Study of the Similarities in Scale Models of a Single-Layer ...downloads.hindawi.com/journals/sv/2017/9181729.pdfexplosion in a single-layer spherical lattice shell structure by using

Research ArticleStudy of the Similarities in Scale Models ofa Single-Layer Spherical Lattice Shell Structure underthe Effect of Internal Explosion

WenbaoWang, Xuanneng Gao, and Lihui Le

College of Civil Engineering, Huaqiao University, Xiamen 361021, China

Correspondence should be addressed to Xuanneng Gao; [email protected]

Received 5 November 2016; Revised 1 January 2017; Accepted 28 February 2017; Published 16 March 2017

Academic Editor: Isabelle Sochet

Copyright © 2017 Wenbao Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The similarity of each scale model is verified based on the theory of similarity, deriving the similarity law of internal explosionsin a single-layer spherical lattice shell structure via dimensional theory, calculated based on models with scaling coefficients of 1,0.8, 0.6, 0.4, 0.2, and 0.1. The results show that the shock wave propagation characteristics, the distribution of the overpressure onthe inner surface, the maximum dynamic response position, and the position at which the earliest explosion venting occurs are allsimilar to those of the original model. With the decrease of scaling coefficients, the overpressure peak value of the shock waves ofeach scale model, and the specific action time of the positive pressure zone, as well as specific impulse are increasingly deviatedfrom the original model values; when the scaling coefficient is 0.1, the maximum relative error between the overpressure peak valueat the measurement point and the specific action time of the positive pressure zone as well as the specific impulse and the originalmodel value is 4.9%. Thus, it is feasible to forecast the internal explosion effect of the original structure size model by using theexperiment results of the scale model with scaling coefficient 𝜆 ≥ 0.1.

1. Introduction

In recent years, worldwide terrorist activities have becomeincreasingly frequent, with an increasing number of bombingassaults. On January 24, 2016, the Moscow Airport in theRussian capital exploded under a terrorist attack; 35 peoplewere killed and hundreds of people were injured. On March22, 2016, an explosion occurred at the Brussels Airport inBelgium, killing 17 people. On June 28, 2016, an explosionoccurred at the Turkey Ataturk International Airport, killing41 people. It can be seen that steel structures with largespace, high visitor flowrate, and significant symbolic valueare more likely to become a target of explosion attack, andonce subject to a terrorist attack, heavy casualties, propertydamage, and adverse social impacts are likely [1–3]. Theexplosion effect of explosives in the structure is much morecomplex than that of an external explosion, primarily becauseof the multiple reflection and diffraction phenomena thatoccur during the transmission of shock waves when meetingthe roof, walls, and a variety of structures and constructionswithin the structure; such phenomena cause continuous

superposition or offset of the reflected waves and incidentwaves in the structure, with gas flow fields within the buildingexhibiting irregular movement, thus making it difficult todetermine the shock wave pressure field acting on the roof[4–6]. Therefore, it is necessary to conduct experimentalstudy on the single-layer spherical lattice shell structureunder the action of internal explosion; however, becauseof the constraints of the experimental techniques, researchfunds, and other conditions, it is difficult to conduct aninternal explosion experiment in a full-scale model. A scalemodel experiment is a commonly used experimental methodcurrently, and it is necessary to study whether the scalemodelexperiment of a single-layer spherical lattice shell structureunder the action of internal explosion will meet the similaritylaw and whether it will be affected by the size effect.

Currently, scholars have conducted experimentalresearch studies on the damage in a structure or componentunder explosive impact. In foreign studies, Zyskowski et al.[7] studied the distribution of explosion load in structures.Ngo et al. [8] conducted a large 5000 kg explosion experimentin southern Australia and conducted a quantitative analysis

HindawiShock and VibrationVolume 2017, Article ID 9181729, 13 pageshttps://doi.org/10.1155/2017/9181729

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2 Shock and Vibration

of the parameters obtained. Spranghers et al. [9] studieda numerical simulation of the dynamic response of analuminium plate under free-space explosion, conductedexperimental verification, and studied the effect of differentparameters on the simulation results. Remennikov andUy [10] conducted a near-field explosion experiment.Dey and Nimje [11] conducted experimental researchon a full-size sandwich panel explosion and performednumerical simulation analysis of the results. Ichino et al. [12]conducted a 1/20 scale model internal explosion experimentin an underground ammunition depot; the characteristicsof the air shock waves and the ground vibration wereanalysed and verified via a 1/10 scale model experiment. Indomestic research, Yang et al. [13, 14] simulated the shockwaves propagation law of an internal explosion in closed-wall vault tunnel operation via a numerical computationmethod and determined the propagation law of an airshock wave along the tunnel and obtained the formulasof the shock wave overpressure and action time in thetunnel; the calculated results were found to be in goodagreement with the experimental results. Wei et al. [15]studied the load law of an internal explosion experimentby establishing a small volume simplified chamber. Fanet al. [16] fitted out the forecast formula of the explosiveshock waves characteristic parameters of the centre ofthe inner structure through numerical simulation andanalysis under the same working conditions as those ina straight tunnel explosion experiment. Cheng et al. [17]conducted theoretical analysis and experimental study ofthe large plastic deformation response of rectangular steelplates with four-sided restraints under explosive shockwaves; the theoretical calculations were compared withthe experimental results and the numerical calculations,revealing that the elastic-plastic analysis method had bettercalculation accuracy and applicability. Gao et al. [1, 4, 18, 19]conducted a series of studies on the shock wave propagationcharacteristics, the distribution of the pressure field, thestructural dynamic response, and the explosion ventingmeasures under internal explosion of large-space cylindricalreticulated shells, spherical reticulated shells, and otherstructures; the effect of various parameters, such as structureheight, span, structure-to-span ratio, TNT equivalentweight, location of explosion, and hole arrangement onthe explosion, was studied. Experimental researches ondestruction of structures or components under explosionhave gained some achievements. However, there are fewstudies on destruction under explosion in large-space steelstructures, and, limited by such constraints as experimentaltechniques and scientific research funds, it is much hardto carry out an explosion experiment in full-scale model.Therefore, it is necessary to make a similarity research forscaled model experiment.

In this paper, we derived the similarity law of an internalexplosion in a single-layer spherical lattice shell structureby using dimensional theory and constructed a finite ele-ment model with the scaling coefficients of 1, 0.8, 0.6,0.4, 0.2, and 0.1 based on LS-DYNA software. The shockwave propagation characteristics, the distribution law ofthe overpressure peak on the inner surface, the maximum

Figure 1: Layout of single-layer spherical shell structural elements.

dynamic response position, and the reticulated shell explo-sion venting phenomena of the scale models were studied.The similarity of the scale models was verified accordingto the critical TNT quantity of explosion venting at theconnection between the wall and the reticulated shell of themodel structure, providing a reference for internal explosionresearch.

2. Establishment of the Finite Element Modeland Selection of the Material Parameters

2.1. Establishment of the Finite Element Model and Selectionof Material Parameters. A K6 single-layer spherical latticeshell model was established by using ANSYS/LS-DYNAfinite element software, which is mainly composed of thefollowing: air, explosives, ground, reticulated shell bars,beams, columns, connecting structure, and envelope. Theunscaled model is shown in Figure 1; the model has thefollowing characteristics: a span of 40m, vector height of8m, a rise span ratio of 1/5, a frequency number of 5,a lower support structure height of 10m, and a sectionaldimension of the seamless steel tubes of the main bar,cross bar, and diagonal bar of 𝜑140 × 8mm; all the beamcolumns adopt the H-section. Because the air and explosivesare considered as uniform and continuous media, Solid164is chosen as the calculation unit, and because the groundand envelope are regarded as rigid bodies, Shell163 elementis adopted; Beam161 element is adopted for the structuralbars, beams, and columns, and Link160 element is used forthe purlin hanger. The air and explosives adopt the ALEalgorithm, the air, structure, and ground use adopt the fluid-solid coupling algorithm, and the air boundary adopts thetransmission boundary to avoid arithmetic errors caused bythe reflection of shock waves in the air domain boundaryand to improve the calculation accuracy [19]. However, whenterrorist attacks and other explosions occur, the locationof the explosion point is often unascertainable; consideringthe complexity of the explosion analysis, it is assumed thatthe explosion is located in the centre of the structure, 1mabove the ground, and the explosive charge is spherical. Afinite element model with scaling coefficients of 0.8, 0.6, 0.4,0.2, and 0.1 was established for numerical simulation andanalysis.

Air uses the MAT_NULL material model, assumingthat the air is the ideal gas, and the state equation adoptsthe EOS_LINEAR_POLYNOMIAL linear polynomial stateequation [20], referring to the linear relationship between

Shock and Vibration 3

the air pressure and the initial internal energy, for which theexpression is as follows:

𝑃 = 𝐶0 + 𝐶1𝜇 + 𝐶2𝜇2 + 𝐶3𝜇3 + (𝐶4 + 𝐶5𝜇 + 𝐶6𝜇2) 𝐸.𝜇 = 𝜌𝜌0 − 1

(1)

𝜌 is the current air density, 𝜌0 is the reference air density,𝐸 is the internal energy per unit of reference volume air, 𝑉0is the initial relative volume, the specific input parameter 𝜌 =1.290 kg/m3, 𝐸0 = 2.5 × 105 J/m3, real constants 𝐶0 = 𝐶1 =𝐶2 = 𝐶3 = 𝐶6 = 0, and 𝐶4 = 𝐶5 = 0.4.

The explosive adopts the MAT_HIGH_EXPLOSIVE_BURN high explosive material model and the JWL stateequation [21], and the explosive charge is spherical, asspecified with the LS-DYNA initial volume fraction keyword∗INITIAL_VOLUME_FRACTION_GEOMETRY [22].

𝑝 = 𝐴(1 − 𝜔𝑅1𝑉) 𝑒−𝑅

1𝑉 + 𝐵(1 − 𝜔𝑅2𝑉) 𝑒

−𝑅2𝑉

+ 𝜔𝐸0𝑉 ,(2)

where 𝐴 = 540.9GP, 𝐵 = 9.4GP, 𝑅1 = 4.5, 𝑅2 = 1.1, Ω =0.35, the initial internal energy 𝐸0 = 8 × 109 J/m3, and theinitial relative volume 𝑉0 = 1.0.

The steel used in the entire reticulated shell structure isQ235, and the Johnson-Cook constitutive relation and failurecriterion calibrated by the test group as per the test resultsare taken as the material model. The specific parameters areshown in Table 1.

2.2. Model Credibility Verification. The rationality of themodel and material parameters is directly related to theaccuracy of the model calculation; hence, it is necessaryto validate the rationality of the model and the materialparameters. The research group established a free-spaceexplosion model and extracted the simulation results of theoverpressure peak value at the measurement point of theshock waves and then compared the values with those of thecommonly used free-space explosion overpressure empiricalformula [23, 24]; the results show that the simulation resultof the overpressure peak value at the measurement pointof the shock waves is consistent with the variation trendof the empirical formula result and is most similar to thecalculated result of Henrych’s empirical formula [24]. Thus,we can see that the calculation model and parameter valuesare reasonable and the calculation result of the model iscredible [6].

3. Analysis of Similar Models underInternal Explosion in Single-Layer SphericalReticulated Shell Structure

3.1. Similarity Theory Analysis of the Internal Explosion ScaleModel of a Single-Layer Spherical Reticulated Shell Structure.The explosion of explosives inside a structure is often more

Table 1: Steel material parameters.

Parameters Value𝜌𝑆/kg⋅m−3 7850𝐸/MPa 2.1 × 105] 0.3𝐴/MPa 320.755𝐵/MPa 582.102𝑁 0.3823𝐶 0.02548𝐹𝑆 0.25

complicated than an explosion outside the structure, primar-ily because of the reflection and mutual superimposition ofshock waves. The internal explosion shock wave of a single-layer spherical lattice shell structure is a complex movementthat integrates incidence, reflection, diffraction, aggregation,and so forth. [6], despite all this, its explosive features obeythe Hopkinson-Sachs similarity law [25, 26] to some extent.The damage of a reticulated shell structure primarily dependson the strength of the inner surface shock waves; the threeparameters describing the strength of the shock wave arethe peak overpressure value, the action time of the positivepressure zone, and the impulse.Themain factors affecting thestrength of shock wave are as follows:

(1) Explosive parameters: the TNT quantity is 𝑊, theexplosive density is 𝜌𝑇, the unit mass release energy is𝐸𝑇, and the explosion product expansion coefficientis 𝛾𝑇

(2) Air parameters: the initial pressure is 𝑝0, the initialdensity is 𝜌𝐴, and the adiabatic index is 𝛾𝐴

(3) Structural parameters: the span is 𝑙, the vector heightis 𝑓, the wall height is ℎ, the strength is 𝜎, Young’smodulus is 𝐸, and the material density is 𝜌

(4) The distance 𝑅 between the structural surface and theexplosion centre

As per 𝜋 theory of Buckingham [27, 28], the functionalrelationship between the shock wave overpressure on theinner surface and the influencing factors of an internalexplosion of reticulated shell structure is

Δ𝑃 = 𝑓 (𝑊, 𝜌𝑇, 𝐸𝑇, 𝛾𝑇, 𝑝0, 𝜌𝐴, 𝛾𝐴, 𝑙, 𝑓, ℎ, 𝜎, 𝐸, 𝜌, 𝑅) . (3)

Regarding a scale model, if the structural material is thesame as the prototype, then Young’s modulus 𝐸 is not anindependent quantity and can be omitted; 𝛾𝑇 and 𝛾𝐴 aredimensionless quantities (consistent with the dimensionlessprinciple of the similarity law) and can also be omitted.Therefore, formula (3) can be simplified as follows:

Δ𝑃 = 𝑓 (𝑊, 𝜌𝑇, 𝐸𝑇, 𝑝0, 𝜌𝐴, 𝑙, 𝑓, ℎ, 𝜎, 𝜌, 𝑅) . (4)


Based on the principle of dimensional harmony, therelationship between the overpressure peak of the shockwaves and the influencing factors is established:

[Δ𝑃]= [𝑊𝑎 ⋅ 𝜌𝑏𝑇 ⋅ 𝐸𝑐𝑇 ⋅ 𝑝𝑑0 ⋅ 𝜌𝑒𝐴 ⋅ 𝑙𝑓 ⋅ 𝑓𝑔 ⋅ ℎℎ ⋅ 𝜎𝑖 ⋅ 𝜌𝑗 ⋅ 𝑅𝑘] . (5)

