EFFECTS OF BUCKLING AND LOW CYCLE FATIGUE ON SEISMIC PERFORMANCE OF REINFORCING BARS AND MECHANICAL COUPLERS FOR CRITICAL STRUCTURAL MEMBERS

Transcription

1 EFFECTS OF BUCKLING AND LOW CYCLE FATIGUE ON SEISMIC PERFORMANCE OF REINFORCING BARS AND MECHANICAL COUPLERS FOR CRITICAL STRUCTURAL MEMBERS A Technical Report Submitted to the California Department of Transportation under Contract 59A0539 Sashi K. Kunnath Amit Kanvinde Yan Xiao* Guowei Zhang* * University of Southern California June 2009 Department of Civil and Environmental Engineering Structural Engineering and Structural Mechanics University of California at Davis

2 STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION TECHNICAL REPORT DOCUMENTATION PAGE TR0003 (REV. 10/98) 1. REPORT NUMBER 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT S CATALOG NUMBER CA/UCD-SESM TITLE AND SUBTITLE Effects of Buckling and Low Cycle Fatigue on Seismic Performance of Reinforcing Bars and Mechanical Couplers for Critical Structural Members 7. AUTHOR(S) Sashi Kunnath, Amit Kanvinde, Guowei Zhang, Yan Xiao 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil Engineering University of California, Davis Davis, CA SPONSORING AGENCY AND ADDRESS California Department of Transportation Engineering Services Center th Street Sacramento, CA SUPPLEMENTAL NOTES 5. REPORT DATE June PERFORMING ORGANIZATION CODE CA/UCD-SESM PERFORMING ORGANIZATION REPORT NO. CA/UCD-SESM WORK UNIT NUMBER 11. CONTRACT OR GRANT NUMBER 59A TYPE OF REPORT AND PERIOD COVERED Final Technical Report ( ) 14. SPONSORING AGENCY CODE ABSTRACT Since modern provisions for the design of reinforced concrete (RC) bridge columns require high degree of confinement, the inelastic action in these regions can lead to low-cycle fatigue failure of the reinforcement. Current seismic provisions are based on cyclic tests of scaled columns with smaller bar diameters. Results of previous testing suggest that fatigue life may be influenced by bar diameter and other material parameters. This research seeks to investigate the low-cycle fatigue characteristics of large diameter reinforcing bars and couplers that experience inelastic strain reversals in the plastic hinge regions of critical structural members. While the original scope of work included testing of #18 reinforcing bars and typical couplers used to splice bars in bridge columns, the available budget and initial difficulties in designing the test setup and finalizing the test protocols precluded testing of #18 bars and couplers. Therefore, the results presented in this report is limited to evaluating the cyclic axial stress-strain response of ASTM A-615 #11 and ASTM A-706 #14 steel reinforcing bars to various loading histories. In particular, the low-cycle fatigue behavior of #14 bars for three different heats were established. A special-purpose rebar-grip was designed to facilitate large-amplitude cyclic testing of bars. Issues related to fatigue testing and the development of fatigue-life expressions for reinforcing bars is identified. Another objective of this study is to develop a cyclic testing protocol that accurately represents the expected strain history during a design event. A series of analytical simulations of typical bridge columns were conducted to determine the rotation or curvature histories of potential plastic hinge (yielding) zones and corresponding strain histories have been established. These studies will provide the basis for establishing the range of strain amplitudes as well as the number of cycles that typical reinforcing bars experience under earthquake loads. The proposed research will provide fundamental knowledge on the inelastic properties of reinforcing bars and the research outcomes will aid in developing new cyclic testing protocols, provide a more rational approach to postearthquake damage assessment and enhance the safety of highway bridge construction in California. 17. KEY WORDS Experimental testing; low-cycle fatigue; reinforcing bars 18. DISTRIBUTION STATEMENT No restrictions. This document is available to the public through the National Technical Information Service, Springfield, VA SECURITY CLASSIFICATION (of this report) Unclassified 20. NUMBER OF PAGES 69 + viii Reproduction of completed page authorized 21. PRICE ii

3 Disclaimer The results, recommendations and opinions presented in this report are those of the authors and do not necessarily reflect the viewpoints of the California Department of Transportation or the State of California. iii

4 Acknowledgements Funding for this study provided by the California Department of Transportation (Caltrans) under Contract No.59A0539 is gratefully acknowledged. Input and comments from Peter Lee and Issam Noureddine throughout the project are sincerely appreciated. We also acknowledge the donation of the reinforcing bars from TAMCO, Rancho Cucamonga, CA. iv

5 Abstract Since modern provisions for the design of reinforced concrete (RC) bridge columns require high degree of confinement, the inelastic action in these regions can lead to low-cycle fatigue failure of the reinforcement. Current seismic provisions are based on cyclic tests of scaled columns with smaller bar diameters. Results of previous testing suggest that fatigue life may be influenced by bar diameter and other material parameters. This research seeks to investigate the low-cycle fatigue characteristics of large diameter reinforcing bars and couplers that experience inelastic strain reversals in the plastic hinge regions of critical structural members. While the original scope of work included testing of #18 reinforcing bars and typical couplers used to splice bars in bridge columns, the available budget and initial difficulties in designing the test setup and finalizing the test protocols precluded testing of #18 bars and couplers. Therefore, the results presented in this report is limited to evaluating the cyclic axial stress-strain response of ASTM A-615 #11 and ASTM A-706 #14 reinforcing bars to various loading histories. In particular, the low-cycle fatigue behavior of #14 bars for three different heats were established. A special-purpose rebar-grip was designed to facilitate large-amplitude cyclic testing of bars. Issues related to fatigue testing and the development of fatigue-life expressions for reinforcing bars is identified. Another objective of this study is to develop a cyclic testing protocol that accurately represents the expected strain history during a design event. A series of analytical simulations of typical bridge columns were conducted to determine the rotation or curvature histories of potential plastic hinge (yielding) zones and corresponding strain histories have been established. These studies will provide the basis for establishing the range of strain amplitudes as well as the number of cycles that typical reinforcing bars experience under earthquake loads. The proposed research will provide fundamental knowledge on the inelastic properties of reinforcing bars and the research outcomes will aid in developing new cyclic testing protocols, provide a more rational approach to post-earthquake damage assessment and enhance the safety of highway bridge construction in California. v

