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Torsional behavior of steel fiber reinforced concrete beams
Construction and Building Materials 28 (2012) 269–275
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Torsional behavior of steel fiber reinforced concrete beams Fuad Okay ⇑, Serkan Engin Kocaeli University, Department of Civil Engineering, 41380 Kocaeli, Turkey
a r t i c l e
i n f o
Article history: Received 1 February 2009 Received in revised form 10 December 2010 Accepted 16 August 2011 Available online 13 October 2011 Keywords: Reinforced concrete Torsion Beam Steel fiber Twist angle Torsional model
a b s t r a c t Torsion of structural members and the behavior of steel fiber reinforced concrete became the area of interest of many researchers in the past and it is still newsworthy. In this study, 12 reinforced concrete (R/C) beams with Steel Fiber Reinforced Concrete (SFRC) were tested to observe the failure under torsional moments. The volumetric steel fiber content, fiber aspect ratio, and the longitudinal reinforcement were the variables of the investigation. Unit torsional angle of twist versus torsional moment (torque) response of each specimen was monitored during the experiments, and the effect of above variables on this response was critically investigated. It was observed that not only the torque capacity of R/C beam is modified by the addition of Steel Fiber Reinforcement (SFR) but also the energy absorption capacity is significantly affected by the SFR addition. Besides, an empirical equation relating the torque to twist for SFRC beams is proposed and tested against the test data. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Beams are the structural members, which mainly support the transverse loading through the flexural moment and the vertical shear. However, in some cases torsional response of beams may also control the overall structural behavior [1,2]. For this reason, the torsional behavior of beams should be studied and comprehended as well as its shear or flexural behavior. This is especially important since the cracked torsional stiffness of a reinforced concrete beam may be much smaller than its uncracked stiffness. Realizing the importance of the problem, many researches have been conducted on torsional behavior of reinforced concrete beams. The earliest experimental studies have investigated the effect of the presence of the reinforcement, both transverse and longitudinal, on torsional capacity and stiffness of the concrete beams. Thus, the torsional behavior of reinforced concrete beams has been compared with that of unreinforced (plain) concrete companions. It has observed that the torsional behavior of concrete beams is linear until the first cracking torque. It has also been observed that the ‘‘uncracked’’ torsional stiffness of the beam, that is the slope of linear part of the torque-twist (T /) diagram, is independent of the presence and amount of the reinforcement. In other words, up to the cracking point, the torque-twist response of identical concrete and reinforced concrete beams is comparable. Surely, the reinforcement becomes effective after cracking, providing additional ductility, and even additional capacity, if proper reinforcement detailing is supplied [3]. The behavior of high strength ⇑ Corresponding author. Tel.: +90 262 303 32 74; fax: +90 262 303 30 03. E-mail address: [email protected] (F. Okay). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.08.062
concrete beams is observed with reinforced and unreinforced specimens [4,5]. Since the torsional capacity, i.e., the cracking torque, of an unreinforced concrete beam provides a limit state for torsional behavior of reinforced concrete beams, many analytical studies have also been conducted to define the cracking torque. In these studies, empirical expressions were proposed for the cracking torque level in terms of material and cross-sectional properties of the beam [6–8]. Advances in the material technology have pointed out the addition of steel fibers in concrete to improve the main characteristics of concrete, such as, stiffness, toughness, and ductility. Some compression tests using normal strength concrete with fiber reinforced specimens show that the addition of fibers may cause a decrease in compressive strength. However, a considerable amount of increase in the tensile strength of the fiber reinforced specimens is observed in split cylinder tests [9]. When steel fibers are added to high strength concrete, the increase in fiber volumetric ratio also results in an increase in the compressive strength of the concrete, as well. However, when this addition exceeds a certain volumetric level, the increase in the strength becomes less [10,11]. Energy absorption capacity of fiber reinforced high strength concrete also increases linearly with the increasing volumetric ratio of steel fibers. This phenomenon shows that high strength concrete, which is known as a brittle material, behaves sort of a ductile manner with the addition of steel fibers [12]. It is reported that energy absorption of concrete in bending grows up with the increase in fibers’ aspect ratio. Volumetric ratio of fiber also increases energy absorbing capacity in bending [13]. It should be noted that these improvements closely affects the torsional capacity of a normal strength concrete beam. Thus, researches started to investigate
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Nomenclature b C1, C2 dl dw df E ETcr ETf ETu EU E/cr fck fctf fcts fyk G h L lf T Tcr Tcr-eq Tcr-test Tf
small dimension of the cross section (m) torsion coefficient depending of h/b ratio of the cross section diameter of the longitudinal reinforcement (mm) diameter of the shear reinforcement (mm) fiber diameter (mm) modulus of elasticity of concrete (N/mm2) percent error between the cracking torques obtained from test data and elastic theory percent error between the final torques obtained from test data and proposed analytical model percent error between the ultimate torques obtained from test data and the proposed analytical model percent error between the energies obtained from test data and proposed analytical model percent error between the unit angle of twist obtained from experiment and elastic theory characteristic compressive strength of concrete (N/mm2) flexural tensile strength of concrete (N/mm2) split cylinder tensile strength of concrete (N/mm2) the average yield strength for both longitudinal and web reinforcement (N/mm2) shear modulus of concrete (N/mm2) longer dimension of the cross-section (m) length of test region (m) fiber length (mm) torque (kN m) cracking torque (kN m) cracking torque calculated by using the elastic theory (kN m) cracking torque obtained from the test data (kN m) torque value when experiment is terminated (kN m)
whether the torsional capacity of a normal strength concrete beam can be increased with the addition of steel fibers in concrete, or not. Researches adding steel fiber into concrete in various aspect ratios and volume content give the corresponding stress strain curves relating to test specimens. As the volume content of the fiber increases, ductility under constant load increases in the diagrams. In other words, no difference is observed up to crack, but ductility increases afterwards with the increase in volume content of steel fibers [14–16]. In pure torsion experiments performed with unreinforced high strength concrete, the specimens fail with the formation of the first crack because of the lack of the reinforcement. Torsional capacity increases according to the strength of concrete, but the failure is observed to be more sudden and brittle. The data obtained from the experiments are compared with the equations derived in theories of elasticity, plasticity, and skew bending, shows that tests give comparable results with the theories based on split cylinder strength of concrete [5]. Some differences in the values of cracking torques are observed according to the compressive strength of concrete in the experiments using normal and high strength reinforced concrete. However, the variation of the amount of stirrups and longitudinal reinforcement has no effect on these differences in cracking torques. But the change in the ratio of longitudinal reinforcement to shear reinforcement results considerable differences in the cracked torsional capacities of elements. It increases with the increase in ratio of longitudinal reinforcement to web reinforcement [3]. The unreinforced normal strength concrete beams reach their capacity when their unit angle of twist fall in the range of 0.002–0.003 rad/m. The unit angle of twist increases up to 0.006–0.007 rad/m when the concrete contains fiber
Tf-eq Tf-test Tu Tu-eq Tu-test Umodel Utest Vf V0 b
m ql qw / /cr /cr-eq /cr-test /f /u
the final torque that corresponds to /f calculated by model (kN m) the final torque that corresponds to /f obtained from experiment (kN m) ultimate torque (kN m) ultimate torque predicted by the proposed analytical model (kN m) ultimate torque carried by the sections in the experiments (kN m) energy absorbed per unit length due to torsion as calculated by the proposed analytical model (kN m/m) energy absorbed per unit length due to torsion as obtained from test data (kN m/m) volumetric fiber ratio (%) coefficient used in the model, related to the amounts of longitudinal and shear reinforcement torsion coefficient depending of the dimensions of the section Poisson’s ratio of the concrete volumetric longitudinal reinforcement ratio of the section volumetric shear reinforcement ratio of the section unit angle of twist (rad/m) unit angle of twist at crack (rad/m) unit angle of twist calculated by using the elastic theory (rad/m) unit angle of twist at the moment of cracking obtained from the experiment (rad/m) unit angle of twist at which the experiments are terminated, (120 103 rad/m) unit angle of twist at ultimate torque (rad/m)
reinforcement [17]. Crack distribution of fiber added reinforced concrete beams show differences compared to beams without fiber. As the fiber content increases, the number of cracks increases but the widths of the cracks decrease [14,18]. Various theoretical and empirical equations are proposed for the torsional capacity of fiber added reinforced concrete beams. In those equations, the total capacity is determined by the addition of the capacity of concrete, reinforcement, and steel fiber separately [18]. Simple torsion experiments have been conducted on normal strength reinforced concrete beams with single type of fiber reinforcement and its observed that the addition of fiber reinforcement gives satisfactory results on the mentioned specimens [19]. After the use of carbon fiber sheets as structural strengthening materials, in torsion tests performed with the material mentioned above, increases in torsional capacities are observed [20–22]. The present paper is about steel fibers and torsion of reinforced concrete beams, which are both still topical. It also focuses the influence of volumetric steel fiber ratio and the aspect ratio of fibers in relation to the variations. In the experiments in which four different steel fiber aspect ratios and two longitudinal reinforcement ratios were used, the torsional ductility and the torsional capacity of reinforced concrete beams are investigated, with the variation of test parameters except the web reinforcement. 2. Experimental study 2.1. Materials Concrete mixtures are prepared by using crushed stone as coarse aggregate, sand as fine aggregate, and cement with standard compressive strength of 42.5 MPa. In order to obtain the desired compressive strength, 30 MPa, superplast-
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F. Okay, S. Engin / Construction and Building Materials 28 (2012) 269–275 Table 1 Nomenclature and fiber content of specimens.
