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Study on explosion resistance performance experiment and damage assessment model of high-strength reinforcement concrete beams
International Journal of Impact Engineering 133 (2019) 103362
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Study on explosion resistance performance experiment and damage assessment model of high-strength reinforcement concrete beams ⁎
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T
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Zhen Liaoa, Degao Tanga, , Zhizhong Lia, , Yulong Xueb, , Luzhong Shaoa a b
State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing, Jiangsu 210000, China Research Institute for National Defense Engineering of Academy of Military Science PLA China, Beijing 100850, China
A R T I C LE I N FO
A B S T R A C T
Keywords: High-strength RC beam Explosion resistance performance Experimental study Pressure-impulse curves Assessment model
To investigate the explosion resistance performance of high-strength reinforced concrete (RC) beams and analyze the effect of high strength reinforcement on the dynamic response and damage characteristics of RC beams, comparative experimental research of three groups of high-strength RC beams and ordinary RC beams under different uniformly distributed blast loads are conducted in this paper. The P–I damage curves of two kinds of RC beams are established with the maximum tensile reinforcement strain as the damage criterion. The damage assessment of RC beams under various test conditions is carried out, and the assessment results are compared with the test results. The results show that high strength reinforcement can effectively improve the explosion resistance performance of RC beams under blast loads by reducing the deformation of components as well as the length and width of cracks. The damage evaluation results of RC beams according to P–I curves agree well with the experimental results, which indicates that the P–I curves established by using the steel strain of mid-span section as the damage criterion of RC beams can accurately evaluate the failure of the high-strength RC beams under the explosion loading. The research results provide references for the application of HTB700 high strength reinforced concrete beams in the explosion resistant field of civil engineering and the damage assessment of components.
1. Introduction Compared with other structural forms, RC structure stands out with its heavy mass and brilliant explosion resistance performance, thus it has become a first choice for anti-explosion design of engineering projects worldwide and it is also the primary research object in the field of blast resistance and explosion protection for engineering structures [1]. In recent years, large number of studies have been conducted on the dynamic response and damage mode of RC structures under blast loads [2–5], and a series of research results have been obtained. However, when RC beam is concerned, both domestic and foreign scholars focus more on the effect of concrete strength grade [6], reinforcement ratio [7], stirrup ratio [8], boundary conditions [9], and blast loading [10] on its blast resistance performance, few of them have studied the blast resistance performance of RC beams with different strength grade of reinforcement, especially the RC beams using high strength reinforcement. High-strength reinforcement has the characteristics of high strength, which can not only improve the bearing capacity of components, but also reduce the reinforcement ratio of the structure so as to effectively avoid the brittle failure of ordinary RC
⁎
structures with high reinforcement ratio under blast loading. It also saves steel. Therefore, it is of great significance to study the explosion resistance performance of high-strength reinforced concrete structures subjected to blast loads. In terms of damage prediction and evaluation of RC structures under explosion loading, the overpressure-impulse (P–I) damage curve is one of the most scientific and effective methods to describe the blast resistance performance of structures at present and is also a widelyadopted means in structure damage assessment. The P–I curve was initially applied to the damage assessment of civil buildings such as residential buildings, office buildings and industrial plants in the UK after World War II, and was later used to establish damage assessment models for human body or some specific organs (such as eardrum and lungs) under blast loading [13–15]. In 1951, the researchers at Stanford Research Institute also used P–I curves to evaluate the damage degree of aircraft structures subjected to blast loading [16]. At present, P–I curves can be used to predict and evaluate the damage degree of structural members with different material models, boundary conditions and even different damage modes under explosion loading. For example, Hamra [17] studied the P–I curves of frame beam under blast
Corresponding authors. E-mail addresses: [email protected] (D. Tang), [email protected] (Z. Li), [email protected] (Y. Xue).
https://doi.org/10.1016/j.ijimpeng.2019.103362 Received 9 October 2018; Received in revised form 22 July 2019; Accepted 22 July 2019 Available online 23 July 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Impact Engineering 133 (2019) 103362
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according to the calculation principle of reinforced concrete beams. The test conditions are listed in Table 1, and the general mechanical parameters of reinforcements according to the static tensile test are shown in Table 2.
loading through theoretical derivation, conducted parametric analysis on the P–I curves of the frame beam, and performed the influence of lateral restraint, axial compression ratio and other factors on the P–I curves. Fallah [18] studied the influence of blast loading shape on the P–I curves of the continuous beam based on the dimensional analysis theory. By introducing dimensionless parameters, the P–I curves of elastic and elastic-plastic continuous beams which is independent of load shape were obtained. Xu [19] utilized single-degree-of-freedom (SDOF) system to analyze the direct shear failure of RC slabs under blast loading. The P–I curves for evaluating the direct shear failure of slabs and the simplified function for quickly determining the damage curves were obtained. To study the explosion resistance performance of high-strength RC beams under uniformly distributed blast loading and to analyze the influence of high-strength reinforcement on the dynamic response and failure characteristics of RC beams, comparative experimental study on high-strength RC beams and ordinary RC beams under different explosion loading is carried out. The P–I curves of the two kinds of RC beams are established adopting the strain of mid-span tensile reinforcement as the failure criterion, and are used to evaluate the damage of RC beams in the explosion experiments. The structure of this paper is as follows: Section 2 introduces the explosion resistance performance tests of high-strength RC beams; Section 3 analyzes the experimental results in details; Section 4 describes the analysis method of P–I curves; Section 5 establishes the P–I curves of the test beams and conducts the damage assessment, and the final section is conclusion.
2.2. Experiment principle and test measurement The experiment was carried out in a cylindrical blast pressure simulator (hereinafter referred to as blast simulator), as shown in Fig. 2. The inner diameter of the blast simulator is 1.9 m, which mainly consists of three parts, namely the explosion section, the transition section and the test section. At the beginning of the test, an electric detonator was used to initiate the detonation of detonating cord. The detonation products produced by detonator cord could not be rapidly diffused due to the good tightness of the device, therefore the explosion cavity was immediately filled with a large number of disorderly high-pressure air masses. When the high-pressure air passed the grid plate system installed in transition section, a relatively uniform blast wave is formed under the damping effect of small holes. Two RC beams were simply supported under the grid plate of blast simulator with their vicinities being compacted with fine sand. The upper end of the beams was leveled with the surrounding sand surface to ensure that the loads applied on the RC beams was uniformly distributed blast loads. Two wall pressure sensors were respectively installed on both ends of the specimens to measure the uniformly distributed blast loading, as shown in Fig. 3. According to the test loading conditions, the pressure sensor adopts two ranges of 0.5 MPa and 1 MPa with a sensitivity of 28.038 mV/MPa. The upper measuring limits of the two sensors were 0.5 MPa and 1 MPa with sensitivity of 28.038 mV/MPa. In order to analyze the mechanical characteristics of high-strength steel bars and ordinary steel bars under blast loading, the steel strain at the mid-span section of the RC beams has been measured. Two strain gauges of the same type were attached to each tensile reinforcement at mid-span section to ensure the validity of the test data. All the experimental data mentioned above were obtained by DH5922N dynamic signal test acquisition and analysis system.
