by Andrew W. Taylor, William C. Stone and Sashi K. Kunnath
This article presents research resulting from NCEER's Highway Project, task 106-E-5.3. The article is taken from a task technical paper submitted for inclusion in an annual report to the Federal Highway Administration summarizing Research Year 3 of the Seismic Vulnerability of Existing Highway Construction project. Comments and questions should be directed to Andrew Taylor, National Institute of Standards and Technology, at (301) 975-6078.
The objective of this project task is to develop performance-based criteria for the seismic design, retrofit, and repair of reinforced concrete bridge columns. Phase 1 of this task focused on the experimental behavior of typical California Department of Transportation (Caltrans) circular columns subjected to fatigue and random loading. The objectives of the current phase are to use the experimental data obtained in Phase 1 to evaluate and calibrate existing analytical damage models for reinforced concrete bridge columns, derive improved damage models, and develop methods using these damage models in practical design applications.
In earthquake engineering studies of reinforced concrete bridge columns, laboratory test specimens have traditionally been loaded with a controlled, cyclic lateral displacement pattern with increasing amplitudes. However, in actual earthquakes, bridge columns are exposed to random cyclic lateral loading patterns, which are much different from the typical laboratory loading patterns. Current American Association of State Highway and Transportation Officials (AASHTO) and Caltrans seismic design provisions are based almost entirely on tests in which traditional, controlled cyclic loading patterns at increasing levels of ductility have been applied. The differences in the effects of these two types of loading ?controlled, cyclic lateral loads, and random earthquake type loads ?have never been explored systematically. In this study, both types of loading were applied to a series of circular, cantilever flexural columns. Additionally, some specimens were subjected to constant-amplitude cycling to derive fatigue relationships for seismically-detailed columns.
The research is being conducted in two phases: Phase 1, now completed, consisted of monotonic, cyclic, constant-amplitude and random load tests on twelve quarter-scale circular bridge columns; and Phase 2, still in progress, is concerned with the utilization of the experimental results of the previous phase to develop damage-based seismic design guidelines. Several existing damage models are applied to observed data from the laboratory tests to verify their potential in damage prediction. A new fatigue relationship for seismically detailed flexural columns is being developed for use in performance-based design of bridge columns.
Summary of Phase 1 Tasks
The test program was designed to keep material, geometric and section variables to a minimum. Only flexural failure modes were considered in this study. Specimen and equivalent prototype details are listed in Table 1. A unique test facility was designed to expedite the testing process. Removable end-blocks were anchored to the specimen base through post-tensioned bolts. In keeping with the main objectives of the study, the primary variables considered were the amplitude, sequence and type of loading pattern. Specimens were labeled A1 through A12. Specimen A1 was loaded monotonically and unidirectionally up to failure; specimen A2 was subjected to a standard quasi-static cyclic load; Specimens A3 - A6 were subjected to constant amplitude cycles of 2.0, 3.0, 4.0 and 5.0 times the yield displacement, respectively.
Table 1: Details of Prototype and Model
|Long. Steel||24 # 11||21 # 3||p = 2 %|
|Spirals||# 5||wire = 4.04 mm dia.|
|Spiral Pitch||76 mm||19 mm|
|Spiral Yield Strength||414 mPa||379-448 mPa|
|Column Diameter||1219 mm||305 mm||Scale 1:4|
|Column Length||5486 mm||1372 mm||"|
|Cover||51 mm||13 mm||"|
|Embedment Lenght||Tension = 1359 mm |
Comp. = 711 mm
|Tension = 330 mm |
Comp. = 178 mm
|Specimen Embedment = 864 mm|
|Axial Load||3218 kN||201 kN||0.1 f'c Ag|
|Lateral Load Capacity||1446 kN||90 kN||Vp = Mp/L|
|Spacing of Long. Steel||102 mm||31 mm|
Random displacement histories used on specimens A7 - A12 were developed from separate analytical simulation studies using IDARC (Kunnath et al., 1992). A summary of the imposed displacement histories is presented in Table 2.
Summary of Phase 2 Tasks
The data generated in Phase 1 was used in a damage evaluation study to identify potential models for use in developing a damage-based design criteria. Four different damage models characterizing different damage measures were selected from the literature for detailed evaluation. The first model selected for evaluation is a modified form of the system softening index (Lybas and Sozen, 1977). This is representative of damage models that consider only a single structural parameter to quantify damage. The change in structural stiffness is associated with system degradation which alters the fundamental period of the system. An additional advantage of this model is that it can be monitored in actual structures. In the present study, the following normalized expression is used to quantify damage:
D = (k m - k o )/(k f - k o )
where k m is the stiffness of the structure at the maximum induced displacement, k f is the pre-established stiffness at failure of the system (typically under monotonic loads), and k o is the initial stiffness prior to loading.