6 TABLE OF CONTENTS TECHNICAL REPORT DOCUMENTATION PAGE... ii Acknowledgements... iv Abstract... v Table of Contents... vi List of Figures... vii List of Tables... viii 1 INTRODUCTION Scope of work DESIGN OF EXPERIMENTAL PROGRAM Test Parameters and Protocol Experimental Setup Design And Fabrication Of Rebar Grip TESTS ON ASTM A-615 #11 REINFORCING BARS Control Tests Cyclic Fatigue Testing Phase II Fatigue Testing RESULTS OF TESTING OF ASTM A-706 #14 BARS Control Testing Low Cycle Fatigue Testing Investigation of Model Properties Summary of Findings Observed Reinforcing Steel Behavior from Monotonic Tests Observed Reinforcing Steel Behavior from Cyclic Tests IMPLICATIONS OF TEST RESULTS ON SEISMIC DESIGN OF BRIDGE COLUMNS Strain Demands in Reinforcing Bars in Typical Bridge Columns Ground Motions Strain Histories Transforming Random Histories To Equivalent Cycles At Constant Amplitude CONCLUSIONS Recommendations for future work REFERENCES vi

7 LIST OF FIGURES Figure 1-1: Effect of bar size on fatigue life of reinforcing bars (Brown and Kunnath, 2004). 2 Figure 2-1: Elevation and plan view of test setup... 7 Figure 2-2: Actuator and Guiding System... 8 Figure 2-3: View of one section of the gripping device... 9 Figure 2-4: Photograph showing one end of grip with reinforcing bar in place... 9 Figure 2-5: Complete grip with specimen secured to loading system... 9 Figure 2-6: Final test setup Figure 4-1: Fatigue life model for #14 bars (Heat 1) Figure 4-2: Fatigue life model for #14 bars (Heat 2) Figure 4-3: Fatigue life model for all #14 bars (average from both heats) Figure 4-4: Computation of effective single cycle strain for bars subjected to positive strains Figure 4-5: Fatigue life model for #14 bars subjected to positive strains only Figure 4-6: State of fractured specimens (N14H3F3S1 & N14H3F3S2) Figure 4-7: Monotonic response in tension and compression of reinforcing steel Figure 4-8: Cyclic response of reinforcing bar demonstrating (a) shrinking yield plateau, hardening and growth of curvature; (b) close-up of yield plateau region Figure 4-9: Low-cycle fatigue response (a) strain history; (b) stress-strain response Figure 4-10: Stress relaxation under fatigue loading at varying strain amplitudes Figure 4-11: Cyclic loading of bars including buckling Figure Elevation and sectional details of typical over-crossing in California Figure 5-2 Computer modeling of typical over-crossing Figure Time history of axial strain in #11 reinforcing bar (EQ # 2) Figure Time history of axial strain in #11 reinforcing bar (EQ # 4) Figure Time history of axial strain in #11 reinforcing bar (EQ # 9) Figure Time history of axial strain in #11 reinforcing bar (EQ # 12) Figure Time history of axial strain in #14 reinforcing bar (EQ # 2) Figure Time history of axial strain in #14 reinforcing bar (EQ # 4) Figure Time history of axial strain in #14 reinforcing bar (EQ # 9) Figure Time history of axial strain in #14 reinforcing bar (EQ # 12) vii

8 LIST OF TABLES Table 1-1: Original Test Matrix... 4 Table 3-1: Summary of #11 Bars (Single heat only) Table 3-2: Summary of Chemical Composition and Specified Strengths of #11 Bars 12 Table 4-1: Summary of Chemical Composition and Specified Strengths of #14 Bars 21 Table 4-2: Summary of Monotonic Testing of #14 Bars Table 4-3: Summary of Cyclic Testing of #14 Bars (Heats 1 & 2) Table 4-4: Low-Cycle Fatigue Testing of #14 Bars (Heat 3) Table 4-5: Summary of Model Parameter Testing of #14 Bars viii

9 1 INTRODUCTION Current provisions for the design of reinforced concrete (RC) members subjected to severe seismic loading rely on proper detailing of well-defined plastic hinge regions where most of the inelastic deformation is expected to occur. A great deal of experimental work in the past has focused on improving seismic detailing to prevent loss of confinement and to avoid buckling of longitudinal reinforcement in compression members. The inelastic action in the plastic hinge regions of critical components results in significant tension and compression strain reversals in the longitudinal reinforcing steel. Therefore, the critical failure mode of a well-detailed reinforced concrete structural member may eventually be controlled by low-cycle fatigue of the longitudinal steel. When a reinforced concrete bridge column is subjected to cyclic loading, the concrete cover will typically spall at fairly small strains (below the yield strain of the reinforcing steel). As the spalling progresses, the reinforcing steel is exposed to the air. The cyclic response of exposed reinforcing bars in the inelastic strain range can be reasonably reproduced in a fatigue test of an individual bar without the presence of concrete. During cyclic reversals, the bar will eventually buckle under compression. This buckling, however, can extend beyond single hoop spacing. Buckling leads to weakening (embrittlement) of the material which in turn translates into reduced fatigue life. Hence, low-cycle fatigue failure of longitudinal reinforcement is a critical failure mode that deserves more attention. Most fatigue tests carried out in the past have been medium- to high-cycle fatigue with strain amplitudes less than 1% and failure occurring between complete cycles (Helgason, 1976). These studies revealed that both bar diameter and grade of bar influenced the finite-life fatigue strength of reinforcing bars. More importantly, however, was the finding that the lug geometry factor r/h (lug base radius to lug height) played an important role in altering the fatigue resistance of the steel. Rolled-on deformations lead to regions of stress concentration which then become initiators of fatigue cracks. Deformation patterns were also found to influence fatigue life (Kokubu and Okamura, 1969; MacGregor et al., 1971). Other high-cycle fatigue studies on reinforcing bars have identified the effect of bar size on fatigue life with larger diameter bars displaying decreased fatigue resistance (Hanson et al., 1968; Pfister and Hognestad, 1964). Many of these findings will also be relevant in studies of low-cycle fatigue. 1

10 A series of low-cycle fatigue tests were performed at the State University of New York, Buffalo (Mander et al., 1994). The unsupported length (s) of the specimens tested was equivalent to six or larger bar diameters (d b ). Results indicated that s/d b ratios larger than six led to a reduction in strength below the yield value as large compressive strains (i.e. deformations normalized by unsupported length approximately equal to 0.06) were imposed. This was a result of severe inelastic buckling. However, Mander's tests were limited to small bar sizes (#5) not typically used for longitudinal reinforcement. The most recent published work on fatigue behavior of reinforcing bars is the work supervised by the lead PI at NIST (Brown and Kunnath, 2004). Additionally, there exists unpublished data by manufacturers of reinforcing steel on fatigue life characteristics of these components. The tests by Brown and Kunnath indicate that bar size is a factor that influences fatigue life (as shown in Figure 1-1). These findings are based on a limited study of bar sizes # 6 through #9. If the fatigue life of longitudinal bars used in bridge construction in high seismic zones is influenced by the diameter of the bar, this effect must be investigated and documented # 6 # 7 # 8 # Half cycles to failure Total Strain, εa 1 Figure 1-1: Effect of bar size on fatigue life of reinforcing bars (Brown and Kunnath, 2004) Similarly, current seismic provisions for transverse reinforcement in plastic hinge zones are mostly based on testing done for scaled columns containing small diameter bars. While some tests on large scale and full scale columns do exist, they are inadequate to fully quantify lowcycle fatigue effects on reinforcing bars for the range of parameters of interest. 2

11 A clearer understanding of the low-cycle fatigue behavior of reinforcing steel used in RC bridge construction in California is crucial to the development of design criteria for performance assessment of RC bridges in seismic applications. The proposed study will provide new experimental and analytical data on the buckling and low-cycle fatigue behavior of reinforcing bars and consequently provide a basis for developing design criteria as well as a methodology for damage assessment of bridge columns after a major seismic event. 1.1 SCOPE OF WORK The scope of the original work included testing of #18 reinforcing bars and typical couplers used to splice bars in bridge columns. The proposed test schedule is shown in Table 1-1. However, the available budget and initial difficulties in designing the test setup and finalizing the test protocols precluded testing of #18 bars and couplers. Some of the issues related to developing an appropriate test protocol in terms of stress and strain control are discussed in this report. The work presented in this report is limited to preliminary tests on ASTM A-615 #11 bars so as to verify the adequacy of the test setup, additional testing on ASTM A-706 #14 bars to establish acceptable testing protocols and finally low-cycle fatigue testing of three sets of A-706 #14 bars from different heats. 3

12 Table 1-1: Original Test Matrix REBARS COUPLERS Heat A Heat B Heat C Rebar/Coupler Upper Limit Strength Lower Limit Strength Test Strain (%) Min. No. Tests #18 Bars Min. No. Tests #14 Bars Min. No. of Tests (Possible) Max. No. of Tests Intermediate Strength TOTALS(Bars) Bar Splice BPI-GRIP XL Bar Splice Taper Thread Grip-Twist Dayton/Richmond US/MC Dextra Bartec Dayton Superior Bar-Lock XL Erico/Lenton Plus Std. Coupler A12 Erico/Lenton Plus Position Coupler P14L HRC Std. Coupler HRC 410/420 HRC Position Coupler HRC 410/490 HRC Std. Coupler Xtender N/A N/A N/A

13 2 DESIGN OF EXPERIMENTAL PROGRAM The experimental setup comprises two parts: the design and construction of the test frame for physical testing of the reinforcing bars, and the design and fabrication of the special grips to be used in anchoring the ends of the reinforcing bars without altering the deformations on the bar. 2.1 TEST PARAMETERS AND PROTOCOL 1. L/d ratio: In the case of deformed reinforcing bars without couplers, tests will be conducted on unaltered (non-machined) specimens with a fixed aspect ratio. While the hoop spacing in RC members in potential plastic hinge zones is specified in design, the actual buckled profile may spread beyond one hoop spacing. Based on preliminary analysis and observed experimental data, the L/d ratio selected for the testing is Strain amplitude: The amplitude of the imposed strain will vary between 1.0% and 4.0% at intervals of 1.0%. 3. Strain rates: Imposed strain rates will be realistic and correspond to typical frequencies in RC bridge columns. The proposed testing will include direct tension, compression and cyclic tests. Control tests pure tension only: The stress-strain response under direct tension will first be established. Compression tests: The importance of performing pure compression tests is to gain a better understanding of the buckling behavior of bars. Additionally, the stress-strain response can be compared to the behavior in pure tension. Model parameter tests: These tests were carried out to establish certain critical cyclic model parameters to enable development of analytical constitutive models of reinforcing steel. Constant amplitude cyclic tests: These tests provide critical fatigue life relationships for use in modeling fracture and failure. 5

14 2.2 EXPERIMENTAL SETUP The experiments were performed in the Structural Testing laboratory at the University of Southern California. A schematic of the test frame is shown in Figure 2-1. The setup is composed of a steel reaction frame and a concrete block rigidly attached to the frame as shown in the figure. The actuators are mounted to the concrete block. For testing of #14 bars, a single 300 kip actuator is adequate while testing of #18 bars would require that two actuators (as shown in Figure 2-1) be used in parallel. The specimen mounted on the rebar grips is placed between the actuator and an L-frame which is rigidly connected to the heavy floor beam. Both vertical and lateral supports are provided to maintain stability of the system during loading. Following preliminary tests of the system it was found that an additional constraint beam was necessary to prevent out-of-plane movement of the setup in the horizontal plane. Hence, the testing apparatus was modified by adding a guiding system, and a reaction mechanism for the steel box on the other side of specimen. The guiding system restrains the actuator joints from out-of-plane movement during loading process, allowing the specimen to be loaded in a purely axial manner. The key components of the guiding system are a steel C channel welded on backing plate and a steel bar welded on the attachment of the actuator. Figure 2-2 illustrates how the guiding system works during loading. The steel bar welded on actuator attachment is placed inside the C channel, transferring the total weight of actuator and its attachment to the C channel and then to the back plates. Approximately 400 lbf friction was observed from a free run of the actuator. Measures to counter this minimal friction included: adding Teflon pads between the sliding bar and the channel; adding some lubricant oil to the contact area inside the C channel. The total force lost to friction is estimated to be less than 0.1% of the applied force and hence considered to be negligible. 6

15 Notes: 1. Vertical supports are provided to keep the actuators horizontal during loading process. 2. Lateral supports are provided to avoid lateral instability of the actuators during loading process. Concrete block Reaction frame L shape frame Stackable grips Loading beam Actuator Specimen Floor beam SIDE VIEW Figure 2-1: Elevation and plan view of test setup 7

16 Figure 2-2: Actuator and Guiding System 2.3 DESIGN AND FABRICATION OF REBAR GRIP A special purpose grip consisting of high-strength stackable blocks and grooved wedges was designed and fabricated for the testing of large diameter bars. A view of the grip for one of the bar ends is shown in Figure 2-3. A similar grip is used at the other end to complete the specimen attachment to the test assembly shown previously in Figure 2-1. Figure 2-4 shows a view of the reinforcing bar secured to one end of the grip. The complete grip with the bar firmly secured after tightening the bolts and mounted on the loading system is shown in Figure

17 Figure 2-3: View of one section of the gripping device Figure 2-4: Photograph showing one end of grip with reinforcing bar in place Figure 2-5: Complete grip with specimen secured to loading system 9

18 The final view of the test setup is shown in Figure 2-6. Figure 2-6: Final test setup 10

19 3 TESTS ON ASTM A-615 #11 REINFORCING BARS Preliminary testing was carried out on #11 reinforcing bars to check the robustness of the test setup and to examine issues related to cyclic testing of bars using strain and stress control as well as issues related to strain measurements in the presence of bar buckling. Since a reinforcing bar will buckle in compression and the stress at the initiation of buckling can vary from cycle to cycle, it is difficult to predict the buckling stress prior to testing. Furthermore, the measured strain is a function of the gage length. Standard strain measurements are based on a gage length of 2 inches. However, these strains become erratic when the bar begins to buckle accompanied by a reduction in measured stress. Table 3-1 presents a summary of the 11 bars (from the same heat) that were tested in this phase. The chemical composition of the bar material is listed in Table 3-2. Detailed information on the test results is presented in subsequent sections of this chapter. Table 3-1: Summary of #11 Bars (Single heat only) Specimen Average strain Test type L/D Designation* (%) 1 N11T1 Monotonic N11T2 Monotonic N11T3 Cyclic N11Y1 Cyclic N11Y2 Cyclic N11Y3 Cyclic 7 5 to 3 7 N11Y4 Cyclic 7 ±2 and ±4 8 N11Y5 Cyclic 7 2 to 0 9 N11Y6 Cyclic N11Y7 Cyclic N11Y8 Cyclic 7 ±2 and 0 to 4 * Notation: First 3 letters Bar No (e.g. N11 Number 11 reinforcing bar) Next letter Type of test: T = Monotonic tension test; Y = Cyclic Next number Test or specimen number Example: N11T1 = No.11 bar subjected to monotonic tension (Specimen #1) N11Y2 = No.11 bar subject to cyclic (Specimen #5) 11

20 Table 3-2: Summary of Chemical Composition and Specified Strengths of #11 Bars Strength (ksi) Yield: 70.5 Ultimate: Chemical Composition C Mn P S Si Cr Ni Cu Mo V CE CONTROL TESTS The first series of tests consisted of control tests to establish the typical monotonic stressstrain response for the batch of reinforcing bars. The obtained stress-strain relationship will then form the basis of determining the protocols for the low-cycle fatigue testing. Measured yield stress values for the three specimens were: 70.1, 71.5 and 70.0ksi, respectively with corresponding strains of , and The mean values for the material properties of the specimens are summarized below: Yield Strain: Yield Stress: 70.7 ksi; Peak Stress: 95.2 ksi Ultimate strain at failure: Specimen name: N11T1 HEAT #1 Stress Strain (10-2 ) relationship Summary: The rebar was tested under pure axial tension with strains measured across a gage length of 2 at the center of the specimen. Rupture of the bar occurred outside the measured gage length; hence the measured strain beyond yield does not precisely indicate the inelastic behavior. 12

21 Specimen name: N11T2 HEAT #1 Stress Strain (10-2 ) relationship Summary: The bar ruptured within the measured gage length at a strain corresponding to approximately 21%. Specimen name: N11T3 HEAT #1 Stress Strain (10-2 ) relationship Summary: The bar ruptured within the measured gage length at a strain corresponding to 24.6%. However, it is possible that there was some slip after 15% strain; hence the failure strain value is not entirely reliable. 13

22 3.2 CYCLIC FATIGUE TESTING The next series of tests investigated the fatigue behavior of #11 bars. As discussed at the start of this section, numerous problems were encountered in the cyclic tests as a result of compression buckling. A summary of the test results are presented in this section. Specimen: N11Y1 HEAT #1 Stress-strain (10-2 ) relationship Overall view of final condition of the bar Summary: The rebar was tensioned to an average strain of 0.10 and then subjected to compressive load. The rebar began to buckle when the tension strain reduced to After the compressive stress increased to about 60 ksi, the strain data from extensometer indicated that the deformation in the gage length across which the strain was being measured did not change any further because of seriously buckling. Testing was stopped at this stage since it was not possible to acquire reliable data beyond this point. 14

23 Specimen name: N11Y2 HEAT #1 Stress Strain (10-2 ) relationship Overall view of final condition of the bar Summary: As in the case of the previous test, the reinforcing bar was subject to tensile force up to a recorded strain of 0.10 (10%) and then subjected to compressive loading. Once again, the rebar buckled when the tension strain reduced to Serious buckling (as shown in the figure above) prevented the test from continuing since the total deformation across the bar length and the measured strains in the strain gage was not correlated to the overall buckling deformation. 15

24 3.2.1 PHASE II FATIGUE TESTING The bar was subjected to an initial tension strain of 0.02 and then subjected to compression until buckling was initiated. The load was reversed from compression to tension each time that buckling was visibly observed on the compression side. Specimen name: N11Y3 HEAT #1 Stress (ksi) Strain (10-2 ) relationship Overall view of final condition of the bar (side view) Observations: Note that the compressive stress at buckling reduced with each successive cycle this is partly due to the fact that the measured strain across the buckled region is not accurate. After 6 cycles, the rebar ruptured during the tension cycle. 16

25 Specimen name: N11Y4 HEAT #1 Stress (ksi) Strain (10-2 ) relationship Final condition of the bar Summary: The rebar was tested using the strains measured across the entire specimen length including buckling deformations. It was original compressed to a strain and then tensioned to a deformation After eight constant cycles from to 0.02, the imposed strain was increased to After three constant cycles from to 0.04, the rebar ruptured during the final tension cycle. 17

26 Specimen name: N11Y5 HEAT #1 Stress (ksi) Strain (10-2 ) relationship Summary: Once again, the bar was tested using measured strains across the entire bar length. Cyclic strain between 0.02 in tension and zero strain in compression were applied. After 35 constant cycles, the tensile strain was increased to At this stage, control of the linear potentiometer was lost and the test had to be stopped. Specimen name: N11Y6 HEAT #1 Stress (ksi) strain (10-2 ) relationship Summary: This test explored the idea of using either stress or the total bar strain as a control method. At first, the bar was subjected to an axial tension of 1% based on the strain measured by the strain gage. The experiment was then controlled using the measured strain in the linear potentiometer across the full gage length of the specimen. It was observed that the average strains across the entire bar length gradually increased though the strain in the extensometer was nearly constant. The loading was reversed from compression to tension either when the strain reached zero strain or when any sign of buckling was observed. The bar ruptured after 22 cycles. 18

27 Specimen name: N11Y7 HEAT #1 Stress (ksi) Strain (10-2 ) relationship Summary: In an attempt to establish some correlation between strains measured in the extensometer (gage length = 2 ) and strains in the linear potentiometer (gage length = 7 ), the bar was initially subjected to cyclic strains of +/- 2% followed by strains corresponding to 4% tension and 0% compression (for an average +/- strain corresponding to 2%). Rupture of the bar occurred after 5 full cycles. Specimen name: N11Y8 HEAT #1 Stress (ksi) Strain (10-2 ) relationship Summary: The specimen was subjected to full cyclic strains corresponding to +/- 5% strain. Strain measurement was based on the strain in the extensometer across a 2 gage length. Failure occurred during the 3 rd cycle in tension. 19

28 4 RESULTS OF TESTING OF ASTM A-706 #14 BARS Findings from the preliminary series of tests on #11 reinforcing bars highlighted the following issues: The fatigue life is influenced by bar buckling that occurs during compression loading Bar buckling leads to errors and inconsistencies in measured strain across a gage length that is fixed at the center of the specimen Hence, it was necessary to conduct testing of the bars using multiple measurements. Initially, the project was conceived, in accordance with Caltrans specifications at the start of the project, to be conducted under stress control. However, the two concerns listed above and additional issues that arose during testing of the bars required repetition of the lowcycle fatigue testing. Results of the testing and relevant findings are described in the following sections. 4.1 CONTROL TESTING The first series of five tests were designed to establish the basic monotonic stressstrain behavior of #14 bars. This was followed by 21 low-cycle fatigue tests at various strain amplitudes and various strain ranges. Finally, four additional tests were carried out the assist in modeling the cyclic behavior of reinforcing bars under random cyclic loading. The actual strain histories experienced by a reinforcing bar in a bridge column during a seismic event is random and therefore, results from this set of tests is expected to enable improved modeling of rebars and facilitate inelastic response prediction of highway bridges subjected to severe seismic loads. Table 4-1 provides the chemical composition of the reinforcing bars obtained from three different heats. In the first phase of testing, three types of tests were conducted: the first set comprised direct tension tests, the next set consisted of limited pure compression tests followed finally by some controlled cyclic tests to establish model parameters for describing the cyclic constitutive behavior of the bars. Details of the test protocols are provided in Table

29 Table 4-1: Summary of Chemical Composition and Specified Strengths of #14 Bars Heat Strength # (ksi) 1 Yield: 74.0 Ultimate: Yield: 68.5 Ultimate: Yield: 73.0 Ultimate: 97.5 Chemical Composition C Mn P S Si Cr Ni Cu Mo V CE Table 4-2: Summary of Monotonic Testing of #14 Bars Heat No. Ultimate No. Test type L/D Strain (%) 1 N14H1T1 1 Tension test N14H1T2 1 Tension test N14H2T1 2 Tension test N14H2T2 2 Tension test N14H3T1 3 Tension test N14H3T2 3 Tension test N14H1C1 1 Compression test N14H1C2 1 Compression test N14H1C3 1 Compression test Bar Designation: N##H#Q# Notation - N##: Bar Number H#: Heat Number Q#: Test type (T= Tension; C= Compression; Y = Cyclic) & specimen # Example: N14H1T2 (No.14 bar from heat #1 subject to direct tension, test #2) N14H1C3 (No.14 bar from heat #1 subject to direct compression, test #9) 21

30 The first series of tests were conducted to establish the backbone stress-strain relationships in both tension and compression. Results of the monotonic testing are presented in this section beginning with the tension tests. Two specimens were tested for each heat in the case of tension tests. Specimen name: N14H1T1 Heat #1, Specimen #1 Stress (ksi) Strain relationship Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: The rebar was tested under control of the extensometer. Tensile force was applied at a constant rate until failure. The location of rupture was outside the gage length of the extensometer. Hence, during necking there was no increase in strain as the stress level started to drop. 22

31 Specimen name: N14H1T2 Heat #1, Specimen #2 Stress (ksi) Strain relationship Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: The rebar was tested under control of the extensometer. As in the case of the previous specimen, a tensile force was applied until failure of the specimen. Tensile rupture occurred as a measured strain of approximately 28%. Results of the tensile tests for the bars from heats #2 and #3 are presented in the next set of figures. The curves in some of these tests have been smoothed to remove the noise in the data which results from machine vibrations and also the fact that the imposed strains on the specimens was carried out in stages to avoid sudden failure and damage to the extensometer. 23

32 Stress-Stress Relationship Specimen name: N14H2T1, Specimen name: N14H2T2, Heat #2, Specimen #1 Heat #2, Specimen #2 Stress (ksi) Strain relationship Specimen name: N14H3T1 Specimen name: N14H3T2, Heat #3, Specimen #1 Heat #3, Specimen #2 24

33 The next series of specimens (all from Heat #1) were subjected to purely compressive forces to establish the stress-strain relationship in compression. Results are presented in the next three figures. Specimen name: N14H1C1 Stress (ksi) Strain relationship Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: The length of the rebar between the two grips was 9.75 inch which corresponds to an L/d ratio of approximately 5.5. Buckling initiated at a stress of about 82 ksi. When the compressive strength dropped to 60 ksi (corresponding to a strain of 0.14), the extensometer was removed since the measured strains were not representative of material strain and the test was stopped. The rebar ruptured when the compressive force was unloaded. 25

34 Specimen name: N14H1C2 Stress (ksi) Strain relationship Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: This specimen also had a similar L/d ratio as the previous specimen. The measured peak stress was 82.0 ksi followed by buckling of the bar. When the compressive strength dropped to 62.7 ksi at a strain of 0.135, the test was stopped. The rebar ruptured when the compressive force was unloaded. 26

35 Specimen name: N14H1C3 Stress (ksi) Strain relationship Summary: The length of the rebar between two grips was reduced to 6.5 inch to avoid buckling of the specimen. To protect the extensometer, the test was stopped when the peak stress reached about 110 ksi at a compressive strain of The rebar ruptured when the compressive force was unloaded. 27

36 4.2 LOW CYCLE FATIGUE TESTING Low-cycle fatigue tests were carried out on #14 bars under constant strain amplitudes. The test protocol was derived based on the experience and observations from the cyclic tests on the #11 bars. The following procedure was adopted: The bar was initially stressed in tension up to the target strain using the extensometer as the control instrument. The strain at this stage in the linear potentiometer was recorded across the full gage length of the reinforcing bar. Control of the test was then transferred to the potentiometer. The cyclic testing then continued by removing the extensometer and using strain control for the remainder of the experiment. Details of the test specimens are provided in Table 4-3. Table 4-3: Summary of Cyclic Testing of #14 Bars (Heats 1 & 2) No. Heat No Strain range Cycles to (%) failure 1 N14H1F1S1 1 ± N14H1F1S2 1 ± N14H1F2S1 1 ± N14H1F2S2 1 ± N14H1F4S1 1 ± N14H1F4S2 1 ± N14H2F6S1 1 ± N14H2F6S2 1 ± N14H2F1S1 2 ± N14H2F1S2 2 ± N14H2F2S1 2 ± N14H2F2S2 2 ± N14H3F4S1 2 ± N14H3F4S2 2 ± N14H3F6S1 2 ± N14H3F6S2 2 ±6 1.0 Notation: N##H#F#S# Bar#, Heat#, Fatigue test strain amplitude in %, Specimen # Example: N14H2F4S1 = Bar #14, Heat 2, Fatigue test at strain of 4%, specimen 1 28

37 Results of the low-cycle fatigue tests provided the following fatigue life expressions: Heat 1: ( ) 2.02 N f = ε Heat 2: ( ) 2.00 N f = ε m m (4-1) (4-2) In the above expression, is the number of cycles to failure and is the peak or total strain (equal amplitude in both tension and compression). Figure 4-1 shows the life curve in log space for the #14 bars corresponding to Heat 1. A similar curve for Heat 2 is displayed in Figure 4-2. Figure 4-1: Fatigue life model for #14 bars (Heat 1) Figure 4-2: Fatigue life model for #14 bars (Heat 2) 29

38 If the average response of all #14 bars from both heats is considered, the fatigue life expression will transform to the following equation: N f = ( ε ) 2.03 m (4-3) y = x Cycles to failure Strain 0.1 Figure 4-3: Fatigue life model for all #14 bars (average from both heats) Sample plots and figures of typical stress-strain responses for the low-cycle fatigue tests and the failure state of the ruptured bar are in the following section. 30

39 Specimen name: N14H1F1S1 Heat 1 Stress Strain Response Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: Constant amplitude strain between and was applied to the specimen. For the first half cycle the rebar was tensioned under the control of the extensometer. After the average strain reached the expected value, the test was continued under the control of linear potentiometer (which measured strain across the full gage length). The bar ruptured after 45 cycles of loading. 31

40 Specimen name: N14H1F2S1 Heat 1 Stress Strain Response Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: The L/d ratio of the specimen was 5.5. The gage length for strain measurement was the full bar length.constant amplitude strain between and was applied to the specimen. The bar ruptured after 16 full cycles. 32

41 Specimen name: N14H1F4S1 Heat 1 Stress Strain Response Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: The L/d ratio of the specimen was 5.5. Constant amplitude strain between and was applied to the specimen. The bar ruptured after 4 full cycles. The gage length was the full bar length. 33

42 Specimen name: N14H1F6S1 Heat 1 Stress Strain Response Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: Constant amplitude strain between and was applied to the specimen. The bar ruptured after a single cycle of loading. 34

43 Specimen name: N14H2F1S1 Heat 2 Stress Strain Response Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: Constant amplitude strain between and was applied to the specimen. For the first half cycle the rebar was tensioned under the control of the extensometer. After the average strain reached the expected value, the test was continued under the control of linear potentiometer (which measured strain across the full gage length). The bar ruptured after 42 cycles of loading. 35

44 Specimen name: N14H2F2S1 Heat 2 Stress Strain Response Summary: Constant amplitude strain between and was applied to the specimen. For the first half cycle the rebar was tensioned under the control of the extensometer and the test was continued under the control of linear potentiometer (which measured strain across the full gage length). The bar ruptured after 14 cycles of loading. Specimen name: N14H2F4S1 Heat 2 Stress Strain Response Summary: The rebar was tested under the control of the linear potentiometer hence the gage length was equal to the total bar length between the grips. Constant amplitude strain between and was applied to the specimen. The bar ruptured after almost 5 cycles of loading. 36

45 Specimen name: N14H2F6S1 Heat 2 Stress Strain Response Summary: Constant amplitude strain between and was applied to the specimen. The bar ruptured after 1 cycle of loading. In the case of Heat 3, the bars were cycled between zero strain in compression and the target strain in tension. This was done to possibly avoid the significant effects of localized strain resulting from buckling of the specimen. Since the average tensile strain is larger in this case than in the previous sets of experiments, these results and the resulting fatigue life models are presented separately. Table 4-4 summarizes the specimens tested in this phase of the project. Table 4-4: Low-Cycle Fatigue Testing of #14 Bars (Heat 3) No. Heat No Strain range Cycles to (%) failure 1 N14H3F2S N14H3F2S N14H3F3S N14H3F3S N14H3F4S N14H3F4S Notation: N##H#F#S# Bar#, Heat#, Fatigue tension strain in %, Specimen # Example: N14H3F4S1 = Bar #14, Heat 2, Max tension strain of 4%, specimen 1 37

46 In developing the fatigue-life expression for the tests conducted in this phase, the effective single cycle strain was set equal to half the imposed strain 2ε t (see Figure 4-4). The resulting fatigue-life expression for these tests is given below: N = ( ε ) 2.4 f t (4-4) is the number of cycles to failure and is the peak tensile strain (zero strain in compression). Figure 4-5 shows the resulting low-cycle fatigue life curve. 2ε t Figure 4-4: Computation of effective single cycle strain for bars subjected to positive strains Figure 4-5: Fatigue life model for #14 bars subjected to positive strains only 38

