Behavior of Reinforced Concrete Columns Under Variable Axial Loads: Analysis
Effects of the magnitude and pattern of the axial load on the lateral response of bridge piers are analyzed and compared with the experimental findings reported in a companion paper. A computer program developed to address the analytical needs of the research was used for the purpose. It included a fiber-based moment-curvature analysis, a plastic hinge method addressing load variations, material models calibrated against test data, and new hysteretic rules.
The program was able to predict the column response reasonably well under various loading patterns. Predictions in close agreement with the experimental evidence revealed that the effects of the magnitude and path of axial load were significant. These effects should be addressed in design practice where seismic excitation in lateral and vertical directions is a concern. Analytical tools, including monotonic and hysteresis material models, may be refined further for a better simulation of the behavior under various loading patterns.
Keywords: analysis; column; load; moment.
(ProQuest Information and Learning: … denotes formula omitted.)
Loading history has a considerable influence on the behavior of reinforced concrete structural elements and in particular columns. Hysteretic characteristics including stiffness, strength, ductility, and energy absorption as important factors in design of bridge piers or building columns are affected by the loading history. As an example, the lateral displacement or force capacity of a column designed based on a constant axial load could become unsatisfactory depending on the previous history of the loading. It has been observed that for the same level of axial load and deflection (or curvature for a section), the flexural capacity is significantly different depending on the past history of the loading pattern.1 Considering these effects is important to the design of new bridges and the retrofit of existing deficient structures.
Columns in bridges and various types of structures are subject to a combination of loading patterns in lateral and axial directions when exposed to a dynamic excitation of any source. This is more pronounced for earthquake excitations, especially in a near-fault situation.
Effects of a near-source earthquake excitation on buildings and highway structures in general, and bridges and tall buildings in particular, consist of, but are not limited to, high accelerations of significant vertical as well as horizontal ground motions, large velocity pulses, directional effects, repetitive pulse effects, and aftershocks. The frequency, amplitude, and phasing of the vertical and lateral excitations imposed on the structure may vary depending on the source of the earthquake, distance, site, and structure properties.2-4 Also, due to the so-called overturning moment, columns in multi-column bents in bridges or the exterior columns in buildings would be subjected to variable axial loads typically proportional to the corresponding lateral forces. The induced axial load by the lateral forces inflicted by wind has a similar characteristic.
Generally, the effects of dynamic excitations on a structure, which leads to specific loading paths on structural members, should be investigated from both the demand and capacity viewpoints. Characterization of the effect of such excitations and the corresponding loading pattern analytically involves advanced mechanics principles and a detailed nonlinear dynamic analysis that will be of interest in terms of demandcapacity relationship. This paper, however, is on the capacity side of the problem, and as a complementary part of the experimental phase of these studies examines simple, commonly used analytical models and methods in simulating the behavior of reinforced concrete members, especially bridge piers, under various loading patterns. This prediction is necessary for the capacity and performance assessment of reinforced concrete bridge columns subjected to the combined effect of uncoupled variations of lateral and axial load. To this extent, analytical performance of a reinforced concrete column under several loading patterns is compared with the test results in this paper. It should be noted that most of the data, such as axial loading patterns and related conclusions referred in this paper are included in the experimental paper and are not repeated herein.
This paper examines the accuracy of the analytical methods in simulating the performance of bridge piers subjected to various nonsequential types of loading paths as observed in experiment. Analysis of the effects of variable axial loads on the capacity and performance of piers is required for a more realistic characterization of the performance and the actual capacity of bridge piers under various loading patterns. Exploring the applicability and accuracy of the commonly used analytical tools in capturing these effects contributes to the information needed for future performancebased design guidelines and design implementation of the effects of various loading paths.
To predict the performance of reinforced concrete columns under various loading paths, a computer program was developed for nonlinear analysis of reinforced concrete columns under arbitrary lateral and axial loading histories.5-6 The program is an application with a friendly interface and various options in terms of the input data, analytical methods and models, and the output data.
Analysis is based on fiber modeling in which the section is divided into uniaxially stressed fibers along the longitudinal axis. This model has been used effectively by others in analysis of reinforced concrete columns subjected to reversed loading paths.7-8 The effect of confinement is considered in the monotonic stress-strain relationship of concrete confined by the lateral reinforcement. The monotonic curve serves as the envelope for the hysteretic response. Strain hardening of the steel was implemented in the material model used in the program.
Summary of analysis process
For a moment-curvature analysis, the section is defined in terms of its geometry, reinforcement type, and arrangement in the longitudinal and transverse directions; material properties are set for steel and concrete in terms of their monotonic and cyclic responses; and moment-curvature analysis, as discussed later, is performed based on the selected loading condition. For a monotonic or cyclic curvature, the axial load can be constant or variable. The variation in axial load can be independent, or defined as a function of the moment. A proportionally variable axial load with a pre-defined proportionality factor with respect to moment is one of the loading cases with a dependent axial load. For a displacement controlled analysis, the input data at each step will be the “curvature and axial load” and the corresponding moment is found through an iterative process in which the history of each element on the section is traced and updated at each step. For a force-controlled analysis, the curvature is found for a given “moment and axial load” at each step.
For a force-deflection analysis, in addition to the section and material data as mentioned for the moment-curvature analysis, the length of the column and the model for the curvature distribution along the column, explained later, are defined. The monotonic or cyclic displacement at the tip of the column and the corresponding axial load serve as the input data for a displacement-controlled analysis. For a force-controlled analysis, the input data are the lateral force and the corresponding axial load. Axial load can be constant or variable. A variable axial load can change independently or can be defined as a function of lateral force. Based on the assumption on the curvature distribution, the moment-curvature of several sections along the column are monitored and used in an iteration process to evaluate the lateral force for a given displacement and axial load in a displacementcontrolled analysis, or the displacement for a given lateral force and axial load in a force-controlled case.
Monotonic stress-strain model for steel-A relatively simple model was developed for the monotonic stress-strain relationship of steel. This model, with four parameters: K1, K^sub 2^, K^sub 3^, and K^sub 4^, is versatile and can be tuned to simulate the behavior of different types of steel. The parameters are illustrated in Fig. 1. The model was calibrated against material test results conducted at the University of Southern California structural laboratories on samples of the steel used in the test columns. The parameters are as follows:
1. K^sub 1^ is the ratio of the strain at start of the strain hardening to the yield strain;
2. K^sub 2^ is the ratio of strain at peak stress to yield strain;
3. K^sub 3^ is the ratio of ultimate strain to yield strain; and
4. K^sub 4^ is the ratio of the peak stress to yield stress.
A quadratic curve joins the point at the start of strain hardening, the peak stress and the rupture point. The mathematical formulation of this part of the model for K^sub 1^ε^sub y^ ≤|ε|
The values used as the input data for the monotonic stressstrain curve of steel in the analysis were chosen based on the material test results conducted on the samples of the bars used for construction of the columns. These values are as follows: f^sub y^ = 469 MPa (68 ksi); E = 200,000 MPa (29,000 ksi); K^sub 1^ = 4.0; K^sub 2^ = 25.0; K^sub 3^ = 40.0; and K^sub 4^ = 1.3.
Hysteretic stress-strain model for steel-The model developed and used for the hysteretic behavior of steel has the three major parts common in all hysteretic rules. Before any strain reversal, the stress and strain follow the monotonic stress-strain curve of steel as described in the monotonic stress-strain curve for steel. At the turning point (strain reversal) the modulus of elasticity is assumed to be the same as the initial modulus of elasticity of steel. The Bauschinger effect is considered in the model by changing the stiffness of steel to a portion of the initial stiffness beyond a certain stress in hysteretic response. For a more realistic implementation of strain hardening of steel, this ratio and the level at which the change occurs are different in the first and third quarters from their corresponding values in the second and fourth quarters of the coordinate plane. Figure 2 shows an instance of the hysteretic stress-strain model, and the five parameters P^sub 1^, P^sub 2^, P^sub 3^, R^sub 1^, and R^sub 2^, implemented in the model. The values used for these parameters were 0.333, 0.17, 0.9, 0.2, and 0.1, respectively, in the analysis.
