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**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

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**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

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**Many Elements Involved**

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

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**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

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**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

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**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

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**Condition at end of tests**

Buckled Bars After sixth repetition of Maximum Run - Olive View Fractured Spiral Fractured Bar 2002 PEER Annual Meeting

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**Long Duration Excitations**

1985 Llolleo, Chile Record 2002 PEER Annual Meeting

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**Peak Displacement Response**

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

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**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

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**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

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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

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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

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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

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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

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**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

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**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

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**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

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**Concentrated Plastic Hinge Models**

Bilinear Stiffness Degrading Maximum Credible Lateral Direction Stiffness Degrading Bilinear 2002 PEER Annual Meeting

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**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

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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

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**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

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**Model Performance vs. Test**

2002 PEER Annual Meeting

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**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

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**Damage Idecies at First Bar Fracture**

2002 PEER Annual Meeting

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**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

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**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

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**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

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**Moment-Curvature Analysis**

Also determine c (Neutral Axis Depth) 2002 PEER Annual Meeting

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**Idealized Force-displacement Model**

Deformation Force Exact Behavior Idealized Fy K Kh dult Fu Also determine Lp and T 2002 PEER Annual Meeting

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**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

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**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

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**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

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**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

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**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

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**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

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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

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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

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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

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