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Damage mechanism and mode of square reinforced concrete slab subjected to blast loading
Theoretical and Applied Fracture Mechanics 63–64 (2013) 54–62
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Damage mechanism and mode of square reinforced concrete slab subjected to blast loading C.F. Zhao ⇑, J.Y. Chen Institute of Earthquake Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China
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Article history: Available online 15 March 2013 Keywords: Reinforcement concrete slab Blast loading Arbitrary Lagrange Euler Damage mechanism Numerical simulation
a b s t r a c t Reinforcement concrete is the principle material for military engineering and nuclear power plant. However, impact and explosions could completely destroy such structures, causing tremendous casualties and property loss. Therefore, analyzing the damage mechanism and mode of the structures under blast loading is significant. The objective of this paper is to investigate the dynamics response and damage mechanism of three 1000 mm 1000 mm 400 mm reinforcement concrete slabs subjected to 400 mm standoff distance close-in explosions using LS-DYNA software and compare with experimental tests. A three-dimensional material model including explosive, air and reinforcement concrete slab with separated modeling method taking into account Arbitrary Lagrange–Euler, multiple materials algorithms and fluid–structure coupling interaction, is created to simulate the experiments. The sophisticated concrete and reinforcement bar material models, considering the strain rate effects, dynamic increasing factor and equation of state for concrete, are applied in simulating the damage mechanism and dynamic response. From the numerical results and comparison of the experimental data, it can be seen that the numerical results and experimental data shows a favorable agreement and the present model can still give a reliable prediction of the damage characteristic for the reinforcement concrete slabs. It also observed that the increase of the amount of the explosive can change the failure mode of the slab. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Reinforced concrete is the principal material used for the hardened structures, such as military engineering and containment of nuclear power plants. The behavior, analysis and design of hardened structures for protection against short duration dynamic loading effects, such as that induced by air blast, is a subject of extensive studies in the last decades. The intensive dynamic loading by detonations should be considered in the structural design for both military and civilian structures and facilities in such cases [1,2]. Damage due to blast loading is a general topic that are of interest to researchers in different field. A number of papers conducted in depth studies on the damage of the reinforced concrete structures attributable to air blast loadings and issues regarding projectiles impact on metal target at high velocities have produced in recent years. For example, The work [1] provided an efficient analytical model to obtain pressure-impulse diagram of one way reinforced concrete slabs under different shapes of air blast loading using single degree of freedom method. The work [3] in addressed the scaling of the dynamic response of one-way square reinforced concrete slabs under close-in blast loading. A procedure was inves⇑ Corresponding author. Tel./fax: +86 411 84708549. E-mail address: [email protected] (C.F. Zhao). 0167-8442/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tafmec.2013.03.006
tigated to estimate how the explosive charge weight and standoff distance impose certain levels of damage on reinforced concrete slabs [4]. The work [5] in evaluated the effectiveness of fiber sheet reinforcement on the explosive resistant performance of concrete plate. The propagation law of a blast pressure wave and dynamic response of reinforced concrete slab under explosive pressure wave effect were analyzed by using proper state material parameters and equations and nonlinear finite element analysis software [6]. The response of stiffened slabs subjected to gas explosion was studied by experimental and numerical [7]. A high-fidelity physics based computer program, LS-DYNA were utilized to provide numerical simulation of the dynamic responses and residual axial strength of reinforced concrete columns under short standoff distance blast load [8]. Damage mechanism and response of the reinforced concrete containment structure under internal blast loading were investigated [9]. Fragment size distribution resulting from concrete spallation attributable to air blast loads was estimated [10]. Current design guidelines, such as TM5-1300, on the damage to reinforced concrete slabs provide combinations of estimated explosive charge weights and standoff distances that are likely to generate a certain damage level to concrete [1]. The localized impactor–armor interaction approach was applied to determine the order of the plates in the armor that provided the maximum ballistic resistance of the armor, and the optimal arrangement of the plates in a multi-layered shield was also
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investigated [11,12]. The effect of deviation from a conical shape on ballistic properties of space shield using two-term localized interaction model was investigated. It was found that the effect was insignificant and caused small changes in the magnitude of the ballistic limit velocity and energy absorbed by a shield [13]. Numerical calculation manner was adopted to develop the analytical model that determined the residual velocities and penetration depths of the ogival node projectiles during normal impact with double-layered composite target of varying thicknesses. The analytical model predictions showed well agreement with ballistic tests [14]. However, few papers have been conducted to investigate the damage modes and damage mechanism of one way square reinforced slabs under blast load. The purpose of the paper is to numerical simulate and investigate the concrete damage mode and mechanism of three 1000 mm 1000 mm 40 mm slabs subjected to different explosive charges blast loading located at a 400 mm standoff distance above the slabs. The complicated material and reinforced concrete models are applied in the simulation considering the strain rate effects and the appropriate coupling at the air–solid interface. The criterion technique is applied to capture the fracture and material separation process during the detonation of the explosive. A comparative analysis is performed based on the predicted results from the numerical simulation and the experimental results [3].
2. Experiment setup and structural geometry In the present study and Ref. [3], the reinforced concrete slabs were constructed with a 6 mm diameter bar meshing spaced at a distance of 75 mm from one another in the major bending lane (q = 1.43%) and at a distance of 75 mm from one another in the other plane (q = 1.43%), where q is the reinforcement ratio. the thickness of the concrete cover was 20 mm, the cylinder compressive strength, tensile strength and Young’s modulus of concrete are 39.5 MPa, 4.2 MPa and 28.3 GPa, respectively. the reinforcement had a yield strength of 600 MPa and Young’s modulus of 200 GPa. Fig. 1 shows the geometry and section detail of the test specimens. All the slab specimens have a cross-sectional dimension of 1000 1000 mm and a vertical thickness of 40 mm. The installation arrangement of the specimen test is shown in Fig. 2. TNT explosive is employed in the experiment as it is a standard high explosion and the mass of explosive is 0.2–0.46 kg. As shown in Fig. 2, the explosive charge was suspended above the test specimens at a specific standoff distance of 400 mm, the detail information of the experiment in the literature [3] and Table 1.
Fig. 2. The installation arrangement of the experiment.
