1. Seismic Performance Modeling of Reinforced Concrete Bridges
Stephen Mahin
Byron and Elvira Nishkian Professor of Structural Engineering
Mahmoud Hachem, Brian Buckman and Colin Cook Graduate Student Reseachers
University of California at Berkeley
2002 PEER Annual Meeting

2. PEER Bridge Program Focus on:
Monolithic reinforced concrete bridge construction
New rather than older construction detailing
Representative of:
Viaducts
Overcrossings
Major interchanges
2002 PEER Annual Meeting

3. Many Elements Involved
Approaches
Abutments
Foundations
Movement Joints
Columns/Piers
Superstructure
Nonstructural Features
Thrust Area 5
Structural Performance
2002 PEER Annual Meeting

4. Spirally Reinforced Column Tests
Test Matrix
Loading history
Traditional cyclic
Pulse initiated cyclic
Variable axial load
Shaking table testing
Loading rates:
Fast and quasi-static
Aspect ratios:
Moderate and low
Cross-section
Circular and interlocking spirals
2002 PEER Annual Meeting

5. Discuss: Shaking Table Tests
Objectives:
Data to validate analytical models
Compare performance for near-fault and long-duration excitations
Assess effects of multiple components of ground motion
Assess cumulative damage models
Effect of cross-sectional geometry
Circular sections with spirals
Noncircular with interlocking spirals
2002 PEER Annual Meeting

6. After First Maximum Level Event (m=6)
Column Performance
After Design Level Event (R=4)
After First Maximum Level Event (m=6)
2002 PEER Annual Meeting

7. Condition at end of tests
Buckled Bars
After sixth repetition of Maximum Run - Olive View
Fractured Spiral Fractured Bar
2002 PEER Annual Meeting

8. Long Duration Excitations
1985 Llolleo, Chile Record
2002 PEER Annual Meeting

9. Peak Displacement Response
Bi-directional input has limited effect and in the cases considered extends life of column
2002 PEER Annual Meeting

10. Bi-directional Response
Ground motion characteristics have a large effect on:
Nature of bi-directional response
Sensitivity of maximum displacements to intensity
Residual displacements
Currently design criteria, for ideal conditions and without significant P-D effects or eccentric gravity loads, result in well-performing columns with significant reserve capacity
Displacement, in.
Displacement, in.
2002 PEER Annual Meeting

11. Verification of Analytical Models
Response quantities:
Analytical Models:
Elastic Analysis with equivalent sectional Stiffness (EI e)
Concentrated hinge models with equivalent plastic hinge properties
Fiber models with distributed section properties with equivalent material properties
Global:
Displacements,
Residual Displacements,
Forces, Moments
Local:
Curvatures,
Strains,
Slip Rotations, …
Cumulative Damage
An important objective of the shaking table tests was the verification of current analytical models in terms of their ability to predict response quantities. Those quantities can be divided into:
2002 PEER Annual Meeting

12. Elastic Models
Various assumptions for approximating “effective” section stiffness EI
EIe as defined by Caltrans gives reasonable results for maximum displacement
EIe
Test
Maximum Credible
2002 PEER Annual Meeting

13. Elastic Models
Various assumptions for approximating “effective” section stiffness EI
EIe as defined by Caltrans gives reasonable results for maximum displacement
Not always
EIe
Test
Maximum Credible
Maximum Credible
Lateral
Direction
2002 PEER Annual Meeting

14. Elastic Models
Various assumptions for approximating “effective” section stiffness EI
EIe as defined by Caltrans gives reasonable results for maximum displacement
Not always
EIe
Test
Maximum Credible
Design Level
2002 PEER Annual Meeting

15. Elastic Models
Various assumptions for approximating “effective” section stiffness EI
EIe as defined by Caltrans gives reasonable results for maximum displacement
Not always
No information on residual displacements
Other engineering demand parameters inferred from pushover analyses
EIe
Test
Maximum Credible
Residual Displacement
Design Level
2002 PEER Annual Meeting

16. Concentrated Plastic Hinge Models
Various methods for estimating equivalent properties for concentrated plastic hinge (Lp, M-f, etc.)
Various idealized hysteretic models
Bilinear vs. Stiffness Degrading
Coupled and uncoupled
Bilinear
Stiffness Degrading
Maximum Credible
2002 PEER Annual Meeting

17. Concentrated Plastic Hinge Models
Most models provide adequate estimate of maximum displacement
Bilinear
Stiffness Degrading
Maximum Credible
Bilinear
Stiffness Degrading
Design Level
2002 PEER Annual Meeting

18. Concentrated Plastic Hinge Models
Most models provide adequate estimate of maximum displacement
Nonlinear models provide indication of yielding and degradation on wave form and residual displacement
Estimates are often poor
Stiffness degrading models generally better
Bilinear
Stiffness Degrading
Maximum Credible
Lateral Direction
Stiffness Degrading
Bilinear
2002 PEER Annual Meeting

19. Concentrated Plastic Hinge Models
Bilinear
Stiffness Degrading
Maximum Credible
Lateral Direction
Stiffness Degrading
Bilinear
2002 PEER Annual Meeting

20. Concentrated Plastic Hinge Models
Local information on strains, bar buckling, fatigue, etc. must be inferred from detailed analysis of member
Problem under cyclic loads?
2002 PEER Annual Meeting

21. Fiber Models
Useful for well confined members controlled by ductile yielding
Approximations at material level, number of fibers used to model section, manner in which member is discretized longitudinally
Concentrated
Hinge Models
Bilinear
Stiffness Degrading
Fiber Model
Maximum Credible
2002 PEER Annual Meeting

22. Fiber Models Generally, much better fidelity
Maximum Credible
Generally, much better fidelity
Results, especially for residual displacement and local deformations (strain) sensitive to modeling of section
Fixed end rotations due to bar pullout not yet accounted for in OpenSees
2002 PEER Annual Meeting

23. Model Performance vs. Test
2002 PEER Annual Meeting

24. Performance Evaluation
Damage Indices:
Park & Ang:
Bar Fatigue Damage Index:
where 2Nf is the number of half cycles to failure at a plastic strain
In order to evaluate the performance of each column, several damage indices and performance measures were used.
Section Fatigue, spalling, bar buckling, residual displacement, etc.
2002 PEER Annual Meeting

25. Damage Idecies at First Bar Fracture
2002 PEER Annual Meeting

26. Parametric Study Design multiple columns with varying: Diameter (Dcol)
Axial Load (Pr)
Hence, we design a number of columns having varying: Diameter, ….. Axial load, ….. and aspect ratio. This allows us to construct spectra-like plots showing for example the effects of Axial Load and Aspect Ratio on ductility demand for a given column diameter as one would use constant strength spectra. The difference is that these plots would correspond to actual columns designed according to code requirements, instead of just having random strength and periods. H
Aspect Ratio (ar)
2002 PEER Annual Meeting

27. Design Procedure Used Given Dcol, ar, Pr Determine rsp from BDS
Solve iteratively for rl that would result in the required strength:
For each rl value:
Perform M-f analysis, determine My, EIeff, Mu and fu (failure reached when ec>ecu or es>esu)
2002 PEER Annual Meeting

28. Column Design according to ARS
This is an example of how each column was designed. The plot shows the 70%g ARS spectra for type B soil. Each of the other lines represents the process of designed a specific columns using different amounts of longitudinal reinforcement ranging from 1 to 8%. Since changing the amount of reinforcement changes both the strength and stiffness of the column, the lines have those strange shapes. For each column, a longitudinal reinforcement ratio was chosen so that the period and strength of column fall as close to the ARS curve as possible, as represented by the red circles.
2002 PEER Annual Meeting

29. Moment-Curvature Analysis
Also determine c (Neutral Axis Depth)
2002 PEER Annual Meeting

30. Idealized Force-displacement Model
Deformation
Force
Exact Behavior
Idealized Fy K Kh
dult Fu
Also determine Lp and T
2002 PEER Annual Meeting

