Experimental investigation on the shear capacity of prestressed concrete beams using digital image correlation

Engineering Structures 82 (2015) 82–92

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Experimental investigation on the shear capacity of prestressed concrete beams using digital image correlation K. De Wilder a,⇑, P. Lava b, D. Debruyne b, Y. Wang b, G. De Roeck a, L. Vandewalle a a b

KU Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, 3001 Leuven, Belgium KU Leuven, Department of Metallurgy and Materials Engineering, Kasteelpark Arenberg 44, 3001 Leuven, Belgium

a r t i c l e

i n f o

Article history: Received 28 October 2013 Revised 16 October 2014 Accepted 17 October 2014

Keywords: Mechanical behavior Prestressed concrete Shear Digital image correlation

a b s t r a c t Despite more than a century of continuous effort, shear still remains one of the few areas of research into fundamentals of the behavior of concrete structures where dispute remains amongst researchers about the mechanisms that enable the force flow through a concrete member and across cracks. Due to our incomplete understanding and the brittle failure modes associated with shear, current codes of practice tend to propose highly conservative shear design provisions. This paper investigates the shear capacity and mechanical behavior of prestressed concrete beams. The results of 9 full-scale I-shaped prestressed concrete beams subjected to a four-point bending test until failure are presented. Two stereo-vision digital image correlation (DIC) systems were used to discretely measure three-dimensional displacements and in-plane deformations in both zones where a shear force exists. A numerical technique has been adopted to generate optimized patterns for DIC and the resulting speckle pattern was applied onto each specimen using a stencil printing technique. Using the sectional shear design procedure found in Eurocode 2, a severe underestimation of the experimentally observed shear capacity was found. Direct strut action was the main bearing mechanism of the reported prestressed concrete members with shear reinforcement whereas the presented specimens without shear reinforcement failed when the principal tensile stress reached the tensile strength limit. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Despite more than a century of continuous research effort [1], shear is one of a few areas into fundamentals of the behavior of concrete structures where dispute remains amongst researchers about the mechanisms that enable the force flow through a concrete member and across cracks. The main reason for this contention can be attributed to the complexity of the shear phenomenon in structural concrete members. Firstly, various interrelated shear transfer mechanisms contribute to the overall shear capacity of a structural concrete member. The 1973 ASCE-ACI Committee 426 report [8] identified four different mechanisms of shear transfer apart from the shear reinforcement contribution: (a) shear stresses in uncracked concrete, i.e. compressive zone; (b) interface shear transfer (also referred to as aggregate interlock or crack friction); (c) dowel action of the longitudinal reinforcement; and (d) arch action. Since that report was issued, a new mechanism was identified, namely (e) residual tensile stresses transmitted directly across cracks. Secondly, many parameters influence each shear transfer ⇑ Corresponding author. Tel.: +32 16321987. E-mail address: [email protected] (K. De Wilder). http://dx.doi.org/10.1016/j.engstruct.2014.10.034 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

mechanism separately. Due to the complexity of the shear phenomenon, a generally accepted theoretical basis to model shear in structural concrete members is still absent within the research community. Therefore, a multitude of analytical models dealing with shear in structural concrete members can be found in literature. Based on the crack pattern typically observed during beam tests, Ritter [17] idealized the flow of forces in a cracked structural concrete member by means of a parallel chord truss consisting of compressive diagonals inclined at 45° and vertical tension ties, as illustrated in Fig. 1. This approach has ever since been referred to as the truss model approach and was later extended to the case of torsion in concrete members by Mörsch [11]. The necessary amount of shear reinforcement per unit length follows from the calculation of the axial force in the vertical tension ties. The truss analogy is on one hand easy to understand and highly didactic but on the other hand a very simple representation of the actual structural behavior. It is thus clear that more refined models are needed in order to optimize and economize the overall structural design of reinforced and prestressed concrete beams. An excellent overview of different theoretical approaches to shear in concrete members is given in the review paper by the Joint

K. De Wilder et al. / Engineering Structures 82 (2015) 82–92

Fig. 1. Truss analogy as proposed by Ritter [17].

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each specimen except for beams B103, B106 and B109. Shear reinforcement consisted of single legged stirrups with a nominal diameter of 6 mm and a center-to-center distance equal to 150 mm, leading to a shear reinforcement ratio qw equal to 0.0027 which is approximately 2.5 times the required minimum shear reinforcement ratio according to EC 2 [3]. The chosen shear reinforcement ratio allowed for a safe margin to avoid failure immediately after the onset of diagonal cracking and failure due to bending. Splitting reinforcement was provided at both ends of each specimen with a nominal diameter of 8 mm and a centerto-center distance equal to 50 mm. The geometry and reinforcement layout of all specimens are shown in Fig. 2(a–f). 2.2. Materials

ACI-ASCE Committee 445 [16]. However, up to date, none of the existing theoretical approaches for shear in structural concrete elements has been able to fully explain the mechanics behind the problem of shear failure. The on-going debate in the literature on how to deal with shear in concrete members is also reflected by the code provisions. Current codes of practice all recommend very different approaches that result in different design shear capacities and take parameters affecting the shear capacity into account in a different way. Due to our incomplete understanding and the brittle modes of failure associated with shear, current international codes of practice tend to propose highly conservative shear design provisions. The model found in Eurocode 2 [3,13] is derived from the aforementioned truss analogy and is often referred to as the variable angle truss model (VATM). This model relates the entire shear capacity of a structural concrete member with shear reinforcement to the amount of provided shear reinforcement per unit length. Hence, for members with shear reinforcement, the inherent capacity to resist shear is ignored. This is particularly disadvantageous for prestressed concrete elements. This paper therefore presents the results of an extensive experimental campaign consisting of 9 full-scale shear-critical prestressed concrete beams. In the first section, the experimental program is elaborated. The specimen properties and the experimental setup are reported. Specific attention is paid to the use of the stereo-vision digital image correlation (DIC) technique as an optical-numerical full-field measurement technique for full-scale tests. In the second part of this paper, the experimental results are presented and compared to analytical predictions of the shear capacity obtained using the current shear design provisions found in Eurocode 2. Based on the DIC measurements, an investigation is performed of the shear carrying mechanisms of the reported test beams.

2. Experimental research 2.1. Specimen design This paper presents the results of 9 factory-made prestressed concrete I-shaped beams. The beams were labeled with the descriptive letter B followed by a number ranging from 101 to 109. Each specimen was 7000 mm long and 630 mm high with a flange width equal to 240 mm. The web of each specimen was 70 mm wide. Specimens B101–B106 were provided with eight 7-wire strands at the bottom whereas specimens B107–B109 were prestressed at the bottom using four 7-wire strands. The nominal strand diameter was equal to 12.5 mm. To counteract the induced bending moment due to the eccentric prestressing force and thus avoid cracking at the top of the beam, each specimen was also prestressed with two 7-wire strands at the top with a nominal strand diameter of 9.3 mm. Shear reinforcement was placed in

The concrete mixture was designed to have a characteristic cylindrical compressive strength equal to 50 MPa. Cement was specified as CEM I 52.5R, whereas the coarse aggregate consisted of 12 mm maximum size limestone gravel. Limestone filler and a high-range water reducer were also used. The concrete mixture composition is listed in Table 1. Tensile tests were performed on both shear and splitting reinforcement bars to determine the modulus of elasticity Es , the yield and ultimate stress, f ym respectively f tm , and the strain at failure su . The same characteristics of the prestressing strands were taken from the manufacturer. The results are summarized in Table 2. 2.3. Specimen construction Concrete mixtures were made in volumes of 2 m3 enabling the construction of three specimens with one mixture. Together with each beam, cubes (sides: 150 mm), cylinders (height/diameter: 300 mm/150 mm) and prisms (150  150  600 mm3) were cast to determine the mean compressive strength on cubes and cylinders, f cm;cube respectively f cm , the mean secant modulus of elasticity Ecm and the mean flexural tensile strength f ctm;fl at the day of testing. A summary of the results per specimen group is given in Table 3. The mean mixture density qm and the age of each specimen at the day of testing is also indicated in the aforementioned Table 3. From the results presented in Table 3, it can firstly be seen that a relatively large scatter was found on both the experimentally measured cube and cylinder compressive strengths. Moreover, it can be seen that the cylinder compressive strength is lower than the cube compressive strength for specimen set B101–B103 whereas the opposite is true for the remaining sets B104–B106 and B107–B109. In general, the cube compressive strength is higher than the cylinder compressive strength due to the confinement originating from friction stresses between the specimen and the testing device. For standard size concrete specimens, this effect tends to be larger for cubes than for cylinders, hence resulting in higher cube compressive strengths. However, for high strength concrete, which is the case in the reported study, the effect of confinement becomes less effective [12]. The day after casting, demountable mechanical strain gauges (DEMEC) were glued onto the concrete side surface to determine the immediate and time-dependent stress losses in the prestressing reinforcement. The location of the aforementioned DEMECpoints was already presented in Fig. 2(a–b). Each strand of specimens B101–B103 and B107–B109 was given an initial prestrain equal to 0.0075 mm/mm (rpm;0 ¼ 1488 MPa) whereas each strand of specimens B104–B106 was given an initial prestrain equal to 0.0038 mm/mm (rpm;0 ¼ 750 MPa). While it is uncommon in the industry to reduce stress levels below the allowable, the stresses were varied to isolate the effect of varying the prestressing force while keeping the longitudinal reinforcement ratio ql constant. At the day of testing, it was found that the stress losses in

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Fig. 2. Geometry, splitting and shear reinforcement (if applicable) layout of (a) specimens B101–B102, B104–B105 and B107–B108; (b) specimens B103, B106 and B109; cross section and reinforcement layout of (c) specimens B101–B102 and B104–B105; (d) specimens B103 and B106; (e) specimens B107–B108; (f) specimen B109 (location of DEMEC-points indicated with ).

Table 2 Reinforcement properties. Table 1 Concrete mixture composition.

Reinf. type

Material

Amount (kg/m3)

CEM I 52.5R Limestone gravel 2/12 Sand 0/2 Water Limestone filler High-range water reducer

380.0 1075.0 652.0 155.0 150.0 6.0

Type

Top prestress. reinf. Bot. prestress. reinf. Shear reinf. Splitting reinf.

7-wire 7-wire Cold worked Cold worked

dp a

Ep

f p0:1m

f pm

ds (mm)

Es (GPa)

f ym (MPa)

f tm (MPa)

9.3 12.5 6.0 8.0

198.0 198.0 210.0 203.0

1737 1737 608 542

1930 1930 636 603

pu su (%) 5.20 5.20 2.73 5.97

a Subscript p and s denote properties of prestressing respectively conventional reinforcement types.

Table 3 Concrete material properties for reported specimens. Specimens B101–B103 B104–B106 B107–B109 a b

Ecm (GPa)

qm (kg/m3)

f cm;cube (MPa) (#a, sb)

f cm (MPa) (#, s)

(#, s)

f ctm;fl (MPa) (#, s)

(#, s)

87.1 (8, 6.8) 82.8 (9, 10.5) 74.6 (9, 9.6)

77.5 (5, 10.7) 88.9 (6, 10.2) 89.3 (6, 14.2)

43.4 (3, 2.3) 43.5 (6, 8.0) 42.2 (6, 4.6)

5.8 (6, 0.6) 6.5 (6, 1.0) 5.7 (6, 1.1)

2399 (6, 25.3) 2369 (6, 15.8) 2376 (6, 7.5)

Number of tested specimens. Standard deviation.

Age (days) 28-233-393 412-404-407 428-424-412

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the bottom and top prestressed reinforcement were approximately equal to 20% respectively 10% of the initial stress levels. These stress level values will be adopted further on for the reported calculations.

(shear span-to- effective depth ratios 62.5) while remaining a safe margin to avoid a bending failure mode.

2.4. Experimental setup

Apart from linear variable differential transformers (LVDT), refer to Fig. 3(a), full-field three-dimensional displacements were recorded using two stereo-vision digital image correlation (DIC) systems for specimens B103–B109. As an optical-numerical fullfield measurement technique, DIC has proven to be an ideal tool for a wide range of applications, including the identification of the mechanical material behavior through inverse modeling [4,5], structural health monitoring [18] and the study of the deformation characteristics of a wide range of materials [7,20]. The basic principle behind this technique is to calculate the displacements on the surface of an object by taking images of a random speckle pattern in undeformed and deformed state. There are three main steps in the DIC method: (1) capture images, (2) correlation process and (3) post-processing phase. In the reported study, both zones where a shear force occurs, were investigated by using two stereo-vision DIC systems. The size of each zone was approximately 1500 mm by 630 mm, refer to Fig. 3(a). Each system consisted of two 8-bit CCD cameras (AVT F201B; 1628 pixels by 1236 pixels resolution) with lenses having a focal length equal to 12 mm. Both cameras were mounted on a rigid tripod. The cameras were located at a perpendicular distance of approximately 2700 mm from the web of the beam, as shown in Fig. 3(b–c). To ensure good lighting conditions and small exposure times, two 500 W quartz iodine lamps were placed per investigated zone, as depicted in Fig. 3(b–c). The image acquisition rate of each camera was equal to 2 Hz with an exposure time of 20 ms. All images were transferred to a desktop computer and synchronized with the analogue data (applied force and corresponding displacement) of the hydraulic press. A subset-based method is applied to correlate two speckle patterns using the MatchID software package [10]. This method considers a pixel and its neighborhood (subset size ð2M þ 1Þ pixels by ð2M þ 1Þ pixels) in the undeformed image f ðxi ; yj Þ and searches the same subset in the deformed image gðxi ; yj Þ at a certain time t (i.e. load step) by adopting an optimization routine for a degree of similarity expressed via a correlation criterion. Here, the ZeroNormalized Sum of Squared Differences (ZNSSD) correlation criterion is adopted in combination with a bicubic spline subpixel interpolation scheme. The size of the subset can be set prior to the evaluation. In this work, the dimensions of each subset were 27 by 27 pixels (stepsize: 3 px) where each pixel has the physical dimension of approximately 1 mm. The subset size can be chosen freely prior to the evaluation. The subset size should on one hand be chosen small enough to allow for a reasonable linear interpolation of the displacement field but on the other hand large enough to avoid

All specimens were subjected to a four-point bending test as shown in Fig. 3(a). The tests were carried out in load-control using a hydraulic press (Instron, maximum capacity of 2.5 MN). The force from the hydraulic press was converted to two point loads using a steel transfer beam (HEB 400). The total load was monotonically increased at 0.250 kN/s (i.e. shear force rate V_ ¼ dV ¼ 0:125 kN/s) dt until failure occurs. The distance between the support points was 5000 mm. Referring to Fig. 2(a–b), the distance outside the support points was therefore equal to 1000 mm. This setup firstly enables the authors to study shear outside the length needed for the prestressing force to gradually develop over the member’s height and secondly prevents failure due to debonding of the prestressing strands. The shear span a, i.e. the distance between the support point and the nearest point load, is equal to 1600 mm for specimens B101, B104 and B107 and 2000 mm for the remaining specimens. An overview of the investigated parameters per specimen is given in Table 4. With the given experimental setup, shear span-to-effective depth ratios between 2.91 and 3.91 are obtained, refer to Table 4. In this way, the authors avoid less useful setups

Fig. 3. Schematic representation of experimental setup for DIC measurements: (a) front view; (b) side view; and (c) top view (note: units in millimeter).

2.5. Digital image correlation (DIC)

Table 4 Overview of investigated parameters per specimen.

a b

Specimen

d (mm)

rpm;0 (MPa)

a (mm)

a d

B101 B102 B103 B104 B105 B106 B107 B108 B109

511 511 511 511 511 511 550 550 550

1488 1488 1488 750 750 750 1488 1488 1488

1600 2000 2000 1600 2000 2000 1600 2000 2000

3.13 3.91 3.91 3.13 3.91 3.91 2.91 3.64 3.64

(–)

ql ¼ bAwslda (–)

qw ¼ bAwswsb (–)

0.0208 0.0208 0.0208 0.0208 0.0208 0.0208 0.0097 0.0097 0.0097

0.0027 0.0027 0 0.0027 0.0027 0 0.0027 0.0027 0

Longitudinal reinforcement ratio with Asl : the area of longitudinal reinforcement, bw : the web width and d: the effective depth. Shear reinforcement ratio with Asw : the area of shear reinforcement and s: the shear reinforcement spacing.

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correlation problems due to non-uniqueness of the subset information. The size of the chosen subset followed from a parameter study of the influence of the subset size on the measurement displacement resolution which can be related to the standard deviation of the displacement components. The results, obtained from an autocorrelation in undeformed state, are summarized in Fig. 4(a–c) for specimen B104. From Fig. 4(a–c), it can be clearly seen that smaller subset sizes severely increase the standard deviation of the displacement components whereas larger subset sizes only marginally decrease the standard deviation of the displacement components. As the subset size increases, any accuracy that is gained from better correlation has to be weighed against the localized averaging of displacement behavior that comes from using such large subsets. In this work, specific attention has been paid to the quality and reproducibility of the adopted speckle pattern. To study the influence of speckle patterns on the displacement measurement accuracy of DIC as well as to assess the quality of a speckle pattern, various parameters, including both local and global parameters, have been proposed [19]. Local parameters, such as the subset entropy proposed by Sun et al. [19] and the sum of square of subset intensity gradient (SSSIG) proposed by Pan et al. [15] both assess the local speckle pattern quality of each subset separately. However, for most speckle patterns used in DIC, the speckle granules contained in the image are evenly distributed, thus the local parameters computed for various subsets are normally of very little difference. Due to this reason, a global parameter rather than the aforementioned local parameter will be more convenient and useful for practical use. Pan et al. [14] therefore propose to use the mean intensity gradient to evaluate the quality of the entire speckle pattern. Recently, Crammond et al. [6] proposed a robust morphological methodology using edge detection to evaluate the physical properties of different speckle patterns. The main disadvantage of the aforementioned procedures is that a speckle pattern is a priori required to evaluate its suitability for DIC measurements. Therefore, to be able to generate suitable DIC speckle patterns, a

numerical technique based on Fourier transform and morphological image processing, as recently proposed by Bossuyt [2], was adopted. The resulting patterns are optimized on one hand for sensitivity (i.e. measurement precision) and on the other hand for correlation robustness (i.e. measurement accuracy). The same pattern is then applied onto each specimen where the DIC technique has been used, i.e. specimens B103–B109, using a stencil printing technique. The printed speckles have a precalculated oversampling of at least 5 pixels in order to avoid aliasing effects in the obtained results. This is of crucial importance when working with displacement and strain fields of this small magnitude. Here, a speckle size of at least 5 mm is required (considering the field of view, the camera resolution and the aforementioned oversampling) which is nearly impossible to obtain using traditional speckling techniques (e.g. spray painting). Fig. 5(a) depicts a detail of the numerically generated pattern whereas Fig. 5(b) shows the same pattern detail applied onto specimen B105. Fig. 5(c) shows the experimental setup of beam B105 indicating both speckled zones whereas Fig. 5(d) illustrates the adopted area of interest (AOI). Since full-field displacement data is available, strain data can be derived from the aforementioned displacement data. Therefore, the displacement data is smoothed over a certain zone to damp out the effect of noise and local uncertainties. This is a commonly adopted method during the process of strain determination [9]. In this step of the analysis, a bilinear or quadratic plane is fitted through the displacement values in the points around the center of the strain window (CSW) where displacements are available. This analytical approximation makes the calculation of the fullfield strain information straightforward. The number of points that are taken into account is referred to as the strain window size (SWS). Here, a relatively large SWS was used since strains were to be determined of a highly heterogeneous material which also exhibited a profound cracking pattern. An overview of the measurement information for the performed DIC tests per specimen is given in Table 5.

Fig. 4. Parameter study of the influence of the subset size on the displacement resolution for specimen B104 with indication of chosen subset size (vertical dotted line): (a) horizontal in-plane displacement ux ; (b) vertical in-plane displacement uy ; and (c) out-of-plane displacement uz .

Fig. 5. (a) Detail of numerically generated speckle pattern; (b) detail of speckle pattern shown in (a) applied onto specimen B105; (c) experimental setup of specimen B105 showing both speckled zones; (d) indication of area of interest (AOI) and subset size in red. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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K. De Wilder et al. / Engineering Structures 82 (2015) 82–92 Table 5 Measurement information for the DIC tests on specimens B103–B109.

a b c d

Unit

B103

B104

B105

B106

B107

B108

B109

Subset Step Measurement pointsa Temporal resolution Camera distance

(px) (px) (–) (fps) (mm)

27 3 68327 2 2700

27 3 67482 2 2700

27 3 67819 2 2700

27 3 70290 2 2700

27 3 68843 2 2700

27 3 68868 2 2700

27 3 75864 2 2700

Displacement Spatial resolutionb In-plane resolution Out-of-plane resolution

(mm) (mm) (mm)

31.2 0.018 0.125

30.9 0.019 0.120

29.9 0.034 0.165

29.5 0.027 0.167

29.9 0.020 0.147

30.8 0.021 0.149

28.4 0.023 0.152

Strain Smoothing method Spatial resolutiond Resolution

(–) (mm) (lm/m)

b.q.c 176.8 97

b.q. 175.1 102

b.q. 169.4 164

b.q. 167.2 169

b.q. 169.4 154

b.q. 174.5 150

b.q. 160.9 124

Number of data points in one measurement field. Physical dimension of subset. Bilinear quadrilateral. Physical dimension of strain window.

3. Results and discussion In this subsection, the experimentally observed structural behavior is reported. Fig. 6(a–c) depict the measured load-displacement response curves for all investigated specimens. The

onset of diagonal (M) and bending () cracking is also shown in Fig. 6(a–c). All specimens but beam B107 and B108 exhibited a brittle shear failure mode. The aforementioned beams exhibited severe diagonal cracking leading to excessive yielding and rupture of the shear reinforcement. Crushing of the diagonal struts was not

Fig. 6. Load–displacement response curves measured at 1200 mm from the support point for specimens B101–B102 (obtained from LVDT data) and B103–B109 (obtained from DIC data) with indication of first bending () and diagonal (M) cracks.

Fig. 7. Typical experimentally observed failure modes: (a) diagonal tension failure mode of specimen B102; (b) bending failure mode of specimen B107; (c) detail of diagonal tension failure mode of specimen B104; and (d) detail of ruptured prestressing strands of specimen B107.

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(a) B101

(b) B102

(c) B103

(d) B104

(e) B105

(f) B106

(g) B107

(h) B108

(i) B109

Fig. 8. Experimentally observed cracking pattern for the reported test specimens.