Table 1 Experimental program [3]. Test Slab Dimension (mm) T1 T2 T3
S1 S2 S3
Explosive Scale distance Damage level charge (kg) (m/kg1/3)
1000 1000 40 0.2 1000 1000 40 0.31 1000 1000 40 0.46
0.684 0.591 0.518
Low damage Moderate damage Moderate damage
3. LS-DYNA analysis of slab damage under blast load Numerical simulation using finite element model program has become a powerful means in the design process of a structure, as well as in the investigation of physical and failure mechanisms. To simulate and predict concrete spallation more realistically under various charge weights and standoff distances, a three dimensional numerical simulation is adopted necessarily. In the current study, a numerical model is employed to investigate the global effect and local concrete damage and spallation. The explicit nonlinear finite element model program LS-DYNA is used in this study to set up a numerical model to investigate the global structural effect and local concrete spallation as its proven effectiveness in geometric modeling and impact analysis. The description of modeling includes relevant material models and comparison of numerical results and experimental results. 3.1. Material model for concrete To study the interactive mechanism and dynamic response of reinforced concrete slab under blast loads, a proper and reliable
Fig. 1. Geometry of the reinforcement concrete slab.
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Fig. 5. Comparison between the present compaction model and experimental results in the literature [19] for normal concrete of 50 MPa.
Fig. 3. Pressure versus volumetric curve.
dynamic damage model that reflects the characteristics of the concrete material behavior at high strain rate is needed. The finite element code LS-DYNA, which is used in this study, contains several material models that can be used to represent concrete, namely, material type 5 (Soil and Crushable Foam), material type 14 (Soil and Crushable Foam Failure), material type 16 (Pseudo Tensor), material 25 (Geological Cap Model), material type 72 (Concrete Damage), material 84 (Winfrith Concrete), and type 96 (Brittle Damage). Material type 72RW3 (Mat_Concrete_Damage_Rel3), is the third release of Karagozian and Case (K&C) concrete model. It is a plasticity-based model, using three shear failure surfaces and including damage and strain-rate effect. The model has a default parameter generation function based on the unconfined compressive strength of the concrete [8,15–17]. In this model, the stress tensor is expressed as the sum of the hydrostatic stress tensor and the deviatoric stress tensor. The hydrostatic tensor changes the concrete volume and the deviatoric stress tensor controls the shape deformation. For the hydrostatic stress tensor, the compaction model is a multiple approximation in internal energy. Pressure is defined as:
p ¼ Cðem Þ þ cTðem ÞE
ð1Þ
where E is the initial energy per initial volume, c is the ratio of specific heats. The volume strain (ev) is given by the natural logarithm of the relative volume. As shown in Fig. 3 the model contains an elastic limit. When tension is greater than the hydrostatic tension cut-off, tensor failure occurs. When the volumetric strain exceeds the elastic limit, compaction occurs and the concrete turns into a granular kind of material. The bulk unloading modulus is a function of volumetric strain. Uploading occurs along the uploading bulk
modulus to the pressure cut-off. Reloading always follows the unloading path to the point where uploading began and continue on the loading path [8]. A three-curve model is adopted to analyze the deviatoric stress tensor, as shown in Fig. 4, where the upper curve represents the maximum strength curve, the middle curve is the initial yield strength curve and the lower curve is the failed material residual strength curve. As known to all, when the reinforced concrete structures are subjected to blast loads, both concrete and steel may respond at very high strain rate of up to an order of 10–1000 s1 or even higher. At such strain rates, the apparent strength of these materials can increase significantly by more than 100% for concrete in compression, and more than 600% in tension for concrete, by more than 50% for the reinforcing steel [18]. Therefore, the strain rate effect for concrete and steel needs be considered for the reliable simulation of structural response to blast loads. In order to consider the fact that under higher loading rates concrete exhibited increased strength, a dynamic increase factor (DIF), namely the ratio of the dynamic-to-static strength versus strain rate, is employed in this study. The expressions proposed by Malvar and Crawford are utilized [2,19]. The DIF of compressive strength is given by CEB expressed as follows [2]:
8 1:026a < e_ d 0 ; e_ d 6 30 s1 fcd e_ s CDIF ¼ 0 ¼ 1 fcs : cðe_ d Þ3 ; e_ d > 30 s1
ð2Þ
0 where fcd is the dynamic compressive strength at the strain rate e_ d , e_ s ¼ 30 106 s1 , log c = 6.15a 0.492, a = (5 + 3fcu/4)1, fcu0 is the
Fig. 4. Strength model for concrete [15]: (a) failure surfaces in concrete material model and (b) concrete constitutive model.
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Fig. 9. 1/4 Three-dimensional numerical model.
Fig. 6. DIF for the tensile strength of concrete [19].
Fig. 10. Reinforcement bar arrangement.
Fig. 7. Dynamic increase factor of reinforcement bar.
static compressive strength and fcu is the static cube compressive strength in MPa. Fig. 5 shows the comparison between the experimental results and the CEB empirical results in literature [20] for normal concrete of 50 MPa. CEB also recommends the use of the DIF to determine the tensile strength of concrete. However, it was noted during comparing available experimental results with the CEB empirical results, that the tensile DIF recommended by CEB was substantially underestimated as illustrated in Fig. 6. For this
reason, the modified CEB formulation [21] is applied to estimate the DIF of tensile strength as follows:
8 d > e_ d ftd < e_ s ; TDIF ¼ ¼ > e_ d 13 fts : b ; e_ s
e_ d 6 1 s1 ð3Þ
e_ d > 1 s1
where ftd is the dynamic tensile strength at the strain rate e_ d , e_ s ¼ 106 s1 , log b = 6d 2, d ¼ ð1 þ 8fc0 =fco0 Þ1 , fc0 is the static 0 compressive strength in MPa and fco ¼ 10 MPa. The strain rate effects in Eqs. (2) and (3) are only in the strain rate range of 106–1000 s1, DIFs for 1000 s1 are used to avoid overestimation
Fig. 8. 1/2 Three-dimensional finite element model.
