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Blast responses of concrete beams reinforced with GFRP bars: Experimental research and equivalent static analysis
Composite Structures 226 (2019) 111271
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Blast responses of concrete beams reinforced with GFRP bars: Experimental research and equivalent static analysis ⁎
T
⁎
Sanfeng Liua, Yinzhi Zhoua, , Jiannan Zhoua, Bei Zhanga,d, Fengnian Jina, Qing Zhengb,c, , ⁎ Hualin Fanb,c, a
State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing 210007, China Research Center of Lightweight Structures and Intelligent Manufacturing, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China c State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China d Institute of Defense Engineering, AMS, PLA, Luoyang 471023, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: GFRP composite bar Concrete beam reinforced with GFRP bars Blast loading Experiments Damage
The glass fiber reinforced polymer (GRFP) bars have advantages in corrosion resistance and high load bearing capacity. They can be used as reinforcements in concrete protective structures to replace the steel bars. In this paper, the GRFP bar reinforced concrete beams (GRCBs) were designed, casted and tested. As a reference, the steel bar reinforced concrete beams (SRCBs) having the same flexural rigidity with the GRCBs were also designed, casted and tested. Under close-in explosions, the GRCBs have smaller displacement than the SRCBs, because the GFRP bars are still in elastic state while the steel bars enter into plastic deformation. Cracks, spalling and cratering are the main damage modes of the concrete beams. Through the quasi-static four-point-bending experiments, the residual load capacity of each beam has been evaluated, which quantitatively reveals the damage degree of the exploded concrete beams. In current research the explosion produces negligible damage to the GRCBs, while the load capacity the SRCBs reduces at a level of 28% at the most. The research proves that the GRCB has greater blast-resistance.
1. Introduction Generally, fiber reinforced polymers (FRPs) have high strength and they can be applied as reinforcements of concrete structures [1–6] and metallic structures [7–9]. Steel corrosion greatly reduces the durability of concrete structures. Replacing steel bars by FRP bars is an effective method to prevent the steel corrosion [10,11]. Yu and Kodur [12] investigated factors governing the fire response of the FRP-rebar reinforced concrete beams. Duic et al. [13] studied the performance of concrete beams reinforced with the basalt fiber composite rebars. Sovják et al. [14] discussed the long-term behavior of concrete slabs pre-stressed by the carbon fiber composite rebars. Fava et al. [15] explored the bond between the GFRP rebars and the concrete, as well as Rolland et al. [16]. Wu et al. [17] applied calcium sulfoaluminate to improve the bond between the GFRP rebars and the concrete. FRP rebars are preferable for ocean protective structures, which have high demand of corrosion resistance and blast resistance. With the increase of terrorist attacks and explosion accidents, the
research on the blast resistance of structures is becoming important in recent years [18]. Although the blast experiment on steel bar reinforced concrete (RC) structures is complex and expensive, there are several literatures focused on the blast experiments of the RC beams. Liu et al. [19] conducted an experimental research on the blast behavior of the RC beams and columns. With the decrease of the scaled distance, the damage patterns of the RC beams evolved from a small number of cracks on the surface to large spalling area on the bottom and crushing on the top. Zhang et al. [20] conducted experimental and numerical researches on the behaviors of the RC beams under close-in explosions. With the increase of the explosive mass, the damaged mode of RC beam changes from overall bending to plastic hinge at mid-span, with severe exfoliation at the bottom surface around the mid-span. Yao et al. [21] launched an experimental investigation on the dynamic response of the RC beams with different stirrup ratios. The displacement-to-thickness ratio is inversely proportional to the stirrup ratio. Wang et al. [22] conducted experimental studies on the blast resistance of the one-way RC slabs under closed-in explosion, and obtained the failure patterns of
⁎ Corresponding authors at: Research Center of Lightweight Structures and Intelligent Manufacturing, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (Q. Zheng) and (H. Fan). E-mail addresses: [email protected] (Y. Zhou), [email protected] (Q. Zheng), [email protected] (H. Fan).
https://doi.org/10.1016/j.compstruct.2019.111271 Received 31 December 2018; Received in revised form 16 June 2019; Accepted 26 July 2019 Available online 27 July 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
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to 0.0262. The elastic modulus is 39 GPa. The steel bar is elastic perfectly-plastic. Its yield strength is 458 MPa and the elastic modulus is 193 GPa. The grade of the concrete material is C40. The mixture ratio of cement: sand: pebble: water is 1: 1.19: 2.31: 0.42. The Portland cement has a strength grade of 42.5 MPa. The concrete was cured under room temperate and the average compressive strength is 51.5 MPa, obtained by six cubes with a dimension of 150 mm × 150 mm × 150 mm. The density of the concrete material is 2.5 g/cm3.
the RC slabs under different scaled distance. To enhance the anti-blast capacity of the RC structures, highstrength concrete (HSC) and high-strength reinforcing bars have been applied. Yang et al. [23] investigated the blast responses of the highstrength reinforced concrete beams. However, the effect of the highstrength concrete on the blast performance of the RC beam is limited. To improve the anti-blast properties of the RC structures furtherly, Yang and Hassan [24] took another blast experiment on HSC beams by replacing the normal steel bars with high-strength ASTM A1035 bars whose yield strength is 900 MPa. The results show that the highstrength bars can effectively enhance the anti-blast capacity of the RC beam. As the FRP bars also have high strength as mentioned above, the concrete members reinforced with the FRP bars might have a better anti-blast capacity than those reinforced with normal steel bars. Feng et al. [1] has revealed the blast responses of the slabs reinforced with FRP bars through explosion experiments. It is found that the slabs reinforced with FRP bars have strong deformation recovering ability and the spalling area of the concrete under shock wave is greatly reduced [1]. After explosion, the static load carrying ability of the exploded slab is still higher than that of the slab reinforced with steel bars. As the slab is not thick, the deformation of the FRP bars under explosion is limited, far below their rupture strain. The slab fails at spalling and breaching. For deep concrete beams, the FRP bars will sustain much larger deformation and the beams have great probability to fail at FRP rupture. The feasibility of using FRP bars to reinforce deep protective concrete beams is worthy of investigation. In this research, the concrete beams reinforced with GFRP bars were designed and casted. Their blast-resistant performances were checked by close-in explosion experiments and compared with those of the concrete beams reinforced with steel bars which were designed of the same stiffness.
2.2. Concrete beams reinforced by GFRP composite bars Two types of concrete beams with equal flexural rigidity were designed and casted, including the steel bar reinforced concrete beam (S12) and the GFRP bar reinforced concrete beam (G22). The width and height of the beam are 220 mm × 300 mm and the length is 2000 mm, as shown in Fig. 2. The compression reinforcements of all these beams are two HRB400 steel bars with a diameter of 12 mm. For beam S12 the tensile reinforcements are two HRB400 steel bars with a diameter of 12 mm. For beam G22 the tensile reinforcements are three GFRP bars with a diameter of 22 mm. The stirrups are made up of C8@80 mm and C8@ 150 mm. The thickness of the protective layer is 20 mm. 3. Dynamic responses under blast loading 3.1. Protocol of explosion experiments Close-in explosion experiments were carried out, as shown in Fig. 3. The beam was simply-supported on a steel frame fixed on the ground, with 100 mm hanging over at each end. The square TNT cartridge was placed over the mid-span of the beam and detonated by an electric detonator. Three displacement sensors were placed below the beam. Strain gauges were pre-adhered on the surfaces of the tensile bars. Pressure sensors were placed on the upper surface of the beam at mid-span. A free-field pressure sensor was set at the same height with the TNT block, but 1.5 m away from the TNT block. All the data were collected by a dynamic system D5922N. The explosion experiment scheme is listed in Table 1 in details.
2. Concrete beams reinforced by GFRP bars 2.1. Material properties The reinforcement of the GFRP bar is boron-free glass fiber and the matrix is vinylester resin. The diameter of the GFRP bars is 22 mm, as shown in Fig. 1. The GFRP bars were produced by Nanjing Fenghui Composite Material Co., Ltd. through pultrusion method with a fiber volume fraction of 65%. The surface was ribbed before the resin solidifies, without sand coating. Provided by manufacturer, the ultimate tensile strength is more than 600 MPa, and the elastic modulus is ranged from 40 GPa to 45 GPa, the ultimate shear strength is ranged from 110 MPa to 150 MPa. The steel bars used in this experiment is HRB400 with a diameter of 12 mm. Tensile properties of the GFRP bar and the steel bar were tested by a 2000 kN MTS system at a loading rate of 0.6 mm/min. The strain–stress relationships of bar are shown in Fig. 1. The GFRP bar has linear elastic deformation. The ultimate strength is 1027 MPa where the strain is up
3.2. Free-field pressure and reflective overpressure The measured maximum free-field pressure under 0.5 kg TNT with a stand-off of 1.5 m is 0.220 MPa. The measured maximum reflective overpressure under 0.1 kg TNT with a stand-off of 0.5 m is 2.935 MPa. These data were compared with those calculated by CONWEP [25–28], a calculation program for explosion parameters of conventional weapons issued by US Department of the Army. As the CONWEP is based on a large number of explosive experiments, its accuracy is world-widely recognized. The CONWEP calculation process is listed as follows. Firstly, select the calculation process of weapon effect. Secondly, choose the explosive type of “air-blast” and “aboveground detonation”. Thirdly, input the weight of TNT and distance between the explosive point and the specimen. Finally, choose the output data as pressure and duration and then the free-field pressure and the reflective overpressure history curves can be obtained. The maximum free-field pressure under 0.5 kg TNT with a stand-off of 1.5 m calculated by CONWEP is 0.215 MPa. The maximum reflective overpressure under 0.1 kg TNT with a stand-off of 0.5 m calculated by CONWEP is 3.490 MPa. The curves are compared in Fig. 4 and they are in good agreement, indicating that the measurements are credible. Detailed information is summarized in Table 2. There may be a lot of uncertainty in the explosion experiments. For beams S12-1-3 and S12-3, the blast loads were so strong that the stain gauges were broken and no strain data were obtained. Damage phenomena of each beam are
1200 Steel rebar GFRP rebar
1000
Stress (MPa)
800 600 400 200 0 0.000
0.015
0.030
0.045
0.060
Strain
(a)
(b)
(c)
Fig. 1. (a) Steel bar, (b) GFRP bar and (c) their tensile properties. 2
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(a)
(b) Fig. 2. Reinforcing diagram of (a) steel bar reinforced concrete beam S12 and (b) GFRP bar reinforced concrete beam G22 (Unit: mm).
described in Table 3 in details.
