Influence of Vertical Surroundings on the Dynamic Response of Fixed Square Plate Subjected to Blast Loading

,(School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240,China)

Abstract:The dynamic response of fixed square plates with different heights of vertical surroundings subjected to blast loading was studied by using numerical simulation.The covering angle concept was proposed to express the change of the surroundings height.By using the control variable method,the dynamic response of 264 square plates with vertical surroundings was analyzed,and the influence of the covering angle,explosive mass,plate thickness as well as relative detonation distance on the explosion impulse and midpoint deflection of the square plates was investigated.The results show that the existence of vertical surroundings will increase the explosion impulse acting on the target plate.When the height of the surroundings reaches up to a critical value,the value of the explosion impulse will reach its maximum and keep a constant.An empirical formula was proposed to predict the dimensionless impulse Iˉ,which is only related to the covering angle and the relative detonation distance.Moreover,another empirical formula was proposed to predict the midpoint deflection-thickness ratio of the plate considering the influence of the vertical surroundings,and its applicability was discussed.

Key words:vertical surroundings;covering angle;deflection-thickness ratio;dynamic response;blast loading

0 Introduction

A solid plate structure is a basic engineering component widely applied in aerospace engineering,marine and offshore infrastructures,which is likely to be subjected to air blast loading caused by missiles or dynamites during its service[1].Therefore,it is vital to gain an understanding on the nonlinear dynamic response of the plate under blast loading,which has been a research hotspot over the past years[2-3].

The study on the dynamic response of the plate under blast loading mainly focuses on two general topics according to the difference of impact environment:free air blast and inner blast.For the free air blast,the shock wave only causes a one-time impact on the plate before it disperses and vanishes,which has a relatively simple loading process and the research results are relatively rich[4-8].In contrast,the inner blast occurs in a closed space.It means that the shock wave will be reflected by the boundaries several times.Therefore,the loading process of an inner blast should be much more complicated.In addition,during the inner blast,a high pressure,which is called quasistatic pressure,would be maintained over a relatively long time in the closed space.Consequently,the investigation of the dynamic response of structures under inner blast is a tough issue.Related researches mainly focus on the inner blast loading[9-10],deformation[11-13]and failure modes[9,14]of classical structures and the prediction of quasi-static pressure[15]by conducting numerical and experimental investigations.However,theoretical studies in these fields are relatively rare[16].

In engineering practice,there exists a circumstance that the blast loading is generated in semiclosed field.For instance,the target plate or the detonation point is next to some barriers,as illustrated in Fig.1.A container is partially disclosed under the inner blast,as illustrated in Fig.2.A typical example found in engineering practice is shown in Fig.3,where the explosive is detonated at the vicinity of the stiffeners.The existence of the stiffeners impedes the dispersion of the shock wave,thereby increasing the impact load on the plate[17-18].According to the unified facilities criteria design manual(UFC-3-340-02)[19],this scenario is defined as a fully-vented blast.Similar to an inner blast,multiple reflection and interaction of the shock wave may occur due to the existence of obstacles or other boundaries.But the quasi-static pressure will be absent under the fully-vented blast as the space is not fully confined.The dynamic response of plate under fully-vented blast is worth studying due to its practical significance.However,few studies have been published in this field.Geretto[11]studied the influence of the degrees of confinement on the final deformation of square mild steel plates subjected to blast loading.Equations for predicting the final midpoint deflection of the plates in three degrees of confinement(namely free air,fully vented and fully confined)were presented.Curry[20]investigated the influence of charge backing on the impact loading and deformation of plates subjected to blast waves both numerically and experimentally.It was shown that the explosion impulse applied on the plate increased 5 times when the charge was metal-backed.But the increment of the plate deformation was much less than the one of impulse.

Fig.1 Barriers near detonation point

Fig.2 Structure with one or more sides open

Considering the possible impact environment as shown in Figs.1-3,the present work aims to study the dynamic response of fixed square plates with rigid vertical surroundings subjected to fully-vented blast loading.The influence of the surroundings height,plate thickness,explosive mass and detonation distance on the explosion impulse and final plastic deformation of the plate is systematically analyzed.Empirical expressions for the explosion impulse and midpoint deflection are proposed based on the parametric study.

1 Simulation

1.1 Model

1.1.1 Geometry

Fig.4 shows the structure of an explosion model which consisted of a square plate with a thickness oftand a half-width ofL=100 mm as well as vertical surroundings with a height ofh.A spherical explosive was placed above the center point of the target plate with a vertical heightH.

Fig.4 Explosion model

For convenience,the following parameters were defined:

Relative detonation distance

Relative height of surroundings

The covering angleθis the angle between the vertical direction and the line from the detonation point to the upper end of the vertical surroundings as shown in Fig.4.