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mesh. In current study, when the concrete response in an element reaches principle tensile strains exceed 10%, the element is immediately failure [16]. 3.2. Material model for reinforcement bar
Fig. 11. Pressure contours for half model.
of the strain rate effect. Once the DIFs are calculated, a dynamic compressive strength and a dynamic tensile strength are obtained by multiplying the DIF with the respective static strength. The element erosion function, while not a material property or physics based phenomena provide a powerful method to simulate the spall of concrete and produce response plots that provide a more realistic graphical representation of the actual blast event. The erosion feature is characterized by a physical separation of the eroded solid element from the rest of the
The Reinforcement steel within the slab is represented by MATPLASTIC_KINEMATIC material model from LS-DYNA in this study, which is a strain sensitive uniaxial elastic–plastic material to account for its strain rate sensitivity as well as stress–strain history dependence. The expressions proposed by Malvar on strain rate effect are applied in this paper [21]. Fig. 7 shows this proposed DIF for both yield and ultimate stress. The DIF for steel can be expressed as follows:
DIF ¼
e_
a ð4Þ
104
where for the yield stress, a = afy is expressed as follows:
afy ¼ 0:074 0:040
fy 414
ð5Þ
and for the ultimate stress, a = afu is expressed as follows:
afu ¼ 0:019 0:009
fy 414
ð6Þ
Fig. 12. Comparison of the numerical and experimental results for reinforcement concrete slab S1: (a)numerical results on the upper surface, (b) experimental results on the upper surface [3], (c) numerical results on the bottom surface and (d) experimental results on the bottom surface[3].
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where the strain rate equation is in s1 and fy is the yield stress in MPa. Eq. (4) is valid with yield stresses between 290 and 710 MPa, and for strain rates between 104 and 225 s1. 3.3. Material model for explosive and air The Arbitrary–Lagrange–Euler (ALE) approach was utilized to model the interface between the air and structure. LS-DYNA applies a fluid–structure coupled algorithm to allow an optimum numerical simulation. Using this approach, different domains of physical problem such as structures and fluids can be modeled simultaneously using Lagrange and Euler approaches. These different domains are then coupled together in space and time. The features make this computer program especially suitable for the study of interaction problems involving multiple materials of fluids and structures. In this numerical model, high explosives (TNT) are typically modeled by using the Jones–Wilkins–Lee (JWL) equation of state (EOS), which models the pressure generated from the chemical energy in an explosion. It can be expressed as follows:
x R1 V x R2 V xE P ¼A 1 þB 1 þ e e V R1 V R2 V
ð7Þ
where A, B are linear explosion parameters; x, R1 and R2 are nonlinear explosion parameters; V is relative volume and E is specific
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internal energy of every unit of mass, P is the pressure of the detonation products of high explosives. In the present simulation, for a TNT explosive charge, A, B, R1, R2, E and x are 373.77 GPa, 3.7471 GPa, 4.15, 0.9, 4905 kJ/kg and 0.35, respectively.Air is assumed to ideal gas that modeled by linear-polynomial EOS and linear in internal energy. The pressure related to the energy can be expressed as follows:
P ¼ C 0 þ C 1 l þ C 2 l2 þ C 3 l3 þ ðC 4 þ C 5 l þ C 6 l2 ÞE0
ð8Þ
where E0 is the specific initial energy, and l = q/q0 1, Ci (i = 0–6) are the coefficients. For the ideal gases, the coefficients in the EOS are setting as C0 = C1 = C2 = C3 = C6 = 0, and C4 = C5 = c 1, c is the polytropic ratio of specific heats. The pressure is then given by:
P ¼ ðc 1Þ
q E q0 0
ð9Þ
where q/q0 the ratio of current density to reference density, c is the ratio of specific heats, q0 is the initial density of air, and q is the current density of air. E0 is the specific initial energy, with the gamma law EOS under standard atmospheric pressure and c = 1.4, its initial energy is E0 = 2.5 105 J/kg.
Fig. 13. Comparison of the numerical and experimental results for reinforcement concrete slab S2: (a) numerical results on the upper surface, (b) experimental results on the upper surface [3], (c) numerical results on the bottom surface and (d) experimental results on the bottom surface[3].
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4. Numerical simulation 4.1. Numerical model In the numerical simulation, the initial detonation and blast wave propagation with an axial symmetric three dimensional model as shown in Fig. 8. As symmetry of the structure, only 1/4 of the slab is considered. Figs. 9–11 show the 1/4 three-dimensional model and pressure contour of the half model, respectively. In this study, multiple materials coupling algorithm is applied to simulate reinforcement concrete slab under blast loading. In the numerical model, the concrete slab is modeled by Lagrange mesh, in which the coordinates moves with material; while the air and explosive are modeled by Euler mesh, in which the mesh is fixed and material flows through it. The Euler Lagrange interface interaction is considered. The Lagrange mesh imposes a geometric constraint on the Euler mesh whereas the Euler mesh provides a pressure boundary to the Lagrange mesh. The boundary condition of the Euler mesh is set as an outflow boundary [22]. In the supporting area of the slab, upper and back supports are created, and all the displacements of the supports are fixed. Solid elements with single integration point are used to model concrete and reinforcement bar. As a result, the reinforcement bar and concrete are linked by shared nodes and they are assumed to be perfectly
bonded without any slip. In the central part of the model, which is within the area of 500 500 mm2, the element size is 5 5 5 mm3 for concrete, air and explosive, the element size for reinforcement bar is 5 mm, sufficiently small to obtain reliable results. The element number for the reinforcement concrete slab is 81,600 whereas the element number is 326,700 for the air and explosive. The simulation stopped at 6 ms because the residual velocities and stress in the slab are extremely small. 4.2. Model calibration It is said in the paper [23], ‘‘every finite method has methodical inherent errors of discretization in space and time, namely, mesh size sensitively and timestep’’. The convergence theorem is ‘‘If a partial differential equation is discretized by means if finite elements or finite differences, the numerical solution must monotonically converge to the exact solution if and only if the element size and the timestep rend to zero’’. In the present numerical simulation, the minimum mesh size has been employed as much as possible, but further reduction of mesh size will cause memory overflow owing to the computer used. the accuracy of the numerical model is checked by comparing modeled peak pressure with that obtained from the empirical method in TM5-1300 [22,24]. From TM5-1300, the peak pressure for TNT charge case can be
Fig. 14. Comparison of the numerical and experimental results for reinforcement concrete slab S3: (a) numerical results on the upper surface, (b) experimental results on the upper surface [3], (c) numerical results on the bottom surface and (d) experimental results on the bottom surface [3].
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estimated as 20.7458 MPa, while in the numerical simulation, the recorded peak pressure is 21.097 MPa. The result agrees well with the empirical prediction.
4.3. Comparison between numerical and experimental results Fig. 12 shows the comparison of the damage areas obtained from the numerical simulation with those from the reinforced concrete slab S1 under a 0.2 kg TNT charge in the blast test of the paper [3]. From Fig. 12a, it can be found that there is no evident damage at the center of the slab upper surface. It also can be seen from the Fig. 12b that only a small indentation in the center of the back surface which is caused by the high pressure from the explosion and the fixed support. A small damage area on the back surface of the slab is shown in Fig. 12c, while in Fig. 12d there is a small crack in the center of the back surface of the slab. In the simulation, the damage part at the center is slightly greater than the experiment during to the solid fixed support. Fig. 13 shows the comparison of the damage areas obtained from the numerical simulation with those from the reinforcement concrete slab test T2 under 0.31 kg TNT in the paper [3]. The numerical results (Fig. 13a) show that there is some damage on the upper surface of the slab, the damage area is consistent with experiment test that shown in Fig. 12b. In the numerical simulation, on the bottom surface of the reinforcement concrete slab, there is also some tensile damage area during to the tension by the blast loading can be found in Fig. 13c. Similarly Fig. 13d shows the spalling crater damage on the bottom surface of the slab for the experiment test and the radius of the damage area is approximately 90 mm. However, the radius of the damage area in the numerical simulation is about 100 mm, which is slightly greater than the experimental damage area, the slab exhibits moderate damage. Fig. 14 shows the comparison between the numerical and test results of reinforcement concrete slab under 0.46 kg TNT. The numerical result in Fig. 14a shows that there is a square damage area at the center of the slab, the test result in Fig. 14b shows that some cracks and damage area are observed on the upper surface of the slab, which is consistent with those numerical results. Similarly, it can be found the calculated damaged area on the bottom of slab subjected blast loading from Fig. 14c. Spalling occurred on the bottom of the slab in the test is shown in Fig. 14d, which is consistent with the numerical results, and the damage area of the experiment is approximately 45 216 mm2. But in the numerical simulation, the damage area of the slab is about 53 066 mm2, which is slightly greater than the experimental damage area. The slab also exhibits moderate damage. Form the numerical and experimental results of the slab, it can be found that the damage of the slab calculated by numerical simulation is slightly greater than the experiments. The reason of these may be the relatively softer conditions in the experimental test, which makes the real frequency of the slab lower than the model, while the boundary condition of the slab in the numerical simulation is stronger. Another factor may be the material con-
Table 2 Comparison results between numerical simulations and experimental results. Test
T1 T2 T3
Slab
S1 S2 S3
Experiment [3]
Numerical simulation
Central deflection (mm)
Damage radius (mm)
Central deflection (mm)
Damage radius (mm)
10 15 35
– 90 120
8.8 12.7 31.1
– 100 130
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stants, which are not based on the real material tests, but rather are based on the reasonable assumptions. Table 2 shows the comparison results of deflections between the numerical simulation and experimental test. The central defections of the slab obtained from numerical results are less than those from the experiments. The difference of the deflections may be caused by the boundary condition, which is fixed at all the time in the numerical simulation, whereas loosened in the experiment after the explosion. The damage areas of the slab in the numerical simulations are slightly greater than those in the experiment, which may be due to the material constants. However, it should be noted that the differences caused by the boundary condition is insignificant due to the initial stress wave propagation in the slab. As a whole, the numerical results show a favorable agreement with the experimental tests in terms of fracture spreading and the damage area. Therefore, the numerical results are credible. 5. Conclusion A three-dimensional material model including explosive, air and reinforcement concrete slab with separated modeling method using nonlinear finite element analysis software LS-DYNA has been employed in this study. By using this model, three numerical simulations of the one-way square reinforced concrete slabs subjected to close-in blast loading from different TNT charge mass detonations has been carried out and compared with the blast tests in literature[3]. The sophisticated concrete and reinforcement bar material model considering the strain rate effects, dynamic increase factor, Arbitrary Lagrange–Euler and fluid–structure coupling model were applied into simulate the damage mechanism and dynamic response. From the numerical results and comparison of the experiment in Ref. [3], it can be seen that the present model can still give a reliable prediction of the damage characteristic for the reinforcement concrete slabs. For the reinforcement concrete slab dynamic response and the damage process show that under close-in blast pressure effect, tensile failure and spalling may occur on the bottom center of the slab due to the shock wave converted into tensile wave because of the concrete low tensile strength. It also observed that the increase of the amount of the explosive can change the failure mode of the slab. Acknowledgements This study was financial supported by the Specialized Research Fund for Doctoral Program of Higher Education Institutions China (Grant No. 20110041110012) and the State Key Program of National Natural Science of China (Grant No. 51138001). References [1] W. Wang, D. Zhang, F. Lu, S.-C. Wang, F. Tang, Experimental study and numerical simulation of the damage mode of a square reinforced concrete slab under close-in explosion, Eng. Fail. Anal. 27 (2013) 41–51. [2] S.H.P.P.H. Bischoff, Compressive behaviour of concrete at high strain rates, Mater. Struct. 24 (1991) 425–450. [3] W. Wang, D. Zhang, F. Lu, S. Wang, F. Tang, 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. 49 (2012) 158–164. [4] P.F. Silva, B. Lu, Improving the blast resistance capacity of RC slabs with innovative composite materials, Compos. Part B: Eng. 38 (2007) 523–534. [5] K. Ohkubo, M. Beppu, T. Ohno, K. Satoh, Experimental study on the effectiveness of fiber sheet reinforcement on the explosive-resistant performance of concrete plates, Int. J. Impact Eng. 35 (2008) 1702–1708. [6] Y.S. Tai, T.L. Chu, H.T. Hu, J.Y. Wu, Dynamic response of a reinforced concrete slab subjected to air blast load, Theoret. Appl. Fract. Mech. 56 (2011) 140–147.
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Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Damage mechanism and mode of square reinforced concrete slab subjected to blast loading C.F. Zhao ⇑, J.Y. Chen Institute of Earthquake Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China
a r t i c l e
i n f o
Article history: Available online 15 March 2013 Keywords: Reinforcement concrete slab Blast loading Arbitrary Lagrange Euler Damage mechanism Numerical simulation
a b s t r a c t Reinforcement concrete is the principle material for military engineering and nuclear power plant. However, impact and explosions could completely destroy such structures, causing tremendous casualties and property loss. Therefore, analyzing the damage mechanism and mode of the structures under blast loading is significant. The objective of this paper is to investigate the dynamics response and damage mechanism of three 1000 mm 1000 mm 400 mm reinforcement concrete slabs subjected to 400 mm standoff distance close-in explosions using LS-DYNA software and compare with experimental tests. A three-dimensional material model including explosive, air and reinforcement concrete slab with separated modeling method taking into account Arbitrary Lagrange–Euler, multiple materials algorithms and fluid–structure coupling interaction, is created to simulate the experiments. The sophisticated concrete and reinforcement bar material models, considering the strain rate effects, dynamic increasing factor and equation of state for concrete, are applied in simulating the damage mechanism and dynamic response. From the numerical results and comparison of the experimental data, it can be seen that the numerical results and experimental data shows a favorable agreement and the present model can still give a reliable prediction of the damage characteristic for the reinforcement concrete slabs. It also observed that the increase of the amount of the explosive can change the failure mode of the slab. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Reinforced concrete is the principal material used for the hardened structures, such as military engineering and containment of nuclear power plants. The behavior, analysis and design of hardened structures for protection against short duration dynamic loading effects, such as that induced by air blast, is a subject of extensive studies in the last decades. The intensive dynamic loading by detonations should be considered in the structural design for both military and civilian structures and facilities in such cases [1,2]. Damage due to blast loading is a general topic that are of interest to researchers in different field. A number of papers conducted in depth studies on the damage of the reinforced concrete structures attributable to air blast loadings and issues regarding projectiles impact on metal target at high velocities have produced in recent years. For example, The work [1] provided an efficient analytical model to obtain pressure-impulse diagram of one way reinforced concrete slabs under different shapes of air blast loading using single degree of freedom method. The work [3] in addressed the scaling of the dynamic response of one-way square reinforced concrete slabs under close-in blast loading. A procedure was inves⇑ Corresponding author. Tel./fax: +86 411 84708549. E-mail address: [email protected] (C.F. Zhao). 0167-8442/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tafmec.2013.03.006
tigated to estimate how the explosive charge weight and standoff distance impose certain levels of damage on reinforced concrete slabs [4]. The work [5] in evaluated the effectiveness of fiber sheet reinforcement on the explosive resistant performance of concrete plate. The propagation law of a blast pressure wave and dynamic response of reinforced concrete slab under explosive pressure wave effect were analyzed by using proper state material parameters and equations and nonlinear finite element analysis software [6]. The response of stiffened slabs subjected to gas explosion was studied by experimental and numerical [7]. A high-fidelity physics based computer program, LS-DYNA were utilized to provide numerical simulation of the dynamic responses and residual axial strength of reinforced concrete columns under short standoff distance blast load [8]. Damage mechanism and response of the reinforced concrete containment structure under internal blast loading were investigated [9]. Fragment size distribution resulting from concrete spallation attributable to air blast loads was estimated [10]. Current design guidelines, such as TM5-1300, on the damage to reinforced concrete slabs provide combinations of estimated explosive charge weights and standoff distances that are likely to generate a certain damage level to concrete [1]. The localized impactor–armor interaction approach was applied to determine the order of the plates in the armor that provided the maximum ballistic resistance of the armor, and the optimal arrangement of the plates in a multi-layered shield was also
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investigated [11,12]. The effect of deviation from a conical shape on ballistic properties of space shield using two-term localized interaction model was investigated. It was found that the effect was insignificant and caused small changes in the magnitude of the ballistic limit velocity and energy absorbed by a shield [13]. Numerical calculation manner was adopted to develop the analytical model that determined the residual velocities and penetration depths of the ogival node projectiles during normal impact with double-layered composite target of varying thicknesses. The analytical model predictions showed well agreement with ballistic tests [14]. However, few papers have been conducted to investigate the damage modes and damage mechanism of one way square reinforced slabs under blast load. The purpose of the paper is to numerical simulate and investigate the concrete damage mode and mechanism of three 1000 mm 1000 mm 40 mm slabs subjected to different explosive charges blast loading located at a 400 mm standoff distance above the slabs. The complicated material and reinforced concrete models are applied in the simulation considering the strain rate effects and the appropriate coupling at the air–solid interface. The criterion technique is applied to capture the fracture and material separation process during the detonation of the explosive. A comparative analysis is performed based on the predicted results from the numerical simulation and the experimental results [3].
2. Experiment setup and structural geometry In the present study and Ref. [3], the reinforced concrete slabs were constructed with a 6 mm diameter bar meshing spaced at a distance of 75 mm from one another in the major bending lane (q = 1.43%) and at a distance of 75 mm from one another in the other plane (q = 1.43%), where q is the reinforcement ratio. the thickness of the concrete cover was 20 mm, the cylinder compressive strength, tensile strength and Young’s modulus of concrete are 39.5 MPa, 4.2 MPa and 28.3 GPa, respectively. the reinforcement had a yield strength of 600 MPa and Young’s modulus of 200 GPa. Fig. 1 shows the geometry and section detail of the test specimens. All the slab specimens have a cross-sectional dimension of 1000 1000 mm and a vertical thickness of 40 mm. The installation arrangement of the specimen test is shown in Fig. 2. TNT explosive is employed in the experiment as it is a standard high explosion and the mass of explosive is 0.2–0.46 kg. As shown in Fig. 2, the explosive charge was suspended above the test specimens at a specific standoff distance of 400 mm, the detail information of the experiment in the literature [3] and Table 1.
Fig. 2. The installation arrangement of the experiment.
Table 1 Experimental program [3]. Test Slab Dimension (mm) T1 T2 T3
S1 S2 S3
Explosive Scale distance Damage level charge (kg) (m/kg1/3)
1000 1000 40 0.2 1000 1000 40 0.31 1000 1000 40 0.46
0.684 0.591 0.518
Low damage Moderate damage Moderate damage
3. LS-DYNA analysis of slab damage under blast load Numerical simulation using finite element model program has become a powerful means in the design process of a structure, as well as in the investigation of physical and failure mechanisms. To simulate and predict concrete spallation more realistically under various charge weights and standoff distances, a three dimensional numerical simulation is adopted necessarily. In the current study, a numerical model is employed to investigate the global effect and local concrete damage and spallation. The explicit nonlinear finite element model program LS-DYNA is used in this study to set up a numerical model to investigate the global structural effect and local concrete spallation as its proven effectiveness in geometric modeling and impact analysis. The description of modeling includes relevant material models and comparison of numerical results and experimental results. 3.1. Material model for concrete To study the interactive mechanism and dynamic response of reinforced concrete slab under blast loads, a proper and reliable
Fig. 1. Geometry of the reinforcement concrete slab.
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Fig. 5. Comparison between the present compaction model and experimental results in the literature [19] for normal concrete of 50 MPa.
Fig. 3. Pressure versus volumetric curve.
dynamic damage model that reflects the characteristics of the concrete material behavior at high strain rate is needed. The finite element code LS-DYNA, which is used in this study, contains several material models that can be used to represent concrete, namely, material type 5 (Soil and Crushable Foam), material type 14 (Soil and Crushable Foam Failure), material type 16 (Pseudo Tensor), material 25 (Geological Cap Model), material type 72 (Concrete Damage), material 84 (Winfrith Concrete), and type 96 (Brittle Damage). Material type 72RW3 (Mat_Concrete_Damage_Rel3), is the third release of Karagozian and Case (K&C) concrete model. It is a plasticity-based model, using three shear failure surfaces and including damage and strain-rate effect. The model has a default parameter generation function based on the unconfined compressive strength of the concrete [8,15–17]. In this model, the stress tensor is expressed as the sum of the hydrostatic stress tensor and the deviatoric stress tensor. The hydrostatic tensor changes the concrete volume and the deviatoric stress tensor controls the shape deformation. For the hydrostatic stress tensor, the compaction model is a multiple approximation in internal energy. Pressure is defined as:
p ¼ Cðem Þ þ cTðem ÞE
ð1Þ
where E is the initial energy per initial volume, c is the ratio of specific heats. The volume strain (ev) is given by the natural logarithm of the relative volume. As shown in Fig. 3 the model contains an elastic limit. When tension is greater than the hydrostatic tension cut-off, tensor failure occurs. When the volumetric strain exceeds the elastic limit, compaction occurs and the concrete turns into a granular kind of material. The bulk unloading modulus is a function of volumetric strain. Uploading occurs along the uploading bulk
modulus to the pressure cut-off. Reloading always follows the unloading path to the point where uploading began and continue on the loading path [8]. A three-curve model is adopted to analyze the deviatoric stress tensor, as shown in Fig. 4, where the upper curve represents the maximum strength curve, the middle curve is the initial yield strength curve and the lower curve is the failed material residual strength curve. As known to all, when the reinforced concrete structures are subjected to blast loads, both concrete and steel may respond at very high strain rate of up to an order of 10–1000 s1 or even higher. At such strain rates, the apparent strength of these materials can increase significantly by more than 100% for concrete in compression, and more than 600% in tension for concrete, by more than 50% for the reinforcing steel [18]. Therefore, the strain rate effect for concrete and steel needs be considered for the reliable simulation of structural response to blast loads. In order to consider the fact that under higher loading rates concrete exhibited increased strength, a dynamic increase factor (DIF), namely the ratio of the dynamic-to-static strength versus strain rate, is employed in this study. The expressions proposed by Malvar and Crawford are utilized [2,19]. The DIF of compressive strength is given by CEB expressed as follows [2]:
8 1:026a < e_ d 0 ; e_ d 6 30 s1 fcd e_ s CDIF ¼ 0 ¼ 1 fcs : cðe_ d Þ3 ; e_ d > 30 s1
ð2Þ
0 where fcd is the dynamic compressive strength at the strain rate e_ d , e_ s ¼ 30 106 s1 , log c = 6.15a 0.492, a = (5 + 3fcu/4)1, fcu0 is the
Fig. 4. Strength model for concrete [15]: (a) failure surfaces in concrete material model and (b) concrete constitutive model.
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Fig. 9. 1/4 Three-dimensional numerical model.
Fig. 6. DIF for the tensile strength of concrete [19].
Fig. 10. Reinforcement bar arrangement.
Fig. 7. Dynamic increase factor of reinforcement bar.
static compressive strength and fcu is the static cube compressive strength in MPa. Fig. 5 shows the comparison between the experimental results and the CEB empirical results in literature [20] for normal concrete of 50 MPa. CEB also recommends the use of the DIF to determine the tensile strength of concrete. However, it was noted during comparing available experimental results with the CEB empirical results, that the tensile DIF recommended by CEB was substantially underestimated as illustrated in Fig. 6. For this
reason, the modified CEB formulation [21] is applied to estimate the DIF of tensile strength as follows:
8 d > e_ d ftd < e_ s ; TDIF ¼ ¼ > e_ d 13 fts : b ; e_ s
e_ d 6 1 s1 ð3Þ
e_ d > 1 s1
where ftd is the dynamic tensile strength at the strain rate e_ d , e_ s ¼ 106 s1 , log b = 6d 2, d ¼ ð1 þ 8fc0 =fco0 Þ1 , fc0 is the static 0 compressive strength in MPa and fco ¼ 10 MPa. The strain rate effects in Eqs. (2) and (3) are only in the strain rate range of 106–1000 s1, DIFs for 1000 s1 are used to avoid overestimation
Fig. 8. 1/2 Three-dimensional finite element model.