2. Experimental testing 2.1. Experimental model and test conditions To study the difference of explosion resistance between high strength RC beams and ordinary RC beams under uniformly distributed blast loading, three high-strength RC beams (longitudinal reinforcement is HTB700) and three ordinary RC beams (longitudinal reinforcement is HRB400) are designed and fabricated. HRB400 is the abbreviation for hot-rolled ribbed bars with standard yield strength of 400 MPa, while HTB700 refers to hot-rolled ribbed bars with yield strength standard value of 700 MPa. The dimension of two types test beams is 1700 mm × 150 mm × 300 mm, and the section shape is haunched T-shaped section, as shown in Fig. 1. The concrete strength grade is C40, and concrete cover thickness is 25 mm. The strength grade of stirrup is HRB400 at the stirrup ratio of 0.38%. The longitudinal reinforcement diameter is 12 mm at the reinforcement ratio of 0.56%. 6 test beams were divided into 3 groups for comparative explosion tests. Each group contained one ordinary RC beam and one high-strength RC beam. In order to obtain the damage response of the two types of RC beams under different uniformly distributed blast loading, three blast loading of different strength are used in the tests. The peak overpressure is 0.25 MPa, 0.35 MPa and 0.45 MPa respectively. Among them, 0.25 MPa and 0.45 MPa are respectively the overpressure peaks corresponding to the yield failure of the two kind of RC beams subjected to blast loads
3. Experimental result and analysis 3.1. Blast shock waves According to the corresponding relationship between the length of the detonating cord and the peak overpressure value of the blast loading generated within blast simulator, the quantity of the detonating cord of each shot is determined, and the peak overpressure of the measured blast loading is listed in Table 3. It can be seen from Table 3 that the relative error of the overpressure peaks measured by the two pressure sensors of the same shot is below 2.4%, indicating that the blast loading acting on the top surface of the beams caused by the detonating cord is uniformly distributed. To reduce the test error, the average calculation was performed on the two measured load curves of the same shot, and the overpressure time-history curves of the blast loading after filtering are shown in Fig. 4. It can be seen from Fig. 4 that the blast loading generated by the
Fig. 1. Dimensions and reinforcement schematic diagram of RC beams. 2
International Journal of Impact Engineering 133 (2019) 103362
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Table 1 Test conditions. Test no.
1
Specimen no. Type of longitudinal bar Planned blast loading /MPa
HB1-1 HRB400 0.25
2 HB1-2 HTB700
3
HB2-1 HRB400 0.35
HB2-2 HTB700
HB3-1 HRB400 0.45
HB3-2 HTB700
Table 2 The general mechanical parameters of reinforcement. Grades of reinforcement
Nominal diameter /mm
Yield strength /MPa
Tensile strength /MPa
Elastic modulus /GPa
Maximum elongation rate /%
HRB400 HTB700
12 12
433 756
594 951
199.3 201.3
13.7 8.3
Table 3 The measured peak overpressure values of blast loading. Experiment no.
Detonating cord length/m
Peak overpressure/ MPa
Average/MPa
1 2 3
7 16 21
0.242,0.248 0.368,0.376 0.475,0.479
0.245 0.372 0.477
Fig. 2. Blast pressure simulator.
detonating cord achieves the peak overpressure in an instant, and then decays exponentially. The positive overpressure duration is about 1.5 s, which is almost free from the impact of the peak overpressure of the blast loading.
3.2. Damage future analysis A intuitive and effective method to judge the failure degree and mode of RC beams is to observe the crack distribution pattern. The comparison of the cracks of the RC beams after the explosion test are shown in Fig. 5. The crack characteristics of the test RC beams in each group are described and the damage degree was preliminarily determined, as shown in Table 4. It can be summarized from Fig. 5 and Table 4 that: (1) when the peak overpressure of the uniformly distributed blast loading is 0.245 MPa, the crack propagation and distribution characteristics of the ordinary RC beam (HB1-1) and the high-strength RC beam (HB1-2) are basically the same. (2) With the increase of blast loading, the crack development and distribution characteristics of high-strength RC beams
Fig. 4. The overpressure time history curves of blast loading.
are significantly different from ordinary RC beams. The number of cracks of high-strength RC beam is large and the distribution is relatively uniform, and the length and width of the cracks are significantly smaller than that of the ordinary RC beam. This means the high strength reinforcement can effectively reduce the damage degree of the RC beam and significantly inhibits the crack development. (3) Under the high peak overpressure blast loading, the deformation of the
Fig. 3. Schematic diagram of the experiment. 3
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Fig. 5. Comparison of ordinary RC beams and high-strength RC beams in failure mode.
strain and residual reinforcement strain of high-strength RC beam are 4611με and 1090με, respectively. The high-strength reinforcement under this loading condition is approaching the yield point, while the mid-span strain gauge of ordinary RC beam is already broken, proving that reinforcement strain of high-strength RC beam is far lower than that of ordinary RC beam. Considering the strength enhancement effect of materials under the explosion loading, the comprehensive strength enhancement coefficient of both HRB400 and HTB700 reinforcements were supposed to be 1.2 [20], and the corresponding yield strain would be 2598με and 4536με, respectively. Combined with the measured reinforcement strain values of beam HB1-1 and beam HB3-2, it can be seen that the peak overpressure values of blast loads corresponding to yield failure of ordinary RC beam and high-strength RC beam are 0.245 MPa and 0.477 MPa, respectively.
ordinary RC beam is concentrated in several main cracks near the midspan, and obvious plastic hinge is formed in mid-span when it is damaged. However, when the high-strength RC beam is damaged, many cracks with similar length are distributed along the beam span, which makes the deformation of high-strength RC beams relatively uniform and has better explosion resistance performance. 3.3. Stress characteristics analysis To analyze the stress characteristics of two kinds of RC beams subjected to blast loading and reveal the anti-explosion mechanism of high-strength RC beams, the strain of tensile reinforcement at the midspan section is measured. The measured strain time-history curves of reinforcement are demonstrated in Fig. 6. It should be pointed out that reinforcement strain gauges of ordinary RC beam (HB3-1) in Fig. 6(c) is broken due to the severe damage, so no effective data are acquired. As shown in Fig. 6, the strain values of both HTB700 and HRB400 reinforcement increase to the peak value within a very short time and then decays rapidly with multiple oscillations. When the peak overpressure of blast loading is 0.245 MPa, the maximum mid-span strain of ordinary RC beam (HB1-1) is 2216με which is a bit higher than that of high strength RC beam (HB1-2), but their residual reinforcement strain values are basically the same. Combined with Fig. 5 and Table 4, we can see that the two RC beams have almost the same mechanical and deformation characteristics at this time. When the peak overpressure of blast loading is 0.372 MPa, the maximum mid-span strain and residual strain of high strength RC beam are 3521με and 709με, respectively, which account for only 28.6% and 20.0% of that of the ordinary RC beam. This is because although the HTB700 high-strength reinforcement has lower ductile than HRB400 reinforcement but with higher yield strength so that it is still in the stage of elastic deformation when HRB400 reinforcement already yields. When the peak overpressure of blast loading is 0.477 MPa, the maximum mid-span reinforcement
4. P–I damage curves analysis 4.1. Basic characteristics of P–I curves A typical P–I curve is illustrated in Fig. 7, which divides the coordinate plane of first quadrant to be damaged zone and undamaged zone. The coordinates of points on the curve represent the peak overpressure and impulse of blast loading applied on the structures. For a certain structure or component, all the points on the same P–I curve indicate the equal damage degree, thus the P–I curve is also called isodamage curve. Different damage degrees of the structure, such as completely damaged, severely damaged, and slightly damaged, can be represented by a cluster of P–I curves. The two points A (ia, pa) and B (ib, pb) on the curve are the demarcation points of three different blast loading regions (impulse load region, dynamic load region and quasistatic load region). In the impulse load zone, damage of structural components is only related to impulse of blast loading, and is
Table 4 Failure characterization of test beams. Experiment no.
Specimen no.
