Analytical Study Regarding the Seismic Response of a Moment-Resisting (MR) Reinforced Concrete (RC) Frame System with Reduced Cross Sections of the RC Beams

Abstract

In the last few decades, a series of earthquakes were recorded which pointed out several deficiencies regarding the ductile seismic response of MR RC frame structures. Thus, the research problem centres around the failure mechanisms registered by the structures, which differ from the general notions of seismic response commonly found in current design standards and norms regarding seismic actions. In these conditions, in the present paper—by using comparative methods—the analytical validation of the solution of plastic hinge concentration and seismic energy dissipation in the marginal beam areas is proposed. Therefore, the RC beam sections were reduced (weakened) in the marginal areas which exhibit a plastic deformation potential, as well as in the corner areas of concrete slabs with vertical rectangular holes. The significant outcomes of this research imply the partial “guiding” of plastic hinges in the zones adjacent to beam ends. Furthermore, a reduction of both the negative effects of horizontal rigidization of the beams and the cracking and plastic deformation effects of beam-column frame joints was observed. With these technical implications, a complex mechanism of plastic deformation of MR RC frame models is registered in which all lateral elements (including RC columns) participate in the dissipation of seismic energy, without the occurrence of the “weak storey” mechanism for any of the analytical RC frame models. Furthermore, it is possible to observe the partial formation of the global plastic mechanism “Strong Columns—Weak Beams” (SCWB) for some of the structural models. Finally, the analytically studied innovative element regarding the improvement of the seismic response of pure MR RC frame structures is successfully validated.

1. Introduction

The Moment-Resisting (MR) Reinforced Concrete (RC) frame structures designed according to the ductile concept represent a class of lateral systems exhibiting a complex seismic response to severe dynamic loads.

The general mechanisms of structural deformation imply plastic deformations in the marginal beam areas (considered as main dissipative elements) and in the inferior areas for ground storey columns (see Figure 1b). There also exists the possibility of plastic hinges occurrence in superior end areas for columns situated at the top storey. It is also possible for the plastic hinges to appear at both columns’ ends for intermediate storeys, as long as the storey itself fulfils the stability conditions [1,2,3]. In these types of structures, the columns are considered non-dissipative lateral elements with local, minor (insignificant) plastic deformations. The same specifications are attributed both to beam-column frame joints and to RC slabs. In these conditions, the beam-column frame nodes form a common body with RC columns and the RC slab ensures the transfer of storey inertial forces through the beams and towards the vertical structural elements [1,2,3,4,5,6,7,8,9].

This capacity design concept, commonly used for current seismic design norms for pure MR RC frame structures [1,2], leads to the development of the global seismic response mechanism known as “Strong Columns—Weak Beams” (SCWB) [3,4,5,6,7].

Thus, in the following chapters, conceptual aspects that highlight the non-correspondence between the seismic response of pure MR RC frame systems and the elements specified in current seismic design norms for studies with several implications are discussed as follows:

• Theoretical and analytical ones;
• Experimental ones;
• Real ones for RC frame structures which collapsed as a result of in situ severe seismic loading.

As such, the objective of the current study is to underline, through available literature sources [8,9], the incapacity of the current MR RC frame systems—designed according to the ductile concept—to dissipate the seismic energy in accordance with the current seismic norms and to analytically validate a novel solution of plastic hinge concentration and seismic energy dissipation in the marginal beam areas.

2. Methodology

The research methodology pertaining to the current research paper implies four distinct stages:

I.

The establishment of the seismic response of MR RC frame systems through analytical and theoretical studies;

II.

The establishment of the seismic response of MR RC frame systems through the real, recorded seismic response of in situ structures subjected to severe earthquake actions;

III.

The obtainment of relevant conclusions and their overlap with the specifications of the current seismic design norms [1,2];

IV.

The proposal of solutions for improving the seismic response of existing and future MR RC frame systems by means of reducing the section of the beams in marginal areas.

The obtained results were studied by employing the comparative method.

3. Seismic Response of the MR RC Frame Structures through Theoretical and Analytical Implications

Within a complex (research) analytical study, carried out by Sococol et al. [10,11,12,13,14,15] and performed with ATENA 3D computer software [16], the seismic response for a series of MR RC frame models, reduced at a scale of 1:2 according to similitude criteria (rules), was studied [17,18,19]. Within this study, the longitudinal reinforcement percentages of the beams and columns, the reinforcement percentages of the slabs, the transversal beam and slab sections, the concrete strength classes, etc., were all varied. However, the inter-axis dimensions and the height regime of the openings, for all types of structural models, were preserved.

