Demand and Capacity Factor Design: A Performance-based Analytic Approach to Design and Assessment

1. 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. Probabilistic Performance-Based Earthquake Engineering 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.

3. 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. The performance objective can be stated in terms of the mean annual frequency of exceeding a limit state, e.g., collapse lLS is the mean annual frequency of exceeding a limit state P0 is the allowable frequency level is also known as the limit state probability or probability of failure Probabilistic Performance Objective

5. Probabilistic Performance Objective in terms of Structural Parameters • The probabilistic performance objective can be stated in terms of the mean annual frequency of demand exceeding capacity for structural limit state LS • CLS is the structural capacity for limit state LS • D is the structural demand

6. 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. Alternative Probabilistic Representations of Earthquake Ground Motion • Direct Probabilistic Representation of the Ground Motion • Implicit Probabilistic Representation of the Ground Motion

8. Alternative Direct Probabilistic Representations of Ground Motion Uncertainty • Probabilistic Representation of Ground Motion using Intensity Measures (IM-Based, FEMA-SAC Guidelines, PEER Methodology) • Complete Probabilistic Representation of the Ground Motion Time History

9. Direct Probabilistic Representation of Ground Motion Using Intensity Measure • It is assumed that the spectral acceleration is a sufficient intensity measure. • A sufficient intensity measure renders the structural response (e.g., qmax) independent of ground motion parameters such as M and R.

10. 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. Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CA Attenuation law: Abrahamson and Silva, horizontal motion on soil

11. Implicit Probabilistic Representation of Ground Motion in Current Seismic Design and Assessment Procedures 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., Po=10% probability in 50 years). PGA design Mean Annual Frequency of Exceeding PGAAlso Known as PGA Hazard Curve

12. The spectral acceleration at the small-amplitude fundamental period of the structure denoted by or simply, Sa is adopted as the intensity measure (IM). Choice of IM

13. 105 105 105 105 105 106 157 241 241 241 Choice of Structural Response Parameter We have chosen the maximum inter-story drift angle, , a displacement-based structural response, as the structural response parameter.

14. 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. 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. PDF for Structural Response given IM CDF for Structural Capacity given Response Seismic Hazard for IM Probabilistic Representation of Ground Motion using Intensity Measures Probabilistic performance objective: IM-based presentation of the probabilistic performance objective:

17. Seismic Hazard (Direct Probabilistic Representation) for the Ground Motion Intensity Measure (IM)

18. source i: San Andreas Fault (M,R) site: Van Nuys Faults of Los Angeles region Seismic Hazard Model Ground motion and site parameters: magnitude, distance and/or additional variables

19. 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. Seismic Hazard for IM The mean annual rate of exceeding a given spectral acceleration value, also known as spectral acceleration hazard can be calculated as follows: attenuation relation summation over all the surrounding seismic zones all the possible earthquake event scenarios that can take place on seismic zone i and which produce spectral acceleration larger than x. mean annual rate that an earthquake event of interest takes place at seismic zone i

21. Spectral Acceleration Hazard Curve Spectral acceleration hazard curve for: T=0.85sec - Van Nuys, CA Attenuation law: Abrahamson and Silva, horizontal motion on soil

22. Probabilistic Representation for Structural Demand given IM Implementing Non-Linear Dynamic Analysis Methods

23. 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. Probabilistic Representation for Demand The mean annual frequency of exceeding a given value of the structural demand parameter:

25. Drift Hazard Curve

26. Probabilistic Representation for Limit State Capacity Implementing Non-Linear Dynamic Analysis Methods

27. 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. Estimating using nonlinear dynamic analyses A Probabilistic Representation for Structural Limit State Capacity The record-to-record variability in structural capacity can be expressed by the complementary cumulative distribution function (CCDF) of capacity for a given .

29. Demand and Capacity Factored Design (DCFD)

30. Factored Demand (Po) Factored Capacity Demand and Capacity Factor Design (DCFD) 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, Po , can be derived:

31. Main Assumptions Leading to a Closed-form Expression for (DCFD)

32. The spectral acceleration hazard curve can be described by a power-law function (a linear function in the logarithmic scale).

33. Demand (given spectral acceleration) can be described by a lognormal distribution with constant standard deviation and power-law median.

34. Median capacity is described by a lognormal distribution with constant median and standard deviation.

35. A Closed-Form Analytical Solution the Annual Frequency of Exceeding Limit State Capacity is the spectral acceleration corresponding to median capacity.

36. Factored Demand (Po) Factored Capacity Closed-Form Presentation of DCFD Format 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. 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. Factored Demand (Po) Factored Capacity A Graphic Presentation of DCFD format: Drift hazard curve - closed form P0 lLS F.C. F.D.

39. 300 300 300 300 300 300 300 400 600 600 200 Structural Model: A Generic 8-Storey RC Frame Structure Displacement-based Non-Linear Beam-Column Fiber Element Model in OPENSEES

40. k=3 Approximating the Hazard Curve with a Line in the Region of Interest

41. b=1 1 Approximating Structural Demand as a Power-Law Function of Spectral Acceleration

42. Factored Demand Calculating factored demand for the tolerable probability, Po=0.002:

43. Evaluating Structural Capacity for the Limit State of Global Dynamic Instability

44. Factored Capacity Calculating factored capacity for global dynamic instability limit state:

45. ? Factored Capacity Factored Demand (0.002) 0.023 0.052 Finally the “checking” moment:

46. x DCFD Formulation Taking into Account the Structural Modeling Uncertainty(FEMA/SAC Formulation) 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. blLSrepresents the uncertainty in the limit state probability due to the presence of structural modeling uncertainty. kx

47. DCFD Formulation Taking into Account the Structural Modeling Uncertainty(FEMA/SAC Formulation) • After some algebraic manipulations the DCFD format can be presented as: where: bUT represents the uncertainty in the demand and capacity due to structural modeling uncertainty.

48. 105 105 105 105 105 106 157 241 241 241 Structural Model: An Existing RC Frame Structure in Los Angeles Area Beam-column model with stiffness and strength degradation in shear and flexure using DRAIN2D-UW by J. Pincheira et al.

49. Approximating the Hazard Curve with a Line in the Region of Interest

50. Estimating the factored demand for the tolerable probability, Po=0.0084: