- Email: [email protected]
RETRACTED: Response of ship hull laminated plates to close proximity blast loads
Computational Materials Science 52 (2012) 197–203
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Response of ship hull laminated plates to close proximity blast loads Ionel Chirica ⇑, Doina Boazu, Elena-Felicia Beznea University Dunarea de Jos of Galati, 47, Domneasca Str., Galati 800008, Romania
a r t i c l e
i n f o
Article history: Received 1 January 2011 Received in revised form 18 July 2011 Accepted 20 July 2011 Available online 20 August 2011 Keywords: Composite plates Ship hull Blast loads
a b s t r a c t This paper presents selected results of a study concerning the protective capacity of ship hull structures made of composite materials subjected to an explosion of a spherical charge. The analysis of the effects of explosion on ship structure made of composite materials is presented. The main application of these structures lies in design of ship structures of great importance, which should also be protected against exceptional loads of this kind. In this study, a non-linear analysis with the finite-element computer code COSMOS/M was done. The methodology for the blast pressure charging and the mechanism of the blast wave in free air are given. The space pressure variation is determined by using Friedlander exponential decay equation. Various scenarios (parametric calculus) to evaluate the behavior of the ship structure laminated plate to blast loading are presented: explosive magnitude, distance from source of explosion, plate thickness. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Blast loads induced by explosion produced by accidents or intentionally by terrorist attacks within or immediately nearby ship hull can cause catastrophic damage on the ship structure and shutting down of critical life safety systems. Loss of life and injuries to crew and passengers can result from many causes, including direct blast-effects, structural collapse, debris impact, fire and smoke [1]. The analysis and design of structure subjected to blast loads require a detailed understanding of blast phenomena and the dynamic response of various structural elements. The most important way to diminish the injuries due to blast loading is to secure sufficient stand off distance between the explosion source and the target structure and reduce the magnitude of the blast impact wave so that the structure to be not highly damaged. To achieve these targets it is necessary to do various scenarios to evaluating the behavior of the ship structure to blast loading (explosive magnitude, distance from source of explosion, structure scantling etc.). Transient response of the shells from ship hull structures to high intensity loads is often investigated in the context of sonic booms, explosive blasts, and other shock type pressure loads. Sonic booms and explosions that are initiated sufficiently far from a structure are often idealized as pressure waves arriving at all points on the structure’s surface simultaneously. In close proximity to an explosion, the ship hull structures are loaded by a high intensity-short duration pressures that vary in time and space.
⇑ Corresponding author. Tel.: +40 722 383282. E-mail address: [email protected] (I. Chirica). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.07.036
Most part of the literature available concerning plate response to short-duration, high intensity pressures make this assumption. Experimental evidence supporting this assumption is provided in [2]. In many of these works, the time history of the spatially uniform pressure is described by step-pulse, N-pulse, or Friedlander equations [3]. Finite element modeling and analysis for the blast-loaded cylindrical shell are also presented in [4]. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. An in-house developed, verified and fully validated threedimensional finite element code, with rate dependent damage evolution equations for anisotropic bodies is presented in [5]. The code is used to numerically ascertain the damage developed in a fiberreinforced composite due to shock loads representative of those produced by an underwater explosion. Special emphasis are focused on the evolution of mid-point displacements, and plastic strain energy as is done in [6] where numerical solutions are obtained by using the finite element method and the central difference method for the time integration of the non-linear equations of motion. In [7] a robust framework for computational modeling of the response of composite laminates to blast loads. The numerical test-bed for the simulations is the explicit finite element code, LS-DYNA. Delaminations are modeled using a cohesive type tiebreak interface introduced between sublaminates while intralaminar damage mechanisms are captured using a continuum damage mechanics approach. The response of elastic structures to time-dependent external excitations, such as sonic boom and blast loadings, is a subject of
198
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
much interest in the design of marine vehicles [8,9]. For the case of blast loadings, various analytical expressions have been proposed and discussed [10]. The particular equation used for the pressure–time history of the load is often chosen to best match the particular phenomenon; considered. The time history of overpressures due to explosions is often represented by the modified Friedlander exponential decay equation [10]. In the references listed thus far, the blast pressures are of arbitrary magnitude and are applied uniformly across the structure. Methods presented in [10] use experimental data from explosive tests to develop expressions for the blast overpressure as a function of time and distance from the blast, as well as charge weight and other important blast parameters. Very few papers make use of such a realistic blast load. Most of the literature available concerning impulsively loaded plates considers a linear solution for isotropic plates. There are also many linear solutions available for impulsively loaded composite plates, and some of the references listed thus far are of this type [11]. The rapid expansion of the detonation products creates a shock wave in the surrounding medium, which for simplicity in this paper is assumed as being in air. This shock wave in air is known as a blast wave. Similar to the detonation wave there is for practical purposes, a discontinuous increase in pressure, density, temperature and velocity across a blast wave. The shock-induced compression of the ambient air also leads to an increase in temperature behind the shock front. The pre and post shock states are described by conservation equations for mass, momentum and energy, collectively known as Rankine–Hugoniot Jump equations ([12,13]). In [14] a systematic reliability-based design approach is suggested where system reliability is used to compute the probability of failure of a composite laminate. A case study for the design of a five-layer composite laminate incorporating carbon nanotubes and subjected to uncertain blast event is demonstrated. Predicting the structural response to such an explosion requires accurate prediction of the applied pressures and a solution procedure that is adequate for such transient phenomena. The work presented here focuses on the structural response to such close proximity explosions. In particular, the structures considered include orthotropic composite and contacting plates subject to mine blasts. Both linear and a non-linear solution is developed for these simulations.
2. Blast wave characteristics Nevertheless, a huge or close-to-structure explosion will cause extreme damage of main structural components and the entire loss of load-carrying capacity of structural components. Blast resistant design of ship hull frames generally provides sufficient toughness of components and structural system capable of limiting the possibility of hull collapse. Therefore, simulation of blast loading and estimation of structure behavior and damage under blast loading are very important phase of research to evaluate the resistance and safety of hull structure against direct and consequential blast damage. Explosion dissipates energy forming light, sound, and very dense and high pressure wave with initial expansion at very high velocities. Typical explosive detonations in the free field create a suddenly rising and rapidly decaying pressure to satisfy equilibrium with surrounding water, or a shock wave with very short duration. The range in which the risen pressure decays back to normal pressure is defined as a positive phase (see Fig. 1). As the wave front expands, a negative pressure phase occurs when the pressure is lower than ambient pressure. The negative phase has a little effect on the response of structures [13].
p negative phase
positive phase
pso
t T
T +T
Fig. 1. Typical pressure time–history.
Details on the idealization of the pressure wave according to the upper presentation are shown in Fig. 2, where the variation of the pressure acting on the plate surface is presented as a number of rings. Blast wave is reflected and amplified when the incident pressure wave is transferred through air or fluid and contact any structure, causing reflected pressure. The reflected pressure usually varies through the weight and type of explosive, standoff distance (or distance from the detonation) and the incident angle of the wave. The reflected pressure is a blast load on a structure considered in a design and analysis. In addition, the duration and amount of amplification of the reflected pressure can be affected by the shape of target structure or other objects, such as ground, which can produce another reflected blast wave. When a high order explosion is initiated, a very rapid exothermic chemical reaction occurs. As the reaction progresses, the solid or liquid explosive material is converted to very hot, dense, highpressure gas. The explosion products initially expand at very high velocities in an attempt to reach equilibrium with the surrounding air, causing a shock wave. A shock wave consists of highly compressed air, traveling radially outward from the source at supersonic velocities. Only one-third of the chemical energy available in most high explosives is released in the detonation process. The remaining two-thirds is released more slowly as the detonation products mix with air and burn. This afterburning process has little effect on the initial blast wave because it occurs much slower than the original detonation. However, later stages of the blast wave can be affected by the afterburning, particular for explosions in confined spaces. As the
Fig. 2. Details on the pressure wave.
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
shock wave expands, pressures decrease rapidly (with the cube of the distance) because of geometric divergence and the dissipation of energy in heating the air. Pressures also decay rapidly over time (i.e., exponentially) and have a very brief span of existence, measured typically in thousandths of a second, or milliseconds. An explosion can be visualized as a ‘‘bubble’’ of highly compressed air that expands until reaching equilibrium with the surrounding air. Explosive detonations create an incident blast wave, characterized by an almost instantaneous rise from atmospheric pressure to a peak overpressure. As the shock front expands pressure decays back to ambient pressure, a negative pressure phase occurs that is usually longer in duration than the positive phase as it is shown in Fig. 1. The negative phase is usually less important in a design than the positive phase. When the incident pressure wave impinges on a structure that is not parallel to the direction of the wave’s travel, it is reflected and reinforced, producing what is known as reflected pressure. The reflected pressure is always greater than the incident pressure at the same distance from the explosion. The reflected pressure varies with the angle of incidence of the shock wave. When the shock wave impinges on a surface that is perpendicular to the direction it is traveling, the point of impact will experience the maximum reflected pressure. When the reflecting surface is parallel to the blast wave, the minimum reflected pressure or incident pressure will be experienced. In addition to the angle of incidence, the magnitude of the peak reflected pressure (Pr) is dependent on the peak incident pressure (Pi), which is a function of the net explosive weight and distance from the detonation. Typical reflected pressure coefficients (Cr = Pr/Pi) versus the angle of incidence for different peak incident pressures are presented in [10]. The reflected pressure coefficient equals to the ratio of the peak reflected pressure to the peak incident pressure (Cr = Pr/Pi). The reflected pressures for explosive detonations can be almost 13 times greater than peak incident pressures and, for all explosions, the reflected pressure coefficients are significantly greater closer to the explosion. Impulse is a measure of the energy from an explosion imparted to a building. Both the negative and positive phases of the pressure–time waveform contribute to impulse. Fig. 1 shows how impulse and pressure vary over time from a typical explosive detonation. The magnitude and distribution of blast loads on a structure vary greatly with several factors: – Explosive properties (type of material, energy output, and quantity of explosive). – Location of the detonation relative to the structure. – Reinforcement of the pressure pulse through its interaction with the ground or structure (reflections). The reflected pressure and the reflected impulse are the forces to which the building ultimately responds. These forces vary in time and space over the exposed surface of the building, depending on the location of the detonation in relation to the building. Therefore, when analyzing a structure for a specific blast event, care should be taken to identify the worst case explosive detonation location. In the context of other hazards (e.g., earthquakes, winds, or floods), an explosive attack has the following distinguishing features:
– Explosive pressures decay extremely rapidly with distance from the source. Therefore, the damages on the side of the building facing the explosion may be significantly more severe than on the opposite side. As a consequence, direct air-blast damages tend to cause more localized damage. In an urban setting, however, reflections off surrounding buildings can increase damages to the opposite side. – The duration of the event is very short, measured in thousandths of a second, or milliseconds. This differs from earthquakes and wind gusts, which are measured in seconds, or sustained wind or flood situations, which may be measured in hours. Because of this, the mass of the structure has a strong mitigating effect on the response because it takes time to mobilize the mass of the structure. By the time the mass is mobilized, the loading is gone, thus mitigating the response. This is the opposite of earthquakes, whose imparted forces are roughly in the same timeframe as the response of the structures mass, causing a resonance effect that can worsen the damage. 3. Blast load parameters As it was clearly established, the blast wave reaches the peak value in such a short time that the structure can be assumed to be loaded instantly. In certain cases, due to the relative small dimensions of the structure when compared to the blast and sonic boom wave front, it may also be assumed that the pressure is more or less uniformly distributed over the structure. The first step in blast related research is to predict blast loads on the structure. For the purpose of blast resistant design, experimentbased direct load-calculating methods have been mostly used to describe the blast pressure on a structure. The variables of a function to represent a blast pressure time history are peak pressure, impulse, arrival time, and the duration of the pressure. These parameters are generally determined by experimental results. Many documents in this field provide graphical form displaying the values of the blast wave parameters as a function of scaled distance after an explosive weight is converted to TNT equivalent mass [13]. All blast parameters are primarily dependent on the amount of energy released by a detonation in the form of a blast wave and the distance from the explosion. A universal normalized description of the blast effects can be given by scaling distance relative to (E/P0)1/3 and scaling pressure relative to P0, where E is the energy release (kJ) and P0 the ambient pressure (typically 100 kN/m2). For convenience, however, it is general practice to express the basic explosive input or charge weight W as an equivalent mass of TNT. Results are then given as a function of the dimensional distance parameter (scaled distance)
Z ¼ R=W 1=3 ;
ð1Þ
where R is the actual effective distance from the explosion and W is generally expressed in kilograms. Scaling laws provide parametric correlations between a particular explosion and a standard charge of the same substance. Blast wave parameters for conventional high explosive materials have been the focus of a number of studies during the 1950s and 1960s. Estimations of peak overpressure Pso (in kPa) due to spherical blast based on scaled distance Z, were introduced in [13] as: – for Pso > 1000 kPa
– The intensity of the pressures acting on a targeted building can be several orders of magnitude greater than these other hazards. It is not uncommon for the peak incident pressure to be in excess of 689,475 Pa on a building in an urban setting for a vehicle weapon parked along the curb. At these pressure levels, major damages and failure are expected.
