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Punching-shear behaviors of RC-column footings with various reinforcement and strengthening details
Engineering Structures 151 (2017) 282–296
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Punching-shear behaviors of RC-column footings with various reinforcement and strengthening details Gia Toai Truong, Kyoung-Kyu Choi ⇑, Hee-Seung Kim School of Architecture, Soongsil Univ., 369 Sangdo-ro, Dongjak-gu, Seoul 06978, South Korea
a r t i c l e
i n f o
Article history: Received 27 February 2017 Revised 7 May 2017 Accepted 13 August 2017
Keywords: Column footings Punching shear Springs Soil pressure Amorphous metallic fiber (AMF) Shear reinforcement
a b s t r a c t In this study, various reinforcement and strengthening details for the improvement of the punchingshear behavior of concrete footings were developed. Eight test specimens of reinforced-concrete (RC) footings were constructed and tested to investigate the punching-shear behavior of the concrete footings with the developed details. The test parameters include the additional placement of a longitudinal reinforcement around the column-footing connections, the inclination of the shear reinforcement, an additional cast of high-strength concrete, and an additional concrete cast with amorphous metallic fibers (AMFs). The test specimens were supported on a bed of steel car springs that simulate the elastic behavior of soil. The test results show that the punching strength was significantly enhanced by the additional placement of the longitudinal reinforcement around the column-footing connections and by the addition of the retrofit materials (high-strength concrete and AMF-reinforced concrete). Contrarily, the shear reinforcement that was placed around the connections did not significantly affect the punching-shear strength of the footings. The test results were also compared with strength predictions for which the current design codes and the existing theoretical models were used. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The gravity loads of structures need to be transferred to footings resting on soil. Since the soil pressure that develops beneath the footings is dependent on the soil type and stiffness, it is difficult to predict the structural behavior of the footings. In addition, the punching-shear failure of concrete footings is brittle, which could finally lead to a progressive collapse of whole structures [1,2]. Therefore, extensive experimental and theoretical studies have been performed to investigate the punching-shear behavior of concrete-column footings. Coduto [3] and Hegger et al. [4] investigated the effects of the soil type and rigidity (or stiffness) on the actual distribution of the soil pressure beneath concrete footings that were subjected to a vertically concentrated load. In Fig. 1(a) and (b), for a footing resting on loose noncohesive soil (e.g., sandy soil), a convex pressure distribution develops, and for a footing resting on cohesive soil (e.g., clay), a concave pressure distribution develops. Because of the complexity of the soil-pressure distribution, a linear soilpressure distribution beneath the footings is usually assumed for design purposes (Fig. 1(c)). ⇑ Corresponding author. E-mail address: [email protected] (K.-K. Choi). http://dx.doi.org/10.1016/j.engstruct.2017.08.037 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
Hegger et al. [5] and Bonic and Folic [6] performed punching tests on reinforced-concrete (RC) footings using real sand. The test results indicated that the punching-shear capacity increased with the increasing of the effective depth and the decreasing of the shear-span/depth ratio, and the angles of the observed punchingshear cracks in the footing tests are steeper than those of the slender slabs. In addition, the sand stiffness (loose or dense) did not significantly affect the distribution of the soil pressure underneath the footing. In Winkler’s soil model [7], the soil underneath the footings could be simulated by a number of independent elastic springs with a constant spring stiffness ks , known as the ‘‘modulus of soil reaction”. According to Meda et al. [8] and Bowles [9], the value of ks ranges from 0.0048 N/mm3 to 0.128 N/mm3. Based on Winkler’s theory, Kent and Math [10] and Richart [11] performed punching tests using a bed of car springs that simulated a uniform soil-pressure distribution. From the test results, the punchingshear surfaces are close to the column surfaces, and the specimens with a higher longitudinal-reinforcement ratio exhibited a higher punching-shear capacity [10]. In addition, the footings for which the reinforcing bars with hooks at the end are used did not show considerable differences in the bond resistance compared with those for which the reinforcing bars without hooks are used; the fact is due to the hooks did not significantly prevent end slips [11].
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P
P
(a) Loose cohesive-less soil
P
(b) Cohesive soil
(c) Linear pressure distribution
Fig. 1. Soil pressure distribution beneath an elastic footing.
For the present study, various steel-reinforcing details for the new construction of concrete footings and the strengthening details for the retrofit of existing concrete footings were developed. Eight half-scale RC footings with the developed details were constructed and tested. The main parameters of the test specimens are the additional placement (or concentrated layout) of longitudinal rebars around the column-footing connections, the inclination of the shear reinforcement, an additional cast of high-strength concrete, and an additional cast of amorphous metallic fiber (AMF)-reinforced concrete. In this test, a bed of steel car springs that simulate the elastic behavior of soil was used to support the test specimens. The effectiveness of the var-
ious reinforcing and strengthening techniques was compared in terms of the strength, crack pattern, and strain profiles of the footings. Moreover, the test results were compared with the strength predictions from the current design codes and the existing theoretical models.
Fig. 2. Details of specimen F1.
Fig. 3. Details of specimens F2 and F3.
2. Experimental program 2.1. Test specimens Eight reinforced concrete-footing specimens with a scale factor of 1/4 of a common footing were constructed, as follows: One is a control specimen, and the other specimens are different from the control specimen in terms of the steel-reinforcement details and the additional cast of the retrofit materials (concrete or AMFreinforced concrete). The geometries and reinforcement details of the test specimens are classified into three series, as presented in Figs. 2–5 and Table 1. The first series are the three footing specimens with a uniform or concentrated layout of the longitudinal rebars around the column-footing connections, which are denoted
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Fig. 4. Details of specimens FS1 and FS2.
Fig. 5. Details of specimens FC1, FC2, and FC3.
as ‘‘F series”, and simulate new footing specimens. The second series are two footing specimens that have been strengthened with shear rebars, denoted as ‘‘FS series”. The third series are three footing specimens that have been strengthened with an additional cast of high-strength concrete or AMF-reinforced concrete, which are denoted as ‘‘FC series”. Fig. 2 shows the details of the control specimen F1, which was designed to fail in punching shear. The main properties of the control specimen F1 are as follows: effective depth (d) is 100 mm; shear span (=L=2 c=2)-to-depth ratio (a=d) is equal to 4; longitudinal reinforcing bars are D10 with a diameter of 10 mm; and the longitudinal reinforcement ratio (q) is 0.61%; here, L and c are the
sizes of the footings and columns, respectively. The specimens F2 and F3 differ from the specimen F1 in the amounts of the bottom and top longitudinal reinforcements, respectively (Fig. 3). It should be noted that in the specimens F2 and F3, additional longitudinal rebars were placed only in the critical section zone (=c þ 2d, refer to ACI314R-11 [12]). In particular, in the case of the specimen F3, additional rebars were arranged as a top reinforcement, and the length was cut down by 3.2d with a hook length of 100 mm (see Fig. 3). The specimens of the FS and FC series are the retrofitted specimens of the control specimen F1; therefore, the specimens of the FS and FC series have the same material and geometrical properties
Table 1 Details of test specimens.
a b c d e f g
Specimens
d (mm)
c (mm)
l (mm)
a/da
q (%)
f’c (MPa)
fyb (MPa)
Av (mm2)
s (mm)
Feature
F1 F2 F3 FS1 FS2 FC1 FC2 FC3
100 100 100 100 100 100 (150)c 100 (150)c 100 (150)c
100 100 100 100 100 100 100 100
900 900 900 900 900 900 900 900
4 4 4 4 4 4 (2.67)d 4 (2.67)d 4 (2.67)d
0.61 0.96 0.79 0.61 0.61 0.61 (0.41)e 0.61 (0.41)e 0.61 (0.41)e
29.74 29.74 29.74 29.74 29.74 29.74 29.74 29.74
435 435 435 435 435 435 435 435
–f –f –f 78.5 78.5 –f –f 78.5
–f –f –f 80 (85)g 100 (60)g –f –f 50 (30)g
Control specimen Addition of bottom re-bars Addition of top re-bars Inclined shear re-bars Shear re-bars Additional cast of high strength concrete Additional cast of AMF reinforced concrete Additional cast of AMF RC and shear re-bars
Shear span to depth ratio (=(l c)/(2d)). Yield strength of longitudinal re-bars and shear re-bars. Effective depth considering retrofit layer. Shear span to depth ratio considering retrofit layer. Longitudinal reinforcement ratio considering retrofit layer. Not applicable. Spacing between column face and first row of shear reinforcement.
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Flexural strength, σ (MPa)
8
P/2
P/2
a
6
Δ
Table 2 Geometrical and material properties of AMFs.
b
a=100 h b=100 h=100 Unit: mm
3 a P 2 bh 2
Parameters Fiber thickness Fiber length Aspect ratioa Fiber width Yield tensile strength Modulus of elasticity Density
Amorphous metallic FRC
4
a
Plain concrete
2
0 0
0.5
1
1.5
2
2.5
Deflection (mm) Fig. 6. Flexural strength-deflection curves obtained by Truong [16]
as that of the control specimen F1, except for the retrofitted parts. Fig. 4 shows the details of the specimens FS1 and FS2. In the figure, the specimen FS1 has been strengthened with inclined shear rebars (D10) having a diameter of 10 mm, and the angle between the inclined shear rebar and the horizontal axis is 45°. The use of the inclined shear reinforcement was referred to the study by Hallgren et al. [13]. Similar to FS1, the specimen FS2 was also strengthened with shear rebars (D10), but the angle is 90° (or a vertical shear reinforcement). In the FC series, a high-strength concrete layer for the FC1 and an AMF-reinforced concrete layer for the FC2 and FC3 were casted. After the retrofitting, the effective depth of the FC-series specimens is 150 mm. For all of the footing specimens, the footing size is 900 mm 900 mm in length and width, respectively, and the column studs are of a 100 100 mm cross-section and a 100 mm height. The longitudinal reinforcement of the columns consists of four D16 steel rebars, and a transverse reinforcement consists of D10 steel rebars with four legged and closed hoops (refer to Fig. 2). It is noted that, for the installation of the test specimens, four hooks were installed at the corners of the specimens, but they were intentionally not presented in the figures. For this test, the specimens of the FS and FC series were constructed according to the actual retrofit procedures. In the FS series (Fig. 4), the locations of the holes were drilled as per the designed configurations, as follows: 90 mm 20 mm (depth diameter) and the angle of 45° for the specimen FS1; and 105 mm 20 mm and the angle of 90° for the specimen FS2. The holes were then cleaned carefully, followed by the installation of the shear rebars through the holes with a length of 90 mm for FS1 and 100 mm for FS2. For each specimen, a total of 16 shear rebars were used and arranged into a two-row cross in each direction. Lastly, the remaining gaps of the holes were filled with non-shrinkage mortar. In the specimens of the FC series, the top parts of the concrete (10 mm) were first chiseled out. The work was carefully carried out to avoid damaging the internal concrete. After that, the holes were drilled according to the designed configuration, and the steel rebars were then installed through the holes with a length of 30 mm for FC1 and FC2 and 100 mm for FC3. It is noted that the steel rebars were bent into an L-shape for a more-effective connection between the old and new concrete layers. In total, 36 steel rebars were used including six rows in each direction and four additional steel rebars around the column stubs. In FC1 and FC2, short steel rebars were used and were considered as anchor bars, which contributed to the bonding behavior between the old and new concrete layers. Meanwhile, in FC3, long steel rebars were
Amorphous metallic fibers (AMFs) (lm) (mm) (mm) (MPa) (MPa) (kg/m3)
29 30 1034 1.6 1700 140,000 7200
Aspect ratio is the ratio of fiber length to fiber thickness.
used and were considered as shear reinforcement, which contributed to not only the bonding behavior between the old and new concrete layers, but also the shear strength of the specimen. After this stage, the formwork was assembled to ensure the additional concrete-layer dimensions, as follows: 600 mm 600 mm 60 mm. Then, the high-strength concrete layer for the specimen FC1 was cast, followed by the casting of the AMFreinforced concrete layer for the specimens FC2 and FC3. 2.2. Materials In this test, the concrete compressive test that was carried out on the loading date of each specimen is according to the KS F 2405 test standard [14]. The average compressive strength is 29.7 MPa at the 107th day, which is the loading date. In addition, the average compressive strength of the additional concrete is 41.2 MPa, while that of the AMF-reinforced concrete is 52.7 MPa at the loading date. In the case of the AMF-reinforced concrete, the tensile strength was determined based on the flexural test, which was performed according to the ASTM C 1609/1609M-12 test standard [15]. The peak tensile strength of the AMFreinforced concrete that was obtained from the test results is 5.46 MPa. The test results were almost the same as that of Truong et al.’s test [16]. It is noted that in the study by Truong et al. [16], the design compressive strength was also 50 MPa, and the same concrete-mix properties were used as that of present test. Fig. 6 presents the flexural strengthdeflection curves in the flexural test of the specimens that had been reinforced with the AMFs [16]. In Fig. 6, it is obvious that the AMF-reinforced concrete with the fiber-volume fraction of 0.75% exhibited a considerably greater peak strength than that of the plain concrete. The geometrical and material characteristics of the AMFs are presented in Table 2, and can be also found elsewhere [17]. Tensile testing of the longitudinal rebars was performed according to the KS B 0802 [18] and KS B 0814 test standards [19]. The diameter of the steel rebars denoted as D10 is 10 mm, and their specified yield strength is 400 MPa. From the test results, the measured yield tensile stress of D10 is 435 MPa, which exceeds the specified yield strength of 400 MPa, and the corresponding yield strain is 0.0022. In terms of the evaluation of the test results hereafter, the material strengths that were measured from the test were used instead of the specified strengths of the concrete and the steel rebars. 2.3. Test setup Each column footing was subjected to a concentrated axial load through a column stub using a hydraulic actuator with a capacity of 500 kN. Fig. 7 shows the test setup of the column-footing tests. In Fig. 7(a) and (b), a strong main steel frame was assembled on a strong floor where it could support the reaction forces from the
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Actuator Steel frame
A Specimen
LVDT frame Bearing steel blocks
Car springs
Strong floor (a) Test setup
(b) Photo of test setup Hinging plates
Load cell LVDT frame
Three dimensional hinge
LVDTs Springs Specimen
Protecting steel blocks (d) Details of supports
(c) Detail of A region
(e) Three dimensional hinge Fig. 7. Test setup.
hydraulic actuator. Fig. 7(c) and (d) present the details of the connection parts of the actuator and the specimen, and the supports. Around the bed of the steel car springs, steel blocks were assembled to avoid the lateral movement of the specimens (see Fig. 7 (d)). In addition, a steel three-dimensional (3 D) hinge was placed between the actuator head and the specimens to transfer the axial force during the loading process (see Fig. 7(c) and (e)). The axial load was applied to the specimens in a displacement-controlled manner at a loading rate of 0.05 mm/s. The test specimens were supported on a bed of nine steel car springs. As stated previously, the car springs were used to simulate the elastic behavior of soil. Fig. 8 shows the layout of the steel car springs in square grids with a center-to-center spacing of 300 mm. In the figure, each car spring would replace an area of 300 300 (mm2) of the soil beneath the footing. Thus, for the given test setup, the modulus of soil reaction (ks) was estimated as 0.016 N/mm3; this is equal to the spring stiffness (1,400 N/mm)
divided by the relevant area of soil (300 mm 300 mm) replaced by a spring. It is noted that the modulus of soil reaction of 0.016 N/mm3 represents the clayey soil (refer to [9]). Each steel spring has a 45 mm spring diameter, a 200 mm outer diameter, a 600 mm height, and the spring stiffness retains a constant of 1400 N/mm within an elastic limit of deformation (190 mm). The springs used in this test exhibit the same behavior of those used by Lee et al. [20]. In addition, the installation of the car-spring system is also referred to in [20]. As shown in Fig. 8 (b), a 200 mm diameter circular hinge, which can rotate in all directions, was placed on the top of each spring. Then, a steel plate with the dimensions of 250 mm 250 mm 20 mm was placed on top of the hinge to distribute the reactive force of the spring to the bottom surface of the footings during the loading. A steel-bearing plate of 300 mm 300 mm 20 mm was welded to the spring at the bottom and connected to a strong floor to maintain the stability and to prevent the spring from overturning during the loading.
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Steel plate (250x250)
900
300
250
250
300
Steel bearing plate (300x300)
200
300
250
Springs
287
show the arrangement of the steel- and concrete-strain gauges. In Fig. 9, to measure the longitudinal rebar strains, the four steel-strain gauges that were used in the test specimens F1, FS1 to FS2, and FC1 to FC3 are shown (Fig. 9(a)). Further, 10 steelstrain gauges were used in the specimen F2 (Fig. 9(b)), and six steel-strain gauges were used in the specimen F3 (Fig. 9(c)). In the case of the specimens FS1, FS2, and FC3 (refer to Fig. 10(a)), the strains of the shear rebars were also measured at the midheight, and eight steel-strain gauges were installed for each specimen. In Fig. 10(b), the concrete strains for each specimen were measured with the use of four concrete-strain gauges. In addition, four line linear variable-displacement transducers (LVDTs; see Fig. 10(b)) were installed to measure the deflection of the specimens.
3. Experiment results
250 300
250 300
250 300
3.1. Failure mode and crack pattern
900
(a) Plane Test specimen Steel plate (250x250)
600
Spring
725
Hinge
Steel bearing plate (300x300) Strong floor
(b) Elevation
(c) Photo of spring system
Unit: mm
Fig. 8. Spring-type soil system.
2.4. Measurement instrumentations Steel- and concrete-strain gauges were attached onto the longitudinal and shear rebars and the top concrete surface to measure the steel and concrete strains, respectively. Figs. 9 and 10
Most of the test specimens failed in the punching shear of the footing, except for the specimen FC1, which failed in the concrete crushing of its column stub. The failure loads are given in Table 3, wherein the shear demand (V flex ) corresponding to the flexural failure of the footings, which were evaluated according to Simoes et al. [21] (see Appendix A.1), are also presented to investigate the failure mode of the test specimens. It is obvious that the punching-shear capacities that were obtained from the test results are lower than the shear demand (V flex ) corresponding to the flexural failure of the footings. This finding indicated that the shear demand (V flex ) of the footings had not been reached, confirming that the failure occurred due to the punching shear. This observation was also verified using the yield-line analysis, which was suggested by Gesund and Kaushik [22] (see Appendix A.2). In Table 3, the load-carrying capacity (V yieldline ) predicted by the yield-line analysis is considerably greater than the test results (V exp ), and this indicates that no flexural yield line developed in the test specimens. It is noted that in the yield-line analysis of the specimens FC2 and FC3, the additional cast of the AMFreinforced concrete was considered in the evaluation of the effective depth and the longitudinal-reinforcement ratio (see Table 1). In the case of the specimen FC1, the column-axial strength, which was calculated based on ACI 318-14 [23], is 582.1 kN, and this value is greater than the column-failure loads that were observed in the tests (see Table 4). As mentioned previously, however, the column stub of FC1 failed locally in the concrete crushing, it suddenly lost the specimen stability, and then the test was intentionally terminated. Fig. 11 shows the crack pattern at the end of the testing. In general, at the bottom surfaces of the test specimens, several thin cracks that run toward the corners of the specimens first appeared. Then, numerous cracks immediately occurred along the layout of the reinforcing bars; furthermore, several radial cracks appeared and propagated toward the edges (but not the corners) of the specimens, and this is particularly the case in the specimen F2 (see Fig. 11(b)). Lastly, a closed or half-closed punching-failure line was developed, but the shape is different in each specimen. At the punching failure, a significant concrete crushing was also observed. In Fig. 11, the bold line indicates the punching cone at the punching failure of the test specimens. In the case of the specimen FC1 (Fig. 11(f)), the punching-failure line is not observable, but radial cracks developed; here, the observed cracks are thin, and they are not considered as twoway yield lines because the column was crushed ahead of the specimen failure.
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(a) F1, FS1-2, and FC1-3
(b) F2
(c) F3
Steel strain gauge Fig. 9. Arrangement of steel strain gauges attached on longitudinal re-bars.
3.2. Effect of the additional placement of the longitudinal reinforcement around the column-footing connections
300
20
300
Line LVDTs
Column
20
Column
Steel strain gauges
Concrete strain gauges
(b) Location of concrete strain gauges and LVDTs for all specimens
(a) Location of steel strain gauges on shear re-bars of specimens FS1, FS2, and FC3
Unit: mm Fig. 10. Arrangement of steel strain gauges attached on shear re-bars, concrete strain gauges, and line LVDTs.
Fig. 12 shows the measured load versus the center deflection of the column footings for all of the test specimens. In this figure, it is obvious that all of the specimens, except for FC1, failed in the brittle punching failure. Most of the specimens immediately showed a sudden load drop after they reached the peak load; this is attributed to the reaction forces that were provided by the car springs, which cause a sudden failure and a load drop like that of the test in a force-controlled manner. In Fig. 12, the peak load showed variations according to the test parameters. Fig. 13 (a) presents a comparison of the loaddeflection curves of the footings F1, F2, and F3 for the investigation of the effect of the additional placement of the longitudinal reinforcements. The peak load and the deflection at the failure of the control specimen F1 are 208.7 kN and 23.92 mm, respectively. In the specimen F2 that had a greater bottom longitudinal
Table 3 Load-carrying capacity predicted based on flexural failure mode and yield line analysis. Specimens
F1
F2
F3
FS1
FS2
FC1c
FC2
FC3
Mean
COV
V exp (kN) V flex a (kN) V yieldline b (kN)
208.7 307.8 236.6 0.68
250.7 483.7 344.9 0.52
225.3 280.6 302.2 0.80
207.6 307.8 236.6 0.67
208.8 307.8 236.6 0.68
250.8 –d –d –d
319.9 461.7 363.6 0.69
287.9 461.7 363.6 0.62
0.67
0.083
0.88
0.73
0.75
0.88
0.88
–d
0.88
0.79
0.83
0.085
KCI 2012 [26]
ACI 318–14 [23]
EC2 [25]
1.33 1.34 1.30 1.32 1.32 –c 1.14 1.03
1.37 1.65 1.48 1.36 1.37 –c 1.14 1.03
0.84 0.87 0.84 0.84 0.84 –c 0.84 0.76
1.25 0.10
1.35 0.14
0.84 0.03
V exp V flex V exp V yieldline a b c d
V flex is evaluated according to Simoes et al. [21]. V yieldline is evaluated according to Gesund and Kaushik [22]. column failed. not applicable.
Table 4 Punching shear capacity of footings predicted by current design codes. Specimens
V exp a (kN)
V frc (kN)
V n;KCI (kN)
V n;ACI (kN)
V n;EC (kN)
F1 F2 F3 FS1 FS2 FC1d FC2 FC3
208.7 250.7 225.3 207.6 208.8 250.8 319.9 287.9
–c –c –c –c –c –c 113.7 113.7
145.3 172.6 160.2 145.3 145.3 –c 267.3e 267.3e
144.8 144.8 144.8 144.8 144.8 –c 266.8e 266.8e
228.4 265.6 248.9 228.4 228.4 –c 350.4e 350.4e
Mean COV a b c d e
V net =V code b
Values before subtracting the load transferred to the columns. V net is net applied shear force neglecting upward uniform pressure beneath concrete columns, which is defined differently in each design code. Not applicable. Column failed. V n is punching shear capacity predicted by the summation of V c and V frc .
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(a) F1
(b) F2
(c) F3
(d) FS1
(e) FS2
(f) FC1
(g) FC2
(h) FC3
Applied load (kN)
350 300 250 200 150 100 50 0
Applied load (kN)
350 300 250 200 150 100 50 0
Applied load (kN)
Fig. 11. Crack pattern at failure of test specimens.
350 300 250 200 150 100 50 0
350 300 250 200 150 100 50 0
(a) F1 Peak load: 208.7 kN Deflection: 23.92 mm
0
10
20
30
40
50
Peak load: 207.6 kN Deflection: 23.40 mm
10
20
30
40
50
Peak load: 319.9 kN Deflection: 39.02 mm
10
20
10
30
40
50
350 300 250 200 150 100 50 0
Peak load: 250.7 kN Deflection: 28.26 mm
20
30
40
350 300 250 200 150 100 50 0
Peak load: 208.8 kN Deflection: 23.35 mm
10
20
30
40
(c) F3
0
50
(e) FS2
0 350 300 250 200 150 100 50 0
(g) FC2
0
0 350 300 250 200 150 100 50 0
(d) FS1
0
(b) F2
50
10
Peak load: 225.3 kN Deflection: 26.15 mm
20
30
40
50
30
40
50
(f) FC1* Peak load: 250.8 kN Deflection: 34.87 mm
0
10
20
Deflection (mm) (h) FC3 Peak load: 287.9 kN Deflection: 41.49 mm
0
10
20
30
40
50
Deflection (mm) Deflection (mm) * In the case of specimen FC1, the column stub was failed before punching shear failure may occur. Fig. 12. Load-deflection curves of test specimens.
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Applied load (kN)
350
F1 F2 F3
300 250 200 150 100 50 0
0
10
20
30
40
50
Deflection (mm) (a) Layout of longitudinal reinforcement
Applied load (kN)
350
3.4. Effects of the additional casts of the high-strength concrete and the AMF-reinforced concrete
F1 FS1 FS2
300 250 200 150 100 50 0 0
10
20
30
40
50
Deflection (mm) (b) Shear reinforcement
Applied load (kN)
350
F1 FC1 FC2 FC3
300 250
rebars at the first row) are relatively steep with a high inclination, which was expected to be more than 45°, and therefore did not cross the shear rebars. This phenomenon is opposite to the findings of Hegger et al. [24], where the shear reinforcement significantly affected the shear strength of the footings. In addition, the use of shear reinforcement without head or nuts in the thin concrete footings in this study could lead to poor anchor and thus might reduce the effect of shear reinforcement. Further research is necessary to obtain a better understanding of this behavior. In the evaluation of the test results hereafter, the contribution of the shear reinforcement to the punching-shear resistance of the footings is not included.