Table 1 Blast protocol of concrete beams.
3.3. Blast responses of steel bar reinforced concrete beams (SRCBs) 3.3.1. Dynamic deformations Beam S12 has almost no displacement when the scaled distance is 1.0772 m/kg1/3, as listed in Table 1. When the scaled distance is 0.5 m/ kg1/3, the maximum displacement was 3.31 mm and the rebound is 1.77 mm. When the scaled distance is 0.5159 m/kg1/3, the maximum displacement is 4.89 mm and the rebound is 5.52 mm. In these cases, there is little residual deformation after explosion, as shown in Fig. 5(a). When the scaled distance is 0.4507 m/kg1/3, the maximum displacement is 11.14 mm, while the rebound is only 5.515 mm, just half of the downward displacement. The reason is that the steel bars flow into plastic state. As shown in Fig. 5(a), the beam remains 5.50 mm residual displacement after explosion.
Beam
Compression tendon
Tensile tendon
Stirrup
TNT (kg)
Blast distance (m)
Scaled distance (m/kg1/3)
S12-1-1 S12-1-2 S12-1-3 S12-2 S12-3 S12-4 G22-1-1 G22-1-2 G22-2 G22-3
2C12
2C12
C8@ 150 and C8@80
2C12
3 × F3
C8@ 150 and C8@80
0.1 1 3 2 3 4 0.5 3 2 4
0.50 0.50 0.50 0.65 0.65 0.65 0.80 0.65 0.65 0.65
1.0772 0.5000 0.3467 0.5159 0.4507 0.4095 1.0079 0.4507 0.5159 0.4095
almost no deformation, as shown in Fig. 5(b). When the scaled distance is 0.5000 m/kg1/3, the maximum strain of steel bar is 2710 με and the residual strain is 501 με. When the scaled distance is 0.5159 m/kg1/3, the maximum strain is 3226 με and the residual strain is 634 με. In these
3.3.2. Dynamic strains When the scaled distance is 1.0772 m/kg1/3, the steel bars have
Fig. 3. Field explosion experiment protocol of concrete beams. 3
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Fig. 4. Typical (a) free-field pressure under 0.5 kg TNT with a stand-off of 1.5 m and (b) reflective overpressure under 0.1 kg TNT with a stand-off of 0.5 m.
residual displacement is 1.82 mm. Under 0.4507 m/kg1/3, the maximum displacement is 10.53 mm and the rebound is 9.35 mm. The residual displacement is 3.38 mm. The rebound is close to the downward displacement, representing that the GFRP bars are not broken.
cases, the steel bars are in elastic state. The residual strains of the steel bars are also very small, as shown in Fig. 5(b). When the scaled distance is 0.4095 m/kg1/3, the maximum strain is 3515 με and the residual strain is 1352 με. As the steel bars flow into plastic stage, the beams have large deflections. The SRCBs have obvious residual plastic deformations when the scaled distance is small.
3.4.2. Dynamic strains When the scaled distance is 1.0079 m/kg1/3, the maximum strain of GFRP bar is 369 με. When the scaled distance is 0.5159 m/kg1/3, the maximum strain is 3466 με and the residual strain is 728 με, as shown in Fig. 7(b). Under 0.4507 m/kg1/3, the maximum strain is 5867 με and the residual strain is 1022 με. Under 0.4095 m/kg1/3, the maximum strain is 5975 με and the residual strain is 799 με. All these strains are much smaller than the rupture strain of the GFRP (about 25,000 με in Fig. 1), indicating that the GFRP bars still work in elastic state.
3.3.3. Damage modes When the scaled distance is 1.0772 m/kg1/3, the damage of beam S12-1 can be ignored, as shown in Fig. 6. When the scaled distance is 0.5000 m/kg1/3, the beam is slightly damaged, with one main vertical crack of 240 mm long cross the beam. Other cracks are relatively short, about 90 mm long. When the scaled distance is 0.3467 m/kg1/3, the beam is severely damaged, with a spalling area of 770 cm2 at the lower surface. Oblique shear cracks cross through each other and form a triangular peeling region. When the scaled distance is 0.5159 m/kg1/3, beam S12-2 is slightly damaged. There are three cracks, as shown in Fig. 6. The length of the long transverse crack is about 300 mm and that of the short transverse crack is about 150 mm. When the scaled distance is 0.4507 m/kg1/3, beam S12-3 is slightly damaged. There are seven cracks, including two 300 mm long vertical main cracks, three 200 mm long transverse cracks, and two 250 mm long oblique cracks. When the scaled distance is 0.4095 m/kg1/3, beam S12-4 is severely damaged. The spalling area on the lower surface is 1100 cm2. There are eight 200 mm long oblique cracks, forming an inverted triangular damage region.
3.4.3. Damage modes When the scaled distance is 1.0079 m/kg1/3, beam G22-1 is almost intact. When the scaled distance is 0.4507 m/kg1/3, the beam is severely damaged with five vertical cracks and one transverse crack. The width and the length of the transverse crack are 2.4 mm and 400 mm, respectively, as shown in Fig. 8. In fact, a spalling plate is coming into being. When the scaled distance is 0.5159 m/kg1/3, beam G22-2 is slightly damaged with 13 cracks, including three 300 mm long vertical cracks, five 120 mm long vertical cracks, one 360 mm long transverse crack and four 80 mm long transverse cracks. When the scaled distance is 0.4095 m/kg1/3, beam G22-3 is severely damaged. A peeling region with an area of 225 cm2 is observed on the upper surface. The spalling region on the lower surface has an area of 770 cm2. There are eleven 350 mm long oblique cracks, forming a large inverted triangle region.
3.4. Blast responses of GFRP bar reinforced concrete beams (GRCBs) 3.4.1. Dynamic deformations Beam G22 has almost no displacement when the scaled distance is 1.0079 m/kg1/3. Under 0.5159 m/kg1/3, the maximum displacement is 4.81 mm and the rebound is 4.03 mm, as shown in Fig. 7(a). The Table 2 Blast responses of concrete beams. Beam
Scaled distance (m/kg1/3)
Maximum displacement (mm)
Rebound displacement (mm)
Maximum Strain (10 −6)
residual strain (10 −6)
Damage mode
S12-1-1 S12-1-2 S12-1-3 S12-2 S12-3 S12-4 G22-1-1 G22-1-2 G22-2 G22-3
1.0772 0.5000 0.3467 0.5159 0.4507 0.4095 1.0079 0.4507 0.5159 0.4095
0.49 3.31 24.30 4.89 11.14 24.36 0.86 11.25 4.90 18.61
– 1.77 – 5.52 5.52 – – 9.35 4.03 –
47 2710 – 3227 – 3515 369 5867 3466 5975
– 501 – 634 – 1352 – 1022 728 799
Intact Slight damage Severely damaged Slightly damaged Damaged Severely damaged Intact Damaged Severely damaged Severely damaged
4
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Table 3 Damage information of blast loaded beams. Beam
Damage
S12-1-1 S12-1-2 S12-1-3 S12-2 S12-3 S12-4
Intact Cracks Spalling Cracks Cracks Spalling
G22-1-1 G22-1-2 G22-2 G22-3
Intact Cracks Cracks Spalling
Phenomena
Two mid-span cracks. One crosses through the height and the other is 80 mm long Three wide through-thickness cracks. Slight spalling at the front surface and spalling area 770 cm2 at the back One 300 mm long vertical crack. One 300 mm long horizontal crack. One 150 mm long horizontal crack Two 300 mm long vertical cracks. Three 200 mm long horizontal cracks. Two 250 mm long inclined cracks Severe spalling with a spalling area of 1100 cm2 at the back. Slight spalling at the front surface with a spalling area of 175 cm2. Eight 200 mm long inclined cracks One 400 mm long horizontal crack and the crack is wide. Four 200 mm long vertical cracks Three through-thickness cracks. One 360 mm long horizontal crack. Four 80 mm long horizontal cracks. Five 120 mm long vertical cracks Spalling at the back surface with a spalling area of 770 cm2. Spalling at the front surface with a spalling area of 225 cm2. Eleven 350 mm long inclined cracks
layer peels off from the beam. Beam S12-1 under 0.3467 m/kg1/3, beam S12-4 under 0.4095 m/kg1/3, and beam G22-3 under 0.4095 kg/m1/3 were in such damage grade, as listed in Table 2.
3.5. Comparison between GFRP and steel bars reinforced beams 3.5.1. Displacement As all the beams are designed with equal stiffness, they have close maximum downward displacements under the same scaled distance, as shown in Fig. 9. The difference lies in the rebound and the residual displacement. As the steel bars will enter into plastic region in a relative small deformation, beam S12 always has much smaller rebound but much larger residual displacement. As GFRP bars can keep elastic in most cases, GRCBs always have elastic responses. Their rebounds are usually comparable with the maximum downward displacements. The residual displacement is usually very small.
4. Damage evaluation through determination of the residual load capacity 4.1. Four-point bending experiments on reference beams Under four-point-bending, the GRCBs and the SRCBs exhibit different deformation behaviors and failure modes, as shown in Figs. 10 and 11, respectively. The span of the beam is 1800 mm and the span of the load distribution beam is 600 mm. To obtain the damage degree of the concrete beams after explosion, the un-exploded reinforced concrete beams, named S12-5 for steel bar reinforced concrete beam and G22-4 for GFRP bar reinforced concrete beam, are tested firstly as the reference beams. Reference beam SRCB S12-5 has elastic perfectlyplastic deformation, with yielding plateau staying at 107 kN, as listed in Table 4. Reference beam GRCB G22-4 has quasi-linear deformation with a failure load of 409 kN, as listed in Table 5. These two beams have identical flexural rigidity of 0.53 kN/mm. At failure, the strain of the GFRP bar is 0.0138 and the displacement of the beam is 34.0 mm, while the strain of the steel rebar is only 0.0029 and the displacement is 39.9 mm. Under four-point-bending, the failure mode of SRCB S12-5 is flexural failure, with three vertical cracks round mid-span. The failure mode of GRCB G22-4 is the combination of two vertical cracks below the load point and concrete crushing on the upper surface at mid-span. It is a hybrid flexural-shearing failure mode.
3.5.2. Damage levels Through the explosion experiments, the damage can be divided into five grades: intact; slight damage; medium damage; severe damage and spalling. Intact means no cracks are observed. For all the beams under 1.0772 m/kg1/3, the damage grade is intact, as listed in Table 2. Slight damage means the beam has few cracks. The sparse cracks have little influence on the load carrying ability of the beam. In Table 2, when the scaled distance is greater than 0.5 m/kg1/3, the damage grade is slight damage. Medium damage indicates that the beam has visible vertical cracks but no transverse fracture zone along the interface of the tensile rebar, such as beam G22-2 under 0.5159 kg/m1/3. Severe damage indicates that the beam has dense cracks and a wide transverse fracture zone along the interface of the tensile rebar. A potential falling block is visible. Beam S12-3 under 0.4507 m/kg1/3 and beam G22-1–2 under 0.4507 kg/m1/3 are severely damaged, as listed in Table 2. Spalling indicates the fracture zone along the interface of the tensile bars is well developed and the concrete block below the reinforcement
4.2. Residual capacity of blast loaded beams The residual load capacity under four-point-bending is adopted to
6
4000 0.1kg-0.5m 1kg-0.5m 2kg-0.65m 4kg-0.65m
3 0 -6
Strain ( 10 )
Displacement (mm)
3200
-3 -6 1/3
1.0077 m/kg 1/3 0.5000 m/kg 1/3 0.5159 m/kg 1/3 0.4507 m/kg
-9 -12 0.00
0.03
0.06
2400
1600
800
0.09
0.12
0 0.000
0.15
0.005
0.010
0.015
Time (s)
Time (s)
(a)
(b)
Fig. 5. Dynamic (a) displacement and (b) strain curves of typical SRCBs. 5
0.020
0.025
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S12-1-1 (1.0772 m/kg1/3)
S12-2 (0.5159 m/kg1/3)
S12-1-2 (0.5000 m/kg1/3)
S12-3 (0.4507 m/kg1/3)
S12-1-3 (0.3467 m/kg1/3)
S12-4 (0.4095 m/kg1/3)
Fig. 6. Damage modes of typical steel bar reinforced concrete beams.
Another important reason may lies in the load patterns. In explosion, the beam suffers hybrid flexural-shear deformation, but in four-pointbending the central beam only suffers flexural deformation. The spalling demonstrates beam S12-4 has higher damage degree but spalling concrete has little influence on the flexural strength of the beam. As shown in Fig. 13, the exploded GRCBs have a hybrid failure mode of flexural-shearing and concrete crushing. Different from the SRCBs, the explosion has little influence on the residual load capacity of the exploded GRCBs, as listed in Table 5. It is also surprising that the load capacity of the exploded GRCBs is still much greater than that of the unexploded SRCB, as compared in Fig. 13, indicating that the GFRP bar has more advantages in anti-explosion.
quantitatively evaluate the damage degree of the exploded beams, as shown in Figs. 12 and 13 and listed in Tables 4 and 5. The exploded SRCBs have flexural failure mode, with vertical cracks at the mid-span. Their peak load reduces at a maximum proportion of 28.04% compared with that of the reference beam, as listed in Table 4. Spalled beams, such as S12-3 and S12-4, have more serious damage degrees, as they have suffered higher blast loads. Compared with beam S12-4, beam S12-3 bears a lower blast load, but has a higher damage degree according to the static residual load capacity. This may be a random phenomenon, because there are many uncertainties in the explosion experiments and the damage has randomness, especially when the tested sample is not sufficient restricted by the expensive test fees. 12 8
1/3
1.0079 m/kg 1/3 0.5159 m/kg 1/3 0.4507 m/kg 1/3 0.4049 m/kg
5200
4 -6
Strain ( 10 )
Displacement (mm)
6500
1/3
1.0079 m/kg 1/3 0.5159 m/kg 1/3 0.4507 m/kg
0 -4
3900 2600 1300
-8 0
-12 0.00
0.03
0.06
0.09
0.12
0.15
0.00
Time (s)
0.01
0.02
0.03
Time (s)
(a)
(b)
Fig. 7. Dynamics (a) displacement and (b) strain curves of typical GRCBs. 6
0.04
0.05
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G22-1-1 (1.0079 kg/m1/3)
G22-2 (0.5159 kg/m1/3)
G22-1-2 (0.4507 kg/m1/3)
G22-3 (0.4095 kg/m1/3)
Fig. 8. Damage modes of typical GFRP bar reinforced concrete beams.
12
0.5 m/kg
1/3
9
Displacement (mm)
5. Analyses
G22 S12
5.1. Static ultimate load prediction
6
For SRCB S12-5, the yield moment, Mys , is given by
3
nσys As ⎞ 1 Py (L − S ) Mys = nσys Ac ⎛⎜h − t − ⎟ = 2 βbσ 4 cy ⎠ ⎝
0 -1.97 mm -3.53 mm -5.32 mm
-3 -6
where b and h denote the width and thickness of the beam, respectively. L and S denote the spans of the beam and the load distribution beam, respectively. As is the cross-section area of a steel bar. n represents the number of the bars. t is the thickness of the protective layer. σys is the yield strength of the steel bar. σcy is the yield strength of the concrete, with a tested value of 51.5 MPa. β = 0.85. Py is the yield load of the SRCB. The yield load of SRCB S12-5 is 95 kN in this situation. For GRCB G22-4, when the flexural failure mode occurs, the ultimate moment, Muc , is given by
-9 -12 0.00
0.03
0.06
0.09
0.12
0.15
0.18
Time (s) Fig. 9. Displacement comparisons between S12 and G22 under 0.5 m/kg1/3. The strains of the steel bars reveal that the SRCBs usually work in plastic state under explosion, while the GRCBs work in elastic state. All these beams have similar damage modes, developing from intact, sparse cracks, dense cracks to spalling in sequence when the scaled distance changes from 1.0772 m/kg1/3 to 0.4095 m/kg1/3.
nσuc Ac ⎞ 1 Muc = nσuc Ac ⎛⎜h − t − Pu (L − S ) ⎟ = 2βbσcy ⎠ 4 ⎝
1.75 A f bh 0 + f yvk sv h 0 λ + 1.0 tk s
500
12.0
450
400
9.6
360
300
7.2
270
200
4.8
180
100
2.4
90
0
9
18
27
36
(2)
where Ac is the cross-section area of a GFRP bar. σuc is the rupture strength of the GFRP. Pu is the ultimate load of the GRCB. The rupture load of G22-4 is 857 kN. When the shear failure occurs, the ultimate shear load, Vuc , is given by
Vuc =
0
(1)
0.0 45
0 0.000
(a)
0.006
0.012
0.018
(3)
0.024
(b)
Fig. 10. (a) Mid-span displacement curves and (b) strain curves of the bars under four-point-bending. 7
0.030
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Fig. 11. Damage modes of (a) SRCBs and (b) GRCBs under four-point-bending.
where h 0 denotes the valid height of the beam. ftk is the axial tensile strength of the concrete. λ is the value of the length of shear area divided by the valid height of the beam. f yvk denotes the standard axial extension strength of the steel. Asv is the area of all stirrups in the crosssection. s is the spacing distance between two stirrups in the shear area. The ultimate load of G22-4 is 449 kN in this situation. When the concrete crushing failure mode occurs, the ultimate mo′ , is given by ment, Muc
Table 4 Residual load capacity of exploded steel bar reinforced concrete beams. Beam
Scaled distance (m/kg1/3)
Peak load (kN)
Damage
S12-2 S12-3 S12-4 S12-5
0.5159 0.4507 0.4095 –
98 77 91 107
8.41% 28.04% 14.95% 0
x ′ = α1 σcy bx ⎛h 0 − ⎞ Muc 2⎠ ⎝
Table 5 Residual load capacity of exploded GFRP bar reinforced concrete beams. Beam
Scaled distance (m/kg1/3)
Peak load (kN)
G22-1 G22-2 G22-3 G22-4
1.0079 + 0.5000 0.5159 0.5159 –
403 407 432 409
(4)
where α1 is the coefficient of equivalent moment-row stress pattern. fck is the compressive strength of the concrete. x is the equivalent height of the compression area. Equilibrium equation of the internal forces is given by
α1 σcy bx = Ef εf Af
(5)
where Ef is the Young’s modulus of the GFRP bar. Af is the cross-section area of all the GFRP bars. εf is the strain of the GFRP when the concrete 8
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125
100
75
50
S12-2
25
0 0
7
14
21
28
35
42
Mid-span displacements
S12-3
S12-5
S12-4
Fig. 12. Flexural behaviors of exploded steel bar reinforced concrete beams.
450
360
270
180
G22-1 90
0 0
9
18
27
36
45
Mid-span displacements G22-2
G22-4
G22-3
Fig. 13. Flexural behaviors of exploded GFRP bar reinforced concrete beams.
close to the experimental value, 107 kN. Therefore, it is a flexural failure mode. The predicted rupture load of GRCB G22-4 is 395 kN, close to the experimental value, 409 kN. Therefore, it is a concrete-crushing failure mode.
crushing failure occurs, which can be calculated based on the planesection assumption,
εf εuc
=
β h0 − x 0 = 1 h0 - 1 x0 x
(6)
where εuc is the ultimate strain of the concrete. x 0 is the height of the compression area. The ultimate load of GRCB G22-4 is 395 kN in this situation. Accordingly, the predicted ultimate load of SRCB S12-5 is 95 kN,
5.2. Equivalent static load method The equivalent static load method is proposed to predict the failure 9
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(a)
(b)
(c)
(d)
Fig. 14. Peak reflective pressure distribution on the front surface of the beam under (a) 0.5 kg TNT with a stand-off of 0.8 m, (b) 2 kg TNT with a stand-off of 0.65 m, (c) 3 kg TNT with a stand-off of 0.65 m and (d) 4 kg TNT with a stand-off of 0.65 m.
of the beams under close-in explosion. The pressure impacted on the beam is equivalent to a uniform static load. The fundamental frequency of a simply-supported beam is given by
π2 ω= 2 l
B m ¯
Table 6 Constants of pressure acting on beam G22 calculated by CONWEP.
