# Energy-Based Design of Buckling-Restrained Steel Braced Frames for Concurrent Occurrences of Earthquake and Wind

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Studies on Dual Hazards on Structures

#### 1.2. Buckling Restrained Braces

## 2. Research Objectives

- The steel frames are in an undamaged state when the dual hazards strike.
- The excitations caused by earthquakes and wind have the same duration and their maximum effects occur within this time span, i.e., the structure is experiencing its most severe loading condition during this time interval.
- The FEMA-recommended drift limits [11] for the performance-based seismic design (PBSD) of moment and braced steel frames are used in the present study to assess the adequacy of these frames when subject to dual wind–earthquake excitations. This is because in the proposed methodology, the dual hazard effect is represented by a time-varying excitation like that of an earthquake.

## 3. Dual Earthquake–Wind Hazard Power Spectrum

#### 3.1. Earthquake Power Spectrum

^{2}) and its PSD (in (m/s

^{2})

^{2}/Hz) can be computed using the equation:

#### 3.2. Wind Power Spectrum

^{3}, and the standard value used in this study was 1.224 kg/m

^{3}; and ${C}_{D}$ is the nondimensional drag coefficient. For a bluff body, ${C}_{D}$ is dependent on the shape of the body [18]. If the bluff body has a rectangular cross-section with B/D (ratio of the along-wind to cross-wind dimension) larger than 2, ${C}_{D}$ stabilizes to a value of 1.05. Since the frames used in the present study satisfy this requirement, ${C}_{D}$ = 1.05 is used. $A$ denotes the area in m

^{2}exposed in the along-wind direction. Note that the term that involves the square of $u\left(z,t\right)$ when Equation (2) is substituted into Equation (4) is omitted because it is relatively small when compared to the other terms.

^{2}/Hz; ${S}_{a\_wind}\left(f\right)$ denotes the PSD of the fluctuating wind acceleration in (m/s

^{2})

^{2}/Hz; $m$ denotes the mass in kg; $q=\rho {C}_{D}A\overline{U}$ [17], in kg/s; and ${S}_{u}\left(f\right)$ is the PSD of the fluctuating component of wind velocity in (m/s)

^{2}/Hz.

^{2}/Hz is given by:

#### 3.3. Proposed Dual Hazard Power Spectrum

^{2})

^{2}/Hz is obtained by combining the PSDs of earthquakes (given by Equation (1)) and wind (given by Equation (6)) using the SRSS combination rule, i.e.,

^{2}); ${a}_{dual}$ is the dual hazard acceleration (in m/s

^{2}) that corresponds to the fluctuating part of the wind load; and ${\overline{a}}_{wind}=\frac{\frac{1}{2}\rho {C}_{D}A{\overline{U}}^{2}\left(z\right)}{m}$ (in m/s

^{2}) is the dual hazard acceleration caused by the steady part of the wind load. The plus or minus sign in Equation (9) accounts for the possibility that wind can blow in either direction, and so the steady wind component is added to or subtracted from the fluctuating component, in order to determine the more severe condition for the analysis.

## 4. Earthquake and Wind Data Characterizations

#### 4.1. Earthquake Data

#### 4.2. Wind Data

## 5. Dual Earthquake–Wind Hazard Excitations

^{2}along-wind surface area and 21,000 kg mass—is shown in Figure 3b. By using Equation (8), the dual earthquake–wind power spectrum is shown in Figure 3c. Applying IFFT to this spectrum and incorporating the steady component of wind using Equation (9), the resulting dual earthquake–wind excitations are shown in Figure 3d. Note that the plus and minus signs in Equation (9) lead to two excitations: one with ${\overline{a}}_{wind}$ added to and one with ${\overline{a}}_{wind}$ subtracted from ${a}_{dual}$. Therefore, for a given structure, two analyses need to be performed to determine which would result in a more severe condition.

