Aci Structural Journal Jan.-feb. 2015 v. 112 No. 01

V. 112, NO. 1 JANUARY-FEBRUARY 2015

ACI STRUCTURAL

J O U R N A L

A JOURNAL OF THE AMERICAN CONCRETE INSTITUTE

CONTENTS Board of Direction

ACI Structural Journal

President William E. Rushing Jr.

January-February 2015, V. 112, No. 1

Vice Presidents Sharon L. Wood Michael J. Schneider Directors Roger J. Becker Dean A. Browning Jeffrey W. Coleman Alejandro Durán-Herrera Robert J. Frosch Augusto H. Holmberg Cary S. Kopczynski Steven H. Kosmatka Kevin A. MacDonald Fred Meyer Michael M. Sprinkel David M. Suchorski Past President Board Members Anne M. Ellis James K. Wight Kenneth C. Hover Executive Vice President Ron Burg

Technical Activities Committee Ronald Janowiak, Chair Daniel W. Falconer, Staff Liaison JoAnn P. Browning Catherine E. French Fred R. Goodwin Trey Hamilton Neven Krstulovic-Opara Kimberly Kurtis Kevin A. MacDonald Jan Olek Michael Stenko Pericles C. Stivaros Andrew W. Taylor Eldon G. Tipping

Staff Executive Vice President Ron Burg

a journal of the american concrete institute an international technical society

3 A Global Integrity Parameter with Acoustic Emission for Load Testing of Prestressed Concrete Girders, by Francisco A. Barrios and Paul H. Ziehl 13 Analysis of Rectangular Sections Using Transformed Square Cross Sections of Unit-Length Side, by Girma Zerayohannes 23 Evaluation of Post-Earthquake Axial Load Capacity of Circular Bridge Columns, by Vesna Terzic and Bozidar Stojadinovic 35 Shear Behavior of Reinforced Concrete Columns with High-Strength Steel and Concrete, by Yu-Chen Ou and Dimas P. Kurniawan 47 Three-Parameter Kinematic Theory for Shear Behavior of Continuous Deep Beams, by Boyan I. Mihaylov, Bradley Hunt, Evan C. Bentz, and Michael P. Collins 59 Investigation of Bond Properties of Alternate Anchorage Schemes for Glass Fiber-Reinforced Polymer Bars, by Lisa Vint and Shamim Sheikh 69 Stress-Transfer Behavior of Reinforced Concrete Cracks and Interfaces, by Ali Reza Moradi, Masoud Soltani, and Abbas Ali Tasnimi 81 Condition Assessment of Prestressed Concrete Beams Using Cyclic and Monotonic Load Tests, by Mohamed K. ElBatanouny, Antonio Nanni, Paul H. Ziehl, and Fabio Matta 91 Crack Distribution in Fibrous Reinforced Concrete Tensile Prismatic Bar, by Yuri S. Karinski, Avraham N. Dancygier, and Amnon Katz Glass Fiber-Reinforced Polymer-Reinforced Circular Columns 103 under Simulated Seismic Loads, by Arjang Tavassoli, James Liu, and Shamim Sheikh 115 Discussion

Engineering Managing Director Daniel W. Falconer

Bond Strength of Spliced Fiber-Reinforced Polymer Reinforcement. Paper by

Managing Editor Khaled Nahlawi

Behavior of Epoxy-Injected Diagonally Cracked Full-Scale Reinforced

Ali Cihan Pay, Erdem Canbay, and Robert J. Frosch Concrete Girders. Paper by Matthew T. Smith, Daniel A. Howell, Mary Ann T. Triska, and Christopher Higgins

Staff Engineers Matthew R. Senecal Gregory M. Zeisler Jerzy Z. Zemajtis Publishing Services Manager Barry M. Bergin Editors Carl R. Bischof Tiesha Elam Kaitlyn Hinman Kelli R. Slayden

Discussion is welcomed for all materials published in this issue and will appear ten months from this journal’s date if the discussion is received within four months of the paper’s print publication. Discussion of material received after specified dates will be considered individually for publication or private response. ACI Standards published in ACI Journals for public comment have discussion due dates printed with the Standard. Annual index published online at http://concrete.org/Publications/ACIStructuralJournal. ACI Structural Journal Copyright © 2015 American Concrete Institute. Printed in the United States of America. The ACI Structural Journal (ISSN 0889-3241) is published bimonthly by the American Concrete Institute. Publication office: 38800 Country Club Drive, Farmington Hills, MI 48331. Periodicals postage paid at Farmington, MI, and at additional mailing offices. Subscription rates: $166 per year (U.S. and possessions), $175 (elsewhere), payable in advance. POSTMASTER: Send address changes to: ACI Structural Journal, 38800 Country Club Drive, Farmington Hills, MI 48331. Canadian GST: R 1226213149. Direct correspondence to 38800 Country Club Drive, Farmington Hills, MI 48331. Telephone: +1.248.848.3700. Facsimile (FAX): +1.248.848.3701. Website: http://www.concrete.org.

ACI Structural Journal/January-February 2015

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Contributions to ACI Structural Journal

MEETINGS 2015 JANUARY 6-9—Building Innovation 2015 Conference & Expo, Washington, DC, www.nibs.org/?page=conference2015 11-15—TRB 94th Annual Meeting, Washington, DC, www.trb.org/ annualmeeting2015/annualmeeting2015. aspx 20-22—National Association of Home Builders International Builders’ Show, Las Vegas, NV, www.buildersshow.com/ Home FEBRUARY 1-6—2015 Mason Contractors Association of America Convention, Las Vegas, NV, www.masoncontractors.org/convention/ index.php 2-3—The International Concrete Polishing and Staining Conference, Las Vegas, NV, www.icpsc365.com/icpsc2015 2-6—World of Concrete, Las Vegas, NV, www.worldofconcrete.com 14-17—Interlocking Concrete Pavement Institute 2015 Annual Meeting, San Antonio, TX, www.icpi. org/2015annualmeeting 15-18—Geosynthetics 2015, Portland, OR, http://geosyntheticsconference.com

20-22—National Concrete Masonry Association Annual Convention, San Antonio, TX, http://ncma.org/events MARCH 1-3—2015 National Ready Mixed Concrete Association Annual Convention, Orlando, FL, www.nrmca.org/conferences_ events/annualconvention 4-8—Precast/Prestressed Concrete Institute Winter Conference, Orlando, FL, www.pci.org/pci_events/pci_winter_ conference 5-7—The Precast Show 2015, Orlando, FL, http://precast.org/theprecastshow 17-21—International Foundations Congress & Equipment Exposition 2015, San Antonio, TX, www.ifcee2015.com 25-27—International Concrete Repair Institute 2015 Spring Convention, New York City, NY, www.icri.org/ebents/ upcomingevents.asp MARCH/APRIL 30-2—Concrete Sawing & Drilling Association Convention and Tech Fair, St. Petersburg, FL, www.csda.org/events/ event_details.asp?id=444478&group APRIL 13-15—BEST Conference Building Enclosure Science & Technology™, Kansas City, MO, www.nibs.org/?page=best

18-19—2015 ICON-Xchange, San Antonio, TX, http://iconxchange.org

UPCOMING ACI CONVENTIONS The following is a list of scheduled ACI conventions: 2015—April 12-16, Marriott & Kansas City Convention Center, Kansas City, MO 2015—November 8-12, Sheraton Denver, Denver, CO 2016—April 17-21, Hyatt & Wisconsin Center, Milwaukee, WI For additional information, contact: Event Services, ACI 38800 Country Club Drive, Farmington Hills, MI 48331 Telephone: +1.248.848.3795 e-mail: [email protected]

ON COVER: 112-S03, p. 27, Fig. 6—Axial test setup.

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The ACI Structural Journal is an open forum on concrete technology and papers related to this field are always welcome. All material submitted for possible publication must meet the requirements of the “American Concrete Institute Publication Policy” and “Author Guidelines and Submission Procedures.” Prospective authors should request a copy of the Policy and Guidelines from ACI or visit ACI’s website at www.concrete.org prior to submitting contributions. Papers reporting research must include a statement indicating the significance of the research. The Institute reserves the right to return, without review, contributions not meeting the requirements of the Publication Policy. All materials conforming to the Policy requirements will be reviewed for editorial quality and technical content, and every effort will be made to put all acceptable papers into the information channel. However, potentially good papers may be returned to authors when it is not possible to publish them in a reasonable time. Discussion All technical material appearing in the ACI Structural Journal may be discussed. If the deadline indicated on the contents page is observed, discussion can appear in the designated issue. Discussion should be complete and ready for publication, including finished, reproducible illustrations. Discussion must be confined to the scope of the paper and meet the ACI Publication Policy. Follow the style of the current issue. Be brief—1800 words of double spaced, typewritten copy, including illustrations and tables, is maximum. Count illustrations and tables as 300 words each and submit them on individual sheets. As an approximation, 1 page of text is about 300 words. Submit one original typescript on 8-1/2 x 11 plain white paper, use 1 in. margins, and include two good quality copies of the entire discussion. References should be complete. Do not repeat references cited in original paper; cite them by original number. Closures responding to a single discussion should not exceed 1800-word equivalents in length, and to multiple discussions, approximately one half of the combined lengths of all discussions. Closures are published together with the discussions. Discuss the paper, not some new or outside work on the same subject. Use references wherever possible instead of repeating available information. Discussion offered for publication should offer some benefit to the general reader. Discussion which does not meet this requirement will be returned or referred to the author for private reply. Send manuscripts to: http://mc.manuscriptcentral.com/aci Send discussions to: [email protected]


ACI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title No. 112-S01

A Global Integrity Parameter with Acoustic Emission for Load Testing of Prestressed Concrete Girders by Francisco A. Barrios and Paul H. Ziehl Structural evaluation of existing infrastructure has become a critical subject in civil engineering. In recent years, significant efforts have been placed on developing nondestructive techniques such as acoustic emission to monitor and effectively assess the integrity of a structure without causing significant damage. However, acoustic emission methods face challenges regarding the subjectivity of associated performance evaluation criteria and a lack of measurable parameters directly related to the mechanical response of the system. It has been previously suggested that an integrated approach of the cyclic load testing method with acoustic emission techniques may overcome these difficulties and constitute a more effective, nondestructive load testing methodology. The current investigation analyzes experimental data gathered from flexural testing of six full-scale prestressed girder specimens (lightweight and normalweight) and presents a potential approach for damage detection and assessment within the minor to intermediate damage zones based on acoustic emission data. Keywords: girder; prestress; structural load test.

INTRODUCTION Acoustic emission (AE) evaluation used in combination with cyclic load testing is a promising nondestructive, or minimally destructive, technique for the assessment of existing reinforced or prestressed concrete structures (Colombo et al. 2005; Ridge and Ziehl 2006; Galati et al. 2008; Ziehl et al. 2008; Liu and Ziehl 2009; Barrios and Ziehl 2011, 2012; Xu et al. 2013). Attention has been placed on reducing the subjectivity of the AE criteria and quantifying, in terms of structural damage, the changes in measured AE activity. Both AE and cyclic load testing methods have achieved promising results (ACI Committee 437 2007; JSNDI 2000; Barrios and Ziehl 2012), and an integrated approach may offer advantages and decrease the challenges that each method faces for practical implementation (Ziehl et al. 2008). Due to limited research data, however, an integrated standardized approach is lacking and quantification of damage with AE remains somewhat subjective. The approach outlined herein presents the integration of AE data with cyclic load testing for the specific case of prestressed concrete girders within the minor to intermediate damage zones. Potential applications for this approach include prestressed double tees such as those used in parking garages, one- and two-way passively reinforced or post-tensioned building slab systems (Galati et al. 2008; Ziehl et al. 2008), and prestressed and post-tensioned bridge girders. In the case of bridge girders, the loading profile would be simplified and the load magnitude reduced (Ziehl et al. 2009a). RESEARCH SIGNIFICANCE Objective structural integrity evaluation has proven to be a complex subject. Challenges arise from the wide range of potential structural responses due to differing construction ACI Structural Journal/January-February 2015

materials and techniques combined with a need for rapid and reliable integrity assessment. Acoustic emission techniques have demonstrated very high sensitivity for sensing damage, but challenges remain for coherent integrity evaluation during load testing. This investigation presents a methodology to relate acoustic emission data to the level of damage present in prestressed concrete girders loaded in flexure, with a particular focus on the transition from the minor to intermediate damage zone. BACKGROUND Integrity assessment with cyclic load test method With the cyclic load test (CLT) method performance of the system is evaluated through the three criteria of deviationfrom-linearity index, permanency ratio, and residual deflection (ACI Committee 437 2012). The loading pattern is grouped in loadsets and is executed in a stepped fashion. Each loadset consists of two identical load cycles (Cycle A and Cycle B). For the current investigation, the deviationfrom-linearity index was used to define boundaries between minor, intermediate, and heavy damage. This index is defined as follows

I DL = 1 −

tan (a i )

(

tan a ref

)

(1)

where tan(αi) is the secant stiffness of any point i on the increasing loading portion of the load-deflection envelope; and tan(αref) is the slope of the reference secant line for the load-deflection envelope (Fig. 1). A summary of the CLT evaluation results for all six specimens is addressed in a previous publication (Barrios and Ziehl 2012). Integrity assessment with acoustic emission Calm ratio versus load ratio (NDIS-2421)—The Japanese nondestructive inspection standard, NDIS-2421 (JSNDI 2000) proposes a graphical representation of the calm ratio (CR) and load ratio (LR) values to classify the level of damage in reinforced concrete flexural members. This method was first studied and proposed for passively reinforced concrete (Ohtsu et al. 2002); however, it has also been studied for prestressed (Xu 2008; Xu et al. 2013) and post-tensioned (Ziehl et al. 2008) concrete members. The terminology used ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2011-056.R3, doi: 10.14359/51687294, was received February 6, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

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Fig. 1—Deviation from linearity (after ACI Committee 437 [2012]). in NDIS-2421 is described in the following, and additional terms related to AE are included at the end of this paper. Load ratio (LR)—Also known as Felicity ratio or concrete beam integrity (CBI) ratio, the load ratio (LR) is a critical parameter for AE evaluation and monitoring. This ratio generally decreases with the accumulation of damage. It is inversely related to the reduction in AE activity during loading until the material is stressed beyond its previous stress level, known as the Kaiser effect. Calm ratio (CR)—The calm ratio (CR) is related to the AE activity during the unloading portion of the loading cycles (Ohtsu et al. 2002). For the purposes of this paper, the CR is calculated as the ratio of the total cumulative AE activity throughout the unloading phase to the total AE activity during the entire cycle. The CR can be calculated on either the loading cycle or the reloading cycle of a particular loadset. Recent experimental evidence suggests that the former computation may provide a more consistent correlation with the accumulation of damage (Barrios and Ziehl 2011). A crucial aspect for the computation of both the LR and CR is the AE parameter selected for evaluation. Some authors have used the number of “hits” during loading and unloading for evaluation of the AE data (Ohtsu et al. 2002; Colombo et al. 2005). While this approach has shown promise, inspection of the experimental data from the current investigation indicates that “signal energy” or “signal strength” may be more appropriate for evaluation of the AE data than hits (Barrios and Ziehl 2011). This can be explained by the observation that the extent of damage is more closely related to the amount of energy associated with each hit. Based on inspection of trends in the data, the AE parameter of signal energy was found to be promising and was therefore used in the current investigation. The methodology for damage classification described in NDIS-2421 is a graphical representation of CR versus LR for each loadset (or load cycle) divided into four damage zones. This method of damage classification is generally consistent with damage trends. However, some challenges with this approach may exist. First, the manner in which the graph is partitioned allows for two distant damage levels (minor and heavy) in very close proximity to one another, which does not generally correspond to the nature of damage progression. Second, previous investigations have generally relied on the 4

assistance of additional data, such as crack mouth opening displacement (CMOD), to aid in locating the damage zones (Ohtsu et al. 2002). This requires the installation of gauges across existing cracks. To achieve this, existing cracks must be present and visible, which may not be the case for prestressed elements. Furthermore, the CMOD calibration does not address other considerations such as crack density, yielding, slippage of reinforcement, and concrete crushing. These can be important for assessing integrity, particularly for nonflexural modes such as shear and bond failure. Based on CMOD data, threshold values for the CR and LR have been derived for passively reinforced concrete flexural members involved in a previous study (Ohtsu et al. 2002). Other authors have located the boundaries intuitively so that beams with similar damage plotted inside the corresponding damage zone (Liu and Ziehl 2009; Colombo et al. 2005). Experimental data from the current investigation indicate that for the specific case of full-scale prestressed girders, CR values do not increase proportionately with the accumulation of damage. Rather, as the plastic deformation of the member increases, cracks remain open, reducing the AE activity during unloading, hence decreasing the CR value. This behavior complicates the classification of damage for prestressed flexural members. An assessment methodology that combines the CLT criteria and the AE criteria, referred to as a global performance index IG, has been proposed (Ziehl et al. 2008). The approach combines these criteria in a weighted manner and includes a multiplier to account for knowledge of the structure (load history, previous load tests, and reinforcement configuration), and the number of members being tested compared to the total number of similar members in the system. The AE portion of this index relies on the CR-versusLR index ICRLR. This approach has merit but does not eliminate the need for an external criterion to assess the level of damage. In the absence of load-versus-displacement data, it is not possible to quantify the magnitude of damage from the ICRLR assessment. The global integrity parameter (GIP) is a proposed loadtesting evaluation method (Barrios and Ziehl 2012) that aims to quantify the amount of damage based only on deviation from linearity IDL. The GIP identifies damage levels as fixed percentages of the IDL at the nominal capacity of the member. The GIP approach builds on the CLT evaluation criteria, but differs in that the GIP approach uses specific member properties such as the fully cracked moment of inertia and elastic stiffness of the member. The preceding discussion provides an overview of recent studies involving the evaluation of concrete members with AE during in-place load testing, along with the GIP evaluation approach, which is based on load-versus-displacement behavior. The discussion that follows is based on the specific experimental results obtained during this investigation and is focused on minor to intermediate damage in prestressed flexural members. For this study, the damage level zones are defined as: minor damage for IDL < 15%; intermediate damage for 15% < IDL < 35%; and heavy damage for IDL > 35%. For IDL below 15%, cracks were barely visible and slight nonlinearity was observed in the measured load-versusACI Structural Journal/January-February 2015

displacement response; at 35%, IDL cracks started to remain open after unloading, and nonlinear behavior was clearly present (Barrios and Ziehl 2012). EXPERIMENTAL PROCEDURE Experimental data were gathered during four-point bending load testing of six full-scale prestressed girder specimens. The first set of specimens consisted of two girders constructed using self-consolidating concrete (SCC-1 and SCC-2) and one girder constructed with high-early-strength concrete (HESC). The second set of specimens consisted of two girders with self-consolidating lightweight concrete (SCLC-1 and SCLC-2) and one girder with high-earlystrength lightweight concrete (HESLC). Specimen nomenclature is: “concrete type (SCC, HESC, SCLC, or HESLC) specimen number associated with concrete type.” All girders were AASHTO Type III with a span of 58 ft 2 in. (17.7 m) and a depth of 45 in. (1140 mm). A concrete mixture with a design compressive strength of 8000 psi (55.2 MPa) was used to cast the girders, and a 4000 psi

Fig. 2: Photograph and cross-section details (after Barrios and Ziehl [2012]).

(27.6 MPa) concrete deck was cast in the laboratory prior to testing. The concrete deck was 96 in. (2440 mm) wide and 8 in. (203 mm) deep for the normalweight girders, and 30 in. (762 mm) wide and 19 in. (483 mm) deep for the lightweight girders (Fig. 2). All girders were prestressed using 18 bottom and four top (all straight profile) low-relaxation strands with a 0.5 in. (12.7 mm) nominal diameter and a tensile strength of 270 ksi (1860 MPa). A 31,000 lb (138 kN) prestressing force was applied to each strand prior to release. A summary of the girder specimens is provided in Table 1 and further information can be found in related South Carolina Department of Transportation reports (Ziehl et al. 2009b, 2010). Internal and external instrumentation Five vibrating-wire strain gauges were installed within the girders to monitor the long-term strain for prestress loss evaluation and other variables. Instrumentation was outfitted externally on both the girder and deck to measure and record displacement and concrete strain data. Strain gauges were adhered to the top of the deck and at the bottom of the girder. Cracks were marked on the girders with permanent markers as they developed so that cracking patterns at different load levels could be identified. Two draw-wire transducers placed at midspan were used to measure deflection of the girders. Acoustic emission activity was monitored continuously during load testing. Six AE sensors (resonant in the vicinity of 60 kHz with integral pre-amplification of 40 dB) were mounted on one side of each girder specimen. The sensor layout was symmetric about an axis located at the midpoint between the loading points and a 48 dB test threshold was used for all specimens (Fig. 3). Loading protocol The CLT load profiles for the normalweight girders are shown in Fig. 4(a) and those for the lightweight girders are shown in Fig. 4(b). Damage levels as determined through the deviation-from-linearity index, as described previously, combined with visual observations of cracking are shown in the same figures. The resulting load-versus-displacement behavior is shown in Fig. 5 and resulting cracking patterns are shown in Fig. 6 (at 67% and 57% of nominal capacity Pn for normalweight and lightweight specimens, respectively, and also at the conclusion of the CLT loading protocols). The loading profile was developed based on the calculated nominal capacity of the normalweight girder specimens. A total test load (maximum load applied) of 160 kip (712 kN) was selected, which is approximately 73% of the measured

Table 1—Summary of girder specimens Specimen name

Age of girder at Approximate age release, days of girder at testing, days

Approximate age of deck at testing, days

Ultimate load achieved, kip (kN)

Failure mode

SCC-1

2

270

200

226 (1010)

Strand rupture

SCC-2

2

900

300

226 (1005)

Strand rupture

HESC

2

470

100

224 (997)

Strand slip

SCLC-1

16

315

50

290 (1290)

Strand yield followed by deck failure

SCLC-2

16

555

90

276 (1228)


HESLC

16

490

60

260 (1156)



5

capacity of these girder specimens. The maximum load level was high enough to exert significant damage while avoiding failure before the application of a 24-hour load test based on Chapter 20 of ACI 318-11 (ACI Committee 318 2011). The number of applied loadsets and their magnitudes was intended to provide information regarding the behavior of the girders at different deterioration levels. Prior to loading according to Fig. 4(a), Girders SCC-2 and SCLC-2 were subjected to cyclic fatigue loading for 2 million cycles to investigate the potential effect of simulated field loading on the results. The lightweight girders had a higher calculated nominal capacity (Pn equal to 280 kip [1245 kN]) in comparison to the normalweight specimens due to the modified deck geometry. Therefore, for SCLC-1 and the HESLC girder, the maximum test load applied corresponded to approximately 57% of the nominal capacity. Because significant damage, as judged by the deviation-from-linearity index, did not occur at this level of loading, the test load magnitude for Girder SCLC-2 was increased. For this specimen, Loadsets 11 and 12 (at 68% and 80% of Pn, respectively) were added to the loading protocol. Further descriptions of the loading protocols and results are provided elsewhere (Barrios 2010; Barrios and Ziehl 2012).

Fig. 3—Acoustic emission sensor layout (after Barrios and Ziehl [2011]).

EXPERIMENTAL RESULTS The results of the CLT evaluation and the 24-hour load test in terms of load-versus-displacement behavior are described in a previous publication (Barrios and Ziehl 2012). The discussion that follows is focused on the treatment of the AE data and corresponding results. Normalweight girder specimens Calm ratio-versus-load ratio plots for the normalweight prestressed girder specimens are shown with the damage zone classification quadrants as determined from the load-versus-displacement behavior (Barrios and Ziehl 2012) in Fig. 7(a). Loadset 3 fell within the minor damage zone, Loadset 5 within the intermediate damage zone, and Loadset 7 within the heavy damage zone. For purposes of comparison and to enable discussion, Fig. 7(b) shows the damage quadrants from an investigation related to reducedscale (maximum span of 23 ft 0 in. [7.0 m]) prestressed concrete tee-beam specimens (Xu 2008; Xu et al. 2013). The agreement between the damage quadrants from these two studies conducted at very different scales is reasonable. Lightweight girder specimens Results were similar for the lightweight girder specimens (Fig. 8(a)). The classification zones shown in Fig. 8(a) were developed based on the measured load-versus-displacement behavior as for the normalweight specimens (Barrios 2010; Barrios and Ziehl 2012). For comparison, Fig. 8(b) again shows the damage quadrants from an investigation related to reduced-scale prestressed concrete tee-beam specimens (Xu et al. 2013). In this case, the agreement between the two investigations is not as good. To summarize, the CR and LR quadrants from different investigations should not be used without modification; rather, they are specific to the member being tested. Further-

Fig. 4—Loading profiles (after Barrios and Ziehl [2011]). (Note: 1 kip = 4.45 kN.) 6


Fig. 5—Load versus displacement due to CLT loading protocol (after Barrios and Ziehl [2012]). (Note: 1 in. = 25.4 mm.) more, they may vary between similar girders (for example, Loadset 7 for the lightweight prestressed girder specimens [Fig. 8(a)]). However, AE data can aid in the classification of damage for prestressed concrete flexural specimens when an external parameter, such as CMOD or the deviation-fromlinearity index, is used for calibration of the damage quadrants. PROPOSED METHODOLOGY Modified damage classification based on CRversus-LR plot A modified approach for damage classification using the CR-versus-LR plot is proposed and described as follows. The first modification is for convenience: the LR criterion is replaced with unity minus the load ratio (1.0 – LR), referred to hereafter as the complementary load ratio (CLR), so the curves increase upward and to the right (Fig. 9). In this figure, the values of load are labeled where significant changes in slope occur. This is mentioned for later comparison with the IDLversus-load curves shown in Fig. 10, also referred as structural integrity loops (SIL) (Barrios 2010; Barrios and Ziehl 2012). In the CR-versus-CLR plots, hollow squares represent points that were interpolated using curves obtained for CR and CLR independently (Barrios 2010). Values were computed at the ends and at a minimum of two intermediate positions within each linear segment. The plots shown in Fig. 9 illustrate a trend in the AE data with accumulation of damage in these prestressed girder specimens. The curves plotted for the normalweight prestressed ACI Structural Journal/January-February 2015

Fig. 6—Cracking patterns (after Barrios and Ziehl [2012]). girder specimens can be divided into three segments based on changes in slope. The figures indicate that the CR is more sensitive within very early stages of damage, while the CLR is more sensitive in later stages of damage. This behavior continues up to the level of load where cracks no longer close during unloading. At this point, the CR values begin to rapidly decrease (Loadsets 11 and 12 for SCLC-2). This behavior is believed to be specific to prestressed specimens. It is hypothesized that each change in slope in the CR-versus-CLR plot corresponds to a transition between damage zones. An external check is required to confirm this hypothesis. Furthermore, to successfully compare and potentially merge AE evaluation with the deviation-from-linearity index IDL, a coherent description of the damage process is needed. To investigate the hypothesis, load values at the transition points on the CR-versus-CLR plots (Fig. 9) are located on the structural integrity loops (IDL versus load plots [Fig. 10]). The corresponding IDL values are then compared to those obtained from a previous published investigation (Barrios and Ziehl 2012), wherein damage is based on loadversus-displacement behavior. The damage classification zones identified using the CR-versus-CLR plots correspond closely with those found through the load-versusdisplacement behavior. This implies that changes in slope on the CR-versus-CLR plots represent physical changes in the specimens and correlate with the damage process. 7

Fig. 7—Damage quadrants (normalweight specimens).

Fig. 8—Damage quadrants (lightweight specimens).

An interesting effect similar to the Kaiser effect is noticeable in the plots shown in Fig. 10. Girder SCC-1 starts with an initial slope up to 115 kip (512 kN), where there is an abrupt positive change. At this point, the IDL begins to increase more rapidly, crossing from the minor to the intermediate damage zone until it reaches the maximum load for that loadset at 128 kip (569 kN). On the next loadset, the member reaches the same load level over a straight line, and hence, a new higher load must be attained to produce a positive change in the trend. The same trend occurs later for SCC-2, where the IDL increases linearly with load up to 135 kip (601 kN). At this level, a change in slope occurs until it reaches the new maximum load of 160 kip (712 kN). It can also be noted that loadsets applied at similar states of damage tend to group closely until a new abrupt change in slope is produced, generating a new cluster. Lightweight girder specimens, SCLC-1 and HESLC, were only loaded up to 160 kip (57% of Pn at Loadsets 7 and 9). Therefore, the AE information gathered was not sufficient to establish a trend. For SCLC-2 only one damage threshold (intermediate to heavy at 160 kip [712 kN]) was located from the information gathered in the CR-versus-CLR plot (Fig. 9), resulting in the merging of the minor and intermediate zones into one segment. In flexural members the minor damage region is dominated by cracking and, therefore, there is a good match between the damage zones identified with load-versus-displacement behavior (Barrios 2010; Barrios and Ziehl 2012) and those obtained from the AE data. It is possible to find more than three changes in slope in a CR-versus-CLR plot, and the external parameter of IDL is helpful for discriminating the AE data into the three levels of damage (minor, intermediate, and heavy) generally used for structural evaluation.

Proposed AE damage descriptor (arc of damage) Acoustic-emission-based plots of CR versus CLR offer useful information that can be related to the damage state of prestressed flexural members when subjected to the CLT loading protocol. Due to its extreme sensitivity to crack growth, the AE method may also have the potential to offer earlier detection of damage in comparison to methods based on load-versus-displacement behavior. To take advantage of this potential for early damage detection in prestressed flexural members, a new AE damage descriptor is proposed that builds on a previously proposed damage descriptor developed from load testing of two building slab systems (Ziehl et al. 2008). In Fig. 11, this new descriptor, referred to as the arc of damage, is shown graphically. In this figure, results from Loadsets 3, 5, and 7 of an example specimen are plotted. The point where the slope of the CR-versus-CLR curve crosses the horizontal axis (located at calm ratio = 0.0, complementary load ratio = 0.17 in this case) is used as a reference point. The angular measure (in radians) from the reference vertical line drawn through this point to any loadset is then used as an angular measure of damage (θi, where the subscript i refers to a particular loadset); and the linear distance from this reference point to any loadset is used as a linear measure of damage di. A numerical value for the arc of damage AD descriptor is then calculated for each loadset as the product of these two measures: θi × di. To illustrate the influence of these two damage descriptors, values of angular measure θi and linear distance di are plotted separately in Fig. 12(a) and (b) for the normalweight specimens and in Fig. 13(a) and (b) for the lightweight specimens. The resulting arc of damage AD for both specimen types is plotted in Fig. 12(c) and 13(c), respectively.

8


Fig. 9—CR-versus-CLR plots. (Note: 1 kip = 4.4 kN.)

Fig. 10—Structural integrity loops and damage thresholds (after Barrios and Ziehl [2012]). (Note: 1 kip = 4.4 kN.) For the normalweight girder specimens, the arc of damage AD (Fig. 12(c)) provides a more consistent damage representation than either linear distance (Fig. 12(a)) or angular measure (Fig. 12(b)) alone. The linear descriptor grows faster at lower levels of damage (between Loadsets 3 and 5) while the angular measure is more sensitive to damage in the intermediate and heavy damage zones. For the lightweight specimens, the linear distance descriptor from the reference point (Loadset 5) increases rapidly up to the theoretical minor-intermediate threshold corresponding to Loadset 7 (Fig 13(a)). At that point, the linear distance measurement increases at a reduced rate for a wide range of the load value (62% to 87% of nominal capacity). This leads to the observation that the linear distance descriptor is more effective within the minor damage zone, while the angular measure descriptor is more sensitive to damage within the intermediate and heavy damage zones. This descriptor ACI Structural Journal/January-February 2015

increases steadily after Loadset 7 and follows a closely linear pattern for the same load range (Fig. 13(b)). The scatter observed at Loadset 5 for the normalweight specimens and at Loadset 7 for the lightweight specimens in both the linear and the angular descriptors is reduced for the resulting AD, as shown in Fig. 12(c) and 13(c). This supports combining the linear and angular descriptors, resulting in the proposed AD descriptor, to increase the reliability of assessment. One key characteristic of the AD descriptor is that it estimates deterioration of the member through a single numerical parameter, thereby facilitating a comparison of damage levels in differing members. Proposed assessment method within minor damage zone, GIPAE To take advantage of the sensitivity of AE, a new damage criterion that is specifically focused on the minor damage 9

Fig. 11—Definition of arc of damage.

Fig. 13—Arc of damage versus percent of Pn (lightweight specimens). where for lightweight concrete

 P − PO  β = 0.001 + 0.145  T (3)  Pmi − PO 

and for normalweight concrete

Fig. 12—Arc of damage versus percent of Pn (normalweight specimens). zone for prestressed flexural elements is proposed. The criterion is based solely on AE data as load-versus-displacement data may lack the necessary sensitivity for reliable damage assessment in the transition between the minor to intermediate damage zone. Classification within the intermediate and heavy damage zones, on the other hand, can be approached with a number of different approaches including the arc of damage (AD), the previously published GIP method (Barrios and Ziehl 2012), or the CLT method (ACI Committee 437 2012). The proposed evaluation criterion, referred to as GIPAE (for global integrity parameter based on AE), is based on the AD and is described as follows 10

GIPAE = ADβ–1 ≤ 1.0

(2)

 P − PO  β = 0.001 + 0.035  T (4)  Pmi − PO 

where AD is the arc of damage for any loadset; PT is the target load at which the damage criterion should reach unity; Po is the peak load at the loadset with the lowest CR; and Pmi is the load at the theoretical minor to intermediate damage threshold. These equations were derived independently for each set of girders (lightweight and normalweight) using a linear interpolation of the average values of AD within the minor damage region. The same procedure can be followed to generate similar equations for prestressed concrete girders with different span, section geometry, and concrete strength. A minimum number of two loadsets should be included within the minor damage zone to calibrate the GIPAE criterion. However, in a case where structural damage needs to be minimized and even minor cracking in the members is not permitted, a β factor equal to 0.001 would be a conservative value. Results from: a) the CLT criteria (ACI Committee 437 2007); b) the GIP criterion (Barrios and Ziehl 2012); and c) the GIPAE method discussed previously are shown in Fig. 14. For the specimens tested, the GIPAE method provides the most sensitive damage criterion. This is neither intrinsically ACI Structural Journal/January-February 2015

ACI member Paul H. Ziehl is a Professor in the Department of Civil and Environmental Engineering at the University of South Carolina. He is a member of ACI Committee 437, Strength Evaluation of Existing Concrete Structures.

ACKNOWLEDGMENTS Portions of this work were sponsored by the South Carolina Department of Transportation and the ACI Concrete Research Council and their financial support is greatly appreciated. Portions were performed under the support of the U.S. Department of Commerce, National Institute of Standards and Technology, Technology Innovation Program, Cooperative Agreement Number 70NANB9H9007, and their support is likewise appreciated.

REFERENCES

Fig. 14—Values at damage detection. good nor bad, but in some cases, it may be beneficial and desirable to avoid damage caused by the load test itself. The GIPAE method reduces the load required for damage detection to approximately 50% of nominal capacity Pn in the lightweight girder specimens and 52% of nominal capacity in the normalweight girder specimens. These values are comparable to the calculated cracking load (47% of Pn for the lightweight specimens and 48% of Pn for the normalweight specimens). It is possible that the approach could be tailored to further reduce load magnitude by decreasing the AE threshold and/or installing a denser array of sensors. SUMMARY AND CONCLUSIONS One new damage descriptor—the arc of damage AD—and one proposed damage criterion—GIPAE—have been developed based on data from flexural testing of six full-scale prestressed girder specimens. Both are based on AE data. The arc of damage addresses changes in AE behavior that are related to differing damage states and is applicable to minor, intermediate, and heavy damage states. The GIPAE criterion incorporates mechanical properties of the member and is specifically targeted to the transition between minor to intermediate damage, thereby using the high sensitivity of AE for crack initiation and extension. This may be applicable for prestressed applications, particularly for cases where it is desired to minimize damage due to the load-testing procedure itself. For further research, results from field testing of prestressed girders are desirable for refinement of the methods described. AUTHOR BIOS Francisco Barrios is an Assistant Professor in the Department of Civil Engineering at Universidad del Magdalena, Santa Marta, Colombia. He received his BS from the Universidad del Norte, Barranquilla, Colombia; his MS from Tulane University, New Orleans, LA; and his PhD from the University of South Carolina, Columbia, SC. His research interests include structural health monitoring, damage diagnosis, load testing, and nonlinear computer modeling of concrete structures.


ACI Committee 318, 2011, “Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary,” American Concrete Institute, Farmington Hills, MI, 503 pp. ACI Committee 437, 2007, “Load Tests of Concrete Structures: Methods, Magnitude, Protocols, and Acceptance Criteria (ACI 437.1R-07),” American Concrete Institute, Farmington Hills, MI, 38 pp. ACI Committee 437, 2012, “Code Requirements for Load Testing of Existing Concrete Structures and Commentary (ACI 437-12) (ACI Provisional Standard),” American Concrete Institute, Farmington Hills, MI, 34 pp. ASTM E1316-13d, 2013, “Standard Terminology for Nondestructive Examinations,” ASTM International, West Conshohocken, PA, 38 pp. Barrios, F., 2010, “Acoustic Emission Techniques and Cyclic Load Testing for Integrity Evaluation of Self-Consolidating Normal and Lightweight Prestressed Concrete Girders,” PhD dissertation, Department of Civil Engineering, University of South Carolina, Columbia, SC. Barrios, F., and Ziehl, P., 2011, “Effect of Loading Pattern on the Acoustic Emission Evaluation of Prestressed Concrete Girders,” Journal of Acoustic Emission, V. 29, pp. 42-56. Barrios, F., and Ziehl, P., 2012, “Cyclic Load Testing for Integrity Evaluation of Prestressed Concrete Girders,” ACI Structural Journal, V. 109, No. 5, Sept.-Oct., pp. 615-623. Colombo, S.; Forde, M.; Main, I.; and Shigeishi, M., 2005, “Predicting the Ultimate Bending Capacity of Concrete Beams from the ‘Relaxation Ratio’ Analysis of AE Signals,” Construction and Building Materials, V. 19, No. 10, pp. 746-754. doi: 10.1016/j.conbuildmat.2005.06.004 Galati, N.; Nanni, A.; Tumialan, J. G.; and Ziehl, P. H., 2008, “In-Situ Evaluation of Two Concrete Slab Systems. I: Load Determination and Loading Procedure,” Journal of Performance of Constructed Facilities, V. 22, No. 4, pp. 207-216. doi: 10.1061/(ASCE)0887-3828(2008)22:4(207) JSNDI, 2000, “Recommended Practice for In Situ Monitoring of Concrete Structures by Acoustic Emission,” NDIS 2421, Japanese Society for Nondestructive Inspection, Tokyo, Japan, 6 pp. Liu, Z., and Ziehl, P., 2009, “Evaluation of Reinforced Concrete Beam Specimens with Acoustic Emission and Cyclic Load Test Methods,” ACI Structural Journal, V. 106, No. 3, May-June, pp. 288-299. Ohtsu, M.; Uchida, M.; Okamoto, T.; and Yuyama, S., 2002, “Damage Assessment of Reinforced Concrete Beams Qualified by Acoustic Emission,” ACI Structural Journal, V. 99, No. 4, July-Aug., pp. 411-417. Ridge, A., and Ziehl, P., 2006, “Nondestructive Evaluation of Strengthened Reinforced Concrete Beams: Cyclic Load Test and Acoustic Emission Methods,” ACI Structural Journal, V. 103, No. 6, Nov.-Dec., pp. 832-841. Xu, J., 2008, “Nondestructive Evaluation of Prestressed Concrete Structures by Means of Acoustic Emission Monitoring,” PhD dissertation, Department of Civil Engineering, University of Auburn, Auburn, AL. Xu, J.; Barnes, R.; and Ziehl, P., 2013, “Evaluation of Prestressed Concrete Beams Based on Acoustic Emission Parameters,” Materials Evaluation, V. 71, No. 2, pp. 176-185. Ziehl, P.; Galati, N.; Nanni, A.; and Tumialan, J., 2008, “In Situ Evaluation of Two Concrete Slab Systems. II: Evaluation Criteria and Outcomes,” Journal of Performance of Constructed Facilities, ASCE, V. 22, No. 4, pp. 217-227. doi: 10.1061/(ASCE)0887-3828(2008)22:4(217) Ziehl, P. H.; Engelhardt, M.; Fowler, T. J.; Ulloa, F. V.; Medlock, R. D.; and Schell, E., 2009a, “Design and Field Evaluation of a Hybrid FRP/ Reinforced Concrete Bridge Superstructure System,” Journal of Bridge Engineering, ASCE, V. 14, No. 5, pp. 309-318. doi: 10.1061/(ASCE) BE.1943-5592.0000002 Ziehl, P.; Rizos, D.; Caicedo, J.; Barrios, F.; Howard, R.; and Colmorgan, A., 2009b, “Investigation of the Performance and Benefits of Lightweight SCC Prestressed Concrete Bridge Girders and SCC Materials,” Final Report submitted to the South Carolina Department of Transportation, 182 pp. Ziehl, P.; Rizos, D.; Caicedo, J.; Colmorgan, A.; Howard, R.; and Barrios, F., 2010, “Investigation of the Performance and Benefits of Self-Consolidating Concrete for Prestressed Bridge Girders,” Final Report submitted to the South Carolina Department of Transportation, 212 pp.

11

NOTES:

12



TECHNICAL PAPER

Title No. 112-S02

Analysis of Rectangular Sections Using Transformed Square Cross Sections of Unit-Length Side by Girma Zerayohannes This paper deals with the analytical proof of the equivalence between the relative biaxial bending resistance of a rectangular solid reinforced concrete section and the biaxial bending resistance of the transformed solid square section of unit-length side. Similar proofs are also derived for rectangular hollow sections. The results of the analytical proof show that the relative biaxial bending resistance of a rectangular solid section is identical to the biaxial bending resistance of the transformed square solid section of unit-length side. These results only occur when the concrete fiber and reinforcing bar coordinates in the transformed section are in conformity with the transformation that maps the rectangular section into a square cross section of unit-length side. The concrete and steel stresses in the transformed section comply with the resulting stress transformation and the area of reinforcement in the transformed section must comply with the resulting area transformation. The proof also shows the equivalence in rectangular hollow sections, provided that similar transformation-related conditions are met. Keywords: analysis; biaxial bending; cross section; hollow sections; homogeneous transformation; solid sections; transformed sections; unitside length.

INTRODUCTION The results of cross-section analysis1,2 show that the relative biaxial bending resistance of a rectangular solid section is identical to the biaxial bending resistance of the transformed square cross section of unit-length side, provided that: 1) the concrete fiber and reinforcing bar coordinates in the transformed section are in conformity with the transformation that maps the rectangular section into a square cross section of unit-length side; 2) the concrete and steel stresses in the transformed section comply with the resulting stress transformation; and 3) the area of reinforcement in the transformed section comply with the resulting area transformation. The results of cross-section analysis1,2 also show the equivalence between the relative biaxial bending resistance of a rectangular hollow section and the biaxial bending resistance of the transformed square hollow cross section of unit-length side, provided that similar transformation related conditions are met. While comparisons of cross-section analysis results have shown the equivalence, analytical proof for its justification is hardly available in the literature. More recently, however, Cedolin et al.3 used the square cross section of unit-length side to calculate interaction diagrams for load eccentricities along axes parallel to the axes of symmetry and to a diagonal of a solid rectangular cross section for the derivation of approximate analytical expressions of the moment contours based on the ACI 318-05.4 Analytical proof of the equivalence between the dimensionless expressions for the rectanACI Structural Journal/January-February 2015

gular solid section with four-corner reinforcement and the ultimate bending moments and axial force of an equivalent square cross section of unit length-side is also provided in Cedolin et al.3 RESEARCH SIGNIFICANCE This research deals with the development of a new approach for the analytical proof of the equivalence between the related biaxial bending resistances of rectangular solid and hollow sections, and biaxial bending resistances of the transformed square solid and hollow sections of unit-length sides. The proposed method: 1) covers a wider range of reinforced concrete sections with arbitrary reinforcement arrangement; and 2) facilitates the calculations of biaxial interaction diagrams because it allows the use of a single value—unity—as strength input data for the design strengths of all classes of concrete, and the use of unit side lengths as geometric input data representing all rectangular solid and hollow sections. Equivalence between relative biaxial bending resistance of rectangular section and biaxial bending resistance of transformed square cross section of unit-length side Biaxial interaction diagrams for solid rectangular cross section made of reinforced concrete are presented as load contours with the design normal force and biaxial bending resistance expressed in non-dimensional form as

ν Rd =

µ Rd y =

µ Rd z =

N Rd (1) f cd ⋅ b ⋅ h M Rd y f cd ⋅ b ⋅ h 2 M Rd z f cd ⋅ h ⋅ b 2

(2)

(3)

where νRd, μRd y, and μRd z are the relative values of the combined design axial load and biaxial bending resistance of the rectangular cross section; NRd, MRd y, and MRd z are the ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2012-395.R3, doi: 10.14359/51687295, received April 2, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

13

combined design axial load and biaxial bending resistance of the rectangular cross section; fcd is the design compressive strength of concrete; and b and h are the side lengths of the rectangular cross section. Consider a linear homogeneous transformation T represented by the matrix (Eq. (4)), that maps the “original section” defined in Eq. (1) to (3) to its image, which will be referred to as the “transformed section”. The transformation constitutes a two-way dilation, of factor k1 along the y-axis and of factor k2 along the z-axis.

 k1 Tk1 , k2 =  0

0 (4) k2 

The relative values of the combined design axial force and biaxial moment resistances of the transformed section are determined using the expressions in Eq. (5) to (7).

νTRd =

µTRd y =

µTRd z =

T N Rd (5) f ⋅ bT ⋅ hT T cd

T M Rd y

2

(6)

2

(7)

( )

f cdT ⋅ bT ⋅ hT T M Rd z

( )

f cdT ⋅ hT ⋅ bT

combined design axial force and biaxial bending resistance T of the transformed section; f cd is the design compressive strength of concrete in the transformed section; and bT and hT are the side lengths of the transformed section. The proof is carried out by first assuming that the relative values of the combined design axial force and biaxial bending resistance is invariant under the transformation and later proving that the assumption is valid. Thus

ν Rd =

µ Rd y =

µ Rd z =

f cdT ⋅ bT ⋅ hT = 1 (11)

f cdT ⋅ bT ⋅ (hT ) 2 = 1 (12)

f cdT ⋅ hT ⋅ (bT ) 2 = 1 (13)

Equations (11) to (13) represents a system of independent simultaneous equations in bT, hT, and f cdT . The solutions are bT = 1, hT = 1, and f cdT = 1

T N Rd (8) f ⋅ bT ⋅ hT

T M Rd y

( )

f cdT ⋅ bT ⋅ hT M

2

T Rd z

( )

f cdT ⋅ hT ⋅ bT

2

(14)

Because the products in Eq. (11) to (13) are dimensionless, it T follows that each of f cd , bT, and hT are also dimensionless. Therefore, the unit values determined as solutions for bT, hT, T and f cd in Eq. (14) are constants without dimension. With the solutions for bT and hT determined, the elements of the transformation matrix can be readily determined as k1 = 1/b and k2 = 1/h, so that the transformation matrix associated with the special conditions in Eq. (11) to (13) is given by Eq. (15). 1/b 0  T =  (15)  0 1/h 

For a more general transformation condition where the unit values on the right-hand sides of Eq. (11) to (13) are replaced by constants q1, q2, and q3, respectively (q1, q2, and q3 are real numbers greater than zero), the solutions are given by Eq. (16) bT = q3/q1, hT = q2/q1, and f cdT = q13 ( q2 ⋅ q3 ) (16) The solutions indicate a linear homogeneous transformation represented by the transformation matrix in Eq. (4), where the elements k1 and k2 are given by k1 = (q3/q1) · (1/b) and k2 = (q2/q1) · (1/h). The design resistance of the transformed section is then related to the normalized design resistance of the original rectangular section through the relationships shown in Eq. (17) to (19).

T cd

ν Rd =

µ Rd y =

µ Rd z =

T N Rd (17) q1

(9) T M Rd y

q2

(18)

(10)

It can be seen from Eq. (8) to (10) that the combined design axial force and biaxial bending resistance of the transformed T T T section ( N Rd , M Rd y , M Rd ) satisfying the transformation z 14

T T T where ν Rd , µ Rd y, and µ Rd z are the relative values of the combined design axial force and biaxial bending resistance T T T of the transformed section; N Rd , M Rd y , and M Rd z are the

conditions laid down in Eq. (11) to (13) is equal to the relative values of the combined design axial force and biaxial bending resistance (νRd, μRd y, μRd z) of the original section. The conditions state that the results of the products are all unity and dimensionless.

T M Rd z

q3

(19)

Therefore, the equivalent square cross section of unit-length side is a special case resulting from the special condition q1 = q2 ACI Structural Journal/January-February 2015

Fig. 1—(a) Rectangular solid section; and (b) square solid section of unit-length side.

Fig. 2—(a) Rectangular hollow section; and (b) square hollow section of unit-length side. = q3 = 1. Finally, it can be concluded that the relative values of the combined design axial force and biaxial bending resistance of a rectangular section is equal to the combined design axial force and biaxial bending resistance of a square cross section of unit-length side that satisfies the transformation requirements described prevously and associated transformations that will be described in more detail in the following sections. Coordinate transformation Rectangular solid cross sections—Previously it was shown that a homogeneous linear transformation with two-way dilation of factors (1/b) and (1/h) along the y- and z-axes, respectively, transforms a rectangular cross section into an equivalent square cross section of unit-length side. The transformation was represented by the transformation matrix shown in Eq. (15). The matrix is referred to as coordinate transformation matrix to emphasize its use in the determination of the coordinates of any desired point such as corner concrete fibers and reinforcing bar locations in the transformed sections. As an example, the transformation matrix is used in Eq. (20) and (21) to map corner concrete fiber and reinforcing bar coordinates in the first quadrant of the original cross section (Fig. 1(a)), to the images in the equivalent square cross section of unit length-side (Fig. 1(b)).

1/b 0  b / 2  0.5  0 1/h  h / 2  = 0.5 (20)     

1 b 0   (b / 2) − b ′  0.5 − (b ′ /b)   0 1 h  (h / 2) − h ′  = 0.5 − (h ′ /h)  (21)     

The term “homogeneous” is used to indicate that the origin is an invariant point under the transformation. ACI Structural Journal/January-February 2015

Other invariant variables under the transformation include geometric reinforcement ratio ρ, mechanical reinforcement ratio ω, combined related design axial force and biaxial bending resistance (νRd, μRd y, μRd z), and strains of corresponding fibers in the original and transformed sections. The proof of invariance of each variable will be discussed in subsequent sections. Rectangular hollow cross sections—Figures 2(a) and (b) show the actual rectangular hollow section with uniformly distributed reinforcement along the edges and the transformed square hollow section of unit-length side, respectively. The latter is determined using the transformation described in the following. Biaxial interaction diagrams for hollow rectangular cross section made of reinforced concrete are presented in nondimensional form as

ν Rd =

µ Rd y =

µ Rd z =

N Rd (22) f cd ⋅ a ⋅ b ⋅ h M Rd y f cd ⋅ a ⋅ b ⋅ h 2 M Rd z f cd ⋅ a ⋅ h ⋅ b 2

(23)

(24)

where α is the fraction of the solid part of the cross section, which will be referred to as “solidity ratio” in short and the definitions of other variables are as in Eq. (1) to (3). The relative values of the combined design axial force and biaxial moment resistance of the transformed section are determined using the expressions in Eq. (25) to (27).

15

T N Rd (25) f cdT ⋅ aT ⋅ bT ⋅ hT

νTRd =

µTRd y =

T Rd z

µ

T M Rd y

f cdT ⋅ aT ⋅ bT ⋅ (hT ) 2 =

T M Rd z

f cdT ⋅ hT ⋅ (bT ) 2

(26)

(27)

where αT is the solidity ratio of the transformed section that will be shown to be invariant under the transformation— that is, αT = α. The other variables are the same as for solid cross sections. Using similar assumptions made in the solid cross sections regarding invariance under the transformation of the relative values of the combined design axial force and biaxial bending resistance, Eq. (28) to (30) hold. The validity of the assumption will be proved subsequently. ν Rd =

µ Rd y =

µ Rd z =

T N Rd (28) f ⋅ a ⋅ bT ⋅ hT

T M Rd y

f cdT ⋅ a ⋅ bT ⋅ (hT ) 2

1/b 0  b / 2  0.5  0 1/h   h / 2  = 0.5 (35)     

1/b 0  0.4b  0.4   0 1/h  0.3h  = 0.3 (36)     

T M Rd z

f cdT ⋅ a ⋅ hT ⋅ (bT ) 2

(29)

(30)

f cdT ⋅ a ⋅ bT ⋅ hT = 1 (31)

f cdT ⋅ a ⋅ bT ⋅ (hT ) 2 = 1 (32)

f cdT ⋅ a ⋅ hT ⋅ (bT ) 2 = 1 (33)

The solutions are bT = 1, hT = 1, and f cdT = 1/α

The locations of individual reinforcing bar in the square hollow section of unit-length side are also determined using the same transformation matrix. Area and stress transformation Rectangular solid sections—In a linear transformation T with the matrix given by Eq. (15), the magnitude of the determinant (determinant = (1/b) · (1/h) – 0 · 0 = 1/(b · h)) is equal to the ratio of the area of the new shape to the area of the original shape. Therefore, in a transformation with two-way dilation of factors (1/b) and (1/h) parallel to y- and z-axes, respectively, the area of the image undergoes a dilation of factor (1/b) · (1/h) = 1/(b · h). As a result, the transformed area of the compression zone and transformed area of the reinforcement in the square cross section of unitlength side are given by Eq. (37) and (38), respectively.

ΩT =

Ω (37) b⋅h

AsT =

As (38) b⋅h

(34)

Following the same argument as in solid sections, it follows T that each of f cd , bT, and hT are dimensionless. Moreover, because the solutions for bT and hT remain unchanged, the derivations leading to and including the transformation matrix in Eq. (15) apply for hollow cross sections. The argument about the more general transformation conditions can 16

T cd

It can be seen from Eq. (28) to (30) that the combined design axial force and biaxial bending resistance of the transformed T T T section ( N Rd , M Rd y , M Rd z ) satisfying the transformation conditions laid down in Eq. (31) to (33) is equal to the relative values of the combined design axial force and biaxial bending resistance (νRd, μRd y, μRd z) of the original section. The conditions state that the results of the products are all unity and dimensionless.

also be directly extended to rectangular hollow sections to show that the transformed square hollow cross section of unit-length side is a special case resulting from the special condition (q1 = q2 = q3 = 1) in Eq. (31) to (33). Finally, it can be concluded that the relative values of the combined design axial force and biaxial bending resistance of a rectangular hollow section is equal to the combined design axial force and biaxial bending resistance of a square hollow cross section of the unit-length side that satisfies the transformation requirements described previously. The associated transformation requirements will be described in more detail in subsequent sections. As an example, the coordinate transformation matrix is used in Eq. (35) and (36) to map the outer and inner concrete fiber coordinates in the first quadrant of the original cross section (Fig. 2(a)), to the images in the equivalent square hollow sections of unit-length side (Fig. 2(b)). In the example, the aspect ratio and relative wall thicknesses of the original section are arbitrarily chosen as b/h = 2.0 and wb = wh = 0.2h.

where ΩT is the area of the compression zone in the transformed section, and Ω is the area of the compression zone in the original section. Similarly, because the transformation conditions in Eq. (11) to (13) have caused the transformation of the design compressive strength into unity, the transformation factor for stresses in concrete and reinforcement is 1/fcd. As a result, ACI Structural Journal/January-February 2015

Fig. 3—Stress-strain diagrams in original and transformed solid sections: (a) concrete; and (b) reinforcing steel. the stress-strain relationships of concrete and steel in the original solid section are transformed into the stress-strain relationships of concrete and steel in the transformed section (Fig. 3(a) and (b)). In particular, the transformed design yield strength of reinforcement, f ydT , is given by Eq. (39) f ydT =

f yd f cd

(39)

The stress-strain curves for materials and reduction factors in the original sections are according to Eurocode 2.5 In Eurocode 2,5 the stress-strain curve for concrete is idealized by a parabolic function followed by a plateau. The steel stress-strain is idealized as elastic-perfectly plastic. The design strengths are obtained from the characteristic (nominal) values using constant reduction factors (partial factors for concrete, γc, and steel, γs). It is possible to derive the transformation factor for stress resultants after having derived the transformation factor for stresses, and the dilation factors for areas and length measurements along the y- and z-axes. It is also possible to prove the assumption made previously with regard to the invariance under the transformation of the relative values of the combined axial force and biaxial bending resistance of the original section. The combined design axial force and biaxial bending resistance of the transformed section is determined by calculating the stress resultants at the ultimate limit state using Eq. (40) to (42).

T N Rd = ∫ σTc d ΩT + ∑ AsiT σTsi (40)

T T T T T T T M Rd y = ∫ σ c z d Ω + ∑ Asi σ si z si (41)

T T T T T T T M Rd z = ∫ σ c y d Ω + ∑ Asi σ si ysi (42)

ΩT

ΩT

ΩT

i

i

i

where ΩT is the area of the compression zone in the transT formed section; σ c is the compressive stress on an elemental area of concrete in the compression zone; AsiT is the area of steel reinforcement bar i; σTsi is the steel stress in reinforcement bar i; yT and zT are the moment arms of the elemental area of concrete about z- and y-axes, respectively; and ysiT ACI Structural Journal/January-February 2015

T

and zsi are the moment arms of the reinforcement bar i about z- and y-axes, respectively. Because ΩT = Ω/(b · h) from Eq. (37), it follows that dΩT = (1/(b · h))dΩ

(43)

Using the change of variable indicated in Eq. (43) and the transformation factors described previously, Eq. (40) to (42) can be rewritten as

T N Rd =

1 1   ⋅  ∫ σ c d Ω + ∑ Asi σ si  (44) i f cd b ⋅ h  Ω

T M Rd y =

1 1 1   ⋅ ⋅  ∫ σ c zd Ω + ∑ Asi σ si zsi  (45) i f cd h b ⋅ h  Ω

T M Rd z =

1 1 1   ⋅ ⋅  ∫ σ c yd Ω + ∑ Asi σ si ysi  (46) f cd b b ⋅ h  Ω

The quantities in brackets on the right-hand sides of Eq. (44) to (46) are expressions for the stress resultants in the original section. Whether or not the stress resultants are in the ultimate limit state similar to the transformed section needs further discussion. Figures 4(a) and (b) show the strain and stress distributions in the original and transformed sections, respectively. The strain distributions in the transformed section represent an ultimate limit state. The location of the neutral axis in the original section can be determined from the intercepts k1y0 and k2z0 in the transformed section using the dilation factors k1 and k2 along the y- and z-axes, respectively. The intercepts in the original section are thus k1y0/k1 = y0, and k2z0/k2 = z0, as shown in Fig. 4(a). Moreover, it can be verified using geometry that any given fiber parallel to the neutral axis in the transformed section is an image of a corresponding fiber parallel to the neutral axis in the original section. It can also be shown that the ratio of the neutral axis depth in the transformed section to the neutral axis depth in the original section can be expressed in terms of k1, k2, y0, and z0. Let it be designated by k3. Fiber strains on the neutral axis or other fibers parallel to the neutral axis in the original section can be determined from corresponding fiber stresses in the transformed section using the stress transformation and stress-strain diagrams 17

Fig. 4—(a) Strain and stress distribution at ultimate limit state: (a) in original section; and (b) in square cross section of unitlength side. (Fig. (3a)). As an example, the fiber stresses along the neutral T axis in the original section are σNA = σ NA· fcd = 0 · fcd = 0 and using the stress-strain diagrams, the strains are εNA = 0. For parallel fibers in the transformed section that are closer to the neutral axis than the one with strain εc2, at reaching the maximum strength—that is, (1)—the stresses are less than 1 (Fig. 4(b)). As stated previously, the stresses in the corresponding fibers in the original section can be determined using the stress transformation. Although the stresses in the two fibers are different, the strains are the same as shown in the strain stress diagram (Fig. 3(a)). In addition from geometry, the strain in the most compressed fiber in the transformed section can be shown to be equal to the strain in the corresponding fiber in the original section. Therefore, the strain distribution in the original section is one in the ultimate limit state and the stress resultants are the design resistance of the section. The equality of strains holds for all corresponding fibers, including reinforcing bars, leading to the conclusion that strains of corresponding fibers are invariant under the transformation. Thus, Eq. (44) to (46) can be rewritten as

T N Rd =

T M Rd y =

T M Rd z =

N Rd = ν Rd (47) f cd ⋅ b ⋅ h M Rd y f cd ⋅ b ⋅ h 2 M Rd z f cd ⋅ h ⋅ b 2

= µ Rd y (48)

= µ Rd z (49)

Therefore, the transformation factors for the design resistance of the original cross section (NRd, MRd y, MRd z) are 1/(fcd · b · h), 1/(fcd · b · h2), and 1/(fcd · b2 · h), respectively. Because

µTRd y =

µTRd z =

T T N Rd N Rd T (50) = = N Rd f cdT ⋅ bT ⋅ hT 1⋅1⋅1 T M Rd y

f cdT ⋅ bT ⋅ (hT ) 2 T M Rd z

f cdT ⋅ hT ⋅ (bT ) 2

=

=

T M Rd y

1⋅1⋅12 T M Rd z

1⋅1⋅12

T = M Rd y (51)

T = M Rd z (52)

it follows that

νTRd = ν Rd (53)

µTRd y = µ Rd y (54)

µTRd z = µ Rd z (55)

Therefore, the assumption that the relative values of the combined axial force and biaxial bending resistance is invariant under the transformation is valid, and the derivations based on this assumption are appropriate. Further, the geometric reinforcement ratio in the transformed section is

18

νTRd =

rT =

AsT (56) bT ⋅ hT


T Substituting bT = hT = 1 and the expression for As from Eq. (38) in Eq. (56)

A rT = s = r (57) b⋅h

Equation (57) indicates that the geometric reinforcement ratio is invariant under the transformation. Similarly, the mechanical reinforcement ratio in the transformed section is

ωT = rT ⋅

f ydT f cdT

(58)

Substituting f cdT = 1

ωT = rT ⋅ f ydT (59)

Substituting further for ρT and f ydT from Eq. (57) and (39)

ωT = r ⋅

f yd f cd

= ω (60)

Equation (60) indicates that ω is also invariant under the transformation. Finally, from Eq. (39), (56), (59), and (60)

AsT = ω ⋅

f cd (61) f yd

Equation (61) gives the transformed area of steel in the square cross section of unit-length side in terms of the mechanical reinforcement ratio ω, the design compressive strength of the concrete, and yield strength of the reinforcement in the original cross section. This same amount of concrete area is to be deducted if the analysis would be based on net cross section. Usually, analysis is based on gross cross sections, as the use of net cross sections does not affect the result significantly. The effect of the displaced amount of concrete on the cross section capacity may, however, be significant if the high strength of concrete is used, requiring analysis on the basis of net cross section.6,7 T The transformed area of reinforcement, As , can also be expressed in terms of the transformed design yield strength of reinforcement, f ydT , as

AsT =

ω (62) f ydT

Additional analytical advantage can be gained by setting f ydT = 1, because it allows the direct substitution of the reinforcement data by the mechanical reinforcement ratio ω. It is to be noted that this is not a consequence of the transforma-


tions discussed so far. It is rather an isolated action that allows the substitution of the amount of reinforcement AsT in the square cross section of unit-length side by ω, provided T that f yd = 1. The direct use of ω as reinforcement data can be used advantageously in the calculation of biaxial interaction diagrams where it can be systematically varied to cover the practical range of the mechanical reinforcement ratio. Rectangular hollow sections—Hollow sections are treated as the result of two solid components made up of the full cross section and the hollow part with positive and negative areas, respectively. Previously, it was shown that the transformation matrix T for hollow sections and solid sections are the same. Therefore, the transformed area of the compression zone and transformed area of the reinforcement in the square hollow cross section of unit-length side are given by Eq. (37) and (38), respectively. In Eq. (37), the area of the compression zone, Ω, has now two components made up of positive and negative areas associated with the actual solid and hollow component of the compression zone. Further, the transformed area of the solid part of the cross T section, Ac , can be determined using the area transformation factor as AcT =

Ac a ⋅b⋅ h = = a (63) b⋅h b⋅h

The solidity ratio αT in the transformed section is

aT =

AcT a a = T T = = a (64) T T 1⋅1 b ⋅h b ⋅h

Therefore the solidity ratio α is invariant under the transformation. Similarly, because the transformation conditions in Eq. (31) to (33) have caused the transformation of the design compressive strength into 1/α, the transformation factor for stresses in concrete and reinforcement is 1/(α · fcd). As a result, the stress-strain relationships of concrete and reinforcing steel in the original hollow section are transformed into the stress-strain relationships of concrete and reinforcing steel in the transformed section, as shown in Fig. 5(a) and (b). In particular, the transformed design yield strength of reinforcement, f ydT , is given by Eq. (65). f ydT =

f yd

(a ⋅ fcd )

(65)

Following the same argument that led to Eq. (40) to (42) in the solid sections and noting the stress transformations in hollow sections described previously, Eq. (44) to (46) take the form

T N Rd =

1 1   ⋅  ∫ σ c d Ω + ∑ Asi σ si  (66) i (a ⋅ fcd ) b ⋅ h Ω 19

Fig. 5—Stress-strain diagrams in original and transformed hollow sections: (a) concrete; and (b) reinforcing steel.

T M Rd y =

1 1 1   ⋅ ⋅  ∫ σ c zd Ω + ∑ Asi σ si zsi  (67) i (a ⋅ fcd ) h b ⋅ h Ω

T M Rd z =

1 1 1   ⋅ ⋅ ∫ σ yd Ω + ∑ Asi σ si ysi  (68) (a ⋅ fcd ) b b ⋅ h  Ω c

Equations (66) to (68) result in Eq. (69) to (71) after following the same argument as in the solid cross sections. T N Rd =

T M Rd y =

T M Rd z =

N Rd = ν Rd (69) a ⋅ f cd ⋅ b ⋅ h

Substituting bT = hT = 1 and the expression for AsT from Eq. (38) rT =

Equation (76) indicates that the geometric reinforcement ratio is invariant under the transformation. Similarly, the mechanical reinforcement ratio in the transformed section is ωT = rT ⋅

M Rd y a ⋅ f cd ⋅ b ⋅ h 2 M Rd z a ⋅ f cd ⋅ h ⋅ b 2

As = r (76) a ⋅b⋅ h

= µ Rd y (70)

f ydT f cdT

(77)

Substituting f cdT = 1 a = µ Rd z (71)

ωT = αT · ρT · f ydT (78)

Substituting further for ρT and f ydT from Eq. (76) and (65) Because νTRd

µTRd y =

µ

T Rd z

=

T NT N Rd = T T Rd T T = = N T (72) f cd ⋅ a ⋅ b ⋅ h (1 a ) ⋅ a ⋅1⋅1 Rd T M Rd y T cd

T

T

( )

f ⋅a ⋅b ⋅ h T M Rd z T cd

T

T

T 2

=

T M Rd y

(1 a ) ⋅ a ⋅1⋅12

=

T M Rd z

T

2

20

rT =

f yd f cd

= ω (79)

AsT = a ⋅ ω ⋅

f cd (80) f yd

T = M Rd z (74)

it follows that ν Rd = νRd, µ Rd y= μRd y, and µ Rd z = μRd z. Therefore, the assumption that the relative values of the combined axial force and biaxial bending resistance is invariant under the transformation is valid, and the derivations based on this assumption are appropriate. The geometric reinforcement ratio in the transformed section is T

ωT = r ⋅

Equation (79) indicates that ω is also invariant under the transformation. Finally, from Eq. (65), (75), (78), and (79)

( ) (1 a) ⋅ a ⋅1⋅1

f ⋅a ⋅ h ⋅ b

T 2

T = M Rd y (73)

T

AsT (75) a ⋅ bT ⋅ hT

Equation (80) gives the transformed area of steel in the square hollow section of unit-length side in terms of the solidity ratio, mechanical reinforcement ratio ω, design compressive strength of the concrete, and yield strength of the reinforcement in the original rectangular hollow section. This same amount of concrete area is to be deducted in the transformed section if the analysis would be based on net cross section.6,7 T The transformed area of reinforcement, As , can also be expressed in terms of the transformed design yield strength of reinforcement, f ydT , as


AsT =

ω (81) f ydT

Thus, the amount of reinforcement in the square hollow cross section of unit-length side can be replaced by ω by setting f ydT = 1. CONCLUSIONS The following conclusions can be drawn from this study: 1. The paper deals with the analytical proof of the equivalence between the relative biaxial bending resistance of a rectangular solid reinforced concrete section, and the biaxial bending resistance of the transformed square solid sections of unit-length side. The results of the analytical proof show that the relative biaxial bending resistance of a rectangular solid section is identical to the biaxial bending resistance of the transformed square solid section of unit-length side. These results only occur when the concrete fiber and reinforcing bar coordinates in the transformed section are in conformity with the transformation that maps the rectangular section into a square cross section of unit-length side. The concrete and steel stresses in the transformed section comply with the resulting stress transformation and the area of reinforcement in the transformed section must comply with the resulting area transformation. 2. The results of the analytical proof also show the equivalence between the relative biaxial bending resistance of a rectangular hollow section and the biaxial bending resistance of the transformed square hollow section of unit-length side, provided that similar transformation-related conditions are met. 3. The results of the analytical proof has shown that the geometric reinforcement ratio ρ, the mechanical reinforcement ratio ω, the relative values of the combined design axial force and biaxial bending resistances (νRd, μRd y, μRd z), and strains of corresponding fibers in the original and transformed sections are invariant under the transformation. 4. Transformed square cross section of unit-length side fulfilling the aforementioned conditions ((1) and (2)) are used in the calculation of biaxial interaction diagrams for rectangular solid and rectangular hollow sections.2 5. Relative values of combined design axial load and biaxial bending resistances are calculated on the basis of net cross section for high-strength concrete by deducting the amount of the transformed area of steel (AsT = ω · fcd/fyd) and (AsT = α · ω · fcd/fyd) from the square solid and square hollow section of unit-length side.2 AUTHOR BIOS ACI member Girma Zerayohannes is an Associate Professor at Addis Ababa University, Addis Ababa Institute of Technology, Addis Ababa, Ethiopia. He received his BSc and MSc from Addis Ababa University in 1979 and 1984, respectively, and his PhD from the University of Kaiserslautern, Germany, in 1995. His research interests include nonlinear analysis of reinforced concrete structures.

ACKNOWLEDGMENTS

Germany. The author wishes to gratefully acknowledge the financial support by the German Academic Exchange Services (DAAD).

NOTATION AsT AsiT b, h bT, hT fcd fcdT

= area of reinforcement in transformed section = area of steel reinforcement bar i = side lengths of rectangular cross section = side lengths of transformed section = design compressive strength of concrete = design compressive strength of concrete in transformed section fyd, f ydT = design yield strength of reinforcing steel in original and transformed sections, respectively k1, k2 = dilation factors along y- and z-axes, respectively k3 = ratio of neutral axis depth in transformed section to that of original section NRd, MRd y, MRd z = combined design axial load and biaxial bending resistance of rectangular cross section T T , M Rd , T = combined design axial force and biaxial bending N Rd y M Rd z resistance of transformed section yT, zT = moment arms of elemental area of concrete about zand y-axes, respectively = moment arms of the reinforcement bar i about z- and ysiT , zsiT y-axes, respectively α, αT = solidity ratio in original and transformed section, respectively εc2 = concrete strain at reaching maximum strength fcd and/ or 1 νRd, μRd y, μRd z = relative values of combined design axial load and biaxial bending resistance of rectangular cross section νTRd , µTRd y , µTRd z = relative values of combined design axial force and biaxial bending resistance of transformed section ρ, ω = geometric and mechanical reinforcement ratio, respectively ρT , ω T = geometric and mechanical reinforcement ratios in transformed section, respectively = compressive stress on elemental area of concrete in σTc compression zone σTsi = steel stress in reinforcement bar i T Ω = area of compression zone in transformed section Ω = area of compression zone in original section

REFERENCES 1. Busjäger, D., and Quast, U., “Programmgesteuerte Berechnung beliebiger Massivbauquerschnitte unter zweiachsiger Biegung mit Längskraft (Computer Program for the Analysis and Design of Arbitrarily Shaped Reinforced Concrete Sections under Axial Load and Biaxial Bending),” Deutscher Ausschuss für Stahlbeton, Heft 415, Beuth Verlag, Berlin, Germany, 1990, 212 pp. (in German) 2. Zerayohannes, G., “Bemmesungsdiagramme für Schiefe Biegung mit Längskraft nach DIN 1045-1: 2001-07 (Interaction Diagrams for Biaxial Bending with Axial Load on the Basis of the German Code DIN 1045-1: 2001-07),” Schriftenreihe der Fachgebiete Baustofftechnologie und Bauschadenanalyse, Massivbau und Baukonstruktion und Stahlbau des Studienganges Bauingenieurwesen der Technischen Universität Kaiserslautern, Band 4, 2006, 270 pp. (in German and English) 3. Cedolin, L.; Cusatis, G.; Eccheli, S.; and Rovda, M., “Capacity of Rectangular Cross Sections under Biaxially Eccentric Loads,” ACI Structural Journal, V. 105, No. 2, Mar.-Apr. 2008, pp. 215-224. 4. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05),” American Concrete Institute, Farmington Hills, MI, 2005, 430 pp. 5. Eurocode 2, “Part 1-1: General Rules and Rules for Building,” European Committee for Standardization, Brussels, Belgium, 2003, 225 pp. 6. Zilch, K.; Jähring, A.; and Müller, A., “Erläuterungen zu DIN 1045-1 Explanation on DIN 1045-1),” Deutscher Ausschuss für Stahlbeton, Heft 525, Beuth Verlag, Berlin, Germany, 2003. (in German) 7. DIN 1045-1:2001-07, “Tragwerke aus Beton, Stahlbeton und Spannbeton. Teil 1: Bemessung und Konstruktion (Concrete, Reinforced and Prestressed Concrete Structures – Part 1: Design),” Normenausschuss Bauwesen (NABau) im DIN (Deutsches Institut für Normung) e.V. Beuth Verlag, Berlin, Germany, July 2001. (in German)

This investigation was conducted at the Institute of Concrete Structure and Building Construction, Technical University of Kaiserslautern,


21

NOTES:

22



TECHNICAL PAPER

Title No. 112-S03

Evaluation of Post-Earthquake Axial Load Capacity of Circular Bridge Columns by Vesna Terzic and Bozidar Stojadinovic Objective evaluation of the capacity of a bridge to carry selfweight and traffic loads after an earthquake is essential for a safe and timely reopening of the bridge. The ability of a bridge to function depends directly on the remaining capacity of the bridge columns to carry gravity and lateral loads. An experimental study on models of modern circular reinforced concrete bridge columns was performed to investigate the relationship between earthquake-induced damage in bridge columns and the capacity of the columns to carry axial load in a damaged condition. The earthquake-like damage was induced in the column specimens in bidirectional, quasi-static, lateral load tests. The damaged column specimens were then recentered to eliminate the residual drifts and tested in compression to failure to evaluate their remaining axial load strength. It was found that well-confined modern bridge columns lose approximately 20% of their axial load capacity after sustaining displacement ductility demands of 4.5. Keywords: axial tests; earthquake; post-earthquake lateral stiffness; quasistatic tests; reinforced concrete.

INTRODUCTION Modern highway bridges in California designed using the Caltrans Seismic Design Criteria1 (SDC) are expected not to collapse during both frequent and rare earthquake events. Currently, design provisions aimed at preventing structural collapse are supported by numerous experimental data points and calibrated computer models.2,3 However, there is no evidence that the bridge systems were tested for the remaining traffic load capacity after some damage was induced under lateral loading. Still, attempts were made toward analytical evaluation of the ability of a highway overpass bridge4 or bridge columns5 to carry traffic load after an earthquake. Due to the lack of the validated quantitative guidelines for estimating the remaining traffic load-carrying capacity of bridges after an earthquake, bridge inspectors and maintenance engineers provide an estimate of the capacity of the bridge to function based on qualitative observations, with each judgment founded on personal experience. Such subjective evaluation can be significantly improved if a model to provide a quantitative estimate of the remaining load-carrying capacity of bridge columns after an earthquake was developed and calibrated. A combined experimental and analytical research program was performed to investigate the relationship between earthquake-induced damage in reinforced concrete bridge columns and the capacity of the columns in such damaged condition.6 This program comprised one axial load test, three quasi-static cyclic tests, and two hybrid model earthquake response simulations on scaled models of typical circular ACI Structural Journal/January-February 2015

Table 1—Test matrix Specimen designation

Ductility target

Test sequences

Base0

0

Axial

Base15

1.5

Lateral and axial

Base30

3.0

Lateral and axial

Base45

4.5

Lateral and axial

bridge columns used in modern bridges in California. In this paper, the outcomes of the quasi-static cyclic part of the experimental program are presented. In the first stage of the quasi-static testing procedure, three column specimens were tested by applying a bidirectional quasi-static incremental lateral displacement protocol with circular orbits of displacement up to the predetermined displacement ductility targets of 1.5, 3, and 4.5. In the second stage of the testing procedure, an undamaged column specimen and the three damaged specimens with no permanent drifts were subjected to a monotonically increasing axial force up to failure. The specimens are listed in Table 1. These results support evaluations of post-earthquake traffic load capacities of bridges with well-confined reinforced concrete columns. RESEARCH SIGNIFICANCE Reliable evaluation of the capacity of a bridge to carry self-weight and traffic loads is essential for a safe and timely re-opening of the bridge after an earthquake. Columns of modern California bridges are designed to develop significant flexural deformation ductility without shear failure and prevent bridge collapse. An experimental and analytical evaluation of earthquake-damaged modern bridge columns is used to quantify their axial load capacity and to develop reliable models for objective evaluation of the ability of a modern bridge to perform as intended after an earthquake: continue to safely carry traffic load. EXPERIMENTAL INVESTIGATION Ketchum et al.7 developed a series of highway overpass bridges designed in accordance with the Caltrans SDC1 in a recent PEER Center study. Bridge Type 11 (shown in Fig. 1)—typical for tall overpass bridges—was chosen as a prototype for this experimental study. The bridge is a fiveACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-075.R1, doi: 10.14359/51687296, received March 5, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

23

Fig. 1—Prototype Caltrans bridge.7 span single-column-bent overpass with 120 ft (36.58 m) edge spans, 150 ft (45.72 m) inner spans, a 39 ft (11.89 m) wide deck, and 50 ft (15.24 m) tall circular columns. The two principal parameters that affect the remaining axial capacity of bridge columns are the column aspect ratio H/D and the column shear strength (or transverse reinforcement ratio ρt).8 Different possible values of these two parameters, bounded by the provisions of the Caltrans SDC,1 were investigated.6 Values of H/D = 8 and ρt = 0.75% were selected for the columns of the prototype bridge. The columns are modeled with specimens referred to herein as the Base specimens. The Base specimens are cantilever columns representing the bottom half of the prototype bridge columns. The specimens were tested in a single-curvature-bending-using loading pattern that will induce displacement ductilities observed in columns of the prototype bridge for the two bridge directions: transverse and longitudinal. Specimen geometry, reinforcement, and materials The geometry and the reinforcement of a Base specimen are detailed in Fig. 2. The specimen has a 73.75 in. (1.875 m) tall, 16 in. (0.4 m) diameter circular column, and a square (84 x 84 in. [2.13 x 2.13 m]), 24 in. (0.61 m) high foundation block. The effective height of the specimen column, from its base to the level where lateral load is applied, is 64 in. (1.625 m), giving it an aspect ratio of L/D = 4. The 9.75 in. (0.25 m) extension accommodates the installation of the 0.5 in. (13 mm) thick, 16 in. (0.4 m) tall steel jacket to attach the actuators. The column has 12 longitudinal Grade 60 (nominal yield stress in tension is 60 ksi [420 MPa]) No. 4 [Ø13] reinforcing bars placed around its perimeter. The transverse steel reinforcement is a high-strength A82 (nominal yield stress in tension is 80 ksi [550 MPa]) W3.5 (0.211 in. 24

[5.4 mm]) continuous spiral with a center-to-center spacing of 1.25 in. [31.75 mm]. The concrete cover is 0.5 in. (13 mm). The specified unconfined compressive strength of the concrete was 5 ksi (34.5 MPa). Table 2 shows the specified and the actual strengths of the longitudinal steel, the spiral steel, and the concrete. Loading protocol The loading pattern for the Base specimens was selected to represent, as closely as possible, the motion experienced by a column of the prototype overpass bridge in an earthquake. The motion of the bridge column excited by different three-component ground motions is examined using a finite element model of the prototype bridge made in OpenSees.9 Two ground motion suites with 20 records per suite, representing near-field and far-field ground motions, were used. The displacement orbits of the tops of the prototype bridge columns were traced during the nonlinear time history response analyses for the 40 selected ground motions. Because of the different bridge column boundary conditions for bending in the longitudinal (fixed-fixed) and transverse (fixed-free) directions of the bridge, appropriately scaled displacement histories applied to the cantilever bridge column model would not reproduce the deformation state of the prototype. To achieve the close correspondence of deformation states between the model and the prototype, the displacement history of the prototype bridge column had to be normalized by its yield displacements, different in different bridge directions. Therefore, the displacement orbits applied on the model were expressed in terms of displacement ductility. Although the displacements of the tops of the bridge columns were larger in the transverse direction than in the longitudinal bridge direction, the ductility orbits were proportional in the two bridge directions for most of the ACI Structural Journal/January-February 2015

Table 2—Bridge column specimen material properties Specified1, ksi (MPa)

Actual, ksi (MPa)

Material

Yield

Ultimate

Maximum stress

Yield

Ultimate

Test

Maximum stress

Steel, longitudinal

60 (420)

80 (550)

—

70.7 (487)

120 (830)

—

—

Steel, spiral

80 (550)

—

—

95 (655)

106 (730)

Concrete

—

5.0 (34.5)

—

—

—

Base15-L

5.05 (34.82)

Base30-L

4.96 (34.2)

Base45-L

5.09 (35.09)

Base0

5.48 (37.8)

—

Fig. 2—Geometry and reinforcement of Base specimens.

Fig. 3—Displacement orbits at top of bridge column: (a) absolute displacements; and (b) normalized displacements. ground motions. Normalization of the displacement orbits for one ground motion is shown in Fig. 3. A circular loading pattern was selected for this experimental program because this loading pattern imposes larger sustained displacement ductility demand than was observed in any of the considered ground motions.6 Given that the goal of the experimental study is to establish the remaining


load-carrying capacity of damaged bridge columns, imposing the demand with a circular loading pattern is conservative. The selected circular loading pattern is defined by two cycles at each displacement level. In the first cycle, starting from the initial position O, the specimen control point (the center of the column cross section at the level of the actuators) is displaced toward position A, followed by motion that traces a full circle clockwise until point B (black line in Fig. 4). The specimen control point is then moved back to the initial position O to finish the first cycle. The second cycle is a counterclockwise path O-C-D-O (gray line in Fig. 4). The maximum displacement ductility demand imposed on a column specimen in this study was set at 4.5. It is selected to be slightly larger than Caltrans SDC1 design target displacement ductility of 4 and is on the conservative side. Caltrans SDC1 design is based on experimental evidence that well-confined reinforced concrete circular column can sustain a displacement ductility demand of 4 without developing significant flexural or shear damage.10-12 Two additional displacement ductility demand targets of 3.0 and 1.5 were selected to uniformly sample the demand space and evaluate the remaining axial capacity of less-damaged specimens. 25

Table 3—Displacement ductility levels of primary cycles

Fig. 4—Bidirectional displacement pattern. The magnitudes of displacement demand increments for the quasi-static tests were defined following the recommendations in ACI 374.1-0513 and SAC/BD-00/1014 for a major far-field earthquake event. For the Base45 specimen, the increments in the magnitude of the displacement ductility were: 0.08, 0.2, 0.4, 1.0, 1.5, 2.0, 3.0, and 4.5. The pre-yield displacement levels include: a displacement level prior to cracking; two levels between cracking and yielding; and a level approximately corresponding to the first yield of the longitudinal reinforcement. After the yield level, the displacement ductility magnitude of each subsequent primary cycle is 1.25 to 1.5 times larger than its predecessor to provide data of the damage accumulation. The selected two-cycle displacement pattern provides data on specimen strength degradation due to sustained displacement demand. After yielding, each primary displacement ductility demand level was followed by a small displacement level equal to one-third of the previous primary displacement level to evaluate specimen stiffness degradation. The displacement histories for the Base15 and Base30 specimens were obtained by scaling the displacement history for the Base45 specimen by 0.3 and 0.6, respectively. This way, the number of primary cycles in the loading history was the same for all tests to maintain similitude with respect to the duration and number of excursions imposed by real ground motions. Displacement ductility levels of primary cycles for the three lateral displacement tests are given in Table 3. After completing the cycles at the target displacement ductility level, the specimens are cycled through a series of small deformation cycles decreasing in magnitude to zero to eliminate residual lateral forces and deformations and recenter the specimens. This was necessary for the subsequent axial load capacity tests on damaged specimens. The lateral deformation tests were conducted with the column specimen under a constant axial load equal to 10% of the column nominal axial load capacity. This axial load magnitude is consistent with typical bridge column gravity load magnitudes, and slightly larger than the gravity load magnitude in the columns of the prototype bridge.

26

Cycles

Base15

Base30

Base45

Cycle 1

0.02

0.05

0.08

Cycle 2

0.06

0.10

0.20

Cycle 3

0.12

0.25

0.40

Cycle 4

0.30

0.60

1.00

Cycle 5

0.45

1.00

1.50

Cycle 6

0.60

1.25

2.00

Cycle 7

1.00

1.80

3.00

Cycle 8

1.50

3.00

4.50

Test setup In the first phase of the test, lateral and axial loads were applied at the top of the column. The lateral displacement pattern was applied using the two servo-controlled hydraulic actuators, as shown in Fig. 5. An axial load of 100 kip (445 kN), equal to 10% of the column’s nominal axial load capacity, was applied through a spreader beam using pressure jacks and post-tensioning rods placed on each side of the column (Fig. 5). Spherical hinges (threedimensional swivels) were provided at both ends of the rods to avoid bending of the rods during circular motion of the column top in the horizontal plane. A hinge connection (twodimensional hinge) was placed between the spreader beam and the column such that the spreader beam remained horizontal in the plane of the rods during lateral motion of the column to avoid buckling of the rods. Geometry of the axial load application apparatus was monitored throughout the test in order to subtract the horizontal components of the force in the post-tensioned rods from the forces applied by the actuators and compute the actual lateral resistance of the column. In the second phase of the test, the three laterally damaged column specimens and one undamaged column specimen were compressed axially to induce axial failure in the columns. A compression-tension axial load machine with a capacity of 1814 tonnes (4 million lb) and a constant rate of loading was used to accomplish this (Fig. 6). Longitudinal reinforcement strain measurements were used to evaluate presence of bending moment in the specimens during the axial load test based on which the extent of geometric imperfections was estimated. ANALYTICAL INVESTIGATION The experimental results, the hysteretic curves from quasistatic tests, and the axial force-deformation responses from the compression tests were numerically simulated using the force-based fiber beam-column element15 of OpenSees.9 The force-based beam-column element is a line element discretized using the Gauss-Lobatto integration scheme with the integration points at the ends of the element and along the element length. Fiber cross sections are assigned to the integration points. The cross sections of the element are represented as assemblages of longitudinally oriented, unidirectional steel and concrete fibers. Each material in the cross section has a uniaxial stress-strain relation assigned ACI Structural Journal/January-February 2015

Fig. 5—Lateral test setup: (a) plain view; (b) elevation (A-A); and (c) photo of test setup. to it. The deformation compatibility of the cross-section fibers is enforced assuming that plane sections remain plane after deformation. In a flexibility-based formulation of this element, nodal loads imposed on the element ends are used to calculate axial force and moment distribution along the length of the element. Given the moment and axial load values at each integration point, the curvature and the axial deformation of a section are subsequently computed. Because the response of the cross-section fiber materials may be nonlinear, deformation state determination of the cross section may be iterative. The deformation of the element is finally obtained through weighted integration of the section deformations along the length of the member. A non-shear-critical column with hardening section behavior was modeled using five integration points16 along its length. The cross sections of the beam-column element had 132 fibers (24 for unconfined cover, 96 for confined core, and 12 for reinforcing steel) distributed nonuniformly16 and arranged as shown in Fig. 7. To model the reinforced concrete section, the fiber section that accounts for the axial-bending interaction was divided into three parts: concrete cover; concrete core; and reinforcing steel. Fibers of the concrete cover (unconfined concrete) and concrete core (confined concrete) were modeled using the OpenSees ACI Structural Journal/January-February 2015

Fig. 6—Axial test setup. Concrete01 uniaxial material that uses the Kent-Scott-Park model17 to represent the stress-strain relationship of concrete in compression. Reinforcing steel fibers (longitudinal bars) were modeled using the OpenSees Steel02 uniaxial mate27

Fig. 7—Nonuniform arrangement of fibers in column section.16

Fig. 9—Accuracy in prediction of column axial strength as function of ratio of initial to post-peak degrading slope of confined concrete (k). Table 4—Steel02 material model parameters Material Reinforcing steel

fy, ksi (MPa) Es, ksi (GPa) 70.7 (490)*

b

R0

cR1

cR2

29,000 (200) 0.025 15 0.925 0.15

*

From coupon tests.

Table 5—Concrete01 material model parameters Material

fc′

ε0

Concrete cover

fc′

Concrete core

fcc′

* ‡

fcu

εcu

2fc′/Ec

0

0.0055

2fcc′/Ecc†

0.2fcc′

εcu§

†

*

From test results on concrete cylinders (Table 2).

†

Fig. 8—Calibration of the Concrete01 material using data from compression test of concrete cylinders. rial that uses the Giuffre-Manegotto-Pinto uniaxial strainhardening material model.18 Transverse reinforcement was not modeled directly but its effect was accounted for through uniaxial stress-strain relationship of the confined concrete core19 assigned to core fibers. The parameters of the Steel02 and Concrete01 uniaxial materials are given in Tables 4 and 5, respectively. The initial moduli of elasticity used to model plain Ec and confined concrete Ecc were calibrated from the concrete cylinder compression tests performed on the on the day of the tests (for example, Fig. 8). To define the confined concrete model, the maximum compressive strength fcc′ was calculated using Mander’s equations,19 the strain at the maximum compressive strength ε0 was calculated from the initial modulus of elasticity for the Concrete01 material (Ecc = 2fcc′/ε0), the concrete crushing strength fcu was set to 0.2fcc′1, and the strain at crushing strength of concrete εcu was calculated using Eq. (1) and is a function of the post-peak degrading slope of concrete (kEcc). Equation (1) can only be used in conjunction with Concrete01 material, as it is derived for the strain at the maximum compressive strength ε0 of 2fcc′/Ecc

28

e cu =

f cc′ (1 + 2k ) − f cu (1) kEcc

Ec, Ecc are initial moduli of elasticity (calibrated to match test results of concrete cylinders).

‡

Equation from Mander et al.19

§

Equation (1).

The post-peak degrading slope of concrete fibers has a significant effect on the accuracy of prediction of the remaining axial load strength of columns with the earthquake type of damage. If, during a lateral load test simulation, a concrete fiber strain exceeds the strain at the maximum compressive strength (ε0), the peak strength that fiber can attain during the axial loading simulation phase is the strength that corresponds to the maximum strain reached during the lateral load test. This strength is smaller than the peak strength and depends on the post-peak degrading slope of the concrete fiber stress-strain model. Furthermore, the post-peak behavior is different for different fibers of the damaged section. While the post-peak degrading slope for the fibers of the concrete cover can be calibrated from the concrete cylinder compression tests (Fig. 8), the post-peak degrading slope for the fibers of the concrete core can be calibrated from the axial compression tests of the laterally damaged column specimens. For the tested column specimens and concrete modeled with the Concrete01 OpenSees material, the post-peak degrading slope kEcc of 0.014Ecc was found to provide the best predictions of the axial load capacities (Fig. 9). The specimen models were cantilevers with displacements restrained to zero at the bottom node. The two horiACI Structural Journal/January-February 2015

Fig. 10—Lateral force-deformation response curves in two major directions, and state of specimens after the quasi-static tests: (a) Base15; (b) Base30; and (c) Base45. zontal displacements at the top of the specimens matched the displacements commanded during the test. The vertical force at the top of the element matched the vertical force applied and measured during the tests. The moments on the rotation and torsion degrees of freedom on the top node of the model were set to zero. The response of the specimen model was computed using nonlinear analysis, Newton-Raphson integration algorithm, and geometric transformation to account for P-Δ effect. RESULTS AND DISCUSSION Phase 1: Lateral displacement tests The performance of the Base specimens during the bidirectional quasi-static tests is presented in Fig. 10. Experimental and analytical lateral force-displacement response curves for the two major directions of loading (X and Y) are accompanied by the final damage state of specimens for the three lateral tests. Responses obtained from the analytical model are in good agreement with the experimental results. ACI Structural Journal/January-February 2015

The Base15 specimen was laterally displaced up to the displacement ductility level of 1.5. Longitudinal reinforcement yielded first during the 1.0 ductility cycle. At the end of the test, the specimen was only slightly cracked. The horizontal cracks, uniformly distributed along the bottom half of the column, were less than 1/32 in. (0.8 mm) wide and approximately 6 in. (152 mm) apart (Fig. 10(a)). A bridge column in such a light damage state would be classified as being in Damage State 0 (defined by Mackie et al.20) and would likely require no repairs. The Base30 specimen experienced significant yielding and strain hardening of the longitudinal column reinforcement and initiation of spalling of concrete in the plastic hinge region (Fig. 10(b)). After the longitudinal reinforcement began to yield (at a nominal displacement ductility of 0.9), the lateral resistance of the specimen slightly increased due to strain hardening of reinforcing bars, while its stiffness decreased with each subsequent test cycle. In the plastic hinge region of the column (the bottom 12 in. [305 mm]), the 29

distance between the cracks was 3 in. (76 mm) on average and the maximum width of the cracks during the test was approximately 1/16 in. (1.6 mm). Outside the plastic hinge region, the distance between the cracks was 6 in. (152 mm) on average with the widths of the cracks less than 1/32 in. (0.8 mm). Figure 10(b) shows horizontal cracks, vertical cracks, and some spalling of concrete at the bottom 8 in. (203 mm) of the column at the end of the test. Such a moderate damage state would put the column in Damage State 1 (defined by Mackie et al.20). The bridge columns with such earthquake damage are likely to require repairs such as epoxy injection into the plastic hinge cracks and cover patching. The target displacement ductility demand imposed on specimen Base45 (4.5) slightly exceeds the Caltrans SDC design target (4.0). This specimen experienced extensive yielding of the reinforcing steel, spalling of the concrete cover, as well as a crushing and reduction in volume of the concrete core in the plastic hinge region. First yielding of a reinforcing bar occurred at a displacement corresponding to nominal displacement ductility of 0.75. The specimen response was highly nonlinear (Fig. 10(c)), with the expected gradual stiffness degradation and gradual strength increase. Based on the crack distribution along the height of the column during the test, the column was divided into three regions: 1) the plastic hinge region (the bottom 12 in. [305 mm] of column); 2) the intermediate region (12 in. [305 mm] of the column next to the plastic hinge region); and 3) the elastic region (the top 40 in. [1.02 m] of the column). In the plastic hinge region, the distance between the cracks was 3 in. (76 mm) on average, and the maximum width of the cracks during the test was approximately 1/8 in. (3.2 mm). Very extensive spalling of concrete and a reduction in volume of the concrete core were observed. In the intermediate region, the distance between the cracks was 4 in. (102 mm) on average, with the widths of the cracks less than 1/16 in. (1.6 mm). In the elastic region, the distance between the cracks was 6 in. (152 mm) on average, with the widths of the cracks less than 1/32 in. (0.8 mm). Such column damage would be classified into Damage State 2 (defined by Mackie et al.20), requiring significant repairs but not requiring replacement. To analyze bridges for an aftershock, it is important to know effective stiffness keff of bridge columns after the main shock. This stiffness is computed using the response data measured during the small-displacement test cycles that followed the primary cycles. It represents the tangent slope at zero force of force-displacement curve for the smalldisplacement test cycles. The effective stiffness at yield keff,y of tested columns, representing the slope of forcedisplacement curve between origin and the point designating the first reinforcing bar yield, is used as a reference to measure stiffness degradation during the quasi-static tests. The ratio of column effective stiffness over column stiffness at yield keff/keff,y is given in Table 6 for each specimen. The effective stiffness of the damaged column decreases such that, at displacement ductility level of 4.5, it is approximately half that of the effective stiffness of the same column at yield.

30

Table 6—Ratio of column effective stiffness to column stiffness at yield Ductility

Base15

Base30

Base45

1.0

1.00

1.00

1.00

1.2

—

0.85

—

1.5

0.83

—

0.75

1.8

—

0.73

—

2.0

—

—

0.63

3.0

—

0.54

0.52

4.5

—

—

0.44

Phase 2: Axial load tests The experimentally measured and numerically simulated axial force-deformation curves for one undamaged and three damaged specimens are shown in Fig. 11. Because the tests are performed using a force-controlled compression machine, the axial force-displacement relationships are realistic up to the peak force point. The experimental and analytical axial load strengths, the remaining axial strength after damage was induced during the Phase 1 of lateral deformation tests, and the errors in predicting the axial strengths are summarized in Table 7. Testing of the Base0 column specimen was performed to establish the axial strength of an undamaged column specimen: it was 1459 kip (6490 kN). The axial failure resulted from the formation of the shear failure plane in the bottom half of the Base0 specimen column (Fig. 11(a)). The analytical model predicted the axial strength of the undamaged specimen to be 1446 kip (6434 kN) (error is 0.9%). An equally accurate prediction of the axial strength can be achieved using Eq. (2)

Po = fcc′ · (Aeff – Ast) + fy · Ast (2)

if Mander’s equations19 are used to calculate the area of the confined core Aeff and the compressive strength of confined concrete fcc′, based on measured strengths of plain concrete, reinforcing bars, and spiral. Using Eq. (2), the axial strength of the column specimen, Po, is estimated to be 1455 kip (6472 kN), resulting in the ratio of estimated to measured strength (Po/Pm) of 0.997. However, if confinement of the column is not accounted for and the axial strength is calculated following Eq. (3) (per ACI 31821 and Caltrans BDS22) Pn = 0.85 · [0.85 · fc′ · (Ag – Ast) + fy · Ast] (3) The estimated axial strength of the column is significantly smaller than the measured axial strength (Pn/Pm = 0.57). The remaining axial load strength of the Base15 column specimen was 1137 kip (5057 kN)—78% of the original axial strength. Longitudinal reinforcement strain measurements during the axial load tests indicated a bending moment corresponding to a lateral drift of approximately 1%. A posttest inspection of the specimen indicated that the specimen was not accurately leveled when it was installed for axial load testing. The resulting second-order bending moment and the corresponding shear caused a shear crack in the top ACI Structural Journal/January-February 2015

half of the Base15 column specimen (Fig. 11(b)). To numerically simulate the axial strength of the damaged column with the presence of residual drift, the top of the column model was first laterally displaced following the loading pattern of the quasi-static test, then laterally displaced to the observed drift ratio of 1%, and lastly was axially compressed (pushunder analysis) to induce the axial failure. The analytically predicted axial strength of the Base15 column specimen was 1141 kip (5075 kN)—only 0.38% greater than the experimentally measured. The remaining axial load strength of the Base30 column specimen was 1355 kip (6027 kN)—93% of the original axial strength. The specimen was properly leveled before the axial load tests; lateral drift was not present during this phase of testing. The axial failure resulted from the formation of the shear failure plane in the bottom half of the Base30 specimen column (Fig. 11(c)). The analytically predicted axial strength was 1217 kip (5455 kN)—10.2% smaller than the experimentally measured. The remaining axial load strength of the Base45 column specimen was 1170 kip (5204 kN)—80% of the original axial Table 7—Remaining axial strengths following quasi-static tests: experimental versus analytical Test

Experiment, kip (kN)

Analytical, kip (kN)

Numerical error, %

P/Po

Base0

1459 (6490)

1446 (6434)

0.9

1.0

Base15

1137 (5058)

1141 (5075)

0.38

0.78

Base30

1355 (6027)

1217 (5413)

10.17

0.93

Base45

1170 (5204)

1173 (5217)

0.24

0.80

strength. The specimen was properly leveled before the axial load tests but the head of the testing machine was not accurately attached to the specimen: it had a small angle relative to the specimen. Thus, the force imposed on the column had two components: a dominant axial component and a small horizontal component. This initiated a failure slightly earlier than what would be anticipated if there were no flaws in the test setup. The axial failure started at the top of the column and progressed toward the bottom of the column, resulting in the formation of the shear failure plane along the total height of the column (Fig. 11(d)). The misalignment of the test machine head was not measured prior to test and therefore could not be included in analytical simulation of the axial load test. Although the analytical model provides a good estimate (1173 kip [5217 kN], 0.24% error), it is hard to anticipate the magnitude of the error if the flaws in the test setup were included in the analysis. Axial load degradation curve with respect to displacement ductility is developed based on experimental results. It is a bilinear function (Fig. 12) fitted through experimental data from compression tests on column specimens with lateral damage and no geometric imperfections of the damaged specimens. It includes the result of the Base45 specimen, thus providing a conservative estimate of the residual axial strength. It is assumed that there are no losses in the axial strength in the columns if the lateral displacement ductility is less than 1.5. This assumption is based on the damage state of the Base15 specimen after the lateral test (Damage State 0) and on minor loss in axial strength (7%) of the Base30 specimen. The bilinear fit is

Fig. 11—Axial force-displacement relationships and state of specimens after axial load tests: (a) Base0; (b) Base15; (c) Base30; and (d) Base45. ACI Structural Journal/January-February 2015

31

Fig. 12—Column axial load capacity degradation data and a bilinear fit (Eq.(4)).

if µ < 1.5 P 1 (4) = Po 1.09 − 0.0615 ⋅ µ if µ ≥ 1.5

CONCLUSIONS Tests designed to evaluate the axial load capacity of tall modern bridge columns damaged in bidirectional quasistatic cyclic tests up to nominal displacement ductility levels of 1.5, 3, and 4.5 were performed. The following conclusions are drawn: 1. The axial strength and stiffness of a column degrade with the increase in the amount of damage induced by lateral displacement of the column. Well-confined modern bridge columns with no residual post-earthquake lateral drifts lose approximately 20% of their axial load capacity after sustaining displacement ductility demand of 4.5, which is slightly larger than the Caltrans SDC design target displacement ductility demand of 4.0. Therefore, modern bridge columns designed according to Caltrans SDC1 will not experience a significant loss of axial load-carrying capacity after a design-level earthquake. No axial load capacity loss is expected for displacement ductility demands less or equal to 1.5. Axial load capacity loss may conservatively be assumed to vary linearly with increasing displacement ductility demand. The residual post-earthquake displacements have a significant effect on the axial capacity of the column: the column that sustained the displacement ductility demand of 1.5 with no significant local damage but with a residual lateral drift of 1% experienced a reduction in axial load capacity of 22%. 2. Damage states observed during bidirectional lateral displacement tests correspond well to Damage State descriptions defined by Mackie et al.20 Namely, virtually no damage observed at displacement ductility demand of 1.5 corresponds to Damage State 0; moderate damage characterized by cover spalling and pronounced yielding of longitudinal reinforcement at displacement ductility demand of 3.0 corresponds to Damage State 1; significant damage to the cover and core concrete, very pronounced yielding of longitudinal and transverse reinforcement, however without any rein-

32

forcement fractures, at ductility demand of 4.5 corresponds to Damage State 2. These data can be used to calibrate repair cost and time models for modern reinforced concrete bridge columns. 3. The effective lateral stiffness of a damaged column decreases with increasing displacement ductility demand. For example, the effective stiffness of the column that experienced a displacement ductility demand of 4.5 is slightly less than half of the undamaged column effective stiffness. This information can be used to estimate the dynamic characteristics of damaged bridges for aftershock response analysis. 4. The circular bidirectional displacement ductility pattern developed in this study imposes sustained constant displacement ductility demands on the column that very likely exceed the displacement ductility demands imposed by the recorded ground motions. Therefore, this load pattern is a conservative estimate of actual ground motion demands and could be used to investigate residual post-earthquake capacities or function capabilities of damaged structures. 5. A non-shear-critical bridge column in the range of strain-hardening response can be modeled using fiber cross-section force-based beam-column element with distributed plasticity. Two OpenSees material models are recommended to model the steel and concrete unconfined cover and confined core fibers that define a cross section of the element: the Steel02 material model for reinforcing steel, and the Concrete01 material model for concrete fibers. Mander’s equations19 are recommended to calculate the compressive strengths of confined concrete. If the material test data for concrete are available, it is recommended to calibrate the Concrete01 material model to match the test data. The post-peak degrading slope of the confined concrete core (kEcc) has a great influence on the accuracy of the response predictions. For the tested column specimens, the post-peak degrading slope kEcc of 0.014Ecc was found to provide accurate predictions of the axial load capacities. The analytical models of the bridge columns presented in this paper were validated through hybrid simulations of the seismic response of an entire bridge followed by the truck load and the axial compression tests of damaged bridge columns.23 Validated model of a typical bridge was used in an extensive parametric study to evaluate its post-earthquake truck load capacity. The parametric study examined the effects of different ground motions and bridge modeling parameters, including boundary conditions imposed by the bridge abutments, the location of the truck load on the bridge, the amount of bridge column damage, and the amount of bridge column residual drift. Envelopes of bridge responses developed for ranges of the considered parameters were used to evaluate bridge safety and ability to function following an earthquake.6 AUTHOR BIOS Vesna Terzic is an Assistant Professor at the Department of Civil Engineering and Construction Engineering Management, California State University, Long Beach, CA. She received her BS from the University of Belgrade, Belgrade, Serbia; her MS from Saints Cyril and Methodius University of Skopje, Skopje, Macedonia; and her PhD from the University of California, Berkeley, Berkeley, CA. Her research interests include performance-based design and evaluation of civil infrastructure.


ACI member Bozidar Stojadinovic is a Professor and Chair of structural dynamics and earthquake engineering at the Department of Civil, Environmental and Geomatic Engineering at Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. He received his Dipl. Ing. degree in civil engineering from the University of Belgrade; his MS in civil engineering from Carnegie Mellon University, Pittsburgh, PA; and his PhD in civil engineering from the University of California, Berkeley. He is a member of ACI Committees 341, Earthquake-Resistant Concrete Structures; 349, Concrete Nuclear Structures; and 374, Performance-Based Seismic Design of Concrete Structures; and Joint ACI-ASCE Committee 335, Composite and Hybrid Structures. His research interests include probabilistic performance-based seismic design of transportation and energy infrastructure.

ACKNOWLEDGMENTS

These data and findings presented herein stem from the work supported by the California Department of Transportation through Project 04-EQ042 and the Pacific Earthquake Engineering Research (PEER) Center. This support, as well as engineering advice from M. Mahan and C. Whitten of Caltrans and S. Takhirov of the PEER Center, is gratefully acknowledged. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and may not be those of the project sponsors.

REFERENCES 1. Caltrans, “Seismic Design Criteria,” State of California Department of Transportation, Sacramento, CA, 2006. 2. Fenves, G. L., and Ellery, M., “Behavior and Failure Analysis of a Multiple-Frame Highway Bridge in the 1994 Northridge Earthquake,” Report PEER 98/08, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 1998. 3. Arici, Y., and Mosalam, K., “System Identification of Instrumented Bride Systems,” Earthquake Engineering & Structural Dynamics, V. 32, No. 7, 2003, pp. 999-1020. doi: 10.1002/eqe.259 4. Mackie, K., and Stojadinovic, B., “Fragility Basis for California Highway Overpass Bridge Decision Making,” Report PEER 2005/02, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2005. 5. Saiidi, M. S., and Ardakani, S. M., “An Analytical Study of Residual Displacements in RC Bridge Columns Subjected to Near-Fault Earthquakes,” Bridge Structures, V. 8, 2012, pp. 35-45. 6. Terzic, V., and Stojadinovic, B., “Post-Earthquake Traffic Capacity of Modern Bridges in California,” Report PEER 2010/103, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2010. 7. Ketchum, M.; Chang, V.; and Shantz, T., “Influence of Design Ground Motion Level on Highway Bridge Costs,” Report PEER 6D01, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2004. 8. Mackie, K., and Stojadinovic, B., “Fragility Basis for California Highway Overpass Bridge Decision Making,” Report PEER 2005/02, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2005. 9. McKenna, F., and Fenves, G. L., “Open System for Earthquake Engineering Simulation (OpenSees).” Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2004.


10. Lehman, D. E., and Moehle, J. P., “Seismic Performance of Well-Confined Concrete Bridge Columns,” Report PEER 1998/01, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 1998. 11. Lehman, D. E., and Moehle, J. P., “Behavior of Reinforced Concrete Bridge Columns Having Varying Aspect Ratios and Varying Lengths of Confinement,” Report PEER 2000/08, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2000. 12. Hose, Y., and Seible, F., “Performance Evaluation Database for Concrete Bridge Components and Systems under Simulated Seismic Loads,” Report PEER 1999/11, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 1999. 13. ACI Committee 374, “Acceptance Criteria for Moment Frames Based on Structural Testing (ACI 374.1-05) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2006, 9 pp. 14. Krawinkler, H.; Gupta, A.; Medina, R.; and Luco, N., “Loading Histories for Seismic Performance Testing SMRF Components and Assemblies,” Report No. SAC/BD-00/10, SAC Joint Venture, 2000. 15. Taucer, F. F.; Spacone, E.; and Filippou, F. C., “A Fiber BeamColumn Element for Seismic Response Analysis of Reinforced Concrete Structures,” Report No. UCB/EERC-91/17, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 1991. 16. Berry, M. P., and Eberhard, M. O., “Performance Modeling Strategies for Modern Reinforced Concrete Bridge Columns,” Report PEER 2007/07, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2007. 17. Kent, D. C., and Park, R., “Flexural Members with Confined Concrete,” Journal of Structural Engineering, ASCE, V. 97, 1971, pp. 1969-1990. 18. Menegotto, M., and Pinto, P. E., “Method of Analysis for Cyclically Loaded R.C. Plane Frames Including Changes in Geometry and Nonelastic Behaviour of Elements under Combined Normal Force and Bending,” Proceedings of the Symposium on the Resistance and Ultimate Deformability of Structures Acted on by Well Defined Repeated Loads, International Association for Bridge and Structural Engineering, Zurich, Switzerland, 1973, pp. 15-22. 19. Mander, J. B.; Priestley, M. J. N.; and Park, R., “Theoretical Stress-Strain Model for Confined Concrete,” Journal of Structural Engineering, ASCE, V. 114, No. 8, 1988, pp. 1804-1826. doi: 10.1061/ (ASCE)0733-9445(1988)114:8(1804) 20. Mackie, K.; Wong, J. M.; and Stojadinovic, B., “Integrated Probabilistic Performance-Based Evaluation of Benchmark Reinforced Concrete Bridges,” Report PEER 2007/09, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2007. 21. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2005, 430 pp. 22. Caltrans, “Bridge Design Specifications,” State of California Department of Transportation, Sacramento, CA, 2004. 23. Terzic, V., and Stojadinovic, B., “Hybrid Simulation of Bridge Response to Three-Dimensional Earthquake Excitation Followed by a Truck Load,” Journal of Structural Engineering, ASCE, V. 140, 2013. doi: 10.1061/(ASCE)ST.1943-541X.0000913

33

NOTES:

34



TECHNICAL PAPER

Title No. 112-S04

Shear Behavior of Reinforced Concrete Columns with High-Strength Steel and Concrete by Yu-Chen Ou and Dimas P. Kurniawan Eight shear-critical, large-scale, high-strength reinforced concrete columns under low axial load were tested. The specified concrete compressive strengths were 70 and 100 MPa (10,000 and 14,500 psi). The specified yield strengths were 685 and 785 MPa (100,000 and 114,000 psi) for the longitudinal and transverse reinforcement, respectively. Test results show that all the columns exhibited shear failure before longitudinal reinforcement yielding. The transverse reinforcement did not yield at the peak applied load for all the columns. In total, test data for 43 high-strength columns from this study and literature were collected. A comparison to the ACI 318 shear-strength equations show that the ACI simplified shear-strength equation provided conservative estimation for all the columns. However, the ACI detailed shear-strength equation yielded unconservative prediction for 19 columns. This study proposes modifications to the ACI detailed shear-strength equation to address this issue. Keywords: column(s); cyclic loading; double curvature; high-strength concrete; high-strength reinforcement; reinforced concrete.

INTRODUCTION High-strength steel and concrete have gained increasing attention recently in reinforced concrete (RC) buildings due to the need to limit the size of lower-story columns in highrise buildings to increase available floor area. Moreover, the use of high-strength steel reduces reinforcement congestion in the plastic hinge regions in seismic design. With recent advancements in material production technology in Taiwan, deformed reinforcement SD685 with a specified yield strength of 685 MPa (100,000 psi) for longitudinal reinforcement (Fig. 1(a)), deformed reinforcement SD785 with specified yield strength of 785 MPa (114,000 psi) for transverse reinforcement (Fig. 1(b)), and high-strength concrete with a specified compressive strength of 100 MPa (14,500 psi) are commercially available. The SD685 reinforcement has a lower and upper limit on actual yield strength—685 and 785 MPa (100,000 and 114,000 psi), respectively—and a minimum ratio of actual ultimate strength to actual yield strength—1.25, conforming to the ACI 318 seismic design provisions.1 It also has a lower limit of 0.014 on the strain corresponding to a stress equal to the upper limit of yield strength (ensuring a sufficient yield plateau), and a lower limit of 0.1 on elongation (Fig. 1(a)). The SD785 has requirements on minimum yield strength and ultimate strength—785 and 930 MPa (114,000 and 135,000 psi), respectively—and a lower limit of 0.08 on elongation (Fig. 1(b)). Such high-strength materials, when used in columns for shear design, are not allowed by the ACI 318 Code1 to use their full strengths. The yield strength of deformed reinforcing bars for shear design is limited to 420 MPa ACI Structural Journal/January-February 2015

Fig. 1—High-strength reinforcement: (a) SD685; and (b) SD785. (Note: 1 MPa = 145 psi.) (60,900 psi) (ACI 318-11, 11.4.2). Moreover, concrete compressive strength for shear design of columns is limited to 70 MPa (10,000 psi) (ACI 318-11, 11.1.2). Note that this limit can be removed for beams with the minimum web reinforcement (ACI 318-11, 11.1.2.1), but not for columns. The purposes of the yield strength limit in the design of shear reinforcement are to control diagonal crack width and to ensure bar yielding before shear failure.2 Lee et al.,2 who tested 27 beams with fyt of 379 to 750 MPa (55,000 to 109,000 psi), concluded that maximum crack width and fyt are not strongly correlated. Test results for 87 beams ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-134.R1, doi: 10.14359/51686822, was received September 8, 2013, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

35

Table 1—Specimen design

Column

Axial load ratio, %

Transverse reinforcing bar spacing mm (in.)

A-1

Concrete compressive strength, MPa (ksi) fcs′

fc′

70 (10.15)

92.5 (13.41)

100 (14.5)

99.9 (14.49)

70 (10.15)

96.9 (14.05)

100 (14.5)

107.1 (15.53)

70 (10.15)

108.3 (15.71)

100 (14.5)

125.0 (18.13)

70 (10.15)

112.9 (16.37)

100 (14.5)

121.0 (17.55)

Longitudinal reinforcing bar D32 (No. 10) yield strength, MPa (ksi)

Transverse reinforcing bar D13 (No. 4) yield strength, MPa (ksi)

fyls

fyl

fyts

fyt

685 (100)

735 (106.6)

785 (114)

862 (125)

450 (17.72) A-2 10 A-3 260 (10.24) A-4 B-1

15 450 (17.72)

B-2

18

B-3 20 B-4

260 (10.24)

with fyt ranging from 484 to 1454 MPa (70,000 to 211,000 psi)2-4 indicated that beams may fail in shear before shear reinforcement yielding when fyt is very high; for example, fyt = 1454 MPa (211,000 psi), or when used with normalstrength concrete, for example, fyt ≥ 700 MPa (102,000 psi) with fc′ < 40 MPa (5800 psi). Test results for 42 columns with fyt ranging from 846 to 1447 MPa (123,000 to 210,000 psi)3-5 showed that only a few columns had shear failure after shear reinforcement yielding. This suggests that a stress limit should be imposed for shear reinforcement in columns. The limit on fc′ in the ACI 318 Code is due to a lack of test data and practical experience with fc′ ≥ 70 MPa (10,000 psi).1 In total, 42 beams specimens3,6-9 without shear reinforcement and with high-strength concrete with fc′ of 72.8 to 104.2 MPa (10,600 to 15,000 psi) were found in literature. The ACI 318 simplified (Eq. (11-3)) and detailed (Eq. (11-5)) shear-strength equations overestimated measured diagonal cracking strength for two and four specimens, respectively. The overestimation was generally within 10%. When one considers measured ultimate shear strength, only the shear strength of one specimen was overestimated. In total, 35 column specimens3,5,10,11 that were designed with high-strength concrete with fc′ of 73.5 to 130 MPa (10,700 to 19,000 psi) and were reported with diagonal cracking strength were found in literature. Most test results were not evaluated using the ACI 318 shear-strength equations. Moreover, most specimens were small. By testing eight large columns, the objectives of this study are to examine the following: the shear behavior of the columns under low axial load with the high-strength steel and concrete materials mentioned previously; the strains of shear reinforcement at peak column shear; and the applicability of ACI 318 concrete shear-strength equations for high-strength columns. RESEARCH SIGNIFICANCE The use of high-strength materials can decrease the size of lower-story columns in high-rise buildings, increasing available floor area. Additionally, this can also decrease the consumption of aggregate and steel, promoting environmental sustainability. By testing eight large-scale, highstrength concrete columns and by examining test data of 43 high-strength columns, this research studied the applicability of shear-strength equations in the ACI 318 Code on 36

columns with material strengths higher than the code limits. Based on the study results, a modification is proposed for the detailed shear-strength equation of the ACI 318 Code for high-strength columns. EXPERIMENTAL PROGRAM Specimen design Eight large-scale columns with a clear height of 1800 mm (70.87 in.) and a square cross section of 600 x 600 mm (23.62 x 23.62 in.) were tested. Table 1 and Fig. 2 show column design details. The columns were designed to have elastic shear failure; that is, shear failure before longitudinal reinforcement yielding. The columns were reinforced with D32 (No. 10) longitudinal reinforcement and with D13 (No. 4) transverse reinforcement. Each transverse reinforcement layer consisted of a closed hoop with 135-degree hook anchorage and a tie along each principal direction with 90-degree hook at one end and 135-degree hook at the other end. The clear cover to the outer edge of the transverse reinforcement was 40 mm (1.57 in.). The SD685 high-strength deformed bars were used for the longitudinal reinforcement and had specified and actual yield strengths of 685 and 735 MPa (100,000 and 106,600 psi), respectively. The SD785 high-strength deformed bars were used for the transverse reinforcement and had specified and actual yield strengths of 785 and 862 MPa (114,000 and 125,000 psi), respectively. Three study variables—axial load, concrete compressive strength, and amount of transverse reinforcement—were examined. Two levels of axial load ratios—10% (Column A series) and 20% (Column B series)—were examined. The axial load ratio is the ratio of applied axial load to fc′Ag. The value of fc′ was obtained from the average of three 150 x 300 mm (6 x 12 in.) concrete cylinders. Two levels of fcs′—70 and 100 MPa (10,000 and 14,500 psi)—were investigated. Two levels of transverse reinforcement spacing— 450 and 260 mm (17.72 and 10.24 in.)—were studied. Note that actual axial load ratios applied for Specimens B-1 and B-2 were 15% and 18%, respectively, due to the inclusion of relatively low cylinder-strength test results, which were later eliminated when determining actual concrete compressive strength.


Fig. 2—Specimen design: (a) A-1, A-2, B-1 and B-2; (b) A-3, A-4, B-3 and B-4; and (c) cross section. Test setup and loading protocol The columns were tested at the National Center for Research on Earthquake Engineering (NCREE), Taiwan, using the Multi-Axial Testing System (MATS) (Fig. 3), which has maximum axial and lateral load capacities of 60,000 and 7000 kN (13,489 and 1574 kip), respectively. Lateral force was applied by hydraulic actuators placed at the bottom of the MATS using a displacement control loading history (Fig. 4). Axial loading was constant during testing. TEST RESULTS AND DISCUSSION Crack pattern and general behavior Figures 5 and 6 show the lateral force-displacement relationships for Column A series (10% axial load ratio) and Column B series (20% axial load ratio), respectively. The P-Δ effect has been removed from the figures. All specimens exhibited elastic shear failure, where shear failure occurred without any longitudinal reinforcement yielding, as indicated by strain gauge measurements. The Column B series, with higher axial loading, was more brittle. Figure 7 shows the crack patterns of specimens at peak applied load. Flexural cracking appeared first at early drift. Shear cracking initially developed as flexural-shear cracking with an angle of approximately 45 degrees (relative to a column’s longitudinal axis). As the load increased, web-shear cracks appeared. The crack angles decreased as drift increased (Fig. 8). Each point in this figure is the average of dominant diagonal crack angles at that drift. At peak applied load, the average diagonal crack angles (critical diagonal crack angle) were 27, 26, 27, and 24 degrees for Columns A-1, A-2, A-3, and A-4, respectively. They were 25, 21, 20, and 21 degrees for Columns B-1, B-2, B-3, and B-4, respectively. The average values of the critical crack angles were 26 and 22 degrees for Column A and B series, respectively. These angles were smaller than 30 degrees, meaning the commonly used angle ACI Structural Journal/January-February 2015

Fig. 3 —Multi-Axial Testing System (MATS). of 45 degrees is conservative when predicting the shear capacity of the columns. Explosive sounds were typically heard during testing when critical cracks appeared. Figure 9 clearly shows several aggregates were cut through by a crack. This is a typical feature of a failure surface for highstrength concrete. Figure 10 shows the damage distribution of each specimen at test end. Note that Column A-1 testing was terminated when lateral force dropped by more than 50% (Fig. 5(a)), which was earlier than for other specimens, which were tested until the lateral force dropped to nearly zero. Column A series showed shear failure in the middle region of a column, which belongs mainly to the B-region (regions where conventional beam theory applies) as compared to the D-region (regions where conventional beam theory does not apply) at the two ends of a column (extending from each end 37

approximately one member depth). The failure patterns were similar among Column A series. As the axial load increased, the size of the failure region increased as observed from Column B series. Moreover, the failure turned from gradual spalling of concrete to a more brittle, sudden crushing failure of concrete. Columns B-2 and B-3 showed signs of inclined crushing failure (Fig. 10(f) and (g)).

Fig. 4—Loading protocol.

Shear contribution of steel and concrete Figure 11 shows the relationships between maximum stress of transverse reinforcement and column drift for all columns. Transverse reinforcement stress increased slowly in the early drift levels until diagonal cracks appeared. The second column in Table 2 lists the drifts when diagonal crack appeared; the third column lists the corresponding transverse reinforcement stress determined from the regression line obtained from regression analysis of the data points for each column in Fig. 11. The stress levels at diagonal cracking were very small. After the shear cracks appeared, stress increased rapidly. The increase rate of reinforcement stress was higher in columns with high axial load (Column B series) than those with low axial load (Column A series). The sixth column in Table 2 lists drift at maximum applied load; the seventh column lists the corresponding transverse reinforcement stress. Transverse reinforcement did not yield at maximum applied shear for all columns. Note that reinforcement stress increased as drift increased. Because more transverse reinforcement delayed shear failure to a larger drift (Columns A-3 and A-4 compared to Columns A-1 and A-2, and similarly for Column B series), a higher level of transverse reinforcement stress developed. This can be observed by comparing the transverse reinforcement stresses of Columns A-3 and A-4 to those of Columns A-1 and A-2,

Fig. 5—Hysteretic behavior of specimens with 10% axial load ratio: (a) A-1; (b) A-2; (c) A-3; and (d) A-4. 38


Fig. 6—Hysteretic behavior of specimens with 20% axial load ratio: (a) B-1; (b) B-2; (c) B-3; and (d) B-4.

Fig. 7—Crack pattern at the peak applied load for specimens: (a) A-1; (b) A-2; (c) A-3 (d) A-4 (e) B-1; (f) B-2; (g) B-3; and (h) B-4. and by comparing those of Columns B-3 and B-4 to those of Columns B-1 and B-2. Based on the strain measurement of longitudinal reinforcement, when longitudinal reinforcement of Column A and B series reached yielding, drifts were predicted to be approximately 1.6% and 1.85%, respecACI Structural Journal/January-February 2015

tively. Thus, if transverse reinforcement exceeding that in this study is provided, transverse reinforcement may achieve yielding at shear failure before longitudinal reinforcement yielding (Fig. 11).

39

Fig. 8—Drift ratio versus crack angle.

Fig. 9—Crack cutting through aggregates.

Fig. 10—Damage distribution at the end of test: (a) A-1; (b) A-2; (c) A-3 (d) A-4 (e) B-1; (f) B-2; (g) B-3; and (h) B-4. Experimental shear strength provided by transverse reinforcement (Vs_test) was calculated using Eq. (1). In Eq. (1), σst was determined by the stress-drift relationships (Fig. 11), and θ was determined by measuring crack angle (Fig. 8). Experimental shear strength provided by concrete (Vc_test) was calculated using Eq. (2). Two conditions were considered when calculating Vs_test and Vc_test: diagonal cracking and ultimate conditions. The diagonal cracking condition is defined as when a diagonal shear crack first appears. The ultimate condition corresponds to peak applied shear.

Vs _ test =

Av σ st d cot q (1) s

Vc_test = Vtest – Vs_test (2)

Effect of axial load Figure 12 shows a representative relationship between column drift and Vs_test, Vc_test, and Vtest for the effect of axial load. Test results for Columns A-3 and B-3 with axial load ratios of 10% and 20%, respectively, are compared. A higher axial load increased Vc_test and the increase rate for Vs_test. The higher increase rate of Vs_test is due to a smaller shear crack angle under a higher axial load. However, the increase in axial load caused a much more rapid decline in Vc_test after 40

the peak of Vc_test, leading to more brittle behavior. Note that the peak of Vtest may not coincide with the peak of Vc_test (Columns A-3 and B-3) (Fig. 12). Effect of concrete compressive strength Figure 13 shows a representative relationship between column drift and Vs_test, Vc_test, and Vtest to illustrate the effect of compressive strength. Test results for Columns A-3 and A-4 with actual concrete compressive strength of 97 and 107 MPa (14,000 and 15,500 psi), respectively, are compared. Although the two columns were designed with a difference in concrete compressive strength of 30 MPa (4400 psi), the actual difference was only 10 MPa (1500 psi) (Table 1). No significant difference in behavior existed with such a difference in concrete compressive strength (Fig. 13). Effect of amount of transverse reinforcement Figure 14 shows a representative relationship between column drift and Vs_test, Vc_test, and Vtest for the effect of amount of transverse reinforcement. Test results for Columns A-1 and A-3 with transverse reinforcement spacing of 450 and 260 mm (17.72 and 10.24 in.), respectively, are compared. The decrease in transverse reinforcement spacing, that is, an increase in the amount of transverse reinforcement, did not have a significant effect on the peak of Vc_test. However, due to the increase in Vs_test for a given drift, Vtest still increased ACI Structural Journal/January-February 2015

Table 2—Experimental shear strengths contributed by concrete and steel Diagonal cracking shear strength

Ultimate shear capacity

Column

Drift ratio, %

σst, MPa

Vs_test, kN

Vc_test, kN

Drift ratio, %

σst, MPa

Vs_test, kN

Vc_test, kN

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

A-1

0.35

19

9

1255

0.57

243

150

1428

A-2

0.33

6

3

1283

0.53

235

150

1488

A-3

0.32

16

13

1266

0.75

359

413

1359

A-4

0.33

14

10

1288

0.79

418

447

1334

B-1

0.45

18

10

1852

0.59

223

165

1913

B-2

0.41

20

11

1996

0.50

183

195

2103

B-3

0.40

16

17

2081

0.54

214

411

2007

B-4

0.42

18

14

2089

0.64

380

522

2006

Notes: 1 MPa = 145 psi; 1 kN = 0.225 kip.

Fig. 11—Stress of shear reinforcement.

Fig. 13—Drift ratio versus applied shear of Specimens A-3 and A-4. and (5) are detailed equations for Vc, but should not be taken greater than Eq. (6). Equation (7) defines Vs.  Nu  Vc = 0.17  1 +   13.8 Ag 

Fig. 12—Shear strength development of Specimens A-3 and B-3. after the peak of Vc_test, thus increasing the peak of Vtest and the stress in the transverse reinforcement at the peak of Vtest. EXAMINATION OF ACI 318 SHEARSTRENGTH EQUATIONS According to the ACI 318 Code,1 nominal shear strength Vn is contributed by two components: shear strength provided by concrete (Vc) and shear strength provided by shear reinforcement (Vs). In the ACI 318 Code, Eq. (3) is the simplified equation for Vc when axial compression exists. Equations (4)

(3) fc′bw d (psi)

 V d Vc =  0.16 fc′ + 17rw u  bw d Mm  

(MPa)

 V d Vc =  1.9 fc′ + 2500rw u  bw d Mm  

(psi)

M m = Mu − N u

Vc = 0.29 fc′bw d 1 +


 Nu  Vc = 2  1 +   2000 Ag 

fc′bw d (MPa)

(4)

( 4h − d ) (5) 8

0.29 N u Ag

Nu Vc = 3.5 fc′bw d 1 + 500 Ag

(MPa) (6) (pssi) 41

Vc =

Nu ft (max) bw d 1 + (8) F2 ft (max)bw d

Based on beam test data (solid circular dots in Fig. 15), ft(max)/F2 was set as 0.29√fc′ (MPa) or 3.5√fc′ (psi) (horizontal dashed line in Fig. 15). ft(max) was assumed to be 0.62√fc′ (MPa) or 7.5√fc′ (psi). Thus, Eq. (8) becomes Eq. (9). Vc = 0.29 fc′bw d 1 +

1.6 N u fc′bw d

(MPa)

or equal to Fig. 14—Drift ratio versus applied shear of Specimens A-1 and A-3. Mu is taken as moment at distance d from the section of maximum moment when the ratio of shear span to effective depth is greater than 2, or moment at the center of shear span when the ratio of shear span to effective depth is less than 2.12

Vs =

Av f yt d

(7)

s

The ACI concrete shear-strength equations were derived based on shear corresponding to diagonal cracking12 even though they are defined as nominal concrete shear strength in the ACI Code. The experimental concrete shear strength at the diagonal cracking and at the ultimate shear condition (peak applied load) are compared to shear strength predicted by the ACI simplified (Eq. (3)) and detailed (Eq. (4) to (6)) concrete shear-strength equations (Table 3). In the calculation using detailed equations, the values of Mm in Eq. (5) for all columns are negative, meaning that normal tensile stress due to moment effect is small, and diagonal cracking strength should be governed by Eq. (6). Note that when using Eq. (3) to (6), the ACI limit on fc′ (≤ 70 MPa [10,000 psi]) for concrete shear strength was removed. Results of comparison show that Eq. (3) yields conservative predictions for all columns. Equation (6) does not yield conservative estimates of diagonal cracking strength for all columns. Even for ultimate shear strength, Eq. (6) results are not conservative for all columns. This is discussed further in later paragraphs. For shear strength provided by steel reinforcement, Eq. (7) does not yield conservative results for all columns (last column in Table 3). This is expected, as stress in transverse reinforcement was far from yield at the ultimate condition (Table 2). As stated previously, if a higher amount of transverse reinforcement than that used in this study is provided, transverse reinforcement stress at the ultimate condition may be further increased. Further study is needed to examine transverse reinforcement stress when the amount of transverse reinforcement is higher than that used in this research. Equation (6) is based on Eq. (8),12 which was derived based on principal tensile stress at diagonal cracking with an assumption that neglects tensile stress due to moment effect.

42

Vc = 3.5 fc′bw d 1 +

(9) 0.133 N u fc′bw d

(psi)

For simplicity, a constant value of 0.29 (MPa) (0.002 [psi]) replaced 1.6/√fc′ (MPa) (0.133/√fc′ [psi]), which corresponds to fc′ equal to approximately 30 MPa (4400 psi). With this simplification and by approximating bwd as Ag, Eq. (9) becomes Eq. (6). The simplification that sets fc′ equal to 30 MPa (4400 psi) in 1.6/√fc′ (MPa) (0.133/√fc′ [psi]) overestimates mathematically the shear strength of members with fc′ exceeding 30 MPa (4400 psi). The degree of overestimation increases as fc′ increases and can be as high as 28% for the case of 20% axial load ratio and fc′ = 100 MPa (14,500 psi), which are design parameters for Columns B-2 and B-4. To assess the applicability of Eq. (6) for columns with fc′ ≥ 70 MPa (10,000 psi), a test database of 43 high-strength columns from this study and literature3,5,10,11 was established (Table 4). The database includes only columns with fc′ ≥ 70 MPa (10,000 psi) and with data for diagonal cracking strength available. The values of Mm (Eq. (5)) for all 43 columns are negative. Therefore, the shear strength of the columns is governed by Eq. (6), not Eq. (4), for the detailed shear-strength calculation. The eighth column in Table 4 shows the outcome of comparing the measured diagonal cracking strength to predictions by Eq. (3). Predictions are conservative for all columns. On the other hand, predictions by Eq. (6) for 23 columns are not conservative (ninth column in Table 4). If Eq. (9) is used instead of Eq. (6), the number of unconservative results is reduced to 6 (tenth column in Table 4). To investigate the applicability of using ft(max)/F2 equal to 0.29√fc′ (MPa) (3.5√fc′ [psi]) in Eq. (8) for high-strength columns, the values of ft(max)/F2 for all the 43 columns were back-calculated by substituting the measured diagonal cracking strength for Vc in Eq. (8) and solving for ft(max)/F2. Figure 15 shows calculation results (data points bounded by dashed lines). Because the Mm values for all columns are negative, an infinity value was assumed for the horizontal coordinate of the data points of the columns in Fig. 15. Columns from different studies were separated into different groups in Fig. 15 for comparison. Comparison of the distribution of the column data points and that of the beam data points against the horizontal dashed line reveals that ft(max)/F2 equal to 0.29√fc′ (MPa) (3.5√fc′ [psi]) ACI Structural Journal/January-February 2015

Table 3—Ratio of test results to shear-strength prediction using ACI 318 without strength limitation Diagonal cracking shear strength Vc _ test

Vc _ test

Vc _ test

Vc _ test

Vs _ test

Column

VEq (3)

VEq (6 )

VEq (3)

VEq (6 )

VEq ( 7 )

(1)

(2)

(3)

(4)

(5)

(6)

A-1

1.63

0.81

1.86

0.93

0.43

A-2

1.51

0.76

1.75

0.88

0.43

A-3

1.58

0.79

1.70

0.85

0.68

A-4

1.47

0.74

1.52

0.76

0.74

B-1

1.72

0.89

1.77

0.92

0.47

B-2

1.41

0.77

1.48

0.82

0.56

B-3

1.55

0.85

1.50

0.82

0.68

B-4

1.44

0.80

1.39

0.77

0.86

provides predictions of diagonal cracking strength for highstrength columns as reasonable as those for the beams. Note that the original intention of ft(max)/F2 equal to 0.29√fc′ (MPa) (3.5√fc′ [psi]) is to capture the average behavior of beam data points in the right side of Fig. 15, rather than to provide a conservative estimation, because beams in this category typically have ultimate concrete shear strength that is significantly higher than diagonal cracking strength. However, columns carry axial load, which generally delays diagonal cracking, reducing the difference between diagonal cracking strength and ultimate concrete shear strength (Table 2). Therefore, it is suggested to lower ft(max)/F2 to 0.25√fc′ (MPa) (3√fc′ [psi])—nearly the lower bound of the column data points in Fig. 15. The new equation is Eq. (10). The eleventh column in Table 4 shows the predictions using Eq. (10); conservative results were obtained for all columns except for Column A-4, which has a ratio of measured to predicted diagonal cracking strength of 0.99.

Vc = 0.25 fc′bw d 1 +

1.6 N u fc′bw d

(MPa)

or equal to Vc = 3 fc′bw d 1 +

Ultimate shear capacity

(10)

0.133N u fc′bw d

(psi)

A comparison of columns from different studies (Table 4) shows that columns tested in this study have lower diagonal cracking strengths than those in literature. One possible reason is the size effect. The diameter of the columns tested in literature ranges from 200 to 400 mm (7.87 to 15.75 in.), much smaller than that in this study. A large member size generally has low strength due to the inclusion of more places of imperfection than in a small member. CONCLUSIONS An experimental study on eight large-scale columns with high-strength steel and concrete was carried out using double-curvature cyclic loading. Test results are presented and compared to predictions by the ACI shear-strength equaACI Structural Journal/January-February 2015

Fig. 15—Relationship between test data and Eq. (4) and (6). tions. The conclusions drawn from this study are summarized as follows. 1. All columns tested exhibited shear failure before longitudinal reinforcement yielding. The critical diagonal shear crack angles with respect to the column longitudinal axis were, on average, 26 and 22 degrees for columns with 10% and 20% axial load ratios, respectively. Axial load increased concrete shear strength in the range of axial load examined, but also increased brittleness. The amount of transverse reinforcement did not significantly affect concrete shear strength. Transverse reinforcement stress increased with drift. An increased amount of transverse reinforcement delayed shear failure to a higher drift, resulting in a higher level of stress in the transverse reinforcement at shear failure. With the amount of transverse reinforcement used in this study, the transverse reinforcement did not yield at peak applied load for all columns tested. 2. Predictions by the ACI simplified and detailed concrete shear-strength equations were compared to test results of 43 columns from this study and the literature. Comparison results showed that the ACI simplified shear-strength equation yielded a conservative estimate of diagonal cracking strength for all columns. On the other hand, the ACI detailed 43

Table 4—High-strength columns test data

Column

bw, mm

d, mm

fc′, MPa

Axial load ratio

a/d

ρt, %

Vc _ test

Vc _ test

Vc _ test

Vc _ test

VEq (3)

VEq (6 )

VEq ( 9 )

VEq (10 )

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

C13

400

320

93.5

0.39

1.25

0.00

1.83

1.14

1.32

1.54

C23

400

320

93.5

0.39

1.25

0.16

1.67

1.04

1.21

1.40

C43

400

320

77

0.24

1.25

0.40

2.33

1.25

1.39

1.61

3

C5

400

320

93.5

0.39

1.25

0.40

1.74

1.08

1.26

1.46

C63

400

320

77

0.48

1.25

0.40

1.96

1.22

1.36

1.58

3

C7

400

320

93.5

0.39

1.25

0.62

1.90

1.18

1.37

1.59

C83

400

320

93.5

0.39

1.25

0.80

1.92

1.20

1.39

1.61

H-0.6-0.15

200

160

128

0.15

1.25

0.60

1.85

1.00

1.23

1.42

H-0.6-0.35

200

160

125

0.3

1.25

0.60

1.64

1.03

1.27

1.48

5

5

H-0.6-0.6

200

160

120

0.6

1.25

0.60

1.27

0.98

1.22

1.41

HS-0.6-0.35

200

160

128

0.3

1.25

0.60

1.74

1.09

1.37

1.58

5

HS-0.6-0.6

200

160

128

0.6

1.25

0.60

1.11

0.88

1.11

1.29

HS-1.2-0.65

200

160

129

0.6

1.25

1.20

1.32

1.04

1.32

1.53

5

H-0.3-0.6

200

160

128

0.6

1.25

0.30

1.08

0.85

1.08

1.25

H-1.2-0.65

200

160

121

0.6

1.25

1.20

1.50

1.16

1.44

1.67

5

H-1.8-0.6

200

160

130

0.6

1.25

1.80

1.44

1.14

1.45

1.68

H-0.3-0.35

200

160

130

0.3

1.25

0.30

1.68

1.06

1.33

1.54

5

H-1.2-0.3

200

160

121

0.3

1.25

1.20

1.65

1.02

1.26

1.46

H-1.8-0.35

200

160

121

0.3

1.25

1.80

2.04

1.27

1.56

1.81

U-0.4-0.65

200

160

130

0.6

1.25

0.40

1.17

0.93

1.18

1.37

5

U-0.7-0.6

200

160

129

0.6

1.25

0.70

1.25

0.99

1.26

1.46

C6110

400

320

113.8

0.17

1.88

1.19

1.62

0.88

1.05

1.22

10

C62

400

320

113.8

0.17

1.88

0.53

1.73

0.94

1.12

1.30

C6310

400

320

113.8

0.17

1.88

1.19

1.59

0.86

1.03

1.20

10

C31

400

320

113.8

0.33

1.88

1.19

1.66

1.04

1.27

1.47

C3210

400

320

113.8

0.33

1.88

0.53

1.52

0.96

1.16

1.35

10

C33

400

320

113.8

0.33

1.88

1.19

1.60

1.00

1.22

1.42

6-111

300

240

73.5

0.17

1.88

0.53

1.78

0.92

1.00

1.15

6-211

300

240

73.5

0.17

1.88

1.19

2.16

1.11

1.21

1.40

6-311

300

240

73.5

0.17

1.88

0.53

1.78

0.92

1.00

1.15

11

6-4

300

240

73.5

0.17

1.88

1.19

2.16

1.11

1.21

1.40

3-111

300

240

73.5

0.33

1.88

0.53

1.65

0.93

1.02

1.19

11

3-2

300

240

73.5

0.33

1.88

1.19

1.73

0.98

1.08

1.25

3-311

300

240

73.5

0.33

1.88

0.53

1.65

0.93

1.02

1.19

11

3-4

300

240

73.5

0.33

1.88

1.19

1.73

0.98

1.08

1.25

A-1

600

480

93

0.1

1.88

0.16

1.61

0.81

0.91

1.06

A-2

600

480

103

0.1

1.88

0.16

1.49

0.76

0.87

1.01

A-3

600

480

97

0.1

1.88

0.28

1.55

0.79

0.89

1.04

A-4

600

480

107

0.1

1.88

0.28

1.44

0.74

0.85

0.99

B-1

600

480

108

0.15

1.88

0.16

1.69

0.89

1.05

1.22

B-2

600

480

125

0.18

1.88

0.16

1.39

0.77

0.95

1.10

B-3

600

480

113

0.2

1.88

0.28

1.53

0.85

1.02

1.19

B-4

600

480

121

0.2

1.88

0.28

1.42

0.80

0.98

1.13

shear strength upper-bound equation (Eq. (6)) produced unconservative predictions of diagonal cracking strength for 23 columns, including all columns tested in this study. If 44

Eq. (9) was used instead of Eq. (6), the number of columns with unconservative prediction was reduced to 6. Furthermore, because the difference between diagonal cracking ACI Structural Journal/January-February 2015

strength and ultimate concrete shear strength for highstrength columns can be insignificant, it is suggested to base the concrete shear-strength equation on the lower-bound of column test data and, hence, Eq. (10) is recommended for high-strength columns. AUTHOR BIOS ACI member Yu-Chen Ou is an Associate Professor in the Department of Civil and Construction Engineering at the National Taiwan University of Science and Technology, Taipei, Taiwan. He received his PhD from the State University of New York at Buffalo, Buffalo, NY. He is the Vice President of ACI Taiwan Chapter. His research interests include reinforced concrete structures, steel-reinforced concrete structures, and earthquake engineering. ACI member Dimas P. Kurniawan is a Research Assistant in the Department of Civil and Construction Engineering at the National Taiwan University of Science and Technology. He received his BS from Bandung Institute of Technology, Indonesia, and MS from the National Taiwan University of Science and Technology.

ACKNOWLEDGMENTS The authors would like to thank National Center for Research on Earthquake Engineering (NCREE), Taiwan, and the Excellence Research Program of National Taiwan University of Science and Technology for their financial support.

NOTATION Ag Av a bw d F2

= = = = = =

fc′ = fcs′ = ft(max) = fu = fy = fyl = fyls = fyt = fyts = M m = Mu = Nu = s = Vc =

gross area of concrete cross section total cross-sectional area of shear reinforcement shear span effective web width effective depth ratio of shear stress at cracking point to average shear stress of effective cross section concrete compressive strength specified concrete compressive strength concrete principal tensile strength ultimate strength of steel yield strength of steel yield strength of longitudinal reinforcement specified yield strength of longitudinal reinforcement yield strength of shear reinforcement specified yield strength of shear reinforcement applied moment modified to consider effect of axial compression applied moment applied axial load (positive in compression) spacing of shear reinforcement nominal shear strength provided by concrete


Vc_test Vn Vs Vs_test Vtest Vu εu θ ρt ρw σst

= = = = = = = = = = =

experimental shear strength provided by concrete nominal shear strength nominal shear strength provided by shear reinforcement experimental shear strength provided by shear reinforcement experimental shear strength applied shear ultimate strain of steel shear crack angle to column longitudinal axis, deg transverse reinforcement ratio longitudinal tension reinforcement ratio shear reinforcement stress

REFERENCES 1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2011, 503 pp. 2. Lee, J. Y.; Choi, I. J.; and Kim, S. W., “Shear Behavior of Reinforced Concrete Beams with High-Strength Stirrups,” ACI Structural Journal, V. 108, No. 5, Sept.-Oct. 2011, pp. 620-629. 3. Sakaguchi, N.; Yamanobe, K.; Kitada, Y.; Kawachi, T.; and Koda, S., “Shear Strength of High-Strength Concrete Members,” Second International Symposium on High-Strength Concrete, SP-121, W. T. Hester, ed., American Concrete Institute, Farmington Hills, MI, 1990, pp. 155-178. 4. Watanabe, F., and Kabeyasawa, T., “Shear Strength of RC Members with High-Strength Concrete,” High-Strength Concrete in Seismic Regions, SP-176, C. W. French and M. E. Kreger, eds., American Concrete Institute, Farmington Hills, MI, 1998, pp. 379-396. 5. Maruta, M., “Shear Capacity of Reinforced Concrete Column Using High Strength Concrete,” Invited Lecture in the 8th International Symposium on Utilization of High-Strength and High-Performance Concrete, Tokyo, Japan, Oct. 27-29, 2008. 6. Mphonde, A. G., and Frantz, G. C., “Shear Tests of High- and Low-Strength Concrete Beams Without Stirrups,” ACI Journal, V. 81, No. 4, July-Aug. 1984, pp. 350-357. 7. Ahmad, S. H.; Khaloo, A. R.; and Poveda, A., “Shear Capacity of Reinforced High-Strength Concrete Beams,” ACI Journal, V. 83, No. 2, Mar.-Apr. 1986, pp. 297-305. 8. Thorenfeldt, E., and Drangsholt, G., “Shear Capacity of Reinforced High-Strength Concrete Beams,” Second International Symposium on High-Strength Concrete, SP-121, W. T. Hester, ed., American Concrete Institute, Farmington Hills, MI, 1990, pp. 129-154. 9. Xie, Y.; Ahmad, S. H.; Yu, T.; Hino, S.; and Chung, W., “Shear Ductility of Reinforced Concrete Beams of Normal and High-Strength Concrete,” ACI Structural Journal, V. 91, No. 2, Mar.-Apr. 1994, pp. 140-149. 10. Kuramoto, H., and Minami, K., “Experiments on the Shear Strength of Ultra-High Strength Reinforced Concrete Columns,” Proceedings of the Tenth World Conference on Earthquake Engineering, Madrid, Spain, July 1992, pp. 3001-3006. 11. Aoyama, H., Design of Modern High-Rise Reinforced Concrete Structures, Imperial College Press, London, UK, 2001, 391 pp. 12. ACI-ASCE Committee 326, “Shear and Diagonal Tension,” ACI Journal Proceedings, V. 59, No. 2, Feb. 1962, pp. 1-124.

45

NOTES:

46



TECHNICAL PAPER

Title No. 112-S05

Three-Parameter Kinematic Theory for Shear Behavior of Continuous Deep Beams by Boyan I. Mihaylov, Bradley Hunt, Evan C. Bentz, and Michael P. Collins This paper presents a theory for predicting shear strengths, deformations, and crack widths near failure of continuous deep beams. To describe the deformations in continuous deep beams, the theory uses a three-degree-of-freedom kinematic model (3PKT), which is an extension of an earlier two-degree-of-freedom model (2PKT) for members in single curvature. The extended model is validated with the help of measured local and global deformations taken during loading to failure of a large continuous deep beam. The accuracy of the shear-strength predictions given by the theory is evaluated using a database of 129 published tests of continuous deep beams. The theory enables the load distribution and failure loads of continuous deep beams subject to differential settlement to be evaluated. Keywords: deep beams; differential settlement; kinematics; redistribution; shear strength; ultimate deformations.

INTRODUCTION While continuous reinforced concrete deep beams, such as the transfer girder shown in Fig. 1(a), perform more critical load-carrying functions than slender beams, their safety is more difficult to assess. Shear forces in such members are more sensitive to differential settlement of footings and because longitudinal strains vary nonlinearly over beam depth, traditional design procedures for slender beams are not appropriate. The ACI Building Code1 suggests that either the nonlinear distribution of longitudinal strains be taken into account or that strut-and-tie models be used. While finite element programs, such as VecTor2,2 which was used to produce Fig. 1(b), account both for nonlinear distributions of strain and nonlinear material response and can predict both failure loads and deformations, their use requires considerable engineering time and expertise. Strut-and-tie models (Fig. 1(c)), on the other hand, by approximating regions of high compressive stress in concrete and high tensile stress in reinforcement, can usually provide conservative estimates of strength after a few relatively simple calculations. As these models concentrate on statics, they do not provide accurate assessments of deformation patterns close to failure and post-peak behavior. This also applies for most analytical models for deep beams in the literature3 with exceptions focusing entirely on deformation capacity.4 Information on ultimate deformations can be critical in, for example, evaluating the safety of transfer girders damaged by earthquakes or by large differential settlements. It is the purpose of this paper to present a kinematic model (Fig. 1(d)) capable of predicting both strength and deformation patterns near failure of reinforced concrete continuous deep beams. Within each shear span this model uses just three parameters: the average tensile strain in the ACI Structural Journal/January-February 2015

Fig. 1—Modeling of transfer girder. top longitudinal reinforcement, the average tensile strain in the bottom reinforcement, and the vertical deformation of the critical loading zone (CLZ). This three-parameter kinematic theory (3PKT) is an extension to the two-parameter kinematic theory (2PKT) proposed by Mihaylov at el.5,6 for simply supported deep beams. With a relatively small number of hand calculations, the 3PKT can be used to determine the shear failure load, crack widths, deformed shape, and support reactions (accounting for differential settlements) of a continuous deep beam near failure. As part ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-150.R1, doi: 10.14359/51687180, was received March 24, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

47

Fig. 2—Test specimen CDB1 and instrumentation. (Note: 1 mm = 0.0394 in.; 1 kN = 0.225 kip; 1 MPa = 145 psi = 0.145 ksi.)

Fig. 3—Measuring deformation patterns at LS9. of the development of the 3PKT, detailed measurements of deformations were taken on a large beam tested at the University of Toronto, and results of this test will be used to introduce deformation patterns of continuous deep beams. RESEARCH SIGNIFICANCE The novel three-parameter kinematic theory presented in this paper enables engineers to accurately and efficiently evaluate the shear strength and deformation capacity of continuous deep beams such as the transfer girders. The effect of support settlements on the safety of transfer girders can also be assessed using the method. An example demonstrates that transfer girders can be sensitive to support settlements. EXPERIMENTAL PROGRAM A large continuous reinforced concrete deep beam (refer to Fig. 2) was loaded to failure to investigate deformation patterns. The beam, called CDB1, was a one-third-scale model based on the transfer girder in Fig. 1. It had a rectangular cross section with a depth of 1200 mm (47.2 in.), 48

a width of 300 mm (11.8 in.), and two equal spans of 3475 mm (11 ft 5 in.). Both top and bottom longitudinal reinforcement consisted of four 25M bars and one 35M bar. The transverse reinforcement consisted of No. 3 stirrups spaced at 235 mm (9.25 in.) that provided a reinforcement ratio of 0.20%. The compressive strength of the concrete when the beam was tested 80 days after casting was 29.7 MPa (4300 psi). Equal point loads, P, were applied near the middle of each span using a spreader beam which was, in turn, loaded by a manually controlled testing machine (Fig. 3). The specimen incorporated short columns to provide realistic loading and support conditions. The middle support was pinned while the external supports and the two loading columns were provided with rollers to allow for free movement of the specimen. Load cells were placed under the external support columns. Statics requires that the two external support reactions be equal and hence the shears, Vext, in the two external shear spans must also be equal. As the two applied loads, P, are equal, the shears in the interior spans, Vint, also equal each other. However, the ratio Vint/P is not dictated by statics but depends on compatibility of deformations, kinematics. Figure 2 also shows the layout of linearly variable differential transformers (LVDTs) attached to the north, bottom, and top faces of the specimen to measure relative displacements on the concrete surface, as well as the locations of electric resistance strain gauges used to monitor the strains in the longitudinal bars and the stirrups. The south side of the specimen was equipped with a 300 x 300 mm (11.8 x 11.8 in.) grid of targets for demountable displacement transducers, which were used to measure displacement patterns at a number of stages during the test (refer to Fig. 3). At such load stages, LS, crack patterns were marked and crack widths measured with crack comparators. After the formation of major diagonal cracks, crack slip displacements parallel to the cracks were also measured with crack comparators.


EXPERIMENTAL RESULTS The measured load-displacement response of the specimen is shown in Fig. 4. The thick lines show the load P, while the thin lines plot the shear force Vint. Midspan deflections of the two spans were measured by LVDTs 2E and 2W with respect to a steel bar going between the ends of the specimen (refer to Fig. 2). The test was performed in two phases. First the beam was loaded until there was clear evidence of shear failure in one of the spans (refer to Fig. 4(a)). After strengthening the failed shear span with four external clamps, the second phase involved reloading until the second span failed (refer to Fig. 4(b)). Shown in Fig. 4 are four key load stages (LS5, LS7, LS9, and LS11) where crack widths and deformation patterns were measured and these are shown in Fig. 5. In Phase I, first flexural cracking was observed under the east load at P = 422 kN (95 kip), while at P = 546 kN (123 kip), cracks under the west load and over the central column were observed. By LS5, the crack pattern resembled three fans with cracks radiating out from the west load, the central support, and the east load (refer to Fig. 5). The first major diagonal crack occurred in the east inner shear span at P = 1033 kN (232 kip). By LS7 there were diagonal cracks up to 2.5 mm (0.010 in.) wide across the inner shear spans. The maximum value of Vint in the east span was reached just after LS7 and had a value of 813 kN (183 kip) with an east midspan deflection of 3.70 mm (0.146 in.). At the maximum shear, the inner shear to applied load ratio, Vint/P, was 0.619, as opposed to the calculated value of 0.675 from flexural theory ignoring shear strains. By LS9, the 5% increase in applied force caused the east displacement to more than double. While the load P increased, the shear, Vint, decreased by 7.5%, causing Vext to increase by 25%. Concrete crushing was visible near the inner faces of the east load and central support and large displacements (9.28 mm [0.37 in.]) were measured for LVDTs 11E and 13E, indicating post-peak behavior of the CLZ regions (refer to Fig. 2). Because of concerns with irreparably damaging the specimen, the loads were reduced to approximately 200 kN (45 kip) after LS9 and external clamps were applied to the east inner shear span. At the start of Phase II loading, there was a gap of approximately 1 mm (0.04 in.) between the bottom of the central column and the support, which closed upon reloading. The load-deformation response (refer to Fig. 4(b)) was nearly linear up to the peak load of 1604 kN (361 kip) at LS11. The maximum shear of 916 kN (206 kip) also occurred at LS11 at a west deflection of 4.23 mm (0.167 in.). Crushing of the concrete at the top and bottom ends of the critical crack was observed as deformations increased between LS11 and LS13. The test was terminated at a deflection of approximately 18.5 mm (0.73 in.) after a sudden drop of resistance caused by rupture of stirrups in the west inner shear span. A photograph of the west side of the beam taken after the test is shown in Fig. 6. Note the external clamps on the east inner shear span and the critical diagonal crack in the west inner shear span. Based on Vint/P = 0.675 (the elastic value), the moment over the central support should be 1.17 times the moment under the load. Figure 7(a) shows the elastic bending moment diagram for LS5 along with the moment diagrams calculated ACI Structural Journal/January-February 2015

Fig. 4—Measured load-displacement response. (Note: 1 mm = 0.0394 in.; 1 MPa = 145 psi.) from the observed Vint/P ratios for five selected load stages. Load redistribution had a significant effect on the magnitude of the moment over the central support. For example, from LS5 to LS9, the applied loads increased by 37% but the moment over the central support reduced by 39%. At the end of the test, LS13, the moment over the central column had changed sign because the large deformations in the west inner shear span had significantly reduced its shear resistance. Figure 7(b) shows the distribution of strains in the longitudinal reinforcement for the five load stages. It can be seen that, due to redistribution of bending moments, the bottom bars were on the verge of yielding at LS11, the peak load, and had yielded by LS13. After LS5, the measured strains in the bottom bars were tensile along the whole length, while in the top bars they were tensile along almost all of the inner shear spans. Figure 7(c) shows the distribution of transverse strains along the length of the beam as measured by strain gauges on the stirrups and LVDTs mounted across the beam depth. It can be seen that the strains in the outer shear spans remained below yield strain. The central regions of the inner shear spans saw high transverse strains prior to

49

Fig. 5—Measured cracks and deformation patterns (maximum deflection equivalent to 1/25 of the span). (Note: 1 mm = 0.0394 in.; 1 kN = 0.225 kip.) failure but the regions near the loads displayed only small transverse strains. Apart from giving information on crack patterns, crack widths, and crack slips, Fig. 5 also provides detailed measured displacement patterns at four load stages. These deformed shapes were calculated from the readings of the demountable displacement transducers7 and are magnified so that for each diagram the maximum beam deflection plots 50

as if it was 1/25 of the span. It can be seen that at LS5 the deformations are primarily flexural, curving down under the loads and up over the central support. At LS7, significant shear deformations caused by the diagonal crack can be seen in the east span near the inner edge of the central support. By LS9, these shear deformations had greatly increased due to the opening and slipping of the critical diagonal crack. As these large displacements developed, the east part of ACI Structural Journal/January-February 2015

Fig. 6—West span of specimen after failure. the span rotated counterclockwise to comply with the east support. This almost-rigid body motion is associated with a constant slip displacement along the critical diagonal crack and crack widths that increase linearly from the top to the bottom of the crack. A confirmation of this kinematic pattern is provided by the crack measurements at LS9. The red numbers show that the crack slip was on average approximately 5 mm (0.197 in.) while the maximum crack width of 8.5 mm (0.335 in.) was measured near the bottom of the beam. The deformation pattern for LS11, corresponding to maximum shear in the west inner shear span, is dominated by the residual deformations in the clamped east shear span. However, it is evident that at this stage significant shear deformations are occurring near the top of the critical diagonal crack in the west inner shear span. Because of safety concerns, detailed displacement patterns were not measured for LS13. KINEMATICS OF SHEAR SPANS UNDER DOUBLE CURVATURE Predicting deformation patterns such as those shown in Fig. 5 is necessary to understand and predict the shear behavior of continuous deep beams. 2PKT has already been presented5,6 for the deformations in external shear spans subject to only single-curvature bending. This model uses a superposition of two deformation patterns, each of which depends on just a single kinematic parameter (degree of freedom) (refer to Diagrams 1 and 2 in Fig. 8). The first pattern involves the elongation of the bottom longitudinal reinforcement or tie (average strain et1,avg) while the second pattern involves the transverse displacement, Dc, of the critical loading zone (CLZ). The modeling of the inner shear spans of continuous girders requires a third pattern, which involves the elongation of the top longitudinal reinforcement (average strain et2,avg) (refer to Diagram 3 in Fig. 8). The complete kinematic model for the general case of double curvature is shown in Box 1 of Fig. 9. In this figure, the first and the third patterns from Fig. 8 are combined into a single deformation pattern related to the elongation of the top and bottom longitudinal reinforcement. This combined pattern ACI Structural Journal/January-February 2015

Fig. 7—Measured response at five load stages. (Note: 1 mm = 0.0394 in.) consists of two “fans” of rigid radial struts pinned together at the loading and support points and connected to the bottom and top longitudinal ties, respectively. Two main diagonal cracks appear in this pattern: one runs from the inner edge of the bottom support to the top load at the location where the shear force is zero or a minimum. The other crack extends from the inner edge of the loading element to the bottom of the section at the support where the shear is zero. Only the more critical of these two cracks develops the transverse displacement Dc. If deformations et1,avg, et2,avg, and Dc are known, the complete displacement field of the shear span can be obtained from Eq. (8) to (11). These equations are the same as those of the 2PKT,5,6 except that Eq. (10) and (11) contain an extra term accounting for the radial cracks above the critical crack. The 51

accuracy of Eq. (8) to (11) is evaluated in Fig. 5 by using 10 measured values of the degrees of freedom to predict the complete deformed shape of the two-span beam at each of the four load stages. LVDTs 7E/7W to 9E/9W measure the average strains in the longitudinal ties and LVDTs 11E/11W and 13E/13W measure the transverse displacements in the critical loading zones of the inner shear spans (refer to Fig. 2). It can be seen from Fig. 5 that the green circles indicating the predicted locations of the 90 targets of the grid match very well the vertices of the triangles representing the measured location of these targets. Note that at LS5 the major diagonal cracks going from the loads to the central support have not yet formed and hence the Dc values are zero and the predicted deformed shape is flexural. Major diagonal cracks have formed in both the east and the west spans by LS7 with the measured Dc values and resulting shear deformations in the east being approximately twice those in the west. The post east span shear failure LS9 involved Dc values approximately Fig. 8—Three degrees of freedom of kinematic model.

Fig. 9—3PKT for deep beams under single- and double-curvature bending.17 52


10 times larger than those at LS7 but the deformation patterns, which are scaled to the same maximum deflection, look very similar. At maximum shear in the west inner span, LS11, the shear deformations near the top face by the load are prominent. By this stage, the east span had been reinforced with external clamps, which required removal of the LVDTs measuring the kinematic parameters required for calculating the predicted deformation pattern. From Fig. 5, it can be seen that if the three kinematic parameters for each shear span are known, then the complete deformed shape of continuous deep beams can be determined with reasonable accuracy. The three equations in Box 1 of Fig. 9 enable the three kinematic parameters for a shear span to be calculated from the end moments and geometric properties of the shear span. The transverse displacement of the critical loading zone at shear failure, Dc, is calculated as in the 2PKT for single curvature6 but with an additional coefficient kc. This coefficient accounts for the decreased deformation capacity of the CLZ due to the tensile strains et2,min in the top reinforcement. This compression softening effect8 does not occur in members under single curvature because in such members the top zone remains uncracked. In Eq. (3), e1 is the principal tensile strain in the CLZ estimated from compatibility of deformations. CALCULATION OF SHEAR STRENGTH The shear strength of a deep beam can be predicted when equations for the degrees of freedom of the kinematic model are combined with equations for geometry of the model, equations for compatibility of deformations, and constitutive relationships for the components of shear resistance (refer to Fig. 9). The derivation of the basic equations has been discussed in detail elsewhere.6 The procedure will be illustrated by giving the key steps required to calculate the shear resistance of beam CDB1 assuming that Vint/P has the measured value at east span shear failure of 0.619. The effective length of load, lb1e, transmitting shear to the inner shear span (refer to Eq. (4)) is 186 mm (7.32 in.) and hence the angle, a1, of the critical crack going from the inner face of the support to this zero shear location is 35.8 degrees. The dowel length, lk, for the 25M bars from Eq. (5) is 220 mm (8.7 in.), while for the 35M bar, it is 260 mm (10.2 in.). The final geometric term required is the effective area of the stirrups in the shear span, which from Eq. (7), is 606 mm2 (0.939 in.2) if a weighted average of 233 mm (9.17 in.) is used for the reinforcing bar dowel length. Shear-strength calculations commence with an estimate of the shear failure load—for example, 850 kN (191 kip). At this load, M1 = 889 kNm (656 ft-kip) and the average strain in the bottom longitudinal reinforcement from Eq. (1) is 1.505 × 10–3. The moment M2 = 619 kNm (457 ft-kip) and the average strain in the top reinforcement from Eq. (2) is 1.048 × 10–3. The minimum strains, near the support for the bottom bars and near the load for the top, are taken as 75% of these values. From the minimum tensile top bar strain near the load and the angle of the critical crack, the compression softening factor, kc, is calculated as 0.840 and then Dc from Eq. (3) is 2.27 mm (0.089 in.).


From Eq. (14), the shear carried by the critical loading zone, VCLZ, is calculated as 346 kN (77.9 kip). The shear carried across the crack interface, Vci, depends on the crack width, which from Eq. (12) is 2.07 mm (0.081 in.), resulting in a Vci value from Eq. (16) of 164 kN (36.9 kip). Applying the dowel action expression, Eq. (15), separately to the four 25M bars and the one 35M bar and summing the resistances gives a Vd value of 45 kN (10.1 kip). The final shear-strength component, Vs, depends on the strain in the stirrups, which is calculated from Eq. (13) to be 3.11 × 10–3. As this exceeds the yield strain, the effective stirrups are at yield stress and so they resist a shear of 490 x 606, which is 297 kN (66.8 kip). The calculated shear resistance against failure on this critical crack is thus 346 + 164 + 45 + 297 = 852 kN (192 kip). Because the calculated resistance agrees closely with the initial estimate, this value can be taken as converged. The equations in Fig. 9 have been formulated for the case where the diagonal crack at the inner edge of the bottom support is opening at failure. To check whether this crack is indeed critical, calculations need to be also performed for the case where the diagonal crack at the inner face of the top load opens at failure. That is, the CLZ will now be at the bottom support while the dowel action in the longitudinal bars will now occur near the top load. The effective length of the support lb2e resisting the shear is 150 mm (5.91 in.) and the critical crack thus goes from the center of the supporting column to the inner face of the upper column, which gives an angle a1 of 36.4 degrees. This small change in slope causes the dowel length for the 25M bars to change to 216 mm (8.50 in.) but does not change the dowel length for the 35M bar. Using the reduced effective bearing length and the weighted average dowel length of 231 mm (9.09 in.), the calculated value for Av from Eq. (7) increases to 620 mm2 (0.961 in.2). If the initial estimate of the shear resistance is again taken as 850 kN (191 kip), then the calculated strains in the top and bottom reinforcing bars will stay the same. However, as the CLZ is now at the bottom, it is the minimum strain in the bottom bars, 0.75 × 1.505 × 10–3, which governs the compression softening factor kc, giving it a value of 0.744. This smaller value combined with the smaller effective bearing length means that Dc is reduced to 1.59 mm (0.063 in.) and VCLZ is reduced to 254 kN (57 kip). The calculated crack width, 1.43 mm (0.056 in.), due to the opening of the critical crack near the inner face of the top loading area, is a function of Dc and of the minimum strain in the top bars (0.75 × 1.048 × 10–3). The resulting value of Vci is 221 kN (49.8 kip). The dowel action component, Vd, which has a value of 55 kN (12.3 kip), is also a function of the minimum strain in the top bars. The stirrup component, Vs, on the other hand, is a function of the average strain in the top bars (1.048 × 10–3) and has a value of 266 kN (59.9 kip). The calculated shear resistance against opening of the diagonal crack at the inner face of the load is thus 254 + 221 + 55 + 266 = 796 kN (179 kip). Repeating the calculations for an estimated failure shear of 800 kN (180 kip) gives a calculated resistance of 802 kN (180 kip), which can be taken as the converged value. This is the predicted critical failure shear for the member, as it is approximately 6% smaller than the shear required to open 53

Fig. 11—Deformations in continuous deep girders.

Fig. 10—Comparison between 3PKT and ACI shear-strength predictions for 129 tests of continuous beams. the diagonal crack at the inner edge of the support. Note that the measured shear force at first failure for this member was 812 kN (183 kip), giving a Vexp/Vpred ratio of 1.01. Calculations similar to those described previously were performed for 129 experiments on continuous deep beams described in the literature9-15 and the results are summarized in Fig. 10. In this figure, the Vexp/Vpred ratios are plotted against a/d ratios of the specimens. Details of specimens and shear-strength calculations are given in the Appendix* to this paper. Predicted shear capacity is taken as the larger6,16 of the deep beam capacity and the sectional shear capacity. All 129 beams were governed by the 3PKT predictions but for the ACI predictions, 51 beams were governed by sectional capacity. While the 3PKT method requires somewhat more computational effort, it results in predictions that * The Appendix is available at www.concrete.org/publications in PDF format, appended to the online version of the published paper. It is also available in hard copy from ACI headquarters for a fee equal to the cost of reproduction plus handling at the time of the request.

54

are significantly more accurate across the entire range of a/d values. As in the example calculations for beam CDB1, the shear strength calculations for the 129 experiments assumed that the load distribution Vint/P in the critical shear span at failure was equal to that measured in the experiment. If the load distribution was not reported, then it was calculated based on linear elastic beam theory. The assumed load distribution had a relatively small effect on the predicted shear strength of the test beams. For example, if the shear strength of Beam CDB1 is calculated assuming the elastic value of Vint/P of 0.675, rather than the 0.619 experimental value, then the predicted shear-strength changes from 802 to 828 kN (180 to 186 kip). While using the calculated elastic distribution results in a 3% increase in predicted shear strength, the predicted failure load, P, is reduced from 1296 to 1227 kN (292 to 276 kip)—a 5% reduction. For shear-critical transfer girders subjected to significant differential settlements, the redistribution of reactions may cause a much more substantial reduction in failure load. Procedures based on the 3PKT for calculating load distributions near failure are described in the following. LOAD DISTRIBUTION IN CONTINUOUS BEAMS At shear failure of the symmetrical two-span deep beam shown in Fig. 2 linear elastic theory, assuming uniform stiffness along the beam, predicts 67.5% of the total load will be resisted by the central column. The 3PKT can be used to more accurately access load distribution near failure. Because of symmetry, the central support will not rotate and hence each of the two spans acts as a propped cantilever. As shown in Fig. 11, the deformations near failure in the propped cantilever can be expressed in terms of just three parameters: Dc, the transverse displacement of the critical loading zone in the interior shear span; et2,avg, the average tensile strain in the top longitudinal bars of the interior shear span; and et1,avg, the average tensile strain in the bottom ACI Structural Journal/January-February 2015

longitudinal bars, which from the procedures in Fig. 9 will be equal in both shear spans. The downwards displacement of the tip of the cantilever, Ds, can then be calculated from Eq. (18), the derivation of which is illustrated in Fig. 11. D s = (e t ,2 avg a cot a + D c )

+ (e t ,2 avg − e t ,1avg )aext cot a − e t ,1avg aext cot a ext

(18)

If there is no differential settlement of the supports, then Ds should be zero. If there is differential settlement, then Ds is equal to the settlement of the exterior supports minus the settlement of the central support. The procedure to calculate the distribution ratio Vint/P starts with an estimate for this ratio. For example, the elastic value of 0.675 could be chosen. The predicted shear strength and the three kinematic parameters for the chosen ratio are then calculated as explained in the previous section. For the 0.675 ratio, the predicted shear strength is 828 kN (186 kip), et1,avg is 1.146 × 10–3, et2,avg is 1.343 × 10–3, and Dc is 1.76 mm (0.069 in.). Substituting the values of the three kinematic parameters into Eq. (18) gives Ds = 4.99 + 0.45 – 2.85 = 2.59 mm (0.102 in.) This shows that a Vint/P ratio of 0.675 is appropriate if the external supports settle by 2.59 mm (0.102 in.) more than the central support. In this case, the predicted failure value for P would be 1227 kN (276 kip). If the initial estimate of the ratio Vint/P is 0.619, then the predicted shear strength will be 802 kN (180 kip) and the predicted failure value for P will be 1296 kN (292 kip) while the three kinematic parameters are as follows: et1,avg is 1.417 × 10–3, et2,avg is 0.987 × 10–3, and Dc is 1.63 mm (0.064 in.). From Eq. (18)

Fig. 12—Comparison of predicted and observed response. (Note: 1 kN = 0.2248 kip; 1 mm = 0.0394 in.)

Ds = 4.00 – 0.99 – 3.01 = 0.00 mm As Ds is 0, this shows that the predicted value of Vint/P for the case of no differential settlement is 0.619, which happens to also be the value measured in the experiment. The predicted deflection under the load at failure is equal to the first term in Eq. (18), which in this case is 4.00 mm (0.158 in.). In the experiment, the measured deflection under the load at maximum shear force in the east span was 3.70 mm (0.146 in.). Figure 12 demonstrates that the 3PKT can be used to obtain a reasonably accurate prediction of the entire V-D response of the member, where V is the shear in the critical inner shear span and D is the deflection under the load. The peak shear and corresponding deflection are obtained as discussed previously. If the peak point is shifted to the left by Dc and the new point is connected to the origin of the plot, the predicted response when Dc equals 0 is obtained. The Dc parameter will be 0 until a major web-shear diagonal crack forms. The diagonal cracking shear, Vcr, can be calculated as 520 kN (117 kip) from Eq. (11-7) of the ACI Code.1 At low load levels when both M1 and M2 are less than the flexural cracking moment, Mcr, the response of the beam is ACI Structural Journal/January-February 2015

Fig. 13—Predicted effect of support settlements on failure load of deep continuous girders. well-predicted by traditional linear elastic theory. Assuming a modulus of rupture of 0.63√fc' MPa (7.5√fc' psi),1 Mcr equals 247 kNm (182 ft.kip), which corresponds to a shear of 258 kN (58 kip) at flexural cracking. The points 55

corresponding to flexural cracking, web-shear cracking, and the peak load are joined with straight lines to give a trilinear response up to the peak shear. Finally, the postpeak response is calculated by using the 3PKT equations but with Dc being increased, which increases the deflection but decreases the shear resistance. Also shown in Fig. 12 are the experimentally observed response and the predictions from the VecTor2 model and the ACI strut-and-tie model. In the aforementioned discussion, it was demonstrated that a very small differential settlement of the supports of the test beam could reduce the predicted failure load by approximately 5%. Figure 13 has been prepared to demonstrate how the predicted failure loads change for a wide range of differential settlements. In addition to the loads P applied near the middle of each span, a load P was applied over the central support and loads of P/2 were applied over each exterior support to better simulate the real loading situation shown in Fig. 1(a). The additional load over the central support reduces the effective bearing length, lb2e, and hence reduces somewhat the predicted strength of the inner shear span. For different values of the Vint/P ratio, the failure load P and the associated differential settlement Ds can be calculated from the 3PKT equations. These equations have been developed for the case where the deep beam fails in shear prior to yielding of the longitudinal reinforcement in flexure. For the beam studied in Fig. 13, the longitudinal reinforcement is predicted to remain elastic provided that Ds is less than 10.1 mm (0.40 in.) and greater than –8.7 mm (–0.34 in.). As can be seen in Fig. 13, within this range, the predicted failure loads go from a low of 870 kN (196 kip) to a high of 1486 kN (334 kip) and the predicted changes in failure loads agree closely with the changes predicted by the VecTor2 model. In evaluating the significance of these predictions, it is useful to recall that allowable bearing stresses for spread footings are often based18 on restricting differential settlements to approximately 20 mm (0.75 in.) at service loads, which corresponds to approximately 40 mm (1.5 in.) at structural failure loads. The beam whose response is calculated in Fig. 13 was approximately a one-third-scale model of an actual transfer girder and hence the differential settlements that could be expected at failure for the model beam would be approximately ±40/3 = ±13 mm (±0.5 in.). With such differential settlements, the 3PKT and VecTor2 models both predict that significant reductions in member capacity can occur.

to-predicted-shear-strength ratio of 1.09 with a coefficient of variation (COV) of 16%, as compared to the ACI code provisions that resulted in an average ratio of 1.60 and a COV of 36%. The 3PKT presented in this paper enables an engineer to evaluate the behavior of a deep transfer girder subject to differential settlements with a relatively small number of calculations. For the beam studied in the paper, the 3PKT model with only three degrees of freedom gave results very similar to a nonlinear finite element model with thousands of degrees of freedom. AUTHOR BIOS ACI member Boyan I. Mihaylov is an Assistant Professor in the Department of ArGEnCo at the University of Liege, Liege, Belgium. He received his PhD from the ROSE School, Pavia, Italy, in 2009. Bradley Hunt is a PhD Candidate at Carleton University, Ottawa, ON, Canada. He received his Masters of Engineering degree from the University of Toronto, Toronto, ON, Canada, in 2012. Evan C. Bentz, FACI, is an Associate Professor of civil engineering at the University of Toronto. He is Chair of ACI Committee 365, Service Life Prediction, and a member of Joint ACI-ASCE Committee 445, Shear and Torsion. ACI Honorary Member Michael P. Collins is a University Professor and the Bahen-Tanenbaum Professor of Civil Engineering at the University of Toronto. He is a member of Joint ACI-ASCE Committee 445, Shear and Torsion.

NOTATION As1 As2 a aext ag d1 d2 db Es fc' fv fy fye fyv h k kc l0 lb1 lb1e lb2 lk M1

CONCLUSIONS For continuous deep beams, redistribution of forces caused by differential settlements can significantly reduce failure loads. The three-parameter kinematic theory (3PKT) presented in this paper is a valuable tool for assessing such situations. This approach is based on a kinematic model which accurately describes the deformed shape of each shear span using only three degrees of freedom: the average strains in the top and bottom longitudinal reinforcement and the transverse displacement of the critical loading zone. The shear-strength predictions of the 3PKT were validated against the results from 129 published tests of continuous deep beams. The 3PKT produced an average experimental56

M2 nb P P1/2 smax V VCLZ Vci Vd Vs w a a1 Dc dx

= area of bottom longitudinal reinforcement = area of top longitudinal reinforcement = shear span = external shear span = maximum size of coarse aggregate = effective depth of section with respect to bottom reinforcement = effective depth of section with respect to top reinforcement = diameter of bottom longitudinal bars = modulus of elasticity of steel = concrete cylinder strength = stress in stirrups = yield strength of bottom longitudinal bars = effective yield strength of bottom longitudinal bars = yield strength of stirrups = total depth of section = crack shape factor = compression softening factor = length of heavily cracked zone at bottom of critical diagonal crack = width of loading plate parallel to longitudinal axis of member = effective width of loading plate parallel to longitudinal axis of member = width of support plate parallel to longitudinal axis of member = length of dowels provided by bottom longitudinal reinforcement = absolute value of moment causing tension in bottom reinforcement = absolute value of moment causing tension in top reinforcement = number of bottom longitudinal bars = applied load = applied concentrated load/support reaction = distance between radial cracks along bottom edge of member = shear force = shear resisted by critical loading zone = shear resisted by aggregate interlock = shear resisted by dowel action = shear resisted by transverse reinforcement = crack width = angle of line extending from inner edge of support plate to far edge of tributary area of loading plate = angle of critical diagonal crack = transverse displacement of critical loading zone = displacement along x-axis


dz e1 et1/2,avg et1/2,min ev q rl rv

= = = = = = = =

displacement along z-axis principal tensile strain in critical loading zone average strain along bottom/top longitudinal reinforcement minimum strain along bottom/top longitudinal reinforcement transverse web strain angle of diagonal cracks in uniform stress field ratio of bottom longitudinal reinforcement ratio of transverse reinforcement

REFERENCES 1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2011, 503 pp. 2. Vecchio, F. J., “Disturbed Stress Field Model for Reinforced Concrete: Formulation,” Journal of Structural Engineering, ASCE, V. 127, No. 1, 2001, pp. 12-20. doi: 10.1061/(ASCE)0733-9445(2001)127:1(12) 3. Senturk, A. E., and Higgins, C., “Evaluation of Reinforced Concrete Deck Girder Bridge Bent Caps with 1950s Vintage details: Analytical methods,” ACI Structural Journal, V. 107, No. 5, Sept.-Oct. 2010, pp. 544-553. 4. Hong, S. G.; Hong, N. K.; and Jang, S. K., “Deformation Capacity of Structural Concrete in Disturbed Regions,” ACI Structural Journal, V. 108, No. 3, May-June 2011, pp. 267-276. 5. Mihaylov, B. I.; Bentz, E. C.; and Collins, M. P., “A Two Degree of Freedom Kinematic Model for Predicting the Deformations of Deep Beams,” CSCE 2nd International Engineering Mechanics and Materials Specialty Conference, Ottawa, ON, Canada, June 2011, 10 pp. 6. Mihaylov, B. I.; Bentz, E. C.; and Collins, M. P., “Two-Parameter Kinematic Theory for Shear Behavior of Deep Beams,” ACI Structural Journal, V. 110, No. 3, May-June 2013, pp. 447-456. 7. Mihaylov, B. I., “Behavior of Deep Reinforced Concrete Beams under Monotonic and Reversed Cyclic Load,” doctoral thesis, European School for Advanced Studies in Reduction of Seismic Risk, Pavia, Italy, 2008, 379 pp.


8. Vecchio, F. J., and Collins, M. P., “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI Journal Proceedings, V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231. 9. Moody, K. G.; Viest, I. M.; Elstner, R. C.; and Hognestad, E., “Shear Strength of Reinforced Concrete Beams Part 2—Tests of Restrained Beams without Web Reinforcement,” ACI Journal Proceedings, V. 51, No. 1, Jan. 1955, pp. 417-434. 10. Rogowsky, D. M., and MacGregor, J. G., “Tests of Reinforced Concrete Deep Beams,” ACI Journal Proceedings, V. 83, No. 4, July-Aug. 1986, pp. 614-623. 11. Ashour, A. F., “Tests of Reinforced Concrete Continuous Deep Beams,” ACI Structural Journal, V. 94, No. 1, Jan.-Feb. 1997, pp. 3-11. 12. Asin, M., “The Behaviour of Reinforced Concrete Continuous Deep Beams,” doctoral thesis, Delft University Press, Delft, the Netherlands, 1999, 167 pp. 13. Yang, K.-H.; Chung, H.-S.; and Ashour, A. F., “Influence of Shear Reinforcement on Reinforced Concrete Continuous Deep Beams,” ACI Structural Journal, V. 104, No. 4, July-Aug. 2007, pp. 420-429. 14. Yang, K.-H.; Chung, H.-S.; and Ashour, A. F., “Influence of Section Depth on the Structural Behaviour of Reinforced Concrete Continuous Deep Beams,” Magazine of Concrete Research, V. 59, No. 8, 2007, pp. 575-586. doi: 10.1680/macr.2007.59.8.575 15. Zhang, N., and Tan, K.-H., “Effects of Support Settlement on Continuous Deep Beams and STM Modeling,” Engineering Structures, V. 32, No. 2, 2010, pp. 361-372. doi: 10.1016/j.engstruct.2009.09.019 16. AASHTO, “AASHTO LRFD Bridge Design Specifications,” fourth edition, American Association of State Highway Officials, Washington, DC, 2007, 1526 pp. 17. Bentz, E. C.; Vecchio, F. J.; and Collins, M. P., “Simplified Modified Compression Field Theory for Calculating Shear Strength of Reinforced Concrete Members,” ACI Structural Journal, V. 103, No. 4, July-Aug. 2006, pp. 614-624. 18. Craig, R. F., Soil Mechanics, seventh edition, E&FN Spon, London, UK, 2004, 447 pp.

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NOTES:

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TECHNICAL PAPER

Title No. 112-S06

Investigation of Bond Properties of Alternate Anchorage Schemes for Glass Fiber-Reinforced Polymer Bars by Lisa Vint and Shamim Sheikh A study of existing research shows a need for an investigation of the bond properties of anchorage systems for glass fiber-reinforced polymer (GFRP) bars including mechanical anchor heads and bends. In this research program, the standard pullout test procedure was modified, which improved testing efficiency, accommodated bent bar tests, and reduced the variability of concrete properties across test specimens. Based on this experimental work consisting of a total of 72 specimens, it was concluded that the surface profile of GFRP bars influences only the post-peak phase of the bond stress-slip curve. It was also found that GFRP bars with anchor heads still require a considerable bonded length to develop the bars’ full strength. The bend strengths were determined to be between 58 and 80% of the strength of the straight portion of the same bar in the specimens from three GFRP manufacturers. The straight bar embedment length required to develop full strength of bent bar was found to be approximately five times the bar diameter for all bar types tested. Keywords: anchor; bend; bond; glass fiber-reinforced polymer (GFRP) bars; polymers; reinforced concrete.

INTRODUCTION Glass fiber-reinforced polymer (GFRP) reinforcing bars have been introduced as a lightweight, corrosion-resistant material which offers a viable replacement for traditional steel reinforcing bars, especially when the structures are located in aggressive environments such as coastal regions and those subjected to deicing salts. Extensive experimental work is needed in order to develop reliable and rational guidelines for design if GFRP is to be widely accepted as a practical construction material. One property of importance is the bond between the GFRP bars and the surrounding concrete. This property is crucial, as it has a major effect on the structural performance of a member with regards to cracking, deformability, internal damping, and instability in concrete structures (Gambarova et al. 1998). To develop the required design strength of the bars in tension, it is common practice to introduce mechanical anchor heads at the ends of straight bars in reinforced concrete structures when the available space is limited. In recent years, great improvements have been made in the manufacturing of GFRP reinforcement with the current bars having a much higher ultimate strength and stiffness than the bars of previous generations. With these greater strengths come much longer required development lengths, further increasing the need for anchor heads. Due to the lack of standards for the manufacturing of these anchor heads, the current Canadian design code for FRP bars, CSA S80612, requires that engineers check the results from research ACI Structural Journal/January-February 2015

programs to determine if the anchor heads have been proven to develop at least 1.67 times the required design strength within the given length. Bends in GFRP reinforcement are also used for reducing the required development length as well as in stirrups used as shear reinforcement. The strength at these bends has been reported to be approximately 30 to 60% of the ultimate tensile strength of the straight portion of the bars (Imjai et al. 2007; Ahmed et al. 2010). This reduced strength is due to the bearing action of the concrete on the bend inducing normal stresses in the weak lateral direction of the bars, and due to the longitudinal fibers becoming kinked along the interior radius of the bend, reducing their load-carrying capacity (Ahmed et al. 2010). Results from the research on the behavior of various anchorage types would allow engineers to be more confident when designing with this relatively new and always evolving reinforcing material. RESEARCH SIGNIFICANCE While there have been a multitude of studies recently published on the bond properties of GFRP bars, there is a gap in the literature with respect to the behavior of different anchorage types, such as headed and bent bars, and their relative benefits over straight bars. Because no codified standards for GFRP manufacturing are in place, great variations in the mechanical properties and surface profiles of the available GFRP bars exist. This research addresses these issues and presents results from an extensive study on the bond and anchorage of GFRP bar products from different manufacturers tested in a similar manner and under the same conditions. Performance of straight bars, headed bars, and bent bars is discussed and resistance of each type of anchor is investigated. EXPERIMENTAL INVESTIGATION To minimize the risk of splitting failure in the concrete cylinders, a slab layout was adopted for the traditional direct tension pullout test (DTPT) (fib 2000), of anchors. A total of 72 bond specimens of various anchorage types and bonded lengths were embedded in six concrete slabs each measuring 2400 x 1200 x 320 mm (94.5 x 47.2 x 12.6 in.). The test parameters included the surface profile based on the type of ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-180.R3, doi:10.14359/51687042, was received January 16, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

59

Table 1—Experimental GFRP reinforcement coupon test results Ab, mm2 (in.2)

fu, MPa (ksi)

E, MPa (ksi)

Rupture strain*, %

A

197.9 (0.306)

742 (107.6)

49,500 (7180)

1.50

B

197.9 (0.306)

1277 (185.2)

72,600 (10,530)

1.76

C

201 (0.312)

1228 (178.1)

59,200 (8590)

2.07

A

71.3 (0.1105)

833 (120.8)

54,300 (7880)

1.53

B 126.7 (0.1964)

655 (95.0)

42,000 (6090)

1.56

C

912 (132.3)

57,900 (8400)

1.61

Bar type

Straight

Bent

113.1 (0.1753)

*

Calculated from ultimate stress fu and stiffness E except for C-Bent, which was measured.

Fig. 1—Concrete compressive stress versus time. GFRP bar, anchorage type, and bonded length. Each type of specimen was tested in triplicate. Three surface profiles were investigated which included helically wrapped bars with sand coating (Bar Type A), sand-coated (Bar Type B), and ribbed (Bar Type C). The three anchorage types include straight bar, headed bar, and bent bar. All bar types were tested with bonded lengths equal to five and 10 times the bar diameter, 5db and 10db. Additionally, the headed and bent bars were tested where the bar was debonded using a polyvinyl chloride (PVC) tube up to the start of the anchorage mechanism, referred to as 0db, so that only the strength of the anchor head or bend could be determined. Concrete materials The slabs were cast in two batches using ready mixed concrete provided by a local supplier with specified compressive strength of 25 MPa (3.63 ksi) at 28 days. The concrete was also specified with no air entrainment, 20 mm (0.79 in.) maximum aggregate size, and 100 mm (4 in.) slump, which were each verified at the time of the castings. The concrete compressive and rupture strengths were found to be similar for both batches. At the time of testing of bond specimens, the average of three test cylinders gave concrete compressive strength of 35.8 and 30.2 MPa (5.19 and 4.38 ksi) and modulus of rupture values of 3.65 and 3.98 MPa (0.53 and 0.58 ksi), for Batches 1 and 2, respectively. Figure 1 shows the development of compressive strength with age for the two concretes. Reinforcing bar materials All three bar types are manufactured using similar pultrusion processes for the GFRP bar core. The difference is in the manufacturing process for the bars’ surface profiles. Bar Type A is first sand coated and then wrapped helically with fibers before thermosetting of the resin, creating undulations in the side of the bar. Bar Type B has a sand-coated surface that is applied using an in-line coating process (Ahmed et al. 2008) during the pultrusion of the bar. Lastly, Bar Type C has a ribbed bar surface that is cut into the rod after curing is completed.

60

Both straight bars and the straight portion of bent bars with diameters of 16 mm (5/8 in.), and 10 to 12.7 mm (3/8 to 1/2 in.), respectively, were tested for their ultimate tensile strength and Young’s modulus of elasticity. The standard test method for tensile properties of fiber-reinforced polymer matrix composite bars (ASTM D7205-06) was followed when completing these tests and the average results for three types of bars tested can be found in Table 1. It can be seen in the table that straight Bar Type A has a lower ultimate tensile strength and stiffness than Bar Types B and C, whereas bent Bar Type B has a lower strength and stiffness than the other two bar types. These properties were calculated using the nominal bar areas provided by the bar manufacturers as practiced by designers in most cases. The cross-sectional areas of the straight bars based on the actual diameters of each bar, measured by excluding the sand coatings and ribs, were 195, 241, and 210 mm2 (0.302, 0.374, and 0.326 in.2) for Bar Types A, B, and C, respectively. These compare with 198, 198, and 201 mm2 (0.307, 0.307, 0.312 in.2) based on the nominal diameters provided by the manufacturers. Test setup and instrumentation In lieu of using the traditional DTPT, where each specimen is embedded in a single cylinder and undesirable concrete splitting failures can occur, a slab layout was used. The Standard Test Method for Strength of Anchors in Concrete and Masonry Elements (ASTM E488-96) was used as a guideline when designing the slab layout for this research program. A minimum spacing of 400 mm (16 in.) was provided between each test anchor, which is greater than the 1.5 times the embedment length specified in ASTM E488. The bars with the longest embedment length were placed along the center line of the slab while the shorter embedment lengths were alternated along the outer edges (Fig. 2). Both the spacing and bar placement were designed to minimize the risk of interaction between adjacent test anchors. Because one of the objectives of this research program was to determine the bend strength of the stirrups used in a large beam test series by Johnson and Sheikh (2013), the layout of the bent bars was designed to mimic that of a beam. Therefore, bent Bar Types B and C were the stirrups from the same batch as the reinforcement for the large beam test series, and bent Bar Type A was added for the current test program. Two 16 mm (5/8 in.) Type C bars were used at each ACI Structural Journal/January-February 2015

bend to mimic the longitudinal bars used in a beam (Fig. 2). The 320 mm (12.6 in.) height of the slab in this test series is one half of the height of the beams in the large beam test series. The design and construction of the reaction bridge was also in accordance with ASTM E488, where the minimum spacing between the supports need to be at least four times the embedment length. The spacing provided between supports for all the tests performed was 640 mm (25-1/4 in.) and was four times the longest embedment length of 160 mm (6-1/4 in.). The reaction bridge was made up of two HSS 102 x 102 x 13 mm (4 x 4 x 0.5 in.) and one HSS 178 x 127 x 9.5 mm (7 x 5 x 0.38 in.), supported by four steel bearing plates with a thickness of 44.5 mm (1-3/4 in.) (Fig. 3). In some cases, the reaction points were asymmetric about the bond specimen. These reaction points are located at greater than the minimum distance required by ASTM E488 and they had no influence on the stresses around the specimens. A tensile load was applied manually to the coupler attached to the free of each anchor specimen at a rate of 15 kN/min (3.37 kip/min) using a 60 tonne (66 ton) hollow

plunger cylinder which was attached to a hydraulic load maintainer. The load was measured using a load cell with a capacity of 1000 kN (225 kip). To ensure the load was transferred perpendicularly to the slab surface, a spherical seating was placed on top of the reaction bridge. Custom steel plates were machined to fit each element, keeping the bar centered in the test setup throughout the loading cycle. Three linear voltage displacement transducers (LVDTs) were placed concentrically around the test bar just below the steel coupler. Effect of any load eccentricity can be accounted for by taking the average of the three displacements. Data from the LVDTs showed minimal eccentricity of loading. Strain gauges were placed on either side of the test specimen at the loaded end as well as at the midpoint of the bonded length for the straight and headed bars for one of the three repeated specimens to evaluate the strain distribution along the anchor. Each set of strain gauges was protected by a 50 mm (2 in.) strip of foil tape; this length was considered to be debonded from the surrounding concrete. To determine the strain distribution across the bends, strain gauges were attached parallel to the fibers at the midpoint of the bend in the same plane as the bend. Figure 4 shows locations of the strain gauges. RESULTS AND DISCUSSION Analysis of measurements The measurements obtained by the data acquisition system were used to produce bond-slip relationships for the three bar types. Although the bond stress does vary along the bonded length, it is generally accepted to present the bond stresses as an average shear stress along the bonded length that is calculated by dividing the applied load by the surface area of the bonded length (Eq. (1))

Fig. 2—Typical slab layout for bent and headed bars in Batch 1 slabs. (Note: Dimensions in mm.)

τ=

N (1) pdb lembed

where N is the applied load; db is the nominal bar diameter; and lembed is the length of the bar bonded to the concrete. Bar slip at the loaded end can be calculated by subtracting the elongation of the bar (outside the concrete) between the LVDT holder and the top surface of the concrete (70 mm

Fig. 3—Setup for modified direct tension pullout tests. ACI Structural Journal/January-February 2015

61

Fig. 4—Strain gauge placement for: (a) headed bars; and (b) bent bars.

Fig. 5—Strain distribution along bonded length results from VecTor 2 model of: (a) straight (C-S-10) and headed (C-H-10) bars; and (b) bent bars. [2-3/4 in.]) from the average displacement measured by the three LVDTs. Bar slip at the free end is calculated by subtracting the elongation of the bar along the debonded length and the bonded length from the loaded end slip using the data collected by the strain gauges. Because the strain gauges only measured the strain at the midpoint of the bonded length, the strain near the free end had to be extrapolated using the results from an analytical model of the pullout tests that was created using finite element analysis (FEA) software (Vecchio and Wong 2002). A detailed description of the steps used to create this model can be found elsewhere (Vint 2012). The strain distribution predicted by the model along the bonded length for the straight, headed and bent bars can be found in Fig. 5 for Bar Type C with bonded lengths of 10db. It can be seen that the strain at the free end of the straight bar is nearly zero, which is consistent with the existing literature that states free end strain is theoretically zero. The difference between the theoretical and experimental results could be due to the inaccuracy of the strain gauges. The anchor head does an effective job of transferring the strains to the free end, producing a more constant strain distribution along the bonded length. The strain distribution

62

for the bent bars shows that the strains are small in the first bend and quickly dissipate to zero beyond that point. Experimental results In the discussion of the results for all 72 pullout specimens in this paper, the following nomenclature for the specimens is used: The first letter indicates the bar type. The next letter indicates the anchorage type, where S is straight, H is headed, and B is bent. The first number indicates the ratio of bonded length to bar diameter of 0, 5, or 10. The next number indicates the test number of the specimens within a group of three specimens, where the third specimen has the strain gauges along the bonded length. The last number denotes the concrete batch number. As an example, A-S-10-1-2 designates a bond test specimen of Bar Type A with a straight anchorage type and a bonded length of 10 times the bar diameter. It is the first of the three specimens in that group and was in the second batch of concrete. Straight bars—Results for the straight bars with no end anchorage showed that as the embedment length increased from 5db to 10db the peak average bond stress, τ2, decreased. This trend is consistent with the available literature, indiACI Structural Journal/January-February 2015

Table 2—Bond test results for straight bars Bar type

db, mm (in.)

lembed/db

N2, kN (kip)

f2, MPa (ksi)

f2/fu

τ2, MPa (ksi)

sl2, mm (in.)

sf2, mm (in.)

15.9 (0.626)

5

50.6 (11.38)

255 (37.0)

0.346

12.67 (1.838)

1.738 (0.068)

0.420 (0.0165)

A

10

78.7 (17.69)

396 (57.4)

0.537

9.85 (1.429)

2.99 (0.118)

0.692 (0.027)

15.9 (0.626)

5

62.3 (14.01)

314 (45.5)

0.246

15.59 (2.26)

1.410 (0.056)

0.656 (0.026)

10

104.8 (23.6)

528 (76.6)

0.413

13.12 (1.903)

1.919 (0.076)

0.352 (0.0139)

5

50.9 (11.44)

253 (36.7)

0.206

12.65 (1.835)

1.565 (0.062)

0.551 (0.022)

10

90.2 (20.3)

449 (65.1)

0.365

11.22 (1.627)

1.970 (0.078)

0.357 (0.0141)

B

C

16 (0.630)

Fig. 7—Experimental bond-slip curves for straight bars A, B, and C. Fig. 6—Failed specimens from left to right: A-S-10-1-2, B-S-10-1-2, and C-S-10-1-2. cating an increase in peak load of approximately 70% for all bar types as the bonded area is doubled. A summary of the results for the straight bars can be found in Table 2, where the values given are the average of the three repeated specimens. The sand-coated Bar Type B displayed the largest bond stresses of the three types of bars at both embedment lengths. Bar Types A and C gave similar peak average bond stresses at shorter embedment lengths, but ribbed Bar Type C outperformed helically wrapped Bar Type A at a longer embedment length. It should be noted that nominal dimensions were used in calculating the bond stresses shown in Table 2. If actual sizes of the bars are used in the analysis, the bond strength values for all three types of the bars are almost equal. All bars failed by bar pullout; however, the shearing interface is varied depending on the surface profile. When Bar Type A was pulled out of the concrete the sand coating was sheared off while the undulations remain intact, resulting in a ductile post-peak response. When Bar Type B is pulled out of the concrete, the sand coating is completely sheared off. This suggests the shearing interface is between the core of the bar and the sand coated surface profile. For Bar Type C, the concrete is sheared between the ribs when pullout occurs. Thus the shearing interface is between the ribs of the bar and the concrete. The location of the shearing interface suggests that the connection between the core ACI Structural Journal/January-February 2015

and the surface profile of the three bars is strongest for Bar Type C, even though Bar Type B was able to sustain the highest peak average bond stress. This observation indicates that the thickness of ribs in Bar Type C can be optimized to increase the bond strength. The shearing interfaces can be clearly seen in the failed bars found in Fig. 6. Figure 7 shows the average bond stress vs. free end slip for the three bars with 10db bonded length. It can be seen in the figure that Bar Type A, although lower in bond strength, is capable of maintaining higher strength over larger range of slip than Bar Types B and C. Headed bars—Table 3 shows a summary of results for two types of headed bars in which each value represents an average of three readings. Bar Type A headed bars were not available at the time of these tests. Results show that the bars can develop a substantial tensile force as a result of head anchorage alone. Bar Type B headed bars showed an average peak stress of 605 MPa (87.7 ksi) while Bar Type C displayed a stress of 507 MPa (73.5 ksi) without any bonded length. Again, if actual diameters of the bars are used, the stress level would be almost equal. The diameter of the heads of the bases in Bar Types B and C bars were 48 and 40 mm (1.89 and 1.57 in.), respectively. The bar stresses that developed in both types of headed bars with a bonded length of 10db are approximately 50% higher than the bar stresses developed in straight bars with bonded length of 10db. When a bonded length of 5db is added to the anchor heads, the increase in bar stress is only approximately 100 to 150 MPa 63

Table 3—Bond test results for headed bars Bar type

db, mm (in.)

B

15.9 (0.626)

C

16 (0.630)

lembed/db

N2, kN (kip)

f2, MPa (ksi)

f2/fu

τ2, MPa (ksi)

sl2, mm (in.)

sf2, mm (in.)

0

120.2 (27.0)

605 (87.7)

0.474

—

3.33 (0.131)

1.522 (0.060)

5

141.7 (31.9)

714 (103.6)

0.559

35.5 (5.15)

3.71 (0.146)

1.736 (0.068)

10

161.3 (36.3)

812 (117.8)

0.636

20.2 (2.93)

4.24 (0.167)

1.699 (0.067)

0

101.9 (22.9)

507 (73.5)

0.413

—

2.75 (0.108)

0.945 (0.037)

5

133.2 (29.9)

663 (96.2)

0.540

33.1 (4.80)

3.45 (0.136)

1.152 (0.045)

10

141.4 (31.8)

703 (102.0)

0.573

17.6 (2.55)

4.01 (0.158)

1.249 (0.049)

Fig. 8—Typical bar stress-slip relationship for headed bars: (a) Bar Type B; and (b) Bar Type C. (14.5 to 21.8 ksi) in the two types of bars. A bonded length of 10db only adds approximately 200 MPa (29.0 ksi) bar stress. While both headed Bar Type B and C were able to develop similar peak ultimate average bond stresses, their bond stress-slip reponses varied in the post-peak phase just like their straight bar behaviors. This is due to the different manufacturing processes that are used for each bar type. For Bar Type B, the anchor head is attached to a specially prepared surface which consists of O-rings spaced every 5 mm (0.2 in.) (Drouin 2012). Whereas for Bar Type C, the anchor head is attached directly to the 16 mm (5/8 in.) GFRP bar. It can be seen from the bond stress-slip curve of both headed bar types (Fig. 8) that the behavior of the anchor head for Bar Type B has multiple peaks and valleys, while the failure of headed Bar Type C was mostly singular as the stress drops significantly only once. The same failure mode was observed for the Bar Type B headed bars for all bonded lengths. Multiple loud bangs were observed (O-rings failing) in Bar Type B headed specimens with no bonded length. The peak and valley behavior seems to be due to the load sharing mechanism in these specimens and requires further investigation. Due to the high ultimate strength of the bars tested, the bonded length required to develop the full strength of the GFRP bars would be relatively large. All headed bars at all bonded lengths failed by pullout of the bar from the head connection as seen in Fig. 9, where the unruptured bar pulled out of the anchor head, which remained intact and inside 64

Fig. 9—Failed headed bar specimen in concrete cylinder with anchor head intact. the concrete slabs. Note that the photo shown in Fig. 9 is from a bond test series in which the bars were embedded in concrete cylinders (Vint 2012). While it was relatively more manageable to cut cylinders to investigate failed bars, the failure modes of similar specimens in the two series were found to be identical. The mechanical anchor heads are required to develop at least 1.67 times the required level ACI Structural Journal/January-February 2015

Table 4—Bond test results for bent bars Bar type

A

B

Fig. 10—Experimental and analytical strain variation along bonded length for specimen (C-H-S-3-1).

C

Fig. 11—Peak loaded end-slip values for straight and headed bars for all bar types. of tensile load in the bar (CSA S806-12). This would result in the design bar stress to be 360 MPa (52.2 ksi) for Bar Type B, and 304 MPa (44.1 ksi) for Bar Type C based on the results for anchors only (0db). The tested anchor heads with reasonable bonded length should thus be adequate for developing stresses similar to or larger than what would be used for steel reinforcing bars, which usually have a yield stress of 400 MPa (58.0 ksi). The experimental results confirmed what was found in the finite element model, developed in the FEA software, showing that the anchor head effectively stiffened the bars near the free end (Fig. 10). The strain gauges at the midpoint of the bonded length indicated that the strain at this point was nearly the same as at the loaded end of the bar at peak loads. This suggests that for the headed bars, the bond stress is low and evenly distributed along the bonded length. This even distribution is similar to the response that is often observed with steel bars, which generally have somewhat higher bond strength due partly to their higher stiffness (Mosley et al. 2008). The higher loads and constant strain distributions observed for the headed bars resulted in higher loaded end slips for these bars when compared to straight bars of the same bonded length (Fig. 11). Bent bars—The strength of the bar at bends is lower than the strength of the straight portion of the bar due to ACI Structural Journal/January-February 2015

db, mm (in.)

9.43 (0.371)

11.93 (0.470)

13.00 (0.512)

rb/db

5.4

3.0

1.75

lembed/ f2, MPa db (ksi)

sl2, mm (in.)

f2/fu, %

Rupture location

0

555 (80.5)

4.58 (0.180)

66.7

Bend

5

775 (112.4)

4.74 (0.187)

93.1

Bend

10

785 (113.9)

3.63 (0.143)

94.2

Coupler

0

522 (75.7)

6.19 (0.244)

79.6

Bend

5

608 (88.2)

4.97 (0.196)

92.9

External SG

10

612 (88.8)

4.63 (0.182)

93.4

External SG

0

531 (77.0)

6.32 (0.249)

58.3

Bend

5

816 (118.4)

8.50 (0.335)

93.6

Interior Straight

10

724 (105.0)

6.04 (0.238)

79.4

External SG

the relatively lower strength in the transverse direction as well as the kinking of the longitudinal fibers along the interior bend. To determine this reduced strength, the bent bars embedded in concrete were debonded to the top of the loaded end of the bend. The results from these tests showed that the bend strength is similar for all three bar types with peak bar stresses of 555, 522, and 531 MPa (80.5, 75.7, and 77.0 ksi) for Bar Types A, B, and C, respectively. All bar types pass the requirement by CSA S807-10 that the bend strength of a stirrup should be at least 45% of the strength of the straight portion (Table 4). The difference in bend geometry due to manufacturing constraints should be noted as it has been shown to have an influence on bend strengths. Bar Type A produced the greatest bend strength, but it also has the smallest bar diameter, 10 mm (3/8 in.) and the largest bend radius, rb, of 5.4db. Whereas Bar Types B and C had larger bar diameters of 12.7 and 12 mm (0.5 and 0.472 in.) and bend radii-bar diameter ratios of 3.0 and 1.75, respectively. These bars produced slightly lower bend strengths than Bar Type A most likely due to the tighter bend radii. When the bonded length is increased to five and 10 times the bar diameter, all bent bars ruptured along the straight portion of the stirrup indicating that bond was fully developed. This is verified by the fact that when the bonded length was increased from 5db to 10db the bars did not pick up additional loads, as shown in Fig. 12. For the case of Bar Type C, the peak stress actually dropped slightly as the bonded length increased. This is because the observed failure mode in the specimen with 10db bonded length was bar rupture at the location of the external strain gauges where the bar area had been reduced for the purpose of gauge installation, thus causing lower failure load. From the bar stress-slip relationships for the bent bars in Fig. 13, it can be observed that the failure mode was brittle because the bar stress decreased dramatically after peak stresses were observed. This is consistent with the rupture failure of GFRP bars. The testing 65

method for the bent bars was designed to mimic the behavior of a GFRP stirrup in a large beam and all bar types were found to be fully developed within the given slab height. In these specimens, the requirement of CSA S806-12, that web reinforcement should have sufficient development length to develop its design stress at midheight of the member, is satisfied. The rupture location varied between the specimens with the most common location indicated in Table 4 and Fig. 14. In many cases, the failure was observed in the bars where the bar area was reduced as a result of surface preparation for the strain gauges. In some cases, the test was terminated prematurely because of the slip of couplers. Three additional bent specimens without strain gauges were tested in a Batch 2 slab and it was determined that bond could still be developed within 5db, as the bar ruptured along the straight portion of the stirrup inside the concrete.

be found in Fig. 15, where the unknown parameters α1, p, α2, and β can be solved for using the method described as follows. The values for peak average bond stress, τ2, and the free end slip at the peak average bond stress, s2, are taken as averages from the experimental results. The method used to determine α1 was adopted from Cosenza et al. (1997) where the area under the experimental curve is equated to the area under the theoretical curve (Eq. (2) and (3)) for s ≤ s2 to solve for α1.

Analytical modeling of bond-slip relationship The modified Bertero-Eligehausen-Popov, m-BEP, model similar to that used by Cosenza et al. (1997), Gambarova et al. (1998), Focacci et al. (2000), and Kadam (2006) was used to develop a constitutive bond stress-slip (τ-s) relationship for the straight bars (Fig. 15). This model can be divided into two or three parts: the ascending branch up to the peak stress; the descending branch after the peak stress; and a constant stress, τ3. The equations used for the different branches can

From the experimental stress-slip curves for the three bar types shown in Fig. 7, it can be seen that the post-peak response varies depending on the bar type. As mentioned previously, this difference is due to each bar type’s surface profile. The soft wave shape seen in Fig. 16(a) can be attributed to the undulations found on the surface of Bar Type A, which remain as the bar is pulled out. The steep decrease in bond stress seen in Fig. 16(b) is due to the brittle bond failure due to the shearing off of the sand coating on Bar Type B. Finally, the initial post-peak response of Bar Type C was also brittle due to the shearing off of the concrete between the bar’s ribs. However, in this case, higher and more constant residual stresses were observed due to the more uniform friction plane. Due to these varying post-peak responses, different descending branch shapes were used in the analysis for the three bar types. For Bar Types A and C, a linear decreasing branch with a slope of p · τ2/s2 was observed and can be modeled using (2) in Fig. 15, where p is the unknown parameter that is solved by using linear regression of the experimental results. Bar Type B had a nonlinear post-peak response that can be modeled using (3) in Fig. 15, where α2 is determined using nonlinear regression techniques. The constant residual stress, τ3, was observed for Bar Type C and was calibrated using a reduction factor, β, that was solved using the experimental results. The values for known parameters τ2 and s2 and unknown parameters α1, p, α2, and β can be found in Table 5. A comparison of the analytical curves and experimental results for each

Fig. 12—Peak bar stress with increasing bonded length.

 s s s Aτ1 = ∫0 2 τ ( s ) ds = ∫0 2 τ 2    s2  a1 =

a1

ds =

τ 2 s2 (2) 1 + a1

τ 2 s2 − 1 (3) Aτ1

Fig. 13—Typical bar stress-slip relationship for bent bars with all bonded lengths, for Bar Types (a) A; (b) B; and (c) C. 66


Fig. 14—Failed bend specimens with typical rupture locations for Bars A, B, and C, from left to right.

Fig. 15—Modified Bertero-Eligenhausen-Popov model for bond stress-slip relationship (Consenza et al. 1997; Focacci et al. 2000; Kadam 2006; Quayyum 2010). (Note: 1 mm = 0.0394 in.; 1 MPa = 145 psi.) Table 5—Mean values and coefficients of variation for unknown parameters τ2, MPa (ksi)

s2, mm (in.)

α1

p

α2

β

A

11.26 (1.633) [0.447]

0.556 (0.0219) [0.1483]

0.0622 [0.785]

0.0131 [0.514]

—

—

B

14.36 (2.08) [0.312]

0.504 (0.01984) [0.1109]

0.210 [0.918]

—

–0.255 [0.167]

—

C

11.50 0.420 (1.668) (0.01654) [0.0710] [0.1741]

0.110 [0.778]

0.0296 [0.224]

—

0.434 [0.286]

Bar type

Note: Values in brackets are coefficients of variation.


Fig. 16—Experimental and theoretical curves for straight Bar Types A, B, and C with an embedment length of 5db. bar type is shown in Fig. 16. The values for α1 were found to increase with increasing bar stiffness. These values changed with relatively high coefficients of variation, primarily due to the displacement measurements being extremely small during the pre-peak behavior, which can be difficult to capture with high precision. The p values correspond to the post-peak slope, and average values of 0.0131 and 0.0296 were observed for Bar Types A and C, respectively. A higher p value indicates a steeper descending branch, such as for Bar Type C, where failure was due to the concrete shearing off between the ribs. No comparisons can be made for the α2 value, as this post-peak response was unique to Bar Type B; however, the low COV of 0.1666 indicates that the response was consistent for this bar type. CONCLUSIONS Results are presented from an experimental and analytical program in which 72 bond specimens were tested under direct pullout. Three different bar types were used. Bond

67

properties of straight bars, headed bars, and bent bars were investigated. The main purpose of the study was to evaluate the relative performance of GFRP bars and also compare these results with the minimum requirements of design codes. The following conclusions may be made from the results of this study: 1. The surface profile of the tested GFRP bars was found to influence the post-peak bond behavior of the bars. Where Bar Type A is sand-coated and helically wrapped, it produced a more ductile failure mode with higher residual stresses due to the high friction forces between the remaining undulations and the surrounding concrete. Bar Types B and C had less-ductile post-peak responses due to the shearing that occurred between the bar core and the sand coating, or between the rib deformations and the concrete for Bar Types B and C, respectively. 2. The reinforcement bar heads, investigated here, were able to develop tensile stress between 500 and 600 MPa (72.5 and 87.0 ksi) in 16 mm (5/8 in.) diameter bars, which is higher than the tensile strength of traditional steel reinforcing bars. With a bonded length of 10db, the anchor heads were able to develop peak bar stresses in excess of 800 MPa (116.0 ksi). 3. The bend strength of the three GFRP stirrups were found to be similar with strengths of 555, 522, and 531 MPa (80.5, 75.7, and 77.0 ksi) for Bar Types A, B, and C, respectively. All bent bars met the minimum requirements of CSA S806-12 for bend strength. 4. The bent bars with 5db bonded length were able to develop the full tensile strength of the bar as evidenced by their rupture in the straight portion for all bar types. 5. The theoretical bond-slip models that were derived using equilibrium equations as well as linear and nonlinear regression techniques were found to correlate well with the experimental results. Different surface profiles influenced the model shape required for the post-peak response. AUTHOR BIOS ACI member Lisa Vint is an EIT registered with APEGA, working at Williams Engineering Canada. She received her BEng in civil engineering from McGill University, Montreal, QC, Canada, and her MASc in civil engineering from the University of Toronto, Toronto, ON, Canada. Her research interests include reinforced concrete structures and the application of glass fiber-reinforced polymer bars as internal reinforcement in civil engineering structures. Shamim Sheikh, FACI, is a Professor of civil engineering at the University of Toronto. He is a member and Past Chair of Joint ACI-ASCE Committee 441, Reinforced Concrete Columns, and a member of ACI Committee 374, Performance-Based Seismic Design of Concrete Buildings. In 1999, he received the ACI Structural Research Award. His research interests include earthquake resistance and seismic upgrade of concrete structures, confinement of concrete, and use of fiber-reinforced polymer in concrete structures.

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ACKNOWLEDGMENTS The authors wish to express their gratitude and sincere appreciation to the sponsors of this research program for their financial and technical support. These include FACCA Inc., Schoeck Canada Inc., Pultrall Inc., Hughes Brothers Inc., and Vector Construction Group. Additionally, the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.

REFERENCES Ahmed, E. A.; El-Salakawy, E. F.; and Benmokrane, B., 2008, “Tensile Capacity of GFRP Postinstalled Adhesive Anchors in Concrete,” Journal of Composites for Construction, ASCE, V. 12, No. 6, pp. 596-607. doi: 10.1061/(ASCE)1090-0268(2008)12:6(596) Ahmed, E. A.; El-Sayed, A. K.; El-Salakawy, E.; and Benmokrane, B., 2010, “Bend Strength of FRP Stirrups: Comparison and Evaluation of Testing Methods,” Journal of Composites for Construction, ASCE, V. 14, No. 1, pp. 3-10. doi: 10.1061/(ASCE)CC.1943-5614.0000050 ASTM D7205-06, 2006, “Standard Test Method for Tensile Properties of Fiber Reinforced Polymer Matrix Composite Bars,” ASTM International, West Conshohocken, PA, 12 pp. ASTM E488-96, 1996, “Standard Test Methods for Strength of Anchors in Concrete Elements,” ASTM International, West Conshohocken, PA, 8 pp. Cosenza, E.; Manfredi, G.; and Realfonzo, R., 1997, “Behavior and Modeling of Bond of FRP Rebars to Concrete,” Journal of Composites for Construction, ASCE, V. 1, No. 2, pp. 40-51. doi: 10.1061/ (ASCE)1090-0268(1997)1:2(40) CSA S806-12, 2012, “Design and Construction of Building Structures with Fibre Reinforced Polymers,” Canadian Standards Association, Mississauga, ON, Canada, 206 pp. CSA S807-10, 2010, “Specification for Fibre-Reinforced Polymers,” Canadian Standards Association, Mississauga, ON, Canada, 44 pp. Drouin, B., personal communication with L. Vint, May 1, 2012. fib, 2000, “Bond of reinforcement in concrete: state-of-the-art report,” Bulletin 10, Fédération Internationale du Béton Lausanne, Switzerland, 434 pp. Focacci, F.; Nanni, A.; and Bakis, C. E., 2000, “Local Bond-Slope Relationship for FRP Reinforcement in Concrete,” Journal of Composites for Construction, ASCE, V. 4, No. 1, pp. 24-31. doi: 10.1061/ (ASCE)1090-0268(2000)4:1(24) Gambarova, P. G.; Rosati, G. P.; and Schumm, C. E., 1998, “Bond and Splitting: A Vexing Question,” Bond and Development of Reinforcement—A Tribute to Dr. Peter Gergely, SP-180, R. Leon, ed., American Concrete Institute, Farmington Hills, MI, pp. 23-43. Imjai, T.; Guadagnini, M.; and Pilakoutas, K., 2007, “Mechanical Performance of Curved FRP Rebars—Part I: Experimental Study,” Asia-Pacific Conference on FRP in Structures, S. T. Smith, ed., International Institute for FRP in Construction, Kingston, ON, Canada, pp. 333-338. Johnson, D. T. C., and Sheikh, S. A., 2013, “Performance of Bent Stirrup and Headed Glass Fibre Reinforced Polymer Bars in Concrete Structures,” Canadian Journal of Civil Engineering, V. 40, No. 11, pp. 1082-1090. doi: 10.1139/cjce-2012-0522 Kadam, S., 2006, “Analytical Investigation of Bond-Slope Relationship Parameters between Fiber Reinforced Polymer (FRP) Bars and Concrete,” master’s thesis, University of Missouri - Kansas City, Kansas City, MO, 193 pp. Mosley, C. P.; Tureyen, A. K.; and Frosch, R. J., 2008, “Bond Strength of Nonmetallic Reinforcing Bars,” ACI Structural Journal, V. 105, No. 5, Sept.-Oct., pp. 634-642. Vecchio, F., and Wong, P., 2002, VecTor 2 and FormWorks Manual, University of Toronto, Toronto, ON, Canada, http://www.civ.utoronto.ca/ vector/ user_manuals.html. (last accessed July 17, 2014) Vint, L. M., 2012, “Investigation of Bond Properties of Glass Fibre Reinforced Polymer (GFRP) Bars in Concrete under Direct Tension,” master’s thesis, University of Toronto, Toronto, ON, Canada, 213 pp.



TECHNICAL PAPER

Title No. 112-S07

Stress-Transfer Behavior of Reinforced Concrete Cracks and Interfaces by Ali Reza Moradi, Masoud Soltani, and Abbas Ali Tasnimi This paper aims at determining the stress-transfer capabilities of reinforced concrete (RC) cracks and interfaces by adopting proper constitutive laws for aggregate interlock and dowel action mechanisms. The framework of the Contact Density Model, which was developed in earlier research, is adopted for aggregate interlock. The original assumptions are enhanced and modified using the available experimental data and also by introducing four new parameters for both normal- and high-strength concrete. The proposed contact density function considers crack roughness and is probabilistically idealized according to concrete strength, aggregate size, crack width, and asperity degradation in a unified manner. Dowel action behavior of bars is simulated by the model, which was proposed in other research. Stress transfer across RC cracks and interfaces is determined by a combination of the previously mentioned mechanisms and their consistent interactions. The systematic experimental verification shows reliability and versatility of the suggested model and assumptions. Keywords: aggregate interlock; cyclic loading; dowel action; shear.

INTRODUCTION The local and global behavior of reinforced concrete (RC) members and structures may considerably be affected by cracked concrete and the stress-transfer capabilities of cracks and interfaces. Stress transfer across cracks is a major problem in seismic assessment and design of RC structures because the ductility and energy absorption of members are mainly affected by energy consumption along cracks. Different approaches and specimens have been designed to investigate the capabilities of stress transfer across the cracked concrete experimentally. Some studies used precracked specimens and the initial crack width was kept constant during loading.1-3 However, others performed tests with constant normal stress and measured corresponding crack width or applied complicated loading paths.4-11 During the past years, extensive empirical and analytical models have been proposed to investigate the shear transfer behavior of cracked concrete. Bažant and Tsubaki12 pointed out the importance of considering the opening of a crack together with the compressive stress transferred when the shear displacement occurs.5 Considering the dilatancy, Bažant and Gambarova13 identified a nonlinear elastic model by optimizing the fits of Paulay and Loeber’s3 experimental results, taking into account the influence of the maximum aggregate and concrete strength. Walraven and Reinhardt6 idealized the crack surface as a set of circular aggregates. The aggregates and surrounding mortar were modeled to be rigid-plastic bodies, where the effect of aggregate grading was considered. Yoshikawa et al.14 proposed a pathindependent model which clearly classified the shear transfer phenomena into four independent components: ACI Structural Journal/January-February 2015

interface shear stiffness, crack dilatancy, frictional contact, and confining stiffness. Investigation into the complex nature of stress transfer in concrete needs an analytical tool that is accurate and suitable to fundamental characteristics of the stress transfer. In this aspect, Li et al.15 made a great deal of contribution by thoroughly investigating various models, including those which use microscopic physical models that simulate stress transfer based on anisotropic crack surface geometry. They proposed the original Contact Density (OCD) Model, which is simple and very successful in dealing with nonlinearity, shear dilatancy, and path-dependent characteristics of stresstransfer problems. It was proposed that the complicated asperity of a crack surface can be divided into infinitely small pieces, defined as contact units, with various global inclinations. They suggested a simple trigonometric formula called contact density function (CDF) to represent the directional distribution of the contact units which are supposed to be independent of size and grading of aggregates.5,15 Based on extensive experimental investigations, Bujadham and Maekawa9,10 developed a relatively complicated model called Universal Shear Transfer model for generalized paths under static and cyclic loading.10 Following the same approach, Ali and White16 proposed a model that introduced friction in the CDF. Additionally, the roughness of the interface was correlated with the fracture energy of concrete to enable the prediction of shear friction (aggregate interlock) capacity of normal-strength concrete (NSC) as well as highstrength concrete (HSC).16 Based on the experimental observations, Gebreyouhannes11 extended the universal shear transfer model to capture the behavior of cracked concrete under long term loading paths. This paper investigates the stress-transfer capabilities of RC cracks and interfaces by adopting resisting mechanisms (such as aggregate interlock and dowel action). The model proposed by Moradi et al.17 is used to simulate dowel action. To simulate the stress-transfer behavior of cracked concrete, the OCD framework is adopted. The original assumptions and proposals of the OCD are reviewed and the accuracy of the model is shown through comparison with the corresponding experimental results. It will be concluded that the CDF, suggested by Li et al.,15 is the main disadvantage of the OCD model and has a considerable role in the amount of transferred stress. Then the original CDF will be enhanced by ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-204.R2, doi: 10.14359/51687297, received May 25, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

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component of the local compressive deformation ωθ of each contact unit that can be computed from the compatibility of crack deformations (Δ, ω) (Eq. (7)).

Fig. 1—Original assumptions and proposals for OCD suggested by Li et al.15: (a) definition of contact units; (b) contact stress direction for contact unit; (c) histogram of contact direction; and (d) elasto-plastic model for contact compressive stress. suggesting a consistent CDF which considers crack roughness, concrete compressive strength, aggregate size, the rate of asperity degradation and the applied loading paths. RESEARCH SIGNIFICANCE The deformational behavior and the ultimate capacity of RC structures may be considerably affected by weak joints and cracks and also the corresponding interaction between concrete and reinforcement. Local and global behavior of RC members and structures may be affected by cracks and the capabilities of stress transfer. Constitutive models can be proper tools to assess, analyze, and design RC members and structures while considering stress-transfer mechanisms. The proposed model can predict the shear capacity of an interface or cracks while considering the stress-transfer mechanisms. Furthermore, it can simulate the stress-deformation relation of RC cracks under complicated loading paths. AGGREGATE INTERLOCK OCD model The shape of a crack surface is often rough and when loads (shear and normal displacements) are applied to the crack, the two crack surfaces touch each other and the corresponding normal and shear stresses are transferred (Fig. 1(a)). The OCD model was developed based on the extensive experimental observations at the University of Tokyo. The first as well as the main proposal is that the direction of contact stress Z coincides with the orientation of the initial contact unit and remains constant during loading (Fig. 1(b)).15 By measuring the two-dimensional projection of a crack profile in NSC (Fig. 1(c)), Li et al.15 proposed a simple trigonometric CDF, Ω, that can describe the distribution of contact units inclinations across the crack surface (Eq. (1)). It was assumed that the CDF is independent of size and grading of aggregates Gmax, as well as crack width ω. Under a certain applied loading path, some contact units come into contact while the rest of the crack surfaces have no contribution to the transferred stresses (Eq. (2)). Figure 1(d) shows that the contact stress σcon is assumed to be dependent only on the 70

Ω(q) =

1 cos q (1) 2

ωθ = Δsinθ – ωcosθ

(2)

ω max − ω lim , ω q > ω qp ω qp =  (3) 0, ω q ≤ ω qp  Rs =

f y′ ω lim

(4)

f y′ = 13.7 f c1/ 3 (5)

ωlim = 0.04 mm

(6)

σcon = Rs (ωθ – ωθp)

(7)

At =

4 (8) p

where Rs is the elastic rigidity per length; ωθp is the local plasticity in θ direction, which has path-dependent characteristics (Fig. 1(d)); and At is the whole surface area per unit projected crack area. ωθp can be computed by transforming the local system defined on each contact unit (Fig. 1(b)). The effective ratio of contact K(ω) was introduced to take into account the effect of crack roughness size on the mechanical behaviors of stress transfer, apart from the effect of crack shape expressed by the CDF (Eq. (9)). The function for K(ω) was proposed to be related to the maximum size of coarse aggregate Gmax and crack width ω. The CDF Ω(θ), proposed to represent the directional distribution of the contact units θ, which is independent of the maximum size of aggregate.15

 G  K (ω ) = 1 − exp 1 − max  ≥ 0 (9)  2ω 

Z = K(ω)AtΩ(θ)σcon (10)

To satisfy the equilibrium condition, the summation of contact forces Z for all contact units must balance external shear and normal stresses transferred across a crack as15 p

τ = ∫ 2p Z sin qd q (11) −

2

p 2 p − 2

σ = ∫ Z cos qd q (12) ACI Structural Journal/January-February 2015

Feenstra et al.18,19 implemented the five different crack-dilatancy models in interface elements to model discrete cracking. Ample attention was given to issues like formulation of the nonsymmetrical tangential stiffness relation and the stability of the models, including eigenvalues analyses. They concluded that the OCD model resulted in the best correlation with experiments at the lowest computational cost. Herein, the OCD is adopted to simulate the

Fig. 2—Comparison of OCD model with experimental results reported by Maekawa et al.5: (a) test setup; (b) shear stress versus shear displacement under repeated load path; and (c) shear stress versus shear displacement. (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.)

response of cracked concrete. In fact, it is modified and generalized to capture the behavior of the aggregate interlock mechanism. To have a better insight into the accuracy of the OCD model, the results were compared with some experimental studies under different loading paths. A comparison of the test and predicted results for two different crack widths under repeated loading path demonstrates a good agreement, as shown in Fig. 2.15 The OCD can predict the general trend and the amount of transferred shear. Figure 3 shows the comparison between the experimental study conducted by Paulay and Loeber3 and the results of the OCD. It can be found that the accuracy of the model is reduced by increasing the corresponding crack width. Increasing the crack width leads to relatively poor correlation between the model and the experiment. It seems that the accuracy and the reliability of the OCD depend on the specific loading paths where the crack width remains constant or varies little. However, in some structures, loading paths involve both shear slip and crack opening at the same time. To investigate stress-transfer behavior under such deformational paths, Bujadham and Maekawa9,10 performed a series of stress-transfer experiments on precracked concrete specimens in which crack deformation could be arbitrarily controlled. Figures 4(a) and (b) show the applied loading paths and the results of the corresponding experiments, respectively. Bujadham and Maekawa9,10 used the step-type loading paths for checking and examining the applicability of their assumptions. The analytical results of the OCD model are also shown in the figure and except for the first loading step, the model predictions tend to be overestimated (Fig. 4(b) and (d)). It can be concluded the CDF has a considerable role on the accuracy of the model. As can be seen, the proposed formula for the CDF by Li et al.15 just depends on crack unit orientations (Eq. (1)). But, some comparisons and analyses revealed its drawbacks (Fig. 3 and 4). In fact, it does not depend on the Gmax, ω, or crack asperity degradation α, so it can be concluded that the previously mentioned parameters should be considered in the proposed generalized formulation (Eq. (13)). Herein, we assume that the generalized form of the modified CDF can be written as a normal distribution function with following standard deviation.

Fig. 3—Comparison of OCD model with experimental results reported by Paulay and Loeber3: (a) test specimens; and (b) shear stress versus shear displacement. (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.) ACI Structural Journal/January-February 2015

71

Fig. 4—Comparison of OCD with experimental investigation reported by Maekawa et al.5: (a) step-type loading path; (b) shear stress versus shear displacement; (c) step-type loading path; and (d) shear stress versus shear displacement. (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.) sΩ(θ, fc, Gmax, ω, α) = sF1(θ, fc) × F2(Gmax) × F3(ω) × F4(α)

(13)

Equation (13) shows the general form of the proposed equation of standard deviation of the modified CDF. The first term describes the contact units’ orientations and the contribution of concrete compressive strength to shear strength. The second term represents the role of the maximum aggregate size on the shear capabilities of cracked concrete. The last two terms are characterized loading path (crack width) and crack roughness degradation rate, respectively. Each term will be described in the next section. Modified contact density function Concrete compressive strength and contact units’ inclinations—Laible et al.4 experimentally simulated shear transfer capability across a concrete interface, called Interface Shear Transfer (IST). Experimental observation revealed that crack surface roughness has a considerable effect on the amount of transferred shear stress.4 In the past, extensive experimental and analytical investigations have been carried out to investigate joint roughness, especially in rock mechanics framework. Some efforts have been devoted to employ normal distribution (Gaussian), while some others used gamma distribution to determine joints roughness.20 Also, some models adopt a non-periodic function similar to a joint profile to approximate the joint profile by using summation of some periodic function. This means that the joint profile in the whole joint length regarded as a periodic function can be divided into wavelength, amplitude, and phase.21 As it was mentioned, Li et al.15 measured the two-dimensional projection of a crack plane that was experimentally scanned to get some ideas about the crack surface geometry. Li et al.15 proposed Eq. (1) to account for the probabilistic distribution of the contact units orientations. Shear behavior of HSC was experimentally investigated by Bujadham and 72

Maekawa.9,10 They observed that the crack pattern of HSC is completely different from NSC because HSC has a smoother crack surface. Bujadham and Maekawa9,10 probabilistically idealized the geometry of HSC crack surface by using a normal distribution function (NDF)

Ω (q) =

 5  q exp  −21   p 6 

2

  (14) 

It seems that any kind of crack roughness can be described probabilistically by using a proper NDF. In fact, different types of joints—such as NSC, HSC, construction joints, and masonry joints—can be defined uniformly by a simple NDF. The general form of an NDF can be written as the following equation (Eq. (15)). It can be defined by two parameters, m and σvar

F1 (q, m, f c ) =

1 σ var

 1  q − m 2  exp  −    (15) 2p  2  σ var  

where m is the mean; and σvar is known as the standard deviation. Li et al.15 showed that the CDF is a zero-mean function, so in Eq. (16), σvar is the only unknown parameter. Knowing σvar, the roughness probabilistic distribution for any kinds of joints can be determined

m = 0 → F1 (q, f c ) =

1 σ var

 1  q 2 exp  −    (16) 2p  2  σ var  

Equation (1) can be used to determine σvar for NSC.

p/2

E q 2  = ∫− p / 2 Ω(q)q 2 d q =

p2 − 2 (17) 4


where E[θ2] is the mean square value of θ. Equation (18) shows the standard deviation of contact units inclinations for NSC.

σ var = E q 2  =

p2 − 2 (18) 4

Knowing σvar for NSC, Eq. (16) can be described and plotted easily. Similarly, σvar for HSC can be derived as follows

Ω (q) =

2  5  q  exp  −21   → σ var = 0.48 (19)  p   6 

Equation (16) for NSC, HSC, and the original CDF are plotted and compared in Fig. 5. The figure reveals that by increasing the strength of concrete fc, σvar is decreased. The original CDF and the proposed equation for NSC (Eq. (16)) are relatively close. The original CDF has higher values, almost along 90 degrees ≥ |θ| ≥ 30 degrees, but Eq. (16) has higher values, between |θ| and 30 degrees. This means that Eq. (16) has a lower contribution in 90 degrees ≥ |θ| ≥ 30 degrees than the original CDF. But Eq. (19) has different values for all inclinations. It can be seen that the corresponding curve has higher values between |θ| < 30 degrees. In fact, the probabilistic contribution of horizontal surfaces is more than the vertical units. This states that the crack surface has a smoother surface than NSC because it has a smaller σvar (Eq. (19)). It can be concluded that any kind of surface geometry is described by a simple NDF. Effect of maximum aggregate size—Due to the roughness of the crack faces, stress can be transferred from concrete to concrete. This mechanism is based upon the fact that in NSC, the aggregates have a much higher strength than the matrix material.1 Therefore, a crack runs through the matrix and along the interface between aggregates and cement paste. As a consequence, the stiff aggregates cause the crack surfaces to roughen. Decreasing the maximum aggregate size makes crack profiles smoother, as in HSC, and subsequently reduces the shear transfer capability. An experimental study carried out by Thom22 stated that shear strength provided by the aggregate interlocking increased to some extent with increasing maximum aggregate size. Experimental results reported by Li et al.15 as well as Wattar23 expressed that there are no noticeable differences in crack profiles for specimens with aggregate sizes of 15 and 25 mm (0.6 and 1.0 in.). Thus, it seems that for Gmax < 15 mm (0.6 in.) and NSC, the original CDF should be modified. Equation (22) expresses the reduction of stress-transfer capability due to the size of coarse aggregates

η=

Gmax (20) 15

  exp(−a Ω ) η ψ ( η, a Ω ) = 1 −  (21)  1 + exp ( −a Ω ) − 1 η 


Fig. 5—Comparison between proposed model for distribution of contact units’ inclinations for NSC and HSC and CDF suggested by Li et al. 15

Fig. 6—Variation of F2(Gmax) with respect to αΩ. (Note: 1 mm = 0.039 in.)

F2(Gmax) = [1 – ψ(η, αΩ)]

(22)

where ψ controls the contribution of maximum aggregate size in the stress-transfer capabilities of cracked concrete (Eq. (21) and (22)). In fact, ψ provides a family of descending curves for F2 depending on the values of αΩ (Fig. 6). A linearly decreasing relationship is obtained in the particular case of αΩ = 0. Figure 6 shows the variation of F2(Gmax) versus Gmax. For Gmax > 15 mm (0.6 in.), there is no reduction in F2(Gmax) because the experimental results showed no differences; but for Gmax < 15 mm (0.6 in.) and according to αΩ, different kinds of variation and reduction will be obtained. Proposing a unique formulation for ψ requires the proper and sufficient experimental data. Effect of crack width (loading path)—Experimental studies have shown that the amount of shear transfer across cracks directly depends on crack width. Also, applied normal stress can be regarded as an external restraint to adjust the corresponding crack width and have a reasonable effect on shear transfer capability. Figure 7 shows a typical trend of shear stress versus slip response with the probable sequence of occurrence of the different mechanisms. The relative participation of these mechanisms depends on crack width. Figures 7(a) and (b) indicate that for a larger initial crack width, a lower shear strength will be expected, and the participation of the first zone in Fig. 7(b) grows consequently. In fact, for a wider crack width, there will be fewer 73

Fig. 7—Stress-transfer mechanism across cracked concrete: (a) role of initial crack width in shear capacity of crack; (b) schematic representation of interface shear; and (c) variation of proposed relation with respect to crack width. (Note: 1 mm = 0.039 in.) contact units to activate and engage in contact with, and, hence, the shear capacity will be lower.22 According to the consistency of the crack deformational path, the range of active contact units can be detected ω q = D sin qc − ω cos qc = 0

 D qc = cot −1    ω

(23)

where θc defines the lower inclinations of the active contact units at each loading step (Δ, ω). It can be seen that increasing ω leads to a decrease in the active contact units’ range, and subsequently, the amount of transferred shear is reduced. As Fig. 3 depicts, the OCD cannot capture the effect of loading path on the behavior of cracked concrete because it depends on θ. Figure 3 states that for ω = 0.13 mm (0.005 in.), the OCD model has fair accuracy, but it gradually decreases. Consequently, a proper formulation to account for the loading path can be proposed (Fig. 7(c))

 1  F3 (ω ) = exp  (24)  50ω 

Effect of roughness degradation rate—Li et al.15 proposed the CDF to idealized crack surface roughness and geometry probabilistically and it is supposed to be independent of crack asperity deterioration. The OCD is considered a CDF constant during loading, unloading, and reloading processes. On the other hand, it was assumed that there is no fracture and roughness degradation and the surface area is constant. Tassios and Vintzeleou25 carried out block-type experiments exploring influence of surface roughness (smooth, sand blasted, and rough) upon shear strength while the normal restraint stress was kept constant during loading. The rough-

74

ness of the shear plane was measured before and after the actual shear test. For the rough interface, the roughness, defined as half the height of the protruding asperities, was 1.75 mm (0.069 in.) before and 1.45 mm (0.058 in.) after testing due to the deterioration of the crack face.25 Some efforts have been made to measure and study asperity degradation for different types of joints.15,22 It can be concluded that the typical features of rough joint behavior under different loading paths, such as peak shear strength and nonlinear dilation, are significantly affected by the degradation of joint asperities.26-29 Some models considered first- and second-order asperities. The first is a Patton-type model consisting of sawtooth asperity surfaces which degrade, and the second is a sinetooth asperity surface model in which the irregularities are idealized as a series of continuous sine functions which degrade (Fig. 8(a) and (b)).26 The joint asperity was formulated as the variation of the initial asperity angle, which would be evaluated by secant or tangential slope of dilation curves.26-29 Plesha28 and Dowding et al.29 proposed Eq. (25) and (26) to represent the degradation of asperity angle at each load step. In this formula, β0 is the initial asperity angle, a controls the rate of asperity deterioration, and W is the work or energy dissipated the frictional sliding. It seems that the crack width has an effective contribution in joint shear mechanism as well as shear slip. Equations (27) and (28) are suggested to account for work dissipation across crack with respect to the corresponding loading path. As was stated before, the proposed NDF idealized the crack surface considering all inclinations; that is, –π/2 ≤ θ ≤ π/2. Joint roughness degradation is expressed by an exponential formulation which reflects the variation of the standard deviation. σvar0 is the initial standard deviation; α controls the rate of asperity degradation; and Wcr is the work spent on fracture processes during loading


Fig. 8—Crack roughness definitions: (a) assumptions in rock mechanics framework; (b) crack angle variation during loading to represent roughness degradation; (c) variation of standard deviation of proposed formulation; and (d) variation of Eq. (28) to simulate asperity degradation. t

dW = τd D → W = ∫ dWdt (25) 0

β = [1 – exp(–αW)]β0 (26)

dW cr = τd d + σd ω → W cr = ∫ dW cr dt (27)

t

0

σ var = F4 (a ) = exp( −a W cr ) (28) σ var 0

Equation (28) determines the variation of initial standard deviation based on work dissipation during loading. Figure 8(c) shows the variation of the standard deviation with respect to Wcr and Fig. 8(d) explains the effect of the standard deviation variation on the proposed NDF (Eq. (16)) qualitatively. As can be seen, loading causes a reduction in the initial standard deviation and the corresponding NDF becomes narrower, which means that the crack surface asperities deteriorate. Now, knowing all terms, Eq. (13) can be rewritten as σ Ω (q, f c , Gmax , ω, a ) = σ var

    exp ( −a Ω ) η × 1 − 1 −    1 + (exp( −a Ω ) − 1) η   (29)  1  × exp  × exp( −aW cr )  50ω 

Equation (29) shows the complete and final form of the proposed modified CDF applied at the framework of the ACI Structural Journal/January-February 2015

OCD. The suggested formulation expresses and quantifies all contributing aspects of stress-transfer capabilities. DOWEL ACTION Moradi et al.17 proposed a macroscale model to simulate dowel behavior of crossing bar across cracks. The proposed model is established based on the experimental program and the available experimental results. The model formulation is based on the beam on elastic foundation (BEF) theory and extended to the beam on inelastic foundation (BIF) theory by proposing a consistent formula for subgrade springs. The effect of concrete cover splitting as well as the effect of axial stress of deformed bars is considered.17 Herein, the model is adopted to express the dowel behavior of bars under different loading paths. More details are available in Reference 17. STRESS-TRANSFER MECHANISM ACROSS RC CRACKS AND INTERFACES To simulate the behavior of different kinds of RC cracks, interfaces, and construction joints, reliable constitutive models are necessary. As the shear displacement applies at the crack plane, roughness of crack surface tends to widen the crack width (dilatancy). This crack opening increases the axial stress of the bar Asss, while shear displacement causes flexure effect in the bar. The overall stress state in reinforcing bar and surrounding concrete governs the crack opening and slip, which can control the stress transfer across the crack plane. The proposed procedure for finding crack stresses and openings starts by satisfying the equilibrium (Eq. (30)) normal to the crack (Fig. 9). So for any given shear displacement, the unknown stresses and crack opening can be determined in an iterative way (Fig. 10). Figure 9 shows the schematic behavior and stress-transfer mechanism of a single RC crack subjected to shear force V, due to 75

σ = σ(Δ, ω)

σ s = σ s ( S , d ) (32)

(31)

where δ is the transverse displacement of the bar. The compatibility between the normal and transverse displacement of the concrete and for the reinforcing bar is expressed by Maekawa et al.5 as

Fig. 9—Shear-transfer mechanisms across reinforced concrete crack: (a) deformational and mechanical characteristics of RC interface; (b) equilibrium condition for bars perpendicularly crossing crack plane; and (c) equilibrium condition for inclined bars crossing crack plane.

Δ = 2δ; ω = c(2S) (33)

where c is a factor that takes into account the variation in the crack width from the bar surface to the concrete surface (Fig. 9(a)).5 Once the displacement path (Δ, ω) satisfies the equilibrium, the constitutive models for concrete shear τ and steel τs contributions determine the corresponding mechanisms, and the total transferred shear can be computed as follows

τt = τ + ρτs (34)

τ = τ(Δ, ω)

(35)

τs = τs(S, δ)

(36)

The flowchart for solving (Eq. (30)) across RC cracks is shown in Fig. 10, starting from an assumed crack opening and adopting an iterative-based approach until it can satisfy the equilibrium. The only parameter that should be sought is the crack opening because shear displacement is known as input.

Fig. 10—Flowchart of computing shear transfer across crack plane. which a relative shear displacement, Δ, results. To determine the shear transfer across the crack plane, the equilibrium of stresses at a crack can be written as

σ=

N + rσ s (30) Ac

where ρ is the reinforcement ratio; and N is externally applied force defined positive in compression. In fact, by means of equilibrium (Eq. (30)), it can be computed as normal force due to aggregate interlock σ and the axial bar stress σ s . Normal stress of concrete (due to dilatancy) and bar axial stress (due to transverse displacement and axial slip) is as follows 76

EXPERIMENTAL VERIFICATION Systematic and extensive experimental verification is conducted for clarifying the versatility of the proposed model and assumptions. In first part, the proposed model for aggregate interlock mechanism is compared with some experimental works under different loading paths. Then the stress-transfer behavior of a single crack is examined by some experimental studies considering aggregate interlock and dowel action mechanisms. Comparisons are shown in the framework of shear stress-shear displacement predictions as well as the ultimate shear strength of RC cracks. Also, the contribution of the aforementioned mechanisms against the applied shear is determined for different reinforcement ratios and bar diameters. Aggregate interlock To verify the proposed and the suggesting assumptions, the experimental and the computed transferred stress provided by aggregate interlock mechanisms under different kinds of loading paths are compared. Paulay and Loeber3 performed tests on precracked pushoff-type specimens. The upper part of the specimens could slide along the shear plane of the lower part, which was fixed. The comparison of the analysis and the experimental results are shown in Fig. 11(a) for ACI Structural Journal/January-February 2015

Fig. 11—Comparison of modified contact density model and test results: (a) Paulay and Loeber3; and (b) Thom.22 (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.) different initial crack width. Agreement between the calculated and the experimental values seems reasonable. Thom22 experimentally simulated the capability of cracked concrete and the role of the maximum aggregate size in shear. The comparison of the corresponding shear stressshear displacement curves is given in Fig. 11(b). Both specimens have almost the same compressive strength, while the maximum size of the aggregate for the second specimen is half of the first one. Figure 11(b) shows that increasing Gmax changes the general trend of the curve and also reduces the shear strength of cracked concrete at a particular shear displacement (that is, Δ = 0.6 mm [0.02 in.]). The contribution of aggregate size on the amount of transferred shear is well predicted. Li et al.15 experimentally investigated the basic proposals of the OCD under reversed cyclic loading. Figure 12 shows a reasonable correlation between analytical and experimental results for reversed cyclic shear loading under constant crack width of 0.3 and 0.5 mm (0.01 and 0.02 in.) in terms of shear stress-shear displacement and also shear stress-normal stress. The step-type loading paths, which were introduced in the previous section, are used to show the accuracy of the proposed model (MCD) and are compared with the OCD model. Figure 13 shows the shear stress-shear displacement of the first loading path and it can be seen that the MCD has a fair correlation with the corresponding experimental results. It indicates the proper assumptions for the suggested CDF. Good agreement between the model and test results is confirmed for the second step-type loading path (Fig. 13). Verification of stress-transfer model Stress transfer takes place through aggregate interlock and dowel action across cracked RC surfaces. To simulate this, it is necessary for each of those mechanisms to have correct performance. Prediction for the behavior and the entire response of RC interfaces will be verified in the framework of shear stress-shear displacement. In addition to comparing the shear stress-shear displacement relationship, the ultimate shear strength of precracked as well as uncracked push-off and pulloff specimens are compared with the results of the proposed model. Maekawa and Qureshi30 tested beam-type specimens and determined the contribution of each mechanism separately (Fig. 14). Predictions for the cases described in Reference 5 are shown in Fig. 14 and are in fair agreement with the experimental results obtained. Cracks and damages in the ACI Structural Journal/January-February 2015

Fig. 12—Comparison of numerical and experimental results reported by Li et al.5 for reversed cyclic loading under constant crack width. (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.)

Fig. 13—Comparison of behavior under step-type loading path: (a) applied loading path; (b) transferred shear-stress responses; (c) applied loading path; and (d) transferred shearstress responses.5 (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.) subgrade concrete, reduction in the bond between concrete and reinforcement, and the localized nonlinearity in the steel are the main sources of the nonlinear behavior. The total behavior of the specimens are well simulated by the proposed model. Besides the comparison between the total response of the model and the experiment, the contribution of each mechanism from the models is shown. It can be concluded that the contribution of dowel action depends on the reinforcement ratio ρ, concrete strength fc, bar diameter db, and the roughness of the interface. Mattock and Hawkins31 investigated the shear transfer capabilities of RC cracks and interfaces using uncracked pulloff specimens. The comparison between the ultimate shear capacity of the specimens and the corresponding results of the model are shown in Table 1. Also, the details of specimens and materials are shown in the table. The ratios of the analysis to the experiment shear capacity show the fair accuracy of the model to predict the ultimate shear strength of the specimens where both aggregate interlock and dowel action are the main shear mechanisms. Similarly, the comparison with the experimental program reported by Sagaseta and Vollum32 and the results of the proposed model are shown in Table 1. In this table, ω0 is the initial crack width of the specimens. They use precracked 77

Table 1—Comparison with experimental results reported by Mattock and Hawkins31 and Sagaseta and Vollum32

Mattock and Hawkins

Sagaseta and Vollum

Experiment

Analysis

Specimen No.

db, mm

ω0, mm

fy, MPa

fc, MPa

ρfy

τu, MPa

τu, MPa

Ratio

7.1

9.5

0

341.3

33.4

2.65

5.87

5.5

0.94

7.2

9.5

0

341.3

35.3

3.97

6.25

6.9

1.10

7.3

9.5

0

341.3

34.8

5.3

6.72

7.8

1.16

PL2

8.0

0.132

550

53.1

2.31

4.85

5.14

1.06

PL2b

8.0

0.093

550

53.1

2.31

5.82

5.71

0.98

PL3

8.0

0.123

550

53.1

3.52

5.55

5.52

0.99

PL4

8.0

0.12

550

53.1

4.68

7.1

7.55

1.06

PG2

8.0

0.273

550

31.7

2.31

3.67

4.06

1.11

PG3

8.0

0.081

550

31.7

3.52

4.91

4.67

0.95

Average

1.04

Coefficient of variation

7.45%

Notes: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.

strength, surface geometry, maximum aggregate size, the rate of asperity degradation, and applied loading path. The NDF was developed to express the distribution of contact units’ inclinations based on the compressive strength and material properties. By using the NDF, the rate of roughness degradation can be considered. Also, the role of aggregate size and loading paths were adopted consistent with the OCD model framework. The model proposed by Moradi et al.15 was adopted to consider the dowel action mechanism across RC cracks and interfaces. The model formulation was developed by using the BEF analogy and was extended to the BIF by suggesting a simple formula for subgrade spring stiffness. By combining the aforementioned mechanisms and also considering the nonlinear interaction of reinforcement and surrounding concrete, the response and behavior of stress transfer across RC cracks is verified. This model can easily be implemented for not only discrete but also smeared crack approach to simulate the behavior of RC members and structures. Fig. 14—Predicted and experimental shear stress—associated displacement relation at interface (Maekawa and Qureshi30) and test setup.5 (Note: 1 mm = 0.039 in.; 1 MPa = 0.145 ksi.) push-off specimens to investigate the influence of aggregate fracture on shear transfer through cracks. The average ratio of the analysis ultimate shear to the corresponding value of the experiment depicts the versatility of the model in determining the shear capacity of the RC cracks and interfaces. CONCLUSIONS Stress-transfer capabilities across RC cracks were investigated. The OCD model resulted in fair correlation with experiments at low computational cost and it is also very attractive for its simplicity. The original CDF depends on the contact units inclinations and it was modified to incorporate the effective parameters. It was shown that the modified CDF can be described based on the concrete compressive 78

AUTHOR BIOS Ali Reza Moradi received his PhD from the Department of Civil and Environmental Engineering at Tarbiat Modares University, Tehran, Iran. His research interests include nonlinear analysis and design of concrete structures and development of constitutive models. Masoud Soltani is an Associate Professor of civil engineering at Tarbiat Modares University. He received his PhD from the University of Tokyo, Tokyo, Japan. His research interests include nonlinear mechanics and constitutive laws of reinforced concrete, numerical modeling of masonry structures, and seismic response assessment and rehabilitation of structures. Abbas Ali Tasnimi is a Professor of civil engineering at Tarbiat Modares University. He received his PhD from the University of Bradford, Bradford, UK. His research interests include nonlinear mechanics and constitutive laws of reinforced concrete, seismic nonlinear analysis, numerical modeling of reinforced concrete and masonry structures, and seismic response assessment of structures.

REFERENCES 1. Pruijssers, A. F., “Aggregate Interlock and Dowel Action under Monotonic and Cyclic Loading,” PhD thesis, Delft University of Technology, Delft, the Netherlands, 1988, 193 pp.


2. Houde, J., and Mirza, M. S., “A Finite Element Analysis of Shear Strength of Reinforced Concrete Beams,” Shear in Reinforced Concrete, SP-42, American Concrete Institute, Farmington Hills, MI, 1974, pp. 103-128. 3. Paulay, T., and Loeber, P. J., “Shear Transfer by Aggregate Interlock,” Shear in Reinforced Concrete, SP-42, American Concrete Institute, Farmington Hills, MI, 1974, pp. 1-16. 4. Laible, J. P.; White, R. N.; and Gergely, P., “Experimental Investigation of Seismic Shear Transfer Across Cracks in Concrete Nuclear Containment Vessels,” Reinforced Concrete Structures in Seismic Zones, SP-53, N. M. Hawkins and D. Mitchell, eds., American Concrete Institute, Farmington Hills, MI, 1977, pp. 203-226. 5. Maekawa, K.; Pimanmas, A.; and Okamura, H., Nonlinear Mechanics of Reinforced Concrete, first edition, SPON Press, London, UK, 2003, 768 pp. 6. Walraven, J. C., and Reinhardt, H. W., “Theory and Experiments on the Mechanical Behavior of Cracks in Plain and Reinforced Concrete Subjected to Shear Loading,” HERON, V. 26, 1981, pp. 1-68. 7. Millard, S. G., and Johnson, R. P., “Shear Transfer across Cracks in Reinforced Concrete due to Aggregate Interlock and to Dowel Action,” Magazine of Concrete Research, V. 36, No. 126, 1984, pp. 9-21. doi: 10.1680/macr.1984.36.126.9 8. Divakar, M. P.; Fafitis, A.; and Shah, S. P., “Constitutive Model for Shear Transfer in Cracked Concrete,” Journal of Structural Engineering, ASCE, V. 113, No. 5, 1987, pp. 1046-1062. doi: 10.1061/ (ASCE)0733-9445(1987)113:5(1046) 9. Bujadham, B., and Maekawa, K., “Qualitative Studies on Mechanisms of Stress Transfer across Cracks in Concrete,” Proceedings of JSCE, V. 17, No. 451, 1992, pp. 265-275. 10. Bujadham, B., and Maekawa, K., “The Universal Model for Stress Transfer across Cracks in Concrete,” Proceedings of JSCE, V. 17, No. 451, 1992, pp. 277-287. 11. Gebreyouhannes, E. F., “Local-Contact Damage Based Modeling of Shear Transfer Fatigue in Cracks and its Application to Fatigue Life Assessment of Reinforced Concrete Structures,” PhD thesis, The University of Tokyo, Tokyo, Japan, 2006, 148 pp. 12. Bažant, Z. P., and Tsubaki, T., “Concrete Reinforcing Net: Optimum Slip-Free Limit Design,” Journal of the Structural Division, ASCE, V. 105, No. 2, 1979, pp. 327-346. 13. Bažant, Z. P., and Gambarova, P., “Rough Cracks in Reinforced Concrete,” Journal of the Structural Division, ASCE, V. 106, No. 4, 1980, pp. 819-842. 14. Yoshikawa, H.; Wu, Z.; and Tanabe, T., “Analytical Model for Shear Slip of Cracked Concrete,” Journal of Structural Engineering, ASCE, V. 115, No. 4, 1989, pp. 771-788. doi: 10.1061/(ASCE)0733-9445(1989)115:4(771) 15. Li, B.; Maekawa, K.; and Okamura, H., “Contact Density Model for Stress Transfer across Cracks in Concrete,” Journal of the Faculty of Engineering, the University of Tokyo, V. 40, No. 1, 1989, pp. 9-52. 16. Ali, M. A., and White, R. N., “Enhanced Contact Model for Shear Friction of Normal and High-Strength Concrete,” ACI Structural Journal, V. 96, No. 3, May-June 1996, pp. 348-361. 17. Moradi, A. R.; Soltani, M. M.; and Tasnimi, A. A., “A Simplified Constitutive Model for Dowel Action across RC Cracks,” Journal of Advanced Concrete Technology, V. 10, No. 8, 2012, pp. 264-277. doi: 10.3151/jact.10.264


18. Feenstra, P.; de Borst, R.; and Rots, J. G., “Numerical Study on Crack Dilatancy Part I: Models and Stability Analysis,” Journal of Engineering Mechanics, ASCE, V. 117, No. 4, 1991, pp. 733-753. doi: 10.1061/ (ASCE)0733-9399(1991)117:4(733) 19. Feenstra, P.; de Borst, R.; and Rots, J. G., “Numerical Study on Crack Dilatancy Part II: Applications,” Journal of Engineering Mechanics, ASCE, V. 117, No. 4, 1991, pp. 754-769. doi: 10.1061/ (ASCE)0733-9399(1991)117:4(754) 20. Misra, A., “Effect of Asperity Damage on Shear Behavior of Single Fracture,” Engineering Fracture Mechanics, V. 69, No. 17, 2002, pp. 19972014. doi: 10.1016/S0013-7944(02)00073-5 21. Yang, Z.; Taghichian, A.; and Li, W., “Effect of Asperity Order on the Shear Response of Three-Dimensional Joints by Focusing on Damage Area,” International Journal of Rock Mechanics and Mining Sciences, V. 47, No. 6, 2010, pp. 1012-1026. doi: 10.1016/j.ijrmms.2010.05.008 22. Thom, C. M., “The Effect of Inelastic Shear on the Seismic Response of Structures,” PhD thesis, University of Auckland, Auckland, New Zealand, 1988, 203 pp. 23. Wattar, S. W., “Aggregate Interlock Behavior of Large Crack Width Concrete Joints in PCC Airport Pavements,” PhD thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 2001, 556 pp. 24. Divakar, M. P., and Fafitis, A., “Micromechanics-Based Constitutive Model for Interface Shear,” Journal of Engineering Mechanics, ASCE, V. 118, No. 7, 1992, pp. 1317-1337. doi: 10.1061/ (ASCE)0733-9399(1992)118:7(1317) 25. Tassios, T. P., and Vintzeleou, E. N., “Concrete-to-Concrete Friction,” Journal of Structural Engineering, ASCE, V. 113, No. 4, 1987, pp. 832-849. doi: 10.1061/(ASCE)0733-9445(1987)113:4(832) 26. Lee, H. S.; Park, Y. J.; Cho, T. F.; and You, K. H., “Influence of Asperity Degradation on the Mechanical Behavior of Rough Rock Joints under Cyclic Shear Loading,” International Journal of Rock Mechanics and Mining Sciences, V. 38, No. 7, 2001, pp. 967-980. doi: 10.1016/ S1365-1609(01)00060-0 27. Hutson, R. W., and Dowding, C. H., “Joint Asperity Degradation during Cyclic Shear,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, V. 27, No. 2, 1990, pp. 109-119. doi: 10.1016/0148-9062(90)94859-R 28. Plesha, M. E., “Constitutive Models for Rock Discontinuities with Dilatancy and Surface Degradation,” International Journal for Numerical and Analytical Methods in Geomechanics, V. 11, No. 4, 1987, pp. 345-362. doi: 10.1002/nag.1610110404 29. Dowding, C. H.; Zubelewicz, A.; O’Connor, K. M.; and Belytschko, T. B., “Explicit Modeling of Dilation, Asperity Degradation and Cyclic Seating of Rock Joints,” Computers and Geotechnics, V. 11, No. 3, 1991, pp. 209-227. doi: 10.1016/0266-352X(91)90020-G 30. Maekawa, K., and Qureshi, J., “Stress Transfer across Interface in Reinforced Concrete Due to Aggregate Interlock and Dowel Action,” Proceedings of JSCE, V. 34, No. 557, 1997, pp. 159-172. 31. Mattock, A. H., and Hawkins, N. M., “Shear Transfer in Reinforced Concrete—Recent Research,” PCI Journal, V. 17, No. 2, 1972, pp. 55-75. doi: 10.15554/pcij.03011972.55.75 32. Sagasta, J., and Vollum, R. L., “Influence of Aggregate Fracture on Shear Transfer through Cracks in Reinforced Concrete,” Magazine of Concrete Research, V. 63, No. 2, 2011, pp. 119-137.

79

NOTES:

80



TECHNICAL PAPER

Title No. 112-S08

Condition Assessment of Prestressed Concrete Beams Using Cyclic and Monotonic Load Tests by Mohamed K. ElBatanouny, Antonio Nanni, Paul H. Ziehl, and Fabio Matta Eight prestressed T-shaped beams were tested using the cyclic load test (CLT) method as proposed by ACI 437-12 followed by the ACI 318-11 monotonic (24-hour) load test method. The objective of the study is to assess the ability of these methods to evaluate damage in prestressed concrete (PC) beams. The test matrix included both pristine beams (subjected to no prior loading) as well as beams that were cracked and artificially predamaged using accelerated corrosion techniques, impressed current, and wet/dry cycles, prior to load testing. Deflections, crack widths, and slipping of the prestressing strands were recorded during the load tests. The load at which the monotonic test was conducted was chosen to be greater than the service load of Class U PC members, which does not allow cracking. This ensured that at the time of the monotonic load test the specimens were significantly damaged. However, the acceptance criteria associated with this test methodology were still met. Only one index in the CLT acceptance criteria (deviation from linearity) identified the condition of the specimens. The deviation from linearity index is found to correlate to the opening and widening of cracks. Keywords: corrosion; cyclic load test (CLT); monotonic (24-hour) load test; prestressed concrete (PC).

INTRODUCTION The economy of developed countries is heavily reliant on the built infrastructure, and the deterioration of concrete buildings and bridges is a major concern to both owners and users. A gap exists between the annual investment needed to improve the conditions of the U.S. infrastructure and the amount currently spent.1 Proper assessment of the integrity of concrete structures is key to help owners to efficiently prioritize maintenance. If the integrity of a structure is in question, load tests may be used for condition assessment.2-4 The American Concrete Institute (ACI) addresses two methods of load testing: 1) a monotonic (24-hour) load testing per ACI 318-115; and 2) cyclic load test (CLT) per ACI 437.1R-07.6 Currently, the CLT method is available as a provisional standard under the leadership of ACI Committee 437.7 The two documents (that is, ACI 318-11 and ACI 437-12) have different condition assessment criteria based on the load-deflection response. It is noted that the applicability of the monotonic load test on modern structures may be questioned, as its acceptance criteria are consistent with design principles and material properties used in the 1920s.4,6 The CLT method is fairly recent; therefore, more data is needed to assess the ability of this method to determine the condition of in-service structures. Furthermore, most of the research conducted on both load testing methods dealt with passively reinforced ACI Structural Journal/January-February 2015

concrete (RC) structures as opposed to prestressed concrete (PC) structures,4,8,9 a type of RC where the reinforcing steel is used in active fashion. This paper describes load tests conducted on eight prestressed T-shaped beams. The CLT (ACI 437-12)7 was performed first followed by the ACI 318-115 monotonic load test. Five specimens were precracked and predamaged using impressed current or wet/dry cycles to simulate the behavior of deteriorating structures and the effect of corrosion, common in coastal areas or where deicing salts are used, on the results of the load tests. The results are used to assess the sensitivity of monotonic (24-hour) load test and CLT methods to structural damage. It was shown that the monotonic load test method failed to identify damage in the specimens while the deviation from linearity index of the CLT was more sensitive to damage. In uncracked (pristine) specimens, the criterion of the deviation from linearity index is not met when the transition from uncracked to cracked condition takes place; thus, permanent damage occurs in the specimen. A modification to the current deviation from linearity acceptance criterion is proposed for the evaluation of PC flexural members. RESEARCH SIGNIFICANCE This study aims to evaluate the performance of current load testing methods when used on PC flexural members. Results indicate that the monotonic (24-hour) load test prescribed by ACI 318-115 is not suitable for condition assessment of PC members. The deviation from linearity index of the CLT method yields better results; however, the current acceptance limits of this index may be associated with permanent damage in the tested members. This study proposes a modification to the current CLT acceptance criteria for the case of PC flexural members. LOAD TESTING METHODS Monotonic load test This method is described in Chapter 20 of ACI 318-115 and has been used for decades. The structure is loaded for 24 hours and deflections are monitored and recorded continuously or intermittently during the test. The load magnitude of the test is determined using Section 20.3.2 of ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-236.R1, doi: 10.14359/51687181, received March 14, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

81

Fig. 1—Schematic of load-versus-deflection curve for: (left) two load cycles (similar to ACI 437.1R-07, Fig. 6.1); and (right) three load sets (similar to ACI 437.1R-07, Fig. 6.2). ACI 318-11,5 where the applied load is slightly less than the required strength (80 to 90% of the required strength) and is considered appropriate compared to load combinations and strength reduction factors. The structure passes the test if the measured deflections satisfy either Eq. (1a) or (1b)

D1 ≤

lt2 (1a) 20, 000 h

Dr ≤

D1 4

(1b)

In Eq. (1a) and (1b), Δ1 (in.) is the maximum deflection recorded during the 24-hour load hold; lt (in.) is the free span of the member under load test; h (in.) is the overall height of the member; and Δr (in.) is the residual deflection measured following load removal (Δr is the difference between initial deflections, measured not more than 1 hour before the test, and final deflections, after load removal). The final response of the structure should be measured 24 hours after load removal. If the measured response does not satisfy either equation, the test may be repeated no sooner than 72 hours from the removal of the test load. The structure passes the repeated test if Eq. (1c) is satisfied where Δr2 (in.) and Δ2 (in.) are the residual deflection and maximum deflection measured during the repeated test.5

Dr 2 ≤

D2 (1c) 5

The objective of the method is to assess the ability of the structure to sustain a load level near the required strength. The test method has three notable drawbacks: 1) the long duration (24 hours + 24 hours); 2) the significant damage that can be imparted to the structure; and 3) it serves primarily as a proof test, with limited ability to assess the condition of the structure under investigation. Cyclic load test (CLT) This method is described in ACI 437.1R-076 and ACI 437-12.7 The method was proposed by ACI Committee 437 as an alternative to the monotonic load 82

test.7-13 The test includes a series of load sets where each load set includes two load cycles with similar load magnitude, Load Cycles A and B, as illustrated in Fig. 1. The maximum load is applied in five load steps with a minimum load hold time at each step. These steps are applied in both the loading and unloading phases. The evaluation criteria are based on three indexes: 1) repeatability; 2) permanency ratio; and 3) deviation from linearity.6 These indexes are calculated using load and deflection measurements. Figure 1 shows a schematic for calculating the CLT parameters.6 1. Repeatability—This parameter represents the ratio between deflections in two subsequent load cycles (Eq. (2)), where the residual deflection (D rcycle) is subtracted from the 6 maximum deflection (D cycle max ) in each cycle. This parameter 6 was included in ACI 437.1R-07 but is not included in ACI 437-12.7 It is included in this paper for completeness. The repeatability criterion is not met if the index is less than 95%6

Repeatability =

B − D rB D max × 100% (2) A − D rA D max

2. Permanency ratio—The calculation of this parameter was modified in ACI 437-12.7 In Eq. (3a), permanency ratio Ipr is calculated for each load set using the two load cycles included in the load set where Ipi (Eq. (3b)) and Ip(i+1) (Eq. (3c)) are the permanency indexes; D ir and D (ri +1) are the i +1) residual deflections; and D imax and D (max are the maximum deflections during the i-th and (i + 1)-th cycles, respectively (first and second load cycles of a particular load set). The permanency ratio criterion is not met if the index exceeds 50%7

Permanency ratio ( I pr ) =

I p (i +1) I pi

× 100% (3a)

D ir (3b) D imax

I pi =

I p (i +1) =

D r(i +1) (3c) i +1) D (max


3. Deviation from linearity—This parameter measures nonlinear response and is assessed in all load cycles. As shown in Eq. (4), deviation from linearity IDL is calculated as 1 minus the ratio between the secant line for the loadversus-deflection plot of a particular cycle tan(αi), and the slope of the reference point from the load versus deflection plot (secant stiffness), tan(αref). The reference point is determined in the first load cycle. The deviation from linearity criterion is not met if the index exceeds 25%.7

 tan(a i )  Deviation from linearity (I DL ) =  1 −  × 100 % (4)  tan(a ref ) 

EXPERIMENTAL PROGRAM Test specimens Eight PC T-section beams were used. Each specimen was reinforced with two 13 mm (0.5 in.) low-relaxation prestressing strands. The strands were prestressed to 68% of their nominal breaking strength (fpu = 1860 MPa [270 ksi]). The flange contained four Φ10 (No. 3) bars. The beams had a length of 4.98 m (16.3 ft) and were designed to fail in

Fig. 2—Cross section of specimens showing dimensions and reinforcement. (Note: 1 mm = 0.0394 in.)

flexure. A schematic of the cross section and reinforcement layout is shown in Fig. 2. Shear reinforcement was provided using Φ10 (No. 3) stirrups with spacing of 240 mm (9.5 in.). The beams were cast in two groups. The first group included two beams with a 28-day concrete compressive strength fc′ of 29.0 MPa (4200 psi). The water-cement ratio (w/c) used in the mixture design was 0.4. All strands used in casting the first group were slightly precorroded with a uniform mass loss of 33 g/m (0.075 lb/ft) (4% per unit length). Predamage was attained by immersing the strands in NaCl solution and leaving them to corrode freely at ambient temperature. One of the beams in this first set was exposed to a chloride solution using 3-day wet/4-day dry cycles to accelerate corrosion. This beam was preloaded to 80% of the nominal capacity, considering tendon mass loss resulted from precorrosion, to achieve cracks having a width of 0.8 mm (0.032 in.) to facilitate the penetration of chlorides. This was achieved in a four-point bending setup, with a constant moment zone of 0.90 m (3 ft). Plastic inserts were secured into the cracks to prevent them from complete closure. The beam was positioned flange down and a 1.22 m (4 ft) long acrylic dike was built over the midsection of the beam to force corrosion to occur in the maximum moment zone, as shown in Fig. 3. A copper plate was immersed in the chloride solution to form a galvanic cell. The corrosion phase continued for 140 days. The second group, identical in prestressing force and design, included six beams. The concrete had a 28-day compressive strength of 40.7 MPa (5900 psi) and a w/c of 0.4. Four of the beams were preloaded to 60% of the nominal capacity to achieve cracks with a width of 0.4 mm (0.016 in.). The tendons of the preloaded beams underwent accelerated corrosion using applied current for different durations to achieve the desired sectional mass and area losses, as shown in Table 1. The theoretical mass losses were calculated using Faraday’s equation, as shown in Eq. (5), where i is the galvanic current in Amperes and t is the time in seconds Mass loss =

i × t × 55.827 (5) 2 × 96, 487

Fig. 3—Corrosion test setup. (Note: 1 mm = 0.0394 in.) Table 1—Description of specimens

Strand condition prior to casting

Eq. (5)

Eq. (6)

Experimental mass loss, % (area loss, %)

—

Corroded (4% per unit length)

0

0

0 (0)

U2

—

Pristine

0

0

0 (0)

U3

—

Pristine

0

0

0 (0)

0.4 (0.016)

Pristine

15 (15.4)

5.9 (6.4)

6.3 (6.8)

C2-0.4

0.4 (0.016)

Pristine

15 (15.4)

7.7 (8.2)

10.2 (10.7)

C3-0.4

0.4 (0.016)

Pristine

30 (30.4)

12.3 (12.8)

12.8 (13.3)

Specimen U1

Compressive strength, Crack width, MPa (psi) mm (in.)

Theoretical sectional mass loss, % (area loss, %)

29.0 (4200)

C1-0.4 40.7 (5900)

C4-0.4 C5-0.8

29.0 (4200)

0.4 (0.016)

Pristine

30 (30.4)

12.5 (13.0)

12.8 (13.3)

0.8 (0.032)

Corroded (4% per unit length)

16 (16.5)

6.9 (7.4)

4.9 (5.4)


83

Fig. 4—Overview of test setup: (a) photograph showing Specimen C5-0.8 in place; and (b) schematic of test setup. The test matrix is shown in Table 1. An “X-Y-Z” format was used, where “X” indicates the condition of the specimen (“U” for uncracked uncorroded, and “C” for cracked corroded); “Y” is the specimen number; and “Z” indicates the precrack width. Experimental setup and instrumentation The specimens were tested in four-point bending, as shown in Fig. 4. Specially designed roller supports at the top of each reaction stand were used to ensure simple support conditions. Each support included (from bottom to top) a steel plate, steel roller, steel plate, and neoprene pad. A reaction frame was built to hold a 245 kN (55 kip) MTS hydraulic actuator. The load was distributed onto the girder through two roller supports 0.90 m (3 ft) apart. A stiffened steel spreader beam was used to transfer the load from the actuator to the roller supports. Two string potentiometers per specimen were used to measure the midspan deflection. Two displacement transducers were placed at the supports to measure settlement and two more displacement transducers (four total) were placed at either end of each prestressing strand to measure strand slip. After visual detection of cracking in the CLT, two crack mouth opening gauges were mounted across selected cracks to measure crack widths during the remaining part of the test. Load intensity The self-weight of the specimen and the loading apparatus were determined and subtracted from the applied loads. The test targeted three loading levels: 1) service load Ps; 2) cracking load Pcr; and 3) required strength (0.9Pn [nominal capacity]). To obtain these loads, the prestress losses were calculated according to ACI 318-11 and the effective prestress fes was determined. The loading protocol was designed such that the concrete tensile stress was limited with the intent of minimizing or eliminating tensile cracking. The service load level Ps was calculated as the load that would 84

cause zero stress at the bottom concrete fiber of the beam. The cracking load was calculated as the load developing tensile stress at the bottom concrete fiber equal to ft, the maximum allowable tensile stress for Class U prestressed members5 equal to 7.5√fc′. Both service and cracking loads were calculated using gross section properties. The nominal capacity was determined using cracked-section properties. The loads for uncracked (pristine) specimens were calculated directly using section properties. For the first group (Specimen U1), the nominal load was calculated to be 103.2 kN (23.2 kip), while for the second group (Specimens U2 and U3), the nominal load was 112.5 kN (25.3 kip). For cracked-corroded specimens, the presence of corrosion leads to a mass loss in the exposed area of the prestressing strands, which in turn leads to a localized reduction in cross-sectional area of the strand and reduces the nominal capacity of the section. The sectional mass loss was calculated by assuming that the distance of exposure in each strand is equal to the crack width plus the distance of chloride penetration at either side of the crack. Because Faraday’s equation is only valid on standalone metals immersed in solution, a modification to this equation was used to estimate the sectional mass loss, as shown in Eq. (6).14 The sectional area loss was estimated by assuming that the density of the corroded section is equal to the density of the original section, as shown in Table 1.

Mass loss = 0.4651 ×

i × t ×55.827 − 0.5624 (6) 2 × 96, 487

The extent of chloride penetration was estimated based on a study by Mangat and Molloy.15 More information regarding the effect of corrosion on the specimens can be found in ElBatanouny16 and ElBatanouny et al.17 Test procedure A CLT protocol was calculated for all the specimens with a loading rate of 0.90 kN/s (0.2 kip/s). The applied loads in each specimen differed based on the degree of corrosion present. Tests were composed of a series of load sets, each containing twin cycles. A typical plot of the CLT loading protocol is presented in Fig. 5. The maximum load of each test in Cycle 1 (P1) was equal to 75% of the service load Ps, where service load was calculated by setting the resultant stress at the extreme tensile concrete fiber to zero. The load level in Cycle 2 (P2) was equal to 90% of the load level in Cycle 1 (0.9P1). These two cycles did not contain any load steps; therefore, the load was ramped at a constant rate to the desired level. The hold period at the top of each load cycle was set to 4 minutes. These cycles are not typical of the CLT protocol as proposed by ACI 437-127 and are introduced for the purposes of acoustic emission evaluation described in ElBatanouny et al.17 The CLT indexes were, however, also calculated in these cycles. The maximum load in the second load set, Load Cycles 3 and 4, was equal to the service level load Ps. This load set contained five load steps at each cycle in both the loading and unloading phases. The load holds in both cycles (time ACI Structural Journal/January-February 2015

between load steps) were equal to 2 minutes at the intermediate load steps and 4 minutes at the peak load for Cycle 3. For Load Cycle 4, the 4-minute hold was employed following the fourth load step. Changes in load hold times are not necessarily used in CLT protocol and were inserted to aid in the acoustic emission evaluation.16,17 The maximum load in the third load set, Load Cycles 5 and 6, was equal to the cracking load Pcr. This was similar to the first load set, in that the load level in Cycle 6 was equal to 90% of the load level in Cycle 5 (0.9Pcr). The remainder of the CLT testing protocol was conducted with a number of stepped load sets similar to load set 2 (Cycles 3 and 4). The load test protocol contained seven load sets (14 load cycles), exceptions being Specimens C5-0.8 and C3-0.4, where eight load sets (16 load cycles) and six load sets (12 load cycles) were applied, respectively. The load levels for all the specimens are shown in Table 2 as a percentage of ultimate experimental capacity Pu of each specimen while the second row in this table shows the theoretical load level. A minimum load of 2.2 kN (0.5 kip) was maintained through the test to keep the actuator engaged. The targeted load value in the last load set was equal to 90% of the nominal capacity (0.9Pn). The loading protocol for Specimen U1 is shown in Fig. 5. All load hold times were 2 minutes except at the maximum load for the initial cycles and at the fourth step in the repeated cycles where the hold time is 4 minutes.

Following the completion of the CLT or after a specific load set, all specimens were loaded using the monotonic (24-hour) load test criteria per ACI 318-11,5 with the exception of Specimen C3-0.4, which exhibited spalling prior to commencing the test. The maximum previous load observed by a specimen during the latest CLT load set (prior to the monotonic load test) was used as the test load magnitude for the monotonic load test, as shown in Table 3. After both tests were completed, the specimens were loaded to failure to experimentally determine ultimate capacity. It is noted that the two load tests were applied sequentially, with the CLT being commenced first, which is not the recommended procedure per ACI 437-12,7 where only one test is sufficient for performance assessment. RESULTS AND DISCUSSION The experimental mass loss due to corrosion damage was measured following the failure of the specimens, as shown in Table 1. From this table, it is clear that the experimental mass loss has a better agreement with the theoretical sectional mass loss calculated using Eq. (6) as compared to Eq. (5). The final failure load of the specimens is shown in Table 3 and the failure modes are shown in Table 4. All the specimens, except C3-0.4, which failed prematurely due to concrete spalling, failed in one of two modes: 1) strand rupture in the first group of specimens; and 2) failure due to excessive residual deflection (permanent deformation) in the second group of specimens. Failure as a result of excessive residual deflection was determined once the residual deflection exceeded that permissible per ACI 318-11.5 The difference in failure mode is attributed to the effect of uniform corrosion in the strands used in the first group of specimens, which enhanced the development length and prevented strand slipping, as discussed in ElBatanouny et al.17 A more detailed analysis regarding failure modes, strand slipping results, and effect of corrosion can be found in ElBatanouny et al.17 Cyclic load test (CLT) The CLT acceptance criteria were also used to assess the condition of the specimens, as shown in Table 5. Detailed values for the acceptance criteria of each specimen are

Fig. 5—Specimen U1 loading protocol.

Table 2—Applied load levels for each specimen as percentage of ultimate capacity Pu

*

Specimen

Loadset 1 Cycle 1, 2

Loadset 2 Cycle 3,4






Theoretical load level

0.75Ps

Ps

Pcr

0.60Pn

0.70Pn

0.80Pn

0.90Pn

Final failure load Pu, kN

U1

0.24

0.32

0.50

0.61

0.71

0.81

0.88

103.7

U2

0.25

0.32

0.52

0.60

0.70

0.80

0.90

113.9

U3

0.24

0.32

0.52

0.61

0.70

0.80

0.89

113.9

C1-0.4

0.20

0.27

0.48

0.52

0.69

0.77

0.87

101.0

C2-0.4

0.20

0.26

0.48

0.52

0.69

0.78

0.87

100.3

C3-0.4

0.26

0.35

0.64

0.69

0.81

0.92

NA

76.5

C4-0.4

0.18

0.24

0.45

0.48

0.72

0.81

0.90

89.8

C5-0.8*

0.29

0.33

0.43

0.49

0.54

0.78

0.89

89.9

Specimen had additional load set (load set 8) with load = 0.95Pu.

Note: 1 kip = 4.448 kN; NA is not available.


85

Table 3—Results of monotonic load test Specimen

Maximum load experienced

Test load magnitude

Dl, mm, (Eq. (1a))

Dr, mm (Eq. (1b))

Performance

Final failure load Pu, kN

U1

0.88Pu

0.88Pu

25.8 > 2.9

3.9 < 6.4

Pass

103.7

U2

0.80Pu

0.80Pu

45.4 > 2.9

7.5 < 11.4

Pass

113.9

U3

0.80Pu

0.80Pu

66.9 > 2.9

14.6 < 16.7

Pass

113.9

C1-0.4

0.69Pu

0.69Pu

18.6 > 2.9

1.4 < 4.7

Pass

101.0

C2-0.4

0.78Pu

0.78Pu

42.5 > 2.9

5.6 < 10.6

Pass

100.3

C3-0.4

NA

76.5

C4-0.4

0.72Pu

0.72Pu

20.2 > 2.9

1.9 < 5.1

Pass

89.8

C5-0.8

0.54Pu

0.70Pu

27.0 > 2.9

1.6 < 6.7

Pass

89.9

Notes: 1 in. = 25.4 mm; 1 kip = 4.448 kN; NA is not available.

Table 4—Failure mode Specimen

Maximum deflec- Residual deflection, mm (in.) tion, mm (in.)

Failure mode

U1

46.5 (1.83)

NA

Strand rupture

U2

103.6 (4.08)

46.2 (1.82)

Excessive deflection

U3

93.2 (3.67)

38.6 (1.52)


C1-0.4

86.1 (3.39)

40.9 (1.61)


C2-0.4

78.8 (3.09)

39.1 (1.54)


C3-0.4

90.9 (3.58)

NA

Concrete spalling

C4-0.4

66.5 (2.62)

30.9 (1.22)


C5-0.8

48.5 (1.91)

NA

Strand rupture

Note: NA is not available.

available in ElBatanouny.16 The maximum load achieved during the test in all specimens was near 0.9Pu (Table 2). As shown in Table 5, the repeatability index did not fail for any of the specimens. For all specimens, 88% of the load sets conducted had a repeatability index exceeding 100%, which indicates that they are far from failing the index (only seven load sets of the 56 had a value between 95 and 100%, and none had a value below 95%). Four specimens did not meet the permanency ratio criterion toward the end of the test at a load of 77% of ultimate capacity Pu, which shows that the acceptance criterion of this parameter is not sensitive to damage. The only index that was capable of assessing damage in all the specimens was deviation from linearity. By definition, deviation from linearity measures the deviation in the load-deflection slope from a reference slope calculated at the beginning of the test in the linear portion of the load-deflection plot. Therefore, the index can capture significant changes in the slope attributed to damage such as that caused by yielding, slippage of the reinforcement, and crack opening or widening. The different initial conditions, cracked or uncracked, of the specimens affected the load at which the specimens did not meet the deviation from linearity criterion. For the uncracked specimens (U1, U2, and U3), the acceptance criterion was not met when cracking occurred at a load magnitude equal to 70% of ultimate capacity (Pu). This is highlighted in Fig. 6, showing the load-deflection relation of Specimen U1, which represents a typical plot for uncracked (pristine) specimens. During the last step in Cycle 9, the 86

deflection increased significantly, almost doubling its value, resulting in a considerable decrease in the load-deflection slope in this cycle as compared to the reference slope. For the cracked specimens, deviation from linearity criterion was not met at a load magnitude ranging from 0.45 to 0.52 of the ultimate capacity, excluding Specimen C3-0.4, which failed prematurely due to concrete spalling. If compared to uncracked (pristine) specimens, cracked specimens did not meet this criterion at a lower percentage of the ultimate load. This is attributed to the presence of cracks in these specimens, which caused nonlinear behavior to initiate at a lower level of load compared to uncracked (pristine) specimens. This is opposite to the case of passively RC members, where it is expected that the deviation from linearity index will not be met at a higher level of load in cracked specimens if compared to uncracked specimens. Figure 7 shows the load-deflection relationship for Specimen C1-0.4, representing the typical behavior of cracked specimens. The specimen exhibited signs of significant damage at Cycles 5 and 6; however, deviation from linearity criterion was not met at Cycles 7 and 8. Further deviation from the initial slope showed that nonlinearity increased with the increase of load magnitude through the remainder of the cycles. Monotonic load test In all specimens, the CLT was performed first followed by the monotonic (24-hour) load test. The residual deflection was measured prior to commencing the monotonic test and the residual deflection criterion per ACI 437-127 was checked; then the load was applied for 24 hours. The residual deflection prior to the monotonic load test was not included in the calculation of the acceptance criteria to simulate field conditions where the residual deflection is not necessarily known. The final residual deflection was measured 24 hours after removal of the load. A summary of the results from the monotonic load test is shown in Table 3. The load magnitude varied in the specimens with a range of 69 to 88% of measured ultimate capacity Pu. All specimens did not meet the first evaluation criterion (Eq. (1a)) and passed the second evaluation criterion (Eq. (1b)). Therefore, the performance of all specimens was found to be satisfactory per ACI 318-11.5 It is noted that the load magnitude applied during the 24-hour load test significantly exceeded the service load level for PC members, Class U.5 During the monotonic load test the specimens were noticeably cracked ACI Structural Journal/January-February 2015

Table 5—Load at which CLT criteria was not met Specimen

Repeatability (Eq. (2))

Permanency ratio (Eq. (3a))

Deviation from linearity (Eq. (4))

Load

U1

All cycles passed

Was not met at load set 7 (Cycles 13 and 14)

Was not met at load set 5 (Cycle 9)

0.70Pu

U2

All cycles passed

All load sets passed


0.70Pu

U3

All cycles passed



0.70Pu

C1-0.4

All cycles passed



0.52Pu

C2-0.4

All cycles passed



0.52Pu

C3-0.4

All cycles passed



0.60Pu

C4-0.4

All cycles passed



0.45Pu

C5-0.8

All cycles passed



0.50Pu

Fig. 6—Load-deflection relation Specimen U1 with slopes for even cycles shown. (Note: 1 mm = 0.0394 in.)

Fig. 7—Load-deflection relation Specimen C1-0.4 with slopes for even cycles shown. (Note: 1 mm = 0.0394 in.) with crack widths exceeding 1 mm (0.04 in.). Because significant cracking is not allowed in Class U PC members, all specimens were considered damaged during the monotonic load test. Therefore, the monotonic load test failed to assess the condition of the specimens and served only as a proof test for ultimate capacity. For PC members, a reduced monotonic test load magnitude and revisions to both the deflection and recovery criteria should be considered. Discussion An acceptance criterion that did identify damage in all specimens was deviation from linearity index, which is a CLT-based damage index. This was true regardless of the initial condition of each specimen, which only affected the ACI Structural Journal/January-February 2015

Fig. 8—Crack widths and load versus time; Specimen C1-0.4. (Note: 1 mm = 0.0394 in.) level of load at which damage was assessed. In uncracked (pristine) specimens, deviation from linearity criterion was not met when cracks occurred. This agrees with the design procedure of Class U PC members, where the presence of cracks is not allowed. It is noted that upon failing the deviation from linearity index, the cracks measured in the uncracked (pristine) specimens had a width greater than 0.33 mm (0.013 in.). Ideally, the load test should be halted to prevent undesirable permanent damage. In cracked specimens, the deviation from linearity likewise identified damage as a result of the opening of existing cracks. For this purpose, a crack comparator tool was used.18 This tool was used as a guideline for identifying the condition of the specimens tested in this study. Based on the crack comparator tool standards, the cracks are divided into three categories: 1) Level 1 (hairline cracks), visible cracks with width less than 0.33 mm (0.013 in.); 2) Level 2, cracks with widths between 0.33 and 0.63 mm (0.013 and 0.025 in.); and 3) Level 3, cracks with widths greater than 0.63 mm (0.025 in.). Figure 8 shows measured crack widths and load magnitude versus time for Specimen C1-0.4. The crack widths were measured through Cycles 1 to 8 and at Cycle 10. It can be seen that the specimens did not meet the deviation from linearity criterion when the cracks obtained a width greater than 0.33 mm (0.013 in.). These crack widths fall within Level 2 of the crack comparator tool. The nonlinear behavior in the specimen increased when the crack widths exceeded 0.63 mm (0.025 in.) (Level 3 cracks) at Cycles 9 and 10 (Fig. 8). This behavior was observed in all speci87

mens that were precracked with crack widths of 0.4 mm (0.016 in.). For the heavily cracked specimen (C5-0.8), crack widths exceeded 0.33 mm (0.013 in.) in Cycle 3. Bearing in mind that the specimen was initially cracked, in the precracking test, to crack widths of 0.8 mm (0.032 in.); the reference load-deflection slope, calculated at Cycle 1, was smaller than the reference slope in the slightly cracked specimens (crack widths equal to 0.4 mm [0.016 in.]). The aforementioned results highlight the ability of deviation from linearity index to evaluate damage in cracked specimens based on initial crack widths with reasonable comparison to the established crack comparator tool. This shows promise toward the further work of calibrating the deviation from linearity index. Results of the monotonic load test showed that the acceptance criteria were always met regardless of the level of damage. This is notable because in some cases the specimens were loaded to 88% of the measured ultimate capacity prior to conducting the monotonic load test and were significantly cracked. Compliance with the criteria was in all cases due to the insensitivity of the recovery criterion. Chapter 20 of ACI 318-115 states that excessive cracking may be considered as evidence of failure. This provision relies heavily on engineering judgment, contradicting the purpose of load tests that should standardize performance assessment. This is significant in that this study is aimed at evaluating load test methods and acceptance criteria, the scope of this work does not consider proof testing in which the ability of a structure to withstand ultimate loading is considered. The final decision regarding suitability of the structure is made by the licensed design professional. However, the load test evaluation criteria should provide as much useful information as possible to aid in this decision. Given the passing of all monotonic load tests as described previously, it is suggested that the acceptance criteria may therefore be revisited for PC elements. This study included the load testing of individual members with simple supports and the interpretation of results. The behavior of complex systems will differ due to the presence of alternative load paths and other mechanisms such as moment redistribution. However, the acceptance criteria of load tests should be applicable to simple supports, fixed supports, and other conditions. Recommendations Although the deviation from linearity index provides a warning for damage in the specimens, this warning is provided after undue damage results in the uncracked specimens. As such, the specimen in good condition prior to testing is damaged beyond design parameters (uncracked [pristine] specimens) as a result of the test. In the case of pretensioned beams such as those tested herein, a cracked condition may result in excessive deflections and/or an avenue for chloride penetration. As a result of the load test itself, which is ideally intended to evaluate the suitability of a structure without imparting significant damage, damage to the structure may occur. Therefore, a modification to the acceptance limit of the deviation from linearity index is proposed for PC members for the case of pristine (uncracked) specimens. The 88

results indicate that the deviation from linearity index can be given a passing grade when this index is less than 10%. This limit has been determined by considering the results at and beyond the theoretical cracking load. The application of this limit will cause the test to stop prior to imposing permanent damage. The load limit at such point, in this test, is equal to the theoretical cracking load, which exceeds the maximum allowable test load magnitude for the CLT method7 for the case of Class U PC members. This is different than the case of passively RC beams where cracked conditions are expected under service conditions. For the cracked specimens, the deviation from linearity criterion was not met at the theoretical cracking load. Therefore, modifications to the current practice are not proposed for the case of prestressed members other than Class U (deviation from linearity can be given a passing grade if less than 25%). The proposed limits were based on results from beams that failed in flexure due to strand rupture or excessive slipping as discussed in ElBatanouny.16 Therefore, further investigation of these limits should be undertaken for specimens with different failure modes. SUMMARY AND CONCLUSIONS Both monotonic and cyclic load testing were conducted on eight PC beams. The specimens had different characteristics in terms of presence and width of cracks, as well as corrosion damage. The measured load-deflection response was used to evaluate the condition of the specimens. The findings are summarized as follows: 1. The monotonic load test acceptance criteria gave all tested specimens a “pass” grade regardless of the level of damage. Although the beams were loaded beyond the design criteria (Class U PC members) and were heavily cracked, all specimens passed the recovery portion of the acceptance criteria, thereby passing the load test. 2. Considering the passing of all monotonic load tests in the simplest of conditions, the evaluation of the specimen (or structure) is then delegated to the judgment of the Engineer of Record without having sufficient information from the test—a subjective opinion. 3. The repeatability index failed to determine the condition of the specimens and was insensitive to damage. Therefore, this parameter appears to be unsuitable for PC members. 4. The permanency ratio index criterion was not met in half the specimens toward the end of the test. This indicates that the limit of this index needs to be redefined for the case of PC members. The permanency index relies on residual or permanent deformation, which is affected by the prestressing. The current criterion may be more sensitive for passively RC members. 5. The deviation from linearity index succeeded in identifying damage in all the specimens. The critical crack width at which the criterion was not met in both pristine and slightly cracked specimens (crack width equal to 0.4 mm [0.016 in.]), was found to be 0.33 mm (0.013 in.), which is considered Level 2 cracks according to the crack comparator tool used in this research. However, the pristine specimens were damaged at the point of test failure due to the formation of cracks. ACI Structural Journal/January-February 2015

6. The proposed acceptance limit for deviation from linearity enables one to determine a point at which the acceptance criteria is not met in pristine specimens prior to the occurrence of permanent damage (cracking). The new limit is suitable for application in uncracked PC members, Class U, as the load level associated with this limit exceeds the service load of such members. This suggests that the test load magnitude associated with the CLT method should be revised for PC members such that the maximum test load magnitude is equal to service level load. 7. Overall, the CLT method proposed by ACI Committee 437 provides better assessment than the monotonic (24-hour) load test. The findings are limited to PC structures with flexural failure modes and should not be extended to structures that may exhibit shear failure without further investigation. AUTHOR BIOS ACI member Mohamed K. ElBatanouny is a Postdoctoral Fellow in the Department of Civil and Environmental Engineering at the University of South Carolina, Columbia, SC. He received his BS from Helwan University, Cairo, Egypt, and his PhD from the University of South Carolina in 2008 and 2012, respectively. He is a member of ACI Committee 444, Structural Health Monitoring and Instrumentation. His research interests include structural health monitoring using acoustic emission and load testing. Antonio Nanni, FACI, is a Professor and Chair in the Department of Civil, Architectural, and Environmental Engineering at the University of Miami, Coral Gables, FL, and a Professor of structural engineering at the University of Naples–Federico II, Naples, Italy. He is a member of ACI Committees 437, Strength Evaluation of Existing Concrete Structures; 440, Fiber-Reinforced Polymer Reinforcement; 544, Fiber-Reinforced Concrete; 549, Thin Reinforced Cementitious Products and Ferrocement; and 562, Evaluation, Repair, and Rehabilitation of Concrete Buildings. His research interests include construction materials and their structural performance and field application. ACI member Paul H. Ziehl is a Professor in the Department of Civil and Environmental Engineering at the University of South Carolina. He received his PhD from the University of Texas at Austin, Austin, TX. He is a member of ACI Committee 437, Strength Evaluation of Existing Concrete Structures, and Joint ACI-ASCE Committee 335, Composite and Hybrid Structures. His research interests include structural health monitoring and load testing. ACI member Fabio Matta is an Assistant Professor in the Department of Civil and Environmental Engineering at the University of South Carolina. He received his PhD from Missouri S&T, Rolla, MO. He is a member of ACI Committee 440, Fiber-Reinforced Polymer Reinforcement, and Joint ACI-ASCE Committee 446, Fracture Mechanics of Concrete. His research interests include monitoring and assessment of constructed facilities.

ACKNOWLEDGMENTS Special thanks are extended to the personnel of the University of South Carolina Structures and Materials Laboratory, in particular to W. Velez, B. Zarate, and W. McIntosh, and to Mistras Group, particularly M. Gonzalez, for the technical support provided. E. Deaver of Holcim Cement is thanked for support and technical input.


REFERENCES 1. American Society of Civil Engineers (ASCE), “Report Card for America’s Infrastructure,” Reston, VA, 2009. 2. ElBatanouny, M.; Mangual, J.; Barrios, F.; Ziehl, P.; and Matta, F., “Acoustic Emission and Cyclic Load Test Criteria Development for Prestressed Girders,” Structural Faults and Repair 2012, Edinburgh, Scotland, UK, 2012, 9 pp. 3. Ziehl, P. H.; Galati, N.; Nanni, A.; and Tumialan, J. G., “In Situ Evaluation of Two Concrete Slab Systems. II: Evaluation Criteria and Outcomes,” Journal of Performance of Constructed Facilities, ASCE, V. 22, No. 4, 2008, pp. 217-227. doi: 10.1061/(ASCE)0887-3828(2008)22:4(217) 4. Galati, N.; Nanni, A.; Tumialan, J. G.; and Ziehl, P. H., “In Situ Evaluation of Two Concrete Slab Systems. Part I: Load Deformation and Loading Procedure,” Journal of Performance of Constructed Facilities, ASCE, V. 22, No. 4, 2008, pp. 207-216. doi: 10.1061/(ASCE)0887-3828(2008)22:4(207) 5. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2011, 503 pp. 6. ACI Committee 437, “Test Load Magnitude, Protocol, and Acceptance Criteria (ACI 437.1R-07),” American Concrete Institute, Farmington Hills, MI, 2007, 38 pp. 7. ACI Committee 437, “Code Requirements For Load Testing of Existing Concrete Structures (ACI 437-12),” American Concrete Institute, Farmington Hills, MI, 2012, 25 pp. 8. Galati, N.; Casadei, P.; Lopez, A.; and Nanni, A., “Load Test Evaluation of Augspurger Ramp Parking Garage in Buffalo, N.Y.,” RB2C Report, University of Missouri-Rolla, Rolla, MO, 2004. 9. Casadei, P., and Parretti, R., “Nanni, A.; and Heinze, T., “In-Situ Load Testing of Parking Garage RC Slabs: Comparison Between 24-h and Cyclic Load Testing,” Practice Periodical on Structural Design and Construction, ASCE, V. 10, No. 1, 2005, pp. 40-48. doi: 10.1061/ (ASCE)1084-0680(2005)10:1(40) 10. Gold, W. J., and Nanni, A., “In-Situ Load Testing to Evaluate New Repair Techniques,” Proceedings, NIST Workshop on Standards Development for the Use of Fiber Reinforced Polymers for the Rehabilitation of Concrete and Masonry Structures, National Institute of Standards and Technology, Gaithersburg, MD, 1998, pp. 102-112. 11. Nanni, A., and Gold, W., “Evaluating CFRP Strengthening Systems In-situ,” Concrete Repair Bulletin, V. 11, No. 1, Jan.-Feb. 1998, pp. 12-14. 12. Nanni, A., and Gold, W. J., “Strength Assessment of External FRP Reinforcement,” Concrete International, V. 20, No. 6, June 1998, pp. 39-42. 13. Mettemeyer, M., and Nanni, A., “Guidelines for Rapid Load Testing of Concrete Structural Members,” Report No. 99-5, Center for Infrastructure Engineering Studies, University of Missouri-Rolla, Rolla, MO, 1999. 14. Auyeung, Y.; Balaguru, B.; and Chung, L., “Bond Behavior of Corroded Reinforcement Bars,” ACI Materials Journal, V. 97, No. 2, Mar.-Apr. 2000, pp. 214-220. 15. Mangat, P. S., and Molloy, B. T., “Prediction of Long Term Chloride Concentration in Concrete,” Materials and Structures, V. 27, No. 6, 1994, pp. 338-346. doi: 10.1007/BF02473426 16. ElBatanouny, M. K., “Implementation of Acoustic Emission as a Non-Destructive Evaluation Method for Concrete Structures,” PhD dissertation, Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC, 2012, 184 pp. 17. ElBatanouny, M. K.; Ziehl, P.; Larosche, A.; Mangual, J.; Matta, F.; and Nanni, A., “Acoustic Emission Monitoring for Assessment of Prestressed Concrete Beams,” Construction and Building Materials, V. 58, 2014, pp. 46-53. doi: 10.1016/j.conbuildmat.2014.01.100 18. Lovejoy, S., “Acoustic Emission Testing of Beams to Simulate SHM of Vintage Reinforced Concrete Deck Girder Highway Bridges,” Structural Health Monitoring, V. 7, No. 4, 2008, pp. 327-346. doi: 10.1177/1475921708090567

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NOTES:

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TECHNICAL PAPER

Title No. 112-S09

Crack Distribution in Fibrous Reinforced Concrete Tensile Prismatic Bar by Yuri S. Karinski, Avraham N. Dancygier, and Amnon Katz The behavior of fibrous concrete containing conventional steel reinforcement under axial tension is analyzed. The current study reveals that, while distances between cracks in plain concrete are equal, this is not the case for fibrous concrete. It is shown that the crack patterns in conventionally reinforced concrete with and without fibers are qualitatively different, even when distribution of fibers is uniform. The paper proposes a model for the behavior of a reinforced fibrous concrete bar subjected to increasing axial tension load. The model was verified against experimental results from two different sources. Based on the proposed model, an algorithm is presented to calculate the tensile forces that cause cracking and to determine the intervals between the cracks. Keywords: cracking pattern; fibrous reinforced concrete; tensile behavior.