df
lf Fig. 1. Geometry of the steel fibers used in the test.
icizer is also used in concrete mixtures. Bent ended steel fibers having four different aspect ratios lf/df = 40, 55, 67, and 80, corresponding to lf and df dimensions of 30/ 0.75, 30/0.55, 60/0.90, and 60/0.75 were used. Where lf denotes fiber length and df denotes diameter of fibers. Average yield strength for the steel fibers given by the manufacturer was fyf = 1200 MPa. Geometry for steel fibers is given in Fig. 1. dw = 8 mm deformed bars are used as web reinforcement, and dl = 8 mm and dl = 12 mm diameter deformed bars are used as longitudinal reinforcement. The average yield strength for both longitudinal and web reinforcement is found to be fyk = 460 MPa.
Beam designation
Longitudinal reinforcement
lf/df
Vf (%)
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
4 4 4 4 4 4 4 4 4 4 4 4
– 40 40 55 55 67 67 80 80 – 40 80
– 0.3 0.6 0.3 0.6 0.3 0.6 0.3 0.6 – 0.3 0.3
No. No. No. No. No. No. No. No. No. No. No. No.
of of of of of of of of of of of of
8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia 12 mm dia. 12 mm dia. 12 mm dia.
2.2. Test specimens The cross-sectional dimension of the test specimens are 150 200 mm with a length of 1900 mm. The first variable is the longitudinal reinforcement ratio ql, which is chosen as either 0.0067 (4U8) or 0.015 (4U12). Web reinforcement diameter is 8 mm and their center to center spacing is 200 mm, which corresponds to shear reinforcement ratio of qw = 0.006 for all specimens. The shear reinforcement is set to be constant, and the contribution of longitudinal reinforcement and the other variables to capacity is investigated. The tie-bar spacing outside the test region is decreased in order to force the failure of specimen to the test region. In those regions, the tie-bar spacing is 50 mm from center to center. The dimension of the test specimen and reinforcement layout is given in Fig. 2. The volumetric steel fiber ratio Vf is set to 0.3% and 0.6% for all specimens. With these two SFR ratios, it is aimed to observe the behavior of concrete with increasing volumetric content of fibers. It is seen that volumetric ratios that are higher than the ratios chosen in this study are used in some existing studies. In order not to have a settlement problem while preparing the specimens that have small distances between the reinforcement bars, the volumetric ratio of steel fibers is chosen to be 0.3% and 0.6%. For the designation of the specimens, the following procedure is followed: The first character block starting with L designates the diameter of the longitudinal reinforcement of the beam. Thus, L08 is used for longitudinal reinforcement with diameter dl = 8 mm and L12 for dl = 12 mm. Block F designates for the fiber aspect ratio F40, F55, F67, F80 designate for the aspect ratios 40, 55, 67, 80 respectively. Block V designates the volumetric content of the fibers added, where V3 shows 0.3% and V6 shows 0.6% volumetric content. F block is null and V is 0 while naming the specimens without SFR. Specimens’ designation is given in Table 1. It should also be noted that for each test specimen, 12 standard cylindrical specimens with 150 mm diameter and 300 mm height are prepared and tested to obtain the compressive strength, split tensile strength, and modulus of elasticity of concrete. In addition to the cylindrical specimens, five prismatic specimens with the dimensions 100 100 500 mm are prepared and tested to obtain flexural tensile strength of the concrete.
Fig. 3. Test setup, loading, and measurement systems.
the load that is applied to the specimen by a 300 kN capacity hydraulic cylinder pump set ((7) in Fig. 3) is measured by a 100 kN capacity electronic load cell. In this setup, the load that is applied by the hydraulic cylinder pump ((7) in Fig. 3) is shared equally by the ends of short beams ((5) in Fig. 3) that make the lever arm of the twisting moment. The unit angle measurements are read by an electronic gauge ((4) in Fig. 3) with a 100 mm capacity and a 0.02 mm accuracy which touches the arms ((8) in Fig. 3) which are located at the ends of the ‘‘test’’ region (Fig. 3). Test data are collected to a computer with the help of a data acquisition system, and torque versus unit angle of twist graph is obtained during the test. The data are taken every 0.01 rad/m unit angle of twist intervals, and the corresponding torque and crack widths are measured. Loading is terminated when the total unit angle of rotation value is reached to 0.12 rad/m.
2.3. Test setup, loading, and measurement systems Fig. 3 illustrates the test setup where the test specimens are subjected to uniform torsion. As shown in the figure, in order not to restrain the ends of the specimen from free rotation, extension or contraction, the specimen is placed on roller supports aligned with the specimen. Roller supports ((2) in Fig. 3) are settled at two ends of the specimen ((3) in Fig. 3) in order to maintain free rotation and elongation. The specimen is joined to the upper arms ((5) in the Fig. 3) and plates ((1) in Fig. 3), which are fixed to the roller supports by the help of bolts. Torque is obtained by applying a load to the center of the spreader beam ((6) in Fig. 3). The variation of
50
For each test specimens, cracking and ultimate torques, designated respectively as Tcr and Tu, and the corresponding unit angle of twists, designated as /cr and /u, are determined and listed in
200
200
190
140
3. Test results and discussions
150
450
Test Region = 1000
450
Fig. 2. Dimensions of the test specimen and the reinforcement layout (all dimensions are in mm).
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Table 2 Test data. Beam designation
fck (MPa)
fcts (MPa)
fctf (MPa)
E (GPa)
Tcr (kN m)
/cr (rad/m) 103
Tu (kN m)
/u (rad/m) 103
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
34.8 33.4 31.3 31.0 30.9 32.7 29.5 31.9 30.0 34.8 31.7 31.6
3.51 3.55 3.35 3.08 3.41 3.42 3.12 3.46 3.10 3.51 3.58 3.56
4.54 4.96 5.02 4.82 4.57 4.39 4.39 4.56 4.82 4.54 4.46 4.87
31.4 30.0 29.9 27.9 29.9 31.2 28.5 27.9 28.7 31.4 29.6 27.5
4.93 4.58 4.62 4.93 5.10 4.85 4.91 4.80 4.51 4.45 4.61 4.47
3.20 3.22 3.52 3.80 3.95 3.56 3.60 3.36 3.60 3.16 3.68 3.60
4.93 4.58 5.68 4.94 5.87 4.92 5.88 4.85 5.49 5.07 6.01 6.25
3.20 3.22 64.80 3.87 59.27 5.63 65.94 4.09 63.41 34.60 89.74 93.45
3.1. Crack patterns Number of cracks observed in reference specimens is less than the number of cracks in fiber reinforced beams when test is terminated; and failure becomes with the excessive widening of one of these cracks. When the unit angle of twist value reaches / = 0.06 rad/m, crushing of concrete in perpendicular direction to cracks is observed and this crushing is increase to unit angle of twist becomes 0.12 rad/m. The failure of the specimens whose volumetric content Vf = 0.3% is very similar to the reference specimens. Compressive crushing starts with the value from 0.06 to 0.07 rad/m of unit angle of twist and it is intensified toward the end of the experiment. Number of cracks is considerably increased while crack widths are decreased in the specimens whose volumetric content is Vf = 0.6%. Photographs of the specimens that show the above-mentioned behavior are given in Fig. 4.
3.2. Torsional strength The torsional strength of the test specimens without fibers is observed to be (i.e., Tu = Tcr) equal to the cracking torque of the specimens. The torque carried by the section decreases while the angle of twist increases, beyond /cr. On the other hand, the torque capacity continues to increase after cracking up to a certain value (i.e., Tu > Tcr) in specimens that contain steel fibers. This behavioral change is more apparent in the specimens with Vf = 0.6% compared to those with Vf = 0.3% (Fig. 5). L12 specimens, i.e., the specimens that have longitudinal reinforcement with dl = 12 mm, give more satisfactory results with the addition of Vf = 0.3% fibers compared to the specimens with longitudinal reinforcement with dl = 8 mm (Figs. 5 and 6). Test 8 7
Torque (kN.m)
designated. Table 2 also summarizes the average values of 28th day compressive strength fck, split cylinder tensile strength fcts and flexural tensile strength fctf of each test specimen. There is no considerable increase in split cylinder tensile strength fcts and flexural tensile strength fctf values in material tests, but remarkable increases in ultimate twisting moment Tu and corresponding unit angle of twist /u are observed in torsion tests with the increase in volumetric fiber ratio. In the material tests, reference specimens rupture with the first crack while the fiber reinforced specimens continue to sustain the load. This phenomenon confirms the increase in energy absorption of reinforced concrete beams under torsion.
6 5 4 3 2
L08F00V0 L08F40V3 L08F40V6
1 0
0
20
40
60
80
100
120
Unit Angle of Twist (rad/m) x10
140
-3
Fig. 5. The effect of volume content with fiber lf/df = 40.
8
Torque (kN.m)
7 6 5 4 3 2
L12F40V3 L12F80V3 L12F00V0
1 0
0
20
40
60
80
100
Unit Angle of Twist (rad/m) *10 Fig. 4. Crack patterns for (a) L08F00V0, (b) L08F40V3, (c) L08F40V6.
120 -3
Fig. 6. Torque-twist variation in specimens with dl = 12 mm.
140
273
4. Model proposed for torsional behavior A model is proposed using the data obtained in the tests. ‘‘Twisting moment versus unit angle of twist’’ graphics behave linearly up to cracking of the section and then this relationship is not seen. Cracking torque and cracking unit angle of rotation of all test specimens generally attain the same value regardless of the amount of longitudinal reinforcement. Torsional stiffness of the beams is significantly reduced after cracking. Since the specimens behave linearly elastic before cracking, cracking torque of the specimens can be computed from the following expression Eq. (4.1) derived by using elasticity theory [23]. Test results clearly show that the torsional behavior of test specimens is linear in the region up to cracking, in which Saint Venant equation is valid, beyond which the linear behavior is lost [19]. This type of behavior is consistent with the studies in the literature [3]. The fact that the cracking values (i.e., Tcr and /cr) of all specimens are almost the same indicates that the cracking characteristics of test specimens are affected neither from the amount of longitudinal reinforcement nor from the presence of steel fibers. 2
T cr ¼ C 1 hb fctf
T cr L 3
C 2 hb G
ð4:2Þ
where L is the length of testing region, C2 is a coefficient that takes values between 0.1406 and 0.333 depending on the dimensions of the section. In this study, C2 is taken to be 0.179 according to the dimensions of the specimens used. G is the shear modulus of the concrete and is obtained by Eq. (4.3), using the well-known relation,
G¼
E 2ð1 þ mÞ
ð4:3Þ
where E is the modulus of elasticity of concrete, which can be obtained from material tests. Poisson’s ratio of concrete m can be taken as 0.2 [24]. Ultimate torque and the corresponding unit angle of rotation of the sections take different values depending on the amount of the longitudinal reinforcement of specimens. For the specimens with dl = 8 mm and Vf = 0.6%, the ultimate torque values are observed to be around 6 kN m and the corresponding unit angle of rotation value is /u = 60 103 rad/m. The cracking torque Tcr, ultimate torque Tu, and the torque value at the termination of the experiments Tf and the corresponding values of the unit angle of rotations /cr, /u, and /f, respectively, are plotted in Fig. 7. In this figure, triangular dots stand for Tcr, rectangular dots for Tu, and diamond ones stand for Tf values. As it is shown from Fig. 7, the ultimate torque values are equal to the cracking torque values of the specimens whose longitudinal reinforcement diameter is dl = 8 mm and volumetric fiber ratio is Vf = 0.3%; whereas torque values of the specimens with Vf = 0.6% increase after cracking and reach their capacities approximately at the same value irrespective of aspect ratio. When the terminating unit angle of rotation value is reached, the specimens that have no fibers and having a volumetric fiber ratio of 0.3% posses the
Tu
Tcr
0
20
Tf
40
60
80
100
120
140
-3
Fig. 7. Tcr, Tu, and Tf values of the specimens with dl = 8 mm.
7 6
Tu
Tcr
5
Tf
4 3 2 1 0
0
20
40
60
80
100
120
140
Unit Angle of Twist (rad/mx10 -3 )
ð4:1Þ
where h and b are longer and shorter sizes of the rectangular concrete section, respectively, fctf is the flexural tensile strength of the concrete, and C1 is a coefficient taking values between 0.208 and 0.333 depending on h/b ratio of the section. In this study, C1 is taken to be 0.224 based on the dimensions of the specimens used. Similarly, elasticity theory provides the following Eq. (4.2) expression for the unit angle of rotation at cracking:
/cr ¼
7 6 5 4 3 2 1 0
Unit Angle of Twist (rad/m) x10
Torque (kN.m)
results show that the aspect ratio of the steel fiber has no significant effect on the capacity and ductility of the beam.
Torque (kN.m)
F. Okay, S. Engin / Construction and Building Materials 28 (2012) 269–275
Fig. 8. Tcr, Tu, and Tf values of the specimens with db = 12 mm.
Table 3 V0, /u, and b coefficient according to longitudinal reinforcement ratio.
