Simplified Model for Strengthening Design of Beam–Column Internal Joints in Reinforced Concrete Frames

The beam-column joints are very restricted areas in which the internal forces, generated by boundary elements, act on the concrete core and reinforcing bars with a very high gradient. They are the link between horizontal and vertical structural elements, and therefore, they are directly involved in the transfer of seismic forces. Thus, they are crucial to study the seismic behavior of reinforced concrete (RC) structures. To fully understand the seismic performances and failure modes of beam-column joints in RC buildings, a simplified analytical model of joint behavior is proposed and theoretical simulations are performed. The aim of the model, focusing on internal perimetric joints, is to identify the strength hierarchy in terms of capacity for different failure modes (namely failure of cracked joint, bond failure of passing through bars, flexural/shear failures of columns or beams). It could represent a tool for the designers of new joints to quantify the performance of new structures, but also as a tool for the designers of external strengthening of existing joints in order to calculate the benefits of the retrofit and pushing the initial failure to a more desirable failure mode. Further, some experimental results of tests available in the scientific literature are reported, analyzed and compared. Polymers 2015, 7 1733


Introduction
The behavior of the beam-column joint is a crucial aspect in a new or existing reinforced concrete (RC) moment resisting frame and needs to be designed and detailed properly. During earthquakes, the failure of beam-column joints is governed by bond and shear failure mechanisms which are usually brittle. Beam-column joints having deficient reinforcement details are expected to respond poorly, even when subjected to moderate seismic action. Beam-column joints in a RC moment resisting frame are circumscribed portions of the structures where high loads transfer between the connecting elements (i.e., columns and beams) in the structure. This aspect can be particularly crucial in the case of seismically-resistant frames where this high demand mobilizes the inelastic capacity of RC members to dissipate seismic energy while joints are poorly designed, jeopardizing the entire structure, even if it is correctly designed (Manfredi et al.) [1].
Under certain seismic actions, the beams connecting into a joint are subjected to moments in the same (clockwise or counter-clockwise) direction. Under these moments, the bars at the same level are pulled or pushed in the same direction at both sides of the joint panel. If the column is not wide enough, or if the strength of concrete in the joint is too low, the bond between steel bars and concrete cannot balance this stress request (Lignola et al.) [2]. In such cases, the reinforcement bars slip inside the joint region, and beams lose their load capacity. Furthermore, under cyclic actions, joints undergo the diagonal push and pull actions and concrete diagonally cracks into the joint panel.
Stirrups in the joint panel provide the crucial shear strength and confinement pressure to preserve joint panels from premature brittle failures. A proper quantity of transverse reinforcement allows the stresses to be appropriately transmitted between the beams and columns. Nonetheless, the deficiency of transverse reinforcement in joint panels occurs often in structural systems designed for gravity loads or according to outdated seismic codes (Masi et al.) [3]. Hence, current research efforts concentrated on developing cost-effective and sound retrofit techniques and strategies by means of composite materials. Former rehabilitation techniques, such as RC or steel jackets, have been well investigated in the past (e.g., Alcocer and Jirsa [4], Ghobarah et al. [5], Tsonos [6], and Hakuto et al. [7] among others). In recent years, Fiber-reinforced polymer (FRP) has been widely studied as a novel strengthening system, and it has been extensively applied in practice due to its benefits, such as light weight, high strength, corrosion resistance, and fast constructability.
From a designer point of view, current building codes for an earthquake-resistant design of RC moment frame structures provide an empirical limit on joint shear force. Earlier models, namely, traditional truss and strut models to explain the shear resistance mechanism, were proposed by Paulay et al. [14]. Further studies established shape factors given as empirical factors based on laboratory testing to take into account the variation of beam-column joint configurations. Such studies converged in different seismic code provisions and a review of them for joints, from main codes of practice around the world, can be found in Uma and Jain [15]. However they lack consideration for the effects of other critical factors such as the magnitude of column axial load or the capacity of the members' framing to the joint. Shiohara, in 2001 [16], made some of the first efforts to identify all the crucial factors by means of a mechanical model, as described in the following, being the basis of the improvements provided by the proposed novel model. Re-evaluating available experimental results, the idea was not to limit the joint failure to an exceedance of a joint shear stress capacity (because joint shear stress does not degrade as the storey shear does during experimental tests).
Lowes and Altoontash, in 2003 [17], and in the following years with further improvements and calibrations, proposed a nonlinear model for beam-column joints to be used in numerical analyses to evaluate the impact of joints on RC frame earthquake performance and their deformability, including bond-slip behavior. Increasing the complexity, further studies involved the full nonlinear finite element modeling (FEM) of joints clarifying the role of many factors in the behavior of beam column joints (e.g., Manfredi et al. [1] and Lignola et al. [2], among others).

Research Significance
The aim of the present paper is to investigate the internal RC beam-column joints in order to study seismic behavior and to establish the strength hierarchy in the case of failure. It aims to contribute to capacity design (and the subsequent strength hierarchy) principles. In fact, these modern design principles are strongly subordinate to the beam-column joint panels' behavior which can reduce substantially the global ductility, if the joint is subjected to a premature failure. Theoretical simulations and analytical models have been presented and discussed, and theoretical outcomes have been compared to experimental results. Presented theoretical approach aims to represent a practical tool to be used by practitioners in the FRP-strengthening design process of beam column joints.

Basis of Theoretical Model
The study focuses on internal joints which can be identified in a perimetric moment resisting frame. Perimetric frames have been selected because they have less beneficial effect of confinement provided by out-of-plane members (i.e., transverse beams or slabs). Internal joints present two beams framing into the sides of a column.
Experimental joint failure databases (e.g., Kim [18] collected data on 341 subassemblies) highlight typical shear failures in conjunction with, but also without, yielding of longitudinal beam reinforcement. Influencing parameters for joint shear behavior are linked to key points on capacity curves displaying the most distinct stiffness changes. Concrete cracking is usually coupled to yielding or bond failure of longitudinal reinforcement of beams when the most different changes in stiffness, for both overall and local behavior are triggered, up to the point of initiation of joint shear failure (maximum experimental storey shear).
The examined parameters for joint behavior are material properties, joint panel geometry, reinforcement in joint panel, column axial load, and reinforcement bond condition. Furthermore, the effect of external strengthening, e.g., by means of externally-bonded FRP, but not limited to it, is another debated parameter. Obviously, shear and flexural capacity of intersecting elements, namely beams and columns, are also of interest for the capacity design, hence they are considered herein, but not analyzed in detail, since they are considered almost well established, both in terms of as-built and retrofitted capacity.
The proposed model moves from the so called quadruple flexural resistances model [16] sketched in Figure 1. It considers the kinematics of the four segments divided by diagonal cracks in the joint panel, rotating due to bending moment and shear coming from beams and columns. The equilibrium of internal forces in steel and concrete, and external forces acting on beam ends and column ends is taken into account, whereas the compatibility condition is not necessarily satisfied. Dowel effects of bars and shear friction along diagonal cracks were neglected in the original formulation. In fact, the model, as it was originally proposed, presents some essential assumptions, described in the following lines, mainly because of the undetermined system of equations having fewer equations than unknowns.
Polymers 2015, 7 4 parameter. Obviously, shear and flexural capacity of intersecting elements, namely beams and columns, are also of interest for the capacity design, hence they are considered herein, but not analyzed in detail, since they are considered almost well established, both in terms of as-built and retrofitted capacity. The proposed model moves from the so called quadruple flexural resistances model [16] sketched in Figure 1. It considers the kinematics of the four segments divided by diagonal cracks in the joint panel, rotating due to bending moment and shear coming from beams and columns. The equilibrium of internal forces in steel and concrete, and external forces acting on beam ends and column ends is taken into account, whereas the compatibility condition is not necessarily satisfied. Dowel effects of bars and shear friction along diagonal cracks were neglected in the original formulation. In fact, the model, as it was originally proposed, presents some essential assumptions, described in the following lines, mainly because of the undetermined system of equations having fewer equations than unknowns. For instance in the interior joint, accounting for symmetry, given twelve equations defining equilibrium (three per rigid body), the number of independent equations representing the equilibrium reduces to five. Conversely, the unknowns are at least eight; three of them were assumed equal to experimental recorded data in experimental validation or directly assigned in the design phase. For instance yielding of joint stirrups was assumed (according to typical experimental outcomes after joint cracking) and also yielding of all beam bars, or alternatively yielding of tensile bars only and then the stress in compressed bars was derived by bond capacity limitation in the joint panel.
The model was also split to analyze potential beam failure or joint failure (evidently, beam column joint capacity is the smallest one of the two failure loads). However to avoid assuming a priori many unknowns, either supplied by experimental outcomes or considered at their maximum capacity (e.g., steel at yielding point, which actually is not always yielded at joint failure, when looking at experimental databases), modified unknowns, altered assumptions, and different solutions are herein proposed.
The basic idea is to include the beam mode into a unified joint model (adding apart a separate check not only of beam flexural failure, but also of column flexural failure, and for both, checking shear failure also). Doing so, it is possible to merge concrete compression and steel compression in a single compression force resultant at both beam and column ends. This assumption to merge compression in concrete and steel reinforcement has almost no effect on equilibrium of beams, while has minor effect on columns where neutral axis is deeper (Cosenza et al.) [19]. Finally, considering column shear not as For instance in the interior joint, accounting for symmetry, given twelve equations defining equilibrium (three per rigid body), the number of independent equations representing the equilibrium reduces to five. Conversely, the unknowns are at least eight; three of them were assumed equal to experimental recorded data in experimental validation or directly assigned in the design phase. For instance yielding of joint stirrups was assumed (according to typical experimental outcomes after joint cracking) and also yielding of all beam bars, or alternatively yielding of tensile bars only and then the stress in compressed bars was derived by bond capacity limitation in the joint panel.
The model was also split to analyze potential beam failure or joint failure (evidently, beam column joint capacity is the smallest one of the two failure loads). However to avoid assuming a priori many unknowns, either supplied by experimental outcomes or considered at their maximum capacity (e.g., steel at yielding point, which actually is not always yielded at joint failure, when looking at experimental databases), modified unknowns, altered assumptions, and different solutions are herein proposed.
The basic idea is to include the beam mode into a unified joint model (adding apart a separate check not only of beam flexural failure, but also of column flexural failure, and for both, checking shear failure also). Doing so, it is possible to merge concrete compression and steel compression in a single compression force resultant at both beam and column ends. This assumption to merge compression in concrete and steel reinforcement has almost no effect on equilibrium of beams, while has minor effect on columns where neutral axis is deeper (Cosenza et al.) [19]. Finally, considering column shear not as an unknown, but as the main driving parameter to study, it is possible to plot the evolution of all other internal force unknowns with the column shear, V c , and to match the number of equations and unknowns. In this way, according to basic capacity design principles, column shear capacity is provided for each failure mode (e.g., column shear corresponding to concrete compression failure, is found on the curve of concrete in compression at a certain column shear value, V c , intersecting concrete capacity; similarly column shear, V c , corresponding to bond failure, is found on the bond demand curve while intersecting the bond capacity threshold). The model has been particularized for internal joints.

Mathematical Formulation
Joint scheme has been identified by setting geometric parameters and stresses, as shown in Figure 2. It is enough to analyze a single load direction, since load reversal yields to the same stress state, because of the symmetry of the joint. Symmetry of the beam column joint is obvious if the joint is exactly at L b /2 and L c /2, as it is usually in experimental and design schemes; however, this does not mean that steel reinforcement bars having as resultant force, T 1 , at the bottom of the right beam and at the top of the left beam have equal cross section or total area. Internal joints present a symmetry in terms of internal and external forces, but not necessarily in terms of reinforcement bars. Force resultants, F i , (e.g., for the depicted case of clockwise moments on columns, F 1 and F 2 are in compression) are also shown in Figure 2. Accounting for symmetry, the number of nonlinear independent equations is five, in five unknowns (C, F 1 , F 2 , F 3 and F 4 ): Furthermore in this system, column and beam shear, V c and V b , as well as the flexural moment at column or beam, M c or M b , respectively, interface with joint satisfy also these relations:

Failure Modes
The paper focuses on internal perimetric joint failure; however, to fully understand joint behavior, it is recommended to also evaluate beam and column peculiar failure modes (i.e., flexural and shear capacity, according to classical structural analysis approaches, Cosenza et al. [19]). These failure modes are expressed in terms of column shear according to Equations (6)- (8), thus providing the first four possible column shear, Vc, capacity values corresponding to failure of the beam-column joint system. Eventual deficiencies of beams or columns in terms of flexural and shear capacities could be solved by means of FRP strengthening interventions, which are out of the scope of present paper and almost consolidated in the technical community.
Focusing on joint panel, three different failure modes can be expected: failure of concrete strut due to crushing, conventional failure due to yielding of longitudinal bars, or bond failure of longitudinal bars (in fact, the joint panel has limited dimensions to anchor the high stress demand and gradients from the bars).
The former failure mode involves the attainment of concrete crushing in the diagonal strut, so concrete contact force, C, should be limited to the compression strength of strut: If shear friction is assumed along cracks, the concrete contact force, C, can be assumed inclined not as the diagonal (i.e., ϑ ≠ θ), thus requiring a shear friction check where deviation from diagonal (i.e., |ϑ−θ|) should be limited to a threshold value. Afterwards, conventional failure of reinforcement in tension is checked, assuming different bars at each longitudinal reinforcement level i, having cross section Ak, and yielding stress fy,k. So each Fi should be limited to axial strength: Finally, the latter failure mode requires splitting the compression resultant force into steel and concrete contributions ( Figure 3). The basic idea is to evaluate (according to classical structural analysis

Failure Modes
The paper focuses on internal perimetric joint failure; however, to fully understand joint behavior, it is recommended to also evaluate beam and column peculiar failure modes (i.e., flexural and shear capacity, according to classical structural analysis approaches, Cosenza et al. [19]). These failure modes are expressed in terms of column shear according to Equations (6)- (8), thus providing the first four possible column shear, V c , capacity values corresponding to failure of the beam-column joint system. Eventual deficiencies of beams or columns in terms of flexural and shear capacities could be solved by means of FRP strengthening interventions, which are out of the scope of present paper and almost consolidated in the technical community.
Focusing on joint panel, three different failure modes can be expected: failure of concrete strut due to crushing, conventional failure due to yielding of longitudinal bars, or bond failure of longitudinal bars (in fact, the joint panel has limited dimensions to anchor the high stress demand and gradients from the bars).
The former failure mode involves the attainment of concrete crushing in the diagonal strut, so concrete contact force, C, should be limited to the compression strength of strut: If shear friction is assumed along cracks, the concrete contact force, C, can be assumed inclined not as the diagonal (i.e., ϑ ‰ θ), thus requiring a shear friction check where deviation from diagonal (i.e., |ϑ´θ|) should be limited to a threshold value. Afterwards, conventional failure of reinforcement in tension is checked, assuming different bars at each longitudinal reinforcement level i, having cross section A k , and yielding stress f y,k . So each F i should be limited to axial strength: Finally, the latter failure mode requires splitting the compression resultant force into steel and concrete contributions ( Figure 3). The basic idea is to evaluate (according to classical structural analysis approaches, Cosenza et al. [19]) the neutral axis depth, recalling linearity of the strain diagram of the cross section; in the meantime, compression and tensile resultants are in equilibrium, according to the previous systems of equations. Figure 3 also remembers that the neutral axis changes with increasing flexural loads due to concrete nonlinearity. The neutral axis, c, can be evaluated, for instance in the elastic field in a beam, without axial load, equating the first order moment to zero (Equation (11)) and, once the neutral axis, c, is known, the compressive force of reinforcements, S 1 , is derived: Finally, estimating the effective bond length, L eb , the upper bound bond capacity of concrete (based on uniform bond stress capacity, τ) is compared to the bond demand, given by the anchoring of longitudinal bars, even with different longitudinal diameters, Φ k , of bars at the same level, e.g., due to the two resultants S 1 + F 4 : Three different bond stress capacities were considered according to Model Code 90 [20]: the maximum bond capacity, τ max , is equal to 2.5¨?f c (good bond) or a medium value, τ med is equal to 1.25¨?f c all others conditions) for ribbed bars. The minimum value, τ min , equal to 0.3¨?f c can be used in the case of smooth bars. Different bond capacities' detailed analysis is out of the scope of present work, however they can be evaluated and inserted in the proposed model as bond capacity thresholds, e.g., (S 1 + F 4 ) max , to estimate corresponding column shear, V c .
Polymers 2015, 7 7 approaches, Cosenza et al. [19]) the neutral axis depth, recalling linearity of the strain diagram of the cross section; in the meantime, compression and tensile resultants are in equilibrium, according to the previous systems of equations. Figure 3 also remembers that the neutral axis changes with increasing flexural loads due to concrete nonlinearity. The neutral axis, c, can be evaluated, for instance in the elastic field in a beam, without axial load, equating the first order moment to zero (Equation (11)) and, once the neutral axis, c, is known, the compressive force of reinforcements, S1, is derived: Finally, estimating the effective bond length, Leb, the upper bound bond capacity of concrete (based on uniform bond stress capacity, τ) is compared to the bond demand, given by the anchoring of longitudinal bars, even with different longitudinal diameters, Φk, of bars at the same level, e.g., due to the two resultants S1 + F4: Three different bond stress capacities were considered according to Model Code 90 [20]: the maximum bond capacity, τmax, is equal to 2.5 (good bond) or a medium value, τmed is equal to 1.25· (all others conditions) for ribbed bars. The minimum value, τmin, equal to 0.3· can be used in the case of smooth bars. Different bond capacities' detailed analysis is out of the scope of present work, however they can be evaluated and inserted in the proposed model as bond capacity thresholds, e.g., (S1 + F4)max, to estimate corresponding column shear, Vc.

Behavior of Proposed Simplified Model
To evaluate the behavior of proposed simplified model, a parametric analysis is performed, on an internal perimetric joint, to highlight the peculiarities of this joint.
The joint analyzed to perform the parametric analysis is made of a beam of dimensions (30 cm × 50 cm, increased in a case) and a column 30 cm × 30 cm. The considered beam has length equal to 3.60 m,

Behavior of Proposed Simplified Model
To evaluate the behavior of proposed simplified model, a parametric analysis is performed, on an internal perimetric joint, to highlight the peculiarities of this joint.
The joint analyzed to perform the parametric analysis is made of a beam of dimensions (30 cmˆ50 cm, increased in a case) and a column 30 cmˆ30 cm. The considered beam has length equal to 3.60 m, reinforced by five D16 longitudinal bars at upper side and three D16 at the lower one. The length of the column is L c = 3.40 m, reinforced symmetrically by two D16 longitudinal bars at upper and lower sides. Both of them have been transversally reinforced by using D8 stirrups (two legs) at 20 cm. Concrete cover value considered for both beam and column is 2.6 cm (see Figure 4).
Polymers 2015, 7 8 reinforced by five D16 longitudinal bars at upper side and three D16 at the lower one. The length of the column is Lc = 3.40 m, reinforced symmetrically by two D16 longitudinal bars at upper and lower sides. Both of them have been transversally reinforced by using D8 stirrups (two legs) at 20 cm. Concrete cover value considered for both beam and column is 2.6 cm (see Figure 4). Scientific literature presents numerous experimental test results, conducted on different kinds of geometry, evaluating stress values leading to joint failure. By using the proposed model, for each geometry, the values of column shear, Vc, have been calculated for each of the failure modes for nodes (and named Vc1 to Vc11 for each considered failure mode, see Table 1).Starting from the reference condition, geometric variations have been performed leading to new column shear results related to each of the failure modes. In particular, variations about the number (two and four) of legs for stirrups and about the number and type of FRP layers have been considered. Four different Carbon CFRP (Young's modulus Ef = 230 GPa) configurations have been considered: one horizontal uni-axial layer (tf = 0.333 mm), one quadri-axial layer, two quadri-axial layers, and four quadri-axial layers (each layer of quadri-axial has a thickness of 0.053 mm, hence combining the overlapping of inclined layer it is equivalent to 0.1279 mm in horizontal and vertical directions). FRP strain is assumed equal to debonding value (e.g., 0.4% according to Italian guidelines CNR DT200 R1 2013 [21]).
In order to understand which failure mode is independent from the reinforcement and which is improvable by using reinforcement, results were grouped and compared to the reference case, to represent values of column shear (for each failure mode) related to the amount of reinforcement. The main considered parameters were: axial load on column, Nc; concrete strength, fc; stirrups in the joint; FRP externally bonded on the joint panel; longitudinal reinforcement ratio in beams and columns; and beam height, hence joint panel dimensions. Variability of considered parameters is depicted in a numerical test matrix (in Table 2 at every case is associated an ID number). The identification of the Scientific literature presents numerous experimental test results, conducted on different kinds of geometry, evaluating stress values leading to joint failure. By using the proposed model, for each geometry, the values of column shear, V c , have been calculated for each of the failure modes for nodes (and named V c1 to V c11 for each considered failure mode, see Table 1).Starting from the reference condition, geometric variations have been performed leading to new column shear results related to each of the failure modes. In particular, variations about the number (two and four) of legs for stirrups and about the number and type of FRP layers have been considered. Four different Carbon CFRP (Young's modulus E f = 230 GPa) configurations have been considered: one horizontal uni-axial layer (t f = 0.333 mm), one quadri-axial layer, two quadri-axial layers, and four quadri-axial layers (each layer of quadri-axial has a thickness of 0.053 mm, hence combining the overlapping of inclined layer it is equivalent to 0.1279 mm in horizontal and vertical directions). FRP strain is assumed equal to debonding value (e.g., 0.4% according to Italian guidelines CNR DT200 R1 2013 [21]).
In order to understand which failure mode is independent from the reinforcement and which is improvable by using reinforcement, results were grouped and compared to the reference case, to represent values of column shear (for each failure mode) related to the amount of reinforcement. The main considered parameters were: axial load on column, N c ; concrete strength, f c ; stirrups in the joint; FRP externally bonded on the joint panel; longitudinal reinforcement ratio in beams and columns; and beam height, hence joint panel dimensions. Variability of considered parameters is depicted in a numerical test matrix (in Table 2 at every case is associated an ID number). The identification of the failure mode allowed to establish if and how it is possible to change and improve the hierarchy of the failure modes. In many cases, the method led to switch a brittle failure mode into ductile failure modes, even if some mechanisms are independent from the reinforcement inserted into the joint.
As expected, those are all the mechanisms external to the joint (e.g., failure for bending and shear of column or beam) not incorporated in the system of equations previously presented. Furthermore, thanks to the symmetry of the joint, the values for yield strength in upper and lower bars of the column are the same for each analysis (for this reason V c7 was omitted because it is equal to V c6 ).
It is clear from Table 1 that in the case of zero axial load on column (ID1), the failure of the system is due to the flexural failure of column in bending (i.e., V c2 = 37.14 kN is the lower column shear triggering the failure mode). Increasing the axial load (ID2 and ID3) the failure moves to the joint due to yielding of bars in beam (i.e., V c5 = 55.91 kN is the lower column shear triggering the failure mode). The increase of concrete strength (ID4 and ID5) simply increases slightly the column shear leading to the same failure mode of the joint.
Any improvement of the joint reinforcement (ID6 to ID8 in the case of internal reinforcement of new structures, or ID9 to ID12 in the case of external retrofit of existing structures) moves the failure mode from the brittle failure of the joint due to yielding of bars in the beam to the flexural failure of the column. Only an exception is represented by one ply of quadri-axial reinforcement (ID9), not sufficient in the horizontal direction to avoid a joint failure due to yielding of beam bars at a slightly lower load. Increasing the dimensions of the joints (ID13) allows preventing of joint failure, hence moving to the flexural failure of the column (at a slightly higher load, not because joint dimensions increase the ultimate moment of the column, but because the lever arms slightly changes; see Equation (7)).
Assuming a friction along the diagonal cracks in the joint (ID14 and ID15 and assuming the maximum of both) improves slightly the capacity of the joint pushing failure to the flexural capacity of the column.
Since the flexural failure of the column is not yet the optimum failure mode, a further intervention could be foreseen for the column, hence increasing the flexural capacity up to a switch of the failure mode to the beam flexural failure, the optimal ductile failure of such systems. It is worth noting that once increasing the flexural capacity of a member, in this case the column, a shear check should always be coupled and performed to evaluate the need for a combined shear strengthening, because shear loads on the member could increase with the flexural improvement. However this latter aspect of column strengthening is out of the scope of present work.
This discussion was performed assuming a good bond condition. However, in the case of smooth bars, with poor bond conditions, the failure mode would always occur at a much lower column shear with joint failure due to the bond of the longitudinal bars inside the joint. The fifteen analyses (Tables 2 and 3) allow assessing the influence of different parameters, both in terms of failure mode and column shear, V c . In some cases, a parameter has no influence on the effective failure mode, however it has an influence on other possible failure modes, but not triggered first in that case. The effect of axial load is mainly on column reinforcement forces and flexural and shear capacity (see Figure 5 where the force resultant F 3 reaches its yielding value at increasing column shear, V c , when increasing axial load on column, N c ). In Figure 5, F 3 (V c ) curves have different origins because the higher is the axial load, N c , the higher is initial compression level at zero lateral load, V c . Similarly, column shear corresponding to column flexural failure increases from 37.14 kN to 79.46 kN (+114%) (in square brackets the percentage increase has been reported), yet not being triggered as the first failure mode. Axial load has negligible effect on joint failure. The effect of concrete strength is mainly related to shear and flexural capacity of beams and columns.