## 6. Analysis Results for Moment Resisting Frame Responses to Dual Earthquake and Wind Hazards

#### 6.1. Three-Story Frame

- Los Angeles:

- Charleston:

#### 6.2. Nine-Story Frame

- Los Angeles:

- Charleston:

## 7. Modeling of Buckling-Restrained Braces

## 8. Energy-Based Design of Buckling-Restrained Braced Frames

#### 8.1. Energy Capacity of a BRB

^{2}, N/m

^{2}and N/m

^{2}, respectively.

#### 8.2. Energy Demand from the Dual Hazards

#### 8.3. Proposed Energy-Based Design Procedure for BRB

- Step 1: Construct elastic response spectra for the dual excitations.Pseudo (or spectral) acceleration response spectra (often expressed in m/s
^{2}or in terms of acceleration due to gravity, g) are plots of pseudo accelerations S_{a}against system periods T (in seconds). These spectra can be constructed for the dual excitations using software such as Bispec, SeismoSoft, OpenSees, etc. - Step 2: Run a modal analysis to obtain (or estimate) the fundamental period for the intermediate moment frame.
- Step 3: Calculate the base shear using the equation

- Step 4: Determine the equivalent lateral force (ELF) and story shear at story $i$ using the equation:

- Step 5: Perform a linear static analysis on the intermediate moment resisting frame subject to this ELF and determine its maximum inter-story displacement response $\mathsf{\Delta}{u}_{el}$. The maximum displacement response of the frame accounting for inelasticity $\mathsf{\Delta}{u}_{m}$ can be estimated using the equation:

- Step 6: Use the energy equation to determine the required BRB cross-section area for each story. The required BRB yielding core area can be obtained by equating Equation (11) or Equation (14) with Equation (17) and solving for ${{A}_{brb}}_{i}$ to give

## 9. Analysis Results for Buckling-Restrained-Braced-Frame Responses to Dual Earthquake and Wind Hazards

#### 9.1. Three-Story Frame

#### 9.2. Nine-Story Frame

## 10. Results and Discussion

## 11. Summary and Conclusions

- When compared to results obtained for earthquake only or wind only excitation, the two steel frames used in the present study were shown to experience peak and residual inter-story and roof drift ratios that were noticeably higher under the dual earthquake–wind excitations.
- From Table 6 and Table 7, it can be seen that dynamic responses due to the combined earthquake–wind dual excitations cannot be obtained just by adding the dynamic responses due to earthquake only and wind only excitations. This is because of the presence of inelasticity. Once the structure experiences yielding, inelastic deformations will increase rapidly with the applied forces (as illustrated in Figure 10 and Figure 12).
- By retrofitting these frames with BRBs using the proposed energy-based design methodology presented in Section 8, the drift ratios of these frames were drastically reduced, and they all fell below or came very close to the FEMA 356 drift limits.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Abbreviations

ASCE | American Society of Civil Engineers |

BRB | Buckling Restrained Brace |

BRBF | Buckling-Restrained-Braced Frame |

DBE | Design Based Earthquakes |

EDP | Engineering Demand Parameters |

ELF | Equivalent Lateral Force |

FEMA | Federal Emergency Management Agency |

FFT | Fast Fourier Transform |

IBC | International Building Code |

IFFT | Inverse Fast Fourier Transform |

IO | Immediate Occupancy |

LS | Life Safety |

MCE | Maximum Considered Earthquakes |

MRF | Moment Resisting Frame |

PSD | Power Spectral Density |

RSN | Record Sequence Number |

SEI | Structural Engineering Institute |

SRSS | Square Root of the Sum of Squares |

## Appendix A

RSN | Event | Year | Station Name | Magnitude | Vs_{30} (m/s) |
---|---|---|---|---|---|