200 Column failure
150
Fig. 13(c) compares the load-deflection curves of the specimens FC1, FC2, and FC3 after they had been strengthened with the highstrength concrete, the AMF-reinforced concrete, and both the AMFreinforced concrete and the shear reinforcement, respectively, with that of the control specimen F1. In this figure, both the strength and the deflection at the failure of the retrofitted specimens are higher than those of F1; the strength increased by 20.2–53.3%, and the deflection at the failure increased by 45.8–73.5%. It is obvious that the additional cast of the retrofit materials significantly enhanced the punching behavior, especially regarding the strength and the deflection at the failure. Regarding the specimens FC2 and FC3, the shear reinforcement did not cause a significant increase of the punching-shear strength of the footings in this study (see Fig. 14(c)). 3.5. Strain profiles of the longitudinal rebars and concrete
100 50 0 0
10
20
30
40
50
Deflection (mm) (c) Additional cast of high strength concrete and AMF reinforced concrete Fig. 13. Effects of test parameters on load-deflection curves.
reinforcement around the connection, the peak load and the deflection at the failure of the specimen F2 are 250.7 kN and 28.26 mm, respectively. To the contrary, even though the specimen F3 also had a greater longitudinal reinforcement, the top rebars showed a relatively less-robust increment compared with that of F2. 3.3. Effect of the shear reinforcement Fig. 13(b) compared the load-deflection curves of the footings F1, FS1, and FS2 to investigate the effect of the shear reinforcement. In the figure, no significant difference was observed for the three specimens. The shear reinforcement did not considerably contribute to the shear resistance of the columns footings for a given test condition. The strain profiles of the shear rebars of the specimens FS1 and FS2 were obtained from the test results and are presented in Fig. 14(a) and (b). In this figure, it is obvious that the strains of the shear rebars are much lower than their yielding strain of 0.0022 when the punching failure occurred. The shear rebars close to the column stubs exhibited relatively great strains, but their yield strain could not be reached. This finding indicates that the inner cracks (between the column stub and the shear
Fig. 15 presents the typical strain profiles in terms of the longitudinal rebars in the vicinity of the column stubs for the test specimens F1, F3, FS1, and FC2. In this figure, most of the longitudinal rebars reached their yield strain before attaining the peak load. For all that, the flexural failure load according to Simoes et al. [21] and yield line analysis [22] as mentioned above was not reached confirming punching failure of the test specimens. Moreover, in the specimen F3, the strains of S3.1, S3.3, S3.4, and S3.6 measured in the top longitudinal reinforcement but did not reach the yield strain; this indicates that the top longitudinal reinforcement did not affect the punching behavior in particular, the punching strength, which is also confirmed in Figs. 12 and 13(a). Fig. 16 shows the typical strain profiles at the top concrete surface for the specimens F1, FS1, and FC2. As shown in the figure, the concrete strains are less than the expected values, but this was inevitable due to the crushing at the concrete surfaces. 4. Investigation of the punching-shear capacity by comparing the test results with the current design codes and theoretical models In Appendices B and C, the evaluation results of the punchingshear resistance of the footings (V c ) based on the various design codes and theoretical models are presented. Each design code and model showed a significant difference in the evaluation of the effects of the major parameters on the punching-shear strength; this is because these design codes and models are based on different mechanisms and assumptions [23,25,26]. It is also noted that the current design codes use the same design equations in the strength prediction of two-way slabs and footings even though those members are different in terms of the effective depth, shear span to depth ratio (slenderness), and boundary condition
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300 250
350 Yield strain
SS4.1-4, SS4.6-8
Max. load: 207.6 kN
200 SS4.5
150 100
SS4.1 SS4.5 SS4.8
50 0 -0.0005
0
SS4.3 SS4.6
SS4.4 SS4.7
0.0005 0.001 0.0015 0.002 0.0025
Strain (ε) (a) Specimen FS1
Applied load (kN)
350
SS8.1, SS8.3, SS8.8
300
Applied load (kN)
Applied load (kN)
350
300 250
Yield strain
SS5.1, SS5.3 SS5.6-8
Max. load: 208.8 kN
200 SS5.4
150 100 50 0 -0.0005
SS5.1 SS5.6 0
SS5.3 SS5.7
SS5.4 SS5.8
0.0005 0.001 0.0015 0.002 0.0025
Strain (ε) (b) Specimen FS2
Max. load: 287.9 kN Yield strain
250
SS4.1 SS4.2
SS8.6
200 150 100 50 0 -0.0005
SS8.1 SS8.6 0
SS8.3 SS8.8
SS4.4 SS4.3 SS4.5
SS4.6
SS4.8
SS4.7
0.0005 0.001 0.0015 0.002 0.0025
Strain (ε) (c) Specimen FC3
(d) Typical location of steel strain gauges in shear reinforcement
Fig. 14. Strain profiles in shear reinforcement of specimens FS1, FS2, and FC3.
[6]. Fig. 17 compares the areas of the control sections for each design code. In the codes, it is assumed that the soil pressure immediately beneath the column will directly transfer to the column; therefore, in the strength comparison, the failure load V exp is subtracted from the soil pressure within a distance of 0.5d from the column faces according to the ACI 318-14 [23], and within a distance of 0.75d from the column faces according to the KCI 2012 [26]. In this study, the test results were compared with the predictions of the various design codes and theoretical models. 4.1. Strength contribution of the AMF-reinforced concrete The current design codes and theoretical models do not consider the effect of AMF-reinforced concrete. In this study, the strengthening of the punching-shear resistance of the footings with the AMF-reinforced concrete (V n ) was evaluated using Eq. (1), as follows:
V n ¼ V c þ V frc ;
ð1Þ
where V c is the punching-shear resistance of normal concrete. The contribution of the cast of the AMF-reinforced concrete (V frc ) on the normal concrete was evaluated in accordance with the work of Choi et al. [27,28], as follows:
V frc ¼ f pc Afrc cos /;
ð2Þ
Afrc ¼ ½2c1 þ 2c2 þ 4dfrc cot /dfrc = sin /;
ð3Þ
where Afrc is the inclined cross-sectional area that was calculated using the punching-crack angle / (in this study, the cracking angle / is assumed as 45°); dfrc is the depth of the AMF-reinforcedconcrete layer; c1 and c2 are the column sizes, respectively; and f pc is the average tensile strength of the AMF-reinforced concrete.
Based on the test results of this study (Fig. 6), the value of f pc is 3.79 MPa, which is equivalent to approximately 70% of the peak strength. This observation is similar to the post-cracking tensile strength that was investigated in the study by Gu [29]. Fig. 18 presents a method for the determination of the average tensile strength of AMF-reinforced concrete. In the figure, the area under the flexural strength-deflection curve was first evaluated up to a deflection of Lc =150 according to the ASTM C 1609/1609M-12 [15], where Lc is the distance between the supports of the specimens (=300 mm). Then, the average tensile strength was determined so that the shaded area that is generated by actual flexural strength-deflection curve is equivalent to the area that is generated by the average tensile strength. 4.2. Comparison of the strength prediction according to the current design codes and theoretical models Table 4 presents the punching-shear resistance of the calculated footings based on the current design codes including the ACI 31814 [23], Eurocode 2 [25], and KCI 2012 [26]. In the FC series, as shown in Table 4, the punching-shear resistances agreed well with the test results (V exp: ) when the contribution of the AMF-reinforced concrete (V frc ) was added to the prediction by the current design codes (V c;ACI and V c;KCI ). In general, for the current test results, the ratio of V exp =V n;ACI has a mean of 1.35 and a coefficient of variation (COV) of 0.14; meanwhile, the ratio of V exp =V n;KCI exhibited lower values with a mean of 1.25 and a coefficient of variation of 0.10. However, in the cases of the F and FS series, the punching-shear strength that was predicted by the ACI 318-14 [23] is conservative and lower than that predicted by the KCI 2012 [26]. This is because the ACI 318-14 [23] does not consider the effect of the longitudinal-reinforcement ratio on the punching-shear strength, unlike the KCI 2012 [26].
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350
300
Yield strain
250
Applied load (kN)
Applied load (kN)
350
Max. load: 208.7 kN
200 150
S1.2
S1.4
100 50 0 -0.001
S1.1
S1.3
0.001
0.003
S1.1 S1.3 0.005
S1.2 S1.4
0.007
S3.1, S3.4, S3.6 S3.3
300 250
Max. load: 225.3 kN
200 S3.5
150 100
S3.1 S3.5
50 0 -0.001
0.009
0.001
Strain (ε) (a) Specimen F1
0.003
S3.3 S3.6
0.005
S3.4
0.007
0.009
Strain (ε) (b) Specimen F3 350
350
Applied load (kN)
300 S4.1-4
250
Max. load: 207.6 kN
200 150 100
Yield strain S4.1 S4.3
50 0 -0.001
0.001
0.003
0.005
S4.2 S4.4
0.007
300
Max. load: 319.9 kN S7.2 S7.1
250 200 150
S7.4 S7.3
100
S7.1 S7.3
0 -0.001
0.009
0.001
S1.1 S1.2
FS1
S3.4
S3.1 S3.2
S1.4
0.003
0.005
S7.2 S7.4
0.007
0.009
Strain (ε) (d) Specimen FC2
F3
F1
Yield strain
50
Strain (ε) (c) Specimen FS1
FC2 S7.1 S7.2
S4.1
S3.3
S4.2
S4.4
S7.4
S3.5 S1.3
S4.3
S7.3
S3.6
(e) Strain gauge location Fig. 15. Strain profiles in longitudinal re-bars of specimens F1, F3, FS1, and FC2.
350
300 250
Max. load: 208.7 kN
C1.2
Applied load (kN)
Applied load (kN)
350 C1.3-4
200 150 100 50 0 -0.0006
C1.1 C1.1 C1.3
C1.2 C1.4
-0.0004
-0.0002
0
0.0002
300 250
C4.1, C4.4 Max. load: 207.6 kN
200 150
C4.3 C4.2
100 50 0 -0.0006
C4.1 C4.3
C4.2 C4.4
-0.0004
-0.0002
0
0.0002
Strain (ε) (b) Specimen FS1
Strain (ε) (a) Specimen F1 350
Applied load (kN)
Applied load (kN)
Yield strain
300 250
Max. load: 319.9 kN C7.1
C1.4
200 150 100 50 0 -0.0006
C7.3 C7.1 C7.3
C1.3
C7.4
C7.2 C7.2 C7.4
-0.0004
-0.0002
Strain (ε) (c) Specimen FC2
C1.2 0
C1.1
0.0002
(d) Typical location of concrete strain gauges
Fig. 16. Strain profiles at top concrete surface of specimens F1, FS1, and FC2.
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Rectangular column
Circular column
2d 0.2dcot DBroms
0.5d
Simoes et al. [20] KCI 2012 [25] and ACI 318-14 [22] Eurocode 2 [24] Note: DBroms is defined in Broms [29]
Broms [29]
Fig. 17. Control perimeters specified in prediction models.
Table 5 presents the punching-shear resistance that was predicted by Broms [30] and Simoes et al. [21]. Similarly, as shown in Table 5, the punching-shear resistances of the specimens FC2 and FC3 in the FC series agreed well with the test results (V exp: ) when the contribution of the AMF-reinforced concrete (V frc ) was added to the prediction by theoretical models [21,30]. In Table 5, the COVs of the models of Broms [30] and Simoes et al. [21] are 0.09 and 0.11, respectively, indicating that the strut-and-tie model that was developed by Broms [30] and the kinematic theorem-of-limit analysis model that was developed by Simoes et al. [21] could describe the structural behavior well enough to estimate the punching-shear capacity of footings. In general, the research shows that the theoretical models are accurate in the prediction of the punching-shear strength of the footings with the means of 1.21 and 1.18 for Broms [30] and Simoes et al. [21], respectively, which are lower than the existing design codes. 5. Conclusions Eight one-quarter scale reinforced-concrete (RC) footing specimens were constructed with various steel-reinforcement and retrofit details. The test specimens were supported on a bed of steel car springs that simulated the elastic behavior of soil, and they were tested under a concentrated axial load that was applied through the column stubs. The main parameters of this study are the additional placement (or concentrated layout) of longitudinal rebars around the column-footing connections, the inclination of the shear reinforcement, the additional cast of high-strength concrete, and the additional cast of amorphous metallic fiber (AMF)reinforced concrete. From the test results, the primary findings are as follows:
Fig. 18. Determination of average tensile strength (f pc ).
The punching-shear resistance of the footings that was evaluated by the Eurocode 2 [25] is also presented in Table 4. In this table, the predicted resistances are significantly different from the test results. The ratio of V exp =V n;EC2 has a mean of 0.84 and a COV of 0.03, and this because the Eurocode 2 [25] evaluated the punching shear within a relatively large critical perimeter; that is, it is located at a distance of 2d from the column faces. However, according to the observation in the current footing tests, the developed punching-shear cracks are closer to the column faces than the location of the Eurocode 2 critical perimeter [25].
