1. Management of Earthquake Risk using Condition Indicators Determination of vulnerability functions
Jens Ulfkjaer

2. Outline What is the project MERCI about
What is a vulnerability function?
Six steps for determination of the vulnerability function for a structural class.
Choice of the Attenuation function and intensity of the load
Simulation of a set of time histories
Simulation of a set of materials parameters
Structural analysis
Determination of the structural damage
Calculations of the damage probabilities and the vulnerability function
Example

3. What is a vulnerability function?
A vulnerability function is a function that describes the probability of obtaining a certain level of damage dependent on the intensity of the load.

4. Deterministic example

5. Stochastic problem The load in it self is stochastic
The structure class will compromise many structures with different sizes, irregularities, and material parameters
Even at the single house level the strength parameters are only known with a certain probability

6. In general

7. 1) Choice of the Attenuation function and intensity of the load
The are many different available attenuation relationships in the literature and particular attention should be paid to the choice.
For the moment, as a rough choice, we have chosen Boore et al

8. 2) Set of acceleration time histories
Basically two types exist: real records and simulated records

9. 3) Simulation of a set of materials parameters
The material and the structural parameters should be simulated with certain statistical parameters.

10. 4) Structural analysis
The structural response can be calculated in many different ways: analytically and numerically
We are at the moment using the Finite Element Method and the program openSees

11. 5) Determination of the structural damage
The damage should be determined from the response calculated by openSees.
Many possibilities could be imagined but as a beginning we will use the maximal drift during the earthquake.

12. The damage will be categorized in different levels (normally four)
6) Calculations of the damage probabilities and the vulnerability function
The damage will be categorized in different levels (normally four)
The vulnerability is then the probability of having a certain level of damage

13. Example The following house is used 6m 4m mass1 Sec1 Sec2 B y x Mass 1
2.5cmmm
Section A-A:
25cmmm
45cmmm
As = 493mm2
A’s = 226mm2 y z
All rebars 16 (201mm2)
Section B-B:
35cmmm
2.5cmmm z x
Mass 1
Mass x-direction : 1.53 tons

14. Concrete fc (MPa) epsc fu (MPa) epsu Cover Concrete 21 0.002 0.01
0.015
Sec1 Core
24.2
2.98x10-3
21.3
6.89 x10-3
Sec2 Core
26.3
3.83 x10-3
23.8
8.88 x10-3

15. Longitudinal Reinforcement Steel
fy (MPa)
E0 (MPa) B
Longitudinal Reinforcement Steel
420
2x105

16. Material properties variation
The steel yielding strength been assumed to be normal distributed and 250 displacement time series has been calculated (with example3.3 from openSees manual)

20. Load 3 intensities and 3 distances are used
Intensities: M1=5.5 M2=6.5 M3=7.5
Distances:R1=20 km R2=40 km R3=80 km
For each of these nine combinations 10 time series has been generated.

22. Conclusions
The software PreOpenSeesPost developed for calculating vulnerability functions in 2D and 3D is almost finished.
More functions to interact with the user needs to be developed.
More complex structures should be tested
More structural element types as masonry has to be implemented