31. Ground Motions Used 20 LMSR motions and 50 LMLR motions LMSR motions
In this example, a bin consisting of 20 ground motions was used for the analysis. The motions have a relatively large magnitude of around 7, and are recorded at km. The ground motions were similarly scaled so that their mean matches the ARS spectrum.
2002 PEER Annual Meeting

32. Mean Results 2002 PEER Annual Meeting
The peak spectral values for each of the earthquake are determined for each of the columns, and statistical information is computed. The mean values are shown here for the ductility, spectral displacement, Bar fatigue index, and Park&Ang damage index. Some useful results can be concluded. For example, the ductility demand is higher for columns with lower aspect ratio, and is only about 2 for columns with an aspect ratio of 10, which have the highest displacement demands. Bar fatigue index shows the highest values for shorter columns with low axial load. On the other hand, Most columns seem to have high values of the Park&Ang index, except for columns with high aspect ratio and low axial load.
2002 PEER Annual Meeting

33. Mean+1 SD Results 2002 PEER Annual Meeting
Looking at results corresponding to the mean + standard deviation, values are significantly higher by up to times in some cases. The bar fatigue index exhibits especially high variability. Talking about means and standard deviations leads into the probability of exceedance of performance limit state under a specific hazard, which is best described using fragility curves.
2002 PEER Annual Meeting

34. Fragility Curves Compute Fragility curves for:
Park & Ang Index (Minor and Significant damage)
Fatigue Index
Spalling (|ecu| > 0.009)
Fragility curves are computed for a single column by running a suite of ground motions…. …
Assumes analysis model and preformance criteria are correct
2002 PEER Annual Meeting

35. Fragility Curves for Events with Large Magnitude at Small Distance
Minor
Damage
(Park&Ang)
Spalling
Significant
Damage
(Park&Ang)
This plot shows fragility curves computed for a column with a 5’ diameter, aspect ratio of 6, and axial load ratio of 0.2. The X-axis represents ground motion intensity which is taken as the ratio of the scaled motions’ spectral acceleration to that obtained from the ARS spectrum at the column’s period. The Y-axis represents the probability of exceeding a certain limit state (such as spalling) for a given intensity. For example, considering the curve corresponding to significant damage (according to the Park&Ang index), the probability of exceeding that damage level given that the ground motion’s spectral acceleration matches the design spectral accelearation is about 0.3. Then opposite can be done, ie, having a target probability, one can figure out the corresponding intensity.
Fatigue Failure
2002 PEER Annual Meeting

36. Fragility Curves for Events with Large Magnitude at Large Distance
Minor
Damage
(Park&Ang)
Spalling
Significant
Damage
(Park&Ang)
The same can be done for a different bin of ground motions, in this case for ground motions recorded at larger distances. The resulting probabilities are slightly less for the same intensity levels. For example, it’s 0.2 vs. 03. in the previous slide.
Fatigue Failure
2002 PEER Annual Meeting

37. Summary
New design and detailing criteria for circular columns generally result in performance consistent on average with performance objectives
Ground motion characteristics effect:
Maximum response
Bi-directional response characteristics
Residual displacements
2002 PEER Annual Meeting

38. Summary
Analytical models involve significant levels of judgement to get adequate prediction of performance
Nearly all models with reasonable stiffness estimates can predict max. displacements
Small diameter, low aspect ratio (low periods), high loads, P-D effects and gravity load eccentricities potential problems
Fiber models provide best fidelity, but need further assessment and refinement
Residual displacements and local deformations (spalling, bar buckling, steel fracture, etc.) sensitive to modeling
2002 PEER Annual Meeting

39. Summary
Parametric and fragility analyses provide useful basis for understanding behavior, but integration into overall PEER assessment methodology essential
Additional shaking table tests will be carried out along with analytical studies to:
get data on more complex bridge systems requiring significant redistribution of load once yielding occurs
Identify damping and strain rate effects
2002 PEER Annual Meeting