1. Based on the test observations and the yield-line analysis, all of the test specimens failed in the punching shear, except for FC2, the column stub of which was crushed. 2. Due to the addition of a bottom longitudinal reinforcement around the column-footing connections, the peak load and the deflection at the failure of the specimen F2 apparently increased up to approximately 20.1% and 18.1%, respectively. 3. The additional casts of the high-strength concrete and the AMFreinforced concrete increased the punching-shear strength of the footings. The peak load and the deflection at the failures of the specimens FC1, FC2, and FC3 significantly increased up to approximately 53.3% and 73.5%, respectively.
Table 5 Punching shear capacity of footings predicted by existing theoretical models. Specimens
V exp a (kN)
V frc (kN)
V n;Broms (kN)
V n;Simoes (kN)
V net =V model b Broms [30]
Simoes et al. [21]
F1 F2 F3 FS1 FS2 FC1d FC2 FC3
208.7 250.7 225.3 207.6 208.8 250.8 319.9 287.9
–c –c –c –c –c –c 113.7 113.7
149.1 186.1 169.4 149.1 149.1 –c 271.1e 271.1e
154.1 184.9 169.5 154.1 154.1 –c 276.1e 276.1e
1.29 1.24 1.23 1.29 1.29 –c 1.12 1.01
1.25 1.25 1.23 1.24 1.25 –c 1.11 0.99
1.21 0.09
1.18 0.11
Mean COV a b c d e
Values before subtracting the load transferred to the columns. V net is net applied shear force neglecting upward uniform pressure beneath concrete columns, which is defined differently in each theoretical model. Not applicable. Column failed. V n is punching shear capacity predicted by the summation of V c and V frc .
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4. For a given test condition, the placement of a shear reinforcement around the column-footing connections did not affect the punching-shear strength of the footings. On the contrary, in the study by Menetrey and Bruhwiler [32], it was demonstrated that the addition of shear reinforcement could significantly increase the punching-shear capacity. Further research is necessary for a better understanding of this behavior. 5. Considering the contribution of the AMF-reinforced concrete to the punching-shear resistance of the footings, most of the design codes and theoretical models show sound predictions of the punching strength.
Acknowledgments This research was supported by the Basic Science Program (2014R1A1A2053499) through the National Research Foundation of Korea (NRF), funded by the Ministry of Education. Appendix A. Flexural-strength evaluation and yield-line analysis Fig. 19. Yield line pattern considered in punching shear failure of a square footing with a square column.
A.1. Flexural strength of footings Simoes et al. [21] developed an equation to determine the flexural strength of footings (V flex ) based on the yield-line theory. Once the flexural strength is higher than the punching-shear strength, the footings are governed by the punching failure; under the converse condition, the footings are governed by the flexural failure.
V flex ¼ 2pmR
rs r 2s ; r q r c r 2s r 2c
" 2 # 0 fy d 0 þ q ðd d Þ ; mR ¼ f cp d 2d f cp rq ¼
R¼
ðA2-2Þ 1:5pA
c1 þ c2
p
ðc1 þ c2 Þ
!1=3 0:5 2
;
ðA2-3Þ
ðA1-1Þ
where mR is the moment capacity per unit length, R is the radius of the failure zone, and A is the area of the footing.
ðA1-2Þ
Appendix B. Evaluation of the punching-shear strength by the current design codes
ðA1-3Þ
where mR is the moment capacity per unit of length; rq is the radius of the reaction resultant; r c and r s are the radius of the circular column and the circular footings, respectively; d is the effective slab 0 depth; d is the effective depth of the top reinforcement; q is the tension-reinforcement ratio; and f cp is the plastic-concrete compressive strength. A.2. Yield-line analysis
h
2pmR 2 2 i ; p R2 2 þ c1pþc 1 3A 1 þ c1pþc R R
The punching-shear resistance (V c ) for a slab without a shear reinforcement is evaluated as the minimum of three values, as follows:
8 9 qffiffiffiffi 0 > > 4 > > b d f 0:166 2 þ 0 > > c b > > > > < = qffiffiffiffi 0 a d s V c ¼ min 0:083 b þ 2 ; b f 0d c 0 > > > > > > qffiffiffiffi > > > > : 0:332 f 0 b0 d ; c
ðB1-1Þ
where b is the ratio of the long-to-short side of the loading area;
Fig. 19 shows the yield-line pattern for a square footing that has been subjected to a concentrated load through the square column at the center of the footing. The bottom plane of the footing will consequently be loaded with a uniform soil-reaction force, which is produced by the concentrated load in the column. The yieldline pattern was determined based on the studies by Gesund and Kaushik [22] and Gilbert et al. [31]. In this figure, the yield-line pattern is slightly complex and the moment distribution at the column is nearly symmetric. It is noted that the reinforcement is assumed as isotropic. From the yield-line analysis, Gesund and Kaushik [22] derived an equation to determine the punching-shear capacity of footings, as follows: 2 1 c1pþc R
2
B.1. ACI 318-14 [23]
2 ðr3s r3c Þ ; 3 ðr2s r2c Þ
V yieldline ¼
mR ¼ qf y d ;
ðA2-1Þ
as = 40 for the internal loads, 30 for the edge columns, and 20 for the corner columns; and b0 ¼ 2c1 þ 2c2 þ 4d. B.2. Eurocode 2 [25] The punching-shear resistance (V c ) for a slab without a shear reinforcement is evaluated as follows: 0 1=3
V c ¼ 0:18nð100qf c Þ n¼1þ
rffiffiffiffiffiffiffiffiffi 200 ; d
2d b0 d; a
ðB2-1Þ
ðB2-2Þ
where n is the size-effect factor; a is the distance from the periphery of the column to the control perimeter; and b0 ¼ 2c1 þ 2c2 þ 16d. In this design code, the net applied shear force is calculated using Eq. (B2-3), as follows:
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V Ed;red ¼ V Ed DV Ed ;
ðB2-3Þ
where V Ed is the column load and DV Ed is the net upward force within the considered control perimeter; that is, the upward uniform pressure from the soil minus the self-weight of the footing. B.3. KCI 2012 [26] The punching-shear resistance (V c ) is evaluated using Eq. (B31), as follows:
V c ¼ v c b0 d;
ðB3-1Þ
v c ¼ kks kb
ðB3-2Þ
0
f t cot /ðcu =dÞ;
b0 ¼ 2c1 þ 2c2 þ 4 cot /:cu ;
ðB3-3Þ
where v c is the average shear stress; k is the coefficient of the lightweight concrete (k ¼ 1); ks is the size-effect factor [Eq. (B3-4)]; kb0 is an aspect-ratio factor [Eq. (B3-5)]; and cu is the depth of the compression zone [Eq. (B3-6)].
rffiffiffiffiffiffiffiffiffi 4 300 6 1:0; ks ¼ d
ðB3-4Þ
4 kb0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi 6 1:25; as ðbd0 Þ sffiffiffiffi
q
cu ¼ d 25
cot / ¼
0
fc
300
ðB3-5Þ
q
!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 f t ðf t þ 2f c =3Þ ft
;
0
fc
ðB3-6Þ
;
ðB3-7Þ
where as ¼ 1 for the interior column. In the KCI 2012 [26], the soil pressure within the critical perimeter at a distance of 0:75d from the column faces is neglected from the applied load. Appendix C. Evaluation of the punching-shear strength by the theoretical models C.1. Broms [30]
1=3 0:150 V c ¼ rc t sinðcÞu ; t
ðC1-1Þ u x0
2
rc ¼ f 0c 0:6 þ 0:9ð1 0:007 Þ 6 1:2f 0c ; t¼
x0 ; 2 cosðcÞ
u¼p Bþ
x0 x0 þ ; tanð2cÞ 2 tanðcÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 1þ 1 d; x0 ¼ n0 q n0 q
n0 q ¼
D Es 1 þ ln c0 q; Ec 1 þ ln DB
where
rc is the compression strength of the capital; x0 is the com-
ðC1-2Þ ðC1-3Þ ðC1-4Þ
ðC1-5Þ
ðC1-6Þ
pression zone; t is the depth of the compression strut; ð0:150=tÞ1=3 is the size-effect factor with the dimension t (in meters); 0.0150 is the diameter of the standard test-cylinder specimen (in meters); Es and Ec are the elastic modulus of the steel rebars and the concrete, respectively; and D and B are the footing and column sizes, respectively. Moreover, a part of the uniformly distributed load within the critical shear section that does not affect the punching-shear capacity of footings is neglected. In this theoretical model, the perimeter of the critical shear section (c0 ) was determined using Eq. (C1-7), as follows:
2d c0 ¼ p B þ ; tan / tan / ¼
ðC1-7Þ
1:4d P 1; D=3 B=2
ðC1-8Þ
where / is the inclination angle of the shear crack. C.2. Simoes et al. [21] Simoes et al. [21] developed a mechanical model in consideration of the failure mechanisms that is based on both a vertical penetration of the punching cone and a rotation of the outer part of the footing, and this allows for a consideration of the roles of both the bottom and top reinforcements on the failure load regarding the prediction of the punching strength of footings. In the model, the punching-shear failure mode and its interaction were investigated by means of the kinematic theorem-of-limit analysis. In Simoes et al. [21], the punching-shear resistance of footings (V c ) was calculated using Eq. (C2-1), as follows:
1 f b0 d; 0:9 þ rds cp
ðC2-1Þ
b0 ¼ 2p½r c þ 0:2d cotðbÞ;
ðC2-2Þ
Vc ¼
b¼
In Broms [30], a mechanical model was proposed to evaluate the punching-shear strength of footings. The hypothesis of Broms [30] states that the punching occurs when the capital fails in a compression, thereby forming a diagonal shear crack. In addition, the angle to the horizontal of the punching crack near the column (2c) was assumed as 50°. The punching-shear capacity of the footings was then determined using Eq. (C1-1), as follows:
295
p=2 0:8 þ 0:5 da
½rad;
0
f cp ¼ f c ge gfc ;
gfc ¼
f c0 0 fc
ðC2-3Þ ðC2-4Þ
!1=3 6 1;
ðC2-5Þ
where b is the secant inclination of the failure surface; a is the shear span (=rs rc ); ge is the reduction factor that accounts for the presence of the transverse strains and depends on the amounts of the longitudinal mechanical-reinforcement ratio (x ¼ qf y =f cp ) refer to Simoes et al. [21] to determine the value of ge ; gfc is the reduction factor that accounts for the brittleness of high-strength concrete; and f c0 is the reference compressive strength (=30 MPa). In this proposal model, a part of the uniform soil reaction that acts inside the failure zone and does not influence the punchingshear capacity of footings was determined using Eq. (C2-6), as follows:
c0 ¼ p½rc þ d cotðbÞ : 2
ðC2-6Þ
It is noted that the proposal models developed by Broms [30] and Simoes et al. [21] are used for circular footings. To apply such models to square footings, the square columns should be replaced
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by the equivalent circular footings with the same perimeter, and the square footings should be replaced by the equivalent circular footings with the same area. References [1] Hatcher DS, Sozen MA, Siess CP. Test of a reinforced concrete flat slab. J Struct Div (ASCE) 1969;95(ST6):1051–72. [2] Lee S-S, Moon J, Park K-S, Bae K-W. Strength of footing with punching shear preventers. Sci World J 2014. http://dx.doi.org/10.1155/2014/474728. Article ID 474728, 15 pages. [3] Coduto DP. Foundation design: principles and practices. 2nd ed. New Jersey: Prentice-Hall; 2001 [chapter 6]. [4] Hegger J, Ricker M, Ulke B, Ziegler M. Investigations on punching shear behaviour of reinforced concrete footings. Eng Struct 2007;29:2233–41. [5] Hegger J, Sherif AG, Ricker M. Experimental investigations on punching behavior of reinforced concrete footings. ACI Struct J 2006;103(4):604–13. [6] Bonic Z, Folic R. Punching of column footings – comparison of experimental and calculation results. Gradevinar 2013;65:887–99. [7] Aron C, Jonas E. Structural element approaches for soil-structure interaction. Master thesis. Goteborg (Sweden): Chalmers University of Technology; 2012. [8] Meda A, Plizzari GA, Riva P. Fracture behavior of SFRC slabs on grade. Mater Struct 2004;37:405–11. [9] Bowles JE. Foundation analysis and design. 5th ed. Singapore: McGraw-Hill; 1996. [10] Kent ER, Math ER. Tests of concrete and reinforced concrete column footings Thesis for the Bachelor of Science. USA: University of Illinois; 1911. [11] Richart FE. Reinforced concrete wall and column footings – Part 2. ACI J 1948;20(3):237–60. [12] ACI 314R-11. Guide to simplified design for reinforced concrete buildings. ACI Committee 314. Farmington Hills (USA): American Concrete Institute; 2011. [13] Hallgren M, Kinnunen S, Nylander B. Punching shear tests on column footings. Nordic Concrete Res 1998;21(3):1–22. [14] KS F 2405. Standard test method for compressive strength of concrete. Korean Standard Information Center; 2010. [15] ASTM C 1609/1609M-12. Standard test method for flexural performance of fiber-reinforced concrete (using beam with third-point loading). Annual Book of ASTM Standards, ASTM Committee, C09.42; 2012.
