# Low-Damage Friction Connections in Hybrid Joints of Frames of Reinforced-Concrete Buildings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Beam-to-Column Connections

_{d}has to be known. In this case, the feasibility study is conducted assuming M

_{d}= 110 kNm. If we assume an overstrength factor of 1.5, then the positive moment strength of the beam that belongs to the sub-assemblage results in M

_{Rd}= 165 kNm.

_{i}:

_{s,Rd,i}, can be calculated as:

_{res,i}, n

_{bi}, and f

_{ub,i}are the transversal area and number of bolts, and their ultimate strength, respectively, while n

_{s}and μ are the first, the number of planes where friction develops and the second, the slip factor. It is noteworthy to remark that this feasibility study is conducted by adopting metal-to-metal contact as first approach. However, it is known that the international code ASCE07-22 [24] does not allow this type of contact in friction dampers; conversely, the introduction of friction pads is prescribed. In this regard, the authors are currently developing further insights for analyzing best-performing friction materials. Some preliminary results can be found in Pagnotta et al. [25].

_{s}= 2 and μ = 0.4. Moreover, it has to be noted that Equation (3) is the function of the preloading force of each bolt, F

_{pc,i}= 0.7·f

_{ub,i}·A

_{res,i}.

_{s,i}can be introduced for setting the stress level of each preloaded bolt [26]. If t

_{s,i}is assumed as the ratio between the sliding force reported in Equation (1) and the code-compliant slip resistance of Equation (2), then the effective preload design value is:

#### 2.1. BCC-1: Device with Crossed-Slotted Holes

_{1}is equal to 399 mm, while the angle α

_{1}between the sliding-force application point and the axis of the beam measures 61°.

^{2}for the beam and 300 × 400 mm

^{2}for the column are assumed. In the joint model, the structural elements’ length is 2500 mm for the beam and 3000 mm for the column. The latter is pinned at the ends, while a concentrated vertical force applies a cyclic displacement to the tip of the beam. The FE type used for the concrete parts is the first-order hexahedron named C3D8R, while the steel elements which constitute the connection are modeled with linear tetrahedra, type C3D4. This FE model is conducted under the simplified hypotheses of elastic behavior of all components and by neglecting the steel reinforcement of beam and column, thus, focusing only on the mechanism of the friction device. To this scope, the ideal fixed encastre is modelled for connecting the damping device to both beam and column while realistic constraints and interaction properties are simulated for the mechanism of the main top pin and for the sliding plates. A multi-point constraint of type “beam” is assumed between the longitudinal pin axes and the internal surfaces of the holes, while a friction mechanism is introduced in the sliding plates, with a friction coefficient of 0.4. Rigid normal contacts are also modeled between the bolt shanks and the internal surfaces of both circular and slotted holes.

#### 2.2. BCC-2: Curved-Slotted Hole System

_{2}= 380 mm, and the angle α

_{2}= 68°. The FE types used in this model are first-order tetrahedra (C3D4) for the steel elements of the connection, with the exception of the C-profile used for connecting the upper T-stub to the column. This C-profile is modeled with linear hexahedra (C3D8R), as well as the concrete column. Moreover, two-node linear beam elements are used for modelling the HSTCB reinforcement (longitudinal and diagonal rebars, and vertical and inclined stirrups). The FE model developed is more detailed because it takes into account the nonlinear behavior of the materials [17]. The main mechanical parameters of concrete are: f

_{c}= 25 MPa (compressive strength), E

_{0}= 28,960 MPa (elastic modulus), f

_{t}= 2.56 MPa (tensile strength), and G

_{F}= 0.13 N/mm (fracture energy). The modelled steel is class B450C for rebars embedded in concrete and S355 for plates, with elastic modulus E

_{s}= 210,000 MPa. The constitutive model of steel considers an elastic perfectly plastic behavior. Moreover, the FE model considers a perfect bond between the concrete core and the steel truss and reinforcement of the beam. The other main contact properties are assumed as follows: frictionless sliding between the bottom plate of the beam and the concrete core; frictional behaviour between the sliding steel plates, with a friction coefficient equal to 0.4; and ideal rigid contacts in all normal local directions. The results of a cyclic analysis with imposed displacement of ±100 mm are reported in Figure 2b; it can be noted that this device does not present the negative sliding effect previously due to the clearance in the top hinge; here, the center of rotation develops near the weakest cross-section of the T-stub (Figure 1c), allowing a major dissipation capacity, even though a slight shifting of the rotation center is noticed during the analysis. The responses for the positive and negative bending moments are almost symmetric, and limited damage is experienced during the loading–unloading cycles.

#### 2.3. BCC-3: Curved-Slotted Hole Device Combined with Perfobond Connectors

_{3}= 374 mm and α

_{3}= 68°. In this model, the HSTCB is modeled with linear tetrahedra (C3D4) while all other FE types are solid linear bricks (C3D8R) and two-node beams are used for the steel reinforcement of the column. Cohesive interaction is used for modeling steel-to-concrete bond in the beam according to the analytical model proposed by Eligehausen et al. [28], while a perfect bond is assumed in the RC column. Frictional behavior is set for the damping device components, adopting the same parameters used in BCC-2 model.