V0 /u (rad/m) 103 b
0.85ql < qw
0.85ql > qw
0.005 60
0.002 90 7410Vf + 2.62
635; 278V 2f 450:83V f þ 2:56
same Tf values. Similarly, there are no significant differences among the Tf values of specimens having a volumetric ratio of 0.6%. Tu values are observed to be approximately 6 kN m, and /u values are observed to be 90 103 rad/m when dl = 12 mm and Vf = 0.3% as it is seen Fig. 8. Cracking torque values of the specimen having dl = 12 mm are observed to be identical irrespective of the volumetric ratio whereas Tu and Tf values differ. Behavior of the specimens after cracking can be expressed by the following empirical relationship Eq. (4.4) between the torque and the unit angle of rotation:
Fig. 9. Proposed torsional behavior model for reinforced concrete beams with/ without steel fibers.
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Table 4 Comparison of the Tcr and /cr values obtained from model and test results. Beam designation
Tcr-test
Tcr-eq (kN m)
ETcra (%)
/cr-test (rad/m) 103
/cr-eq (rad/m) 103
E/cr b (%)
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
4.93 4.58 4.62 4.93 5.10 4.85 4.91 4.80 4.51 4.45 4.61 4.47
4.58 5.00 5.06 4.86 4.61 4.43 4.43 4.60 4.86 4.58 4.50 4.91
7.2 9.2 9.5 1.4 9.7 8.8 9.9 4.2 7.7 2.8 2.5 9.8
3.20 3.22 3.30 3.58 3.75 3.56 3.60 3.36 3.34 3.18 3.28 3.37
3.13 2.91 2.94 3.13 3.24 3.08 3.12 3.05 2.87 2.83 2.93 2.84
2.1 9.6 11.0 12.4 13.5 13.4 13.3 9.2 14.2 11.0 10.6 15.7
a
ET cr ¼
b
E/cr ¼
T cr-eq T cr-test . T cr-test /cr-eq /cr-test /cr-test
100.
Table 5 Comparison of the Tu and /u values obtained from model and test results. Beam designation
Tu-test (kN m)
Tu-eq (kN m)
ETua (%)
Tf-test (kN m)
Tf-eq (kN m)
E
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
4.93 4.58 5.68 4.94 5.87 4.92 5.88 4.85 5.49 5.07 6.01 6.25
4.58 5.00 6.42 4.86 5.97 4.43 5.79 4.60 6.22 4.58 6.74 7.15
7.2 9.2 13.1 1.6 1.7 10.1 1.5 5.2 13.3 9.7 12.1 14.4
2.74 3.12 3.90 2.96 4.02 2.42 3.54 3.05 3.86 3.45 4.75 5.61
3.08 3.38 3.83 3.24 3.39 2.81 3.18 2.98 3.63 3.97 5.39 5.80
12.4 8.4 1.8 9.5 15.7 15.9 10.2 2.3 6.0 15.1 13.5 3.4
a
ET u ¼
b
ET f ¼
b
(%)
T u-eq T u-test 100. T u-test T f -eq T f -test 100. T f -test
Table 6 Comparison of the absorbed energy obtained from model and test results.
a
Tf
Beam designation
Utest (N m/m)
Umodel (N m/m)
EUa (%)
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
454.57 497.92 644.79 480.83 607.32 429.53 577.32 450.25 625.84 507.38 707.01 762.72
424.08 442.84 598.55 460.18 524.69 377.12 593.53 458.14 596.03 484.72 631.53 660.85
7.2 12.4 7.7 4.5 15.7 13.9 2.7 1.7 5.0 4.7 12.0 15.4
U test EU ¼ U model 100. U test
T ¼ T cr þ bð/ /cr ÞðV f V 0 Þ 103 3bh/ /u ihV f V 0 i 103
ð4:4Þ
where of relevant V0, /u, and b values are given Table 3. The V0, /u, and b coefficient in Eq. (4.4) is obtained from a regression analysis. The brackets h i in Eq. (4.4) are used to designate singularity functions. Thus, if the value inside the brackets is greater than zero, the brackets behave as regular parenthesis; if not the value becomes zero [23]. The torsional model expressed by Eq. (4.4) is plotted in Fig. 9. In this model, Eq. (4.1) is valued in the initial elastic region (Region I) and Eq. (4.4) can be used in the following plastic regions (Regions II and Regions III). According to this model, the test specimens show
the same behavior up to cracking regardless of the amount of longitudinal reinforcement and presence, amount, size of steel fibers observed during the tests. After cracking, the specimens with no fibers and those with fiber volumetric ratio of 0.3% follow the dashed path if 0.85ql < qw, while the specimens with Vf = 0.6% follow the solid path in Fig. 9. If 0.85ql > qw, the specimens with no fibers follow the dashed line; while those with 0.3 percent fiber ratio follow the solid line. For all the specimens that follow the dashed line, the ultimate torque Tu equals to the cracking torque Tcr. In Tables 4 and 5, the results obtained from the model compared to the values of the test and the differences between those values are accepted to be errors. As it is seen from the tables, the largest difference percent between the models and the test result is less than 16%. The energy per unit length absorbed by the element during the test is obtained from the area under the unit angle of rotation versus torque plots of both the tests and models are given and compared in Table 6. 5. Conclusions Based on the limited experiments results in which different aspect ratios of steel fibers are used in pure torsion test of reinforced concrete beams, the following conclusions have been obtained: It is observed that torsional behavior of normal strength concrete beams is changed positively with the addition of steel fibers. Although 0.3% volumetric content fiber addition does
F. Okay, S. Engin / Construction and Building Materials 28 (2012) 269–275
not make significant effect on this behavior, 0.6% volumetric content fiber addition increases the torsional strength of the test specimens with an amount of 10–60%. The specimens having insufficient longitudinal reinforcement do not show any increase in torsional strength with the addition of volumetric content steel fibers in 0.3%. Sufficient addition of steel fibers to concrete causes a decrease in the widths of cracks, whereas an increase in the number of cracks. The simple model proposed in this study yielded comparative results with the data produced in this study.
Acknowledgements Authors gratefully acknowledge the funding from Kocaeli University Research Fund and Beksa-Turkey. References [1] Ersoy U. Reinforced concrete. Middle East Technical University Press; 1999. [2] Nilson AH, Winter G. Design of concrete structures. 11th ed. McGraw-Hill; 1991. [3] Hsu TTC. Torsion of structural concrete – behavior of reinforced concrete rectangular members. Torsion of structural concrete, SP – 18. Detroit: American Concrete Institute; 1968. p. 261–306. [4] Fang I-K, Shaiu J-K. Torsional behaviour of normal and high strength concrete beams. ACI Struct J 2004;101(3):304–13. [5] Atef HB, Faisal FW, Ali AA. Torsional behavior of plain high-strength concrete beams. ACI Struct J 1990;87(5):583–8. [6] MacGregor JG, Ghoneim MG. Design for torsion. ACI Struct J 1995;92(2):211–8. [7] Bhatti MA, Almughrabi A. Refined model to estimate torsional strength of reinforced concrete beams. ACI Struct J 1996;93(5):614–22.
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[8] Rasmussen LJ, Baker G. Torsion in reinforced normal and high-strength concrete beams – Part 2: theory and design. ACI Struct J 1995;92(2):149–56. [9] El-Niema EI. Fiber reinforced beams under torsion. ACI Struct J 1993;90(50):489–95. [10] Song PS, Hwang S. Mechanical properties of high-strength steel fiberreinforced concrete. Constr Build Mater 2004;18(9):669–73. [11] Mohammadi Y, Singh SP, Kaushik SK. Properties of steel fibrous concrete containing mixed fibres in fresh and hardened state. Constr Build Mater 2008;22:956–65. [12] Faisal FW, Samir AA. Mechanical properties of high-strength fiber reinforced concrete. ACI Mater J 1992;89(5):449–55. [13] Jianming G, Wei S, Keiji M. Mechanical properties of steel fiber-reinforced, high-strength, lightweight concrete. Cement Concr Compos 1997;19(4):307–13. [14] Rao TDG, Seshu DR. Analytical model for the torsional response of steel fiber reinforced concrete members under pure torsion. Cement Concr Compos 2005;27(4):493–501. [15] Craig RJ, James AP, Germain E, Mosquera V, Kamilares S. Fiber reinforced beams in torsion. ACI Struct J 1986;83(6):934–42. [16] Engin S. Torsional behaviour of steel fiber reinforced concrete beams. MSc thesis, Department of Civil Engineering, Faculty of Engineering, Kocaeli University; 2005 [in Turkish]. [17] Rao TDG, Seshu DR. Torsion of steel fiber reinforced concrete members. Cem Concr Res 2003;33(11):1783–8. [18] Narayanan R, Karem-Palanjian AS. Torsion in beams reinforced with bars and fibers. J Struct Eng 1986;112(1):53–66. [19] Rao TDG, Seshu DR. Torsional response of fibrous reinforced concrete members: effect of single type of reinforcement. Constr Build Mater 2006;20:187–92. [20] Jing M, Raongjant W, Li Z. Torsional strengthening of reinforced concrete box beams using carbon fiber reinforced polymer. Compos Struct 2007;78:264–70. [21] Chalioris CE. Torsional strengthening of rectangular and flanged beams using carbon fibre-reinforced-polymers – experimental study. Constr Build Mater 2008;22:21–9. [22] Chalioris CE, Karayannis CG. Effectiveness of the use of steel fibres on the torsional behaviour of flanged concrete beams. Cement Concr Compos 2009;31:331–41. [23] Beer FP, Johston ER. Mechanics of materials. McGraw-Hill; 1992. p. 114–82. [24] Neville AM. Properties of concrete. Prentice Hill; 2000. p. 421–42.
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Torsional behavior of steel fiber reinforced concrete beams Fuad Okay ⇑, Serkan Engin Kocaeli University, Department of Civil Engineering, 41380 Kocaeli, Turkey
a r t i c l e
i n f o
Article history: Received 1 February 2009 Received in revised form 10 December 2010 Accepted 16 August 2011 Available online 13 October 2011 Keywords: Reinforced concrete Torsion Beam Steel fiber Twist angle Torsional model
a b s t r a c t Torsion of structural members and the behavior of steel fiber reinforced concrete became the area of interest of many researchers in the past and it is still newsworthy. In this study, 12 reinforced concrete (R/C) beams with Steel Fiber Reinforced Concrete (SFRC) were tested to observe the failure under torsional moments. The volumetric steel fiber content, fiber aspect ratio, and the longitudinal reinforcement were the variables of the investigation. Unit torsional angle of twist versus torsional moment (torque) response of each specimen was monitored during the experiments, and the effect of above variables on this response was critically investigated. It was observed that not only the torque capacity of R/C beam is modified by the addition of Steel Fiber Reinforcement (SFR) but also the energy absorption capacity is significantly affected by the SFR addition. Besides, an empirical equation relating the torque to twist for SFRC beams is proposed and tested against the test data. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Beams are the structural members, which mainly support the transverse loading through the flexural moment and the vertical shear. However, in some cases torsional response of beams may also control the overall structural behavior [1,2]. For this reason, the torsional behavior of beams should be studied and comprehended as well as its shear or flexural behavior. This is especially important since the cracked torsional stiffness of a reinforced concrete beam may be much smaller than its uncracked stiffness. Realizing the importance of the problem, many researches have been conducted on torsional behavior of reinforced concrete beams. The earliest experimental studies have investigated the effect of the presence of the reinforcement, both transverse and longitudinal, on torsional capacity and stiffness of the concrete beams. Thus, the torsional behavior of reinforced concrete beams has been compared with that of unreinforced (plain) concrete companions. It has observed that the torsional behavior of concrete beams is linear until the first cracking torque. It has also been observed that the ‘‘uncracked’’ torsional stiffness of the beam, that is the slope of linear part of the torque-twist (T /) diagram, is independent of the presence and amount of the reinforcement. In other words, up to the cracking point, the torque-twist response of identical concrete and reinforced concrete beams is comparable. Surely, the reinforcement becomes effective after cracking, providing additional ductility, and even additional capacity, if proper reinforcement detailing is supplied [3]. The behavior of high strength ⇑ Corresponding author. Tel.: +90 262 303 32 74; fax: +90 262 303 30 03. E-mail address: [email protected] (F. Okay). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.08.062
concrete beams is observed with reinforced and unreinforced specimens [4,5]. Since the torsional capacity, i.e., the cracking torque, of an unreinforced concrete beam provides a limit state for torsional behavior of reinforced concrete beams, many analytical studies have also been conducted to define the cracking torque. In these studies, empirical expressions were proposed for the cracking torque level in terms of material and cross-sectional properties of the beam [6–8]. Advances in the material technology have pointed out the addition of steel fibers in concrete to improve the main characteristics of concrete, such as, stiffness, toughness, and ductility. Some compression tests using normal strength concrete with fiber reinforced specimens show that the addition of fibers may cause a decrease in compressive strength. However, a considerable amount of increase in the tensile strength of the fiber reinforced specimens is observed in split cylinder tests [9]. When steel fibers are added to high strength concrete, the increase in fiber volumetric ratio also results in an increase in the compressive strength of the concrete, as well. However, when this addition exceeds a certain volumetric level, the increase in the strength becomes less [10,11]. Energy absorption capacity of fiber reinforced high strength concrete also increases linearly with the increasing volumetric ratio of steel fibers. This phenomenon shows that high strength concrete, which is known as a brittle material, behaves sort of a ductile manner with the addition of steel fibers [12]. It is reported that energy absorption of concrete in bending grows up with the increase in fibers’ aspect ratio. Volumetric ratio of fiber also increases energy absorbing capacity in bending [13]. It should be noted that these improvements closely affects the torsional capacity of a normal strength concrete beam. Thus, researches started to investigate
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Nomenclature b C1, C2 dl dw df E ETcr ETf ETu EU E/cr fck fctf fcts fyk G h L lf T Tcr Tcr-eq Tcr-test Tf
small dimension of the cross section (m) torsion coefficient depending of h/b ratio of the cross section diameter of the longitudinal reinforcement (mm) diameter of the shear reinforcement (mm) fiber diameter (mm) modulus of elasticity of concrete (N/mm2) percent error between the cracking torques obtained from test data and elastic theory percent error between the final torques obtained from test data and proposed analytical model percent error between the ultimate torques obtained from test data and the proposed analytical model percent error between the energies obtained from test data and proposed analytical model percent error between the unit angle of twist obtained from experiment and elastic theory characteristic compressive strength of concrete (N/mm2) flexural tensile strength of concrete (N/mm2) split cylinder tensile strength of concrete (N/mm2) the average yield strength for both longitudinal and web reinforcement (N/mm2) shear modulus of concrete (N/mm2) longer dimension of the cross-section (m) length of test region (m) fiber length (mm) torque (kN m) cracking torque (kN m) cracking torque calculated by using the elastic theory (kN m) cracking torque obtained from the test data (kN m) torque value when experiment is terminated (kN m)
whether the torsional capacity of a normal strength concrete beam can be increased with the addition of steel fibers in concrete, or not. Researches adding steel fiber into concrete in various aspect ratios and volume content give the corresponding stress strain curves relating to test specimens. As the volume content of the fiber increases, ductility under constant load increases in the diagrams. In other words, no difference is observed up to crack, but ductility increases afterwards with the increase in volume content of steel fibers [14–16]. In pure torsion experiments performed with unreinforced high strength concrete, the specimens fail with the formation of the first crack because of the lack of the reinforcement. Torsional capacity increases according to the strength of concrete, but the failure is observed to be more sudden and brittle. The data obtained from the experiments are compared with the equations derived in theories of elasticity, plasticity, and skew bending, shows that tests give comparable results with the theories based on split cylinder strength of concrete [5]. Some differences in the values of cracking torques are observed according to the compressive strength of concrete in the experiments using normal and high strength reinforced concrete. However, the variation of the amount of stirrups and longitudinal reinforcement has no effect on these differences in cracking torques. But the change in the ratio of longitudinal reinforcement to shear reinforcement results considerable differences in the cracked torsional capacities of elements. It increases with the increase in ratio of longitudinal reinforcement to web reinforcement [3]. The unreinforced normal strength concrete beams reach their capacity when their unit angle of twist fall in the range of 0.002–0.003 rad/m. The unit angle of twist increases up to 0.006–0.007 rad/m when the concrete contains fiber
Tf-eq Tf-test Tu Tu-eq Tu-test Umodel Utest Vf V0 b
m ql qw / /cr /cr-eq /cr-test /f /u
the final torque that corresponds to /f calculated by model (kN m) the final torque that corresponds to /f obtained from experiment (kN m) ultimate torque (kN m) ultimate torque predicted by the proposed analytical model (kN m) ultimate torque carried by the sections in the experiments (kN m) energy absorbed per unit length due to torsion as calculated by the proposed analytical model (kN m/m) energy absorbed per unit length due to torsion as obtained from test data (kN m/m) volumetric fiber ratio (%) coefficient used in the model, related to the amounts of longitudinal and shear reinforcement torsion coefficient depending of the dimensions of the section Poisson’s ratio of the concrete volumetric longitudinal reinforcement ratio of the section volumetric shear reinforcement ratio of the section unit angle of twist (rad/m) unit angle of twist at crack (rad/m) unit angle of twist calculated by using the elastic theory (rad/m) unit angle of twist at the moment of cracking obtained from the experiment (rad/m) unit angle of twist at which the experiments are terminated, (120 103 rad/m) unit angle of twist at ultimate torque (rad/m)
reinforcement [17]. Crack distribution of fiber added reinforced concrete beams show differences compared to beams without fiber. As the fiber content increases, the number of cracks increases but the widths of the cracks decrease [14,18]. Various theoretical and empirical equations are proposed for the torsional capacity of fiber added reinforced concrete beams. In those equations, the total capacity is determined by the addition of the capacity of concrete, reinforcement, and steel fiber separately [18]. Simple torsion experiments have been conducted on normal strength reinforced concrete beams with single type of fiber reinforcement and its observed that the addition of fiber reinforcement gives satisfactory results on the mentioned specimens [19]. After the use of carbon fiber sheets as structural strengthening materials, in torsion tests performed with the material mentioned above, increases in torsional capacities are observed [20–22]. The present paper is about steel fibers and torsion of reinforced concrete beams, which are both still topical. It also focuses the influence of volumetric steel fiber ratio and the aspect ratio of fibers in relation to the variations. In the experiments in which four different steel fiber aspect ratios and two longitudinal reinforcement ratios were used, the torsional ductility and the torsional capacity of reinforced concrete beams are investigated, with the variation of test parameters except the web reinforcement. 2. Experimental study 2.1. Materials Concrete mixtures are prepared by using crushed stone as coarse aggregate, sand as fine aggregate, and cement with standard compressive strength of 42.5 MPa. In order to obtain the desired compressive strength, 30 MPa, superplast-
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F. Okay, S. Engin / Construction and Building Materials 28 (2012) 269–275 Table 1 Nomenclature and fiber content of specimens.
df
lf Fig. 1. Geometry of the steel fibers used in the test.
icizer is also used in concrete mixtures. Bent ended steel fibers having four different aspect ratios lf/df = 40, 55, 67, and 80, corresponding to lf and df dimensions of 30/ 0.75, 30/0.55, 60/0.90, and 60/0.75 were used. Where lf denotes fiber length and df denotes diameter of fibers. Average yield strength for the steel fibers given by the manufacturer was fyf = 1200 MPa. Geometry for steel fibers is given in Fig. 1. dw = 8 mm deformed bars are used as web reinforcement, and dl = 8 mm and dl = 12 mm diameter deformed bars are used as longitudinal reinforcement. The average yield strength for both longitudinal and web reinforcement is found to be fyk = 460 MPa.
Beam designation
Longitudinal reinforcement
lf/df
Vf (%)
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
4 4 4 4 4 4 4 4 4 4 4 4
– 40 40 55 55 67 67 80 80 – 40 80
– 0.3 0.6 0.3 0.6 0.3 0.6 0.3 0.6 – 0.3 0.3
No. No. No. No. No. No. No. No. No. No. No. No.
of of of of of of of of of of of of
8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia. 8 mm dia 12 mm dia. 12 mm dia. 12 mm dia.
2.2. Test specimens The cross-sectional dimension of the test specimens are 150 200 mm with a length of 1900 mm. The first variable is the longitudinal reinforcement ratio ql, which is chosen as either 0.0067 (4U8) or 0.015 (4U12). Web reinforcement diameter is 8 mm and their center to center spacing is 200 mm, which corresponds to shear reinforcement ratio of qw = 0.006 for all specimens. The shear reinforcement is set to be constant, and the contribution of longitudinal reinforcement and the other variables to capacity is investigated. The tie-bar spacing outside the test region is decreased in order to force the failure of specimen to the test region. In those regions, the tie-bar spacing is 50 mm from center to center. The dimension of the test specimen and reinforcement layout is given in Fig. 2. The volumetric steel fiber ratio Vf is set to 0.3% and 0.6% for all specimens. With these two SFR ratios, it is aimed to observe the behavior of concrete with increasing volumetric content of fibers. It is seen that volumetric ratios that are higher than the ratios chosen in this study are used in some existing studies. In order not to have a settlement problem while preparing the specimens that have small distances between the reinforcement bars, the volumetric ratio of steel fibers is chosen to be 0.3% and 0.6%. For the designation of the specimens, the following procedure is followed: The first character block starting with L designates the diameter of the longitudinal reinforcement of the beam. Thus, L08 is used for longitudinal reinforcement with diameter dl = 8 mm and L12 for dl = 12 mm. Block F designates for the fiber aspect ratio F40, F55, F67, F80 designate for the aspect ratios 40, 55, 67, 80 respectively. Block V designates the volumetric content of the fibers added, where V3 shows 0.3% and V6 shows 0.6% volumetric content. F block is null and V is 0 while naming the specimens without SFR. Specimens’ designation is given in Table 1. It should also be noted that for each test specimen, 12 standard cylindrical specimens with 150 mm diameter and 300 mm height are prepared and tested to obtain the compressive strength, split tensile strength, and modulus of elasticity of concrete. In addition to the cylindrical specimens, five prismatic specimens with the dimensions 100 100 500 mm are prepared and tested to obtain flexural tensile strength of the concrete.
Fig. 3. Test setup, loading, and measurement systems.
the load that is applied to the specimen by a 300 kN capacity hydraulic cylinder pump set ((7) in Fig. 3) is measured by a 100 kN capacity electronic load cell. In this setup, the load that is applied by the hydraulic cylinder pump ((7) in Fig. 3) is shared equally by the ends of short beams ((5) in Fig. 3) that make the lever arm of the twisting moment. The unit angle measurements are read by an electronic gauge ((4) in Fig. 3) with a 100 mm capacity and a 0.02 mm accuracy which touches the arms ((8) in Fig. 3) which are located at the ends of the ‘‘test’’ region (Fig. 3). Test data are collected to a computer with the help of a data acquisition system, and torque versus unit angle of twist graph is obtained during the test. The data are taken every 0.01 rad/m unit angle of twist intervals, and the corresponding torque and crack widths are measured. Loading is terminated when the total unit angle of rotation value is reached to 0.12 rad/m.
2.3. Test setup, loading, and measurement systems Fig. 3 illustrates the test setup where the test specimens are subjected to uniform torsion. As shown in the figure, in order not to restrain the ends of the specimen from free rotation, extension or contraction, the specimen is placed on roller supports aligned with the specimen. Roller supports ((2) in Fig. 3) are settled at two ends of the specimen ((3) in Fig. 3) in order to maintain free rotation and elongation. The specimen is joined to the upper arms ((5) in the Fig. 3) and plates ((1) in Fig. 3), which are fixed to the roller supports by the help of bolts. Torque is obtained by applying a load to the center of the spreader beam ((6) in Fig. 3). The variation of
50
For each test specimens, cracking and ultimate torques, designated respectively as Tcr and Tu, and the corresponding unit angle of twists, designated as /cr and /u, are determined and listed in
200
200
190
140
3. Test results and discussions
150
450
Test Region = 1000
450
Fig. 2. Dimensions of the test specimen and the reinforcement layout (all dimensions are in mm).
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Table 2 Test data. Beam designation
fck (MPa)
fcts (MPa)
fctf (MPa)
E (GPa)
Tcr (kN m)
/cr (rad/m) 103
Tu (kN m)
/u (rad/m) 103
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
34.8 33.4 31.3 31.0 30.9 32.7 29.5 31.9 30.0 34.8 31.7 31.6
3.51 3.55 3.35 3.08 3.41 3.42 3.12 3.46 3.10 3.51 3.58 3.56
4.54 4.96 5.02 4.82 4.57 4.39 4.39 4.56 4.82 4.54 4.46 4.87
31.4 30.0 29.9 27.9 29.9 31.2 28.5 27.9 28.7 31.4 29.6 27.5
4.93 4.58 4.62 4.93 5.10 4.85 4.91 4.80 4.51 4.45 4.61 4.47
3.20 3.22 3.52 3.80 3.95 3.56 3.60 3.36 3.60 3.16 3.68 3.60
4.93 4.58 5.68 4.94 5.87 4.92 5.88 4.85 5.49 5.07 6.01 6.25
3.20 3.22 64.80 3.87 59.27 5.63 65.94 4.09 63.41 34.60 89.74 93.45
3.1. Crack patterns Number of cracks observed in reference specimens is less than the number of cracks in fiber reinforced beams when test is terminated; and failure becomes with the excessive widening of one of these cracks. When the unit angle of twist value reaches / = 0.06 rad/m, crushing of concrete in perpendicular direction to cracks is observed and this crushing is increase to unit angle of twist becomes 0.12 rad/m. The failure of the specimens whose volumetric content Vf = 0.3% is very similar to the reference specimens. Compressive crushing starts with the value from 0.06 to 0.07 rad/m of unit angle of twist and it is intensified toward the end of the experiment. Number of cracks is considerably increased while crack widths are decreased in the specimens whose volumetric content is Vf = 0.6%. Photographs of the specimens that show the above-mentioned behavior are given in Fig. 4.
3.2. Torsional strength The torsional strength of the test specimens without fibers is observed to be (i.e., Tu = Tcr) equal to the cracking torque of the specimens. The torque carried by the section decreases while the angle of twist increases, beyond /cr. On the other hand, the torque capacity continues to increase after cracking up to a certain value (i.e., Tu > Tcr) in specimens that contain steel fibers. This behavioral change is more apparent in the specimens with Vf = 0.6% compared to those with Vf = 0.3% (Fig. 5). L12 specimens, i.e., the specimens that have longitudinal reinforcement with dl = 12 mm, give more satisfactory results with the addition of Vf = 0.3% fibers compared to the specimens with longitudinal reinforcement with dl = 8 mm (Figs. 5 and 6). Test 8 7
Torque (kN.m)
designated. Table 2 also summarizes the average values of 28th day compressive strength fck, split cylinder tensile strength fcts and flexural tensile strength fctf of each test specimen. There is no considerable increase in split cylinder tensile strength fcts and flexural tensile strength fctf values in material tests, but remarkable increases in ultimate twisting moment Tu and corresponding unit angle of twist /u are observed in torsion tests with the increase in volumetric fiber ratio. In the material tests, reference specimens rupture with the first crack while the fiber reinforced specimens continue to sustain the load. This phenomenon confirms the increase in energy absorption of reinforced concrete beams under torsion.
6 5 4 3 2
L08F00V0 L08F40V3 L08F40V6
1 0
0
20
40
60
80
100
120
Unit Angle of Twist (rad/m) x10
140
-3
Fig. 5. The effect of volume content with fiber lf/df = 40.
8
Torque (kN.m)
7 6 5 4 3 2
L12F40V3 L12F80V3 L12F00V0
1 0
0
20
40
60
80
100
Unit Angle of Twist (rad/m) *10 Fig. 4. Crack patterns for (a) L08F00V0, (b) L08F40V3, (c) L08F40V6.
120 -3
Fig. 6. Torque-twist variation in specimens with dl = 12 mm.
140
273
4. Model proposed for torsional behavior A model is proposed using the data obtained in the tests. ‘‘Twisting moment versus unit angle of twist’’ graphics behave linearly up to cracking of the section and then this relationship is not seen. Cracking torque and cracking unit angle of rotation of all test specimens generally attain the same value regardless of the amount of longitudinal reinforcement. Torsional stiffness of the beams is significantly reduced after cracking. Since the specimens behave linearly elastic before cracking, cracking torque of the specimens can be computed from the following expression Eq. (4.1) derived by using elasticity theory [23]. Test results clearly show that the torsional behavior of test specimens is linear in the region up to cracking, in which Saint Venant equation is valid, beyond which the linear behavior is lost [19]. This type of behavior is consistent with the studies in the literature [3]. The fact that the cracking values (i.e., Tcr and /cr) of all specimens are almost the same indicates that the cracking characteristics of test specimens are affected neither from the amount of longitudinal reinforcement nor from the presence of steel fibers. 2
T cr ¼ C 1 hb fctf
T cr L 3
C 2 hb G
ð4:2Þ
where L is the length of testing region, C2 is a coefficient that takes values between 0.1406 and 0.333 depending on the dimensions of the section. In this study, C2 is taken to be 0.179 according to the dimensions of the specimens used. G is the shear modulus of the concrete and is obtained by Eq. (4.3), using the well-known relation,
G¼
E 2ð1 þ mÞ
ð4:3Þ
where E is the modulus of elasticity of concrete, which can be obtained from material tests. Poisson’s ratio of concrete m can be taken as 0.2 [24]. Ultimate torque and the corresponding unit angle of rotation of the sections take different values depending on the amount of the longitudinal reinforcement of specimens. For the specimens with dl = 8 mm and Vf = 0.6%, the ultimate torque values are observed to be around 6 kN m and the corresponding unit angle of rotation value is /u = 60 103 rad/m. The cracking torque Tcr, ultimate torque Tu, and the torque value at the termination of the experiments Tf and the corresponding values of the unit angle of rotations /cr, /u, and /f, respectively, are plotted in Fig. 7. In this figure, triangular dots stand for Tcr, rectangular dots for Tu, and diamond ones stand for Tf values. As it is shown from Fig. 7, the ultimate torque values are equal to the cracking torque values of the specimens whose longitudinal reinforcement diameter is dl = 8 mm and volumetric fiber ratio is Vf = 0.3%; whereas torque values of the specimens with Vf = 0.6% increase after cracking and reach their capacities approximately at the same value irrespective of aspect ratio. When the terminating unit angle of rotation value is reached, the specimens that have no fibers and having a volumetric fiber ratio of 0.3% posses the
Tu
Tcr
0
20
Tf
40
60
80
100
120
140
-3
Fig. 7. Tcr, Tu, and Tf values of the specimens with dl = 8 mm.