With𝑊, 𝜌𝑇, and 𝐸𝑇 as the basic quantity, there is𝑎 = −13 (𝑓 + 𝑔 + ℎ + 𝑘) ,𝑏 = 1 − 𝑑 − 𝑒 + 13 (𝑓 + 𝑔 + ℎ + 𝑘) − 𝑖 − 𝑗,𝑐 = 1 − 𝑑 − 𝑖;

(6)

therefore,

[Δ𝑃] = (𝜌𝑇𝐸𝑇) ⋅ (𝜌−1𝑇 𝐸−1𝑇 𝑝0)𝑑 ⋅ (𝜌−1𝑇 𝜌𝐴)𝑒⋅ (𝑊−1/3𝜌1/3𝑇 𝑙)𝑓 ⋅ (𝑊−1/3𝜌1/3𝑇 𝑓)𝑔 ⋅ (𝑊−1/3𝜌1/3𝑇 ℎ)ℎ⋅ (𝜌−1𝑇 𝐸−1𝑇 𝜎)𝑖 ⋅ (𝜌−1𝑇 𝜌)𝑗 ⋅ (𝑊−1/3𝜌1/3𝑇 𝑅)𝑘

Δ𝑃𝜌𝑇𝐸𝑇 = 𝑓[ 𝑝0𝜌𝑇𝐸𝑇 ,𝜌𝐴𝜌𝑇 ,

𝑙(𝑊/𝜌𝑇)1/3 ,

𝑓(𝑊/𝜌𝑇)1/3 ,

ℎ(𝑊/𝜌𝑇)1/3 ,

𝜎𝜌𝑇𝐸𝑇 ,𝜌𝜌𝑇 ,

𝑅(𝑊/𝜌𝑇)1/3] .

(7)

The same type of charge is adopted under the same airconditions, and the structural materials remain unchanged.When 𝑊1/3𝑃 /𝑊1/3𝑚 = 𝑙𝑝/𝑙𝑚 = 𝑓𝑝/𝑓𝑚 = ℎ𝑝/ℎ𝑚 = 𝑅𝑝/𝑅𝑚 =1/𝜆, as per the law of copying proportion,Δ𝑃𝑝 = Δ𝑃𝑚, wherethe subscript 𝑝 indicates the prototype,𝑚 indicates the scalemodel, and 𝜆 indicates the scaling coefficient.

Similarly, the shock wave impulse 𝑖 on the inner surfacein a reticulated shell structure internal explosion satisfies thesimilarity law of

𝑖𝑊1/3𝜌2/3𝑇 𝐸1/2𝑇 = 𝑓[ 𝑝0𝜌𝑇𝐸𝑇 ,

𝜌𝑎𝜌𝑇 ,𝑙

(𝑊/𝜌𝑇)1/3 ,𝑓

(𝑊/𝜌𝑇)1/3 ,ℎ

(𝑊/𝜌𝑇)1/3 ,𝜎𝜌𝑇𝐸𝑇 ,

𝜌𝜌𝑇 ,𝑅

(𝑊/𝜌𝑇)1/3] .(8)

The same type of charge is adopted under the sameair conditions, and structural materials remain unchanged.When 𝑊1/3𝑃 /𝑊1/3𝑚 = 𝑙𝑝/𝑙𝑚 = 𝑓𝑝/𝑓𝑚 = ℎ𝑝/ℎ𝑚 = 𝑅𝑝/𝑅𝑚 =1/𝜆, as per the law of copying proportion, 𝑖𝑝/𝑖𝑚 = 𝑊1/3𝑝 /𝑊1/3𝑚 = 1/𝜆.

Similarly, 𝑇+𝑝 /𝑇+𝑚 = 𝑊1/3𝑝 /𝑊1/3𝑚 = 1/𝜆.3.2. Analysis of Similarity Law under Internal Explosion inSingle-Layer Spherical Reticulated Shell Structure. The explo-sive similarity law proposed by Hopkinson [25] points out

0 2 4 6 8 10

20 kg60 kg

100 kg140 kg

0.02

0.04

0.06

ΔP

(MPa

)

0.08

0.10

ΔP = −0.01599R + 0.014114

R (m·kg−1/3)

Figure 2: Relationship between reflection overpressure and scaleddistance of typical measuring points.

that if the scaled distances (𝑅/𝑊1/3) are equal in condensedexplosives of the same type, their pressure values of explosivewave with equal initial pressure would be equivalent. Liter-ature [6] shows that shock waves may focus on the internalsurface of spherical reticulated shell under internal explosion,so positions 1, 5, 11, and 19 in Figure 3 were selected as typicalmeasuring points to verify whether the explosion satisfies thesimilarity law when explosive charge changes and structuresize remains unchanged. Samples with TNT amount of 20 kg,60 kg, 100 kg, and 140 kg were selected to form the diagramclarifying the relationship between reflection overpressureand scaled distance of typical measuring points, that is,Figure 2. As shown in Figure 2, the reflection overpressure ofmeasuring points generally follows the similarity law if TNTamount is below the mass of structural explosion venting.Reflection overpressure has linear relationship with scaleddistance and their fitting formula is

