggregates forming the concrete.

The most important aspect in the study of the stress–strain state is the modeling of the failure behavior for concrete. One of the effective ways for this purpose is a discrete element method (DEM) as a generalization of FEM. In [23], concrete is represented as a four-phase material, one of whose phases determines the propagation of cracks. Under dynamic loads, the mechanical characteristics of concrete should be determined, taking into account its dynamic properties, which are shown in [24] on tests of Brazilian disc specimens.

A large group of studies of the lateral impact on corrosion-free and corrosion-damaged reinforced concrete columns focuses on experiments. In the course of such studies, new experimental units are designed to identify the effects of impact resistance and formation of failure mechanisms, for example, in [25]. These units were used to identify the nature of the failure of a reinforced concrete column when a car hits a pier structure. In this case, the failure was triggered by the shear along an inclined crack. In addition, patterns of deformations, changes in the impact strength, and lateral displacement were studied in the above-mentioned experiment-focused works. In [26], an important regularity was identified: the failure mechanism switches from bending to shearing in case of high-velocity impacts, accompanied by compression. A number of experimental studies address composite reinforced concrete columns strengthened by carbon, fiberglass, and CFRP [27,28]. This method can greatly raise the resistance of structures to low-velocity lateral impacts, which are similar to an accidental impact from a car.

This paper proposes a simplified engineering method for evaluating the ultimate bearing capacity of corrosion-damaged columns under lateral impacts. It takes into account the rectangular shape of the column cross-section, the presence of corrosion damage in concrete and rebars, various column restraints, and column flexibility. The impact is assumed to be soft. The calculation results obtained using the engineering method are based on provisions of [29,30].

The purpose of this paper is to evaluate the mechanical safety of reinforced concrete columns with corrosion damage in the event of a horizontal impact accident. In this case, a preliminary assessment for engineering calculations and designs should be carried out by engineers without the use of cumbersome finite element models. To achieve this goal, an analytical model is constructed to establish a relationship between the magnitude of horizontal impact and compressive force in the presence of different corrosion spot locations. In essence, the calculation methodology is based on the construction of formulas for calculations at two levels of stress–strain state. The first one assumes the possibility of plastic fracture, and the second one at high load intensities—brittle fracture. The equations of equilibrium of internal forces are used for these constructions. The advantages of this approach include simplicity and low computational requirements, while the disadvantages include incomplete consideration of the nonlinear behavior of the structure in the pre-fracture stage. Engineering methods for such a problem are currently insufficiently developed or absent.

The novelty of the work lies in the proposal of an analytical engineering method of calculation of corrosion-damaged columns for horizontal impact load such as pulse. The practical significance of the research is that an approach to the preliminary assessment of the safety of reinforced concrete columns without labor-intensive finite element procedures is proposed, including when changing the location of the corrosion focus on the height of the structure.

2. Materials and Methods

2.1. The Problem Statement and the Analytical Model

The stress–strain state of traditional and flexible reinforced concrete columns is studied in terms of flexibility. Each type has the same cross-section, strength class of concrete and rebars, and reinforcement scheme. A flexible column is a structure where flexibility exceeds the value of $\lambda =50$. Columns are assumed to be part of the structural scheme of a building; they are the end columns, but they are not the corner columns. Adjacent spans of a building are assumed to be of equal size, and the column is mainly subjected to compression and minor bending caused by arbitrary eccentricity. The cross-section is assumed to be square, and the reinforcement is symmetrical.

The following limitations are used in the development of the engineering approach to the analysis of bearing capacity:

-

The geometrically linear behavior of the object is not accounted for in the calculation of deflections from horizontal loading in the absence of loss of stability from compression by the longitudinal force;

-

It is considered that the structure loses strength when the stresses in concrete reach the value of design resistance at failure in the direction normal to the longitudinal axis;

-

To estimate the strength of concrete in the sloping direction, the value of the cohesive stresses is used, which takes into consideration the confinement of concrete according to the Mander model and the shear-bearing capacity of concrete and rebars;

-

The dynamic behavior of the structure is accounted for by the use of a dynamic coefficient for the load in the quasi-static analytical calculation, and the Rayleigh damping model is used in the FEM calculation;

-

Formulas (1)–(3), (10), (12), and (13) are obtained for the problem to be solved on the basis of the normative formulae [31]; (4) and (11) on the basis of consideration of the static equilibrium equations; (5) from length similarity; (6)–(9) on the basis of the basic formulas of the resistance of materials; (14) and (16) on the basis of the provisions of the finite element method; and (15) on the basis of the law of conservation of impulses.

In the theoretical model, the bearing capacity is evaluated on the assumption of the truth of the following conjunction of the strength conditions:

$$\left(N(t)\le N_{ult}^{cur}(t)\right)\wedge \left(P(t)\le P_{ult}^{cur}(t)\right)$$

(1)

where $N(t)$ and $P(t)$ are actual longitudinal and lateral loads taken up by the column during corrosion damage time t; $N_{ult}^{cur}(t)$ and $P_{ult}^{cur}(t)$ are the limit values of loads that can be taken up by the column without failure.

These forces (1) can be determined by making the following limit curve of the bearing capacity (LCBC)

$$N_{ult}^{cur}(t)=f(P_{ult}^{cur}(t))=f\left(\phi ,R_{cc}(t),A_c(t),R_{sc}(t),A_{sc}(t),z,\left\{C\right\}\right)$$

(2)

where A_c(t) and A_sc(t) are the areas of concrete and compressed reinforcement bars, respectively; φ is the longitudinal bending coefficient; R_cc(t) and R_sc(t) are design compressive strength values of these materials, taking into account actual corrosion damage; z is the coordinate of the point of application of force pulse $P{t}_0$; $\left\{C\right\}$ are options of constraints for the top and bottom nodes of the column.

Values of R_cc(t) and R_sc(t) in Equation (2) can be identified experimentally using the accelerated corrosion method or extracted from the findings of earlier experimental studies that need reduction factors to find more accurate initial values of R_cc and R_sc.

It is known that if columns have stirrups, concrete is stronger due to its confinement by reinforcement bars; concrete strength is calculated using the Mander model [32]. However, it is experimentally and theoretically proved in [33] that a tensile concrete zone in the section significantly reduces the effect of concrete confinement. Since the section that takes up loads follows the eccentric compression pattern under the action of lateral impact, no strength increase due to concrete confinement is taken into account in the engineering method. In numerical modeling, the effect of confined concrete is conveyed by higher cohesion stress for the Drucker–Prager model. In addition, strength reduction can be corrected in the calculation by changing the elastic modulus for a portion of the material. The LCBC function is plotted using characteristic points A, Bi, and C (Figure 1). The sequence of steps determining the construction of the LCBC is shown in the flowchart in Figure 2.

Here, LCBC plotting is considered (Figure 1a). Initially, the coordinates of the first point (A) are identified; the same is performed for point (C) and points Bi. Afterward, the hyperbolic approximation of these points can be made, and expression (2) can be obtained in the analytical form. The curve plotting procedure is described below.

The Euler’s load, triggering stability loss (curve 1 in Figure 1) of flexible columns, and the condition of material strength (curve 2 in Figure 1) of conventional columns with flexibility $\lambda \le 50$ are used to determine the first point (A):

$$N_{cr,A}(\lambda )=\left\{\begin{array}{c}\frac{{\pi }^{2}E{J}_{red}}{l_0^2}, \lambda >50\\ \phi \left[R_{cc}(t)A_c(t)+R_{sc}(t)A_{sc}(t)\right], \lambda \le 50\end{array}\right., P{t}_0{}_1=0,$$

(3)

where $\phi$ is the longitudinal bending coefficient depending on the actual value of $E{J}_{red}$, which is the bending stiffness of the column reduced to values typical for concrete; $l_0$ is the design length of the column, taking into account the type of support constraints.

The third point (C) is determined for the condition of (1) the absence of compressive force and (2) the maximum value of lateral pulse $P(t)$, acting for one second. This pulse value can be found for a symmetrically reinforced column from the equation of equilibrium for a beam in the absence of a substantial longitudinal force. A symmetrically reinforced section is described by the expression in which bending moment M, triggered in the column by the lateral load, is equal to ultimate moment $M_{ult}$. Let the pulse be applied at point $z=0.5l$. Constants, comprising set $\left\{C\right\}$, convey the hinge support of both nodes of the rod. Then, the following equation is made:

$$M_{ult}^{cur}(t)=P_{ult}^{cur}(t)l/4=R_{sc}(t)A_{sc}(t)(h_0-a^{\prime }), h_0=h_{cor}-a$$

(4)

where $h_{cor}$ is the height of the corrosion-damaged column section; if the surface layer of concrete is not destroyed, then $h_{cor}=h$, and $a, a^{\prime }$ is the minimum distance between the tensile fiber (or the compressed fiber in case of $a^{\prime }$) of the section and the center of gravity of tensile rebars (or compressed rebars in case of $a^{\prime }$). When the layer, with thickness z, fails, as shown in Figure 1b, the value of $h_0$ is measured relative to the inner boundary of layer z.

If LCBC is available, the value of $P_{ult}^{cur}(t)$ can be determined for the section with coordinate $y=0.5l$ in case of corrosion damage there. If the corrosion spot does not coincide with the y coordinate, for which the bearing capacity curve is made, the value of force M for the corrosion spot, calculated using Formula (5), should be divided by the coefficient $k_{sp}$:

$$k_{sp}=\frac{l-x_{sp}}{l}$$

(5)

where $x_{sp}$ is the coordinate of the corrosion spot, measured from the support to the middle of the span; $l$ is the geometric length of the column. Let the lateral pulse be applied in the middle of the column length, and let the corrosion spot be located at a distance of one-fourth of this length. Then, the left part of Equation (4) must be multiplied by $1.(3)$ times at point $k_{sp}=l/\left(l-0.25l\right)=1.(3)$.