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mesh. In current study, when the concrete response in an element reaches principle tensile strains exceed 10%, the element is immediately failure [16]. 3.2. Material model for reinforcement bar
Fig. 11. Pressure contours for half model.
of the strain rate effect. Once the DIFs are calculated, a dynamic compressive strength and a dynamic tensile strength are obtained by multiplying the DIF with the respective static strength. The element erosion function, while not a material property or physics based phenomena provide a powerful method to simulate the spall of concrete and produce response plots that provide a more realistic graphical representation of the actual blast event. The erosion feature is characterized by a physical separation of the eroded solid element from the rest of the
The Reinforcement steel within the slab is represented by MATPLASTIC_KINEMATIC material model from LS-DYNA in this study, which is a strain sensitive uniaxial elastic–plastic material to account for its strain rate sensitivity as well as stress–strain history dependence. The expressions proposed by Malvar on strain rate effect are applied in this paper [21]. Fig. 7 shows this proposed DIF for both yield and ultimate stress. The DIF for steel can be expressed as follows:
DIF ¼
e_
a ð4Þ
104
where for the yield stress, a = afy is expressed as follows:
afy ¼ 0:074 0:040
fy 414
ð5Þ
and for the ultimate stress, a = afu is expressed as follows:
afu ¼ 0:019 0:009
fy 414
ð6Þ
Fig. 12. Comparison of the numerical and experimental results for reinforcement concrete slab S1: (a)numerical results on the upper surface, (b) experimental results on the upper surface [3], (c) numerical results on the bottom surface and (d) experimental results on the bottom surface[3].
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where the strain rate equation is in s1 and fy is the yield stress in MPa. Eq. (4) is valid with yield stresses between 290 and 710 MPa, and for strain rates between 104 and 225 s1. 3.3. Material model for explosive and air The Arbitrary–Lagrange–Euler (ALE) approach was utilized to model the interface between the air and structure. LS-DYNA applies a fluid–structure coupled algorithm to allow an optimum numerical simulation. Using this approach, different domains of physical problem such as structures and fluids can be modeled simultaneously using Lagrange and Euler approaches. These different domains are then coupled together in space and time. The features make this computer program especially suitable for the study of interaction problems involving multiple materials of fluids and structures. In this numerical model, high explosives (TNT) are typically modeled by using the Jones–Wilkins–Lee (JWL) equation of state (EOS), which models the pressure generated from the chemical energy in an explosion. It can be expressed as follows:
x R1 V x R2 V xE P ¼A 1 þB 1 þ e e V R1 V R2 V
ð7Þ
where A, B are linear explosion parameters; x, R1 and R2 are nonlinear explosion parameters; V is relative volume and E is specific
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internal energy of every unit of mass, P is the pressure of the detonation products of high explosives. In the present simulation, for a TNT explosive charge, A, B, R1, R2, E and x are 373.77 GPa, 3.7471 GPa, 4.15, 0.9, 4905 kJ/kg and 0.35, respectively.Air is assumed to ideal gas that modeled by linear-polynomial EOS and linear in internal energy. The pressure related to the energy can be expressed as follows:
P ¼ C 0 þ C 1 l þ C 2 l2 þ C 3 l3 þ ðC 4 þ C 5 l þ C 6 l2 ÞE0
ð8Þ
where E0 is the specific initial energy, and l = q/q0 1, Ci (i = 0–6) are the coefficients. For the ideal gases, the coefficients in the EOS are setting as C0 = C1 = C2 = C3 = C6 = 0, and C4 = C5 = c 1, c is the polytropic ratio of specific heats. The pressure is then given by:
P ¼ ðc 1Þ
q E q0 0
ð9Þ
where q/q0 the ratio of current density to reference density, c is the ratio of specific heats, q0 is the initial density of air, and q is the current density of air. E0 is the specific initial energy, with the gamma law EOS under standard atmospheric pressure and c = 1.4, its initial energy is E0 = 2.5 105 J/kg.
Fig. 13. Comparison of the numerical and experimental results for reinforcement concrete slab S2: (a) numerical results on the upper surface, (b) experimental results on the upper surface [3], (c) numerical results on the bottom surface and (d) experimental results on the bottom surface[3].
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4. Numerical simulation 4.1. Numerical model In the numerical simulation, the initial detonation and blast wave propagation with an axial symmetric three dimensional model as shown in Fig. 8. As symmetry of the structure, only 1/4 of the slab is considered. Figs. 9–11 show the 1/4 three-dimensional model and pressure contour of the half model, respectively. In this study, multiple materials coupling algorithm is applied to simulate reinforcement concrete slab under blast loading. In the numerical model, the concrete slab is modeled by Lagrange mesh, in which the coordinates moves with material; while the air and explosive are modeled by Euler mesh, in which the mesh is fixed and material flows through it. The Euler Lagrange interface interaction is considered. The Lagrange mesh imposes a geometric constraint on the Euler mesh whereas the Euler mesh provides a pressure boundary to the Lagrange mesh. The boundary condition of the Euler mesh is set as an outflow boundary [22]. In the supporting area of the slab, upper and back supports are created, and all the displacements of the supports are fixed. Solid elements with single integration point are used to model concrete and reinforcement bar. As a result, the reinforcement bar and concrete are linked by shared nodes and they are assumed to be perfectly
bonded without any slip. In the central part of the model, which is within the area of 500 500 mm2, the element size is 5 5 5 mm3 for concrete, air and explosive, the element size for reinforcement bar is 5 mm, sufficiently small to obtain reliable results. The element number for the reinforcement concrete slab is 81,600 whereas the element number is 326,700 for the air and explosive. The simulation stopped at 6 ms because the residual velocities and stress in the slab are extremely small. 4.2. Model calibration It is said in the paper [23], ‘‘every finite method has methodical inherent errors of discretization in space and time, namely, mesh size sensitively and timestep’’. The convergence theorem is ‘‘If a partial differential equation is discretized by means if finite elements or finite differences, the numerical solution must monotonically converge to the exact solution if and only if the element size and the timestep rend to zero’’. In the present numerical simulation, the minimum mesh size has been employed as much as possible, but further reduction of mesh size will cause memory overflow owing to the computer used. the accuracy of the numerical model is checked by comparing modeled peak pressure with that obtained from the empirical method in TM5-1300 [22,24]. From TM5-1300, the peak pressure for TNT charge case can be
Fig. 14. Comparison of the numerical and experimental results for reinforcement concrete slab S3: (a) numerical results on the upper surface, (b) experimental results on the upper surface [3], (c) numerical results on the bottom surface and (d) experimental results on the bottom surface [3].
