# Cyclic Behavior of Hollow Section Beam–Column Moment Connection: Experimental and Numerical Study

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## Abstract

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## 1. Introduction

^{®}ConXL

^{TM}moment connection using a finite element model (FEM). The behavior of concrete filled hollow columns was examined. The results obtained showed a satisfactory performance according to requirements in [14]. In this sense, different ConXtech

^{®}ConXL

^{TM}configurations of the connection without concrete filling in the column were tested obtaining a ductile failure mode in a beam for axial load less than 0.4 the capacity of the column. However, non-ductile failure modes were obtained with increases in the axial load level.

## 2. Hollow Structural Section (HSS) Moment Connection

#### 2.1. Description of HSS Moment Connection

#### 2.2. Proposed End-Plate Design

**s**and equated to zero, as follows:

_{p}) can be defined by Equation (10).

_{f}, replacing mp (plastic moment) by the plastic moment capacity of a unit length of plate, and applying the strength factor ϕ

_{d}, Equation (12) can be obtained. The expression to calculate the required end-plate thickness (t

_{p}) given by Equation (14) is found,

_{f}, where (1.11) is a factor to avoid the prying in the end-plate. Additionally, applying the strength factor ϕ

_{d}to the nominal strength, the flexural strength given by Equation (15) is obtained,

## 3. Experimental Study

#### Experimental Results

_{u}/θ

_{y}(θ

_{u}, is the rotation capacity at the point where the flexural strength has dropped to 0.8 Mp and θ

_{y}, is the value corresponding to the intersection between the line representing the initial stiffness and a horizontal line located in the point of maximum resistance). From this criteria, ductilities of μ

_{test1}= 1.5, μ

_{test2}= 1.48 and μ

_{test3}= 1.54 were obtained in test 1, test 2 and test 3 respectively.

_{eq}= E

_{d}/4πE

_{so}, where E

_{d}is the dissipated energy and E

_{so}is the strain energy, as defined in [29]. Limited dissipation energy was obtained in tests performed in comparison to tests reported in [16], despite having similar dimensions and columns in both studies. Furthermore, the equivalent damping values were lower than 2.5% at 0.02 [rad]. Values greater than 10% at 0.04 [rad] were reached; however, these values are not representative for their use in seismic design of steel structures because they were calculated for large levels of deformation.

## 4. Numerical Study

#### 4.1. General Characteristics of the Numerical Model

- Length of the column is equal to inflection points at mid-height of each story.
- Washers are not included in the model for simplicity, considering that inelastic incursion is manifesting in the beam exclusively [16].
- Diameter of the bolt holes is assumed to be equal to the diameter of the bolts, avoiding rigid body movements that could affect the convergence of the model [16].

#### 4.2. Element Type and Mesh

#### 4.3. Boundary Conditions and Loading

#### 4.4. Material Modeling

#### 4.5. Results of Numerical Model

## 5. Cyclic Response of HSS Moment Connection with Other Configurations

_{p}= F

_{y}× Z

_{x}= 43.42 [kN·m]), maximum moment and maximum rotation reached in tests and numerical model are reported. Tests and the FE model reached a flexural resistance greater than nominal plastic moment according to [26,28]. In Figure 16, the normalized moment-rotation curves of the experimental and FE model are compared, obtaining an acceptable level of adjustment. Therefore, from the calibrated numerical model, other configurations with different sizes of beams and columns were studied.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Definition |

b_{f} | Flange width of the beam |

bp | Width of end-plate |

d | Overall depth of beam |

d_{e} | Column bolt edge distance |

F_{y} | Specified minimum yield stress of the yielding element |

g_{1} | Horizontal distance (gage) between fastener lines |

g_{2} | Horizontal distance (gage) between fastener lines below cover plate |

h_{i} | Distance from centerline of compression flange to the centerline of the ith tension bolt row |

h_{o} | Distance from centerline of compression flange to the tension-side outer bolt row in EP-HSS moment connection |

l_{i} | length of yield line |

m_{p} | plastic momento of beam |

m_{pi} | plastic moment internal of beam |

M | Moment obtained |

M_{f} | Probable maximum moment at face of column |

M_{n} | Nominal flexural strength of beam |

M_{p} | Plastic moment of beam |

p_{fi} | Vertical distance from inside of a beam tension flange to nearest inside bolt row |

p_{fo} | Vertical distance from inside of a beam tension flange to nearest outside bolt row |

s | Distance from centerline of most inside or most outside tension bolt row to the edge of a yield line pattern |

t_{p} | Thickness of end-plate |

W_{E} | External work |

W_{i} | Internal work |

Y_{p} | End-plate yield line mechanism parameter |

δ_{1} | Virtual displacement for yield line 2 |

δ_{2} | Virtual displacement for yield line 5 |

δ_{3} | Virtual displacement for yield line 3, 7 and 8 |

ε_{u} | Ultimate deformation |

ε_{y} | Yielding deformation |

ϕ_{d} | Resistance factor for ductile limit states |

θ | Rotation angle due to moment of beam |

θ_{i} | Rotation internal |

σ_{u} | Ultimate stress |

σ_{y} | Yielding stress |

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**Figure 2.**(

**a**) View of end-plate yield line mechanisms of HSS moment connection, (

**b**) lateral view of connection to calculate the conditions of plastic moment and deformation using the virtual work performed.

