Seismic Resistance of Reinforced Concrete Building Frames Based on Interval Assessment of the Coefficient of Permissible Damage

Abstract

The main method for assessing the seismic resistance of buildings in the standards of most countries is the linear-spectral method. This method allows for the calculation of the spatial model of a building for seismic load in the elastic range without resorting to direct integration of the equations of motion. Nonlinear characteristics of reinforced concrete structure materials are usually considered integrally using the reduction factor. However, the values of this factor in the Russian standards are not sufficiently substantiated, as the later studies show. To determine the coefficient of permissible damage (reduction factor), six reinforced concrete frames were considered, with different parameters such as span length, number of spans, and number of floors. The design parameters of beams and columns (section sizes, reinforcement, etc.) were preliminarily selected based on the calculation using the linear-spectral method. In the second stage, numerical modeling was carried out in the OpenSEES PC to implement the pushover analysis procedure. Then, the coefficient of permissible damage was estimated by processing the capacity curves obtained on the basis of nonlinear static calculation. The value of the sought coefficients is practically not affected by the number of stores of the frame; however, with an increase in the number of spans, the coefficient K₁ increases, which is explained by a decrease in the plasticity of the system. On average, for the frames under consideration, the coefficient K₁ was 0.526, which is 1.5 times greater than the coefficient proposed in modern Russian standards, K₁ = 0.35. The results obtained on the basis of pushover analysis are compared with the coefficients K₁ determined through the values of the average degree of damage (d) of the buildings according to the modified seismic scale MMSK-86. For various types of reinforced concrete frame buildings, K₁ = 0.51 was obtained. It is recommended that the coefficient K₁ for reinforced concrete frame buildings should be increased to a value of at least K₁ = 0.5 in the Russian standard.

1. Introduction

Thousands of earthquakes occur worldwide every year, some of which can have catastrophic consequences, causing numerous casualties and significant economic damage [1]. The cause of such consequences is the collapse of and fires on buildings and structures. Therefore, the main task of scientists and engineers is to ensure that the seismic resistance of buildings is designed in earthquake-prone areas [2,3,4]. During an earthquake, the oscillatory movement of the soil at the base of the building excites kinematic vibrations from building structures. Modal response spectrum analysis is the main analysis method in seismic codes for the design of building structures [5].

In this case, it is important to correctly consider the nonlinear behavior of materials, as a result of which the seismic load on the structural elements of the building is reduced. This is implicitly considered by (a) the behavior factor q in EC8 [6], (b) the structural performance factor D in Building Code of Japan (BCJ) [7], (c) the reduction factor R in the American ACI 318-05 [8] and ASCE 7-22 [9], and (d) the permissible damage factor K₁, in the Russian SP 14.13330 [10]. Modern software packages offer calculation procedures that allow you to specify the reduction factor. Dynamic analysis in the time domain provides a large amount of information on the behavior of building-bearing structures during an earthquake, but it is very difficult to implement and places high demands on the quality and detail of calculation models [11,12]. Nonlinear static pushover analysis is easier to implement and a well-proven alternative [13,14].

The pushover method is suitable to obtain the collapse mechanism of the structure, the locations of inelastic behavior, the capacity curve of the building, and a general reduction factor for the building [15]. A number of researchers recommend this method for use in reinforced concrete buildings [16,17], the behavior of which inelastic deformations play a significant role. It can be used both for assessing the seismic resistance of existing buildings [18,19] and for the design of new buildings to determine the steel reinforcement details [20,21].

The pushover analysis methodology is actively developing. In order to overcome the well-known limitations characterizing nonlinear static analysis, the recent upgrades of some building codes, such as the Italian one, provide different rules to employ in nonlinear static procedures. Among these new provisions, one regards the possibility of using a horizontal load profile proportional to the story floor forces derived from a response spectrum analysis in all cases [22].

In general, in seismic codes, the reduction factor R is the product of two parameters: the coefficient R_μ, which takes into account the reserves of seismic resistance due to the ductility of the system, and the overstrength coefficient R_u:

$$R_{\mu }={\displaystyle \frac{H_{el}}{H_{pl}}}$$

(1)

$$R_u={\displaystyle \frac{H_u}{H_{1st-hinge}}}$$

(2)

where H_el is the base shear developed in the ideal infinitely elastic system with stiffness equal to the initial stiffness (slope of the 1st elastic branch in a bilinear representation of the capacity curve) of the nonlinear system, H_pl is the base shear in the generalized yield of the nonlinear system (crossing point in the bilinear representation of the capacity curve), H_u is the base shear at the point of development of a full plastic mechanism, and H_1st-hinge is the base shear at the 1st observed strength exceedance (1st plastic hinge) which is approximately equal to the design base shear.

In Russian norms, SP 14.13330.2018 [10], the concept of the K₁ coefficient is somewhat different from the similar and inverse value of the reduction factor R. The K₁ coefficient is inversely related to the R_µ coefficient, which takes into account the reserves of seismic resistance due to the ductility of the system (Seismic-force-modification Ductility Factor).

$$K_1=1/R_{\mu }={\displaystyle \frac{H_{pl}}{H_{el}}}$$

(3)

where H_pl is the base shear of the inelastic system developed at the ultimate allowable damage (in a bilinear representation of the capacity curve), and H_el is the base shear of the equivalent ideal infinitely elastic system (Figure 1).

According to Table 5.2 of SP 14.13330.2018 [10], the coefficient K₁ is introduced for structural systems in which residual deformations and damages that impede operation may be allowed while ensuring the safety of people and equipment. The same table gives the values of coefficients K₁ for different bearing systems. For RC buildings, the following values are given:

• K₁ = 0.25—buildings with RC walls;
• K₁ = 0.3—buildings of RC panel-block constructions;
• K₁ = 0.35—RC frame without shear walls or steel bracing;
• K₁ = 0.4—RC frame with masonry infills;
• K₁ = 0.3—RC frame with shear walls or steel bracing.

The numerical value of the reduction factor R_μ depends on the type of the building’s structural system, the ductility class, and the regularity of the rigidities along the height and in the plan [23]. In essence, the reduction factor is a characteristic of the ultimate state of the structure. There are a number of criteria on the basis of which the reduction factor can be assigned [24].

The ductility factor μ can be taken as the main criterion for seismic resistance at the ultimate state of reinforced concrete frames.

The criterion of N. Newmark can be used to calculate the reduction factor [25]. According to this approach, the reduction factor depends on the period of the lowest mode of vibrations of the system T₁ and the ductility factor μ:

$$R_{\mu }={\displaystyle \frac{1}{K_1}}=\left\{\begin{array}{c}1, if T_1<0.1 \mathrm{s}\\ \sqrt{2{\mu }_{tot}-1}, if 0.1\le T_1\le 0.5 \mathrm{s}\\ {\mu }_{tot}, if T_1>0.5 \mathrm{s}\end{array}\right.$$

(4)

The Newmark theory applies because of the dominant role of the fundamental, translational period in planar frames or symmetric structures (MDoFs systems) but not in general in all kinds of structures.

The coefficients R_µ and K₁ are assigned for the entire structure as a whole and implicitly consider the elastic–plastic properties of individual structural elements. In the current version of Russian standards, the coefficient K₁ is fixed for a various building structural system (steel or reinforced concrete frame, reinforced concrete wall system, etc.). As shown by numerous studies [26], this approach to reinforced concrete buildings can lead to significant errors and decrease the reliability of the design.

The authors of [15] used a pushover analysis of a 17-story building with RC walls; the K₁ coefficient was 0.37, which is 1.48 higher than the normative value (K₁ = 0.25). The authors of [27] studied an RC frame with shear walls. In the beginning, the calculation was carried out using spectral methodology at the normative value of K₁ = 0.3. In the second stage, where the calculation was carried out with response history analysis, the deficit of strength on the frame was obvious. For the steel frame in article [28], the value of the K₁ coefficient was obtained, which exceeded the normative one by almost 100%. The authors of [29] proposed a refinement of the Newmark criterion for determining the K₁ coefficient in the presence of constructions with brittle failure mechanisms.

Obviously, depending on the design parameters of the frame reinforced concrete frame, the collapse mechanism changes, and, as a result, buildings have different ultimate ductility. For reinforced concrete frames, the collapse mechanism under horizontal load is associated with the successive formation of plastic hinges in the beams and columns of the frame until the capacity is completely exhausted.

In frame structures, the design procedure following seismic codes leads to the development of a beam-sway mechanism at the edge of the nonlinear area just before the collapse. This is achieved by imposing the Strong Column-Weak Beam capacity rule in the design procedure. Additionally, capacity design calculations also apply to shear design in order to prevent brittle failure. This type of ductile frame can fully exploit the ability of a large number of horizontal structural elements to develop plastic deformations at their end-sections until their capacities are exhausted, avoiding the development of plastic hinges in columns (except at the base) which will compromise the stability of the frame structure at an early stage. Calculations according to standards of different countries for the same frame will yield slightly different section parameters, and, as a consequence, the collapse mechanism of the building will be slightly different [30].

The Russian standard SP 14.13330.2018 [10] specifies the principles of designing frame systems. When selecting structural schemes, preference should be given to schemes in which plasticity hinges occur in the horizontal elements of the frame (beam, contour beams, etc.). In the Russian standard, there is no division into Ductility Classes. Requirements for the design and detailing of structural elements of RC frames are uniform. The beam–column joint at a distance equal of 1.5 h (h is the height of the beam (column) section) should be reinforced with transverse reinforcement (hoops), but at least every 100 mm.

In this paper, the K₁ coefficient for RC-frames is investigated, taking into account the real mechanism of their plastic collapse on the basis of pushover analysis. The obtained coefficients are compared with the normative ones. The variability of the K₁ coefficient for different design parameters, including span, number of spans, and stories, is also investigated. This study is carried out using numerical modeling in OpenSEES.

2. Models and Methods

Several reinforced concrete frames are considered (Figure 2), with varying design parameters: span of beams, number of spans, and number of stories.

The modeling and calculation of reinforced concrete frames were performed in the OpenSees v3.5.0 PC. The GiD + OpenSees v.2.9.6 graphical environment [31] was used to build the calculation model and analyze the results.

The frames examined in this study were planar frames with rigid joints of the beams with the columns (Figure 3). The frame was supported at the base under fixed conditions. The floor height for all frames was constant at 3.5 m. The distance between the column axes was 6 and 7.5 m. The columns had a square cross-section of 500 × 500 mm; the beams had a rectangular cross-section of 350 × 600 mm.

All floors were considered rigid diaphragms. Thus, all joints inside each floor will present equal displacements in analysis. The increased stiffness due to the rigidity of joints was considered in the mathematic analysis model. The P-Δ effects were also considered in nonlinear analysis to take into account geometric nonlinearities.

During the seismic analysis of the building, vertical permanent and temporary loads were considered. Permanent loads are the dead weight of the building’s structural elements, the weight of the floor and roof structure, and the weight of the masonry infills.

The dead weight of the structure was considered automatically in the OpenSees PC. The unit weight of concrete was assumed to be equal to 25 kN/m³.

The total uniformly distributed load on the beams was assumed to be 40 kN/m. This is the vertical load in the seismic combination, which combines dead and live (temporary) loads. The load from the external masonry infills was 30 kN. This load was applied to the outermost nodes of the frame on each floor.

The mass on each floor M_1,f, and total frame mass M_tot are as follows:

• P-6-3-5 M_1,f = 97.3 ton and M_tot = 486.5 ton;
• P-6-3-7 M_1,f = 97.3 ton and M_tot = 681.1 ton;
• P-6-5-5 M_1,f = 156.5 ton and M_tot = 782.7 ton;
• P-7.5-3-5 M_1,f = 118.0 ton and M_tot = 590.0 ton;
• P-7.5-3-5 M_1,f = 118.0 ton and M_tot = 826.0 ton;
• P-7.5-3-5 M_1,f = 191.1 ton and M_tot = 955.7 ton.

Pushover analysis was used to determine the actual level of system ductility. In this analysis, the nonlinear model of the frame was under the action of progressively increasing lateral floor loads. The method led to the drawing of the capacity curve of the system in terms of base shear and roof displacement.

The selection of the necessary parameters for the reinforced concrete beams and columns was carried out on the basis of seismic impact calculations using the linear-spectral method according to the methodology of SP 14.13330.2018 [10].

The seismic load was determined by the following formula:

$$S_{0\mathit{ik}}^j=m_k^{j}A{\beta }_i{K}_{\Psi }{\eta }_{\mathrm{ik}}^{\mathrm{j}}$$

(5)

where $m_k^j$ is the mass of the building or the moment of inertia of the corresponding mass of the building, referred to as point k along the generalized coordinate j, determined considering the design loads on the structures.

A is the acceleration value at the foundation level; for this calculation, it was taken to be 2.0 m/s².

β_i is the dynamic coefficient corresponding to the i-th mode of natural oscillations of buildings (Figure 4). The values used for the coefficients β_i are presented in

Table 1. The coefficient β_i for reinforced concrete frames.

Frame	Parameters	Model Number
		1	2	3	4	5
P-6-3-5	Ti (s)	0.707	0.224	0.124	0.082	0.062
	βi	1.88	2.5	2.5	2.23	1.93
P-6-3-7	Ti (s)	0.990	0.320	0.180	0.121	0.088
	βi	1.59	2.5	2.5	2.5	2.32
P-6-5-5	Ti (s)	0.705	0.225	0.125	0.083	0.064
	βi	1.88	2.5	2.5	2.25	1.96
P-7.5-3-5	Ti (s)	0.846	0.265	0.143	0.092	0.069
	βi	1.72	2.5	2.5	2.38	2.04
P-7.5-3-7	Ti (s)	1.186	0.380	0.212	0.140	0.100
	βi	1.45	2.5	2.5	2.5	2.5
P-7.5-5-5	Ti (s)	0.845	0.267	0.145	0.095	0.071
	βi	1.72	2.5	2.5	2.43	2.07

.

K_Ψ is the coefficient considering the ability of buildings and structures to dissipate energy. According to Table 5.3 of SP 14.13330.2018 [10], it was taken as 1.3.

η_ik^j is the coefficient depending on the deformation mode of the building or structure during its natural oscillations according to the i-th mode, on the nodal point of application of the calculated load, and the direction of the seismic impact.

For seismic loading in the X-axis direction, the coefficients η_ik are determined according to the following formula:

$${\mathsf{\eta }}_{ik}={\displaystyle \frac{X_{ik}\sum _{j=1}^n{m}_j{X}_{ij}}{\sum _{j=1}^n{m}_j{X}_{ij}^2}}$$

(6)

where X_ik(j) is the displacement of mass k (j) at i-th mode.

The design seismic load is as follows:

$$S_{\mathit{ik}}^j=K_0{K}_1{S}_{0\mathit{ik}}^j$$

(7)

where K₀ is the importance factor. According to Table 4.2 in SP 14.13330.2018 [10], it was taken as 1.

K₁ is a permissible damage factor, inversely related to the R_µ coefficient (Seismic-force-modification Ductility Factor).

After performing the calculation using the linear-spectral method, the reinforcement in the columns and beams was assigned. The column sections were divided into several sections: a cover layer, a concrete core, and reinforcement. The cover layer has the mechanical properties of unconfined concrete, and the core has the mechanical properties of confined concrete (Figure 5). For beams, the entire section has the properties of unconfined concrete.

The following material properties are specified:

• ordinary (unconfined) concrete: R_b = 14.5 MPa, E_b = 9667 MPa, ε_b1 = 0.0015, ε_b2 = 0.0035;
• confined concrete: R_b,tr = 18,91 MPa, ε_b3,tr = 0.00504;
• reinforcement: R_s = 435 MPa, E_s = 200,000 MPa, ε_so = 0.002175, ε_s2 = 0.025.

The theory of Mander et al. [32] was used to assign mechanical properties to confined concrete.

It is worth distinguishing between the initial modulus of elasticity of concrete E_bo and the reduced modulus E_b. The reduced modulus E_b takes into account the stiffness degradation in structural elements caused by cracking due to nonlinear deformations. The values of mechanical properties of concrete and reinforcement are taken according to the Russian standard SP 63.13330.2018 [33].

The modeling of sections was performed using the Fiber Section tool; the section was divided into separate rectangular fibers and layers of reinforcing bars. The size of one fiber was 10 × 10 mm. Each fiber was assigned the characteristics of the corresponding material.

Nonlinear analysis of reinforced concrete frames consists of two stages. First, a nonlinear static calculation of the frame for vertical loads was performed. The law of load application was linear. The number of loading steps was 50. The solution algorithm corresponds to Newton’s method; integration was performed with force control.

In the second stage, a pushover analysis of the frame for horizontal load was performed. The number of loading steps was 100. Integration was performed with displacement control of the upper right frame node in the direction of the OX axis. The stepping displacement was set equal to 45 mm. The horizontal load was applied according to the triangular law with increasing values along the height of the frame (Figure 6). The base shear was considered equal to F_o = 1 kN. The load F_i at the i-th floor level was calculated using the following formula:

$$F_i=F_o{y}_i/\sum y_i$$

(8)

where y_i is the distance from the base to the i-th floor.

In the following stages, the initial value of the load increased until the ultimate deformation was reached.

Based on the results of the pushover analysis, a bearing capacity curve was constructed, and the actual value of the system’s plasticity coefficient μ was estimated:

$$\mu ={\displaystyle \frac{{\Delta }_{pl}}{{\Delta }_y}}$$

(9)

where Δ_pl is the ultimate deformation of the building corresponding to the selected level of permissible damage.

Δ_y is the yield displacement of the frame corresponding to the transition to the elastic–plastic section of the approximating bilinear diagram.

The displacements Δ_pl and Δ_y are determined by the bilinear diagram, which was obtained by approximating the bearing capacity curve (Figure 7). The bilinearization procedure is described in code EC8 Appendix B [6]. It was assumed that A₁ = A₂.

According to SP 14.13330.2018 [10], the stiffness of the corresponding elastic system is determined by the linear-elastic behavior of the beams and columns, i.e., at K₁ = 1. Next, a linear diagram was constructed for the corresponding elastic system with stiffness equal to the initial stiffness of the bearing capacity curve K_o. Using the principle of equality of the work of elastic and nonlinear systems, the base shear H_el and displacement Δ_el were determined. The corresponding values for the nonlinear system H_pl and Δ_pl were found as the coordinates of the extremum of the bearing capacity curve.

The total energy was determined by the following formula:

$$E=E_{pl}=E_{el}={\displaystyle \frac{1}{2}}{H}_{el}{\Delta }_{el}$$

(10)

In this study, the reduction coefficient was estimated in two ways: according to Formula (3), “by definition”, and according to Formula (4), the Newmark criterion.

3. Results

The obtained reinforcement of columns and beams for the scheme in Figure 8 is presented in

Table 2. Results of calculation by the linear-spectral method.

Frame	Span, m	Number of Spans	Number of Floors	Period T1, s	As1	Floor	As2	As3
P-6-3-5	6	3	5	0.707	3ø28	1,2	3ø25	3ø32
						3	3ø18	3ø28
						4,5	3ø16	3ø25
P-6-3-7	6	3	7	0.990	3ø36	1,2,3	3ø25	3ø36
						4,5	3ø20	3ø32
						6,7	3ø16	3ø25
P-6-5-5	6	5	5	0.705	3ø28	1,2	3ø25	3ø32
						3	3ø18	3ø28
						4,5	3ø16	3ø25
P-7.5-3-5	7.5	3	5	0.846	3ø28	1,2	3ø25	3ø36
						3	3ø20	3ø32
						4,5	3ø16	3ø28
P-7.5-3-7	7.5	3	7	1.186	3ø36	1,2,3	3ø25	3ø36
						4,5	3ø20	3ø32
						6,5	3ø16	3ø28
P-7.5-5-5	7.5	5	5	0.846	3ø28	1,2	3ø25	3ø36
						3	3ø20	3ø32
						4,5	3ø16	3ø28

.

The distribution of elastic–plastic deformations (chord rotations) in the collapse stage is shown in Figure 9. Zones of plastic deformations were concentrated on the extreme sections of beams and columns (plastic hinge). All frames reached the ultimate state in accordance with the concept of “Strong column-Weak beam”. At first, plastic hinges were formed in beams at the ultimate state in the columns of the first floor and at the base and top sections. After this, the system became geometrically unstable. From the analysis of deformations, it follows that the full ductility of the frame building was exploited.

The capacity curves for the frames under consideration, constructed based on the results of pushover analysis, are shown in Figure 10.

Table 3. Results of calculations of the coefficient K₁ for reinforced concrete frames.

Frame	Period T1, s	Hpl, kN	Hel, kN	Δpl, mm	Δel, mm	Δy, mm	E, kN·m	μ	K1 f. (3)	K1 f. (4)
P-6-3-5	0.707	1226.6	2359.5	217	167	112	197.1	1.93	0.520	0.516
P-6-3-7	0.990	1388.8	2693.6	310	240	153	323.2	2.02	0.516	0.494
P-6-5-5	0.705	1967.4	3464.2	194	158	110	273.7	1.76	0.568	0.568
P-7.5-3-5	0.846	1233.5	2519.4	272	204	128	257.0	2.12	0.490	0.470
P-7.5-3-7	1.186	1307.2	2576.9	371	285	181	478.0	2.05	0.507	0.488
P-7.5-5-5	0.845	1964.8	3546.9	222	177	123	313.9	1.81	0.554	0.552

shows the results of determining the coefficient K₁. See Figure 11 for a calculation example for frame P-6-3-5.

The highest base shear H_pl was obtained from 5-span frames P-6-5-5 and P-7.5-5-5: 1967.4 kN and 1964.8 kN. The base shear of 3-span frames (P-6-3-5, P-6-3-7, P-7.5-3-5, and P-7.5-3-7) was almost independent of the number of stories; the H_pl was in the range of 1226.6–1388.8 kN. The highest displacements were obtained for 7-story frames; the top displacements were 310 mm for P-6-3-7 and 371 mm for P-7.5-3-7. With an increase in the span length from 6 m to 7.5 m, the ultimate deformations increased for 3-span frames by 19.7–25.3% and for 5-span frames by 14.4%.

Coefficients K₁ for reinforced concrete frames varied in the range of 0.490–0.568 (according to Formula (3)) and 0.470–0.568 (according to Formula (4)). The results of calculating K₁ using these two formulas differed by no more than 4.3% (for the P-6-3-7 frame). The coefficients obtained using the Newmark criterion (Equation (4)) were slightly less than those obtained using Formula (3).

The average value of the coefficient K₁ when considering all frames was 0.526; the average reduction coefficient R_μ was 1.901.

The results obtained in this paper correlated quite well with the results of other authors who studied K₁ coefficients for different systems [15,28]. In all cases, higher values for the K₁ coefficients were obtained relative to the values specified in the Russian norms.

4. Discussion

The ability of a structural system to effectively redistribute forces at the nonlinear range is determined by its collapse mechanism. In the case of reinforced concrete frame systems, damage occurs gradually with the successive formation of plastic hinges in beams and columns. As a result, the system successively becomes less hyperstatic and, at the ultimate state, becomes unstable. It is necessary that deformations in sections of RC columns and beams do not exceed the ultimate deformations before the nonlinear mechanism is fully developed.

For frame structures, the full exploit of ductility is achieved by implementing the collapse mechanism called “Strong column-Weak beam”. In accordance with this mechanism, plastic hinges are successively formed in the beams of the lower floors first until all the beams of the frame pass into the nonlinear range of behavior. At the final stage, plastic hinges are formed in the first floor columns at the base level and further at the floor level. After this, the frame becomes geometrically unstable and is not able to resist horizontal loads.

It follows from the calculation results that the frames are damaged in full accordance with the concept described above, which is evident from the analysis of the elastic–plastic deformation diagrams (Figure 9).

Since the coefficients R_µ and K₁, according to the formula of N. Newmark (4), directly depend on the ductility coefficient µ, the implemented collapse mechanism affects the level of force reduction under seismic impact. The results of determining the coefficients K₁ using Formulas (3) and (4) confirm the validity of the Newmark criterion for systems with the first vibration mode T₁ > 0.5 s. The coefficient K₁ with full implementation of the nonlinear mechanism for frames R-6-3-5, R-6-3-7, R-7.5-3-5, and R-7.5-3-7 was, on average, 0.508 by Equation (3) and 0.492 by Equation (4), and, for 5-span frames R-6-5-5 and R-7.5-5-5, it was 0.561 by Equation (3) and 0.560 by Equation (4), i.e., 10.4% (Equation (3)) and 13.8% (Equation (4)) more. This is explained by the greater rigidity of 5-span frames in the horizontal direction.

At the same time, an increase in the number of stories has a less significant effect on the value of the coefficient K₁. The deviation of the coefficient K₁ for 7-story frames R-6-3-7 and R-7.5-3-7 and 5-story frames R-6-3-5 and R-7.5-3-5 was no more than 3.5%, despite the fact that the period of the first mode for the first pair was 1.4 longer. Theoretically, it is also possible to increase the K₁ values further with deviations from the effective “Strong column-Weak beam” collapse mechanism when plastic hinges are formed in columns at an earlier stage. According to EC8, plastic hinges are allowed to develop at one of four columns in a story.

That is, for the same structural system, depending on the design features of its elements (section dimensions, reinforcement, thickness of the concrete protective layer, etc.), different plasticity should be observed. It is possible to identify the maximum possible level of force reduction due to nonlinear behavior in a frame structural system with the fine-tuning of the design parameters, namely, when the beam failure mechanism described above is developed.

For reinforced concrete frames without masonry infills and vertical ties involved in the work, for which the period of the first oscillation mode T₁ exceeds 0.5 s, it is recommended to take the value of 2 for the reduction coefficient R_µ or a K₁ value at least equal to 0.5. In the case of frames that are not designed according to the capacity concept, it is necessary to clarify these coefficients by nonlinear static analysis in order to identify deviations from the beam collapse mechanism.

The authors of the article propose the following formula for determining the reduction coefficient R:

$$R_{\mathsf{\mu }}=\mu ={\Delta }_{pl}/{\Delta }_y\le 2,\mathrm{if}{T}_1>0.5 \mathrm{s},$$

(11)

and the coefficient K₁:

$$K_1={\Delta }_y/{\Delta }_{pl}\ge 0.5,\mathrm{if}{T}_1>0.5 \mathrm{s},$$

(12)

where Δ_y is the deformation of the system at which the transition to the nonlinear horizontal section of the diagram occurs for the bilinear approximation (Figure 7), and Δ_pl is the deformation of the system at which the system passes into a geometrically unstable state or the ultimate deformations are reached in one of the elements of the system.

On average, for the frames under consideration, the coefficient K₁ was 0.526, which was 1.5 times greater than the coefficient proposed in SP 14.13330.2018 and earlier SNiPs. An unreasonable assignment of the level of permissible damage for reinforced concrete structural systems is the reason for the underestimation of the forces arising in structures at the design stage.

The value of the coefficient of permissible damage depends on the type of load-bearing structures and on the level of permissible damage. The international modified seismic scale MMSK-86 [34] considers a five-degree damage scale for buildings, which is characterized by the value of the average degree of damage d. We have proposed the following formula linking K₁ with the average degree of damage d in buildings:

$${\mathrm{K}}_1=e^{-0.37d}$$

(13)

The graph of this dependence is shown in Figure 12.

Taking into account Formulas (4) and (13), we obtained the values of the average degree of damage d for damaged buildings from the ductility coefficient µ:

$$d=\left\{\begin{array}{c}1, if T_1<0.1 \mathrm{s}\\ \ln {\left(2{\mu }_{tot}-1\right)}^{1.35}, if 0.1\le T_1\le 0.5 \mathrm{s}\\ \ln {{\mu }_{tot}}^{-2.7}, if T_1>0.5 \mathrm{s}\end{array}\right.$$

(14)

Table 4. Damage to earthquake-resistant buildings.

Type of Buildings	Initial Values [35]			Received Values
	Moderate Damage d	Standard σ	Coefficient of Variation ν	Average Value of Average Damage Degree <d>	Average Value of Standard σ’	Average Value of the Variation Coefficient ν’
Frame civil buildings type IIS-04	2.2	0.83	0.38	1.81	0.53	0.26
Frame industrial buildings type IIS-04	2.2	0.97	0.44	1.82	0.57	0.30
Frame civil buildings type IIS-20	2.3	0.86	0.38	1.83	0.54	0.29

presents estimates of the damage levels during earthquakes for frame buildings of some types, obtained using the computer program “C-TATM” written in the Python 3.12.5 programming language by the authors of this article.

For each type of building, using the Monte Carlo method, 100 synthetic databases were created, consisting of 200 elements characterized by the values of the average degree of damage d, the standard σ, and the values of the variation coefficient ν taken from the work [35]. Note that for the types of frame buildings under consideration, the variation coefficients have large values.

If the difference between the observed earthquakes (J) and the earthquake resistance of buildings (Jc) does not exceed 2, then the probability of buildings receiving damage of degrees 4 and 5 is zero. The probability of buildings receiving different degrees of damage is given in

Table 5. Probabilities of buildings receiving varying degrees of damage.

J-Jc	Degree of Damage
	0	1	2	3	4	5
0	0.9	0.1	-	-	-	-
1	0.4	0.5	0.1	-	-	-
2	0.1	0.3	0.5	0.1	-	-

.

Assuming that the number of elements in the database with high damage levels is overestimated, we changed the number of such elements by decreasing the values of some of their shares. At the same time, as a result of the transformation, the total number of elements in the database did not change. Then, we obtained the average values of d and σ. The d values decreased for civil buildings of the IIS-04 type by 1.222 times, industrial buildings of the IIS-04 type by 1.21 times, and civil buildings of the IIS-20 type by 1.26 times (Table 4). Note that in this case, the values of the variation coefficients ν are less than 0.3.

Using the results of Table 4 for the d values and Formula (13), we obtained the value K₁ = 0.51 for the types of buildings under consideration, which was close to the value we obtained above by calculation.

5. Conclusions

In Russian codes, the concept of the K₁ coefficient is somewhat different from the similar and inverse value of the reduction factors R of other seismic codes (such as American codes or Eurocodes). The K₁ coefficient is inversely related to the coefficient R_µ (Seismic-force-modification Ductility Factor), which accounts only for ductility. However, the values of this factor in the Russian standards are not sufficiently substantiated, as the later studies show. To determine the coefficient K1, six reinforced concrete frames with different parameters were considered:

• To calculate the coefficient K₁, a calculation was performed using the pushover analysis, and the bearing capacity curves were constructed. The obtained values of the coefficients K₁ exceed the values of the coefficient according to SP 14.13330.2018 (K₁ = 0.35) by 1.4-1.49 times for the 3-span frames P-6-3-5, P-6-3-7, P-7.5-3-5, and P-7.5-3-7, and by 1.58-1.62 times for the 5-span frames P-6-5-5 and P-7.5-5-5. Thus, the design standards overestimate the plastic behavior of reinforced concrete frames;
• The 3-span frames (P-6-3-5, P-6-3-7, P-7.5-3-5, and P-7.5-3-7), which have greater flexibility in the horizontal direction, demonstrate a more plastic behavior than 5-span frames (P-6-5-5 and P-7.5-5-5). The ductility coefficient µ for the former was, on average, 13.8% higher;
• A fairly accurate correspondence was obtained between the values of the K₁ coefficients based on the pushover analysis and the K₁ coefficients obtained using the relationship between the values of the average degree of damage d according to the MMSK-86 scale for frames of different types. The average K₁ value found by the pushover analysis was 0.526, and the average K₁ value, taking into account the dependence on d, was 0.51;
• The obtained values of the coefficients of permissible damage K₁ indicate the need to revise the suggested values of K₁ for frames according to the current standards of SP 14.13330.2018 towards an increase. It is proposed to increase the coefficient K₁ for reinforced concrete frames to a value of at least K₁ = 0.5 with mandatory clarification of the actual level of reduction using a nonlinear static method. This recommendation should be imposed in case frames are not designed according to the capacity concept.