Points B_i are determined for values of $N_i=N_{i-1}+\Delta$, starting from point $N_1=\Delta$. For example, if $\Delta =100 \mathrm{kN}$, there is eccentric compression when longitudinal force N₁ acts with eccentricity $e_1$, triggered by deflection $f_{x1}$ caused by lateral loading $P_1$ and random eccentricity $e$. The lateral load is applied at point $y=0.5l$. Let us calculate the eccentricity value for the case of the symmetrical reinforcement of the column:

$$e_1={\eta }_1\left(e+f_x+\frac{h_0-a^{\prime }}{2}\right),$$

(6)

$${\eta }_i=1/(1-N_i/N_{cr,A}),$$

(7)

$$f_x=\frac{P_1{l}^3}{48E(t)J_{red}},$$

(8)

$$J_{red}=\frac{b_{cor}{h}_{cor}^3}{12}+2\frac{E_s(t)}{E_b(t)}{A}_s{\left(\frac{h_0-a^{\prime }}{2}\right)}^2,$$

(9)

$$e=\max \left(h/30, l/600, 0.01\right),$$

(10)

where $E_s(t)$ and $E_b(t)$ are moduli of elasticity of rebars and concrete; ${\eta }_1$ is the coefficient of the column axis deviation in the horizontal direction due to the action of force N_i; $f_x$ is the maximum horizontal deflection from the action of the pulse of force $P_1$; $J_{red}$ is the axial moment of inertia of the section reduced to concrete; $e$ is random eccentricity (m) determined by the column length l and the size of its section h.

Let the condition of the strength of the normal section, derived from the static equilibrium equation of internal forces (see Figure 1b and the column constraint condition (1) in Figure 1a), be formulated as follows:

$$k_{sp}{P}_{i}l/4+N_i{e}_i=\left[R_{cc}(t)b_{cor}{x}_{cor}(h_0-0.5x_{cor})+R_{sc}(t)A_{sc}(t)(h_0-a^{\prime })\right]$$

(11)

where $x_{cor}$ is the height of the concrete compression zone in the section. Its value is determined by the value of ${\xi }_R$ for the corrosion-damaged element according to the following formula:

$${\xi }_R=\frac{0.9}{1+\frac{R_s}{E_s{\epsilon }_{b2}}}$$

(12)

where $E_s(e),{\epsilon }_{b2}$ is the modulus of elasticity of rebars and ultimate plastic deformations in concrete, assumed to be equal to 0.0035; b is the width of the rectangular section of the column; the value, equaling 0.9, is determined from the condition of the static equilibrium in the section of the column, taking into account the displacement of the compressed zone of concrete towards the central axis in case of corrosion damage.

Hence, the value of $x_{cor}$ is determined:

$$\begin{array}{ccc}{x}_{cor,i}=& \left\{\begin{array}{c}\frac{N_i}{\phi R_{cc}(t)b_{cor}},\\ \frac{N_i}{R_{cc}(t)b_{cor}-a},\frac{x_{cor}}{h_0}<{\xi }_R\end{array}\right.& 1\ge \frac{x_{cor}}{h_0}\ge {\xi }_R,\end{array}$$

(13)

Here $\phi$ is the function shown in

Table 1. The value of coefficient $\phi$.

$l_0$/h	6	10	15	20
$\phi$	0.92	0.9	0.8	0.6

.

To determine the condition to be employed, it is necessary to calculate $x_{cor}$ using one of the formula (13), and then, if necessary, the value must be validated using the other formula. By substituting the calculated value of $x_{cor}$ into (11), taking into account (6)–(10), one can obtain the value of $P_1$ for a preset value of N₁, thus, determining point B₁ of LCBC for i = 1. Similar calculations are performed for other points of B_i. To determine the final value of the pulse load, it is necessary to divide the obtained static value by the dynamic factor. Calculations show that this coefficient is in the range of 1.05–1.30.

2.2. Numerical Model of the Stress–Strain State Evaluation

2.2.1. Modeling of Loads

Loads $F(t)$ are determined by multiplying constant $P$ by the function of the unit load change in time $\overline{P}(t)$

$$F(t)=P\overline{P}(t)$$

(14)

In the dynamic formulation, the loads, for which no time law is preset, are considered as suddenly applied. If, in the course of normal operation, these loads trigger a condition in which the structure has less than a 25% safety margin, a sudden pulse load may lead to its failure. The nature of horizontal displacements of the column depends on the pulse shape shown in Figure 3a,b. Vertical loads are represented as N functions shown in Figure 3c,d. The action of lateral pulse load is modeled by the function shown in Figure 3c,d in red color.