**Figure 3.**(

**a**) Dimensions of end-plate [mm], (

**b**) lateral view of the HSS moment connection [mm], (

**c**) plan view of HSS moment connection [mm].

**Figure 4.**(

**a**) Schematic view of test assembly, units in [mm] and (

**b**) location of linear variable differential transformer (LVDT) in test assembly.

**Figure 5.**(

**a**) Load vs. displacement hysteresis curve of test 1, (

**b**) view of damage in test 1, (

**c**) load vs. displacement hysteresis curve of test 2, (

**d**) view of damage in test 2, (

**e**) load vs. displacement hysteresis curve of test 3, (

**f**) view of damage in test 3.

**Figure 6.**Normalized moment rotation hysteresis curves: (

**a**) test 1, (

**b**) test 2 and (

**c**) test 3. Note: Mmax, is the maximum moment reached in the test and Mp, is the plastic moment.

**Figure 8.**(

**a**) Secant stiffness vs. rotation curve of test 1, (

**b**) Secant stiffness vs. rotation curve of test 2, (

**c**) Secant stiffness vs. rotation curve of test 3.

**Figure 9.**(

**a**) Dissipated energy vs. rotation curve of test 1, (

**b**) equivalent damping vs. rotation curve of test 1, (

**c**) dissipated energy vs. rotation curve of test 2, (

**d**) equivalent damping vs. rotation curve of test 2, (

**e**) dissipated energy vs. rotation curve of test 3, (

**f**) equivalent damping vs. rotation curve of test 3.

**Figure 12.**Relation stress-strain of materials used in experimental and numerical study: (

**a**) ASTM-A-36 material (stiffeners and end-plates), (

**b**) ASTM-A-500 Gr. C material (beam), (

**c**) ASTM-A-500 Gr. B material (column) and (

**d**) ASTM-A-325 material (bolts).

**Figure 13.**(

**a**) Distribution of von Mises stress and (

**b**) plastic strain, at maximum displacement of numerical model.

**Figure 14.**(

**a**) Load vs. displacement curve, (

**b**) normalized moment vs. rotation curve in numerical model.

**Figure 15.**(

**a**) Secant stiffness vs. rotation curve, (

**b**) dissipated energy vs. rotation curve, (

**c**) equivalent damping vs. rotation in numerical model.

**Figure 17.**Equivalent stress distribution: (

**a**) P1 model, (

**c**) P2 model, (

**e**) P3 model, (

**g**) P4 model, (

**i**) P5 model, (

**k**) P6 model, (

**m**) P7 model, (

**o**) P8 model, (

**q**) P9 model, and Plastic strain distribution: (

**b**) P1 model, (

**d**) P2 model, (

**f**) P3 model, (

**h**) P4 model, (

**j**) P5 model, (

**l**) P6 model, (

**n**) P7 model, (

**p**) P8 model, (

**r**) P9 model.

**Figure 18.**Normalized moment vs. rotation curves: (

**a**) P1 model, (

**b**) P2 model, (

**c**) P3 model, (

**d**) P4 model, (

**e**) P5 model, (

**f**) P6 model, (

**g**) P7 model, (

**h**) P8 model, (

**i**) P9 model.

Yield Line | Length (li) | Rotation (θi) |
---|---|---|

1 | ${b}_{p}$ | $\theta $ |

2 | $\left({b}_{p}-{b}_{f}\right)/2$ | $\theta \left({h}_{i}/s\right)$ |

3 | ${p}_{fi}+s$ | $2\theta \left(\raisebox{1ex}{$hi$}\!\left/ \!\raisebox{-1ex}{$\left(g2-bf\right)$}\right.\right)$ |

4 | $\left({b}_{p}-{b}_{f}\right)/2$ | $\theta \left(\frac{{h}_{i}}{{p}_{fi}}\right)$ |

5 | ${b}_{p}$ | $\theta \left(\frac{{h}_{o}}{{p}_{fo}}\right)$ |

6 | ${b}_{p}$ | $\theta \left(\frac{{h}_{o}}{{p}_{fo}}-1\right)$ |

7 | ${l}_{7}$ | $\frac{\theta {h}_{i}}{{l}_{7}}\left(\frac{{g}_{2}-bf}{2s}+\frac{2s}{{g}_{2}-bf}\right)$ |

8 | ${l}_{8}$ | $\frac{\theta {h}_{i}}{{l}_{8}}\left(\frac{{g}_{2}-bf}{2{p}_{fi}}+\frac{2{p}_{fi}}{{g}_{2}-bf}\right)$ |

9 | $\left({b}_{p}-{g}_{2}\right)/2$ | $\theta \left(\frac{{h}_{i}}{{p}_{fi}}+\frac{{h}_{i}}{s}\right)$ |