observed. Beams B107 and B108 exhibited a ductile bending failure mode. The observed failure modes are shown in Fig. 7(a–d). The cracking pattern prior to failure is depicted in Fig. 8(a–i) for all reported specimens. The critical crack, i.e. the crack where eventually failure was initiated, is depicted in bold. From Fig. 6(a–c) it can firstly be seen that, as expected, the stiffness in the elastic regime is comparable for all specimens with the same shear span. Indeed, prior to cracking, the load–displacement response for beams with the same shear span is entirely governed by the bending stiffness EI. Referring to Table 3 and Fig. 2, it can be concluded that the bending stiffness is comparable for all specimens since the influence of the amount of longitudinal reinforcement on the second moment of area (I) is rather limited. Secondly, Fig. 6(a–c) show that all test beams but specimen B101 exhibited bending cracks prior to the onset of diagonal cracking. The load at which diagonal cracking occurs is function of the amount of prestressing and the tensile strength respectively. This corresponds well with the experimental data if the onset of diagonal cracking of specimens B101–B103 and B104–B109 are compared, refer to Fig. 6(a–c). After cracking, beams with shear reinforcement failing due to shear (B101, B104 and B102, B105) exhibit a similar post-cracking stiffness eventually resulting in a brittle failure. Beams B107 and B108 however, are on one hand characterized by a lower post-cracking stiffness but on the other hand by a more ductile behavior in comparison to specimens B104 respectively B105. Finally, Fig. 6(c) clearly shows that the overall behavior of beams B103, B106 and B109 remains nearly perfectly linear-elastic. Since no shear reinforcement is provided, no redistribution of internal forces is possible after the occurrence of the first diagonal crack. The width of the critical shear crack, refer to Fig. 8(a–i), measured at mid-depth and perpendicular to the crack face is shown in Fig. 9(a and c) as a function of the applied shear force.

Fig. 9(b and d) show the measured vertical strain y (derived from the displacement data, base length L0 ¼ 100 mm) of the critical shear crack at mid-depth. No data is available for specimen B102. Specimens B107 and B108 failed due to bending and as such do not exhibit a critical shear crack. The inclined crack with a maximum crack width is therefore analyzed. Beams B103, B106 and B109 failed immediately after the onset of diagonal cracking and are therefore not included in Fig. 9(a–d). From Fig. 9(a and c) it can be clearly seen that a similar maximum crack width w? was

Fig. 9. Crack width w? (a and c) perpendicular to the crack face (with indication of onset of yielding of shear reinforcement ) and vertical strain y (b and d) of the critical shear crack at mid-depth of specimens B101, B104–B105 and B107–B108 (note: sy represents the yield strain).

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observed for specimens B101, B104 and B105. At failure, the corresponding vertical strain y of the aforementioned test beams is nearly equal to the ultimate strain of the shear reinforcement as determined from uni-axial tensile tests, reported in Table 2. This clearly indicates that these specimens failed due to diagonal tension (yielding and rupture of the shear reinforcement). The maximum observed crack width, and hence also the vertical strain, of specimens B107–B108 is relatively small in comparison to the other reported beams in Fig. 9. At failure, the crack width was found to be equal to 0.70 mm and 0.63 mm for specimens B107 respectively B108. Table 6 summarizes the experimentally observed shear failure load and failure mode for each specimen. The measured failure load can be compared to analytical calculations using the shear design procedure found in Eurocode 2 (EC 2) [3,13]. In the case of prismatic structural concrete elements with vertical shear reinforcement bars, the design value of the shear capacity should be determined using Eq. (1).

V Rd ¼ min



V Rd;s

ð1Þ

V Rd;max

In Eq. (1), V Rd;s and V Rd;max represent the shear force required to obtain yielding of the shear reinforcement and the shear force required to obtain crushing of the compressive struts respectively, which are to be determined using Eqs. (2) and (3) respectively.

V Rd;s ¼ V Rd;max

Asw zfywd cot h s acw m1 f cd bw z cot h ¼ 1 þ cot2 h

ð2Þ ð3Þ

In Eq. (2), Asw denotes the area of shear reinforcement per unit s length, z is the internal lever arm equal to 0:9d and f ywd is the design value of the yield strength of the shear reinforcement. In Eq. (3), acw is a factor taking into account the stress distribution of the compressive chord depending on the amount of prestressing. For the   r reported specimens, the value of acw is equal to 1 þ f cp , with rcp cd

representing the average normal stress due to the applied prestressing force and f cd denoting the design value of the characteristic cylindrical compressive strength f ck . The design value f cd can be derived from the mean cylindrical compressive strength as

f cm 8

cc

[3] with cc the partial safety factor for concrete equal to 1.5. The factor m1 in Eq. (3) is a reduction factor for the compressive strength of cracked concrete. Indeed, in cracked concrete, large tensile strains

perpendicular to the principal compressive direction significantly reduce the concrete compressive strength. According to EC 2 [3], the value of m1 is equal to 0:6½1  f ck =250. Finally, bw is the web width and h denotes the angle between the horizontal axis and the inclined compressive stresses and can be chosen freely between 1 6 cot h 6 2:5. In practice, this maximum value will always be chosen for the analysis or design of a prestressed structural concrete member. In the Belgian national application document NBN EN 1992-1-1 ANB [13], a purely empirical extension is proposed to the aforementioned limits of cot h which allows the engineer to choose an angle h so that 1 6 cot h 6 cot hmax . The maximum value of cot h can be obtained using Eq. (4) [13].

cot hmax ¼

! 0:15rcp bw d 2 þ Asw 3 zfywd s

ð4Þ

For highly prestressed concrete members, a slightly lower angle (cot hmax ¼ 3 corresponding to hmin ¼ 18:4 ) can be chosen in comparison to the general formulation found in EC 2 [3] (cot hmax ¼ 2:5 corresponding to hmin ¼ 21:8 ). Eq. (4) will be used further on to estimate the shear capacity of test beams with shear reinforcement. In the case of structural concrete members without shear reinforcement, the design value of the shear capacity is to be determined using Eqs. (5) and (6) for sections uncracked respectively cracked in bending.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ibw 2 ðf ctd Þ þ al rcp f ctd S " # rffiffiffiffiffiffiffiffiffi! 1 0:18 200 3 ð100ql f ck Þ þ 0:15rcp bw d ¼ 1þ d cc

V Rd ¼ V Rd;c ¼

ð5Þ

V Rd ¼ V Rd;c

ð6Þ

In Eq. (5), I represents the second moment of area whereas S denotes the first moment of area above and about the centroidal axis. f ctd is the design value of the characteristic uni-axial tensile strength f ctk;0:05 . The latter is equal to 0:7f ctm [3] with f ctm the mean uni-axial tensile strength. The value of f ctm can be derived from the experimentally determined mean flexural tensile strength and is equal to 0:67f ctm;fl [12]. al is a factor taking into account the bond characteristics of the longitudinal reinforcement which can here be taken equal to 1:0. If the aforementioned Eqs. (1)–(6) are to be used to estimate the actual failure load, average material strength properties should be used rather than design values and partial safety factors should be omitted. Due to the relatively low shear reinforcement ratio qw of

Table 6 Experimental failure load and failure mode properties compared to analytical calculations according to EC 2 [3,13]. Specimen

B101 B102 B103 B104 B105 B106 B107 B108 B109

Experiment

Eurocode 2 [3,13]

Failure load V u;exp (kN)

Failure mode

Failure load V u;EC2 (kN)

V u;exp =V u;EC2 (–)

Failure mode

V u;bend (kN)

377.7 321.6 262.8 281.8 251.2 179.7 271.3 213.8 181.0

S-DTa S-DT S-DT S-DT S-DT S-DT Bb B S-DT

165.4 165.4 260.6 142.0 142.0 91.3 157.1 157.1 87.5

2.28 1.94 1.01 1.99 1.77 1.97 1.73 1.36 2.07

S S S S S S S S S

412.2 329.6 329.6 406.6 325.3 325.3 236.5 189.2 189.2

V u;exp c V u;EC2 d

1.79

COV

a b c d

Shear (S) failure mode due to Dagional Tension (DT). Bending (B) failure mode. Mean experimental-to-predicted failure load ratio. Coefficient of variation.

21.8 %

(Eq. (Eq. (Eq. (Eq. (Eq. (Eq. (Eq. (Eq. (Eq.

(7)) (7)) (8)) (7)) (7)) (9)) (7)) (7)) (9))

90

K. De Wilder et al. / Engineering Structures 82 (2015) 82–92

specimens with shear reinforcement (i.e. specimens B101–B102, B104–B105 and B107–B108), the shear capacity of all aforementioned beams is entirely governed by the amount of shear reinforcement per unit length, i.e. the theoretical shear force required to obtain yielding and subsequent rupture of the shear reinforcement bars is much smaller than the theoretical shear force required to obtain crushing of the compressive struts. Hence, the shear capacity of specimens with shear reinforcement is to be estimated using Eq. (7).

V R;s ¼

Asw zftm cot hmax s

ð7Þ

In Eq. (7), f tm is the average ultimate tensile strength of the shear reinforcement bars, refer to Table 2 and cot hmax is determined using Eq. (4) with f ywd replaced by f tm . The failure load of specimens without shear reinforcement (i.e. beams B103, B106 and B109) should be determined using Eq. (8) or Eq. (9) respectively.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Ibw  2 0:67f ctm;fl þ 0:67f ctm;fl rcp f ctm þ rcp f ctm ¼ S # rffiffiffiffiffiffiffiffiffi! 1 200 3 ð100ql f cm Þ þ 0:15rcp bw d V R ¼ 0:18 1 þ d

VR ¼

Ibw S "

ð8Þ ð9Þ

The results of the calculation procedures outlined in Eqs. (7)–(9) are summarized in Table 6 and compared to the experimental failure loads. The load, required to obtain the theoretical bending capacity (V u;bend ¼ MaR ) derived from a plane section analysis, is also listed in Table 6. From Table 6, two preliminary conclusions can be drawn: 1. Increasing the prestressing force while keeping the longitudinal reinforcement ratio constant, increases the shear capacity for specimens with (B101–B104, B102–B105) and without (B103–B106) shear reinforcement. 2. Table 6 clearly indicates that current shear design expressions found in Eurocode 2 severely underestimate the failure load of specimens with shear reinforcement (B101–B102, B104– B105 and B107–B108) and without shear reinforcement (B106 and B109). A nearly perfect estimate of the shear capacity of beam B103 is found. However, since the behavior of the aforementioned beam remains nearly perfectly linear-elastic until failure, it can be expected that the linear-elastic expression, given by Eq. (8), yields accurate results for this specific beam. Moreover, Table 6 shows that the predicted shear strength of specimen B103 (without shear reinforcement) is higher than the analytically calculated shear capacity of the equivalent beam B102 (with shear reinforcement). This observation follows from the fact that current shear design expressions found in EC 2 [3,13] relate the entire shear capacity of a structural concrete member with shear reinforcement to the provided amount of shear reinforcement per unit length, hence neglecting the concrete contribution to the overall shear capacity. From Eqs. (5) and (6) and Eqs. (8) and (9), it can be seen that increasing the prestressing force, i.e. increasing rcp , increases the concrete contribution to the shear capacity. However, as stated above, this concrete contribution is neglected for specimens with shear reinforcement. This explains the lower predicted shear capacity of specimen B102 in comparison to the analytically calculated shear strength of beam B103. The average ratio V u;exp V u;EC2

In the following, an investigation is made of the discrepancy between the reported experimental results and analytical calculations. Firstly, it must be stressed that Eq. (9) was used to estimate the shear capacity of specimens B106 and B109 whereas Eq. (8) was adopted for the determination of the shear capacity of specimen B103, refer to Table 6. However, from Fig. 8(c, f, and i), it can be clearly seen that the critical shear crack, i.e. the crack where eventually failure was initiated, for specimens without shear reinforcement is a diagonal web crack rather than an inclined bending crack. Therefore, it is suggested to also use Eq. (8) for the quantification of the shear capacity of beams B106 and B109. The results of the aforementioned calculation procedure are presented in Table 7. From Table 7 it can be seen that a fairly good correlation is found between the experimentally measured and analytically calculated failure load obtained using Eq. (8) in comparison to the results presented in Table 6 for specimens without shear reinforcement. Table 7 also indicates that, in the case of the presented test beams, current shear design provisions found in EC 2 [3,13] predict a higher shear capacity for specimens without shear reinforcement if compared to the analytically calculated shear capacity of the equivalent test beams with shear reinforcement, refer to Table 6. As already mentioned above, according to EC 2 [3,13], the shear capacity of structural concrete members with shear reinforcement is entirely governed by the provided amount of shear reinforcement per unit length, i.e. the concrete contribution to the overall shear capacity of specimens with shear reinforcement is neglected. This explains the lower predicted shear strength of specimens with shear reinforcement (i.e. beams B102, B105 and B108) in comparison to the analytically calculated shear strength of their equivalent beams without shear reinforcement (i.e. specimens B103, B106 and B109). According to Eurocode 2 [3,13] and mathematically expressed by Eq. (7), the shear capacity of structural concrete members with vertical shear reinforcement is entirely determined by the amount of shear reinforcement per unit length. The variable in Eq. (7) that can be chosen freely and is not depending on material or geometry characteristics, is the angle h. In Fig. 10, the principal compressive

Table 7 Experimentally observed failure load and analytically predicted shear capacity using Eq. (8) for I-shaped specimens without shear reinforcement. Specimen

Experiment V u;exp (kN)

B103 B106 B109

262.8 179.7 181.0

Eurocode 2 [3,13] using Eq. (8) Failure mode S-DT S-DT S-DT

a

V u;pred (kN)

Failure mode

V u;exp V u;pred

260.6 223.7 212.2

S-DT (Eq. (8)) S-DT (Eq. (8)) S-DT (Eq. (8))

1.01 0.80 0.85

V u;exp b V u;pred c

0.89

COV

a b c

(–)

12.1%

Shear failure mode due to diagonal tension. Average experimental-to-predicted failure load ratio. Coefficient of variation.

is equal to 1.79 (COV: 21.8%) and a relatively low correla-

tion is found between experiment and calculations according to EC 2 (qEXPEC2 ¼ 0:48). Current design provisions found in EC 2 [3,13] thus perform poorly in estimating the failure load of the reported specimens.

Fig. 10. Principal compressive strain angle determined at mid-depth of specimen B104 as a function of the distance from the support point at () 80 % and () 95 % of the experimental failure load with indication of the adopted angle hEC2 in the shear strength determination of specimen B104.

K. De Wilder et al. / Engineering Structures 82 (2015) 82–92

strain angle measured at mid-depth is shown as a function of the distance from the support point and load level for specimen B104. The angle adopted in the above presented shear strength calculations, hEC2 , is also indicated in the aforementioned Fig. 10. According to the VATM, a compression field with constant inclination is activated in the web of the beam. Therefore, the angle between the horizontal and the principal compressive strain is assumed to be constant along the member’s length. Similar results were obtained for the remaining reported specimens where the DIC measurement technique has been adopted. From Fig. 10, it can be seen that near failure, the principal compressive strain angle correlates well with the adopted value for hEC2 over a certain length in the vicinity of the middle of the shear span. Fig. 10 also indicates that the value of the principal compressive strain angle is not constant along the shear span. The minimum value is found near mid-span whereas the highest values are observed near the support and loading point. It is thus clear that the VATM, as proposed by EC 2 [3,13], does not capture the actual mechanical behavior of the reported test specimens with shear reinforcement. In Fig. 11(a–b), the direction and magnitude of the experimentally measured full-field principal compressive strain field is shown for the right measurement field (where eventually failure occurred) of specimen B104 at 80% and 95% of the experimental failure load respectively. Similar observations were made for the remaining specimens with shear reinforcement where the DIC measurement technique has been adopted. From Fig. 11(a–b) it can be firstly concluded that, due to the low shear reinforcement ratio, the value of the principal compressive strain remained relatively small near failure. Secondly, given the direction of the principal compressive strains, it can be concluded that a part of the applied load is transferred directly towards the support point via strut action and the remaining part of the applied load fans out towards the bottom of the beam. The possibility of the direct strut action provides a plausible explanation for the observed discrepancy between experimental results and analytical calculations for the reported test specimens with shear reinforcement. In view of the mechanical behavior described above, the experimentally observed failure mode can be understood as a splitting failure mode due to the spreading out of the compressive force in the web of the beam. Finally, the influence of the amount of prestressing on the experimentally observed mechanical behavior can be investigated. Fig. 12(a–b) show the experimentally measured horizontal displacement ux in the top flange (y ¼ 580 mm) near the experimental failure load as a function of the horizontal x-coordinate for specimens B101 and B104. Both aforementioned specimens are identical in terms of geometry, reinforcement layout and loading

Fig. 11. Direction and magnitude of the experimentally measured full-field principal compressive strain field at (a) 80% and (b) 95% of the experimental failure load (note: x ¼ 6000 mm corresponds to the location of the support point).

91

Fig. 12. Measured horizontal displacement ux of the top flange (y ¼ 580 mm) of specimens (a) B101 and (b) B104 at 95% of the experimental failure load as a function of the horizontal x-coordinate (6000 mm corresponds to the right support point) with indication of a polynomial fit (solid line) and 95% prediction interval (dashed line).

procedure. The sole difference lies in the amount of prestressing (rp;m0;B101 ¼ 1488 MPa and rp;m0;B104 ¼ 750 MPa). The horizontal strain x is the derivative of ux with respect to the horizontal coordinate. From Fig. 12(a–b), it can be seen that the horizontal strain x of specimen B101 rapidly decreases to low strain values at x ’ 5200 mm (800 mm from the support) whereas at failure, the top flange of specimen B104 remains strained (and thus stressed) over the entire length of the shear span. This indicates that apart from the direct strut action from the loading point to the support point, the fan region also carries a significant amount of the applied shear force for specimen B104 whereas the direct strut carries most of the applied load for specimen B101. 4. Conclusions This paper presented the results of 9 full-scale I-shaped prestressed concrete beams subjected to a four-point bending test until failure. The main investigated parameters were (a) the amount of prestressing; (b) the amount of longitudinal and shear reinforcement; and (c) the shear span-to-effective depth ratio respectively. During seven of the nine presented tests, threedimensional displacements and in-plane deformations were recorded using two stereo-vision digital image correlation (DIC) systems. In this study specific attention has been devoted to the quality and reproducibility of the applied speckle pattern to obtain a similar level of measurement precision and accuracy for all specimens employing this measurement technique. A numerical technique to generate optimized speckle patterns for DIC has therefore been adopted and the resulting speckle pattern has been applied onto each specimen where the DIC measurement technique has been used by adopting a stencil printing technique. All specimens were designed to fail in shear. Seven of the nine reported specimens failed due to excessive yielding leading to rupture of the shear reinforcement. The two remaining specimens exhibited a ductile bending failure mode. The experimental results were compared to analytical calculations using current shear design provisions found in EC 2. Based on the work presented in this paper, the following conclusions can be drawn. 1. The adopted full-field displacement and deformation measurement technique proved to be a valuable tool in assessing the mechanical behavior of the reported prestressed concrete beams over the entire loading range. Due to the controlled way of applying the required speckle pattern, comparable levels of measurement accuracy and precision were found for specimens where the DIC technique has been adopted. 2. EC 2, using the variable angle truss model, severely underestimates the shear capacity of all reported specimens with shear reinforcement. Concerning the specimens without shear reinforcement, a good estimate of the failure load was found for

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specimen B103 where, according to EC 2, the shear capacity was reached when the principal tensile stress reached the tensile strength limit. The semi-empirical design expression for specimens without shear reinforcement and cracked due to bending, should theoretically be used for beams B106 and B109 which resulted in poorly predicted shear capacities for the aforementioned test beams. Omitting all partial safety factors and using average material strength properties rather than characteristic values, an average experimental-to-predicted failure load ratio equal to 1.79 was found with a coefficient of variation equal to 21.8%. 3. Given the experimentally observed cracking pattern, the shear capacity of beams B106 and B109 without shear reinforcement was also reached when the principal tensile stress reached the tensile strength limit. Determination of the shear capacity based on the aforementioned failure criterion, yielded fairly good results for beams B106 and B109. 4. Rather than relating the entire shear capacity solely to the amount of provided shear reinforcement per unit length, it was found that the shear force applied on prestressed members with shear reinforcement is partly resisted by a direct compression strut and partly resisted by a fan region spreading out from the support and loading point to the top respectively bottom of the beam. Based on the experimental observations, the ultimate shear strength of beams failed by diagonal cracking and subsequent rupture of shear reinforcement was found to be depending on the amount of prestressing.

References [1] Balázs G. A historical review of shear. In: Bulletin 57: Shear and punching shear in RC and FRC elements. Fédération Internationale du béton (fib), 2010. p. 1–13. [2] Bossuyt S. Optimized patterns for digital image correlation. In: SEM international conference and exposition on experimental and applied mechanics, Costa Mesa, California USA, 2012. Society for Experimental Mechanics.

[3] CEN. Eurocode 2: Design of concrete structures – part 1-1: general rules and rules for buildings; 2004. [4] Cooreman S, Debruyne D, Coppieters S, Lecompte D, Sol H. Identification of the mechanical material parameters through inverse modelling. In: Emerging technologies in non-destructive testing; 2008. p. 337–342. [5] Cooreman S, Lecompte D, Sol H, Vantomme J, Debruyne D. Elasto-plastic material parameter identification by inverse methods: calculation of the sensitivity matrix. Int J Solids Struct 2007;44(13):4329–41. [6] Crammond G, Boyd SW, Dulieu-Barton JM. Speckle pattern quality assessment for digital image correlation. Optics Lasers Eng 2013;51(12):1368–78. [7] Ivanov D, Ivanov S, Lomov S, Verpoest I. Strain mapping analysis of textile composites. Optics Lasers Eng 2009;47(3–4):360–70. [8] Joint ASCE-ACI Task Committee 426 on Shear and Diagonal Tension of the Committee on Masonry and Reinforced Concrete of the Structural Division. The shear strength of reinforced concrete members. J Struct Division – Asce 1973;99(6):1091–87. [9] Lava P, Cooreman S, Debruyne D. Study of systematic errors in strain fields obtained via dic using heterogeneous deformation generated by plastic fea. Optics Lasers Eng 2010;48(4):457–68. [10] Lava P, Debruyne D. MatchID; 2013. [Online; accessed 22.09.13]. [11] Mörsch E. Der Eisenbetonbau – Seine Theorie und Anwendung. Stuttgart: Verlag von Konrad Wittwer; 1908. [12] Müller H, Breiner R, Anders I, Mechterine V, Curbach M, Speck K, et al. Codetype models for concrete bahviour, Bulletin. Fédération Internationale du Béton, Lausanne (Switzerland), vol 30, 2013. [13] NBN. Nbn en 1992-1-1 anb: Design of concrete structures – part1-1: general rules and rules for buildings; 2010. [14] Pan B, Lu ZX, Xie HM. Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation. Optics Lasers Eng 2010;48(4):469–77. [15] Pan B, Xie HM, Wang ZY, Qian KM, Wang ZY. Study on subset size selection in digital image correlation for speckle patterns. Opt Exp 2008;16(10):7037–48. [16] Ramirez JA, French CW, Adebar PE, Bonacci JF, Collins MP, Darwin D, et al. Recent approaches to shear design of structural concrete. J Struct Eng – Asce 1998;124(12):1375–417. [17] Ritter W. Die bauweise hennebique. Schweizerische Bauzeitung 1899;33:41–61. [18] Sas G, Blanksvard T, Enochsson O, Taljsten B, Elfgren L. Photographic strain monitoring during full-scale failure testing of ornskoldsvik bridge. Struct Health Mon – Int J 2012;11(4):489–98. [19] Sun Y, Pan H. Study of optimal subset size in digital image correlation of speckle pattern images. Optics Lasers Eng 2007;45(9):967–74. [20] Van Paepegem W, Shulev AA, Roussev IR, De Pauw S, Degrieck J, Sainov VC. Study of the deformation characteristics of window security film by digital image correlation techniques. Optics Lasers Eng 2009;47(3–4):390–7.