Description of failure characteristics
Damage degree
1
HB1-1 HB1-2 HB2-1
Four vertical cracks around the mid-span with a height being 2/3 of the beam depth Three vertical micro-cracks around mid-span with a height being 2/5 of the beam depth Three major vertical cracks in 1/4 span area around the mid-span, extending to the top surface of the beam with a width of about 1.5 mm Seven vertical fine cracks at uniform interval in the 2/3 of span around the mid-span with a height being 1/2 of the beam depth Three vertical cracks in 1/4 of span area around the midspan, among which 2 main cracks extend to the top surface of beam with a width of about 3 mm Nine cracks in the whole span area of the beam, among which an inclined crack appear on both sides of the support. The vertical cracks height near the middle span is 5/6 of the beam depth with a width of about 0.5 mm
Slightly damaged Undamaged Moderately damaged
2
HB2-2 3
HB3-1 HB3-2
4
Slightly damaged Severely damaged Moderately damaged
International Journal of Impact Engineering 133 (2019) 103362
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Fig. 6. The measured strain time-history curves of reinforcement.
damage criterion to obtain P–I curves for analysis. For example, Li [27] et al. established the numerical calculation model for typical RC slabs under blast loading using dynamic analysis software Ls-dyna, and proposed the damage criterion based on the flexural residual bearing capacity at the mid-span section. Finally, a mathematical formula for P–I curve that can be used to evaluate the damage of RC slabs under any kind of blast loading was fitted. Similarly, Shi [28] et al. used numerical analysis method to analyze the anti-explosion performance of a typical RC column under uniformly distributed blast loading. Taking the residual axial bearing capacity of RC column as the damage criterion to judge the degree of damage, the P–I curve of RC column was established. In the assessment of actual structure, the P–I curve established on the maximum displacement criterion or maximum rotation angle criterion is susceptible to be affected by the movement or rotation of the support itself. When the support moves or rotates under the effect of blast loading, difference may arise between the measured value and actual value of the displacement or rotation angle. In such case, the accuracy of the component damage assessment will be affected. As for the P–I curve based on residual bearing capacity demands a large number of numerical calculations established with numerical analysis software, and the analysis process is also quite complicated. In view of the problems above, the P–I damage assessment curves of RC beams are established based on the tensile reinforcement strain criterion at midspan section to assess the damage degree of RC beams under different experimental conditions, and the damage assessment results are compared with the test results.
Fig. 7. Schematic diagram of typical P–I curve.
independent of the peak overpressure. On the contrary, in the quasistatic load zone, the structural damage is independent of the impulse value, only determined by peak overpressure value [11]. Between the above two is the dynamic load zone, where the structural damage is determined by impulse and peak overpressure together. As shown in Fig. 7, there are two vertical asymptotes parallel to the coordinate axis in the P–I curve, which are used to determine the minimum peak overpressure value and the minimum impulse value for structural components under blast loading to reach a certain degree damage. Those two asymptotic lines are also referred to as overpressure asymptote and impulse asymptote. Whether the structure subjected to blast loading can achieve the maximum deformation during the load acting stage is determined by the relationship between overpressure duration and structural vibration period. The blast loads of any attenuation forms can be divided into different load zones as the dimensionless parameter τ, which can be expressed as follows [21]:
τ=
t Meq/ K eq
4.3. General asymptotic equation The simply supported beam under uniformly distributed blast loading is in the stage of elastic deformation before yield failure, and the section deformation conforms to the plane section assumption. The deflection equation of simply supported beam is:
(1)
y (x ) =
where t is positive overpressure duration time, Meq is the mass of equivalent SDOF system, Keq is the stiffness of equivalent SDOF system. Meq = KMLM, M is the total mass of the component, KML is the mass-load coefficient which is supposed to be 0.78 for the simply supported beam under uniformly distributed loads in the elastic stage [22].
16y0 3 [l x − 2lx 3 + x 4] (0 ≤ x ≤ l) 5l 4
(2)
where l is the length of beam, y0 is the maximum mid-span displacement of the beam under uniformly distributed load q. The bending moment of arbitrary cross section is calculated according to the following formula:
4.2. Damage criterion
M (x ) = −EJ The reasonableness of the damage criterion of P–I curve is directly related to the accuracy of structural components damage assessment. At present, the establishment of P–I curve of SDOF model is mostly based on the maximum displacement criterion or maximum rotation angle criterion of the support [23–26]. For the typical load-bearing members, some researchers also used the axial residual bearing capacity as the
192EJy0 d 2y =− (lx − x 2) dx 2 5l 4
(3)
where E is elastic modulus and J is the inertia moment of the cross section. The strain energy U, kinetic energy V and external work W of the simply supported beam under blast loading can be figured out through the formula below: 5
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2
U=
V=
∫ M2EJ(x ) dx =
24.576EJy02 l3
(4)
i 2b2l
1 2 mv = 2 2ρA
(5)
l
W=
∫ pby (x ) dx = 0.64pbly0 0
(6)
where ρ is the density, A is the cross section area of the beam, v is the vibration speed, b is the beam width, p is the peak overpressure of blast loading, and i is the impulse of the blast loading. Since the duration time of the impulse type blast loading is much smaller than the natural vibration period of the structure (τ≪1), the blast loading has already decayed to zero before the structure is significantly deformed. Therefore, it can be approximately assumed that the structure obtains the initial velocity in an instant. When the structure reaches the stage of maximum deformation, the kinetic energy V is completely transformed into strain energy U:
The bending stiffness of RC beam can be figured out through corresponding formula in GB 50010-2010 Code for design of concrete structures [29], as shown below:
y0 ibl = 0.143 l ρAEJ
EJ =
Fig. 8. The resistance curve of an ideal elastoplastic model.
The linear slope of the resistance function in the elastic phase is:
K eq = 384EJ /5l3
(7)
The strain of tensile reinforcement before the yield deformation of the structure can be expressed as:
192y0 (h 0 − x 0) M (x )(h 0 − x 0) (lx − x 2) ε= = 5l 4 EJ
(8)
(9)
The duration time of quasi-static load is much larger than the natural vibration period of the structure (τ≫1). Similarly, the relationship between the strain ε of mid-span cross section and the peak overpressure p can be established according to the external work equal to the strain energy.
pbl 2 =
4EJε h0 − x 0
(10)
To deduct the dimensionless P–I curve, EW is introduced to nondimensionalize the Formulas (9) and (10). Finally, the asymptotic equations of P–I curve for simply supported beam is obtained. impulse asymptotic equation
X = ie =
ib EJ 0.73Jε = (h 0 − x 0) W ρA EW
pbl 2 4Jε = EW (h 0 − x 0) W
0.2 + 1.15ψ +
6αE ρ 1 + 3.5γ ′ f
(14)
5.2. Demarcation points on the P–I curve of test beams As a key parameter to distinguish the three blast loading regions of P–I curve, the magnitude of nondimensional parameter τ determines the relative locations of Points A and B on the curve in Fig. 7. For a given structural failure criterion, the values of τ1 and τ2 are related to the accuracy requirements. By studying the P–I curve of SDOF system subjected to triangular blast loading, Mays [30] et al. found that the analytic accuracy requirement could be well satisfied when τ1 = 0.4 and τ2 = 40. Li [24] et al. found in the literature [30] that the values of τ1 and τ2 are not only affected by the error precision, but also related to the shape of the blast loads, and the parameter values under different load shapes are given. In this paper, the two demarcation point coordinates on the P–I curve of the test RC beams are calculated by the recommended values in the literature [24] (τ1 = 1.3 and τ2 = 32.4). The impulse of the triangular blast loading has the following relationship with the peak overpressure:
(11)
overpressure asymptotic equation
Y = pe =
Es As h 02
where Es and As is the elastic modulus and area of the reinforcement, respectively; h0 is the effective section height; αE = Es/Ec is the elastic modulus ratio of reinforcement to concrete; ρ is the reinforcement ratio; ψ is the nonuniform coefficient of tensile reinforcement strain between cracks, and γf′ is the Enhancement factor of compression flange. The calculation methods of the parameters ψ and γf′ are detailed in the Code [29]. According to Section 3.3, the yield strains of HRB400 and HTB700 reinforcements under blast loading would be 2598με and 4536με, respectively. The ratio of elastic modulus of the two materials is used to convert the reinforcement area into concrete material, and the equivalent section expressed with concrete is finally obtained. The height of the compression zone x0 = 0.134 m is determined according to the condition that the area moment of the compression zone and tensile zone is equal to the neutral axis. Table 5 gives the impulse and overpressure asymptotes for yield failure of ordinary RC beams and high-strength RC beams according to Formulas (11), (12), (13) and (14).
where, h0 is the effective height of the beam and x0 is depth of compression. Substituting Eq. (7) into Eq. (8), the following expression is obtained at the mid-span cross section of the beam:
ib EJ 0.73EJε = h0 − x 0 ρA
(13)
(12)
where W is the bending-resistance modulus, and the meanings of other symbols are shown above. 5. Establishment of the P–I curves and damage assessment of test beams 5.1. Asymptote of P–I curves of the test beams
i=
The occurrence of cracks in RC flexural members under blast loading will cause their bending stiffness to change continuously, and the resistance function is highly nonlinear [22]. Therefore, the resistance function of under-reinforced beams can be simplified to an ideal elastoplastic model in theoretical analysis, as shown in Fig. 8.
