# Substructure Hybrid Simulation Boundary Technique Based on Beam/Column Inflection Points

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Boundary Technique for the HS

_{m,1}and O

_{m,2}are inflection points of the m story; h

_{I,m}is the height from the inflection point to the roof of the m story; ${R}_{\mathrm{b}}$ is the applied force by the actuator at the boundary; and ${R}_{m}$ is the applied force by the actuator at the mth-story. Noting that the dynamics of the frame are represented in the computer and using the moment equilibrium equation, the axial force can be calculated by taking moments about point O

_{1}or O

_{2}. In the same way, the axial force of each story of the physical substructure can be calculated from the results of the upper story. Then, the axial forces of the physical substructure can be obtained. Subsequently, the prediction axial displacement at the boundary can be calculated using Equation (1), i.e.,

## 3. Implement of Boundary Technique

#### 3.1. HS Component Interaction

#### 3.2. Equivalent Force Control Method

_{EQ,i+1}, which is used for the equivalent force (EF) command, can be calculated by the information before the i + 1 step. Therefore, the F

_{EQ}

_{,i+1}is available and remains unchanged at i + 1 step. The F

_{EQ,i+1}, which is used for equivalent force (EF) feedback, is obtained by the three terms: (i) the reaction of the experimental substructure ${\mathit{R}}_{\mathrm{E}}\left[{\mathit{d}}_{{}_{\mathrm{E},i+1}}^{\mathrm{c}}\left(t\right)\right]$, which is measured by the load cells; (ii) the reaction of the numerical substructure ${\mathit{R}}_{\mathrm{N}}\left[{\mathit{d}}_{{}_{\mathrm{E},i+1}}^{\mathrm{c}}\left(t\right)\right]$, which is obtained from the results of the numerical simulation; and (iii) the pseudo-dynamic force, which is linearly related to displacement by

**K**

_{PD}. The displacement command, ${\mathit{d}}_{i+1}^{\mathrm{c}}\left(t\right)$, is determined from the force signal using the force-displacement conversion matrix

**C**

_{F}. The equivalent force controller is used to make certain that the EF feedback can track the EF command quickly without overshoot. This paper uses a PD controller as the equivalent force controller. The parameters of the PD controller are obtained as discussed in Reference [39].

#### 3.3. Basic Process

- Step 1: The initial stiffness matrix, the initial mass matrix, the initial damping matrix, other initial parameters, and the ground motion record are obtained. The ground motion record may be scaled in amplitude to accommodate the goal of different HS tests.
- Step 2: The EF command is calculated. The displacement command is then calculated using the force-displacement conversion matrix and the EF controller.
- Step 3: The pseudo-dynamic force is calculated using the pseudo-dynamic stiffness
**K**_{PD}, which can be determined from the initial parameters. Concurrently, the predicted axial deformation is obtained using Equation (1). Before the next step, all calculations are realized using Matlab. After this step, the calculated displacement command of the numerical substructure and the prediction axial deformation at the boundary is applied to the numerical substructure modeled by OpenSEES. - Step 4: The calculated displacement command of the numerical substructure and the prediction axial deformation at the boundary are sent to the numerical substructure. Subsequently, the reaction force at the boundary is calculated. Similarly, the data calculated by OpenSEES are sent to the Matlab program and subsequently wait to receive the data calculated by Matlab for the next step.
- Step 5: The calculated reaction force in the vertical direction and the displacement command in the horizontal direction are sent to the actuator controllers for the physical substructure. Then, the reaction of the physical substructure is obtained by: (i) the sensors in the associated loading devices; or (ii) OpenSEES for the virtual test.
- Step 6: The EF feedback is obtained by summing up the pseudo-dynamic force, and the reaction of both the numerical substructure and the physical substructure. Then, compare the EF feedback with the EF command to obtain the error between them. If the error between the EF feedback with the EF command is less than the specified tolerance, go the next step. If not, repeat the steps from the third step to the sixth step.
- Step 7: The responses of the entire structure are obtained. Then, repeat the above-mentioned steps until the entire ground motion record has been processed.

## 4. Numerical Simulation

#### 4.1. Numerical Simulation Model Based on the Uniform Design

^{n}for a full factorial design. The number of samples is about m

^{2}for an orthogonal design, which takes both the orthogonality and uniformity of the design space into consideration. However, the number of experiments is about m for the uniform design, which only highlights the uniformity of the design space.

^{s}), which is similar to the orthogonal design table, was constructed in advance, where the symbol U is the label for the uniform design, n is the number of test, s is the maximum number of parameters, and q is the number of levels for each parameter. The deviation D of the uniform design table is taken as the index to evaluate the uniformity of the table. The smaller is the value of D, the better is the uniformity of the design table. By considering the number of stories, the beam–column’s linear stiffness ratio, and the peak ground acceleration (PGA) of the ground motion record, the uniform design table U*

_{15}(15

^{7}) was chosen. The symbol * means that the design table has more uniformity [40]. The deviation D of the selected uniform table is 0.1361, which meets the precision required. More detailed parameters are listed in Table 1.

#### 4.2. Numerical Simulation Results Analysis

_{i}. Then, the cumulative error in the numerical simulation for the different boundary simulation techniques is given by

#### 4.2.1. Displacements Analysis

_{i}for tests No. 2, No. 8, and No. 13 for the traditional and proposed boundary simulation techniques. In Figure 6, we can see clearly that the error in the displacement for the proposed boundary technique is smaller than the traditional technique. The simulation results for the other numerical tests shown in Table 1 followed similar trends. Additionally, the maximum displacement errors and the cumulative errors for the different boundary techniques were calculated for the 15 tests shown in Table 1. The mean of the maximum displacement error and the mean of the cumulative error over all 15 tests was calculated for both the traditional technique and the proposed technique. Both error measures were found to be an order of magnitude smaller for the proposed approach, as compared with the traditional technique. The maximum displacement error ratio between two different techniques had an average of 9.18 with a maximum of 13.00 and a minimum of 2.66. The cumulative error ratio between two different techniques had an average of 9.87 with a maximum of 18.04 and a minimum of 3.12.

#### 4.2.2. Internal Force Analysis

## 5. Experimental Validation

^{2}were sequentially applied to the test structure.

#### 5.1. Test Model

_{1–5}= 88,000 kg, and the mass of the topper story m

_{6}= 84,000 kg. The record of the ground motion is the El Centro. Rayleigh Damping is chosen, with parameters a is 0.115 and b is 0.0024, corresponding to 5% damping in the first two modes.

#### 5.2. Displacement–Force Mixed Control Technique

#### 5.2.1. Vertical Loading Devices

#### 5.2.2. Horizontal Loading devices

#### 5.3. Arrangement of the Measurement Devices

#### 5.3.1. Arrangement of Displacement Measuring Points

#### 5.3.2. Arrangement of Strain Gauge

#### 5.4. Test Results and Analysis

^{2}, respectively. Note that, for PGA levels above 620 cm/s

^{2}, the response of the HS test model is significantly nonlinear, resulting in failure of most of the strain gauges; as a result, the internal forces calculated using the stain response are not available above a PGA of 620 cm/s

^{2}.

_{i}is measured by the stain gauge at the section, ξ

_{1}and ξ

_{8}are two compressed strain gauges at the flange of the section, ξ

_{4}and ξ

_{5}are two tensile strain gauges at the flange of the section, E is the elasticity modulus, I is the sectional inertia moment, A is the sectional area, y is the distance between the point for the stain gauge to the center of the section, N is the axial force on the section, and M is the bending moment. Because the measure of the stain gauge is only recorded at the step when the vertical force changes, Figure 18 and Figure 19 give the calculated values of the bending moment and the axial force at the recorded step.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the different boundary techniques: (

**a**) Full model; (

**b**) traditional technique; and (

**c**) proposed technique. d

_{i}denotes the displacements of each level on the full models. i ∈ [1, 5]. d

_{b}denotes the displacements at the inflection point. δ

_{1}and δ

_{2}denote the axial deformation on interfaces. N

_{G}denotes the constant representative value of gravity load calculated by the numerical substructure. N

_{b1}and N

_{b2}denote the axial forces resulted from the overturning moments on physical substructure transmitted from numerical substructure on each interface.

**Figure 4.**The block diagram of EFC method [39].

**Figure 6.**The time history of the displacement errors for the different boundary techniques at the first story: (

**a**) Test 2; (

**b**) Test 8; and (

**c**) Test 13.

**Figure 7.**Time history of bending moment errors for different boundary technique at the top of the column of the first story: (

**a**) Test 2; (

**b**) Test 8; and (

**c**) Test 13.

**Figure 8.**Time history of shear force errors for different boundary technique at the top of the column of the first story: (

**a**) Test 2; (

**b**) Test 8; and (

**c**) Test 13.

**Figure 9.**Time history of axial force errors for different boundary technique at the top of the column of the first story: (

**a**) Test 2; (

**b**) Test 8; and (

**c**) Test 13.

**Figure 10.**Sketch of the 6-story steel frame: (

**a**) entire frame; (

**b**) numerical substructure; and (

**c**) physical substructure.

**Figure 11.**The physical substructure test model of the steel frame structure: (

**a**) the design drawing of the physical substructure test model; and (

**b**) the manufactured test model at the lab.

**Figure 13.**Loading device: (

**a**) vertical hydraulic jacks; and (

**b**) horizontal hydraulic servo actuators.

**Figure 16.**Arrangement of the strain gauges: (

**a**) arrangement of the sections for pasting the strain gauge; and (

**b**) strain gauges at each section.

**Figure 17.**The comparison curves of the displacement time history between the test results and the simulation results with various PGA: (

**a**) 140 cm/s

^{2}of the PGA; (

**b**) 310 cm/s

^{2}of the PGA; (

**c**) 620 cm/s

^{2}of the PGA; (

**d**) 800 cm/s

^{2}of the PGA; (

**e**) 1200 cm/s

^{2}of the PGA; and (

**f**) 1600 cm/s

^{2}of the PGA.

**Figure 18.**The comparison curves of the internal force time history between the test results and the simulation results at the 310 cm/s

^{2}of PGA: (

**a**) axial force at the bottom of column; and (

**b**) bending moment at the bottom of column.

**Figure 19.**The comparison curves of the internal force time history between the test results and the simulation results at the 620 cm/s

^{2}of PGA: (

**a**) axial force at the bottom of column; and (

**b**) bending moment at the bottom of column.

Test Number | Number of Story | LSR of Beam–Column | PGA (cm/s^{2}) | Number of Story for the Physical Substructure | Beam Section (mm) | Column Section (mm) |
---|---|---|---|---|---|---|

1 | 3 | 0.9 | 540 | 1 | 300 × 550 × 18 × 12 | 400 × 400 × 21 × 13 |

2 | 4 | 1.4 | 420 | 2 | 300 × 600 × 24 × 14 | 400 × 400 × 21 × 13 |

3 | 5 | 1.9 | 300 | 1 | 300 × 700 × 24 × 12 | 400 × 400 × 21 × 13 |

4 | 6 | 0.8 | 180 | 1 | 300 × 500 × 20 × 14 | 400 × 400 × 21 × 13 |

5 | 7 | 1.3 | 55 | 1 | 300 × 600 × 22 × 14 | 400 × 400 × 21 × 13 |

6 | 8 | 1.8 | 580 | 3 | 300 × 700 × 22 × 12 | 400 × 400 × 21 × 13 |

7 | 9 | 0.7 | 460 | 1 | 300 × 500 × 17 × 12 | 400 × 400 × 21 × 13 |

8 | 10 | 1.2 | 340 | 3 | 350 × 550 × 22 × 14 | 400 × 400 × 21 × 13 |

9 | 11 | 1.7 | 220 | 1 | 450 × 800 × 28 × 16 | 500 × 500 × 28 × 18 |

10 | 12 | 0.6 | 100 | 1 | 450 × 550 × 22 × 14 | 500 × 500 × 28 × 18 |

11 | 13 | 1.1 | 620 | 3 | 450 × 650 × 28 × 18 | 500 × 500 × 28 × 18 |

12 | 14 | 1.6 | 500 | 2 | 450 × 800 × 26 × 16 | 500 × 500 × 28 × 18 |

13 | 15 | 0.5 | 380 | 1 | 400 × 500 × 24 × 16 | 500 × 500 × 28 × 18 |

14 | 16 | 1 | 260 | 1 | 450 × 600 × 31 × 18 | 500 × 500 × 28 × 18 |

15 | 17 | 1.5 | 140 | 1 | 450 × 800 × 24 × 16 | 500 × 500 × 28 × 18 |

Theoretical Thickness of Steel Plate (mm) | Real Thickness of Steel Plate (mm) | Yield Strength (Mpa) | Elasticity Modulus (10^{5} Mpa) | Ultimate Strength (Mpa) |
---|---|---|---|---|

7 | 6.8 | 220.5 | 2.02 | 390.4 |

10 | 9.85 | 202.3 | 1.90 | 323.2 |

12 | 10.5 | 216.0 | 2.04 | 345.6 |

16 | 15.7 | 206.5 | 1.91 | 339.6 |

Peak Acceleration (cm/s^{2}) | Axial Force at the Bottom of the Column | Bending Moment at the Bottom of the Column | Bending Moment at the End of Beam |
---|---|---|---|

310 | 2.5% | 3.7% | 6.1% |

620 | 6.7% | 9.3% | 4.0% |

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## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Yan, X.; Wang, H.; Zhu, X.; Spencer, B.F.
Substructure Hybrid Simulation Boundary Technique Based on Beam/Column Inflection Points. *Sustainability* **2018**, *10*, 2655.
https://doi.org/10.3390/su10082655

**AMA Style**

Chen Z, Yan X, Wang H, Zhu X, Spencer BF.
Substructure Hybrid Simulation Boundary Technique Based on Beam/Column Inflection Points. *Sustainability*. 2018; 10(8):2655.
https://doi.org/10.3390/su10082655

**Chicago/Turabian Style**

Chen, Zaixian, Xueyuan Yan, Hao Wang, Xingji Zhu, and Billie F. Spencer.
2018. "Substructure Hybrid Simulation Boundary Technique Based on Beam/Column Inflection Points" *Sustainability* 10, no. 8: 2655.
https://doi.org/10.3390/su10082655