Zhengxiang Yi, Ph.D., Lynne Burks, Ph. D., Youngsuk Kim, Ph.D., Jaskanwal Chhabra, Ph.D., Shabaz Patel.
Introduction – Soft Story Building Vulnerability
If you rent or own a home on the US west coast, earthquake risk is an inevitable topic. Homeowners have many concerns on earthquake risk, such as seismic capacity of the house, the possibility of an earthquake happening, and earthquake insurance. In this article, we will briefly address common questions on earthquake risk by introducing the vulnerability of wood-frame houses and the Los Angeles Ordinance Retrofit program, which could serve as an example for other municipalities attempting to increase earthquake resilience
A wood frame building is the most widely used construction type for residential buildings in the United States. In engineering practice, it is common to have fewer walls on the first story of a building in order to create more open space. This open space, known as a “soft story”, is often used for parking spaces or retail space. You have most likely seen or lived in one of such buildings.
However, such a widely applied construction type could be a huge threat to life safety under earthquakes. The open space on the first story leads to discontinuity in building strength and lateral stiffness along the building height. As a result, during the earthquake, the weak story cannot provide enough resistance to earthquake loading and lead to partial or complete damage. Figure 1 is showing a representative example of a soft story collapsed building under earthquake excitation. Many collapsed buildings due to soft-story damage have been observed in prior earthquakes including the 1971 San Fernando , 1989 Loma Prieta , and 1994 Northridge Earthquake .
Figure 1. Collapsed Apartment Building in Loma Prieta Earthquake, caused by a soft-story 
How to Resolve the Issue – Los Angeles Ordinance Retrofit
The vulnerability of soft-story buildings could potentially lead to substantial fatalities and financial losses during an earthquake, but policy actions are being taken to address such issues. Los Angeles is one of several cities in California that has established policies to address the seismic risk to soft-story wood frame buildings. The Los Angeles Soft-Story Ordinance, which was signed into law in 2015 and later amended, was part of the city’s Resilience by Design initiative . The ordinance mandates the retrofitting of multi-family (more than three units) wood frame buildings originally constructed prior to 1978, and with soft-story issues. According to the Los Angeles Times, there are approximately 13,400 soft story buildings distributed in the city of Los Angeles (see Figure 2). You can search whether your property needs seismic retrofit from their website: https://graphics.latimes.com/soft-story-apartments-needing-retrofit/.
Figure 2. SWOF Building Distribution in the City of Los Angeles
According to the program compliance, property owners must submit proof of the previous retrofit or plans to retrofit or demolish within the first 2 years of the plan and obtain construction permits within 3.5 years. All construction work is anticipated to finish in 7 years. As of Mar 1st, 2021 (6 years after the program kicked off), 92% of properties have submitted plans, 65% of building owners have obtained construction permits, and 45% have completed the retrofitting construction work (see Figure 3).
Figure 3. Soft Story Retrofit Program Status as of Mar 1st 2021
In most cases, seismic retrofitting of soft story buildings involves adding additional structural components (e.g., bracing system to the soft story to supply the building with additional lateral force resistance. For example, in Figure 4, the soft-story used for parking initially contains very few structural panels. Steel moment frames and braces are added to the open space to increase earthquake resistance. The braces allow the lateral forces to be delivered from the existing wood panels to the exterior walls and columns more easily. As a result, the building will retain its integrity under earthquake excitation.
Figure 4. Example of Soft Story Building Retrofitting with Moment Frame 
How vulnerable is your home?
A fragility curve is the most common representation of building seismic performance. It is used for quantifying the probability of collapse or being in damage levels conditioned on the earthquake intensities. Fragility curves are developed by running numerical analysis at multiple different earthquake hazard intensity levels and (historical or artificial) ground motion records will be used as input excitation to the numerical models.
We created numerical models in OpenSees for 96 different types of existing (32) and corresponding retrofitted buildings (64) . These types differentiate from each other by floor plan, building size, number of stories, building materials, and retrofit designs. Each component, including exterior and interior wall panels, diaphragm, and retrofit components was explicitly modeled in OpenSees. Incremental dynamic analyses were performed using 44 pairs of ground motion specified in FEMA-P695  to obtain the building seismic responses under multiple levels of ground shaking intensities. We will show an example for one of the analyses.
The selected example is a 60 ft by 30 ft 2-story building, with gypsum and stucco as exterior wall, gypsum wallboard as interior walls. The example has a similar soft-story configuration to the building shown in Figure 5, where the right side of the first floor is open for parking spaces.
Figure 5. An example of 2-story gypsum building 
Figure 6 shows the collapse fragility curve, which gives the probability of collapse at different earthquake shaking intensities for the existing and retrofitted example buildings . Let’s look at the x-axis and find the place where the spectral acceleration of 0.25 sec equals 1.5g, which roughly corresponds to the 500-year return period ground motions in LA . Comparing the fragility curves at 1.5g, the probability of collapse for the existing building (blue curve) is about 0.71. After being retrofitted, the probability of collapse decreases to 0.35. Retrofit significantly improves the building’s resistances under earthquakes and prevents fatalities led by building collapse.
Figure 6. Fragility curve of the existing and retrofitted example building (adapted from )
Understanding Earthquake Hazard from the Quantitative Perspective – What are we talking about when we say a 500-year earthquake?
The public is always curious if the experts can predict an earthquake. Unfortunately, there is no model or theory that can accomplish such sophisticated and uncertain predictions. Instead, probabilistic seismic hazard assessment (PSHA) is developed to interpret seismic activities probabilistically rather than predicting an earthquake occurrence. Within this framework, the chance of a certain number of earthquakes happening in a certain period of time is described by the Poisson process, which is a statistical model mostly used for modeling the number of events occurring in an interval of time or space. For a Poisson process, the probability of an event happens k times in a unit time/space interval is
(k) = k (-)k!