As such, several different local and global structural deformation mechanisms were observed (see Figure 1c–f), with a different plasticisation mechanism presented in current seismic design norms [1,2] for seismically resistant RC frame structures.

Thus, for all RC frame models, plastic deformations were registered in end (marginal) zones of the columns, with intense concrete cracking and early reinforcement “yielding” (in tensioned areas) [10]. Consequently, the columns are presented as dissipative lateral elements which can lead to the occurrence of the unwanted “weak storey” mechanism (see Figure 1c–f).

The RC beams form a common rigid block with the slab and beam-column joints, “beams-slab-frame nodes”, which influences the direction and occurrence of plastic hinges in marginal zones of the columns, in the area immediately adjacent to beam-column frame nodes. Additionally, the cracking process of the slab was observed. This process “decides” the marginal deformation length of the concrete beam [10,11,12,13,15]. Moreover, the RC slab controls the plastic rotation mode of the beams as well as the concentration of main tensions and specific deformations in concrete and reinforcement bars [10].

In these circumstances, the beams are presented as lateral dissipative elements but not as main seismic energy dissipative elements. Consequently, no (global) ductile mechanism “Strong Columns—Weak Beams” (SCWB), characteristic of pure seismic resistant RC frame structures, was registered (observed) [10,11,12,13,15] (Figure 2).

The RC beam-column joints did not have the performance specified in current seismic design norms, such as: nodes do not crack, their behaviour is in the linear elastic domain, they are a component part of the columns [1,2]. Rather, they crack intensively and become a “common body” with the beams and the RC slab [10,11,12,13,15].

Basically, for all the pure MR RC frame models on which non-linear static analysis (with ATENA 3D computer software, version 5.1.2.11514, Cervenka Consulting Ltd., Prague, Czech Republic [16]) was performed, a complex seismic energy dissipation mechanism was registered. This mechanism is formed through an active participation of all the lateral elements (beams, slabs, beam-column frame joints, columns) to the global seismic response. As such, reinforcement yielding and concrete cracking can be observed, together with concentration effects of the principal and tangential tensions and together with ultimate specific deformations of both concrete and reinforcement bars in the marginal areas of RC columns. The effects of these complex structural deformation processes lead to the occurrence of “weak storey” mechanisms.

Other non-linear static and dynamic studies [24,25,26] present the same local mechanisms of plastic hinge formation in marginal areas of the columns, which are:

• considering the rigid slab in the horizontal plane (for rigid-type floor) [27];
• not considering (using) the RC slab in extreme load conditions (having, as an effect, the fragile failure of the beams and the brittle failure of the columns) [28,29], etc.

Therefore, in most analytical and experimental studies, as well as in situations where real earthquakes occurred (see Figure 2 and collapsed RC frame structures for Wenchuan earthquake and Chi-chi earthquake in the research study conducted by Hu et al. [30], etc.), the following aspects were observed, regarding the seismic response of RC frame structures:

• fragile failure mechanisms of the structural system, through active plasticisation of the RC columns from a certain storey, in their superior and inferior (marginal) areas (Figure 1c,d and Figure 2);
• hybrid failure mechanisms (which have, as a specificity, the random formation/development of plastic hinges at beam and column ends), which can take a multitude of forms (ex.: Figure 1e,f and Figure 2).

Related to the analytical studies, in seismic design codes, several suggestions and conclusions regarding the capacity design concept of pure MR RC frame structures were observed:

Thus, in the FEMA P-2012 norm [28,31], “two important suggestions regarding Strong Columns—Weak Beams (SCWB) seismic energy dissipation mechanism (Figure 1b) for pure Moment Resisting (MR) Reinforced Concrete (RC) frame systems are specified” [32]:

• there is a possibility that the RC beams have a superior bending stiffness compared to the RC columns [11];
• it is proposed to use a higher base shear force than the value resulting from the calculation [33], in order to minimise the consequences of design deficiencies [34].

These suggestions and conclusions found in FEMA P-2012 [31] were taken into account and were practically applied through the seismic design methodology of pure MR steel frame systems, in accordance with P100-1 Romanian seismic design norms [1].

Thus, the steel columns are computed with a different set of lateral seismic forces (generated from the load groupings stage) than the ones for the frame beams, this represents a theoretical solution for the reduction of seismic effects on these types of structures (MR steel frames and MR RC frame systems) [32].

A final aspect to consider regarding the non-ductile seismic response of pure MR RC frame systems is that these types of structures are retrofitted in Japan. Thus, a shift is made by ”changing the structural system from RC frames to coupled RC walls” [35]. In effect, the existing columns are integrated into walls named “wing walls” [35].

4. Real Seismic Response of MR RC Frame Structures and Experimental Research

“The seismic activity from the last two decades” [36] proves, for a large number of situations, the incapacity of development for the ductile deformation mechanism (represented in Figure 1b) for MR RC frame structures.

When confronted with real earthquakes, the issues regarding the capacity design concept and the seismic response of these types of structures become more acute when faced with the fact that their in situ structural response remains the main objective source of data regarding the performance of their seismic design.

Thus, for MR RC frame structures, designed with current seismic regulation norms, fragile seismic energy dissipation mechanisms were recorded (see Figure 1c,d and Figure 2), which manifested through the brittle failure of columns at a certain storey (see Figure 2a,b,d) or through a combination of hybrid failure mechanisms (see Figure 1e,f and Figure 2c)—“weak storey”.

In essence, “the seismic activity from the last two decades” [36] revealed the necessity for the re-evaluation of the capacity design concept for the aforementioned type of structural system.

In this context, by performing experimental studies on seismic platforms or in other technical conditions, checks of the failure mechanisms identified in MR RC frame structures were attempted to be performed.

These studies were made based on structural models scaled according to similitude criteria or based on Reinforced Concrete (RC) frame prototypes.

Thus, the main experimental study in this technical research branch is “Seismic platform testing of a P + 9E seismic resistant RC dual system (with mainly RC frames)—prototype in Japan” [37].

The purpose of this experimental study was twofold: from a theoretical perspective, the checking of the global seismic response specified in seismic design norms was performed; from a practical (real) perspective, the in situ study of the seismic response aspects was performed on real structures from the same category of structural systems (dual type with mainly RC frames).

The main conclusions regarding the seismic response of this reinforced concrete prototype are:

• Extreme cracking of the RC beam-column joint;
• Deformation of marginal areas of RC columns;
• Slab cracking in the areas of interaction with the RC walls;
• Structural deformation of the ground floor walls and in other areas along the height of the structure, etc.

In these circumstances, no global plastic hinge mechanism was registered on beam ends in the way that the idealised mode of structural deformation is presented in current seismic design norms (see Figure 1b) [1,2].

Other experimental studies performed by Wang et al. [38], Taheri et al. [39], Li et al. [40], Zembaty et al. [41], Kamath et al. [42], and Rizwan et al. [43] prove the same manner of concentration of the plastic deformations in the end zones of RC columns. The cracking mode, the confinement effect (concrete transversal reinforcement) and the local concrete deformation in the adjacent beam-column frame node are all highlighted.

In addition, the bending rigidisation effect of the RC beams, due to the presence of the concrete slab in its real form and with real geometrical dimensions, was observed in an experimental study carried out by Pohoryles et al. [44].

Three experimental studies carried out by Li et al. [45], Hou et al. [46], and Zhang et al. [47] on seismic platforms for RC frame structural models and for dual system models with mainly RC frames, reduced to scale according to similitude criteria, and noted the same effects regarding the incapacity of the plasticisation process of the beams in their marginal areas.

Important observations regarding the seismic behaviour of the beam-column frame node, which is a part of the MR RC frame system, were also made in the experimental studies of Liu et al. [48] and Hu et al. [30].

Thus, important cracks and degradations were observed in the beam-column frame node, which occurred under the incidence of the active concentration effects of the plastic deformations in marginal column areas and partially in the end areas of the RC beams.

5. Conclusions That Specify and Reinforce the Need for the Current Analytical Study

Analytical, experimental studies and the effects of earthquakes in the last two decades regarding the seismic response of MR RC frame structures demonstrate a fragile global mechanism of failure by brittle rupture of RC columns in marginal end areas. The RC columns are dissipative elements which crack intensively and do not perform in the linear elastic domain.

The RC beams form a common body with the slab and with the beam-column frame node, “lending” from the bending rigidity of the slab in the horizontal plane, thus rotating and cracking as long as the slab rotates. This leads to the formation of a common rigid block RC “beams-slab-frame nodes”, which, during the seismic energy dissipation process, directs the formation of plastic hinges towards the marginal areas of the RC columns.

The RC beam-column frame nodes form a common body with the beams and the slab and intensely crack together with the end zones of the columns.

The “Strong Columns-Weak Beams” (SCWB) global ductile mechanism presented in the current seismic design norms for pure MR RC frame structures exposed to earthquake actions does not occur for the case of analytical and theoretical studies, nor for real-life structures undergoing moderate to severe earthquakes. As such, weak storey mechanisms are registered, with random deformation patterns.