199
Pso ¼ 670=Z 3 þ 100; – for 10 < Pso < 1000 kPa
Pso ¼ 97:5=Z þ 145:5=Z 2 þ 585=Z 3 1:9
200
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
The shape of blast wave can be represented by linear decay using an approximate triangular equivalents or more realistic exponential decay shown in Fig. 1 based on Friedlander equation which intends to agree with experimental values of blast pressure [1]. A modified Friedlander’s equation is as follows
pðtÞ ¼ pso
0 t T a A tT T0 e 1 T0
ð2Þ
where p(t) is blast pressure at time t, pso is peak incident pressure, T0 is positive phase duration, Ta is arrival time, and A is a decay coefficient. Peak reflected pressure is given as a function of variables such as peak incident pressure, angle of wave incidence to the surface of an object and shock front velocity, etc. Then, reflected impulse proportional to the calibrated peak reflected pressure and the corresponding duration of reflected pressure will be determined [10]. In accordance with above mentioned references the overpressure associated with the blast pulses can be described in terms of the modified Friedlander exponential decay equation [1], as
t atp0 t e py ðs; z; tÞ ¼ pm 1 tp
ð3Þ
where the negative phase of the blast is included. In Eq. (3), pm denotes the peak reflected pressure in excess of the ambient one; tp denotes the positive phase duration of the pulse measured from the time of impact of the structure and a0 denotes a decay parameter which has to be adjusted to approximate the overpressure signature from the blast tests. As concerns the sonic-boom loading, this can be modeled as an N-shaped pressure pulse arriving at a normal incidence. Such a pulse corresponds to an idealized far field overpressure produced by an aircraft flying supersonically in the earth’s atmosphere or by any supersonic projectile rocket or missile. The overpressure signature of the N-wave shock pulse can be described by
( pðs; z; tÞ ¼
for 0 < t < rt p pm 1 ttp 0
ð4Þ
for t < 0 or t > rt p
As it is shown in Fig. 3, some of the assumptions and parameters involved in blast loading are presented. Also, a finite element model of the plate structure is presented. Once the blast distance is determined (Fig. 3), elements within 45° of the blast normal vector are divided into groups based upon their average distance to the center node. There is described the application of a blast load to a finite element mesh. Only elements within the 45° cone are loaded. The area within the cone is divided into a number of rings to determine the pressure acting on the elements of the mesh. The distribution of the blast load on the plate using 10 load rings is shown in Fig. 4.
According to the methods used in this paper an individual pressure–time history to each element based on its distance from the blast is assigned. Each ring from Fig. 4, has its own pressure–time history as it is shown in Fig. 5. In the application, described in this paper, a rectangular plate having the side dimension of 500 mm was used. For the studied plate, ten discretization rings were used to load the panel with the mine blast at the distance h. Parametric calculus was done for equivalent TNT mass (W = 0.1 kg, 0.2 kg and 0.5 kg) placed at the distance h = 0.2 m from the plate surface. The modality to calculate the pressure on the plate is described in Figs. 2 and 4. The spatial distributions and pressure profiles are illustrated in Fig. 5 (the circle 1 is the circle from the middle of the plate). As it is shown in the figure, the time of arrival, Ta is 0.01 s and T0 is equal to 0.01 s.
4. FEM analysis The applied impulse is calculated using the plate mesh. The area of each of the ring zones is easily calculated from the formula for the cross sectional area of a cylinder. These individual areas are then multiplied by the corresponding impulse per unit area to obtain the total applied impulse. Using the ideal and applied impulse values, a percent error for the applied impulse can be determined. The studies were carried out on a square plate from the side shell placed between two pairs of web stiffeners. So, the plate can be considered as being clamped on the all sides. The material is E-glass/polyester having the symmetric stack. The stack of the shell is according to the topologic code [A/B]3 s. The layers made of material A, have the thickness of 0.25 mm and characteristics:
Ex ¼ 80 GPa; Ey ¼ 80 GPa; Gxy ¼ 10 GPa;
lxy ¼ 0:2
The layers made of material B, have the thickness of 0.1 mm and characteristics:
Ex ¼ 3:4 GPa; Ey ¼ 3:4 GPa; Gxy ¼ 1:3 GPa;
lxy ¼ 0:3:
Due to the double symmetry, one quarter of plate was studied. In the non-linear calculus, Tsai–Wu criterion was considered for the limit state stresses evaluation. The dynamic calculus was done by direct integration with Newmark method (c = 0.5; b = 0.25).
Explosion point Blast normal distance vector Center node
h
45o
Circle of loaded elements
R RG Loaded surface
Fig. 3. Application of the blast load to the finite element mesh.
Fig. 4. Distribution of the blast load on the plate using 10 load rings.
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
201
displacement [mm]
Fig. 5. Pressure time histories for all rings.
15 10 5 0 0
1000
2000
3000
4000
number of elements Fig. 6. Maximum displacement versus number of elements in convergence study.
Fig. 8. Maximum von Mises stress on the sides of the plate without damping.
Fig. 7. Maximum von Mises stress on the sides of the plate with damping.
4.1. Convergence study A convergence study to choose the optimum mesh was done. The finite element mesh contains one variable (n – number of divisions along the plate side) that affect the number of elements in the model. Varying n affects the accuracy of the results. The elements throughout the convergence study have aspect ratio 1:1. The model used in this study is a square plate with the side of 0.5 m, thickness of 2.1 mm, 4.2 mm and 6.3 mm and fixed on sides. The plate is loaded to a static pressure linear varying within the loading circle (about the same as blast load). For each mesh considered, the maximum deflection in the middle is compared. The results of the convergence study are shown in Fig. 6.
Fig. 9. Variation of the maximum transversal displacement in the case h = 0.2 m.
Finally, the mesh with 2809 SHELL4L elements (2911 nodes) was chosen since all deflections are less than 1% different from the results for the 2916 element mesh.
