The influence of the beam-column connection on the global results of multi-storey reinforced concrete structures

O efeito do grau de conexão viga-pilar na análise global de estruturas de concreto armado de múltiplos pavimentos

Abstracts

Abstract

The effect of the semi-rigid connection between beams and columns on the global behavior of reinforced concrete (RC) structures was investigated. The relative rotations in the nodes of the spatial frames were determined using the analytical approach proposed by Alva and El Debs. The rotational restraint factor (${\mathrm{\alpha }}_{\mathrm{r}}$) was defined and applied in each node of the spatial frames. RC spatial frames with total heights of 21, 42, and 63 m were considered. It was found that the semi-rigid connection increases the lateral displacements, the 2^nd order effects, and the ${\mathrm{\gamma }}_{\mathrm{z}}$ values. The lower the stiffness of the nodes, the lower the influence of ${\mathrm{\alpha }}_{\mathrm{r}}$ on the global results. The total fixed beam-column connections can lead to an unrealistic behavior of the spatial frame and thus to global results.

Keywords:
reinforced concrete structures; beam; column; connections

Resumo

O efeito da conexão semirrígida entre vigas e pilares no resultado global de pórticos de estruturas de concreto foi investigado. As rotações relativas foram determinadas usando o método analítico de Alva e El Debs. O fator de restrição de rotação (${\mathrm{\alpha }}_{\mathrm{r}}$) foi aplicado em cada nó do pórtico. A pesquisa foi feita em estruturas com alturas totais de 21.0, 42.0 e 63.0 m. Concluiu-se que as ligações semirrígidas aumentam os deslocamentos laterais, efeitos de 2ª ordem e coeficiente ${\mathrm{\gamma }}_{\mathrm{z}}$. Quanto menor a rigidez das vigas e pilares, menor a influência do fator ${\mathrm{\alpha }}_{\mathrm{r}}$ nos resultados. O uso de conexões viga-coluna perfeitamente engastadas pode levar a um comportamento global não realístico de estruturas de concreto armado.

Palavras-chave:
estruturas de concreto; vigas; pilares; ligações

1. INTRODUCTION

The real behavior of the frames must be reflected in the numerical model used for structural analysis in the design of reinforced concrete (RC) high-rise buildings. The stiffness of the connection between beams and columns influences how efforts are distributed in the structural model. According to Alva and El Debs [¹] research, the stiffness of the connection is primarily determined by the relative slip of the flexural (longitudinal) steel reinforcement in the connection area (at the node of the spatial frame) and the slip caused by the cracking of the concrete at the ends of the beam. It is influenced by several factors, including the geometry of the cross-sectional inertia of the beam, the strength of the concrete, the total area of the reinforcement, and its location in the cross-section of the structure. All these data affect the degree of the beam-column connection (nodes) and, consequently, the global results, such as the global displacement, second-order bending moments, and others.

In most RC structural designs, it is usually assumed that the frame nodes are rigid, i.e., that there is no relative rotation between the beam and column connections, and that the transfer of bending moments is complete (i.e., a fixed or perfect connection). However, experimental investigations [²], [³] show the presence of relative rotations in these connections. Alva et al. [⁴] found that in the results proposed by Alva [³], the degree of the beam-column connection was between 76% and 82%, indicating a partial connection between them. Conventional approaches generally assume that the beam-column connections are perfectly pinned or fixed, whereas in practice the real connections are semi-rigid [⁵]. The design of structural frames are governed by the degree of rotational stiffness of the beams and columns connections [⁶], but the semi-rigid approach results in more real solutions [⁷].

Alva and El Debs [¹] have developed a theoretical analysis model to consider the influence of the reinforcement on the sliding induced by the bending moment in these regions (i.e., in the structural nodes). This allows the calculation of the moment-rotation curve of the beam-column connection in RC structures, and then the estimation of the influence on the global behavior of the frames.

The model assumes that the bending deformation results are due to two main mechanisms, referred to as A and B, which generate relative rotations between the beam and the column. Mechanism A is the result of relative rotations caused by the sliding of the tensile reinforcement of the beam adjacent to the column, and mechanism B is the result of rotations caused by flexural concrete cracking formed in the beam close to the column along the length L_p. Combining the rotations of mechanisms A and B, the total rotation for the elastic phase (i.e., M<M_y) is obtained according to Equation 1, where Equation 2 and Equation 3 need to be assumed.

$\mathrm{\theta }={\mathrm{C}}_1\times {\mathrm{M}}^2+{\mathrm{C}}_2\times \left(\frac{1}{\mathrm{r}}\right)$ (1)

${\mathrm{C}}_1=\frac{\mathrm{Ø}}{{\mathrm{E}}_{\mathrm{s}}\times {\mathrm{\tau }}_{\mathrm{b}\mathrm{y}}\times 8\times \left(\mathrm{d}-\mathrm{x}\right)\times {\mathrm{A}}_{\mathrm{s}}^2\times {\mathrm{z}}^2}$ (2)

${\mathrm{C}}_2=\mathrm{0,5}\times \left({\mathrm{L}}_{\mathrm{p}}+{\mathrm{s}}_{\mathrm{R}}\right)$ (3)

According to NBR 9062 [⁸], the stiffness in relation to the bending moment of the connection can be defined from the moment-rotation curve. Regarding the non-linear behavior of the connections, a linear analysis was performed for simplicity, when the secant stiffness (Rsec) is used. According to Barlati [⁹], the semi-rigid behavior of the connection depends on the relationship between the beam stiffness (${EI}_{sec}$) and the secant stiffness to the bending moment ($R_{sec}$). Santos [¹⁰] explains that the rotation restriction factor allows for evaluating how near the connection is to be perfectly fixed (${\mathrm{\alpha }}_{\mathrm{r}}=1.0$) or pinned (${\mathrm{\alpha }}_{\mathrm{r}}=0$). The rotation constraint factor is calculated by dividing the structural element end rotation (${\mathrm{\theta }}_1$) by the combined rotation of the element and the connection (${\mathrm{\theta }}_2$) caused by the moment in the connection according to Equation 4.

${\mathrm{\alpha }}_{\mathrm{r}}=\frac{{\mathrm{\theta }}_1}{{\mathrm{\theta }}_2}$ (4)

The ${\mathrm{\theta }}_1$ is equal to the rotation of the simply supported beam as shown in Equation 5, and ${\mathrm{\theta }}_2$ the total rotation (rotation of the simply supported beam added to the rotation of the connection) as shown in Equation 6, then Equation 4 can be rewritten using Equation 5 and Equation 6 according to Equation 7.

${\mathrm{\theta }}_1=\left(\frac{\mathrm{M}\times \mathrm{L}}{3\times \mathrm{E}\times \mathrm{I}}\right)$ (5)

${\mathrm{\theta }}_2={\mathrm{\theta }}_{\mathrm{l}\mathrm{i}\mathrm{g}}+{\mathrm{\theta }}_1=\frac{\mathrm{M}}{{\mathrm{R}}_{\mathrm{s}\mathrm{e}\mathrm{c}}}+\left(\frac{\mathrm{M}\times \mathrm{L}}{3\times \mathrm{E}\times \mathrm{I}}\right)$ (6)

${\mathrm{\alpha }}_{\mathrm{r}}=\frac{{\mathrm{\theta }}_1}{{\mathrm{\theta }}_2}=\frac{\left(\frac{\mathrm{M}\times \mathrm{L}}{3\times \mathrm{E}\times \mathrm{I}}\right)}{\frac{\mathrm{M}}{{\mathrm{R}}_{\mathrm{s}\mathrm{e}\mathrm{c}}}+\left(\frac{\mathrm{M}\times \mathrm{L}}{3\times \mathrm{E}\times \mathrm{I}}\right)}=\frac{1}{1+\frac{3{\left(\mathrm{E}\mathrm{I}\right)}_{\mathrm{s}\mathrm{e}\mathrm{c}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{e}\mathrm{c}}+{\mathrm{L}}_{\mathrm{e}\mathrm{f}}}}$ (7)

Equation 7 is the same one presented in NBR 9062 [⁸].

Based on this rotation restriction factor, also named as ${\mathrm{\alpha }}_{\mathrm{r}}$ factor, NBR 9062 classifies the connection as pinned (when ${\mathrm{\alpha }}_{\mathrm{r}}<0.15$), semi-rigid (${0.15\le \mathrm{\alpha }}_{\mathrm{r}}<0.85$) or rigid (${\mathrm{\alpha }}_{\mathrm{r}}\ge 0.85$). This classification defines the magnitude of the effort transfer between the beam and the column. Although this is more common in precast structures, it can inspire studies related to cast-in-place concrete structures. The proposed classification has also been adopted by other similar research [¹¹] –^[23].

According to Gernay and Franssen [²⁴], the structural capacity and failure mode of spatial frames depend on the degree of rotational restraint at the beam-column connections. Thai et al. [²⁵] describe that the effect of semi-rigid connections on structural analysis does not only change the moment distribution along the beams and columns, but also increases the frame deformability and stresses due to second-order effects. As suggested in the literature review by Burle [²⁶], all studies agree that the rotational behavior of the beam-column connection should be considered when performing a more realistic structural analysis. However, Nguyen et Kim’s [²⁷] results show that the use of semi-rigid connection models does not significantly affect the behavior of frames. Nonlinear semi-rigid connections reduce deflection due to energy dissipation, and consonance does not occur in semi-rigid frames.

Some researchers have already studied the effects of reducing the stiffness of the connection, such as [²⁸], who reduced ${\mathrm{\alpha }}_{\mathrm{r}}$ by 15%, resulting in a 31% increase in lateral displacements of the structure (global displacements). Saraiva and Boito [²⁹] gradually reduced the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor from 5% to 25%, which led to an increase in ${\mathrm{\gamma }}_{\mathrm{z}}$ results from 1.06 to 1.09. Both studies reduced the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor in all frame nodes, using the same reduction for all nodes. Santos [¹⁰] explains that the ${\mathrm{\alpha }}_{\mathrm{r}}$ for each node of the frame changes depending on the internal forces (i.e., bending moment) and the arrangement of the reinforcement at the beam-column interface. However, using the same ${\mathrm{\alpha }}_{\mathrm{r}}$ (the stiffness reduction in beam-to-column connection) in all frames is therefore an approach to speed up the structural analysis (the time required), but it has shortcomings.

Santos [¹⁰] used the TQS software to carry out simulations in two frames (5 and 19 floors), assuming a total fixed connection as the initial condition, and defining the ${\mathrm{\alpha }}_{\mathrm{r}}$ using Alva and El Debs [¹] analytical procedure. As a result, an increase in lateral displacements of the order of 27% was found for the 19-floor model and 23% for the 5-floor model. However, Santos [¹⁰] applied the same ${\mathrm{\alpha }}_{\mathrm{r}}$ obtained for one floor to all the model. This is a rational simplification to reduce the time required, but we know that the values change on each floor, which has not yet been evaluated.

The problem often arises in the analysis of steel structures. Studies on steel and composite structures serve as inspiration, as there are few studies on reinforced concrete frames. Yang [³⁰] investigated two types of prefabricated steel structures with a new type of semi-rigid connection. A semi-rigid connection test was conducted to evaluate the rotational properties, and the initial cross-sections of the structural components were optimized using optimization software. The structures were significantly affected by the bending properties of the semi-rigid connections, and the stiffness of the steel frame was found to be much more significant than that of the modular structures. Movaghati and Abdelnaby [³¹] investigated the effects of bearing-type deformations on the nonlinear behavior of bolted beam-column connections, showing that the slip between the beams and columns governs the deformations and the ultimate connection rotation increases by up to 19%.

Rigi et al. [¹⁷] investigated the behavior of rigid and semi-rigid steel frames. The results show that semi-rigid frames have lower shear stresses and higher structural deformations, but a higher energy absorption capacity. Considering the stiffness in the beam-column composite can affect the performance of steel moment frames, which should not be ignored when designing these structures. Mu et al. [¹¹], on the other hand, presented a novel proposal for an assembled bolted connection between a precast reinforced concrete column and a composite reinforced concrete beam.

In the last decade, a larger number of studies have been carried out to evaluate the effects of the connection between beams and columns on the results of global frames, mainly focused on steel structures [¹¹], [¹³], [³²] –^[54]. It should be noted that there are few similar studies on reinforced concrete structures, which underlines the uniqueness of the proposed research.