6. Complex Static Non-Linear Analysis of the Representative Pure Moment Resisting (MR) Reinforced Concrete (RC) Frame Model

6.1. General Aspects

The results, conclusions and observations pertaining to analytical and experimental studies, as well as to the experience obtained following the study of the MR RC frame structures, thus pointed towards a non-satisfactory seismic response of such structures. Obviously, the necessity of finding practical solutions which can lead to the improvement of the “ductile seismic response” [49] for such structural systems was recognised.

Consequently, a series of numerical simulations using the computer program ATENA 3D [16,50], were performed, having as a representative analytical model the K_7 MR RC frame system (see Figure 3) (

Table 1. Principal characteristic parameters considered in numerical analyses of the Moment Resisting (MR) Reinforced Concrete (RC) frame models.

NSC	CSC	LSRT	TSRT	LSR RC C [CS:15 × 15 cm]	LSR RC LB [CS:15 × 20 cm]	LSR RC TB [CS:15 × 20 cm]	TSR RC C	TSR RC LB and TB	R RC S [hs = 7 cm]	GR
K_7	C20/25	Bst 500S	Bst 500M	4ϕ14	4ϕ8	4ϕ8	1ϕ4/1 CS	1ϕ4/1 CS	ϕ6	Figure 8(a2)
K_7_3_A	C20/25	Bst 500S	Bst 500M	4ϕ14	4ϕ8	4ϕ8	1ϕ4/1 CS	1ϕ4/1 CS	ϕ6	Figure 8(b2)
K_7_2_A	C20/25	Bst 500S	Bst 500M	4ϕ14	4ϕ8	4ϕ8	1ϕ4/1 CS	1ϕ4/1 CS	ϕ6	Figure 8(c2)
K_7_1_A	C20/25	Bst 500S	Bst 500M	4ϕ14	4ϕ8	4ϕ8	1ϕ4/1 CS	1ϕ4/1 CS	ϕ6	Figure 8(d2)
K_7_S_B_1	C20/25	Bst 500S	Bst 500M	4ϕ14	4ϕ8	4ϕ8	1ϕ4/1 CS	1ϕ4/1 CS	ϕ6	Figure 8(e2)

Note: NSC—Numerical Simulation Code; CSC—Concrete Strength Class; LSRT—Longitudinal Steel Reinforcement Type; TSRT—Transverse Steel Reinforcement Type; LSR—Longitudinal Steel Reinforcement; RC—Reinforced Concrete; C—Columns; CS—Cross Section; LB—Longitudinal Beams; TB—Transverse Beams; TSR—Transverse Steel Reinforcement; R—Reinforcement; S—Slabs; h_s—slabs thickness; GR—Graphical Representation.

) [10,15]. Based on this structural system, the other analytical models were generated, which had as a necessary condition (and main purpose), the modification of the seismic behaviour characteristics of three reinforced concrete structural elements from the make-up of the structural system:

• Beams;
• Slabs;
• Columns.

In the present paper, the structural modifications pertaining to the RC beams are analysed and discussed.

Therefore, for the analytical MR RC frame models, the cross section of the beam was reduced through the vertical mechanical drilling process in the zones which have a plastic deformation potential. The obtained seismic response from these structures was overlapped by means of the comparative method, both with the seismic response of the representative K_7 MR RC frame model—which does not have any reduced beam cross section—as well as with the seismic response of the K_7_S_B_1 MR RC frame model—for which the cross sections of the beams and slabs were reduced through the vertical mechanical drilling process.

The MR RC frame models thus considered were laterally loaded with equivalent static forces (see Figure 3b,f), consequently allowing the obtainment of the “F-D” capacity curves together with the other types of curves. In addition, numerical values in a tabular form were obtained together with graphical representations of the deformation mode for each analytical RC frame model.

In essence, the present paper aims to validate the mechanical weakening (of resistance and lateral stiffness) method of dissipative structural elements—reinforced concrete beams—in the design stage, highlighting the emergence of the common rigid block “beams-slab-frame nodes” [51].

6.2. Input Data Consideration

The analytical MR RC frame models, reduced to a ½ scale according to “similitude criteria” [17,18,19] from the present paper, were developed based on the K_7 representative structural system (see Figure 3a,c,d) and were laterally loaded with static forces at each storey level (Figure 3b,f). The numerical simulations for each structural model were made with the ATENA 3D computer program [16], having as input data the parameters specified in Table 1 and

Table 2. Principal aspects regarding the cross section reducing method of the RC beams through vertical drilling (mechanical) process in the marginal areas with potential plastic deformation for analytical MR RC frame models.