100 | “Hollister-03” | 1974 | “San Juan Bautista_ 24 Polk St” | 5.14 | 336 |

187 | “Imperial Valley-06” | 1979 | “Parachute Test Site” | 6.53 | 347 |

280 | “Trinidad” | 1980 | “Rio Dell Overpass-FF” | 7.2 | 312 |

292 | “Irpinia_ Italy-01” | 1980 | “Sturno (STN)” | 6.9 | 382 |

313 | “Corinth_ Greece” | 1981 | “Corinth” | 6.6 | 361 |

725 | “Superstition Hills-02” | 1987 | “Poe Road (temp)” | 6.54 | 317 |

832 | “Landers” | 1992 | “Amboy” | 7.28 | 383 |

1119 | “Kobe_ Japan” | 1995 | “Takarazuka” | 6.9 | 312 |

1762 | “Hector Mine” | 1999 | “Amboy” | 7.13 | 383 |

2093 | “Nenana Mountain_ Alaska” | 2002 | “TAPS Pump Station #09” | 6.7 | 383 |

5865 | “El Mayor-Cucapah_ Mexico” | 2010 | “Palm Springs Airport” | 7.2 | 312 |

6911 | “Darfield_ New Zealand” | 2010 | “HORC” | 7 | 326 |

RSN | Event | Year | Station Name | Magnitude | Vs_{30} (m/s) |
---|---|---|---|---|---|

26 | “Hollister-01” | 1961 | “Hollister City Hall” | 5.6 | 198.8 |

35 | “Northern Calif-06” | 1967 | “Hollister City Hall” | 5.2 | 198.8 |

163 | “Imperial Valley-06” | 1979 | “Calipatria Fire Station” | 6.53 | 206 |

314 | “Westmorland” | 1981 | “Brawley Airport” | 5.9 | 209 |

462 | “Morgan Hill” | 1984 | “Hollister City Hall” | 6.19 | 198.8 |

718 | “Superstition Hills-01” | 1987 | “Imperial Valley Wildlife” | 6.22 | 179.0 |

1931 | “Anza-02” | 2001 | “El Centro Array #10” | 4.92 | 203 |

1992 | “Gulf of California” | 2001 | “Calipatria Fire Station” | 5.7 | 206 |

4100 | “Parkfield-02_ CA” | 2004 | “Parkfield-Cholame 2WA” | 6 | 173.0 |

4462 | “L’Aquila_ Italy” | 2009 | “Avezzano” | 6.3 | 199.0 |

180 | “Imperial Valley-06” | 1979 | “El Centro Array #5” | 6.53 | 206 |

726 | “Superstition Hills-02” | 1987 | “Salton Sea Wildlife Refuge” | 6.54 | 191.1 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | ${{\mathit{A}}_{\mathit{b}\mathit{r}\mathit{b}}^{\mathit{r}}}_{\mathit{i}}\left[\mathbf{c}{\mathbf{m}}^{2}\right]$ |
---|---|---|---|---|---|---|---|---|

3 | 0.941 | 0.261 | 5101 | 5075 | 0.557 | 2121 | 416 | 616 |

2 | 0.952 | 0.264 | 4694 | 4724 | 0.312 | 1189 | 647 | 960 |

1 | 0.946 | 0.263 | 4694 | 4694 | 0.1305 | 497 | 505 | 750 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | ${{\mathit{A}}_{\mathit{b}\mathit{r}\mathit{b}}^{\mathit{r}}}_{\mathit{i}}\left[\mathbf{c}{\mathbf{m}}^{2}\right]$ |
---|---|---|---|---|---|---|---|---|

3 | 1.422 | 0.395 | 5101 | 5089 | 0.558 | 3205 | 857 | 337 |

2 | 1.435 | 0.398 | 4694 | 4723 | 0.312 | 1792 | 1331 | 523 |

1 | 1.426 | 0.396 | 4694 | 4694 | 0.1303 | 749 | 908 | 357 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | ${{\mathit{A}}_{\mathit{b}\mathit{r}\mathit{b}}^{\mathit{r}}}_{\mathit{i}}\left[\mathbf{c}{\mathbf{m}}^{2}\right]$ |
---|---|---|---|---|---|---|---|---|

3 | 0.435 | 0.1207 | 5101 | 4223 | 0.513 | 1017 | 96.7 | 143 |

2 | 0.523 | 0.1453 | 4694 | 4678 | 0.342 | 679 | 169.1 | 251 |

1 | 0.525 | 0.1458 | 4694 | 4694 | 0.1445 | 286 | 136.8 | 203 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | |
---|---|---|---|---|---|---|---|---|