A clear effect is also on bond capacity; Figure 6 shows bond demand F 4 + S 1 reaching its capacity value at increasing column shear, V c , when changing bond capacity (in good bond conditions) due to concrete strength τ max (f c ).
To evaluate the failure mode for each considered configuration case it is possible to refer to values reported in Table 3.             The effect of stirrups is meaningful, as it has many consequences. The most relevant is on longitudinal reinforcement bars, reducing their load, thus bearing a portion of horizontal load demand. Stirrups are assumed to yield, and for simplicity, their tensile force (T 9 ) is assumed equal to yielding force from the very beginning; for this reason, in Figure 7 the F 4 (V c ) curves have different origins. However the presence of stirrups increases column shear corresponding to joint failure due to beam tensile reinforcement yielding from 55.91 kN (no stirrups and actual failure of ID2 case) to 64.01 kN (+14.48%) (two D8 stirrups with two legs, ID6 case) and 86.14 kN (+54.06%) (eight D8 stirrups with two legs), but in these cases the failure mode switches to column flexural failure. Thus, it can be concluded that, in this case, two D8 stirrups are enough to change the failure mode from brittle joint failure to more desirable ductile column flexural failure.
Polymers 2015, 7 12 The effect of stirrups is meaningful, as it has many consequences. The most relevant is on longitudinal reinforcement bars, reducing their load, thus bearing a portion of horizontal load demand. Stirrups are assumed to yield, and for simplicity, their tensile force (T9) is assumed equal to yielding force from the very beginning; for this reason, in Figure 7 the F4(Vc) curves have different origins. However the presence of stirrups increases column shear corresponding to joint failure due to beam tensile reinforcement yielding from 55.91 kN (no stirrups and actual failure of ID2 case) to 64.01 kN (+14.48%) (two D8 stirrups with two legs, ID6 case) and 86.14 kN (+54.06%) (eight D8 stirrups with two legs), but in these cases the failure mode switches to column flexural failure. Thus, it can be concluded that, in this case, two D8 stirrups are enough to change the failure mode from brittle joint failure to more desirable ductile column flexural failure. Similar comments can be repeated for externally-bonded FRP. The role of the reinforcement is similar to the insertion of stirrups; however, benefits can also be achieved on an existing joint, even if, originally, they have no internal stirrups. From a numerical point of view, the presence of FRP is similar to stirrups (note that one ply of uni-axial FRP is not enough to change the failure mode, remaining the joint failure with yielding of beam bars). FRP and stirrups both provide an increase of horizontal, T9, (or vertical, T10, if vertical fibers are also applied) load carrying capacity. FRP tensile force is assumed equal to the debonding force from the very beginning. For this reason, in Figure 8 the F4(Vc) curves have different origins. However, the presence of FRP increases the shear column value, Vc, corresponding to system failure, from 55.91 kN (yet joint failure due to beam tensile reinforcement yielding) to 65.00 kN (+16.25%) (two plies of quadri-axial FRP, ID11 case). FRP was able to change the failure mode from brittle joint failure to more desirable ductile column flexural failure, with the only exception of ID9.
Longitudinal reinforcement has a low influence on the evaluation of force resultants, e.g., Figure 9 shows the F4(Vc) curves, they are overlapping; however, the yielding capacity is different, thus leading to much different column shear values, Vc, corresponding to joint failure due to beam tensile reinforcement yielding. These column shear values, Vc, reduces from 55.91 kN (actual failure of ID2 case) to 38.66 kN (−30.85%] (2+2D16 bars in the beams), or increases to 144.53 kN (+158.5%) (4+4D32 bars in the beams) yet not reaching yielding of bars, however in this latter case the failure mode is a different Similar comments can be repeated for externally-bonded FRP. The role of the reinforcement is similar to the insertion of stirrups; however, benefits can also be achieved on an existing joint, even if, originally, they have no internal stirrups. From a numerical point of view, the presence of FRP is similar to stirrups (note that one ply of uni-axial FRP is not enough to change the failure mode, remaining the joint failure with yielding of beam bars). FRP and stirrups both provide an increase of horizontal, T 9 , (or vertical, T 10 , if vertical fibers are also applied) load carrying capacity. FRP tensile force is assumed equal to the debonding force from the very beginning. For this reason, in Figure 8 the F 4 (V c ) curves have different origins. However, the presence of FRP increases the shear column value, V c , corresponding to system failure, from 55.91 kN (yet joint failure due to beam tensile reinforcement yielding) to 65.00 kN (+16.25%) (two plies of quadri-axial FRP, ID11 case). FRP was able to change the failure mode from brittle joint failure to more desirable ductile column flexural failure, with the only exception of ID9.
Longitudinal reinforcement has a low influence on the evaluation of force resultants, e.g., Figure 9 shows the F 4 (V c ) curves, they are overlapping; however, the yielding capacity is different, thus leading to much different column shear values, V c , corresponding to joint failure due to beam tensile reinforcement yielding. These column shear values, V c , reduces from 55.91 kN (actual failure of ID2 case) to 38.66 kN (´30.85%] (2+2D16 bars in the beams), or increases to 144.53 kN (+158.5%) (4+4D32 bars in the beams) yet not reaching yielding of bars, however in this latter case the failure mode is a different one, the column flexural failure, at V c = 62.18 kN. Conversely, the increase of beam dimensions has two main effects: the first one is the increase of beam capacity, both in terms of flexure and shear; and the second one is the increase of joint panel dimensions.
Polymers 2015, 7 13 one, the column flexural failure, at Vc = 62.18 kN. Conversely, the increase of beam dimensions has two main effects: the first one is the increase of beam capacity, both in terms of flexure and shear; and the second one is the increase of joint panel dimensions.  The increase of joint panel dimensions reduces the demand in terms of resultant forces in reinforcements, as shown in Figure 10 where the bond demand is lower for the higher beam. The same figure also highlights the influence of bond capacity: the same beam-column joint, if reinforced with smooth bars, presents a much lower capacity. In the case of beams of dimensions 70 cm × 30 cm, potential failure due to bond occurs at 97.45 kN in the case of good bond for ribbed bars, and it drops dramatically to 12.62 kN (−87.05%) in the case of smooth bars. It is a "potential" failure mode in the case of a good bond with ribbed bars, because the actual failure mode is column flexural failure at a shear column value, Vc = 66.79 kN.   The increase of joint panel dimensions reduces the demand in terms of resultant forces in reinforcements, as shown in Figure 10 where the bond demand is lower for the higher beam. The same figure also highlights the influence of bond capacity: the same beam-column joint, if reinforced with smooth bars, presents a much lower capacity. In the case of beams of dimensions 70 cm × 30 cm, potential failure due to bond occurs at 97.45 kN in the case of good bond for ribbed bars, and it drops dramatically to 12.62 kN (−87.05%) in the case of smooth bars. It is a "potential" failure mode in the case of a good bond with ribbed bars, because the actual failure mode is column flexural failure at a shear column value, Vc = 66.79 kN. The increase of joint panel dimensions reduces the demand in terms of resultant forces in reinforcements, as shown in Figure 10 where the bond demand is lower for the higher beam. The same figure also highlights the influence of bond capacity: the same beam-column joint, if reinforced with smooth bars, presents a much lower capacity. In the case of beams of dimensions 70 cmˆ30 cm, potential failure due to bond occurs at 97.45 kN in the case of good bond for ribbed bars, and it drops dramatically to 12.62 kN (´87.05%) in the case of smooth bars. It is a "potential" failure mode in the case of a good bond with ribbed bars, because the actual failure mode is column flexural failure at a shear column value, V c = 66.79 kN. Compared to counterpart case ID2 (differing only for beam height), the increase of joint panel dimensions increases the failure load. Conversely, if smooth bars where used, the predicted failure mode of the system would have occurred due to bond failure at a very low shear column value, Vc = 12.62 kN. This leads one to conclude that bond capacity has a meaningful effect on beam-column joint strength.

Model Validation
Theoretical predictions of the proposed simplified model were also compared to some experimental tests available in the literature (Hakuto et al. [22], Prota et al. [23] and Zaid et al. [24]) to validate the proposed theoretical model. Following, some beam-column joint specimens will be discussed in detail showing the entire table of "potential" failure modes. Actual failure load is the smallest of the potential failure modes. Discussion of the results considers two limit bond performances: maximum and minimum (due to smooth bars) bond conditions. This approach allows one not only to better understand the strength hierarchy, but also to calibrate strengthening design.
The basic idea is to avoid all undesired failure modes (e.g., brittle shear failure of joint, but also of beams and columns) to push failure mode to more desirable ductile beam flexural failure. Once the desired failure mode is selected, than all failure mode mechanisms presenting lower column shear values should be improved and strengthening (on existing structures) or design improvements (in structures still under design) can be calibrated to exceed the column shear, Vc corresponding to desired failure mode (i.e., according to "capacity design" approach).
Six retrofit configurations have been implemented, and "applied" to literature experimental results. In particular they have been supposed (as retrofit method): (a) one quadri-axial ply of FRP of thickness, t = 0.1279 mm; (b) two quadri-axial plies of FRP of thickness, t = 0.2559 mm; (c) four quadri-axial plies of FRP of thickness, t = 0.5116 mm; (d) one uni-axial horizontal ply of FRP of thickness, t = 0.333 mm; (e) two stirrups (two legs) of diameter 12 mm; and (f) four stirrups (two legs) of diameter 12 mm, simulated into the model by appropriate values of F9 and F10.
In all the cases FRP has a Young's Modulus of 230 GPa and is assumed to debond at 0.4%. Table 4 shows concrete characteristics and beam and column longitudinal and transversal reinforcements for all considered specimens of experimental tests available in scientific literature. Compared to counterpart case ID2 (differing only for beam height), the increase of joint panel dimensions increases the failure load. Conversely, if smooth bars where used, the predicted failure mode of the system would have occurred due to bond failure at a very low shear column value, V c = 12.62 kN. This leads one to conclude that bond capacity has a meaningful effect on beam-column joint strength.

Model Validation
Theoretical predictions of the proposed simplified model were also compared to some experimental tests available in the literature (Hakuto et al. [22], Prota et al. [23] and Zaid et al. [24]) to validate the proposed theoretical model. Following, some beam-column joint specimens will be discussed in detail showing the entire table of "potential" failure modes. Actual failure load is the smallest of the potential failure modes. Discussion of the results considers two limit bond performances: maximum and minimum (due to smooth bars) bond conditions. This approach allows one not only to better understand the strength hierarchy, but also to calibrate strengthening design.
The basic idea is to avoid all undesired failure modes (e.g., brittle shear failure of joint, but also of beams and columns) to push failure mode to more desirable ductile beam flexural failure. Once the desired failure mode is selected, than all failure mode mechanisms presenting lower column shear values should be improved and strengthening (on existing structures) or design improvements (in structures still under design) can be calibrated to exceed the column shear, V c corresponding to desired failure mode (i.e., according to "capacity design" approach).
Six retrofit configurations have been implemented, and "applied" to literature experimental results. In particular they have been supposed (as retrofit method): (a) one quadri-axial ply of FRP of thickness, t = 0.1279 mm; (b) two quadri-axial plies of FRP of thickness, t = 0.2559 mm; (c) four quadri-axial plies of FRP of thickness, t = 0.5116 mm; (d) one uni-axial horizontal ply of FRP of thickness, t = 0.333 mm; (e) two stirrups (two legs) of diameter 12 mm; and (f) four stirrups (two legs) of diameter 12 mm, simulated into the model by appropriate values of F 9 and F 10 .
In all the cases FRP has a Young's Modulus of 230 GPa and is assumed to debond at 0.4%. Table 4 shows concrete characteristics and beam and column longitudinal and transversal reinforcements for all considered specimens of experimental tests available in scientific literature. However more details about experimental programs and results can be found in relevant cited publications. Table 5 shows results obtained by the resolution of the experimental configuration model (for each experimental program and each specimen).
The validation is proposed in Table 6 where the agreement between experimental outcomes and model predictions (both in terms of failure modes and column shear capacity) is usually satisfactory and on the safe side.

Simulation of Experimental Tests in Research from Hakuto et al.
Both specimens tested by Hakuto et al. [22], have a failure mode due to yielding of bars of the beam, with a value of V c5 equal to 130.20 kN and 76.03 kN, for Unit 5 and Unit 1 (considering maximum bond), respectively. The proposed model satisfactorily underestimates the actual capacity of the tested joint. By applying the retrofit methods (a) to (d) at Unit 5, the failure mode does not change (both in case of minimum or maximum bond); instead, applying methods (e) and (f), the failure mode in case of maximum bond, moves from yielding of bars of column to yielding of bars of column, but not one of the considered retrofits provides a ductile failure mode. Unit 1 has a different behavior. Considering the maximum bond case, by applying the retrofit methods (a) and (b) failure does not change. In case of maximum bond, retrofit methods (c) to (f) allow to move to a failure mode due to bending moment of the beam (the most desirable one) applying three plies of FRP (t = 0.33 mm), so the joint is upgradable. Results of the specimen named Unit 1 are shown in Table 7. Bold numbers indicate failure considering maximum bond. Italic numbers indicate failure considering minimum bond.

Simulation of Experimental Tests in Research from Prota et al.
In this paper [23], four specimens of a wider experimental program on perimetric joints have been considered, namely H1, L1, L2, and H2. For all of them, column length, L c , is 264 cm, beam length, L b , is 305 cm, distance between reinforcement layers in the column, h * c = 12.4 cm, distance between reinforcement layers in the beam, h * b = 27.9 cm, and concrete cover of beam and column is 4 cm. The specimen named L1 does not present external FRP reinforcement. Considering the case of maximum bond the failure mode can be changed by applying one ply of uni-axial FRP (t = 0.13 mm) from bond failure to the bending moment of the column. Similar results can be obtained in cases of medium or minimum bond, increasing the number of FRP plies or the number of stirrups. In any case, the retrofit yields to a failure due to the bending moment of the column. Nevertheless, to avoid this kind of joint failure, column bending capacity needs to be improved, but this kind of retrofit is out of the scope of present paper. Results are shown in Table 8.
Concrete strength of specimen H1 (31.70 MPa) and H2 (36.50 MPa) is lower than previous cases (specimen L1 f c = 38.60 MPa), so it is more difficult to obtain every desirable failure mode.
Considering maximum bond, the failure mode moves to flexural column failure by application of retrofit named (f) in both cases, even if V c values of specimen H2 are higher than values of specimen H1. Furthermore, the reduced concrete strength reduces the possibilities of FRP retrofit since the concrete strut crushing (i.e., V c11 ) could occur at column shear close to the desired flexural failure of beams (i.e., V c1 ). All potential failure modes and relevant column shears are reported in Tables 9 and 10.
Specimen L2 presents results comparable with specimen L1.

Simulation of Experimental Tests in Research from Zaid et al.
In this case [24], column length, L c , is 147 cm, beam length, L b , is 270 cm, distance between reinforcements layer in the column, h * c = 23 cm, distance between reinforcements layer in the beam, h * b = 23 cm, concrete cover of beam and column is 3.5 cm and concrete contact force inclination is equal to ϑ = 0.785 rad.
Results obtained by applying retrofits (a) to (f), and not reported here for brevity, show that it is not possible to move from the joint failure mode due to yielding of bars of beam by retrofitting the joint only with the considered retrofit configurations. In this case the reinforcement ratio of the beams is high; hence the flexural capacity is relatively high to move to the desired ductile flexural beam failure.

Concluding Remarks
The beam-column joints behavior can strongly influence the seismic performance of RC buildings. A simplified analytical/mechanical model of joint behavior is discussed and theoretical simulations are performed in order to fully understand the mechanical behavior and the failure modes. Compared to previous models and theories the proposed one does not require FEM or complex nonlinear numerical analyses of the structure or sub-assemblage and does not require the calibration of empirical factors to account for the many parameters affecting joint seismic response and outlined in the paper.
The model moves from a previous one, but it improves and alters deeply the main unknowns, assumptions, and solutions. Among the main parameters affecting the performance of joints, namely material property, joint panel geometry, reinforcement in joint panel, column axial load, and reinforcement bond condition; axial load on columns showed negligible influence, while meaningful influence was given by bond performance. Similarly, joint (transverse) reinforcement, either external FRP or internal stirrups, provides many benefits pushing failure mode from brittle joint shear to ductile beam flexural.
The identification of the failure mode establishes if and how it is possible to change and improve the hierarchy of the failure modes to switch a brittle failure mode into ductile failure modes, even if some mechanisms are independent from the reinforcement inserted into the joint, and should be treated independently.
Once the desired failure mode is selected, than all failure mode mechanisms presenting lower column shear values should be improved and strengthening (on existing structures) or design improvements (in structures still under design) can be calibrated to exceed the column shear, V c corresponding to desired failure mode. Hence, the proposed tool fully fits the "capacity design" approach for strengthening interventions design.

Conflicts of Interest
The authors declare no conflict of interest.

Appendix
In this appendix a numerical worked example is proposed analyzing the experimental test S3 by Zaid et al., [24]. The main system of Equations (1) to (5) is solved adopting as input the geometrical parameters and material characteristics reported in the last line of Table 4 for the relevant experimental S3 data of Zaid et al., [24]. F 9 = 88.22 kN is the capacity of the 4 stirrups (2 legs D6) in the joint. In this way it is possible to calculate the main five unknowns F i (V c ) [N], with i equal to 1 to 4 and diagonal contact force C(V c ) [N].
F 1 " 0.5ˆ91, 908`V c (1A) F 2 " 5.56ˆ10´3ˆp9, 000, 000`49¨V c q (2A) C " 1.38ˆ10´1 5ˆp 6.58ˆ10 20´1 .41ˆa2.16ˆ10 41´1 .63ˆ10 36¨V c q (5A) Previous equations trace the evolution of forces in the joint as function of column shear, V c , hence it is possible to evaluate the relevant V c,i triggering each considered failure mode. In particular, looking at the joint failure, column shear corresponding to the yielding failure of bars in tension into the beam or column, at the upper or lower side, can be evaluated by equating F 4 to minimum yielding force of beam reinforcement and F 3 to minimum yielding force of column reinforcement, respectively.
Such yielding forces of bars have to be calculated in advance. Since the reinforcement is symmetric, the minimum yielding force is simply evaluated by accounting for 5D16 (A sb = 1004.8 mm 2 ) in the beam and 4D19 (A sc = 1133.54 mm 2 ) in the column, so that yielding forces are: Hence the column shear corresponding to the failure due to the yielding of bars in beam, V c5 is equal to 121.73 kN and the column shear corresponding to the failure due to the yielding of bars in column, V c6 is equal to 128.33 kN, as it can be found in Table 5.
To evaluate the column shear leading to the failure due to reinforcement debonding, the bond capacity should be calculated. The bond capacity can be calculated as F bond : where n is the number of longitudinal bars, Φ is bar diameter, L eb is the effective bond length and τ is the uniform bond stress capacity. Three conditions of bond have been considered. The case of maximum, medium and minimum bond capacities, namely τ max = 13.23 MPa, τ med = 6.61 MPa and τ min = 1.59 MPa, calculated according model Code 1990 [20]. Hence F bondmax , F bondmed , F bondmin are evaluated substituting the values of τ max , τ med and τ min into the Equation (8A): F bondmax " 5πˆ16ˆ230ˆ13.23 " 764.31 kN (8Aa) F bondmed " 5πˆ16ˆ230ˆ6.61 " 382.15 kN (8Ab) F bondmin " 5πˆ16ˆ230ˆ1.59 " 91.72 kN (8Ac) To evaluate bond demand F 4 + S 1 as a function of column shear V c , the bar force in compression, S 1 , is extracted from the global compression force F 1 , as previously described. The whole section is assumed in the elastic range, hence it is possible to assess the neutral axis, c, by considering the static moment of the cracked section equal to zero (see Equation (11)). Compression force S 1 is evaluated according to Equation (12) as: where neutral axis, c, is equal to 47 mm. In this way, equating the bond demand to the bond capacity previously evaluated in the three bond conditions, it is possible to evaluate the column shears V c8 , V c9 and V c10 , respectively, as reported in Table 5 for the relevant simulation case.
Similarly V c11 corresponding to joint failure due to crushing of concrete strut is evaluated by equating C to C max = 1,188 kN, in this case (see Equation (9)).
It is now possible to evaluate failure modes due to bending and shear failures of beam and column. According to technical theory of flexure and shear capacity at ultimate state of RC beams and columns (e.g., in this case according to Italian practice [19]), the relevant values can be expressed as functions of the column shear (see Equations (6)-(8)). In the present case, relevant capacities are: beam flexural, M b = 142.50 kNm, column flexural, M c = 90.19 kNm, beam shear, V b = 423.23 kN and column shear, V c = 328.58 kN, directly equal to V c4 . These values yield to column shears V c1 to V c4 , respectively ( Table 5). Due to changes of sign of fibers in tension and compression, the capacities are those minimum, hence evaluated accounting for the minimum reinforcement (in this case reinforcement is symmetric).
To identify the actual failure mode, the lowest column shear identifies the failure, and in this case (in good bond conditions) it is V c5 , hence a joint failure mode with yielding of bars in tension in the beam (V c = 121.5 kN as reported in Table 6, slightly underestimating the experimental capacity, that is 130 kN, with the same reported joint failure mode).
As a concluding remark, to design a retrofit intervention, failure mode should be moved to beam flexural failure, hence joint should be improved to avoid a joint failure for a V c smaller than V c1 = 171.3 kN (see Table 5). In any cases, having a clear summary of the column shears activating different failure modes for a beam column joint, it is possible to design a proper and effective retrofit intervention according to capacity design approach.