202
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
Fig. 14. The variation of ultimate stress (Fail 1 – tension) for t = 6.3 mm, in the case W = 0.7 kg.
Fig. 10. Variation of the maximum transversal displacement in the case h = 0.15 m.
without damping characteristics). As it is seen, the maximum stress in the case of damping is 2.5 times bigger than the stress in the case of the plate without damping. In Figs. 9 and 10 are presented time variation of the displacement of the central point of the plate without damping, in two cases of values of blast normal distance h, for W = 0.5 kg. 5. Concluding remarks
Fig. 11. Variation of maximum stress versus equivalent TNT mass W for t = 6.3 mm; h = 0.25 mm.
Fig. 12. The variation of ultimate stress (Fail 1 – tension) for t = 2.1 mm, in the case W = 0.7 kg.
Fig. 13. The variation of ultimate stress (Fail 1 – tension) for t = 4.2 mm, in the case W = 0.7 kg.
4.2. FEM analysis The FEM parametric calculus was done for three groups of values for: equivalent TNT mass W (0.1 kg, 0.2 kg and 0.5 kg), blast normal distance vector h (0.15 m, 0.2 m and 0.25 m) and for total thickness of the composite plate (2.1 mm, 4.2 mm and 6.3 mm). For all cases the calculus was done so for material without damping and for damping material according to the equation
C ¼ 0:1K þ 0:1M
ð5Þ
Time variation of the maximum von Mises stress obtained in the point placed on the middle of the side plate are presented in Fig. 7 (material with damping characteristics) and Fig. 8 (material
For high-risks facilities such as navy and commercial ships, design considerations against extreme events (bomb blast, high velocity impact) are very important. It is recommended that guidelines on abnormal load cases and provisions on progressive collapse prevention should be included in the current ship hull structure design norms. Requirements on ductility levels also help improve the structure performance under severe load conditions. Dynamic calculus was done with COSMOS/M soft package using specific elements SHELL4L. According to the parametric calculus, the material damping model used in the analysis leads to the decreasing of the maximum stress occuring in the plate up to 0.5 from the value obtained in the model without damping. For example, in Fig. 11 the variation of maximum stress versus equivalent TNT mass W for plate thickness of 6.3 mm and blast normal distance h of 0.25 mm for the both cases of damping is presented. In all analyzed cases, for equivalent TNT mass W lesser than 0.2 kg, the stress obtained in the plate is almost constant, for various distances h and various plate thicknesses t. The blast wave is instantaneously increases to a value of pressure above the ambient atmospheric pressure. This is referred to as the side-on overpressure that decays as the shock wave expands outward from the explosion source. After a short time, the pressure behind the front may drop below the ambient pressure (Fig. 2). During such a negative phase, a partial vacuum is created and air is sucked in. This is also accompanied by high suction winds that carry the debris for long distances away from the explosion source. In the paper this phase is considered as equal to zero. The values of the pressure acting on the plate have small differences for W = 0.1 kg and W = 0.2 kg. For the values of the equivalent TNT mass W lesser than 0.7 kg, the fails do not occur in the material and so the integrity of the plate is not affected. In the case of W = 0.7 kg in all cases of distance h and thickness t the variations of failure criterion for tension (Fail 1) are presented in Figs. 12–14. As it is seen, generally speaking, the damping ‘‘is helping’’ the plate integrity: the stress when the Fail 1 (case of tension) is occurring in the case of damping is greater than the plate without damping. It is critical for accurate analytical predictions to obtain pressure maps and time histories for muzzle blast from a particular blast explosion. This data should then be fit to the blast pressure model to obtain duration, delay, and maximum pressure as a function of location for each exposed element in a finite element model.
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
Use large safety factors (obviously more than 2.0) when sizing structure due to uncertainties in blast pressure characteristics and structure dynamic behavior is design requesting. Structure subjected to blast explosion should be as compliant as possible while maintaining structural integrity to minimize transferred loads. Internal shear joints require special attention to ensure adequate strength for transferred loads. According to the analysis, the developed blast simulation model and optimal design system can enable the prediction, design and prototyping of blast-protective composite structures for a wide range of damage scenarios in various blast events, ranging from plate damage, localized structural failure. From the studies, the proposal of a composite structure with special damping system can help the structure to sustain blast load. The inclusion of a damping material in the composite structure can absorb energy under blast load and help to reduce the force transmitted to the main structure. Also, the damping material helps to reduce stress concentration in the plate material. Acknowledgment The work has been performed in the scope of the Romanian Project PN2 – IDEI, Code 512/2008 (2009–2011).
203
References [1] W. Baker, Explosions in Air, University of Texas Press, Austin, 1973. [2] L. Librescu, A. Nosier, AIAA Journal 23 (2) (1990) 345–352. [3] M.J. Crocker, R.R. Hudson, Journal of Sound and Vibration 9 (3) (1969) 454–468. [4] H.S. Türkmen, Journal of Sound and Vibration 249 (4, 24) (2002) 663–678. [5] R.C. Batra, N.M. Hassan, Composites: Part B 38 (2007) 448–468. [6] A. Kadid, Journal of Civil Engineering and Management 14 (3) (2008) 155–161. [7] A. Forghani, C. McGregor, S. McClennan, R. Vaziri, F. Ellyin, A. Poursartip, A. Bouamoul, Modelling of damage development in blast loaded composite panels, in: The 16th International Conference on Composite Materials, Kyoto, 2007, pp. 1–8. [8] R. Houlston, J.E. Slater, N. Pegg, C.G. Des Rochers, Computers and Structures 21 (1985) 273–289. [9] A.D. Gupta, F.H. Gregory, R.L. Bitting, Proceedings of XIII Southestern Conference on Theoretical and Applied Mechanics 1 (1985) 385–390. [10] ⁄⁄⁄ Risk Management Series, Reference Manual to Mitigate Potential Terrorist Attacks Against Buildings. Providing Protection to People and Buildings, FEMA, 426, 2003. [11] P. Cooper, Explosive Engineering, Wiley-VCH, New York, 1996. [12] G. Kinney, K. Graham, Explosive Shocks in Air, second ed., Springer Verlag, New York, 1985. [13] H.L. Brode, Numerical Solution of Spherical Blast Waves, American Institute of Physics, New York, 1955 (Journal of Applied Physics). [14] A.B. Altunc, J.J. Kim, M. Al-Haik, M.M. Reda Taha, Composite Structures 93 (8) (2011) 2042–2048.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Response of ship hull laminated plates to close proximity blast loads Ionel Chirica ⇑, Doina Boazu, Elena-Felicia Beznea University Dunarea de Jos of Galati, 47, Domneasca Str., Galati 800008, Romania
a r t i c l e
i n f o
Article history: Received 1 January 2011 Received in revised form 18 July 2011 Accepted 20 July 2011 Available online 20 August 2011 Keywords: Composite plates Ship hull Blast loads
a b s t r a c t This paper presents selected results of a study concerning the protective capacity of ship hull structures made of composite materials subjected to an explosion of a spherical charge. The analysis of the effects of explosion on ship structure made of composite materials is presented. The main application of these structures lies in design of ship structures of great importance, which should also be protected against exceptional loads of this kind. In this study, a non-linear analysis with the finite-element computer code COSMOS/M was done. The methodology for the blast pressure charging and the mechanism of the blast wave in free air are given. The space pressure variation is determined by using Friedlander exponential decay equation. Various scenarios (parametric calculus) to evaluate the behavior of the ship structure laminated plate to blast loading are presented: explosive magnitude, distance from source of explosion, plate thickness. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Blast loads induced by explosion produced by accidents or intentionally by terrorist attacks within or immediately nearby ship hull can cause catastrophic damage on the ship structure and shutting down of critical life safety systems. Loss of life and injuries to crew and passengers can result from many causes, including direct blast-effects, structural collapse, debris impact, fire and smoke [1]. The analysis and design of structure subjected to blast loads require a detailed understanding of blast phenomena and the dynamic response of various structural elements. The most important way to diminish the injuries due to blast loading is to secure sufficient stand off distance between the explosion source and the target structure and reduce the magnitude of the blast impact wave so that the structure to be not highly damaged. To achieve these targets it is necessary to do various scenarios to evaluating the behavior of the ship structure to blast loading (explosive magnitude, distance from source of explosion, structure scantling etc.). Transient response of the shells from ship hull structures to high intensity loads is often investigated in the context of sonic booms, explosive blasts, and other shock type pressure loads. Sonic booms and explosions that are initiated sufficiently far from a structure are often idealized as pressure waves arriving at all points on the structure’s surface simultaneously. In close proximity to an explosion, the ship hull structures are loaded by a high intensity-short duration pressures that vary in time and space.
⇑ Corresponding author. Tel.: +40 722 383282. E-mail address: [email protected] (I. Chirica). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.07.036
Most part of the literature available concerning plate response to short-duration, high intensity pressures make this assumption. Experimental evidence supporting this assumption is provided in [2]. In many of these works, the time history of the spatially uniform pressure is described by step-pulse, N-pulse, or Friedlander equations [3]. Finite element modeling and analysis for the blast-loaded cylindrical shell are also presented in [4]. Kinematically admissible displacement functions are chosen to represent the motion of the clamped cylindrical shell and the governing equations are obtained in the time domain using the Galerkin method. An in-house developed, verified and fully validated threedimensional finite element code, with rate dependent damage evolution equations for anisotropic bodies is presented in [5]. The code is used to numerically ascertain the damage developed in a fiberreinforced composite due to shock loads representative of those produced by an underwater explosion. Special emphasis are focused on the evolution of mid-point displacements, and plastic strain energy as is done in [6] where numerical solutions are obtained by using the finite element method and the central difference method for the time integration of the non-linear equations of motion. In [7] a robust framework for computational modeling of the response of composite laminates to blast loads. The numerical test-bed for the simulations is the explicit finite element code, LS-DYNA. Delaminations are modeled using a cohesive type tiebreak interface introduced between sublaminates while intralaminar damage mechanisms are captured using a continuum damage mechanics approach. The response of elastic structures to time-dependent external excitations, such as sonic boom and blast loadings, is a subject of
198
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
much interest in the design of marine vehicles [8,9]. For the case of blast loadings, various analytical expressions have been proposed and discussed [10]. The particular equation used for the pressure–time history of the load is often chosen to best match the particular phenomenon; considered. The time history of overpressures due to explosions is often represented by the modified Friedlander exponential decay equation [10]. In the references listed thus far, the blast pressures are of arbitrary magnitude and are applied uniformly across the structure. Methods presented in [10] use experimental data from explosive tests to develop expressions for the blast overpressure as a function of time and distance from the blast, as well as charge weight and other important blast parameters. Very few papers make use of such a realistic blast load. Most of the literature available concerning impulsively loaded plates considers a linear solution for isotropic plates. There are also many linear solutions available for impulsively loaded composite plates, and some of the references listed thus far are of this type [11]. The rapid expansion of the detonation products creates a shock wave in the surrounding medium, which for simplicity in this paper is assumed as being in air. This shock wave in air is known as a blast wave. Similar to the detonation wave there is for practical purposes, a discontinuous increase in pressure, density, temperature and velocity across a blast wave. The shock-induced compression of the ambient air also leads to an increase in temperature behind the shock front. The pre and post shock states are described by conservation equations for mass, momentum and energy, collectively known as Rankine–Hugoniot Jump equations ([12,13]). In [14] a systematic reliability-based design approach is suggested where system reliability is used to compute the probability of failure of a composite laminate. A case study for the design of a five-layer composite laminate incorporating carbon nanotubes and subjected to uncertain blast event is demonstrated. Predicting the structural response to such an explosion requires accurate prediction of the applied pressures and a solution procedure that is adequate for such transient phenomena. The work presented here focuses on the structural response to such close proximity explosions. In particular, the structures considered include orthotropic composite and contacting plates subject to mine blasts. Both linear and a non-linear solution is developed for these simulations.
2. Blast wave characteristics Nevertheless, a huge or close-to-structure explosion will cause extreme damage of main structural components and the entire loss of load-carrying capacity of structural components. Blast resistant design of ship hull frames generally provides sufficient toughness of components and structural system capable of limiting the possibility of hull collapse. Therefore, simulation of blast loading and estimation of structure behavior and damage under blast loading are very important phase of research to evaluate the resistance and safety of hull structure against direct and consequential blast damage. Explosion dissipates energy forming light, sound, and very dense and high pressure wave with initial expansion at very high velocities. Typical explosive detonations in the free field create a suddenly rising and rapidly decaying pressure to satisfy equilibrium with surrounding water, or a shock wave with very short duration. The range in which the risen pressure decays back to normal pressure is defined as a positive phase (see Fig. 1). As the wave front expands, a negative pressure phase occurs when the pressure is lower than ambient pressure. The negative phase has a little effect on the response of structures [13].
p negative phase
positive phase
pso
t T
T +T
Fig. 1. Typical pressure time–history.
Details on the idealization of the pressure wave according to the upper presentation are shown in Fig. 2, where the variation of the pressure acting on the plate surface is presented as a number of rings. Blast wave is reflected and amplified when the incident pressure wave is transferred through air or fluid and contact any structure, causing reflected pressure. The reflected pressure usually varies through the weight and type of explosive, standoff distance (or distance from the detonation) and the incident angle of the wave. The reflected pressure is a blast load on a structure considered in a design and analysis. In addition, the duration and amount of amplification of the reflected pressure can be affected by the shape of target structure or other objects, such as ground, which can produce another reflected blast wave. When a high order explosion is initiated, a very rapid exothermic chemical reaction occurs. As the reaction progresses, the solid or liquid explosive material is converted to very hot, dense, highpressure gas. The explosion products initially expand at very high velocities in an attempt to reach equilibrium with the surrounding air, causing a shock wave. A shock wave consists of highly compressed air, traveling radially outward from the source at supersonic velocities. Only one-third of the chemical energy available in most high explosives is released in the detonation process. The remaining two-thirds is released more slowly as the detonation products mix with air and burn. This afterburning process has little effect on the initial blast wave because it occurs much slower than the original detonation. However, later stages of the blast wave can be affected by the afterburning, particular for explosions in confined spaces. As the
Fig. 2. Details on the pressure wave.
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
shock wave expands, pressures decrease rapidly (with the cube of the distance) because of geometric divergence and the dissipation of energy in heating the air. Pressures also decay rapidly over time (i.e., exponentially) and have a very brief span of existence, measured typically in thousandths of a second, or milliseconds. An explosion can be visualized as a ‘‘bubble’’ of highly compressed air that expands until reaching equilibrium with the surrounding air. Explosive detonations create an incident blast wave, characterized by an almost instantaneous rise from atmospheric pressure to a peak overpressure. As the shock front expands pressure decays back to ambient pressure, a negative pressure phase occurs that is usually longer in duration than the positive phase as it is shown in Fig. 1. The negative phase is usually less important in a design than the positive phase. When the incident pressure wave impinges on a structure that is not parallel to the direction of the wave’s travel, it is reflected and reinforced, producing what is known as reflected pressure. The reflected pressure is always greater than the incident pressure at the same distance from the explosion. The reflected pressure varies with the angle of incidence of the shock wave. When the shock wave impinges on a surface that is perpendicular to the direction it is traveling, the point of impact will experience the maximum reflected pressure. When the reflecting surface is parallel to the blast wave, the minimum reflected pressure or incident pressure will be experienced. In addition to the angle of incidence, the magnitude of the peak reflected pressure (Pr) is dependent on the peak incident pressure (Pi), which is a function of the net explosive weight and distance from the detonation. Typical reflected pressure coefficients (Cr = Pr/Pi) versus the angle of incidence for different peak incident pressures are presented in [10]. The reflected pressure coefficient equals to the ratio of the peak reflected pressure to the peak incident pressure (Cr = Pr/Pi). The reflected pressures for explosive detonations can be almost 13 times greater than peak incident pressures and, for all explosions, the reflected pressure coefficients are significantly greater closer to the explosion. Impulse is a measure of the energy from an explosion imparted to a building. Both the negative and positive phases of the pressure–time waveform contribute to impulse. Fig. 1 shows how impulse and pressure vary over time from a typical explosive detonation. The magnitude and distribution of blast loads on a structure vary greatly with several factors: – Explosive properties (type of material, energy output, and quantity of explosive). – Location of the detonation relative to the structure. – Reinforcement of the pressure pulse through its interaction with the ground or structure (reflections). The reflected pressure and the reflected impulse are the forces to which the building ultimately responds. These forces vary in time and space over the exposed surface of the building, depending on the location of the detonation in relation to the building. Therefore, when analyzing a structure for a specific blast event, care should be taken to identify the worst case explosive detonation location. In the context of other hazards (e.g., earthquakes, winds, or floods), an explosive attack has the following distinguishing features:
– Explosive pressures decay extremely rapidly with distance from the source. Therefore, the damages on the side of the building facing the explosion may be significantly more severe than on the opposite side. As a consequence, direct air-blast damages tend to cause more localized damage. In an urban setting, however, reflections off surrounding buildings can increase damages to the opposite side. – The duration of the event is very short, measured in thousandths of a second, or milliseconds. This differs from earthquakes and wind gusts, which are measured in seconds, or sustained wind or flood situations, which may be measured in hours. Because of this, the mass of the structure has a strong mitigating effect on the response because it takes time to mobilize the mass of the structure. By the time the mass is mobilized, the loading is gone, thus mitigating the response. This is the opposite of earthquakes, whose imparted forces are roughly in the same timeframe as the response of the structures mass, causing a resonance effect that can worsen the damage. 3. Blast load parameters As it was clearly established, the blast wave reaches the peak value in such a short time that the structure can be assumed to be loaded instantly. In certain cases, due to the relative small dimensions of the structure when compared to the blast and sonic boom wave front, it may also be assumed that the pressure is more or less uniformly distributed over the structure. The first step in blast related research is to predict blast loads on the structure. For the purpose of blast resistant design, experimentbased direct load-calculating methods have been mostly used to describe the blast pressure on a structure. The variables of a function to represent a blast pressure time history are peak pressure, impulse, arrival time, and the duration of the pressure. These parameters are generally determined by experimental results. Many documents in this field provide graphical form displaying the values of the blast wave parameters as a function of scaled distance after an explosive weight is converted to TNT equivalent mass [13]. All blast parameters are primarily dependent on the amount of energy released by a detonation in the form of a blast wave and the distance from the explosion. A universal normalized description of the blast effects can be given by scaling distance relative to (E/P0)1/3 and scaling pressure relative to P0, where E is the energy release (kJ) and P0 the ambient pressure (typically 100 kN/m2). For convenience, however, it is general practice to express the basic explosive input or charge weight W as an equivalent mass of TNT. Results are then given as a function of the dimensional distance parameter (scaled distance)
Z ¼ R=W 1=3 ;
ð1Þ
where R is the actual effective distance from the explosion and W is generally expressed in kilograms. Scaling laws provide parametric correlations between a particular explosion and a standard charge of the same substance. Blast wave parameters for conventional high explosive materials have been the focus of a number of studies during the 1950s and 1960s. Estimations of peak overpressure Pso (in kPa) due to spherical blast based on scaled distance Z, were introduced in [13] as: – for Pso > 1000 kPa
– The intensity of the pressures acting on a targeted building can be several orders of magnitude greater than these other hazards. It is not uncommon for the peak incident pressure to be in excess of 689,475 Pa on a building in an urban setting for a vehicle weapon parked along the curb. At these pressure levels, major damages and failure are expected.
199
Pso ¼ 670=Z 3 þ 100; – for 10 < Pso < 1000 kPa
Pso ¼ 97:5=Z þ 145:5=Z 2 þ 585=Z 3 1:9
200
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
The shape of blast wave can be represented by linear decay using an approximate triangular equivalents or more realistic exponential decay shown in Fig. 1 based on Friedlander equation which intends to agree with experimental values of blast pressure [1]. A modified Friedlander’s equation is as follows
pðtÞ ¼ pso
0 t T a A tT T0 e 1 T0
ð2Þ
where p(t) is blast pressure at time t, pso is peak incident pressure, T0 is positive phase duration, Ta is arrival time, and A is a decay coefficient. Peak reflected pressure is given as a function of variables such as peak incident pressure, angle of wave incidence to the surface of an object and shock front velocity, etc. Then, reflected impulse proportional to the calibrated peak reflected pressure and the corresponding duration of reflected pressure will be determined [10]. In accordance with above mentioned references the overpressure associated with the blast pulses can be described in terms of the modified Friedlander exponential decay equation [1], as
t atp0 t e py ðs; z; tÞ ¼ pm 1 tp
ð3Þ
where the negative phase of the blast is included. In Eq. (3), pm denotes the peak reflected pressure in excess of the ambient one; tp denotes the positive phase duration of the pulse measured from the time of impact of the structure and a0 denotes a decay parameter which has to be adjusted to approximate the overpressure signature from the blast tests. As concerns the sonic-boom loading, this can be modeled as an N-shaped pressure pulse arriving at a normal incidence. Such a pulse corresponds to an idealized far field overpressure produced by an aircraft flying supersonically in the earth’s atmosphere or by any supersonic projectile rocket or missile. The overpressure signature of the N-wave shock pulse can be described by
( pðs; z; tÞ ¼
for 0 < t < rt p pm 1 ttp 0
ð4Þ
for t < 0 or t > rt p
As it is shown in Fig. 3, some of the assumptions and parameters involved in blast loading are presented. Also, a finite element model of the plate structure is presented. Once the blast distance is determined (Fig. 3), elements within 45° of the blast normal vector are divided into groups based upon their average distance to the center node. There is described the application of a blast load to a finite element mesh. Only elements within the 45° cone are loaded. The area within the cone is divided into a number of rings to determine the pressure acting on the elements of the mesh. The distribution of the blast load on the plate using 10 load rings is shown in Fig. 4.
According to the methods used in this paper an individual pressure–time history to each element based on its distance from the blast is assigned. Each ring from Fig. 4, has its own pressure–time history as it is shown in Fig. 5. In the application, described in this paper, a rectangular plate having the side dimension of 500 mm was used. For the studied plate, ten discretization rings were used to load the panel with the mine blast at the distance h. Parametric calculus was done for equivalent TNT mass (W = 0.1 kg, 0.2 kg and 0.5 kg) placed at the distance h = 0.2 m from the plate surface. The modality to calculate the pressure on the plate is described in Figs. 2 and 4. The spatial distributions and pressure profiles are illustrated in Fig. 5 (the circle 1 is the circle from the middle of the plate). As it is shown in the figure, the time of arrival, Ta is 0.01 s and T0 is equal to 0.01 s.
4. FEM analysis The applied impulse is calculated using the plate mesh. The area of each of the ring zones is easily calculated from the formula for the cross sectional area of a cylinder. These individual areas are then multiplied by the corresponding impulse per unit area to obtain the total applied impulse. Using the ideal and applied impulse values, a percent error for the applied impulse can be determined. The studies were carried out on a square plate from the side shell placed between two pairs of web stiffeners. So, the plate can be considered as being clamped on the all sides. The material is E-glass/polyester having the symmetric stack. The stack of the shell is according to the topologic code [A/B]3 s. The layers made of material A, have the thickness of 0.25 mm and characteristics:
Ex ¼ 80 GPa; Ey ¼ 80 GPa; Gxy ¼ 10 GPa;
lxy ¼ 0:2
The layers made of material B, have the thickness of 0.1 mm and characteristics:
Ex ¼ 3:4 GPa; Ey ¼ 3:4 GPa; Gxy ¼ 1:3 GPa;
lxy ¼ 0:3:
Due to the double symmetry, one quarter of plate was studied. In the non-linear calculus, Tsai–Wu criterion was considered for the limit state stresses evaluation. The dynamic calculus was done by direct integration with Newmark method (c = 0.5; b = 0.25).
Explosion point Blast normal distance vector Center node
h
45o
Circle of loaded elements
R RG Loaded surface
Fig. 3. Application of the blast load to the finite element mesh.
Fig. 4. Distribution of the blast load on the plate using 10 load rings.
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
201
displacement [mm]
Fig. 5. Pressure time histories for all rings.
15 10 5 0 0
1000
2000
3000
4000
number of elements Fig. 6. Maximum displacement versus number of elements in convergence study.
Fig. 8. Maximum von Mises stress on the sides of the plate without damping.
Fig. 7. Maximum von Mises stress on the sides of the plate with damping.
4.1. Convergence study A convergence study to choose the optimum mesh was done. The finite element mesh contains one variable (n – number of divisions along the plate side) that affect the number of elements in the model. Varying n affects the accuracy of the results. The elements throughout the convergence study have aspect ratio 1:1. The model used in this study is a square plate with the side of 0.5 m, thickness of 2.1 mm, 4.2 mm and 6.3 mm and fixed on sides. The plate is loaded to a static pressure linear varying within the loading circle (about the same as blast load). For each mesh considered, the maximum deflection in the middle is compared. The results of the convergence study are shown in Fig. 6.
Fig. 9. Variation of the maximum transversal displacement in the case h = 0.2 m.
Finally, the mesh with 2809 SHELL4L elements (2911 nodes) was chosen since all deflections are less than 1% different from the results for the 2916 element mesh.
202
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
Fig. 14. The variation of ultimate stress (Fail 1 – tension) for t = 6.3 mm, in the case W = 0.7 kg.
Fig. 10. Variation of the maximum transversal displacement in the case h = 0.15 m.
without damping characteristics). As it is seen, the maximum stress in the case of damping is 2.5 times bigger than the stress in the case of the plate without damping. In Figs. 9 and 10 are presented time variation of the displacement of the central point of the plate without damping, in two cases of values of blast normal distance h, for W = 0.5 kg. 5. Concluding remarks
Fig. 11. Variation of maximum stress versus equivalent TNT mass W for t = 6.3 mm; h = 0.25 mm.
Fig. 12. The variation of ultimate stress (Fail 1 – tension) for t = 2.1 mm, in the case W = 0.7 kg.
Fig. 13. The variation of ultimate stress (Fail 1 – tension) for t = 4.2 mm, in the case W = 0.7 kg.
4.2. FEM analysis The FEM parametric calculus was done for three groups of values for: equivalent TNT mass W (0.1 kg, 0.2 kg and 0.5 kg), blast normal distance vector h (0.15 m, 0.2 m and 0.25 m) and for total thickness of the composite plate (2.1 mm, 4.2 mm and 6.3 mm). For all cases the calculus was done so for material without damping and for damping material according to the equation
C ¼ 0:1K þ 0:1M
ð5Þ
Time variation of the maximum von Mises stress obtained in the point placed on the middle of the side plate are presented in Fig. 7 (material with damping characteristics) and Fig. 8 (material
For high-risks facilities such as navy and commercial ships, design considerations against extreme events (bomb blast, high velocity impact) are very important. It is recommended that guidelines on abnormal load cases and provisions on progressive collapse prevention should be included in the current ship hull structure design norms. Requirements on ductility levels also help improve the structure performance under severe load conditions. Dynamic calculus was done with COSMOS/M soft package using specific elements SHELL4L. According to the parametric calculus, the material damping model used in the analysis leads to the decreasing of the maximum stress occuring in the plate up to 0.5 from the value obtained in the model without damping. For example, in Fig. 11 the variation of maximum stress versus equivalent TNT mass W for plate thickness of 6.3 mm and blast normal distance h of 0.25 mm for the both cases of damping is presented. In all analyzed cases, for equivalent TNT mass W lesser than 0.2 kg, the stress obtained in the plate is almost constant, for various distances h and various plate thicknesses t. The blast wave is instantaneously increases to a value of pressure above the ambient atmospheric pressure. This is referred to as the side-on overpressure that decays as the shock wave expands outward from the explosion source. After a short time, the pressure behind the front may drop below the ambient pressure (Fig. 2). During such a negative phase, a partial vacuum is created and air is sucked in. This is also accompanied by high suction winds that carry the debris for long distances away from the explosion source. In the paper this phase is considered as equal to zero. The values of the pressure acting on the plate have small differences for W = 0.1 kg and W = 0.2 kg. For the values of the equivalent TNT mass W lesser than 0.7 kg, the fails do not occur in the material and so the integrity of the plate is not affected. In the case of W = 0.7 kg in all cases of distance h and thickness t the variations of failure criterion for tension (Fail 1) are presented in Figs. 12–14. As it is seen, generally speaking, the damping ‘‘is helping’’ the plate integrity: the stress when the Fail 1 (case of tension) is occurring in the case of damping is greater than the plate without damping. It is critical for accurate analytical predictions to obtain pressure maps and time histories for muzzle blast from a particular blast explosion. This data should then be fit to the blast pressure model to obtain duration, delay, and maximum pressure as a function of location for each exposed element in a finite element model.
I. Chirica et al. / Computational Materials Science 52 (2012) 197–203
Use large safety factors (obviously more than 2.0) when sizing structure due to uncertainties in blast pressure characteristics and structure dynamic behavior is design requesting. Structure subjected to blast explosion should be as compliant as possible while maintaining structural integrity to minimize transferred loads. Internal shear joints require special attention to ensure adequate strength for transferred loads. According to the analysis, the developed blast simulation model and optimal design system can enable the prediction, design and prototyping of blast-protective composite structures for a wide range of damage scenarios in various blast events, ranging from plate damage, localized structural failure. From the studies, the proposal of a composite structure with special damping system can help the structure to sustain blast load. The inclusion of a damping material in the composite structure can absorb energy under blast load and help to reduce the force transmitted to the main structure. Also, the damping material helps to reduce stress concentration in the plate material. Acknowledgment The work has been performed in the scope of the Romanian Project PN2 – IDEI, Code 512/2008 (2009–2011).
203
References [1] W. Baker, Explosions in Air, University of Texas Press, Austin, 1973. [2] L. Librescu, A. Nosier, AIAA Journal 23 (2) (1990) 345–352. [3] M.J. Crocker, R.R. Hudson, Journal of Sound and Vibration 9 (3) (1969) 454–468. [4] H.S. Türkmen, Journal of Sound and Vibration 249 (4, 24) (2002) 663–678. [5] R.C. Batra, N.M. Hassan, Composites: Part B 38 (2007) 448–468. [6] A. Kadid, Journal of Civil Engineering and Management 14 (3) (2008) 155–161. [7] A. Forghani, C. McGregor, S. McClennan, R. Vaziri, F. Ellyin, A. Poursartip, A. Bouamoul, Modelling of damage development in blast loaded composite panels, in: The 16th International Conference on Composite Materials, Kyoto, 2007, pp. 1–8. [8] R. Houlston, J.E. Slater, N. Pegg, C.G. Des Rochers, Computers and Structures 21 (1985) 273–289. [9] A.D. Gupta, F.H. Gregory, R.L. Bitting, Proceedings of XIII Southestern Conference on Theoretical and Applied Mechanics 1 (1985) 385–390. [10] ⁄⁄⁄ Risk Management Series, Reference Manual to Mitigate Potential Terrorist Attacks Against Buildings. Providing Protection to People and Buildings, FEMA, 426, 2003. [11] P. Cooper, Explosive Engineering, Wiley-VCH, New York, 1996. [12] G. Kinney, K. Graham, Explosive Shocks in Air, second ed., Springer Verlag, New York, 1985. [13] H.L. Brode, Numerical Solution of Spherical Blast Waves, American Institute of Physics, New York, 1955 (Journal of Applied Physics). [14] A.B. Altunc, J.J. Kim, M. Al-Haik, M.M. Reda Taha, Composite Structures 93 (8) (2011) 2042–2048.