1. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Fatemeh Jalayer Assistant Professor Department of Structural Engineering University of Naples Federico II

2. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 One of the main attributes distinguishing performance-based earthquake engineering from traditional earthquake engineering is the definition of quantifiable performance objectives. Performance objectives are quantified usually based on life-cycle cost considerations, which encompass various parameters affecting structural performance, such as, structural, non-structural or contents damage, and human casualties. Probabilistic performance-based engineering can be distinguished by defining probabilistic performance objectives. Probabilistic Performance-Based Earthquake Engineering

3. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Performance objectives There is uncertainty in the future ground motion that is going to take place at the site of the engineering project. There is uncertainty in determining the parameters and building the mathematical model of the real structure.

4. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Performance Objective The performance objective can be stated in terms of the mean annual frequency of exceeding a limit state, e.g., collapse LS is the mean annual frequency of exceeding a limit state P 0 is the allowable frequency level is also known as the limit state probability or probability of failure

5. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 The probabilistic performance objective can be stated in terms of the mean annual frequency of demand exceeding capacity for structural limit state LS C LS is the structural capacity for limit state LS Dis the structural demand Probabilistic Performance Objective in terms of Structural Parameters

6. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Earthquake Ground Motion the Major Source of Uncertainty The uncertainty in the prediction of earthquake ground motion significantly contributes to the uncertainty in demand and capacity.

7. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Alternative Probabilistic Representations of Earthquake Ground Motion ADirect Probabilistic Representation of the Ground Motion BImplicit Probabilistic Representation of the Ground Motion

8. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Alternative Direct Probabilistic Representations of Ground Motion Uncertainty AProbabilistic Representation of Ground Motion using Intensity Measures (IM-Based, FEMA-SAC Guidelines, PEER Methodology) BComplete Probabilistic Representation of the Ground Motion Time History

9. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 It is assumed that the spectral acceleration is a sufficient intensity measure. A sufficient intensity measure renders the structural response (e.g.,  max ) independent of ground motion parameters such as M and R. Direct Probabilistic Representation of Ground Motion Using Intensity Measure

10. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CA Attenuation law: Abrahamson and Silva, horizontal motion on soil Direct Probabilistic Representation of Ground Motion Using Intensity Measure (IM) -- IM Hazard Curve A probabilistic representation of the ground motion intensity measure be stated in terms of the mean annual frequency of exceeding a given ground motion intensity level. This quantity is also known as the IM hazard curve.

11. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Current seismic design procedures (FEMA 356, ATC-40) take into account the uncertainty in the ground motion implicitly by defining “design earthquakes” with prescribed probabilities of exceeding given peak ground acceleration (PGA) values in a given time period (e.g., P o =10% probability in 50 years). Implicit Probabilistic Representation of Ground Motion in Current Seismic Design and Assessment Procedures Mean Annual Frequency of Exceeding PGA Also Known as PGA Hazard Curve PGA design

12. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Choice of IM The spectral acceleration at the small-amplitude fundamental period of the structure denoted by or simply, S a is adopted as the intensity measure (IM).

13. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Choice of Structural Response Parameter We have chosen the maximum inter-story drift angle,, a displacement- based structural response, as the structural response parameter. 241241241 157 105 105 105 106 105 105

14. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Structural Limit States The limiting states for which the assessments are done depend on the performance objectives. Here, we focus on the onset of global dynamic instability in the structure that can be considered as an indicator of imminent collapse in the structure. A non-linear dynamic analysis procedure called the incremental dynamic analysis can be used to determine the onset of global dynamic instability.

15. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Structural Limit State: Global Dynamic Instability Similar to a pushover curve that maps out the structural behavior for increasing lateral loads, an IDA curve maps out the structural response for incrementally increasing ground motion intensity.

16. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Representation of Ground Motion using Intensity Measures Probabilistic performance objective: IM-based presentation of the probabilistic performance objective: Seismic Hazard for IM PDF for Structural Response given IM CDF for Structural Capacity given Response

17. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Seismic Hazard (Direct Probabilistic Representation) for the Ground Motion Intensity Measure (IM)

18. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 source i: San Andreas Fault (M,R) site: Van Nuys Faults of Los Angeles region Ground motion and site parameters: magnitude, distance and/or additional variables Seismic Hazard Model

19. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Representation for IM for a given M and r The relation between IM and ground motion parameters, such as magnitude and distance, can be expressed in the following generic form: The spectral acceleration for a given magnitude and distance can be described by a log-normal distribution. The parameters of this distribution, namely, mean and standard deviation, are predicted by the ground motion prediction relation:

20. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Seismic Hazard for IM summation over all the surrounding seismic zones mean annual rate that an earthquake event of interest takes place at seismic zone i all the possible earthquake event scenarios that can take place on seismic zone i and which produce spectral acceleration larger than x. The mean annual rate of exceeding a given spectral acceleration value, also known as spectral acceleration hazard can be calculated as follows: attenuation relation

21. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CA Attenuation law: Abrahamson and Silva, horizontal motion on soil Spectral Acceleration Hazard Curve

22. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Representation for Structural Demand given IM Implementing Non-Linear Dynamic Analysis Methods

23. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Representation for Demand given Spectral Acceleration The record-to-record variability in structural demand for a given intensity level can be expressed by the conditional probability density function (PDF) of for a given level. Estimating using nonlinear dynamic analyses

24. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Representation for Demand The mean annual frequency of exceeding a given value of the structural demand parameter:

25. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Drift Hazard Curve

26. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Probabilistic Representation for Limit State Capacity Implementing Non-Linear Dynamic Analysis Methods

27. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Incremental Dynamic Analysis (IDA) The IDA curve provides unique information about the nature of the structural response of an MDOF system to a ground motion record.

28. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Estimating using nonlinear dynamic analyses The record-to-record variability in structural capacity can be expressed by the complementary cumulative distribution function (CCDF) of capacity for a given. A Probabilistic Representation for Structural Limit State Capacity

29. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Demand and Capacity Factored Design (DCFD)

30. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Demand and Capacity Factor Design (DCFD) Factored Demand (P o ) Factored Capacity The probabilistic performance objective: After algebraic manipulations and making a set of simplifying assumptions, an LRFD-like probabilistic design criterion for a given allowable probability level, P o, can be derived:

31. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Main Assumptions Leading to a Closed-form Expression for (DCFD)

32. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 The spectral acceleration hazard curve can be described by a power-law function (a linear function in the logarithmic scale).

33. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Demand (given spectral acceleration) can be described by a lognormal distribution with constant standard deviation and power- law median.

34. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Median capacity is described by a lognormal distribution with constant median and standard deviation.

35. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 A Closed-Form Analytical Solution the Annual Frequency of Exceeding Limit State Capacity is the spectral acceleration corresponding to median capacity.

36. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Closed-Form Presentation of DCFD Format Factored Demand (Po) Factored Capacity After algebraic manipulations and making a set of simplifying assumptions, an LRFD-like probabilistic design criterion for a given allowable probability level, Po, can be derived:

37. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Where is the spectral acceleration corresponding to median demand y. A Closed-Form Analytical Solution the Annual Frequency of Exceeding Structural Demand (Also Known as Drift Hazard)

38. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 P0P0 LS F.D. F.C. Drift hazard curve - closed form A Graphic Presentation of DCFD format: Factored Demand (Po) Factored Capacity

39. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Displacement-based Non-Linear Beam-Column Fiber Element Model in OPENSEES Structural Model: A Generic 8-Storey RC Frame Structure 600 200 600 400 300 300 300 300 300 300 300

40. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Approximating the Hazard Curve with a Line in the Region of Interest k=3

41. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Approximating Structural Demand as a Power-Law Function of Spectral Acceleration 1 b=1

42. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Calculating factored demand for the tolerable probability, P o =0.002: Factored Demand

43. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Evaluating Structural Capacity for the Limit State of Global Dynamic Instability

44. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Calculating factored capacity for global dynamic instability limit state: Factored Capacity

45. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Finally the “checking” moment: ? Factored Capacity Factored Demand (0.002) 0.023 0.052

46. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 In the presence of structural modeling uncertainty the statement for the performance objective can be written as: Where x the level of confidence in the statement of the performance objective.  LS represents the uncertainty in the limit state probability due to the presence of structural modeling uncertainty. DCFD Formulation Taking into Account the Structural Modeling Uncertainty (FEMA/SAC Formulation) x kxkx

47. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 DCFD Formulation Taking into Account the Structural Modeling Uncertainty (FEMA/SAC Formulation) After some algebraic manipulations the DCFD format can be presented as: where:  UT represents the uncertainty in the demand and capacity due to structural modeling uncertainty.

48. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Beam-column model with stiffness and strength degradation in shear and flexure using DRAIN2D-UW by J. Pincheira et al. 241 241 241 157 105 105 105 106 105 105 Structural Model: An Existing RC Frame Structure in Los Angeles Area

49. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Approximating the Hazard Curve with a Line in the Region of Interest

50. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Estimating the factored demand for the tolerable probability, P o =0.0084:

51. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Factored capacity estimation for the limit state of global dynamic instability: Getting help from the IDA's …

52. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Finally the “checking” moment: ? Factored Capacity Factored Demand (0.0084) 0.026 0.02

53. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 If the variability in response due to structural uncertainty can be represented by: And the factored demand to capacity ratio is equal to: There is 90% confidence associated with the statement of performance objective. x=90% k x =-1.31

54. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Conclusions Probabilistic performance-based engineering is based on quantifiable and probabilistic performance objectives. The probabilistic nature of the performance objectives is due to the uncertainties in the prediction of the future ground motion and also in the structural modeling. The uncertainty in the future ground motion input is the dominant source of uncertainty in the performance assessments.

55. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 Conclusions (Continued) DCFD is an analytical format for structural performance assessments that is based on probabilistic performance objectives. Non-linear dynamic analyses can be used to make structural performance assessments in the framework of the DCFD taking into account ground motion uncertainty. The uncertainty in structural model can be taken into account in the form of a confidence factor in the statement of the probabilistic performance objective.

56. Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment Sharif University of Technology, 25 April 2011 This Presentation is Prepared Based on the Following References: Cornell C. A., Jalayer F., Hamburger R. O., and Foutch D. A. (2002), “The probabilistic basis for the 2000 SAC/FEMA steel moment frame guidelines’’, ASCE Journal of Structural Engineering, April, 2002. Jalayer F., Franchin P. and Pinto P.E. (2007), “A scalar decision variable for seismic reliability analysis of RC frames”, Special issue of Earthquake Engineering and Structural Dynamics on Structural Reliability, Vol. 36 (13): 2050-2079, June 2007. Jalayer F., and Cornell C. A. (2009), “Alternative nonlinear demand estimation methods for probability-based seismic assessments”, Earthquake Engineering and Structural Dynamics, 38: 951-972, 2009. Jalayer F., and Cornell C. A. (2003), “A Technical Framework for Probability-Based Demand and Capacity Factor Design (DCFD) Seismic Formats”, PEER Report 2003/08. Jalayer F. (2003), “Direct Probabilistic Seismic Analysis: Implementing Non-linear Dynamic Assessments”, Ph.D. Dissertation, Department of Civil and Enviromental Engineering, Stanford University, California.

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