The second model considered in the evaluation is the Kratzig model (Kratzig, 1987), since it incorporates only energy-related terms in its formulation. In developing his model, Kratzig defines a primary half cycle (PHC) as the energy contained in the half cycle at the maximum deformation point. Additional cycles with displacement amplitudes less than the peak deformation are accumulated as follower half cycles (FHC). Positive and negative deformations are treated separately. Accumulated damage for the positive portions of the response is defined as:
where E+p,i is the energy in an PHC, E+i is the energy in an FHC and Ef is the energy absorbed in a monotonic test to failure. A similar expression is computed for negative deformations, and the two quantities are normalized as follows:
The inclusion of the follower cycles in the numerator and denominator of equation (2) suggest that their contribution to damage is small, or less significant than deformations that extend the response envelope.
The next model considered in the study is the Park-Ang model (Park and Ang, 1985). This model represents a hybrid model, and was included in the evaluation partly because of its ease in implementation and partly because it is one of the most widely used damage models today. The model is used in its original form as follows:
The constant "B" was identified directly from the standard cyclic test conducted on Specimen A2.
Table 2: Summary of Displacement Histories Used in Testing
Specimen Label Load Description Remarks A1 Monotonic Load A2 Standard Cyclic Load 3 cycles at ?1, 1.5, 2, 3, 4, 5, and 6
delta y with an intermediate small amplitude cycle at
?0.5 delta y between each increase in amplitude A3 Quasi-Fatigue Constant amplitude cycling at ?2.0 delta y A4 Quasi-Fatigue Constant amplitude cycling at ?3.0 delta y A5 Quasi-Fatigue Constant amplitude cycling at ?4.0 delta y A6 Quasi-Fatigue Constant amplitude cycling at ?5.0 delat y A7 Random Loading Loma Prieta, Presidio (1989) @ 1.2g + Imperial Valley, Superstition Mt. (1979) @ 0.34 g + San Fernando,
2011 Zonal Ave. (1971) @ 0.1 g + San Fernando, 455 S Figueroa St. (1971) @ 0.54g to represent damaging
event + two minor events + severe event A8 Random Loading Imperial Valley, Superstition Mt (1979) @ 0.34 g + San Fernando, 2011 Zonal Ave. (1971) @ 0.1 g + Loma
Prieta, Presidio (1989) @ 1.2g + San Fernando, 455 S Figueroa St. (1971) @ 0.54g to represent two minor
events and two major events, respectively A9 Random Loading San Fernando, Orion Blvd.(1971) @ 1.43 g + San Fernando, 2011 Zonal Ave. (1971) @ 0.1 g + El Centro
(1940) @ 0.35 g + San Fernando, 455 S Figueroa St (1971) @ 0.15 g + San Fernando, Orion Blvd.(1971)
@ 1.43 g to represent a damaging event, an after-shock, a moderate event, a minor event and a severe event. A10 Random Loading San Fernando, 2011 Zonal Ave. (1971) @ 0.1 g + El Centro (1940) @ 0.35 g + San Fernando, 455 S.
Figueroa St (1971) @ 0.15 g + San Fernando, Orion Blvd.(1971) @ 1.43 g + San Fernando, Orion Blvd. (1971)
@ 1.43 g to represent two minor events and a moderate event, followed by two severe events A11 Random Loading Northridge, VA Hospital (1994) @ 0.42 g + Northridge, Griffith (1994) @ 0.26 g + Taft (1952) @ 0.36 g +
Mexico City SCT (1985) @ 0.17 g to represent a major event, two moderate events and a
severe event A12 Random Loading Northridge, Griffith (1994) @ 0.26 g + Taft (1952) @ 0.36 g + Mexico City SCT (1985) @ 0.17 g Northridge,
VA Hospital (1994) @ 0.42 g to represent two moderate and two severe events
The final model selected for investigation was derived from principles of low-cycle fatigue. The fatigue behavior of the longitudinal steel under reversed cyclic loading is formulated in terms of the Coffin-Manson (Manson, 1953) equation:
|ep||=||plastic strain amplitude|
|e1f||=||a material constant to be determined from fatigue testing|
|(2Nf)||=||number of complete cycles to failure|
|c||=||a material constant to be evaluated experimentally|
An experimental fit to this expression was obtained by Mander et al. (1994):
Using fundamental relationships between curvature and strain, a relationship between curvature (or rotation) and cycles to failure (Nf) can be derived. If the plastic hinge length is defined as lp, an expression for the plastic strain in terms of plastic curvature (or rotation) can be established (Paulay and Priestley, 1992) assuming that the plastic rotation, Rotp, takes place about the center of the plastic hinge:
which can be used directly to define the number of cycles to failure for a given plastic strain or a given plastic deformation. Cumulative damage is then defined as the Summation of:
The four models described above were applied to the data obtained from the experimental testing of Phase 1. A comparative study of the damage models was carried out to assess their capability in seismic damage prediction.