In the case of global analysis, the TQS software incorporates a field of forces that takes into account all floors in the structural design. In other words, regardless of the floor, the entire structure is designed with the load combination that causes the critical case of the frames. For the design of the beams of the frame, Santos [¹⁰] used data generated from the force field of all floors (i.e., of the entire structure). The ${\mathrm{\alpha }}_{\mathrm{r}}$ factors used in your frame were generated based on the results of this beam design (i.e., assuming the field of forces through the frame). This raises an optimization problem because, despite its flexibility, the measure does not accurately reflect reality (the actual value of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor that exists in each node).

The TQS software allows the user to specify a partial connection factor between the beam and column interface. This factor reflects the rotational constraint between the beam and the column. These are the same criteria listed in the NBR 9062 [⁸]. To approximate the effects of these connections on the spatial frame (global analysis), TQS reduces the values of the beam-column connection at the ends of the beam bars (according to the factor assumed by the user), resulting in a redistribution of forces. The rotational constraint factor between the beam-column connection is set to 1 (i.e., completely fixed) by default in the TQS software. However, it is known that the beam-column connection is not perfectly fixed and a critical analysis must be performed in this sense.

To fill the gap, this investigation examined the influence of semi-rigid connections between beam-column interface on the global results (i.e., ${\mathrm{\gamma }}_{\mathrm{z}}$ values, 2^nd order bending moments and lateral displacement) of RC frames designed according to NBR 6118 [⁵⁵] procedures. For this proposal, the arrangement and cross-sectional area of reinforcement of the beam, their stiffness (modulus of elasticity and inertia), and the concrete strength (fck) were taken into account. This was done using the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor according to Alva and El Debs [¹]. The research evaluates how the total height of the building and also the stiffness of the beams and columns affect the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor and then the results of the global structural analysis. Based on this study, an attempt is made to understand the influence on the global results when more realistic criteria for the connection of beams and columns are adopted. The study was justified by the lack of research that sought to optimally evaluate the influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor on the design of RC structures.

2. METHODS

2.1 Frame characteristics

Frames with 7, 14, and 21 floors were assumed, named, respectively, F7, F14, and F21 (see Table 1). The total height of these structures was 21.0, 42.0, and 63.0 m, respectively. These structures were modeled with a single typical floor (see Figure 1), varying only the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor at each beam-column interface (frame node). The ${\mathrm{\alpha }}_{\mathrm{r}}$ was calculated for each floor and repeated for the two adjacent floors to maximize the time required in the numerical resolution. For example, the factor for the first floor was calculated and extrapolated to the second and third floors. A new factor was calculated for the fourth floor and extrapolated to the fifth and sixth floors. And so on for the other floors. This allowed the range of forces used in the design of the beams to be limited to three floors. The ${\mathrm{\alpha }}_{\mathrm{r}}$ factor was then calculated for each force field (i.e., all three floors), which brings the models studied closer to reality. Each group of three-floor plans was named GF (group floor, see also Figure 2).

Table 1
– Nomenclature and characteristics of the frames

Frame number	Number of floors	Total height (m)
F7	7	21
F14	14	42
F21	21	63

Figure 1
– Floor plan assumed to the structural frames analysis.

Figure 2
– Structural frames assumed in the research.

Each node of the structure (i.e., frame) would have a restriction factor every three floors that was calculated using the maximal field of forces for these three floors.

Frames F7, F14, and F21 each consist of a total of 16 columns (C), 8 beams (B), and 9 slabs (S), as shown in Figure 1. For all frames, 12 cm thick slabs were used, meeting the minimum thickness criteria according to NBR 6118 [⁵⁵]. The beams of the F7, F14 and F21 models have a typical cross-section of 19x60 cm. The dimensions of the columns (C) for each frame F can be found in Table 2.

Table 2
– Number of columns and their cross-section characteristics

Column number	Structural frame
	F7	F14	F21
C1	19x49	30x70	45x90
C2	49x19	70x30	90x45
C3	19x49	30x70	45x90
C4	49x19	70x30	90x45
C5	49x19	70x30	90x45
C6	49x19	70x30	90x45
C7	19x49	30x70	45x90
C8	19x49	30x70	45x90
C9	19x49	30x70	45x90
C10	19x49	30x70	45x90
C11	49x19	70x30	90x45
C12	49x19	70x30	90x45
C13	49x19	70x30	90x45
C14	19x49	30x70	45x90
C15	49x19	70x30	90x45
C16	19x49	30x70	45x90

Table 2 shows that the cross-sectional geometry of the columns in each numerical model varied according to the total number of floors in the proposed spatial frame (the frames F7 to F21 given in Table 1). For example, the cross-sectional dimension of C3 was different in each spatial frame (19x49 in F7, 30x70 in F14 and 45x90 in F21, see Table 2). The cross-sectional dimensions of these columns were kept constant over the entire height of the spatial frame. The proposed criterion was necessary to ensure that the analyses of the assumed spatial frames have columns that are proportional to the height of the building and thus have the same degree of mechanical stress. Using columns with the same cross-sectional geometries in a building with 7 and 21 floors, for example, would be irrational and the models would not be comparable. This criterion was chosen so that the three models have approximately the same relative lateral displacement (i.e., h/x), with model F7 having h/2314, model F14 having h/2330, and model F21 having h/2319 (h is the total height of the frame).

The GF-I was the group floor, which consisted of the first, second and third floors. The GF-II was a group floor composed of the fourth, fifth and sixtieth floors. GF-III comprises the seventh, eighth and ninth floors, when applicable. The GF-IV represents the tenth, eleventh and twelfth floors. If applicable, the GF-V with the thirteenth, fourteenth and fifteenth floors. The GF-V is on the sixteenth, seventeenth and eighteenth floors. And the new tenth, twentieth and twenty-first floors for the GF-VII.

2.2 Frame submodels characteristics

Two subcases have been developed from the F14 to test the influence of the stiffness of the elements (beams and columns) in semi-rigid connections. The first reduces the height of the beams by 10% and 20%, resulting in submodels F14-B10 and F14-B20. In the second method, the largest side of the columns is reduced by 10% and 20%, respectively, resulting in F14-C10 and F14-C20 models. In this sense, as shown in Table 3, four submodels were generated, which were derived from the F14.

Table 3
– Number of columns and their cross-section characteristics

Frame number	Number of floors	Total height (m)	Beams dimensions	Columns dimensions
F14	14	42	19x60 cm	30x70 cm
F14-B10	14	42	19x54 cm	30x70 cm
F14-B20	14	42	19x48 cm	30x70 cm
F14-C10	14	42	19x60 cm	30x63 cm
F14-C20	14	42	19x60 cm	30x56 cm

2.3 Iterative analysis process

The beam-column connections that make up all frames (see Table 1 and also Table 3) were evaluated according to two cases, designated as rigid (R) and semi-rigid (SR). In the R case, fixed connections were used, whereas in the SR, the relative rotation of each beam-column was calculated using the analytical model developed by Alva and El Debs [¹] and the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor for each connection was solved using Equation 7.

Using TQS software, the ${\mathrm{\alpha }}_{\mathrm{r}}$ was added to the numerical model. Since the analytical procedure of Alva and El Debs [¹] depends on the cross-sectional area of the reinforcements, their location in the cross-section of the beam, and also on the forces acting on the beam-column connection, the addition of ${\mathrm{\alpha }}_{\mathrm{r}}$ to the model resulted in a redistribution of effort and then a new structural design (and then reinforcements) was required, making the process iterative. The interactive process ceases when all connections show a maximum change of 5% in the value of the ${\mathrm{\alpha }}_{\mathrm{r}}$ about to the previous process. This was the criterion defined by the authors.

Some thoughts on the studies by Alva and El Debs should be clarified. The analytical model proposed by these authors is based on the assumption that the columns are continuous and the beams are discontinuous and the slabs are not coupled to the beams. This criterion was then included in the proposed study.

2.4 Materials

The concrete compressive strength assumed was fc=30 MPa to meet the durability criteria of NBR 6118 [⁵⁵] for CAA II (ambient aggression class #2, as nomenclature in Portuguese), which is typical of urban situations. CA-50 and CA-60 steel (according to NBR 7480 [⁵⁶]) with ${\mathrm{f}}_{\mathrm{y}}$=500 MPa and ${\mathrm{f}}_{\mathrm{y}}$=600 MPa, respectively, were assumed for longitudinal and transversal (stirrups) reinforcement. According to NBR 7480 [⁵⁶], both have a longitudinal modulus of elasticity of 210 GPa.

To reduce the number of variables in the numerical models (as can be seen in the [⁵⁷]–^[59]), changes in concrete cover or structural cross-section elements due to fire requirements (according to the NBR 15200 [⁶⁰]) were not taken into account. The concrete cover was assumed the same for all frames shown in Tables 1 and 3.

2.5 Applied load

Vertical loads were chosen by NBR 6123 [⁶¹]. For the definition of the dead load of the structure, 25 kN/m³ was assumed. A load of 3 kN/m² was applied to the slabs. For the roofing floor, it was 1 kN/m². Wind efforts created the horizontal loads. The wind characteristics were calculated using NBR 6123 [⁶²] and estimated to be 45 m/s in all frames (Table 1 and 3). No other loads were used in the numerical model (such as wall loads and mortar coating on the floor).

2.6 Structural analysis and design procedures

The proposed structure is built in RC. Slabs, beams and columns were calculated according to ABNT NBR 6118 [⁵⁵] and solved using the TQS software. The global deformation analysis and also the ${\gamma }_Z$ coefficient were determined using the NBR 6118 [⁵⁵] calculation criteria. The computational frames assumed in this research are shown in Figure 3.

Figure 3
– Structural frames assumed in the research.

3. RESULTS

The results of the research are shown below.

3.1 General comments

Except for the F7, which required three connections, all frames needed just two interactions to converge (see Section 2.3). The nodes of the F7 frame utilized a total of 576 ${\mathrm{\alpha }}_{\mathrm{r}}$ factors, the F14, F14-B10, F14-B20, F14-C10, and F14-C20 models used 672 ${\mathrm{\alpha }}_{\mathrm{r}}$ factors, and the F21 frame used 864 ${\mathrm{\alpha }}_{\mathrm{r}}$ factors.

It was found that the inner beam-column connections of the structures, especially the nodes of columns C6, C7, C10, and C11, had the lowest values for the estimated rotation limiting factors. That is, in the connections where the beams were continuous and had more negative bending moments, lower values of ${\mathrm{\alpha }}_{\mathrm{r}}$ were obtained, indicating that these nodes had a lower proportion of the total pinned connections than the nodes at the end of the structure (i.e., external columns).

The results show that the calculated rotation limiting factors have their lowest values on floors that are between 40% and 80% of the total height of the structure. As a result, the connections on the first and last floors have a higher ${\mathrm{\alpha }}_{\mathrm{r}}$ (tendency of fixed) than the connections on the intermediate floors. It occurs because the intermediate floors, especially those located between 1/3 and 2/3 of the total height (h) of the frame, have larger negative moments in the connections due to a frame effect caused by lateral loading (wind load). It happens because the deformation in the spans of the columns in this region is greater in relation to the other floors, which increases the moments.

The average values of the ${\mathrm{\alpha }}_{\mathrm{r}}$ for each column are shown in Tables 4, 5, and 6 for F7, F14, and F21, respectively. These results represent the average of all values obtained at the beam-column interface of each column. The values did not show fluctuations greater than 12.1% (up or down)

Table 4
–${\alpha }_r$ factors of F7 frames

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors (variability in %)
	F7
	GF-I	GF-II	GF-III	Average GF-I to III
C1	0.84	0.87	0.91	0.87
C2	0.83	0.84	0.92	0.86
C3	0.86	0.86	0.95	0.89
C4	0.84	0.84	0.92	0.87
C5	0.86	0.86	0.95	0.89
C6	0.79	0.80	0.92	0.83
C7	0.79	0.80	0.92	0.84
C8	0.83	0.84	0.92	0.86
C9	0.83	0.84	0.92	0.86
C10	0.79	0.80	0.92	0.83
C11	0.79	0.80	0.92	0.84
C12	0.86	0.86	0.86	0.89
C13	0.84	0.84	0.92	0.87
C14	0.86	0.86	0.95	0.89
C15	0.83	0.86	0.92	0.87
C16	0.84	0.86	0.91	0.87
Average C1-16	0.83	0.84	0.93

Table 5
–${\alpha }_r$ factors of F14 frames

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors
	F14
	GF-I	GF-II	GF-III	GF-IV	GF-V	Average GF-I to V
C1	0.80	0.80	0.77	0.80	0.90	0.81
C2	0.81	0.78	0.78	0.78	0.88	0.80
C3	0.80	0.80	0.79	0.82	0.87	0.82
C4	0.80	0.80	0.77	0.79	0.90	0.81
C5	0.80	0.80	0.79	0.82	0.87	0.82
C6	0.80	0.80	0.78	0.79	0.82	0.80
C7	0.80	0.80	0.79	0.78	0.82	0.80
C8	0.81	0.78	0.78	0.78	0.88	0.80
C9	0.81	0.78	0.78	0.78	0.88	0.80
C10	0.80	0.80	0.79	0.78	0.82	0.80
C11	0.80	0.80	0.79	0.78	0.83	0.80
C12	0.80	0.80	0.79	0.82	0.87	0.82
C13	0.80	0.80	0.77	0.79	0.90	0.81
C14	0.80	0.80	0.79	0.82	0.87	0.82
C15	0.81	0.78	0.78	0.78	0.88	0.80
C16	0.80	0.80	0.77	0.81	0.90	0.81
Average C1-16	0.80	0.80	0.78	0.80	0.87

Table 6
–${\alpha }_r$ factors of F21 frames

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors
	F21
	GF-I	GF-II	GF-III	GF-IV	GF-V	GF-VI	GF-VII	Average GF-I to VII
C1	0.83	0.83	0.82	0.78	0.78	0.77	0.82	0.80
C2	0.84	0.84	0.84	0.81	0.77	0.76	0.82	0.81
C3	0.83	0.83	0.83	0.80	0.77	0.78	0.84	0.81
C4	0.83	0.83	0.82	0.78	0.77	0.76	0.83	0.80
C5	0.83	0.83	0.83	0.80	0.7	0.76	0.84	0.81
C6	0.84	0.84	0.84	0.81	0.79	0.76	0.80	0.81
C7	0.84	0.84	0.84	0.81	0.79	0.76	0.80	0.81
C8	0.84	0.84	0.83	0.81	0.77	0.76	0.83	0.81
C9	0.84	0.84	0.84	0.81	0.77	0.77	0.83	0.81
C10	0.84	0.84	0.84	0.81	0.79	0.76	0.80	0.81
C11	0.84	0.84	0.84	0.81	0.79	0.77	0.80	0.81
C12	0.83	0.83	0.83	0.80	0.77	0.76	0.84	0.81
C13	0.83	0.83	0.82	0.78	0.77	0.76	0.83	0.80
C14	0.83	0.83	0.83	0.80	0.79	0.78	0.84	0.81
C15	0.84	0.84	0.84	0.81	0.77	0.77	0.83	0.81
C16	0.83	0.83	0.82	0.78	0.79	0.77	0.83	0.81
Average C1-16	0.83	0.83	0.83	0.80	0.78	0.77	0.82

3.2 Discussions of F7, F14 and F21 frames

Due to a reduction in the stiffness of the connections (and thus an increase in flexibility) of the frame, the consideration of semi-rigid connections led to an increase in the maximum lateral displacement in all models.

When compared to Rigid (i.e., perfectly fixed connections) to Semi-rigid (i.e., with ${\mathrm{\alpha }}_{\mathrm{r}}$ factor applied in the nodes), the lateral displacement of the F7, F14 and F21 increased by 8.8%, 14.4% and 16.5%, respectively. The F7 had the smallest increase in total lateral displacement. Semi-rigid connections influenced the total stiffness of the nodes of the frame, resulting in flexibility in the beam-to-column connection. This flexibility affects the mechanical interaction between the bars (beam and the columns), reducing the beam's role in global stability.

Table 7 shows the results of all fixed beam-column connections (Rigid) and with the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor applied (Semi-rigid). The increasing displacement values between both cases are also shown.

Table 7
– Maximum global lateral displacement of the frames

Frame number		Maximum lateral total (global) displacement
		Rigid (R) (cm)	Semi-rigid (SR) (cm)	Increase (SR in relation to R) (%)
F7		0.91	0.99		8.1
F14		1.81	2.07		14.4
Subcases	F14-B10	2.03	2.31		13.8
	F14-B20	2.34	2.65		13.3
	F14-C10	2.07	2.31		11.6
	F14-C20	2.42	2.68		10.7
F21		2.72	3.17		16.5

Despite the increase in the value of the total lateral displacement, Table 7 shows a decrease in the percentage of the total lateral displacement when the height of the beams increases. As Barlati [⁹] justified, the semi-rigid behavior of the connection depends on the relationship between the beam stiffness (${EI}_{sec}$) and the secant stiffness to the bending moment of the connection ($R_{sec}$). In this sense, the reduction of the stiffness of the beams led to an increase in the calculated ${\alpha }_r$. Therefore, the relative rotations between the beams and the columns were closer to a pinned situation in the case of the beams with lower stiffness than in the models with the more rigid beams.

The reason for the increase in ${\alpha }_r$ corresponding to the decrease in beam stiffness is related to the results of $R_{sec}$ and ${EI}_{sec}$, as proposed in Equation 7. The $R_{sec}$ values depend on the bending moment to which the connection is subjected and also the total rotation of the connection. The reduction of the beam height is compensated by an increase in the bending reinforcement required for the beam. In this sense, the rotation of the beam increases significantly despite the reduction in its height, causing the $R_{sec}$ values to decrease. However, this reduction is insignificant compared to the ${EI}_{sec}$ results. In this case, the cross-sectional inertia depends on the height of the beam, which is increased by the exponent three. Therefore, it can be seen that the influence of ${EI}_{sec}$ is more pronounced in relation to the $R_{sec}$ values. As the ${EI}_{sec}$ results are inversely proportional to the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor, a reduction in the beam height leads to an increase in the ${\mathrm{\alpha }}_{\mathrm{r}}$ results.

Figure 4 shows the displacements of the three frames used (SR case).

Figure 4
– Total displacement in the spatial frames

The increase in the total height of the building is related to the increase in displacements. As the height of the building increases, the degree of beam-column connection increases the total global displacement. Figure 5 shows the relative lateral displacement h/x, which correlates the structure's height with the lateral total displacement, where “x” is the relationship between the total height “h” and the maximum lateral displacement “$\mathrm{\delta }$ “ of the structure (see also Table 7). This research was done since NBR 6118 [⁵⁵] specifies the maximum displacement “$\mathrm{\delta }$” of a structure as $\mathrm{\delta }$=h/1700, and then x=1700. The structure is less flexible with an increase in the value of “x”. For the structures assumed in this research, the values of “x” are shown in Equation 8 to 13 and also in Figure 5.

Figure 5
– Relative lateral displacement (x values)

${\mathrm{x}}_{\mathrm{F}7-\mathrm{R}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}=\frac{\mathrm{h}}{\mathrm{\delta }}=\frac{2110\mathrm{}\mathrm{c}\mathrm{m}}{0.91\mathrm{}\mathrm{c}\mathrm{m}}=2314.4$ (8)

${\mathrm{x}}_{\mathrm{F}14-\mathrm{R}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}=\frac{\mathrm{h}}{\mathrm{\delta }}=\frac{4210\mathrm{}\mathrm{c}\mathrm{m}}{1.81\mathrm{}\mathrm{c}\mathrm{m}}=2330.1$ (9)

${\mathrm{x}}_{\mathrm{F}21-\mathrm{R}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}=\frac{\mathrm{h}}{\mathrm{\delta }}=\frac{6310\mathrm{}\mathrm{c}\mathrm{m}}{2.72\mathrm{}\mathrm{c}\mathrm{m}}=2319.1$ (10)

${\mathrm{x}}_{\mathrm{F}7-\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}=\frac{\mathrm{h}}{\mathrm{\delta }}=\frac{2110\mathrm{}\mathrm{c}\mathrm{m}}{0.99\mathrm{}\mathrm{c}\mathrm{m}}=2128.5$ (11)

${\mathrm{x}}_{\mathrm{S}14-\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}=\frac{\mathrm{h}}{\mathrm{\delta }}=\frac{4210\mathrm{}\mathrm{c}\mathrm{m}}{2.07\mathrm{}\mathrm{c}\mathrm{m}}=2038.2$ (12)

${\mathrm{x}}_{\mathrm{F}21-\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}=\frac{\mathrm{h}}{\mathrm{\delta }}=\frac{6310\mathrm{}\mathrm{c}\mathrm{m}}{3.17\mathrm{}\mathrm{c}\mathrm{m}}=1987.9$ (13)

In the R case, all frames presented a very similar relative lateral displacement in the order of h/2300. However, in the SR case, a difference was observed between them. As the structure height increases, the variation between case R and SR case also increases. This illustrates that decreasing the stiffness of connections has a greater impact on higher constructions, and also on the lateral displacement. As expected, the flexibility of the structure increases with its height. The influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ increases with the increment in the total height of the building, which is not observed in the case of all fixed beam-column connections (“Rigid” case).

The chosen frames have a linear rise in the number of floors (i.e., F7, F14, and F21 with total heights of 21.0, 42.0, and 63.0 m, respectively), however, the variation of lateral displacement is not linear with respect to the height increase in the “Semi-rigid” frame case. The TQS software confirmed an increase in 2^nd order efforts due to the increase in lateral displacements (frames with ${\mathrm{\alpha }}_{\mathrm{r}}$ factor). The 2^nd order analysis was performed using the coefficient ${\mathrm{\gamma }}_{\mathrm{Z}}$ proposed in NBR 6118 [⁵⁵].

Comparing the respective R and SR cases, model F7 showed a 13.2% rise in 2^nd-order bending moments, model F14 showed a 19.2% increase, and model F21 showed a 22.4% increase. In this sense, similar to the lateral displacements shown above, the 2^nd order bending moments presented non-linear results with the increase of the ${\mathrm{\alpha }}_{\mathrm{r}}$ in the beam-column connections. Even if the columns' dimensions are changed to maintain the same relative lateral displacement between F7, 14 and 21 in the R case, the insertion of the ${\mathrm{\alpha }}_{\mathrm{r}}$ increases the lateral displacements and the 2^nd order effects as the total height of the frame increases. In this sense, the increase in the 2^nd order bending moments caused an increase in the coefficient ${\mathrm{\gamma }}_{\mathrm{Z}}$ of the models, as shown in Table 8.

Table 8
– coefficient ${\gamma }_Z$ to each frame

Frame number		coefficient ${\mathrm{\gamma }}_{\mathrm{Z}}$
		Rigid (R)	Semi-rigid (SR)	Increase (SR in relation to R) (%)
F7		1.075	1.085		0.9
F14		1.073	1.088		1.4
Subcases	F14-B10	1.084	1.100		1.5
	F14-B20	1.099	1.117		1.6
	F14-C10	1.081	1.095		1.3
	F14-C20	1.092	1.108		1.5
F21		1.076	1.094		1.7

3.3 Subcases of S14 frames (decrease in beam stiffness)

3.3.1 Decrease in beam stiffness

In addition to Table 5, Table 9 and Table 10 show the ${\mathrm{\alpha }}_{\mathrm{r}}$ for this case.

Table 9
–${\alpha }_r$ factors of F14-B10 frame

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors
	F14-B10
	GF-I	GF-II	GF-III	GF-IV	GF-V	Average GF-I to V
C1	0.81	0.83	0.78	0.77	0.87	0.81
C2	0.83	0.82	0.78	0.78	0.85	0.81
C3	0.81	0.82	0.80	0.80	0.84	0.81
C4	0.81	0.81	0.78	0.77	0.86	0.81
C5	0.81	0.82	0.80	0.80	0.85	0.82
C6	0.84	0.84	0.81	0.79	0.80	0.81
C7	0.84	0.84	0.81	0.79	0.80	0.81
C8	0.83	0.82	0.80	0.78	0.85	0.82
C9	0.83	0.82	0.80	0.78	0.85	0.82
C10	0.84	0.84	0.81	0.79	0.80	0.81
C11	0.84	0.84	0.81	0.79	0.80	0.81
C12	0.81	0.82	0.80	0.80	0.85	0.82
C13	0.81	0.81	0.78	0.77	0.86	0.81
C14	0.81	0.82	0.80	0.80	0.84	0.81
C15	0.83	0.82	0.80	0.78	0.85	0.82
C16	0.81	0.83	0.78	0.77	0.86	0.81
Average C1-16	0.82	0.82	0.80	0.79	0.84

Table 10
– ${\alpha }_r$ factors of F14-B20 frame

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors
	F14-B20
	GF-I	GF-II	GF-III	GF-IV	GF-V	Average GF-I to V
C1	0.86	0.85	0.83	0.79	0.84	0.83
C2	0.86	0.85	0.83	0.79	0.82	0.83
C3	0.85	0.85	0.81	0.81	0.85	0.83
C4	0.86	0.85	0.83	0.81	0.83	0.84
C5	0.85	0.85	0.81	0.79	0.85	0.83
C6	0.86	0.86	0.84	0.81	0.79	0.83
C7	0.86	0.86	0.84	0.81	0.77	0.83
C8	0.86	0.85	0.83	0.79	0.82	0.83
C9	0.86	0.85	0.83	0.79	0.83	0.83
C10	0.86	0.86	0.84	0.81	0.79	0.83
C11	0.86	0.86	0.84	0.81	0.79	0.83
C12	0.85	0.85	0.81	0.82	0.84	0.83
C13	0.86	0.84	0.83	0.79	0.84	0.83
C14	0.85	0.84	0.81	0.80	0.85	0.83
C15	0.86	0.85	0.83	0.79	0.82	0.83
C16	0.86	0.85	0.82	0.78	0.87	0.84
Average C1-16	0.86	0.85	0.83	0.80	0.83

This section discusses the variants of the F14 frames, i.e. the case of F14-B10 and F14-B20. As shown in F14, frames F14-B10 and F14-B20 increased their lateral displacement when the SR was assumed. Frame F14 showed a 14.4% increase in displacement between the SR and R cases. For F14-B10 and F14-B20, SR showed an increase in lateral displacement of the order of 13.8% and 13.2%, respectively, compared to the R case. As already shown in Table 7, the percentual of increase in lateral displacement decrease in relation to the reduction of the beam stiffness. The smaller the beam’s height (i.e., their cross-sectional stiffness), the less influence the ${\mathrm{\alpha }}_{\mathrm{r}}$ has on the global displacement results.

According to Barlati [⁹], this is because the SR beam-column connections depend proportionally on the stiffness of the beam cross-section ($\mathrm{E}{\mathrm{I}}_{\mathrm{s}\mathrm{e}\mathrm{c}}$) and also on the stiffness of the secant bending moment of the beam-column connection (${\mathrm{R}}_{\mathrm{s}\mathrm{e}\mathrm{c}}$) as already shown in Equation 7. Table 4 to Table 6 confirm this hypothesis: the lower the stiffness of a beam cross-section, the higher the value of the ${\mathrm{\alpha }}_{\mathrm{r}}$-and then the closer it comes to the values of the fully rigid state (i.e., R case). This demonstrates that the relative rotations between the beam-columns were closer to an R case (fully fixed) in the models with lower stiffness of the beam cross-section than in the models with higher stiffness.

The decrease in the stiffness of the cross-section of the beams led to a decrease in the stiffness of the spatial frame. In this sense, the frames became more displaceable and then the 2^nd order moments increased. However, the lower the stiffness of the beams, the lower the influence of the beam-column connection on the results of the lateral displacement of the global frame. Compared to the R case, the frames F14, F14-B10, and F14-B20 with the SR criterion show an increase in 2^nd-order moments of 19.2%, 18.1%, and 16.%, respectively.

The ${\mathrm{\gamma }}_{\mathrm{Z}}$ coefficient, on the other hand, did not follow the same logic. For spatial frames F14, F14-B10, and F14-B20, the case of R and SR showed an increase in ${\mathrm{\gamma }}_{\mathrm{Z}}$ from 1.073 to 1.088; 1.084 to 1.10; and 1.099 to 1.117, respectively. In this sense, insert the ${\mathrm{\alpha }}_{\mathrm{r}}$ in the frames F14, F14-B10, and F14-B20 shown and increase the ${\mathrm{\gamma }}_{\mathrm{Z}}$ by 1.4%, 1.5%, and 1.6%, respectively. Therefore, to the same frame, the lower the stiffness of the beam cross-section, the higher the value of the ${\mathrm{\gamma }}_{\mathrm{Z}}$. In the R case, the reduction of 10 and 20% of the beam stiffness produced an increase in ${\mathrm{\gamma }}_{\mathrm{Z}}$ coefficient of 1.01% and 2.42%, respectively. In the SR case, 1.10 and 2.66%, respectively. There is an inverse relationship: the lower the stiffness of the beam's cross-section, the smaller the influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor on the ${\mathrm{\gamma }}_{\mathrm{Z}}$ results.

Decreasing the cross-section stiffness of the beams, the 2^nd order effects increase. However, the importance and influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor on these results decreases with the decrease in the beam stiffness. That is, the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor tends to interfere less with 2^nd order moments with low-height beams.

According to Carvalho and Pinheiro [⁶³], the ${\mathrm{\gamma }}_{\mathrm{Z}}$ correlates the 2^nd order of vertical bending moments (i.e., generated by the displacement of vertical loads on each floor) with the lateral bending moment (by horizontal loads). Since the lateral bending moment remained the same as the three models (F14, F14-B10, and F14-B20) have the same total height, floor plan geometry, and horizontal loads, and since the variation of the 2^nd order vertical bending moment increased as the stiffness of the beams decreased, the coefficient ${\mathrm{\gamma }}_{\mathrm{Z}}$ then increased.

3.3.2 Decrease in column stiffness

In addition to Table 5, Table 11 and Table 12 show the ${\mathrm{\alpha }}_{\mathrm{r}}$ for this case.

Table 11
– ${\alpha }_r$ factors of F14-C10 frame

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors
	F14-C10
	GF-I	GF-II	GF-III	GF-IV	GF-V	Average GF-I to V
C1	0.81	0.80	0.78	0.79	0.90	0.82
C2	0.81	0.81	0.77	0.78	0.88	0.81
C3	0.81	0.80	0.82	0.82	0.88	0.83
C4	0.81	0.80	0.78	0.81	0.90	0.82
C5	0.79	0.80	0.82	0.82	0.88	0.82
C6	0.82	0.80	0.80	0.78	0.83	0.81
C7	0.83	0.80	0.80	0.79	0.83	0.81
C8	0.81	0.81	0.77	0.78	0.87	0.81
C9	0.81	0.81	0.77	0.78	0.88	0.81
C10	0.82	0.80	0.80	0.79	0.83	0.81
C11	0.82	0.80	0.80	0.79	0.81	0.80
C12	0.79	0.80	0.82	0.82	0.88	0.82
C13	0.81	0.80	0.78	0.81	0.90	0.82
C14	0.81	0.80	0.82	0.82	0.88	083
C15	0.81	0.81	0.77	0.78	0.88	0.81
C16	0.81	0.80	0.78	0.79	0.90	0.81
Average C1-16	0.81	0.80	0.79	0.80	0.87

