# Estimation of Uniform Risk Spectra Suitable for the Seismic Design of Structures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{y}) so that a structure reaches a specific performance target. These spectra are obtained as a function of the dynamic characteristics of soil and the inelastic dynamic response of several single degree of freedom (SDOF) systems. One example are uniform ductility spectra, which are obtained by using reduction factors over elastic response or design spectra [24,25,26,27,28,29,30,31,32]; these factors are generally obtained from the statistical analysis and mathematical regressions of the response of instrumented buildings. These factors are easily implemented in building codes but, unfortunately, since only important buildings have been instrumented, only a few buildings are available to provide information.

## 2. Methodology

#### 2.1. Stage 1

#### 2.2. Stage 2

## 3. Application Example

#### 3.1. Defining the Seismic Hazard

_{o}= 39.7 years [39,60]. Figure 3 shows the estimated exceedance rate of magnitudes curve for the Guerrero Gap.

#### 3.2. Computing the Nonlinear Response in Terms of Loss

_{TJ}) damage index [62] and, subsequently, classified by magnitude. The computed nonlinear response of several SDOF systems, with vibration periods between 0.01 and 5 s, which cover the typical period range of response spectra, for the record set considered was used to define the damage spectra considering pre-established combinations of stiffness and strength (T and $S{a}_{y}$) for a damping ratio of 0.05 and a post-yielding rate of α = 0.05 [63].

_{PA}) [73] with the associated repair action and its repair cost ratio.

_{TJ})—structural loss ratio (β) for conventional RC structures. Figure 6 shows a fitted curve obtained through nonlinear regressions (Equation (2)); this curve was obtained by relating the midpoint of each DI interval and the corresponding loss ratio. In the same figure, it is possible to see that the physical characterization of damage does not have a linear relationship with the characterization of the economic loss. Reparation costs exhibit high variability; however, uncertainty between the repair cost and damage index was not considered in this study.

#### 3.3. Estimating the Exceedance Probability of Loss

#### 3.4. Computing the Exceedance Rate of Loss

## 4. Accuracy of the Uniform Risk Spectra

#### 4.1. Summary of Mechanical Characteristics

^{2}(21,707 MPa); f’c =250 kg/cm

^{2}(24.5 MPa); Es = 2,010,000 kg/cm

^{2}(197,113 MPa); fy = 4200 kg/cm

^{2}(412 MPa). Table 3 summarizes the main structural characteristics for the analysis, and Figure 11 shows the structural typology for each structure.

#### 4.2. Risk Assessment Process

_{max}) for each structure; the gray lines represent the IDA curves for each seismic record, and the black line represents the average curve.

_{y}) were consistent with those of the analyzed structures. For this purpose, the values for T of the SDOFs were taken equal to the fundamental periods of the MRF structures, and the Sa