In Figure 3, $X_{d\max }$ is the maximum dynamic displacement; $X_d$ is the displacement following the decay of vibrations in the interval $\left[t_2;t_3\right]$ in Figure 3d at constant pulse intensity $F(t)$; $X_{ds}$ is the displacement value after the pulse action; $X_{ds}$ is the displacement value during dynamic relaxation following vertical loading; $\left[0;t_0\right]$ is the interval of vertical loading; $\left[0;t_0\right]$ is the interval of vibration decay after the application of this load; $\left[t_2;t_3\right]$ in Figure 3c is the interval of pulse application; $\left[0;t_4\right]$ is the total analyzed time of the dynamic transient process.

Within a relatively small time frame of pulse loading, if the indenter mass is much smaller than the mass of the column, the inertial properties of the indenter can be described as follows. The mass of the indenter m acts on the column at finite velocity $\overrightarrow{V}$, transferring the body pulse $m\overrightarrow{V}$ to the column. This pulse is equal to the pulse of a sudden equivalent static force $2\overrightarrow{H}\Delta t$, $\Delta t=1$ s. In this case, if the actual time of loading is t, the static equivalent force used in the calculation can be found by transforming vectors into scalars for horizontal loading in the following formula:

$$P=H_x=0.5m{V}_{x}t$$

(15)

2.2.2. Equation for Dynamics, Materials, and Kinematic Constraints

The following system of resolving equations can be written within the framework of FEM in case of a lateral impact on a reinforced concrete column:

$$\begin{array}{c}\left[M\left(\left\{Z\right\}\right)\right]{\left\{\ddot{Z}\right\}}^{}+\left[C\left(\left\{Z\right\}\right)\right]\left\{\dot{Z}\right\}+\left[\left[K_c\left(\left\{Z\right\}\right)\right]+\left[K_r\left(\left\{Z\right\}\right)\right]\right]\left\{Z\right\}\\ =\left\{F(t)\right\}+\left\{G\right\}\chi (t)+\left\{H(t)\right\},\end{array}$$

(16)

where $\left\{Z\right\}$, $\left\{\dot{Z}\right\}$, and $\left\{\ddot{Z}\right\}$ are vectors of nodal displacements, velocities, and accelerations, respectively; $\left[M\left(\left\{Z\right\}\right)\right]$ and $\left[C\left(\left\{Z\right\}\right)\right]$ are matrices of finite element masses and damping; $\left[K_c\left(\left\{Z\right\}\right)\right], \left[K_r\left(\left\{Z\right\}\right)\right]$ are stiffness matrices of concrete and rebars, respectively; $\left\{F(t)\right\}, \left\{H(t)\right\}$ are vectors of nodal vertical and horizontal loads; $\left\{G\right\}\chi (t)$ is the product of gravity forces and the Heaviside function.

In this case, the damping matrix is determined using the simplified Rayleigh scheme, in which only structural damping is taken into account. Here, this function is defined as the product of the total stiffness matrix and the $\beta$ coefficient. This coefficient can be accurately determined by means of an experiment, but for a reinforced concrete structure in a state that is close to the limit one, the task becomes considerably more difficult. The difficulty is related to cracking, crack propagation, and opening in the process of vibration or changes in the material structure and the damping coefficient itself. For the frame structures with no initial damages, the initial value of the $\beta$ coefficient is assumed to be equal to 0.05. However, experimental studies show that this coefficient can be assumed to be equal to 0.1 due to imperfections of nodal joints and other factors.

Calculations take into account the concrete deformation diagram with regard to concrete confinement in accordance with the Mander model modified for the case of eccentric compression [34]. Initially, the resistance of confined concrete is calculated as follows:

$$R_{cc}=R_c\left(-1.254+2.254\sqrt{1+7.94\frac{R_l}{R_c}}-2\frac{R_l}{R_c}\right), R_l=k_e\frac{2R_{sc}{A}_{sc}}{sd},$$

(17)

where $R_c$ is the compressive strength of regular concrete; $k_e$ is the coefficient characterizing the ratio of areas of effective confinement to core confinement; $s, d$ is the spacing between the stirrups that ensure the height-wise confinement and the length of the stirrup parallel to one of the column sides.

Afterward, cohesion stress C (Figure 1b) is corrected for the Drucker–Prager model by adding the factor equal to the $R_{cc}/R_c$ ratio. In the course of modeling, an approach is applied by virtue of which rebar is preset by beam elements and concrete is preset by volumetric hexahedra. In this case, rebars can be connected to concrete by means of friction contact, rigid or gap elements. Calculations show that within the framework of this problem, the actual ductility of the connection between rebars and concrete does not greatly affect modeling results. That is why this connection is assumed to be rigid. Longitudinal rebars and lateral stirrups deform in accordance with the Prandtl diagram. Concrete cracking and crack propagation in the concrete, which is initially not damaged by corrosion, is taken into account by the reduced modulus of elasticity of concrete. A reduction in the initial value of the modulus of elasticity of concrete to 0.1 is assumed for the compressed zone in bending.

2.2.3. Model of Corrosion Damage and Its Implementation for Numerical Methods

Let us consider corrosion as a corrosion spot with size $z_1$ along the column height. In the most dangerous case, the entire column is in the adverse medium. When concrete corrosion is modeled, the layer-by-layer degradation of mechanical properties is taken into account. Layer A (highlighted in yellow in Figure 4) is considered to be the most heavily damaged; its modulus of elasticity does not exceed 8% of the initial value. Its strength is also substantially reduced. Layer B (highlighted in magenta) features transient processes of corrosion development deeper into the concrete; in this case, a change in the modulus of elasticity over the layer thickness will be represented as a parabolic function $K(i)$. Here, i is the number of layers into which layer B is divided. In accordance with the caption of Figure 4a and [35]:

$$\begin{gathered} E_{c(z)}=0.08E_c;E_{c(z+i)}=E_{c(z)}+\Delta E_{c}K(i); E_{c(z+\delta )}=E_c; \\ K(i)=1+z_{Bi}/\delta +z_{Bi}^2/{\delta }^2, \end{gathered}$$

(18)

where $z_{Bi}$ are the coordinates of the i-th layer in layer B.

Values of moduli of elasticity in layer B, if i = 4, are calculated in Figure 4b.

As a result of electrochemical corrosion, the rebar steel softens, and in addition, the effective area of longitudinal rebars decreases as a result of electrolytic dissociation. This reduction can be taken into account for an individual case of corrosion at some point in time by applying coefficients $w_0, w_1$ (see Figure 4a). The values of these coefficients depend on the duration of corrosion processes and can be taken as 0.75–0.95 according to the results of experimental studies. In case of compression accompanied by bending in the corrosion spot, the position of the neutral axis is determined by the value of p in such a way that dimensions of the compressed concrete zone $x=\delta +p$, or the damaged concrete layer, are not taken into account when the value of x is calculated. As a result, the compressed zone of concrete is displaced closer to the central axis of inertia, and the bearing capacity of the section is reduced. The calculation of stresses ${\sigma }_{cc}, {\sigma }_{cc1}, {\sigma }_{ct}$ in the form of analytical dependence is problematic. Therefore, to evaluate the bearing capacity, in particular, the value of $x$, a simplification is applied. This simplification is provided in Formula (12) and described as the failure caused by the action of the compressive force and lateral pulse loading and related to the failure of concrete due to compression and the deterioration of cohesion in the direction of inclined sections.

3. Results

3.1. The Case of LCBC Made for a Reinforced Concrete Column with and without Corrosion Damage

The height of the bearing capacity of the column is 4 m, its cross-section is 0.4 × 0.4 m, and it is subjected to 1.000 kN compression, it is evaluated to prove the efficiency of the proposed engineering method. The mechanical force pulse, equal to 120 kN s, is applied to the column in the middle of the span, according to Figure 3. Further, Equations (5)–(9) are used in the calculations. The column constraints correspond to (2) shown in Figure 1a, where the bottom support node is rigidly fixed, and the top one is hinge fixed. The rebars of the column are symmetrical; longitudinal rebars 4d28 are located in the corners of the section, and stirrups d10 have a spacing of 0.25 along the height of the column. The column has a corrosion spot in the middle of the span that has height $y_1=0.5 \mathrm{m}$. In this corrosion spot, concrete and rebars are corrosion-damaged along the column perimeter, as shown in Figure 3a. The corrosion spot is located in the middle of the column height. Mechanical characteristics of undamaged and corrosion-damaged column materials are provided in

Table 2. Initial data and parameters used in the calculations.

Parameter	Value (Normal Conditions)	Value (Corrosion Damage)	Reduction Factor
Initial modulus of elasticity of concrete	27.5 GPa	16.5 GPa	0.6
Modulus of elasticity with account for cracking	0.275 GPa	0.165 GPa	0.6
Modulus of elasticity of rebars	200 GPa	180 GPa	0.9
Design resistance of rebars, concrete	435 MPa; 11.5 MPa	391.5 MPa; 9.20 MPa	0.9; 0.8
Coefficient of design length	0.7	0.7	-
Effective section height $h_0$	0.35 m	0.325 m	0.89 (3)
Area of rebars in tension and compression	12.32 cm2	11.088 cm2	0.9; 0.9
Distance a (Figure 1)	0.05 m	0.05 m	-
Dimensions of zones z and $\delta$ (Figure 2)	-	0.05 m; 0.1 m	-
Cohesion	3.3 MPa	1.32 MPa for zone (z); 2.31 MPa for zone ($\delta$)	0.36 0.7

. Other parameters needed for analytical and numerical modeling of the stress–strain state of the column are also provided here.

Initially, LCBC is made for the system without corrosion damage. Geometric characteristics of the column section, reduced to concrete, are calculated:

-

The sectional area:

$$A_{red}=hb+\frac{E_s}{E_b}\left(A_{sc}+A_s\right)=0.4\cdot 0.4+\frac{200}{27.5}\left(12.23+12.23\right)\cdot {10}^{-4}=0.1754 {\mathrm{m}}^2;$$

-

The axial moment of inertia:

$$J_{red}=\frac{b{h}_{}^3}{12}+2\frac{E_s}{E_b}{A}_s{\left(\frac{h_0-a}{2}\right)}^2=\frac{0.4\cdot {0.4}^3}{12}+2\frac{200}{14.5}12.32\cdot {10}^{-4}\cdot 0.325\cdot 0.5=0.002682 {\mathrm{m}}^4.$$

The column flexibility:

$$\lambda =l_0/i=\mu l/\sqrt{J_{red}/A_{red}}=0.7\cdot 4/\sqrt{0.002682/0.1754}=22.64<50.$$

The second equation from the system (3) is used to identify the coordinates of point A: $N_{cr,A}(\lambda )=\phi \left[R_{cc}{A}_c+R_{sc}(A_{sc}+A_s)\right]=0.91\left[11.5\cdot 0.16\cdot {10}^3+435\cdot 2\cdot 12.32/10\right]=2649 \mathrm{k}\mathrm{N}.$ $P=0$.

Coordinates of point C (see Figure 1a) are identified using the following condition: $\left(N=0\right)\wedge \left(P\to max\right)$; the maximum bending moment will be in the rigid support node for the preset scheme of schematic constraints under the action of force P in the middle of the span. It is equal to $M_{\max }=3Pl/16$. However, for a symmetrically reinforced column with some safety margin, the ultimate moment is determined by the right part of Equation (4), then

$$P_{ult}^{cur}=16R_s{A}_s(h_0-a)/3l=16\cdot 435\cdot 12.32/10\cdot (0.35-0.05)/12=214.36 \mathrm{k}\mathrm{N}$$

Now let us find the coordinates of points B_i. Having analyzed expression (13), we can see that these points will form two curves depending on ${\xi }_R$. If there is no corrosion damage, the value of 0.8 will be used in Formula (12) instead of 0.9. Then, ${\xi }_R=0.8\left(1+R_s/(E_s{\epsilon }_{b2}\right)=0.4933$. Let us consider the calculation of values for one of the LCBC points, if $\xi <{\xi }_R$, and for one of the points, if $\xi \ge {\xi }_R$.

Let us assume that $N_1=500$ kN. The verification is made using Formula (13): $x_0=N/R_{cc}(b-a)=500/(11.5\cdot (0.4-0.05)\cdot 1000)=0.1242$ m, $\xi =0.1242/0.35=0.3548<{\xi }_R$.

The condition is fulfilled; $x=x_0$ in Equation (11). Random eccentricity is calculated as follows: $e=\max \left(h/30, l/600, 0.01\right)=\max (0.01(3),0.00(6),0.01)=0.01(3).$

The horizontal deflection of the column is determined using Simpson’s method. If some calculations are omitted, the following result is obtained:

$$f_x=P_1{l}^3/(192E_b{J}_{red})=P_1\cdot 4^3/(192\cdot 27.5\cdot {10}^3\cdot 0.002682)=P_1\cdot 0.004519 \mathrm{m}.$$

The coefficient, which takes into account the effect of the longitudinal force on deflection, is calculated as follows:

${\eta }_1=1/(1-N_1/N_{cr,A})=1/\left(1-500/2649\right)=1.2326$, and then eccentricity is calculated as follows:

$$\begin{array}{cc}{e}_1={\eta }_1\left(e+f_x+0.5\left(h_0-a^{\prime }\right)\right)\hfill & =1.2326\left(0.01(3)+0.004519P_1+0.5(0.35-0.05\right)\hfill \\ & =0.2013+0.00557P_1.\hfill \end{array}$$

Let us formulate the condition of strength (11) with respect to this case of kinematic constraints of supports, if $k_{sp}$ = 1, since in this case corrosion damage is not taken into account. Then, $3P_{1}l/16+N_1{e}_1=\left[R_{cc}bx(h_0-0.5x)+R_{sc}{A}_{sc}(h_0-a^{\prime })\right]$. By substituting the expression for $e_1$ into this formula, the following is obtained: $3P_{1}l/16+N_1(0.2013+0.00557P_1)=\left[R_{cc}bx(h_0-0.5x)+R_{sc}{A}_{sc}(h_0-a^{\prime })\right]$. $P_1=\frac{\left[R_{cc}bx(h_0-0.5x)+R_{sc}{A}_{sc}(h_0-a^{\prime })\right]-N_{1}0.2013}{3l/16+0.00557N_1}$.

By substituting $N_1=500$ kN and previously calculated values into this expression, the limiting value of the static lateral load is obtained. In this case, the column will have the required bearing capacity. The following value is obtained: $P_1=\frac{\left[11.5\cdot 0.4\cdot 0.1242(0.35-0.5\cdot 0.1242)+435\cdot 12.32\cdot 0.1\cdot (0.35-0.05)\right]-500\cdot 0.2013}{3\cdot 4/16+0.00557\cdot 500}=295.8$ kN.

Now let us consider the case when $\xi \ge {\xi }_R.$ We assume that $N_1=1000$ kN. According to Formula (13), we verify $x_0=N/R_{cc}(b-a)=1000/(11.5\cdot (0.4-0.05)\cdot 1000)=0.2448$ m, which is used to find the following value: ${\xi }_0=0.2448/0.35=0.6994>{\xi }_R.$ In this case, the calculation is made using the first equation from Formula (13): $x=N_2/(\phi R_{cc}b)=1000/(0.91\cdot 11.5\cdot 0.4\cdot 1000)=0.2388$. $\xi =0.682>{\xi }_R.$

The values of variables are calculated similarly to the sequence of actions described above: $f_x=P_2\cdot 0.009038$. ${\eta }_2=1.6064$. $e_2=0.2623+0.01451P_2$. $P_2=204.55$ kN. Hence, for the above ranges of $\xi$ values, the LCBC curve can be plotted both with and without corrosion damage. The results are shown in Figure 5.

According to Figure 5, if at $\xi <{\xi }_R$, the calculation results of the analytical model are in good agreement with the results of numerical analysis in the 3D formulation. The curve shows the points obtained by calculation. The tail end of the function in the analytical model differs greatly from curve 1, but it is acceptable because minimum values of longitudinal forces for this column in the condition of its real-life operation cannot be below 500 kN. In the $\xi \ge {\xi }_R$ range, the analytical calculation allows for a high safety margin if compared with the finite element analysis, and it is also acceptable.

For this structure, if N = 1000 kN, the calculation is made in the dynamic formulation according to Figure 3d, with the value of the horizontal pulse being equal to 204.55 kN s. As a result, the structure collapsed. Then, the pulse value successively decreased. The structure retained its bearing capacity at $P_{ult}^{}=185$ kN s. The dynamic ratio k_d was 1.1.

Now let us make calculations in accordance with the problem statement that encompasses the presence of the corrosion spot. Numerical modeling is carried out by taking into account the corrosion damage according to Figure 3 and using Equations (13)–(17). Hence, the values of the model characteristics are reduced, as shown in Table 2. The curves, illustrating the adequacy of the analytical model, are shown in Figure 6.

LCBC allows verifying condition (1) by determining whether the point, characterized by quasi-static forces N, P, is below the bearing capacity curve if corrosion damage is taken into account. Therefore, P = F(t) k_d = $120\cdot 1.1=132$ kN, and the longitudinal force is N = 1000 kN.

It is evident from Figure 6 that under the preset loading conditions that encompass lateral pulse loading, the column retains its bearing capacity if it is not subjected to corrosion damage. If the column is corrosion damaged, the finite element analysis and the engineering method prognosticate its failure (point T lies above curves 3 and 4; in fact, it is the verification for condition (1)). In the case of this type of corrosion damage, the bearing capacity can only be guaranteed if lateral loading equals 25 kN, which corresponds to the lateral pulse of 22.7 kN s.

3.2. Numerical Analysis of SSS for Corrosion Damaged Square Section Columns: Comparison of Calculations and Experimental Results

Experiment data and conditions of an experiment designed for the LCBC verification are taken from [35]. In this work, a 1.5 m long column specimen is considered. The column is symmetrically reinforced using four longitudinal rebars d12 in the angular zones of the section with design resistance $R_s=440$ MPa. Stirrups ensure the strength of inclined sections near the column supports under the action of the horizontal pulse. These tests are made without corrosion damage. Characteristics of tested specimens are summarized in

Table 3. Characteristics of specimens applied in the experiment.

Specimen Grade	Section Dimensions, m	Section Height, h0, m	Z0, m	${\mathit{R}}_{\mathit{b}}$, MPa	N, kN	P, kN D/S
B12-D	0.240 × 0.150	0.214	0.180	30.6	0	114/98
K12-D-1	0.245 × 0.150	0.200	0.170	27.7	120	180/146
K12-D-2	0.246 × 0.156	0.212	0.190	27.7	240	246/178
K12-D-3	0.240 × 0.155	0.220	0.195	28.2	480	274/200

. Here, $Z_0$ is the distance between centroids of effective rebars; D/S is the identifier of the testing mode, where D stands for dynamic testing, and S stands for static testing. To find out whether the proposed engineering method is in agreement with the experimental studies, corrosion-free LCBC is made.

This effort takes into account the loading scheme implemented in the experimental unit, for which the bending moment is equal to PA/2 if unaffected by any lateral force. Here, A = 0.5 m is the distance from the dynamometric support to the two-point load cross-arm. The value of a is not taken into account to calculate the value of the compressed zone x in Formula (13) in the absence of corrosion damage b_cor = b, and the longitudinal force is multiplied by coefficient $\eta$ instead of the multiplier $1/\phi$. The results of the comparison between the engineering method and the experiment are shown in Figure 7.

When the convergence evaluation of the finite element method procedure was used, control criteria such as discrepancy on nodal forces 0.001, discrepancy on displacements 0.0005, and discrepancy on external and internal force work 0.00001 were used. This indicates the stability of the solution at the adopted mesh size. In total, the model contained 6561 nodes, 5120 hexahedral, and 752 rod elements. At the same time, the calculation of the column on a larger mesh of volumetric elements is inexpedient because it is impossible to model the zone of corrosion damage. Calculation on a finer mesh leads to huge time consumption (days) and does not change the results, which indicates the feasibility of developing the engineering approach proposed in the paper.

4. Discussion and Further Investigation Prospects

As studies have shown, the proposed analytical methodology and calculations based on the finite element method have a discrepancy. This discrepancy in calculations based on the engineering method leads to the design of the structure with a safety margin, which is acceptable, and in some cases, can be positive. The fact that rate of corrosion development in depth is a very complex and difficult-to-predict process. If the corrosion by the time t will be more, the proposed analytical technique will give a more reliable result, and if less, the result on reliability will be the same. The discrepancy in the results is due to the fact that the finite element method in the considered problem formulation takes into account more factors of the real behavior of the structure, especially nonlinearities, which are described below.

Having analyzed the LCBCs, obtained using different approaches to the analysis of the stress–strain state, one can find out that there is an area (for example, the area between curves 3 and 4) that can lead to different predictions if actual loads are within its boundaries. Namely, the finite element analysis makes a positive prediction in terms of the bearing capacity at point P = 75 kN, N = 1000 kN, while the engineering method predicts failure. This result is the one that is anticipated because finite element modeling takes into account the yielding of rebars and the plastic failure of concrete. In the engineering method, these phenomena are simplified, and only stresses are recorded in concrete and rebars. Nevertheless, the proposed engineering method may be used to make approximate predictions of the bearing capacity. It is applicable because, in an emergency situation, the engineering method guarantees some safety margin, while it is inappropriate to save materials in this case. The main purpose is to ensure safety.

In the case when predicted values of the bearing capacity, obtained using different methods, differ, the only way to check the trustworthiness of the prediction is to conduct a full-scale experiment. However, it is very difficult to carry out this experiment because, in addition to the geometry of the full-scale model, it is necessary to obtain the values of mechanical characteristics and reproduce the corrosion damage. Therefore, this paper compares the engineering method with the actual experiment without taking corrosion into account.

The prospects for further research into the subject under consideration lie in the area of refinement of analytical dependencies needed for the engineering method in case of large values of the longitudinal force and small values of the lateral pulse: this is the LCBC portion shown in the Figures by a dashed line. In the presence of substantial corrosion damage, the most accurate evaluation of the bearing capacity in this area is an important issue. In addition, relevant challenges accompany the development and application of the engineering method, taking into account the dimensions of the corrosion spot, the presence of two corrosion spots, secondary geometric imperfections, as well as other aspects that may affect the real technical condition of the columns in the aggressive medium. The LCBC approach can be effectively applied to evaluate the stress–strain state within the framework of optimization algorithms applied to bearing structures [36,37,38] in the case of repeated calculations and changing parameters.

5. Conclusions

• An analytical engineering method was developed for the approximate evaluation of the bearing capacity of axially compressed corrosion-damaged reinforced concrete columns subjected to lateral pulse loading as a result of accidents. The efficiency of the analytical method is confirmed by a comparison between the results of finite element modeling and a full-scale experiment.
• The model of corrosion damage, designed for a specific moment in time, is developed for 3D finite element schemes. This model is based on the parabolic law of deteriorating mechanical characteristics of concrete depending on depth-wise corrosion propagation.
• It is established that the shape of the pulse has a great effect on a corrosion-damaged column. In this case, given that the same kinetic energy is transferred to the structure, the most dangerous is the pulse with the smallest peak and the longest duration.
• The best evaluation result, obtained in terms of the ultimate bearing capacity of columns, is generated if the engineering method is applied in the range of 0.4–0.8 of the ultimate value of the horizontal pulse in case of compression by a force of 0.25–0.7 of the ultimate value. Deviations in results are caused by an increase in the safety margin of the structure, which allows for its use for design purposes.
• It is established by the calculation that an increase in corrosion damage of concrete leads to its brittle failure under accidental impacts, while an increase in the depth of corrosion damage under substantial compressive loads leads to the local loss of stability of rebars in the zone of action of the lateral pulse and a substantial reduction in the column resistance to progressive collapse.