Δ𝑃 = −0.01599𝑅 + 0.014114. (9)

In the formula, Δ𝑃 is the reflection overpressure (MPa)and 𝑅 is the scaled distance (m⋅kg−1/3).4. Similarity Analysis of the InternalExplosion Scale Model of the Single-LayerSpherical Reticulated Shell Structure

4.1. Similarity Analysis of Shock Wave Propagation Character-istics. The internal explosion in reticulated shell structure isoftenmore complicated than outside the structure [29]; whenthe centre explodes, the shock waves will converge on the topof the reticulated shell and the connection of the wall and thereticulated shell and finally converge at the midpoint of theground, forming a new round of reflection and convergence[6]. The whole-process shock wave overpressure nephogramwith model scaling coefficients of 1, 0.8, 0.6, 0.4, 0.2, and 0.1is shown in Figure 3.


(a) 𝜆 = 1

(b) 𝜆 = 0.8

(c) 𝜆 = 0.6

(d) 𝜆 = 0.4

(e) 𝜆 = 0.2

(f) 𝜆 = 0.1

Figure 3: Shock wave overpressure nephogram of each scale model.

As shown in Figure 3, the whole-process shock waveoverpressure nephogram of each scale model has the samevariation trend. The shock wave propagation characteristicsare as follows: an obvious Mach-wave effect occurs nearthe ground [30] after explosive detonation, and, with thecontinuation of shock wave propagation, it first reaches thetop of reticulated shell and then converges on the top ofthe reticulated shell and the connection of the wall and thereticulated shell before finally converging at the midpointof the ground. After the ground reflection, it has a secondcontact with the top of the reticulated shell, starting a newround of reflection and convergence [6, 31]; as a result, theshock wave propagation characteristics of the scale model aresimilar to the original model. The source model in this paperrefers to the model with scaled coefficient 𝜆 = 1.4.2. Similarity Analysis of the Distribution of Overpressure onthe Inner Surface. To study the distribution characteristicsof the overpressure peak value of shock waves on theinternal surface of the internal explosion in a reticulatedshell structure, the unit overpressure at several locations onthe inner surface of the structure is analysed, as shown inFigure 4.

The overpressure peak values at measure points of eachscale model are shown in Table 2.

As indicated from Table 2, the overpressure peak valuesat the top of the reticulated shell and the connection of thewall and the reticulated shell as well as the bottom of thewall of each scale model are larger than those of other parts.Figure 5 shows that the distribution characteristics of the unitoverpressure peak values of each scale model are as follows:the distribution of overpressure peak values of the shockwaves in the reticulated shell and the side wall is large on bothends and small in the middle, indicating that the distributionof the peak surface unit overpressure values of each scalemodel is similar to the original model.

4.3. Similarity Analysis of the Dynamic Response of the Retic-ulated Shell Structure. The nodal displacement and memberbar stress are chosen as the dynamic response parame-ters [19] of the single-layer spherical lattice shell structureunder the effect of internal explosion, and the maximumdisplacement and maximum stress response of each scalemodel are shown in Table 3. Table 3 indicates that themaximum displacement response of each scale model underthe effect of the internal explosion occurs at the centre node


TNT

1 2 3 4 56

78

910111213141516171819

Figure 4: Schematic diagram of the unit positions.

Table 2: Peak overpressure values at each unit position.

Unit position Peak overpressure values at each unit positionΔ𝑃/Pa Δ𝑃/Pa Δ𝑃/Pa Δ𝑃/Pa Δ𝑃/Pa Δ𝑃/Pa1 105910 106130 106580 107500 106010 1268602 71480 71440 71280 71440 71460 789103 66560 66560 66230 66550 66560 753804 69610 69480 69280 69520 69580 770205 69210 69090 69130 69120 69120 768106 71510 71440 71410 71450 71450 795107 75070 74710 74710 74760 74780 832508 74180 74100 74000 74080 74080 814009 73950 73930 73700 73950 73930 8321010 82960 82950 82880 82930 82940 9060011 96500 96490 96450 96500 96500 10765012 74300 74280 74020 74270 74260 8089013 74150 74140 73890 74140 74090 8221014 79120 79110 78950 79110 78970 8683015 77770 77770 77500 77770 77760 8528016 80470 80470 80110 80470 80380 8909017 85630 85680 85420 85680 85700 9471018 85680 85690 85610 85690 85690 9678019 96250 96230 96000 96220 96210 107180Scaling coefficient 𝜆 1 0.8 0.6 0.4 0.2 0.1TNT quantity/kg 140 71.69 30.24 8.96 1.12 0.14

of the spherical shell, and the maximum stress responseoccurs near the midspan member bar, indicating that theoccurring location of maximum displacement response andmaximum stress response of each scale model is similar tothe original model. From Figures 6 and 7, we can knowthat time-history curves of maximum nodal displacementand maximum member stress in scaled models fail to obeythe similarity law; when the scaled coefficient 𝜆 ≥ 0.6, thetwo curves have high similarity with the source model andwhen the scaled coefficient 𝜆 < 0.6, the similarity lowers.This is mainly because the dynamic response of latticed shellnot only is related to the explosive shock wave but alsodepends on the material properties of the structure itself.Thescaled model in this paper is not subject to full scaling buta scaled model with structural material unchanged and sizechanged.