_{y}values were defined as the spectral ordinates Sa of the design URS for the corresponding periods. Later, the damage index was mapped to loss using Equation (2). Figure 14 shows the obtained relationships for each structure.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- WEF. The Global Risk Report 2019. In Insight Report, 14th ed.; World Economic Forum: Geneva, Switzerland, 2019. [Google Scholar]
- Federal Emergency Management Agency (FEMA). Seismic Performance Assessment of Buildings-Methodology; Fema P-58-1; FEMA: Washington, DC, USA, 2012; Volume 1, p. 278.
- Moehle, J.; Stojadidinovic, B.; Kiureghian, A.; Yang, T. An Application of PEER Performance-Based Earthquake Engineering Methodology; Pacific Earthquake Engineering Research Centre: Berkeley, CA, USA, 2005; pp. 2–5. [Google Scholar]
- Günay, S.; Mosalam, K. PEER Performance-based earthquake engineering methodology, revised. J. Earthq. Eng.
**2013**, 17, 829–858. [Google Scholar] [CrossRef] - Bazzurro, P.; Cornell, A.; Menun, C.; Motahari, M.; Luco, N. Advanced Seismic Assessment Guidelines; Pacific Earthquake Engineering Research Center: Berkeley, CA, USA, 2006. [Google Scholar]
- Esteva, L. Bases para la Formulacion de Decisiones de Diseno Sismico. Ph.D. Thesis, Universidad Autonoma Nacional de Mexico, Mexico City, Mexico, 1968. [Google Scholar]
- Esteva, L. Seismic risk and seismic design decisions. In Proceedings of the MIT Symposium, Seismic Design For Nuclear Power Plants, Cambrige, MA, USA, 15 March 1970. [Google Scholar]
- McGuire, R. Probabilistic seismic hazard analysis and design earthquakes. Bull. Seismol. Soc. Am.
**1995**, 85, 1275–1284. [Google Scholar] [CrossRef] - McGuire, R. Probabilistic seismic hazard analysis: Early history. Earthq. Eng. Struct. Dyn.
**2007**, 37, 329–338. [Google Scholar] [CrossRef] - Noreña, F.; Castañeda, C.; Iglesias, J. The Mexico Earthquake of September 19, 1985—Evaluation of the Seismic Capacity of Buildings in Mexico City. Earthq. Spectra
**1989**, 5, 19–25. [Google Scholar] [CrossRef] - Hail, F.; Beck, J. Structural damage in Mexico City. Geophys. Res. Lett.
**1986**, 13, 589–592. [Google Scholar] [CrossRef] - Bruneau, M. Preliminary report of structural damage from Loma Prieta (San Francisco) earthquake 1989 and pertinence to Canadian structural engineering practice. Can. J. Civ. Eng.
**1990**, 17, 198–208. [Google Scholar] [CrossRef] - EERI. Northridge Earthquake, January 17, 1994. Preliminary Reconnaissance Report; 94-01; Earthquake Engineering Research Institute: Oakland, CA, USA, 1994. [Google Scholar]
- Muguruma, H.; Nishiyama, M.; Watanabe, F. Lessons learned from the Kobe earthquake—A Japanese perspective. PCI J.
**1995**, 40, 28–42. [Google Scholar] [CrossRef] - Mera, W.; Vera, X.; La Tegola, A.; Ponce, G. April 2016 Ecuador Earthquake of Moment Magnitude Mw7.8: Overview and Damage Report. Key Eng. Mater.
**2017**, 747, 662–669. [Google Scholar] [CrossRef] - Reinoso, E.; Quinde, P.; Buendía, L.; Ramos, S. Intensity and damage statistics of the September 19, 2017 Mexico earthquake: Influence of soft story and corner asymmetry on the damage reported during the earthquake. Earthq. Spectra
**2021**, 37, 1875–1899. [Google Scholar] [CrossRef] - Park, R.; Paulay, T. Reinforced Concrete Structures; John Wiley and Sons: New York, NY, USA, 1976. [Google Scholar]
- Xiang, N.; Shahria, M. Displacement-based seismic design of bridge bents retrofitted with various bracing devices and their seismic fragility assessment under near-fault and far-field ground motions. Soil Dyn. Earthq. Eng.
**2019**, 119, 75–90. [Google Scholar] [CrossRef] - Moehle, J. Displacement-based of RC structures subjected to earthquakes. Earthq. Spectra
**1992**, 8, 403–428. [Google Scholar] [CrossRef] - Priestley, M.; Calvi, G. Concepts and procedures for direct displacement-based design and assessment. In Seismic Design Methodologies for the Next Generation of Codes; Fajfar, P., Krawinkler, H., Eds.; Routledge: Rotterdam, The Netherlands, 1997; pp. 171–182. [Google Scholar]
- López, S.; Ayala, G. Displacement-based seismic design method for RC frames. Rev. Mex. Ing. Sísmica