_{c}= 30.31 MPa and the Young’s modulus E

_{c}= 30,683 Mpa. The steel elements are modelled adopting a multi-linear behaviour with hardening phases and material rupture [29,30]. Three cyclic analyses are performed for 0.5 M

_{d}, M

_{d}, and M

_{rd}= 1.5 M

_{d}, defining cycles with amplitude ±Θ

_{y}, ±4Θ

_{y}, and ±Θ

_{max}= ±7Θ

_{y}, the yielding chord rotation being Θ

_{y}= 7 mrad. This loading scheme is chosen according to the code ACI 374.2R-13 [31] with the aim of achieving multiples of the yielding chord rotation. However, only the three aforementioned amplitudes have been adopted in order to contain the computational time demand of the analysis. The results are reported in Figure 3b. It can be observed that the stiffer top connection is able to prevent the shifting of the rotation center, allowing more stable hysteresis cycles. The cycles are wide and proportional to the different values of preload in the bolts. Figure 3c reports the strain state at the maximum rotation, for the cycles developed at M

_{d}value. The figure shows that the steel elements of the friction damper are elastic, with localized plastic strain concentrations at the end of the intersection between the bottom plate of the truss and the sliding holed plate. The upper T-stub assumes higher inelastic strain values in the section devoted to develop the rotation center of the system.

## 3. Column-Base Connections

## 4. Seismic Analysis of a Two-Storey MRF

^{2}and it is reinforced with 10 20 mm diameter rebars. The HSTCBs have an upper chord comprised of three rebars with diameter of 16 mm, a lower chord made with a 5 mm thick steel plate and diagonal rebars of 12 mm of diameter and 300 mm spacing. The HSTCBs are slab-thick beams with cross-section of 300 × 250 mm

^{2}. First of all, a traditional frame (TF) is modeled as a benchmark; then, the innovative frame (IF) is modelled by introducing BCC-2 at joints. Conversely, the behavior of the frame endowed with BCC-3 is still under investigation by the authors. Furthermore, the performance of the IF is also investigated in the presence of CBC-1; this frame is indicated as an innovative frame with threaded bars and disk springs (IF-TB). Finally, the effects of CBC-2 are analysed, and the frame is indicated as an innovative frame endowed with threaded bars and disk springs with friction devices (IF-FD). Non-linear time history analyses (NLTHAs) are performed adopting 30 artificial accelerograms [34] whose duration is 30 s, whit a strong motion phase of 20 s. The generated accelerograms are response-spectrum-compatible and they are obtained through the process proposed initially by Vanmarcke and Gasparini [35] and successively modified by Cacciola et al. [36], who adopted the modulating function by Jennings et al. [37]. As an example, Figure 5 reports one of the generated artificial accelerograms adopted for the analysis. At the end of the seismic action, five more seconds of free oscillation is added to the analysis for assessing the residual drift. Moreover, the intensity of the adopted spectrum-compatible accelerograms was amplified with a scale factor of 1.7 in order to produce an extensive cracking of the panel zones and the loss of bond of the steel reinforcement, other than the development of plastic hinges. The behavior factor adopted is q = 3. The vertical loads applied to the frame consist of 14.5 kN/m

^{2}of dead and live loads while 38 kN/m are added for considering the seismic masses due to the transversal frames of a spatial structure.

#### 4.1. Model of the Traditional RC Frame Panel Zone

#### 4.2. Implementation of Friction BCCs

_{i}= 10

^{6}kNm/rad; yielding moment M

_{y}= 110 kNm; and post-yield stiffness ratio compared to the elastic stiffness r = 10

^{−4}.

#### 4.3. Modeling of the Re-Centering Systems

_{d}

^{FD}, design value of bending moment; M

_{0}, decompression moment given by two contributions, i.e., the threaded bars and the axial force; M

_{1}, decompression moment provided by the friction pads, if present; M

_{u}

^{FD}, value of the ultimate bending moment; Θ

_{u}, ultimate rotation of the system; k

_{Θ,1}

^{FD}, initial rotational stiffness; and k

_{Θ,2}

^{FD}, stiffness after the attainment of M

_{d}

^{FD}.

_{d}

^{FD}= 200 kNm (i.e., 75% the column’s bending moment strength); M

_{0}= 120 kNm (i.e., 60% the total bending strength value); M

_{1}= 80 kNm (moment strength given by the flange and the web friction pads); M

_{u}

^{FD}= 290 kNm; Θ

_{u}= 40 mrad; and k

_{Θ,1}

^{FD}= 10

^{6}kNm/rad; k

_{Θ,2}

^{FD}= 2261 kNm/rad.

_{d}

^{TB}= 120 kNm, which is given by the threaded bars and the axial force; k

_{Θ,2}

^{TB}= k

_{Θ,2}

^{FD}= 2261 kNm/rad; and M

_{u}

^{TB}= 210 kNm.

## 5. Discussion of the Results

#### 5.1. Comparison in Capacity from Push-Over Analysis

_{TF,max}≈ 0.65 in the graph. For drift values higher than 5%, the friction devices attain their ultimate rotation, which is consistent with the design provisions.

#### 5.2. Global and Local Response from NLTHAs

## 6. Conclusions

## 7. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Solution BCC-1: (

**a**) detail of the system; (

**b**) cyclic response; and (

**c**) stress state at peak.

**Figure 2.**Solution BCC-2: (

**a**) detail of the system; (

**b**) cyclic response; and (

**c**) plastic deformations at peak.

**Figure 3.**Solution BCC-3: (

**a**) detail of the system; (

**b**) cyclic response; and (

**c**) plastic strains at maximum rotation for M

_{d}= 165 kNm.

**Figure 10.**Moment-rotation curves of IF vs. TF: (

**a**) beam end section; (

**b**) column base section; (

**c**) panel zone; and (

**d**) friction device BCC-2.

**Table 1.**Design sliding force, stress-level coefficient, and effective preload for the proposed BCCs.

Connection | F_{d,i} (kN) | t_{s,i} (%) | F_{pc,d,i} (kN) |
---|---|---|---|

BCC-1 | 275.7 | 52.3 | 57.5 |

BCC-2 | 289.5 | 53.8 | 72.3 |

BCC-3 | 294.1 | 42.9 | 73.5 |

**Table 2.**Analysis parameters for modelling the panel zone and the bar-slip phenomenon of the traditional frame.

Panel Zone | Bar Slip | |||
---|---|---|---|---|

Backbone curve parameters | Initial rotational stiffness | 245,000 (kNm/rad) | Elastic stiffness | 65,000 (kNm/rad) |

Cracking moment | 49 (kNm) | Yield moment | 174/−169 (kNm) | |

Yield moment | 298 (kNm) | Rotation at peak strength | 0.02/−0.02 (rad) | |

Yield rotation | 0.006 (rad) | Peak moment strength | 184/−179 (kNm) | |

Ultimate rotation | 0.2 (rad) | Residual moment strength | 1.86/−1.73 (kNm) | |

Post-yield stiffness ratio | 0.001 (-) | |||

Hysteresis shape parameters | Stiffness degradation | 4 (-) | Pinching factor | 0.5 (-) |

Ductility-based strength decay | 0.6 (-) | Deterioration factor | 0.6 (-) | |

Hysteretic energy-based strength decay | 0.6 (-) | |||

Slip parameter | 0.5 (-) |

Panel Zone | ||
---|---|---|

Backbone curveparameters | Initial rotational stiffness | 330,000 (kNm/rad) |

Cracking moment | 66 (kNm) | |

Yield moment | 406 (kNm) | |

Yield rotation | 0.006 (rad) | |

Ultimate rotation | 0.2 (rad) | |

Post-yield stiffness ratio | 0.001 (-) | |

Hysteresis shapeparameters | Stiffness degradation | 4 (-) |

Ductility-based strength decay | 0.6 (-) | |

Hysteretic energy-based strength decay | 0.6 (-) | |

Slip parameter | 0.5 (-) |

MIDR | RIDR | ||||||||
---|---|---|---|---|---|---|---|---|---|

TF (%) | IF/TF | IF-TB/TF | IF-FD/TF | TF (%) | IF/TF | IF-TB/TF | IF-FD/TF | ||

First storey | avg. | 2.72 | 0.94 | 1.17 | 0.94 | 0.17 | 2.76 | 0.59 | 0.53 |

CV (%) | 13.27 | 18.14 | 12.61 | 13.86 | 77.99 | 79.74 | 86.60 | 70.89 | |

Second storey | avg. | 2.91 | 0.98 | 1.07 | 0.91 | 0.14 | 4.21 | 1.71 | 1.36 |

CV (%) | 11.16 | 16.58 | 17.04 | 17.91 | 78.71 | 78.15 | 79.57 | 76.96 |

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**MDPI and ACS Style**

Colajanni, P.; La Mendola, L.; Monaco, A.; Pagnotta, S.
Low-Damage Friction Connections in Hybrid Joints of Frames of Reinforced-Concrete Buildings. *Appl. Sci.* **2023**, *13*, 7876.
https://doi.org/10.3390/app13137876

**AMA Style**

Colajanni P, La Mendola L, Monaco A, Pagnotta S.
Low-Damage Friction Connections in Hybrid Joints of Frames of Reinforced-Concrete Buildings. *Applied Sciences*. 2023; 13(13):7876.
https://doi.org/10.3390/app13137876

**Chicago/Turabian Style**

Colajanni, Piero, Lidia La Mendola, Alessia Monaco, and Salvatore Pagnotta.
2023. "Low-Damage Friction Connections in Hybrid Joints of Frames of Reinforced-Concrete Buildings" *Applied Sciences* 13, no. 13: 7876.
https://doi.org/10.3390/app13137876