The behavior of the model is symmetric with respect to the origin as a symmetrical monotonic stress-strain curve has been assumed for steel. Considering the limitations on the length of this paper, description of the behavior is provided in the flowchart shown in Fig. 3. Definitions of the symbols can be found besides the chart or in the list of notations. In the model, failure flag (FF) is set to one when the element fails and the plastic return flag (PRF) is raised when a strain reversal occurs for a strain more than the steel yield strain.
Monotonic stress-strain model for confined concreteThe model for the monotonic stress-strain relationship of confined concrete proposed by Mander, Priestley, and Park9 was used in this analysis. To model the cover concrete, the same model is used without any lateral reinforcement as proposed in the model. Based on the material test results on the concrete samples used in the experimental phase of this research, the unconfined concrete strength was taken as f’^sub c^ = 49.3 MPa (7.15 ksi) in the analysis.
It should be noted that using a different model for confined concrete will lead to slightly different analytical predictions for cases with a constant low axial load. This difference increases with axial load level, and is noticeable for constant axial load values more than 0.3A^sub g^ f’^sub c^ . In another study by the first author,10 it was shown that some recent models with a better representation of the confined concrete stress-strain behavior provide a better prediction of the member response under various loading patterns.
Hysteretic stress-strain model for concrete-The monotonic stress-strain curve serves as the envelope for the hysteretic stress-strain model of concrete developed and used in the analysis. As shown in Fig. 4, the hysteretic curve follows a parabolic path at a strain reversal. The curve is concave-upward for a decreasing strain and has a slope of E^sub c2^ on the envelope curve. The stress decreases to zero when the tensile strength is ignored, or will decrease to the tensile strength f^sub t^ with a slope of E^sub ct^ after the sign change of the stress. At a strain reversal with an increasing strain, the stress remains zero up to the latest strain corresponding to zero stress ε^sub z^ and then it grows on a concave-downward parabola which has a slope of E^sub c1^ on the strain axis. The stress increases up to the envelope curve and then follows that. It should be added that stiffness and strength degradation of concrete may be implemented in the model by linking the values of E^sub c1^, E^sub c2^, and E^sub ct^ to the strain history. The mathematical description of the concrete hysteretic rules to find the stress σ for a new strain of ε, with a previous strain and stress of ε^sub p^ and σ^sub p^, respectively, can be found in the flowchart shown in Fig. 5. Most of the symbols are defined in the chart or can be found in the list of notations.
Each element on the section has two flags, cracking flag (CRF) and crushing flag (CUF), associated with the first tensile failure and the first compression failure, respectively. Initially, an element is uncracked and uncrushed and ε^sub z^ = ε^sub p^ = σ^sub p^ = 0.0. The deformation history of individual elements is tracked and updated at each step. An element will not have any tensile strength after the first crack and no compressive strength after the first crush in compression.
While E^sub c1^ and E^sub c2^ can be different in this model, these values have been chosen to be identical to E^sub cc^, the initial stiffness of the concrete in the present analysis. The computer application provides a friendly interface to examine the hysteretic response of the material to be used in the analysis.5 In a simplified version, the reversals can be linear with the same modulus of elasticity as the initial value of the stiffness.
In general, for a fiber-based section analysis, concrete on the section of the model column is divided into elements in two directions to consider biaxial independent moments along with an arbitrary axial load, and steel bars are considered at their actual locations as shown in Fig. 6(d), (e), and (f). Because of a unidirectional moment in this analysis, the section was divided as shown in Fig. 6(b) and (c). For a displacement controlled analysis, the neutral axis location is found for a given curvature and axial load level, with a predetermined accuracy having the strain and stress history of each element on the section; and then the corresponding moment is evaluated followed by updating the stress and strain state of each element. For a force controlled case, where at each step the moment and corresponding axial load are the input data more, computational effort is required for convergence of the iteration process to a desired level of accuracy. In any case, the hysteretic response of the section is evaluated by tracing the history of strain and stress of each single element on the section during analysis.
Deflection at the tip of the column, where the horizontal force was applied during the test, is considered as a combination of the elastic deflection associated with the elastic portion of the column, and plastic deflection associated with deformation within the plastic hinge region and also the rotation induced by the pull-out action of the reinforcing bar at the footing-column interface. Distribution of curvature within the plastic region is important especially for deflections beyond the maximum flexural strength of the member under a certain axial load. Most of the plastic hinge assumptions such as the one proposed by Priestley and Park11 are useful for monotonic deflections under a constant axial load. Experimental observations have revealed the effect of variation of axial load and cyclic deflections on the plastic hinge length. To address these effects, a method was developed that can be used for any loading and displacement case including cyclic cases with both a variable or constant axial load. The assumption for the curvature distribution considers variation of the hinge length as observed during tests with a cyclic or monotonic lateral displacement and variable axial load. Figure 7 shows various regions of the column in terms of the curvature distribution. The value l^sub p1^, with a uniform curvature of [straight phi]^sub u^, is assumed to be equal to the depth of the column section in the direction of analysis. For columns with a length to depth ratio of more than 12.5, l^sub p1^ = 0.08l where l is the column length. The value l^sub p2^ = 0.15f^sub s^d^sub b^ (or 0.022f^sub s^d^sub b^ [SI]) where f^sub s^ is the maximum tensile stress on the section located at the column-footing interface and d^sub b^ is the diameter of the longitudinal bar with the maximum tensile stress. The value l^sub p2^, with a uniform curvature of [straight phi]^sub u^, varies at each step based on the stress profile on the section. The value l^sub trans^ is not constant and increases as the location of the section experiencing the first yield moves upward. So, the portion of the column remaining within the elastic range is not constant and changes based on the loading and deflection condition. Portions of the column experiencing a deformation beyond the yield deformation in any step will fall out of this linearelastic length for the rest of analysis. Initially, the whole column is elastic. As the lateral displacement increases and depending on the axial load level, the section marked as the end of elastic region will move. The four regions on the column including the linear-elastic length, transition length, plastic length, and the stress penetration or pull-out action length and their corresponding curvature distributions are updated at each step of analysis. Moment-curvature of two sections, one at the column-footing interface and the other at the end of elastic region, are monitored in this method. Note that for the latter, the location of the section changes based on the loading condition. A detailed description of the method can be found elsewhere.6
COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS
Analytical predictions were compared with measured results from five model columns tested under various quasistatic loading conditions. The model columns had a circular section with a diameter of 406 mm, and a total height of 2083 mm above the top of the footing. The effective length of the column, measured from the top of the footing to the application point of the lateral force, was 1829 mm. The longitudinal reinforcement consisted of 12 No. 13 (nominal diameter = 12.5 mm) Grade 410 (ASTM G 60) bars evenly distributed in a circle. Details of the columns, reinforcement, material properties, instrumentation, test setup, loading program, and evaluation of the experimental moment curvature and force deflection can be found in the paper on the experimental part of this study.1
Loading cases considered in the analytical studies were as follows:
1. Case 1: cyclic lateral displacement with a constant axial load;
2. Case 2: cyclic lateral displacement with an axial load varied proportionally with the lateral force, simulating the effects of overturning moment in a two column bent;
3. Case 3: monotonic lateral displacement (push-over) without any axial load;
4. Cases 4 and 5: monotonic lateral displacement with two different nonproportional axial loading patterns.
Moment curvature at different heights and force-deflection response of the tip of the columns extracted from test data were compared with their corresponding analytical predictions.
