Effects of loading rate and notch-to-depth ratio of notched beams on flexural performance of ultra-high-performance concrete

Accepted Manuscript Effects of loading rate and notch-to-depth ratio of notched beams on flexural performance of ultra-high-performance concrete Weina Meng, Yiming Yao, Barzin Mobasher, Kamal Henri Khayat PII:

S0958-9465(16)30786-7

DOI:

10.1016/j.cemconcomp.2017.07.026

Reference:

CECO 2876

To appear in:

Cement and Concrete Composites

Received Date: 28 November 2016 Revised Date:

11 March 2017

Accepted Date: 28 July 2017

Please cite this article as: W. Meng, Y. Yao, B. Mobasher, K.H. Khayat, Effects of loading rate and notch-to-depth ratio of notched beams on flexural performance of ultra-high-performance concrete, Cement and Concrete Composites (2017), doi: 10.1016/j.cemconcomp.2017.07.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Effects of Loading Rate and Notch-to-Depth Ratio of Notched Beams on Flexural

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Performance of Ultra-High-Performance Concrete

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Weina Meng1, Yiming Yao2, Barzin Mobasher2, Kamal Henri Khayat1,* 1

Department of Civil, Architectural, and Environmental Engineering, Missouri University of

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Science and Technology, 500 W. 16th Street, Rolla, MO 65409, USA 2

School of Sustainable Engineering and Built Environment, Arizona State University, Tempe, AZ 85287-5306, USA

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7 Abstract

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This paper addresses the effects of loading rate and notch-to-depth ratio on flexural properties of

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ultra-high-performance concrete (UHPC) notched beam specimens, in order to enable use of

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standardized laboratory test data to predict flexural properties of UHPC structures that have

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different dimensions and are subjected to a range of loading rates. UHPC notched beams were

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tested in three-point bending to study the effects of three notch-to-depth ratios of 1/6, 1/3, and

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1/2 at five loading rates of 0.05, 0.50, 1.25, 2.50, and 5.00 mm/min on flexural performance. Test

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results indicate that loading rate and notch-to-depth ratio have significant effects on flexural

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properties of the UHPC notched beams. The flexural strength is shown to increase with the

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loading rate and the notch-to-depth ratio. The fracture energy increases with the loading rate but

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decreases with the notch-to-depth ratio. The changes of flexural properties with the loading rate

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are also dependent on the notch-to-depth ratio. Regression analyses to correlate flexural

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properties associated with the loading rate and notch-to-depth ratio were conducted to obtain

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parameters for UHPC structures.

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Keywords: Flexural properties; Loading rate; Notch-to-depth ratio; Post-cracking behavior;

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Ultra-high-performance concrete (UHPC)

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Corresponding author. Email: [email protected] Page 1

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1. Introduction Ultra-high-performance concrete (UHPC) is a new class of fiber reinforced cementitious

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composite with a low water-to-cementitious materials ratio (w/cm < 0.25). With appropriate

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combination of portland cement and supplementary cementitious materials, adequate sand

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gradation, fiber reinforcement, and use of high-range water reducer (HRWR), UHPC can be

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produced to achieve superb flowability, compressive strength, tensile strength, energy absorption

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capacity, high durability, and extended service life [1, 2]. In particular, with adequate use of steel

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fibers, UHPC can exhibit high resistance to cracking and high tensile/flexural strengths. Cracked

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UHPC can carry higher loads, exhibiting “strain-hardening” behavior and multiple cracks [3].

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For these reasons, UHPC has been gaining increasing interests in highway and pedestrian bridge

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applications in the forms of precast girders [4], in-fill deck joints [5], railroad track slab [6], as

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well as thin elements, such as pavement overlay [7], full-depth deck panels [8], stay-in-place

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formwork [9, 10], and functionally-graded composites [11].

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In addition to quasi-static loads, UHPC can be used to resist short-duration loads or impact

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loads due to earthquake, wind, blast, or collision [12]. Ranade et al. [13] studied the effects of

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strain rate on the direct tensile properties of a high-strength, high-ductility concrete. As the

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tensile strain rate was increased from 10-4 /s to 10 /s, the cracking strength and the tensile

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strength were increased by 53% and 42%, respectively. Zhang et al. [14] investigated the effects

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of loading rate on the flexural tensile strength and fracture energy of a steel fiber reinforced

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concrete. As the increasing rate of mid-span deflection was increased from 10-3 mm/s to 103 mm

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/s, the flexural tensile strength and fracture energy were increased by 248% and 152%,

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respectively. Pyo et al. [15] and Tran et al. [16] studied the effects of tensile strain rate on the

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direct tensile strength and fracture energy of UHPC mixtures. Smooth and twisted steel fibers

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were blended at different volume ratios in the UHPC mixtures. Pyo et al. [15] reported that the

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tensile strength and fracture energy of the UHPC mixtures were respectively increased by up to

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36% and 96%, as the strain rate was increased from 10-4 /s to 10-1 /s. Tran et al. [16] reported that

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the tensile strength and fracture energy were respectively increased by up to 190% and 920%, as

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the strain rate was increased from 5 /s to 92 /s.

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While the structural dimensions in many projects vary significantly, the practical sizes of

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test specimens in the laboratory are limited to a relatively small range. However, mechanical

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properties of fiber reinforced composites correlate with the geometry of the test specimen [17–

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19]. In general, the flexural tensile strength of fiber reinforced composites decreases with the

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depth or thickness of the specimen [20]. The depth and width are the primary geometrical

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parameters for unnotched specimens, while the notch-to-depth ratio (N/D) is a primary

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geometrical parameter for notched specimens with a specified cross section [21]. Size effects of

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fiber reinforced composites have been studied by different researchers, aiming at utilizing the

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mechanical properties obtained from standardized laboratory tests to predict structural behavior

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of fiber reinforced composites with different dimensions. Reineck and Frettlöhr [22] reported

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that the width-to-depth ratio of UHPC prism was the key parameter that influenced the flexural

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strength. Spasojevic et al. [23] claimed that thick UHPC specimens were more sensitive to size

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effect than thin specimens in flexure. The abovementioned studies were all based on prism

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specimens loaded at a single loading rate. Size effect for UHPC specimens has not been fully

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investigated at different loading rates.

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Three-point bending tests using notched beam specimens are easy to conduct and allow

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more stable cracking with reproducible results compared with unnotched beams, since the notch

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helps localize the fracture plane. As a crack is initiated at the notch tip and propagates through

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the beam section, the load-carrying capacity of the section gradually diminishes and the carried

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load decreases with the increasing deflection. Thus, in this study, notched beam specimens are Page 3

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used to evaluate the post-cracking behavior of fiber reinforced composites. However, different

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specimen geometries are recommended in different codes. For example, the RILEM committee

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issued RILEM TC 162-TDF [24], which specifies concrete beams of 150 × 150 mm cross

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section with a minimum length of 550 mm and a notch to depth (N/D) ratio of 1/6; the loading

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rate for deflection control is 0.2 mm/min. The specimen dimension recommended for normal

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concrete might be inappropriate for UHPC, which is relatively homogeneous due to removal of

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coarse aggregates. In addition, the ultra-high strength property of UHPC increases the

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requirement of load-carrying capacity of the test setup for large specimens. A wide range of

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specimen size and notch depth are permitted in Japan [25], in order to accommodate various

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fiber-reinforced concretes that have different maximum aggregate sizes. The depth (D) of the

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beam specimen shall be no less than four times of the maximum aggregate size; the loading rate

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shall be 0.0005D to 0.001D/min; the N/D is 0.3. However, the loading rate is too low to

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characterize the flexural/tensile behavior of UHPC, which has high impact resistance property.

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European Committee for Standardization issued EN 14651:2005+A1 [26], which gives

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recommendations that are similar to the RILEM recommendations.

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In this study, notched UHPC beams with the dimensions in accordance with the JCI

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recommendations were used [25]. The notched beam specimens had a 75 × 75 mm cross section

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and a 400 mm length, with a 5 mm wide notch at mid-span. Three notch-to-depth ratios were

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respectively applied, which are 1/6, 1/3, and 1/2, corresponding to recommendations by RILEM,

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JCI, and thin UHPC element application perspective, respectively. A baseline loading rate of

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0.05 mm/min, (in the range of 0.0375 to 0.075 mm/min by JCI recommendations [25]), was used

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and extended to 0.50, 1.25, 2.50, and 5.00 mm/min to study the effect of loading rate. The main

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objective of this study is to investigate the effects of loading rate and N/D on the flexural

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performance of UHPC notched beams. The direct tensile and compressive properties of a UHPC

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mixture were characterized. Single fiber pull-out tests of straight and hooked steel fibers, which

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are partially embedded in the UHPC matrix, were conducted to assess the post-cracking

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behaviors of the UHPC mixture.

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2. Experimental program

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2.1. Materials, specimens, and curing regime

An optimized UHPC mixture as shown in Table 1 was used [27]. The w/cm of 0.20, by mass,

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and sand-to-cementitous materials ratio of 1.0, by volume, were used. The cementitious materials

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consisted of tertiary blend of 45% Type III portland cement, 50% Class C fly ash, and 5% silica

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fume, by volume. The Blaine finenesses of the cement and fly ash were 560 and 465 m2/kg,

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respectively. The silica fume has an average diameter of 0.15 µm, a SiO2 content of 96%, and a

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specific surface area of 18,200 m2/kg, determined using the BET method. The sand consisted of

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30% masonry sand (0–2 mm, BSG=2.63), 52.5% Missouri River sand (0–4.75 mm, BSG=2.64),

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and 17.5% lightweight sand (0–4.75 mm, BSG=1.78), by mass. The lightweight sand was pre-

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treated to secure saturated surface-dry condition to an internal absorption of 17% prior to

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batching.

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Table 1. Optimized UHPC mixtures

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Raw materials UHPC (kg/m3) Cement 663 Silica fume 42 Fly ash 367 River sand (SSD) 452 Masonry sand (SSD) 308 Lightweight sand (SSD) 120 Mixing water 214 HRWR (including water) 40 Air detrainer 9 Straight steel fibers 78 Hooked-end steel fibers 78

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A polycarboxylic-based high-range water reducer (HRWR) with a solid mass content of 23% and specific gravity of 1.05 was used to enhance workability. The HRWR dosage was adjusted

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to ensure an initial mini-slump flow of 280 ± 10 mm, in accordance with ASTM C 230/C 230M

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[28] thus ensuring self-consolidation. The flow time was measured using a mini-V funnel in

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accordance with the EFNARC [29]. An air-detrainer admixture with a specific gravity of 0.97

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was employed to decrease the volume of entrapped air during mixing and casting in the viscous

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UHPC mixture.