Preliminary Results and Conclusions
There exists a so called "threshold" ductility level for well-confined flexural circular columns designed by current Caltrans specifications beyond which severe degradation of stiffness and strength take place. For the bridge columns tested in this study, this ductility level occurs between 2Dy and 4Dy. Specimen A3, which was cycled 150 times at a ductility of 2Dy, showed no significant signs of damage or deterioration. Specimen A5, which was cycled at a displacement ductility of 4Dy, failed in less than 10 cycles. It may, therefore, be inferred that seismically detailed bridge columns subjected to earthquakes which impose ductility demands less than 2.0 can survive a series of similar events without undergoing any significant structural damage. When the ductility demand approaches 4.0, the likelihood of moderate to severe damage is high and depends on the number of such inelastic cycles experienced by the structure.
Under a sequence of low amplitude cycles, it is most likely that the confining spiral will fail prior to low-cycle fatigue failure of the longitudinal reinforcing bars. Conversely, if the bridge column is subjected to high amplitude inelastic cycles, it is most likely that the longitudinal bars will rupture before confinement failure occurs. In the present study of flexural columns, it was found the threshold "low-amplitude" cycle is approximately 2-3% drift, while high-amplitude cycles are those in excess of 4%.
A large database of displacement histories was produced for the bridge column specimens using dozens of recorded ground motions at different soil profiles. A significant finding of this research study, based on these numerous analytical simulations, is that typical earthquakes produce few large amplitude cycles, hence based on the preceding paragraph, failure is generally governed by confinement. The constant amplitude and random cyclic testing clearly indicates that the energy capacity of a member at failure is strongly path dependent. Proof of this observation is shown in figures 2 and 3 which show plots of cumulative energy dissipated for all specimens tested in both phases of the study. If specimen A2, tested using standard cyclic displacement amplitudes, is referred to as the benchmark energy capacity, it is evident that the energy capacity of the columns varies considerably depending on the displacement amplitude and the path or previous load history.
Damage models evaluated in this study indicate that most non-fatigue based theories are incapable of adequately reproducing observed damage. Models based on the degradation of a single structural parameter, such as the softening index evaluated in this study, are sensitive in the early stage of damage progression and show little variation beyond this point to failure, making them difficult to calibrate. Energy-based models which do not account for the level of ductility consistently over-predict damage. The Park-Ang model is essentially a ductility-based model since the energy term is not adequately represented: energy damage is sometimes overestimated at small inelastic amplitudes and underestimated at large inelastic cycles. On the other hand, existing fatigue theories, using a Coffin-Manson rule in combination with Miner's hypothesis, account only for low-cycle fatigue of steel. It appears that a model which combines low-cycle fatigue failure with confinement deterioration could yield excellent results.
A cumulative fatigue model was developed based on experimental fitting of the Coffin-Manson fatigue expression using results from testing of columns A3, A4, A5 and A6. As a result of evaluating the model coefficients for plastic strain vs. the number of half cycles to failure, the following expression is obtained:
The difference in the constant term that appears in the above expression to that obtained by Mander is due to the contribution of damage resulting from loss of confinement and concrete fatigue. The predicted damage using the above fatigue expression in conjunction with the formulation described in equations (5) - (8) for specimens A7 - A12 is shown in figure 4.
In an attempt to correlate visually observed damage during testing with damage limit states, all of the recorded test data were evaluated carefully to develop a correlation chart to be used along with the proposed ATC-33 effort to define damage states. The summary is presented in table 3.
This task is concerned with the development of performance-based guidelines for design and repair of highway bridge columns. Performance criteria will be based on damage limit states that have been correlated to observed experimental response. The guidelines can be utilized in seismic vulnerability assessment of existing highway bridge columns and can also be extended to evaluate repair and retrofit options for damaged columns.
Table 3: Correlation of Damage Limit States with Visual Observations
|Damage Indicator||Damage State||Description||Visual |
Based on Current
|N||None||No visible damage, either cosmetic or structural||No visible cracks|
|I||Insignificant||Damage requires no more than cosmetic repair. No structural repairs necessary||Hair-line cracks |
No exposed reinforcement
|M||Moderate||Repairable structural damage has occurred. The existing elements can be repaired essentially in place, without substantial demolition or replacement of elements||Excessive spalling|
No buckling of longitudinal bars
No necking of spirals
|H||Heavy||Damage is so extensive that repair of elements is either not feasible or requires major demolition or replacement.||Buckling/fracture of longitudinal bars|
Necking/rupture of spirals
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