The covering angle reflects the relationship amongh,HandL,which is more efficient than using the absolute heighthalone in the parametric study.The larger the value ofθis,the more the blast wave will act on the whole model.The covering angle without surroundings(h=0)was defined as the air blast covering angleθ0.

1.1.2 Material

A bilinear elasto-plastic model was employed to estimate the relationship between the true stress and the equivalent plastic strain of the material.The yielding stress is dependent on the dynamic strain rate following the Cower-Symonds model[21]:

whereσdis the dynamic yield stress,σythe static yield stress,ε˙the equivalent plastic strain rate,and the materials constantsD=40/s andP=5.The strain hardening is taken into account as following[21]:

whereσ0is the initial yield stress,Eis the elastic modulus,Ehis the hardening modulus andεpis the equivalent plastic strain.

The modelling parameters consist of the densityρ=7.85×103kg/m3,the elastic modulusE=2.1×105MPa,the Poisson’s ratioμ=0.3,the static yield stressσy=235 MPa,the hardening modulusEh=250 MPa and the failure strainδ=0.28.

The air condition is described by the ideal-gas equation,also known as the Gamma equation[21].

wheree=2.1×105J/kg is the specific internal energy of air,ρ=1.25 kg/m3is the density of air,andγ=1.4 is the specific heat ratio.The explosive charge is simulated as high energy density air,with a density of 1 600 kg/m3and an energy density of 4.2×106J/kg.

1.2 Validation of the nonlinear finite element results with experiments

Geretto[11]presented the experimental results of square mild steel plates subjected to air blast loading.The target plates(200 mm×200 mm×3.1 mm)were clamped and bolted between two clamp frames which were then attached to a ballistic pendulum.The blast loads were generated by detonating different mass(30-70 g)of spherical plastic explosives at a constant distance of 100 mm from the target plate.

Experiments in Ref.[11]were numerically simulated in order to determine a suitable finite element mesh size and validate the accuracy of the present numerical method.The finite element model of the square plate was established by using MSC.Patran.All the six degrees of freedom of four edges were restricted.The Eulerian element was used to simulate explosive charge and air,and the Lagrangian element was applied to simulate the plate.The duration of the simulation time was 0.003 s.The numerical calculations were conducted on MSC.Dytran.The coupling mode was formulated using General Coupling Algorithm,and the ROE solver was used for the solution due to its high calculation precision.It should be noted that the explosive was PE4 in Ref.[11],while the equivalent mass of TNT is used in present simulations.

The sensitivity of the mesh size is firstly studied.Five different mesh sizes(10 mm×10 mm,8 mm×8 mm,5 mm×5 mm,4 mm×4 mm and 3 mm×3 mm)are considered here.The explosion impulse acting on the plate with five different mesh sizes are 33.35 N·s,35.16 N·s,37.98 N·s,38.95 N·s and 39.96 N·s respectively.The midpoint deflection values are 13.74 mm,13.68 mm,13.63 mm,13.60 mm and 13.58 mm respectively.In order to ensure the result accuracy while the calculation is not time-consuming,the model with the mesh size of 4 mm×4 mm is chosen.The geometry properties and coordinate system of the ship hull plate are shown in Fig.5.The longitudinal,transverse and vertical directions are denoted asx,yandz.

Fig.5 FE model of the square plate

The time histories of the explosion impulse and the midpoint deflection of the plate under each case are illustrated in Figs.6-7.The comparison of the numerical results and the experimental results is summarized in Tab.1 and Figs.8-9.It should be noted that the impulse in Tab.1 is the total impulse acting on the square plate when the midpoint deflection of the square plate reaches the maximum value,and the midpoint deflection in Tab.1 is the average value from 0.002 s to 0.003 s.

Fig.6 Time history of the explosion impulse

Fig.7 Time history of the midpoint deflection

Fig.8 Impulse versus the mass of explosive

Tab.1 Comparison of simulation and experiment results

Fig.9 Midpoint deflection versus the mass of explosive

As shown in Tab.1,the errors between the numerical and experimental results are about 2%,indicating that the present numerical method can be applied due to its high accuracy.Then,the verified numerical model was used to study the influence of vertical surroundings on the response of fixed square plate subjected to blast loading.

1.3 Simulation case design

Various combinations of relative surrounding height-h,explosive massm,plate thicknesstand relative detonation distance-Hwere considered.A total of 12 cases and 264 FE models were designed since each case had 22 FE models with different relative surroundings heights(-h=0,0.05,0.1,0.2,0.3,…,1.8,1.9,2.0).The simulation cases are detailed in Tab.2.

Tab.2 Simulation cases

2 Results and discussion

2.1 Impulse analysis

Fig.10 shows the variation of the explosion impulseIwith the covering angleθwhencan be observed that the influence of the covering angleθon the explosion impulse with different explosive masses is similar.The results show that the value ofIincreases with the increase ofθwhenθ＜2.2 rad and then approaches a constant value whenθ＞2.2 rad.In other words,the explosion impulse increases with the height of surroundings.When the height of surroundings reaches up to a critical value,the value of the explosion impulse reaches the maximum and keeps a constant.This is because the blast wave will interact with the vertical surroundings,causing the wave to reflect and converge to the target square plate,which increases the impulse of the plate.With higher surroundings,there is more reflected and converged shock wave,thereby leading to a greater impulse.However,when the height of the surroundings is larger than the charge height,the reflected shock wave no longer propagates to the target plate,and the converging effect reduces gradually.As the height of the surroundings reaches up to a critical value(θ=2.2 rad),the amount of the reflected and converged shock wave acting on the target plate will not further increase,and the value of the explosion impulse of the target plate will eventually be constant.Fig.10 also reveals that the explosion impulse is dependent on the explosive mass but independent of the plate thickness.

Fig.10 Impulse versus covering angle under different masses of explosive(

In order to investigate the influence of surroundings height on the explosion impulse,a dimensionless impulse as following is defined:

whereI0is the explosion impulse without surroundings(air blasth=0),Iθis the explosion impulse with covering angleθ.Therefore,the dimensionless impulserepresents the enhancement effect of the vertical surroundings on the explosion impulse.

The variation of the dimensionless impulsewithθwhenis shown in Fig.11.It is seen that all the values of the curves in Fig.11 are almost the same at the sameθafter non-dimensional-ization,which means the dimensionless impulseis independent of explosive mass.This is because the explosive mass affects the absolute value ofIθandI0,but the dimensionless impulseepresents the enhancement ofIθcompared withI0,which is equivalent to an amplification coefficient.Therefore,the enhancement effect of the vertical surroundings on the explosion impulse with different explosive mass is the same once the covering angleθis fixed.

Results of the dimensionless impulseversus the covering angleθwitht=3 mm andm=60 g for different relative detonation distanceare presented in Fig.12.It is seen that the value ofincreases with the increase ofwhenθis fixed.But the trend of the dimensionless impulsewith the covering angleθf(wàn)or differentis similar to that shown in Fig.11.This indicates thatcan be expressed as a function ofθand

Fig.11 Dimensionless impulse versus covering angle with different mass of explosive

Fig.12 Dimensionless impulse versus covering angle with t=3 mm and m=60 g for different

The expression for the dimensionless impulsecan be derived based on Fig.12.Whenθ＜2.2 rad,is a power function ofθ;whenθ≥2.2 rad,is a constant.

This gives the following expression:

whereAare coefficients related to the relative detonation distance

The coefficients for differentcan be obtained by using data-fitting method based on data points in Fig.12,and the corresponding equations and fitting curves are shown in Fig.13.

Fig.13 Fitting curves and corresponding equations for different(t=3 mm,m=60 g)

According to data-fitting results,the coefficients in Eq.(9)can be expressed as:

The final form of the dimensionless impulse is obtained by substituting Eq.(10)into Eq.(9).

The obtained empirical formula Eq.(11)was used to predict the dimensionless impulse of the fixed square plate when1.6 with differentThe dimensionless impulseobtained from simulation and empirical formula results are compared in Tab.3.It can be found that the error of the results obtained from the two methods is relatively large whenis equal to 0.05(about 9%)and 0.1(about 10%),but is less than 6%in other data points.This indicates that Eq.(11)can meet the accuracy requirements of engineering application.Therefore,the proposed empirical formula can be used to predict the explosion impulse of the square plate with vertical surroundings subjected to blast loading.Once the explosion impulseI0under air blast(without surroundings)is estimated by simulation,experiment or theoretical method,the proposed empirical formula which is only related to explosion impulseI0can be used to predict the explosion impulse.

Tab.3 Comparison ofbetween the empirical formula predictions and the simulation results with=1.6

Tab.3 Comparison ofbetween the empirical formula predictions and the simulation results with=1.6

?

Tab.3(Continued)

2.2 Midpoint deflection

Based on experimental results,Nurick and Martin[2-3]proposed a dimensionless damage numberφqto characterize the explosion impulse received by a rectangular plate under air blast.

whereIis the explosion impulse of the plate,tis the plate thickness,LandBare the plate length and plate width respectively,ρis the density andσyis the static yield stress of the material.The relation between midpoint deflection-thickness ratio(δ/t)and the dimensionless damage numberφqis as follow:

The midpoint deflection-thickness ratio is linearly proportional toφq,so a more general form is

Therefore,the coefficientsKcan be expressed as

Eq.(15)was used to predict the value ofK.The variation of the coefficientKwith the covering angleθf(wàn)or different relative detonation distanceis shown in Fig.14.It can be seen that the value ofKdecreases with the increase ofθandWhenKis independent of the explosive mass,but is just a function ofθ.This indicates thatKcan be expressed as a function ofθand

According to Fig.14,the coefficientKcan be expressed as

Eventually,substituting Eq.(17)into Eq.(14),the final form of the empirical formula for midpoint deflection-thickness ratio is as follow:

sults agreed well with the simulation results,as shown in Fig.15 and Tab.4.It can be seen from Tab.4 that the maximum error ofδ/tby using the empirical formula and simulation is about 7%,implying a satisfactory prediction accuracy can be obtained by use of proposed empirical formula for computing the midpoint deflection of a plate subjected to fully-vented air blast.

Tab.4 Comparison of δ/t between the empirical formula predictions andthe simulation results with=1.6

Tab.4 Comparison of δ/t between the empirical formula predictions andthe simulation results with=1.6

-h 0 δ/t Error/%1.195-2.602-7.177-4.598 1.560 5.210 6.024 5.190 3.293 1.179-0.770-2.247-2.885-2.530-1.353-0.010 0.480 0.215-0.023-0.215-0.377-0.532 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 θ/rad 0.559 0.582 0.607 0.663 0.729 0.806 0.896 1.002 1.123 1.261 1.412 1.571 1.730 1.881 2.018 2.140 2.246 2.336 2.413 2.478 2.535 2.583 FE method 2.635 3.032 3.410 4.089 4.794 5.597 6.551 7.618 8.765 9.904 10.955 11.838 12.477 12.844 12.982 13.009 13.011 13.010 13.010 13.009 13.008 13.009 Empirical formula 2.666 2.953 3.165 3.901 4.868 5.888 6.946 8.013 9.053 10.021 10.871 11.572 12.117 12.519 12.806 13.007 13.073 13.038 13.007 12.981 12.959 12.940

Fig.15 Comparison of K between the empirical formula predictions and the simulation results with

2.3 Scope of applicability

The coefficientKobtained from simulation results and that from Eq.(17)with different explosive masses whenare compared in Fig.16.It is shown that Eq.(17)predicts the coefficientK

Fig.16 Comparison of K between the empirical formula and the simulation results for different explosive masses

with a good accuracy over a certain range of explosive mass,e.g.m=50 g,60 g,and 70 g.When the mass of explosive is small,the error becomes significant.For instance,the relative large error can be seen at a low covering angle(θ＜0.9 rad)whenm=40 g.Whenm=30 g,the prediction result is significantly unreliable withθ＜1 rad,but is accurate when the covering angleθis larger than 1 rad.

Fig.17 shows the relationship between the midpoint deflection-thickness ratio and the dimen-sionless damage number of the fixed square plate without surroundings under air blast withIt can be found that the deformation of the square plate can be divided into two stages with the increase of impact load:bending deformation and tensile deformation.An exponential relationship between bending deformation and dimensionless damage number can be drawn,while the relationship between the tensile deformation and dimensionless damage number is linear as shown in Eq.(14).This explains why the empirical formula underestimates the value of the coefficientKwhen the explosive mass is small.This is because the plate deformation is in the bending deformation state when the impact load is small.The linear relationship in Eq.(14)can not predict the exponential relationship accurately.Therefore,the proposed formula Eq.(18),which is based on Eq.(14),is only suitable for the tensile deformation stage when the impact load is relatively large.For the bending deformation stage,the problem needs to be treated separately and another formula should be derived.

Fig.17 Ratio of deflectionand thicknessversus dimensionlessdamage number(1.0,h=0)

3 Conclusions

A series of simulations were carried out to investigate the effect of vertical surroundings on the dynamic response of a fixed square plate subjected to fully-vented blast.The influence of the parameters including covering angleθ,explosive massm,plate thicknesstand relative detonation distanceon the explosion impulse and midpoint deflection were evaluated.Based on the investigations,the following conclusions are drawn:

(1)The existence of vertical surroundings leads to an increase of the explosion impulse acting on the target plate.The explosion impulseIincreases with the height of the surroundings.Once the height reaches up to a critical value(θ=2.2 rad),the value ofIreaches its maximum value and keeps a constant.It should be noted that the effect of plate thickness on the impulseIcan be ignored.

(2)The dimensionless explosion impulseis a function of the covering angleθand the relative detonation distancebut is independent of the explosive mass and plate thickness.An empirical formula was proposed to predict dimensionless explosion impulse.

(3)An improved empirical formula considering the influence of covering angleθand the relative detonation distanceis proposed to predict the midpoint deflection-thickness ratio of the fixed square plate with vertical surroundings under fully-vented blast load.

(4)The applicability of the proposed deflection-thickness ratio empirical formula is discussed.The deformation of the plate can be divided into two stages with the increase of impact load:bending deformation and tensile deformation.The present empirical formula is only applicable in the tensile deformation stage.