[16] Truong GT, Kim J-C, Choi K-K. Seismic performance of reinforced concrete retrofitted by various methods. Eng Struct 2017;134:217–35. [17] Truong GT. Shrinkage and creep behavior of amorphous metallic fiberreinforced concrete. Master thesis. Seoul (Korea): Soongsil University; 2014. [18] KS B 0802. Method of tensile test for metallic materials. Korean Standard Information Center; 2013. [19] KS B 0814. Method of tensile test for metallic materials. Korean Standard Information Center; 2001. [20] Lee CD, Lee SH, Ko KS, Yang J-M. Structural performance of SFRC slab-on-grade supported on elastic spring system. Mag Concrete Res 2017;69(15):757–71. [21] Simoes JT, Faria DMV, Fernandez RM, Muttoni A. Strength of reinforced concrete footings without transverse reinforcement according to limit analysis. Eng Struct 2016;112:146–61. [22] Gesund H, Kaushik YP. Yield line analysis of punching failures in slabs. Mermoir Int Assoc Bridge Struct Eng 1970;30(I):41–60. [23] ACI 318-14. Building code requirements for structural concrete. ACI Committee 318. Farmington Hills (USA): American Concrete Institute; 2014. [24] Hegger J, Ricker M, Sherif AG. Punching strength of reinforced concrete footings. ACI Struct J 2009;106(5):706–16. [25] Eurocode 2. Design of concrete structures – Part 1–1: General rules and rules for buildings. Brussels (Belgium): European Committee for Standardization (CEN); 2004. [26] KCI 2012. KCI concrete design code. Korea: Korea Concrete Institute; 2012. [27] Choi K-K, Truong GT, Kim S-D, Choi I-R. Punching shear behavior of shear reinforced concrete slabs. Struct Build 2015;168(6):402–20. [28] Choi K-K, Taha MMR, Park H-G, Maji AK. Punching shear strength of interior concrete slab-column connections reinforced with steel fibers. Cem Concrete Compos 2007;29(5):409–20. [29] Gu DO. Flexural behaviors of amorphous metallic fiber-reinforced concrete. Master thesis. Seoul (Korea): Soongsil University; 2013. [30] Broms CE. Concrete flat slabs and footings – design method for punching and detailing for ductility. Doctor thesis. Stockholm (Sweden): Royal Institute of Technology; 2005. [31] Gilbert M, He L, Pritchard T. The yield-line method for concrete slabs: automated at last. Struct Engineer 2015;93(10):44–8. [32] Menétrey Ph, Brühwiler E. Shear strengthening of existing reinforced concrete slabs: an experimental investigations. Innsbruck: Composite Construction– Conventional and Innovative; 1997. p. 920–1.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Punching-shear behaviors of RC-column footings with various reinforcement and strengthening details Gia Toai Truong, Kyoung-Kyu Choi ⇑, Hee-Seung Kim School of Architecture, Soongsil Univ., 369 Sangdo-ro, Dongjak-gu, Seoul 06978, South Korea
a r t i c l e
i n f o
Article history: Received 27 February 2017 Revised 7 May 2017 Accepted 13 August 2017
Keywords: Column footings Punching shear Springs Soil pressure Amorphous metallic fiber (AMF) Shear reinforcement
a b s t r a c t In this study, various reinforcement and strengthening details for the improvement of the punchingshear behavior of concrete footings were developed. Eight test specimens of reinforced-concrete (RC) footings were constructed and tested to investigate the punching-shear behavior of the concrete footings with the developed details. The test parameters include the additional placement of a longitudinal reinforcement around the column-footing connections, the inclination of the shear reinforcement, an additional cast of high-strength concrete, and an additional concrete cast with amorphous metallic fibers (AMFs). The test specimens were supported on a bed of steel car springs that simulate the elastic behavior of soil. The test results show that the punching strength was significantly enhanced by the additional placement of the longitudinal reinforcement around the column-footing connections and by the addition of the retrofit materials (high-strength concrete and AMF-reinforced concrete). Contrarily, the shear reinforcement that was placed around the connections did not significantly affect the punching-shear strength of the footings. The test results were also compared with strength predictions for which the current design codes and the existing theoretical models were used. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The gravity loads of structures need to be transferred to footings resting on soil. Since the soil pressure that develops beneath the footings is dependent on the soil type and stiffness, it is difficult to predict the structural behavior of the footings. In addition, the punching-shear failure of concrete footings is brittle, which could finally lead to a progressive collapse of whole structures [1,2]. Therefore, extensive experimental and theoretical studies have been performed to investigate the punching-shear behavior of concrete-column footings. Coduto [3] and Hegger et al. [4] investigated the effects of the soil type and rigidity (or stiffness) on the actual distribution of the soil pressure beneath concrete footings that were subjected to a vertically concentrated load. In Fig. 1(a) and (b), for a footing resting on loose noncohesive soil (e.g., sandy soil), a convex pressure distribution develops, and for a footing resting on cohesive soil (e.g., clay), a concave pressure distribution develops. Because of the complexity of the soil-pressure distribution, a linear soilpressure distribution beneath the footings is usually assumed for design purposes (Fig. 1(c)). ⇑ Corresponding author. E-mail address: [email protected] (K.-K. Choi). http://dx.doi.org/10.1016/j.engstruct.2017.08.037 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
Hegger et al. [5] and Bonic and Folic [6] performed punching tests on reinforced-concrete (RC) footings using real sand. The test results indicated that the punching-shear capacity increased with the increasing of the effective depth and the decreasing of the shear-span/depth ratio, and the angles of the observed punchingshear cracks in the footing tests are steeper than those of the slender slabs. In addition, the sand stiffness (loose or dense) did not significantly affect the distribution of the soil pressure underneath the footing. In Winkler’s soil model [7], the soil underneath the footings could be simulated by a number of independent elastic springs with a constant spring stiffness ks , known as the ‘‘modulus of soil reaction”. According to Meda et al. [8] and Bowles [9], the value of ks ranges from 0.0048 N/mm3 to 0.128 N/mm3. Based on Winkler’s theory, Kent and Math [10] and Richart [11] performed punching tests using a bed of car springs that simulated a uniform soil-pressure distribution. From the test results, the punchingshear surfaces are close to the column surfaces, and the specimens with a higher longitudinal-reinforcement ratio exhibited a higher punching-shear capacity [10]. In addition, the footings for which the reinforcing bars with hooks at the end are used did not show considerable differences in the bond resistance compared with those for which the reinforcing bars without hooks are used; the fact is due to the hooks did not significantly prevent end slips [11].
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P
P
(a) Loose cohesive-less soil
P
(b) Cohesive soil
(c) Linear pressure distribution
Fig. 1. Soil pressure distribution beneath an elastic footing.
For the present study, various steel-reinforcing details for the new construction of concrete footings and the strengthening details for the retrofit of existing concrete footings were developed. Eight half-scale RC footings with the developed details were constructed and tested. The main parameters of the test specimens are the additional placement (or concentrated layout) of longitudinal rebars around the column-footing connections, the inclination of the shear reinforcement, an additional cast of high-strength concrete, and an additional cast of amorphous metallic fiber (AMF)-reinforced concrete. In this test, a bed of steel car springs that simulate the elastic behavior of soil was used to support the test specimens. The effectiveness of the var-
ious reinforcing and strengthening techniques was compared in terms of the strength, crack pattern, and strain profiles of the footings. Moreover, the test results were compared with the strength predictions from the current design codes and the existing theoretical models.
Fig. 2. Details of specimen F1.
Fig. 3. Details of specimens F2 and F3.
2. Experimental program 2.1. Test specimens Eight reinforced concrete-footing specimens with a scale factor of 1/4 of a common footing were constructed, as follows: One is a control specimen, and the other specimens are different from the control specimen in terms of the steel-reinforcement details and the additional cast of the retrofit materials (concrete or AMFreinforced concrete). The geometries and reinforcement details of the test specimens are classified into three series, as presented in Figs. 2–5 and Table 1. The first series are the three footing specimens with a uniform or concentrated layout of the longitudinal rebars around the column-footing connections, which are denoted
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Fig. 4. Details of specimens FS1 and FS2.
Fig. 5. Details of specimens FC1, FC2, and FC3.
as ‘‘F series”, and simulate new footing specimens. The second series are two footing specimens that have been strengthened with shear rebars, denoted as ‘‘FS series”. The third series are three footing specimens that have been strengthened with an additional cast of high-strength concrete or AMF-reinforced concrete, which are denoted as ‘‘FC series”. Fig. 2 shows the details of the control specimen F1, which was designed to fail in punching shear. The main properties of the control specimen F1 are as follows: effective depth (d) is 100 mm; shear span (=L=2 c=2)-to-depth ratio (a=d) is equal to 4; longitudinal reinforcing bars are D10 with a diameter of 10 mm; and the longitudinal reinforcement ratio (q) is 0.61%; here, L and c are the
sizes of the footings and columns, respectively. The specimens F2 and F3 differ from the specimen F1 in the amounts of the bottom and top longitudinal reinforcements, respectively (Fig. 3). It should be noted that in the specimens F2 and F3, additional longitudinal rebars were placed only in the critical section zone (=c þ 2d, refer to ACI314R-11 [12]). In particular, in the case of the specimen F3, additional rebars were arranged as a top reinforcement, and the length was cut down by 3.2d with a hook length of 100 mm (see Fig. 3). The specimens of the FS and FC series are the retrofitted specimens of the control specimen F1; therefore, the specimens of the FS and FC series have the same material and geometrical properties
Table 1 Details of test specimens.
a b c d e f g
Specimens
d (mm)
c (mm)
l (mm)
a/da
q (%)
f’c (MPa)
fyb (MPa)
Av (mm2)
s (mm)
Feature
F1 F2 F3 FS1 FS2 FC1 FC2 FC3
100 100 100 100 100 100 (150)c 100 (150)c 100 (150)c
100 100 100 100 100 100 100 100
900 900 900 900 900 900 900 900
4 4 4 4 4 4 (2.67)d 4 (2.67)d 4 (2.67)d
0.61 0.96 0.79 0.61 0.61 0.61 (0.41)e 0.61 (0.41)e 0.61 (0.41)e
29.74 29.74 29.74 29.74 29.74 29.74 29.74 29.74
435 435 435 435 435 435 435 435
–f –f –f 78.5 78.5 –f –f 78.5
–f –f –f 80 (85)g 100 (60)g –f –f 50 (30)g
Control specimen Addition of bottom re-bars Addition of top re-bars Inclined shear re-bars Shear re-bars Additional cast of high strength concrete Additional cast of AMF reinforced concrete Additional cast of AMF RC and shear re-bars
Shear span to depth ratio (=(l c)/(2d)). Yield strength of longitudinal re-bars and shear re-bars. Effective depth considering retrofit layer. Shear span to depth ratio considering retrofit layer. Longitudinal reinforcement ratio considering retrofit layer. Not applicable. Spacing between column face and first row of shear reinforcement.
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Flexural strength, σ (MPa)
8
P/2
P/2
a
6
Δ
Table 2 Geometrical and material properties of AMFs.
b
a=100 h b=100 h=100 Unit: mm
3 a P 2 bh 2
Parameters Fiber thickness Fiber length Aspect ratioa Fiber width Yield tensile strength Modulus of elasticity Density
Amorphous metallic FRC
4
a
Plain concrete
2
0 0
0.5
1
1.5
2
2.5
Deflection (mm) Fig. 6. Flexural strength-deflection curves obtained by Truong [16]
as that of the control specimen F1, except for the retrofitted parts. Fig. 4 shows the details of the specimens FS1 and FS2. In the figure, the specimen FS1 has been strengthened with inclined shear rebars (D10) having a diameter of 10 mm, and the angle between the inclined shear rebar and the horizontal axis is 45°. The use of the inclined shear reinforcement was referred to the study by Hallgren et al. [13]. Similar to FS1, the specimen FS2 was also strengthened with shear rebars (D10), but the angle is 90° (or a vertical shear reinforcement). In the FC series, a high-strength concrete layer for the FC1 and an AMF-reinforced concrete layer for the FC2 and FC3 were casted. After the retrofitting, the effective depth of the FC-series specimens is 150 mm. For all of the footing specimens, the footing size is 900 mm 900 mm in length and width, respectively, and the column studs are of a 100 100 mm cross-section and a 100 mm height. The longitudinal reinforcement of the columns consists of four D16 steel rebars, and a transverse reinforcement consists of D10 steel rebars with four legged and closed hoops (refer to Fig. 2). It is noted that, for the installation of the test specimens, four hooks were installed at the corners of the specimens, but they were intentionally not presented in the figures. For this test, the specimens of the FS and FC series were constructed according to the actual retrofit procedures. In the FS series (Fig. 4), the locations of the holes were drilled as per the designed configurations, as follows: 90 mm 20 mm (depth diameter) and the angle of 45° for the specimen FS1; and 105 mm 20 mm and the angle of 90° for the specimen FS2. The holes were then cleaned carefully, followed by the installation of the shear rebars through the holes with a length of 90 mm for FS1 and 100 mm for FS2. For each specimen, a total of 16 shear rebars were used and arranged into a two-row cross in each direction. Lastly, the remaining gaps of the holes were filled with non-shrinkage mortar. In the specimens of the FC series, the top parts of the concrete (10 mm) were first chiseled out. The work was carefully carried out to avoid damaging the internal concrete. After that, the holes were drilled according to the designed configuration, and the steel rebars were then installed through the holes with a length of 30 mm for FC1 and FC2 and 100 mm for FC3. It is noted that the steel rebars were bent into an L-shape for a more-effective connection between the old and new concrete layers. In total, 36 steel rebars were used including six rows in each direction and four additional steel rebars around the column stubs. In FC1 and FC2, short steel rebars were used and were considered as anchor bars, which contributed to the bonding behavior between the old and new concrete layers. Meanwhile, in FC3, long steel rebars were
Amorphous metallic fibers (AMFs) (lm) (mm) (mm) (MPa) (MPa) (kg/m3)
29 30 1034 1.6 1700 140,000 7200
Aspect ratio is the ratio of fiber length to fiber thickness.
used and were considered as shear reinforcement, which contributed to not only the bonding behavior between the old and new concrete layers, but also the shear strength of the specimen. After this stage, the formwork was assembled to ensure the additional concrete-layer dimensions, as follows: 600 mm 600 mm 60 mm. Then, the high-strength concrete layer for the specimen FC1 was cast, followed by the casting of the AMFreinforced concrete layer for the specimens FC2 and FC3. 2.2. Materials In this test, the concrete compressive test that was carried out on the loading date of each specimen is according to the KS F 2405 test standard [14]. The average compressive strength is 29.7 MPa at the 107th day, which is the loading date. In addition, the average compressive strength of the additional concrete is 41.2 MPa, while that of the AMF-reinforced concrete is 52.7 MPa at the loading date. In the case of the AMF-reinforced concrete, the tensile strength was determined based on the flexural test, which was performed according to the ASTM C 1609/1609M-12 test standard [15]. The peak tensile strength of the AMFreinforced concrete that was obtained from the test results is 5.46 MPa. The test results were almost the same as that of Truong et al.’s test [16]. It is noted that in the study by Truong et al. [16], the design compressive strength was also 50 MPa, and the same concrete-mix properties were used as that of present test. Fig. 6 presents the flexural strengthdeflection curves in the flexural test of the specimens that had been reinforced with the AMFs [16]. In Fig. 6, it is obvious that the AMF-reinforced concrete with the fiber-volume fraction of 0.75% exhibited a considerably greater peak strength than that of the plain concrete. The geometrical and material characteristics of the AMFs are presented in Table 2, and can be also found elsewhere [17]. Tensile testing of the longitudinal rebars was performed according to the KS B 0802 [18] and KS B 0814 test standards [19]. The diameter of the steel rebars denoted as D10 is 10 mm, and their specified yield strength is 400 MPa. From the test results, the measured yield tensile stress of D10 is 435 MPa, which exceeds the specified yield strength of 400 MPa, and the corresponding yield strain is 0.0022. In terms of the evaluation of the test results hereafter, the material strengths that were measured from the test were used instead of the specified strengths of the concrete and the steel rebars. 2.3. Test setup Each column footing was subjected to a concentrated axial load through a column stub using a hydraulic actuator with a capacity of 500 kN. Fig. 7 shows the test setup of the column-footing tests. In Fig. 7(a) and (b), a strong main steel frame was assembled on a strong floor where it could support the reaction forces from the
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Actuator Steel frame
A Specimen
LVDT frame Bearing steel blocks
Car springs
Strong floor (a) Test setup
(b) Photo of test setup Hinging plates
Load cell LVDT frame
Three dimensional hinge
LVDTs Springs Specimen
Protecting steel blocks (d) Details of supports
(c) Detail of A region
(e) Three dimensional hinge Fig. 7. Test setup.
hydraulic actuator. Fig. 7(c) and (d) present the details of the connection parts of the actuator and the specimen, and the supports. Around the bed of the steel car springs, steel blocks were assembled to avoid the lateral movement of the specimens (see Fig. 7 (d)). In addition, a steel three-dimensional (3 D) hinge was placed between the actuator head and the specimens to transfer the axial force during the loading process (see Fig. 7(c) and (e)). The axial load was applied to the specimens in a displacement-controlled manner at a loading rate of 0.05 mm/s. The test specimens were supported on a bed of nine steel car springs. As stated previously, the car springs were used to simulate the elastic behavior of soil. Fig. 8 shows the layout of the steel car springs in square grids with a center-to-center spacing of 300 mm. In the figure, each car spring would replace an area of 300 300 (mm2) of the soil beneath the footing. Thus, for the given test setup, the modulus of soil reaction (ks) was estimated as 0.016 N/mm3; this is equal to the spring stiffness (1,400 N/mm)
divided by the relevant area of soil (300 mm 300 mm) replaced by a spring. It is noted that the modulus of soil reaction of 0.016 N/mm3 represents the clayey soil (refer to [9]). Each steel spring has a 45 mm spring diameter, a 200 mm outer diameter, a 600 mm height, and the spring stiffness retains a constant of 1400 N/mm within an elastic limit of deformation (190 mm). The springs used in this test exhibit the same behavior of those used by Lee et al. [20]. In addition, the installation of the car-spring system is also referred to in [20]. As shown in Fig. 8 (b), a 200 mm diameter circular hinge, which can rotate in all directions, was placed on the top of each spring. Then, a steel plate with the dimensions of 250 mm 250 mm 20 mm was placed on top of the hinge to distribute the reactive force of the spring to the bottom surface of the footings during the loading. A steel-bearing plate of 300 mm 300 mm 20 mm was welded to the spring at the bottom and connected to a strong floor to maintain the stability and to prevent the spring from overturning during the loading.
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Steel plate (250x250)
900
300
250
250
300
Steel bearing plate (300x300)
200
300
250
Springs
287
show the arrangement of the steel- and concrete-strain gauges. In Fig. 9, to measure the longitudinal rebar strains, the four steel-strain gauges that were used in the test specimens F1, FS1 to FS2, and FC1 to FC3 are shown (Fig. 9(a)). Further, 10 steelstrain gauges were used in the specimen F2 (Fig. 9(b)), and six steel-strain gauges were used in the specimen F3 (Fig. 9(c)). In the case of the specimens FS1, FS2, and FC3 (refer to Fig. 10(a)), the strains of the shear rebars were also measured at the midheight, and eight steel-strain gauges were installed for each specimen. In Fig. 10(b), the concrete strains for each specimen were measured with the use of four concrete-strain gauges. In addition, four line linear variable-displacement transducers (LVDTs; see Fig. 10(b)) were installed to measure the deflection of the specimens.
3. Experiment results
250 300
250 300
250 300
3.1. Failure mode and crack pattern
900
(a) Plane Test specimen Steel plate (250x250)
600
Spring
725
Hinge
Steel bearing plate (300x300) Strong floor
(b) Elevation
(c) Photo of spring system
Unit: mm
Fig. 8. Spring-type soil system.
2.4. Measurement instrumentations Steel- and concrete-strain gauges were attached onto the longitudinal and shear rebars and the top concrete surface to measure the steel and concrete strains, respectively. Figs. 9 and 10
Most of the test specimens failed in the punching shear of the footing, except for the specimen FC1, which failed in the concrete crushing of its column stub. The failure loads are given in Table 3, wherein the shear demand (V flex ) corresponding to the flexural failure of the footings, which were evaluated according to Simoes et al. [21] (see Appendix A.1), are also presented to investigate the failure mode of the test specimens. It is obvious that the punching-shear capacities that were obtained from the test results are lower than the shear demand (V flex ) corresponding to the flexural failure of the footings. This finding indicated that the shear demand (V flex ) of the footings had not been reached, confirming that the failure occurred due to the punching shear. This observation was also verified using the yield-line analysis, which was suggested by Gesund and Kaushik [22] (see Appendix A.2). In Table 3, the load-carrying capacity (V yieldline ) predicted by the yield-line analysis is considerably greater than the test results (V exp ), and this indicates that no flexural yield line developed in the test specimens. It is noted that in the yield-line analysis of the specimens FC2 and FC3, the additional cast of the AMFreinforced concrete was considered in the evaluation of the effective depth and the longitudinal-reinforcement ratio (see Table 1). In the case of the specimen FC1, the column-axial strength, which was calculated based on ACI 318-14 [23], is 582.1 kN, and this value is greater than the column-failure loads that were observed in the tests (see Table 4). As mentioned previously, however, the column stub of FC1 failed locally in the concrete crushing, it suddenly lost the specimen stability, and then the test was intentionally terminated. Fig. 11 shows the crack pattern at the end of the testing. In general, at the bottom surfaces of the test specimens, several thin cracks that run toward the corners of the specimens first appeared. Then, numerous cracks immediately occurred along the layout of the reinforcing bars; furthermore, several radial cracks appeared and propagated toward the edges (but not the corners) of the specimens, and this is particularly the case in the specimen F2 (see Fig. 11(b)). Lastly, a closed or half-closed punching-failure line was developed, but the shape is different in each specimen. At the punching failure, a significant concrete crushing was also observed. In Fig. 11, the bold line indicates the punching cone at the punching failure of the test specimens. In the case of the specimen FC1 (Fig. 11(f)), the punching-failure line is not observable, but radial cracks developed; here, the observed cracks are thin, and they are not considered as twoway yield lines because the column was crushed ahead of the specimen failure.
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(a) F1, FS1-2, and FC1-3
(b) F2
(c) F3
Steel strain gauge Fig. 9. Arrangement of steel strain gauges attached on longitudinal re-bars.
3.2. Effect of the additional placement of the longitudinal reinforcement around the column-footing connections
300
20
300
Line LVDTs
Column
20
Column
Steel strain gauges
Concrete strain gauges
(b) Location of concrete strain gauges and LVDTs for all specimens
(a) Location of steel strain gauges on shear re-bars of specimens FS1, FS2, and FC3
Unit: mm Fig. 10. Arrangement of steel strain gauges attached on shear re-bars, concrete strain gauges, and line LVDTs.
Fig. 12 shows the measured load versus the center deflection of the column footings for all of the test specimens. In this figure, it is obvious that all of the specimens, except for FC1, failed in the brittle punching failure. Most of the specimens immediately showed a sudden load drop after they reached the peak load; this is attributed to the reaction forces that were provided by the car springs, which cause a sudden failure and a load drop like that of the test in a force-controlled manner. In Fig. 12, the peak load showed variations according to the test parameters. Fig. 13 (a) presents a comparison of the loaddeflection curves of the footings F1, F2, and F3 for the investigation of the effect of the additional placement of the longitudinal reinforcements. The peak load and the deflection at the failure of the control specimen F1 are 208.7 kN and 23.92 mm, respectively. In the specimen F2 that had a greater bottom longitudinal
Table 3 Load-carrying capacity predicted based on flexural failure mode and yield line analysis. Specimens
F1
F2
F3
FS1
FS2
FC1c
FC2
FC3
Mean
COV
V exp (kN) V flex a (kN) V yieldline b (kN)
208.7 307.8 236.6 0.68
250.7 483.7 344.9 0.52
225.3 280.6 302.2 0.80
207.6 307.8 236.6 0.67
208.8 307.8 236.6 0.68
250.8 –d –d –d
319.9 461.7 363.6 0.69
287.9 461.7 363.6 0.62
0.67
0.083
0.88
0.73
0.75
0.88
0.88
–d
0.88
0.79
0.83
0.085
KCI 2012 [26]
ACI 318–14 [23]
EC2 [25]
1.33 1.34 1.30 1.32 1.32 –c 1.14 1.03
1.37 1.65 1.48 1.36 1.37 –c 1.14 1.03
0.84 0.87 0.84 0.84 0.84 –c 0.84 0.76
1.25 0.10
1.35 0.14
0.84 0.03
V exp V flex V exp V yieldline a b c d
V flex is evaluated according to Simoes et al. [21]. V yieldline is evaluated according to Gesund and Kaushik [22]. column failed. not applicable.
Table 4 Punching shear capacity of footings predicted by current design codes. Specimens
V exp a (kN)
V frc (kN)
V n;KCI (kN)
V n;ACI (kN)
V n;EC (kN)
F1 F2 F3 FS1 FS2 FC1d FC2 FC3
208.7 250.7 225.3 207.6 208.8 250.8 319.9 287.9
–c –c –c –c –c –c 113.7 113.7
145.3 172.6 160.2 145.3 145.3 –c 267.3e 267.3e
144.8 144.8 144.8 144.8 144.8 –c 266.8e 266.8e
228.4 265.6 248.9 228.4 228.4 –c 350.4e 350.4e
Mean COV a b c d e
V net =V code b
Values before subtracting the load transferred to the columns. V net is net applied shear force neglecting upward uniform pressure beneath concrete columns, which is defined differently in each design code. Not applicable. Column failed. V n is punching shear capacity predicted by the summation of V c and V frc .
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(a) F1
(b) F2
(c) F3
(d) FS1
(e) FS2
(f) FC1
(g) FC2
(h) FC3
Applied load (kN)
350 300 250 200 150 100 50 0
Applied load (kN)
350 300 250 200 150 100 50 0
Applied load (kN)
Fig. 11. Crack pattern at failure of test specimens.
350 300 250 200 150 100 50 0
350 300 250 200 150 100 50 0
(a) F1 Peak load: 208.7 kN Deflection: 23.92 mm
0
10
20
30
40
50
Peak load: 207.6 kN Deflection: 23.40 mm
10
20
30
40
50
Peak load: 319.9 kN Deflection: 39.02 mm
10
20
10
30
40
50
350 300 250 200 150 100 50 0
Peak load: 250.7 kN Deflection: 28.26 mm
20
30
40
350 300 250 200 150 100 50 0
Peak load: 208.8 kN Deflection: 23.35 mm
10
20
30
40
(c) F3
0
50
(e) FS2
0 350 300 250 200 150 100 50 0
(g) FC2
0
0 350 300 250 200 150 100 50 0
(d) FS1
0
(b) F2
50
10
Peak load: 225.3 kN Deflection: 26.15 mm
20
30
40
50
30
40
50
(f) FC1* Peak load: 250.8 kN Deflection: 34.87 mm
0
10
20
Deflection (mm) (h) FC3 Peak load: 287.9 kN Deflection: 41.49 mm
0
10
20
30
40
50
Deflection (mm) Deflection (mm) * In the case of specimen FC1, the column stub was failed before punching shear failure may occur. Fig. 12. Load-deflection curves of test specimens.
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Applied load (kN)
350
F1 F2 F3
300 250 200 150 100 50 0
0
10
20
30
40
50
Deflection (mm) (a) Layout of longitudinal reinforcement
Applied load (kN)
350
3.4. Effects of the additional casts of the high-strength concrete and the AMF-reinforced concrete
F1 FS1 FS2
300 250 200 150 100 50 0 0
10
20
30
40
50
Deflection (mm) (b) Shear reinforcement
Applied load (kN)
350
F1 FC1 FC2 FC3
300 250
rebars at the first row) are relatively steep with a high inclination, which was expected to be more than 45°, and therefore did not cross the shear rebars. This phenomenon is opposite to the findings of Hegger et al. [24], where the shear reinforcement significantly affected the shear strength of the footings. In addition, the use of shear reinforcement without head or nuts in the thin concrete footings in this study could lead to poor anchor and thus might reduce the effect of shear reinforcement. Further research is necessary to obtain a better understanding of this behavior. In the evaluation of the test results hereafter, the contribution of the shear reinforcement to the punching-shear resistance of the footings is not included.
200 Column failure
150
Fig. 13(c) compares the load-deflection curves of the specimens FC1, FC2, and FC3 after they had been strengthened with the highstrength concrete, the AMF-reinforced concrete, and both the AMFreinforced concrete and the shear reinforcement, respectively, with that of the control specimen F1. In this figure, both the strength and the deflection at the failure of the retrofitted specimens are higher than those of F1; the strength increased by 20.2–53.3%, and the deflection at the failure increased by 45.8–73.5%. It is obvious that the additional cast of the retrofit materials significantly enhanced the punching behavior, especially regarding the strength and the deflection at the failure. Regarding the specimens FC2 and FC3, the shear reinforcement did not cause a significant increase of the punching-shear strength of the footings in this study (see Fig. 14(c)). 3.5. Strain profiles of the longitudinal rebars and concrete
100 50 0 0
10
20
30
40
50
Deflection (mm) (c) Additional cast of high strength concrete and AMF reinforced concrete Fig. 13. Effects of test parameters on load-deflection curves.
reinforcement around the connection, the peak load and the deflection at the failure of the specimen F2 are 250.7 kN and 28.26 mm, respectively. To the contrary, even though the specimen F3 also had a greater longitudinal reinforcement, the top rebars showed a relatively less-robust increment compared with that of F2. 3.3. Effect of the shear reinforcement Fig. 13(b) compared the load-deflection curves of the footings F1, FS1, and FS2 to investigate the effect of the shear reinforcement. In the figure, no significant difference was observed for the three specimens. The shear reinforcement did not considerably contribute to the shear resistance of the columns footings for a given test condition. The strain profiles of the shear rebars of the specimens FS1 and FS2 were obtained from the test results and are presented in Fig. 14(a) and (b). In this figure, it is obvious that the strains of the shear rebars are much lower than their yielding strain of 0.0022 when the punching failure occurred. The shear rebars close to the column stubs exhibited relatively great strains, but their yield strain could not be reached. This finding indicates that the inner cracks (between the column stub and the shear
Fig. 15 presents the typical strain profiles in terms of the longitudinal rebars in the vicinity of the column stubs for the test specimens F1, F3, FS1, and FC2. In this figure, most of the longitudinal rebars reached their yield strain before attaining the peak load. For all that, the flexural failure load according to Simoes et al. [21] and yield line analysis [22] as mentioned above was not reached confirming punching failure of the test specimens. Moreover, in the specimen F3, the strains of S3.1, S3.3, S3.4, and S3.6 measured in the top longitudinal reinforcement but did not reach the yield strain; this indicates that the top longitudinal reinforcement did not affect the punching behavior in particular, the punching strength, which is also confirmed in Figs. 12 and 13(a). Fig. 16 shows the typical strain profiles at the top concrete surface for the specimens F1, FS1, and FC2. As shown in the figure, the concrete strains are less than the expected values, but this was inevitable due to the crushing at the concrete surfaces. 4. Investigation of the punching-shear capacity by comparing the test results with the current design codes and theoretical models In Appendices B and C, the evaluation results of the punchingshear resistance of the footings (V c ) based on the various design codes and theoretical models are presented. Each design code and model showed a significant difference in the evaluation of the effects of the major parameters on the punching-shear strength; this is because these design codes and models are based on different mechanisms and assumptions [23,25,26]. It is also noted that the current design codes use the same design equations in the strength prediction of two-way slabs and footings even though those members are different in terms of the effective depth, shear span to depth ratio (slenderness), and boundary condition
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300 250
350 Yield strain
SS4.1-4, SS4.6-8
Max. load: 207.6 kN
200 SS4.5
150 100
SS4.1 SS4.5 SS4.8
50 0 -0.0005
0
SS4.3 SS4.6
SS4.4 SS4.7
0.0005 0.001 0.0015 0.002 0.0025
Strain (ε) (a) Specimen FS1
Applied load (kN)
350
SS8.1, SS8.3, SS8.8
300
Applied load (kN)
Applied load (kN)
350
300 250
Yield strain
SS5.1, SS5.3 SS5.6-8
Max. load: 208.8 kN
200 SS5.4
150 100 50 0 -0.0005
SS5.1 SS5.6 0
SS5.3 SS5.7
SS5.4 SS5.8
0.0005 0.001 0.0015 0.002 0.0025
Strain (ε) (b) Specimen FS2
Max. load: 287.9 kN Yield strain
250
SS4.1 SS4.2
SS8.6
200 150 100 50 0 -0.0005
SS8.1 SS8.6 0
SS8.3 SS8.8
SS4.4 SS4.3 SS4.5
SS4.6
SS4.8
SS4.7
0.0005 0.001 0.0015 0.002 0.0025
Strain (ε) (c) Specimen FC3
(d) Typical location of steel strain gauges in shear reinforcement
Fig. 14. Strain profiles in shear reinforcement of specimens FS1, FS2, and FC3.
[6]. Fig. 17 compares the areas of the control sections for each design code. In the codes, it is assumed that the soil pressure immediately beneath the column will directly transfer to the column; therefore, in the strength comparison, the failure load V exp is subtracted from the soil pressure within a distance of 0.5d from the column faces according to the ACI 318-14 [23], and within a distance of 0.75d from the column faces according to the KCI 2012 [26]. In this study, the test results were compared with the predictions of the various design codes and theoretical models. 4.1. Strength contribution of the AMF-reinforced concrete The current design codes and theoretical models do not consider the effect of AMF-reinforced concrete. In this study, the strengthening of the punching-shear resistance of the footings with the AMF-reinforced concrete (V n ) was evaluated using Eq. (1), as follows:
V n ¼ V c þ V frc ;
ð1Þ
where V c is the punching-shear resistance of normal concrete. The contribution of the cast of the AMF-reinforced concrete (V frc ) on the normal concrete was evaluated in accordance with the work of Choi et al. [27,28], as follows:
V frc ¼ f pc Afrc cos /;
ð2Þ
Afrc ¼ ½2c1 þ 2c2 þ 4dfrc cot /dfrc = sin /;
ð3Þ
where Afrc is the inclined cross-sectional area that was calculated using the punching-crack angle / (in this study, the cracking angle / is assumed as 45°); dfrc is the depth of the AMF-reinforcedconcrete layer; c1 and c2 are the column sizes, respectively; and f pc is the average tensile strength of the AMF-reinforced concrete.
Based on the test results of this study (Fig. 6), the value of f pc is 3.79 MPa, which is equivalent to approximately 70% of the peak strength. This observation is similar to the post-cracking tensile strength that was investigated in the study by Gu [29]. Fig. 18 presents a method for the determination of the average tensile strength of AMF-reinforced concrete. In the figure, the area under the flexural strength-deflection curve was first evaluated up to a deflection of Lc =150 according to the ASTM C 1609/1609M-12 [15], where Lc is the distance between the supports of the specimens (=300 mm). Then, the average tensile strength was determined so that the shaded area that is generated by actual flexural strength-deflection curve is equivalent to the area that is generated by the average tensile strength. 4.2. Comparison of the strength prediction according to the current design codes and theoretical models Table 4 presents the punching-shear resistance of the calculated footings based on the current design codes including the ACI 31814 [23], Eurocode 2 [25], and KCI 2012 [26]. In the FC series, as shown in Table 4, the punching-shear resistances agreed well with the test results (V exp: ) when the contribution of the AMF-reinforced concrete (V frc ) was added to the prediction by the current design codes (V c;ACI and V c;KCI ). In general, for the current test results, the ratio of V exp =V n;ACI has a mean of 1.35 and a coefficient of variation (COV) of 0.14; meanwhile, the ratio of V exp =V n;KCI exhibited lower values with a mean of 1.25 and a coefficient of variation of 0.10. However, in the cases of the F and FS series, the punching-shear strength that was predicted by the ACI 318-14 [23] is conservative and lower than that predicted by the KCI 2012 [26]. This is because the ACI 318-14 [23] does not consider the effect of the longitudinal-reinforcement ratio on the punching-shear strength, unlike the KCI 2012 [26].
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350
300
Yield strain
250
Applied load (kN)
Applied load (kN)
350
Max. load: 208.7 kN
200 150
S1.2
S1.4
100 50 0 -0.001
S1.1
S1.3
0.001
0.003
S1.1 S1.3 0.005
S1.2 S1.4
0.007
S3.1, S3.4, S3.6 S3.3
300 250
Max. load: 225.3 kN
200 S3.5
150 100
S3.1 S3.5
50 0 -0.001
0.009
0.001
Strain (ε) (a) Specimen F1
0.003
S3.3 S3.6
0.005
S3.4
0.007
0.009
Strain (ε) (b) Specimen F3 350
350
Applied load (kN)
300 S4.1-4
250
Max. load: 207.6 kN
200 150 100
Yield strain S4.1 S4.3
50 0 -0.001
0.001
0.003
0.005
S4.2 S4.4
0.007
300
Max. load: 319.9 kN S7.2 S7.1
250 200 150
S7.4 S7.3
100
S7.1 S7.3
0 -0.001
0.009
0.001
S1.1 S1.2
FS1
S3.4
S3.1 S3.2
S1.4
0.003
0.005
S7.2 S7.4
0.007
0.009
Strain (ε) (d) Specimen FC2
F3
F1
Yield strain
50
Strain (ε) (c) Specimen FS1
FC2 S7.1 S7.2
S4.1
S3.3
S4.2
S4.4
S7.4
S3.5 S1.3
S4.3
S7.3
S3.6
(e) Strain gauge location Fig. 15. Strain profiles in longitudinal re-bars of specimens F1, F3, FS1, and FC2.
350
300 250
Max. load: 208.7 kN
C1.2
Applied load (kN)
Applied load (kN)
350 C1.3-4
200 150 100 50 0 -0.0006
C1.1 C1.1 C1.3
C1.2 C1.4
-0.0004
-0.0002
0
0.0002
300 250
C4.1, C4.4 Max. load: 207.6 kN
200 150
C4.3 C4.2
100 50 0 -0.0006
C4.1 C4.3
C4.2 C4.4
-0.0004
-0.0002
0
0.0002
Strain (ε) (b) Specimen FS1
Strain (ε) (a) Specimen F1 350
Applied load (kN)
Applied load (kN)
Yield strain
300 250
Max. load: 319.9 kN C7.1
C1.4
200 150 100 50 0 -0.0006
C7.3 C7.1 C7.3
C1.3
C7.4
C7.2 C7.2 C7.4
-0.0004
-0.0002
Strain (ε) (c) Specimen FC2
C1.2 0
C1.1
0.0002
(d) Typical location of concrete strain gauges
Fig. 16. Strain profiles at top concrete surface of specimens F1, FS1, and FC2.