3 | 0.625 | 0.1735 | 5101 | 4380 | 0.521 | 1452 | 127.5 | 50 |

2 | 0.732 | 0.2033 | 4694 | 4722 | 0.338 | 943 | 223 | 88 |

1 | 0.728 | 0.2021 | 4694 | 4694 | 0.1413 | 394 | 162.5 | 64 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | |
---|---|---|---|---|---|---|---|---|

9 | 0.433 | 0.1202 | 5248 | 5247 | 0.273 | 1482 | 222 | 330 |

8 | 0.456 | 0.1265 | 4856 | 5108 | 0.215 | 1168 | 502 | 744 |

7 | 0.453 | 0.1257 | 4856 | 5077 | 0.1681 | 914 | 621 | 921 |

6 | 0.450 | 0.1249 | 4856 | 5043 | 0.1271 | 691 | 631 | 936 |

5 | 0.446 | 0.1240 | 4856 | 5007 | 0.0916 | 498 | 763 | 1131 |

4 | 0.443 | 0.1231 | 4856 | 4968 | 0.0618 | 336 | 792 | 1174 |

3 | 0.439 | 0.1220 | 4856 | 4924 | 0.0376 | 205 | 801 | 1187 |

2 | 0.435 | 0.1208 | 4856 | 4875 | 0.01930 | 104.9 | 770 | 1143 |

1 | 0.433 | 0.1203 | 4954 | 4954 | 0.00707 | 38.4 | 1138 | 1440 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | |
---|---|---|---|---|---|---|---|---|

9 | 0.690 | 0.1918 | 5248 | 5245 | 0.275 | 2368 | 483 | 190 |

8 | 0.717 | 0.1991 | 4856 | 5039 | 0.213 | 1840 | 1146 | 450 |

7 | 0.714 | 0.1982 | 4856 | 5016 | 0.1674 | 1443 | 1387 | 545 |

6 | 0.710 | 0.1973 | 4856 | 4993 | 0.1267 | 1093 | 1342 | 527 |

5 | 0.707 | 0.1963 | 4856 | 4967 | 0.0915 | 789 | 1665 | 654 |

4 | 0.703 | 0.1952 | 4856 | 4939 | 0.0619 | 533 | 1712 | 672 |

3 | 0.698 | 0.1939 | 4856 | 4907 | 0.0378 | 326 | 1716 | 674 |

2 | 0.693 | 0.1925 | 4856 | 4871 | 0.01943 | 167.5 | 1617 | 635 |

1 | 0.691 | 0.1919 | 4954 | 4954 | 0.00712 | 61.4 | 2437 | 816 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | |
---|---|---|---|---|---|---|---|---|

9 | 0.252 | 0.0699 | 5248 | 3879 | 0.205 | 875 | 81.1 | 120 |

8 | 0.403 | 0.1119 | 4856 | 5744 | 0.245 | 1049 | 262 | 389 |

7 | 0.392 | 0.1088 | 4856 | 5583 | 0.1879 | 803 | 362 | 537 |

6 | 0.380 | 0.1054 | 4856 | 5411 | 0.1385 | 592 | 384 | 569 |

5 | 0.366 | 0.1018 | 4856 | 5224 | 0.0971 | 415 | 470 | 698 |

4 | 0.352 | 0.0978 | 4856 | 5019 | 0.0634 | 271 | 490 | 727 |

3 | 0.336 | 0.0934 | 4856 | 4794 | 0.0372 | 159.1 | 495 | 734 |

2 | 0.319 | 0.0886 | 4856 | 4546 | 0.01829 | 78.1 | 475 | 705 |

1 | 0.341 | 0.0946 | 4954 | 4954 | 0.00718 | 30.7 | 702 | 889 |

Story | ${\mathit{S}}_{\mathit{a}}\left[\mathbf{g}\right]$ | ${\mathit{C}}_{\mathit{s}}$ | $\mathit{W}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{W}}_{\mathit{e}\mathit{f}\mathit{f}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{C}}_{\mathit{v}}$ | ${\mathit{F}}_{\mathit{i}}\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathit{I}\mathit{E}}_{\mathit{i}}[\mathbf{k}\mathbf{N}\xb7\mathbf{m}]$ | |
---|---|---|---|---|---|---|---|---|