7 6
Tu
Tcr
5
Tf
4 3 2 1 0
0
20
40
60
80
100
120
140
Unit Angle of Twist (rad/mx10 -3 )
ð4:1Þ
where h and b are longer and shorter sizes of the rectangular concrete section, respectively, fctf is the flexural tensile strength of the concrete, and C1 is a coefficient taking values between 0.208 and 0.333 depending on h/b ratio of the section. In this study, C1 is taken to be 0.224 based on the dimensions of the specimens used. Similarly, elasticity theory provides the following Eq. (4.2) expression for the unit angle of rotation at cracking:
/cr ¼
7 6 5 4 3 2 1 0
Unit Angle of Twist (rad/m) x10
Torque (kN.m)
results show that the aspect ratio of the steel fiber has no significant effect on the capacity and ductility of the beam.
Torque (kN.m)
F. Okay, S. Engin / Construction and Building Materials 28 (2012) 269–275
Fig. 8. Tcr, Tu, and Tf values of the specimens with db = 12 mm.
Table 3 V0, /u, and b coefficient according to longitudinal reinforcement ratio.
V0 /u (rad/m) 103 b
0.85ql < qw
0.85ql > qw
0.005 60
0.002 90 7410Vf + 2.62
635; 278V 2f 450:83V f þ 2:56
same Tf values. Similarly, there are no significant differences among the Tf values of specimens having a volumetric ratio of 0.6%. Tu values are observed to be approximately 6 kN m, and /u values are observed to be 90 103 rad/m when dl = 12 mm and Vf = 0.3% as it is seen Fig. 8. Cracking torque values of the specimen having dl = 12 mm are observed to be identical irrespective of the volumetric ratio whereas Tu and Tf values differ. Behavior of the specimens after cracking can be expressed by the following empirical relationship Eq. (4.4) between the torque and the unit angle of rotation:
Fig. 9. Proposed torsional behavior model for reinforced concrete beams with/ without steel fibers.
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Table 4 Comparison of the Tcr and /cr values obtained from model and test results. Beam designation
Tcr-test
Tcr-eq (kN m)
ETcra (%)
/cr-test (rad/m) 103
/cr-eq (rad/m) 103
E/cr b (%)
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
4.93 4.58 4.62 4.93 5.10 4.85 4.91 4.80 4.51 4.45 4.61 4.47
4.58 5.00 5.06 4.86 4.61 4.43 4.43 4.60 4.86 4.58 4.50 4.91
7.2 9.2 9.5 1.4 9.7 8.8 9.9 4.2 7.7 2.8 2.5 9.8
3.20 3.22 3.30 3.58 3.75 3.56 3.60 3.36 3.34 3.18 3.28 3.37
3.13 2.91 2.94 3.13 3.24 3.08 3.12 3.05 2.87 2.83 2.93 2.84
2.1 9.6 11.0 12.4 13.5 13.4 13.3 9.2 14.2 11.0 10.6 15.7
a
ET cr ¼
b
E/cr ¼
T cr-eq T cr-test . T cr-test /cr-eq /cr-test /cr-test
100.
Table 5 Comparison of the Tu and /u values obtained from model and test results. Beam designation
Tu-test (kN m)
Tu-eq (kN m)
ETua (%)
Tf-test (kN m)
Tf-eq (kN m)
E
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
4.93 4.58 5.68 4.94 5.87 4.92 5.88 4.85 5.49 5.07 6.01 6.25
4.58 5.00 6.42 4.86 5.97 4.43 5.79 4.60 6.22 4.58 6.74 7.15
7.2 9.2 13.1 1.6 1.7 10.1 1.5 5.2 13.3 9.7 12.1 14.4
2.74 3.12 3.90 2.96 4.02 2.42 3.54 3.05 3.86 3.45 4.75 5.61
3.08 3.38 3.83 3.24 3.39 2.81 3.18 2.98 3.63 3.97 5.39 5.80
12.4 8.4 1.8 9.5 15.7 15.9 10.2 2.3 6.0 15.1 13.5 3.4
a
ET u ¼
b
ET f ¼
b
(%)
T u-eq T u-test 100. T u-test T f -eq T f -test 100. T f -test
Table 6 Comparison of the absorbed energy obtained from model and test results.
a
Tf
Beam designation
Utest (N m/m)
Umodel (N m/m)
EUa (%)
L08F00V0 L08F40V3 L08F40V6 L08F55V3 L08F55V6 L08F67V3 L08F67V6 L08F80V3 L08F80V6 L12F00V0 L12F40V3 L12F80V3
454.57 497.92 644.79 480.83 607.32 429.53 577.32 450.25 625.84 507.38 707.01 762.72
424.08 442.84 598.55 460.18 524.69 377.12 593.53 458.14 596.03 484.72 631.53 660.85
7.2 12.4 7.7 4.5 15.7 13.9 2.7 1.7 5.0 4.7 12.0 15.4
U test EU ¼ U model 100. U test
T ¼ T cr þ bð/ /cr ÞðV f V 0 Þ 103 3bh/ /u ihV f V 0 i 103
ð4:4Þ
where of relevant V0, /u, and b values are given Table 3. The V0, /u, and b coefficient in Eq. (4.4) is obtained from a regression analysis. The brackets h i in Eq. (4.4) are used to designate singularity functions. Thus, if the value inside the brackets is greater than zero, the brackets behave as regular parenthesis; if not the value becomes zero [23]. The torsional model expressed by Eq. (4.4) is plotted in Fig. 9. In this model, Eq. (4.1) is valued in the initial elastic region (Region I) and Eq. (4.4) can be used in the following plastic regions (Regions II and Regions III). According to this model, the test specimens show
the same behavior up to cracking regardless of the amount of longitudinal reinforcement and presence, amount, size of steel fibers observed during the tests. After cracking, the specimens with no fibers and those with fiber volumetric ratio of 0.3% follow the dashed path if 0.85ql < qw, while the specimens with Vf = 0.6% follow the solid path in Fig. 9. If 0.85ql > qw, the specimens with no fibers follow the dashed line; while those with 0.3 percent fiber ratio follow the solid line. For all the specimens that follow the dashed line, the ultimate torque Tu equals to the cracking torque Tcr. In Tables 4 and 5, the results obtained from the model compared to the values of the test and the differences between those values are accepted to be errors. As it is seen from the tables, the largest difference percent between the models and the test result is less than 16%. The energy per unit length absorbed by the element during the test is obtained from the area under the unit angle of rotation versus torque plots of both the tests and models are given and compared in Table 6. 5. Conclusions Based on the limited experiments results in which different aspect ratios of steel fibers are used in pure torsion test of reinforced concrete beams, the following conclusions have been obtained: It is observed that torsional behavior of normal strength concrete beams is changed positively with the addition of steel fibers. Although 0.3% volumetric content fiber addition does
F. Okay, S. Engin / Construction and Building Materials 28 (2012) 269–275
not make significant effect on this behavior, 0.6% volumetric content fiber addition increases the torsional strength of the test specimens with an amount of 10–60%. The specimens having insufficient longitudinal reinforcement do not show any increase in torsional strength with the addition of volumetric content steel fibers in 0.3%. Sufficient addition of steel fibers to concrete causes a decrease in the widths of cracks, whereas an increase in the number of cracks. The simple model proposed in this study yielded comparative results with the data produced in this study.
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