Here, could be interpreted as the average number of events happening within a unit interval. In practice, return periods are commonly defined with respect to attributes of earthquake events, for example, local attributes like exceeding a shaking intensity at a given location, or more commonly, regional attributes like exceeding a certain regional loss threshold. It should however be noted that when the return period is defined with respect to local attributes like exceeding a shaking intensity at a given location, no single earthquake event can result in the same return period intensity at multiple locations.
The Poisson process describes the probability of a certain number of events occurring within a unit time span. From the perspective of earthquake engineering, practitioners are more interested in the chances of any such earthquakes happening within the building design life span such that the buildings could be designed to resist the earthquake loading to meet specific performance targets such as collapse prevention.
Deriving from the Poisson process, with the rate , the probability of at least a single earthquake happening within a period of time t could be calculated using the following equation.
(T<t) = 1-(-t)
Intuitively, the longer a time span is, the higher the probability of observing one or more events. Other than the frequency of earthquakes, the civil engineers are of more interest in the shaking intensity at an individual site. The ground shaking intensity could be used for designing new facilities or evaluating existing facilities. The shaking intensity of an earthquake event is correlated with earthquake source characteristics, earthquake magnitude, site-to-source distance, soil condition, among others.
How is the above knowledge applied to engineering practice and what does it inform us? The above framework provides a consistent language to decision makers and stakeholders to understand, communicate and mitigate the seismic risk. For example, the Los Angeles Ordinance Retrofit program  requires the retrofitting component to be designed for 75% of the 500-year (more precisely 475-year) return period earthquake hazard. Using the equation above, the 500-year return period hazard has a 10% probability of exceedance in a 50 year time span. The design for such an intensity level emphasizes on collapse prevention and ensures life safety.
What’s the financial benefit of retrofitting for homeowners?
Clear estimates of financial loss from earthquakes are very valuable for homeowners.Equipped by PSHA, experts can have an estimation of the possible financial losses. Experts use three scenarios to analyze the earthquake-induced losses: 1) the building collapse and the owner suffers total building loss; 2) the building doesn’t collapse but cannot be either used or repaired, and the owner also suffers total loss; 3) the building doesn’t collapse but some parts of components or contents are damaged, thus the owner needs to pay for the reparations. At each shaking intensity level, the losses and associated probabilities of happening for the three scenarios could be calculated using the structural responses obtained from numerical analyses. The outcome of this step would be loss curves, which show the estimated financial losses given an earthquake intensity. Then, the loss curves could be aggregated with probabilities of earthquake occurrence to come up with an expected annual loss. An expected annual loss describes the annualized potential financial losses caused by earthquake events, and it could be used for estimating earthquake insurance premiums.
Then, how do we know how much the house owner can save by retrofitting their property? The benefit of retrofitting your property could be split into two parts: reduced earthquake losses and premium savings. For the former, the buildings’ capacities to resist future earthquakes are strengthened such that the potential financial losses and cost for repairing could be reduced. As for the latter, the California Earthquake Authority offers insurance premium discounts ranging from 10% to 25% for seismically retrofitted buildings .
Let’s continue with the previous 2-story building example. Following the aforementioned procedure, the expected annual loss due to earthquake damage of the building without being retrofitted is 3.8% of its own value. As for the same building applying retrofit, the expected annual loss is decreased to 1.4%. The direct annual saving from reduced earthquake loss is about 2.4% of building value. In the following figure, we show the cumulative earthquake-related cost for existing and retrofitted buildings. The annual cost includes the annual earthquake loss and earthquake insurance premium. Based on a report, the premiums of earthquake insurance generally range from $800 to $5,000 , and here we are using the lower ($800) and upper bound ($5,000) to do the calculation. Based on these data, Figure 7 is developed to present the cumulative earthquake-related cost of existing and retrofitted prototype buildings. The orange bar shows a one-time retrofit cost of $8,000, which is estimated based on the size of the building, construction conditions, soft-story type, and retrofitting method. For the two cases, the savings from reduced earthquake-induced losses and insurance discounts could cover the one-time retrofitting cost within the first 2 years after the construction has been finished. For all the analyzed archetypes, it takes on average 5 years for the benefits to cover the retrofit cost. Though the break-even time for a couple of cases could be as long as 30 to 35 years, overall, it is cost-effective.
Figure 7. Accumulated cost versus time for existing and retrofitted building with (a) $800 and (b) $5,000 earthquake insurance premium 
In this article, we briefly introduced the concepts and methodology of quantifying earthquake risk, building seismic fragility, and building earthquake-induced losses using the example of Los Angeles soft-story building retrofit.
The city of Los Angeles initiated the Ordinance Retrofit program to resolve the vulnerability issues of soft-story buildings which are extremely vulnerable during earthquakes if they are properly designed and constructed. As shown in this article, we can anticipate significant benefits in finance and life safety from the program. For the example 2-story residential building shown in the article, the probability of collapse could be reduced from 71% to 35% under 500-year return period ground motions in LA. Retrofitting the building is not as costly as expected from the financial perspective, because the benefit from reduced earthquake losses and insurance discount could compensate for the one-time retrofit cost in 5 years after the retrofit work has been completed in this case study.
Soft-story buildings are not an issue only observed on the west coast of the U.S. Soft story vulnerability is a threat to life safety in all earthquake active regions. Our mission will motivate further efforts on expanding our capacity to a more comprehensive understanding and larger scale of evaluation of the soft-story risk and mitigation.
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