Research ArticleA Predictive Model for Damage Assessment and Deformation inBlast Walls Resulted by Hydrocarbon Explosions

Majid Aleyaasin

Lecturer School of Engineering University of Aberdeen Aberdeen AB24 3UE UK

Correspondence should be addressed to Majid Aleyaasin eng780abdnacuk

Received 18 February 2019 Revised 7 May 2019 Accepted 10 June 2019 Published 4 July 2019

Academic Editor Chiara Bedon

Copyright copy 2019 Majid Aleyaasin )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper a new method is developed to find the ductility ratio in blast walls resulted by hydrocarbon explosions In thismethod only the explosion energy and distance from the centre of explosion are required to find the damage by using simplepredictive models in terms of empirical-type formulas )e explosion model herein is a TNO multiphysic method )is providesthe maximum overpressure and pulse duration in terms of the explosion length and distance from explosion centre )ereafterthe obtained results are combined with the SDOFmodel of the blast wall to determine the ductility ratio and the damage By usingadvanced optimisation techniques two types of predictive models are found In the first model the formula is found in terms of 2parameters of explosion length and distance from explosion centre However the 2nd model has 3 parameters of explosion lengthdistance and also the natural period of the blast wall )ese predictive models are then used to find explosion damages andductility ratio )e results are compared with FEM analysis and pressure-impulse (P-I) method It is shown that both types ofmodels fit well with the outputs of the simulation Moreover results of both models are close to FEM analysis )e comparisontables provided in this paper show that in the asymptotic region of P-I diagrams results are not accurate )erefore this newmethod is superior to classical pressure-impulse (P-I) diagrams in the literature Advantage of the newmethod is the easy damageassessment by using simple empirical-type formulas )erefore the researchers can use the method in this paper for damageassessment in other types of blast resistive structures

1 Introduction

Blast walls are sacrificial barriers to protect offshore struc-tures when subjected to hydrocarbon explosions Substantialresearch has been performed to develop a code of practicefor design of such structures [1] )e theoretical foundationsfor designing blast-resistive structures and blast walls can befound in [2 3]

An analytical method based on plate theory for blastwall design [4] is rarely used in the literature since theresults of those studies cannot be used directly as a designcode However linear and nonlinear finite elements havebeen used significantly (for example [5 6]) )ey areapplicable in cases where batch simulations enable cost-benefit analysis [7]

Presently the dominant approach is a single degree Offreedom (SDOF) method [1ndash3] and leads to some designcurves known as Biggrsquos chart )ey appeared first in a

well-known book [8] but originated from the initial at-tempt by Newmark [9] )is SDOF method enables thefamous pressure-impulse (P-I) diagrams which was firstintroduced in [9] to be constructed [1ndash3]

)ese P-I diagrams strongly depend on pressure versustime expression (pulse shape) of the explosion [10] andtogether with SDOF modelling they are used to find theblast response of complex of structures such as cable-supported facades [11] Both SDOF-type model [12] andcontinuous beam model [13] are used for developing P-Idiagrams Recently it is shown that batch finite elementsimulations [14] cannot lead to P-I diagram unless pre-liminary information regarding SDOF parameters isavailable

Regardless of importance of the P-I diagrams in thedamage assessment they are not straightforward and thedesigner needs substantial information about the calculationof the explosive loads and pulse shape to be able to use P-I

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 5129274 13 pageshttpsdoiorg10115520195129274

diagrams in damage assessment)ere is not any attempt (ornew method) that directly connects intensity of explosion tothe resulted damage and deflection in the blast walls Re-cently the author looked at this important issue where inthe vicinity of box girders [15] TNT explosions may occurSince the possibility of hydrocarbon explosions are muchhigher than any terrorist activities blast walls are used inmany offshore structures )erefore any research regardingthis topic is justifiable

In this article the explosive physics known as themultienergy method known as TNO [16] and furthermodels fitted into it [17] is combined with the SDOFmethod for deformation of the blast walls )ereafter thedeformation and ductility for both rigid plastic models andelastic-plastic models are determined in each distance andexplosion length )en outcomes of the batch simulationsare exported to advanced optimisation programs to developtwo types of predictive models expressed by using simpleempirical-type formulas

Using any of the models in this paper the designer canfind the deformation (or ductility) from the intensity ofexplosions (explosion length) distance of the blast wall fromexplosion centre and natural period of the blast wall As faras the author is aware this new method is the easiest one forpredicting the damage in the blast wall thereby declaring theexplosion resistance)e knowledge about explosion physicsis embedded in the formulas)erefore it is an excellent toolfor preliminary analysis of the blast wall

In a case study in the asymptotic region of the P-Idiagram it is shown that while P-I provides inaccurateresults this method leads to accurate results when it iscompared with FEM simulation of the blast wall )ereforethe approach herein can be extended to other types ofstructures in future to replace P-I diagrams (or FEM) forpredicting the damage

2 Overpressure History in Explosions

When hydrocarbon mass mc (in kg) with heat energy ΔHc(Joulekg) causes an explosion with efficiency η the resultedexplosive energy E0 will be

E0 ηmcΔHc (1)

In the TNO multienergy method [16] an explosionlength is defined by

R0 E0

p01113888 1113889

13

(2a)

where p0 is the atmospheric pressure (in Pa) therefore R0truly has units of the length (m) If Rs is the distance from theexplosion centre (m) dimensionless R will be defined by

R Rs

R0

E0

p01113888 1113889

minus13

Rs (2b)

)en the overpressure pmax (in bar ie dimensionless)and explosion pulse duration t+ (dimensionless) can befound from TNO charts [16] In those charts the over-pressure and duration can be found from the curves

designated by the level of the explosion )e charts aredeveloped from computer simulations performed ineighties and are strongly applicable to hydrocarbon ex-plosions Due to the importance of the TNO charts re-searchers produced curve fitted formulas for the data inthose charts )ese formulas are given in [17] via thefollowing equation

06leRle 30 pmax 00605Rminus099

pmaxt+ 00605Rminus099 Level 3

06leRle 100 pmax 0301Rminus111

pmaxt+ 0114Rminus103 Level 6

2leRle 100 pmax 0318Rminus113

pmaxt+ 0114Rminus103 Level 9

(3)

It should be reminded that another valuable software isprovided for blast waves (for example [18 19]) but notreformulated for designers yet (such as (3)) In the aboveexpressions t+ is dimensionless overpressure pulse durationgiven in [16]

t+ tdC0

R0 (4)

where td is the overpressure duration in sec and C0 is thesound velocity at atmospheric conditions in msec Majorityof explosions will fall into all of the three levels in (3) It isrecommended that the overpressure and duration should becomputed in each level and the average value should betaken into consideration [17]

)e author herein produced the overpressure contoursin terms of R0 and Rs which are two important parameters inany explosions )ey are shown in Figure 1 and are used inthe next part of the paper for developing the new method

3 SDOF Model for Blast Walls

)e typical geometry of the cross section of a blast wall [1] issimilar to (a) in Figure 2

)e finite element analysis shows [5] the deformationpattern resulted by an explosion by using shell elementswhich is similar to Figure 3

)e front view of a typical blast wall [1] is shown in (b) inFigure 2 )e main parameter is the pitch p that is shown in(a))ewall is connected to the structure by upper and lowersupports shown in (c) When overpressure pmax is applied tothe wall with uniform distribution the upper and lowersupports with thicknesses tU and tL (in m) (see (c) inFigure 2) have equivalent lengths LU and LL shown in(Figure 15) )ey will yield since they have limited yieldstress flowasty (Pa) )e total length is L (in m) and (McRd)U and(McRd)L are the yield moments (per length ie in N) of theupper and lower supports and are given by the followingequations [1ndash3]

McRd1113872 1113873U t2Uflowasty

4

McRd1113872 1113873L t2Lflowasty

4

(5)

McRd or the plastic bending moment (per unit length) ofthe main wall is given by (6) It depends on the details of the

2 Advances in Civil Engineering

cross section in (a) in Figure 2 which are designated by twoparametersWply (plastic section modulus) and flowasty (materialyield stress) of the cross section

McRd Wplyf

lowastyKFKVM

p (6)

KF and KVM in (6) are attening and shear correctionfactors described in [1] e equivalent length of the blastwall LE is less than the total length L and can be found by

LE 2L

1 + McRd( )LMcRd( )radic

+1 + McRd( )UMcRd( )radic

(7)

Derivation of (7) is shown in Appendix A and instead oftotal length LE will be used in all calculations regarding theblast wall For example the stiness per unit length will begiven as shown in [1ndash3] as follows

k 384EI5L3Ep

(8)

e corrected stiness of wall kR is recommended in [1]to correct (8) resulted from beam theory which is

kR kLE

16Lminus 06LE (9)

Equations (8) (9) and others that follow are true whenthe SDOF method is chosen as a route of the analysis wherethe beam simplication and can be justied is is alsocurrent practice for the preliminary design of blast walls

[1ndash3] However for the detail of the buckling pattern similarto Figure 3 the beam model simplication is not appro-priate According to rigid plastic theory in structures themaximum resistance of a beam cross section Rm [2 3] isgiven by

Rm 8McRd

LE (10)

is Rm is dened for nding maximum elastic de-formation of the wall yel [1ndash3] by using the followingformula

yel 8McRd

kRLE (11)

However if the maximum blast load F1 given by thefollowing equation exceeds Rm the wall deforms plastically

F1 Aspmax (12)

In (12) As is the projected blast area per pitch in Figure 4For further clarication this area with the pressure pmaxapplied to it is shown in Figure 4

e deformation is allowed up to the ductility limit eductility μ is very important in design of structures underextreme and blast loading [2 3 8] and is the ratio ofmaximum plastic deformation to the elastic limit yel givenby

μ ymax

yel (13)

024534

042362

0601

907

8018

0958

46

1136

7

1315

1493

316

716

1849

820

281

5

10

15

20

25

30

35

40

45

50

Expl

osio

n le

ngth

R0 (

m)

10 155Distance Rs (m)

(a)

50

15

R0 (m)

R s (m)10

0 5

0

05

1

15

2

25

3

Ove

rpre

ssur

e in

bar

(b)

Figure 1 Overpressure in the TNO model for explosion (a) Overpressure contours in bar (b) Average value of levels 3 6 and 9

Advances in Civil Engineering 3

e backbone of the SDOF model relies on the naturalperiod of free structural vibration T [2 3 8] of the blast wallwhich will be given by

T 2π

MKLM

pkR

radic

(14)

In (14)M is the blast wall mass (for one pitch) andKLMis the correction factor for the distributed mass In Ap-pendix B it shows that for rigid plastic theory based onplastic hinge assumption [2 3] we nd that KLM 0333However in the current practice [1] designers use highervalues without any justication Part of this article

(a)

(c)

(b)

Plate thickness

pmax

tu

tL

LNeutral axis

Top girder

Lower deck

Corrugated profile

yФ

p

Figure 2 (a) Cross section of the (b) blast wall (front view) and (c) upper and lower supports

Figure 3 Deformation pattern from FEM analysis

Pmax

Pmax

Pmax

Pmax

φ

Pmax

Figure 4 Applied pressure on the wall surface

4 Advances in Civil Engineering

investigates how this apparent inconsistency can aect theductility results

e SDOF modelling is well known by Biggsrsquo chart sinceit appeared in a famous book [8] However the initial re-search is done by Newmark who is one of the pioneers instructural dynamic He summarised Biggrsquos chart a decadebefore it is seen in [8] in his famous paper [9] by using thefollowing formula

F1

Rm

2μminus 1radic

tdT( )π+(1minus(12μ)) tdT( )

tdT( ) + 07 (15)

All the parameters in (15) are described in previousformulas When an explosion with length R0 occurs atdistance Rs one can nd the preliminary ductility curvesFor a particular blast wall that is designed by a manufacturergeometrical and material details are available erefore theductility contour can be constructed easily from (15)without using the pressure-impulse diagram of the blastwall

4 Numerical Example

For a steel blast wall with pitch p 12 meter the cross-sectional dimensions are shown in Figure 5 It is one of theexisting proles of the blast wall that is described in [1]

e second moment of the cross section I 8767 times 10minus5 m4 the section modulusWply 437times 10minus4 m3mass per pitchM 410 kg thicknesses of the upper and lowersupports tU 12mm and tL 10mm and Youngrsquos modulusE 210GPa and yield stress flowasty 400MPa the lengthL 3m and the correction factors [1] KF 09 andKVM 095 In Figure 6 the ductility is shown which is theresult of substantial simulations of the SDOF model for thisblast wall

Figure 6 is prepared for KLM 085 as recommended in[1] and is not the result of rigid plastic theory Figure 6 isdrawn in range 15ltR0 lt 25 and 5ltRs lt 10 and the con-tours seem linear and visible However for higher rangesvisibility and linearity cannot be observed

5 Model with Two Parameters

A nonlinear predictive model of Figure 6 with two pa-rameters R0 and Rs (both explosion related) can be suggestedin this form

μ Cμ2Rα0R

βs (16)

For example the higher range estimation of ductility forcan be replaced by the following approximate expression

μ 10008R424340 Rminus62520s 20ltR0 lt 50 10ltRs lt 15

(17)In Figure 7 the computed ductility ratio and the esti-

mated ductility ratio in (17) are drawn together It can beconcluded that in higher ductility ratios where severeplastic deformation occurs the estimated ductility is veryclose to the computed ductility In (17) only explosion-

related parameters are used ree parameter models will bediscussed as well

6 Rigid Plastic Modelling

Rigid plastic theory [2 3] assumes plastic hinge at themidlength of the blast wall In appendix B it is shown thatin such situation the equivalent mass Me M3 andKLM 0333 e damage calculation will be straightfor-ward because the calculations regarding overpressure andduration remain the same as the ones used for producingFigure 7 Obviously if we assume KLM 0333 the resultswill change which is shown in Figure 8e region in whichductility ratio is below 1 remains elastic and by producingsuch contour maps the pressure-impulse diagram is notrequired If we compare Figure 6 in which peak de-formation ymax 375yel with Figure 8 in whichymax 503yel we can conclude that considering KLM 0333 (rigid plastic model) provides conservative estima-tion for ductility

7 Model with Three Parameters

A nonlinear predictive model with three parameters R0 Rs(explosion related) and T in (14) which are blast wall relatedcan be suggested as in the following form

μ Cμ3Rα0R

βsT

c (18)

e parameters Cμ3 α β and c in (18) can be found bytaking the logarithm for that expression that will change itinto

log(μ) log Cμ3( ) + α log R0( ) + β log Rs( ) + c log(T)(19)

e above expression enables the linear regressiontechniques to be implemented for nding the parametersCμ3 α β and c ese parameters can be found by usingnonlinear regression analysis Moreover the powerfulNelderndashMead algorithm [20] which is built in MATLAB isalso used to nd the fractional powers α β and c in (18)Finally the numerical expression of (18) when KLM 0333(rigid plastic modelling) will be in the following form

μ 10008R444830 Rminus61082s T02698 20ltR0 lt 50 10ltRs lt 15

(20)

t = 5 mm

5631deg

pitch = 1200 mm

300 mm

200 mm200 mm400 mm200 mm200 mm

Figure 5 A typical cross section (one pitch) of a blast wall [1]

Advances in Civil Engineering 5

In Figure 9 the computed ductility ratio and the estimatedductility ratio in (20) are drawn together It can be concludedthat in higher ductility ratios where severe plastic de-formation occurs the estimated ductility is very close to thecomputed ductility In (20) explosion-related parameters plusblast wall natural period are used ree-parameter modelsuse KLM 0333 (rigid plastic modelling) because of itsconservativeness in estimation of the maximum ductility

e author has suggested many other forms for theregression analysis using advanced optimisation techniques

[20] and so far he has not found better forms than (20) forthe 3-parameter-type model and (17) for the 2-parameter-type model It is quite possible that some other forms withclosest t may be found by further research

8 Comparison of the Results

Consider that an explosion with eective energyE0 9500MJ occurs at distance Rs 12m from the ex-plosion centre According to parameters (2a) (2b) and (3)

Ductility ratio for mass factor 085

12291

1458216874

1916521456

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

2

Duc

tility

ratio

9

Ductility ratio for mass factor 085

R0 (m)20 8

Rs (m)

4

7615 5

(b)

Figure 6 Contours for the ductility ratio

050

2

4

15

6

Duc

tility

ratio

40

8

14

Estimated (two parameters) versus computed surface

Explosion length (m)

10

13

Distance from centre (m)

12

30 1211

20 10

Figure 7 Computed versus estimated ductility ratio

6 Advances in Civil Engineering

e overpressure is the average value of the explosion levels3 and 9 and 6 in (3)

R0 45629m

R 0263

pmax 0997 bar

(21)

e elastic deformation from (11) is yel 78mmwhereas themaximumdeection at themiddle section ymaxin (12) can be found by knowing about the ductility ratio

Since the velocity of sound in the room temperaturecondition is C0 340msec from formula (4) we haveduration of the explosion pulse td 74msec whereas the

natural period of the blast wall herein which is given by using(14) is T 161msec

e pressure-impulse curve that introduced before is stillused for damage assessment for many structures ey are aseries of the asymptotic curves inscribed in the vertical andhorizontal asymptotes To nd the points on the curves eitherwe use analytical methods [21 22] or numerical methods [23]and sometimes FEManalysis [24] In the x-y plane the verticalaxis displays FmaxKeyel whereas horizontal axis displaysx Iyel

KeMeradic

I is the impulse and Fmax is the maximumexplosion forces With uniform overpressure they are

Fmax Aeffpmax

I 05Fmaxtd(22)

In [2 3] it can be shown that the equations of the verticaland horizontal asymptotes are in terms of the ductility ratioμ that is dened in (13) ie

I

yelKeMeradic

2μminus 1radic

Fmax

Keyel2μminus 12μ

(23)

Typical curves for elastic-plastic structures are shown inFigure 10 in which the ductility ratio can be found via in-terpolation e snapshot designated by the point shows thecoordinates Iyel

KeMeradic

13622 and FmaxKeyel 103that correspond to this particular explosion and we can ndthe ductility μ 724 as a result of this explosion

However the direct simulation in this paper shows thatμ 2414 It shows that the P-I method particularly in as-ymptotic ends are signicantly inaccurate e two

Ductility ratio for mass factor 0333

12516

12516

1503317549

2006522581

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

5

Duc

tility

ratio

9

Ductility ratio for mass factor 0333

R0 (m)20 8

Rs (m)

10

7615 5

(b)

Figure 8 Contours for the ductility ratio in the rigid plastic model

5015

40 14Explosion length (m)

13

Distance from centre (m)30 1211

20 10

0

2

4

6

8

10

12

Estimated (three parameters) versus computed surface

Duc

tility

ratio

Figure 9 Computed versus estimated ductility ratio

Advances in Civil Engineering 7

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

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cross section in (a) in Figure 2 which are designated by twoparametersWply (plastic section modulus) and flowasty (materialyield stress) of the cross section

McRd Wplyf

lowastyKFKVM

p (6)

KF and KVM in (6) are attening and shear correctionfactors described in [1] e equivalent length of the blastwall LE is less than the total length L and can be found by

LE 2L

1 + McRd( )LMcRd( )radic

+1 + McRd( )UMcRd( )radic

(7)

Derivation of (7) is shown in Appendix A and instead oftotal length LE will be used in all calculations regarding theblast wall For example the stiness per unit length will begiven as shown in [1ndash3] as follows

k 384EI5L3Ep

(8)

e corrected stiness of wall kR is recommended in [1]to correct (8) resulted from beam theory which is

kR kLE

16Lminus 06LE (9)

Equations (8) (9) and others that follow are true whenthe SDOF method is chosen as a route of the analysis wherethe beam simplication and can be justied is is alsocurrent practice for the preliminary design of blast walls

[1ndash3] However for the detail of the buckling pattern similarto Figure 3 the beam model simplication is not appro-priate According to rigid plastic theory in structures themaximum resistance of a beam cross section Rm [2 3] isgiven by

Rm 8McRd

LE (10)

is Rm is dened for nding maximum elastic de-formation of the wall yel [1ndash3] by using the followingformula

yel 8McRd

kRLE (11)

However if the maximum blast load F1 given by thefollowing equation exceeds Rm the wall deforms plastically

F1 Aspmax (12)

In (12) As is the projected blast area per pitch in Figure 4For further clarication this area with the pressure pmaxapplied to it is shown in Figure 4

e deformation is allowed up to the ductility limit eductility μ is very important in design of structures underextreme and blast loading [2 3 8] and is the ratio ofmaximum plastic deformation to the elastic limit yel givenby

μ ymax

yel (13)

024534

042362

0601

907

8018

0958

46

1136

7

1315

1493

316

716

1849

820

281

5

10

15

20

25

30

35

40

45

50

Expl

osio

n le

ngth

R0 (

m)

10 155Distance Rs (m)

(a)

50

15

R0 (m)

R s (m)10

0 5

0

05

1

15

2

25

3

Ove

rpre

ssur

e in

bar

(b)

Figure 1 Overpressure in the TNO model for explosion (a) Overpressure contours in bar (b) Average value of levels 3 6 and 9

Advances in Civil Engineering 3

e backbone of the SDOF model relies on the naturalperiod of free structural vibration T [2 3 8] of the blast wallwhich will be given by

T 2π

MKLM

pkR

radic

(14)

In (14)M is the blast wall mass (for one pitch) andKLMis the correction factor for the distributed mass In Ap-pendix B it shows that for rigid plastic theory based onplastic hinge assumption [2 3] we nd that KLM 0333However in the current practice [1] designers use highervalues without any justication Part of this article

(a)

(c)

(b)

Plate thickness

pmax

tu

tL

LNeutral axis

Top girder

Lower deck

Corrugated profile

yФ

p

Figure 2 (a) Cross section of the (b) blast wall (front view) and (c) upper and lower supports

Figure 3 Deformation pattern from FEM analysis

Pmax

Pmax

Pmax

Pmax

φ

Pmax

Figure 4 Applied pressure on the wall surface

4 Advances in Civil Engineering

investigates how this apparent inconsistency can aect theductility results

e SDOF modelling is well known by Biggsrsquo chart sinceit appeared in a famous book [8] However the initial re-search is done by Newmark who is one of the pioneers instructural dynamic He summarised Biggrsquos chart a decadebefore it is seen in [8] in his famous paper [9] by using thefollowing formula

F1

Rm

2μminus 1radic

tdT( )π+(1minus(12μ)) tdT( )

tdT( ) + 07 (15)

All the parameters in (15) are described in previousformulas When an explosion with length R0 occurs atdistance Rs one can nd the preliminary ductility curvesFor a particular blast wall that is designed by a manufacturergeometrical and material details are available erefore theductility contour can be constructed easily from (15)without using the pressure-impulse diagram of the blastwall

4 Numerical Example

For a steel blast wall with pitch p 12 meter the cross-sectional dimensions are shown in Figure 5 It is one of theexisting proles of the blast wall that is described in [1]

e second moment of the cross section I 8767 times 10minus5 m4 the section modulusWply 437times 10minus4 m3mass per pitchM 410 kg thicknesses of the upper and lowersupports tU 12mm and tL 10mm and Youngrsquos modulusE 210GPa and yield stress flowasty 400MPa the lengthL 3m and the correction factors [1] KF 09 andKVM 095 In Figure 6 the ductility is shown which is theresult of substantial simulations of the SDOF model for thisblast wall

Figure 6 is prepared for KLM 085 as recommended in[1] and is not the result of rigid plastic theory Figure 6 isdrawn in range 15ltR0 lt 25 and 5ltRs lt 10 and the con-tours seem linear and visible However for higher rangesvisibility and linearity cannot be observed

5 Model with Two Parameters

A nonlinear predictive model of Figure 6 with two pa-rameters R0 and Rs (both explosion related) can be suggestedin this form

μ Cμ2Rα0R

βs (16)

For example the higher range estimation of ductility forcan be replaced by the following approximate expression

μ 10008R424340 Rminus62520s 20ltR0 lt 50 10ltRs lt 15

(17)In Figure 7 the computed ductility ratio and the esti-

mated ductility ratio in (17) are drawn together It can beconcluded that in higher ductility ratios where severeplastic deformation occurs the estimated ductility is veryclose to the computed ductility In (17) only explosion-

related parameters are used ree parameter models will bediscussed as well

6 Rigid Plastic Modelling

Rigid plastic theory [2 3] assumes plastic hinge at themidlength of the blast wall In appendix B it is shown thatin such situation the equivalent mass Me M3 andKLM 0333 e damage calculation will be straightfor-ward because the calculations regarding overpressure andduration remain the same as the ones used for producingFigure 7 Obviously if we assume KLM 0333 the resultswill change which is shown in Figure 8e region in whichductility ratio is below 1 remains elastic and by producingsuch contour maps the pressure-impulse diagram is notrequired If we compare Figure 6 in which peak de-formation ymax 375yel with Figure 8 in whichymax 503yel we can conclude that considering KLM 0333 (rigid plastic model) provides conservative estima-tion for ductility

7 Model with Three Parameters

A nonlinear predictive model with three parameters R0 Rs(explosion related) and T in (14) which are blast wall relatedcan be suggested as in the following form

μ Cμ3Rα0R

βsT

c (18)

e parameters Cμ3 α β and c in (18) can be found bytaking the logarithm for that expression that will change itinto

log(μ) log Cμ3( ) + α log R0( ) + β log Rs( ) + c log(T)(19)

e above expression enables the linear regressiontechniques to be implemented for nding the parametersCμ3 α β and c ese parameters can be found by usingnonlinear regression analysis Moreover the powerfulNelderndashMead algorithm [20] which is built in MATLAB isalso used to nd the fractional powers α β and c in (18)Finally the numerical expression of (18) when KLM 0333(rigid plastic modelling) will be in the following form

μ 10008R444830 Rminus61082s T02698 20ltR0 lt 50 10ltRs lt 15

(20)

t = 5 mm

5631deg

pitch = 1200 mm

300 mm

200 mm200 mm400 mm200 mm200 mm

Figure 5 A typical cross section (one pitch) of a blast wall [1]

Advances in Civil Engineering 5

In Figure 9 the computed ductility ratio and the estimatedductility ratio in (20) are drawn together It can be concludedthat in higher ductility ratios where severe plastic de-formation occurs the estimated ductility is very close to thecomputed ductility In (20) explosion-related parameters plusblast wall natural period are used ree-parameter modelsuse KLM 0333 (rigid plastic modelling) because of itsconservativeness in estimation of the maximum ductility

e author has suggested many other forms for theregression analysis using advanced optimisation techniques

[20] and so far he has not found better forms than (20) forthe 3-parameter-type model and (17) for the 2-parameter-type model It is quite possible that some other forms withclosest t may be found by further research

8 Comparison of the Results

Consider that an explosion with eective energyE0 9500MJ occurs at distance Rs 12m from the ex-plosion centre According to parameters (2a) (2b) and (3)

Ductility ratio for mass factor 085

12291

1458216874

1916521456

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

2

Duc

tility

ratio

9

Ductility ratio for mass factor 085

R0 (m)20 8

Rs (m)

4

7615 5

(b)

Figure 6 Contours for the ductility ratio

050

2

4

15

6

Duc

tility

ratio

40

8

14

Estimated (two parameters) versus computed surface

Explosion length (m)

10

13

Distance from centre (m)

12

30 1211

20 10

Figure 7 Computed versus estimated ductility ratio

6 Advances in Civil Engineering

e overpressure is the average value of the explosion levels3 and 9 and 6 in (3)

R0 45629m

R 0263

pmax 0997 bar

(21)

e elastic deformation from (11) is yel 78mmwhereas themaximumdeection at themiddle section ymaxin (12) can be found by knowing about the ductility ratio

Since the velocity of sound in the room temperaturecondition is C0 340msec from formula (4) we haveduration of the explosion pulse td 74msec whereas the

natural period of the blast wall herein which is given by using(14) is T 161msec

e pressure-impulse curve that introduced before is stillused for damage assessment for many structures ey are aseries of the asymptotic curves inscribed in the vertical andhorizontal asymptotes To nd the points on the curves eitherwe use analytical methods [21 22] or numerical methods [23]and sometimes FEManalysis [24] In the x-y plane the verticalaxis displays FmaxKeyel whereas horizontal axis displaysx Iyel

KeMeradic

I is the impulse and Fmax is the maximumexplosion forces With uniform overpressure they are

Fmax Aeffpmax

I 05Fmaxtd(22)

In [2 3] it can be shown that the equations of the verticaland horizontal asymptotes are in terms of the ductility ratioμ that is dened in (13) ie

I

yelKeMeradic

2μminus 1radic

Fmax

Keyel2μminus 12μ

(23)

Typical curves for elastic-plastic structures are shown inFigure 10 in which the ductility ratio can be found via in-terpolation e snapshot designated by the point shows thecoordinates Iyel

KeMeradic

13622 and FmaxKeyel 103that correspond to this particular explosion and we can ndthe ductility μ 724 as a result of this explosion

However the direct simulation in this paper shows thatμ 2414 It shows that the P-I method particularly in as-ymptotic ends are signicantly inaccurate e two

Ductility ratio for mass factor 0333

12516

12516

1503317549

2006522581

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

5

Duc

tility

ratio

9

Ductility ratio for mass factor 0333

R0 (m)20 8

Rs (m)

10

7615 5

(b)

Figure 8 Contours for the ductility ratio in the rigid plastic model

5015

40 14Explosion length (m)

13

Distance from centre (m)30 1211

20 10

0

2

4

6

8

10

12

Estimated (three parameters) versus computed surface

Duc

tility

ratio

Figure 9 Computed versus estimated ductility ratio

Advances in Civil Engineering 7

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

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Submit your manuscripts atwwwhindawicom

e backbone of the SDOF model relies on the naturalperiod of free structural vibration T [2 3 8] of the blast wallwhich will be given by

T 2π

MKLM

pkR

radic

(14)

In (14)M is the blast wall mass (for one pitch) andKLMis the correction factor for the distributed mass In Ap-pendix B it shows that for rigid plastic theory based onplastic hinge assumption [2 3] we nd that KLM 0333However in the current practice [1] designers use highervalues without any justication Part of this article

(a)

(c)

(b)

Plate thickness

pmax

tu

tL

LNeutral axis

Top girder

Lower deck

Corrugated profile

yФ

p

Figure 2 (a) Cross section of the (b) blast wall (front view) and (c) upper and lower supports

Figure 3 Deformation pattern from FEM analysis

Pmax

Pmax

Pmax

Pmax

φ

Pmax

Figure 4 Applied pressure on the wall surface

4 Advances in Civil Engineering

investigates how this apparent inconsistency can aect theductility results

e SDOF modelling is well known by Biggsrsquo chart sinceit appeared in a famous book [8] However the initial re-search is done by Newmark who is one of the pioneers instructural dynamic He summarised Biggrsquos chart a decadebefore it is seen in [8] in his famous paper [9] by using thefollowing formula

F1

Rm

2μminus 1radic

tdT( )π+(1minus(12μ)) tdT( )

tdT( ) + 07 (15)

All the parameters in (15) are described in previousformulas When an explosion with length R0 occurs atdistance Rs one can nd the preliminary ductility curvesFor a particular blast wall that is designed by a manufacturergeometrical and material details are available erefore theductility contour can be constructed easily from (15)without using the pressure-impulse diagram of the blastwall

4 Numerical Example

For a steel blast wall with pitch p 12 meter the cross-sectional dimensions are shown in Figure 5 It is one of theexisting proles of the blast wall that is described in [1]

e second moment of the cross section I 8767 times 10minus5 m4 the section modulusWply 437times 10minus4 m3mass per pitchM 410 kg thicknesses of the upper and lowersupports tU 12mm and tL 10mm and Youngrsquos modulusE 210GPa and yield stress flowasty 400MPa the lengthL 3m and the correction factors [1] KF 09 andKVM 095 In Figure 6 the ductility is shown which is theresult of substantial simulations of the SDOF model for thisblast wall

Figure 6 is prepared for KLM 085 as recommended in[1] and is not the result of rigid plastic theory Figure 6 isdrawn in range 15ltR0 lt 25 and 5ltRs lt 10 and the con-tours seem linear and visible However for higher rangesvisibility and linearity cannot be observed

5 Model with Two Parameters

A nonlinear predictive model of Figure 6 with two pa-rameters R0 and Rs (both explosion related) can be suggestedin this form

μ Cμ2Rα0R

βs (16)

For example the higher range estimation of ductility forcan be replaced by the following approximate expression

μ 10008R424340 Rminus62520s 20ltR0 lt 50 10ltRs lt 15

(17)In Figure 7 the computed ductility ratio and the esti-

mated ductility ratio in (17) are drawn together It can beconcluded that in higher ductility ratios where severeplastic deformation occurs the estimated ductility is veryclose to the computed ductility In (17) only explosion-

related parameters are used ree parameter models will bediscussed as well

6 Rigid Plastic Modelling

Rigid plastic theory [2 3] assumes plastic hinge at themidlength of the blast wall In appendix B it is shown thatin such situation the equivalent mass Me M3 andKLM 0333 e damage calculation will be straightfor-ward because the calculations regarding overpressure andduration remain the same as the ones used for producingFigure 7 Obviously if we assume KLM 0333 the resultswill change which is shown in Figure 8e region in whichductility ratio is below 1 remains elastic and by producingsuch contour maps the pressure-impulse diagram is notrequired If we compare Figure 6 in which peak de-formation ymax 375yel with Figure 8 in whichymax 503yel we can conclude that considering KLM 0333 (rigid plastic model) provides conservative estima-tion for ductility

7 Model with Three Parameters

A nonlinear predictive model with three parameters R0 Rs(explosion related) and T in (14) which are blast wall relatedcan be suggested as in the following form

μ Cμ3Rα0R

βsT

c (18)

e parameters Cμ3 α β and c in (18) can be found bytaking the logarithm for that expression that will change itinto

log(μ) log Cμ3( ) + α log R0( ) + β log Rs( ) + c log(T)(19)

e above expression enables the linear regressiontechniques to be implemented for nding the parametersCμ3 α β and c ese parameters can be found by usingnonlinear regression analysis Moreover the powerfulNelderndashMead algorithm [20] which is built in MATLAB isalso used to nd the fractional powers α β and c in (18)Finally the numerical expression of (18) when KLM 0333(rigid plastic modelling) will be in the following form

μ 10008R444830 Rminus61082s T02698 20ltR0 lt 50 10ltRs lt 15

(20)

t = 5 mm

5631deg

pitch = 1200 mm

300 mm

200 mm200 mm400 mm200 mm200 mm

Figure 5 A typical cross section (one pitch) of a blast wall [1]

