Acceptance Criteria for Beams in Reinforced Concrete Frame Structures Under Accidental Design Conditions

Abstract

Localized failures of structural components can lead to serious social, economic, and environmental consequences, such as the collapse of an entire structure or part of it. Therefore, it is important to thoroughly investigate and justify the acceptance criteria for these components, taking into account their performance in extreme conditions. However, the scientific literature lacks a systematic analysis of how various factors can affect the resistance of structures and influence acceptance criteria under extreme conditions. Therefore, this study investigates the typical substructures of reinforced concrete frame buildings in areas that are potentially prone to local collapse. To assess their resistance and structural robustness, an analytical model has been developed. The results of 22 tests on typical substructures of monolithic and precast frames, reported in various research studies, were used to validate this model. Further, this analytical model was used to conduct a parametric study on the impact of various factors on the performance of substructures under extreme conditions. These factors included the depth-to-span ratio of the beam, the strength of the bond between the steel reinforcement and the concrete, the stiffness of the horizontal bracing within the substructure, and the proportion of the effective depth to the total depth of the beam section. It has been found that the ultimate rotation angle in the plastic hinge of beams increases as the ratio of the beam’s cross-sectional depth to the span increases. An increase in the bond strength between the reinforcement and concrete leads to a decrease in the ultimate rotation angles in the plastic hinge at the flexural and arch stages of resistance and, in some cases, to reinforcement rupture without transitioning to the catenary stage of resistance. A decrease in the ratio of the effective depth of the beam section to its overall depth leads to an increase in the load-bearing capacity at the catenary stage of 19%.

1. Introduction

An analysis of the consequences of damage to buildings and structures with reinforced concrete frames under various natural and man-made hazards [1,2] highlights the continued relevance of improving approaches to ensuring the reliability, structural robustness, and safety of these structures. New types of man-made threats, such as attacks on civilian structures by missiles and other combat unmanned aerial vehicles, demonstrate the need to find a balance between the costs of structural measures to prevent disproportionate collapse, on the one hand, and the costs of mitigating the consequences, restoring, and strengthening damaged structures, as well as covering costs associated with human casualties, health damage, economic losses, and other factors [2,3,4,5].

Localized failures of structural components can lead to serious social, economic, and environmental consequences, such as the collapse of an entire structure or part of it. Therefore, it is important to thoroughly investigate and justify the acceptance criteria for these components, taking into account their performance in extreme conditions [6]. Amendment No. 4 to the UFC-4-023-03-2009 code [7] introduced deformation-based acceptance criteria for various types of load-bearing members, borrowed from the design codes for earthquake-resistant structures. For flexural reinforced concrete members, these criteria are defined as the ultimate rotation angles at plastic hinges. They take into account the reinforcement ratios in the tension and compression zones, as well as the magnitude of the relative shear force. However, the tabulated values of these criteria [7] do not account for the presence of compressive arch forces arising in the elements due to constrained deformation within the substructure. Also, they do not account for the unloading effect from second-order moments resulting from the action of axial forces during large deflections. In this regard, a parametric study on the impact of these and other factors on the values of the ultimate rotation angles at a plastic hinge is noteworthy.

The study [8] and Russian standards SP 385.132580 [9] consider ultimate strains in concrete and steel reinforcement as criteria for the ultimate limit state of beams during a robustness check. In addition, these documents provide limitations on the relative deflection of flexural members: 1/30 and 1/50 of the span for non-prestressed and prestressed members, respectively. Ultimate strains have demonstrated their effectiveness in verifying the ultimate states of load-bearing capacity under normal operating conditions, where it is permissible to use the Bernoulli hypothesis, and deflections are small compared to the characteristic dimensions of the cross-sections. This allows calculations to be performed that ignore second-order effects for members in bending.

However, a more complete use of structural resistance reserves requires consideration of performance levels where deflections are comparable to or exceed the typical cross-sectional dimension and where the steel reinforcement operates in the plastic phase. Additionally, the bond strength between the steel reinforcement and concrete is significantly reduced within the length of the plastic hinge. It should be noted that the ultimate relative deflections [8] are derived from an analysis of a limited number of tests and do not account for some peculiarities, such as the reinforcement ratio, asymmetrical spans, the ratio of the member’s cross-section depth to span, etc.

An analysis of a broader range of experimental studies [10,11,12,13,14] indicates that deflections may exceed these values, which is accompanied by the activation of compressive arch action and subsequent tensile catenary action. However, steel reinforcement may fail without a transition to catenary action, which results in structural collapse [15,16]. The studies [17,18,19] examined the resistance of partially restrained frame substructures that represent fragments of the structural frame at facility corners. It was shown that, as a result of the low stiffness of the joint, the deflections of the beams increase, and a loss of load-bearing capacity in the unbraced columns is possible. If the horizontal bracing of the substructure is insufficient, which is typical for a corner column loss, the arch and catenary actions do not occur.

Therefore, the objective of this study was to evaluate the deformation acceptance criteria for typical substructures of reinforced concrete frames in the zone of potential local collapse and to identify patterns in their behavior under various design parameters. The novelty of this study lies in the consideration of the influence of bracing stiffness of the substructure and the analysis of the degree of influence of the following factors: the beam depth-to-span ratio, the bond strength between the reinforcement and the concrete, and the effective depth to overall depth ratio. Although studies [20,21,22,23] have presented the results of analyses of certain factors, a comprehensive study of these factors in the scientific literature appears to be lacking.

2. Analytical Model and Research Methods

2.1. Experimental Basis for Development of Analytical Model of Reinforced Concrete Frame Typical Substructure

Consider a typical substructure of a reinforced concrete frame of a building in an area prone to local collapse, as shown in Figure 1. As in previous studies [24], assume that, if the failure process can be halted within this area, the collapse of the structural block containing the initial local failure will not occur.

According to experimental studies by Pham et al. [17], Tao Y. et al. [16], Feng F.-F. [15], Lew et al. [25], Yu & Tan [26,27], Savin [24], and Kang & Tan [28,29], the following performance stages of frame substructures under accidental conditions can be identified: flexural action (FA), compressive arch action (CAA), and tensile catenary action (TCA). Figure 2 shows the experimental data of maximum relative deflections of the beams (z/h) at various stages of their performance, where h is the depth of the beam’s cross-section.

Until yielding occurs in the tensile reinforcement, the deflections of the beam are small compared to the cross-sectional dimensions. They lie in the range, on average, from 0.1 to 0.2 of the cross-sectional depth, as presented in Figure 2. Thus, the deformed state does not have a significant effect on the stressed state of the members in bending. Consequently, the flexural action stage of the beam’s performance is characterized by satisfactory compliance with the Bernoulli hypothesis and the small-deformation hypothesis. This allows the structural analysis of members in bending at this stage to be performed using the undeformed design scheme.

