1 Introduction

Reinforced Concrete (RC) structures realised in ’60s and ‘70s of the past century are very common, both in Italy and in the whole Mediterranean area. They are widely used for public buildings, such as school, hospitals and administrative institutions (ISTAT 2001), and, then, they should guarantee high seismic performance objectives (M.II.TT. 2008; CEN 2004). However, these structures were often designed according to old codes which were not inspired to the modern concepts of performance-based seismic design criteria (Paulay and Priestley 1992). Therefore, they do not generally meet the safety standards required by the current codes of standards and, hence, they are generally characterised by significant levels of seismic vulnerability (Faella et al. 2006).

The observations of damage occurred during recent seismic events highlight the aforementioned critical aspects in the behaviour of existing RC frames. Specifically, they point out that the response of beam-to-column joints hugely affects their overall seismic behaviour of RC frame structures (Verderame et al. 2009).

Although several capacity models are currently available in the scientific literature for evaluating the capacity of RC members (fib 2003), they are not sufficiently accurate for beam-to-column joints (Lima 2012). In fact, these models for beam-to-column joints are based on analytical or semi-analytical formulations, either derived from mechanical considerations or empirically calibrated on experimental results (Adibi et al. 2018). Although they aim at evaluating the shear strength of RC joints, they cannot reproduce the strength and stiffness degradation induced by the cyclic nature of seismic actions, although this phenomenon is clearly highlighted by the available experimental results (Melo et al. 2015). Moreover, the same models are affected by significant levels of variability in terms of shear strength, as a result of both the different theoretical assumptions on which they are based and the variable parametric fields explored by the Authors who originally calibrated and proposed them (Lima et al. 2012a, b).

This paper aims to provide an original contribution about how to simulate shear strength degradation due to the cyclic actions in beam-to-column joints of RC frames. Specifically, the interpretation of the aforementioned strength degradation is approached by means of the low-cycle fatigue theory (Bathias and Pineau 2010), which is intended as an extension to the inelastic case of the elastic fatigue theory (high-cycle fatigue) (Palmgren 1924; Miner 1945). Therefore, Sect. 2 outlines the fundamental aspects of the low-cycle fatigue theory, which is, then, employed in Sect. 3 for determining the parameters of low-cycle fatigue curves for both interior and exterior RC joints, on the bases of a database of experimental results collected from the scientific literature. The outcome of this preliminary calibration is proposed in Sect. 4: the influence of geometric layout and design criteria (considering unreinforced, reinforced and EC8-compliant joints) clearly emerges in terms of slope of the corresponding low-cycle fatigue curves identified for the RC joints under consideration.

Finally, it is worth highlighting that, far for providing the final relationships capable to predict the low-cycle fatigue curve based on the properties of RC joints under consideration, the present paper aims at demonstrating that low-cycle fatigue is a promising approach to predicting the decay of shear strength in RC joints subjected to cyclic actions. A wider database of well-documented cyclic tests would be eventually needed to unveil the actual relationship between the low-cycle fatigue curve and the relevant properties of the RC joint under consideration.

2 Outline of the low-cycle fatigue theory

Experimental tests on RC joints subjected to under cyclic actions exhibit a degradation in terms of both strength and stiffness (Melo et al. 2015). Theoretically, this progressive reduction in strength (schematically depicted in Fig. 1) can be interpreted within the general framework of low-cycle fatigue theory (Bathias and Pineau 2010), which is an extension of the theory attributed to Palmgren (1924) and Miner (1945) for describing the failure of mechanical components subjected to a high number of cyclic excitations, though within the elastic range.

Fig. 1
figure 1

Strength degradation of a generic structural component under cyclic displacements with constant amplitude

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As a matter of fact, the concept of Low-Cycle Fatigue is extensively used in earthquake engineering and generally aims at evaluating the number and amplitude of cycles leading to failure of a structural component under cyclic actions (Mander et al. 1994). Specifically, a functional D can be generally defined in order to represent the cumulated damage due to the cyclic actions. According to the assumptions of Palmgreen-Miner, the damage D induced by a history of n1 cycles with constant amplitude V1 can be expressed as follows:

$$D = \frac{{n_{1} }}{{N_{1} }}$$
(1)

where N1 is the number of cycles leading to failure under the same amplitude V1.