Advances in Civil Engineering 5

In Figure 9 the computed ductility ratio and the estimatedductility ratio in (20) are drawn together It can be concludedthat in higher ductility ratios where severe plastic de-formation occurs the estimated ductility is very close to thecomputed ductility In (20) explosion-related parameters plusblast wall natural period are used ree-parameter modelsuse KLM 0333 (rigid plastic modelling) because of itsconservativeness in estimation of the maximum ductility

e author has suggested many other forms for theregression analysis using advanced optimisation techniques

[20] and so far he has not found better forms than (20) forthe 3-parameter-type model and (17) for the 2-parameter-type model It is quite possible that some other forms withclosest t may be found by further research

8 Comparison of the Results

Consider that an explosion with eective energyE0 9500MJ occurs at distance Rs 12m from the ex-plosion centre According to parameters (2a) (2b) and (3)

Ductility ratio for mass factor 085

12291

1458216874

1916521456

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

2

Duc

tility

ratio

9

Ductility ratio for mass factor 085

R0 (m)20 8

Rs (m)

4

7615 5

(b)

Figure 6 Contours for the ductility ratio

050

2

4

15

6

Duc

tility

ratio

40

8

14

Estimated (two parameters) versus computed surface

Explosion length (m)

10

13

Distance from centre (m)

12

30 1211

20 10

Figure 7 Computed versus estimated ductility ratio

6 Advances in Civil Engineering

e overpressure is the average value of the explosion levels3 and 9 and 6 in (3)

R0 45629m

R 0263

pmax 0997 bar

(21)

e elastic deformation from (11) is yel 78mmwhereas themaximumdeection at themiddle section ymaxin (12) can be found by knowing about the ductility ratio

Since the velocity of sound in the room temperaturecondition is C0 340msec from formula (4) we haveduration of the explosion pulse td 74msec whereas the

natural period of the blast wall herein which is given by using(14) is T 161msec

e pressure-impulse curve that introduced before is stillused for damage assessment for many structures ey are aseries of the asymptotic curves inscribed in the vertical andhorizontal asymptotes To nd the points on the curves eitherwe use analytical methods [21 22] or numerical methods [23]and sometimes FEManalysis [24] In the x-y plane the verticalaxis displays FmaxKeyel whereas horizontal axis displaysx Iyel

KeMeradic

I is the impulse and Fmax is the maximumexplosion forces With uniform overpressure they are

Fmax Aeffpmax

I 05Fmaxtd(22)

In [2 3] it can be shown that the equations of the verticaland horizontal asymptotes are in terms of the ductility ratioμ that is dened in (13) ie

I

yelKeMeradic

2μminus 1radic

Fmax

Keyel2μminus 12μ

(23)

Typical curves for elastic-plastic structures are shown inFigure 10 in which the ductility ratio can be found via in-terpolation e snapshot designated by the point shows thecoordinates Iyel

KeMeradic

13622 and FmaxKeyel 103that correspond to this particular explosion and we can ndthe ductility μ 724 as a result of this explosion

However the direct simulation in this paper shows thatμ 2414 It shows that the P-I method particularly in as-ymptotic ends are signicantly inaccurate e two

Ductility ratio for mass factor 0333

12516

12516

1503317549

2006522581

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

5

Duc

tility

ratio

9

Ductility ratio for mass factor 0333

R0 (m)20 8

Rs (m)

10

7615 5

(b)

Figure 8 Contours for the ductility ratio in the rigid plastic model

5015

40 14Explosion length (m)

13

Distance from centre (m)30 1211

20 10

0

2

4

6

8

10

12

Estimated (three parameters) versus computed surface

Duc

tility

ratio

Figure 9 Computed versus estimated ductility ratio

Advances in Civil Engineering 7

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

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investigates how this apparent inconsistency can aect theductility results

e SDOF modelling is well known by Biggsrsquo chart sinceit appeared in a famous book [8] However the initial re-search is done by Newmark who is one of the pioneers instructural dynamic He summarised Biggrsquos chart a decadebefore it is seen in [8] in his famous paper [9] by using thefollowing formula

F1

Rm

2μminus 1radic

tdT( )π+(1minus(12μ)) tdT( )

tdT( ) + 07 (15)

All the parameters in (15) are described in previousformulas When an explosion with length R0 occurs atdistance Rs one can nd the preliminary ductility curvesFor a particular blast wall that is designed by a manufacturergeometrical and material details are available erefore theductility contour can be constructed easily from (15)without using the pressure-impulse diagram of the blastwall

4 Numerical Example

For a steel blast wall with pitch p 12 meter the cross-sectional dimensions are shown in Figure 5 It is one of theexisting proles of the blast wall that is described in [1]

e second moment of the cross section I 8767 times 10minus5 m4 the section modulusWply 437times 10minus4 m3mass per pitchM 410 kg thicknesses of the upper and lowersupports tU 12mm and tL 10mm and Youngrsquos modulusE 210GPa and yield stress flowasty 400MPa the lengthL 3m and the correction factors [1] KF 09 andKVM 095 In Figure 6 the ductility is shown which is theresult of substantial simulations of the SDOF model for thisblast wall

Figure 6 is prepared for KLM 085 as recommended in[1] and is not the result of rigid plastic theory Figure 6 isdrawn in range 15ltR0 lt 25 and 5ltRs lt 10 and the con-tours seem linear and visible However for higher rangesvisibility and linearity cannot be observed

5 Model with Two Parameters

A nonlinear predictive model of Figure 6 with two pa-rameters R0 and Rs (both explosion related) can be suggestedin this form

μ Cμ2Rα0R

βs (16)

For example the higher range estimation of ductility forcan be replaced by the following approximate expression

μ 10008R424340 Rminus62520s 20ltR0 lt 50 10ltRs lt 15

(17)In Figure 7 the computed ductility ratio and the esti-

mated ductility ratio in (17) are drawn together It can beconcluded that in higher ductility ratios where severeplastic deformation occurs the estimated ductility is veryclose to the computed ductility In (17) only explosion-

related parameters are used ree parameter models will bediscussed as well

6 Rigid Plastic Modelling

Rigid plastic theory [2 3] assumes plastic hinge at themidlength of the blast wall In appendix B it is shown thatin such situation the equivalent mass Me M3 andKLM 0333 e damage calculation will be straightfor-ward because the calculations regarding overpressure andduration remain the same as the ones used for producingFigure 7 Obviously if we assume KLM 0333 the resultswill change which is shown in Figure 8e region in whichductility ratio is below 1 remains elastic and by producingsuch contour maps the pressure-impulse diagram is notrequired If we compare Figure 6 in which peak de-formation ymax 375yel with Figure 8 in whichymax 503yel we can conclude that considering KLM 0333 (rigid plastic model) provides conservative estima-tion for ductility

7 Model with Three Parameters

A nonlinear predictive model with three parameters R0 Rs(explosion related) and T in (14) which are blast wall relatedcan be suggested as in the following form

μ Cμ3Rα0R

βsT

c (18)

e parameters Cμ3 α β and c in (18) can be found bytaking the logarithm for that expression that will change itinto

log(μ) log Cμ3( ) + α log R0( ) + β log Rs( ) + c log(T)(19)

e above expression enables the linear regressiontechniques to be implemented for nding the parametersCμ3 α β and c ese parameters can be found by usingnonlinear regression analysis Moreover the powerfulNelderndashMead algorithm [20] which is built in MATLAB isalso used to nd the fractional powers α β and c in (18)Finally the numerical expression of (18) when KLM 0333(rigid plastic modelling) will be in the following form

μ 10008R444830 Rminus61082s T02698 20ltR0 lt 50 10ltRs lt 15

(20)

t = 5 mm

5631deg

pitch = 1200 mm

300 mm

200 mm200 mm400 mm200 mm200 mm

Figure 5 A typical cross section (one pitch) of a blast wall [1]

Advances in Civil Engineering 5

In Figure 9 the computed ductility ratio and the estimatedductility ratio in (20) are drawn together It can be concludedthat in higher ductility ratios where severe plastic de-formation occurs the estimated ductility is very close to thecomputed ductility In (20) explosion-related parameters plusblast wall natural period are used ree-parameter modelsuse KLM 0333 (rigid plastic modelling) because of itsconservativeness in estimation of the maximum ductility

e author has suggested many other forms for theregression analysis using advanced optimisation techniques

[20] and so far he has not found better forms than (20) forthe 3-parameter-type model and (17) for the 2-parameter-type model It is quite possible that some other forms withclosest t may be found by further research

8 Comparison of the Results

Consider that an explosion with eective energyE0 9500MJ occurs at distance Rs 12m from the ex-plosion centre According to parameters (2a) (2b) and (3)