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estimated as 20.7458 MPa, while in the numerical simulation, the recorded peak pressure is 21.097 MPa. The result agrees well with the empirical prediction.
4.3. Comparison between numerical and experimental results Fig. 12 shows the comparison of the damage areas obtained from the numerical simulation with those from the reinforced concrete slab S1 under a 0.2 kg TNT charge in the blast test of the paper [3]. From Fig. 12a, it can be found that there is no evident damage at the center of the slab upper surface. It also can be seen from the Fig. 12b that only a small indentation in the center of the back surface which is caused by the high pressure from the explosion and the fixed support. A small damage area on the back surface of the slab is shown in Fig. 12c, while in Fig. 12d there is a small crack in the center of the back surface of the slab. In the simulation, the damage part at the center is slightly greater than the experiment during to the solid fixed support. Fig. 13 shows the comparison of the damage areas obtained from the numerical simulation with those from the reinforcement concrete slab test T2 under 0.31 kg TNT in the paper [3]. The numerical results (Fig. 13a) show that there is some damage on the upper surface of the slab, the damage area is consistent with experiment test that shown in Fig. 12b. In the numerical simulation, on the bottom surface of the reinforcement concrete slab, there is also some tensile damage area during to the tension by the blast loading can be found in Fig. 13c. Similarly Fig. 13d shows the spalling crater damage on the bottom surface of the slab for the experiment test and the radius of the damage area is approximately 90 mm. However, the radius of the damage area in the numerical simulation is about 100 mm, which is slightly greater than the experimental damage area, the slab exhibits moderate damage. Fig. 14 shows the comparison between the numerical and test results of reinforcement concrete slab under 0.46 kg TNT. The numerical result in Fig. 14a shows that there is a square damage area at the center of the slab, the test result in Fig. 14b shows that some cracks and damage area are observed on the upper surface of the slab, which is consistent with those numerical results. Similarly, it can be found the calculated damaged area on the bottom of slab subjected blast loading from Fig. 14c. Spalling occurred on the bottom of the slab in the test is shown in Fig. 14d, which is consistent with the numerical results, and the damage area of the experiment is approximately 45 216 mm2. But in the numerical simulation, the damage area of the slab is about 53 066 mm2, which is slightly greater than the experimental damage area. The slab also exhibits moderate damage. Form the numerical and experimental results of the slab, it can be found that the damage of the slab calculated by numerical simulation is slightly greater than the experiments. The reason of these may be the relatively softer conditions in the experimental test, which makes the real frequency of the slab lower than the model, while the boundary condition of the slab in the numerical simulation is stronger. Another factor may be the material con-
Table 2 Comparison results between numerical simulations and experimental results. Test
T1 T2 T3
Slab
S1 S2 S3
Experiment [3]
Numerical simulation
Central deflection (mm)
Damage radius (mm)
Central deflection (mm)
Damage radius (mm)
10 15 35
– 90 120
8.8 12.7 31.1
– 100 130
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stants, which are not based on the real material tests, but rather are based on the reasonable assumptions. Table 2 shows the comparison results of deflections between the numerical simulation and experimental test. The central defections of the slab obtained from numerical results are less than those from the experiments. The difference of the deflections may be caused by the boundary condition, which is fixed at all the time in the numerical simulation, whereas loosened in the experiment after the explosion. The damage areas of the slab in the numerical simulations are slightly greater than those in the experiment, which may be due to the material constants. However, it should be noted that the differences caused by the boundary condition is insignificant due to the initial stress wave propagation in the slab. As a whole, the numerical results show a favorable agreement with the experimental tests in terms of fracture spreading and the damage area. Therefore, the numerical results are credible. 5. Conclusion A three-dimensional material model including explosive, air and reinforcement concrete slab with separated modeling method using nonlinear finite element analysis software LS-DYNA has been employed in this study. By using this model, three numerical simulations of the one-way square reinforced concrete slabs subjected to close-in blast loading from different TNT charge mass detonations has been carried out and compared with the blast tests in literature[3]. The sophisticated concrete and reinforcement bar material model considering the strain rate effects, dynamic increase factor, Arbitrary Lagrange–Euler and fluid–structure coupling model were applied into simulate the damage mechanism and dynamic response. From the numerical results and comparison of the experiment in Ref. [3], it can be seen that the present model can still give a reliable prediction of the damage characteristic for the reinforcement concrete slabs. For the reinforcement concrete slab dynamic response and the damage process show that under close-in blast pressure effect, tensile failure and spalling may occur on the bottom center of the slab due to the shock wave converted into tensile wave because of the concrete low tensile strength. It also observed that the increase of the amount of the explosive can change the failure mode of the slab. Acknowledgements This study was financial supported by the Specialized Research Fund for Doctoral Program of Higher Education Institutions China (Grant No. 20110041110012) and the State Key Program of National Natural Science of China (Grant No. 51138001). References [1] W. Wang, D. Zhang, F. Lu, S.-C. Wang, F. Tang, Experimental study and numerical simulation of the damage mode of a square reinforced concrete slab under close-in explosion, Eng. Fail. Anal. 27 (2013) 41–51. [2] S.H.P.P.H. Bischoff, Compressive behaviour of concrete at high strain rates, Mater. Struct. 24 (1991) 425–450. [3] W. Wang, D. Zhang, F. Lu, S. Wang, F. Tang, 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. 49 (2012) 158–164. [4] P.F. Silva, B. Lu, Improving the blast resistance capacity of RC slabs with innovative composite materials, Compos. Part B: Eng. 38 (2007) 523–534. [5] K. Ohkubo, M. Beppu, T. Ohno, K. Satoh, Experimental study on the effectiveness of fiber sheet reinforcement on the explosive-resistant performance of concrete plates, Int. J. Impact Eng. 35 (2008) 1702–1708. [6] Y.S. Tai, T.L. Chu, H.T. Hu, J.Y. Wu, Dynamic response of a reinforced concrete slab subjected to air blast load, Theoret. Appl. Fract. Mech. 56 (2011) 140–147.
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