**2013**, 88, 91–111. (In Spanish) [Google Scholar] - NZSEE; SESOC; NZGS. The Seismic Assessment of Existing Buildings; Technical Guidelines for Engineering Assessment: Wellington, New Zealand, 2017. [Google Scholar]
- PEER. Guidelines for Performance-Based Seismic Design of Tall Buildings; Report No. 2017/06; PEER & Charles Pankow Foundation: Berkeley, CA, USA, 2017. [Google Scholar]
- Riddell, R.; Newmark, N. Statistical Analysis of the Response of Nonlinear Systems Subjected to Earthquakes; Department of Civil Engineering University of Illinois at Urbana-Champaign: Urbana, IL, USA, 1979. [Google Scholar]
- Riddell, R.; Hidalgo, P.; Cruz, E. Response modification factors for earthquake resistant design of short period buildings. Earthq. Spectra
**1989**, 5, 571–590. [Google Scholar] [CrossRef] - Hidalgo, P.; Arias, A. New Chilean code for earthquake resistant design of buildings. In Proceedings of the 4th U.S. National Conference Earthquake Engineering, Palms Spring, CA, USA, 20–24 May 1990. [Google Scholar]
- Nassar, A.; Krawinkler, H. Seismic Demands for SDOF and MDOF Systems; John Blume Earthquake Engineering Center; Department of Civil Engineering Report 95; Stanford University: Stanford, CA, USA, 1991. [Google Scholar]
- Miranda, E. Site-dependant strength-reduction factors. J. Struct. Eng.
**1993**, 115, 2166–2183. [Google Scholar] - Anand, V.; Kumar, S. Sensitivity of strength reduction factor for structures considering soil-structure interaction. Structures
**2022**, 39, 593–606. [Google Scholar] [CrossRef] - Miranda, E. Estimation of Maximum Inter-Storey Drift Demands in Displacement-Based Design. In Seismic Design Methodologies for the Next Generation of Codes; Fafjar, P., Krawinkler, H., Eds.; Routledge: Rotterdam, The Netherlands, 1997; pp. 253–264. [Google Scholar]
- Zerbin, M.; Aprile, A.; Spacone, E. New formulation of ductility reduction factor or RC frame-wall dual systems for design under earthquake loadings. Soil Dyn. Earthq. Eng.
**2020**, 138, 106279. [Google Scholar] [CrossRef] - Chikh, B.; Mebarik, A.; Laouami, N.; Mehani, Y. Inelastic deformation ratio for seismic assessment of structures. Procedia Eng.
**2017**, 199, 558–563. [Google Scholar] [CrossRef] - Mendoza, E.; Díaz, O.; Esteva, L. Consistent-reliability spectra for some nonlinear SDOF systems. In Proceedings of the ICCOSSAR 89 5th International Conference Structural Safety and Reliability, San Francisco, CA, USA, 10 August 1989. [Google Scholar]
- Gkimprixis, A.; Tubaldi, E.; Douglas, J. Comparison of methods to develop risk-targeted seismic design maps. Bull. Earthq. Eng.
**2019**, 17, 3727–3752. [Google Scholar] [CrossRef] - Avelar, C.; Ayala, G.; de León, A. Design spectra determination for performance based seismic design. In Proceedings of the 9th International Conference Applications Statistics Probability Civil Engineering, Amsterdam, The Netherlands, 6–9 July 2003. [Google Scholar]
- Rivera, J. Design approach based on UAFR spectra for structures with displacement-dependent dissipating elements. Earthq. Spectra
**2007**, 23, 417–439. [Google Scholar] [CrossRef] - Datta, D.; Ghosh, S. Estimating Park-Ang damage index using equivalent systems. In Proceedings of the 14th World Conference Earthquake Engineering, Beijing, China, 12–17 October 2008. [Google Scholar]
- Loth, C.; Baker, J. Rational design spectra for structural reliability assessment using the response spectrum method. Earthq. Spectra
**2015**, 31, 2007–2026. [Google Scholar] [CrossRef] - Niño, M.; Ayala, G.; López, S. Uniform fragility spectra for the performance-based seismic design of structures considering variabilities in structural properties. Earthq. Eng. Struct. Dyn.
**2018**, 47, 1742–1754. [Google Scholar] [CrossRef] - Kennedy, R.; Short, S.A. Basis for Seismic Provisions of DOE-STD-1020; Rep. No. UCRL-CR-111478; Lawrence Livermore National Laboratory, Livermore, Calif., and Rep. No. BNL52418; Brookhaven National Laboratory: Upton, NY, USA, 1994. [Google Scholar]
- Cornell, A. Calculating building seismic performance reliability: A basis for multi-level design norms. In Proceedings of the11th World Conference Earthquake Engineering, Acapulco, Mexico, 23–28 June 1996. [Google Scholar]