Ductility ratio for mass factor 085

12291

1458216874

1916521456

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

2

Duc

tility

ratio

9

Ductility ratio for mass factor 085

R0 (m)20 8

Rs (m)

4

7615 5

(b)

Figure 6 Contours for the ductility ratio

050

2

4

15

6

Duc

tility

ratio

40

8

14

Estimated (two parameters) versus computed surface

Explosion length (m)

10

13

Distance from centre (m)

12

30 1211

20 10

Figure 7 Computed versus estimated ductility ratio

6 Advances in Civil Engineering

e overpressure is the average value of the explosion levels3 and 9 and 6 in (3)

R0 45629m

R 0263

pmax 0997 bar

(21)

e elastic deformation from (11) is yel 78mmwhereas themaximumdeection at themiddle section ymaxin (12) can be found by knowing about the ductility ratio

Since the velocity of sound in the room temperaturecondition is C0 340msec from formula (4) we haveduration of the explosion pulse td 74msec whereas the

natural period of the blast wall herein which is given by using(14) is T 161msec

e pressure-impulse curve that introduced before is stillused for damage assessment for many structures ey are aseries of the asymptotic curves inscribed in the vertical andhorizontal asymptotes To nd the points on the curves eitherwe use analytical methods [21 22] or numerical methods [23]and sometimes FEManalysis [24] In the x-y plane the verticalaxis displays FmaxKeyel whereas horizontal axis displaysx Iyel

KeMeradic

I is the impulse and Fmax is the maximumexplosion forces With uniform overpressure they are

Fmax Aeffpmax

I 05Fmaxtd(22)

In [2 3] it can be shown that the equations of the verticaland horizontal asymptotes are in terms of the ductility ratioμ that is dened in (13) ie

I

yelKeMeradic

2μminus 1radic

Fmax

Keyel2μminus 12μ

(23)

Typical curves for elastic-plastic structures are shown inFigure 10 in which the ductility ratio can be found via in-terpolation e snapshot designated by the point shows thecoordinates Iyel

KeMeradic

13622 and FmaxKeyel 103that correspond to this particular explosion and we can ndthe ductility μ 724 as a result of this explosion

However the direct simulation in this paper shows thatμ 2414 It shows that the P-I method particularly in as-ymptotic ends are signicantly inaccurate e two

Ductility ratio for mass factor 0333

12516

12516

1503317549

2006522581

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

5

Duc

tility

ratio

9

Ductility ratio for mass factor 0333

R0 (m)20 8

Rs (m)

10

7615 5

(b)

Figure 8 Contours for the ductility ratio in the rigid plastic model

5015

40 14Explosion length (m)

13

Distance from centre (m)30 1211

20 10

0

2

4

6

8

10

12

Estimated (three parameters) versus computed surface

Duc

tility

ratio

Figure 9 Computed versus estimated ductility ratio

Advances in Civil Engineering 7

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

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In Figure 9 the computed ductility ratio and the estimatedductility ratio in (20) are drawn together It can be concludedthat in higher ductility ratios where severe plastic de-formation occurs the estimated ductility is very close to thecomputed ductility In (20) explosion-related parameters plusblast wall natural period are used ree-parameter modelsuse KLM 0333 (rigid plastic modelling) because of itsconservativeness in estimation of the maximum ductility

e author has suggested many other forms for theregression analysis using advanced optimisation techniques

[20] and so far he has not found better forms than (20) forthe 3-parameter-type model and (17) for the 2-parameter-type model It is quite possible that some other forms withclosest t may be found by further research

8 Comparison of the Results

Consider that an explosion with eective energyE0 9500MJ occurs at distance Rs 12m from the ex-plosion centre According to parameters (2a) (2b) and (3)

Ductility ratio for mass factor 085

12291

1458216874

1916521456

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

2

Duc

tility

ratio

9

Ductility ratio for mass factor 085

R0 (m)20 8

Rs (m)

4

7615 5

(b)

Figure 6 Contours for the ductility ratio

050

2

4

15

6

Duc

tility

ratio

40

8

14

Estimated (two parameters) versus computed surface

Explosion length (m)

10

13

Distance from centre (m)

12

30 1211

20 10

Figure 7 Computed versus estimated ductility ratio

6 Advances in Civil Engineering

e overpressure is the average value of the explosion levels3 and 9 and 6 in (3)

R0 45629m

R 0263

pmax 0997 bar

(21)

e elastic deformation from (11) is yel 78mmwhereas themaximumdeection at themiddle section ymaxin (12) can be found by knowing about the ductility ratio

Since the velocity of sound in the room temperaturecondition is C0 340msec from formula (4) we haveduration of the explosion pulse td 74msec whereas the

natural period of the blast wall herein which is given by using(14) is T 161msec

e pressure-impulse curve that introduced before is stillused for damage assessment for many structures ey are aseries of the asymptotic curves inscribed in the vertical andhorizontal asymptotes To nd the points on the curves eitherwe use analytical methods [21 22] or numerical methods [23]and sometimes FEManalysis [24] In the x-y plane the verticalaxis displays FmaxKeyel whereas horizontal axis displaysx Iyel

KeMeradic

I is the impulse and Fmax is the maximumexplosion forces With uniform overpressure they are

Fmax Aeffpmax

I 05Fmaxtd(22)

In [2 3] it can be shown that the equations of the verticaland horizontal asymptotes are in terms of the ductility ratioμ that is dened in (13) ie

I

yelKeMeradic

2μminus 1radic

Fmax

Keyel2μminus 12μ

(23)

Typical curves for elastic-plastic structures are shown inFigure 10 in which the ductility ratio can be found via in-terpolation e snapshot designated by the point shows thecoordinates Iyel

KeMeradic

13622 and FmaxKeyel 103that correspond to this particular explosion and we can ndthe ductility μ 724 as a result of this explosion

However the direct simulation in this paper shows thatμ 2414 It shows that the P-I method particularly in as-ymptotic ends are signicantly inaccurate e two

Ductility ratio for mass factor 0333

12516

12516

1503317549

2006522581

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

5

Duc

tility

ratio

9

Ductility ratio for mass factor 0333

R0 (m)20 8

Rs (m)

10

7615 5

(b)

Figure 8 Contours for the ductility ratio in the rigid plastic model

5015

40 14Explosion length (m)

13

Distance from centre (m)30 1211

20 10

0

2

4

6

8

10

12

Estimated (three parameters) versus computed surface

Duc

tility

ratio

Figure 9 Computed versus estimated ductility ratio

Advances in Civil Engineering 7

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

e overpressure is the average value of the explosion levels3 and 9 and 6 in (3)

R0 45629m

R 0263

pmax 0997 bar

(21)

e elastic deformation from (11) is yel 78mmwhereas themaximumdeection at themiddle section ymaxin (12) can be found by knowing about the ductility ratio

Since the velocity of sound in the room temperaturecondition is C0 340msec from formula (4) we haveduration of the explosion pulse td 74msec whereas the

natural period of the blast wall herein which is given by using(14) is T 161msec

e pressure-impulse curve that introduced before is stillused for damage assessment for many structures ey are aseries of the asymptotic curves inscribed in the vertical andhorizontal asymptotes To nd the points on the curves eitherwe use analytical methods [21 22] or numerical methods [23]and sometimes FEManalysis [24] In the x-y plane the verticalaxis displays FmaxKeyel whereas horizontal axis displaysx Iyel

KeMeradic

I is the impulse and Fmax is the maximumexplosion forces With uniform overpressure they are

Fmax Aeffpmax

I 05Fmaxtd(22)

In [2 3] it can be shown that the equations of the verticaland horizontal asymptotes are in terms of the ductility ratioμ that is dened in (13) ie

I

yelKeMeradic

2μminus 1radic

Fmax

Keyel2μminus 12μ

(23)

Typical curves for elastic-plastic structures are shown inFigure 10 in which the ductility ratio can be found via in-terpolation e snapshot designated by the point shows thecoordinates Iyel

KeMeradic

13622 and FmaxKeyel 103that correspond to this particular explosion and we can ndthe ductility μ 724 as a result of this explosion

However the direct simulation in this paper shows thatμ 2414 It shows that the P-I method particularly in as-ymptotic ends are signicantly inaccurate e two

Ductility ratio for mass factor 0333

12516

12516

1503317549

2006522581

5 55 6 65 7 75 8 85 9 95 10Distance Rs (m)

15

20

25

Expl

osio

n le

ngth

R0 (

m)

(a)

025 10

5

Duc

tility

ratio

9

Ductility ratio for mass factor 0333

R0 (m)20 8

Rs (m)

10

7615 5

(b)

Figure 8 Contours for the ductility ratio in the rigid plastic model

5015

40 14Explosion length (m)