9 | 0.354 | 0.0984 | 5248 | 4022 | 0.214 | 1233 | 100.9 | 40 |

8 | 0.531 | 0.1474 | 4856 | 5576 | 0.240 | 1382 | 366 | 144 |

7 | 0.518 | 0.1440 | 4856 | 5447 | 0.1849 | 1064 | 485 | 191 |

6 | 0.505 | 0.1403 | 4856 | 5307 | 0.1370 | 788 | 483 | 190 |

5 | 0.490 | 0.1362 | 4856 | 5155 | 0.0966 | 556 | 613 | 241 |

4 | 0.474 | 0.1318 | 4856 | 4987 | 0.0635 | 366 | 631 | 248 |

3 | 0.456 | 0.1268 | 4856 | 4798 | 0.0376 | 216 | 631 | 248 |

2 | 0.436 | 0.1212 | 4856 | 4585 | 0.01860 | 107.0 | 591 | 232 |

1 | 0.462 | 0.1283 | 4954 | 4954 | 0.00724 | 41.7 | 895 | 300 |

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**Figure 1.**Procedure of generating the dual earthquake–wind hazard power spectrum and time domain excitation function.

**Figure 3.**(

**a**) Power spectrum of a scaled earthquake (RSN 725), (

**b**) power spectrum for wind with 41 m/s basic wind speed, (

**c**) dual earthquake–wind power spectrum and (

**d**) dual earthquake–wind excitations for the performance level of IO in Los Angeles.

**Figure 4.**Story-based time-history excitations and seismic lumped masses used for the three-story MRF.

**Figure 5.**Peak and residual inter-story and roof drift ratios of the three-story MRF in Los Angeles, analyzed for IO.

**Figure 6.**Peak and residual inter-story and roof drift ratios of the three-story MRF in Los Angeles, analyzed for LS.

**Figure 7.**Hysteresis curves of a third story beam for the three-story frame in Los Angeles under the performance level of (

**a**) IO and (

**b**) LS.

**Figure 8.**Peak and residual inter-story and roof drift ratios of the three-story MRF in Charleston, as analyzed for IO.

**Figure 9.**Peak and residual inter-story and roof drift ratios of the three-story MRF in Charleston, as analyzed for LS.

**Figure 10.**Hysteresis curves of a third-story beam for the three-story frame in Charleston under the performance level of (

**a**) IO and (

**b**) LS.

**Figure 11.**Peak and residual inter-story roof drift ratios of the nine-story MRF in Los Angeles, as analyzed for IO.

**Figure 12.**Hysteresis curves of a second-story beam for the nine-story frame in Los Angeles under the performance level of IO.

**Figure 14.**Force–displacement relationship of a chevron and a diagonal BRB under one cycle of loading and unloading.

**Figure 15.**Force–displacement relationships of an inelastic frame and the corresponding elastic frame: (

**a**) energy demand based on maximum drift limit, and (

**b**) energy demand based on residual drift limit.

**Figure 17.**Peak and residual inter-story and roof drift ratios of the three-story BRBF in Los Angeles.

**Figure 18.**Peak and residual inter-story and roof drift ratios of the three-story BRBF in Charleston.

**Figure 19.**Placement of BRBs for the nine-story BRBF: (

**a**) two-bay BRBF, (

**b**) three-bay BRBF and (

**c**) four-bay BRBF.

**Figure 20.**Peak and residual inter-story and roof drift ratios of the nine-story BRBF in Los Angeles.

**Table 1.**Drift limits for IO and LS [11].

Steel Frame Type | Immediate Occupancy (IO) | Life Safety (LS) | ||
---|---|---|---|---|

Peak Drift | Residual Drift | Peak Drift | Residual Drift | |

Moment frames | 0.7% | Negligible | 2.5% | 1% |

Braced frames | 0.5% | Negligible | 1.5% | 0.5% |

Los Angeles | Charleston | |
---|---|---|

Immediate Occupancy (IO) | 41 | 62 |

Life Safety (LS) | 46 | 72 |

Parameters | Tension | Compression |
---|---|---|

Initial (elastic) stiffness of BRB, ${E}_{eq}$ | ${{f}_{sm}E}_{s}$ | ${f}_{sm}{E}_{s}$ |

Stiffness modification factor, ${f}_{sm}$ | 1.39 | 1.39 |

Elastic modulus of steel, ${E}_{s}$ | 200 GPa | 200 GPa |

Yield strength of BRB steel core, ${f}_{y}$ | ${{\gamma}_{mo}f}_{ys}$ | ${\gamma}_{mo}{f}_{ys}$ |

Material overstrength factor, ${\gamma}_{mo}$ | 1.11 | 1.11 |

Yield strength of steel, ${f}_{ys}$ | 248 MPa | 248 MPa |

Kinematic hardening ratio, ${b}_{k}$ | 0.5% | 2.5% |

Shape parameter *, ${R}_{0}$ | 26 | 26 |

Shape parameter *, ${r}_{1}$ | 0.91 | 0.89 |

Shape parameter *, ${r}_{2}$ | 0.10 | 0.02 |

Initial hardening ratio for isotropic materials, ${b}_{i}$ | 0.25% | 0.6% |

Saturated hardening ratio for isotropic materials, ${b}_{l}$ | 0.01% | 0.03% |

Intersection point between ${b}_{i}$ and ${b}_{l}$, ${\rho}_{i}$ | 0.8 | 0.3 |

Isotropic transition parameter, ${R}_{i}$ | 3.0 | 3.0 |

Length of the yield plateau, ${l}_{yp}$ | 1.0 | 1.0 |

Ultimate strength of BRB steel core, ${f}_{u}$ | $1.65{f}_{y}$ | $2.0{f}_{y}$ |

Kinematic hardening to perfectly plastic transition parameter, ${R}_{u}$ | 2.0 | 2.0 |

Model | ${\mathit{A}}_{\mathit{b}\mathit{r}\mathit{b}}\left[\mathbf{c}{\mathbf{m}}^{2}\right]$ | Configuration |
---|---|---|

Los Angeles for IO | 387 | 2-bay |

Los Angeles for LS | 206 | 2-bay |

Charleston for IO | 206 | 1-bay |

Charleston for LS | 71 | 1-bay |

Model | ${\mathit{A}}_{\mathit{b}\mathit{r}\mathit{b}}\left[\mathbf{c}{\mathbf{m}}^{2}\right]$ | Configuration | |||
---|---|---|---|---|---|

Story | 1st | 2nd–5th | 6th–8th | 9th | |

Los Angeles for IO | 84 | 219 | 310 | 361 | 4-bay |

Los Angeles for LS | 65 | 181 | 232 | 284 | 3-bay |

Charleston for IO | 42 | 168 | 245 | 310 | 3-bay |

Charleston for LS | 21 | 90 | 123 | 155 | 2-bay |

**Table 6.**Comparison of drift ratios for single and dual hazards for the three-story frame without BRBs.

City | Performance Level | Hazard | Peak Inter-Story | Residual Inter-Story | Peak Roof | Residual Roof |
---|---|---|---|---|---|---|

Los Angeles | IO | Earthquake | 3.25% | 1.06% | 2.47% | 0.92% |

Wind | 0.36% | ≈0% | 0.31% | ≈0% | ||

Dual | 4.10% | 2.24% | 3.46% | 2.08% | ||

LS | Earthquake | 6.80% | 4.52% | 5.94% | 4.32% | |

Wind | 0.47% | ≈0% | 0.40% | ≈0% | ||

Dual | 11.63% | 10.40% | 10.99% | 10.08% | ||

Charleston | IO | Earthquake | 1.14% | 0.083% | 0.96% | 0.06% |

Wind | 0.93% | 0.11% | 0.78% | 0.09% | ||

Dual | 1.93% | 0.93% | 1.65% | 0.75% | ||

LS | Earthquake | 1.75% | 0.32% | 1.39% | 0.24% | |

Wind | 1.35% | 0.42% | 1.14% | 0.36% | ||

Dual | 3.57% | 2.62% | 3.27% | 2.42% |

**Table 7.**Comparison of drift ratios for single and dual hazards for the nine-story frame without BRBs.

City | Performance Level | Hazard | Peak Inter-Story | Residual Inter-Story | Peak Roof | Residual Roof |
---|---|---|---|---|---|---|