If there are restrictions on the horizontal displacement of the beam’s support sections, as shown in Figure 2, a thrust develops as the deflection of the beam increases under confined deformation conditions. This generates second-order moments of opposite sign relative to the support first-order bending moments caused by the lateral load. Although a small thrust arises as early as the flexural action stage, its effect becomes noticeable only as deflections increase, reaching a maximum at deflections averaging 0.3 to 0.4 times the depth of the beam’s cross-section, as shown in Figure 2. After this, the compressed concrete in the beam’s support sections crushes, which results in a transitional deformation stage between CAA and TCA. Tensile catenary action begins when the beam deflection is in the range of 0.8 to 1 times the cross-sectional depth, as can be seen in Figure 2. Figure 2 shows the ratios of deflections to the cross-sectional depth of the substructures at the moments when the ultimate load-bearing capacity is reached for the considered resistance stages of the substructure. The data in the figure can be used as a first approximation to determine the deflections corresponding to the attainment of the ultimate load-bearing capacity at each resistance stage within the considered range of h/L values.

In experimental studies with a deformation-controlled action, a relatively smooth descending branch of the load–displacement curve was obtained. However, for real building frames subjected to gravitational loads, the transition stage is characterized by an abrupt drop with dynamic effects regardless of the nature of the initial local failure (brittle or plastic). This is because, from the moment the maximum load-bearing capacity is reached in the compressive arch action until equilibrium is achieved in the tensile catenary action, the floor structures are practically in a state of free fall. In the experimental study by Pham and Tan [17], the dynamic load increase factor during tensile catenary action exceeds 2.0.

Figure 3 shows the relative changes in the load-bearing capacity of typical frame substructures at various stages of performance, based on experimental studies by Pham et al. [17], Tao Y. et al. [16], Feng F.-F. [15], Lew et al. [25], Yu & Tan [26,27], and Kang & Tan [28,29]. The results presented in the figure are normalized to the load-bearing capacity at the compressive arch action stage. As the depth-to-span ratio of the beams, h/L, increases, a decrease in the load-bearing capacity at the tensile catenary action stage is observed. A slightly less noticeable decrease in the load-bearing capacity is observed in the transitional stage of beam performance as the depth-to-span ratio h/L increases.

Figure 4 presents the values of the rotation angles in plastic hinges corresponding to the ultimate load-bearing capacity of the substructure at various resistance stages under an accidental design situation. The average values of the plastic hinge rotation angle were 0.03 rad for the arch resistance stage, 0.09 rad for the transition stage, and 0.23 rad for the catenary resistance stage.

According to experimental studies, the relative deflection at maximum load-bearing capacity was found to range from 1/91 to 1/43 of the beam span during the compressive arch action stage, from 1/31 to 1/16 during the transition stage, and from 1/13 to 1/7 during the tensile catenary stage. Thus, the relative deflection criterion of 1/30, established in SP 385.132580 for flexural members with non-prestressed reinforcement, is a conservative assessment of the resistance during the transitional stage of frame substructure performance when the compressive arch action transitions to the tensile catenary action. In fact, it does not account for the ultimate load-bearing capacity during the tensile catenary action of the frame substructure. However, such a restriction can be considered reasonable in the absence of adequate and sufficiently simple models for performing engineering calculations of structures in the tensile catenary action stage, as well as due to the lack of specified acceptance criteria for performance stages under extraordinary conditions. It is also reasonable to note that, in a number of the tests considered, failure occurred due to reinforcement rupture before the implementation of the tensile catenary action stage [15,16].

2.2. Analytical Model of a Reinforced Concrete Frame Substructure Under an Accidental Design Situation

The analytical models of a frame substructure developed in previous studies [6] and experimentally validated through tests on scaled frames of multi-story buildings [24] were adopted to analyze how structural design and detailing affect the load-bearing and deformation capacities of such substructures under accidental conditions. The parameters assigned to these models have been refined based on the experimental results from Pham et al. [17], Tao et al. [16], Feng [15], Lew et al. [25], Yu & Tan [26,27], and Kang & Tan [28,29], as discussed in the previous section.

The frame substructure shown in Figure 1b illustrates a situation where, as a result of sudden column loss, the area of the tension reinforcement in one of the beam’s support sections becomes smaller than that of the compression reinforcement due to a change in the sign of the moment. To determine the ultimate bending moment in such a section, one uses a bilinear concrete strain diagram and assumes a trapezoidal stress distribution in compressed concrete, as presented in Figure 5. Assume that, by the time the ultimate bending capacity is reached during the flexural or compressive arch action, strains in the compressive concrete and tensile steel reinforcement have reached ε_cu and ε_s,el, respectively. The ultimate bending moment at the flexural action is as follows:

$$M_{u,FA}=f_c^{’}b\lambda x\left(h_0-{\displaystyle \frac{\lambda x}{2}}\right)+f_c^{’}b{\displaystyle \frac{1-\lambda }{2}}x\left(h_0-{\displaystyle \frac{2\lambda +1}{3}}x\right)+{\epsilon }_{sc}{E}_{s}A{’}_s\left(h_0-a^{\prime }\right);$$

(1)

where λ is the ratio of the depth of the compression zone over which the concrete is in the plastic region to the total depth of the compression zone, as follows from Equation (2):

$$\lambda ={\lambda }_u={\displaystyle \frac{x_{pl,u}}{x}}=1-{\displaystyle \frac{{\epsilon }_{c,red}}{{\epsilon }_{cu}}}=1-{\displaystyle \frac{0.0015}{0.0035}}=0.571,$$

(2)

Compressive reinforcement strain ε_sc based on the Bernoulli assumption for composite cross-sections:

$${\epsilon }_{sc}={\displaystyle \frac{{\epsilon }_{cu}+{\epsilon }_{s,el}}{h_0}}\left(x-a\right)={\varphi }_{bal}\left(x-a\right),$$

(3)

where, for standard heavy-duty concrete with strength classes up to and including B60:

$$x={\displaystyle \frac{\frac{{\epsilon }_{cu}+{\epsilon }_{s,el}}{h_0}a{E}_{s}A{’}_s+f_y{A}_s}{f_c^{’}b\frac{1+\lambda }{2}+\frac{{\epsilon }_{cu}+{\epsilon }_{s,el}}{h_0}{E}_{s}A{’}_s}}={\displaystyle \frac{{\varphi }_{bal}a{E}_{s}A{’}_s+f_y{A}_s}{0.786f_c^{’}b+{\varphi }_{bal}{E}_{s}A{’}_s}},$$

(4)

in which ${\varphi }_{bal}$ is the curvature of the cross-section at which the strains of the extreme compression fiber of the concrete and the tensile reinforcement are equal to ε_cu and ε_s,el, respectively.