In the general cases of cycles characterised by variable amplitudes (Fig. 2), the expression of the damage functional D defined in Eq. (1) can be generalised as follows (Miner 1945):

$$D = \frac{{n_{i} }}{{N_{i} }} + \frac{{n_{2} }}{{N_{2} }} + \frac{{n_{3} }}{{N_{3} }} \cdots$$
(2)

where ni is the actual number of cycles with amplitude Vi and Ni is the corresponding number of equal amplitude cycles leading to failure. In principle, in this case failure is expected to occur (conventionally) when the functional D is equal to the unit:

$$D = \sum\limits_{i}^{k} {\frac{{n_{i} }}{{N_{i} }}} = 1 .$$
(3)

in which k represents the number of cycle groups of equal amplitude.

Fig. 2
figure 2

Hysteretic response under cyclic actions with variable amplitude

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Four seismic performance levels (i.e., fully operational, immediate occupancy, life safety, and collapse prevention) are generally considered to describe the expected performance of buildings, or alternatively, the expected damage, and economic loss. Therefore, according to the scientific literature (Cosenza and Manfredi 2000) structural collapse is supposed to occur as D > 1, whereas structural damage is still reparable for D < 0.5. In the case of 0.5 < D < 1.0, collapse does not occur, but the building (or the structural member under consideration) is not reparable. Moreover, for D < 0.2 damage is supposed to be negligible.

Jiang et al. (2011) suggested specific values of D related to the Limit States generally considered in seismic analysis of structures. In particular, damage index D less than 0.05 implies fully operational capabilities; from 0.05 to 0.15, immediate occupancy capability; from 0.15 to 0.45, life safety; and from 0.45 to 1.0, incipient collapse.

Based on a generalised Palmgren–Miner rule (Hashin 1979), the amplitude Vi leading to failure of the element subjected to a defined number of cycles Ni is related to the strength Vmon under monotonic action as follows:

$$\log \left( {V_{i} } \right) = \log (V_{mon} ) - \frac{1}{m}\log (N_{i} ) .$$
(4)

where m is a parameter defining the degradation effect.

The Eq. (4) assumes a linear relationship (in a log–log plane) between ultimate shear strength and available number of cycles up to failure (Fig. 3). Specifically, the (logarithm of) strength decreases linearly with the (logarithm of) the number of cycles.

Fig. 3
figure 3

Relationship between shear strength and number of cycles needed for acting joint failure

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If the parameter m and the monotonic strength Vmon are known, the number of cycles Ni needed for inducing failure due to an action of given amplitude Vi < Vmon can be evaluated by solving Eq. (4) with respect to Ni:

$$N_{i} = \left( {\frac{{V_{i} }}{{V_{mon} }}} \right)^{ - m} .$$
(5)

In case of cyclic protocols characterised by variable amplitudes, the following relationship can be determined by replacing Eq. (5) in Eq. (2) and fixing the limit condition dictated by Eq. (3):

$$\sum\limits_{i = 1}^{k} {\frac{{n_{i} }}{{\left( {\frac{{V_{i} }}{{V_{mon} }}} \right)^{ - m} }}} = 1 ,$$
(6)

where k, representing the number of cycle groups of equal amplitude, can be recognised throughout the load history up to the specimen failure.

Referring to RC joints analysed in this paper, Eq. (6) includes 2 unknown terms:

  • the parameter m;

  • the shear strength under monotonic loads Vmon.

Therefore, at least two nominally identical specimens tested under both monotonic and cyclic loads would be needed to identify the aforementioned low-cycle fatigue curve: Vmon could be determined from the former, whereas the cyclic test performed with action Vi < Vmon could be considered for estimating the corresponding number of m from Eq. (6) with k = 1. This condition is not achieved for any tests in the selected database, which only collects cyclic tests. Hence, an alternative estimation of the monotonic shear strength is required for deriving the fatigue curve.

Based on the comprehensive assessment and comparison proposed by Lima et al. (2012a), the model by Kim et al. (2009) exhibited the highest accuracy and reliability among those considered in a comparative study (Lima et al. 2012b): hence, that model is employed herein for determining Vmon based on the geometric and mechanical properties available for each RC joints mentioned within the database.

Therefore, once the monotonic shear strength is estimated, Eq. (6) can be solved for determining the parameter m, as described in the following Sect. 3.

3 Evaluation of the low-cycle fatigue curves

3.1 Shear strength under monotonic actions

As already mentioned in Sect. 2, the empirical model by Kim et al. (2009) is employed hereafter for estimating the monotonic shear strength Vmon of RC joints under consideration which were tested under cyclic conditions. It was originally developed on the basis of a wide database collecting experimental tests performed on both interior and exterior joint. A first version of the model (Kim et al. 2007) took into account joints with shear reinforcement only, while shear strength equal to zero was provided for unreinforced joints. Two years later, Kim et al. (2009) recalibrated the model on the basis of a database wider than the one used in 2007 including unreinforced joints in order to extend the formulation to the case of joints without stirrups.