Table 12
– ${\alpha }_r$ factors of F14-C20 frame

Column number (node)	${\mathrm{\alpha }}_{\mathrm{r}}$ factors
	S14-C20
	GF-I	GF-II	GF-III	GF-IV	GF-V	Average GF-I to V
C1	0.81	0.80	0.77	0.79	0.90	0.81
C2	0.81	0.81	0.77	0.79	0.88	0.81
C3	0.81	0.80	0.81	0.82	0.88	0.82
C4	0.81	0.80	0.77	0.79	0.91	0.82
C5	0.81	0.79	0.81	0.82	0.88	0.82
C6	0.82	0.81	0.80	0.78	0.83	0.81
C7	0.82	0.81	0.80	0.78	0.83	0.81
C8	0.81	0.81	0.77	0.77	0.87	0.81
C9	0.81	0.81	0.77	0.79	0.88	0.81
C10	0.82	0.81	0.80	0.78	0.83	0.81
C11	0.82	0.81	0.80	0.79	0.83	0.81
C12	0.81	0.79	0.81	0.82	0.88	0.82
C13	0.81	0.80	0.77	0.79	0.91	0.82
C14	0.81	0.80	0.81	0.83	0.88	0.82
C15	0.81	0.81	0.77	0.78	0.88	0.81
C16	0.81	0.80	0.77	0.79	0.90	0.81
Average C1-16	0.81	0.80	0.79	0.79	0.87

Table 5, Table 11 and Table 12 show the ${\mathrm{\alpha }}_{\mathrm{r}}$ defined for the connections of each frame. Note that the calculated values are similar for the three cases (F14, F14-C10, and F14-C20), as the method of Alva and El Debs [¹] is hardly affected by the change of the column cross-section. Only a few variables change, such as the negative bending moments and the corresponding negative reinforcement, as the lower stiffness of the columns affects the negative moment of the beams and also their effective span.

As shown in Table 7, the F14-B10, F14-B20, F14-C10, and F14-C20 show similar results when the total lateral displacement is described. As noted in the previous analysis (see Section 3.3.1 – Decrease in Beam Stiffness), the lateral displacement increases as the column cross-section decreases, but the influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ on the results decreases.

In comparison to the case R, the SR to the beam-column connection exhibits a 14.4% increase in total lateral displacement in frame F14. In the F14-C10 and F14-C20 frames, the increase in lateral displacement between R and SR is 11.6 and 10.7%, respectively. This shows that structural spatial models with a lower stiffness of the column cross-section have more displaceable frames. However, the lower the stiffness of the column cross-section, the lower the influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ on the results.

When the stiffness of the cross-section of the columns decreases, the lateral displacement of the frame increases. However, comparing case R with case SR, the lower the stiffness of the column, the smaller the percentage increase of the 2^nd order effects in case SR. Compared to case R, the 2^nd order bending moments in case SR were lower by 19.2%, 16.4% and 15.5% for frames F14, F14-C10 and F14-C20, respectively. In this sense, the more the stiffness is reduced, the smaller the increase of the 2^nd order effects between case R and SR. By reducing the stiffness of the columns, the lateral deformation of the frame increases, but the magnitude of the 2^nd order effects decreases. The lower the stiffness of the columns, the less the increase of the 2^nd order effects is affected by the factor ${\mathrm{\alpha }}_{\mathrm{r}}$.

F14, F14-C10, and F14-C20 exhibit ${\mathrm{\gamma }}_{\mathrm{Z}}$ coefficients of 1.073, 1.081, and 1.092, respectively, for R connections. Coefficient ${\mathrm{\gamma }}_{\mathrm{Z}}$ in the case of SR was 1.088, 1.095, and 1.108, respectively. In each frame, the difference between SR and R case was 1.4, 1.3, and 1.5%, respectively.

4 CONCLUSIONS

The general conclusions of this paper are:

• The smaller the stiffness of the cross-section of the beams, the smaller the influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor (i.e., degree of connection between beams and columns) on the global displacement results;

• The lower the stiffness of the beam or column cross-section, the larger the structure's lateral displaceability, but the less influence the connection between beam and column (i.e., ${\mathrm{\alpha }}_{\mathrm{r}}$ factor) has on the lateral displacement of the spatial frame.;

• The lower the stiffness of the cross-section of beams and columns, the lower the influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$-factor on the global results.;

• The lower the stiffness of the beam cross-section, the higher the value of the ${\mathrm{\gamma }}_{\mathrm{Z}}$ and the spatial displacements, but the lower the influence of the beam-column connection (${\mathrm{\alpha }}_{\mathrm{r}}$ factor) on these spatial results;

• Decreasing the cross-section stiffness of the beams, the 2^nd-order effects increase. However, the importance and influence of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor on these results decreases;

• The use of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor in the beam-column connection may change the procedure for the design of spatial frames. Some frames that assumed the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor showed non-negligible 2^nd order effects (i.e., ${\mathrm{\gamma }}_{\mathrm{Z}}>1.10$);

• In relation to the columns, the stiffness of the beam cross-section showed the greatest influence on the results of the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor;

• Since this was the average value of the most conservative results obtained throughout this investigation, it is suggested that ${\mathrm{\alpha }}_{\mathrm{r}}$ factor equal to 0.80 be used in the design of RC structures (in the absence of more extensive research). However, this value was obtained only in the cases tested in this study and cannot be generalized.;

• For future research, it is proposed to study the connection between the beam and column in case of fire. As the structure heats and then shows mechanical damage, the beam-column connection becomes more flexible, the ${\mathrm{\alpha }}_{\mathrm{r}}$ factor probably decreases and then the global effects may be greater. These authors (FLB) have already started research in this area.

• Financial support: None.
• Data Availability: The data that support the findings of this study are available from the corresponding author, [L. M.], upon reasonable request.
• How to cite: L. Manoel, E. C. Pachla, and F. L. Bolina, “The influence of the beam-column connection on the global results of multi-storey reinforced concrete structures,” Rev. IBRACON Estrut. Mater., vol. 17, no. 6, e17615, 2024, https://doi.org/10.1590/S1983-41952024000600015

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Edited by

Editors: Leandro Trautwein, Daniel Carlos Taissum Cardoso.

Data availability

Data Availability: The data that support the findings of this study are available from the corresponding author, [L. M.], upon reasonable request.

Publication Dates

• Publication in this collection
  10 May 2024
• Date of issue
  2024

History

• Received
  10 Aug 2023
• Reviewed
  03 Mar 2024
• Accepted
  04 Mar 2024