13

Distance from centre (m)30 1211

20 10

0

2

4

6

8

10

12

Estimated (three parameters) versus computed surface

Duc

tility

ratio

Figure 9 Computed versus estimated ductility ratio

Advances in Civil Engineering 7

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

approximatedmodels in this paper that are expressed by (17)and (20) to replace the P-I method give much closer resultse comparison is shown in Table 1

Further comparison can be done by using FEM tech-nique via ABAQUS modelling [25] of the blast wall in thisexample e meshing is shown by a snapshot in Figure 11In this model 6500-shell-type S4R elements each with nineinternal integration point are used Obviously substantialFEM outputs including the local buckling details in bottomanges are available However the one that can be comparedwith ymax in (12) has been extracted Since ductility ratio isnot dened in ABAQUS Table 2 is provided to compare theymax (maximum deection) in each approach

e last row of Table 2 is found from history of thedisplacement of the middle of the top ange of the blast wallis history for U V and A is shown in Figure 12 It isobvious that velocity in mms and acceleration in ms2 arebig numbers since T in (14) is very low

Figure 12 is prepared by using history of nodesHowever the history of stress and strain in any location ofthe blast wall can be prepared by element output lesSimilar to Figure 11 Figure 13 shows the Mises stress mapthat is scaled in Pa

Obviously the yield stress is flowasty 400MPa and thematerial is assumed elastic-perfectly plastic (E-P-P) allsimilar to the SDOF model Since the blast wall is modelledwith shell elements Poissonrsquos ratio of the material υ 03 is

also required e history of the Mises stress and also themaximum principal strain can be found from the elementle To do this the shell element corresponding to middleof the top ange of the blast wall is chosen e history lefor stress and stain for that location is shown in Figure 14 Itis obvious that stress does not exceed 400MPa Howeverfor strain after quick jump at the beginning of the ex-plosion the uctuations are not signicant From themodel in this paper we can check and verify the dis-placement as shown in Table 2 is suits the purpose ofthis paper in developing a simple and accurate model forchecking high-delity FEM analysis

In Table 3 the material properties and also maximumvelocity acceleration and stress and strain are shown emaximum displacement is shown in Table 2 for the com-parison purposes e maximum stress in Table 3 exceededslightly above 400MPa because the E-P-P material model isABAQUS which is expressed via a very low plastic Youngrsquosmodulus (not zero)

0

1

2

3

4

5

6

7

8

9

10

4 6 8 10 12 142

0

02

04

06

08

1

12

14

12 13 14 15 1611

micro = 1micro = 2micro = 3

micro = 4micro = 5micro = 6

micro = 8micro = 7

F max

Ke y

el

Iyel KeMe

Figure 10 Pressure-impulse diagram for elastic-plastic structures

Table 1 Comparison of the results (ductility ratio)

e method used Ductility ratioTNO+SDOF simulation μ 2414Pressure-impulse curves μ 724Two-parameter empirical formula (17) μ 1968ree-parameter empirical formula (20) μ 2019

8 Advances in Civil Engineering

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

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Navigation and Observation

International Journal of

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wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

e outcomes of this section are shown in Table 4 istable compares the advantages and disadvantage of eachmethod that is discussed It can be seen that there are manyadvantages of using the method in this paper particularlywhen we compare with the pressure-impulse diagramHowever it should be used together with high-delity FEManalysis to achieve more details about the response of theblast wall to the explosion

9 Conclusions and Remarks

It this paper a new method for damage assessments in blastwalls are developed It is much easier than the classical methodof the pressure-impulse diagram and FEM analysis As shownin [21ndash24] and also in this paper the high-delity analytical orFEM models cannot predict explosion response withoutknowledge about explosion overpressure and pulse duration

U magnitude+8333e ndash 02

+7638e ndash 02

+6944e ndash 02

+6249e ndash 02

+5555e ndash 02

+4861e ndash 02

+4166e ndash 02

+3472e ndash 02

+2778e ndash 02

+2083e ndash 02

+1389e ndash 02

+6944e ndash 03

+0000e + 00

YY

ZZ XX

Figure 11 FEM meshing of the blast wall (displacement map in m)

0 002 004 006 008 01 012 014Time (sec)

ndash20

ndash10

0

U (m

m)

(a)

0 002 004 006 008 01 012 014Time (sec)

ndash1

0

1

V (m

ms

ec) times104

(b)

0 002 004 006 008 01 012 014Time (sec)

ndash2

0

2

A (m

sec

2 ) times104

(c)

Figure 12 (a) Displacement (deection) (b) velocity and (c) acceleration history of the top ange

Table 2 Comparison of the results (maximum deection)

e method used Maximum deectionTNO+SDOF simulation ymax 189mmPressure-impulse curves ymax 567mmEmpirical formula (17) ymax 132mmEmpirical formula (20) ymax 158mmFEM analysis via ABAQUS ymax 195mm

Advances in Civil Engineering 9

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

e inaccuracy of the P-I diagram in the asymptoticregion is clearly shown in this paper via Tables 1 and 2Regardless of that the P-I diagram is an active eld of research

for blast-resistive structure as seen in recent publications[22ndash24] erefore an alternative method is required to re-place the P-I diagram in asymptotic region is approach

U misesSNEG(fraction = ndash10)(avg 75)

+3777e + 08

+3464e + 08

+3150e + 08

+2837e + 08

+2524e + 08

+2210e + 08

+1897e + 08

+1583e + 08

+1270e + 08

+9568e + 07

+6434e + 07

+3301e + 07

+1672e + 06

YY

ZZ XX

Figure 13 Mises stress map scaled in Pa

600

400

200

Mise

s str

ess (

MPa

)

00 002 004 006 008

Time (sec)01 012 014

(a)

001

0005

0

ndash0005

ndash001

Max

imum

prin

cipa

l str

ain

0 002 004 006 008Time (sec)

01 012 014

(b)

Figure 14 History of (a) Mises stress and (b) maximum plastic strain of the middle of top ange

Table 3 Summary of FEM analysis (results are for middle of the top ange)

Youngrsquos modulus for E-P-P steel E 210GPaPoissonrsquos ratio υ 03Maximum velocity vmax 6456mmsMaximum acceleration amax 15001ms2Maximum Mises stress σmax 400810000 PaMaximum principal strain εp 00059

Table 4 Comparison table for methods discussed

e method used Advantages Disadvantages

TNO+ SDOF simulation (1) Reliable source for checking FEM (deection)(2) Simple compared to FEM

(1) Low delity compared to FEM analysis(2) Includes the model for history of overpressure

Pressure-impulse curves (1) Simple to nd the ductility and displacement(1) Inaccuracy in asymptotic region

(2) Needs overpressure history(3) Low delity compared to FEM analysis

Empirical formulas (17) and (20)in this paper

(1) No need for history of overpressure(2) Simple to nd the ductility and displacement(3) Reliable source for checking FEM (deection)

(1) Low delity compared to FEM analysis

FEM analysis via ABAQUS(1) High delity of the model

(2) Availability of results in any location(3) Local buckling details

(1) Needs overpressure history(2) Meshing dilaquoculties and model complexity(3) Verication of the results by another method

10 Advances in Civil Engineering

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

should be much easier than FEM analysis and can produce aresult accurate enough to be compared with FEM)e authorbelieves that he has found an alternative in this paper

When overpressure-time history is not available bothof the SDOF and FEM cannot predict the damage )e ad-vantage of this new method is the combination of SDOFmethod and overpressure-time history of explosion Hereinthe TNO method (that provides overpressure history) withSDOF (that provides deflection) has been combined together)ereafter approximate formulas have been produced thateasily predicts the ductility ratio without using P-I diagramsor doing SDOF calculations or FEM analysis)erefore it willbe very useful for preliminary design applications

Symbols

As Cross-sectional areaA B C Constants of the parabolic functionAeff Effective overpressured areaC0 Velocity of soundE0 E Explosive energy and modulus of elasticityflowasty F1 Steel yield stress and total applied forceFmax )e maximum explosion forceI Second moment of cross section

(in bending)I Impulse of the explosion pulsek kR Stiffness and reduced stiffness (in bending)KF KVM Flattening and shear correction factorsKLM Mass correction factorL LE Total length and equivalent lengthLU LL Lengths of the upper and lower supportsM Me Mass and equivalent mass of the blast wallmc Hydrocarbon mass(McRd)U(McRd)L

Yield bending moment in the upper andlower supports

McRd Yield bending moment in the blast wallpmax Maximum overpressurep p0 Projected blast area per pitch and

atmospheric pressureR0 R Explosion length and dimensionless

explosion lengthRm Maximum elastic resistance of beam cross

sectionRs Distance from explosion centret+ td Dimensionless pulse duration and pulse

durationtU and tL )icknesses of the upper and lower

supportsT Natural period of the structureWply Plastic section modulusW0 Maximum deflection of the midspanyel ymax Maximum elastic and maximum plastic

deformationΔHc Heat energyα β c Constants in the predictive modelμ Ductility ratioη Efficiency of explosion