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Rectangular column
Circular column
2d 0.2dcot DBroms
0.5d
Simoes et al. [20] KCI 2012 [25] and ACI 318-14 [22] Eurocode 2 [24] Note: DBroms is defined in Broms [29]
Broms [29]
Fig. 17. Control perimeters specified in prediction models.
Table 5 presents the punching-shear resistance that was predicted by Broms [30] and Simoes et al. [21]. Similarly, as shown in Table 5, the punching-shear resistances of the specimens FC2 and FC3 in the FC series agreed well with the test results (V exp: ) when the contribution of the AMF-reinforced concrete (V frc ) was added to the prediction by theoretical models [21,30]. In Table 5, the COVs of the models of Broms [30] and Simoes et al. [21] are 0.09 and 0.11, respectively, indicating that the strut-and-tie model that was developed by Broms [30] and the kinematic theorem-of-limit analysis model that was developed by Simoes et al. [21] could describe the structural behavior well enough to estimate the punching-shear capacity of footings. In general, the research shows that the theoretical models are accurate in the prediction of the punching-shear strength of the footings with the means of 1.21 and 1.18 for Broms [30] and Simoes et al. [21], respectively, which are lower than the existing design codes. 5. Conclusions Eight one-quarter scale reinforced-concrete (RC) footing specimens were constructed with various steel-reinforcement and retrofit details. The test specimens were supported on a bed of steel car springs that simulated the elastic behavior of soil, and they were tested under a concentrated axial load that was applied through the column stubs. The main parameters of this study are the additional placement (or concentrated layout) of longitudinal rebars around the column-footing connections, the inclination of the shear reinforcement, the additional cast of high-strength concrete, and the additional cast of amorphous metallic fiber (AMF)reinforced concrete. From the test results, the primary findings are as follows:
Fig. 18. Determination of average tensile strength (f pc ).
The punching-shear resistance of the footings that was evaluated by the Eurocode 2 [25] is also presented in Table 4. In this table, the predicted resistances are significantly different from the test results. The ratio of V exp =V n;EC2 has a mean of 0.84 and a COV of 0.03, and this because the Eurocode 2 [25] evaluated the punching shear within a relatively large critical perimeter; that is, it is located at a distance of 2d from the column faces. However, according to the observation in the current footing tests, the developed punching-shear cracks are closer to the column faces than the location of the Eurocode 2 critical perimeter [25].
1. Based on the test observations and the yield-line analysis, all of the test specimens failed in the punching shear, except for FC2, the column stub of which was crushed. 2. Due to the addition of a bottom longitudinal reinforcement around the column-footing connections, the peak load and the deflection at the failure of the specimen F2 apparently increased up to approximately 20.1% and 18.1%, respectively. 3. The additional casts of the high-strength concrete and the AMFreinforced concrete increased the punching-shear strength of the footings. The peak load and the deflection at the failures of the specimens FC1, FC2, and FC3 significantly increased up to approximately 53.3% and 73.5%, respectively.
Table 5 Punching shear capacity of footings predicted by existing theoretical models. Specimens
V exp a (kN)
V frc (kN)
V n;Broms (kN)
V n;Simoes (kN)
V net =V model b Broms [30]
Simoes et al. [21]
F1 F2 F3 FS1 FS2 FC1d FC2 FC3
208.7 250.7 225.3 207.6 208.8 250.8 319.9 287.9
–c –c –c –c –c –c 113.7 113.7
149.1 186.1 169.4 149.1 149.1 –c 271.1e 271.1e
154.1 184.9 169.5 154.1 154.1 –c 276.1e 276.1e
1.29 1.24 1.23 1.29 1.29 –c 1.12 1.01
1.25 1.25 1.23 1.24 1.25 –c 1.11 0.99
1.21 0.09
1.18 0.11
Mean COV a b c d e
Values before subtracting the load transferred to the columns. V net is net applied shear force neglecting upward uniform pressure beneath concrete columns, which is defined differently in each theoretical model. Not applicable. Column failed. V n is punching shear capacity predicted by the summation of V c and V frc .
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4. For a given test condition, the placement of a shear reinforcement around the column-footing connections did not affect the punching-shear strength of the footings. On the contrary, in the study by Menetrey and Bruhwiler [32], it was demonstrated that the addition of shear reinforcement could significantly increase the punching-shear capacity. Further research is necessary for a better understanding of this behavior. 5. Considering the contribution of the AMF-reinforced concrete to the punching-shear resistance of the footings, most of the design codes and theoretical models show sound predictions of the punching strength.
Acknowledgments This research was supported by the Basic Science Program (2014R1A1A2053499) through the National Research Foundation of Korea (NRF), funded by the Ministry of Education. Appendix A. Flexural-strength evaluation and yield-line analysis Fig. 19. Yield line pattern considered in punching shear failure of a square footing with a square column.
A.1. Flexural strength of footings Simoes et al. [21] developed an equation to determine the flexural strength of footings (V flex ) based on the yield-line theory. Once the flexural strength is higher than the punching-shear strength, the footings are governed by the punching failure; under the converse condition, the footings are governed by the flexural failure.
V flex ¼ 2pmR
rs r 2s ; r q r c r 2s r 2c
" 2 # 0 fy d 0 þ q ðd d Þ ; mR ¼ f cp d 2d f cp rq ¼
R¼
ðA2-2Þ 1:5pA
c1 þ c2
p
ðc1 þ c2 Þ
!1=3 0:5 2
;
ðA2-3Þ
ðA1-1Þ
where mR is the moment capacity per unit length, R is the radius of the failure zone, and A is the area of the footing.
ðA1-2Þ
Appendix B. Evaluation of the punching-shear strength by the current design codes
ðA1-3Þ
where mR is the moment capacity per unit of length; rq is the radius of the reaction resultant; r c and r s are the radius of the circular column and the circular footings, respectively; d is the effective slab 0 depth; d is the effective depth of the top reinforcement; q is the tension-reinforcement ratio; and f cp is the plastic-concrete compressive strength. A.2. Yield-line analysis
h
2pmR 2 2 i ; p R2 2 þ c1pþc 1 3A 1 þ c1pþc R R
The punching-shear resistance (V c ) for a slab without a shear reinforcement is evaluated as the minimum of three values, as follows:
8 9 qffiffiffiffi 0 > > 4 > > b d f 0:166 2 þ 0 > > c b > > > > < = qffiffiffiffi 0 a d s V c ¼ min 0:083 b þ 2 ; b f 0d c 0 > > > > > > qffiffiffiffi > > > > : 0:332 f 0 b0 d ; c
ðB1-1Þ
where b is the ratio of the long-to-short side of the loading area;
Fig. 19 shows the yield-line pattern for a square footing that has been subjected to a concentrated load through the square column at the center of the footing. The bottom plane of the footing will consequently be loaded with a uniform soil-reaction force, which is produced by the concentrated load in the column. The yieldline pattern was determined based on the studies by Gesund and Kaushik [22] and Gilbert et al. [31]. In this figure, the yield-line pattern is slightly complex and the moment distribution at the column is nearly symmetric. It is noted that the reinforcement is assumed as isotropic. From the yield-line analysis, Gesund and Kaushik [22] derived an equation to determine the punching-shear capacity of footings, as follows: 2 1 c1pþc R
2
B.1. ACI 318-14 [23]
2 ðr3s r3c Þ ; 3 ðr2s r2c Þ
V yieldline ¼
mR ¼ qf y d ;
ðA2-1Þ
as = 40 for the internal loads, 30 for the edge columns, and 20 for the corner columns; and b0 ¼ 2c1 þ 2c2 þ 4d. B.2. Eurocode 2 [25] The punching-shear resistance (V c ) for a slab without a shear reinforcement is evaluated as follows: 0 1=3
V c ¼ 0:18nð100qf c Þ n¼1þ
rffiffiffiffiffiffiffiffiffi 200 ; d
2d b0 d; a
ðB2-1Þ
ðB2-2Þ
where n is the size-effect factor; a is the distance from the periphery of the column to the control perimeter; and b0 ¼ 2c1 þ 2c2 þ 16d. In this design code, the net applied shear force is calculated using Eq. (B2-3), as follows:
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V Ed;red ¼ V Ed DV Ed ;
ðB2-3Þ
where V Ed is the column load and DV Ed is the net upward force within the considered control perimeter; that is, the upward uniform pressure from the soil minus the self-weight of the footing. B.3. KCI 2012 [26] The punching-shear resistance (V c ) is evaluated using Eq. (B31), as follows:
V c ¼ v c b0 d;
ðB3-1Þ
v c ¼ kks kb
ðB3-2Þ
0
f t cot /ðcu =dÞ;
b0 ¼ 2c1 þ 2c2 þ 4 cot /:cu ;
ðB3-3Þ
where v c is the average shear stress; k is the coefficient of the lightweight concrete (k ¼ 1); ks is the size-effect factor [Eq. (B3-4)]; kb0 is an aspect-ratio factor [Eq. (B3-5)]; and cu is the depth of the compression zone [Eq. (B3-6)].
rffiffiffiffiffiffiffiffiffi 4 300 6 1:0; ks ¼ d
ðB3-4Þ
4 kb0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi 6 1:25; as ðbd0 Þ sffiffiffiffi
q
cu ¼ d 25
cot / ¼
0
fc
300
ðB3-5Þ
q
!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 f t ðf t þ 2f c =3Þ ft
;
0
fc
ðB3-6Þ
;
ðB3-7Þ
where as ¼ 1 for the interior column. In the KCI 2012 [26], the soil pressure within the critical perimeter at a distance of 0:75d from the column faces is neglected from the applied load. Appendix C. Evaluation of the punching-shear strength by the theoretical models C.1. Broms [30]
1=3 0:150 V c ¼ rc t sinðcÞu ; t
ðC1-1Þ u x0
2
rc ¼ f 0c 0:6 þ 0:9ð1 0:007 Þ 6 1:2f 0c ; t¼
x0 ; 2 cosðcÞ
u¼p Bþ
x0 x0 þ ; tanð2cÞ 2 tanðcÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 1þ 1 d; x0 ¼ n0 q n0 q
n0 q ¼
D Es 1 þ ln c0 q; Ec 1 þ ln DB
where
rc is the compression strength of the capital; x0 is the com-
ðC1-2Þ ðC1-3Þ ðC1-4Þ
ðC1-5Þ
ðC1-6Þ
pression zone; t is the depth of the compression strut; ð0:150=tÞ1=3 is the size-effect factor with the dimension t (in meters); 0.0150 is the diameter of the standard test-cylinder specimen (in meters); Es and Ec are the elastic modulus of the steel rebars and the concrete, respectively; and D and B are the footing and column sizes, respectively. Moreover, a part of the uniformly distributed load within the critical shear section that does not affect the punching-shear capacity of footings is neglected. In this theoretical model, the perimeter of the critical shear section (c0 ) was determined using Eq. (C1-7), as follows:
2d c0 ¼ p B þ ; tan / tan / ¼
ðC1-7Þ
1:4d P 1; D=3 B=2
ðC1-8Þ
where / is the inclination angle of the shear crack. C.2. Simoes et al. [21] Simoes et al. [21] developed a mechanical model in consideration of the failure mechanisms that is based on both a vertical penetration of the punching cone and a rotation of the outer part of the footing, and this allows for a consideration of the roles of both the bottom and top reinforcements on the failure load regarding the prediction of the punching strength of footings. In the model, the punching-shear failure mode and its interaction were investigated by means of the kinematic theorem-of-limit analysis. In Simoes et al. [21], the punching-shear resistance of footings (V c ) was calculated using Eq. (C2-1), as follows:
1 f b0 d; 0:9 þ rds cp
ðC2-1Þ
b0 ¼ 2p½r c þ 0:2d cotðbÞ;
ðC2-2Þ
Vc ¼
b¼
In Broms [30], a mechanical model was proposed to evaluate the punching-shear strength of footings. The hypothesis of Broms [30] states that the punching occurs when the capital fails in a compression, thereby forming a diagonal shear crack. In addition, the angle to the horizontal of the punching crack near the column (2c) was assumed as 50°. The punching-shear capacity of the footings was then determined using Eq. (C1-1), as follows:
295
p=2 0:8 þ 0:5 da
½rad;
0
f cp ¼ f c ge gfc ;
gfc ¼
f c0 0 fc
ðC2-3Þ ðC2-4Þ
!1=3 6 1;
ðC2-5Þ
where b is the secant inclination of the failure surface; a is the shear span (=rs rc ); ge is the reduction factor that accounts for the presence of the transverse strains and depends on the amounts of the longitudinal mechanical-reinforcement ratio (x ¼ qf y =f cp ) refer to Simoes et al. [21] to determine the value of ge ; gfc is the reduction factor that accounts for the brittleness of high-strength concrete; and f c0 is the reference compressive strength (=30 MPa). In this proposal model, a part of the uniform soil reaction that acts inside the failure zone and does not influence the punchingshear capacity of footings was determined using Eq. (C2-6), as follows:
c0 ¼ p½rc þ d cotðbÞ : 2
ðC2-6Þ
It is noted that the proposal models developed by Broms [30] and Simoes et al. [21] are used for circular footings. To apply such models to square footings, the square columns should be replaced
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