Element | Designation | Σy [MPa] | εy | σu [MPa] | εu |
---|---|---|---|---|---|

Stiffeners, End-plates | ASTM-A-36 | 380 | 0.0018 | 575 | 0.20 |

Beam 200 × 70 × 4.3 | ASTM-A-500 Gr. C | 450 | 0.0024 | 517 | 0.007 |

Column 220 × 220 × 9 | ASTM-A-500 Gr. B | 496 | 0.0025 | 597 | 0.01 |

Bolt | ASTM-A-325 | 634 | 0.0036 | 848 | 0.14 |

No. | No. of Cycles | Drift Angle (θ) [rad] |
---|---|---|

1 | 6 | 0.00375 |

2 | 6 | 0.005 |

3 | 6 | 0.0075 |

4 | 4 | 0.01 |

5 | 2 | 0.015 |

6 | 2 | 0.02 |

7 | 2 | 0.03 |

8 | 2 | 0.04 |

Specimen | L_{max} [kN] | D_{max} [mm] | M/M_{p} | M_{max} [kN.m] | R_{max} [rad] |
---|---|---|---|---|---|

Test 1 | 60.24 | 75.17 | 2.12 | 90.35 | 0.05 |

Test 2 | 62.25 | 75.18 | 2.20 | 93.37 | 0.05 |

Test 3 | 60.86 | 75.17 | 2.15 | 91.30 | 0.05 |

_{max}) Maximum load, (D

_{max}) Maximum displacement, (M/M

_{p}) Normalized Moment, (M

_{max}) Maximum moment obtained and (R

_{max}) Maximum rotation obtained.

Component | Number of Elements | Number of Nodes |
---|---|---|

End-plates | 4600 | 24,975 |

Column | 2960 | 19,128 |

Beam | 6264 | 43,411 |

Bolts | 10,619 | 18,948 |

Vertical Stiffener | 132 | 1042 |

Horizontal Stiffener | 2250 | 13,444 |

Elements Connection | Contact | Movement in Normal Direction | Movement in Tangential Direction |
---|---|---|---|

Column-Horizontal Stiffeners | Bonded | No separation | No slip |

Column-Vertical Stiffeners | Bonded | No separation | No slip |

Vertical Stiffeners-Horizontal Stiffeners | Bonded | No separation | No slip |

End-plate-Horizontal Stiffeners | Bonded | No separation | No slip |

End-plate-Vertical Stiffeners | Bonded | No separation | No slip |

End-plate-End-plate | Frictional | Separation allowed | Slip allowed |

Beam-End-plate, Bolt-Nut | Bonded | No separation | No slip |

Bolt- End-plate, Nut-End-plate | Frictionless | Separation allowed | Slip allowed |

Specimen | L_{max} [kN] | D_{max} [mm] | M/M_{p} | M_{max} [kN.m] | R_{max} [rad] |
---|---|---|---|---|---|

Test S-01 | 60.24 | 75.17 | 2.12 | 90.35 | 0.05 |

Test S-02 | 62.25 | 75.18 | 2.20 | 93.37 | 0.05 |

Test S-03 | 60.86 | 75.17 | 2.15 | 91.30 | 0.05 |

FEM Model | 61.18 | 90 | 2.16 | 91.77 | 0.06 |

_{max}) Maximum load, (D

_{max}) Maximum displacement, (M/M

_{p}) Normalized Moment, (M

_{max}) Maximum moment obtained and (R

_{max}) Maximum rotation obtained.

Numerical Model | Dimension of Column [mm] | Dimension of Beam [mm] |
---|---|---|

P0 | 220 × 220 × 9 | 200 × 70 × 4.3 |

P1 | 220 × 220 × 9 | 180 × 65 × 4 |

P2 | 220 × 220 × 9 | 220 × 90 × 4.5 |

P3 | 220 × 220 × 9 | 260 × 90 × 5.5 |

P4 | 220 × 220 × 9 | 300 × 100 × 5.5 |

P5 | 220 × 220 × 9 | 300 × 100 × 7 |

P6 | 260 × 260 × 11 | 320 × 120 × 7 |

P7 | 260 × 260 × 11 | 320 × 120 × 9 |

P8 | 260 × 260 × 11 | 350 × 170 × 9 |

P9 | 260 × 260 × 11 | 350 × 170 × 11 |

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**MDPI and ACS Style**

Nuñez, E.; Boainy, N.; González, F.; Torres, R.; Picón, R.; Guerrero, N.
Cyclic Behavior of Hollow Section Beam–Column Moment Connection: Experimental and Numerical Study. *Metals* **2020**, *10*, 1608.
https://doi.org/10.3390/met10121608

**AMA Style**

Nuñez E, Boainy N, González F, Torres R, Picón R, Guerrero N.
Cyclic Behavior of Hollow Section Beam–Column Moment Connection: Experimental and Numerical Study. *Metals*. 2020; 10(12):1608.
https://doi.org/10.3390/met10121608

**Chicago/Turabian Style**

Nuñez, Eduardo, Nwar Boainy, Freddy González, Ronald Torres, Ricardo Picón, and Néstor Guerrero.
2020. "Cyclic Behavior of Hollow Section Beam–Column Moment Connection: Experimental and Numerical Study" *Metals* 10, no. 12: 1608.
https://doi.org/10.3390/met10121608