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Figure 1 shows the steel fibers that had electro-plated brass coating to enhance corrosion

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resistance. The steel fiber’s Young’s modulus and tensile strength are 203 and 1.9 GPa,

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respectively. A blended mixture, containing 50 vol.% straight microfibers (diameter: 0.2 mm,

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length: 13 mm) and 50 vol.% hooked-end fibers (diameter: 0.5 mm, length: 30 mm) were used at

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a total volume content of 2%.

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(a)

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Figure 1. Photograph of steel fibers: (a) straight fibers, and (b) hooked-end fibers.

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The UHPC mixture was blended in a 22-L Hobart mixer following a three-step procedure:

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(1) dry cementitious materials and sand were mixed at 60 rpm for 3 minutes (min); (2) 90% of

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the mixing water and 90% of the HRWR were added and mixed at 120 rpm for 3 min; (3) the

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remaining mixing water and HRWR were added and mixed at 120 rpm for 5 min; (4) steel fibers Page 6

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were gradually added within 1 min at the mixing rate of 60 rpm; and (5) the mixture was mixed

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at 120 rpm for additional 2 min. For each mixture, specimens were cast in one lift without mechanical consolidation. The

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mixtures were poured from one side of the prisms and flowed to the other side to promote fiber

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orientation along the relatively narrow section of the samples. After casting, all specimens were

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cured for 24 hours (h) in mold covered by wet burlap and plastic sheets at 23 ± 1°C. After

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demolding, the specimens were cured in lime-saturated water at 23 ± 1°C for 28 days (d). The

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notch of beam was made by saw cutting in accordance with the JCI recommendations [25].

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2.2. Compressive tests

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The compressive strength was evaluated using 50 mm cube specimens at 28 d at a loading

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rate of 1.6 kN/s, in accordance with ASTM C109 [30]. The modulus of elasticity and Poisson’s

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ratio were measured using 100 × 200 mm cylinders at 28 d in accordance with ASTM C469 [31].

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2.3. Direct tensile tests

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Direct tensile tests were conducted using dog-bone specimens measuring 25 mm in

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thickness [32], as depicted in Figure 2. Each end of the specimen was hold by a fixture gripped

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on a load frame (load capacity: 250 kN) through a ball hinge connection to minimize the bending

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moment applied on the specimen, ensuring the specimen was primarily subjected to tension. Two

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linear variable differential transformers (LVDTs) were used to measure the deformation over a

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gage length of 160 mm. The applied load was recorded by a load cell embedded in the load

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frame. The test was performed at a displacement rate of 0.05 mm/min.

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Figure 2. Flexural test setup and notched beam specimen. Unit: mm. 2.4. Single fiber pull-out tests

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Single fiber pull-out tests were performed using a customized setup [32]. The load and the

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pull-out displacement were measured using a 1 kN load cell and a displacement transducer

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embedded in the load frame. The test was performed under stroke control mode at constant

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displacement rates.

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2.5. Flexural tests

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Figure 3 illustrates the notched beam specimen with notch-to-depth ratios N/D= 1/6, 1/3,

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and 1/2, corresponding to notch depths of 12.5, 25.0, and 37.5 mm, respectively. The mid-span

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deflection was measured as the average of two LVDTs mounted at two sides of the beam. A

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frame with a 250 kN capacity was used and the load measured by an strain gage based load cell.

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The test was controlled by a constant rate of the mid-span deflection. A baseline loading rate of

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0.05 mm/min was used and extended to 0.50, 1.25, 2.50, and 5.00 mm/min to study the effect of

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loading rate on the flexural properties.

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Figure 3. Flexural test setup and notched beam specimen. Unit: mm.

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The specimens and loading protocols are shown in Table 2. A total of 15 test cases and 6

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replicate specimens per case were used to address the effect of loading rate and the N/D on the

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flexural properties of UHPC.

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Table 2. Notched beam specimens and loading rates in flexural tests Code

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3.1. Fresh properties

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S-12.5-0.05 S-12.5-0.50 S-12.5-1.25 S-12.5-2.50 S-12.5-5.00 S-25.0-0.05 S-25.0-0.50 S-25.0-1.25 S-25.0-2.50 S-25.0-5.00 S-37.5-0.05 S-37.5-0.50 S-37.5-1.25 S-37.5-2.50 S-37.5-5.00 3. Experimental results

Notch depth (mm) 12.5 12.5 12.5 12.5 12.5 25.0 25.0 25.0 25.0 25.0 37.5 37.5 37.5 37.5 37.5

Loading rate (mm/min) 0.05 0.50 1.25 2.50 5.00 0.05 0.50 1.25 2.50 5.00 0.05 0.50 1.25 2.50 5.00

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The mini-slump flow of the mixtures for all 90 specimens was adjusted to be 280 ± 10 mm.

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The mini V-funnel flow time was 18 ± 3 s. The air content and unit weight of representative

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UHPC samples were 2.5% and 2450 ± 30 kg/m3, respectively. Page 9

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3.2. Elastic properties, compressive strength, and tensile properties The compressive strength at 28 d was determined to be 158 ± 5 MPa. The modulus of

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elasticity and Poisson’s ratio were 50 ± 2 GPa and 0.20 ± 0.02, respectively. With the steel fibers

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distributed in the UHPC matrix, the specimens gain enhanced ductility in tension. After cracks

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are initiated, the steel fibers crossing the crack interfaces can bridge the cracks, preventing abrupt

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drop in the load-carrying ability. Figure 4 shows the load-elongation curves of three out of six

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tested UHPC specimens and their average. In the other three specimens, the cracks did not occur

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within the gauge length. At the beginning of loading, the applied load increased linearly with

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elongation until cracks were initiated when the tensile stress in matrix reached the cracking limit.

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After cracking, the applied load continued to increase with a decreasing rate until the cracks were

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substantially developed, reaching peak values. Then, the load decreased with further elongation

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of the specimen until the specimen completely failed. In the post-cracking stage, the majority of

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the tensile load was carried by the steel fibers across the cracks via a load transfer mechanism

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that depends on the fiber-matrix interfaces. Therefore, the post-cracking behavior of the UHPC is

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primarily controlled by fiber pull-out mechanisms. Multiple cracks appeared progressively with

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increases in elongation. At the completion of testing, the widths of cracks were in a range of 0.05

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to 1.10 mm.

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Load (kN)

10 8 6 Onset of cracking 4

Specimen 1 Specimen 2 Specimen 3 Average

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1 1.5 Elongation (mm)

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3.3. Single fiber pull-out tests Effect of loading rate on fiber pull-out behavior was studied by means of single fiber pull-

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out tests at the displacement rates of 0.05 and 5 mm/min, respectively. The two values of loading

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rate correspond to the lower and upper limits of the loading rates used for the notched beams.

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The mean results of nine straight microfibers and hooked-end fibers are shown in Figures 5(a)

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and 5(b), respectively. Results indicate that within the range of tested samples, the fiber

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geometry has a significant effect on the pullout response; however the loading rate is not as

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significant in the ranges studied. The peak pull-out loads and areas under the load-slip curves are

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shown to increase with the displacement rate.

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Pull-out load (N)

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180 160 140 120 100 80 60 40 20 0

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Hooked fiber - 0.05 mm/min Hooked fiber - 5 mm/min

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(a) (b) Figure 5. Mean values of nine single fiber pull-out tests for: (a) straight microfiber, and (b)

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hooked-end fiber.

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Figure 6 shows an intact hooked fiber that was not embedded in the UHPC matrix during the

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pull-out testing with uneven and rough surface. This surface roughness can cause substantial

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friction and interlock at the interface between the fibers and the UHPC matrix. After the

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embedded fibers are pulled out from the UHPC matrix, abrasion and scratches can be observed at

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the fiber surface. The hooked end of the fiber is deformed during testing, as shown in Figure 6.

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Figure 6. Surface conditions of steel fibers before and after pull-out testing for two portions of

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each fiber; with left side representing the section embedded in the UHPC matrix.

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3.4. Flexural tests

For the notched beam specimens, cracks are initiated in the vicinity of the notch tips, due to

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the reduced cross section and stress concentration. The steel fibers then bridge the cracks and

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cause multiple cracks, which greatly enhance the ductility of the beams. For this reason, after

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cracks are initiated, the load carried by the beam does not drop abruptly. Instead, higher loads are

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carried, demonstrating a hardening behavior. In the post-cracking stage, the tensile force is

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primarily carried by the steel fibers crossing the crack interfaces, which is mainly associated with

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the fiber pull-out mechanisms. At the completion of testing, the widths of cracks were in a range

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of 0.05–2.50 mm.

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Figures 7(a)–7(c) show representative load-deflection curves of the notched beam specimens.

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As the N/D is increased from 1/6 to 1/2 (i.e. increasing the notch depth from 12.5 to 37.5 mm),

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the peak load is reduced by 58% at the loading rate of 0.05 mm/min, and 78% at the loading rate

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of 5.00 mm/min. The peak load increased with the loading rate at each N/D level; however,

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significant discrepancies among beams with different N/D were observed. As the loading rate Page 12

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was increased from 0.05 to 5.00 mm/min, the peak loads at N/D = 1/6 and 1/2 were respectively

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increased by 10 kN (56%) and 1.1 kN (18%). This indicates that the increase of the N/D

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suppresses the loading rate effect on the load-carrying capacity.

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(b)

1.5 2 2.5 3 3.5 Mid-span deflection (mm)

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25 Load (kN)

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1.5 2 2.5 3 3.5 Mid-span deflection (mm)

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4.1. Flexural strength and residual strengths

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Figure 7. Load-deflection curves of notched beams with N/D of: (a) 1/6, (b) 1/3, and (c) 1/2.

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The flexural strength of UHPC notched beam is calculated using Equation (1) in accordance with RILEM recommendations [24]: fu =

3Pu L 2b( D − N ) 2

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where fu and Pu represent the flexural strength and the peak load in the load-deflection curves; L,

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b, D, and N denote the span, width, depth, and notch depth respectively. The test results of

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flexural strength from six specimens were averaged.

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Figures 8(a) and 8(b) show the correlation of flexural strength with the loading rate and N/D,

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respectively. As the loading rate was increased from 0.05 to 5.00 mm/min, the flexural strength

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for N/D of 1/6, 1/3, and 1/2 was increased by 62%, 71%, and 19%, respectively. The increases in

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the flexural strength due to higher loading rates may be partially attributed to the increase in

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interfacial bond strength as corroborated by the single fiber pull-out tests in Figure 5, which also

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agrees well with the study by Xu et al. [33].

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Flexural strength (MPa)

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Figure 8. Flexural strength versus: (a) loading rate (b) N/D. The results of flexural strength demonstrate interaction coupling of the loading rate and N/D,

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and Figure 8(a) indicates a marginal effect of N/D when the loading rate is not more than 1.25

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mm/min. As the N/D is increased from 1/6 to 1/2, the strength variations at different loading

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rates are up to ±6%. However, at the loading rates of 2.50 and 5.00 mm/min, increasing the N/D

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from 1/6 to 1/2 reduces the flexural strength by 20%–30%, meaning the effect of the N/D on the

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flexural strength can be significantly magnified by applying higher loading rates.

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The residual strengths are calculated by Equation (2) in accordance with RILEM TC 162TDF recommendations [24]:

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fR,i =

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3PR,i L

(2)

2b( D − N )2

where fR,i and PR,i respectively denote the residual strength and residual load corresponding to

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various mid-span deflections δR,i (i = 2, 3, and 4). The deflections δR,2, δR,3, and δR,4 are 1.31,

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2.15, 3.00 mm, respectively, according to the RILEM recommendations [24]. The test results of

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residual strengths from six specimens were averaged.

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The residual strengths obtained from the load-deflection responses as shown in Figures

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9(a)–9(c) observed the coupling between the loading rate and the N/D. At the loading rate of

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0.05 mm/min, the residual strengths increased with the N/D. However, with the increase in the

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loading rate, different tendencies were demonstrated. At the loading rate of 5.00 mm/min, fR,2

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and fR,3 decreased with the N/D, while the value of fR,4 was relatively sustained.

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0.2 0.3 0.4 Notch-to-depth ratio

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Residual flexural strength fR,2 (MPa)

40

1.25

0.5

Residual flexural strength fR,3 (MPa)

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0.05 2.50

5 0 0.1

(b)

0.50 5.00

1.25

0.2 0.3 0.4 Notch-to-depth ratio

273 274

Residual flexural strength fR, 4 (MPa)

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(c)

35 30 25 20 15 10

0.05 2.50

5 0 0.1

0.50 5.00

0.2 0.3 0.4 Notch-to-depth ratio

1.25 0.5

Figure 9. Residual flexural strengths versus N/D: (a) fR,2, (b) fR,3, and (c) fR,4.

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The results of flexural strength and residual strengths are summarized in Table 3. “Average”

277

represents the average value of replicate specimens; “C.O.V.” represents the coefficient of

278

variation, which is defined as the ratio of standard deviation to the average value. Six replicate

279

specimens were prepared for each type of specimen at each loading rate. Five specimens were

280

tested for S-12.5-0.05, and six specimens were tested for other cases. The maximum C.O.V. was

281

5.66% of the flexural strength (fu), 6.22% for the residual strength fR,2, 9.93% for the residual

282

strength fR,3, and 12.01% for the residual strength fR,4.

fR,3 Average C.O.V. (MPa) (%) 14.34 8.39 18.50 6.24 20.27 5.67 25.01 7.92 29.40 7.95 17.18 5.44 19.72 6.89 24.32 9.09 27.12 7.75 31.44 6.53 22.13 8.83 24.43 6.56 26.33 5.50 26.70 9.93 27.78 5.64

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fR,2 C.O.V. Average C.O.V. (%) (MPa) (%) 3.25 16.84 3.01 3.26 21.90 3.32 4.69 25.58 3.92 3.13 30.44 5.68 4.53 33.11 6.11 3.14 20.34 5.31 2.92 25.19 5.66 5.66 26.57 5.17 3.12 32.51 3.43 3.79 35.21 6.22 5.03 24.19 5.33 3.06 25.61 3.95 3.27 28.51 4.84 3.57 27.95 5.61 3.61 29.58 5.74

fR,4 Average C.O.V. (MPa) (%) 11.99 9.68 15.69 7.60 16.93 12.01 22.78 7.61 24.76 6.35 14.76 10.61 16.99 6.13 21.64 9.48 23.44 11.62 27.07 9.87 18.48 10.03 20.87 6.95 24.59 9.12 24.20 8.77 25.13 8.19

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S-12.5-0.05 S-12.5-0.50 S-12.5-1.25 S-12.5-2.50 S-12.5-5.00 S-25.0-0.05 S-25.0-0.50 S-25.0-1.25 S-25.0-2.50 S-25.0-5.00 S-37.5-0.05 S-37.5-0.50 S-37.5-1.25 S-37.5-2.50 S-37.5-5.00

fu Average (MPa) 23.59 28.00 31.43 35.64 38.36 22.01 25.88 29.53 34.22 37.64 25.22 28.68 30.65 30.06 29.90

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Table 3. Summary of flexural test results

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Using a linear fit on the curves in Figures 8(b) and 9(a)–9(c), the sensitivity of the flexural

286

or residual strengths to N/D is measured. The slope is denoted as kfS and mathematically

287

expressed using a differentiation equation as given in Equation (3):

288

k fS =

∂f ∂ ( N / D)

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where f represents the flexural or residual strengths. This equation is applicable to both the

290

flexural tensile strength and the residual strength. When kfS is equal to zero, the strength can be

291

considered to be independent on N/D.

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Noting that the changing rates of the flexural or residual strengths to N/D are different at

293

different loading rates, the changing rates kfS are plotted against loading rate, as shown in Figure

294

10. As the loading rate is increased from 0.05 to 5.00 mm/min, the values of kfS are shown to

295

decrease, indicating that kfS is associated with the loading rate. The relationship between kfS and

296

loading rate can be expressed using a differential equation as given in Equation (4): ∂k fS ∂2 f = ∂γ& ∂( N / D)∂γ&

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k fSL =

297

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(4)

where kfSL is the changing rate of kfS with regard to loading rate. A non-zero value of kfSL

299

indicates that the relationship between the flexural or residual strengths to N/D is associated with

300

the loading rate, i.e. coupling or interactive effect of loading rate and N/D on the flexural or

301

residual strengths. Taking the flexural tensile strength (fu) for example, as the loading rate

302

increased from 0.05 to 5.00 mm/min, the changing rate of the flexural strength to N/D was

303

changed from 4.96 to -25.78 MPa at a rate (kfSL) of -6.48 MPa·min/mm. A larger kfSL indicates a

304

more significant coupling effect.

fu fR,2 fR,3 fR,4

30 20

kfS (MPa)

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305 306

1

2 3 4 Loading rate (mm/min)

5

Figure 10. Coupling effects of loading rate and N/D on flexural strength and residual strengths. Page 17

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4.2. Analytical modelling of flexural responses A modelling approach to simulate the flexural responses based on a developed equivalent

309

tensile stress-strain relationship [34] is also presented. The modelling procedure correlates the

310

flexural load-deflection and tensile stress-strain responses through moment-curvature analysis.

311

Parameterized multi-linear compressive and tensile stress-strain constitutive laws are used in the

312

derivations. The compression law is assumed to be elastic-perfectly plastic governed by Young’s

313

modulus, compressive strength, and ultimate compressive strain. The tension law is characterized

314

by Young’s modulus, tensile strength, and constant residual tensile residual strength. The cross

315

sectional moment-curvature analysis is based on the assumption of plane section remains plane

316

such that the linear distribution of strain is obtained. The stress profiles in linear elastic and

317

cracked stages are therefore derived using the stress-strain constitutive laws. By solving the static

318

equilibrium equations along the cross section, neutral axis depth ratio is calculated which is

319

subsequently used to derive geometric parameters inlcuiding size of tension/compression zones

320

and the moment arm of each force term. The total bending moment is then computed as the

321

summation of the contributions from all force terms while the curvature is related to the strain

322

and neutral axis depth. In order to obtain the full range load-deflection responses, moment-area

323

method is employed to calculate the mid-span deflection at each load step. Details about the

324

constitutive laws and complete derivations can be found in the original work [34]. The procedure

325

has been applied to various types of materials including fiber-reinforced concrete reinforced by

326

polypropylene and glass fiber, textile reinforced concrete reinforced by polypropylene and

327

aramid textiles, high-performance fiber-reinforced concrete reinforced by steel fibers [35–37].

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Figures 11(a), (b), and (c) compare the simulated and experimental flexural stress-deflection

329

responses of the UHPC beams with different notch depths and loading rates. Compressive

330

strength of 158 MPa and modulus of elasticity of 50 GPa measured from compressive tests were Page 18

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used. The tensile material properties illustrated as multi-linear stress-strain diagrams were back-

332

calculated by fitting the experimental flexural responses for N/D of 1/6, as shown in Figure

333

11(d).The results show that in order to fit the experimental loading rate effect, the tensile and

334

residual strength of have to increase from 9.5 to 15.9 MPa (67%) and 3.1 to 5.2 MPa (67%),

335

respectively. The loading rate effects revealed by the increasing strength in tensile stress-strain

336

laws agree with the experimental investigations on tensile properties of UHPC under varying

337

strain rates [10, 13, 14]. The percentages of improvement are consistent with those of flexural

338

strength measured from experiment. It can be seen that the simulated responses agree well with

339

the experimental results for N/D of 1/6 and 1/3, while discrepancies are observed when N/D

340

increases to 1/2. As previously discussed, the loading rate effects on ultimate flexural strength

341

and residual strength are suppressed for the beams with notch of 37.5 mm compared to those

342

with shallow notches. Since the proposed modelling procedure correlates the tension and flexural

343

responses by means of moment-curvature analysis, the back-calculated stress-strain responses

344

from load-deflection curves at different rates are equivalent tension laws and consistent with the

345

loading rate sensitivity of experimental results (N/D=1/6). These may require further

346

investigation into the coupled effects of stress intensity due to deep notch and loading rate. One

347

possible explanation is the relationship between the stress rate and displacement rate. Since the

348

fracture tests in present study were controlled by displacement, the same displacement rates may

349

result in varying stress rates when the bending stiffness changes (due to changing notch depth).

350

The coupled loading rate and notch depth effects on failure mechanisms of UHPC are not

351

considered in the analytical model, which may require further investigation in future study. Since

352

the present tests were performed using displacement control, as the notch depth changes, the

353

bending stiffness of beam specimen decreases quadratically such that the stress rate obtained

354

along the cross section may not be proportional to the displacement rate. The size of the potential

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fracture process zone (FPZ) may be several times larger than what can be possibly developed in

356

the samples as the net section size decreases. This limits the full capability of the material for

357

samples with exceedingly small effective depths.

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On the other hand, the tensile stress-strain laws for simulations are found to overestimate the

359

average stress-strain responses from experiment. This phenomenon may be explained by the

360

loading rate effects and the different mechanisms between tension and flexure tests. While the

361

magnitudes of displacement rates are similar, the loading rates of bending and tension tests are in

362

terms of different parameters. The rate of actuator displacement does not necessarily lead to

363

same strain rates due to the different strain distributions in bending and uniaxial tension samples.

364

In addition, the tensile properties of UHPC was only evaluated at a rate of 0.05 mm/min in this

365

study, while the loading rate of bending test differed by two orders of magnitude. On the other

366

hand, studies have shown that use of uniaxial tension data underpredicts the flexural response for

367

this class of material [34, 35] by a significant margin. This is attributed to differences in the

368

stress distribution profiles of the two test methods. In the tension test, the entire volume of the

369

specimen is a potential zone for transverse crack initiation. Comparatively, in the flexural test,

370

only a small fraction of the tension region is subjected to an equivalent ultimate tensile stress.