The ultimate load and deflection at the flexural action stage are determined from Equations (5) and (6) according to diagrams in Figure 6.

$$P_{FA}={\displaystyle \frac{2\left(M_{lef}+M_{rig}\right)}{l}}-ql,$$

(5)

$$z_{FA}={\displaystyle {\displaystyle \underset{0}{\overset{l}{\int }}}\left(\frac{1}{r}\right)\left(\frac{l}{2}-x\right)dx.}$$

(6)

The deformed state of the element is determined by the strains in the cracked region or, for high-strength concrete, in the region adjacent to the critical crack. As a first approximation and for analytical convenience, we assume that the total length of the cracked sections equals the sum of the plastic hinge lengths at the support sections. The ultimate rotation angle at the flexural stage is then given by Equation (7):

$${\theta }_y\cong 2L_{pl}{\left({\displaystyle \frac{1}{r}}\right)}_{crc}.$$

(7)

In Equation (7), the curvature over the length L_pl is assumed constant and is determined from the average strains in the concrete and reinforcement in either the section between adjacent cracks or the transfer length over which forces are transferred from the reinforcement to the concrete (for isolated cracks in high-strength concrete):

$${\left({\displaystyle \frac{1}{r}}\right)}_{crc}={\displaystyle \frac{{\mathsf{\epsilon }}_{sm}-{\mathsf{\epsilon }}_{bm}}{h_0}}={\displaystyle \frac{M_{u,FA}-0.8M_{crc}}{E_s{A}_s{z}_s{h}_0}}+{\displaystyle \frac{0.9M_{u,FA}}{E_{b,red}{A}_b{z}_s{h}_0}}.$$

(8)

where z_s and A_b are the lever arm of the internal force couple and the compressive concrete area, respectively, determined from ultimate force equilibrium under flexural action.

L_pl is the ultimate plastic hinge length, determined according to [6,30] as follows (see Figure 7):

$$L_{pl}={\displaystyle \frac{f_u-f_y}{{\mathsf{\tau }}_{bm,pl}}}{\displaystyle \frac{d_s}{4}},$$

(9)

where τ_bm,pl = 0.27τ_b,max is the average bond stress in the plastic stage of the reinforcing bar;

d_s is the diameter of the longitudinal tension reinforcing bar in the beam support section (if bars of different diameters are present, the smallest value is taken);

and f_u is the ultimate tensile strength of the steel reinforcement at rupture.

For the compressive arch action stage, the ultimate bending moment equals:

$$\begin{array}{l}{M}_{u,CAA}=f_c^{\prime }b\lambda x\left(h_0-{\displaystyle \frac{\lambda x}{2}}\right)+f_c^{\prime }b{\displaystyle \frac{1-\lambda }{2}}x\left(h_0-\lambda x-{\displaystyle \frac{2\lambda +1}{3}}x\right)+\\ +{\epsilon }_{sc}{E}_s{A}_s^{\prime }\left(h_0-a^{\prime }\right)-N\left(z\right)\left(y_g-a\right);\end{array}$$

(10)

where, the depth of the compression zone is determined by accounting for the axial force N(z) and using the trapezoidal stress distribution in the concrete compression zone, as shown in Figure 5:

$$x={\displaystyle \frac{{\varphi }_{bal}a{E}_{s}A{’}_s+f_y{A}_s+N\left(z\right)}{0.786f_c^{’}b+{\varphi }_{bal}{E}_{s}A{’}_s}}$$

(11)

In Equation (11), the axial force N(z) is calculated based on the deflection of the two-span beam using the following formula:

$$N\left(z\right)=\Delta L{\displaystyle \frac{C_1{C}_2}{C_1+C_2}}=\left[{\left({l_1}^2+2\left(h_0-a^{\prime }\right)z-z^2\right)}^{{\displaystyle \frac{1}{2}}}-l_1\right]{\displaystyle \frac{C_1{C}_2}{C_1+C_2}},$$

(12)

where C₁ and C₂ are the horizontal displacement stiffnesses of the left and right support sections of the hypothetical arch respectively, as shown in Figure 8.

In the case of C₁, the stiffnesses under horizontal displacements of the supports are determined from Equation (13):

$$C_1^{-1}={\displaystyle \sum _{i=0}^n\frac{1}{C_{1,b,i}}}+{\displaystyle \frac{1}{{\displaystyle \sum _{j=0}^m{C}_{1,col,j}}}}+{\displaystyle \frac{1}{{\displaystyle \sum _{k=0}^p{C}_{1,sw,k}}}},$$

(13)

where C_1,b,i is the conventional stiffness of the frame substructure associated with beam longitudinal compression during the compressive arch action stage (upon transition to the tensile catenary action stage, this stiffness must account for crack opening effects);

C_1,col,j is the conventional stiffness of the frame substructure associated with interstory drift of the vertical load-bearing elements;

and C_1,sw,k is the conventional stiffness of the frame substructure associated with interstory drift of the shear walls.

The ultimate load and deflection in the compressive arch action stage are determined, according to Figure 9, from Equation (14):

$$P_{CAA}={\displaystyle \frac{2\left(M_{lef}+M_{rig}\right)}{l}}-ql-N\left(z\right)z_{CAA},$$

(14)

where Z_CAA is equal to the minimum value of the deflection z at which P(z) reaches an extremum, i.e., condition (15) fulfills:

$${\displaystyle \frac{dP}{dz}}=0\Rightarrow z_{CAA}$$

(15)

Then, the rotation angle in the plastic hinge for the compressive arch action stage is as follows:

$${\theta }_{pra,CAA}={\displaystyle \frac{z_{CAA}}{l}}-{\theta }_y.$$

(16)

where Z_CAA/L is the complete angle of rotation, as shown in Figure 9.

Assuming Z_CAA,ult = h as a first approximation for the transition stage from compressive arch action to tensile catenary action [6], we obtain the following:

$$P_{Tr}=h\left({\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\mathsf{\sigma }}_{si}{A}_{si}}}{l_1-{\Delta }_{x1}}}+{\displaystyle \frac{{\displaystyle \sum _{j=1}^m{\mathsf{\sigma }}_{sj}{A}_{sj}}}{l_2-{\Delta }_{x2}}}\right).$$

(17)

Here, l₁ and l₂ are the design spans of the beams to the left and right of the removed vertical element (see Figure 2). Δx₁ and Δx₂ are the horizontal displacements of the beam support sections due to the thrust action (see Figure 8). In the case of a corner column removal, only one term remains in the parentheses in expression (16).

σ_s1 and σ_s2 can be determined according to [31] as follows:

$${\mathsf{\sigma }}_{si,\max }=2.39\sqrt{{\displaystyle \frac{{\mathsf{\tau }}_{b,\max }{E}_s}{d_s}}\Delta L_{Tr}^{1.4}}\le f_y.$$

(18)

where τ_b,max is the maximum bond stress corresponding to elastic behavior of the reinforcement; d_s is the diameter of the longitudinal reinforcing bar; and ΔL_Tr is the maximum elongation of the beam face during the transition from the arch action to the catenary action, determined by the following formula:

$$\Delta L_{Tr}=\sqrt{l^2+h_0^2}-l.$$

(19)

Hence, the ultimate rotation angle in the plastic hinge ${\theta }_{pra,Tr}$ is:

$${\theta }_{pra,Tr}={\displaystyle \frac{h}{l}}-{\theta }_y$$

(20)

where the beam deflection at the transition stage is Z_Tr = h.