Appendix

A Equivalent Lengths and BendingMoment Distribution

According to Figure 15 the total length of the blast wallconsist of 3 parts

L LE + LU + LL (A1)

)e overpressure as a result of explosion produces auniform load that results a parabolic type of bending mo-ment as follows

M(x) Ax2

+ Bx + C (A2)

When we place the origin of the coordinate system at themiddle of the wall then we have

M(0) A times 02 + B times 0x + C McRd⟹C McRd

(A3)

Moreover the shear force at maximum bendingmomentis zero ie

dM

dx 2Ax + B

dM

dx(0) 2A times 0 + B 0⟹B 0

(A4)

)e segment with length LE ltL acts as the simplysupported beam such that in its two ends the bendingmoment is zero such that

L LE

LU

LL

(MCRd)U

MCRd

(MCRd)L

Figure 15 Parabolic bending moment distribution

Advances in Civil Engineering 11

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

LE

21113874 1113875 A times

LE

21113874 1113875

2+ 0 times

LE

2+ McRd 0⟹A

minus4McRd

L2E

(A5)

)en (A2) can simplified into

M(x) McRd 1minus4x2

L2E

1113888 1113889 (A6)

)e upper and lower supports of the blast wall act ascantilevers such that maximum bending moments of thesupports occur at the corners such that

M 05LE + LU( 1113857 McRd 1minus4 05LE + LU( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873U⟹4 05LE + LU( 1113857

2

L2E

1 +McRd1113872 1113873UMcRd

(A7)

M 05LE + LL( 1113857 McRd 1minus4 05LE + LL( 1113857

2

L2E

1113888 1113889

minus McRd1113872 1113873L⟹4 05LE + LL( 1113857

2

L2E

1 +McRd1113872 1113873LMcRd

(A8)

)e expressions (A7) and (A8) can be simplified into

LU LE

2

1 +McRd1113872 1113873UMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

LL LE

2

1 +McRd1113872 1113873LMcRd

11139741113972

minus 1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(A9)

Substituting (A9) into (A1) and after simplification wehave equation (7)

LE 2L

1 + McRd1113872 1113873UMcRd1113872 1113873

1113969+

1 + McRd1113872 1113873LMcRd1113872 1113873

1113969

(A10)

B Rigid-Plastic Beam Model

In rigid plastic type of modelling the plastic hinge occurs atthe middle of the beam where the maximum lateral de-flection W0 will occur (Figure 16)

Obviously the lateral deformation and velocity patternwill be linear and are given by

W W0

(L2)x⟹ _W

W0

(L2)_x (B1)

)en considering the form in (B1) the overall kineticenergy of the beam will be

KE 21113946L

0

12ρA( _W(x))

2dx ρA 1113946

L

0

x

L2_W01113874 1113875

2dx

4ρA _W

20

L2 1113946L2

0x2dx

4ρA _W20L

3

24L2 16

M _W20

(B2)

)e equivalent mass Me located at the plastic hingeposition should possess the same kinetic energy in (B2) ie

KE 12Me

_W20 (B3)

Comparing (B2) with (B3) will result12Me

_W20

16

M _W20⟹Me

13

M (B4)

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e author declares no conflicts of interest

Acknowledgments

)e author appreciates Aberdeen University for the timeprovided to him for doing independent research as part ofhis duties of an academic post

References

[1] Fire and Blast Information Group (FABIG) ldquoDesign guide forstainless steel blast wallsrdquo Technical Note 5 Fire and BlastInformation Group Berkshire (UK) Berkshire UK 1999

[2] M Y H Bangash and T Bangash Explosion-Resistant Build-ings Design Analysis and Case Studies Springer New YorkNY USA 2006

[3] T Krauthammer Modern Protective Structures CRC PressBoca Raton FL USA 2008

[4] J Q Fang P Chung and R W Wolfe ldquoAnalysis of a blast-loaded protective wall for bridge columnsrdquo Bridge Structuresvol 4 no 3-4 pp 135ndash141 2008

[5] L A Louca M Punjani and J E Harding ldquoNon-linearanalysis of blast walls and stiffened panels subjected to hy-drocarbon explosionsrdquo Journal of Constructional Steel Re-search vol 37 no 2 pp 93ndash113 1996

x W0

y

Figure 16 Plastic hinge in the simply supported beam

12 Advances in Civil Engineering

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[6] G K Schleyer T H Kewaisy J W Wesevich andG S Langdon ldquoValidated finite element analysis model ofblast wall panels under shock pressure loadingrdquo Ships andOffshore Structures vol 1 no 3 pp 257ndash271 2006

[7] H Y Lei J C Lee C B Lia et al ldquoCostndashbenefit analysis ofcorrugated blast wallsrdquo Ships and Offshore Structures vol 10no 5 pp 565ndash574 2015

[8] J M Biggs Introduction to Structural Dynamics McGraw-Hill New York NY USA 1964

[9] N M Newmark An Engineering Approach to Blast ResistanceDesign Vol121 University of Illinois Champaign IL USA 1956

[10] Q M Li and H Meng ldquoPulse loading shape effects onpressure-impulse diagram of an elastic-plastic single-degree-of-freedom structural modelrdquo International Journal of Me-chanical Sciences vol 44 no 9 pp 1985ndash1998 2002

[11] C Amadio and C Bedon ldquoViscoelastic spider connectors forthemitigation of cable-supported faccedilades subjected to air blastloadingrdquo Engineering Structures vol 42 pp 190ndash200 2012

[12] J Dragos C Wu and K Vugts ldquoPressure-impulse diagramsfor an elastic-plastic member under confined blastsrdquo In-ternational Journal of Protective Structures vol 4 no 2pp 143ndash162 2013

[13] A S Fallah E Nwankwo and L A Louca ldquoPressure-impulsediagrams for blast loaded continuous beams based on di-mensional analysisrdquo Journal of Applied Mechanics vol 80no 5 article 051011 2013

[14] M H Hedayati S Sriramula and R D Neilson ldquoDynamicbehaviour of unstiffened stainless steel profiled barrier blast wallsrdquoShips and Offshore Structures vol 13 no 4 pp 403ndash411 2018

[15] M Aleyaasin ldquoProtective and blast resistive design of post-tensioned box girders using computational geometryrdquo Advancesin Civil Engineering vol 2018 Article ID 4932987 7 pages 2018

[16] A C Van den Berg ldquo)e multi-energy method a frameworkfor vapour cloud explosion blast predictionrdquo Journal ofHazardous Materials vol 12 no 1 pp 1ndash10 1985

[17] F D Alonso E G Ferradas J F S Perez A M AznarJ R Gimeno and J M Alonso ldquoCharacteristic overpressure-impulse-distance curves for vapour cloud explosions using theTNO multi-energy modelrdquo Journal of Hazardous Materialsvol 137 no 2 pp 734ndash741 2006

[18] P W Sielicki and M Stachowski ldquoImplementation of sapper-blast-module a rapid prediction software for blast wavepropertiesrdquo Central European Journal of Energetic Materialsvol 12 no 3 pp 473ndash486 2015

[19] A Alia and M Souli ldquoHigh explosive simulation using multi-material formulationsrdquo Applied 8ermal Engineering vol 26no 10 pp 1032ndash1042 2006

[20] J C Lagarias J A Reeds M H Wright and P E WrightldquoConvergence properties of the nelder--mead simplexmethodin low dimensionsrdquo SIAM Journal on Optimization vol 9no 1 pp 112ndash147 1998

[21] V R Feldgun D Z Yankelevsky and Y S Karinski ldquoAnonlinear SDOFmodel for blast response simulation of elasticthin rectangular platesrdquo International Journal of ImpactEngineering vol 88 pp 172ndash188 2016

[22] Y Ye L Zhu X Bai T X Yu Y Li and P J TanldquoPressurendashimpulse diagrams for elastoplastic beams subjectedto pulse-pressure loadingrdquo International Journal of Solids andStructures vol 160 pp 148ndash157 2019

[23] R Yu D Zhang L Chen and H Yan ldquoNon-dimensionalpressure-impulse diagrams for blast-loaded reinforced con-crete beam columns referred to different failure modesrdquoAdvances in Structural Engineering vol 21 no 14pp 2114ndash2129 2018

[24] S Chen X Xing Chen G Q Li and Y Lu ldquoDevelopment ofpressure-impulse diagrams for framed PVB-laminated glasswindowsrdquo Journal of Structural Engineering vol 145 no 3article 04018263 2019

[25] A Khennane Introduction to Finite Element Analysis UsingMATLAB and ABAQUS CRC Press Boca Raton FL USA2013

Advances in Civil Engineering 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom