Experimental Testing to Determine Concrete Fracture Energy Using Simple Laboratory Test Setup

Transcription

1 ACI MATERIALS JOURNAL Title no. 104-M63 TECHNICAL PAPER Experimental Testing to Determine Concrete Fracture Energy Using Simple Laboratory Test Setup by Joshua Martin, John Stanton, Nilanjan Mitra, and Laura N. Lowes Nonlinear finite element analysis can complement experimental investigation and provide insight into the behavior of reinforced concrete structures. The development of a finite element model of a concrete structure typically requires specification of the concrete tensile strength, fracture energy, the shape of the post-peak tensile response curve, and other material parameters. Previous research has resulted in a number of different tests for determining fracture energy and post-cracking response. Because test specimens exhibit extremely brittle response, these methods typically require the use of a very stiff, closed-loop test machine so that load can be applied under displacement control. A study recently completed at the University of Washington employed the fracture energy test method recommended by RILEM, with test specimens modified to include counterweights and with an open-loop testing machine. The fracture energy and post-cracking response data generated from these tests fall within the range typically observed for the RILEM-type tests. Additionally, nonlinear finite element analyses performed using these data to calibrate the concrete constitutive models reproduced, with acceptable accuracy, the load-displacement response observed in the laboratory. Keywords: experimental testing; finite element analysis; fracture energy; nonlinear analysis. INTRODUCTION Nonlinear finite element analysis can complement experimental investigation and provide improved understanding of the phenomena that determine the local behavior of reinforced concrete structures. To provide understanding of behavior, it is necessary to employ material models that simulate the observed response of plain concrete, including softening response under tensile loading. To complement an experimental investigation of concrete-steel bond, researchers at the University of Washington are using a commercial nonlinear finite element analysis program and a concrete constitutive model that employs the smeared-crack approach to simulate concrete softening response in tension. 1 Calibration of the smeared-crack model requires specification of concrete tensile strength f t, fracture energy G F, and the form of the softening response curve. The research presented herein focuses on experimental testing to determine concrete fracture energy and postcracking response for use in simulating concrete response in concrete-steel bond zones. REVIEW OF PREVIOUS RESEARCH In 1976, Hillerborg et al. 2 proposed the cohesive crack model for simulation of plain concrete, in which concrete fracture energy characterized the softening response of a cohesive crack that could develop anywhere in a concrete structure. In the ensuing years, a number of methods were proposed for experimental measurement of fracture energy. 3 In 1985, RILEM Technical Committee 50, Fracture Mechanics of Concrete, produced a draft recommendation for a concrete fracture energy testing procedure. 4 Hillerborg 5 provided a theoretical basis for this proposed testing procedure, often referred to as the work-of-fracture method, in which the fracture energy per unit area of concrete is computed as the area under the experimental load-deflection response curve for a notched, plain concrete beam subjected to threepoint bending, divided by the area of fractured concrete. Several additional test methods have been proposed through the years for determining concrete fracture properties from which fracture energy may be computed. In 1990, RILEM 6,7 approved two methods as standard recommendations: 1) the two-parameter method proposed by Jenq and Shah 8 in which concrete fracture is characterized by the critical stress intensity factor and the critical crack tip opening displacement; and 2) the size-effect method proposed by Bažant and Pfeiffer 9 in which the strength of a series of geometrically similar but different sized specimens are used in combination with a size effect law 10 to determine G f, the fracture energy associated with the area under the initial crack-stress opening curve. Fracture energy G F, defined as the total area under the crack stress versus opening curve, may be computed using the results of these tests and an empirical relationship between G f and G F 11 and, for the test proposed by Jenq and Shah, Irwin s equation relating fracture energy and the stress intensity factor. Researchers have continued to address the issue, and the current activities of ACI Committee 446, Fracture Mechanics, include development of a Report on Fracture Toughness Testing of Concrete that is expected to provide recommendations for testing to determine fracture properties. Of the three methods recommended by RILEM, 4,6,7 the work-of-fracture method is the most commonly used and has drawn the most interest from researchers, likely due to its earlier adoption. One issue of interest is the impact of specimen size and geometry on fracture energy as measured using this method. In his original presentation of the method, Hillerborg 5 concluded that concrete fracture energy is a function of specimen size, but that the variation due to sizeeffects is of no greater importance for the calculated strength of a structure than the corresponding size dependency in ordinary strength tests. Studies by Malvar and Warren 12 and Hanson and Ingraffea 13 concluded that fracture energy depends on specimen size and geometry, and increases with increasing size. Additionally, Hanson and Ingraffea 13 concluded that the RILEM-specified specimen sizes were ACI Materials Journal, V. 104, No. 6, November-December MS No. M received June 23, 2006, and reviewed under Institute publication policies. Copyright 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including authors closure, if any, will be published in the September-October 2008 ACI Materials Journal if the discussion is received by June 1, ACI Materials Journal/November-December

2 Joshua Martin is a Bridge Engineer with the Washington State Department of Transportation, Olympia, WA. He received his MS and BS from the University of Washington, Seattle, WA. His research interests include the experimental investigation of concrete-steel bond-zone behavior. ACI member John Stanton is a Professor in the Department of Civil and Environmental Engineering at the University of Washington. He is a member of ACI Committees 318-E, Shear and Torsion (Structural Concrete Building Code); 318-N, Notation and Editorial; and Joint ACI-ASCE Committee 550, Precast Concrete Structures. He received his BA and MA from Cambridge University, Cambridge, MA; his MSCE from Cornell University, Ithaca, NY; and his PhD from the University of California, Berkeley, Berkeley, CA. His research interests include the behavior and design of concrete structures, with an emphasis on prestressed concrete. ACI member Nilanjan Mitra is a Lecturer in the Department of Civil and Environmental Engineering at the California Polytechnic State University, San Luis Obispo, CA. He is a member of Joint ACI-ASCE Committees 352, Joints and Connections in Monolithic Concrete Structures, and 447, Finite Element Analysis of Reinforced Concrete Structures. He received his BE from Bengal Engineering College, Shibpur, India; his MTech from the Indian Institute of Technology, Kharagpur, India; and his PhD from the University of Washington. His research interests include numerical simulation of reinforced concrete structures. ACI member Laura N. Lowes is an Associate Professor in the Department of Civil and Environmental Engineering at the University of Washington. She received her BS from the University of Washington and her MS and PhD from University of California, Berkeley. She is Chair of Joint ACI-ASCE Committee 447, Finite Element Analysis of Reinforced Concrete Structures. Her research interests include numerical simulation of reinforced concrete structures. not large enough to represent infinitely large specimens and, thus, converged values of fracture energy. Qian and Luo, 14 however, show that fracture energy decreases with specimen size for small mortar specimens and argue that the widespread, but anomalous, finding that fracture energy increases with specimen size is due to an error in the way in which specimen self-weight is taken into account. Navalurkar et al., 15 computing fracture energy from load versus crack opening data and an empirical relationship between crack opening and beam deflection, found no dependence on specimen size or geometry and concluded that previous observed dependence was likely due to inaccurate measurement of beam deflection. Several additional studies have addressed other aspects of the test method. Malvar and Warren 12 proposed modifying the method to include an unload-reload cycle of the beam starting when specimen strength had dropped by 5% from the peak. They proposed defining concrete fracture energy as the area under the complete load-deflection curve less the energy dissipated up to unloading of the specimen. They argued that, using this approach, the measured fracture energy would include only that energy dissipated in opening a single, well-defined crack and not energy associated with establishing a single crack. They concluded, however, that their proposed method did not offer any substantial advantages over the originally proposed method and resulted in a substantially more complex testing protocol. Bažant and Becq-Giraudon, 16 on the basis of a statistical analysis of a large data set, conclude that G F data computed from workof-fracture testing have significantly more scatter than do G f data computed using other test methods and suggest that errors in measurement of the tail of the load-displacement response curve may be responsible for much of the scatter. Qian and Luo 14 show that for large beams, specimen selfweight is significantly larger than the test load, with the result that fracture energy is the difference between two large and nearly equal numbers; they suggest that this may result in inaccurate determination of G F. All three RILEM test methods 4,6,7 involve delicate specimens and require careful conduct and very accurate measurements. There are, however, specific details of the 576 methods that should be considered when selecting one of them for use. The size-effect method requires experimental measurement only of the maximum strength of a notched, plain concrete beam subjected to three-point bending, but requires testing of multiple specimens that span a wide range of sizes. The two-parameter method requires testing of only a single specimen, but requires application of a cyclic load history, which necessitates use of a closed-loop testing machine. Additionally, both of these methods require the use of empirical relationships to determine G F, the fracture energy associated with the complete crack stress versus opening response history. The work-of-fracture method: 1) requires the least and simplest laboratory testing; 2) provides the opportunity to use an open-loop testing system, if the response speed can be controlled to achieve stable crack growth; and 3) provides the opportunity to employ counterweights to improve accuracy in measuring the tail of the load-displacement response curve and computing fracture energy. It was therefore used as the basis for this research. RESEARCH SIGNIFICANCE Nonlinear finite element analysis of concrete structures using multi-dimensional constitutive models requires specification of concrete fracture energy and post-cracking softening response. Traditional test methods to determine these material parameters require closed-loop testing equipment. The results of this study provide a method for determining these material parameters using a basic, open-loop testing machine. Comparison with the results of previously proposed test methods and the results of finite element analysis using the data show that the proposed test method produces material parameter data that may be used to calibrate nonlinear finite element models of concrete structures. RESEARCH OBJECTIVES AND SCOPE The primary objective of the research presented herein was to develop a testing protocol for the measurement of concrete fracture energy using an open-loop testing machine. Fracture energy data were validated through comparison with previous research results and with simulated response histories. Response was simulated using commercial finite element analysis software and a smeared-crack concrete constitutive model 1 that was calibrated using the experimental data. The impacts of specimen geometry and concrete mixture properties on measured fracture energy were investigated. The complete research effort is presented as follows: 1) design and construction of test specimens; 2) test setup including boundary conditions, instrumentation and testing procedures; 3) experimental data including material test data, observations of testing, and load-deflection histories; 4) data analysis including computation of fracture energy, comparison of data from different test specimens and comparison with previous research results; 5) simulation of laboratory tests using finite element analysis and measured fracture energy; and 6) summary and conclusions. EXPERIMENTAL SETUP AND PROCEDURES Test specimen selection and design Previous research indicates that fracture energy, as determined using the work-of-fracture test method, is controlled by concrete mixture and test specimen size and shape. Previous research indicates also that specimen self-weight affects measurement of fracture energy. To investigate the impact of these parameters, a series of 12 tests was conducted using the ACI Materials Journal/November-December 2007

3 proposed test method. Counterweights were added to the RILEM-type beam specimen to mitigate the impact of specimen self-weight. The tests comprised specimens with three different geometries and two different concrete mixtures. Figure 1 and Table 1 show the basic configuration and the dimensions of the notched beam specimens used in the tests. Each specimen was given a name that defined its characteristics, for example, FR-42-R-1. Here, the FR indicates fracture specimen, 42 indicates the throat dimensions (bd throat ) in inches, the R or A indicates rounded or angular aggregate, and the final number indicates the specimen number of that type. RILEM gives four standard specimen sizes (Table 2) that depend on the maximum aggregate size D max and provides rules for determining the dimensions of specimens larger than the largest standard one. The standard specimen sizes are not geometrically similar. In all cases, the notch depth/ total depth ratio d notch /d is 0.5, but the width/depth ratio varies from 0.5 to 1.0 and the span/total-depth ratios vary between 4 and 8. These ranges suggest that test results are not particularly sensitive to span/depth and width/depth ratios. Although RILEM specifies a fixed value of 0.5 for d notch /d, others 8,17 have used different values for measuring concrete fracture properties. For the current tests, three different specimen geometries were used. Specimen dimensions were chosen to lie within the range of geometric ratios implied by the RILEM specification, to meet laboratory configuration requirements, and to investigate the impact of specimen geometry on response. The choice of specimen length and maximum depth was influenced by the desire to use a standard loading rig (for ASTM C78-02 Modulus of Rupture beam specimens) and the corresponding steel beam molds. To investigate the impact of specimen geometry on fracture energy, dimensions were chosen such that one group of specimens had approximately the same throat area but different span-depth ratios (Specimens Type FR-42 and FR-33), and a second group of specimens had different throat areas but approximately the same width-depth ratio (Specimens Type FR-63 and FR-42). The maximum aggregate sizes D max used in the tests were 3/4 and 7/8 (19 and 21 mm). Thus, the absolute dimensions used for the specimens differed slightly from the values specified by RILEM but, with the exception of the notch depth/total depth, all the ratios (for example, span-depth) lay within the implied ranges of the RILEM specification. In addition to geometry, the influence of aggregate type on fracture energy was also investigated. Two different concrete mixtures were used. The weights of the components were the same for each mixture, but one mixture used angular (crushed) aggregate (D max = 7/8 [ mm]) and the second used rounded aggregate (D max = 3/4 [19.05 mm]). Table 3 provides details of the mixture design. Specimen construction The beams were made in 6 x 6 x 21 (153 x 153 x 534 mm) steel molds, into which plywood blockouts were inserted to achieve the desired dimensions. The plywood was sealed to prevent moisture absorption. Forms were removed a few days after the concrete was cast. At all times before testing, the specimens were stored at room temperature in sealed plastic bags that contained moist towels. To form the notch at midspan (Fig. 1), RILEM recommends saw-cutting but allows casting. Casting was chosen here because of the potential risk of premature cracking during saw-cutting and handling. To prevent shrinkage cracking during curing, the notch was created using a flexible form composed of a 1/4 (6 mm) thick piece of foam, sandwiched inside a folded piece of sheet metal and wrapped in plastic wrap. The root of the notch was formed by foam covered by plastic wrap protruding from between the metal plates; this produced a rounded end to the notch and reduced the likelihood of stress cracks from forming during the curing process. Notch widths ranged from 1/4 to 3/8 (6 to 9 mm). Experimental test setup The laboratory apparatus used for the fracture energy tests was adapted from a standard three-point loading flexural strength test setup (ASTM C78-02) (Fig. 1). The specimen rested notch downward on two simple supports of which one was a roller and the other was a ball. The load was applied to a single roller Table 1 Specimen dimensions Specimen type L, l, b, d, d notch, d throat, A throat, 2 b/d b/d throat l/d FR FR FR Note: 25.4 mm = 1 Table 2 Specimen dimensions specified in RILEM 4 D max, mm d, mm b, mm L, mm l, mm A throat, mm 2 b/d b/d throat l/d 1 to to , to , to , Note: 25.4 mm = 1 Table 3 Concrete mixture design Component Weight, lb/yd 3 % by weight Cement, Type I Water Coarse aggregate Fine aggregate Note: 1 lb/yd 3 = kg/m 3. Fig. 1 Fracture energy test specimen including counterweights (1 = 25.4 mm). ACI Materials Journal/November-December

4 at midspan via a concentrated load at its mid-width. This arrangement was adopted to avoid introducing torsion into the specimen. Plywood blocking was applied to the interior edges of the rotating supports so that they could rotate outwards (that is, away from the center of the specimen) but could not rotate inward. This ensured that the load remained in the center of the specimen by restricting rigid body translation. After placing the specimen on the supports, concrete blocks were attached to the ends of the specimen to act as counterweights, as shown in Fig. 2. The blocks were attached by thin steel sheets anchored at the top of both the specimen and the counterweights with 1/4 (6 mm) concrete anchors. Steel weights were placed on top of the concrete weights when additional load was necessary. The purpose of the counterweights was to produce a small, negative midspan moment under dead load. With the counterweights in place, the specimen remained in contact with the loading head even after it was cracked, thereby allowing the descending branch of the loaddeflection curve to be followed. The counterweight value was chosen to produce a negative midspan moment that was approximately 1% of the positive moment developed under the peak load expected in the test. Figure 2 shows a specimen, with counterweights attached, in the test setup. The load was applied to the specimen using a 300 kip (1300 kn) universal testing machine. The load was transferred from the test machine to the specimen through a load train composed of threaded steel rods and a 3 kip (13 kn) load cell. For either a closed- or an open-loop system, if the load train is too flexible, a snap-back response will occur post-peak and data will be lost as the system jumps to a stable equilibrium point at a large displacement level. After the first two tests, the stiffness of the load train was measured, and the components of it were changed to reduce the likelihood of such snap-back. Martin 18 provides a detailed discussion of this issue. Instrumentation The instrumentation consisted of a load cell, displacement measurement gauges, and a data collection system. The load cell used in these series of tests was a 3 kip (13 kn) S-type load cell. Four potentiometers were used to measure the midspan displacement of the specimen. Locations are shown in Fig. 1. RILEM requires that the deflection be measured to an accuracy of (0.01 mm) and requires that the deformation be measured with regard to a line between two points on the beam above the supports. No measurement method is specified, however. RILEM allows, as an alternative, Fig. 2 Test setup. direct measurement of the beam deflection, provided that the supports do not deform inelastically by more than (0.01 mm). The latter approach was adopted for the current study, and steel plates were used at the supports to minimize inelastic deformations. The potentiometers were located on a block, the design of which was changed twice during the program to improve system accuracy. Two different data collection systems were used during this test series. For the FR-42 and FR-63 specimens, an HP Datalogger system was used. For the FR-33 specimens, a Labview system was used. The faster reading ratio of the Labview system (20 scans per second) led to better data collection. Test procedures The process of assembling the test rig is summarized here Details are given by Martin, 18 including minor variations in the execution of some of the procedures. During assembly, the specimen was kept wet at all times to avoid cracking due to drying shrinkage. First, holes were drilled in the top of the specimen to accommodate the fasteners for the counterweights. A diamond drill was used to minimize impacts. Machined steel plates were attached to the underside of the beam at the support locations, and sheet metal plates were attached on either side of the notch to provide even bearing for the potentiometers. The specimen was then installed in the test rig and the counterweights were attached, using sheet metal strips on the top of the beam. Small supplementary weights were added to achieve the correct balance, and the potentiometers were installed beneath the beam at midspan. From this time on, the critical region of the specimen around the notch was kept moist using a spray bottle of water. The test machine controls were adjusted to give an initial loading rate of approximately 200 lb/minute (890 N/minute). Because the specimen was much less stiff than the test machine, that load rate corresponded essentially to a constant displacement rate of approximately /minute (0.18 mm/minute). As the peak load was approached, initial cracking softened the specimen and the displacement rate began to increase. Test machine controls were adjusted to maintain the initial displacement rate (for Specimen FR-33-R04, operator error resulted in an increase in the imposed displacement rate in the vicinity of the peak load; this may have affected the behavior of this specimen). In the post-peak regime, the displacement rate increased again as the crack propagated from the top of the notch to near the top of the beam. Test machine controls were adjusted again to reduce the imposed displacement rate; however, it was not possible to maintain the initial displacement rate. Thus, while crack growth was stable, the post-peak displacement rate was faster than the initial rate. Once significant strength loss had occurred, static equilibrium was restored, and test machine controls were readjusted to achieve the original displacement rate. These settings were maintained until the displacement had reached a value approximately three times that at peak load, after which the rate was increased in the interests of completing the test in a reasonable time. Each test typically lasted approximately 15 minutes. TEST RESULTS Table 4 lists fracture energy values computed from test data, as well as other material parameters, for the 12 test specimens included in this study. The first four test specimens listed in Table 4 (the FR-**-A-* specimens) were originally intended merely for fine-tuning and evaluating the test setup 578 ACI Materials Journal/November-December 2007

5 to be used in the subsequent tests, and no material testing was done for them. Their fracture responses, however, differed consistently from those of the subsequent specimens with rounded aggregate, so their fracture energy data are reported in Table 4. Material properties Four 6 x 12 (153 x 305 mm) concrete quality-control cylinders were cast with each of the batches of rounded aggregate concrete. Compression tests (ASTM C39-02), split cylinder tension tests (ASTM C496-02), and elastic modulus tests (ASTM C469-02) were performed for each batch using these cylinders. Material testing was done on or near the day of fracture energy testing. Results are listed in Table 4. Fracture energy test observations The observed behavior was essentially the same in each fracture test. Before cracking, the beam deflections were too small to be seen with the naked eye. To facilitate observation of cracking, the sides of the specimen were sprayed lightly with water, but in no case was a crack observed before the data acquisition system revealed that the peak load had been passed. The first crack to appear was hairline in thickness and occurred without any audible sound. In most specimens, a single crack formed, but in Specimens FR-42-R-2, FR-63-R-1, and FR-33-R-2, the crack bifurcated into several branches. Shortly after the crack formed, it propagated up to approximately 0.5 (13 mm) from the top face of the beam in the span of approximately 15 to 20 seconds. Thereafter, its length remained visibly unchanged until the end of the test. When the load was removed, the specimens