- freely available
Appl. Sci. 2019, 9(24), 5486; https://doi.org/10.3390/app9245486
2. Retrofitting Method Using Viscous Dampers
2.1. Structural Model Used for Verification
2.2. Design Method
2.2.1. Step1: Initial Placements of Dampers to Satisfy Code-Specific Requirements
- For each GM, scale PGA to 0.4 g. Select a layout pattern of the damper placement; e.g., the pattern shown in Figure 3.
- Use a uniform placement from a small damping coefficient, e.g., Ci = 20 kN·(s/m), as shown in Section 2.1 or from any other value based on the designer’s experience.
- Calculate δi for each story. If δi ≤ 1/50 for any story, the initial placement of dampers satisfies code-specific requirements; stop the procedure, otherwise, increase the damping coefficients Ci until the requirement δi ≤ 1/50 is met for story i. This procedure is performed until δi ≤ 1/50 for any story. In such a way, Ci of each story is determined.
2.2.2. Step 2: Optimum Placements of the Dampers at the Same Retrofitting Cost
- By referring to initial scheme 1, perform IDA analysis on the structure to obtain its collapse-resistant capacity CC. Note that the selected GM for initial scheme 1 corresponds to GM 1.
- Scale GM to CC and perform a further dynamic analysis on the initial retrofitted structure. Obtain the inter-story drift ratio (δi) of each story.
- If δi, max is close enough to the pre-defined collapse inter-story drift ratio δcollapse, no iteration is necessary; otherwise, modify the distribution of the story damping coefficient Ci. The story damping coefficient Ci is reduced in stories with δi lower than δcollapse and increased in stories with δi higher than δcollapse (using Equation (2) for the adjustment and Equation (3) to keep the total damping coefficient unchanged). The adjustment is repeated until all the inter-story drift ratios satisfy Equation (4).
- Perform IDA analysis on the structure to obtain its new collapse-resistant capacity CC’. Scale the PGA of GM to CC’ and perform (3) again. Carry out this procedure until any further iteration is unable to increase the CC’. The value of CC’ so obtained is recognized as the optimum result for the selected GM.
- Using the above procedure, 7 new schemes of placements of dampers for the frame can be obtained; these new schemes have the same retrofitting cost as the initial retrofitting schemes obtained in Step 1, but with higher collapse-resistant capacities.
2.2.3. Step 3: Final Check Based on Code-Specific Requirements
- Determine the required minimum total damping coefficient. Note that the schemes obtained in Step 2, which satisfy the code-specific requirements about inter-story drift ratio under the rare earthquake intensity, are taken as schemes to be considered for each GM. In this step, each scheme obtained in Step 2 is analyzed under 7 GMs scaled to PGA = 0.07 g for frequent earthquake and PGA = 0.4 g for rare earthquake, respectively. Then, the average inter-story drift ratio of each scheme is obtained by averaging 7 results. The required minimum total damping coefficient is determined from the scheme average inter-story drift ratios, which satisfies the code requirements under frequent and rare earthquakes (1/550 and 1/50, respectively ).
- Determine an optimum distribution pattern from schemes obtained in Step 2 with total damping coefficient scaled to the required minimum total damping coefficient. This procedure checks which distribution pattern is better under a given minimum total damping coefficient. The verification criteria are the same used in (1).
2.3. Comparison of Failure Pattern and Collapse Fragility
3. Extension to 3D Structures
4. Economic Analysis for the Proposed Method
4.1. Performance Group of Structural and Non-Structural Members
4.2. Economic Benefit Analysis
5. Sustainability Analysis for the Proposed Method
- The proposed retrofitting method can be applied both to satisfy the requirements of design codes and to enhance the structural collapse-resistant capacity of existing structures. An advantage of the proposed method is that the retrofitted structure has a larger earthquake collapse-resistant capacity compared with an ordinary retrofitting method entailing the same retrofitting cost. For a regular 3D frame, the computational cost is low because a 2D frame model can be adopted to determine the optimum placement pattern of the viscous dampers. Similar to other design methods for the placement of dampers, the proposed method is also a procedure based on a series of time-history analyses. However, for the 6-story RC frame structure used in this study, one time-history analysis commonly only takes about three minutes (for the 2D frame model). Therefore, the computational cost of the proposed method is acceptable.