(7)
where l is the span, and m ¯ is the mass per unit length. B is the flexural rigidity and acquired by the four-point-bending test. For GRCB G22-4, ω = 366.1 s−1. The distributions of the peak reflective pressures calculated by CONWEP are shown in Fig. 14. The equivalent uniform static load Ps is given by
Ps = κK d pmax = K d pu
(8)
where κ is a distribution factor of the pressure, K d is the dynamic coefficient, deduced in Appendix A, pmax is the maximum pressure, and pu is the uniformed pressure. The constants calculated by CONWEP are listed in Table 6. When the concrete crushing mode controls the failure, the critical dynamic pressure the beam can sustain is decided by the bending moment, Mu ,
κK d Pm 2 P bl = s bl 2 ≤ Muc 8 8
0.5 kg-0.8 m
2 kg-0.65 m
3 kg-0.65 m
4 kg-0.65 m
pmax (MPa) pu (MPa) I (kPa·ms) td (ms) κ ωtd Kd κK d Ps (MPa) M (kN·m)
4.959 3.112 326.0 0.21 0.628 0.077 0.0384 0.0241 0.1196 48.45
28.78 16.23 999.9 0.12 0.564 0.044 0.0220 0.0123 0.3565 144.39
37.87 21.97 1229 0.11 0.580 0.040 0.0201 0.0117 0.4424 179.17
45.75 26.92 1396 0.10 0.588 0.037 0.0183 0.0107 0.4928 199.58
where Fs is the peak force of the beam in four-point-bending experiment, lp is the span of the loading beam (600 mm in this paper), l is the span of the beam (1800 mm in this paper). Corresponding to the critical strain of concrete, for GRCB G22-4, Muc = 122.7 kN·m. As shown in Fig. 15, the critical scaled distance predicted by equivalent static load method is 0.6271 m/kg1/3 for the GRCB. The prediction is consistent with the experiment, proving that using equivalent static load method, the dynamic response of the GRCB can be correctly predicted.
(9)
or (10)
Ps ≤ Psc
Case(g)
5.3. Utilization ratio of GFRP
with
Muc =
1 ⎛ l − lp ⎞ Fs 2 ⎝ 2 ⎠ ⎜
The utilization ratio of the GFRP bar can be represented by the ratio of the strain of the GFRP bar in hybrid failure with flexural-shearing mode and concrete crushing mode in the four-point-bending
⎟
(11) 10
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250
Bending moment (kN·m)
In this research, ε y′ = 0.0124, close to the experimental value, 0.0138. Av = 47.63%, which is a high utilization ratio of the GFRP bars. Increasing the strength of the concrete or increasing ratio of the stirrups can improve the utilization ratio of the GFRP bars.
Tested data Eq. (11)
Severely damged Damged
200
Slightly damged
6. Conclusions
150
100
In this research, blast responses of the GRCBs were investigated through explosion experiments and compared with those of the SRCBs. It is concluded that:
Intact
Danger
50
Safe
1) Complying with identical bending rigidity design criterion, the GRCBs have much greater load capacity than the SRCBs, while the flexures are comparable. The GRCBs have long elastic deformation while the SRCBs have elastic perfectly-plastic deformation mode. Elasticity improves the load capacity of the GRCBs while yielding limits the load capacity of the SRCBs. 2) In close-in explosions, the beams have several damage modes, including sparse cracks, dense cracks, potential spalling and spalling. Usually, the GRCBs have obvious rebound whose amplitude is close to the maximum displacement, indicating that the GRCBs vibrate elastically. On the contrary, the SRCBs have much smaller rebound, indicating that the SRCBs enter plastic flow state, which can also be reflected by the greater residual displacements. 3) Residual load capacity quantitatively reveals the damage degree of the exploded beams. In current research, the load capacity of the exploded SRCBs is reduced at a maximum proportion of 28.0%, while the load capacity of the exploded GRCBs is not affected by the explosion. The exploded SRCBs still have much greater load capacity than the unexploded SRCB. 4) The experiment research supports that with identical bending rigidity the GRCBs have more excellent blast-resistance and they can be applied in protective structures to replace the SRCBs.
0 0.4
0.6
0.8
1.0 1/3
Scaled distance (m/kg ) Fig. 15. Critical scaled distance for GRCBs predicted by equivalent static load method.
experiment to the ultimate strain in the tensile experiment represents the utilization ratio of the GFRP bar. In the four-point-bending experiment, the unexploded beam fails at hybrid failure with flexuralshearing and concrete crushing with Muc = 122.7 kN·m when the concrete reaches the ultimate strain. Only considering the flexural strain, the bending moment is calculated by
x M = α1 fck bx ⎛h 0 − ⎞ 2⎠ ⎝
(12)
where fck is the compressive strength of the concrete. The equilibrium equation of the internal forces is given by
α1 fck bx = f y′ Af
(13)
where f y′ is the valid tensile strength of the GFRP bars in four-pointbending experiments. Af is the cross-section areas of the GFRP bars. The tensile force is given by
f y′ = ε y′ Ey
Acknowledgements Supports from National Natural Science Foundation of China (51508566, 51478465, 51778622, and 11672130), State Key Laboratory of Disaster Reduction in Civil Engineering (SLDRCE16-01) and State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-0217G03) are gratefully acknowledged.
(14)
where ε y′ is the valid strain of the GFRP bars in the four-point-bending experiments. Ey is the elastic modulus of the GFRP bar. The utilization ratio of the GFRP bars, Av , is defined as
Av =
ε y′ εmax
Data availability statement
× 100%
(15)
The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.
where εmax is the extreme strain of the GFRP. Appendix A
The deduction of the dynamic factor is repeated in Appendix A to supply convenience for readers. In this research, the blast load p (t ) can be simplified as a triangular load without boost time, as shown in Fig. A1, where the td is the duration of the positive pressure. The dynamic coefficient K d is defined as
Kd =
vmax Pmax / B
(A1)
where, vmax is the maximum mid-span displacement of the beam under the blast load p (t ) . The motion equation of beam at mid-span v (t ) in the duration time of blast load is given as
t V¨ (t ) + ω2V (t ) = ω2 ⎛1 − ⎞, when 0 ≤ t ≤ td, t d⎠ ⎝ ⎜
⎟
(A2)
v (t ) . Pmax / B
Therefore, the maximum value of V (t ) , Vmax , is the dynamic coefficient K d . According to the velocity and displacement of the where V (t ) = beam at the initial time are zero, the displacement of beam at mid-span can be written as
V (t ) = 1 −
t sin(ωt ) − cos(ωt ) + td ωtd
(A3)
11
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Fig. A1. Simplified blast pressure.
The blast duration td (about 0.2 ms according to Fig. 4) is much shorter than time when the beam reaches the maximum displacement (about 18 ms according to Fig. 5). When the blast load stops, the displacement and velocity of beam at mid-span are given as
V (td ) =
sin(ωtd ) − cos(ωtd ) ωtd
(A4)
and
V̇ (td ) = ω sin(ωtd ) +
cos(ωtd ) 1 − td td
(A5)
When t > td , the motion equation of the beam at the mid-span V (t ) in the duration time is given by
V¨ (t ) + ω2V (t ) = 0, when t > td.
(A6)
According to the initial velocity and displacement of the beam at time td , the mid-span displacement is given by
V (t ) = A sin[ω (t − td )] + C cos[ω (t − td )]
(A7)
1 ̇ V ω
(td ) and C = V (td ) . When the velocity of the beam is zero, the displacement of the beam reaches the maximum. Time tmax can be where A = calculated by V̇ (t ) = ω {A cos[ω (tmax − td )] − C sin[ω (tmax − td )]} = 0
(A8)
Time tmax is given by
tmax =
1 A tan - 1 ⎛ ⎞ + td ω ⎝C ⎠
(A9)
Taking Eq. A(9) into Eq. A(7), K d is then obtained as 2
Kd =
2
⎡ ωtd − sin(ωtd ) ⎤ + ⎡ 1 − cos(ωtd ) ⎤ ⎢ ⎥ ⎢ ⎥ ωtd ωtd ⎣ ⎦ ⎣ ⎦
(A10)
pre-stressed CFRP tendons. Constr Build Mater 2016;122:607–18. [9] Meng FM, Li WX, Fan HL, Zhou YZ. A nonlinear theory for CFRP strengthened aluminum beam. Compos Struct 2015;131:574–7. [10] Dong Z, Wu G, Xu Y. Experimental study on the bond durability between steel-FRP composite bars (SFCBs) and sea sand concrete in ocean environment. Constr Build Mater 2016;115:277–84. [11] Jabbar Shahad AbdulAdheem, Farid Saad BH. Replacement of steel bars by GFRP rebars in the concrete structures. Karbala Int J Modern Sci 2018;4(2):216–27. [12] Yu B, Kodur VKR. Factors governing the fire response of concrete beams reinforced with FRP rebars. Compos Struct 2013;100:257–69. [13] Duic J, Kenno S, Das S. Performance of concrete beams reinforced with basalt fibre composite rebar. Constr Build Mater 2018;176:470–81. [14] Sovják R, Havlásek P, Vítek J. Long-term behavior of concrete slabs prestressed with CFRP bars subjected to four-point bending. Constr Build Mater 2018;188:781–92. [15] Fava G, Carvelli V, Pisani MA. Remarks on bond of GFRP rebars and concrete. Compos Part B 2016;93:210–20. [16] Rolland A, Quiertant M, Khadour A, Chataigner S, Argoul P. Experimental investigations on the bond behavior between concrete and FRP reinforcing bars. Constr Build Mater 2018;173:136–48. [17] Wu C, Bai Y, Kwon S. Improved bond behavior between GFRP rebar and concrete using calcium sulfoaluminate. Constr Build Mater 2016;113:897–904. [18] Al-Thairy H. A modified single degree of freedom method for the analysis of building steel columns subjected to explosion induced blast load. Int J Impact Eng 2016;94:120–33.
References [1] Feng J, Zhou YZ, Wang P, Wang B, Zhou JN, Chen HL, et al. Experimental research on blast-resistance of one-way concrete slabs reinforced by BFRP bars under close-in explosion. Eng Struct 2017;150:550–61. [2] Zhang X, Wang P, Jiang MR, Fan HL, Zhou JN, Li WX, et al. CFRP strengthening reinforced concrete arches: strengthening methods and experimental studies. Compos Struct 2015;131:852–67. [3] Chen HL, Xie W, Jiang MR, Wang P, Zhou JN, Fan HL, et al. Blast-loaded behaviors of severely damaged buried arch repaired by anchored CFRP strips. Compos Struct 2015;122:92–103. [4] Xie W, Jiang MR, Chen HL, Zhou JN, Yi Xu, Wang P, et al. Experimental behaviors of CFRP cloth strengthened buried arch structure subjected to subsurface localized explosion. Compos Struct 2014;116:562–70. [5] Chen HL, Zhou JN, Fan HL, Jin FN, Xu Y, Qiu YY, et al. Dynamic responses of buried arch structure subjected to subsurface localized impulsive loading: experimental study. Int J Impact Eng 2014;65:89–101. [6] Wang P, Jiang MR, Zhou JN, Wang B, Feng J, Chen HL, et al. Spalling in concrete arches subjected to shock wave and CFRP strengthening effect. Tunnel Undergr Space Technol 2018;74:10–9. [7] Zhu PC, Ai PC, Zhou YZ, Xu LX, Fan HL. Structural analyses of extruded CFRP tendon-anchor system and applications in thin-walled aluminum/CFRP bridge. Thin Wall Struct 2018;127:556–73. [8] Zhu PC, Fan HL, Zhou YZ. Flexural behavior of aluminum I-beams strengthened by
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high-strength ASTM A1035 bars. Int J Impact Eng 2019;130:41–67. [25] US Department of the Army. Fundamentals of Protective Design for Conventional Weapons. Washington D C: Manual TM5-855-1, Nov, 1986. [26] Henchie TF, Chung Kim Yuen S, Nurick GN, Ranwaha N, Balden VH. The response of circular plates to repeated uniform blast loads: an experimental and numerical study. Int J Impact Eng 2014;74(1):36–45. [27] Jiang PF, Tang DG, Wu J. Analysis on penetration characteristics of assembled blast-resistance wall impacted by fragments with high velocities. J Vib Shock 2008;27(7):131–5. [28] Zhang B, He HG, Zhou JN, Zhou Q, Fan HL. Construction and failure analysis of ultra-light GFRP fluted-core sandwich protective structures. Compos Sci Technol 2019;173:73–82.
[19] Liu Y, Yan JB, Huang FL. Behavior of reinforced concrete beams and columns subjected to blast loading. Defence Technol 2018;14(05):550–9. [20] Zhang D, Yao SJ, Lu FY. Experimental study on scaling of RC beams under close-in blast loading. Eng Fail Anal 2013;33:497–504. [21] Yao SJ, Zhang D, Lu FY, Wang W, Chen XG. Damage features and dynamic response of RC beams under blast. Eng Fail Anal 2016;62:103–11. [22] Wang W, Zhang D, Lu FY, Wang SC, Tang FJ. Experimental study on scaling the explosion resistance of a one-way square reinforced concrete slab under a close-in blast loading. Int J Impact Eng 2012;49:158–64. [23] Yang L, Omar A, Hassan A. Response of high-strength reinforced concrete beams under shock-tube induced blast loading. Constr Build Mater 2018;189:420–37. [24] Yang L, Hassan A. Blast response of beams built with high-strength concrete and
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Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Blast responses of concrete beams reinforced with GFRP bars: Experimental research and equivalent static analysis ⁎
T
⁎
Sanfeng Liua, Yinzhi Zhoua, , Jiannan Zhoua, Bei Zhanga,d, Fengnian Jina, Qing Zhengb,c, , ⁎ Hualin Fanb,c, a
State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing 210007, China Research Center of Lightweight Structures and Intelligent Manufacturing, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China c State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China d Institute of Defense Engineering, AMS, PLA, Luoyang 471023, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: GFRP composite bar Concrete beam reinforced with GFRP bars Blast loading Experiments Damage
The glass fiber reinforced polymer (GRFP) bars have advantages in corrosion resistance and high load bearing capacity. They can be used as reinforcements in concrete protective structures to replace the steel bars. In this paper, the GRFP bar reinforced concrete beams (GRCBs) were designed, casted and tested. As a reference, the steel bar reinforced concrete beams (SRCBs) having the same flexural rigidity with the GRCBs were also designed, casted and tested. Under close-in explosions, the GRCBs have smaller displacement than the SRCBs, because the GFRP bars are still in elastic state while the steel bars enter into plastic deformation. Cracks, spalling and cratering are the main damage modes of the concrete beams. Through the quasi-static four-point-bending experiments, the residual load capacity of each beam has been evaluated, which quantitatively reveals the damage degree of the exploded concrete beams. In current research the explosion produces negligible damage to the GRCBs, while the load capacity the SRCBs reduces at a level of 28% at the most. The research proves that the GRCB has greater blast-resistance.
1. Introduction Generally, fiber reinforced polymers (FRPs) have high strength and they can be applied as reinforcements of concrete structures [1–6] and metallic structures [7–9]. Steel corrosion greatly reduces the durability of concrete structures. Replacing steel bars by FRP bars is an effective method to prevent the steel corrosion [10,11]. Yu and Kodur [12] investigated factors governing the fire response of the FRP-rebar reinforced concrete beams. Duic et al. [13] studied the performance of concrete beams reinforced with the basalt fiber composite rebars. Sovják et al. [14] discussed the long-term behavior of concrete slabs pre-stressed by the carbon fiber composite rebars. Fava et al. [15] explored the bond between the GFRP rebars and the concrete, as well as Rolland et al. [16]. Wu et al. [17] applied calcium sulfoaluminate to improve the bond between the GFRP rebars and the concrete. FRP rebars are preferable for ocean protective structures, which have high demand of corrosion resistance and blast resistance. With the increase of terrorist attacks and explosion accidents, the
research on the blast resistance of structures is becoming important in recent years [18]. Although the blast experiment on steel bar reinforced concrete (RC) structures is complex and expensive, there are several literatures focused on the blast experiments of the RC beams. Liu et al. [19] conducted an experimental research on the blast behavior of the RC beams and columns. With the decrease of the scaled distance, the damage patterns of the RC beams evolved from a small number of cracks on the surface to large spalling area on the bottom and crushing on the top. Zhang et al. [20] conducted experimental and numerical researches on the behaviors of the RC beams under close-in explosions. With the increase of the explosive mass, the damaged mode of RC beam changes from overall bending to plastic hinge at mid-span, with severe exfoliation at the bottom surface around the mid-span. Yao et al. [21] launched an experimental investigation on the dynamic response of the RC beams with different stirrup ratios. The displacement-to-thickness ratio is inversely proportional to the stirrup ratio. Wang et al. [22] conducted experimental studies on the blast resistance of the one-way RC slabs under closed-in explosion, and obtained the failure patterns of
⁎ Corresponding authors at: Research Center of Lightweight Structures and Intelligent Manufacturing, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (Q. Zheng) and (H. Fan). E-mail addresses: [email protected] (Y. Zhou), [email protected] (Q. Zheng), [email protected] (H. Fan).
https://doi.org/10.1016/j.compstruct.2019.111271 Received 31 December 2018; Received in revised form 16 June 2019; Accepted 26 July 2019 Available online 27 July 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
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to 0.0262. The elastic modulus is 39 GPa. The steel bar is elastic perfectly-plastic. Its yield strength is 458 MPa and the elastic modulus is 193 GPa. The grade of the concrete material is C40. The mixture ratio of cement: sand: pebble: water is 1: 1.19: 2.31: 0.42. The Portland cement has a strength grade of 42.5 MPa. The concrete was cured under room temperate and the average compressive strength is 51.5 MPa, obtained by six cubes with a dimension of 150 mm × 150 mm × 150 mm. The density of the concrete material is 2.5 g/cm3.
the RC slabs under different scaled distance. To enhance the anti-blast capacity of the RC structures, highstrength concrete (HSC) and high-strength reinforcing bars have been applied. Yang et al. [23] investigated the blast responses of the highstrength reinforced concrete beams. However, the effect of the highstrength concrete on the blast performance of the RC beam is limited. To improve the anti-blast properties of the RC structures furtherly, Yang and Hassan [24] took another blast experiment on HSC beams by replacing the normal steel bars with high-strength ASTM A1035 bars whose yield strength is 900 MPa. The results show that the highstrength bars can effectively enhance the anti-blast capacity of the RC beam. As the FRP bars also have high strength as mentioned above, the concrete members reinforced with the FRP bars might have a better anti-blast capacity than those reinforced with normal steel bars. Feng et al. [1] has revealed the blast responses of the slabs reinforced with FRP bars through explosion experiments. It is found that the slabs reinforced with FRP bars have strong deformation recovering ability and the spalling area of the concrete under shock wave is greatly reduced [1]. After explosion, the static load carrying ability of the exploded slab is still higher than that of the slab reinforced with steel bars. As the slab is not thick, the deformation of the FRP bars under explosion is limited, far below their rupture strain. The slab fails at spalling and breaching. For deep concrete beams, the FRP bars will sustain much larger deformation and the beams have great probability to fail at FRP rupture. The feasibility of using FRP bars to reinforce deep protective concrete beams is worthy of investigation. In this research, the concrete beams reinforced with GFRP bars were designed and casted. Their blast-resistant performances were checked by close-in explosion experiments and compared with those of the concrete beams reinforced with steel bars which were designed of the same stiffness.
2.2. Concrete beams reinforced by GFRP composite bars Two types of concrete beams with equal flexural rigidity were designed and casted, including the steel bar reinforced concrete beam (S12) and the GFRP bar reinforced concrete beam (G22). The width and height of the beam are 220 mm × 300 mm and the length is 2000 mm, as shown in Fig. 2. The compression reinforcements of all these beams are two HRB400 steel bars with a diameter of 12 mm. For beam S12 the tensile reinforcements are two HRB400 steel bars with a diameter of 12 mm. For beam G22 the tensile reinforcements are three GFRP bars with a diameter of 22 mm. The stirrups are made up of C8@80 mm and C8@ 150 mm. The thickness of the protective layer is 20 mm. 3. Dynamic responses under blast loading 3.1. Protocol of explosion experiments Close-in explosion experiments were carried out, as shown in Fig. 3. The beam was simply-supported on a steel frame fixed on the ground, with 100 mm hanging over at each end. The square TNT cartridge was placed over the mid-span of the beam and detonated by an electric detonator. Three displacement sensors were placed below the beam. Strain gauges were pre-adhered on the surfaces of the tensile bars. Pressure sensors were placed on the upper surface of the beam at mid-span. A free-field pressure sensor was set at the same height with the TNT block, but 1.5 m away from the TNT block. All the data were collected by a dynamic system D5922N. The explosion experiment scheme is listed in Table 1 in details.
2. Concrete beams reinforced by GFRP bars 2.1. Material properties The reinforcement of the GFRP bar is boron-free glass fiber and the matrix is vinylester resin. The diameter of the GFRP bars is 22 mm, as shown in Fig. 1. The GFRP bars were produced by Nanjing Fenghui Composite Material Co., Ltd. through pultrusion method with a fiber volume fraction of 65%. The surface was ribbed before the resin solidifies, without sand coating. Provided by manufacturer, the ultimate tensile strength is more than 600 MPa, and the elastic modulus is ranged from 40 GPa to 45 GPa, the ultimate shear strength is ranged from 110 MPa to 150 MPa. The steel bars used in this experiment is HRB400 with a diameter of 12 mm. Tensile properties of the GFRP bar and the steel bar were tested by a 2000 kN MTS system at a loading rate of 0.6 mm/min. The strain–stress relationships of bar are shown in Fig. 1. The GFRP bar has linear elastic deformation. The ultimate strength is 1027 MPa where the strain is up
3.2. Free-field pressure and reflective overpressure The measured maximum free-field pressure under 0.5 kg TNT with a stand-off of 1.5 m is 0.220 MPa. The measured maximum reflective overpressure under 0.1 kg TNT with a stand-off of 0.5 m is 2.935 MPa. These data were compared with those calculated by CONWEP [25–28], a calculation program for explosion parameters of conventional weapons issued by US Department of the Army. As the CONWEP is based on a large number of explosive experiments, its accuracy is world-widely recognized. The CONWEP calculation process is listed as follows. Firstly, select the calculation process of weapon effect. Secondly, choose the explosive type of “air-blast” and “aboveground detonation”. Thirdly, input the weight of TNT and distance between the explosive point and the specimen. Finally, choose the output data as pressure and duration and then the free-field pressure and the reflective overpressure history curves can be obtained. The maximum free-field pressure under 0.5 kg TNT with a stand-off of 1.5 m calculated by CONWEP is 0.215 MPa. The maximum reflective overpressure under 0.1 kg TNT with a stand-off of 0.5 m calculated by CONWEP is 3.490 MPa. The curves are compared in Fig. 4 and they are in good agreement, indicating that the measurements are credible. Detailed information is summarized in Table 2. There may be a lot of uncertainty in the explosion experiments. For beams S12-1-3 and S12-3, the blast loads were so strong that the stain gauges were broken and no strain data were obtained. Damage phenomena of each beam are
1200 Steel rebar GFRP rebar
1000
Stress (MPa)
800 600 400 200 0 0.000
0.015
0.030
0.045
0.060
Strain
(a)
(b)
(c)
Fig. 1. (a) Steel bar, (b) GFRP bar and (c) their tensile properties. 2
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(a)
(b) Fig. 2. Reinforcing diagram of (a) steel bar reinforced concrete beam S12 and (b) GFRP bar reinforced concrete beam G22 (Unit: mm).
described in Table 3 in details.
Table 1 Blast protocol of concrete beams.
3.3. Blast responses of steel bar reinforced concrete beams (SRCBs) 3.3.1. Dynamic deformations Beam S12 has almost no displacement when the scaled distance is 1.0772 m/kg1/3, as listed in Table 1. When the scaled distance is 0.5 m/ kg1/3, the maximum displacement was 3.31 mm and the rebound is 1.77 mm. When the scaled distance is 0.5159 m/kg1/3, the maximum displacement is 4.89 mm and the rebound is 5.52 mm. In these cases, there is little residual deformation after explosion, as shown in Fig. 5(a). When the scaled distance is 0.4507 m/kg1/3, the maximum displacement is 11.14 mm, while the rebound is only 5.515 mm, just half of the downward displacement. The reason is that the steel bars flow into plastic state. As shown in Fig. 5(a), the beam remains 5.50 mm residual displacement after explosion.
Beam
Compression tendon
Tensile tendon
Stirrup
TNT (kg)
Blast distance (m)
Scaled distance (m/kg1/3)
S12-1-1 S12-1-2 S12-1-3 S12-2 S12-3 S12-4 G22-1-1 G22-1-2 G22-2 G22-3
2C12
2C12
C8@ 150 and C8@80
2C12
3 × F3
C8@ 150 and C8@80
0.1 1 3 2 3 4 0.5 3 2 4
0.50 0.50 0.50 0.65 0.65 0.65 0.80 0.65 0.65 0.65
1.0772 0.5000 0.3467 0.5159 0.4507 0.4095 1.0079 0.4507 0.5159 0.4095
almost no deformation, as shown in Fig. 5(b). When the scaled distance is 0.5000 m/kg1/3, the maximum strain of steel bar is 2710 με and the residual strain is 501 με. When the scaled distance is 0.5159 m/kg1/3, the maximum strain is 3226 με and the residual strain is 634 με. In these
3.3.2. Dynamic strains When the scaled distance is 1.0772 m/kg1/3, the steel bars have
Fig. 3. Field explosion experiment protocol of concrete beams. 3
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Fig. 4. Typical (a) free-field pressure under 0.5 kg TNT with a stand-off of 1.5 m and (b) reflective overpressure under 0.1 kg TNT with a stand-off of 0.5 m.
residual displacement is 1.82 mm. Under 0.4507 m/kg1/3, the maximum displacement is 10.53 mm and the rebound is 9.35 mm. The residual displacement is 3.38 mm. The rebound is close to the downward displacement, representing that the GFRP bars are not broken.
cases, the steel bars are in elastic state. The residual strains of the steel bars are also very small, as shown in Fig. 5(b). When the scaled distance is 0.4095 m/kg1/3, the maximum strain is 3515 με and the residual strain is 1352 με. As the steel bars flow into plastic stage, the beams have large deflections. The SRCBs have obvious residual plastic deformations when the scaled distance is small.
3.4.2. Dynamic strains When the scaled distance is 1.0079 m/kg1/3, the maximum strain of GFRP bar is 369 με. When the scaled distance is 0.5159 m/kg1/3, the maximum strain is 3466 με and the residual strain is 728 με, as shown in Fig. 7(b). Under 0.4507 m/kg1/3, the maximum strain is 5867 με and the residual strain is 1022 με. Under 0.4095 m/kg1/3, the maximum strain is 5975 με and the residual strain is 799 με. All these strains are much smaller than the rupture strain of the GFRP (about 25,000 με in Fig. 1), indicating that the GFRP bars still work in elastic state.
3.3.3. Damage modes When the scaled distance is 1.0772 m/kg1/3, the damage of beam S12-1 can be ignored, as shown in Fig. 6. When the scaled distance is 0.5000 m/kg1/3, the beam is slightly damaged, with one main vertical crack of 240 mm long cross the beam. Other cracks are relatively short, about 90 mm long. When the scaled distance is 0.3467 m/kg1/3, the beam is severely damaged, with a spalling area of 770 cm2 at the lower surface. Oblique shear cracks cross through each other and form a triangular peeling region. When the scaled distance is 0.5159 m/kg1/3, beam S12-2 is slightly damaged. There are three cracks, as shown in Fig. 6. The length of the long transverse crack is about 300 mm and that of the short transverse crack is about 150 mm. When the scaled distance is 0.4507 m/kg1/3, beam S12-3 is slightly damaged. There are seven cracks, including two 300 mm long vertical main cracks, three 200 mm long transverse cracks, and two 250 mm long oblique cracks. When the scaled distance is 0.4095 m/kg1/3, beam S12-4 is severely damaged. The spalling area on the lower surface is 1100 cm2. There are eight 200 mm long oblique cracks, forming an inverted triangular damage region.
3.4.3. Damage modes When the scaled distance is 1.0079 m/kg1/3, beam G22-1 is almost intact. When the scaled distance is 0.4507 m/kg1/3, the beam is severely damaged with five vertical cracks and one transverse crack. The width and the length of the transverse crack are 2.4 mm and 400 mm, respectively, as shown in Fig. 8. In fact, a spalling plate is coming into being. When the scaled distance is 0.5159 m/kg1/3, beam G22-2 is slightly damaged with 13 cracks, including three 300 mm long vertical cracks, five 120 mm long vertical cracks, one 360 mm long transverse crack and four 80 mm long transverse cracks. When the scaled distance is 0.4095 m/kg1/3, beam G22-3 is severely damaged. A peeling region with an area of 225 cm2 is observed on the upper surface. The spalling region on the lower surface has an area of 770 cm2. There are eleven 350 mm long oblique cracks, forming a large inverted triangle region.
3.4. Blast responses of GFRP bar reinforced concrete beams (GRCBs) 3.4.1. Dynamic deformations Beam G22 has almost no displacement when the scaled distance is 1.0079 m/kg1/3. Under 0.5159 m/kg1/3, the maximum displacement is 4.81 mm and the rebound is 4.03 mm, as shown in Fig. 7(a). The Table 2 Blast responses of concrete beams. Beam
Scaled distance (m/kg1/3)
Maximum displacement (mm)
Rebound displacement (mm)
Maximum Strain (10 −6)
residual strain (10 −6)
Damage mode
S12-1-1 S12-1-2 S12-1-3 S12-2 S12-3 S12-4 G22-1-1 G22-1-2 G22-2 G22-3
1.0772 0.5000 0.3467 0.5159 0.4507 0.4095 1.0079 0.4507 0.5159 0.4095
0.49 3.31 24.30 4.89 11.14 24.36 0.86 11.25 4.90 18.61
– 1.77 – 5.52 5.52 – – 9.35 4.03 –
47 2710 – 3227 – 3515 369 5867 3466 5975
– 501 – 634 – 1352 – 1022 728 799
Intact Slight damage Severely damaged Slightly damaged Damaged Severely damaged Intact Damaged Severely damaged Severely damaged
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Table 3 Damage information of blast loaded beams. Beam
Damage
S12-1-1 S12-1-2 S12-1-3 S12-2 S12-3 S12-4
Intact Cracks Spalling Cracks Cracks Spalling
G22-1-1 G22-1-2 G22-2 G22-3
Intact Cracks Cracks Spalling
Phenomena
Two mid-span cracks. One crosses through the height and the other is 80 mm long Three wide through-thickness cracks. Slight spalling at the front surface and spalling area 770 cm2 at the back One 300 mm long vertical crack. One 300 mm long horizontal crack. One 150 mm long horizontal crack Two 300 mm long vertical cracks. Three 200 mm long horizontal cracks. Two 250 mm long inclined cracks Severe spalling with a spalling area of 1100 cm2 at the back. Slight spalling at the front surface with a spalling area of 175 cm2. Eight 200 mm long inclined cracks One 400 mm long horizontal crack and the crack is wide. Four 200 mm long vertical cracks Three through-thickness cracks. One 360 mm long horizontal crack. Four 80 mm long horizontal cracks. Five 120 mm long vertical cracks Spalling at the back surface with a spalling area of 770 cm2. Spalling at the front surface with a spalling area of 225 cm2. Eleven 350 mm long inclined cracks
layer peels off from the beam. Beam S12-1 under 0.3467 m/kg1/3, beam S12-4 under 0.4095 m/kg1/3, and beam G22-3 under 0.4095 kg/m1/3 were in such damage grade, as listed in Table 2.
3.5. Comparison between GFRP and steel bars reinforced beams 3.5.1. Displacement As all the beams are designed with equal stiffness, they have close maximum downward displacements under the same scaled distance, as shown in Fig. 9. The difference lies in the rebound and the residual displacement. As the steel bars will enter into plastic region in a relative small deformation, beam S12 always has much smaller rebound but much larger residual displacement. As GFRP bars can keep elastic in most cases, GRCBs always have elastic responses. Their rebounds are usually comparable with the maximum downward displacements. The residual displacement is usually very small.
4. Damage evaluation through determination of the residual load capacity 4.1. Four-point bending experiments on reference beams Under four-point-bending, the GRCBs and the SRCBs exhibit different deformation behaviors and failure modes, as shown in Figs. 10 and 11, respectively. The span of the beam is 1800 mm and the span of the load distribution beam is 600 mm. To obtain the damage degree of the concrete beams after explosion, the un-exploded reinforced concrete beams, named S12-5 for steel bar reinforced concrete beam and G22-4 for GFRP bar reinforced concrete beam, are tested firstly as the reference beams. Reference beam SRCB S12-5 has elastic perfectlyplastic deformation, with yielding plateau staying at 107 kN, as listed in Table 4. Reference beam GRCB G22-4 has quasi-linear deformation with a failure load of 409 kN, as listed in Table 5. These two beams have identical flexural rigidity of 0.53 kN/mm. At failure, the strain of the GFRP bar is 0.0138 and the displacement of the beam is 34.0 mm, while the strain of the steel rebar is only 0.0029 and the displacement is 39.9 mm. Under four-point-bending, the failure mode of SRCB S12-5 is flexural failure, with three vertical cracks round mid-span. The failure mode of GRCB G22-4 is the combination of two vertical cracks below the load point and concrete crushing on the upper surface at mid-span. It is a hybrid flexural-shearing failure mode.
3.5.2. Damage levels Through the explosion experiments, the damage can be divided into five grades: intact; slight damage; medium damage; severe damage and spalling. Intact means no cracks are observed. For all the beams under 1.0772 m/kg1/3, the damage grade is intact, as listed in Table 2. Slight damage means the beam has few cracks. The sparse cracks have little influence on the load carrying ability of the beam. In Table 2, when the scaled distance is greater than 0.5 m/kg1/3, the damage grade is slight damage. Medium damage indicates that the beam has visible vertical cracks but no transverse fracture zone along the interface of the tensile rebar, such as beam G22-2 under 0.5159 kg/m1/3. Severe damage indicates that the beam has dense cracks and a wide transverse fracture zone along the interface of the tensile rebar. A potential falling block is visible. Beam S12-3 under 0.4507 m/kg1/3 and beam G22-1–2 under 0.4507 kg/m1/3 are severely damaged, as listed in Table 2. Spalling indicates the fracture zone along the interface of the tensile bars is well developed and the concrete block below the reinforcement
4.2. Residual capacity of blast loaded beams The residual load capacity under four-point-bending is adopted to
6
4000 0.1kg-0.5m 1kg-0.5m 2kg-0.65m 4kg-0.65m
3 0 -6
Strain ( 10 )
Displacement (mm)
3200
-3 -6 1/3
1.0077 m/kg 1/3 0.5000 m/kg 1/3 0.5159 m/kg 1/3 0.4507 m/kg
-9 -12 0.00
0.03
0.06
2400
1600
800
0.09
0.12
0 0.000
0.15
0.005
0.010
0.015
Time (s)
Time (s)
(a)
(b)
Fig. 5. Dynamic (a) displacement and (b) strain curves of typical SRCBs. 5
0.020
0.025
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S12-1-1 (1.0772 m/kg1/3)
S12-2 (0.5159 m/kg1/3)
S12-1-2 (0.5000 m/kg1/3)
S12-3 (0.4507 m/kg1/3)
S12-1-3 (0.3467 m/kg1/3)
S12-4 (0.4095 m/kg1/3)
Fig. 6. Damage modes of typical steel bar reinforced concrete beams.
Another important reason may lies in the load patterns. In explosion, the beam suffers hybrid flexural-shear deformation, but in four-pointbending the central beam only suffers flexural deformation. The spalling demonstrates beam S12-4 has higher damage degree but spalling concrete has little influence on the flexural strength of the beam. As shown in Fig. 13, the exploded GRCBs have a hybrid failure mode of flexural-shearing and concrete crushing. Different from the SRCBs, the explosion has little influence on the residual load capacity of the exploded GRCBs, as listed in Table 5. It is also surprising that the load capacity of the exploded GRCBs is still much greater than that of the unexploded SRCB, as compared in Fig. 13, indicating that the GFRP bar has more advantages in anti-explosion.
quantitatively evaluate the damage degree of the exploded beams, as shown in Figs. 12 and 13 and listed in Tables 4 and 5. The exploded SRCBs have flexural failure mode, with vertical cracks at the mid-span. Their peak load reduces at a maximum proportion of 28.04% compared with that of the reference beam, as listed in Table 4. Spalled beams, such as S12-3 and S12-4, have more serious damage degrees, as they have suffered higher blast loads. Compared with beam S12-4, beam S12-3 bears a lower blast load, but has a higher damage degree according to the static residual load capacity. This may be a random phenomenon, because there are many uncertainties in the explosion experiments and the damage has randomness, especially when the tested sample is not sufficient restricted by the expensive test fees. 12 8
1/3
1.0079 m/kg 1/3 0.5159 m/kg 1/3 0.4507 m/kg 1/3 0.4049 m/kg
5200
4 -6
Strain ( 10 )
Displacement (mm)
6500
1/3
1.0079 m/kg 1/3 0.5159 m/kg 1/3 0.4507 m/kg
0 -4
3900 2600 1300
-8 0
-12 0.00
0.03
0.06
0.09
0.12
0.15
0.00
Time (s)
0.01
0.02
0.03
Time (s)
(a)
(b)
Fig. 7. Dynamics (a) displacement and (b) strain curves of typical GRCBs. 6
0.04
0.05
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G22-1-1 (1.0079 kg/m1/3)
G22-2 (0.5159 kg/m1/3)
G22-1-2 (0.4507 kg/m1/3)
G22-3 (0.4095 kg/m1/3)
Fig. 8. Damage modes of typical GFRP bar reinforced concrete beams.
12
0.5 m/kg
1/3
9
Displacement (mm)
5. Analyses
G22 S12
5.1. Static ultimate load prediction
6
For SRCB S12-5, the yield moment, Mys , is given by
3
nσys As ⎞ 1 Py (L − S ) Mys = nσys Ac ⎛⎜h − t − ⎟ = 2 βbσ 4 cy ⎠ ⎝
0 -1.97 mm -3.53 mm -5.32 mm
-3 -6
where b and h denote the width and thickness of the beam, respectively. L and S denote the spans of the beam and the load distribution beam, respectively. As is the cross-section area of a steel bar. n represents the number of the bars. t is the thickness of the protective layer. σys is the yield strength of the steel bar. σcy is the yield strength of the concrete, with a tested value of 51.5 MPa. β = 0.85. Py is the yield load of the SRCB. The yield load of SRCB S12-5 is 95 kN in this situation. For GRCB G22-4, when the flexural failure mode occurs, the ultimate moment, Muc , is given by
-9 -12 0.00
0.03
0.06
0.09
0.12
0.15
0.18
Time (s) Fig. 9. Displacement comparisons between S12 and G22 under 0.5 m/kg1/3. The strains of the steel bars reveal that the SRCBs usually work in plastic state under explosion, while the GRCBs work in elastic state. All these beams have similar damage modes, developing from intact, sparse cracks, dense cracks to spalling in sequence when the scaled distance changes from 1.0772 m/kg1/3 to 0.4095 m/kg1/3.
nσuc Ac ⎞ 1 Muc = nσuc Ac ⎛⎜h − t − Pu (L − S ) ⎟ = 2βbσcy ⎠ 4 ⎝
1.75 A f bh 0 + f yvk sv h 0 λ + 1.0 tk s
500
12.0
450
400
9.6
360
300
7.2
270
200
4.8
180
100
2.4
90
0
9
18
27
36
(2)
where Ac is the cross-section area of a GFRP bar. σuc is the rupture strength of the GFRP. Pu is the ultimate load of the GRCB. The rupture load of G22-4 is 857 kN. When the shear failure occurs, the ultimate shear load, Vuc , is given by
Vuc =
0
(1)
0.0 45
0 0.000
(a)
0.006
0.012
0.018
(3)
0.024
(b)
Fig. 10. (a) Mid-span displacement curves and (b) strain curves of the bars under four-point-bending. 7
0.030
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Fig. 11. Damage modes of (a) SRCBs and (b) GRCBs under four-point-bending.
where h 0 denotes the valid height of the beam. ftk is the axial tensile strength of the concrete. λ is the value of the length of shear area divided by the valid height of the beam. f yvk denotes the standard axial extension strength of the steel. Asv is the area of all stirrups in the crosssection. s is the spacing distance between two stirrups in the shear area. The ultimate load of G22-4 is 449 kN in this situation. When the concrete crushing failure mode occurs, the ultimate mo′ , is given by ment, Muc
Table 4 Residual load capacity of exploded steel bar reinforced concrete beams. Beam
Scaled distance (m/kg1/3)
Peak load (kN)
Damage
S12-2 S12-3 S12-4 S12-5
0.5159 0.4507 0.4095 –
98 77 91 107
8.41% 28.04% 14.95% 0
x ′ = α1 σcy bx ⎛h 0 − ⎞ Muc 2⎠ ⎝
Table 5 Residual load capacity of exploded GFRP bar reinforced concrete beams. Beam
Scaled distance (m/kg1/3)
Peak load (kN)
G22-1 G22-2 G22-3 G22-4
1.0079 + 0.5000 0.5159 0.5159 –
403 407 432 409
(4)
where α1 is the coefficient of equivalent moment-row stress pattern. fck is the compressive strength of the concrete. x is the equivalent height of the compression area. Equilibrium equation of the internal forces is given by
α1 σcy bx = Ef εf Af
(5)
where Ef is the Young’s modulus of the GFRP bar. Af is the cross-section area of all the GFRP bars. εf is the strain of the GFRP when the concrete 8
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125
100
75
50
S12-2
25
0 0
7
14
21
28
35
42
Mid-span displacements
S12-3
S12-5
S12-4
Fig. 12. Flexural behaviors of exploded steel bar reinforced concrete beams.
450
360
270
180
G22-1 90
0 0
9
18
27
36
45
Mid-span displacements G22-2
G22-4
G22-3
Fig. 13. Flexural behaviors of exploded GFRP bar reinforced concrete beams.
close to the experimental value, 107 kN. Therefore, it is a flexural failure mode. The predicted rupture load of GRCB G22-4 is 395 kN, close to the experimental value, 409 kN. Therefore, it is a concrete-crushing failure mode.
crushing failure occurs, which can be calculated based on the planesection assumption,
εf εuc
=
β h0 − x 0 = 1 h0 - 1 x0 x
(6)
where εuc is the ultimate strain of the concrete. x 0 is the height of the compression area. The ultimate load of GRCB G22-4 is 395 kN in this situation. Accordingly, the predicted ultimate load of SRCB S12-5 is 95 kN,
5.2. Equivalent static load method The equivalent static load method is proposed to predict the failure 9
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(a)
(b)
(c)
(d)
Fig. 14. Peak reflective pressure distribution on the front surface of the beam under (a) 0.5 kg TNT with a stand-off of 0.8 m, (b) 2 kg TNT with a stand-off of 0.65 m, (c) 3 kg TNT with a stand-off of 0.65 m and (d) 4 kg TNT with a stand-off of 0.65 m.
of the beams under close-in explosion. The pressure impacted on the beam is equivalent to a uniform static load. The fundamental frequency of a simply-supported beam is given by
π2 ω= 2 l
B m ¯
Table 6 Constants of pressure acting on beam G22 calculated by CONWEP.
(7)
where l is the span, and m ¯ is the mass per unit length. B is the flexural rigidity and acquired by the four-point-bending test. For GRCB G22-4, ω = 366.1 s−1. The distributions of the peak reflective pressures calculated by CONWEP are shown in Fig. 14. The equivalent uniform static load Ps is given by
Ps = κK d pmax = K d pu
(8)
where κ is a distribution factor of the pressure, K d is the dynamic coefficient, deduced in Appendix A, pmax is the maximum pressure, and pu is the uniformed pressure. The constants calculated by CONWEP are listed in Table 6. When the concrete crushing mode controls the failure, the critical dynamic pressure the beam can sustain is decided by the bending moment, Mu ,
κK d Pm 2 P bl = s bl 2 ≤ Muc 8 8
0.5 kg-0.8 m
2 kg-0.65 m
3 kg-0.65 m
4 kg-0.65 m
pmax (MPa) pu (MPa) I (kPa·ms) td (ms) κ ωtd Kd κK d Ps (MPa) M (kN·m)
4.959 3.112 326.0 0.21 0.628 0.077 0.0384 0.0241 0.1196 48.45
28.78 16.23 999.9 0.12 0.564 0.044 0.0220 0.0123 0.3565 144.39
37.87 21.97 1229 0.11 0.580 0.040 0.0201 0.0117 0.4424 179.17
45.75 26.92 1396 0.10 0.588 0.037 0.0183 0.0107 0.4928 199.58
where Fs is the peak force of the beam in four-point-bending experiment, lp is the span of the loading beam (600 mm in this paper), l is the span of the beam (1800 mm in this paper). Corresponding to the critical strain of concrete, for GRCB G22-4, Muc = 122.7 kN·m. As shown in Fig. 15, the critical scaled distance predicted by equivalent static load method is 0.6271 m/kg1/3 for the GRCB. The prediction is consistent with the experiment, proving that using equivalent static load method, the dynamic response of the GRCB can be correctly predicted.
(9)
or (10)
Ps ≤ Psc
Case(g)
5.3. Utilization ratio of GFRP
with
Muc =
1 ⎛ l − lp ⎞ Fs 2 ⎝ 2 ⎠ ⎜
The utilization ratio of the GFRP bar can be represented by the ratio of the strain of the GFRP bar in hybrid failure with flexural-shearing mode and concrete crushing mode in the four-point-bending
⎟
(11) 10
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250
Bending moment (kN·m)
In this research, ε y′ = 0.0124, close to the experimental value, 0.0138. Av = 47.63%, which is a high utilization ratio of the GFRP bars. Increasing the strength of the concrete or increasing ratio of the stirrups can improve the utilization ratio of the GFRP bars.
Tested data Eq. (11)
Severely damged Damged
200
Slightly damged
6. Conclusions
150
100
In this research, blast responses of the GRCBs were investigated through explosion experiments and compared with those of the SRCBs. It is concluded that:
Intact
Danger
50
Safe
1) Complying with identical bending rigidity design criterion, the GRCBs have much greater load capacity than the SRCBs, while the flexures are comparable. The GRCBs have long elastic deformation while the SRCBs have elastic perfectly-plastic deformation mode. Elasticity improves the load capacity of the GRCBs while yielding limits the load capacity of the SRCBs. 2) In close-in explosions, the beams have several damage modes, including sparse cracks, dense cracks, potential spalling and spalling. Usually, the GRCBs have obvious rebound whose amplitude is close to the maximum displacement, indicating that the GRCBs vibrate elastically. On the contrary, the SRCBs have much smaller rebound, indicating that the SRCBs enter plastic flow state, which can also be reflected by the greater residual displacements. 3) Residual load capacity quantitatively reveals the damage degree of the exploded beams. In current research, the load capacity of the exploded SRCBs is reduced at a maximum proportion of 28.0%, while the load capacity of the exploded GRCBs is not affected by the explosion. The exploded SRCBs still have much greater load capacity than the unexploded SRCB. 4) The experiment research supports that with identical bending rigidity the GRCBs have more excellent blast-resistance and they can be applied in protective structures to replace the SRCBs.
0 0.4
0.6
0.8
1.0 1/3
Scaled distance (m/kg ) Fig. 15. Critical scaled distance for GRCBs predicted by equivalent static load method.
experiment to the ultimate strain in the tensile experiment represents the utilization ratio of the GFRP bar. In the four-point-bending experiment, the unexploded beam fails at hybrid failure with flexuralshearing and concrete crushing with Muc = 122.7 kN·m when the concrete reaches the ultimate strain. Only considering the flexural strain, the bending moment is calculated by
x M = α1 fck bx ⎛h 0 − ⎞ 2⎠ ⎝
(12)
where fck is the compressive strength of the concrete. The equilibrium equation of the internal forces is given by
α1 fck bx = f y′ Af
(13)
where f y′ is the valid tensile strength of the GFRP bars in four-pointbending experiments. Af is the cross-section areas of the GFRP bars. The tensile force is given by
f y′ = ε y′ Ey
Acknowledgements Supports from National Natural Science Foundation of China (51508566, 51478465, 51778622, and 11672130), State Key Laboratory of Disaster Reduction in Civil Engineering (SLDRCE16-01) and State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-0217G03) are gratefully acknowledged.
(14)
where ε y′ is the valid strain of the GFRP bars in the four-point-bending experiments. Ey is the elastic modulus of the GFRP bar. The utilization ratio of the GFRP bars, Av , is defined as
Av =
ε y′ εmax
Data availability statement
× 100%
(15)
The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.
where εmax is the extreme strain of the GFRP. Appendix A
The deduction of the dynamic factor is repeated in Appendix A to supply convenience for readers. In this research, the blast load p (t ) can be simplified as a triangular load without boost time, as shown in Fig. A1, where the td is the duration of the positive pressure. The dynamic coefficient K d is defined as
Kd =
vmax Pmax / B
(A1)
where, vmax is the maximum mid-span displacement of the beam under the blast load p (t ) . The motion equation of beam at mid-span v (t ) in the duration time of blast load is given as
t V¨ (t ) + ω2V (t ) = ω2 ⎛1 − ⎞, when 0 ≤ t ≤ td, t d⎠ ⎝ ⎜
⎟
(A2)
v (t ) . Pmax / B
Therefore, the maximum value of V (t ) , Vmax , is the dynamic coefficient K d . According to the velocity and displacement of the where V (t ) = beam at the initial time are zero, the displacement of beam at mid-span can be written as
V (t ) = 1 −
t sin(ωt ) − cos(ωt ) + td ωtd
(A3)
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Fig. A1. Simplified blast pressure.
The blast duration td (about 0.2 ms according to Fig. 4) is much shorter than time when the beam reaches the maximum displacement (about 18 ms according to Fig. 5). When the blast load stops, the displacement and velocity of beam at mid-span are given as
V (td ) =
sin(ωtd ) − cos(ωtd ) ωtd
(A4)
and
V̇ (td ) = ω sin(ωtd ) +
cos(ωtd ) 1 − td td
(A5)
When t > td , the motion equation of the beam at the mid-span V (t ) in the duration time is given by
V¨ (t ) + ω2V (t ) = 0, when t > td.
(A6)
According to the initial velocity and displacement of the beam at time td , the mid-span displacement is given by
V (t ) = A sin[ω (t − td )] + C cos[ω (t − td )]
(A7)
1 ̇ V ω
(td ) and C = V (td ) . When the velocity of the beam is zero, the displacement of the beam reaches the maximum. Time tmax can be where A = calculated by V̇ (t ) = ω {A cos[ω (tmax − td )] − C sin[ω (tmax − td )]} = 0
(A8)
Time tmax is given by
tmax =
1 A tan - 1 ⎛ ⎞ + td ω ⎝C ⎠
(A9)
Taking Eq. A(9) into Eq. A(7), K d is then obtained as 2
Kd =
2
⎡ ωtd − sin(ωtd ) ⎤ + ⎡ 1 − cos(ωtd ) ⎤ ⎢ ⎥ ⎢ ⎥ ωtd ωtd ⎣ ⎦ ⎣ ⎦
(A10)
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