NSC	RC Drilled Element Type in the Potentially Plastic Zone	Hole Type Depends on the Geometric Shape (Form)	Variable (V)/Constant (C) Size Holes	Number of Holes	Number of Rows of Holes	Constant (C)/Variable (V) Distance between Holes	Constant (C)/Variable (V) Distance between Rows of Holes	Minimum (Min)/Maximum (Max) Distance between Holes and RC B-C Joint/RC Column	Rows of Vertical Holes Positioning (Zig-Zag, Parallel etc.)	GR
K_7	-	-	-	-	-	-	-	-	-	Figure 8(a1,a3)
K_7_3_A	beam	square holes	C	2 for LB 1 for TB	1 for LB 1 for TB	C	-	Min.	-	Figure 8(b1,b3)
K_7_2_A	beam	square holes	C	3 for LB 2 for TB	1 for LB 1 for TB	C	-	Min.	-	Figure 8(c1,c3)
K_7_1_A	beam	square holes	C	4 for LB 3 for TB	1 for LB 1 for TB	C	-	Min.	-	Figure 8(d1,d3)
K_7_S_B_1	beam and slab	square holes	C	4 for LB 3 for TB 6 for RC slab	1 for LB 1 for TB 2 for RC slab	C	- - C	Min.	- - parallel	Figure 8(e1,e3)

Note: Vertical holes were positioned between RC beams stirrups and RC slabs (steel) reinforcement bars, without structural integrity destruction of these structural elements. NSC—Numerical Simulation Code; RC—Reinforced Concrete; LB—Longitudinal Beams; TB—Transverse Beams; CS—Cross Section; GR—Graphical Representation.

. For reinforcement disposition, the representations in Figure 4, Figure 5, Figure 6 and Figure 7 are to be consulted.

It should be noted that the current analytical study is part of a larger research project which includes the testing of a MR RC frame model on the shake table. Thus, the reduction to scale was performed according to the bearing capacity of the shake table as well as to the geometric in-plane restriction conditions.

For each of the studied analytical models, the “discretization rules” [52,53] (see Figure 3e), “stress-strain relations for concrete” [14,54,55,56,57,58,59] and “stress-strain laws for steel reinforcement” were observed and applied [14,60,61,62,63].

The way the current analytical study was approached, regarding the structural models for which the cross sections of the RC beams are reduced through mechanical drilling in zones where plastic deformation potential exists, stems from the idea of obtaining a mechanism exhibiting the “ductile seismic response” [49] of MR RC frame structures, i.e., with deformable beams in marginal zones and with columns with a linear elastic behaviour (see Figure 1b).

As such, this method may be viewed as a solution for the concentration and direction of plastic deformations in the marginal zones of the beams, such that the global plastic mechanism of MR RC frame structures will develop in its idealised form.

Therefore, for each type of analytical model, the parameters from Table 2 were considered, without taking into account the following specific detail elements:

• The influence of the geometric shape of the drilled holes regarding the local deformation mode of the structural element;
• The influence of the variability of the dimensions of the drilled holes regarding the local deformation mode of the structural element;
• the constant/variable distance between the drilled holes with the specified implicit value and the effect of such distances upon the local deformation mode of the structural element;
• The constant/variable distance between the rows of drilled holes with the specified implicit value and the effect of these distances upon the local deformation mode of the structural element;
• The distance from the node/column with the specified implicit value;
• The influence of the positioning of the rows of drilled holes in a zig-zag/parallel pattern, etc., upon the local deformation mode of the structural element.

The choice and the use of the set of parameters presented in Table 2 was carried out in order to simplify the analytical models generated with ATENA 3D.

Another desired simplification was the generation of the discretisation mesh, such that no issues would intervene regarding the “interaction between concrete and reinforcement bars” [52,53], etc.

The number of vertical drilled holes in the beams was established based on Lpl (where Lpl is the length of the plastic hinge in the RC beams, computed according to the P100-1 norm [1,64,65]) for superior limit for each direction, respectively for each type of beam: longitudinal, see Figure 4 and transversal, see Figure 5).

The same way of considering the vertical holes was used in the case of the RC slabs for the K_7_S_B_1 model.

The vertical holes were positioned between the stirrups of the beams/wire mesh of the slabs at a minimum distance from the beams/beam-column frame nodes (see Table 2). This minimum distance is dictated by the position of the first stirrup of the beams (Figure 4 and Figure 5)/first wire mesh outside of the beam-column frame node (Figure 7).

In Figure 8, the analytical models K_7, K_7_3_A, K_7_2_A, K_7_1_A and K_7_S_B_1 are presented in 3D format, including the zone pertaining to the beam-column frame node and the reinforcement configuration.

7. Analytical Results and Complementary Comments

7.1. Analytical Results

Following the non-linear static analyses for the cases specified in Table 1 and Table 2 and Figure 8, regarding the MR RC frame models, the following data (results) were obtained (see

Table 3. Analytical results in lateral forces, horizontal displacements and specific deformations for K_7, K_7_3_A, K_7_2_A, K_7_1_A and K_7_S_B_1 laterally loaded structural MR RC frame models with equivalent static forces.

NSC	Fu (kN)	du (m)	F*y (kN)	d*y (m)	SPO CB	TSE (CF)	TSE (TF)	GR TSE (CF/TF)	PFSM	GR PFSM
K_7	41.575	0.03288	40	0.0187	Figure 13a	0.002789	0.006118	Figure 14(a3,a4)	0.0413	Figure 14(a1,a2)
K_7_3_A	41.575	0.03109	40	0.0182	Figure 13c	0.002599	0.005614	Figure 14(b5,b6)	0.02547	Figure 14(b1–b4)
K_7_2_A	39.49625	0.0262	38.1	0.0167	Figure 13d	0.002072	0.003713	Figure 14(c5,c6)	0.01824	Figure 14(c1–c4)
K_7_1_A	39.49625	0.02665	38.1	0.0167	Figure 13e	0.002187	0.004011	Figure 14(d5,d6)	0.01959	Figure 14(d1–d4)
K_7_S_B_1	41.575	0.03179	40.4	0.0188	Figure 13b	0.002693	0.00576	Figure 14(e5,e6)	0.02913	Figure 14(e1–e4)

Note: NSC—Numerical Simulation Code; F_u—ultimate lateral force corresponding to global system collapse; d_u—ultimate lateral displacement of the structural system; F*_y—lateral force corresponding to structural yielding of the equivalent SDOF structural system; d*_y –horizontal peak displacement corresponding to structural yielding of the equivalent SDOF structural system; SPO CB—Static Push-Over Curve Bilinearised; TSE—Total Strain Eps zz; CF—Compressive Failure; TF—Tensile Failure; GR—Graphical Representation; PFSM—Principal Fracture Strain Max. Specific deformations values in this table correspond to the final horizontal loading step. SPO curves for all MR RC frame models specified in the current table are graphically represented in Figure 10. Lateral Forces (LF)—PFSM curves for all MR RC frame models specified in the current table are graphically represented in Figure 11. LF—TSE curves for all MR RC frame models specified in the current table are graphically represented in Figure 12.

and Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14):

• Ultimate lateral force (Fu);
• Ultimate lateral displacement of the structural system (du);
• The lateral force corresponding to structural yielding of the equivalent SDOF structural system (F*_y);
• The horizontal displacement at the top level of the structure corresponding to structural yielding of the equivalent SDOF structural system (d*_y);
• The total specific deformations Eps zz;
• The main (maximum) failure deformations;
• The cracking panel for the final step of lateral loading.

The analytical results obtained following the non-linear static calculus are specified both in a tabular form (see Table 3) and as diagrams (see Figure 10, Figure 11 and Figure 12) and were discussed in parallel with the graphical exposure of the cracking mode of each of the studied analytical model (see Figure 14). This was accompanied by the extraction of the effects and of the local seismic energy dissipation mechanisms for each of the structural element, in concert with the global mechanisms (for the entire structure).

The lateral forces F*_y and the lateral displacements d*_y corresponding to the structural yielding of the equivalent SDOF structural system were determined following a bilinearisation of the SPO capacity curves (see Figure 13) in accordance with the prescriptions found in Eurocode 8 [2]. The computer program SPO2FRAG was used in this process [67,68].

The general conclusions regarding the global and local seismic responses of the analysed RC frame models are presented in

Table 4. General aspects (conclusions) regarding the structural deformation response of the analytical MR RC frame models.

NSC	RC Beams Cracking Process		RC Columns Cracking Process		RC Slabs Cracking Process		RC Column-Beam Joint Cracking	Final Rupture—RC Structural Element/Elements	Zone/Zones of Final Rupture	RC Beam Cracking Length Limiting by RC Slab Cracking Area	Risk of the Common Rigid Block RC “Beam-Slab-Frame Node” Formation	Concrete Cracks Migration Process from the Longitudinal Beams to the Transverse Beams in the Adjacent Area of the Frame Node	GR
	Local—in Potential Plastic Zones	On Entire Length	Local—in Marginal Areas	On Entire Height	Local Area	Extended Area
K_7	low	-	intense	low	low	medium to intense	intense	columns and nodes	marginal zones of the columns; entire volume of the nodes	yes	high with practical formation	low to insignificant	Figure 14a
K_7_3_A	medium	-	medium	low	low to medium	medium to intense	medium to intense	beams and nodes	marginal zones of the beams in reduced cross sections and immediately adjacent area of the beam-column joints	yes	medium with partial formation	low to medium	Figure 14b
K_7_2_A	medium to intense	-	medium	low	low to medium	medium to intense	medium to intense	beams and nodes	marginal zones of the beams in reduced cross sections and immediately adjacent area of the beam-column joints	yes	medium with partial formation	low to medium	Figure 14c
K_7_1_A	medium to intense	-	medium	low	medium	medium to intense	medium to intense	beams and nodes	marginal zones of the beams in reduced cross sections and immediately adjacent area of the beam-column joints	yes	medium with partial formation	low to medium	Figure 14d
K_7_S_B_1	medium to intense	-	medium	low	medium to intense	medium to intense	medium	beams, slabs and nodes	marginal zones of the beams in reduced cross sections; corner area for reduced cross section of the slabs; partial volume of the beam-column joints	partial with limited influence	medium with low process formation	medium to high	Figure 14e

Note: NSC—Numerical Simulation Code; RC—Reinforced Concrete; GR—Graphical Representation. Specified conclusions in the current table were developed based on the recorded observations at each lateral loading step for each MR RC frame model. Specified figures in GR section (column) correspond to the seismic response of the MR RC frame systems (considered laterally loaded with equivalent static forces) in the ultimate horizontal loading step.

. The main failure mechanisms and other structural mechanisms which developed and were observed during the lateral loading of the structures are highlighted.

Thus, from the point of view of the seismic response at lateral forces and lateral displacements at the top of the structure, the K_7 and K_7_S_B_1 structural models registered the maximum values (see Table 3; Figure 10). Nevertheless, the structural yielding of the entire system varies for these two analytical models. Regarding the values of the specific failure deformations, the K_7 model is superior to K_7_S_B_1 model (see Figure 11 and Figure 12). The other structural models which vary the number of vertical drilled holes in the end zones of the beams are positioned below the K_7 and K_7_S_B_1 analytical models, from the point of view of the values of the lateral forces, of lateral displacements (Figure 10), of specific failure deformations (Figure 11) and of total specific Eps zz deformations (Figure 12).

On the other hand, when graphically visualising the structural deformation process (mechanism), the model K_7 develops the least favourable mechanism of seismic energy dissipation, because it forms a common rigid block ”slab-beams-frame node” (see Figure 14a). Furthermore, an excessive deformation of the beam-column frame node can be observed, together with large deformations in both marginal zones of the RC columns.

Compared with the seismic response of the K_7 model, the K_7_3_A, K_7_2_A and K_7_1_A models register a concrete cracking mechanism towards the weakened zones, i.e., towards the zones where the vertical drilled holes are present (Figure 14b–d). It can be thus stated that the concentration and direction process of the plastic hinges towards the drilled areas takes place (Figure 14(b3,b4,c3,c4,d3,d4)). For the analytical model which possesses the largest number of vertical drilled holes (K_7_1_A model), the optimum bending deformation shape of the RC beams is achieved (Figure 14d).

Additionally, the structural K_7_1_A and K_7_S_B_1 models exhibit the most favourable local deformation mechanisms for both concrete and steel, which favour the dissipation of the seismic energy.

As such, for these two structures, a mechanism of crack migration was observed. The cracks migrated from the zone of the longitudinal beam, which was weakened by drilling, towards the slab zone and towards the marginal zone (weakened also by drilled holes) of the transversal beam (on the short side of the structural system) (see Figure 14d,e).

In this way, the partial conservation mechanism for structural integrity of the beam-column frame node occurs. This is accompanied by an increase in the redundancy and in the ductility of the structural system, through the implication of a larger number of plastic zones (hinges).

7.2. Complementary Comments

The numerical simulations corresponding to the analytical MR RC frame models prove the importance of the clear graphical specification of the deformation zone for each structural system. In the case where the graphical specification is not clear (regarding the precise trajectory of the cracks) and the obtained conclusions are based only on the capacity curves in Figure 10 or on other types of curves (Figure 11 and Figure 12), a wrong interpretation of the global seismic response of the structure may occur. This is due to the complexity of deformation and structural cracking of reinforced concrete elements.

Moreover, the implicit value of the main failure deformations for any type of studied frame has a lower importance than the zone in which the maximum deformation itself occurs. This aspect can be particularly well observed in the K_7_1_A and K_7_S_B_1 models, for which the marginal zones of the beams which exhibit a plastic deformation potential became plastic zones following the lateral loading (see Figure 14d,e).

In addition to the above comments, the following aspects are specified:

• The number of drilled holes in the longitudinal and transversal RC beams influences the state of cracking/deformation of the zone with a plastic deformation potential, for each step of lateral loading. Consequently, for the K_7_1_A structural model in which the beams have a maximum number of drilled holes, the zones with plastic potential crack much faster (they exhibit important cracks from a lower loading step) and influence the deformation state of the RC slab. As the number of drilled holes in the beams decreases (as is the case for K_7_2_A and K_7_3_A models), the maximum main failure deformations affect the beam-column frame node and the RC column ends with a greater intensity (see Figure 14b,c).
• The number of vertical drilled holes considered in the corner zones of the RC slabs for the K_7_S_B_1 analytical frame model influences not only the structural cracking and deformation state of the respective zone (immediately adjacent to the drilled holes) with a plastic deformation potential, but also the cracking mode of the adjacent zones (i.e., the beam end zones) (Figure 14e). Additionally, the potentially plastic zones with vertically drilled holes in the corners of the slabs have become “migration zones” for the cracks/deformations from the longitudinal beams towards the transversal beams and towards the slab. This mechanism occurs in a shorter time (for a smaller number of lateral loading steps of the analytical model) and far more efficiently in comparison with the situation of the K_7 model. A part of the negative effects pertaining to the beam-column frame node and to the marginal zones of the RC columns is thus mitigated.
• The SPO curves presented in Figure 13 prove the incapacity for a complete visualisation of the global seismic response mode of the structures and can even lead to the obtainment of wrong conclusions. Thus, by analysing the bilinearised SPO curves from Figure 13, a conclusion that the unaffected model K_7 presents a global seismic response superior to the other analytical models may be reached.

The values of F*_y lateral forces and of d*_y lateral displacements corresponding to the structural yielding of the equivalent SDOF structural system for the K_7, K_7_S_B_1, K_7_3_A, K_7_2_A and K_7_1_A RC frame models, which can be found in Table 3, were determined following the bilinearisation of the SPO capacity curves in conformity with the prescriptions found in Eurocode 8 [2], with the SPO2FRAG computer program [67,68]. The representations of the bilinearised curves for each structural model may be consulted in Figure 13.

8. Conclusions

The general conclusions regarding the results of the analytical study pertaining to the applicability and validity of the practical method for improving the global and local seismic response by using the method of the reduction of the cross section of the RC beams through mechanical drilling of vertical holes in the zones with a plastic deformation potential are summarised in Table 4.

The main local and/or global mechanisms of the analytical MR RC frame models studied in the present research paper, which were observed during the lateral step-by-step loading, are as follows:

• The guiding and concentration of the maximum failure deformations (PFSM) of concrete in the marginal (“weakened”) areas of the beams with a reduced section (through the employment of vertically drilled holes) was achieved;
• The migration of cracks from the longitudinal beams to the transversal ones along the path of the corner zones of the drilled slabs was observed; as such, a partial “conservation” of the structural integrity (of strength and stiffness) of the beam-column frame node was accomplished;
• The failure of concrete and the reinforcement yielding in RC beam areas with a reduced section was noted;
• The interruption of the mechanism of development for the common rigid block “beams-slab-frame nodes” was achieved;
• The reduction of the cracking length influence of the slabs upon the cracking length of the reinforced concrete beams was observed;
• The reduction of the local destruction (failure) mechanism (effect) of the beam-column frame node was observed;
• The reduction of the mechanism (effect) of the plastic deformation concentration in the superior and inferior zones of the reinforced concrete columns was noted;
• The occurrence of complex seismic energy dissipation mechanisms through the registering of plastic deformations in beams, slabs and partially in frame nodes and end zones of the reinforced concrete columns was detected.

Despite these important observations, both the SPO curves and the bilinearised curves of the structural models associated with the tabular results with implicit values of the lateral forces and top displacements cannot offer a clear and realistic image regarding the structural seismic response.

Therefore, such a desirable and complete perspective should “capture” and describe the concrete failure mechanisms of the lateral system. This can only be achieved with the graphical knowledge of the accurate cracking and deformation mode at both the structural element level and at the lateral, interconnected elements level.

Hence, the proposed method of modifying the reinforced concrete beams through the reduction of the transversal (cross) section in zones with plastic deformation potential, such that the process of seismic energy dissipation in these specially designed zones will occur, is validated. Moreover, the negative effects induced by the superior bending stiffness of the slabs are limited.

It can be specified that the main objectives of the current analytical study were achieved:

• The integrity of the capacity and global design concept for the type of discussed structural system was achieved;
• The ductile mechanism “Strong Columns-Weak Beams” was partially achieved for the MR RC frame systems;
• The current design norms and regulations of seismic design for MR RC frame systems were respected;
• The base of this research included the real seismic response of reinforced concrete frame structures and the seismic response of the same type of structures discussed in other analytical studies.