Los Angeles | IO | Earthquake | 2.52% | 1.00% | 1.82% | 0.74% |

Wind | 0.77% | ≈0% | 0.61% | ≈0% | ||

Dual | 10.4% | 9.83% | 6.72% | 6.19% | ||

LS | Earthquake | 4.36% | 2.83% | 3.20% | 2.19% | |

Wind | 1.08% | 0.35% | 0.78% | 0.14% | ||

Dual | C * | C * | C * | C * | ||

Charleston | IO | Earthquake | 1.30% | 0.16% | 0.84% | 0.08% |

Wind | 9.40% | 8.70% | 4.92% | 4.38% | ||

Dual | C * | C * | C * | C * | ||

LS | Earthquake | 1.71% | 0.41% | 1.18% | 0.26% | |

Wind | C * | C * | C * | C * | ||

Dual | C * | C * | C * | C * |

**Table 8.**Comparison of drift ratios before and after the addition of BRBs for the three-story frame.

City | Performance Level | Frame Type | Peak Inter-Story | Residual Inter-Story | Peak Roof | Residual Roof |
---|---|---|---|---|---|---|

Los Angeles | IO | MRF | 4.10% | 2.24% | 3.46% | 2.08% |

BRBF | 0.27% | 0.01% | 0.27% | 0.01% | ||

LS | MRF | 11.65% | 10.40% | 10.99% | 10.08% | |

BRBF | 0.80% | 0.14% | 0.52% | 0.06% | ||

Charleston | IO | MRF | 1.93% | 0.93% | 1.65% | 0.75% |

BRBF | 0.27% | ≈0% | 0.25% | ≈0% | ||

LS | MRF | 3.57% | 2.62% | 3.27% | 2.42% | |

BRBF | 0.86% | 0.20% | 0.66% | 0.14% |

City | Performance Level | Frame Type | Peak Inter-Story | Residual Inter-Story | Peak Roof | Residual Roof |
---|---|---|---|---|---|---|

Los Angeles | IO | MRF | 10.40% | 9.83% | 6.72% | 6.19% |

BRBF | 0.64% | 0.05% | 0.56% | 0.05% | ||

LS | MRF | C * | C * | C * | C * | |

BRBF | 1.33% | 0.50% | 0.80% | 0.21% | ||

Charleston | IO | MRF | C * | C * | C * | C * |

BRBF | 0.30% | ≈0% | 0.26% | ≈0% | ||

LS | MRF | C * | C * | C * | C * | |

BRBF | 1.41% | 0.76% | 0.89% | 0.33% |

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## Share and Cite

**MDPI and ACS Style**

Shan, T.; Lui, E.M.
Energy-Based Design of Buckling-Restrained Steel Braced Frames for Concurrent Occurrences of Earthquake and Wind. *CivilEng* **2024**, *5*, 343-377.
https://doi.org/10.3390/civileng5020018

**AMA Style**

Shan T, Lui EM.
Energy-Based Design of Buckling-Restrained Steel Braced Frames for Concurrent Occurrences of Earthquake and Wind. *CivilEng*. 2024; 5(2):343-377.
https://doi.org/10.3390/civileng5020018

**Chicago/Turabian Style**

Shan, Taonian, and Eric M. Lui.
2024. "Energy-Based Design of Buckling-Restrained Steel Braced Frames for Concurrent Occurrences of Earthquake and Wind" *CivilEng* 5, no. 2: 343-377.
https://doi.org/10.3390/civileng5020018