4.4. Similarity Analysis of the Reticulated Shell DestructionPhenomenon. According to the design principle of “strongstructure and weak envelope” [18], the explosion venting ofthe reticulated shell structure is primarily envelope explosionventing. We can see from the propagation characteristicsof shock waves in the internal explosion in the reticulatedshell structure and the distribution characteristics of theoverpressure peak values on the inner surface that the topof the reticulated shell and the connection of the wall andthe reticulated shell are the key parts of explosion venting.Purlin hanger connects the maintenance structure andmajorstructure, as shown in Figure 8; thus, purlin hanger fractureis defined as explosion venting in maintenance structure, asshown in Figure 9. To study the critical position of explosionventing in the internal explosion in the reticulated shellstructure, an explosion venting simulation is conducted for


Table 3: Maximum dynamic response positions of each scale model.

Scaling coefficient 𝜆 Maximum displacement response Maximum stress response𝑊max /mm 𝑡1/𝑠 Node position 𝜎max/MPa 𝑡2/s Member bar position1 140.6 0.126 Midspan 418 0.119 Near the midspan0.8 100 0.102 Midspan 416 0.096 Near the midspan0.6 62 0.082 Midspan 411 0.074 Near the midspan0.4 26 0.058 Midspan 403 0.051 Near the midspan0.2 13 0.023 Midspan 384 0.016 Near the midspan0.1 5.2 0.015 Midspan 377 0.018 Near the midspan

10.80.6

0.40.20.1

2 4 6 8 10 12 14 16 18 200Element

0.06

0.07

0.08

0.09

0.10

0.11

0.12

0.13

ΔZ

(m)

Figure 5: The surface unit peak overpressure values in the reticu-lated shell structure.

1, TNT = 140 kg0.8, TNT = 71.69 kg0.6, TNT = 30.24 kg

0.4, TNT = 8.96 kg0.2, TNT = 1.12 kg0.1, TNT = 0.14 kg

−0.075

−0.050

−0.025

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.05 0.10 0.15 0.200.00t (s)

ΔP

(MPa

)

Figure 6: Time-history curve of maximum nodal displacement inscaled models.

each scale model. The TNT quantities and positions of eachscale model in explosion venting are shown in Figure 10.

As shown in Figure 10, the explosion venting of eachscale model occurs first at the connection of the wall and

0

0.00 0.05 0.10 0.15 0.20

100

200

300

400

500

−100

t (s)

𝜎(M

Pa)

𝜆 = 1, TNT = 140 kg𝜆 = 0.8, TNT = 71.69 kg𝜆 = 0.6, TNT = 30.24 kg

𝜆 = 0.4, TNT = 8.96 kg𝜆 = 0.2, TNT = 1.12 kg𝜆 = 0.1, TNT = 0.14 kg

Figure 7: Time-history curve of maximummember stress in scaledmodels.

Purlin hanger

Structural bars

Envelope

Figure 8: Schematic diagram of connecting component.

the reticulated shell. As the TNT quantity increases, explo-sion venting also occurs at the top of the reticulated shellstructure.The research group has made an internal explosionexperiment in scaledmodel with single-layer spherical latticeshell structure in the field. Experimental model and site areshown in Figure 11, wherein the lattice shell rise is 0.8m,span is 4m, rise span ratio is 1/5, frequency is 4-ring, andthe height of lower supporting structure is 1m. Purlin hangerconnects themaintenance structure and lattice shell structureof experimental model, as shown in Figure 12. The firstsuffered position is the joint of wall and lattice shell in


Figure 9: Fracture of connecting component.

𝜆 = 1, TNT = 160Kg 𝜆 = 0.8, TNT = 81.93Kg 𝜆 = 0.6, TNT = 34.56Kg

𝜆 = 0.4, TNT = 10.24 Kg 𝜆 = 0.2, TNT = 1.28 Kg 𝜆 = 0.1, TNT = 0.15 Kg

Figure 10: Explosion venting positions of the envelope.

Figure 11: Diagram of experimental model and site.


Table 4: Peak overpressure values at each unit position, action time of the positive pressure zone, and impulse.

𝜆Overpressure peak values of the

shock wave, Δ𝑃/Pa Action time of the positive pressurezone, 𝑇+/ms Impulse, 𝑖/Pa⋅ms

1 5 11 19 1 5 11 19 1 5 11 191 131712 83416 118545 118901 37.02 19.03 20.12 15.97 1007019 595442 828065 7708990.8 133139 83276 118520 118869 29.63 15.21 16.07 12.80 804899 475771 662355 6183960.6 133413 83368 118214 118722 22.17 11.38 11.92 9.62 606907 355934 495916 4628840.4 134818 83306 118529 118863 14.82 7.59 7.96 6.37 401788 237994 330996 3092800.2 131962 83307 118501 118835 7.39 3.82 4.03 3.16 196783 119139 165342 1547840.1 143020 81774 117695 119148 3.65 1.85 1.97 1.52 98776 59533 81695 76120

Figure 12: Connection by purlin hanger.