INTRODUCTION The use of fibers in reinforced concrete (RC) elements has become more widespread in recent decades. Furthermore, in the past decade, consideration has been paid in modern codes to the mechanical properties of fibrous concrete used in structural design.1,2 Fibers have been studied for their use in structural members—for example, as part of the shear reinforcement.3-5 Under tension, they affect crack width, spacing, and pattern.6 Therefore, tension stiffening, which is an essential characteristic of reinforced concrete, is enhanced by the presence of fibers that can bridge cracked cross sections.7 Fiber action of this kind leads to additional advantages of using them in concrete mixtures—namely, improved crack control8 and increased material toughness.9-11 Improved crack control has been observed in tension elements containing both steel fibers and conventional steel reinforcing bars at service load, at which thin and closely spaced cracks were reported.12-14 Researchers also noted that, interestingly, at ultimate load, only one or two cracks widened more than the others. A similar phenomenon of localization was also observed in flexural tests of beam specimens.15,16 One reason for this phenomenon is fiber distribution,17,18 which causes a crack pattern in tensile reinforced fiber-reinforced concrete (R/FRC7) elements8 that is different from the relatively uniform pattern in plain RC tension bars.19 Any attempt to explain this phenomenon of non-uniform crack distribution should be based on a proper model of the tensile behavior of fibrous reinforced concrete. This paper presents such a model. The behavior of an RC bar subjected to axial tension is controlled by the steel-concrete bond, the reinforcement ratio, and the concrete tensile strength. The behavior of R/FRC tensile bars depends also on the fiber type and content, especially in the post-peak range.1,20 Cracks develop at locaACI Structural Journal/January-February 2015

tions where the concrete stress reaches its tensile strength. In plain RC (without fibers), as the tensile force increases, new cracks, of similar width, develop midway between existing cracks (subject to strength variation along the concrete bar). Furthermore, all uncracked segments are of the same length.19 This results from the fact that, at relatively small tensile strains, the residual tensile stresses in the cracked concrete are relatively small. In fibrous concrete, however, fibers cause considerable residual stresses across the cracked cross sections, which affect the cracking process.7,21 The bond between the conventional steel and fibrous concrete is another important factor in the cracking process. Some of the existing models assume constant bond stress along the reinforcing bar,22 which lead to a simplified analytical solution. Yet, the bond-slip relation is more complex and is commonly assumed to be nonlinear. Models that apply such a nonlinear relation require the development of a numerical crack analysis procedure.6,7,19,21 For small steel-concrete slip values (as in the initial phase of cracking when transverse cracks are already developed but are still relatively narrow), the bond stress may be assumed to be a linear function, which enables an analytical solution. This paper proposes a simplified model for the behavior of a reinforced fibrous concrete bar while increasing axial tension load. The model predicts the loads that cause new cracks to develop as well as the distances between the cracks at various stages of loading. Note that this phase of the cracking process (crack initiation) occurs within the elastic range of the material behavior. Initiation of a crack can be predicted by using a tensile strength criterion, while further propagation of existing cracks can be handled with fracture mechanics criteria. The current paper deals with the basic problem that refers to the first phase of crack initiation that precedes the phase of crack propagation, which is out of the scope of this paper. RESEARCH SIGNIFICANCE A simplified model is proposed for the assessment of crack distribution in an RC tensile prismatic bar containing fibers. The model refers to the cracking phase, during which cracks are relatively narrow—that is, up to the yield of reinforcing bar in any one of the cracks. An analytical soluACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-290.R2, doi: 10.14359/51687298, received May 25, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

91

σc = Ψσc0, distributed over the entire concrete cross section Ac such that the resultant tension forces for both the actual and equivalent stress distributions are equal Fig. 1—Problem definition: tensile reinforced concrete bar.

∫∫ Ac σ c ( r ) dAc *

Ψ=

σ c 0 Ac

=

∫∫ Ac f ( r ) dAc Ac

(1)

The coefficient Ψ may be different for various types of bars, depending on the concrete cover (that is, on the expected distribution of the transverse stress). Fig. 2—Stress distribution in uncracked concrete.

DIFFERENTIAL EQUATION OF REINFORCING BAR/CONCRETE SLIP The method of obtaining the differential equation of the reinforcing bar/concrete slip is well documented7,21,23 and is presented herein only for the sake of clarity of subsequent derivations. Equilibrium of a differential segment of the considered specimen of length dx (Fig. 3) yields the following system of equations Ac d σ c − Στdx = 0

Fig. 3—Equilibrium in differential segment. tion is enabled using a linear bond-slip relation during this phase. This simplified approach enables one to calculate the entire distribution of distances between cracks that develop in the tensile prismatic bar, while most models refer only to the average, minimum, or maximum crack spacing. In spite of the simplified approach, the model yields predictions that are in good agreement with experimental results. PROBLEM DEFINITION Consider a R/FRC specimen, a long bar of RC that contains fibers that is subjected to tension force T applied at the ends of the reinforcing bar (refer to Fig. 1). This work refers to the cracking phase, during which crack widths are relatively small—that is, before yielding of the reinforcing bar in any of the cracks. The cross-sectional areas of the concrete and steel reinforcing bar are Ac and As, respectively, and the total perimeter of the steel-concrete interface is Σ (note that in the case of a single reinforcing bar with a diameter ϕ, As is equal to πϕ2/4 and Σ = πϕ). This study deals with conventional reinforcement ratios—that is, As << Ac. It is also assumed that the fiber content is not very high and, thus, the fibrous concrete is characterized by post-peak softening after cement matrix cracking.1 Due to the strain distribution along the concrete cross section, the tension stress in the uncracked concrete is generally non-uniform—that is, σc*(r) = σc0 · f(r), where r is the distance from the longitudinal x-axis and σc0 is the stress at the reinforcing bar/concrete interface (refer to Fig. 2). Similar to the approach presented in fib Bulletin 10,23 this research used in the following derivation a uniform stress,

92

(2) rAc d σ s + Στdx = 0

where ρ is the reinforcement ratio (As/Ac); τ is the bond stress; and σs is the steel tension stress. The following relations are valid for small deformations

σ c = Ec

duc du ; σ s = nEc s (3) dx dx

where us and uc are, respectively, the steel and concrete displacements at the steel-concrete interface; Ec is the Young’s modulus of fibrous concrete; and n = Es/Ec (where Es is the Young’s modulus of steel). Additionally, for small values of steel-concrete slip w (as in the initial phase of cracking when transverse cracks are already developed but are still relatively narrow), bond stress is assumed to be the following linear function

τ = –Aw (4)

where

w = us – uc (5)

The minus sign in Eq. (4) conforms to the sign agreement according to Fig. 3. Substituting Eq. (3) into Eq. (2) and summing the result yields

d 2 uc d 2 us + n r = 0 (6) dx 2 dx 2

Equations (5) and (6) yield the following relations between the second derivatives of the displacements and w


d 2 uc nr d 2 w = − nr + 1 dx 2 dx 2

d 2 us 1 d 2w =+ 2 nr + 1 dx 2 dx

(7)

Substituting Eq. (3) into Eq. (2) and subtracting the result yields

A E  d 2u d 2u  τ = c c  2c − nr 2s  (8) 2Σ  dx dx 

Substituting the expressions from Eq. (7) and (4) into Eq. (8) yields the following differential equation for the slip w

d 2w − λ 2 w = 0 (9) dx 2

where λ is given by

λ2 =

AΣ Ac Ec

 1 1 + nr  (10)

CRACK INITIATION Cracks develop at cross sections at which the stress in the concrete (either plain or fibrous) reaches the concrete’s tensile strength fct. According to the present model, concrete stress can be represented by either of the following two ways (Fig. 2): 1) stress closest to the steel bar, σc0 (Eq. (11) and (12)); or 2) equivalent stress σc (Eq. (13)). The first way may be written in the following form

σc0 =

σc = f ct (11) Ψ

which yields

σc = Ψfct (12)

The second way suggests that

σc = fct (13)

The only difference between these two ways is in the choice of the cracking criterion—that is, in choosing the critical value of the calculated stress σc. According to the second way, this value is equal to the concrete tensile strength (Eq. (13)). However, when the first way is applied, this critical value is smaller than the tensile strength and is obtained by multiplying the concrete tensile strength by a coefficient Ψ (smaller than 1.0 [Eq. (12)]). The choice of this criterion depends on the expected distribution of the transverse stress in the concrete (this choice is out of the scope of this paper). In the following derivation, the cracking criterion has the form σc = fct, where fct is an equivalent tensile strength, which is given by either Eq. (12) or (13). ACI Structural Journal/January-February 2015

Fig. 4—Equilibrium adjacent cracks).

in

a

segment

(between

two

Consider an element of length Lc located between two cracks (Fig. 4). The origin of the local coordinate system is set at the center of the element and the reinforcing bar/ concrete slip within the element (–Lc/2 < x < Lc/2) is governed by Eq. (9). Bond analysis of an RC specimen that does not contain fibers commonly assumes that concrete stresses at the crack are zero. In fibrous concrete, however, the fibers provide a residual tensile stress, denoted herein as σf. The current model refers to the initiation of cracks, where this phase occurs within the elastic range of the steel behavior. Furthermore, in concrete reinforced with hooked-end steel fibers, the stress across the crack remains relatively constant for rather large crack opening.24 Thus, the residual stress in this phase may be considered constant and can be obtained from standard tests for determination of the mechanical properties of fibrous concrete (for example, RILEM25). This is an empirically calibrated parameter that represents the overall action of fibers that bridge a crack, including the effects of their quantity, elastic modulus, aspect ratio, bond properties, orientation, and other characteristics of the fibers and the surrounding matrix.20 Note that due to uneven fiber distribution, residual stress σf may vary among different cracks. As a result, steel stress at each crack may also vary to maintain equilibrium along the specimen (the total axial force at each cross section is equal to external force T). The forces in the steel and in the fibrous concrete at the cracked cross sections located at the edges of the element are given by the following expressions

Es As

dus dx

= T − σ ±f Ac x = ± Lc / 2

du Ec Ac c dx

(14)

= σ ±f Ac x = ± Lc / 2

where σf± = σf (x = ±Lc/2) is the residual stress at the lefthand and right-hand edges of the element (refer to Fig. 4). Due to the longitudinal strain distribution along the crack, residual stress is generally non-uniform. Similar to the stress distribution in the uncracked cross section (Fig. 2), a coefficient similar to Ψ (Eq. (1)) allows one to use an equivalent uniform residual stress, represented herein by the term “σf±”. Hence, σf± is applied in the calculations similar to fct, either according to the criterion stated in Eq. (11) and (12) or that stated in Eq. (13). Extracting the expression for 93

dw  dus duc  = − dx x =± Lc / 2  dx dx 

x =± Lc / 2

duc dx

dus dx

from Eq. (14) yields the following boundary conditions for Eq. (9) ±  σf dw T 1  1 1 = − + (15) dx x =± Lc / 2 Ec Ac nr  nr  Ec

Stresses in concrete and steel The solution of Eq. (9) with boundary conditions (Eq. (15)) takes the following form w=−

C1 cosh ( λx ) C2 sinh ( λx ) (16) + λ  λLc  λ  λLc  sinh  cosh   2   2 

C2 =

σ +f − σ −f  1  + 1  2 Ec  nr 

(17)

σ c = Ec

(19)

+ − dus 1 (σ f − σ f ) sinh(λx) = dx nr 2 Ec  λL  sinh  c   2 

−

+ − 1 σ f + σ f T 1  cosh(λx) − ⋅ + D2   nr  2 Ec Ec Ac nr + 1  λLc  cosh   2 

where D1 and D2 are constants of integration. According to Eq. (14), the boundary conditions for the left-hand side of the segment are

94

(20)

nrEc Ac

T 1 ⋅ Ec Ac ( nr + 1)

(21)

+ − duc (σ f − σ f ) sinh(λx) = 2 Ec dx  λL  sinh  c   2 

 σ +f + σ −f T 1  cosh(λx) 1 T + ⋅ + − ⋅  λ L r 1 r + 1) E A n A ( n + 2   c  cosh  c c c  2 

(22)

  σ +f + σ −f T  1 cosh(λx) T nυ  + ⋅ − ⋅ −   Ac nr + 1  λL  Ac (nr + 1)   2 cosh  c   2  

cosh ( λx ) sinh ( λx ) d w = −C1λ + C2 λ (18) 2 λ L dx    λL  sinh  c  cosh  c   2   2 

 σ +f + σ −f T 1  cosh(λx) + D1 + − ⋅  Ec Ac nr + 1  λLc   2 Ec cosh   2 

T − σ −f Ac

 + − dus 1  1 (σ f − σ f ) sinh(λx) σ s = nEc =  dx r  r 2  λL  sinh  c   2  

2

+ − duc (σ f − σ f ) sinh(λx) = 2 Ec dx  λL  sinh  c   2 

Ec

These constants also satisfy the boundary conditions for the right-hand side (refer to Eq. (14) for x =+Lc/2). Equations (19) and (20) yield the concrete and steel stresses as follows

+ −  T 1 σf +σf  1 − + 1  2 Ec  nr  Ec Ac nr

Substituting Eq. (17) and (18) into Eq. (7) and integrating once yields the concrete and steel strains

= x = − Lc / 2

D1 = D2 =

The second derivative of the slip w (Eq. (16)) is given by

x = − Lc / 2

σ −f

Hence, D1 and D2 are given by the following expression

where λ is given in Eq. (10), and C1 and C2 are the following constants C1 =

=

It should be noted that under external tension force T, both concrete and steel stresses at the ends of the element (x = ±Lc/2) must be non-negative. This leads to the evident condition on the residual stresses at the cracked cross sections, 0 ≤ σf± ≤ T/Ac. In addition, residual stresses at the cracked cross sections, σf±, must be lower than the tensile strength of the fibrous concrete fct, yielding the following, final requirement

T  0 ≤ σ ±f ≤ min  , f ct  (23) A  c 

CRACKING PROCESS For an idealized representation of the material, as the external load T increases, the maximum stress within the concrete, σc,MAX, increases as well and a crack develops when (and if) σc,MAX reaches the concrete tensile strength fct. The RC bar specimen is divided by the first crack into two uncracked segments (Fig. 5(a)) and with the increase in T, this cracking process continues within each of the two uncracked segments, whereby each occurrence k of σc,MAX = fct corresponds to 2k uncracked segments and 2k – 1 cracks (for example, refer to Fig. 5(b) for k = 2). Higher ACI Structural Journal/January-February 2015

Fig. 5—Cracking process (first two stages) in tensile reinforced concrete bar. external loads (up to the yield load of the steel bar) correspond to larger number of shorter, uncracked segments. A new crack develops in an uncracked segment only if the maximum value of σc (Eq. (22)) equals the value of fct within the segment (Fig. 6). For plain concrete, in which concrete stress at the cracks drops to zero, the crack spacing is uniform (that is, all uncracked segments have the same length). This, however, is not the case for fibrous concrete, as will be shown below. Zeroing the first derivative of σc (Eq. (22)) leads to the following expression for the location of the maximum concrete stress x* (according to the local coordinate system [Fig. 4 and 6])     1 B  (24) x* ( Lc ) = arctanh  λ  λLc     tanh  2    

where B=

σ +f − σ −f (25) 2T − σ +f − σ −f Ac ( nr + 1)

Fig. 6—Stress distribution in concrete within uncracked segment upon commencement of new crack.

(

(

)

σ +f − σ −f sinh  λx* ( Lc ) T 1 + Ac ( nr + 1) 2  λL  sinh  c   2 

)

*  σ +f + σ −f T 1  cosh  λx ( Lc )  = f ct + − 2 Ac ( nr + 1)   λL    cosh  c   2 

(27)

Note that the condition dσc/dx = Ec(d2uc/dx2) = 0 is satisfied for cross sections for which x = x*. This condition, together with Eq. (6) and (8), imply that for cross section x = x*, the bond stress τ between the concrete and the steel is zero. Therefore, there is no slip in this cross section—that is, w(x*) = 0, and the strains in the concrete and in the steel are equal. The well-known rule of mixtures26 may therefore be applied, and the concrete stress at x = x* is equal to (T/Ac)(1/(nr + 1)). It follows that a crack can exist only when

T ≥ Ac(nρ + 1)fct (28)

It further follows that Eq. (23) may be rewritten as follows If |x (Lc)| > Lc/2, the maximum of the function of σc (Eq. (22)) occurs outside the segment and the function inside the segment is monotonic and less than fct (Eq. (23)). Hence, only –Lc/2 ≤ x*(Lc) ≤ Lc/2 is relevant for the case of further cracking. This range of x*(Lc) and Eq. (24) yield the following condition for the segment length for which further cracking is possible *

Lc ≥

2 arctanh λ

( B ) (26)

It is noted that the condition (Eq. (26)) is required but not sufficient for cracking. The condition σc(x*) = fct (Eq. (22)) yields the following equation for the force T that causes the initiation of a new crack within the segment (of length Lc)

0 ≤ σf± ≤ fct (29)

Substituting x* from Eq. (24) into Eq. (27) yields the following implicit equation for T at cracking

 λL   λL  tanh 2  c  − B 2 = C ⋅ sinh  c  (30)  2   2 

where B is given in Eq. (25) and C is given by

2T − 2 f ct Ac ( nr + 1) C= (31) 2T − σ +f − σ −f Ac ( nr + 1)

Equation (30) always has a solution, as ensured by Condition (Eq.) (26), which represents a physical state of cracking and guarantees that the expression under the square root ACI Structural Journal/January-February 2015

95

in Eq. (30) is non-negative. According to Conditions (Eq.) (28) and (29), C is always positive and so Eq. (30) can be squared, yielding the following biquadratic equation

 λL   λL  tanh 4  c  − 1 + B 2 − C 2 ⋅ tanh 2  c  + B 2 = 0 (32)  2   2 

(

)

Considering the definitions of B and C (Eq. (25) and (31)) and Conditions (Eq.) (28) and (29), it can be shown that all four roots of this equation are real. While two of those solutions are negative, and therefore irrelevant to the current physical problem, the positive solutions are given by 2

( Lc )1,2 = λ arctanh

1 1 + B2 − C 2 ± 2 

(1 + B

2

− C2

)

2

 − 4 B 2  (33) 

It can be shown, based on Conditions (28) and (29), that the discriminant in Eq. (33) is always positive. It can be further shown that only the larger root of Eq. (33) satisfies Condition (26) and so the correct relation between the length of the segment, Lc, and the force T that initiates a new crack in the segment is given by Lc =

2 1 1 + B2 − C 2 + arctanh 2  λ

(1 + B

2

− C2

)

2

 − 4 B 2  (34) 

Equation (34), with B and C given by Eq. (25) and (31), yields the following two solutions for T Ac (1 + nr) 2   (35a)  λL  2 2  λLc  2 Φ sinh  cossh  c   +D  +      2 2 ⋅ ×  σ f + σ −f +  λ L λ L  c 2 2  2  c  ⋅ sinh cosh Φ ± Φ − D        2   2  T=

where Δ = σf+ – σf–, and Φ = 2fct – (σf+ + σf–). The condition that T must go to infinity as Lc approaches zero (that is, lim (T ) = ∞ ) leads to the choice of a minus sign Lc → 0

in the denominator of Eq. (35a); that is T=

Ac (1 + nr) 2

  (35b)  λL   λL  Φ 2 sinh 2  c  + D 2 cossh  c   +   2   2  ⋅ ×  σ f + σ −f +   λLc  2 2  2  λLc   ⋅ sinh cosh Φ − Φ − D    2   2 

When T reaches this value, two new segments appear within the segment in question and their lengths are equal to Lc/2 + x* and Lc/2 – x* (left- and right-hand segments, respectively [Fig. 6]). When σf+ > σf–, x* is positive (Eq. (24) and (25)) and the left-hand segment is larger, and vice versa. In the symmetric case of σf+ = σf– = σf, B is equal to zero and the maximum concrete stress (as well as the minimum steel stress) occurs

96

Fig. 7—Concrete stress distribution at onset of first crack (coordinates along the bar and concrete stress are normalized with respect to the bar’s length and fct, respectively). at the middle of the segment (x* = 0). In this case, the value of the force T is given by

 λL  σ f − f ct cosh  c   2  T = (1 + nr) Ac (36)  λLc  1 − cosh   2 

In a plain RC tension bar in which σf = 0, cracks will appear at equal intervals (because x* = 0 in all segments). In a fibrous RC bar, on the other hand, the stresses σf in the cracked cross sections are greater than zero and are equal to zero only at the bar edges. Residual stresses in the segments located near the left-hand edge of a tension bar are, therefore, σf– = 0 and σf+ = σf > 0; whereas near the right-hand edge of a tension bar, residual stresses are σf+ = 0 and σf– = σf > 0. For these edge segments, x* ≠ 0 and the cracking force must be calculated using Eq. (35b). As a result, crack patterns in conventionally reinforced concrete with and without fibers will be qualitatively different (even if fiber distribution is uniform— that is, σf+ = σf– in all internal segments). The aforementioned residual stress is distributed uniformly along the cracked cross section (Fig. 4) and it represents an average value of the tractions resulting from the local action of the fibers. Because it is reasonable to assume that the average fiber distribution in well-mixed concrete mixtures is essentially the same for all cross sections, the assumption in the following derivation is that residual stresses in all internal, cracked cross sections are equal (σf+ = σf– = σf). Thus, the location x* of the maximum concrete stress in an internal uncracked segment, which is bound by two existing cracks, is equal to zero (Eq. (24) and (25)). Furthermore, for the case of an internal segment like this, the cracking tension force is given by Eq. (36) (unlike the case of edge segments, in which σf+ ≠ σf– and Eq. (35b) applies). The above model refers to a condition of constant concrete tensile strength fct in each of the cross sections. This condition is based on the understanding that for common sizes of RC members, the distribution of concrete properties across each cross section is similar. Therefore, the variation in fct (which is essentially the mean value of concrete tensile strength across a section) along the length of the bar is very limited. In relatively long bars, it is, however, important to take into account the distribution of fct along the bar. This point is illustrated in Fig. 7, which shows the concrete stress distribution σc(x) (Eq. (22)) for a force Tc1 that initiates the ACI Structural Journal/January-February 2015

Fig. 8—Schematic cracking process of fibrous reinforced concrete bar under axial tension. first crack in the segment (Eq. (36) with σf = 0) for two lengths Lc of uncracked bars: a relatively short bar and a bar three times longer (denoted “L” and “3L” in Fig. 7). Figure 7 shows that as the length of the bar increases, the stress distribution along the center of the segment becomes more uniform and extends over a longer portion of the segment—that is, the gradient of σc is relatively small. In such cases, variation of fct along the length of the bar can lead to a crack, which is not necessarily located at the calculated point of the mean concrete tensile strength. ALGORITHM FOR CALCULATING CRACKING PROCESS The general procedure for calculating tensile forces that cause cracking and the intervals between the cracks is as follows: Stage 1. The first cracking force Tc1 is calculated using Eq. (36), with σf = 0 and the length of the entire bar substituted for Lc. The first crack is set at the center of the bar (x* = 0 [Fig. 5(a)]). The following steps refer to the left-hand half of the bar, based on a theoretical symmetric cracking pattern (Fig. 8(a)).


Stage 2. The second cracking force Tc2 is calculated using Eq. (35b), with σf– = 0, σf+ = σf > 0, and half of the bar length substituted for Lc. Parameter B is calculated from Eq. (25), with Tc2, and is then is used in Eq. (24) to calculate x*. The lengths of the two new segments, L2left and L2right (Fig. 8(b)), are given by

Lileft = Lc/2 + x*; Liright = Lc/2 – x*; i = 2, 3, 4... (37)

Note that in this case (σf+ > σf–), x* > 0 and, therefore, L2left > L2right. Hence, the following crack is expected to develop within L2left, which is the (left-hand) end-segment of the bar. Stage 3. The third cracking force Tc3 and the corresponding crack location (x*) are calculated as in Stage 2, with Lc = L2left (σf– = 0, σf+ = σf > 0). The lengths of the two new segments, L3left > L3right (Fig. 8(c)), are again given by Eq. (37). At this stage, each half of the bar is subdivided into three segments of different lengths. (On the left hand, they are L3leftt ≠ L3right ≠ L2right [Fig. 8(c)]). The subsequent stages are calculated as follows: Stage i (i ≥ 4). 1. Find the largest internal segment from the previous stages. Substitute the length of this segment, Lint,max, for Lc in the calculation of cracking force Tci,1 using 97

Fig. 9—Comparison between model and experimental results27 (measured maximum value of 198 mm at strain of 0.00049 is not shown). (Note: 1 mm = 0.039 in.) Eq. (36), with σf ≠ 0 (σf is equal to the residual stress of fibrous concrete). 2. If Lint,max is greater than the length of the edge segment, then this will be the next cracked segment, Tci=Tci,1, and the crack will be set at the center of the segment (x* = 0 and Lileft = Liright [Fig. 8(d)]). 3. Else, calculate the cracking force Tci,2 for the edge segment and its corresponding crack location x2* (Eq. (24)), as in Stage 2 with σf– = 0, σf+ = σf > 0. The next cracking force Tci = min{Tci,1, Tci,2}. If Tci = Tci,2, then the lengths of the two new segments, Lileft > Liright, are given by Eq. (37), with x* = x2*. Else, x* = 0 and Lileft = Liright. Note that this procedure is not valid when T exceeds the yield force of the steel reinforcing bar. A flow chart for the cracking process is given in the Appendix. COMPARISON WITH EXPERIMENTAL RESULTS The above approach is compared both with experimental results reported by Leutbecher and Fehling27 and with results obtained by the authors at the Technion’s National Building Research Institute (NBRI). Comparison with results obtained by Leutbecher and Fehling Leutbecher and Fehling reported on experiments with ultra-high-performance concrete in which 1300 mm (51.2 in.) long concrete bars were subjected to tension. The bars, which had a 70 x 220 mm (2.8 x 8.7 in.) cross section, were reinforced with four 12 mm (0.5 in.) steel bars (St 1470/1620) and contained 0.9% 17 mm (0.7 in.) steel fibers. These authors reported minimum, maximum, and average values of crack spacings for three different levels of reinforcing bar strain (Fig. 9). The calculation algorithm described in the previous section was performed with the following data, according to the values reported by Leutbecher and Fehling: Ac = 15,400 mm2 (23.9 in.2); As = 452 mm2 (0.7 in.2); Σ = 151 mm; (5.9 in.); tensile strengths are 9 MPa (1305 psi); residual strength is 8 MPa (1160 psi); and Ec = 50,000 MPa (7252 ksi) (based on a reported compression strength of 160 to 190 MPa [23.2 to 98

Fig. 10—Effect of parameters A and Ψ. (Note: 1 mm = 0.039 in.) 27.6 ksi]). Additionally, the values of the following parameters were completed from Leutbecher28: A = 665 N/mm3 (2450 kip/in.3), and Es = 205,000 MPa (29,733 ksi). Here we have set a value of 0.5 for Ψ and used the criterion specified by Eq. (11) and (12). This value of Ψ was calibrated to best fit a single experimental result (at steel strain of 0.0045) and then the same value was applied for all other values of steel strain. These data yield n = 4.1, ρ = 0.029, λ = 0.035 mm–1 (0.884 in.–1), fct = 0.5 · 9 = 4.5 MPa (653 psi), σf = 0.5 · 8 = 4 MPa (580 psi). Figure 9 presents a comparison between the measured and calculated values of uncracked segment lengths for the earlier parameters. The figure shows good agreement between the average values of the measured and calculated results (mean relative error and standard deviation of 0.15 and 0.25, respectively), especially for strains greater than 0.0013 (mean relative error and standard deviation of 0.05 and 0.18, respectively). A fair agreement was obtained for the maximum values of uncracked segment lengths (mean relative error and standard deviation of 0.26 and 0.12, respectively), although there is poor agreement for the minimum values. Figure 10 shows a parametric study of the sensitivity of the average crack spacing to A and Ψ. For clarity, the figure also shows the average values that correspond to the various experiments.27 It can be seen that the effect of parameter Ψ is pronounced at small steel strains whereas it is negligible at relatively larger strains (see two graphs of A = 340 N/mm3 [1253 kip/in.3] in Fig. 10). An opposite effect was found for the bond-slip coefficient A, which has a pronounced effect only at relatively larger strains (see Ψ = 0.7 in Fig. 10). Comparison with results obtained at NBRI Experimental procedure—Four types of concrete were tested: normal-strength concrete (NSC) and high-strength concrete (HSC), with or without fibers. The tests considered herein are of those specimens that contained steel fibers: two of HSC and one NSC (a second NSC specimen exhibited technical problems). Table 1 presents the mixture composition of these mixtures together with their mechanical properties. Concrete prisms, 75 x 75 x 670 mm (3 x 3 x 26.4 in.), were prepared with one 12 mm (0.5 in.) deformed steel reinforcing bar located along the center of each prism with both ACI Structural Journal/January-February 2015

Table 1—Mixture composition (kg/m3) and mechanical properties of tested fibrous concretes N-1

H-1

190

160

Cement

255

500

Silica fume

—

40

Fibers†

60

60

1170

880

725

860

Water *

Coarse aggregate Fine aggregate ‡§

Compressive strength at 28 days, MPa

30.7 (1.3)

115 (1.3)

Flexural strength at 28 days, MPa‡||

5.9 (0.9)

13.1 (0.7)

Splitting strength at 28 days, MPa‡

3.7 (0.3)

13.1 (2.5)

*

CEM I 52.5N, complies with EN 196-1.

†

Steel fibers, RC-65/35-BN.

‡

Numbers in parentheses denote standard deviations.

§

Measured on 100 mm (3.93 in.) cubes that were wet-cured for 7 days.

||

Measured on 70 x 70 x 280 mm (2.76 x 2.76 x 11.02 in.) prisms.

Notes: 1 kg/m3 = 0.0624 lb/ft3; 1 MPa = 145 psi.

ends protruding from the concrete to enable it to be mounted on the testing machine, as depicted in Fig. 1. The bar was isolated from the concrete along 60 mm (2.4 in.) using bond breakers at each end of the specimen to avoid stress concentration29 (so that the net length of the RC prism specimen was 550 mm [21.6 in.]). Load was applied gradually to the protruding ends of the steel, and development of cracks along the concrete prism was recorded. Although the tests were performed up to the ultimate state of the specimens, only the service phase of the load, as mentioned earlier (refer to “Problem definition”), is referenced. The calculation algorithm described earlier was performed using the following data: Ac = 5625 mm2 (8.7 in.2); As = 113 mm2 (0.2 in.2); and Σ = 38 mm (1.5 in.). The modulus of elasticity of the steel bar Es = 206,800 MPa [29,994 ksi] was measured directly. High-strength concrete specimens—The modulus of elasticity Ec of the HSC specimens was taken as 47,000 MPa (6817 ksi) based on the measured concrete compressive strength (Table 1) and its corresponding modulus of elasticity, proposed in Table 5.1.7 of the Model Code.1 Tensile strength was set to 7.1 MPa (1030 psi) based on the measured flexural strengths (Table 1) and the ratio of mean flexural strength to axial tensile strength of concrete also proposed in the Model Code, which according to the data in Table 1 is equal to 0.54. A residual strength of 5 MPa (725 psi) was used based on the residual-maximum strength ratio reported by Leutbecher and Fehling,27 and was corrected proportionally to the lower fiber volume ratio of 0.75% (refer to Table 1). In addition, the value of A = 200 N/mm3 (737 kip/in.3) was obtained from bond-slip measurements of direct and flexural pullout tests.30 The value of Ψ was set to 0.7 and the criterion specified by Eq. (11) and (12) was used. This value of Ψ was calibrated to best fit a single experimental result (tensile force at first cracking) and the same value was then applied to all other values of steel strain. These data yielded n = 4.4,


Fig. 11—Comparison between model and experimental results of HSC specimens (NBRI tests). (Note: 1 mm = 0.039 in.; 1 kN = 0.225 kip.) ρ = 0.02, λ = 0.02236 mm–1 (0.56794 in.–1), fct = 0.7 · 7.1 = 4.97 MPa (721 psi), and σf = 0.7 · 5 = 3.5 MPa (508 psi). Figure 11(a) presents a comparison between the results measured for each of the two specimens and the calculated prediction of the average crack spacing. Although the figure reveals good agreement between the measured and calculated results, it also shows a considerable scatter in the experimental results for the two specimens (refer to “H-1-1” versus “H-1-2” in Fig. 11(a)). The scatter in concrete properties also caused these results to diverge from the expected symmetric cracking pattern. It is therefore reasonable to consider average values of (actual) corresponding segments with respect to the axis of symmetry of the entire bar (Fig. 12) for both specimens. Figure 11(b) presents a comparison between these values and the calculated values for the minimum and maximum segment lengths, as well as average lengths of all segments. It is evident that this way of considering the scatter in the results yields better agreement between the experimental and calculated data. The mean relative errors for minimum, average, and maximum values of uncracked segment lengths are 0.02, 0.06 and 0.07, respectively (with corresponding standard deviations of 0.15, 0.10, and 0.06). These values, as well as Fig. 11(b), indicate a very good agreement for all the tested parameters.

99

Table 2—Crack spacing at increasing tensile load (NBRI tests) Crack spacing, mm Load, kN

First Second Third segment segment segment Minimum Average Maximum Calculated

Fig. 12—Corresponding segments in Specimen H-1-1. (Note: The two right-hand cracks developed simultaneously and are considered as one.)

17.0

275

—

—

275

275.0

275

22.7

153

122

—

122

137.5

153

41.87

86

67

122

67

91.7

122

Measured* 17.6

275

—

—

275

275.0

275

20.7

158

117

—

117

137.5

158

43.6

87

71

117

71

91.7

117

3.4%

0.0%

—

—

0.0%

0.0%

0.0%

–9.7%

3.2%

–4.3%

—

–4.3%

0.0%

3.2%

4.0%

1.1%

5.6%

–4.3%

5.6%

0.0%

–4.3%

Relative error (Exp – Cal)/Exp (%)

*

Average values of corresponding left- and right-hand segments with respect to axis of symmetry of entire bar (Fig. 12). Notes: 1 kN = 0.225 kip; 1 mm = 0.0394 in.

Fig. 13—Comparison between model and experimental results of NSC specimens (NBRI tests). (Note: 1 mm = 0.039 in.; 1 kN = 0.225 kip.) Normal-strength concrete specimens—Like in the case of the HSC specimens, the following parameters were used for the NSC specimen calculations: Ec = 28,000 MPa (4061 ksi); tensile and residual strengths of 3.5 and 1.5 MPa (508 and 218 psi), respectively; A = 80 N/mm3 (295 kip/in.3); and Ψ = 0.7. These data yielded n = 7.4, ρ = 0.02, λ = 0.0122 mm–1 (0.3099 in.–1); fct = 0.7 · 3.5 = 2.45 MPa (355 psi) and σf = 0.7 · 1.5 = 1.05 MPa (152 psi). Figure 13 presents a comparison between the measured lengths of the uncracked segments (minimum, maximum, and average) and the calculated predictions. Note that there are some differences in the measured lengths of corresponding left- and right-hand segments. Therefore, each experimental point in Fig. 13 represents the average of these two values. The mean relative errors for minimum, average, and maximum values of uncracked segment lengths are 0.03, 0.02, and 0.02, respectively (with corresponding standard deviations of 0.15, 0.10, and 0.07). These values, as well as Fig. 13, indicate a very good agreement for all the tested parameters. This agreement is demonstrated also in Table 2, which describes the crack distribution at increasing tensile loads, as well as the corresponding relative errors. CONCLUSIONS This paper proposes a simplified model for the behavior of a reinforced fibrous concrete bar when subjected to increasing axial tension load. It refers to the cracking phase during which cracks are relatively narrow—that is, up to the yield of reinforcing bar in any one of the cracks. The model

100

predicts the loads that cause new cracks to develop as well as the distances between the cracks at various stages of loading. Phenomena that occur at the ultimate state, which is related to the cracking pattern, originate from the crack spacing that commences at the service state. The proposed approach was verified against experimental results from two different sources. Based on this model, an algorithm is presented to calculate the tensile forces that cause cracking as well as the intervals between the cracks. The current study shows that, while distances between cracks in plain concrete are equal, this is not the case for fibrous concrete. It is shown that crack patterns in conventionally reinforced concrete with and without fibers are qualitatively different, even when fiber distribution is uniform. This phenomenon is a result of the stress-free edges of the concrete. The model’s predictions of the minimum, average, and maximum of crack spacing (lengths of uncracked segments) were compared with available experimental results. It is evident from these comparisons that there is fair to good agreement, especially for the average values (mean relative error and standard deviation do not exceed 0.06 and 0.18, respectively). A parametric study shows that the effect of the stress distribution in the concrete (indicated by the parameter Ψ) is pronounced at small steel strains whereas it is negligible at relatively larger strains. An opposite effect was found for the bond-slip coefficient A, which has a pronounced effect only at relatively larger strains. AUTHOR BIOS Yuri S. Karinski is a Senior Researcher at the National Building Research Institute, Technion – Israel Institute of Technology (the Technion), Haifa, Israel. He received his BSc, MSc, and PhD at the Faculty of Mathematics and Mechanics, Moscow State University, Moscow, Russia. His research interests include mathematical simulations of the dynamic behavior of elastic-plastic structures and of the dynamic response of buried structures.


ACI member Avraham N. Dancygier is an Associate Professor at the Faculty of Civil and Environmental Engineering and a Researcher at the National Building Research Institute at the Technion. His research interests include static and dynamic behavior of reinforced concrete structures, highstrength/high-performance concrete structures, and the behavior of buried structures under static and dynamic loads. ACI member Amnon Katz is an Associate Professor at the Faculty of Civil and Environmental Engineering at the Technion. He received his BSc, MSc, and PhD at the Technion. He is a member of ACI Committees 440, Fiber-Reinforced Polymer Reinforcement, and 555, Concrete with Recycled Materials. His research interests include advanced concrete technology and environmental aspects of concrete structures.

ACKNOWLEDGMENTS This work was supported by a joint grant from the Centre for Absorption in Science of the Ministry of Immigrant Absorption and the Committee for Planning and Budgeting of the Council for Higher Education under the framework of the KAMEA Program and by the Israeli Ministry of Construction and Housing. The research grants are greatly appreciated. The authors would like to thank also S. Engel, V. Eisenberg, and G. Ashuah for their useful advice and support.

REFERENCES

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12. Redaelli, D., “Testing of Reinforced High Performance Fibre Concrete Members in Tension,” Proceedings of the 6th International Ph.D. Symposium in Civil Engineering, Zurich, Switzerland, 2006, pp. 122-123. 13. Deluce, J. R., and Vecchio, F. J., “Cracking of SFRC Members Containing Conventional Reinforcement,” ACI Structural Journal, V. 110, No. 3, May-June 2013, pp. 481-490. 14. Redaelli, D., and Muttoni, A., “Tensile Behaviour of Reinforced Ultra-High Performance Fiber Reinforced Concrete Element,” Proceedings of fib Symposium: Concrete Structures: Stimulators of Development, Dubrovnik, Croatia, 2007, pp. 267-274. 15. Dancygier, A. N., and Berkover, E., “Behavior of Fibre-Reinforced Concrete Beams with Different Reinforcement Ratios,” Proceedings of fib Symposium: Concrete Engineering for Excellence and Efficiency, Prague, Czech Republic, 2011, pp. 1133-1136. 16. Dancygier, A. N., and Savir, Z., “Flexural Behavior of HSFRC with Low Reinforcement Ratios,” Engineering Structures, V. 28, No. 11, 2006, pp. 1503-1512. doi: 10.1016/j.engstruct.2006.02.005 17. Barragán, B. E.; Gettu, R.; Martín, M. A.; and Zerbino, R. L., “Uniaxial Tension Test for Steel Fibre Reinforced Concrete—A Parametric Study,” Cement and Concrete Composites, V. 25, No. 7, 2003, pp. 767-777. doi: 10.1016/S0958-9465(02)00096-3 18. Gettu, R.; Gardner, D. R.; Saldivar, H.; and Barragfin, B. E., “Study of the Distribution and Orientation of Fibers in SFRC Specimens,” Materials and Structures, V. 38, Jan.-Feb. 2005, pp. 31-37. 19. Yankelevsky, D. Z.; Jabareen, M.; and Abutbul, A. D., “One-Dimensional Analysis of Tension Stiffening in Reinforced Concrete with Discrete Cracks,” Engineering Structures, V. 30, No. 1, 2008, pp. 206-217. doi: 10.1016/j.engstruct.2007.03.013 20. Bentur, A., and Mindess, S., Fibre Reinforced Cementitious Composites, second edition, Taylor & Francis, Oxford, UK, 2007, 624 pp. 21. Chiaia, B.; Fantilli, A. P.; and Vallini, P., “Evaluation of Crack Width in FRC Structures and Application to Tunnel Linings,” Materials and Structures, V. 42, No. 3, 2009, pp. 339-351. doi: 10.1617/s11527-008-9385-7 22. Yuguang, Y.; Walraven, J. C.; and den Uijl, J. A., “Combined Effect of Fibers and Steel Rebars in High Performance Concrete,” Heron, V. 54, No. 2/3, 2009, pp. 205-224. 23. fib Bulletin 10, “Bond of Reinforcement in Concrete,” Fédération internationale du béton (fib), Lausanne, Switzerland, 2000, 434 pp. 24. ASTM C1609/C1609M-05, “Standard Test Method for Flexural Performance of Fiber-Reinforced Concrete (Using Beam With Third-Point Loading),” ASTM International, West Conshohocken, PA, 2005, 8 pp. 25. RILEM TC 162-TDF, “Recommendations—Bending Test,” Materials and Structures, V. 33, Jan.-Feb. 2000, pp. 3-5. 26. Mindess, S.; Young, J. F.; and Darwin, D., Concrete, second edition, Prentice Hall, Upper Saddle River, NJ, 2003, 644 pp. 27. Leutbecher, T., and Fehling, E., “Tensile Behavior of Ultra-HighPerformance Concrete Reinforced with Reinforcing Bars and Fibers: Minimizing Fiber Content,” ACI Structural Journal, V. 109, No. 2, Mar.-Apr. 2012, pp. 253-264. 28. Leutbecher, T., “Rissbildung und Zugtragverhalten von mit Stabstahl und Fasern bewehrtem Ultrahochfesten Beton (UHPC),” PhD thesis, University of Kassel, Kassel, Germany, 2007, 264 pp. 29. Rilem/CEB/FIP, “Bond Test for Reinforcing Steel: 2. Pullout Test,” Materials and Structures, V. 3, No. 15, 1970, pp. 175-178. 30. Dancygier, A. N.; Katz, A.; and Wexler, U., “Bond between Deformed Reinforcement and Normal and High Strength Concrete with and without Fibers,” Materials and Structures, V. 43, No. 6, 2010, pp. 839-856. doi: 10.1617/s11527-009-9551-6

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APPENDIX—FLOWCHART FOR CRACKING PROCESS

102



TECHNICAL PAPER

Title No. 112-S10

Glass Fiber-Reinforced Polymer-Reinforced Circular Columns under Simulated Seismic Loads by Arjang Tavassoli, James Liu, and Shamim Sheikh This paper presents the experimental results of nine large-scale circular concrete columns reinforced with longitudinal and transverse glass fiber-reinforced polymer (GFRP) bars. These specimens were tested under lateral cyclic quasi-static loading while simultaneously subjected to constant axial load. Based on the measured hysteretic loops of moment-versus-curvature and shear-versus-tip deflection relationships, a series of parameters related to ductility, energy dissipation capacity, and flexural strength are used to evaluate the seismic behavior of each column. The results showed that concrete columns reinforced with GFRP bars and spirals can behave in a manner that has stable post-peak response and achieve high levels of deformability. The results indicate that, as a relatively new material with excellent corrosion resistance and high strength-weight ratio, GFRP bars can be successfully used as internal reinforcement in ductile concrete columns. Keywords: concrete column; ductility; experiment; GFRP reinforcement; seismic resistance; strength.

INTRODUCTION A considerable amount of work has been done on the behavior of steel-reinforced concrete columns. Appropriately designed lateral confinement, such as the use of closely spaced transverse steel reinforcement, externally bonded steel jackets, or fiber-reinforced polymer (FRP) wrapping, have proven to significantly improve the ductility, energy dissipation capacity and flexural strength of steel-reinforced concrete columns under seismic loading. The corrosion of steel—especially the lateral reinforcement in structures—has cost billions of dollars in infrastructure repair in North America. It is estimated that $3.6 trillion are needed by 2020 to alleviate potential problems in civil infrastructure.1 Approximately one in nine bridges in the United States are rated as structurally deficient, requiring about $20.5 billion annually to eliminate the bridge deficient backlog by 2028. As a relatively new material with excellent corrosion resistance and a high strength-weight ratio, internal glass fiber-reinforced polymer (GFRP) reinforcement is considered a feasible and sustainable alternative to steel reinforcement for future infrastructure. A number of studies have been carried out on GFRP-reinforced concrete members subjected to flexure and shear. However, only a few experimental studies have been reported on GFRPreinforced concrete columns. Alsayed et al.2 reported that replacing longitudinal steel reinforcement with GFRP bars of the same volumetric ratio resulted in a 13% reduction in the axial load capacity of columns. De Luca et al.3 reported that at low longitudinal reinforcement ratios, the response of GFRP-reinforced columns is very similar to that of steelreinforced columns and the contribution of GFRP bars can ACI Structural Journal/January-February 2015

be neglected in the axial load capacity of columns. They also stated that the relatively lower compressive strength and stiffness of GFRP bars will make FRP-reinforced concrete columns susceptible to instability. Based on tests on eight columns under concentric axial loads, of which five were longitudinally and transversely reinforced with GFRP, Tobbi et al.4 concluded that GFRP bars can be used as main reinforcement in columns provided that closely spaced transverse reinforcement is used. Based on the sectional analysis of GFRP-reinforced columns with a longitudinal reinforcement ratio no less than 3%, Choo et al.5 concluded that no balance point exists in the axial load-moment interaction curves and that the flexural strength tends to increase monotonically with the decrease of axial load. Sharbatdar and Saatcioglu6 tested square columns reinforced with FRP bars under axial and lateral load. They concluded that FRPreinforced columns under 30% of their axial capacity can develop 2 to 3% lateral drift ratios. Aside from this study, experimental work on GFRP-reinforced circular columns subjected to combined axial, shear and flexural loads, especially under cyclic loads simulating seismic forces, is almost nonexistent. The experimental study reported here is part of an ongoing comprehensive research program at the University of Toronto, in which full-scale square and circular concrete columns are tested under simulated seismic load in the same manner to provide comparable results to investigate different variables and design parameters (refer to References 7 and 8). The variables include column types, steel configuration, concrete strength, axial load level, and amounts of steel and FRP confinement. This paper describes the results from nine large-scale circular concrete columns internally reinforced with longitudinal GFRP bars and transverse GFRP spirals. A series of parameters related to curvature ductility, displacement ductility, energy dissipation capacity, and flexural strength are used to evaluate the seismic behavior of columns. RESEARCH SIGNIFICANCE Available research on the behavior of columns internally reinforced with longitudinal and lateral GFRP reinforcement is very limited. Information on their response under large lateral cyclic displacements is almost nonexistent. ACI Structural Journal, V. 112, No. 1, January-February 2015. MS No. S-2013-309.R1, doi: 10.14359/51687227, received April 25, 2014, and reviewed under Institute publication policies. Copyright © 2015, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published ten months from this journal’s date if the discussion is received within four months of the paper’s print publication.

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Table 1—Test results of GFRP-reinforced circular columns Specimen

Spiral diameter, mm (in.) Spiral pitch, mm (in.) Axial load level P/Po Mmax, kN.m (kip.ft) Vmax, kN (kip) Plastic hinge length, mm (in.)

P28-C-12-50

12 (1/2)

50 (2.0)

0.28

224 (165)

70.0 (15.7)

260 (10.2)

P28-C-12-160

12 (1/2)

160 (6.3)

0.28

152 (112)

71.1(16.0)

320 (12.6)

P28-C-16-160

16 (5/8)

160 (6.3)

0.28

123 (91)

59.4 (13.4)

260 (10.2)

P28-B-12-50

12 (1/2)

50 (2.0)

0.28

227 (167)

78.2 (17.6)

270 (10.6)

P42-C-12-50

12 (1/2)

50 (2.0)

0.42

219 (161)

74.9 (16.8)

250 (9.8)

P42-C-12-160

12 (1/2)

160 (6.3)

0.42

162 (119)

58.8 (13.2)

300 (11.8)

P42-B-12-160

12 (1/2)

160 (6.3)

0.42

160 (118)

63.1 (14.2)

270 (10.6)

P42-B-16-160

16 (5/8)

160 (6.3)

0.42

187 (137)

70.0 (15.7)

240 (9.4)

P42-B-16-275

16 (5/8)

275 (10.8)

0.42

154 (113)

69.8 (15.7)

370 (14.6)

Fig. 1—Details of test specimen. (Note: 1 mm = 0.039 in.) This research provides results from large-scale GFRPreinforced concrete columns tested under simulated earthquake loading. The measured ductility and deformability, energy dissipation capacity, and strength of the columns provide valuable parameters in understanding the seismic behavior of these columns. The comparison between similarly tested GFRP- and steel-reinforced columns can also help clarify and further develop some of the existing design provisions that are overly conservative and in some cases vague. EXPERIMENTAL PROGRAM Specimens Nine circular columns were constructed and tested in this study. Each specimen consisted of a 356 mm (14 in.) diameter and 1473 mm (58 in.) long column cast monolithically with a 508 x 762 x 813 mm (20 x 30 x 32 in.) stub (Fig. 1). The column part of the specimen represents the column between the sections of maximum moment and zero moment in a structure while the stub represents a footing or a joint. Both the longitudinal and transverse reinforcement in each column were made out of GFRP. The main variables investigated were the axial load level, the type of GFRP, and size and spacing of GFRP spirals.

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The specimens were tested under constant axial load and lateral cyclic displacement excursions. The axial load was either 0.28Po or 0.42Po, where Po is the nominal axial load capacity of the column, determined by Po = 0.85fcʹ(Ag – AGFRP) + εcʹEGFRPAGFRP; Ag is the gross cross-sectional area of column; fcʹ is the compressive strength of unconfined concrete; εcʹ is the concrete strain at peak strength; and AGFRP and EGFRP are the total cross-sectional area and the modulus of elasticity of the longitudinal GFRP reinforcement in compression, respectively. The specimen details are provided in Table 1. Each specimen’s label has four parts indicating the level of axial load (P28 or P42), the type of GFRP material (B or C), the spiral diameter (12 or 16 mm [0.47 or 0.63 in.]) and the spiral pitch in mm in the 740 mm (29 in.) long test region of columns adjacent to the column-stub interface. The length of the test region was decided based on the previous tests in which failure was observed in that region. To ensure that failure occurs in the test region of columns, a closely spaced steel spiral was used outside the test region which was additionally confined by externally bonded FRP wrapping. Figure 2 shows the column cage and the FRP-wrapped column before the test.


Fig. 3—GFRP material used in this study: 12 mm (0.47 in.) Type B-SP spiral (left); 25 mm (1 in.) Type B-ST longitudinal bar (top right); and 25 mm (1 in.) Type C-ST longitudinal bar (bottom right). Fig. 2—Column cage reinforcement (left) and column before test (right). Concrete The specimens were cast vertically by using two trucks of ready mixed concrete with the maximum aggregate size of 14 mm (0.55 in.) and a slump of around 150 mm (6 in.). One truck was used for the nine stubs and another for the nine columns. The development of concrete strength with age was monitored in accordance with ASTM C39/C39M9 by tests of 150 x 300 mm (6 x 12 in.) concrete cylinders that were cured adjacent to the column specimens under similar conditions. There were at least three cylinders in each group which were tested at 7, 14, and 28 days after casting and throughout the column testing period. Testing of the nine columns started on the 70th day and was completed on the 125th day after casting, during which the concrete strength was measured between 34.1 and 35.4 MPa (4.95 and 5.13 ksi). Hence, fcʹ was taken as 35 MPa (5.08 ksi) for all columns. Glass fiber-reinforced polymers (GFRP) Each column contained six longitudinal GFRP bars of 25 mm (1 in.) diameter, which were embedded completely to near the end of stubs (Fig. 1 and 2). The transverse reinforcement in the test region of the column consisted of 12 or 16 mm (0.47 or 0.63 in.) diameter GFRP spirals at specified spacing. The clear concrete cover was 24 mm (0.95 in.) to the outmost surface of spirals and the area ratio of concrete core, measured to the centerline of spirals, to the gross cross-section of the column was approximately 72%. Two types of GFRP reinforcement from two different manufacturers were used in this program, as shown in Fig. 3. Five columns were reinforced with GFRP longitudinal bars and spirals from manufacturer “C,” while the other four columns contained GFRP reinforcement from the manufacturer indicated as “B.” The mechanical properties of the GFRP bars were obtained by testing three coupons for each bar type. Tensile tests were carried on both longitudinal and transverse GFRP bars, while compressive tests were done on longitudinal GFRP bars only. For the tensile tests of GFRP bars, steel couplers were used on each end of GFRP samples and the gap between


the GFRP bar and the coupler was filled with an expansive cement mortar and capped. Figure 4 shows the GFRP samples during the tension test. The strain was measured using a detachable gauge attached to the bar at midheight with a gauge length of 25 mm (1 in.). To avoid damage to the gauge, it was removed at about a third of the ultimate load. Elastic behavior was assumed to extrapolate the strain data until the ultimate stress. Table 2 summarizes the results for all the GFRP coupon tension tests. The behavior of GFRP bars in compression is still not as well-studied as the tensile response. For GFRP bars with L/d < 6, where L is the free length and d is the diameter of bars, it was reported that their compressive strength can be 10 to 50% of their ultimate tensile strength, depending on fiber content, the manufacturing procedure, and the resin quality.10 Based on tests on FRP bars with L/d < 6, it was reported that the ultimate compressive strength is approximately half of the tensile strength, while the modulus of elasticity was found to be approximately equal in both compression and tension.11 In this study, 15 GFRP bar samples were tested in compression. The test setup is shown in Fig. 5, in which the axial compression was directly applied to the samples through steel cylindrical caps at both ends. A different free length of GFRP bars between the two steel caps was chosen based on the spiral pitch in different columns. Two strain gauges were installed at the midheight of each GFRP coupon on opposite sides to measure the longitudinal compressive strain. The modulus of elasticity was obtained by averaging the two strains during the initial stages of the test before the two strain values deviated from each other due to buckling effect. Table 3 summarizes the results obtained from these tests. Steel Two types of deformed steel bars were used in specimens. In each column outside the 740 mm (29 in.) long test region, the transverse reinforcement consisted of Grade 60 U.S. No. 3 steel spirals at 50 mm (2 in.) pitch (volumetric ratio to the concrete core ρsh = 1.93%). The stubs were reinforced with steel cages made of 10M (area of cross section = 100 mm2 [0.155 in.2]) stirrups spaced at 64 mm (2.5 in.) in both horizontal and vertical directions. The mechanical 105

Table 2—Mechanical properties of GFRP bars in tension Reinforcing bar type*

Nominal diameter, mm (in.)

Actual diameter, mm (in.)

Modulus of elasticity, MPa (ksi)

Ultimate stress, MPa (ksi)

Ultimate strain

C - SP

12 (0.472)

12.62 (0.497)

C - SP

16 (0.630)

16.05 (0.632)

58,399 (8470)

1454 (211)

0.0249

51,224 (7429)

1069 (155)

0.0209

†

0.0165†

C - ST

25 (0.984)

25.11 (0.989)

65,779 (9540)

1087 (158)

B - SP

12.7 (0.5)

12.7 (0.5)

58,948 (8550)

1243 (180)

0.0211

B - SP

15.87 (0.625)

16.01 (0.630)

54,567 (7914)

1159 (168)

0.0213

B - ST

25.4 (1)

28.44 (1.120)

74,270 (10772)

1338 (194)

0.0180

*

The first part indicates type of GFRP material and second part identifies whether straight sample was taken from material used for straight bars (ST) or spirals (SP).

†

These values do not represent ultimate stress and strain. The test was terminated due to slippage of bars in coupler.

Table 3—Mechanical properties of GFRP bars in compression Reinforcing bar type

Free length, mm (in.)

Nominal area, mm2 (in.2)

Actual area, mm2 (in.2)

491 (0.761)

495 (0.768)

50 (2) C

B

Modulus of elasticity, MPa (ksi)

Ultimate stress, MPa (ksi)

Ultimate strain

55,569 (8060)

619 (90)

0.01132

160 (6.3)

56,357 (8174)

602 (87)

0.01066

50 (2)

71,018 (10300)

864 (125)

0.0104

72,165 (10467)

873 (127)

0.00932

72,701 (10544)

759 (110)

0.00901

160 (6.3)

507 (0.786)

275 (10.8)

635 (0.985)

Fig. 4—GFRP coupon test (tension), left to right: 12 mm (0.47 in.) Type B-SP before and failure; and 12 mm (0.47 in.) Type C-SP before and failure, left to right: Bar Type B with free length of 275 mm (10.8 in.) before and after failure; and Bar Type C with free length of 160 mm (6.3 in.) before and after failure.

Fig. 5—GFRP coupon test (compression). 106


Table 4—Mechanical properties of reinforcing steels Reinforcing bar size

Area, mm2 (in.2)

Yield stress fy, MPa (ksi)

Yield strain εy

Modulus of elasticity E, MPa (ksi)

Start of strainhardening εsh

Ultimate stress fu, MPa (ksi)

Ultimate strain εu

10M

100 (0.16)

420 (60.9)

0.0023

187,105 (27,137)

0.0251

542 (78.6)

0.1960

US No. 3

71 (0.11)

485 (70.3)

0.0025

191,570 (27,785)

0.0273

598 (86.7)

0.1632

Fig. 6—Location of strain gauges on GFRP bars and spiral. (Note: 1 mm = 0.039 in.) properties obtained from the tests on a minimum of three samples of each type of steel bar are presented in Table 4. Instrumentation To monitor the deformation of GFRP reinforcement in each specimen during testing, 18 strain gauges were installed on the longitudinal bars and six on the spirals— three on each of the two turns adjacent to the stub face—as shown schematically in Fig. 6. Ten linear variable differential transformers (LVDTs) were installed on one side of the column and light-emitting diode (LED) targets were used on the other side to measure deformation of the concrete core in the potential plastic-hinge region. The LVDTs were mounted on the threaded rods installed inside the columns before concrete casting to measure the inelastic deformation of core concrete. Three LEDs were mounted on a stationary location and were used as reference, while 14 targets were placed on each specimen. In addition to the linear strains, the LED targets provided three-dimensional movements at each location. The lateral deflection along each specimen was measured by six LVDTs. The instrumentation is displayed in Fig. 7. Testing procedure Each column was tested under a constant axial load and quasi-static lateral cyclic displacement excursions. The axial load was applied by a hydraulic jack with a capacity of 10,000 kN (2250 kip), while the cyclic lateral loading was applied using an actuator with a 1000 kN (225 kip) load capacity and approximately 100 mm (4 in.) stroke capacity. ACI Structural Journal/January-February 2015

The test frame is shown in Fig. 8. Two hinges permitted each end of the specimen to rotate freely in plane and kept the axial load path constant throughout testing. Even though the cantilever length of concrete columns was 1473 mm (58 in.), this test setup resulted in an actual shear span of 1841 mm (72.5 in.) measured from the center of the right hinge to the column-stub interface—that is, from the zero moment section to the maximum moment section of columns. Each specimen was strictly aligned before testing, so that its center line coincided with the action line of axial load. At the beginning of each test, the predetermined axial load was firstly applied to the specimen and kept constant throughout testing. The lateral cyclic excursions were then applied in a displacement-control mode following the specified deflection regime shown in Fig. 9 until the column collapsed under the constant axial load. TEST OBSERVATIONS All columns behaved almost elastically during the first two lateral load cycles. Flexural cracks appeared on top and bottom faces of the column in the testing region during the third cycle. For columns under an axial load of 0.42Po, surface cracks in the longitudinal direction appeared prior to flexural cracks close to the column-stub interface. Cover spalling for the majority of specimens initiated during the fourth lateral cycle. For well-confined columns, cover spalling was delayed by a few cycles, while for columns with a higher axial load, complete cover deterioration occurred before the eighth cycle. In general, columns showed a very stable response with large deformability (Table 5). For 107

Fig. 7—Instrumentation: (a) strain gauges on bars and spirals in the test region (left), horizontal LVDTs on one side of the column (top right), and LED targets on the other side of the column (bottom right); and (b) location of LVDTs. (Note: 1 mm = 0.039 in.)

Fig. 9—Lateral loading protocol. (Note: 1 mm = 0.039 in.) Fig. 8—Test setup. instance, specimen P28-B-12-50 underwent 34 cycles of lateral displacement excursions and still maintained applied axial load. Two additional cycles at the maximum stroke limit of the actuator caused crushing of the longitudinal bars in compression and a drop in the axial load. Specimen 108

P28-C-16-160 showed lower ductility and strength than expected due to the honeycombing in the test region of the column, which led to premature failure. The failure of all columns was gradual and mainly due to the crushing and buckling of the longitudinal GFRP bars in compression accompanied by damage of the core concrete. Even after the collapse of the columns, no spiral damage or rupture was observed in any of the tests. It was observed that ACI Structural Journal/January-February 2015

where the inclination θ of the stub-column interface is

Table 5—Ductility parameters Specimen

μΦ

μΔ

δ, %

NΔ80

W80

NΔ

W

Vmax, kN (kip)

Mmax, kN.m (kip.ft)

P28-C>10.0 6.8 7.3 12-50

84

210 159 456

70.0 (15.7)

224 (165)

P28-C12-160

>9.0

3.2 3.0

17

22

99

160

71.1 (16.0)

152 (112)

P28-C16-160

>5.3

2.8 2.6

11

10

86

134

59.4 (13.4)

123 (91)

260 178 671

78.2 (17.6)

227 (167) 219 (161)

P28-B>15.6 9.2 9.0 12-50

94

P42-C>11.8 5.8 4.4 12-50

30

65

142 395

74.9 (16.8)

P42-C12-160

>7.5

3.7 3.0

16

28

50

93

58.8 (13.2)

162 (119)

P42-B12-160

>9.5

2.3 1.9

7

9

70

96

63.1 (14.2)

160 (118)

P42-B>10.7 4.0 3.3 16-160

18

34

119 239

70.0 (15.7)

187 (137)

P42-B16-275

7

8

56

69.8 (15.7)

154 (113)

>8.8

2.5 2.1

60

the failure of each column occurred within the test region. Due to the additional confinement provided by the concrete stubs, the most damaged section of columns was pushed away from the stub-column interface where the moment is the largest. Photos of the failed columns are shown in Fig. 10, while the measured lengths of the most damaged regions are presented in Table 1. RESULTS AND DISCUSSION Test results The columns were tested horizontally; therefore, certain geometric adjustments are needed to convert the measured test data according to the cantilever column model shown in the schematic drawing of the specimen in Fig. 11. The tip displacement Δ is the tangential deviation of the contraflexural Point B, calculated as

D = dL

a+b (1) a

V ′ = PL

a (2) a+b

As discussed by Liu and Sheikh,8 the shear force V at the base of the cantilever column differs from the applied lateral force Vʹ at Point B. As a result, the base shear force V can be determined from the components of the axial load P and lateral force Vʹ, as shown in Eq. (3)

V = Vʹcosθ – Psinθ


D (4) a+b

Moment at the most damaged section is found by summing the moments caused by both the lateral and axial load. The following expression is used to calculate the moment in the most damaged section

M = V (L – Dmd) + PΔ

(5)

where L = H + c = 1841 mm (72.5 in.) is the shear span of the column; and Dmd is the distance from the column-stub interface to the most damaged section. As mentioned earlier, 10M bars were precast in the testing region of the columns to measure core deformations after cover spalling and to obtain curvature. However, as a result of high deformation of the column and considerable concrete crushing in the damaged region, the measured curvature values from these rods were not found to be accurate at large displacements. Thus the curvature results reported here were obtained using strains from the strain gauges installed on the longitudinal bars. In cases where the strain gauges on the outer bars in the most damaged region stopped functioning before the failure of columns, curvature was obtained from the strain gauges located on the inner bars to extend the moment curvature (M-Φ) behavior. This extension is shown in dotted line in Fig. 12, while the horizontal dashed lines in each graph in Fig. 12 represent the nominal moment capacity of the column section (Mn). To evaluate the value of Mn by sectional analysis, the stress-strain relationship of unconfined concrete was used with the ultimate strain of 0.0035. Meanwhile, linearly elastic stress-strain relationship was assumed for longitudinal GFRP reinforcement with ultimate tensile and compressive strengths provided in Tables 2 and 3, respectively. In addition to the M-Φ response of the most damaged section, the shear-versus-tip deflection relationship (V-Δ) is also provided for each column. The dashed lines on the V-Δ curves represent the nominal shear capacity Vn with a decreasing slope caused by secondary effects. Vn is calculated using the following expression

where δL is the deflection at the point of application of lateral load PL. The lateral force Vʹ at the right hinge B can be determined from the applied force PL as

q=

Vn =

M n − PD y L

(6)

where P is the applied axial load; Δy is the yield displacement; and L is the shear span. Although GFRP-reinforced columns do not undergo yielding, the procedure used by Liu and Sheikh8 for steel-reinforced columns was also used in this study to define a hypothetical yield displacement Δy to evaluate the displacement ductility factor. Displacement Δy is determined corresponding to the nominal lateral load capacity Vn along a straight line joining the origin and a point of 65% Vn on the ascending branch of the lateral shearversus-tip deflection curve, which is determined for a specific axial load applied on the column.

(3) 109

Fig. 10—Plastic hinge regions at end of testing.

Fig. 11—Schematic drawing of test specimen.

110

Ductility parameters Several ductility parameters have been used to explain the behavior of reinforced concrete sections in the literature.7 Curvature and displacement ductility factors (μΦ, μΔ), drift ratio (δ), cumulative displacement ductility ratio (NΔ80), and work damage indicator (W80) are used in this study to quantify the deformability of the specimens.6 Figure 13 provides a graphical representation of how the displacement-related ductility parameters are calculated while parameters based on curvature can be defined in a similar manner. The results on ductility parameters are summarized in Table 5. The curvature at the most damaged section is obtained from the strain gauges on the longitudinal bars. Because the strain gauges stopped functioning before the failure of columns, the ultimate curvature ductility factors will be higher in most cases than the values reported here. ACI Structural Journal/January-February 2015

Fig. 12—Moment-curvature and shear-deflection responses of columns. (Note: 1 mm = 0.039 in.; 1 kN = 0.225 kip; 1 kNm = 0.738 kip.ft; 1 rad/km = 305×10–6 rad/ft.)


111

Fig. 13—Definitions of ductility parameters. The values of NΔ80 and W80 are respectively the cumulative displacement ductility ratio and the work damage indicator over a number of cycles until the shear capacity of the column drops to 80% of the peak load. As shown in Fig. 12, the column loses its stiffness (shown by Ki in Fig. 13) as the test progresses and becomes negative for most columns during the last few cycles of testing. Due to the presence of the axial load on a highly deformed column during the last cycles, the shear force has to switch direction to maintain equilibrium. To avoid adding negative energies to the work damage indicator, only cycles with a positive Ki are included in the calculation of NΔ and W. Although this does not capture the full response of the column, the results show a similar trend to the reliable values of NΔ80 and W80. Effect of axial load level Two levels of column axial load (0.27Po and 0.42Po) were used in this study. In steel-reinforced columns, it has been found that increasing the axial load would result in faster deterioration of the core concrete and initiation of longitudinal bar buckling.7 Similar behavior was observed for the GFRP-reinforced concrete columns tested here. The plastic hinge length (most damaged zone) shown in Table 1 is greater for columns with higher axial load indicating wider spread of the damaged zone. The effect of axial load on column ductility is more visible on well-confined specimens. For example, Columns P28-C-12-50 sustained an axial load of 0.28Po and achieved a displacement ductility factor μΔ of 6.8 while the similar Column P40-C-12-50 carrying an axial load of 0.42Po showed μΔ of 5.8. The same observation can also be found by comparing the test results of other columns shown in Table 5 and Fig. 12. An increase of axial load from 0.28Po to 0.42Po did not significantly affect the flexural strength of GFRP-reinforced columns. For example, the measured flexural strengths of columns P28-C-12-50 and P42-C-12-50 were almost identical. This experimental observation confirms the similarity between nominal moment capacities of specimens under

112

Fig. 14—Hysteresis response of steel-confined versus GFRP-reinforced column. (Note: 1 mm = 0.039 in.; 1 kN = 0.225 kip.) different axial loads evaluated using sectional analysis. Similar results were also reported by Choo et al.5 Effect of GFRP bar type In general, there was not a significant difference between the performances of the columns reinforced with two different types of GFRP bars. The maximum moment capacity at the most damaged section, as shown in Table 1, is slightly higher for columns reinforced with GFRP Type B. This can be attributed to the fact that both longitudinal bars and spirals of Type B have considerably larger actual areas compared with the nominal areas than those of GFRP Type C. The length of the damaged region is very similar for columns in pairs such as P28-C-12-50 and P28-B-12-50 or P42-C-12-160 and P42-B-12-160, in which the only difference was the type of GFRP bar. The latter pair of columns showed a similar deflected shape at all stages and failure for both columns occurred at almost the same lateral displacement. Effect of spiral pitch and reinforcement ratio Research on steel-reinforced columns has shown that increasing the transverse reinforcement ratio and decreasing ACI Structural Journal/January-February 2015

the spiral/tie pitch delay column failure by confining the concrete core prevents premature buckling of the longitudinal bars.7,8 Consequently, columns with lower spiral spacing have higher strength and ductility. Similar conclusions can be made for GFRP-reinforced columns using the results obtained in this experimental study. The moment capacity of the column considerably increased as the spiral spacing reduced from 160 to 50 mm (6.2 to 2 in.). The maximum shear capacity is approximately similar for all columns. This is expected because the maximum shear occurs in the first few cycles and is not affected by the spiral configuration. Table 5 shows that the absorbed energy is significantly higher for columns with smaller spiral spacing and the resulting higher transverse reinforcement. Closer spiral spacing resulted in better confinement of the concrete core and delayed the buckling of the longitudinal bars. For instance, the lateral drift ratio for column P28-C-12-50 is more than twice that of column P28-C-12-160. Reducing the spiral spacing from 160 to 50 mm (6.2 to 2 in.) resulted in a 50% increase in the moment capacity of the column. The transverse reinforcement ratio can be adjusted by changing the size of the spiral as well. However, the effect on column response is not as noticeable as adjusting the spiral spacing. The effect of changing the spiral size can be observed by comparing columns P42-B-12-160 and P42-B-16-160. Increasing the spiral size from 12 to 16 mm (0.47 to 0.63 in.) resulted in a 13% increase in the moment capacity of the column and doubled the dissipated energy. Comparison of GFRP-reinforced and steelreinforced columns GFRP-reinforced concrete columns generally show a softer ascending branch of shear-versus-deflection behavior than steel-reinforced concrete columns due to the lower modulus of elasticity of GFRP. The difference in stiffness is significant even though reinforcement represents only a small portion of the column’s properties. A comparison of the shear-versus-lateral tip deflection relationships of two columns is shown in Fig. 14. Column P27-NF-2 had 9.5 mm (No. 3) steel spiral with 100 mm (4 in.) pitch which resulted in ρsh = 0.9%. The longitudinal steel reinforcement ratio for this column was 3.01%. Both these values were similar to those of column P28-C-12-160, as were the column dimensions and testing conditions. The concrete strength for column P-27-NF-2 was 40 MPa while this value for column P28-C-12-160 was 35 MPa (5080 psi). The hysteretic response shows that the steel-reinforced column absorbs more energy in each lateral deflection loop as a result of the yielding and Bauschinger effect of the longitudinal steel reinforcement. It should be noted that a lack of yield plateau in GFRP bars results in a much lower residual deflection when the load is removed. The stiffness of the steelreinforced column is higher than that of the GFRP-reinforced column and the higher shear capacity of column P27-NF-2 is due to the higher modulus of elasticity of the steel reinforcement. The capacity gap could simply be decreased by using a larger amount of the GFRP reinforcement to balance its low stiffness which may reduce deformability of the column.


Table 6—Average spiral strain in most damaged region Maximum spiral strain at 80% peak εfh80%, µε

Maximum spiral strain εfh, µε

Ultimate spiral strain εufh, µε

εfh/εufh

P28-C12-50

4145

4145

24,900

0.167

P28-C12-160

3612

4031

24,900

0.162

P28-C16-160

4450

5605

19,900

0.282

P28-B12-50

9306

9779

21,100

0.463

P42-C12-50

2276

6440

24,900

0.259

P42-C12-160

7835

11,882

24,900

0.477

P42-B12-160

5010

7581

21,100

0.359

P42-B16-160

6597

13,054

21,300

0.613

P42-B16-275

5902

9723

21,300

0.456

Average “B”

6704

10,034

21,200

0.473

Average “C”

4464

6421

23,900

0.269

Specimen

Both columns showed stable post-peak descending branches and high ductility. However, it can be seen that the GFRP-reinforced column has a longer post-peak descending branch before final failure than the steel-reinforced column. This is mainly due to the fact that the steel bars have very low tangent modulus after yielding and therefore are more susceptible to buckling under compression than GFRP bars which maintain their modulus of elasticity throughout the entire duration of loading. GFRP reinforcement does not experience yielding and results in a nearly linear elastic moment-versus-curvature relationship of columns with no post-peak decline, as shown in Fig. 12, unlike the steel-reinforced columns in which the steel yields and the moment curvature response displays a descending branch. The GFRP transverse reinforcement despite being softer than steel at small strains, continue to provide increasing confinement until the column failure. Strain effectiveness of GFRP spirals The maximum strain in the GFRP spiral, εsp,80%, was recorded for each column at the stage when shear capacity dropped to 80% of the peak shear and also just before failure. The results are summarized in Table 6. As expected, in columns with high axial load, a greater hoop strain is observed. The average value of εsp,80% for all nine columns is 0.00546, which is close to the recommended Canadian code12 design value of 0.006. Maximum recorded spiral strains before column failure indicate that GFRP spirals were able to provide increasing confinement to the core while the 113

maximum strain is significantly less than the rupture strain. The average maximum spiral strain for the nine columns is about four times the steel yield strain before failure. The ratio between the maximum spiral strain and ultimate GFRP strain—spiral efficiency—is also listed for all the columns in Table 6. The increase in the moment capacity of the columns, especially those reinforced with a 16 mm (0.63 in.) spiral, over the last few cycles indicates the effective confinement provided by the GFRP spiral. This strength gain over the last cycles can clearly be seen in the momentversus-curvature response of columns P28-C-16-160, P42-B-16-160, and even P42-B-16-275. Comparison with code requirements for confinement The transverse GFRP spirals in specimens P28-C-12-50 and P28-B-12-50 were designed for a targeted lateral drift ratio of 4% according to CAN/CSA-S806-12,12 while specimen P42-C-12-50 was designed to achieve 2.5% drift ratio, and the rest of specimens had only substandard transverse reinforcement in terms of confinement spiral spacing required by this design code. Both columns P28-C-12-50 and P28-B-12-50 outperformed the design expectations, with measured drift ratios of 7.3 and 9.0%, respectively. Column P42-C-12-50 also achieved a lateral drift ratio of 4.4%, which exceeded the designed target of 2.5%. In fact, only columns P42-B-16-275 and P42-B-12-160 had a lateral drift ratio lower than 2.5%. The widely spaced spirals in P42-B-16-275 led to the premature buckling of longitudinal reinforcement and limited deformability. Specimen P42-B-12-160 showed a lateral drift ratio of only 1.9% due to the presence of honeycomb regions in the testing region. The experimental results indicated that, if designed according to CSA-S806-12,12 GFRP-reinforced columns can achieve much higher lateral drift capacity than design expectations. SUMMARY AND CONCLUSIONS To understand the behavior of concrete columns reinforced longitudinally with GFRP bars and transversely with GFRP spirals, nine large-scale specimens were tested under lateral displacement excursions and constant axial load. Experimental results in the form of moment-versus-curvature and shear-versus-tip deflection hysteretic responses and various ductility parameters are presented. The following conclusions can be drawn from this study: 1. The crushing strength of GFRP bars in compression is approximately half of their ultimate tensile strength, and the modulus of elasticity in compression for a GFRP bar was found to be similar to that under tension. 2. Columns reinforced with GFRP bars and spirals showed a stable response, and the type of GFRP material used did not cause a significant change in the behavior of the columns. 3. Columns P28-B-12-50 and P28-C-12-50 were designed for a lateral drift ratio of 4% and they achieved a lateral drift ratio in excess of 7%. Column P42-C-12-50 achieved a lateral drift ratio of 4.4%, which was 1.9% higher than the design value. 4. Columns that were subjected to a higher axial load sustained more damage and displayed lower levels of ductility and deformability. The amount and detailing of 114

transverse reinforcement is more critical for high axially loaded columns. 5. GFRP bars, due to their larger stiffness at larger strains, performed in a more stable manner than steel bars. 6. The transverse steel reinforcement provides effective confinement to the core at early stages; however, as the steel starts to yield, the confinement is less effective, allowing the concrete core to expand. GFRP spirals, on the other hand, provide an increasing level of confinement with increased deformation which delays crushing of the core concrete. AUTHOR BIOS

Arjang Tavassoli is a Project Associate at Parsons Brinckerhoff Halsall Inc. in Toronto, ON, Canada. He received his BASc and MASc in civil engineering from the University of Toronto, Toronto, ON, Canada, in 2011 and 2013, respectively. James Liu is a Structural Engineer with Brown and Company Engineering Ltd. in Toronto, ON, Canada. He received his BS in civil engineering from Tongji University, Shanghai, China, and his PhD from the University of Toronto. His research interests include analysis and retrofit of concrete structures. Shamim Sheikh, FACI, is a Professor of civil engineering at the University of Toronto. He is Chair of ACI Subcommittee 441-E, Columns with Multi-Spiral Reinforcement. He is a former Chair and member of Joint ACI-ASCE Committee 441, Reinforced Concrete Columns, and a member of ACI Committee 374, Performance-Based Seismic Design of Concrete Buildings. In 1999, he received the ACI Chester Paul Seiss Award for Excellence in Structural Research. His research interests include earthquake resistance and design of concrete structures, confinement of concrete, use of fiber-reinforced polymer for sustainable concrete structures.

REFERENCES 1. ASCE, “2013 Report Card for America’s Infrastructure,” American Society of Civil Engineers, www.infrastructurereportcard.org. (last accessed Oct. 15, 2014) 2. Alsayed, S. H.; Al-Salloum, Y. A.; Almusallam, T. H.; and Amjad, M. A., “Concrete Columns Reinforced by Glass Fiber Reinforced Polymer Rods,” 4th International Symposium—Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures, SP-188, C. W. Dolan, S. H. Rizkalla, and A. Nanni, eds., American Concrete Institute, Farmington Hills, MI, 1999, pp. 103-112. 3. De Luca, A.; Matta, F.; and Nanni, A., “Behavior of Full-Scale Glass Fiber-Reinforced PolymerReinforced Concrete Columns under Axial Load,” ACI Structural Journal, V. 107, No. 4, July-Aug. 2010, pp. 589-596. 4. Tobbi, H.; Farghaly, A. S.; and Benmokrane, B., “Concrete Columns Reinforced Longitudinally and Transversally with Glass Fiber-Reinforced Polymer Bars,” ACI Structural Journal, V. 109, No. 4, July-Aug. 2012, pp. 551-558. 5. Choo, C. C.; Harik, I. E.; and Gesund, H., “Strength of Rectangular Concrete Columns Reinforced with Fiber-Reinforced Polymer Bars,” ACI Structural Journal, V. 103, No. 3, May-June 2006, pp. 452-459. 6. Sharbatdar, M. K., and Saatcioglu, M., “Seismic Design of FRP Reinforced Concrete Structures,” Asian Journal of Applied Sciences, V. 2, No. 3, 2009, pp. 211-222. doi: 10.3923/ajaps.2009.211.222 7. Sheikh, S. A., and Khoury, S. S., “Confined Concrete Columns with Stubs,” ACI Structural Journal, V. 90, No. 4, July-Aug. 1993, pp. 414-431. 8. Liu, J., and Sheikh, S. A., “Fiber-Reinforced Polymer-Confined Circular Columns under Simulated Seismic Loads,” ACI Structural Journal, V. 110, No. 6, Nov.-Dec. 2013, pp. 941-952. 9. ASTM C39/C39M-12, “Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens,” ASTM International, West Conshohocken, PA, 2012, 7 pp. 10. ACI Committee 440, “Guide for the Design and Construction of Structural Concrete Reinforced with FRP Bars (ACI 440.1R-06),” American Concrete Institute, Farmington Hills, MI, 2006, 44 pp. 11. Deitz, D.; Harik, I.; and Gesund, H., “Physical Properties of Glass Fiber Reinforced Polymer Rebars in Compression,” Journal of Composites for Construction, ASCE, V. 7, No. 4, 2003, pp. 363-366. doi: 10.1061/ (ASCE)1090-0268(2003)7:4(363) 12. CAN/CSA-S806-12, “Design and Construction of Building Components with Fiber-Reinforced Polymers,” Canadian Standards Association, Mississauga, ON, Canada, 2012, 198 pp.


DISCUSSION Discussion 111-S22/From the March-April 2014 ACI Structural Journal, p. 257

Bond Strength of Spliced Fiber-Reinforced Polymer Reinforcement. Paper by Ali Cihan Pay, Erdem Canbay, and Robert J. Frosch Discussion by José R. Martí-Vargas Professor, ICITECH, Institute of Concrete Science and Technology, Universitat Politècnica de València, València, Spain

The discussed paper presents an interesting experimental study on the bond behavior of unconfined tension lap-spliced reinforcement. Steel-reinforced concrete beams and reinforced beams with ﬁber-reinforced polymer (FRP) bars—glass FRP and carbon FRP—were tested to provide additional experimental data for a better understanding of the bond strength between FRP and concrete. Variables such as splice length, surface condition, modulus of elasticity, axial rigidity, and bar casting position on bond strength were considered. The authors should be complimented for producing a detailed paper with comprehensive information. This is acknowledged by the discusser, who would like to offer the following comments and questions for their consideration and response, mainly about some aspects included in the Bond Strength section and the corresponding conclusions. The authors conclude that bond strength depends on splice length, the modulus of elasticity of reinforcement, and the axial rigidity of reinforcement, among other factors. These conclusions are supported by Fig. 6 to 10, as follows: (a) Fig. 6 (for No. 5 bars) and 7 (for No. 8 bars) depict the effect of splice length on bond strength; (b) Fig. 8 depicts the effect of modulus of elasticity on bond strength; and (c) Fig. 9 (for No. 5 bars) and 10 (for No. 8 bars) depict the effect of axial rigidity on bond strength. As stated by the authors, bond strength rises nonlinearly with increasing splice length, and linearly as the modulus of elasticity and/or the axial rigidity increase. However, despite the values being normalized by the fourth root of concrete compressive strength to eliminate the effect of variations in concrete strength, these figures present the computed reinforcement stress reached at failure ftest instead of average bond stress µavg. Therefore, it seems that some conclusions should correspond to reinforcement stress rather than to bond strength. By way of example, for the glass-sand coated case in Fig. 6, normalized reinforcement stresses approximately range from 30 to 50 ksi (207 to 345 MPa), whereas splice length ranges from 12 to 54 in. (305 to 1372 mm). Then normalized reinforcement stresses increase by 67%, whereas splice length increases by 350%. It is true that the effect is nonlinear. However, as bar size does not vary and the hypothesis of uniform distribution of bond stresses along splice length is assumed, this implies that the average bond stress is lower for the longer splice length case. In other words, normalized reinforcement stress at failure increases when splice length becomes longer, but bond strength decreases. If the discusser is right, this conclusion is the opposite of that reached by the authors. The same interpretation can be made for the remaining cases included in Fig. 6 and 7. Regarding the effect of the modulus of elasticity of the reinforcement on bond strength (Fig. 8, No. 5 bars), one can interpret that for one same deformation, a greater reinforcement stress results for a higher modulus of elasticity. By way of example for the fabric texture case in Fig. 8, normalized reinforcement stresses approximately range from 47 to 62 ksi ACI Structural Journal/January-February 2015

(324 to 427 MPa), whereas modulus of elasticity ranges from 7300 to 18,500 ksi (50.3 to 127.6 GPa). Then, normalized reinforcement stresses increase by 32%, whereas modulus of elasticity increases by 154%. It is true that the effect is linear in terms of the normalized reinforcement stresses. However, as bar size and splice length do not vary and the hypothesis of uniform distribution of bond stresses along splice length is assumed, additional information based on reinforcement deformation is needed to know the effect on average bond stress. This is because one same normalized reinforcement stress can be offered from two different moduli of elasticity if different bar deformations are developed which, in turn, may generate distinct bond-stress relationships based on distinct reinforcement-to-concrete slips. Regarding the effect of the axial rigidity of reinforcement on bond strength, the discusser believes that the same reasoning provided in the previous paragraph based on modulus of elasticity is applicable, which is included in axial rigidity together with the nominal cross-sectional area of the bar, and is more complex in this case given the potential differences in normalized reinforcement stresses and deformations due to both modulus of elasticity and area parameters. AUTHORS’ CLOSURE The authors would like to thank the discusser for his thoughtful comments. Unfortunately, it appears that there was confusion regarding the use of the term “bond strength.” The authors used the term “bond strength” consistent with the terminology commonly used in the field and consistent with that used by Joint ACI-ASCE Committee 408, Bond and Development of Steel Reinforcement. In ACI 408R-03, “Bond and Development of Straight Reinforcing Bars in Tension,” it is stated that the term “bond strength represents the maximum bond force that may be sustained by a bar.” The term “bond force represents the force that tends to move a reinforcing bar parallel to its length with respect to the surrounding concrete.” Therefore, in the paper, when it was commented that the “bond strength” increased, it was referring to the total force or total stress resisted by the bar, not the average bond stress. As discussed for Fig. 6, the normalized reinforcement stress at failure increases as the splice length increases. Therefore, considering the aforementioned terminology, it was concluded that the bond strength increases as splice length increases. It is agreed that the average bond stress (Table 2 provides the values) decreases as the splice length increases due to the nonlinear relationship of bond force to splice length. As discussed for Fig. 8 through 10, the bond strength increases linearly with increasing modulus of elasticity and increasing axial rigidity. In these cases, both the bond strength (bar stress or force) as well as the average bond stress increase considering that the bar surface area and splice length are identical for the cases compared. Even in the case of the steel bar with the hole, the surface area of the 115

bar resisting bond remains the same. While the paper did not discuss average bond stress, the average bond stress for these cases would be directly computed from the bar force divided by the surface area of the bar (pdbls), which is independent of the bar deformation or surface treatment. It is not clear why the discussion indicates that “additional information based on reinforcement deformation is needed to know the effect on average bond stress.” Regardless, bond strength was found to be essentially independent of the surface deformation for the bars tested as discussed in the paper.

Overall, average bond stress is not a good measure to evaluate the bond strength of reinforcing bars. As indicated by the nonlinear relationship with splice length shown in Fig. 6 and 7, bond stresses are nonlinear over the length of the reinforcement. This nonlinearity results in significant variation of the maximum average bond stress, as indicated by the test results in Table 2. For this reason, the conclusions were presented based on the bond strength (maximum bar force or bar stress) rather than average bond stress.

Discussion 111-S25/From the March-April 2014 ACI Structural Journal, p. 291

Behavior of Epoxy-Injected Diagonally Cracked Full-Scale Reinforced Concrete Girders. Paper by Matthew T. Smith, Daniel A. Howell, Mary Ann T. Triska, and Christopher Higgins Discussion by William L. Gamble FACI, Professor Emeritus, University of Illinois, Urbana, IL

This paper is an important contribution to the literature on repair of reinforced concrete girders. However, I am not sure that the test specimens were very representative of 1950s details. The shear reinforcement is quite heavy compared to at least some examples. Gamble (1984) describes a reinforced concrete deck girder (RCDG) bridge that was built in 1957 and suffered major distress in 1967, followed by partial girder replacement and epoxy injection. The bridge had been designed by the 1953 AASHTO Spec (AASHTO 1953) and the provisions of that specification are at least partially responsible for the distress. The allowable service load shear stresses under the 1953 AASHTO specification were extremely high, compared to any current standard. The allowable value of vc was equal to 0.04fc', with no limit on concrete strength. The specified fc' = 4000 psi (27.6 MPa), so the service load shear stress assigned to the concrete was 120 psi (0.83 MPa). Shear reinforcement was required only when this stress was exceeded, and the maximum shear stresses in this bridge were slightly less than the allowable value. Any current Code will assign a shear stress somewhat less than this value, applied to factored loads rather than service loads, and this change combined with other restrictions will lead to shear reinforcement in nearly all reinforced concrete beams. In addition, the 1953 AASHTO specification had almost no minimum limits on shear reinforcement. There was no minimum shear steel ratio Av/bws. There was no maximum spacing limit such as the 24 in. (600 mm) now commonly observed, and the d/2 limit in current codes was invoked only when shear reinforcement was required. ACI 318-51 (ACI Committee 318 1951) was essentially similar with respect to shear design. These defects were addressed in the first AASHTO Strength Design Provisions, but were not addressed in the Working Stress Design provisions until the 1974 Interim Specs. ACI 318-56 (ACI Committee 318 1956) had a partial fix to this set of problems, with a major revision in the 1963 Code. The test specimens had a shear steel ratio of approximately 0.0024. The lowest shear steel ratio in the structure in Gamble (1984), which did not require shear reinforcement by 1953 AASHTO requirements, was approximately 0.00035, and the highest approximately 0.0007. ACI 318-56 required a minimum shear steel ratio of 0.0015 (when web reinforcement was required), and current ACI Codes and the last AASHTO non-LRFD spec require a minimum of 0.00125 for Grade 40 steel and 0.00083 for Grade 60 steel. The LRFD 116

minimum shear reinforcement ratios are somewhat larger but still much smaller than that used in the test specimens. A lower shear steel ratio would have led to larger, less wellcontrolled cracks and would have been more typical of the 1950s. Stirrups spaced at d/2 would also have been more representative. The bridge in Gamble (1984) needs a little additional comment herein. It had a significant problem with concrete strength in the span that failed, but had significant shear cracking in other spans with adequate concrete strengths. It was designed as a rigid frame structure with integral abutments, and as such apparently developed significant axial tensions due to restrained shrinkage in spite of expansion joints in two locations. The bridge was one of approximately 85 similar designs (with different numbers of spans, locations of expansion joints, and foundation details), and approximately half of these had experienced significant cracking requiring repairs by 2002 (personal communication; Gamble [2002]). None of the other similar structures had significantly deficient concrete strengths, so it must be concluded that the combination of very high shear stresses assigned to the concrete in conjunction with axial tensions from restrained shrinkage led to the distress. REFERENCES

AASHTO, 1953, “Standard Specifications for Highway Bridges,” sixth edition, American Association of State Highway Officials, Washington, DC, 328 pp. ACI Committee 318, 1951, “Building Code Requirements for Reinforced Concrete (ACI 318-51),” American Concrete Institute, Farmington Hills, MI, 63 pp. ACI Committee 318, 1956, “Building Code Requirements for Reinforced Concrete (ACI 318-56),” American Concrete Institute, Farmington Hills, MI, 77 pp. Gamble, W. L., 1984, “Bridge Evaluation Yields Valuable Lessons,” Concrete International, V. 6, No. 6, June, pp. 68-74. Personal communication from W. D. Gamble, deceased, 2002.

AUTHORS’ CLOSURE The authors would like to thank the discusser for his thoughtful comments and for reading our paper. These allow us to provide additional detail regarding the design considerations for our specimens that could not be reported in the original paper due to space constraints. The issues related to what many would consider a “shear” problem for 1950s-era RCDGs are several. We respectfully submit Table 3, which highlights the relevant changes in the AASHTO 1944, 1949, and 1953 standards from the period considered. In addition, the AASHTO allowable stress for the transverse reinforcing steel changed from 16 ksi (110 MPa) ACI Structural Journal/January-February 2015

in 1949 to 20 ksi (138 MPa) in 1953. The maximum spacing of transverse steel, when required, was prescribed as threequarters the section depth. While the prescribed allowable concrete stresses for shear are different than those described by the discusser, they do not change the premise of his discussion. We strongly agree that RCDG bridges of the 1950s are under-reinforced for shear by modern standards and that they represent a disproportionately large number of bridges with reported diagonal-tension cracking. However, it is not only the concrete allowable stress and transverse reinforcement which contributes to their deficiencies. Bridge designers now widely recognize the interaction between flexure and shear on the diagonal-tension behavior of reinforced concrete girders and beams. Therefore, one must consider the nexus of the many different changes that were taking place in the design specifications (AASHTO and ACI) as well as in material standardization (ASTM) at the time. One of the most important changes that took place was the standardization of reinforcing steel bars that occurred in 1950 with the adoption of ASTM 305-50T (1950). The newly standardized deformation patterns substantially relaxed the anchorage and bond requirements and design practice changed rapidly from the past as according to ACI 208-58: “Acceptance by the ACI Building Code committee of the ASTM A305 definition of a deformed bar produced an immediate drastic change in both structural and general practice….” Where designers would previously have required hooks and bends to anchor or transition reinforcing steel in flexural tension zones, they could now terminate the steel with straight-bar cutoffs. This was an era of economy of materials and designers worked hard to produce highly efficient designs. As a result, RCDGs designed during the 1950s and early 1960s typically contain poorly detailed flexural reinforcing steel in addition to being lightly reinforced with transverse reinforcing steel. These are the population of bridges that commonly contain diagonal cracks and which require redress. Our research has focused on assessing and remediating shear-moment deficiencies in the population of RCDG bridges from around the 1950s. To do this, we have attempted to mimic the original features of the girders and cross beams in overall geometry, concrete and reinforcing materials and details, as well as loading geometry to control the moment-toshear ratios in the specimens to those that reflect the in-place conditions. When we began this work over a decade ago, my colleagues and I developed a database of the relevant geometry and details (Higgins et al. 2004). We considered RCDG bridges constructed from 1947 to 1962 that were identified as cracked within the inventory contained by the Oregon Department of Transportation, which maintains a large population of this type of bridge. There were 442 bridges in the database with 1487 separate spans. Structural drawings for each span were reviewed and parameters corresponding to overall bridge geometry (span length, indeterminacy, skew, girder spacing,

number of lanes, number of girder lines, size, location, and spacing of diaphragms, as well as deck thickness); material properties (concrete compressive strength and reinforcing steel properties); and member proportions and reinforcement details (cross-section type, web width, member height, haunch, taper, span-depth ratios, flexural reinforcing steel areas in positive and negative regions, stirrup sizes and spacing in high and low shear zones, for example) were recorded in the database. The database was queried to provide summary details for individual parameters and relationships between parameters. Further, dead load magnitudes and live load capacities were developed for comparison with specified load models, weigh-in-motion service-level loads, and permit tables. Using information extracted from the population of cracked RCDG bridge spans contained in the database, laboratory specimens were dimensioned and individual bridges were identified for field investigation. The most common details in high-shear regions for positive- and negative-moment regions (average results which corresponded closely to the mode of the data) were selected as representative. Also using this data set, we conducted experiments at the practical bounds, such as no stirrups and heavy stirrups (equivalent transverse reinforcing ratios of 0.0 and 0.0048, respectively), with and without cutoff flexural steel, the results of which are reported by Higgins et al. (2004, 2007). Based on the detailed investigation of the many different parameters that coalesce around the problem space and in-depth study of large numbers of actual designs, we feel our specimens are indeed representative of high-shear negative moment regions in RCDGs of the 1950s. They were not intended to represent the minimum case, and based on our review of vintage designs, find that the negative-moment region near continuous supports contained higher-than-minimum stirrups (even by modern standards).We acknowledge that there are girders with more and less transverse steel and we further feel that our findings would hold for girders with at least minimum stirrups (as defined by the modern design standards). REFERENCES

AASHTO, 1944, “Standard Specifications for Highway Bridges,” fourth edition, American Association of State Highway Officials, Washington, DC. AASHTO, 1949, “Standard Specifications for Highway Bridges,” fifth edition, American Association of State Highway Officials, Washington, DC. ACI Committee 208, 1964, “Test Procedure to Determine Relative Bond value of Reinforcing Bars (ACI 208-58),” American Concrete Institute, Farmington Hills, MI. ASTM A305-50T, 1950, “Tentative Specifications for Minimum Requirements for the Deformations of Deformed Steel Bars Concrete Reinforcement,” ASTM International, West Conshohocken, PA. Higgins, C.; Yim, S.; Miller, T.; Robelo, M.; and Potisuk, T., 2004, “Remaining Life of Reinforced Concrete Beams with Diagonal-Tension Cracks,” Report No. FHWA-OR-RD-04-12, Research Unit, Oregon Department of Transportation, Salem, OR, 124 pp. Higgins, C.; Potisuk, T.; Farrow III, W. C.; Robelo, M. J.; McAuliffe, T. K.; and Nicholas, B. S., 2007, ”Tests of RC Deck Girders with 1950s Vintage Details,” Journal of Bridge Engineering, ASCE, V. 12, No. 5, pp. 621-631.

Table 3—Comparison of relevant AASHTO-specified concrete shear and bond stresses, noting changes occurring around 1950 Allowable shear stress in concrete, psi With web reinforcement Without web reinforcement Longitudinal bars not Longitudinal bars Longitudinal bars not Longitudinal bars Design specification and year anchored anchored anchored anchored AASHTO 1944 AASHTO 1949 AASHTO 1953

0.046fc′ 0.046fc′ 0.075fc′

0.06fc′ 0.06fc′ 0.075fc′

0.02fc′ 0.02fc′ 0.02fc′

0.03fc′ 0.03fc′ 0.03fc′

Allowable bond stress (straight), psi Structural or intermediate grade

0.033fc′(max 100) 0.05fc′(max 150) 0.10fc′(max 350)

Note: 1 psi = 6.89 kPa.


117

NOTES:

118


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 ORGANIZATIONAL – $1020 PLUS APPLICABLE SHIPPING FEES Includes one set of the Manual of Concrete Practice (MCP), a wall plaque, and a subscription to Concrete International, Concrete Repair Bulletin, and both ACI Materials and Structural Journals. MANUAL OF CONCRETE PRACTICE: – Select a format and include applicable shipping fee:  Seven-volume Set – U.S. add $30 shipping fee; Canada and Mexico add $130 shipping fee; and all others, add $220 shipping fee  CD – no additional shipping costs  Online Subscription – no additional shipping costs  INDIVIDUAL – $230/year Individuals 28 years old or above residing worldwide.  YOUNG PROFESSIONAL – $130/year Individuals under the age of 28 who do not qualify for student membership. E-STUDENT – FREE Join at www.concrete.org/Membership/StudentMembership.aspx.  STUDENT (U.S. AND CANADA) – $40/year  STUDENT (Outside U.S. and Canada) – $80/year Individuals under the age of 28 who are registered full-time students at an educational institution. Full-time students age 28 and above may be granted Student Membership when the request is endorsed by the student’s advisor. SUSTAINING MEMBERSHIP – ACI Sustaining Members receive all membership benefits of Organizational Members plus a free copy of every new ACI publication and increased corporate exposure, positioning them as a leader in the concrete industry, and much more. For complete details or to join, visit www.sustaining.concrete.org or call +1.248.848.3800.

Profile Information (Select all that apply.) Markets Occupations

 Design  Materials  Production  Construction  Testing  Repair  Owner

 Management  Consultant  Engineer  Architect  Contractor  Technical Specialist  Quality Control

 Inspector  Craftsman  Sales & Marketing  Association Employee  Government Employee

Included Subscriptions Your ACI membership includes a hard copy and online subscription to Concrete International and your choice of one of the following:  Digital access to the ACI Materials Journal and ACI Structural Journal  Concrete Repair Bulletin, hard copy

Additional Periodical Options

To add subscriptions in hard copy: please check a circle below and include an additional $47 ($52 for individuals outside U.S. & Canada, $30 for students, and $42 for students outside of the U.S. & Canada) per title.  ACI Structural Journal, hard copy  ACI Materials Journal, hard copy Please allow up to 2 months for delivery outside the U.S.

Optional Online Subscriptions Bundle your ACI membership with these optional online subscriptions for additional ACI resources:  Symposium Papers Subscription, $98  Manual of Concrete Practice, $409

 Researcher  Educator  Student  Other __________

Payment

Fees

CHECK NUMBER

$

Membership dues

 CHECK payable to American Concrete Institute

$

Additional subscriptions

 CREDIT CARD  Visa  MasterCard  American Express

$

Organizational six volume set shipping fees

$

For Journals, Canadian residents add 5% GST (#126213149RT)

$

ACCOUNT NUMBER

EXPIRATION DATE

NAME (AS APPEARS ON CREDIT CARD)

TOTAL (U.S. Funds Only) SIGNATURE

Code: SJ


CALL FOR ACTION ACI Invites You To...

Do you have EXPERTISE in any of these areas? • BIM • Chimneys • Circular Concrete Structures Prestressed by Wrapping with Wire and Strand • Circular Concrete Structures Prestressed with Circumferential Tendons • Concrete Properties • Demolition • Deterioration of Concrete in Hydraulic Structures • Electronic Data Exchange • Insulating Concrete Forms, Design, and Construction • Nuclear Reactors, Concrete Components • Pedestal Water Towers • Pipe, Cast-in-Place • Strengthening of Concrete Members • Sustainability

Then become a REVIEWER for the ACI Structural Journal or the ACI Materials Journal. How to become a Reviewer:

1. Go to: http://mc.manuscriptcentral.com/aci; 2. Click on “Create Account” in the upper right-hand corner; and 3. Enter your E-mail/Name, Address, User ID and Password, and Area(s) of Expertise.

Did you know that the database for MANUSCRIPT CENTRAL, our manuscript submission program, is separate from the ACI membership database? How to update your user account: 1. Go to http://mc.manuscriptcentral.com/aci; 2. Log in with your current User ID & Password; and 3. Update your E-mail/Name, Address, User ID and Password, and Area(s) of Expertise.

QUESTIONS?

E-mail any questions to [email protected].

ACI STRUCTURAL J O U R N A L J O U R N

The American Concrete Institute (ACI) is a leading authority and resource worldwide for the development and distribution of consensus-based standards and technical resources, educational programs, and certifications for individuals and organizations involved in concrete design, construction, and materials, who share a commitment to pursuing the best use of concrete. Individuals interested in the activities of ACI are encouraged to explore the ACI website for membership opportunities, committee activities, and a wide variety of concrete resources. As a volunteer member-driven organization, ACI invites partnerships and welcomes all concrete professionals who wish to be part of a respected, connected, social group that provides an opportunity for professional growth, networking and enjoyment.

Aci Structural Journal Jan.-feb. 2015 v. 112 No. 01

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