1. CE 5603 Seismic Hazard Assessment
CHAPTERS 5-6-7
Probabilistic Seismic Hazard Methodology Design Spectrum Risk Calculations
Design Time Series

2. The basic methodology of PSHA involves computing the how often a specified level of ground motion will be exceeded at the site.
distance from the source to the site
the annual rate of
events that produce a ground motion parameter, Sa, that exceeds a specified level, z
earthquake magnitude
Ni(Mmin) is the annual rate of earthquake with magnitude greater than or equal to Mmin
fm(m), and fr(r) are probability density functions for the magnitude, and distance
annual rate of exceedance
inverse of ν is called the "return period

3. The ground motion variability is contained in the P(Sa>z|M,r) term:
ε is the number of standard deviations of the ground motion
fε(ε) is the probability density function for the number of standard deviations (a standard normal distribution with mean 0 and variance 1)
Including the ground motion variability the hazard equation becomes:

4. An advantage of this from of the hazard integral is that it can be directly related to the deterministic approach. In the deterministic approach, the magnitude, distance, and number of standard deviation of the ground motion need to be specified. The hazard integral is simply constructing a suite of all possible deterministic scenarios in terms of (M, r, ε) triplets. The rate of each scenario is:
The integrals are simply summing up the rates of the scenarios!
Please remember: A rule for PSHA is that all aleatory variability is modeled by PDFs and there should be an integral over each PDF. Each aleatory variable is specified by a probability density function in the equation below!

5. For planar sources:
These four aleatory variables replace the single aleatory variable distance shown in the previous equation!
We need to consider the finite dimension and location of the rupture in to compute the closest distance which is used by the ground motion models. Given the rupture width, W, rupture length, L, location along strike, Locx, and location down dip, Locy, the closest distance from the source to the site can be computed.

6. Hazard From Multiple Sources:
For multiple seismic sources, the total annual rate of events with ground motions that exceed z at the site is the sum of the annual rate of events from the individual sources (assuming that the sources are independent).
Nsource is the total number of fault and areal sources
The rates are summed up over all sources because we are interested in how often severe shaking occurs at the site, regardless of what source caused the ground motion. Distinctions between ground motions from different magnitudes and distances are considered through the deaggregation.

7. To convert the annual rate of events to a probability, we need to consider the probability that the ground motion exceeds test level z at least once during a specified time interval.
Renewal Model
Poisson Process
In renewal model, the probability of an earthquake occurring is computed for a specified time period (e.g. the next 50 years). In this case, it is common to convert the earthquake probability from the renewal model to an equivalent Poisson rate:
For a Poisson process, the probability of at least one occurrence of ground motion level z in T years is given by:
Using this equivalent Poisson rate, the hazard can be computed using standard software with the Poisson assumption.

8. Deaggregation of Hazard:
The hazard curve gives the combined effect of all magnitudes and distances on the
probability of exceeding a given ground motion level. Since all of the sources, magnitudes, and distances are mixed together, it is difficult to get an intuitive understanding of what is controlling the hazard from the hazard curve by itself.
To provide insight into what events are the most important for the hazard at a given ground motion level,
the hazard curve is broken down into its contributions from different earthquake scenarios.
This process is called deaggregation (Bazzurro and Cornell, 1999).

9. Deaggregation of Hazard:
In a deaggregation, the fractional contribution of different scenario groups to the total hazard is computed. The most common form is a two-dimensional deaggregation in magnitude and distance bins. This approach allows the dominant scenario earthquakes (magnitude and distance pair) to be identified
In a hazard calculation, there is a large number of scenarios considered. To reduce this large number of scenarios to a manageable number, similar scenarios are grouped together.
Most hazard studies use equal spacing in magnitude space and distance space.
This may not be appropriate for a specific project. The selection of the grouping of scenarios should be defined by the engineers conducting the analysis of the structure (Abrahamson, 2006).

10. Deaggregation of Hazard:
Notes:
The deaggregation is normalized such that it sums to unity for all scenario groups.
Formally, it is the conditional probability of the ground motion being generated by an earthquake with magnitude in the range M1-M2 and distance in the range R1-R2.)
The deaggregation by magnitude and distance bins allows the dominant scenario
earthquakes (magnitude and distance pair) to be identified.
The results of the deaggregation will be different for different probability levels (e.g. 100 yr vs yr return periods) and for different spectral periods.

11. Deaggregation of Hazard: Example from Davis, CA
Sa at T=2 seconds
Sa at T=0.2 seconds
High frequency ground motions are controlled by nearby moderate magnitude earthquakes
Long period ground motions are controlled by distant large magnitude earthquakes

12. Uniform Hazard Spectrum
A common method for developing design spectra based on the probabilistic approach is to use the Uniform Hazard Spectrum (UHS) (or sometimes called Equal Hazard Spectrum). The UHS is developed by computing the hazard independently at a set of spectral periods and then computing the ground motion for a specified probability level at each spectral period.

13. The term “uniform hazard spectrum” is used because the spectral acceleration value at each period has an equal chance of being exceeded.
Because the hazard is computed independently for each spectral period, the UHS does not represent the spectrum of any single earthquake.
In practice, the hazard analyst often only provides the UHS to the engineer in the hazard report. A hazard report should also include a comparison of the UHS with the spectra from the individual representative events indentified by the deaggregation.
Dominated by M6.5-7
D30-50 km
Dominated by M7.5-8
D km

14. Let’s do a simple calculation for a simple source as example:
Fault A:
Fault length is 100 km, fault width is 12 km
b-value = 0.9
Style of faulting: Strike-slip, dip angle: 90 degrees
Youngs and Coppersmith (1985) Composite Magnitude PDF
Slip rate = 9 mm/year
Step 1: Calculate the parameters that you need!
Mean characteristic magnitude
Accumulated seismic moment
Seismic moment release per earthquake
Activity rate
Step 2: Follow the steps you seen in Chapter 1!
Let’s take a look at the spreadsheet

15. In seismic studies:
the term "hazard" is used for the rate of the ground motion at the site
the term "risk" is used for the rate of the consequences of the ground motion.
the annual probability that a ground motion will exceed 0.3g is a measure of the hazard
the annual probability that a building will collapse is a measure of the risk
Mathematically, the hazard integral is given by:

16. The risk integral is given by:
The probability of the consequence as a function of the ground motion level is called the fragility curve

17. Usually, the "consequence" is given in terms of discrete damage states.
For example, we could consider four possible damage states of a building after an earthquake as green tag, yellow tag, red tag, and collapse. Fragility curves describe the probability of a damage state or worst occurring as a function of the ground motion level.
the fragility curve for a yellow tag would give the probability that the building has a yellow tag, red tag, or collapsed.

18. The probability of the building being in a discrete state is given by the difference
between the fragility curve for that state and the fragility curve for the next worst
state. The probabilities of being in each damage state are:
For very high levels of shaking the probability of having a green tag go to zero, but the fragility curve for green tag goes to unity!

19. To compute the risk:
The hazard curve is first converted to a rate of scenario ground motions.
The probability of the specified consequence, given the ground motion, is computed using a fragility curve.
Multiplying the rate for the scenario ground motion by the probability of the consequence gives the marginal rate of the consequence.
Summing these marginal rates for all of the scenario ground motions gives the total rate for the consequence.
PGA (g)
Annual Hazard
0.001
4.95E-01
0.1
6.39E-02
0.2
2.20E-02
0.3
9.18E-03

20. PGA (g)
Annual Hazard
PGA Range
Rate of
Occurence
0.001
4.95E-01
4.31E-01
0.1
6.39E-02
4.19E-02
0.2
2.20E-02
1.28E-02
0.3
9.18E-03
This part comes from the Hazard Curve!
This part comes from the fragility curve!

21. Finally, combine hazard and fragility:
Multiply Column 2 with Columns 3-5 to get Columns 6-8

22. Non- Linear Dynamic Analysis Linear Dynamic Analysis
In many seismic analyses, ground motion time series are required in addition to the design response spectrum.
Non- Linear Dynamic Analysis
Linear Dynamic Analysis
Procedures for selecting and scaling ground-motion records for a site-specific hazard are described in building codes and have been the subject of much research in recent years.