- The economic benefit analysis is conducted to check the convenience of the proposed retrofitting method. The owners can clearly know the economic benefits under different earthquake intensities that may happen in the future. For the 6-story RC frame structure used in this study, during its lifetime cycle, the economic benefit of retrofitting is not favorable if the structure only suffers an earthquake with a frequent earthquake intensity, while the economic benefit of retrofitting is favorable if the structure suffers an earthquake with an occasional earthquake intensity (depending on the building functions which influence the economic loss by downtime/repair time. The example building used in this study is a middle school, meaning that downtime/repair time will not induce much economic loss. However, the economic loss will be higher if it would be a commercial building.), and the economic benefit of retrofitting is significantly favorable if the structure suffers an earthquake with a rare intensity (the ratio between the cost of retrofitting and the cost of complete replacement is about 45%; the ratio between the repair time of retrofitting and the repair time of complete replacement is about 10%); in this case, the retrofitting will save householders’ lives in case of earthquake. This economic benefit analysis results can be provided to householders who may decide whether to perform retrofitting or not. Besides the economic benefit, environmental sustainability is also discussed by way of a carbon emission calculation. Under a rare earthquake, the original frame needs replacement, which leads to more carbon emissions than a retrofitted frame in the repair process.
Conflicts of Interest
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|Story||Size (mm × mm) Width × Height||Area of Longitudinal Rebars (mm2)/Stirrup|
|Side Bay||Middle Bay||Side Bay||Middle Bay|
|Beam Ends||Midspan||Beam Ends||Midspan|
|1||200 × 50||200 × 300||2250/ϕ[email protected]||1600/ϕ[email protected]||2250/ϕ[email protected]||1350/ϕ[email protected]|
|2||200 × 50||200 × 300||2200/ϕ[email protected]||1600/ϕ[email protected]||2200/ϕ[email protected]||1250/ϕ[email protected]|
|3||200 × 50||200 × 300||2050/ϕ[email protected]||1600/ϕ[email protected]||2050/ϕ[email protected]||1100/ϕ[email protected]|
|4||200 × 50||200 × 300||1800/ϕ[email protected]||1650/ϕ[email protected]||1800/ϕ[email protected]||850/ϕ[email protected]|
|5||200 × 50||200 × 300||1450/ϕ[email protected]||1650/ϕ[email protected]||1450/ϕ[email protected]||750/ϕ[email protected]|
|6||200 × 50||200 × 300||850/ϕ[email protected]||1400/ϕ[email protected]||850/ϕ[email protected]||600/ϕ[email protected]|
|Story||Size (mm × mm)||Area of Longitudinal Rebars (mm2)/Stirrup|
|Side Column||Middle Column||Side Column||Middle Column|
|1||400 × 400||400 × 400||2000/ϕ[email protected]||1400/ϕ[email protected]|
|2||400 × 400||400 × 400||800/ϕ[email protected]||1000/ϕ[email protected]|
|3||400 × 400||400 × 400||800/ϕ[email protected]||1000/ϕ[email protected]|
|4||400 × 400||400 × 400||700/ϕ[email protected]||700/ϕ[email protected]|
|5||400 × 400||400 × 400||700/ϕ[email protected]||700/ϕ[email protected]|
|6||400 × 400||400 × 400||1300/ϕ[email protected]||1000/ϕ[email protected]|
|Tag||Earthquake||NGA ID and Component||Year||Magnitude|
|1||San Fernando, USA||RSN68 SFERN PEL090||1971||6.61|
|2||San Fernando, USA||RSN93 SFERN WND143||1971||6.61|
|3||Tabas, Iran, USA||RSN143 TABAS TAB-T1||1978||7.35|
|4||Imperial valley—06, USA||RSN161 IMPVALL.H H-BRA315||1979||6.53|
|5||Imperial valley—06, USA||RSN178 IMPVALL.H H-E03230||1979||6.53|
|6||Imperial valley—06, USA||RSN180 IMPVALL.H H-E05140||1979||6.53|
|7||Imperial valley—06, USA||RSN183 IMPVALL.H H-E08140||1979||6.53|
|Initial scheme 1||300,000||20,000||250,000||250,000||20,000||20,000||860,000|
|Initial scheme 2||150,000||150,000||50,000||50,000||20,000||20,000||440,000|
|Initial scheme 3||1,300,000||1,300,000||1,000,000||500,000||50,000||20,000||4,170,000|
|Initial scheme 4||1,000,000||1,000,000||800,000||250,000||20,000||20,000||3,090,000|
|Initial scheme 5||1,200,000||1,300,000||750,000||200,000||50,000||50,000||3,550,000|
|Initial scheme 6||800,000||750,000||600,000||100,000||20,000||20,000||2,290,000|
|Initial scheme 7||20,000||20,000||500,000||50,000||20,000||20,000||630,000|
|Optimum Times||Collapse Resistance|
|Initial scheme 1||0.16 g||300,000||20,000||250,000||250,000||20,000||20,000||860,000|
|1st time||0.18 g||499,230||32,883||185,666||127,186||7857||7178||860,000|
|2nd time||0.19 g||849,740||1476||5545||3329||0||0||860,000|
|Optimum scheme 1||0.19 g||849,740||1476||5455||3329||0||0||860,000|
|Optimum scheme 1||849,740||1476||5455||3329||0||0||860,000|
|Optimum scheme 2||262,542||115,899||31,765||20,796||5172||3826||440,000|
|Optimum scheme 3||3,335,742||543,619||164,958||125,681||0||0||4,170,000|
|Optimum scheme 4||2,741,236||186,395||101,966||60,403||0||0||3,090,000|
|Optimum scheme 5||1,610,247||1,339,753||553,541||46,459||0||0||3,550,000|
|Optimum scheme 6||1,604,274||412,497||139,411||133,818||0||0||2,290,000|
|Optimum scheme 7||602,094||3659||22,695||1551||0||0||630,000|
|Group||Fragility ID ||EDP||Cost/Each (Unit: US Dollar)||Number in Each Story||Total Number|
|Table and chair||E2022.020||PFA||14.9||300||1800|
|Construction Items||Construction Period (Unit: Workday)|
|Upper structure project||305|
|Elevator system installation||50|
|Heat-supply system installation||50|
|Air conditioning system installation||65|
|Electric power substation project||40|
|Total required workdays||600|
|Performance Group||Fragility ID ||EDP||Costing Based Upon in PACT ||Number in Each Story||Total Number|
|Structural member||Beam joints||B1041.101b||IDR||1 EA||24||144|
|Non-structural member||Curtain walls||B2022.001||IDR||30 SF||34.82||208.92|
|Wall finishes||C3011.001a||IDR||100 LF||1.34||8.04|
|Cold water piping||D2021.011a||PFA||1000 LF||0.07||0.42|
|Hot water piping||D2022.011a||PFA||1000 LF||0.40||2.40|
|Sanitary waste piping||D2031.011b||PFA||1000 LF||0.21||1.26|
|HVAC equipment||D3041.011a||PFA||1000 LF||0.24||1.44|
|Variable Air Volume (VAV) box||D3041.041a||PFA||10 EA||1.90||11.4|
|Concrete tile roof||B3011.011||PFA||100 SF||32.28||32.28|
|Recessed ceiling lighting||C3033.001||PFA||1 EA||71.21||427.26|
|Independent pendant lighting||C3034.001||PFA||1 EA||71.21||427.26|
|Sprinkler water supply||D4011.021a||PFA||1000 LF||0.85||5.1|
|Low voltage switchgear||D5012.021a||PFA||225 AMP||1.00||6|
|Motor control center||D5012.013a||PFA||1 EA||2.00||2.00|
|Frame||Frequent Earthquake PGA = 0.07 g||Occasional Earthquake PGA = 0.2 g||Rare Earthquake PGA = 0.4 g|
|3D original frame||5||34||482|
|3D retrofitted frame||2||7||53|
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