371

This also explains the fact that the residual strength obtained from the model tension laws

372

underestimates the residual strength (fR,i) measured and calculated from experimental flexural

373

responses (see Table 3). The discrepancies are traced back to the inherent assumption of the

374

standard method by RILEM, which assumes that the neutral axis is at the centroid of the

375

specimen and the stress distribution is linear throughout at cracked stage. But the present model

376

predicts the movement of neutral axis towards compression zone and the reduced flexural

377

stiffness as flexural crack propagates. Details of comparisons and correlations between the two

378

methods can be found elsewhere [35, 36].

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(b)

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(c)

(d)

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Figure 11. Comparisons between simulated and experimental flexural stress-deflection

380

responses for the UHPC beams with N/D of: (a) 1/6, (b) 1/3, (c) 1/2, and the tension laws used

381

for simulation.

382

4.3. Fracture energy

383

Fracture energy, denoted by GF, is the energy necessary to create a crack of unit area and

384

can be calculated from the area under the load-deflection curves by Equation (6), to allow for the Page 21

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self-weight of specimen, in accordance with the RILEM [24] and JCI [25] recommendations. In

386

this study, the specimens were not deformed till complete failure [12], and each specimen was

387

loaded to reach a mid-span deflection of 5 mm, causing a slight underestimation of the fraction

388

energy. The fracture energy is calculated as: GF =

389

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W + mgδ b( D − N )

(6)

where GF is the fracture energy, W is the energy absorption, m is the total mass of the test beam,

391

g is the acceleration of gravity, δ is the mid-span deflection.

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The fracture energy results are plotted and compared in Figures 12(a) and 12(b), which

393

manifest the trends of how the fracture energy changes with the loading rate and N/D,

394

respectively. Figure 12(a) indicates that the fracture energy increased nonlinearly with the

395

loading rate from 0.05 to 5.00 mm/min at a decreasing rate. Figure 12(b) indicates the fracture

396

energy approximately linearly changed with the N/D.

20 15 10 5 0

1

2 3 4 Loading rate (mm/min)

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397 398

(a)

1/3

0.05 0.50 1.25 2.50 5.00

25

1/2

Fracture energy (N/mm)

1/6

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Fracture energy (N/mm)

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20 15 10 5 0

5

0.1

(b)

0.2

0.3 0.4 Notch-to-depth ratio

0.5

Figure 12. Fracture energy versus: (a) loading rate, (b) N/D.

399

As the loading rate increased from 0.05 to 5.00 mm/min, the fracture energy at N/C of 1/2,

400

1/3, and 1/6 increased by 23%, 64%, and 87%, respectively, indicating that the relationship

401

between the fracture energy and loading rate is dependent on N/D. On the other hand, as the N/D

402

is reduced from 1/2 to 1/6, the fracture energy is increased by 23% at a loading rate of 0.05

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403

mm/min, and by 87% at a loading rate of 5.00 mm/min, indicating that the relationship between

404

the fracture energy and N/D is also dependent on loading rate. The slopes of the variations of the fracture energy versus N/D shown in Figure 12(b) are

406

plotted in Figure 13 with the increase of loading rate. The values of the slopes represent the

407

changing rate or sensitivity of the fracture energy to the change of N/D. As the loading rate was

408

increased from 0.05 to 5.00 mm/min, the sensitivity of the fracture energy to the N/D changed

409

from -6.36 to -29.43 N/mm. The negative sign means that the fracture energy decreases with N/D.

410

The absolute value represents the magnitude of the sensitivity, and a larger absolute value

411

indicates a higher sensitivity of the fracture energy to N/D. Figure 13 shows that the magnitude

412

of sensitivity of the fracture energy to N/D increases as the loading rate increases.

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-10 -15 -20

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Changing rate of fracture energy to N/D (N/mm)

-5

-25 -30

0

2 3 4 Loading rate (mm/min)

5

Figure 13. Coupling effect of loading rate and N/D on fracture energy. 4.4. Regression and analysis of variance

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Based on the investigation above, the flexural strength and the fracture energy are found to

417

be associated with the loading rate γ& and N/D, as well as their coupling effects. Based on the

418

experimental results, regression analyses are conducted to formulate the flexural strength and the

419

fracture energy. In this study, for the sake of simplicity of formulation, the flexural strength and

420

fracture energy are assumed to vary linearly with the loading rate and the N/D, as shown in

421

Equation (7). Page 23

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422

f u = af0 + af1 ( N / D) + af2γ& + af3 ( N / D)γ& GF = bG0 + bG1 ( N / D) + bG2γ& + bG3 ( N / D)γ&

(7)

where, af0 and bG0 are respectively constant terms of the flexural tensile strength and fracture

424

energy; af1, af2, and af3 represent the sensitivity coefficients of the flexural strength to N/D,

425

loading rate, and coupling term of N/D and loading rate, respectively; bG0, bG1, bG2, and bG3

426

represent the sensitivity coefficients of the fracture energy to N/D, loading rate, and coupling

427

term of N/D and loading rate, respectively. These coefficients are determined by regression

428

analysis of testing data using the least square method.