- Luco, N.; Ellingwood, B.; Hamburger, R.; Hooper, J.; Kimball, J.; Kircher, C. Risk-targeted versus current seismic design maps for the conterminous United States. In Proceedings of the SEAOC 2007 Convention, Squaw Creek, CA, USA, 26–29 September 2007. [Google Scholar]
- Sewell, R. Damage Effectiveness of Earthquake Ground Motion: Characterizations Based on the Performance of Structures and Equipment; Stanford University: Stanford, CA, USA, 1989. [Google Scholar]
- Borzognia, Y.; Hachem, M.; Campbell, K. Ground motion prediction equation (“attenuation relationship”) for inelastic response spectra. Earthq. Spectra
**2010**, 26, 1–23. [Google Scholar] [CrossRef] - Douglas, J. Ground Motion Prediction Equations 1964–2021; Department of Civil and Environmental Engineering, University of Strathclyde: Glasgow, UK, 2022. [Google Scholar]
- Zizmond, J.; Dolsek, M. Formulation of risk-targeted seismic action for the force- based seismic design of structures. Earthq. Eng. Struct. Dyn.
**2019**, 48, 1406–1428. [Google Scholar] [CrossRef] - Sanchez-Sesma, F.; Chávez-Pérez, S.; Suárez, M.; Bravo, M.; Pérez-Rocha, L.E. The Mexico Earthquake of September 19, 1985—On the Seismic Response of the Valley of Mexico. Earthq. Spectra
**1988**, 4, 569–589. [Google Scholar] [CrossRef] - Bard, P.; Campillo, M.; Chavez-Garcia, F.; Sanchez-Sesma, F. The Mexico Earthquake of September 19, 1985—A theoretical investigation of large- and small-scale amplification effects in the Mexico City valley. Earthq. Spectra
**1988**, 4, 609–633. [Google Scholar] [CrossRef] - Seed, H.; Romo, M.; Sun, J.; Jaime, A.; Lysmer, J. The Mexico earthquake of September 19, 1985—Relationships between soil conditions and earthquake ground motions. Earthq. Spectra
**1988**, 4, 687–729. [Google Scholar] [CrossRef] - Kawase, H.; Aki, K. A study on the response of a soft basin for incident S, P, and Rayleigh waves with special reference to the long duration observed in Mexico City. Bull. Seismol. Soc. Am.
**1989**, 79, 1361–1382. [Google Scholar] - Chavez-Garcia, F.; Bard, P. Site effects in Mexico City eight years after the September 1985 Michoacán earthquakes. Soil Dyn. Earthq. Eng.
**1994**, 13, 229–240. [Google Scholar] [CrossRef] - Singh, S.K.; Quaas, R.; Ordaz, M.; Mooser, F.; Almora, D.; Torres, M.; Vasquez, R. Is there truly a “hard” rock site in the Valley of Mexico? Geophys. Res. Lett.
**1995**, 22, 481–484. [Google Scholar] [CrossRef] - Reinoso, E.; Ordaz, M. Spectral ratios for Mexico City from free-field recordings. Earthq. Spectra
**1999**, 15, 273–295. [Google Scholar] [CrossRef] - Jaimes, M.; Reinoso, E.; Ordaz, M. Comparison of methods to predict response spectra at instrumented sites given the magnitude and distance of an earthquake. J. Earthq. Eng.
**2006**, 10, 887–902. [Google Scholar] [CrossRef] - Jaimes, M.; Gaytán, A.; Reinoso, E. Ground-motion prediction model from intermediate-depth intraslab earthquakes at the hill and lake-bed zones of Mexico City. J. Earthq. Eng.
**2015**, 19, 1260–1278. [Google Scholar] [CrossRef] - Cruz-Atienza, V.; Tago, J.; Sanabria-Gomez, D.; Chaljub, E.; Etienne, V.; Vireux, J.; Quintanar, L. Long duration of ground motion in the paradigmatic valley of Mexico. Nat.-Sci. Rep.
**2016**, 6, 38807. [Google Scholar] [CrossRef] [PubMed] - Cornell, A.; Krawinkler, H. Progress and challenges in seismic performance assessment. PEER Cent. News.
**2000**, 3, 1–4. [Google Scholar] - Reinoso, E.; Jaimes, M. Criteria to obtain design accelerograms in sites affected by several seismic sources. Rev. Mex. Ing. Sísmica
**2009**, 81, 1–18. (In Spanish) [Google Scholar] - Niño, M.; Ayala, G.; Ordaz, M. Ground-Motion Simulation by the Empirical Green’s Function Method with a Source Defined by Two Corner Frequencies and a Two-Stage Summation Scheme. Bull. Seismol. Soc. Am.
**2018**, 18, 901–912. [Google Scholar] [CrossRef] - Ordaz, M.; Miranda, E.; Reinoso, E. Expert system for the seismic loss assessment in Mexico. In Proceedings of the 12th Mexico Conferernce Earthquake Engineering, Morelia, Mexico, 17–20 November 1999. (In Spanish). [Google Scholar]
- Benjamin, J.; Cornell, A. Probability, Statistics and Decision for Civil Engineers; Courier Corporation: Chelmsford, MA, USA, 1970. [Google Scholar]
- Terán, A.; Jirsa, J. A damage model for practical seismic design that accounts for low cycle fatigue. Earthq. Spectra
**2005**, 21, 803–832. [Google Scholar] [CrossRef] - Borzognia, Y.; Bertero, V. Damage spectra: Characteristics and applications to seismic risk reduction. J. Struct. Div.
**2003**, 3, 411–438. [Google Scholar] - Takeda, T.; Sozen, M.; Nielsen, N. Reinforced Concrete Response to Simulated Earthquakes. Struct. Div. ASCE
**1970**, 96, 2257–2573. [Google Scholar] [CrossRef] - Nakashima, M.; Saburi, K.; Tsuji, B. Energy input and dissipation behaviour of structures with hysteretic dampers. Earthq. Eng. Struct. Dyn.
**1996**, 25, 483–496. [Google Scholar] [CrossRef] - Connor, J.; Wasa, A.; Iwata, M.; Huang, Y. Damage-controlled structures I: Preliminary design methodology for seismically active regions. Bull. Seismol. Soc. Am.
**1997**, 12, 423–431. [Google Scholar] [CrossRef] - Jing, J.; Ye, L.; Quian, J. Inelastic seismic response of lumped mass MDOF systems based on energy concept. Eng. Mech.
**2003**, 20, 31–37. [Google Scholar] - Qiang, H.; Feng, P. Seismic responses of postyield hardening single-degree-of-freedom systems incorporating high-strength elastic material. Earthq. Eng. Struct. Dyn.
**2019**, 48, 611–633. [Google Scholar] [CrossRef] - Borzi, B.; Calvi, G.; Elnashai, A.; Faccioli, E.; Boomer, J. Inelastic spectra for displacement-based seismic design. Soil Dyn. Earthq. Eng.
**2001**, 21, 47–61. [Google Scholar] [CrossRef] - Farrow, K.; Kurama, Y. SDOF demand index relationships for performance-based design. Earthq. Spectra
**2003**, 19, 799–838. [Google Scholar] [CrossRef] - Ruiz-García, J.; Miranda, E. Performance-Based Assessment of Existing Structures Accounting for Residual Displacements; Department of Civil and Environmental Engineering Report No. 153; Stanford University: Stanford, CA, USA, 2005. [Google Scholar]
- Fu, Q.; Menun, C. Residual displacement caused by fault-normal near-field ground motions. In Proceedings of the 8th U.S. National Conference Earthquake Engineering, San Francisco, CA, USA, 18–22 April 2006. [Google Scholar]
- Park, Y.; Ang, A.; Wen, Y. Seismic damage analysis of reinforced concrete buildings. J. Struct. Eng.
**1985**, 1, 740–757. [Google Scholar] [CrossRef] - Kanno, R. Strength Deformation and Seismic Resistance of Joints between Steel Beams and Reinforced Concrete Columns; Cornell University: Ithaca, NY, USA, 1993. [Google Scholar]
- Stone, W.; Taylor, A. Seismic Performance of Circular Bridge Columns Designed in Accordance with AASHTO/CALTRANS Standards; NIST Building Sciences Series 170; Federal Highway Administration: McLean, VA, USA, 1993.
- EERI. Expected Seismic Performance of Buildings; Publication Number SP-10; EERI: Oakland, CA, USA, 1994. [Google Scholar]
- Chacón, R.; Paz, I. Seismic Performance Analysis of Typical School Building 780 post 97 on the Peruvian Coast. Master’s Thesis, Catholic University of Peru, San Miguel, Peru, 2016. (In Spanish). [Google Scholar]
- Mexico City Government. Normas Técnicas Complementarias para Diseño y Construcción de Estructuras de Concreto; Gaceta Oficial de la Ciudad de México: Mexico City, Mexico, 2017. (In Spanish)
- Prakash, V.; Powell, G.; Campbell, S. DRAIN, Base Program Description and User Guide; Department of Civil Engineering, University of California: Berkeley, CA, USA, 1993. [Google Scholar]
- Miranda, E. Strength reduction factors in performance-based design. In Proceedings of the Symposium to Honor Vitelmo Vertero, Berkeley, CA, USA, 31 January–1 February 1997. [Google Scholar]
- Vamvatzikos, D.; Cornell, A. Incremental dynamic analysis. Earthq. Eng. Struct. Dyn.
**2001**, 31, 491–514. [Google Scholar] [CrossRef] - Kircher, C.; Nassar, A.; Kutsu, O.; Holmes, W. Development of building damage functions for earthquake loss estimation. Earthq. Spectra
**1997**, 13, 663–682. [Google Scholar] [CrossRef] - Calvi, G. A displacement-based approach for vulnerability evaluation of classes of buildings. J. Earthq. Eng.
**1999**, 3, 411–438. [Google Scholar] [CrossRef] - Panagiotakos, T.; Fardis, M. Deformations of reinforced concrete members at yielding and ultimate. ACI Struct. J.
**2001**, 98, 135–148. [Google Scholar]