23. What does the building code recommends?
CBC (2003) 1659A 4.2 Time Histories
Pairs of appropriate horizontal ground motion time history components shall be selected and scaled from no less than 3 recorded events.
Appropriate time histories shall have magnitudes, fault distances and source mechanisms that are consistent with those that control the design basis earthquake.
Where three appropriate recorded ground motion time histories are not avaliable, appropriate simulated ground motion time history pairs may be used.
UBC 1997 And IBC 2000
Increase the number of selected ground motion pairs and calculate the average response
For each scaled pair pair calculate the SRSS spectrum (square-root-sum of the squares)
(SRSS) Of The Two Horizontal Components – Should Not Be Less Than 1.4 The Design Spectrum Ordinates In The Range From 0.2T To 1.5T
Simply the codes in United States requires:
Use at least 3 ground motion pairs and take the maximum!
If you have more than 7 pairs of ground motion, you can use the average!

24. It is common practice to select emprical recordings of ground motion and scale these ground motions to the level of design spectrum.
Deaggregation of the hazard identifies the main contributors to the hazard
HOW?
The selection of the records and the amount of scaling that can be applied remain controversial!
Typically, the time series is selected from recorded ground motions with similar magnitudes and similar distances.

25. There are two main approaches used to develop design ground motions:
(a) scaling the ground motions
(b) adjusting the ground motions to match a design spectrum.
Scaling refers to multiplying the record by a constant factor at all time points and called time domain approach.
Scaled to match the PGA
Scaled to match the spectral acceleration at a specific period

26. The selection of the starting time histories for use in either scaling or spectral matching is important due to nonlinear response of the soil and structure.
It is common practice to select the initial ground motion time histories based on the seismological properties such as magnitude and distance to the fault.
Similar site conditions, style of faulting and directivity effects may also be considered in the selection process.

27. Additionally, the scale factor required to scale the time series to the design spectrum may also be considered. In general, scale factors closer to unity are preferred and many ground motion experts recommend a limit on the amount of scaling applied.
Recommendations:
Select the records based on seismological properties:
± 0.5 magnitude units (can be extended to ± 1)
Wide distance range (ex:0-30 km)
All styles of faulting earthquakes (but same tectonic setting)
Consider directivity conditions (forward, average, backward)
Consider site classes (for hard rock sites it is best to select the hard rock recordings)
No limits in the amount of scaling!
2. From the suite of candidate recordings:
Use a simple non-linear system as a proxy for more complicated full model of the structure
Select records that give closest to the average response of the simple non-linear system.

28. Example study for bridge response:
Target response spectrum for scaling
AS 1997, Magnitude 7 and Distance 5 km
114 recordings are selected
2 different scaling approaches:
-same scale factor to all components (code approach)
-different scale factors to different components

29. Same scale factor to all components (code approach) Please note that:
There is a large variability in the response for the records that have the similar scale factor.
So, even if the scale factor for a record is near unity, that record may not give a good estimate of the average response.
Variability of the response is not sensitive to the scaling of the record.

30. Different scale factor to different components: Please note that:
Some of the records with large scale factors produce response close to average, whereas, some of the records with scale factors close to unity product response values greater than or less than the average.
It is possible to get unbiased results even for large scale factors

31. RSPMatch software may be used for spectral mathching!
Spectrum Compatible Methods (spectral matching)
Real strong motion records have response spectral peaks and troughs that impact the non-linear response of a structure.
Spectrum compatible time histories are modified in terms of their frequency content to match the entire spectrum.
Ideally, we should use unmodified time histories to sample this behavior, but sampling requires numerious time histories.
If the spectrum compatible methods are used, then a small number of sets of time histories can be used and still provide a reliable estimate of the average response of the structure.
The downside of the spectrum compatible method is the elimination of the variability of response by matching the time history to the target spectrum.
It is a money saving method and depends on the personal feelings of the structural engineers.
RSPMatch software may be used for spectral mathching!

32. Example study for bridge response:
Spectrum compatible recordings:
The scatter in both of the response parameters reduces as expected.
The decrease in the variability of the response is due to the reduced variability in the ground motion records that are modified to match the target spectrum.