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The coefficients for the flexural strength are: af0 = 24.66 MPa, af1 = 4.42 MPa, af2 = 4.24

430

MPa·min/mm, and af3 = -6.48 MPa·min/mm. The coefficients for the fracture energy are: bG0 =

431

14.27 N/mm, bG1 = -9.07 N/mm, bG2 = 2.63 N·min/mm2, and bG3 = -4.57 N·min/mm2. Then,

432

Equation (7) can be rewritten as:

434

f u = 24.66 + 4.42( N / D) + 4.24γ& − 6.48( N / D)γ& GF = 14.23 − 9.07( N / D) + 2.63γ& − 4.57( N / D)γ&

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(8)

where, 1/6 ≤ N/D ≤ 1/2, and 0.05 mm/min ≤ γ& ≤ 5.00 mm/min. The positive values of af1 and af2 indicate that the flexural strength increases with N/D and

436

loading rate. The negative value of af3 indicates that the sensitivity of flexural strength to N/D

437

decreases with loading rate, and vice versa. The negative value of bG1 and the positive value of

438

bG2 indicate that the fracture energy decreases with N/D and increases with loading rate. The

439

negative value of bG3 indicates that the sensitivity of fracture energy to N/D decreases with

440

loading rate, and vice versa.

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The regression analysis results were examined by analysis of variance (ANOVA) using the

442

F-test criteria [38] with a significance level of 5% (α = 0.05). For the regression analysis of the

443

flexural tensile strength, the coefficient of determination (R2) is 0.795, and the standard error is Page 24

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2.48 MPa. The P-value of F-significance is 4.2 × 10-4, which is far smaller than the significance

445

level 0.05, indicating the regressed formula has high significance. For the regression analysis of

446

fracture energy, the coefficient of determination (R2) is 0.909, and the standard error is 1.21

447

N/mm. The P-value of F-significance is 5.2×10-6, which is far smaller than the significance level

448

0.05, indicating the regressed formula has high significance.

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The formulae in Equation (8) are plotted in design charts to facilitate engineers to implement

450

in engineering practice, as shown in Figure 14. Since the formulae were obtained from the test

451

results of specimens with N/D in the range of 1/6 to 1/2 at loading rates in the range of 0.05 to 5

452

mm/min, interpolation and extrapolation of data points should be within the specified ranges, to

453

ensure the validity of the design charts.

40 35 30 25

457

2 3 Loading rate (mm/min)

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1

4

5

N/D=1/6 N/D=1/4 N/D=1/2

20

N/D=1/5 N/D=1/3

15 10 5 0

(b)

1

2 3 4 Loading rate (mm/min)

5

Figure 14. Design charts: (a) flexural strength; (b) fracture energy.

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N/D=1/5 N/D=1/3

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N/D=1/6 N/D=1/4 N/D=1/2

Fracture energy (N/mm)

25

45

(a)

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5. Conclusions

458

Based on the above investigations, main findings are summarized as follows.

459

(1) In general, the flexural strength and residual strengths of UHPC notched beam increase

460

with loading rate and the notch-to-depth ratio. The fracture energy of UHPC notched beam

461

increases with loading rate and decreases with notch-to-depth ratio. The effects of the notch-to-

462

depth ratio are associated with the loading rate. For a notch-to-depth ratio in the range of 1/6 to Page 25

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1/2, the effect of notch-to-depth ratio on the flexural strength is less than 10% at a deflection rate

464

up to 1.25 mm/min. However, as the deflection rate is increased to 2.5 mm/min, the effect of

465

notch-to-depth ratio on the flexural strength of the UHPC mixture is increased to 30%. As the

466

deflection rate is increased from 0.05 to 5.00 mm/min, the fracture energy is increased by 87%

467

for the beam with a notch-to-depth ratio of 1/6, and by 23% for the beam with a notch-to-depth

468

ratio of 1/2.

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(2) Based on the experimental results, the flexural tensile strength and fracture energy of the

470

UHPC notched beams are formulated with the loading rate and the notch-to-depth ratio by

471

regression analyses. The regressed formulae of the flexural tensile strength and fracture energy

472

are evaluated by analysis of variance, and reasonable significance of the regressed formulae are

473

corroborated. The regressed formulae have reasonable quality and are plotted in design charts

474

that can be conveniently used in engineering practice to predict the flexural properties of UHPC

475

that is subjected to different loading rates or used in structures with different dimensions.

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(3) The UHPC mixture proportioned with high-volume fly ash and 2% blend steel fibers

477

demonstrate high ductility in tension and strain-hardening behavior. The steel fibers can bridge

478

cracks and allow the UHPC to carry sustained load and have multiple cracks. The post-cracking

479

performance are primarily dependent on the fiber pull-out mechanisms, which are characterized

480

using single fiber pull-out tests. After debonding is initiated at the interface between the steel

481

fibers and the UHPC matrix, the interface can carry substantial load before the steel fiber is

482

completely pulled out. The bond strength and the post-debonding behaviors are associated with

483

the loading rate.

484

Acknowledgement

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This study was funded by the RE-CAST Tier – 1 University Transportation Center [grant

486

number DTRT13-G-UTC45] and the Energy Consortium Research Center at Missouri S&T Page 26

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[grant number SMR-1406-09]. Barzin Mobasher and Yiming yao sincerely acknowledge the

488

Arizona Department of Transportation (ADOT) for partial funding of this research [grant number:

489

SPR 745].

490

References

491

[1] Yoo DY, Banthia N. Mechanical properties of ultra-high-performance fiber-reinforced

492

concrete: A review. Cem Concr Compos 2016;73:267-280.

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[2] Meng W, Khayat KH. Improving flexural performance of ultra-high-performance concrete by

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