**Figure 1.**3D surface of the relationship among loss ($\beta $), exceedance rate of loss ($v$ ($\beta $)), and lateral strength ($S{a}_{y}$) for a defined vibration period ($T$).

**Figure 4.**Damage spectra for SDOF associated to Sa

_{y}= 100 gals and seismic records with magnitude of (

**a**) 7.2 and (

**b**) 7.3.

**Figure 5.**The Park and Ang damage index and Teran and Jirsa damage index relationship. (Dots are the computed damage indexes, and the black line is the fitted curve).

**Figure 7.**Kolmogorov–Smirnov fitted test applied to the statistical data of the SDOF with T = 2.2 s and lateral strength of (

**a**) 200, (

**b**) 500, and (

**c**) 1000 gals under the action of M 8.0 seismic records.

**Figure 11.**MRF: (

**a**) single bay, single story; (

**b**) three bays, fifteen stories; (

**c**) three bays, twenty-five stories, dimensions in cm.

**Figure 14.**Interstory drift vs. loss (markers) and fitted curve (continuous line) associated to (

**a**) 1 story, (

**b**) 15 stories, and (

**c**) 25 stories for different magnitudes.

**Figure 16.**Exceedance rate of intensities associated to the studied seismic source for structural vibration periods of 0.12, 1.25, and 1.80 s.

**Figure 17.**Comparison between the estimated exceedance rate of loss from the design (URS) and the estimated exceedance rate of loss after the design using the PEER approach.

Park and Ang Damage Index Limits | Reparation Actions | Reparation Cost or Structural Loss Ratio (β) |
---|---|---|

DI = 0.0 | None | 0.00 |

0.00 < DI ≤ 0.10 | Paint restitution and restoration of architectural features with cement made cast in situ. | 0.03 |

0.10 < DI ≤ 0.25 | Repair cracks including the use of epoxy to restore the elements and architectural features. | 0.30 |

0.25 < DI ≤ 0.40 | Mortar covering should be replaced by specialized structural mortar and architectural features. | 0.62 |

0.40 < DI ≤ 1.0 | Restore the detached concrete with structural mortar. Replacement of whole structural elements. | 1.00 |

Park and Ang Damage Index Limits | Teran and Jirsa Damage Index Limits |
---|---|

DI = 0.0 | DI = 0.0 |

0.00 < DI ≤ 0.10 | 0.0 < DI ≤ 0.10 |

0.10 < DI ≤ 0.25 | 0.10 < DI ≤ 0.30 |

0.25 < DI ≤ 0.40 | 0.30 < DI ≤ 0.50 |

0.40 < DI ≤ 1.0 | 0.50 < DI ≤ 1.0 |

Structure ID | Stories | Beams Dimensions (cm) | Columns Dimensions (cm) | Vibration Period (s) |
---|---|---|---|---|

A | 1 | 24 × 40 | 40 × 40 | 0.12 |

B | 15 | 50 × 100 | 115 × 115 | 1.25 |

C | 25 | 80 × 120 | 160 × 160 | 1.70 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Buendía, L.; Niño, M.; Reinoso, E.; González, C.
Estimation of Uniform Risk Spectra Suitable for the Seismic Design of Structures. *Buildings* **2023**, *13*, 2165.
https://doi.org/10.3390/buildings13092165

**AMA Style**

Buendía L, Niño M, Reinoso E, González C.
Estimation of Uniform Risk Spectra Suitable for the Seismic Design of Structures. *Buildings*. 2023; 13(9):2165.
https://doi.org/10.3390/buildings13092165

**Chicago/Turabian Style**

Buendía, Luis, Mauro Niño, Eduardo Reinoso, and Carlos González.
2023. "Estimation of Uniform Risk Spectra Suitable for the Seismic Design of Structures" *Buildings* 13, no. 9: 2165.
https://doi.org/10.3390/buildings13092165