generally remained in one piece until they were moved, at which point they separated into two pieces. Fracture energy test data Figure 3 shows typical, corrected load-versus-deflection data for the test specimens. Before reporting the load-deflection data, the origin of the response history was shifted. First, the value of the residual load, that is, the load required to overcome the negative moment resulting from the counterweights, was subtracted from all load readings. This correction was slightly different for every specimen and was determined by weighing each specimen and the counterweights. Second, a linear trendline was superimposed on the linear portion of the ascending curve for each specimen. It was projected backwards to zero load, and that point was taken as the zero for displacement. This procedure was necessary to eliminate any initial nonlinearity in the curve, which was attributed to settling in at the supports and initial stick-slip in the potentiometers. ANALYSIS AND DISCUSSION OF RESULTS Peak load Nominal peak tensile stresses, computed from the peak loads using principles of mechanics of materials and the throat cross sections, are listed in Table 4 in the column headed σ max. These nominal peak stresses are reported in the interest of simplicity and are not the true peak stresses, which could be computed using fracture mechanics principles 19 to account for the stress concentration effect of the notch. The ratio of computed peak stress σ max to tensile strength measured per ASTM C496-02, f t, was 1.13, with a coefficient of variation of 16%. Figure 4(a) shows peak stress σ max normalized by measured tensile strength f t versus measures of specimen size and geometry. Following laboratory testing, finite element analyses were conducted to validate the experimental data. The results of these analyses, which are described in a following section, indicated that some test specimens were more flexible and weaker than expected. They are identified as having low strength in Fig. 4. This distinction was supported by the fact that those specimens were also less stiff and strong than their nominally identical companion specimens. The data in Fig. 4(a) suggest that peak tensile stress is a function of specimen throat area, with larger throat areas Fig. 3 Typical fracture-energy test load-displacement histories (1 lb = 4.45 N; 1 = 25.4 mm). Table 4 Test specimen and material properties Specimen b, d throat, E c, ksi f c, psi f t, psi σ max, psi G F, lb/ G F (Eq. (1)), lb/ G F (Eq. (2)), lb/ FR-42-A FR-42-A NA NA NA FR-63-A NA NA FR-63-A FR-42-R FR-42-R FR-63-R FR-63-R FR-33-R FR-33-R FR-33-R FR-33-R Note: 25.4 mm = 1 ; 1 ksi = 6.89 MPa; 1 psi = MPa; and 1 lb/ = 175 N/m. ACI Materials Journal/November-December

6 having smaller strengths. This finding supports the sizeeffect theory. 9,10 The FR-63-R specimens, with larger throat areas, however, had stiffnesses and strengths that were substantially smaller than predicted by finite element analysis. Thus, no conclusion can be drawn from the data in Fig. 4(a). Figures 4(b) and (c) show that specimen strength is not affected by the ratio of specimen width to throat depth b/d throat or specimen length to depth l/d within the ranges considered for those variables. Fracture energy Fracture energy G F was computed from the area under the complete, corrected load-deflection curve divided by the cross-sectional throat area. Because, for some specimens, the net load (after subtracting the quantity needed to equilibrate the gravity loads due to self-weight and counter weights) never dropped to exactly zero, even at deflections of approximately 0.25 (6.4 mm), a definition was needed for the upper limit of the deflection to be used in the area calculation. This value was set somewhat arbitrarily to the displacement at which the strength had dropped to 5% of the peak value. The fracture energy data for each specimen are given in Table 4. The experimental fracture energy values for the specimens with angular aggregate range from 0.41 to 0.69 lb/ (72 to 121 N/m). With the exception of Specimen FR-33-R-4, the experimental fracture energy values for specimens with rounded aggregate range from 0.60 to 0.79 lb/ (105 to 138 N/m). The smaller fracture energy, 0.48 lb/ (84 N/m), computed for Specimen FR-33-R-4 may be attributed to the large gaps in the measured displacement history that resulted from the accidentally high load rate used during the test. Table 4 also lists, for specimens with rounded aggregate, fracture energy computed using the empirical equations proposed by Bažant and Becq-Giraudon 16 D max f G F = 2.5α c o w --- c 0.18 N/m (1a) D max f G F = α c o (1b) w --- c 0.18 lb/ and by Comité Euro-International du Béton (CEB) 20 G F = ( ( D max ) 2 0.5D max + 26) (2a) N/m f c G F = ( 0.174( D max ) D max ) lb/ (2b) 1450 f c Fig. 4 Relationship between peak tensile stress and specimen geometry: (a) maximum tensile stress versus A throat ; (b) maximum tensile stress versus b/d throat ; and (c) maximum tensile stress versus l/d (1 2 = 647 mm 2 ). where α o is an aggregate shape factor (α o = 1 for rounded aggregate, α o = 1.12 for angular aggregate), f c is the compressive strength of the concrete (MPa or psi), D max is the maximum aggregate size (mm or ), and w/c is the water-cement ratio of the concrete. Fracture energy is computed using Eq. (1) and (2) only for specimens in this study with rounded aggregate as compressive strength data were not available for specimens with angular aggregate. The experimental values found herein are consistent with those predicted using the equation by Bažant and Becq- Giraudon 16 and somewhat larger than those predicted using the CEB 20 equation. Specifically, on average the measured fracture energy is 2% greater than that predicted using Eq. (1), with a coefficient of variation of 14%, and 19% greater than that predicted using Eq. (2), with a coefficient of variation of 3%. The equation proposed by Bažant and Becq-Giraudon was calibrated using a data set of 161 RILEM work-of-fracture tests 4 ; whereas the equation proposed by CEB was calibrated using a much smaller data. This likely explains why Eq. (1) provides a better fit to the data presented here It should also be noted that the model by Bažant and Becq- Giraudon predicts fracture energy values that are higher for 580 ACI Materials Journal/November-December 2007

7 specimens with angular aggregate than with rounded aggregate, whereas the experimental fracture energy values show the opposite trend. The compositions of the angular and rounded aggregates were nominally the same; however, without measured material data for use in Eq. (1), no formal conclusions can be drawn regarding the observed impact of aggregate shape on fracture energy. Figure 5 shows the experimental fracture energy values plotted versus measures of specimen size and geometry. Specimens with stiffnesses and strengths that were substantially less than predicted by finite element analysis are indicated. If these specimens are ignored, the data in Fig. 5(a) to (c) suggest that fracture energy decreases very slightly as throat area increases, but is not affected by the ratio of specimen width to throat depth b/d throat, or the ratio of specimen length to depth l/d. The data also show that the specimens with low peak strengths had the highest fracture energies. This may be caused by defining the fracture energy as the area under the load-displacement curve out to a strength loss of 95%. Because the tail of the load-displacement curve was relatively flat, a relatively larger area under the tail of the loaddisplacement curve was included in computing fracture energy for specimens with lower peak strength. after the peak load. The G F values were consistent with published values for similar concretes, suggesting that testing with an open-loop machine is viable; 2. Use of a stiff load train helps to reduce any gaps in the load-deflection data. With one exception, the gaps in the data had little impact on the measured value of G F ; 3. Measuring the displacements by placing sensors directly below the specimen, rather than on an instrumentation rig, may have influenced the accuracy of the initial displacement measurements. Direct measurements were chosen because an instrumentation rig, such as that suggested by RILEM, would introduce flexibility by virtue of its geometry and would therefore be expected to lead to measured displacements of limited accuracy. Noncontact displacement sensors, such as linear variable differential transformers (LVDTs), appear preferable to potentiometers, if accurate measurement of very small displacements is necessary; and Initial stiffness Difficulties were experienced in obtaining accurate measurements of initial stiffness. It is worth noting that the accuracy of measurements of the three quantities, stiffness, load, and fracture energy, should be expected to be in that order, because each is essentially the integral of the previous one. In these tests, the greatest difficulties were experienced in obtaining accurate measurements of the very small deflections at the start of the tests. The measured deflections were found to be nonlinear with load. The nonlinearities are attributed to both real behavior and limited instrument precision. The specimens twisted slightly during the initial stages of loading, as revealed by both the potentiometer data and the measurements from a laser level laid across the specimen. Averaging the four measured displacements did not produce a linear load-deflection curve, however, nor was the twist angle linear with load. It is believed that the twisting was caused primarily by slight misalignments between the load and the axis of the supports, and that the nonlinearities were caused by very small local deformations (on the order of [ mm]) as the supports settled The difficulties were exacerbated by limits on the precision of the instrumentation. Potentiometers are contact instruments and, thus, can develop some friction between the slider and the body of the instrument. This friction can cause initial sticking and inaccuracy in measuring very small displacements as the instrument starts to move. Once displacements become relatively large, the impact of friction and initial sticking becomes insignificant. Evaluation of test procedures The tests involve relatively delicate procedures, the details of which may significantly affect the results, especially if an open-loop test machine is used. The most important issues are reviewed as follows: 1. Counterweights were found to provide a feasible method of controlling the progression of deformation and cracking. Attaching them presented no particular problems. They allowed the complete load-displacement curve to be recorded, albeit with a slight gap in the data collection soon Fig. 5 Relationship between fracture energy and specimen geometry (1.0 lb/ = N/mm; 1 2 = 647 mm 2 ). ACI Materials Journal/November-December

8 4. To avoid twisting, great care is needed in aligning the specimen, regardless of the type of test machine used. COMPARISON WITH NUMERICAL SIMULATION Typically, fracture energy testing of concrete is done to generate data for calibration of numerical models. To validate the proposed test method, laboratory tests were simulated numerically using two-dimensional continuum models and the fracture energy, tensile strength, and elastic modulus data generated in the laboratory. Commercial software was used for the analyses. 1 The numerical and laboratory data were compared to determine whether the proposed test method was appropriate for use in generating data for model calibration. In simulating the response of the laboratory test specimens, an initial prototype model was created for each of the test series (FR-42, FR-63, and FR-33) using a commercial nonlinear finite element analysis program. 1 This model included a preferred: 1) level of mesh refinement; 2) element formulation; 3) representation of the specimen geometry, boundary conditions, and loading; and 4) concrete constitutive model. For each of the three test series, the concrete properties used in the simulation were the averages of the values measured in the laboratory. This model was used to generate simulated specimen response, which was then compared with the corresponding measured response. To validate the numerical data and quantify the impact of modeling decisions on simulation data, a series of additional analyses of test Specimen FR-33-R1 were conducted. These analyses are presented in detail by Mitra 21 and considered the impact element formulation and mesh refinement as well as concrete constitutive model parameters including rotating versus fixed cracks, the threshold angle for formation of subsequent cracks in a fixed-crack model, shear stiffness reduction for cracked concrete, the shape of the tensile post-peak softening curve, fracture energy, tensile strength, and modulus of elasticity. The following sections present the prototype model, the response data generated using this model, and a comparison of simulated and observed response histories. Prototype model The prototype model represented the preferred approach to modeling the laboratory test specimens. The results of a series of analyses, in which key modeling decisions were varied, were used to determine the preferred model characteristics. The results of these analyses are presented by Mitra. 21 The prototype model had the following characteristics: 1. An idealized representation of the specimen geometry, in which the counterweights were ignored and the concrete had zero density. This arrangement significantly reduced the number of degrees of freedom (DOFs) needed, but had minimal impact on simulated response, as was verified in a subsequent analysis. 21 Thus, the model consisted of a 21 (534 mm) long, notched concrete beam on an 18 (458 mm) simple span; 2. A mesh comprising 0.25 x 0.25 (6.25 x 6.25 mm) eight-node two-dimensional quadrilateral elements, for which a 2-by-2 reduced integration scheme was used. This level of mesh refinement resulted in a total element count of 849 for the FR-63 specimens, 1273 for the FR-42 specimens, and 1693 for the FR-33 specimens. That this mesh resulted in a converged solution was validated in subsequent analyses of Specimen FR-33-R-1 21 ; 3. Concrete response in compression was assumed to be linear elastic, for which the measured E c was used; and 4. Concrete material response in tension was simulated using the decomposed strain multiple fixed-crack model 1,22 available in the software. This model is a multiple, fixed, smeared-crack model in which concrete strain is decomposed into strain within the concrete continuum and strain due to crack width opening. Concrete cracking is assumed to occur when the maximum principal stress exceeds the concrete tensile strength and the orientation of the crack surface exceeds the threshold angle between crack surfaces. For the prototype analysis, a threshold angle of 60 degrees was used because analyses with larger threshold angles were found to produce overly strong post-peak response. 21 A stressversus-crack width opening model controls response normal to the crack surface; the energy dissipated in opening the crack is assumed equal to the concrete fracture energy. For the prototype model, the Hordijk strength deterioration model 23 was used; this model was found to provide better overall simulation of the experimental response than did the other models available in the software. 21 The Hordijk model assumes that normal stress decays exponentially with crack opening. Once cracking occurs, shear stiffness parallel to the crack s surface is assumed to remain constant and is specified by a shear stiffness reduction ratio. Here a value of was chosen as results by Mitra 21 indicated that values in excess of this resulted in overly stiff post-peak response. Calibration of the model required identification of the concrete tensile strength and fracture energy; these parameters were defined equal to the laboratory measured values. The impact of the concrete constitutive modeling assumptions and the uncertainty in laboratory measurement of concrete material properties were investigated in subsequent analyses and found to have minimal impact. 21 Figure 6 shows the midspan load-deflection response for all of the specimens with rounded aggregate, as simulated using the prototype model and as measured in the laboratory. Analyses were not done for the specimens with angular aggregate as concrete modulus and strength data were not available for those specimens. These data show that the model simulates the observed response of Specimens FR-42-R-2, FR-33-R-1, and FR-33-R-2 well. For these specimens, the model simulates initial stiffness, peak strength, displacement at peak strength, and initial post-peak stiffness with a high level of accuracy. The model does not simulate the tail of the response history because the analysis stops when all elements at midspan of the beam have cracked and lost tensile strength, with the exception of the uppermost element in the beam, which must carry the stresses associated with bending alone. The data in Fig. 6 show a less favorable comparison with the observed response for Specimens FR-42-R-1, FR-63-R*, FR-33-R-3, and FR-33-R-4. For these specimens, the measured strength is as much as 20% lower, and the difference in initial stiffness is greater still. For Specimen FR-33-R-4, the discrepancy is attributed to accidental, rapid loading of the test specimen, as previously discussed. For Specimens FR-42-R-1 and FR-33-R-3, the discrepancies must be attributed to experimental variations, because in each case the measured response differed from that of the nominally identical (same concrete batch, casting and curing conditions, test preparation, and observed specimen condition at the time of testing) specimen (FR-42-R-2 and FR-33-R-1 and 2), which were successfully modeled by the finite element analysis. Given that the model is validated against experimental data, a similar conclusion must be reached for the FR-63-R* specimens. 582 ACI Materials Journal/November-December 2007

9 influence of these factors on maximum concrete stress and fracture energy. To validate the experimental data, commercial finite element analysis software was used to simulate the loaddisplacement response of the fracture energy test specimens. Numerical models were calibrated using measured material properties, including fracture energy. Additionally, the measured fracture energy values were compared with those predicted using previously proposed empirical equations that estimate fracture energy as a function of water-cement ratio, aggregate size, and compressive strength. CONCLUSIONS The results of this study support the following conclusions: 1. The proposed test method allows the fracture energy of concrete to be measured using the work-of-fracture method and an open-loop testing machine. A stiff load train is important for minimizing any gaps in the data caused by rapid strength loss when cracking occurs; 2. Measurements of initial stiffness are very sensitive to minor imperfections in the alignment and support conditions of the specimen and in instrumentation. This is true for both the RILEM work-of-fracture test method 4 and the modified version developed here Use of noncontact displacement sensors, such as LVDTs, is advisable; 3. The use of counterweights allows stable testing to continue out to displacements much larger than that corresponding to peak load. The resulting long tail of the load-displacement curve contributes to the fracture energy, and accurate measurement of it likely contributed to the reduction in scatter in the G F values compared with previous tests. 16 A standard definition is needed for the region of the curve over which the fracture energy should be computed; 4. The presence of the counterweights causes the work due to the self-weight of the specimen to be essentially zero, thereby avoiding the need for detailed calculation of it and consideration of how much of it should be removed from the total work done; 5. Within the range of specimen geometries considered, fracture energy is not affected by the shape of the specimen and only slightly affected by the size of the specimen. Specimen size and geometry may affect nominal maximum tensile stress; however, additional testing is required to verify this conclusion; and 6. Finite element analysis may be used to validate experimental data, as it provides validation of initial stiffness, strength, and post-peak response. Fig. 6 Load-displacement response for test specimens as observed in laboratory and as simulated using prototype numerical model: (a) Specimens FR-42-R*; (b) Specimens FR-63-R*; and (c) Specimens FR-33-R* (1 lb = 4.45 N; 1 = 25.4 mm). SUMMARY A new method for measuring concrete fracture-energy was developed and validated through comparison with previously proposed empirical equations and with the results of finite element analysis. The new test method modifies the original RILEM 4 work-of-fracture test by adding counterweights to offset the midspan moment due to self-weight, thereby making the use of an open-loop testing machine possible. Three different specimen geometries and two different concrete mixtures were used to investigate the RECOMMENDATIONS FOR FUTURE WORK The proposed test method represents a practical method for measuring concrete fracture energy without special equipment. The results of this preliminary study suggest that, within the range of specimen sizes considered, fracture energy is not affected by specimen size and geometry. Additional research is required, however, to verify this and to determine fracture energy values for infinitely large specimens. ACKNOWLEDGMENTS Funding for the project was provided by the National Science Foundation, under Grant No. CMS , and is gratefully acknowledged. The counterweight concept was suggested by Graduate Student Assistant T. Popham. NOTATION A throat = beam throat area b = out-of-plane width of beam as defined in Fig. 1 ACI Materials Journal/November-December

10 D max = maximum dimension of coarse aggregate d = depth of beam as defined in Fig. 1 d notch = depth of notch in beam as defined in Fig. 1 d throat = depth of beam throat as defined in Fig. 1 E c = concrete elastic modulus f c = concrete compressive strength f t = concrete tensile strength, fracture energy G F = concrete fracture energy density defined per RILEM 4 G f = concrete fracture energy density defined by area under initial portion of load-deflection curve measured in RILEM 4 workof-fracture test method L = total length of beam as defined in Fig. 1 l = length of beam between supports as defined in Fig. 1 w/c = water-cement ratio used in concrete mixture α max = nominal maximum tensile stress α o = aggregate shape factor (Eq. (1)) REFERENCES 1. TNO DIANA, DIANA-9 Finite Element Analysis User s Manual, F. C. dewitte, ed., Hillerborg, A.; Modeer, M.; and Petersson, P. E., Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements, Cement and Concrete Research, V. 6, 1976, pp ACI Committee 446, Fracture Mechanics of Concrete: Concepts, Models, and Determination of Material Properties (ACI 446.1R-91), American Concrete Institute, Farmington Hills, MI, 1991, 146 pp. 4. RILEM TC 50-FMC Fracture Mechanics of Concrete, Determination of the Fracture Energy of Mortar and Concrete by Means of Three-Point Bend Tests on Notched Beams, Materials and Structures, V. 18, No. 106, 1985, pp Hillerborg, A., Theoretical Basis of a Method to Determine the Fracture Energy G F of Concrete, Materials and Structures, V. 18, No. 106, 1985, pp RILEM TC 89-FMT Fracture Mechanics of Concrete-Test Methods, Determination of Fracture Parameters (K lc s and CTOD c ) of Plain Concrete Using Three-Point Bend Tests, Materials and Structures, V. 23, 1990, pp RILEM TC 89-FMT Fracture Mechanics of Concrete Test Methods, Size Effect Method for Determining Fracture Energy and Process Zone Size of Concrete, Materials and Structures, V. 23, 1990, pp Jenq, Y., and Shah, S. P., Two-Parameter Fracture Model for Concrete, Journal of Engineering Mechanics, V. 111, No. 10, 1985, pp Bažant, Z. P., and Pfeiffer, P. A., Determination of Fracture Energy from Size Effect and Brittleness Number, ACI Materials Journal, V. 84, No. 6, Nov.-Dec. 1987, pp Bažant, Z. P., Size Effect in Blunt Fracture: Concrete, Rock, Metal, Journal of Engineering Mechanics, ASCE, V. 110, 1987, pp Planas, J.; Elices, M.; and Guinea, G. V., Measurement of the Fracture Energy Using Three-Point Bend Tests: Part 2. Influence of Bulk Energy Dissipation, Materials and Structures, V. 25, 1992, pp Malvar, L. J., and Warren, G. E., Fracture Energy for Three-Point- Bend Tests on Single-Edge-Notched Beams, Experimental Mechanics, V. 28, No. 3, 1988, pp Hanson, J. H., and Ingraffea, A. R., Using Numerical Simulation to Compare the Fracture Toughness Values for Concrete from the Size-Effect Two-Parameter and Fictitious Crack Models, Engineering Fracture Mechanics, V. 70, 2003, pp Qian, J., and Luo, H., Size Effect on Fracture Energy of Concrete Determined by Three-Point Bending, Cement and Concrete Research, V. 27, No. 7, 1997, pp Navalurkar, R. K.; Hsu, C. T. T.; Kim, S. K.; and Wecharatana, M., True Fracture Energy of Concrete, ACI Materials Journal, V. 96, No. 2, Mar.-Apr. 1999, pp Bažant, Z. P., and Becq-Giraudon, E., Statistical Prediction of Fracture Parameters of Concrete and Implications for Choice of Testing Standard, Cement and Concrete Research, V. 32, 2002, pp Kozul, R., and Darwin, D., Effects of Aggregate Type, Size and Content on Concrete Strength and Fracture Energy, Technical Report SM43, The University of Kansas Research, Inc., Lawrence, KS, 1997, 85 pp. 18. Martin, J. S., An Experimental Investigation of Bond in Reinforced Concrete, MSCE thesis, University of Washington, Seattle, WA, 2006, 163 pp. 19. Tada, H.; Paris, P. C.; and Irwin, G. R., The Stress Analysis of Cracks Handbook, Del Research Corp., Hellertown, PA, 1973, 677 pp. 20. Comité Euro-International du Béton, CEB-FIP Model Code 1990, Thomas Telford, London, UK, 1990, 437 pp. 21. Mitra, N., Modeling of Reinforced Concrete Beam-Column Joints, PhD dissertation, University of Washington, Seattle, WA, 2006, 253 pp. 22. de Borst, R., and Nauta, P., Non-Orthogonal Cracks in a Smeared Finite Element Model, Engineering Computations, V. 2, No. 1, 1985, pp Reinhardt, H. W.; Cornelissen, H. A. W.; and Hordijk, D. A., Tensile Tests and Failure Analysis of Concrete, Journal of Structural Engineering, V. 112, No. 11, 1986, pp ACI Materials Journal/November-December 2007