1 pt 2
(15)
According to Formula (1), Formulas (11)–(13) and Formula (15), the coordinates of the demarcation points on the dimensionless P–I curves of the test beams were obtained, as shown in Table 5. 6
International Journal of Impact Engineering 133 (2019) 103362
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Table 5 The asymptotes and demarcation coordinates of the P–I curves of the test beams . Test beam
Impulse asymptote
Overpressure asymptote
Demarcation point between Zones I and II ia pa
Demarcation point between Zones II and III ib pb
Ordinary RC beam High-strength RC beam
0.0022 0.0039
0.0122 0.0213
0.0022 0.0039
0.0199 0.0348
0.0336 0.0595
Fig. 9. Simplified triangular and rectangular overpressure time-history curves.
0.0122 0.0213
Fig. 10. P–I curves of test RC beams for yield failure.
5.3. Test beams damage assessment
Three points A, B and C are distributed on the right of the demarcation point between the dynamic load region and the quasi-static load region, meaning that the blast loading generated by blast simulator belongs to the quasi-static load. A′ is the corresponding point when the measured blast loading with an peak overpressure of 0.245 MPa is simplified into a rectangular blast loading. From the figure we can find that both points A′ and A are located on the P–I curve of the ordinary RC beam with yield failure. This suggests two ways of load simplification produce exactly same assessment result. Therefore, in the theoretical analysis, the quasi-static load can be further simplified to be rectangular blast loads to evaluate the damage of RC beams.
The overpressure of air shock wave generally decreases exponentially with time. The time-history curves of measured blast overpressure in this paper is shown in Fig. 4. Since the positive duration time was quite long (being about 1.5 s), the overpressure curves are simplified according to the principle of tangent simplification. The simplified triangular overpressure time-history curves are shown in Fig. 9. and its characteristic parameters and coordinates on P–I curves are listed in Table 6. According to data in Table 5, the P–I curves for yield failure of highstrength RC beams and ordinary RC beams are plotted. The hyperbolic square relation is used to fit the dynamic load region [12], as shown in Fig. 10. The three points A, B and C in Fig. 10 represent the explosion tests whose peak overpressure value is 0.245 MPa, 0.372 MPa and 0.477 MPa, respectively. The relative position of the three points and P–I curves represents the failure of components, whereas their relative locations to the demarcation points indicate the type of the blast loading. It can be clearly seen from the figure that point A and point C fall on the P–I curve of ordinary RC beam and high-strength RC beam, respectively, with point B being located between them. It indicates that the yield failure of ordinary RC beams occurs when the peak overpressure value of the blast loading is 0.245 MPa, while that of highstrength RC beams only occurs when the peak overpressure value is 0.477 MPa. When the peak overpressure value is 0.372 MPa, the ordinary RC beam is damaged already, but high-strength RC beam remains undamaged. In the meantime, the P–I curve of high-strength RC beam is located at the upper right of the ordinary RC beam, indicating that the high-strength RC beam can resist larger blast loads than the ordinary RC beam and has better blast resistance performance.
6. Conclusion In this paper, three groups of high-strength RC beams and ordinary RC beams are tested and compared under uniformly distributed blast loading. Through the comparative analysis of the failure characteristics and the reinforcement strain of RC beams, the blast resistance performance and damage mechanism of the high-strength RC beam are obtained. The P–I damage curves of two kinds of RC beams are established with the maximum tensile reinforcement strain as the damage criterion. The damage assessment of RC beams under various test conditions is carried out, and the assessment results are compared with the test results. The main conclusions are as follows: (1) The deformation characteristics of high-strength RC beams are basically the same as that of ordinary RC beams under small peak overpressure blast loading. With the increase of blast loads, high strength reinforcement plays a significant role in reducing component deformation and crack length and width, and can effectively improve the blast resistance performance of RC beams.
Table 6 Characteristic parameters and coordinates on P–I curve of blast loading. Numbering
Peak overpressure p/MPa
Positive duration time t/s
Impulse i/ MPa⋅s
ie
pe
A B C
0.245 0.372 0.477
0.163 0.185 0.191
0.0200 0.0344 0.0456
0.0441 0.0758 0.1000
0.011 0.016 0.021
7
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(2) When the peak overpressure of blast loading is 0.245 MPa, the ordinary RC beam is close to yield failure, while the high-strength RC beam begins to yield failure when the peak overpressure value is 0.477 MPa. The damage evaluation results of RC beams according to P–I curves agree well with the experimental results, which indicates that the P–I curves established by using the reinforcement strain of mid-span section as the damage criterion of RC beams can accurately evaluate the failure of the high-strength RC beams under blast loading. (3) When using the P–I curve to evaluate the damage of the RC beams, the quasi-static load can be simplified into a triangular blast load or a rectangular blast load as needed in practical analysis, and the damage evaluation results are identical.
[9] Chen WX, Guo ZK, Ye JH. Dynamic responses and failure modes of reinforced concrete beams with flexible supports under blast loading. Acta Armam 2011;32:1271–7. [10] Zhang XH, Duan ZD, Zhang CW. Analysis for dynamic response and failure process of reinforced concrete beam under blast load. J Northeast For Univ 2009;37:50–3. [11] Wang W. Study on damage effects and assessments method of reinforced concrete structural members under blast loading. Chang Sha: National University of Defense Technology; 2012. p. 24–37. [12] Carta G, Stochino F. Theoretical models to predict the flexural failure of reinforced concrete beams under blast loads. Eng Struct 2013;49:306–15. [13] Smith PD, Hetherington JG. Blast and ballistic loading of structures. London: Butterworth Heinemann; 1994. [14] Jarrett DE. Derivation of British explosives safety distances. Ann. N. Y. Acad. Sci. 1968;152(1):18–35. [15] Baker W.E., Cox P.A., Westine P.S., et al. Explosion hazards and evaluation. New York: Elsevier Scientific Pub. Co.; 1983.p. 121–143. [16] Abrahamson GR, Lindberg HE, Abrahamson GR, et al. Peak load-impulse characterization of critical pulse loads in structural dynamics. Nucl Eng Des 1976;37(1):35–46. [17] Hamra L, Demonceau JF, Denoël V. Pressure–impulse diagram of a beam developing non-linear membrane action under blast loading. Int J Impact Eng 2015;86:188–205. [18] Fallah AS, Nwankwo E, Louca LA. Pressure-impulse diagrams for blast loaded continuous beams based on dimensional analysis. J Appl Mech 2013;80(5):051011. [19] Xu J, Wu C, Li Z. Analysis of direct shear failure mode for RC slabs under external explosive loading. Int J Impact Eng 2014;69:136–48. [20] Zhou ZL, Li XB, Hong L. Underground protective engineering and structure. Chang Sha: Central South University; 2014. p. 28–51. [21] Fallah AS, Louca LA. Pressure-impulse diagrams for elastic-plastic-hardening and softening single-degree-of-freedom models subjected to blast loading. Int J Impact Eng 2007;34:823–42. [22] Fang Q, Liu JC. Underground protective structure. Beijing: China Water &Power Press; 2010. p. 120–34. [23] Li QM, Meng H. Pressure-impulse diagram based on dimensional analysis and single-degree-of-freedom model. J Eng Mech 2002;128(1):87–92. [24] Li QM, Meng H. Pulse loading shape effects on pressure–impulse diagram of an elastic–plastic, single-degree-of-freedom structural model. Int J Mech Sci 2002;44(9):1985–98. [25] Tian ZM, Zhang JH, Jiang SY. Damage assessment for steel-concrete composite beams subjected to blast loading. J Vib Shock 2016;35:42–8. [26] Li TH. Dynamic response and damage assessment of reinforced concrete slabs subjected to blast loading. Xi'an: Chang’ an University; 2012. p. 11–41. [27] Li ZX, Shi YC, Shi XS. Damage analysis and assessment of RC slabs under blast load. J Build Struc 2009;30:60–6. [28] Shi Y, Hao H, Li ZX. Numerical derivation of pressure–impulse diagrams for prediction of RC column damage to blast loads. Int J Impact Eng 2008;35(11):1213–27. [29] Ministry of Housing and Urban-Rural Construction of the People's Republic of China. Code for design of concrete structures (GB 50010-2010). China Architecture & Building Press; 2014. [30] Mays GC, Smith PD. Blast effects on buildings-design of buildings to optimize resistance to blast loading. London: Thomas Telford; 1995.
Acknowledgment This research was funded by the National Natural Science Foundation of China under grant no. 11872072 and 11402304. I thank to Xiaolu Wang and Lu Dong for their support during the experiment. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijimpeng.2019.103362. References [1] Shi YC, Li ZX. State-of-the-art in damage and collapse analysis of RC structures under blast loading. Chin Civil Eng J 2010;43:83–92. [2] Yang JL, Xi F. Experimental and theoretical study of free–free beam subjected to impact at any cross-section along its span. Int J Impact Eng 2003;28(7):761–81. [3] Yan B, Liu F, Song DY, et al. Numerical study on damage mechanism of RC beams under close-in blast loading. Appl Mech Mater 2015;730:55–64. [4] Yan S, Liu L, Qi BX, et al. Dynamic response and failure mode analysis of concrete infilled rectangular steel tube columns under blasting loading. J Disaster Prev Mitig Eng 2011;31:477–82. [5] Thai DK, Kim SE. Numerical investigation of the damage of RC members subjected to blast loading. Eng Fail Anal 2018;92:350–67. [6] Hallgren M, Ansell A, Magnusson J. Air-blast-loaded, high-strength concrete beams. Part I: experimental investigation. Mag Concr Res 2010;62(2):127–36. [7] Li MS, Li J, Li H, et al. Deformation and failure of reinforced concrete beams under blast loading. Expl Shock Wave 2015;35:177–83. [8] Kuang ZP, Yang QH, Cui M. Experiment research and failure modes analyses of RCBeams under blast loading. J Tongji Univ (Nat Sci) 2009;37:1153–6.
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Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Study on explosion resistance performance experiment and damage assessment model of high-strength reinforcement concrete beams ⁎
⁎
T
⁎
Zhen Liaoa, Degao Tanga, , Zhizhong Lia, , Yulong Xueb, , Luzhong Shaoa a b
State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing, Jiangsu 210000, China Research Institute for National Defense Engineering of Academy of Military Science PLA China, Beijing 100850, China
A R T I C LE I N FO
A B S T R A C T
Keywords: High-strength RC beam Explosion resistance performance Experimental study Pressure-impulse curves Assessment model
To investigate the explosion resistance performance of high-strength reinforced concrete (RC) beams and analyze the effect of high strength reinforcement on the dynamic response and damage characteristics of RC beams, comparative experimental research of three groups of high-strength RC beams and ordinary RC beams under different uniformly distributed blast loads are conducted in this paper. The P–I damage curves of two kinds of RC beams are established with the maximum tensile reinforcement strain as the damage criterion. The damage assessment of RC beams under various test conditions is carried out, and the assessment results are compared with the test results. The results show that high strength reinforcement can effectively improve the explosion resistance performance of RC beams under blast loads by reducing the deformation of components as well as the length and width of cracks. The damage evaluation results of RC beams according to P–I curves agree well with the experimental results, which indicates that the P–I curves established by using the steel strain of mid-span section as the damage criterion of RC beams can accurately evaluate the failure of the high-strength RC beams under the explosion loading. The research results provide references for the application of HTB700 high strength reinforced concrete beams in the explosion resistant field of civil engineering and the damage assessment of components.
1. Introduction Compared with other structural forms, RC structure stands out with its heavy mass and brilliant explosion resistance performance, thus it has become a first choice for anti-explosion design of engineering projects worldwide and it is also the primary research object in the field of blast resistance and explosion protection for engineering structures [1]. In recent years, large number of studies have been conducted on the dynamic response and damage mode of RC structures under blast loads [2–5], and a series of research results have been obtained. However, when RC beam is concerned, both domestic and foreign scholars focus more on the effect of concrete strength grade [6], reinforcement ratio [7], stirrup ratio [8], boundary conditions [9], and blast loading [10] on its blast resistance performance, few of them have studied the blast resistance performance of RC beams with different strength grade of reinforcement, especially the RC beams using high strength reinforcement. High-strength reinforcement has the characteristics of high strength, which can not only improve the bearing capacity of components, but also reduce the reinforcement ratio of the structure so as to effectively avoid the brittle failure of ordinary RC
⁎
structures with high reinforcement ratio under blast loading. It also saves steel. Therefore, it is of great significance to study the explosion resistance performance of high-strength reinforced concrete structures subjected to blast loads. In terms of damage prediction and evaluation of RC structures under explosion loading, the overpressure-impulse (P–I) damage curve is one of the most scientific and effective methods to describe the blast resistance performance of structures at present and is also a widelyadopted means in structure damage assessment. The P–I curve was initially applied to the damage assessment of civil buildings such as residential buildings, office buildings and industrial plants in the UK after World War II, and was later used to establish damage assessment models for human body or some specific organs (such as eardrum and lungs) under blast loading [13–15]. In 1951, the researchers at Stanford Research Institute also used P–I curves to evaluate the damage degree of aircraft structures subjected to blast loading [16]. At present, P–I curves can be used to predict and evaluate the damage degree of structural members with different material models, boundary conditions and even different damage modes under explosion loading. For example, Hamra [17] studied the P–I curves of frame beam under blast
Corresponding authors. E-mail addresses: [email protected] (D. Tang), [email protected] (Z. Li), [email protected] (Y. Xue).
https://doi.org/10.1016/j.ijimpeng.2019.103362 Received 9 October 2018; Received in revised form 22 July 2019; Accepted 22 July 2019 Available online 23 July 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.
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according to the calculation principle of reinforced concrete beams. The test conditions are listed in Table 1, and the general mechanical parameters of reinforcements according to the static tensile test are shown in Table 2.
loading through theoretical derivation, conducted parametric analysis on the P–I curves of the frame beam, and performed the influence of lateral restraint, axial compression ratio and other factors on the P–I curves. Fallah [18] studied the influence of blast loading shape on the P–I curves of the continuous beam based on the dimensional analysis theory. By introducing dimensionless parameters, the P–I curves of elastic and elastic-plastic continuous beams which is independent of load shape were obtained. Xu [19] utilized single-degree-of-freedom (SDOF) system to analyze the direct shear failure of RC slabs under blast loading. The P–I curves for evaluating the direct shear failure of slabs and the simplified function for quickly determining the damage curves were obtained. To study the explosion resistance performance of high-strength RC beams under uniformly distributed blast loading and to analyze the influence of high-strength reinforcement on the dynamic response and failure characteristics of RC beams, comparative experimental study on high-strength RC beams and ordinary RC beams under different explosion loading is carried out. The P–I curves of the two kinds of RC beams are established adopting the strain of mid-span tensile reinforcement as the failure criterion, and are used to evaluate the damage of RC beams in the explosion experiments. The structure of this paper is as follows: Section 2 introduces the explosion resistance performance tests of high-strength RC beams; Section 3 analyzes the experimental results in details; Section 4 describes the analysis method of P–I curves; Section 5 establishes the P–I curves of the test beams and conducts the damage assessment, and the final section is conclusion.
2.2. Experiment principle and test measurement The experiment was carried out in a cylindrical blast pressure simulator (hereinafter referred to as blast simulator), as shown in Fig. 2. The inner diameter of the blast simulator is 1.9 m, which mainly consists of three parts, namely the explosion section, the transition section and the test section. At the beginning of the test, an electric detonator was used to initiate the detonation of detonating cord. The detonation products produced by detonator cord could not be rapidly diffused due to the good tightness of the device, therefore the explosion cavity was immediately filled with a large number of disorderly high-pressure air masses. When the high-pressure air passed the grid plate system installed in transition section, a relatively uniform blast wave is formed under the damping effect of small holes. Two RC beams were simply supported under the grid plate of blast simulator with their vicinities being compacted with fine sand. The upper end of the beams was leveled with the surrounding sand surface to ensure that the loads applied on the RC beams was uniformly distributed blast loads. Two wall pressure sensors were respectively installed on both ends of the specimens to measure the uniformly distributed blast loading, as shown in Fig. 3. According to the test loading conditions, the pressure sensor adopts two ranges of 0.5 MPa and 1 MPa with a sensitivity of 28.038 mV/MPa. The upper measuring limits of the two sensors were 0.5 MPa and 1 MPa with sensitivity of 28.038 mV/MPa. In order to analyze the mechanical characteristics of high-strength steel bars and ordinary steel bars under blast loading, the steel strain at the mid-span section of the RC beams has been measured. Two strain gauges of the same type were attached to each tensile reinforcement at mid-span section to ensure the validity of the test data. All the experimental data mentioned above were obtained by DH5922N dynamic signal test acquisition and analysis system.
2. Experimental testing 2.1. Experimental model and test conditions To study the difference of explosion resistance between high strength RC beams and ordinary RC beams under uniformly distributed blast loading, three high-strength RC beams (longitudinal reinforcement is HTB700) and three ordinary RC beams (longitudinal reinforcement is HRB400) are designed and fabricated. HRB400 is the abbreviation for hot-rolled ribbed bars with standard yield strength of 400 MPa, while HTB700 refers to hot-rolled ribbed bars with yield strength standard value of 700 MPa. The dimension of two types test beams is 1700 mm × 150 mm × 300 mm, and the section shape is haunched T-shaped section, as shown in Fig. 1. The concrete strength grade is C40, and concrete cover thickness is 25 mm. The strength grade of stirrup is HRB400 at the stirrup ratio of 0.38%. The longitudinal reinforcement diameter is 12 mm at the reinforcement ratio of 0.56%. 6 test beams were divided into 3 groups for comparative explosion tests. Each group contained one ordinary RC beam and one high-strength RC beam. In order to obtain the damage response of the two types of RC beams under different uniformly distributed blast loading, three blast loading of different strength are used in the tests. The peak overpressure is 0.25 MPa, 0.35 MPa and 0.45 MPa respectively. Among them, 0.25 MPa and 0.45 MPa are respectively the overpressure peaks corresponding to the yield failure of the two kind of RC beams subjected to blast loads
3. Experimental result and analysis 3.1. Blast shock waves According to the corresponding relationship between the length of the detonating cord and the peak overpressure value of the blast loading generated within blast simulator, the quantity of the detonating cord of each shot is determined, and the peak overpressure of the measured blast loading is listed in Table 3. It can be seen from Table 3 that the relative error of the overpressure peaks measured by the two pressure sensors of the same shot is below 2.4%, indicating that the blast loading acting on the top surface of the beams caused by the detonating cord is uniformly distributed. To reduce the test error, the average calculation was performed on the two measured load curves of the same shot, and the overpressure time-history curves of the blast loading after filtering are shown in Fig. 4. It can be seen from Fig. 4 that the blast loading generated by the
Fig. 1. Dimensions and reinforcement schematic diagram of RC beams. 2
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Table 1 Test conditions. Test no.
1
Specimen no. Type of longitudinal bar Planned blast loading /MPa
HB1-1 HRB400 0.25
2 HB1-2 HTB700
3
HB2-1 HRB400 0.35
HB2-2 HTB700
HB3-1 HRB400 0.45
HB3-2 HTB700
Table 2 The general mechanical parameters of reinforcement. Grades of reinforcement
Nominal diameter /mm
Yield strength /MPa
Tensile strength /MPa
Elastic modulus /GPa
Maximum elongation rate /%
HRB400 HTB700
12 12
433 756
594 951
199.3 201.3
13.7 8.3
Table 3 The measured peak overpressure values of blast loading. Experiment no.
Detonating cord length/m
Peak overpressure/ MPa
Average/MPa
1 2 3
7 16 21
0.242,0.248 0.368,0.376 0.475,0.479
0.245 0.372 0.477
Fig. 2. Blast pressure simulator.
detonating cord achieves the peak overpressure in an instant, and then decays exponentially. The positive overpressure duration is about 1.5 s, which is almost free from the impact of the peak overpressure of the blast loading.
3.2. Damage future analysis A intuitive and effective method to judge the failure degree and mode of RC beams is to observe the crack distribution pattern. The comparison of the cracks of the RC beams after the explosion test are shown in Fig. 5. The crack characteristics of the test RC beams in each group are described and the damage degree was preliminarily determined, as shown in Table 4. It can be summarized from Fig. 5 and Table 4 that: (1) when the peak overpressure of the uniformly distributed blast loading is 0.245 MPa, the crack propagation and distribution characteristics of the ordinary RC beam (HB1-1) and the high-strength RC beam (HB1-2) are basically the same. (2) With the increase of blast loading, the crack development and distribution characteristics of high-strength RC beams
Fig. 4. The overpressure time history curves of blast loading.
are significantly different from ordinary RC beams. The number of cracks of high-strength RC beam is large and the distribution is relatively uniform, and the length and width of the cracks are significantly smaller than that of the ordinary RC beam. This means the high strength reinforcement can effectively reduce the damage degree of the RC beam and significantly inhibits the crack development. (3) Under the high peak overpressure blast loading, the deformation of the
Fig. 3. Schematic diagram of the experiment. 3
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Fig. 5. Comparison of ordinary RC beams and high-strength RC beams in failure mode.
strain and residual reinforcement strain of high-strength RC beam are 4611με and 1090με, respectively. The high-strength reinforcement under this loading condition is approaching the yield point, while the mid-span strain gauge of ordinary RC beam is already broken, proving that reinforcement strain of high-strength RC beam is far lower than that of ordinary RC beam. Considering the strength enhancement effect of materials under the explosion loading, the comprehensive strength enhancement coefficient of both HRB400 and HTB700 reinforcements were supposed to be 1.2 [20], and the corresponding yield strain would be 2598με and 4536με, respectively. Combined with the measured reinforcement strain values of beam HB1-1 and beam HB3-2, it can be seen that the peak overpressure values of blast loads corresponding to yield failure of ordinary RC beam and high-strength RC beam are 0.245 MPa and 0.477 MPa, respectively.
ordinary RC beam is concentrated in several main cracks near the midspan, and obvious plastic hinge is formed in mid-span when it is damaged. However, when the high-strength RC beam is damaged, many cracks with similar length are distributed along the beam span, which makes the deformation of high-strength RC beams relatively uniform and has better explosion resistance performance. 3.3. Stress characteristics analysis To analyze the stress characteristics of two kinds of RC beams subjected to blast loading and reveal the anti-explosion mechanism of high-strength RC beams, the strain of tensile reinforcement at the midspan section is measured. The measured strain time-history curves of reinforcement are demonstrated in Fig. 6. It should be pointed out that reinforcement strain gauges of ordinary RC beam (HB3-1) in Fig. 6(c) is broken due to the severe damage, so no effective data are acquired. As shown in Fig. 6, the strain values of both HTB700 and HRB400 reinforcement increase to the peak value within a very short time and then decays rapidly with multiple oscillations. When the peak overpressure of blast loading is 0.245 MPa, the maximum mid-span strain of ordinary RC beam (HB1-1) is 2216με which is a bit higher than that of high strength RC beam (HB1-2), but their residual reinforcement strain values are basically the same. Combined with Fig. 5 and Table 4, we can see that the two RC beams have almost the same mechanical and deformation characteristics at this time. When the peak overpressure of blast loading is 0.372 MPa, the maximum mid-span strain and residual strain of high strength RC beam are 3521με and 709με, respectively, which account for only 28.6% and 20.0% of that of the ordinary RC beam. This is because although the HTB700 high-strength reinforcement has lower ductile than HRB400 reinforcement but with higher yield strength so that it is still in the stage of elastic deformation when HRB400 reinforcement already yields. When the peak overpressure of blast loading is 0.477 MPa, the maximum mid-span reinforcement
4. P–I damage curves analysis 4.1. Basic characteristics of P–I curves A typical P–I curve is illustrated in Fig. 7, which divides the coordinate plane of first quadrant to be damaged zone and undamaged zone. The coordinates of points on the curve represent the peak overpressure and impulse of blast loading applied on the structures. For a certain structure or component, all the points on the same P–I curve indicate the equal damage degree, thus the P–I curve is also called isodamage curve. Different damage degrees of the structure, such as completely damaged, severely damaged, and slightly damaged, can be represented by a cluster of P–I curves. The two points A (ia, pa) and B (ib, pb) on the curve are the demarcation points of three different blast loading regions (impulse load region, dynamic load region and quasistatic load region). In the impulse load zone, damage of structural components is only related to impulse of blast loading, and is
Table 4 Failure characterization of test beams. Experiment no.
Specimen no.
Description of failure characteristics
Damage degree
1
HB1-1 HB1-2 HB2-1
Four vertical cracks around the mid-span with a height being 2/3 of the beam depth Three vertical micro-cracks around mid-span with a height being 2/5 of the beam depth Three major vertical cracks in 1/4 span area around the mid-span, extending to the top surface of the beam with a width of about 1.5 mm Seven vertical fine cracks at uniform interval in the 2/3 of span around the mid-span with a height being 1/2 of the beam depth Three vertical cracks in 1/4 of span area around the midspan, among which 2 main cracks extend to the top surface of beam with a width of about 3 mm Nine cracks in the whole span area of the beam, among which an inclined crack appear on both sides of the support. The vertical cracks height near the middle span is 5/6 of the beam depth with a width of about 0.5 mm
Slightly damaged Undamaged Moderately damaged
2
HB2-2 3
HB3-1 HB3-2
4
Slightly damaged Severely damaged Moderately damaged
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Fig. 6. The measured strain time-history curves of reinforcement.
damage criterion to obtain P–I curves for analysis. For example, Li [27] et al. established the numerical calculation model for typical RC slabs under blast loading using dynamic analysis software Ls-dyna, and proposed the damage criterion based on the flexural residual bearing capacity at the mid-span section. Finally, a mathematical formula for P–I curve that can be used to evaluate the damage of RC slabs under any kind of blast loading was fitted. Similarly, Shi [28] et al. used numerical analysis method to analyze the anti-explosion performance of a typical RC column under uniformly distributed blast loading. Taking the residual axial bearing capacity of RC column as the damage criterion to judge the degree of damage, the P–I curve of RC column was established. In the assessment of actual structure, the P–I curve established on the maximum displacement criterion or maximum rotation angle criterion is susceptible to be affected by the movement or rotation of the support itself. When the support moves or rotates under the effect of blast loading, difference may arise between the measured value and actual value of the displacement or rotation angle. In such case, the accuracy of the component damage assessment will be affected. As for the P–I curve based on residual bearing capacity demands a large number of numerical calculations established with numerical analysis software, and the analysis process is also quite complicated. In view of the problems above, the P–I damage assessment curves of RC beams are established based on the tensile reinforcement strain criterion at midspan section to assess the damage degree of RC beams under different experimental conditions, and the damage assessment results are compared with the test results.
Fig. 7. Schematic diagram of typical P–I curve.
independent of the peak overpressure. On the contrary, in the quasistatic load zone, the structural damage is independent of the impulse value, only determined by peak overpressure value [11]. Between the above two is the dynamic load zone, where the structural damage is determined by impulse and peak overpressure together. As shown in Fig. 7, there are two vertical asymptotes parallel to the coordinate axis in the P–I curve, which are used to determine the minimum peak overpressure value and the minimum impulse value for structural components under blast loading to reach a certain degree damage. Those two asymptotic lines are also referred to as overpressure asymptote and impulse asymptote. Whether the structure subjected to blast loading can achieve the maximum deformation during the load acting stage is determined by the relationship between overpressure duration and structural vibration period. The blast loads of any attenuation forms can be divided into different load zones as the dimensionless parameter τ, which can be expressed as follows [21]:
τ=
t Meq/ K eq
4.3. General asymptotic equation The simply supported beam under uniformly distributed blast loading is in the stage of elastic deformation before yield failure, and the section deformation conforms to the plane section assumption. The deflection equation of simply supported beam is:
(1)
y (x ) =
where t is positive overpressure duration time, Meq is the mass of equivalent SDOF system, Keq is the stiffness of equivalent SDOF system. Meq = KMLM, M is the total mass of the component, KML is the mass-load coefficient which is supposed to be 0.78 for the simply supported beam under uniformly distributed loads in the elastic stage [22].
16y0 3 [l x − 2lx 3 + x 4] (0 ≤ x ≤ l) 5l 4
(2)
where l is the length of beam, y0 is the maximum mid-span displacement of the beam under uniformly distributed load q. The bending moment of arbitrary cross section is calculated according to the following formula:
4.2. Damage criterion
M (x ) = −EJ The reasonableness of the damage criterion of P–I curve is directly related to the accuracy of structural components damage assessment. At present, the establishment of P–I curve of SDOF model is mostly based on the maximum displacement criterion or maximum rotation angle criterion of the support [23–26]. For the typical load-bearing members, some researchers also used the axial residual bearing capacity as the
192EJy0 d 2y =− (lx − x 2) dx 2 5l 4
(3)
where E is elastic modulus and J is the inertia moment of the cross section. The strain energy U, kinetic energy V and external work W of the simply supported beam under blast loading can be figured out through the formula below: 5
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2
U=
V=
∫ M2EJ(x ) dx =
24.576EJy02 l3
(4)
i 2b2l
1 2 mv = 2 2ρA
(5)
l
W=
∫ pby (x ) dx = 0.64pbly0 0
(6)
where ρ is the density, A is the cross section area of the beam, v is the vibration speed, b is the beam width, p is the peak overpressure of blast loading, and i is the impulse of the blast loading. Since the duration time of the impulse type blast loading is much smaller than the natural vibration period of the structure (τ≪1), the blast loading has already decayed to zero before the structure is significantly deformed. Therefore, it can be approximately assumed that the structure obtains the initial velocity in an instant. When the structure reaches the stage of maximum deformation, the kinetic energy V is completely transformed into strain energy U:
The bending stiffness of RC beam can be figured out through corresponding formula in GB 50010-2010 Code for design of concrete structures [29], as shown below:
y0 ibl = 0.143 l ρAEJ
EJ =
Fig. 8. The resistance curve of an ideal elastoplastic model.