The experimental moment-curvature was evaluated independent of the force-deflection response, using the local segmental deformations recorded by the linear displacement transducers installed on the opposite sides of the segments, as detailed in the experimental paper. Reliability of this method was validated by extracting identical responses using the recorded longitudinal strains affixed on the bars within the same segment for several cases. Figure 8 compares the experimental and analytical moment-curvature response of the first specimen under a constant axial load of 0.3A^sub g^ f’^sub c^ . Predicted moments are less than the experimental moments for this case. Lower analytical moments compared to test data have been reported by others12 for similar cases with a high level of constant axial load. Analytical predictions and test results for Case 2, where the specimen was tested under a variable axial load proportional to the lateral force and a reversal lateral displacement, are compared in Fig. 9. The absolute value of the maximum level of axial load in compression and tension was approximately 0.01A^sub g^ f’^sub c^ . This low level of axial load was a result of its proportionality with the lateral force. Behavior of the specimen could be closely predicted by analysis. In the push direction (positive curvature) with a tensile axial load, however, flexural capacity is slightly overestimated and in the pull direction (negative curvature) with a compressive axial load predictions are lower than test results. Experimental moment curvature response of the third case on the first and second 100 mm long segments of the column (the first segment was adjacent to the column-footing interface) is compared with the analytical results in Fig. 10. Although this column was nominally subjected to monotonic load, it was in fact subjected to one complete cycle to ±10% drift ratio. As is evident from the figure, while predictions are reasonably close to test results, analysis underestimates the flexural capacity and curvature ductility. It should be noted that the maximum experimental drift ratio was dictated by the instrumental limitations and the specimen could have achieved a higher drift and force. In this figure, there is a difference between the slope of the experimental curve at the first segment that is adjacent to the column-footing interface, and the slopes of analytical curve and the experimental curve on the second segment. The reason is that the experimental curve for the first segment is plotted based on the recorded data in which the rotation caused by the pullout action of the bars is included. As the curvature increases, the furthermost bar will experience a higher stress, leading to an increase in the rotation caused by the pullout action of the bars. As discussed in the plastic hinge assumption, this difference is zero at the beginning when there is no tensile stress in the bar and increases as the tensile stress is increased.
Analytical and experimental horizontal force-drift ratio responses for the first case are compared in Fig. 11. The analytical predictions are conservative compared to the experimental results. Analysis is conducted using the plastic hinge method developed and used in this study. Using a different plastic hinge assumption based on a constant length will result in a different prediction. This difference is more noticeable and pronounced for cases with a variable axial load compared to cases with a constant axial load such as the present case.
Figure 12 compares the analytical and experimental results for the second case where the axial load was proportional to the lateral force within ±0.01A^sub g^f’^sub c^ . The prediction in the pull direction where a compressive axial load was involved is lower while in the push direction with a negative or tensile axial load the analytical predictions and experimental results are close. Figure 13 shows the analytical predictions and test results for the third case where one cycle of lateral displacement with an approximate maximum drift ratio of 10% in both directions was applied without any axial load.
Figure 14 and 15 show the experimental horizontal forcedrift ratio curves for Cases 4 and 5, with a monotonic lateral displacement and a nonproportionally variable axial force. The level of axial load in these tests fluctuated between 0.3A^sub g^f’^sub c^ (compression) and -0.1A^sub g^f’^sub c^ (tension) for several cycles during a monotonically increasing lateral displacement. The only difference between these two cases was the slight difference in pattern of the axial load as described in the experimental report of this study. This has led to relatively distinct responses in terms of the flexural capacity that could be captured by analysis up to certain accuracy, depending on the material models used in the analysis. Note that in these cases the flexural capacity is underestimated for instances under a tensile axial load and overestimated for instances under a compressive axial load, which is different from constant axial load cases.
Experimental results confirmed by analysis show that at the same level of displacement and identical axial force levels, the flexural capacity is significantly different when the axial force has a different variation history.
In general, comparison of the moment-curvature and force-deflection responses confirms that a reasonable prediction of the performance of the member can be achieved, using relatively simple analytical processes depending on the accuracy of the models used for monotonic and especially hysteretic response of the material. Refinement of the material models can improve analytical predictions. This includes adjustment of material hysteretic parameters against material test data if available, or finding their proper values iteratively using the experimental data through an optimization process that may be different for each parameter, and switching to more realistic monotonic confined concrete models. A comparative study of various models for monotonic stressstrain relationship of confined concrete has shown that confined concrete monotonic stress-strain model can noticeably affect the accuracy of predictions.10
The analysis is also capable to provide detailed strain history of any fiber in the section. The analytical prediction and the experimentally recorded strain history of a gauge affixed on a steel bar located at the mid-depth in the critical section of the first column are compared in Fig. 16. This gauge was selected solely because of the validity of the recorded data at this location for several loading cycles. Gauges affixed on opposite sides exceeded their linear limit in the very first few cycles and the recorded data was not reliable after strain reversals. Figure 16 shows that these types of predictions may be possible with a reasonable accuracy depending on the models used for monotonic and hysteretic behavior of the material.
SUMMARY AND CONCLUSIONS
An analytical program was conducted to simulate the behavior of reinforced concrete columns tested under various loading patterns and reported in the paper on the experimental part of this study. Hysteretic material models and a plastic hinge method were developed and implemented in a fiber-based moment-curvature, and in turn force deflection analysis. Experimental results from five large-scale circular reinforced concrete columns were used to validate the analytical predictions. Analysis could reproduce the test results with a reasonable accuracy. Analytical results, confirmed by the experimental data, show that the axial force level and path play significant roles in the flexural strength and deformation capacity and, in general, the overall performance of the column. For a constant level of axial force, as expected and observed during experimental and analytical phases of this study, an increase in compressive axial load within the analytical balanced level leads to an increase in the flexural capacity, but a decrease in the ductility. A relatively high axial load, either constant or variable, requires more confinement to achieve enough ductility.
Both analysis and test results show that while under a constant axial load, the peak flexural strength and displacement capacity of the column under reversed lateral forces are similar to those for a monotonic loading case; they are different under different variable axial loading paths. In other words, at a certain displacement and axial load level, the flexural capacity is significantly different depending on the history of axial loading path, from the flexural capacity at the same displacement and the same level but constant axial load.
The significant effect of variation pattern of the axial loading on the response of the column can be captured by a relatively simple analysis process as described in this research program. This effect, discussed in more detail in the paper on the experimental part of this research program, needs to be taken into consideration for assessment of the load carrying capacity and deformability of the column.
In general, using proper models and rules for monotonic and hysteretic behavior of material and a reasonable assumption on curvature distribution similar to what developed and used in this study, along with a relatively simple analytical approach, such as fiber model for the present work, can simulate the behavior of the specimen with an acceptable accuracy.
The experimental part of the research described in this paper has been funded by the National Science Foundation, Pacific Earthquake Engineering Research Center (PEER) under contract number 5061999. Kansas State University provided the first author’s support to refine the computer program described in this paper.
E^sub cl^ = initial ascending modulus of elasticity of concrete in hysteretic rules
E^sub c2^ = initial descending modulus of elasticity of concrete in hysteretic rules
E^sub cc^ = initial modulus of elasticity of confined concrete
E^sub ct^ = tensile modulus of elasticity of concrete
E^sub s^ = steel modulus of elasticity
f’^sub c^ = concrete stress
f^sub cc^ = confined concrete maximum strength
f^sub s^ = steel stress
f^sub t^ = tensile strength of concrete
f^sub u^ = peak steel stress
f^sub y^ = yield stress of steel
K^sub 1^ to K^sub 4^ = parameters for monotonic stress-strain relationship of steel
l^sub p1^, l^sub p2^ = length of plastic hinge regions in curvature distribution model
l^sub trans^ = length of transition length in curvature distribution model
P^sub 1^ to P^sub 3^ = parameters in hysteretic stress-strain model of steel (for stress level)
R^sub 1^ to R^sub 2^ = parameters for hysteretic stress-strain model of steel (for stiffness change)
ε = strain
ε^sub c^ = concrete strain
ε^sub cc^ = confined concrete strain at peak stress
ε^sub p^ = strain of steel or concrete at previous point in hysteretic model
ε^sub u^ = ultimate strain (rapture or crash) of steel or concrete
ε^sub y^ = yield strain of steel
ε^sub z^ = strain of concrete at last zero stress[straight phi]^sub t^, [straight phi]^sub u^ = curvature at top of transition region and plastic region, respectively
σ = stress
σ^sub p^ = stress of steel or concrete at previous point in hysteretic model
1. Esmaeily, A., and Xiao, Y., “Behavior of Reinforced Concrete Columns under Variable Axial Loads,” ACI Structural Journal, V. 101, No. 1, Jan.-Feb. 2004, pp. 124-132.
2. Bozorgnia, Y.; Niazi, M.; and Campbell, K. W.; “Characteristics of Free-Field Vertical Ground Motion During the Northridge Earthquake,” Earthquake Spectra, V. 11, No. 4, Nov. 1995, pp. 515-525.
3. Papazoglou, A. J., and Elnashai, A. S., “Analytical and Field Evidence of the Damaging Effect of Vertical Earthquake Ground Motion,” Earthquake Engineering and Structural Dynamics, V. 25, 1996, pp. 1109-1137.
4. Uenishi, K., and Sakurai, S., “Characteristics of the Vertical Seismic Waves Associated with the 1995 Hyogo-Ken Nanbu (KOBE), Japan Earthquake Estimated from the Failure of the Daikai Underground Station,” Earthquake Engineering and Structural Dynamics, V. 29, No. 6, 2000, pp. 813-821.
5. Esmaeily, A., “USC_RC, Moment-Curvature and Force-Deflection Analysis of a Reinforced Concrete Member,” http://www.usc.edu/dept/civil_eng/structural_lab/asad/usc_rc.htm, 2002.
6. Esmaeily, A., and Xiao, Y., “Seismic Behavior of Bridge Columns Subjected to Various Loading Patterns,” Pacific Earthquake Engineering Research Center, PEER 2002/15, Dec. 2002, p. 321.
7. Saadeghvaziri, M. A., “Nonlinear Response and Modeling of RC Columns Subjected to Varying Axial Load,” Engineering Structures, V. 19, No. 6, 1997, pp. 417-424.
8. Prakash, V.; Powell, G.; and Campbell, S., DRAIN 2D User Guide, UCB/SEM-93/17, V 1.10, University of California at Berkeley, Calif., 1993.
9. Mander, J. B.; Priestley, M. J. N.; and Park, R., “Theoretical StressStrain Model for Confined Concrete,” Journal of Structural Engineering, ASCE, V. 114, No. 8, Aug. 1988, pp. 1804-1825.
10. Esmaeily, A., and Lucio, K., Analytical Performance of Reinforced Concrete Columns Using Various Confinement Models, ISCC-2004, Changsha, China, June 2004, p. 12.
11. Priestley, M. J. N., and Park, R., “Strength and Ductility of Concrete Bridge Columns under Seismic Loading,” ACI Structural Journal, V. 84, No. 1, Jan.-Feb. 1987, pp. 61-75.
12. Priestley, M. J. N.; Park, R.; and Potangaroa, R. T., “Ductility of Spirally-Confined Concrete Columns,” Journal of the Structural Division, ASCE, V. 107, No. ST1, Jan. 1981, pp. 181-201.
ACI member Asad Esmaeily is an assistant professor of civil engineering at Kansas State University, Manhattan, Kans. He received his BS and his MS in civil engineering from Tehran University, Iran, an MS in structural engineering, an MS in electrical engineering, and a PhD in civil engineering from the University of Southern California, Los Angeles, Calif. He is an associate member of Joint ACI-ASCE Committee 441, Reinforced Concrete Columns. His research interests include earthquake-resistant design, analysis of reinforced concrete structures, structural materials, and structural control.
ACI member Yan Xiao is an associate professor of civil engineering at the University of Southern California. He also holds the Cheung Kong Scholarship at Hunan University of China. He is a member of ACI Committee 335, Composite and Hybrid Structures, and Joint ACI-ASCE Committee 441, Reinforced Concrete Columns. His research interests include earthquake-resistant design of structures; structural concrete; steel, hybrid, or composite systems; and structural materials.
Copyright American Concrete Institute Sep/Oct 2005
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