Figure 13: First explosion venting position in experimental model.

field experiment, as shown in Figure 13, which is consistentwith the first explosion’s venting position in simulated scaledmodels, showing similarity between explosion venting oflattice shell in scaled models and that in original model.

4.5. Similarity Verification of the Scale Models. The destruc-tion of the reticulated shell envelope mainly depends on thestrength of the shock waves, and there are three parametersthat describe the strength of the shock waves: overpressurepeak, action time of positive pressure zone, and impulse. Asmentioned above, the destruction phenomena of all scalemodels are similar; therefore, the critical TNT quantityundergoing explosion venting at the connection of the walland the reticulated shell in the model structure is taken

as the basis for the calculation of each scale model toanalyse whether the overpressure peak value, action time ofthe positive pressure zone, and impulse under critical TNTquantity obey a similarity law in the scale model.

According to the propagation characteristics of the shockwaves in the internal explosion in the reticulated shell struc-ture and the distribution characteristics of the overpressurepeak values on the inner surface, units 1, 5, 11, and 19are selected as typical locations for a pressure time-historydiagram analysis. The typical location pressure time-historydiagram of each scale model is shown in Figure 14.

Figure 14 reveals that the variation trend of the pressuretime-history curve at the measurement point locations ofeach scalemodel and that of the originalmodel are consistent:when the scaling coefficient 𝜆 ≥ 0.2, the overpressure peakvalue at eachmeasure point does not changemuch comparedwith the originalmodel value; when 𝜆 = 0.1, the overpressurepeak value at each measurement point is greater than thatof the original model value. With the decrease of the scalingcoefficient, the scale model pulse width gradually decreases.The overpressure peak value, action time of positive pressurezone, and impulse of each scale model under the critical TNTquantity are shown in Table 4.

From the above similarity theory analysis of the scalemodel, we can see that Δ𝑃𝑃 = Δ𝑃𝑚, 𝑖𝑝/𝑊1/3𝑝 = 𝑖𝑚/𝑊1/3𝑚 ,and 𝑇+𝑃/𝑊1/3𝑝 = 𝑇+𝑚/𝑊1/3𝑚 ; to facilitate the analysis of scalemodel similarity, Table 4 is converted to Table 5.

Figure 15 shows the relationship diagram between thescaling coefficients and the peak overpressure value of shockwave, specific action time of positive pressure zone, andspecific impulse. As seen from Table 5 and Figure 15, (a) withthe decrease of scaling coefficient 𝜆, the deviation degree ofthe peak overpressure value of shock wave 𝑃, specific actiontime of positive pressure zone 𝑇+𝑊−1/3, and specific impulse𝑖𝑊−1/3 in each scale model and the original model becomelarger with maximum errors of 4.9%, 2.7%, and 2.2%; (b)the deviation degree of the peak overpressure value of shockwaves 𝑃 and the original model value is less than that ofthe specific action time of positive pressure zone 𝑇+𝑊−1/3and specific impulse 𝑖𝑊−1/3; (c) the deviation degree of theoverpressure peak value of shock waves 𝑃, specific actiontime of positive pressure zone 𝑇+𝑊−1/3, and specific impulse𝑖𝑊−1/3 of the reticulated shell top unit 1 and the originalmodel is greater than that of other parts; (d) when scaling


−0.04

−0.02

0.00

0.00 0.05

0.020.040.060.080.10

0.10 0.15 0.20

0.120.14

15

1119

15

1119

15

1119

15

1119

15

1119

15

1119

t (s)

−0.04

−0.02

0.00

0.00 0.02 0.04

0.020.040.060.080.10

0.06 0.08 0.10 0.12

0.120.14

t (s)

−0.04

−0.02

0.00

0.00 0.01

0.020.040.060.080.10

0.02 0.03 0.04

0.120.14

t (s)

−0.05

0.00

0.000 0.005

0.05

0.10

0.010 0.015 0.020

0.15

t (s)

−0.04

−0.02

0.00

0.00 0.02

0.020.040.060.080.10

0.04 0.06 0.08

0.120.14

t (s)

−0.04

−0.02

0.00

0.00 0.04

0.020.040.060.080.10

0.08 0.12 0.16

0.120.14

t (s)

𝜆 = 1 𝜆 = 0.8

𝜆 = 0.6 𝜆 = 0.4

𝜆 = 0.2 𝜆 = 0.1

ΔP

(MPa

)ΔP

(MPa

)ΔP

(MPa

)

ΔP

(MPa

)ΔP

(MPa

)ΔP

(MPa

)

Figure 14: Typical location pressure time-history diagram of each scale model.

coefficient𝜆 = 0.1, the relative errors of the peak overpressurevalues and the original model values of units 1, 5, 11, and 19are 4.9%, 0.9%, 0.4%, and 0.1%, the relative errors of specificaction time of positive pressure zone𝑇+𝑊−1/3 are 0.7%, 0.9%,0, and 2.7%, and the relative errors of specific impulse 𝑖𝑊−1/3are 0.2%, 2.2%, 0.8%, and 0.9%. Below are main possiblereasons: (a) as the scaled coefficient decreases, the action time

in positive pressure region and specific impulse values havemore errors, so their accuracy is less than that of the peakoverpressure; (b) as the scaled coefficient decreases, the errorof grid size effect becomes larger; (c) unit 1 on the top of latticeshell is obviously affected by the secondary wave; (d) as themodel shrinks, the boundary effect on the system is obviouslylarger.Therefore, the peak overpressure value, the action time


Table 5: Overpressure peak values at each unit position, specific action time of positive pressure zone, and specific impulse.

𝜆Overpressure peak value of shock wave,Δ𝑃/Pa

Specific action time of positive pressurezone,𝑇+𝑊−1/3/ms⋅kg−1/3

Specific impulse,𝑖𝑊−1/3/Pa⋅ms⋅kg−1/31 5 11 19 1 5 11 19 1 5 11 19

1 131712 83416 118545 118901 6.82 3.51 3.71 2.94 185495 109681 152531 1420010.8 133139 83276 118520 118869 6.82 3.50 3.70 2.95 185322 109543 152502 1423810.6 133413 83368 118214 118722 6.81 3.49 3.66 2.95 186322 109273 152248 1421070.4 134818 83306 118529 118863 6.82 3.50 3.67 2.93 185025 109597 152425 1424250.2 131962 83307 118501 118835 6.81 3.52 3.71 2.91 181239 109728 152281 1425570.1 143020 81774 117695 119148 6.87 3.48 3.71 2.86 185904 112045 153756 143263

15

1119

0.07

0.09

0.11

ΔP

(MPa

)

0.13

0.15

0.2 0.4 0.6 0.8 10𝜆

(a) Influence of 𝜆 on Δ𝑃

2.6

3.1

3.6

4.1

4.6

5.1

5.6

6.1

6.6

7.1T

+W

−1/3

(ms·k

g−1/3

)

0.2 0.4 0.6 0.8 10𝜆

15

1119

(b) Influence of 𝜆 on 𝑇+𝑊−1/3

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

iW−1/3

(MPa

·ms·k

g−1/3

)

0.2 0.4 0.6 0.8 10𝜆

15

1119

(c) Influence of 𝜆 on 𝑖𝑊−1/3

Figure 15: Relationship diagram between the scaling coefficients and the overpressure peak value of the shock wave, the specific action timeof positive pressure zone, and the specific impulse.

of positive pressure zone, and the impulse under the criticalTNT quantity are similar in the scale models.

5. Conclusion

(1) The calculation model and parameter selection ofthe single-layer spherical internal explosion in thereticulated shell structure based on LS-DYNAare rea-sonable, and the results of the numerical simulationand analysis for large-space structures under internalexplosion are credible.

(2) If the structure size remains unchanged and TNTquantity changes, the reflection overpressure on theinner surface of the reticulated shell is always sub-ject to the similarity law of B. Hopkinson, and the

reflection overpressure is linearly proportional to thescaled distance.

(3) If the structure material remains unchanged and sizechanges, the shock wave propagation characteristicsand the distribution of overpressure on the innersurface of each scale model are similar to those of theoriginal model; positions of maximum displacementresponse and maximum stress response in scaledmodels have similarity with those in the originalmodel; time-history curves of maximum nodal dis-placement and maximum member stress in scaledmodels generally have no similarity with those in theoriginal model and such similarity decreases with thereduction of scaled coefficient; the explosion ventingof each scale model first occurs at the connection of


wall and the reticulated shell has similarity with thatin source model.

(4) Simulation and analysis of each scale model areconducted according to the critical TNT quantity ofexplosion venting at the connection between the walland the reticulated shell of the model structure; theresults show that the overpressure peak value of theshock waves 𝑃, the specific action time of the positivepressure zone 𝑇+𝑊−1/3, and the specific impulse𝑖𝑊−1/3 in each scale model are generally similarto those of the original model, and as the scaledcoefficient reduces, the above three factors have largerdeviation extent and lower similarity. Therefore, itis feasible to forecast the internal explosion effect ofthe original structure size model by adopting scalingcoefficient 𝜆 ≥ 0.1 in a scale model experiment.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article.

Acknowledgments

The authors wish to acknowledge the financial support fromthe Natural Science Foundation of China (no. 51278208), theFujian Province College & University-Industry CooperativeMajor Project in Science and Technology (no. 2012Y4010),the Fujian Natural Science Foundation (no. 2011J01319), andthe Postgraduate Research and Innovation Capacity TrainingPlan of Huaqiao University (no. 1400104003).

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