47 Stress-strain responses for each of the tested specimens are presented below. Specimen: N14H3F2S1 Stress Strain Response Specimen: N14H3F2S2 Summary: The bar was subjected to low-cycle fatigue loading corresponding to a tension strain of 2% and a compression strain of 0%. Specimen 1 ruptured after 17 full cycles of loading while specimen 2 ruptured after 20 cycles of loading. 39

48 Specimen: N14H3F3S1 Stress Strain Response Specimen: N14H3F3S2 Summary: The bar was subjected to low-cycle fatigue loading corresponding to a tension strain of 3% and a compression strain of 0%. Specimen 1 ruptured after 8 full cycles of loading while specimen 2 ruptured after 7 cycles of loading. Figure 4-6: State of fractured specimens (N14H3F3S1 & N14H3F3S2) 40

49 Specimen: N14H3F4S1 Stress Strain Response Specimen: N14H3F4S2 Specimen N14H3F4S1 specimen shape after rupture Summary: The bar was subjected to low-cycle fatigue loading corresponding to a tension strain of 4% and a compression strain of 0%. Specimen 1 failed after 4 full cycles while specimen 2 failed after only 3 cycles of load. Buckling was unavoidable at this strain range as is evident from the bar shape after rupture. One final specimen was tested at a peak tensile strain of 4% and unloaded to 0% on the compression cycle this specimen failed on the return tensile strain. 41

50 4.3 INVESTIGATION OF MODEL PROPERTIES This phase of the study was aimed at investigating the cyclic behavior of reinforcing steel bars under random cyclic loads so as to provide input into improving existing constitutive models. The tests were designed to provide input on three specific aspects of cyclic response: (i) Bauschinger effect; (ii) effect of cyclic strains on the yield plateau; and (iii) strength deterioration under cyclic strains. Table 4-5 lists the 4 specimens Table 4-5: Summary of Model Parameter Testing of #14 Bars No. Test type L/D Strain (%) 1 N14H3M1 Model testing 6 Various 2 N14H3M2 Model testing 6 Various 3 N14H3M3 Model testing 6 Various 4 N14H3M4 Model testing 6 Various Bar Designation: N##H#Mn Notation - N##: Bar #, H#: Heat #, Mn: Model test specimen # Example: N14H3M1 (#14 bar from heat 3 subject to various cyclic strains, specimen #1) Specimen name: N14H3M1 Heat #3 Stress Strain Response (a) Strain across full bar length (b) Strain across gage length of 2 inch Summary: This test was designed to establish the correlation between strains across different gage lengths (linear potentiometer used across full bar length and extensometer mounted in middle of specimen at a 2 inch gage length). 42

51 Specimen name: N14H3M2 Heat #3 Stress Strain Response Summary: This objective of this test was to confirm observe aspects of cyclic behavior of reinforcing steel. Cycling of the steel after yielding has an influence on Bauschinger effect. The rebar was tested under control of extensometer. It was first tensioned to a strain of 10% and then compressed to the initiation of buckling. The load was reversed to achieve approximately the same tensile stress. The process was repeated till failure of the specimen. 43

52 Specimen name: N14H3M3 Heat #3 Stress Strain Response Summary: The rebar was tested under control of the extensometer. To establish the cyclic behavior of the material starting with compression loading, the following loading protocol was applied: 1. Load in compression up to 0.35% 2. Unload in tension up to 0.1% 3. Load in compression up to 0.6% 4. Unload in tension to same 0.1% (as in step 2) 5. Load in compression to approx total strain of 1% strain 6. Unload in tension to same strain as in steps 2 and 4 7. Load in compression up to 2% 8. Unload in tension to -0.5% 9. Reload in compression to 4% 10. Load in tension till failure 44

53 Specimen name: N14H3M4 Heat #3 Stress-Strain Response Photo-1, rupture section (side view) Photo-2, rupture section (top view) Summary: The rebar was tested under control of the extensometer. To establish the cyclic behavior of the material starting with compression, the following strain history was imposed: 1. Load in compression to a strain of 0.35% 2. Unload in tension up to 0.1% 3. Reload in compression to 0.6% 4. Unload in tension to 0.1% strain (as in step 2) 5. Load in compression to 1% strain (about 0.25% after strain hardening) 6. Unload in tension to 0.1% strain (as in steps 2 and 4) 7. Load in compression up to 2% 8. Unload in tension to -0.5% 9. Reload in compression to 4% 10. Load in tension till failure There was necking before the rebar rupture (as seen in figure) 45

54 4.4 SUMMARY OF FINDINGS The following statements summarize the essential findings of the study. Effect of Chemical Composition: As identified in Table 4-1, the chemical composition of the steel for the different heats is quite similar; hence there was no obvious difference in the low-cycle fatigue behavior of reinforcing bars from different heats. In fact, since no clear change in response was observed in the number of cycles to failure between heat #1 and heat #2, the test protocol was changed for the next heat in order to observe the influence of compression buckling on fatigue life. Test Methodology: Stress-controlled testing is not a feasible method for low-cycle fatigue testing since the magnitude of strain is not constant at constant stress as the bar undergoes yielding and buckling. In fact, post-buckling behavior results in a loss of strength, and thus it may be physically unrealistic to apply pre-determined stress amplitudes if buckling is expected. Variations in the magnitude of strain from cycle to cycle during a stress-controlled low-cycle fatigue test will not yield consistent results. Strain controlled testing, on the other hand, is a function of gage length. That is, the measured strain is in fact a pseudo-strain, which reflects normalized deformation, rather than the material strain which may vary over the gage length. It is essential to establish a reasonable gage length (with or without buckling) for measuring strains in a low-cycle fatigue test. Definition of Rebar Strain: Measurement of strain across the full gage length is more reliable since buckling deformations will be incorporated directly; however, since strain is a function of gage length, the estimated fatigue life will also be a function of the gage length across which rebar strains are measured. 46