The linear slope of the resistance function in the elastic phase is:
K eq = 384EJ /5l3
(7)
The strain of tensile reinforcement before the yield deformation of the structure can be expressed as:
192y0 (h 0 − x 0) M (x )(h 0 − x 0) (lx − x 2) ε= = 5l 4 EJ
(8)
(9)
The duration time of quasi-static load is much larger than the natural vibration period of the structure (τ≫1). Similarly, the relationship between the strain ε of mid-span cross section and the peak overpressure p can be established according to the external work equal to the strain energy.
pbl 2 =
4EJε h0 − x 0
(10)
To deduct the dimensionless P–I curve, EW is introduced to nondimensionalize the Formulas (9) and (10). Finally, the asymptotic equations of P–I curve for simply supported beam is obtained. impulse asymptotic equation
X = ie =
ib EJ 0.73Jε = (h 0 − x 0) W ρA EW
pbl 2 4Jε = EW (h 0 − x 0) W
0.2 + 1.15ψ +
6αE ρ 1 + 3.5γ ′ f
(14)
5.2. Demarcation points on the P–I curve of test beams As a key parameter to distinguish the three blast loading regions of P–I curve, the magnitude of nondimensional parameter τ determines the relative locations of Points A and B on the curve in Fig. 7. For a given structural failure criterion, the values of τ1 and τ2 are related to the accuracy requirements. By studying the P–I curve of SDOF system subjected to triangular blast loading, Mays [30] et al. found that the analytic accuracy requirement could be well satisfied when τ1 = 0.4 and τ2 = 40. Li [24] et al. found in the literature [30] that the values of τ1 and τ2 are not only affected by the error precision, but also related to the shape of the blast loads, and the parameter values under different load shapes are given. In this paper, the two demarcation point coordinates on the P–I curve of the test RC beams are calculated by the recommended values in the literature [24] (τ1 = 1.3 and τ2 = 32.4). The impulse of the triangular blast loading has the following relationship with the peak overpressure:
(11)
overpressure asymptotic equation
Y = pe =
Es As h 02
where Es and As is the elastic modulus and area of the reinforcement, respectively; h0 is the effective section height; αE = Es/Ec is the elastic modulus ratio of reinforcement to concrete; ρ is the reinforcement ratio; ψ is the nonuniform coefficient of tensile reinforcement strain between cracks, and γf′ is the Enhancement factor of compression flange. The calculation methods of the parameters ψ and γf′ are detailed in the Code [29]. According to Section 3.3, the yield strains of HRB400 and HTB700 reinforcements under blast loading would be 2598με and 4536με, respectively. The ratio of elastic modulus of the two materials is used to convert the reinforcement area into concrete material, and the equivalent section expressed with concrete is finally obtained. The height of the compression zone x0 = 0.134 m is determined according to the condition that the area moment of the compression zone and tensile zone is equal to the neutral axis. Table 5 gives the impulse and overpressure asymptotes for yield failure of ordinary RC beams and high-strength RC beams according to Formulas (11), (12), (13) and (14).
where, h0 is the effective height of the beam and x0 is depth of compression. Substituting Eq. (7) into Eq. (8), the following expression is obtained at the mid-span cross section of the beam:
ib EJ 0.73EJε = h0 − x 0 ρA
(13)
(12)
where W is the bending-resistance modulus, and the meanings of other symbols are shown above. 5. Establishment of the P–I curves and damage assessment of test beams 5.1. Asymptote of P–I curves of the test beams
i=
The occurrence of cracks in RC flexural members under blast loading will cause their bending stiffness to change continuously, and the resistance function is highly nonlinear [22]. Therefore, the resistance function of under-reinforced beams can be simplified to an ideal elastoplastic model in theoretical analysis, as shown in Fig. 8.
1 pt 2
(15)
According to Formula (1), Formulas (11)–(13) and Formula (15), the coordinates of the demarcation points on the dimensionless P–I curves of the test beams were obtained, as shown in Table 5. 6
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Table 5 The asymptotes and demarcation coordinates of the P–I curves of the test beams . Test beam
Impulse asymptote
Overpressure asymptote
Demarcation point between Zones I and II ia pa
Demarcation point between Zones II and III ib pb
Ordinary RC beam High-strength RC beam
0.0022 0.0039
0.0122 0.0213
0.0022 0.0039
0.0199 0.0348
0.0336 0.0595
Fig. 9. Simplified triangular and rectangular overpressure time-history curves.
0.0122 0.0213
Fig. 10. P–I curves of test RC beams for yield failure.
5.3. Test beams damage assessment
Three points A, B and C are distributed on the right of the demarcation point between the dynamic load region and the quasi-static load region, meaning that the blast loading generated by blast simulator belongs to the quasi-static load. A′ is the corresponding point when the measured blast loading with an peak overpressure of 0.245 MPa is simplified into a rectangular blast loading. From the figure we can find that both points A′ and A are located on the P–I curve of the ordinary RC beam with yield failure. This suggests two ways of load simplification produce exactly same assessment result. Therefore, in the theoretical analysis, the quasi-static load can be further simplified to be rectangular blast loads to evaluate the damage of RC beams.
The overpressure of air shock wave generally decreases exponentially with time. The time-history curves of measured blast overpressure in this paper is shown in Fig. 4. Since the positive duration time was quite long (being about 1.5 s), the overpressure curves are simplified according to the principle of tangent simplification. The simplified triangular overpressure time-history curves are shown in Fig. 9. and its characteristic parameters and coordinates on P–I curves are listed in Table 6. According to data in Table 5, the P–I curves for yield failure of highstrength RC beams and ordinary RC beams are plotted. The hyperbolic square relation is used to fit the dynamic load region [12], as shown in Fig. 10. The three points A, B and C in Fig. 10 represent the explosion tests whose peak overpressure value is 0.245 MPa, 0.372 MPa and 0.477 MPa, respectively. The relative position of the three points and P–I curves represents the failure of components, whereas their relative locations to the demarcation points indicate the type of the blast loading. It can be clearly seen from the figure that point A and point C fall on the P–I curve of ordinary RC beam and high-strength RC beam, respectively, with point B being located between them. It indicates that the yield failure of ordinary RC beams occurs when the peak overpressure value of the blast loading is 0.245 MPa, while that of highstrength RC beams only occurs when the peak overpressure value is 0.477 MPa. When the peak overpressure value is 0.372 MPa, the ordinary RC beam is damaged already, but high-strength RC beam remains undamaged. In the meantime, the P–I curve of high-strength RC beam is located at the upper right of the ordinary RC beam, indicating that the high-strength RC beam can resist larger blast loads than the ordinary RC beam and has better blast resistance performance.
6. Conclusion In this paper, three groups of high-strength RC beams and ordinary RC beams are tested and compared under uniformly distributed blast loading. Through the comparative analysis of the failure characteristics and the reinforcement strain of RC beams, the blast resistance performance and damage mechanism of the high-strength RC beam are obtained. The P–I damage curves of two kinds of RC beams are established with the maximum tensile reinforcement strain as the damage criterion. The damage assessment of RC beams under various test conditions is carried out, and the assessment results are compared with the test results. The main conclusions are as follows: (1) The deformation characteristics of high-strength RC beams are basically the same as that of ordinary RC beams under small peak overpressure blast loading. With the increase of blast loads, high strength reinforcement plays a significant role in reducing component deformation and crack length and width, and can effectively improve the blast resistance performance of RC beams.
Table 6 Characteristic parameters and coordinates on P–I curve of blast loading. Numbering
Peak overpressure p/MPa
Positive duration time t/s
Impulse i/ MPa⋅s
ie
pe
A B C
0.245 0.372 0.477
0.163 0.185 0.191
0.0200 0.0344 0.0456
0.0441 0.0758 0.1000
0.011 0.016 0.021
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International Journal of Impact Engineering 133 (2019) 103362
Z. Liao, et al.
(2) When the peak overpressure of blast loading is 0.245 MPa, the ordinary RC beam is close to yield failure, while the high-strength RC beam begins to yield failure when the peak overpressure value is 0.477 MPa. The damage evaluation results of RC beams according to P–I curves agree well with the experimental results, which indicates that the P–I curves established by using the reinforcement strain of mid-span section as the damage criterion of RC beams can accurately evaluate the failure of the high-strength RC beams under blast loading. (3) When using the P–I curve to evaluate the damage of the RC beams, the quasi-static load can be simplified into a triangular blast load or a rectangular blast load as needed in practical analysis, and the damage evaluation results are identical.
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