Nonlinear analysis of monolithic beam-column connections for reinforced concrete frames

abstract: This paper deals with nonlinear analysis of deformability of monolithic beam-column connections for bending moments in framed reinforced concrete structures. Due to the simplicity, the connections deformability is considered by using an analytical model of moment-rotation curve. Material nonlinearity of the structural elements is considered by using the flexural stiffness obtained in moment-curvature relationship of the sections. The formulation of the analytical model to obtain the relative rotations between beam and column and the formulation to construct moment-curvature curves is deduced and presented to allow the computational implementation in structural analysis software. The numerical simulations carried out in this study indicated that even in the case of monolithic connections, taking into account the bending moment deformability of the connections leads to significantly better results than the hypothesis of fully rigid connections.


INTRODUCTION
In reinforced concrete structures, cracking of concrete, plastification of materials and bond-slip behavior between steel and concrete are responsible for the material nonlinear behavior of these structures.
For Ultimate Limit State procedures in the design of frames structures for buildings, the material nonlinearity of structural elements in global analysis can be considered simply by flexural stiffness reductions of these elements for use in linear analysis, as recommended by NBR 6118 [1] and others international codes. Alternatively, in the need for checking the design based on simplified linear analysis, nonlinear analysis can be employed with the use of momentcurvature relationships of the sections. In this case, the framed structure is discretized, and the flexural stiffness of each section is calculated as a function of its moment-curvature relationship.
The nonlinear effects that occur in beam-column connections of monolithic reinforced concrete structures -such as the slippage of flexural reinforcement of the beams in the joint region and the formation of flexural cracks at the beam extremity -induce the generation of relative rotations between the beam and the column. Thus, monolithic connections, strictly speaking, are not perfectly rigid under bending moment. Evidently, there is greater concern with the bending deformability in precast concrete structures. However, the consideration of bending deformability in monolithic connections brings benefits to the structural analysis justified by the greater precision in obtaining stresses and displacements -as shown in this paper.
Due to the simplicity and the good results they can provide, analytical models are the most attractive way in design procedures to consider the effects of the slippage of the beam reinforcement inside the column and the effects induced by the flexural cracks at the beam extremity on the bending deformability. Examples of these models can be found in Paultre et al. [2], Sezen and Moehle [3], Sezen and Setzler [4], Kwak and Kim [5], Ferreira et al. [6], Alva et al. [7] and Alva and El Debs [8], the latter being emphasized in this study.
The Alva and El Debs [8] model was proposed for exterior beam-column connections. Due to the formatting of the formulation, this model can be easily implemented in software programs for nonlinear analysis of frames that use moment-curvature relationships for the consideration of the material nonlinearity of structural elements.
There are three central objectives of this paper, namely: • Complement the investigations on the efficiency of the model proposed by Alva and El Debs [8] to consider the bending deformability caused by the slippage of flexural reinforcement of the beams in the joint region; • Present an analytical formulation to obtain moment-curvature relationships of reinforced concrete rectangular sections, aiming to consider the material nonlinearity of beams and columns and the application of the analytical model proposed by Alva and El Debs [8] in nonlinear analysis of reinforced concrete frames; • Show the efficiency of the constitutive models employed for considering material nonlinearity of the structural elements (by moment-curvature relationships) and bending deformability in nonlinear analysis of framed reinforced concrete structures.

PREVIOUS STUDIES
Although there are numerous researches in the literature (especially international) on the behavior of monolithic beam-column connections, few studies that focus on analytical models for considering the deformability of connections subjected to bending moment are found.
There are analytical models that exclusively consider the portion of rotation resulting from the slippage of the flexural reinforcement in the anchorage region, such as those found in Paultre et al. [2], Sezen and Moehle [3] and Sezen and Setzler [4]. Paultre et al. [2] used a tri-linear moment-rotation curve with points defined by the cracking of the concrete, yielding of the reinforcement and failure of the beam section. For the calculation of the rotations, the authors used a simplified distribution of bond stresses in the elastic and inelastic ranges (after reinforcement yielding). Sezen and Moehle [3] and Sezen and Setzler [4] proposed an analytical model applicable to the case of slippage of longitudinal tension reinforcement of columns (anchored in foundations) or beams (anchored in beam-column joints). As in Paultre et al. [2], Sezen and Moehle [3] and Sezen and Setzler [4] used a simplified distribution of bond stresses but proposed an additional simplification regarding the distribution of the axial strains of the reinforcement in the anchorage regions.
Among the analytical models that consider both portions of relative rotations -those resulting from slippage of the flexural reinforcement inside the joint and those resulting from cracking at the beam extremity -the following models can be cited: Kwak and Kim [5], Ferreira et al. [6], Alva et al. [7] and Alva and El Debs [8]. Kwak and Kim [5] proposed an analytical model which accounts for the effects of relative rotations by reducing the flexural stiffness along the equivalent plastic length of the beams (extremities). The total rotation calculated by the model is associated with the slippage of flexural reinforcement of the beam inside the column added the rotation induced by the crack at the beamcolumn interface. These rotations are obtained by solving the differential equations which represent the bond-slip behavior. The analytical models presented in Ferreira et al. [6], Alva et al. [7] and Alva and El Debs [8] take into account the two rotation portions, but consider that the slippage induced by flexural cracking occurs in a certain length of the beam extremity, associated with its effective depth. Ferreira, El Debs and Elliott [6] model was proposed for connections between precast elements, being later extended to monolithic connections, as presented in Alva et al. [7]. Subsequently, Alva and El Debs [8] presented a specific analytical model for exterior beam-column monolithic connections. This model has the advantage of including in the formulation parameters not considered in Alva et al. [7], such as bond strength in the joint region and the diameter of the beam reinforcement bars, a parameter that influences the flexural crack widths in this member (beam).

MODEL PROPOSED BY ALVA AND EL DEBS [8]
Based on the conceptual model proposed by Ferreira et al. [6], Alva and El Debs [8] proposed a theoretical model which is capable of representing the bond-slip behavior of the reinforcement without the need for parameters from experimental tests. In addition, it can be easily implemented in software programs for structural analysis. In this model, it is assumed that the bending deformability is the result of two mechanisms, which produce relative rotations between the beam and the column ( where A θ is the rotation due to Mechanism A and B θ is the rotation due to Mechanism B.

Mechanism A
The contribution of Mechanism A is calculated through the model proposed by Sezen and Moehle [3], which assumes the distribution of bond and axial stresses of the steel bar as shown in Figure 2. Bond stresses are divided into two uniformly distributed portions: by τ for the elastic range ( s ε ≤ y ε ) and bu τ for the inelastic range ( s ε > y ε ).
Inelastic range ( s ε > y ε ): where s ε is the steel bar axial strain; s σ is the steel bar axial stress; Ø is the steel bar diameter; y ε is the steel strain at yield strength; y f is the steel yield strength of steel.
Knowing the slip resulting from Mechanism A, it is possible to calculate the respective relative rotation between the beam and column elements: where d is the effective depth of the beam and x is the neutral axis depth of the beam.
As a simplification, Alva and El Debs [8] suggest that constant values of neutral axis depth x be used in each range. In the elastic range, the authors suggest the value II x x = corresponding to Stage II (cracked section), since this value becomes practically constant after crack stabilization. In the inelastic range, the authors suggest the value u x x = corresponding to the ultimate moment, since in Stage III there is a rapid stabilization of the x values between the yielding moment and the ultimate moment.

Mechanism B
The relative rotation related to Mechanism B is caused by the sum of the slips induced by the flexural cracks at the extremity of the beam next to the column along the length p L . As shown in Figure 3, cracks are supposed equally spaced ( R s ) in Alva   Therefore, the total rotation along the length p L induced by a number of cracks n is given by Equation 6. , where i x is the neutral axis depth at the section where the crack occurs (crack width: i w ). In this case, the simplification suggested by the authors can be used ( i II . Assuming small differences between the values of i w along the length p L , it is possible to obtain a single crack opening value in that length by the Equation 7: where r s is the crack spacing; sm cm ε ε − is the difference between the average reinforcement strain and the average concrete strain.
Again ignoring the strain of the concrete in tension and knowing that the spacing between cracks r s allows the evaluation of the probable number of cracks along the length p L , Alva and El Debs [8] deduced the following expression for relative rotation resulting from Mechanism B: x is the neutral axis depth, which can be simplified as suggestion of the authors ( is the average deformation in the reinforcement, considering the contribution of tensioned concrete (tension stiffening); / 1 r is the curvature of the beam section, considering the contribution of tensioned concrete (tension stiffening).
Knowing the bending moment M at the end of the beam, it is possible to find the axial stresses and strains in the reinforcement (Equations 3 and 4) and also the curvature / 1 r of Equation 8. Hence, the model proposed by Alva and El Debs [8] can be deduced, according to Equations 9 and 10.
In the elastic range: In the inelastic range: ( ) The spacing between cracks R s can be evaluated from codes expressions or from formulations found in literature. In this paper, the expression presented in Eurocode 2 [9] was used: . . . .
where ∅ is the diameter of the beam steel reinforcement bars; 1 k is a coefficient which considers for the bond properties of the reinforcement steel bars (equal to 0.8 for high-bond bars and equal to 1.6 for plain surface bars);

MOMENT-CURVATURE RELATIONS
This item presents the analytical formulation for obtaining the moment-curvature relationships necessary to consider the material nonlinearity of the structural members (beam and column). It should be noted that the model proposed by Alva and El Debs [8] uses the curvature value at the beam end (next to the joint region) to calculate the relative rotation component resulting from flexural cracks (Mechanism B). Item 4.1 presents the analytical formulation for the construction of moment-curvature curves of rectangular sections from the integration of the material stresses and the equilibrium and strain compatibility equations, applicable for concrete up to C50. For concretes between C55 and C90, the analytical formulation can be found in Alva [10]. The analytical formulation of item 4.1 was implemented in a computational procedure in FORTRAN language and the results were validated by free and commercial software for structural analysis found in Brazil, as presented in Alva [10]. The computational procedure was used in the examples presented in item 5.

Integration of stresses and equilibrium equations in reinforced concrete section (columns and beams)
To understand the problem of rectangular sections subjected to axial load and bending moment, as well the analytical formulation, it is shown in Figure 5 a generic rectangular section with known (or pre-defined) longitudinal reinforcement. Figure 5 also contains diagrams representing the section strains, the internal resultant forces, the stresses in the concrete and the internal resultant forces in the longitudinal reinforcement.
For the equilibrium of the horizontal forces, the applied axial force Sd N must be equal to the sum of the resultant internal forces of concrete and reinforcement: where cc R is the resultant of the concrete compressive stresses;  : where h is the section height; where: x is the neutral axis depth;   The constitutive models for concrete in compression and non-prestressed steel reinforcement according to NBR 6118 [1] are illustrated in Figure 6.
where b is the section width (constant for rectangular section); c σ is the compressive stress of concrete as a function of the neutral axis depth x . In this case, the stress diagram takes on a parabolic format as shown in Figure 6. The position of the resultant cc R in the section is defined with the calculation of CG y , expressed by: Equation 22 can be deduced from substituting the analytical expression of the parabola that describes the compressive stress of the concrete in the integral of Equation 20: From Equation 18, it is possible to rewrite cc R according to the ordinate y : Solving the integral expressed in Equation 23, results in: In the numerator of Equation 21, the analytical expression of the parabolic curve of the concrete compressive stress is used to calculate the integral: Solving the integral of the numerator of Equation 26 results in: Solving the integrals of Equations 28 and 29 provides the expressions for cc R and CG y : In Equation 29, substituting the parabolic function that describes the compressive stress in concrete c σ and using Equation 18 that relates c ε and y result in:

Case 3: cc c2
ε ε ≤ e x h > In this case, the section is completely compressed, and the concrete has not yet reached its maximum stress (strength). The integrals used for the calculation of cc R and CG y must be calculated within the range of ordinates y that cover the section, according to Equations 32 and 33.
Solving the integrals of Equations 32 and 33 provides the expressions for cc R and CG y : Solving the integrals of Equations 36 and 37 provides the expressions for cc R and CG y :

Tension Stiffening
For considering the contribution of tensioned concrete between cracks (tension stiffening), Torres et al. [11] model was used, which assumes a stress-strain curve for concrete in tension as shown in Figure 9.
where N is the axial compressive force (positive sign for compression); A is the cross-section area; d is the cross-section effective depth; h is the cross section total height; e α is the ratio between the modulus of elasticity of steel to the concrete modulus of elasticity; ρ is the tensile reinforcement ratio (related to the section effective depth). The values 1 α and 2 α are calculated in case of tensile stresses in the section. The value of the tensile resultant in concrete ct R and its position (with the distances t y and t z , as shown in Figure 5) are calculated using the equations of equilibrium and compatibility of the section.

Algorithm for generating the points of the moment-curvature diagrams
In a simplified way, the solution algorithm used in this paper for the generation of each point of the momentcurvature diagram is presented: (1) Set the curvature value /  In the presented algorithm, ErrorN is associated with the relative error in terms of axial force. Tolerance must be defined: values around 0.001 (0.1%) are sufficient to achieve good accuracy.

Equivalent Branson stiffness (beams only)
A simpler alternative than that presented in item 4.1 in beams is the use of the expression suggested by Branson [12] to calculate the equivalent flexural stiffness in Stage II (cracked). Thus, the moment-curvature curve is defined by the cracking moment, by the ultimate moment (strength) -both calculated by usual design of reinforced concrete sections -and by the segment obtained by relations indicated in Equations 42 and 43 corresponding to Stage II: . .
where c E is the concrete modulus of elasticity; The moment-curvature diagrams of the reinforced concrete beam for one of the connections analyzed by numerical simulations of item 5 (LVP1) is shown in Figure 10 for the purpose of comparison between the differences found when using the equilibrium equations of the section (item 4.1) and when using Branson expression in the Stage II.

NUMERICAL SIMULATIONS
In this item, numerical simulations of beam-column connections of reinforced concrete frames are presented for the comparison between theoretical and experimental results. To obtain the theoretical results, the analytical model proposed by Alva and El Debs [8] was applied to account for the deformability under bending moment (item 3). Moment-curvature relationships (as per item 4) were used in the consideration of the material nonlinearity. Figure 11 illustrates the geometry of the beam-column connections analyzed by the numerical simulations, as well as the loading scheme applied and the longitudinal reinforcement of beams and columns of these connections. The mechanical characteristics are summarized in Table 1 (concrete and longitudinal reinforcements). The dimensions of the connections and the area of longitudinal reinforcements of beams and columns are shown in Table 2. Table 3 contains the constants values of the model presented in Alva and El Debs [8] for the connections analyzed by the numerical simulations. It should be noted that all parameters of the analytical model were calculated based on the mechanical properties of the materials (characterization tests).

Beam-column connections: Alva [13]
Alva [13] performed tests on exterior beam-column connections subjected to alternating cyclic loads. The first stage of loading was the same to all connections: application of cyclic loads with amplitude increments of 10 kN up to the value of 60 kN. This loading in the first cycle generated a maximum bending moment corresponding to 60% of the yielding moment. Higher loads were applied at the end of the beam in the second stage of loading until the failure of the connection, as shown in Figure 12.

Figure 12.
Loading history at the last stage of loading -Alva [13] In all connections, the failure occurred by crushing the diagonal strut due to beam-column joint shear forces. In the connections LVP1 and LVP2, the connection failure occurred with yielding of the beam flexural reinforcement. In the connections LVP3 and LVP4, the failure of the connection occurred without the yielding of the beam flexural reinforcement. Further information about the experimental investigation is found in Alva [13] and Alva and El Debs [8].
To obtain the relative rotations between beam and column, horizontal displacement transducers were used, as shown in Figure 13.

T3 300mm
Beam Column H Figure 13. Displacement transducers used to evaluate relative rotations -Alva [13] In this case the relative rotation is calculated by: where 3 δ and 4 δ are the displacements measured by the transducers T3 and T4 and H is the distance between the transducers.
According to Figure 13, the length p L to be used in Alva and El Debs [8] model is equal to 300 mm (see transducers position in relation to the column face).
The moment-rotation curves of the connections tested by Alva [13] are shown in Figures 14 to 17 for the two stages of loading mentioned.   In a general way, it can be concluded that Alva and El Debs [8] analytical model simulates satisfactorily the bending deformability of the connections. For the second loading stage of the connections LVP3 and LVP4, the results provided by the analytical model were less satisfactory, since the shear joint failure did not allow the connection to reach the yielding moment of the beams.

Beam-column connections: Lee et al. [14]
Lee et al. [14] presented experimental results of beam-column connections subjected to seismic loads. Specimen 2 and Specimen 5 were chosen for comparison with the theoretical results. Figure 18 illustrates the structural model used to obtain the theoretical force-displacement curves using the finite element software ANSYS. The moment-rotation behavior of the beam-column connections was simulated by nonlinear springs, using the COMBIN39 element. The joint region was simulated with rigid offsets. Beams and columns were discretized and represented by frame elements, using the finite element BEAM188, which allows the consideration of material nonlinearity by moment-curvature relationships. The points of the moment-curvature curves were obtained by the formulation presented in item 4. The iterative incremental Newton-Raphson method was used for the numerical solution of the nonlinear problem, with convergence criteria based on the residual forces and moments.   Figure 19 contains the theoretical force-displacement curves (fully rigid and deformable connections) and the experimental curve for the first loading cycle. It can be seen from the referred curves that the consideration of the bending deformability led to significantly better results than those obtained by the hypothesis of a fully rigid connection.
There are no experimental results from moment-rotation curves in Lee et al. [14]. Thus, the experimental relative rotations between beam and column were obtained indirectly from the experimental displacements, according to

FINAL CONSIDERATIONS AND CONCLUSIONS
This paper dealt with the question of the deformability of reinforced concrete monolithic beam-column connections in the nonlinear analysis of framed reinforced concrete structures. To consider the deformability under bending moment, Alva and El Debs [8] analytical model was used. To consider the material nonlinearity of structural elements, momentcurvature relationships were used. The entire formulation of the constitutive models was deduced and presented, to allow the implementation of these models in computational procedures.