55 It is necessary to develop different fatigue life expressions for different strain measurements. Influence of Buckling: Finally, since buckling is unavoidable unless a very small gage length is used, the stress concentration resulting from buckling can significantly alter the fatigue life of reinforcing bars Observed Reinforcing Steel Behavior from Monotonic Tests Prior to cyclic tests, monotonic tests were carried out to obtain the tension and compression bounding curves, which can be used as a basis for comparison with the deteriorated responses of reinforcing steel during cyclic loading. Although the cross section and the length between the two grips varies during the loading, the stress and strain response is computed using the initial cross section and length, which results in overestimation in compression and underestimation in tension as shown in Figure 4-7. Evidently, in true stress-strain space, the two curves would be identical before the effect of buckling controls the compressive response of reinforcing steel or the effect of necking controls the tensile response. Figure 4-7: Monotonic response in tension and compression of reinforcing steel 47

56 4.4.2 Observed Reinforcing Steel Behavior from Cyclic Tests Diminishing Yield Plateau: The yield plateau is a very unique feature in the response of reinforcing steel, which never appears in other metals and high carbon steels as observed by some researchers (Bertero and Popov, 1976; Seyed-Ranjbari, 1986; Dodd and Cooke, 1992). In engineering stress-strain space, the plateau is characterized by constant stress with increasing strain. This feature brings about an interesting observation during a cyclic test as shown in Figure 4-8. The dotted line shows the response from a typical cyclic test, where the hardening initiation point is marked with a circle (Figure 4-8b) on the solid line which represents the monotonic tensile envelop curve. During cyclic loading, the elasticperfectly-plastic region (or the size of the yield plateau) on the monotonic envelope curve for the first loading branch is seen to diminish after a few cycles till the accumulated plastic strain reaches a certain limit value. Hence, the onset of hardening occurs earlier, which results in a diminished yield plateau during cyclic loading accompanied by strain hardening. This mechanism is strongly associated with Bauschinger effect and the growth of the curvature during cyclic loading, which are discussed in the following section. (a) Figure 4-8: Cyclic response of reinforcing bar demonstrating (a) shrinking yield plateau, hardening and growth of curvature; (b) close-up of yield plateau region (b) Bauschinger Effect and Growth of Curvature: The Bauschinger effect is associated with the phenomenon of the size of elastic range getting smaller whenever the direction of straining is reversed during the cyclic test of ductile materials, and this is generally observed in most metals. On the other hand, the curvature of the stress-strain curve during each cycle keeps 48

57 getting larger. The degree of the reduction of elastic ranges and the growth of the curvature (Seyed-Ranjbari 1986) varies as a function of the accumulated plastic strain. This mechanism can be described as the process of strain hardening (cold-working) beyond the elastic range of the first cycle. As strain increases in one direction, the atoms of crystalline materials begin to be dislocated on a microscopic scale. Due to piling up of the dislocations, it becomes harder to increase the deformation in that direction, known as strain hardening. While reversal loading in the opposite direction is applied to the same specimen, however, it would be much easier to attract dislocations in the opposite direction, and the strength is reduced not only because of local back stresses but also due to defects during the previous cycles. Therefore, it yields earlier during the reversal loading (Bauschinger effect). In other words, the larger deterioration inside the material influences the reduction of the elastic range (yield surface) and also the growth of curvature depending on the accumulated plastic strain, which results in strain hardening upon loading as shown in Figure 4-8 and strain softening during the reversed loading. Low-Cycle Fatigue and Strength Degradation: Bar rupture due to low-cycle fatigue and the associated strength degradation is one of the common features of reinforcing steel observed during cyclic loading, which the material model ought to be capable to capture, if fatigue crack growth is not explicitly modeled. Repeated loading and unloading reduces the strength of reinforcing steel in each cycle and eventually leads the material to reach its failure limit even if it never reach the ultimate strength as shown in Figure 4-9. The rebar response shown in the figure was subjected to tension and compression cycles in the strain range from 0.2% to -0.2%, and it failed after 33 half cycles. The fatigue strength is mainly controlled by the number of cycles and strain amplitude. Coffin (1954) and Manson (1953) developed the well-known fatigue life formula using two material parameters to facilitate prediction of bar rupture. Also it was demonstrated by Kunnath et al. (2009) that strength degradation can also be expressed using Coffin-Manson s fatigue life expression. Therefore, assuming that cumulative strength degradation and cumulative fatigue damage has linear relationship, this feature was successfully incorporated into a proposed constitutive model for reinforcing steel by Kunnath et al. (2009). 49

58 (a) (b) Figure 4-9: Low-cycle fatigue response (a) strain history; (b) stress-strain response Cyclic Stress Relaxation under Constant Strain Amplitude: In addition to strength degradation due to low-cycle fatigue, another phenomenon associated with constant amplitude loading is stress relaxation. At the microscopic level, once the yield point is reached, metallic bonds are broken, resulting in structural line defects and pile-up of dislocations. The degree of dislocation accumulation controls the process of stress relaxation. To illustrate this concept, consider the stress-strain history shown in Figure 4-10a. The material is first subjected to tensile strain beyond yielding. If the material is now subjected to low-amplitude cyclic loading, upon reversal from a compression cycle a lower stress level is attained for the same strain amplitude. This is because the unloading strain in compression was limited and additional pile-up of dislocations in that direction is avoided thereby reducing the resistance to deformation in the tension cycle. Also, on the compression side, stress hardening is observed since less accumulation of dislocation during the tensile loading makes more room to accumulate dislocations in that direction. This phenomenon is also described in terms of the decrease in the average mean stress by Ma et al. (1976). The average mean stresses denoted by σ m1, σ m2, and σ m3 for each cycle are seen to systematically reduce under the limited strain amplitude as shown in Figure 4-10b. Depending upon the average strain range, the responses will be totally different as shown in Figure 4-10 (d) and (f) even if it has the same type of constant amplitude loading because it depends on the degree of plastic strain accumulation. The solid and dotted lines represent the first and the subsequent cycles respectively in Figure 4-10 (d) and (f). Experimental 50

59 results obtained from the strain history shown in Figure 4-10c is plotted in Figure 4-10d. In the next set of results presented in Figures 4-10 e-f, the specimen is first subjected to 6% strain in tension and then unloaded to zero strain. This is a fairly significant compression strain which leads to buckling of the bar specimen and increased strength degradation. Repeated cycling of the bar at this strain range leads to bar rupture after 13 cycles. (a) (b) (c) (d) (e) Figure 4-10: Stress relaxation under fatigue loading at varying strain amplitudes (f) 51