The assumption regarding the magnitude of the deflection in the transition state is based on an analysis of the experimental data presented in Figure 2. A similar assumption follows from the analysis of the static equilibrium of the calculation scheme shown in Figure 8. As long as the deflection is smaller than the characteristic section height, the external load induces compressive forces in the virtual arch. However, once the deflection exceeds the dimensions of the characteristic section, tensile forces must act in the beams according to the static equilibrium condition. The transition from the arching stage to the catenary stage involves a transient process during which the static equilibrium conditions may be violated until a new equilibrium state is reached in the catenary stage. Analysis of the experimental data presented in Section 2.1 shows that, in dynamic tests, the deflection magnitude in the transition stage is closer to the section size than in static tests, where the deflection may exceed the section size when the sign changes. The presence of compressive force in static tests when the section size is exceeded may be associated with friction forces in the near-joint zones of the beams. Analysis of Formula (17) shows that reducing the deflection used for the analytical analysis of the transition stage leads to a decrease in the calculated value of the load-carrying capacity in this stage.

If reinforcing bars are placed in several rows along the beam section height, as shown in Figure 10, for longitudinal bars located at a distance Z_si from the compressed face of the section, elongation at the level of each reinforcing bar ΔL_si is as follows:

$$\Delta L_{si}={\displaystyle \frac{\Delta L_{Tr}}{h_0}}{z}_{si}.$$

(21)

Substituting ΔL_si for ΔL_Tr in Formula (19), we find the stresses σ_si in all longitudinal reinforcing bars placed in several rows along the section depth.

If the condition σ_s < R_su is satisfied for the deformed state of the structure at the transition stage (P_Tr, Z_Tr), the ultimate deflection Z_TCA at the moment when the most tensioned reinforcing bars rupture in one of the support sections at the catenary stage of resistance can be found according to [6] by taking σ_s = f_u:

$$z_{TCA}=\sqrt{{\left(l+\Delta L_{TCA}\right)}^2-{\left(l-{\Delta }_x\right)}^2},$$

(22)

$$\Delta L_{TCA}={\displaystyle \frac{f_u-f_y}{{\mathsf{\tau }}_{bm,pl}}}{\displaystyle \frac{d_s}{4}}{\displaystyle \frac{{\mathsf{\epsilon }}_{su}}{2}}+{\displaystyle \frac{f_y}{E_s}}2d_s+0.288{\left({\displaystyle \frac{d_s{f}_y^2}{{\mathsf{\tau }}_{b,\max }{E}_s}}\right)}^{0.714}.$$

(23)

The ultimate load-bearing capacity at tensile catenary action stage P_TCA is calculated using Formula (17), with Z_TCA substituted for h:

$$P_{TCA}=z_{TCA}\left({\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\mathsf{\sigma }}_{si}{A}_{si}}}{l_1-{\Delta }_{x1}}}+{\displaystyle \frac{{\displaystyle \sum _{j=1}^m{\mathsf{\sigma }}_{sj}{A}_{sj}}}{l_2-{\Delta }_{x2}}}\right).$$

(24)

In (24), the stress in the tension reinforcement working in the elastoplastic phase is determined by Formula (25):

$${\mathsf{\sigma }}_{si,\max }=f_y+{\displaystyle \frac{f_u-f_y}{{\mathsf{\epsilon }}_{su}}}\left({\mathsf{\epsilon }}_{s,\max }-{\mathsf{\epsilon }}_{su}\right),$$

(25)

in which

$${\mathsf{\epsilon }}_{s,\max }={\left\{{\displaystyle \frac{8{\mathsf{\tau }}_{bm,pl}{\mathsf{\epsilon }}_{su}}{d_s\left(f_u-f_y\right)}}\left[\Delta L_{TCA}-0.288{\left({\displaystyle \frac{d_s{f}_y^2}{{\mathsf{\tau }}_{b,\max }{E}_s}}\right)}^{0.714}-2d_s{\displaystyle \frac{f_y}{E_s}}\right]+{\mathsf{\epsilon }}_{su}^2\right\}}^{0.5},$$

(26)

The rotation angle in the plastic hinge at the tensile catenary action stage of beam performance is:

$${\theta }_{pra,TCA}={\displaystyle \frac{z_{TCA}}{l}}-{\theta }_y,$$

(27)

The validation basis for the considered analytical models is provided by the test results of typical monolithic and precast frame substructures presented in the studies by Pham et al. [17], Tao Y. et al. [16], Feng F.-F. [15], Lew et al. [25], Yu & Tan [26,27], and Kang & Tan [28,29]. The main design parameters of the specimens used in the calculations are summarized in

Table 1. Design parameters of the specimens used in the calculations.

Source	Specimen	Cross-Section Dimensions		Beam Span (Before Initial Localized Failure), mm	Longitudinal Reinforcement at Support Section		Materials’ Properties
		Depth, mm	Width, mm		Top	Bottom	f’c, MPa	fy, MPa	fu, MPa
Pham et al. [17]	FR	180	100	2220	3Ø10	2Ø10	35	554	653
	FD2-F/34	180	100	2220	3Ø10	2Ø10	35	554	653
Tao Y. et al. [16]	RC	250	150	2700	3Ø10	2Ø10	54.8	445	545
Feng F.-F. [15]	PCF-1	250	150	2600	2Ø16	2Ø12	31.6	485	600
	PCF-2	300	180	2600	4Ø10	1Ø10 + 2Ø8	30	554	649
	PCF-3	300	180	2600	4Ø10	1Ø10 + 2Ø8	32.1	554	649
	PCF-4	250	150	2600	2Ø16	2Ø12	32.1	542	659
Lew et al. [25]	IMF	508	711	5385	4Ø25.4	2Ø28.6	32.4	476	648
Yu & Tan [26]	S1	250	150	2750	1Ø13 + 2Ø10	2Ø10	31.2	518	688
	S2	250	150	2750	3Ø10	2Ø10	31.2	511	731
Yu & Tan [27]	S3	250	150	2750	3Ø13	2Ø10	38.2	494	593
	S4	250	150	2750	3Ø13	2Ø13	38.2	494	593
	S5	250	150	2750	3Ø13	3Ø13	38.2	494	593
	S6	250	150	2750	3Ø16	2Ø13	38.2	513	612
	S7	250	150	2150	3Ø13	2Ø13	38.2	494	593
Kang & Tan [28]	MJ-L-0.52/0.35S	300	150	2750	3Ø10	2Ø10	27.9	462	553
	MJ-B-0.88/0.59R	300	150	2750	3Ø13	2Ø13	27.9	471	568
	MJ-L-0.88/0.59R	300	150	2750	3Ø13	2Ø13	27.9	471	568
	MJ-L-1.19/0.59R	300	150	2750	2Ø16 + 1Ø13	2Ø13	27.9	513	679
Kang & Tan [29]	MJ-B-1.19/0.59	300	150	2750	2Ø16 + 1Ø13	2Ø13	40.5	567	679
	EMJ-B1.19/0.59	300	150	2750	2Ø16 + 1Ø13	2Ø13	40.5	567	679
	EMJ-L1.19/0.59	300	150	2750	2Ø16 + 1Ø13	2Ø13	40.5	567	679

.

3. Results and Discussion

3.1. Validation of the Analytical Models of Frame Substructures

The calculation results obtained with the proposed analytical model of frame substructure for the stages of performance under extreme conditions are presented in

Table 2. Comparison of experimental data and calculation results obtained with the proposed model for the flexural and arch action stages of typical frame substructures in the region of potential local failure (substructure parameters are given in Table 1).

Specimen	Flexural Action						Compressive Arch Action
	Zexp, mm	Zcal, mm	Zcal/Zexp	Pexp, kN	Pcal, kN	Pcal/Pexp	Zexp, mm	Zcal, mm	Zcal/Zexp	Pexp, kN	Pcal, kN	Pcal/Pexp
FR	20	19	0.950	19	21.6	1.137	66	49	0.742	27	24.07	0.891
FD2-F/34	16	19	1.188	22	21.6	0.982	52	49	0.942	30	24.07	0.802
RC	11.3	10	0.885	24.6	22.3	0.907	61.9	70	1.131	39.4	43.5	1.104
PCF-1	29	25	0.862	42.3	35.2	0.832	57	65	1.140	50.5	44.8	0.887
PCF-2	23	13	0.565	35.5	39.8	1.121	108	83	0.769	54	55.67	1.031
PCF-3	34	26	0.765	52	39.9	0.767	93	96	1.032	62.5	61.8	0.989
PCF-4	26	33	1.269	31.3	39.1	1.249	90	83	0.922	45.8	42.7	0.932
IMF	70	62	0.886	264.7	211.1	0.798	132	172	1.303	296	358.1	1.210
S1	40	28	0.700	27	25	0.926	73	68	0.932	41.64	37.2	0.893
S2	37	35	0.946	24	24.7	1.029	63	75	1.190	38.38	39.3	1.024
S3	39	32	0.821	34.9	24.1	0.691	79	72	0.911	54.47	36.5	0.670
S4	39	29	0.744	43.2	39.8	0.921	79.5	69	0.868	63.22	52.5	0.830
S5	45	31	0.689	55.8	58	1.039	72	81	1.125	70.33	75.7	1.076
S6	80	30	0.375	59.2	41.2	0.696	127	60	0.472	70.33	47.4	0.674
S7	46	30	0.652	63.5	50.9	0.802	69	70	1.014	82.82	72.8	0.879
MJ-L-0.52/0.35S	60	24	0.400	32.3	27.6	0.854	75	84	1.120	41.36	50.38	1.218
MJ-B-0.88/0.59R	59	31	0.525	44.9	46.4	1.033	98	81	0.827	63.28	63.14	0.998
MJ-L-0.88/0.59R	68	31	0.456	41.8	46.4	1.110	102	81	0.794	53.85	63.14	1.173
MJ-L-1.19/0.59R	68	51	0.750	50.9	50.3	0.988	110	91	0.827	57.37	58.2	1.014
MJ-B-1.19/0.59	65	33	0.508	67.8	56.4	0.832	105	83	0.790	90.4	73.3	0.811
EMJ-B1.19/0.59	62	33	0.532	71	56.4	0.794	109	83	0.761	91.1	73.3	0.805
EMJ-L1.19/0.59	62	33	0.532	71.6	56.4	0.788	104	83	0.798	91.1	73.3	0.805

and

Table 3. Comparison of experimental data and calculation results obtained with the proposed model for the transition and catenary action stages of typical frame substructures in the region of potential local failure (substructure parameters are given in Table 1).

Specimen	Transition Stage						Tensile Catenary Action
	Zexp, mm	Zcal, mm	Zcal/Zexp	Pexp, kN	Pcal, kN	Pcal/Pexp	Zexp, mm	Zcal, mm	Zcal/Zexp	Pexp, kN	Pcal, kN	Pcal/Pexp
FR	230	180	0.783	26	25.4	0.977	550	490	0.891	72	69.9	0.971
FD2-F/34	190	180	0.947	20	25.4	1.270	460	490	1.065	74	69.9	0.945
RC	172	170	0.988	29.4	16.75	0.570	0	0	0	0	0	0
PCF-1	250	250	1.000	41	38.98	0.951	765.5	730	0.954	127.4	112.5	0.883
PCF-2	218	220	1.009	41	30.28	0.739	0	0	0	0	0	0
PCF-3	300	300	1.000	30	39.56	1.319	586	616	1.051	79.3	78.8	0.994
PCF-4	250	250	1.000	42	40.1	0.955	636.7	650	1.021	105.4	104.1	0.988
IMF	508	508	1.000	206.3	207.5	1.006	1090	1170	1.073	547	473.1	0.865
S1	245	250	1.020	15	30.49	2.033	573	550	0.960	68.91	65	0.943
S2	257	250	0.973	19	29.4	1.547	612	580	0.948	67.63	67	0.991
S3	195	250	1.282	24.4	28.3	1.160	729.3	740	1.015	124.37	83.5	0.671
S4	176	250	1.420	45.7	41.7	0.912	614.3	620	1.009	103.68	102.2	0.986
S5	220	250	1.136	52	55.96	1.076	665.9	530	0.796	105.07	114.9	1.094
S6	203	250	1.232	61.4	47.3	0.770	675.3	720	1.066	139.9	135.4	0.968
S7	264	250	0.947	40.5	53.94	1.332	628.5	520	0.827	105.99	108.9	1.027
MJ-L-0.52/0.35S	327	300	0.917	22.5	26.98	1.199	644.4	612	0.950	49.5	53.5	1.081
MJ-B-0.88/0.59R	298	300	1.007	38.8	46.8	1.206	726.2	614	0.845	98.55	93.2	0.946
MJ-L-0.88/0.59R	298	300	1.007	31.4	46.8	1.490	663.8	614	0.925	77.24	93.2	1.207
MJ-L-1.19/0.59R	270	300	1.111	41.8	55.1	1.318	520.9	698	1.340	86.6	128.4	1.483
MJ-B-1.19/0.59	235	300	1.277	36.3	59.2	1.631	452	615	1.361	108.2	117.6	1.087
EMJ-B1.19/0.59	199	300	1.508	51.8	59.2	1.143	430.2	615	1.430	110.3	117.6	1.066
EMJ-L1.19/0.59	197	300	1.523	42.9	59.2	1.380	431.2	615	1.426	88.3	117.6	1.332

. A comparison of the experimental and calculated data is shown in Figure 11. For both the ultimate loads and the ultimate deflections of the typical frame substructure in the region of potential local collapse, the coefficients of determination R² were 0.941 and 0.931, respectively, indicating satisfactory accuracy of the proposed analytical model.

The coefficient of determination R² was calculated using the following formula:

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(y_{cal}-y_{\exp }\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_{cal}-y_m\right)}^2}}}$$

(28)

where y_cal are the calculated values of the load-bearing capacity or deflection of the substructure at various resistance stages, determined for load-bearing capacity using Formulas (5), (14), (17), (24), and for deflection using Formulas (6), (15), (22), or taken as equal to the section height h at the transition stage;

y_exp are the experimental values of the load-bearing capacity or deflection of the substructure at various resistance stages;

and y_m is the mean experimental value of the load-bearing capacity or deflection of the substructure at various resistance stages.

Despite the high overall coefficient of determination R², the discrepancies between individual test results and analytical predictions in Table 2 and Table 3 are substantial for several cases. For specimen S1, the deviation in the transition stage exceeds 100%. For the catenary action stage of specimen S3, the deviation is 33%. Elevated deviations were also identified for some specimens of the MJ and EMJ series during the transition stage. Therefore, consider the design features of these specimens and possible reasons for the deviations.

In specimen S1 [27], the transverse reinforcement in the joint regions was assigned according to seismic design provisions, unlike the other specimens in series S. The spacing of transverse reinforcement in the near-joint zones was half that in the span. The deviation for this specimen may be attributed to the compliance of the structural joints in the tests. This is noted by Yu and Tan [27], whose numerical study obtained a similar discrepancy between experimental data and analytical results when using a rigid joint model that does not allow rotation or distortion. When a component-based joint model was used in their numerical study, the discrepancy decreased. On the other hand, in our opinion, the difference may be due to the non-simultaneous engagement of all longitudinal rebars across the section depth when transitioning to the catenary action stage. Supporting this view is the fact that if the forces in the longitudinal rebars in the compression zone of the section are set to zero, the deviation does not exceed 30%, whereas if compressive forces in the same bars are accounted for, the deviation does not exceed 5% for specimen S1. However, for simplicity, the model proposed in this study assumes that, in the transitional state, the substructure already behaves as a catenary mechanism. It is worth noting that, in dynamic tests [17] in which the catenary action stage was identified, the transition to this stage occurred more abruptly and at smaller deflection values than in static tests with a deformation-controlled loading regime. Accordingly, in dynamic tests, all longitudinal rebars across the section depth are engaged in tension catenary action at smaller deflection values than in static tests.

For specimen S3, a lap splice was used for the bottom longitudinal reinforcement in the middle joint. Lap splices or mechanical couplers are widely used in real structures. Therefore, the effect of the compliance of such splices on the load-carrying capacity and deformation capacity of frame substructures warrants in-depth investigation.

The specimens of series MJ [28] and EMJ [29] were precast, which influenced their behavior in the arching and catenary stages as well as in the transition stage. Such specimens exhibited greater deformation capacity in the arching action stage. It should also be noted that, in the experimental studies considered, for each design variant [15,16,17,26,27,28,29] or loading regime, only one specimen was tested. This does not allow a full statistical analysis of the results for each specific specimen type or test regime to assess the influence of random factors on the outcome. To evaluate the degree of influence of the random nature of the input parameters, a parametric study is presented below.

In addition to the overall diagrams in Figure 11, Figure 12 below presents diagrams for each stage individually, showing the reserve of load-carrying capacity when using the proposed analytical model. In these diagrams, values less than one indicate an overestimation of the load-carrying capacity in the corresponding calculations using the proposed model. Thus, the minimum value for the flexural stage is 0.80; for the arching stage—0.82; for the transition stage—0.50; and for the catenary stage—0.67. Moreover, the following minimum overestimation values of the load-carrying capacity were obtained with 95% probability: for the flexural stage—0.88; for the arching stage—0.85; for the transition stage—0.61; and for the catenary stage—0.75.

Furthermore, Figure 13 shows Pareto diagrams for each stage of resistance of the frame substructures. These diagrams allow evaluation of the distribution of deviations of the calculation results obtained with the proposed model relative to the experimental data and identification of the most likely deviation intervals. Thus, for the flexural stage, 80% of cases lie in the interval [0.69, 1.03]; for the arching stage, 72% lie in the interval [0.69, 1.07]; for the transition stage, 78% lie in the interval [0.57, 1.39]; and for the catenary stage, 88% lie in the interval [0.69, 1.11]. The model has the least predictive capability for the transition stage. It is worth noting that the analytical model for this stage is the most sensitive to the loading regime and the model parameters.

Based on the analysis of the presented graphs, service condition factors for each stage can be recommended as a first approximation. It should be noted here that several model parameters, such as the stiffness of the horizontal restraint of the substructure and the bond between reinforcing bars and concrete, were assigned based on indirect evidence. Therefore, it is necessary to perform a sensitivity assessment of the model to these parameters. A more reliable evaluation of the safety of design solutions developed using the proposed model requires probabilistic modeling that accounts for the random nature of the parameters of loading, concrete and steel strengths, as well as the bond between them. Such probabilistic modeling with systematic risk analysis is one of the priority directions for future research.

For the subsequent parametric studies, the typical frame substructure IMF tested by Lew et al. [25] was adopted. Unlike the other experimental specimens listed in Table 1, the IMF substructure was full scale, thus eliminating scale effects on the test results. Figure 14 shows the load–deflection diagram of the IMF specimen according to the test results and according to the calculation.

The following subsections present the results of parametric studies obtained by varying the relevant parameters of the original experimental configuration: beam depth-to-span ratio; bond strength between reinforcement and concrete; stiffness of horizontal support restraints; ratio of effective depth to total depth of the beam section.

3.2. Beam Depth-to-Span Ratio

Consider the influence of the ratio of the beam cross-sectional depth to its span h/L on the resistance parameters and deformation acceptance criteria of the structural robustness. This study uses the typical IMF frame substructure (see Table 1) tested by Lew et al. [25]. For research purposes, the beam span was varied so that the depth-to-span ratio h/L took values from 0.04 (1/25) to 0.15 (1/7) in increments of 0.01. Figure 15 presents the load–deflection diagrams of the considered typical frame substructures. As the span decreases, the load-carrying capacity increases at all performance stages of the typical substructure, which follows from the analysis of the analytical expressions (5), (14), (17) and (24) for the respective performance stages.

Figure 16 shows the ultimate rotation angles in the plastic hinge for various performance stages of the considered typical frame substructures in the region of potential local collapse if varying the beam span. For the adopted reinforcement and bond parameters, the deformation capacity of the beams increases with increasing beam depth-to-span ratio h/L from 0.008 to 0.027 rad at the compressive arch action; from 0.028 to 0.118 rad at the transition stage; and from 0.13 to 0.21 rad at the tensile catenary action. For comparison, UFC 4 023 03 2009, Amendment No. 4 [7] provides the following values of plastic hinge rotation angles for sections where the area of compression reinforcement exceeds that of tension reinforcement: for the flexural and compressive arch action stages, in the presence of additional detailing measures to improve bond within the plastic hinge length (e.g., reinforcement spacing ≤ 0.3 h₀)—from 0.05 to 0.063 rad; for the transition stage (maximum deformation capacity at the compressive arch action stage)—from 0.08 to 0.10 rad; in the absence of such additional detailing measures—from 0.025 to 0.05 rad for the flexural and compressive arch action stages and from 0.03 to 0.08 rad for the transition stage, respectively.

If the beam depth-to-span ratio reaches h/L = 0.11, for the adopted longitudinal and transverse reinforcement parameters (for transverse reinforcement—regarding its indirect effect on bond resistance), the catenary action stage is not achieved. This is because, as the cross-sectional depth increases, the same plastic hinge rotation angle of the beam leads to a larger pull out of the tension longitudinal bars located near the most tensioned face of the member and, consequently, to larger forces in those bars. This may cause rupture of these bars during the transition stage.

3.3. Bond Between Steel Reinforcement and Concrete

The Model Code [31] provides bond slip diagrams for deformed reinforcing bars depending on the cover thickness, clear spacing between the bars of the main reinforcement, and parameters of transverse or indirect reinforcement that create a confining effect within the linear plastic hinge region of the beam. Some of the distinctions are qualitative, without adequate quantitative criteria. For example, bond resistance under concrete splitting (the most typical case for structural members) is divided into cases of “good bond conditions” and “all other conditions.” However, clear criteria for applying these categories are lacking. At the same time, UFC 4 023 03 2009, Amendment No. 4 [7] distinguishes applicability criteria for beam plastic hinge rotation angles based on whether the transverse reinforcement details satisfy the requirements for seismic design (conforming or nonconforming). Given that transverse and indirect reinforcement affect bond strength, it is of interest to investigate their influence on the resistance stages under extreme conditions. As in the previous case, the typical IMF frame substructure (see Table 1) tested by Lew et al. [25] is considered. For the original experimental specimen, “all other bond conditions” are assumed.

The bond strength is determined according to Model Code [31] using the following formulas:

-

for the pull-out mechanism under good bond conditions:

$${\tau }_{\max }=2.5\sqrt{f_{ck}},$$

(29)

-

for the pull-out mechanism under all other bond conditions:

$${\tau }_{\max }=1.25\sqrt{f_{ck}},$$

(30)

-

for the splitting mechanism under good bond conditions and in the absence of lateral concrete confinement:

$${\tau }_{\max }=7{\left({\displaystyle \frac{f_{ck}}{20}}\right)}^{0.25},$$

(31)

-

for the splitting mechanism under good bond conditions with lateral concrete confinement:

$${\tau }_{\max }=8{\left({\displaystyle \frac{f_{ck}}{20}}\right)}^{0.25},$$

(32)

-

for the splitting mechanism under all other bond conditions and in the absence of lateral concrete confinement:

$${\tau }_{\max }=5{\left({\displaystyle \frac{f_{ck}}{20}}\right)}^{0.25}$$

(33)

-

for the splitting mechanism under all other bond conditions with lateral concrete confinement:

$${\tau }_{\max }=5.5{\left({\displaystyle \frac{f_{ck}}{20}}\right)}^{0.25}$$

(34)

Here, f_ck is the characteristic strength of cylinder specimens. Good bond conditions are achieved for ribbed rebars, whereas for plain rebars, “all other bond conditions” should be used.

In the parametric study, the bond strength varies from 4 MPa for the worst bond conditions according to Formula (33) to 13 MPa for the best bond conditions according to Formula (29), given an average concrete strength f^’_c = 32.4 MPa. Figure 17 and Figure 18 show the load–deflection diagrams of typical substructures for different bond strengths and the plastic hinge rotation angles at various extreme resistance stages, respectively.

Increasing the bond strength reduces crack widths, which leads to smaller deflections at the flexural and arch action stages (see Figure 17). However, at the same time, smaller slip of the reinforcement corresponds to larger forces in the bars, which ultimately reduces the deformation capacity of the structures (decreased deflections and ultimate plastic hinge rotation angles—see Figure 17 and Figure 18, respectively). This leads, in some cases, to rupture of the tension reinforcement without the structure transitioning to catenary action. It is worth noting that, in the calculations, the bond stress over the plastic hinge length is taken as τ_bm,pl = 0.27τ_b,max in accordance with [30]. It appears that this value needs refinement to account for various detailing parameters of the load-bearing members, as well as the effect of axial compression (for segments of reinforcing bars inside joints).

According to studies by Mander, Priestley, and Park [32], a certain spacing of transverse reinforcement creates a confined concrete effect (hoop action), thereby increasing its strength and deformability. The strength of concrete confined by transverse hoops is determined by the following formula:

$$f_{cc}^{’}=f_{co}^{’}+4.1k_e{\rho }_h{f}_{yh}$$

(35)

where:

f′_co is the strength of unconfined concrete under uniaxial compression;

f_yh is the yield strength of the transverse reinforcement;

k_e is the confinement effectiveness coefficient, which accounts for non-uniform confinement in non-circular cross-sections and is taken according to Mander, Priestley, and Park [32];

ρ_h is the volumetric transverse reinforcement ratio.

The strength of the cover concrete is taken without considering the effect of lateral deformation confinement.

Since bond strength depends on the compressive strength of concrete, an increase is expected when using special detailing measures, such as reducing the spacing of transverse reinforcement or using mesh indirect reinforcement. However, in practical cases, the strengthening effect from reduced transverse reinforcement spacing may be insignificant, as in the tests of specimen S1 [27]. On the other hand, an increase in bond strength generally reduces the deformation capacity of the substructure, whereas to sustain a vertical load exceeding the ultimate P_CAA value for the arching stage, the catenary stage requires large deflections on the order of 1/10 of the span.

Furthermore, there are cases when reinforced concrete frames of buildings and facilities are exposed to high-temperature effects as a result of fires [33,34]. High temperatures cause a loss of concrete strength: at 500–600 °C, up to 50% of the original strength. There is also a softening of the reinforcement, which becomes particularly noticeable at temperatures above 400 °C. As a result of thermal deformations, cracking occurs, and the bond between reinforcement and concrete is impaired.

Many studies in this area have been performed on specimens in the form of prisms or cylinders that were loaded after exposure to fire. Such tests did not account for the presence of mechanical cracks in reinforced concrete elements behaving in the arching or catenary stage. Thus, under high-temperature fire effects, in addition to bond, the direct local thermal effect on the reinforcement inside the crack will also affect the load-carrying and deformation capacity. This circumstance requires further in-depth investigation in the future.

3.4. Influence of Horizontal Restraints

To assess the influence of support restraint stiffness against horizontal displacements, the stiffness parameter of the left support C₁ was varied while keeping the right support stiffness constant.

When assessing the influence of horizontal restraint stiffness on the force and deformation parameters of the resistance stages of the substructure, it was assumed that the beams are incompressible in the longitudinal direction and that no stiffening diaphragms are present. Therefore, the stiffness of the horizontal restraint is taken as the sum of the stiffnesses of the columns C_1,col,j. For a column fixed at both ends, the nominal stiffness for a unit displacement of one of the support sections is determined by the following formula:

$$C_{1,col,j}\cong 12{\displaystyle \frac{0.3E_c{I}_c}{l_{col}^3}}=3.6{\displaystyle \frac{E_c{I}_c}{l_{col}^3}},$$

(36)

for a column pinned at one end and fixed at the other:

$$C_{1,col,j}\cong 3{\displaystyle \frac{0.3E_c{I}_c}{l_{col}^3}}=0.9{\displaystyle \frac{E_c{I}_c}{l_{col}^3}},$$

(37)

where E_c and I_c are the initial modulus of elasticity of concrete and the moment of inertia of the concrete section; 0.3 is a factor accounting for stiffness reduction due to cracking and plastic behavior of the structural member.

In this study, the following column parameters were adopted for stiffness calculation: cross-section dimensions 0.7 × 0.7 m, column clear height l_col = 3 m, initial modulus of elasticity of concrete 30,000 MPa.

The horizontal bracing stiffness values C₁ and C₂ adopted for the parametric study, along with their corresponding frame schemes, are given in

Table 4. Horizontal restraint stiffness values adopted for the parametric study.

Restraint Conditions	Scheme of the Frame Under Column Loss Scenario	Stiffness of the Left Restraint C1, kN/m	Stiffness of the Right Restraint C2, kN/m
Full Restraint		600,000	600,000
Partial Restraint		300,000	600,000
No Restraint		60,000	600,000
No Restraint		30,000	600,000
No Restraint		0	600,000

.

To evaluate the influence of support stiffness under horizontal displacements, the stiffness parameter of the left support C₁ was varied while keeping the stiffness of the right support C₂ constant. Thus, this section considers the following characteristic cases:

Full horizontal restraint: C₁ = C₂ = 600,000 kN/m;

Partial restraint: C₁ = 300,000 kN/m, C₂ = 600,000 kN/m;

No restraint when removing a corner column on the lower story of the frame: C₁ = 60,000 kN/m, C₂ = 600,000 kN/m;

No restraint when removing a corner column on the middle story of the frame with reduced column cross-section sizes along the height: C₁ = 30,000 kN/m, C₂ = 600,000 kN/m;

No restraint when removing a corner column on the upper story of the frame: C₁ = 0 kN/m, C₂ = 600,000 kN/m.

Load–deflection diagrams of the typical frame substructures and the ultimate plastic hinge rotation angles at various resistance stages are shown in Figure 19 and Figure 20, respectively. With partial horizontal restraint of the typical substructure, the beam deflection at the compressive arch and tensile catenary action stages is slightly larger than with full restraint. However, the stiffness of the horizontal restraints had a greater effect on the ultimate load-carrying capacity at the compressive arch action stage compared to the case of full restraint. From the graphs in Figure 18, it can be seen that the stiffness of the horizontal restraints did not significantly affect the ultimate plastic hinge rotation angles. For the maximum load-carrying capacity at the compressive arch action stage, the average plastic hinge rotation angle was 0.025 rad; for the transition stage—0.082 rad; and for the tensile catenary stage (with full or partial restraint on both sides)—0.2 rad.

3.5. Influence of the Ratio of Effective Depth to Total Depth

To evaluate the influence of the location of the longitudinal reinforcing bars along the section depth on the parameters of extreme resistance stages, the ratio of the effective depth of the beam section to the total section depth, h₀/h, was varied from 0.5 to 0.9 in steps of 0.05 h. Load–deflection diagrams of the typical frame substructures and the ultimate plastic hinge rotation angles at various resistance stages are shown in Figure 21 and Figure 22, respectively.

From the graphs in Figure 21, it can be seen that reducing the ratio of effective depth to total section depth to h₀/h = 0.5 leads to a significant decrease in the maximum load-carrying capacity at the flexural and arch action stages; however, at the catenary action stage, the load-carrying capacity increases by 19% compared to that at h₀/h = 0.9. Therefore, as an additional measure to increase the load-carrying and deformation capacity of beams at the catenary action stage, it may be recommended to place additional longitudinal reinforcing bars near the mid-height of the cross-section of the members. Such reinforcement will only be activated when the members transition to the catenary action stage. In this case, the increase in load-carrying capacity is also achieved because, as the deflection grows, the ratio between the horizontal and vertical support reactions shifts in favor of the latter. The reduction in horizontal support reaction, in turn, has a positive effect on the resistance of vertical members. The length of such additional reinforcing bars should be determined based on the need to provide the required development length accounting for plastic behavior.

From Figure 22, it can be seen that, regardless of the h₀/h ratio, the ultimate plastic hinge rotation angles at various resistance stages change as follows. For the maximum load-carrying capacity at the compressive arch action stage, the plastic hinge rotation angle ranges from 0.009 rad at h₀/h = 0.5 to 0.019 rad at h₀/h = 0.9; for the transition stage, the plastic hinge rotation angle remained almost constant and averaged 0.083 rad; and for the tensile catenary stage—0.2 rad.

4. Conclusions

• Analytical resistance models have been developed for typical reinforced concrete building frame substructures in the region of potential local failure for various extreme resistance stages. Validation of the proposed model was performed by comparison with experimental data for 22 specimens tested by various researchers. The coefficients of determination R² obtained for the calculated force and deformation parameters of the resistance stages were 0.941 and 0.931, respectively. Using the proposed analytical models, a parametric study was carried out on the influence of the following factors on the extreme resistance: beam depth-to-span ratio, bond strength between reinforcement and concrete, horizontal restraint stiffness of the typical substructure, and the ratio of effective depth to total depth of the beam section.
• It has been established that, for the adopted reinforcement and bond parameters, the ultimate plastic hinge rotation angle of the beams increases with increasing beam depth-to-span ratio h/L from 0.04 to 0.15: from 0.008 rad to 0.027 rad at the arch action stage; from 0.028 rad to 0.118 rad at the transition stage; and from 0.13 rad to 0.21 rad at the catenary action stage.
• It was demonstrated that increasing the bond strength between reinforcement and concrete leads to a reduction in the ultimate plastic hinge rotation angles at the flexural and arch action stages and, in some cases considered in this work, to rupture of the reinforcement without transitioning to the catenary action stage. Moreover, when the catenary action stage is achieved, the load-carrying capacity is inversely proportional to the bond strength.
• It was shown that the stiffness of the horizontal restraints did not significantly affect the ultimate plastic hinge rotation angles. For the maximum load-carrying capacity at the arch action stage, the average plastic hinge rotation angle was 0.025 rad; for the transition stage—0.082 rad; for the catenary stage (with full or partial restraint on both sides)—0.2 rad.
• It was found that reducing the ratio of the effective depth of the beam section to its total depth increases the load-carrying capacity at the catenary stage by 19%. Therefore, as an additional measure to increase the load-carrying and deformation capacity of beams at the catenary action stage, it may be recommended to place additional longitudinal reinforcing bars near the neutral axis of the members at the support sections. Such reinforcement will be activated only when the members transition to the catenary action stage, including at the moment of rupture of the bars located near the most tensioned face of the member.

The presented model applies to cases of initial local failure caused by mechanical actions and does not account for the combined effect of loads and aggressive environments or high temperatures. The obtained research results can be used as recommendations for the design and detailing of reinforced concrete frames within the framework of the Alternative Load Path Method.

Analysis of the results presented in this paper allows the formulation of the following directions for future research:

• Investigation of the resistance stages in the post-yield behavior of deep beams;
• Investigation of the resistance stages considering corrosion damage to reinforcement and concrete and reduction in bond strength during the service life considering wear and corrosion damage;
• Investigation of the resistance stages considering the combined effect of loads and high temperatures that cause a reduction in the strength of concrete and reinforcement and in bond strength;
• Investigation of the parameters of resistance stages of structures in post-yield states with the formation of complex plastic hinges under the combined action of bending and torsional moments, as well as axial compressive or tensile forces.