According to this proposal, the shear strength of reinforced concrete joints can be evaluated by multiplying the specific shear strength vjh for the geometric dimension of the joint panel:

$$V_{jh} = v_{jh} \cdot b_{j} \cdot h_{c} ,$$
(7)

in which hc is the height of the column cross section and bj is the effective width of the joint panel which is provided by the following relationship:

$$b_{j} = \hbox{min} \left( {b_{c} \;;\;\frac{{b_{c} + b_{b} }}{2}} \right) ,$$
(8)

where bc and bb are the width of the column and the beam, respectively.

The joint specific shear strength vjh is evaluated as follows:

$$v_{jh} = \alpha_{t} \cdot \beta_{t} \cdot \eta_{t} \cdot \lambda_{t} \cdot \left( {JI} \right)^{0.15} \cdot \left( {BI} \right)^{0.30} \cdot f_{c}^{0.75} ,$$
(9)

in which αt is a parameter used for describing the in-plane geometry of the beam-to-column joint: it is assumed equal to the unit for interior joints, 0.7 for exterior ones and 0.4 for knee connections (characterised by the absence of the top column). Moreover, the βt coefficient describes the out-of-plane geometry and is equal to 1.0 for joints without transversal beams or 1.18 otherwise, while λt = 1.31 is fixed by the authors (Kim et al. 2009) in order to correlate values of shear strength obtained through Eq. (9) to the ones observed in experimental tests collected in the considered experimental database. The concrete compressive strength is denoted as fc, while other quantities of Eq. (9) are defined in the following:

$$\eta_{t} = \left( {1 - \frac{{e_{b} }}{{b_{c} }}} \right)^{0.67} ,$$
(10)
$$BI = \frac{{\rho_{b} \cdot f_{yb} }}{{f_{c} }} ,$$
(11)
$$JI = \frac{{\rho_{j} \cdot f_{yj} }}{{f_{c} }} \ge 0.0139 ,$$
(12)

where fyb and fyj denote the yield stress of longitudinal bars in the beams and horizontal stirrups within the joint, respectively. The limitation above 0.0139 to JI was introduced later with the aim to cover the case of unreinforced joints (Kim et al. 2009).

The beam reinforcement ratio ρb and the volumetric joint transverse reinforcement ratio ρj introduced in Eqs. (11) and (12) are defined as follows:

$$\rho_{b} = \frac{{A_{sb,\sup } + A_{sb,\inf } }}{{b_{b} \cdot h_{c} }} ,$$
(13)
$$\rho_{j} = \frac{{A_{sjh} \cdot h_{c} }}{{h_{c} \cdot b_{c} \cdot \left( {h_{b} - 2d'} \right)}} ,$$
(14)

where Asb,sup e Asb,inf are the longitudinal reinforcement at the top and bottom of the beam, respectively, Asjh is the area of horizontal stirrups in the joint panel, bc, hc, bb and hb are the geometric dimensions of the cross sections of the beam and the column and d’ is the thickness of the concrete which cover steel bars.

Finally, the parameter ηt [Eq. (10)] takes into account the eccentricity eb between the beam and the column axes, while BI and JI are two parameters which consider the effects of the amounts of steel reinforcement in the beam [Eq. (11)] and the joint panel [Eq. (12)], respectively.

3.2 The experimental database

The behaviour of beam-to-column joints under cyclic loading is highly influenced by several seismic details including the amount of shear reinforcement and the anchorage of rebars in beam. The present study is based on a subset of the database collected as part of the first Author’s Ph.D. Thesis (Lima 2012). The database is partitioned between interior and exterior joints and a further distinction between unreinforced and reinforced members. Test reports providing detailed information about geometric dimensions, materials and the load history are selected within the wide database of experimental tests. In particular, 24 interior joint specimens are selected and analysed, 4 of which were unreinforced and 20 were internally reinforced by stirrups (Table 1). As for exterior joints, 31 experimental tests are selected, collecting 3 and 28 tests on unreinforced and reinforced joint specimens (Table 2).

Table 1 Database: reports about experimental tests on interior joints
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Table 2 Database: reports about experimental tests on exterior joints
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Both tables report the reference to the source article, the original identification code (ID) of each specimen and information about whether the joint is reinforced (R) or not (U): in the case of reinforced joints the tables also clarify whether the amount of steel stirrups complies or not with EC8 provisions (CEN 2004). Moreover, the observed failure mode is also mentioned in the tables, which can be either:

  • Shear failure of the joint with beam and columns in the elastic range (J);

  • Shear failure of the joint after formation of a plastic hinge in the beam (BJ);

  • Shear failure of the joint after formation of a plastic hinge in the column (CJ).

Finally, further aspects about structural details are highlighted for each specimen and described at the bottom of both tables.

The scientific literature collects a limited number of experimental tests performed under cyclic conditions, especially about unreinforced joints. As a matter of fact, this is due to both the low displacement capacity and the significant strength degradation of such subassemblies, which makes difficult to perform tests under loading reversals.

The number of cycles and the shear force at each cycle are derived from the graphs reporting the load–displacement relationship of the specimen considered. The parameter m of each test is evaluated through [Eq. (6)]. Conventionally, failure is attained as soon as a strength decay equal to 20% is observed.

As an example, and Table 3 report the processing of experimental results of the interior joint labelled X1 tested by Durrani and Wight (1982) and described in Fig. 4.

Table 3 Processing experimental results of X1 specimen
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Fig. 4
figure 4

Geometry and structural details of specimen X1 tested by Durrani and Wight (1982)

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Specifically, the sequence of operations performed for evaluating the parameters n1 and N1 needed to back-calculate the exponent m are depicted in Fig. 5. The first complete cycle n1 ranges between 6.35 and − 5.08 mm (Fig. 5) which corresponds to the shear strength V1 = 212.67 kN.

Fig. 5
figure 5

Counting the number of cycles for estimating the damage parameter

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After the first cycle (i = 1), Eq. (6) can be applied obtaining a value of damage D lower than the unit and dependent from the parameter m only (Table 3). The second cycle shows a shear force V2 equal to 992.47 kN, while at the third one V3 = 1016.10 kN is recorded. Also, for the second (i = 2) and third (i = 3) cycles Eq. (6) is applied and the corresponding damage is obtained as a function of the unknown parameter m. The processing operations described above are replicated until the conventional failure (strength decay equal to 20%).

Finally, the parameter m is back-calculated by solving Eq. 6: the result m = 20.58 is actually obtained.

3.3 Interior joints

The procedure described above at the end of Sect. 3.2 is applied to the interior joints collected in the database. Tables 4 and 5 report the m values determined for unreinforced and reinforced interior joints, respectively. The analysis of the experimental results shows that the mean value of the parameter m governing the low-cycle fatigue of unreinforced interior joints (Table 4) is equal to 4.88 with a corresponding value of the standard deviation σ(m) = 1.28.

Table 4 Parameter m evaluated for unreinforced interior joints
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Table 5 Parameter m evaluated for reinforced interior joints
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Figure 6 depicts (in grey) the low-cycle fatigue curves derived for each specimen about unreinforced interior joints. The following equation describes the low-cycle fatigue curves:

$$\frac{V}{{V_{mon} }} = N^{{ - \frac{1}{m}}} .$$
(15)
Fig. 6
figure 6

Low-cycle fatigue curves determined for unreinforced interior joints

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The thick black line in Fig. 6 represents the mean curve associated to m = 4.88.

The mean value of the parameter m for reinforced interior joints is evaluated by considering only 20 cyclic tests and neglecting the two tests providing the maximum and minimum values of m, respectively (Table 5). Specifically, the mean value m = 6.08 is obtained for the 18 processed tests and the corresponding standard deviation is σ(m) = 2.03.

Figure 7 shows the low-cycle fatigue curves derived from the 18 experimental tests under consideration (in grey) and the mean curve related to reinforced interior joints. It is worth observing that the average curve determined for unreinforced joints is slightly steeper than the one obtained for reinforced ones, which means that, as expected, the latter are less prone to strength degradation than the former.

Fig. 7
figure 7

Low-cycle fatigue curves determined for reinforced interior joints

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3.4 Exterior joints

Tables 6 and 7 report the values of the parameter m derived from experimental tests performed on unreinforced and reinforced exterior joints, respectively.

Table 6 Parameter m evaluated for unreinforced exterior joints
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Table 7 Parameter m evaluated for reinforced exterior joints
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The processing of experimental results provides a mean value of the parameter m associated to unreinforced exterior joints equal to 3.85 with a standard deviation σ(m) = 1.29. Figure 8 shows the low-cycle fatigue curves of the three unreinforced exterior joints under consideration (in grey) and the mean curve (thick line).

Fig. 8
figure 8

Low-cycle fatigue curves evaluated for unreinforced exterior joints

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The mean value of m evaluated for reinforced exterior joints (Table 7) is equal to 7.58, while its standard deviation is σ(m) = 2.92.

Figure 9 depicts the low-cycle fatigue curves derived for reinforced exterior joints (in grey). The results are characterised by high value of dispersion if compared with the ones obtained for interior joints.

Fig. 9
figure 9

Low-cycle fatigue curves determined for reinforced exterior joints

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Finally, also for exterior joints the low-cycle fatigue curve obtained for unreinforced joints is steeper than the one for reinforced joints. In this case the difference between the two curves is slightly more pronounced than in the case of interior joints, as shear strength in the former rely more on stirrups than in the latter, which are confined by the two adjacent beams.

4 Results and comparisons

Figure 10 shows the comparison between the mean low-cycle fatigue curves obtained for both unreinforced and reinforced interior joints. Unreinforced beam-to-column joints are characterised by degradation faster than the one observed in reinforced interior connections. The standard deviations evaluated in the previous section equal to 1.28 and 2.03 for unreinforced and reinforced joints, respectively.

Fig. 10
figure 10

Low-cycle fatigue curves for interior joints: comparison between unreinforced and reinforced ones

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The comparison reported in Fig. 10 shows a limited different behaviour between unreinforced and reinforced joints demonstrating that, for interior ones, beams provide a significantly beneficial confinement effects and, consequently, the presence of horizontal stirrups within the panel zone does play a significant role in reducing the strength degradation under cyclic actions. The standard deviations evaluated in the previous section equal to 1.28 and 2.03 for unreinforced and reinforced joints, respectively.

Conversely, as it might have been expected, in exterior joints the absence of horizontal reinforcements in the panel zone leads to a significant difference in terms of damage evolution under cyclic actions (Fig. 11). This apparent difference is “measured” by two significantly different values of the average m determined for unreinforced and reinforced external RC joints. Specifically, unreinforced joints exhibit an average value of m equal 3.85 with a standard deviation σ(m) = 1.29, whereas reinforced joints lead to an average m equal to 7.58 with standard deviation σ(m) = 2.92.

Fig. 11
figure 11

Low-cycle fatigue curves for exterior joints: comparison between unreinforced and reinforced ones

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Finally, Fig. 12 shows that further differences emerge in the mean fatigue curves determined for EC8-compliant and non-compliant reinforced joints (Table 2). Specifically, the slope of the fatigue curve (and, hence, the proneness to damage under cyclic actions) is lower in EC8-compliant joints (m = 10.51) than in reinforced and non-compliant joints (m = 6.77).

Fig. 12
figure 12

Low-cycle fatigue curves for reinforced exterior joints: difference between EC8 compliant and under-designed joints

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5 Conclusions

This study proposes an original interpretation of cyclic behaviour of RC joints. Specifically, it follows an approach based on the Theory of Low-Cycle Fatigue for interpreting the reduction in shear strength observed in experimental tests under cyclic actions. A significant number of experimental tests taken from the scientific literature have been processed with the aim to determine the number and amplitude of cycles leading them to fail in shear. More specifically, both interior and exterior joints are considered and the response of both unreinforced and reinforced joints are analysed separately. The following conclusions are worthy to be highlighted:

  • the different responses observed in the experimental tests considered for interior and exterior joints, and for unreinforced and reinforced, ones is clearly caught by the proposed interpretation in terms of low-cycle fatigue curve;

  • in fact, the influence of reinforcement in terms of the estimated m value is more apparent for exterior joint than for interior ones, as in the former stirrups are the only elements capable to realise a confining effect resulting in a lower strength degradation;

  • furthermore, as expected, the amount of reinforcement plays a role in the resulting proneness to damage under cyclic actions: the slope of the resulting mean fatigue curves is slightly lower for EC8-compliant joints than for under designed ones.

Finally, far from proving readers with the final calibration of fatigue curves for the considered classes of joints (which would require a much wider number of well-documented experimental results), this study demonstrates that the degradation in strength of beam-to-column joints induced by cyclic actions can be regarded within the general framework of low-cycle fatigue theory. Further studies are needed to better calibrate the fatigue curves of interior and exterior joints, taking into account several relevant aspects, such as the amount and structural detailing of rebars and the properties of materials.